Application Specificities of Array Antennas - Cosic

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Application Specificities of Array Antennas - Cosic
KATHOLIEKE UNIVERSITEIT LEUVEN
FACULTEIT INGENIEURSWETENSCHAPPEN
DEPARTEMENT ELEKTROTECHNIEK (ESAT)
AFDELING ESAT-TELEMIC
Kasteelpark Arenberg 10, B-3001 Leuven (Heverlee), België
Application Specificities of Array Antennas:
Satellite Communication and
Electromagnetic Side Channel Analysis
Promotor :
Prof. Dr. Ir. G. Vandenbosch
Prof. Dr. Ir. I. Verbauwhede
Prof. Dr. Ir. P. Coppin
Proefschrift voorgedragen tot
het behalen van het doctoraat
in de ingenieurswetenschappen
door
Wim AERTS
June 2009
“ There’s never time to do it right,
but always time to do it over.”
– Meskimen’s law
“ Der Horizont vieler Menschen
ist ein Kreis mit dem Radius Null
- und das nennen sie ihren Standpunkt.”
“ Research is what I’m doing
when I don’t know what I’m doing.”
“ When elephants fight,
it is the grass that suffers.”
– Albert Einstein
– Wernher von Braun
– While the source of this quote
is lost in the distant past,
the wisdom is as true today
“ Verstandig kiezen is de boodschap.”
– Koen Van Vlaanderen
“ ... where questions make dancers of
people who’s stories aren’t straight.”
– Living in detente / NoMeansNo
“ The years we had together
never we’ll forget.
You’re in my heart as long as
daylight gets me straight.”
– Takes you back / Jazzanova
KATHOLIEKE UNIVERSITEIT LEUVEN
FACULTEIT INGENIEURSWETENSCHAPPEN
DEPARTEMENT ELEKTROTECHNIEK (ESAT)
AFDELING ESAT-TELEMIC
Kasteelpark Arenberg 10, B-3001 Leuven (Heverlee), België
Application Specificities of Array Antennas:
Satellite Communication and
Electromagnetic Side Channel Analysis
Jury :
Prof. Dr. Ir. Y. Willems, voorzitter
Prof. Dr. Ir. G. Vandenbosch, promotor
Prof. Dr. Ir. I. Verbauwhede, promotor
Prof. Dr. Ir. P. Coppin, promotor
Prof. Dr. Ir. E. Van Lil
Prof. Dr. Ir. L. Ligthart
bla (Technische Universiteit Delft, Nederland)
Prof. Dr. Ir. D. Stroobandt
bla (Universiteit Gent)
Prof. Dr. Ir. C. Craeye
bla (Université catholique de Louvain)
Proefschrift voorgedragen tot
het behalen van het doctoraat
in de ingenieurswetenschappen
door
Wim AERTS
U.D.C. 621.39
Wet. Depot: D/2009/7515/70
ISBN 978-94-6018-087-3
June 2009
c
Katholieke
Universiteit Leuven - Faculteit Ingenieurswetenschappen
Arenbergkasteel, B-3001 Leuven (Heverlee), België
Alle rechten voorbehouden. Niets uit deze uitgave mag worden vermenigvuldigd en/of
openbaar gemaakt worden door middel van druk, fotocopie, microfilm, elektronisch
of op welke andere wijze ook zonder voorafgaande schriftelijke toestemming van de
uitgever.
All rights reserved. No part of the publication may be reproduced in any form by
print, photoprint, microfilm or any other means without written permission from the
publisher.
Wet. Depot: D/2009/7515/70
ISBN 978-94-6018-087-3
Voorwoord
Het is bij momenten op het randje geweest, of moet ik zeggen tot aan het randje:
“Zie dat je nog thuis geraakt he Steven”. Het is er bij momenten ook over geweest,
zoals die ene keer dat ik mijn laptop niet mee had genomen naar het toilet. Als ik
had geweten hoe het hier ineen zat, dan was ik hier nooit begonnen. Maar als het
hier niet zo leuk was, dan had ik hier nooit al die jaren gebleven! Ik heb hier altijd
geprobeerd veel bij te leren, leuke dingen te doen en aangenaam met mensen om te
gaan. Op een luttel klein detail is me dat toch redelijk goed gelukt, denk ik.
Op al die jaren, ben ik met zeer veel dingen bezig geweest. Deels omdat ik alles de
moeite vind, deels omdat ik niet nee kan zeggen als iemand me iets vraagt. Bijgevolg
bevat dit boekje stukken afkomstig uit heel veel domeinen. Hopelijk zit er ook een
stukje bij dat U kan boeien. Het merendeel van de kennis die ik vergaard heb, heb ik
erin proberen condenseren, zodat het een aanzet of startpunt kan zijn voor toekomstige onderzoekers of collegae die willen bijlezen over een verwant onderwerp. Ik hoop
dat op die manier, met mijn vertrek, toch niet al mijn kennis mee verdwijnt.
Dat streven, en het feit dat ik graag dingen uitleg en netjes opschrijf, hebben het
aantal bladzijden behoorlijk laten oplopen. Op zich vind ik dat geen slechte zaak.
Het aanbod is er, de keuze is aan de lezer.
Veel plezier ermee!
Wim Aerts
Leuven, juni 2009.
i
invisible filling
Dankwoord
Op die bijna acht jaren dat ik op ESAT werkte, heb ik het geluk gehad met vele
fijne mensen te mogen omgaan. Ik heb van vele mensen steun, hulp of gewoon leuk
gezelschap gekregen.
Toen ik mijn tekst begon te schrijven, zag ik daar echt tegen op. Vooral door de
wilde verhalen die de ronde deden. Een babbeltje tussen pot en pint met Claudia
Diaz heeft me eraan gezet. En eigenlijk viel het allemaal mee, want ik schrijf graag.
Maar na bladzijde 160 begon het wel wat te veel van het goede te worden.
In mijn eerste jaren heb ik ongetwijfeld Servaas Vandenberghe en Peter Delmotte
danig verveeld met al mijn vragen. Het leuke aan de antwoorden van Servaas was,
dat ze me hebben geleerd de handleiding te lezen. Later heb ik genoeg vragen van
jongere collega’s mogen beantwoorden om volgens het pay forward principe uit de
schuld te geraken. Ik vrees alleen dat ik iets te weinig mijn best heb gedaan om hen
te leren lezen.
Tijdens mijn tijd op ESAT, heb ik voor verscheidene projecten met vele mensen
samengewerkt. Vooral Eugene Jansen (Verhaert N.V.), Stefaan Burger (O.M.P.),
Arnold Schoonwinkel (Stellenbosch Universiteit), Keith Palmer (Stellenbosch Universiteit), Dirk Stroobandt (UGent) en Michiel De Wilde (UGent) zijn me bijgebleven
voor de vlotte samenwerking.
Binnen ESAT kon ik altijd terecht bij Jozef Lodeweyckx, Ilja Ocket, Christophe De
Cannière, Danny De Cock, Nele Mentens en Fréderique Gobert voor consulting en
leuke babbels. Met Roel Peeters viel ook leuk te babbelen, zolang het maar over de
besjestheorie ging.
Als meetverantwoordelijke heb ik leuk mogen samenwerken en uitwisselen met Frederik Daenen (MICAS), Luc Pauwels (IMEC) en Robert Roovers (De Nayer), en mogen
afdingen bij o.a. Johan Buschgens (Anritsu) en Veerle Kerkhofs (Agilent/Telogy).
Dat de studenten van vandaag de collega’s van morgen zijn, heb ik aan den lijve mogen
ondervinden. Iedereen die op langere termijn kan denken, weet dus hoe belangrijk
het is om oefenzittingen goed te geven en thesissen goed te begeleiden.
iii
iv
Dankwoord
Ik heb mijn studenten altijd met heel veel enthousiasme, overgave en hartstocht begeleid. Maar ik heb daar ook veel voor teruggekregen. Het was mijn plezier om
Pieter Vandromme, Joris Vankeerbergen, Dieter De Moitié en Sebastiaan Indesteege
te kunnen helpen.
Sebastiaan is daarna ook een fijne collega geworden. Gelukkig waren er wel meer.
Steven Mestdagh en Yves Schols zijn buiten categorie. Onze onvergetelijke wandelingetjes in het prachtige park hebben vele problemen opgelost of voorkomen en waren
noodzakelijk in het uitstippelen van het traject op lange termijn of het ontwikkelen
van de visie. Ook Peter Delmotte en Elke De Mulder waren buitengewone collega’s.
Luc Mombaerts, Rudi Casteels en Bruno Vanham zijn buitengewone technici. Hun
vakmanschap en precisie bewonder ik enorm en is minstens evenwaardig aan de intellectuele prestaties die aan de K.U.Leuven worden geleverd.
Mijn promotoren Guy Vandenbosch, Ingrid Verbauwhede en Pol Coppin wil ik bedanken voor de mogelijkheid om mijn doctoraat (af) te maken en de enorme academische
vrijheid die ik van hen kreeg. Dankzij die vrijheid heb ik kunnen onderzoeken wat
ik interessant vond, me kunnen bijscholen en me helemaal kunnen ontplooien. Zonder die vrijheid had ik nooit zoveel bij kunnen leren en de ingenieur en onderzoeker
worden die ik nu ben. Hun bijdrage van vakkennis, kritische opmerkingen en zinvolle
suggesties hebben de kwaliteit van mijn werk en publicaties, en – niet in het minst –
van dit proefschrift sterk verbeterd. Dit laatste geldt zeker ook voor de feedback van
mijn assessors Emmanuel Van Lil en Leo Ligthart, en van de leden van de leesjury,
Dirk Stroobandt en Christophe Craeye.
In de acht jaar, en zeker ook in het laatste jaar, heb ik eigenlijk enorm veel gewerkt
en bijgevolg de zorg voor Sander en Elin voor ruim meer dan de helft overgelaten
aan Barbara. Zoals eigenlijk altijd het geval is, hangt het slagen van een project niet
alleen af van de persoon die het project uitvoert, maar zeker ook van die mensen die
ongevraagd en in stilte het werk overnemen en zorgen dat alles blijft draaien.
Hoewel dankjewel zeggen nogal goedkoop is, heb ik spijtig genoeg ook moeten vaststellen dat het voor sommigen veel moeite is. Een mens kan het niet te veel zeggen
...
Bedankt!
Wim Aerts
Abstract
Array antennas have numerous applications in every day life. In this work the classical array theory is profoundly reviewed and applied to satellite communication and
electromagnetic side channel analysis.
Satellite communication is a typical communication application. Bandwidths are
generally spoken small and all standard telecommunication engineering methods are
valid. Designing for space, however, requires special attention due to the hostile environment. Consequently, in the design of a system for up link of in-situ collected data
to an earth observation satellite, much effort was spent on material and component
selection. Another interesting peculiarity of the design, was the application of an
analog base band implementation of a technique often used for digital beam forming.
Electromagnetic side channel analysis requires an approach sometimes very different from standard telecommunication engineering methods. When observing direct
radiation of small currents performing cryptographic operations in silicon hardware,
the antennas are designed to be small and sensitive to magnetic fields. Matching is
not performed in order to assure power transfer, but to obtain a high signal-to-noise
ratio. The signal should be digitized with as less quantization error as possible to
allow calculation of correlation with a hypothesis in post-processing. Array antennas
should perform beam forming on very wide band signals and preferably off-line to
allow simultaneous monitoring of different active regions in the chip.
v
vi
Abstract
Roosterantennes vinden hun toepassing in vele aspecten van het dagelijkse leven. In
dit werk wordt de klassieke theorie van roosterantennes grondig herhaald en toegepast
op satellietcommunicatie en electromagnetische nevenkanaalsanalyse.
Satellietcommunicatie is een typische communicatietoepassing. Bandbreedtes zijn
doorgaans klein en alle standaardmethodieken uit het telecommunicatieontwerp mogen toegepast worden. Het ontwerpen voor toepassing in de ruimte vereist echter
speciale aandacht, omwille van het vijandige klimaat. Bijgevolg werd veel aandacht
besteed aan de keuze van materialen en componenten tijdens het ontwerp van een systeem, dat in-situ verzamelde gegevens opzendt naar een satelliet voor aardobservatie.
Een ander interessant aspekt van het ontwerp, was de toepassing van een analoge basisbandimplementatie van een techniek die veelvuldig voor digitale fasesturing wordt
gebruikt.
Electromagnetische nevenkanaalsanalyse vereist een aanpak die soms erg verschilt van
de standaard ontwerptechnieken in de telecommunicatie. Wanneer men de straling
probeert waar te nemen afkomstig van kleine stromen die de cryptografische bewerkingen uitvoeren in een apparaat, moeten de antennes klein zijn en gevoelig voor het
magnetische veld. Aanpassen is hier niet zozeer nodig om maximale vermogensoverdracht te realiseren, maar wel om een zo goed mogelijke signaal-tot-ruisverhouding
te bekomen. Het signaal moet gedigitaliseerd worden met een zo klein mogelijke
quantisatiefout, om achteraf de correlatie met een hypothese te kunnen berekenen.
Roosterantennes moeten hun bundelsturing toepassen op signalen met grote bandbreedte en liefst off-line om toe te laten gelijktijdig verschillende actieve delen van de
chip te kunnen monitoren.
Contents
Voorwoord
i
Dankwoord
iii
Abstract
v
Contents
vii
List of Figures
xv
List of Tables
xxii
List of publications
xxv
Nomenclature
xxix
List of Acronyms
xxxv
Nederlandse samenvatting
xxxix
1 Introduction
1
1.1
Array Antennas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.2
Outline of this Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
1.3
The Thesis at a Glance . . . . . . . . . . . . . . . . . . . . . . . . . .
4
vii
viii
I
Contents
Array Theory
7
2 Beam Forming
13
2.1
Beam Forming in the Receiving Chain . . . . . . . . . . . . . . . . . .
14
2.2
Some Beam Forming Implementations . . . . . . . . . . . . . . . . . .
18
2.2.1
Time Delaying . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
2.2.2
Phase Shifting . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
2.2.3
Frequency Scanning . . . . . . . . . . . . . . . . . . . . . . . .
23
Beam Forming Approximation Effects . . . . . . . . . . . . . . . . . .
24
2.3
3 Phased Array Design
3.1
25
Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
3.1.1
Array Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
3.1.2
Array Radiation Properties . . . . . . . . . . . . . . . . . . . .
27
3.1.3
Example: Subarray of the ATCRBS in Bertem . . . . . . . . .
28
3.2
The Array Factor via a Fourier Transform . . . . . . . . . . . . . . . .
30
3.3
Array Design with Array Factor . . . . . . . . . . . . . . . . . . . . . .
32
3.3.1
Influence of the Array Geometry . . . . . . . . . . . . . . . . .
32
3.3.2
Influence of Excitations . . . . . . . . . . . . . . . . . . . . . .
40
3.3.3
Common Design Techniques . . . . . . . . . . . . . . . . . . . .
47
What if the Assumptions no longer Hold . . . . . . . . . . . . . . . . .
48
3.4.1
Errors that can be solved by Calibrating . . . . . . . . . . . . .
48
3.4.2
Quantization Errors . . . . . . . . . . . . . . . . . . . . . . . .
48
3.4.3
Errors that Void the Theory
. . . . . . . . . . . . . . . . . . .
49
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
3.4
3.5
Contents
II
ix
Application: Satellite Communication
4 Introduction to the Application
4.1
51
55
System Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
56
4.1.1
In-Situ Data Collection . . . . . . . . . . . . . . . . . . . . . .
56
4.1.2
Space-Ground Communication Protocol . . . . . . . . . . . . .
57
4.1.3
Space Segment . . . . . . . . . . . . . . . . . . . . . . . . . . .
59
4.1.4
System Design Choices . . . . . . . . . . . . . . . . . . . . . . .
60
4.2
Orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
60
4.3
Link Budget . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
62
4.3.1
Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
62
4.3.2
Space Ground Trade Off . . . . . . . . . . . . . . . . . . . . . .
64
4.3.3
Example: Link Budget for an Orbit at 600 km . . . . . . . . .
65
Motivation of Electronic Beam Steering . . . . . . . . . . . . . . . . .
66
4.4.1
Note on Vibration . . . . . . . . . . . . . . . . . . . . . . . . .
67
4.4.2
Examples of Phased Arrays on Satellites . . . . . . . . . . . . .
68
Designing Space Instruments . . . . . . . . . . . . . . . . . . . . . . .
70
4.5.1
Space Conditions . . . . . . . . . . . . . . . . . . . . . . . . . .
70
4.5.2
Product Assurance . . . . . . . . . . . . . . . . . . . . . . . . .
76
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
77
4.4
4.5
4.6
5 Array Elements
79
5.1
Element Specifications . . . . . . . . . . . . . . . . . . . . . . . . . . .
79
5.2
Antenna Substrates for Space Application . . . . . . . . . . . . . . . .
80
5.2.1
Tolerances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
80
5.2.2
Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
80
5.2.3
Substrate Materials . . . . . . . . . . . . . . . . . . . . . . . .
82
5.2.4
Metalization
83
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
x
Contents
5.2.5
Space Environment . . . . . . . . . . . . . . . . . . . . . . . . .
85
5.2.6
Material Selection . . . . . . . . . . . . . . . . . . . . . . . . .
88
5.3
Element Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
89
5.4
Selecting Array Geometry . . . . . . . . . . . . . . . . . . . . . . . . .
89
5.4.1
Linear Array . . . . . . . . . . . . . . . . . . . . . . . . . . . .
90
5.4.2
Planar Array . . . . . . . . . . . . . . . . . . . . . . . . . . . .
97
5.5
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
6 Signal Modification and Combination
6.1
6.2
6.3
III
103
Analog Quadrature BB Phase Shifter . . . . . . . . . . . . . . . . . . . 104
6.1.1
Analog Implementation of Digital Technique . . . . . . . . . . 104
6.1.2
Architecture of the Demonstrator Array Antenna . . . . . . . . 105
6.1.3
Measurement Results . . . . . . . . . . . . . . . . . . . . . . . 109
Space Qualified Phase Shifter . . . . . . . . . . . . . . . . . . . . . . . 116
6.2.1
Space Segment System Overview . . . . . . . . . . . . . . . . . 117
6.2.2
Signal Path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
6.2.3
Control Hardware . . . . . . . . . . . . . . . . . . . . . . . . . 119
6.2.4
Software Overview . . . . . . . . . . . . . . . . . . . . . . . . . 119
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
Application: EM Side Channel Analysis
7 Introduction to the Application
123
127
7.1
Example of A Block Cipher . . . . . . . . . . . . . . . . . . . . . . . . 128
7.2
Cryptographic Hardware . . . . . . . . . . . . . . . . . . . . . . . . . . 129
7.2.1
Field Programmable Gate Array (FPGA) . . . . . . . . . . . . 130
7.2.2
Microcontroller (µC) . . . . . . . . . . . . . . . . . . . . . . . . 131
Contents
7.3
xi
Side Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
7.3.1
Timing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
7.3.2
Power Consumption . . . . . . . . . . . . . . . . . . . . . . . . 133
7.3.3
Electromagnetic Radiation . . . . . . . . . . . . . . . . . . . . 134
7.4
Frequency Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
7.5
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
8 Array Elements
8.1
8.2
8.3
8.4
137
Element Specifications . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
8.1.1
Capacitive and Inductive Sensors . . . . . . . . . . . . . . . . . 138
8.1.2
Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
8.1.3
Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
8.1.4
Bandwidth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
8.1.5
Rigidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
8.1.6
Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
8.1.7
Baluns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
Shielded Loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
8.2.1
Loop Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
8.2.2
Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
8.2.3
Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
RFID Loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
8.3.1
Loop Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
8.3.2
Power source and current enhancement . . . . . . . . . . . . . 163
8.3.3
Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
Maximal Resolution Sensor . . . . . . . . . . . . . . . . . . . . . . . . 168
8.4.1
Geometrical Design
. . . . . . . . . . . . . . . . . . . . . . . . 169
8.4.2
Enhancement Design . . . . . . . . . . . . . . . . . . . . . . . . 180
xii
Contents
8.4.3
Practically Implementing a Large Nt . . . . . . . . . . . . . . . 180
8.4.4
Twins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
8.5
Array Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
8.6
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
9 Signal Modification and Combination
187
9.1
Digitization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
9.2
Measurement Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
9.2.1
9.3
Noise Contributions . . . . . . . . . . . . . . . . . . . . . . . . 189
Improving Signal-to-Noise Ratio . . . . . . . . . . . . . . . . . . . . . 191
9.3.1
Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
9.3.2
Avoiding Standing Waves . . . . . . . . . . . . . . . . . . . . . 192
9.3.3
Amplification . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
9.3.4
Comparison of some Setups . . . . . . . . . . . . . . . . . . . . 204
9.4
Digital time shifting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
9.5
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
IV
Conclusions
209
Appendices
217
A Doppler Shift Compensation by Frequency Scanning
217
A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
A.2 Mathematical Description . . . . . . . . . . . . . . . . . . . . . . . . . 218
A.2.1 Doppler Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
A.2.2 Frequency Scanning . . . . . . . . . . . . . . . . . . . . . . . . 218
A.2.3 Doppler Shift Compensation . . . . . . . . . . . . . . . . . . . 220
Contents
xiii
A.3 Applicability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
A.3.1 LEO satellite . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
A.3.2 Future spacecraft and base station . . . . . . . . . . . . . . . . 221
A.4 Conclusion
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
B The Orbit Simulator
223
B.1 The Orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
B.2 The Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
C Pseudo Code of the Array Control Routines
227
D Eavesdropping on Computer Displays
231
D.1 Source of Radiation
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
D.2 Screen Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
E A KeeLoq Transceiver
237
F A Low Cost VNA
243
G RFID Basics
245
G.1 Different Transmission Systems . . . . . . . . . . . . . . . . . . . . . . 245
G.1.1 Inductive Coupling . . . . . . . . . . . . . . . . . . . . . . . . . 246
G.1.2 Capacitive Coupling . . . . . . . . . . . . . . . . . . . . . . . . 247
G.1.3 Back Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
G.1.4 Radio Transmission . . . . . . . . . . . . . . . . . . . . . . . . 248
G.2 Link Budget . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
G.2.1 Electric Far Fields . . . . . . . . . . . . . . . . . . . . . . . . . 249
G.2.2 Electric Near Fields . . . . . . . . . . . . . . . . . . . . . . . . 249
G.2.3 Magnetic Near Fields . . . . . . . . . . . . . . . . . . . . . . . 250
G.3 ISO-14443A RFID Standard . . . . . . . . . . . . . . . . . . . . . . . . 250
xiv
Contents
H Determining the Inductance of a Coil
253
H.1 Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
H.2 Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
I
Standing Waves in Measurement Setup
257
I.1
Oscillations in a parallel RLC circuit . . . . . . . . . . . . . . . . . . . 257
I.2
Reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
I.3
Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
References
263
List of Figures
1.1
The principle of beam steering. . . . . . . . . . . . . . . . . . . . . . .
9
1.2
Spheres to determine main lobe for time delay beam forming. . . . . .
10
1.3
The three parts of a phased array antenna. . . . . . . . . . . . . . . .
11
2.1
Phase steering at different stages in the receiver. . . . . . . . . . . . .
16
2.2
The phase shift can also be applied to the local oscillator signal. . . .
17
2.3
Working principle of the I-Q phase shifter. . . . . . . . . . . . . . . . .
21
2.4
Spectra for BB and IF mixing. . . . . . . . . . . . . . . . . . . . . . .
22
2.5
The I-Q phase shifter for IF.
. . . . . . . . . . . . . . . . . . . . . . .
22
2.6
The shifter of Fig. 2.3(b) for transmission and reception. . . . . . . . .
23
3.1
The geometry of an array antenna. . . . . . . . . . . . . . . . . . . . .
26
3.2
The array factor and the directivity of an array. . . . . . . . . . . . . .
29
3.3
The array of dipoles as inserted in MAGMAS . . . . . . . . . . . . . .
30
3.4
Array factors of a 1D, 2D and 3D array. . . . . . . . . . . . . . . . . .
33
3.5
Array factors (LESA, N = 10, varying d). . . . . . . . . . . . . . . . .
34
3.6
Darray of a uniform LESA varies with d. . . . . . . . . . . . . . . . . .
35
3.7
Array factors (LESA, length 20λ, varying N). . . . . . . . . . . . . . .
36
3.8
Several sparse arrays are combined into one antenna. . . . . . . . . . .
37
3.9
The radiation pattern of a sparse array is symmetrical. . . . . . . . . .
38
xv
xvi
List of Figures
3.10 Array factors of geometric arrays (a > 1, N = 10). . . . . . . . . . . .
39
3.11 Array factors of geometric arrays (a < 1, N = 10). . . . . . . . . . . .
40
3.12 Comparison of some linear non-equally spaced arrays. . . . . . . . . .
41
3.13 Array factor of regular planar array. . . . . . . . . . . . . . . . . . . .
41
3.14 Random arrays have no grating lobes but should be sparse. . . . . . .
42
3.15 Beam steering makes use of a Fourier property. . . . . . . . . . . . . .
43
3.16 Chebychev polynomial mapped to one interval of array factor function.
45
3.17 Any type of tapering results in a lower value for Dmax (N = 7). . . . .
46
3.18 Directivity of the ATCRBS subarray. . . . . . . . . . . . . . . . . . . .
47
4.1
Subsystems of the ground segment. . . . . . . . . . . . . . . . . . . . .
58
4.2
Steps of the space-ground communication protocol . . . . . . . . . . .
59
4.3
Subsystems of the space segment. . . . . . . . . . . . . . . . . . . . . .
59
4.4
User and satellite geometry. . . . . . . . . . . . . . . . . . . . . . . . .
62
4.5
Link budget. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
63
4.6
Signal and noise in the link. . . . . . . . . . . . . . . . . . . . . . . . .
64
4.7
Channel capacity as a function of SNR. . . . . . . . . . . . . . . . . .
65
4.8
Model of the satellite. . . . . . . . . . . . . . . . . . . . . . . . . . . .
67
4.9
The 80 beams of the IRIDIUM antenna. . . . . . . . . . . . . . . . . .
68
4.10 The Alcatel X-band antenna for LEO satellites. . . . . . . . . . . . . .
69
4.11 Overview of the problems due to low pressure.
. . . . . . . . . . . . .
71
4.12 Overview of the problems related to temperature. . . . . . . . . . . . .
73
4.13 Aurora Borealis and Van Allen Belts. . . . . . . . . . . . . . . . . . . .
74
4.14 Overview of the problems related to radiation. . . . . . . . . . . . . .
75
5.1
Gain as a function of r for a probe fed patch antenna. . . . . . . . . .
88
5.2
Gain as a function of dt for a probe fed patch antenna. . . . . . . . . .
89
List of Figures
xvii
5.3
Scaled layout of the dual feed patch antenna. . . . . . . . . . . . . . .
90
5.4
Scattering parameter of the dual feed patch antenna. . . . . . . . . . .
90
5.5
The array topology, with dipoles as array elements. . . . . . . . . . . .
91
5.6
Supergaining on an array factor function graph. . . . . . . . . . . . . .
91
5.7
dopt for LESA with uniform tapering, scanned to φ = 10◦ . . . . . . . .
92
5.8
Gain of the optimal LESA scanned to φ = 10◦ as a function of N . . .
93
5.9
Directivity of several arrays as a function of scan angle (N = 8). . . .
93
5.10 Free space path loss and array gain. . . . . . . . . . . . . . . . . . . .
94
5.11 Directivity as a function of d for N = 8 array. . . . . . . . . . . . . . .
96
5.12 Directivity of current imposed and voltage fed dipole array. . . . . . .
97
5.13 Satellite passing over a ground station. . . . . . . . . . . . . . . . . . .
97
5.14 Two types of planar arrays obtained from linear arrays. . . . . . . . .
99
5.15 Gain as a function of d for concentric circles array. . . . . . . . . . . .
99
5.16 Gain variation as a function of φ for planar arrays. . . . . . . . . . . . 102
6.1
Analog implementation of Eq. (6.1). . . . . . . . . . . . . . . . . . . . 104
6.2
Photograph of the array antenna inside the anechoic chamber. . . . . . 105
6.3
Photograph of the antenna element. . . . . . . . . . . . . . . . . . . . 106
6.4
Influence of ground plane and superstrate on element radiation pattern 107
6.5
Schematic of four phase shifters on one controller PCB. . . . . . . . . 108
6.6
Photograph of PCB with four phase shifters. . . . . . . . . . . . . . . 109
6.7
Implementation of the array control. . . . . . . . . . . . . . . . . . . . 110
6.8
Constellation plot of the phase shifter. . . . . . . . . . . . . . . . . . . 113
6.9
Radiation patterns of uniform eight-by-eight array. . . . . . . . . . . . 114
6.10 Radiation patterns of Chebychev eight-by-eight array . . . . . . . . . . 115
6.11 7 elements concentric circles array. . . . . . . . . . . . . . . . . . . . . 117
6.12 Subsystems of the space segment. . . . . . . . . . . . . . . . . . . . . . 118
xviii
List of Figures
6.13 Implementation of the signal path of the array hardware. . . . . . . . 119
6.14 Implementation of the control of the array hardware. . . . . . . . . . . 120
6.15 Overview of the Routines that are needed on the Satellite. . . . . . . . 121
7.1
Schematic of the KeeLoq block cipher. . . . . . . . . . . . . . . . . . . 129
7.2
Schematic of a typical FPGA. . . . . . . . . . . . . . . . . . . . . . . . 131
7.3
Block diagram of a typical 8051 family µC. . . . . . . . . . . . . . . . 132
8.1
|E|/|H| of a dipole as a function of distance.
8.2
This system is not balanced nor unbalanced if Z1 6= Z2 . . . . . . . . . 144
8.3
A sleeve balun . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
8.4
A practical implementation of a sleeve balun . . . . . . . . . . . . . . 145
8.5
Working principle of a transformer balun. . . . . . . . . . . . . . . . . 145
8.6
Photograph of the shielded loops. . . . . . . . . . . . . . . . . . . . . . 146
8.7
Schematic drawings of the four loop types. . . . . . . . . . . . . . . . . 147
8.8
A shielded loop implementation using biax. . . . . . . . . . . . . . . . 147
8.9
S11 of the loops for 22 MHz − 1 GHz . . . . . . . . . . . . . . . . . . . 150
. . . . . . . . . . . . . . 139
8.10 S11 of the loops for 1 kHz − 50 MHz . . . . . . . . . . . . . . . . . . . 151
8.11 S11 of the mœbius with short on a Smith Chart. . . . . . . . . . . . . 152
8.12 S11 of a mœbius without short with different capacitances. . . . . . . . 153
8.13 Layout of a loop that combines the balanced and mœbius loop. . . . . 153
8.14 Working principle of a shielded loop. . . . . . . . . . . . . . . . . . . . 155
8.15 Reader loop geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
8.16 rl /rd as a function of rd /λ. . . . . . . . . . . . . . . . . . . . . . . . . 159
8.17 Magnetic field as a function of reading distance for several Nt . . . . . 160
8.18 Equivalent circuit of an inductor. . . . . . . . . . . . . . . . . . . . . . 162
8.19 Schematic of the four RLC resonance circuits . . . . . . . . . . . . . . 164
List of Figures
xix
8.20 The RFID reader loops. . . . . . . . . . . . . . . . . . . . . . . . . . . 167
8.21 Resonance circuit used to determine the reading range. . . . . . . . . . 167
8.22 Frequency dependent value of L for the copper tube. . . . . . . . . . . 168
8.23 Contour lines of Eq. (8.39) as a function of rl and Nt . . . . . . . . . . 170
8.24 Nt as function of fL = fH for Z = ∞ and Z = 1 MΩ. . . . . . . . . . 173
8.25 Minimum rl as a function of fL = fH for Z = ∞ and Z = 1 MΩ. . . . 174
8.26 Minimum rl for two loop sensors with varying working frequency band. 175
8.27 Variation of Nswitch as function of rl /rw and d/2rw . . . . . . . . . . . 175
8.28 rl as function of fL = fH for unloaded loops with different d/2rw . . . 176
8.29 Minimum rl as function of fL = fH for Z = 1 MΩ k 13 pF and Z = 50. 176
8.30 rl for Z = 1 MΩ k 13 pF and Z = 50 as function of frequency band. . 177
8.31 Effect of restricted Nt on rl . . . . . . . . . . . . . . . . . . . . . . . . . 177
8.32 rl as function of fL = fH for Z = 1 MΩ k 13 pF and several rw /rl . . . 178
8.33 Graphical representation of the minimum rl search. . . . . . . . . . . . 179
8.34 The difference between Eq. (8.39) and (8.55). . . . . . . . . . . . . . . 181
8.35 Equivalent circuit of the loop sensor connected to a load.
. . . . . . . 182
8.36 V of loop as function of position relative to dipole. . . . . . . . . . . . 183
8.37 A twin loop configuration solves location ambiguity of a dipole source. 183
8.38 V of loop as function of position relative to magnetic dipole.
. . . . . 184
8.39 Twin loops can be used to suppress noise. . . . . . . . . . . . . . . . . 184
9.1
Typical measurement setup for a EM side channel analysis. . . . . . . 189
9.2
Circuit of diode detector. . . . . . . . . . . . . . . . . . . . . . . . . . 192
9.3
The importance of matching . . . . . . . . . . . . . . . . . . . . . . . . 193
9.4
Possible circuit for a matched amplifier with a MAR-6+. . . . . . . . . 194
9.5
Schematics of an ideal class E amplifier . . . . . . . . . . . . . . . . . 195
9.6
The topology of a push-pull class E amplifier. . . . . . . . . . . . . . . 196
xx
List of Figures
9.7
Circuit schematic of the modulation circuit. . . . . . . . . . . . . . . . 197
9.8
Equivalent low frequency scheme of the class E. . . . . . . . . . . . . . 198
9.9
Choke current and drain voltages of transistors of class E. . . . . . . . 198
9.10 Drain voltage of the IRF9530 with and without diode. . . . . . . . . . 199
9.11 Simulation of loop current with and without Rext . . . . . . . . . . . . 200
9.12 Waveforms of a class E amplifier with the complete SPICE model. . . 201
9.13 Picture of the class E amplifier. . . . . . . . . . . . . . . . . . . . . . . 202
9.14 PCB layout of the class E amplifier. . . . . . . . . . . . . . . . . . . . 202
9.15 Gate and drain voltage of the class E transistors. . . . . . . . . . . . . 203
9.16 Field transmitted by the loop antenna driven by the class E amplifier. 204
9.17 Zoom of the signal as received with a magnetic probe at 30 cm. . . . . 204
9.18 Circuit schematic of the modulation circuit with current mirror. . . . . 205
9.19 General schematic of the measurement chain. . . . . . . . . . . . . . . 205
9.20 SN Ros for four different setups. . . . . . . . . . . . . . . . . . . . . . . 207
A.1 The principle of frequency scanning. . . . . . . . . . . . . . . . . . . . 219
A.2 The geometry of a LEO satellite that passes over a ground station. . . 221
A.3 Doppler shift as a function of α for some LEO satellites. . . . . . . . 222
B.1 Satellite orbit and ground station in two coordinate systems. . . . . . 224
B.2 Doppler shift with and without earth rotation. . . . . . . . . . . . . . 226
D.1 Schematic representation of a PSD of a video signal. . . . . . . . . . . 232
D.2 Effect of using a repetition instead of BB version of the video signal. . 233
D.3 Effect of averaging multiple frames on the image quality. . . . . . . . . 234
D.4 Measurement setup for eavesdropping on computer displays. . . . . . . 235
E.1 Photograph of the KeeLoq transceiver. . . . . . . . . . . . . . . . . . . 238
List of Figures
xxi
E.2 Envelope of voltage over transmitting loop sending opcode. . . . . . . 238
E.3 Circuit of the Tx part in the KeeLoq transceiver. . . . . . . . . . . . . 239
E.4 Envelope of voltage over transmitting loop receiving response. . . . . . 239
E.5 Circuit of the Rx part in the KeeLoq transceiver. . . . . . . . . . . . . 240
E.6 Circuit to interface Rx with FPGA in KeeLoq transceiver. . . . . . . . 241
F.1 Setup for the S11 measurements below 45MHz. . . . . . . . . . . . . . 244
G.1 Schematic representation of inductive coupling. . . . . . . . . . . . . . 246
G.2 Schematic representation of capacitive coupling. . . . . . . . . . . . . . 247
G.3 Schematic representation of back scattering. . . . . . . . . . . . . . . . 248
G.4 Schematic representation of radio transmission. . . . . . . . . . . . . . 248
G.5 Specifications of the pause in the PCD signal as defined in ISO14443 . 251
G.6 Schematic of the building blocks of the altered RFID system . . . . . 251
H.1 The integration over the area of the loop. . . . . . . . . . . . . . . . . 254
H.2 Inductance calculated with integral and with Grover’s formula. . . . . 254
I.1
Lowest resonance frequency as function of cable length. . . . . . . . . 259
I.2
Resonance frequency as function of cable length for different k. . . . . 260
I.3
Experimental validation of theory on standing waves. . . . . . . . . . . 262
invisible filling
List of Tables
2.1
Comparison of phase steering at RF, IF and BB. . . . . . . . . . . . .
17
3.1
Shading windows and their properties. . . . . . . . . . . . . . . . . . .
44
3.2
Numeric values of the DFT pair for Chebychev tapering synthesis. . .
45
3.3
Excitation coefficients of the ATCRBS subarray. . . . . . . . . . . . .
47
4.1
Orbit characteristics for Low Earth Orbits. . . . . . . . . . . . . . . .
61
4.2
Numeric example for link budget. Input left, output right. . . . . . . .
66
4.3
Temperature cycles for LEO satellites (αT /T = 1). . . . . . . . . . . .
72
5.1
Some examples of the available r for RF substrates. . . . . . . . . . .
81
5.2
Overview of the different PTFE based substrates. . . . . . . . . . . . .
82
5.3
Overview of some commonly used metals. . . . . . . . . . . . . . . . .
84
5.4
Weights and corresponding heights for copper.
. . . . . . . . . . . . .
84
5.5
Some substrates that meet the specs concerning outgassing. . . . . . .
86
5.6
TCr from data sheets compared with measurements. . . . . . . . . .
87
5.7
∆ue for linear arrays with uniform tapering. . . . . . . . . . . . . . . .
92
5.8
dopt compared to results of an optimization search. . . . . . . . . . . .
94
5.9
Optimal d in dipole arrays from near and far field iterative search. . .
95
5.10 Orbit heights h and corresponding scan angles θmax . . . . . . . . . . .
98
xxiii
xxiv
List of Tables
5.11 Optimal d for arrays of concentric circles scanned to broadside. . . . . 100
5.12 Optimal d for regular grid arrays scanned to broadside. . . . . . . . . 100
6.1
Geometrical and electrical parameters of the patch. . . . . . . . . . . . 106
6.2
Components in signal path and controller. . . . . . . . . . . . . . . . . 120
8.1
Overview of the relevant probe specs. . . . . . . . . . . . . . . . . . . . 138
8.2
Measured DC resistance values for the four loop types. . . . . . . . . . 148
8.3
Advantages and disadvantages of the four loop types. . . . . . . . . . . 149
8.4
Self-resonance frequency of loops with and without insulation. . . . . . 162
8.5
Characteristics overview of the RFID reader loops. . . . . . . . . . . . 168
9.1
Measured noise contributions for an oscilloscope. . . . . . . . . . . . . 190
9.2
Components selected for the class E amplifier . . . . . . . . . . . . . . 201
D.1 Comparison of screen reconstruction methods. . . . . . . . . . . . . . . 234
I.1
The properties of the different cables. . . . . . . . . . . . . . . . . . . 261
List of publications
International Journals
• W. Aerts and G.A.E. Vandenbosch, “Gain Enhancement by Optimizing Interelement Spacing in Linear Array Antennas,” Microwave and Optical Technology Letters, vol. 43, no. 4, 20 November 2004
JCR 2004 Impact Factor 0.456.
• M. Vrancken, Y. Schols, W. Aerts and G.A.E. Vandenbosch, “Benchmark of full
Maxwell 3-dimensional electromagnetic field solvers on prototype cavity-backed
aperture antenna,” AEU - International Journal of Electronics and Communications, doi:10.1016/j.aeue.2006.07.001, July 2006
SCIE 2004 Impact Factor 0.483.
• M. Vrancken, W. Aerts, Y. Schols and G.A.E. Vandenbosch, “Benchmark of
Full Maxwell 3D Electromagnetic Field Solvers on an SOIC8 Packaged and
Interconnected Circuit,” International Journal of RF and Microwave ComputerAided Engineering, vol. 64, issue 6, 1 June 2007
JCR 2007 Impact Factor 0.291.
• W. Aerts, E. De Mulder, B. Preneel, G. Vandenbosch, and I. Verbauwhede,
“Dependence of RFID Reader Antenna Design on Read Out Distance,” IEEE
Transactions on Antennas and Propagation, vol. 56, issue 12, 1 December 2008
ISI 2007 Impact Factor 1.636.
• W. Aerts, P. Delmotte and G. Vandenbosch, “Conceptual Study of Analog Baseband Beam Forming: Design and Measurement of an Eight-by-eight Phased
Array,” IEEE Transactions on Antennas and Propagation, vol. 57, issue 6, 1
June 2009
ISI 2007 Impact Factor 1.636.
xxv
xxvi
List of publications
National Journals
• W. Aerts and G.A.E. Vandenbosch, “Optimal Element Spacing in Linear Arrays
for Satellite Communication,” HF Revue, Belgian Journal of Electronics and
Communications, no. 2, p. 30, 2004.
International Conferences
• W. Aerts and G.A.E. Vandenbosch, “Optimal Inter-Element Spacing in Linear
Array Antennas and its Application in Satellite Communications,” Proc. of
34th European Micorwave Conference (EuMC), Amsterdam, Nederland, 12-14
October 2004
• P. Delmotte, W. Aerts, V. Volski, S. Mestdagh and G.A.E. Vandenbosch, “Measurement Results of a Phased Array with a New Type of Phase Shifter”, Proc.
28th ESA Antenna Workshop on Space Antenna Systems and Technologies, Noordwijk, Nederland, 31 May-3 June 2005
• V. Volski, W. Aerts, A. Vasylchenko and G.A.E. Vandenbosch, “Composite
Textiles Filled with Arbitrarily Oriented Conducting Fibres using a Periodic
Model for Crossed Strips”, Proc. International Conference on Mathematical
Methods in Electromagnetic Theory 2006, Kharkiv , Ukraine, 26 June-1 July
2006
• W. Aerts, E. De Mulder, B. Preneel, G.A.E. Vandebosch and I. Verbauwhede,
“Matching Shielded Loops for Cryptographic Analysis,” Proc. of 1st European
Conference on Antennas and Propagation (EuCAP) 2006, Nice, France, 6-10
November 2006
• W. Aerts and G.A.E. Vandenbosch, “Reflections on Doppler Shift Compensation by Frequency Scanning”, 30th ESA Antenna Workshop on Antennas for
Earth Observation, Science, Telecommunication and Navigation Space Missions,
Noordwijk, Nederland, 27-30 May 2008
• W. Aerts, E. De Mulder, B. Preneel, G.A.E. Vandebosch and I. Verbauwhede,
“Designing Maximal Resolution Loop Sensors for Electromagnetic Cryptographic
Analysis,” Proc. of 3rd European Conference on Antennas and Propagation (EuCAP) 2009, Berlin, Germany, 23-27 March 2009
• E. De Mulder, W. Aerts, B. Preneel, G.A.E. Vandebosch and I. Verbauwhede,
“A Class E Power Amplifier for ISO-14443A,” Proc. of IEEE Symposium on
Design and Diagnostics of Electronic Circuits and Systems (DDECS) 2009,
Liberec, Czech Republic, 15-17 April 2009
List of publications
xxvii
National Conferences
• W. Aerts and G.A.E. Vandenbosch, “The Influence of Array Geometry,” Proc.
K.U.Leuven Faculty of Engineering PhD Symposium, Leuven, Belgium, p. 44,
11 December 2002.
• W. Aerts and G.A.E. Vandenbosch, “Study of Array Factor with Fourier Transform,” Proc. 10th URSI Forum, Brussels, Belgium, 13 December 2002.
• W. Aerts and G.A.E. Vandenbosch, “Optimal Element Spacing in Linear Arrays
for Satellite Communication,” Proc. 11th URSI Forum, Brussels, Belgium, p.
41, 18 December 2003.
• W. Aerts and G.A.E. Vandenbosch, “Choosing Substrates for Space Applications,” Proc. 12th URSI Forum, Brussels, Belgium, p. 50, 10 December 2004.
invisible filling
Nomenclature
αi
Inclination Angle of the Satellite Orbit, page 223
αl
Latitude Coordinate of the Ground Station, page 223
αT
Thermal Absorption Coefficient, page 72
α
Elevation Angle of Satellite above Horizon, page 62
β
N
Wave Number (β = 2π/λ), page 26
Convolution, page 36
∆ue
Half of Width of Main Lobe in u for d = λ, page 91
δ3
3D Dirac Delta Function, page 30
δADC
Step Size of ADC [V], page 189
Dielectric Permittivity ( = 0 r ), page 18
0
Dielectric Permittivity of vacuum (8.85419 × 10−12 F/m), page 18
T
Thermal Emissivity Coefficient, page 71
η
Antenna Efficiency, page 28
ηA
Array Tapering Efficiency, page 46
λ
Wavelength [m], page 63
µ
Magnetic Permeability (µ = µ0 µr ), page 18
µ0
Magnetic Permeability of vacuum (4π × 10−7 N/A2 ), page 18
ωc
Carrier pulsation (ωc = 2πfc ), page 14
ωe
Sidereal Rotation of the Earth (7.2925 × 10−5 rad/s), page 61
⊕
Exclusive Or, page 129
xxix
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Nomenclature
φ
Angle of the Azimuth Beam Direction, page 27
Π(x)
Rectangular Function, page 36
ψ
Arc, page 61
<
Real Part of a Complex Number, page 31
tan δ
Loss Tangent of a Substrate, page 81
θ
Angle of the Elevation Beam Direction, page 27
θDOA Angle between a propagation vector and an array indicating the DOA, page 9
δ̃
Phase Difference between Two consecutive Elements, page 19
~
A
Vector Potential, page 26
~bn
Translation Vector for the nth Element, page 26
~
E
Electric Field, page 27
~
H
Magnetic Field, page 27
~ir
Unit Vector along r-axis (Spherical Coordinates), page 27
J~
Current Density, page 26
r~0
Vector Source Coordinate, page 27
~r
Vector Observation Coordinate, page 27
∗
Complex Conjugate, page 34
A
Area [m2 ], page 63
a
Geometry Factor of the Geometric Array, page 38
an
Complex Excitation Coefficient for the nth element, page 26
Aeff
Effective Area [m2 ], page 63
B
Bandwidth [Hz], page 64
C
Capacitance [F], page 81
c
c0
√
√
Speed of light in a medium, equal to c = 1/ µ = c0 / µr r , page 9
√
Speed of light in vacuum (c0 = 1/ 0 µ0 = 2.997925 × 108 m/s), page 18
Cc
Channel Capacity [bit/s], page 64
Ctt
Inter Turn Capacitance [F], page 161
Nomenclature
d
Inter element spacing, page 9
dt
Thickness of the Substrate Dielectric [m], page 81
Dant
Antenna Directivity, page 27
ddip
Optical Inter Element Spacing in Arrays of Dipoles, page 95
dff
Optical Inter Element Spacing from Far Field Calculation, page 95
dnear
Optical Inter Element Spacing from Near Field Calculation, page 95
dopt
Optical Inter Element Spacing, page 91
Eb
Energy per Bit [J], page 66
F
Array Factor, page 27
fc
Carrier frequency, page 14
fH
Upper Working Frequency of the Sensor [Hz], page 141
fL
Lower Working Frequency of the Sensor [Hz], page 142
fres
Resonance Frequency [Hz], page 161
Fg
Gravitational Force [N], page 60
G
Constant of Gravitation (6.67 × 10−11 Nm2 /kg2 ), page 60
g
Acceleration of Gravity (9,81 m/s2 ), page 70
Gr
Gain of the Receiving Antenna, page 63
Gs
Gain of the Transmitting Antenna, page 63
Gant
Antenna Gain, page 28
Gatm
Attenuation Factor of the Atmosphere, page 63
Gf s
Free Space Path Loss, page 63
h
Height [m], page 61
hp
Planck’s Constant (6.61 × 10−34 Js), page 74
Il
Loop Current [A], page 157
k
Boltzmann’s Constant (1.38 × 10−23 J/K), page 65
L
Loop Inductance [H], page 156
l
Length [m], page 67
xxxi
xxxii
Nomenclature
L
Moment of Inertia of a Cube, page 67
LM
Moment of Inertia of a Prism, page 67
Lchoke Choke Inductance [H], page 194
m
Mass [kg], page 60
Me
Mass of the Earth, page 60
N
Number of Elements in the Array, page 26
N0
Power Spectral Density of Noise [W/Hz], page 65
nb
Number of Bits, page 48
Nt
Number of Turns of a Loop, page 157
NF
Amplifier Noise Figure, page 190
P
Power [W], page 63
Pn
Noise Power [W], page 64
Ps
Transmitted Power [W], page 63
Pin
Incoming Power [W], page 71
Pout
Outgoing Power [W], page 71
QRLC Quality Factor of an RLC Chain, page 164
R
Resistance [Ω], page 164
Rd
Distance Radius [m], page 63
rd
Read Out Distance [m], page 157
rl
Loop Radius [m], page 139
rw
Radius of a Wire [m], page 152
RDS(ON ) Transistor Drain-to-Source Resistance in Saturation [Ω], page 195
Re
Radius of the Earth [m], page 61
Ro
Radius of the Orbit [m], page 61
Rsl
Chebychev Side Lobe Level, page 44
Sxy
Scattering parameter from port x to port y., page 143
T
Temperature [K], page 71
Nomenclature
Tm (u) Chebychev Polynomial of order m, page 45
Tn
Noise Temperature [K], page 65
v
Speed [m/s], page 71
Vl
Voltage over a Loop [V], page 161
vcab
Signal Speed in Cable [m/s], page 189
Vcc
Power Supply Voltage [V], page 195
Vsat
Saturation Voltage of Transistor [V], page 195
w(~r)
Windowing (Shading or Tapering) Function of the Array, page 40
Z0
Free Space Wave Impedance (Z0 ≈ 120π), page 139
Zc
Characteristic Impedance of a Transmission Line [Ω], page 81
Zs
Oscilloscope Input Impedance [Ω], page 257
Zin
Input Impedance of a Port [Ω], page 151
F
Fourier Transformation, page 30
xxxiii
invisible filling
List of Acronyms
ACR Anomalous Component of Galactic Cosmic Radiation
ADC Analog to Digital Converter
ASIC Application Specific Integrated Circuit
ATCRBS Air Traffic Control Radar Beacon System
BAP Battery Assisted Passive RFID Tag
BB BaseBand
BBM Bread Board Model
BER Bit Error Rate
BNC Bayonet Neill-Concelman
CME Coronal Mass Ejection
CMOS Complementary Metal Oxide Semiconductor
COTS Commercial Off-The-Shelf
CPU Central Processing Unit
CRAND Cosmic Ray Albedo Neutron Decay
CSP Chip Scale Package
CTE Coefficient of Thermal Expansion
DAC Digital to Analog Converter
DFT Discrete Fourier Transformation
DOA Direction Of Arrival
DOD Direction of Departure
xxxv
xxxvi
List of Acronyms
EMA ElectroMagnetic Analysis
EMC Electromagnetic Compatibility
EM ElectroMagnetic
EMP Electro Magnetic Pulse
ESA European Space Agency
ESD ElectroStatic Discharge
FEC Forward Error Correction
FM Flight Model
FPGA Field Programmable Gate Array
GCR Galactic Cosmic Radiation
GEO Geostationary Earth Orbit
GSM Global System for Mobile Communications
IDFT Inverse Discrete Fourier Transform
IFF Identify Friend or Foe
IF Intermediate Frequencies
I/O Input Output
IP International Protection Rating
ISI Inter Symbol Interference
JERS Japanese Earth Resources Satellite
LEO Low Earth Orbit
LESA Linear Equally Spaced Array
LNA Low Noise Amplifier
LO Local Oscillator
LOS Line of Sight
LUT LookUp Table
MAGMAS Model for the Analysis of General Multilayered Antenna Structures
MEMS Micro Electro Mechanical System
List of Acronyms
MIMO Multiple Input Multiple Output
MSG Meteosat Second Generation
NASA National Aeronautics and Space Administration
NLF Non Linear Function
NSA National Security Agency
PA Product Assurance
PCB Printed Circuit Board
PE PolyEthylene
PhD Doctor of Philosophy
PIM Passive Inter Modulation
PLL Phase Locked Loop
PoE Power over Ethernet
PPL Preferred Parts List
PSD Power Spectral Density
PTFE PolyTetraFluoroEthyleen (or Teflon)
QAM Quadrature Amplitude Modulation
QEM Quality Engineering Model
QPL Qualified Parts List
QPSK Quadrature Phase Shift Keying
RAM Random Access Memory
RFID Radio-frequency identification
RF Radio Frequencies
ROM Read Only Memory
RTFM Read The Manual
RTF Reader Talks First
Rx Receive
SAA South Atlantic Anomaly
xxxvii
xxxviii
SCADA Supervisory Control And Data Acquisition
SDR Software Defined Radio
SKA Square Kilometer Array
SNR Signal to Noise Ratio
SOIC8 Small Outline Integrated Circuit with 8 pins
SSB Single Side Band
SVGA Super Video Graphics Array
TDMA Time Division Multiple Access
TTC Telemetry, Tracking and Command
TV TeleVision
Tx Transmit
UART Universal Asynchronous Receiver/Transmitter
UV UltraViolet
UWB Ultra Wide Band
VGA Variable Gain Amplifier
VHDL VHSIC Hardware Description Language
VHSIC Very High Speed Integrated Circuit
VNA Vector Network Analyzer
WISE Wide Band Sparse Elements
XOR eXclusive OR
List of Acronyms
Nederlandse samenvatting
Toepassingsspecificiteiten bij Roosterantennes:
Satellietcommunicatie en
Electromagnetische Nevenkanaalsanalyse
Inleiding
In dit werk zullen twee totaal verschillende toepassingen van roosterantennes bestudeerd worden. De bestudeerde roosterantennes zijn klassieke roosterantennes, waar
alle elementen van de roosterantenne translaties zijn van een basiselement en alle
elementen hetzelfde signaal delen. Dit sluit multiple input multiple output (MIMO)
systemen en gekromde roosterantennes uit. De twee toepassingen die hier worden
uitgewerkt, zijn: een roosterantenne voor het oppikken met een satelliet van in-situ
verzamelde meetgegevens, en een roosterantenne voor gebruik bij nevenkanaalsanalyse van cryptografische toestellen. Maar eerst wordt een overzicht gegeven van de
klassieke theorie over roosterantennes.
Theorie van Roosterantennes
De essentiële werking van roosterantennes is uit te leggen doordat alle elementen in de
roosterantenne hetzelfde signaal oppikken, maar lichtjes verschoven in de tijd. Inderdaad zal een signaal op licht verschillende tijdstippen aankomen bij de verschillende
elementen in de roosterantenne. Dit tijdverschil zal afhangen van de richting van
waaruit het signaal komt. Bijgevolg zal het optellen van de verschillende signalen
van de verschillende elementen, na een inverse verschuiving in de tijd, terug het oorspronkelijke signaal opleveren. Bovendien zullen op die manier de signalen afkomstig
uit andere richtingen niet constructief worden opgeteld, maar zullen ze zich eerder
(gedeeltelijk) uitdoven.
xxxix
xl
Nederlandse samenvatting
Dit verschuiven in de tijd is equivalent met een faseverschuiving in het geval het
signaal een sinusfunctie is. Daarom wordt dikwijls gesproken over fasegestuurde roosterantennes en wordt als benadering het gedrag van de roosterantenne bestudeerd
aan de hand van de karakteristieken van de roosterantenne op de draaggolffrequentie.
Deze benadering is alsmaar juister naarmate de bandbreedte van het signaal meer
naar nul gaat.
Zowel het verschuiven in de tijd als het draaien van de fase (van de draaggolf) worden
in de praktijk gebruikt om de signalen klaar te maken alvorens ze op te tellen of samen
te voegen. Hoewel enkel het tijdsverschuiven op de radiofrequenties exact is, wordt,
omwille van kostprijs of ontwerpgemak, ook vaak tijdsverschuiven op intermediaire
frequenties of faseverschuiven op radio-, intermediaire- of basisbandfrequenties gebruikt. Zowel analoge als digitale implementaties zijn in gebruik. De digitale hebben
het voordeel dat ze zeer flexibel zijn en bovendien met off-line verwerking de mogelijkheid bieden om de ganse ruimte te onderzoeken. Het kan dan ook niet verbazen
dat vele militaire radars dit systeem gebruiken.
Naast tijds- en faseverschuivingen, kan ook het variëren van frequentie gebruikt worden om de richting van constructieve interferentie te veranderen. Vermits dit echter
het wijzigen van de draaggolffrequentie inhoudt, moet er een apart kanaal zijn waarover zender en ontvanger hun frequenties op elkaar kunnen afstellen. Het is duidelijk
dat deze beperking het gebruik in de praktijk in de weg staat. We zien dan ook
de techniek vooral opduiken in toepassingen waarbij de zender en ontvanger fysisch
dezelfde locatie delen, zoals bij radar of andere types sensorroosters.
Om een roosterantenne te bestuderen, zoals reeds gezegd, kan in eerste instantie best
de bandbreedte nul worden verondersteld, zodat tijdsverschuiving en faseverschuiving identiek zijn. Uit de wetten van Maxwell blijkt dan dat de eigenschappen van
de roosterantenne in zijn geheel, kunnen worden opgesplitst in een bijdrage afkomstig
van het roosterelement, en een bijdrage afkomstig van de opbouw en aansturing van
het rooster. De invloed van de positie van de roosterelementen en van de amplitude
en fase van het signaal waarmee elk element wordt aangestuurd, kan eenvoudig bestudeerd worden via een Fouriertransformatie. Daaruit blijkt dan dat hoe groter het
gebied is dat ingenomen wordt door de roosterantenne, hoe kleiner de ruimtehoek is
waarbinnen de straling (voor zenden) of gevoeligheid (voor ontvangen) maximaal is.
Verder blijkt dat een kleinere singaalamplitude naar de randen van de roosterantenne
toe resulteert in lagere zijlobniveaus, maar een bredere ruimtehoek voor het maximum. De interpretatie met de kenmerken van de Fouriertransformatie bevestigt ook
de intuitieve uitleg over het wijzigen van de richting van constructieve interferrentie
via het lineair laten oplopen van de fase over de roosterantenne.
Nederlandse samenvatting
xli
Satellietcommunicatie
Bij satellietcommunicatie, waar de afstand tussen zender en ontvanger algauw honderden tot duizenden kilometers bedraagt, is het gebruik van directieve antennes zonder
meer aan te raden. Het heeft immers geen zin om vermogen in alle richtingen uit te
stralen, daar in de meeste richtingen toch alleen maar een lege ruimte gaapt.
Natuurlijk hoeft die directieve antenne niet perse een roosterantenne te zijn. Een
paraboolantenne heeft bijvoorbeeld een hogere winst voor dezelfde ingenomen oppervlakte. Maar het nadeel van dit soort directieve antennes is dat ze mechanisch naar
de zender of ontvanger gericht moeten worden. Dit is in sommige gevallen te traag,
zodat bijvoorbeeld de IRIDIUM satellieten toch voorzien zijn van een roosterantenne.
Bovendien veroorzaakt het slijtage en trillingen, iets wat aan boord van een satelliet
voor (optische) aardobservatie absoluut uit den boze is, omdat het de ruimtelijke
resolutie van de (beeld)sensor aanzienlijk vermindert.
Gezien de hoge kost, van materialen en ontwerp, van elektronica voor toepassing in
de ruimte, omwille van de vijandige atmosfeer en de onmogelijkheid tot reparatie, kan
best getracht worden om zoveel mogelijk complexiteit van het systeem te implementeren in het grondsegment en zo weinig mogelijk in het ruimtesegment. Een duidelijk
voorbeeld is het overhevelen van antennewinst naar de antenne van het grondstation, immers in de radiovergelijking is enkel het product van de winst van zend- en
ontvangstantenne van belang. Echter, in een systeem met zeer veel grondstations (of
aardse gebruikers) kan het economischer zijn om de antenne van het grondstation zo
goedkoop mogelijk te houden, gezien het grote aantal.
Hoewel het voor aardobservatiesatellieten een voordeel is als ze veel gebied overvliegen, is dit voor communicatie niet erg handig. Gezien de grote snelheid zal het
tijdsvenster waarin communicatie mogelijk is, zeer beperkt zijn. Bijgevolg dient de
hoeveelheid informatie die overgebracht moet worden best klein gehouden te worden,
of moet getracht worden de signaal-tot-ruisverhouding zeer groot te maken, met veel
zendvermogen of zeer grote antennewinsten. In het kader van de eerste oplossing,
kan het systeem gezien worden dat in de tekst wordt uitgewerkt: een systeem om
in-situ verzamelde data naar een aardobservatiesatelliet te sturen zodat die meteen
kan gebruikt worden om de gegevens van de beeldsensor te verwerken en te reduceren
vóór transmissie naar het grondstation.
Dit systeem is een typisch communicatiesysteem met een klassieke toepassing van
roosterantennes, op het vereisen van functioneren in de ruimte na, en biedt een mooie
gelegenheid om bijvoorbeeld het berekenen van een linkbudget of ontwerpen van een
sturing voor een roosterantenne te illustreren. De toepassing in de ruimte vereist
speciale aandacht wat betreft het kiezen van het antennesubstraat. Dit substraat moet
immers in vacuüm en bij sterk variërende temperaturen nog steeds zijn eigenschappen
houden en er mag geen breuk optreden, noch van (de metalen vlakken of baantjes op)
het substraat, noch van probes, pootjes van componenten of vias.
xlii
Nederlandse samenvatting
De roosterantenne moet ontworpen worden voor maximale winst onder lage elevatie
van de satelliet ten opzichte van de horizon van het grondstation. Dit behelst eigenlijk het zoeken van de tussenafstand in een roosterantenne waarvoor het totale
uitgestraalde vermogen zo klein mogelijk is voor een zelfde maximum in een bepaalde
richting. Voor lineaire roosterantennes van isotrope stralers kan dit berekend worden.
In essentie wordt de tussenafstand van de elementen in het rooster zo gekozen dat de
hoofdlob zo smal mogelijk is en er juist geen tweede hoofdlob in het zichtbare interval
opduikt. Voor lineaire roosterantennes van dipolen werkt de uitgewerkte techniek ook
nog aanvaardbaar. De techniek is echter niet eenvoudig uit te breiden naar planaire
roosterantennes.
Voor de fasedraaiers werd eerst aan een analoge implementatie van een veelgebruikte
digitale techniek gedacht. Met de steeds sneller en nauwkeuriger wordende digitale
logica, ligt het echter meer voor de hand om rechtstreeks die digitale implementatie
te gaan gebruiken. Enkel als de roosterantenne zeer veel elementen heeft of als de
bandbreedte van het systeem zeer groot is, biedt de analoge uitvoering voordelen.
Electromagnetische Nevenkanaalsanalyse
De hele wetenschap van nevenkanaalsanalyse onstond toen men midden de jaren negentig van vorige eeuw plots besefte dat, hoewel een algoritme perfect wiskundig veilig
kan zijn, het misschien kan gekraakt worden door te kijken naar nevenkanalen, zoals
tijdsduur van de berekeningen, vermogenverbruik of electromagnetische straling van
de implementatie van het algoritme. Cryptografische algoritmes kunnen bijvoorbeeld
geı̈mplementeerd worden op een CPU van een computer, een FPGA, een microcontroller of een ASIC.
Inderdaad zullen stromen die lopen in een logische schakeling om registers te schrijven of logische bewerkingen te doen, ook electromagnetische velden uitstralen. Deze
velden zullen gemiddeld genomen zeer klein zijn in vergelijking met het veld afkomstig van bijvoorbeeld klokken in het systeem. Maar soms kan door een fout tegen de
ontwerpregels om overspraak te beperken, door die kleine stroompjes een veel sterker
signaal worden gemoduleerd. Observatie van dit gemoduleerde signaal kan dan veel
gevoelige informatie opleveren.
Maar zelfs als het ontwerp perfect gebeurt en er dus geen observeerbare modulatie van
informatie op sterkere dragers plaatsvindt, kan nog steeds de directe straling van de
kleine stroompjes die de logische bewerkingen uitvoeren, worden geobserveerd. Voor
dergelijke meting zal de sensor zeer dicht tegen de bron van straling geplaatst moeten
worden. De weinig wetenschappelijke manier waarop deze metingen meestal worden
uitgevoerd, is dat men met de sensor een beetje in het rond beweegt en op enkele
plaatsen een aanval probeert uit te voeren. Het toepassen van een roosterantenne
zou deze tijdrovende voorzoekstap uitsparen en bovendien metingen van verschillende
plaatsen verschaffen die, gecombineerd, waarschijnlijk meer informatie opleveren.
Nederlandse samenvatting
xliii
Maar alvorens tot een rooster te komen, moet eerst het roosterelement ontworpen
worden. Dit moet een antenne zijn die (vooral) gevoelig is aan magnetische velden,
vermits deze het sterkst zullen zijn rondom een cryptografische chip gezien het grote aantal kleine stroomlussen. Het moet, met andere woorden, een lusantenne zijn.
Enkele voorbeelden van lusantennes zijn antennes voor (inductieve) RFID of afgeschermde lusantennes, typisch gebruikt in EMC remediëring en certifiëring. Maar de
antenne in het sensorrooster zal een spiraalantenne zijn. Het aantal wikkelingen wordt
volledig bepaald door de veldsterkte, de gewenste signaalamplitude en bandbreedte.
Het rooster zelf is een honingraat van spiraalsensoren, om ieder stukje van de onderliggende chip te kunnen observeren. De grootste uitdaging zal echter het uitbrengen van
de sensorsignalen voor verdere verwerking zijn. Inderdaad is er fysisch plaats nodig
om signalen over een degelijke golfgeleider naar buiten te brengen. Er zou dus kunnen gedacht worden aan geı̈ntegreerde signaalverwerking om de hoeveelheid kanalen
te reduceren, bijvoorbeeld door tijds-multiplexing of een digitale voorbewerking. Inderdaad moet het signaal immers toch gedigitaliseerd worden om een cryptografische
analyse met behulp van stochastische technieken uit te voeren. Daarbij moet echter
opgelet worden dat niet net die informatie die cruciaal is voor de nevekanaalsanalyse
wordt weggegooid. Bepalen wat dan die cruciale informatie is, is een zeer moeilijke
opgave.
In die context is het ook onduidelijk waaraan een goede meetopstelling voor nevenkanaalsanalyse moet voldoen. Het streven naar een optimale signaal-tot-ruisverhouding
lijkt logisch van het uitgangspunt van informatietheorie. Maar omdat voor vele aanvallen slechts correlatie wordt gezocht op een welbepaald tijdstip, dient eigenlijk enkel
de signaal-tot-ruisverhouding op dat welbepaalde tijdstip geoptimaliseerd te worden.
Dit desnoods door reflecties te introduceren, wat resulteert in een lagere signaal-totruisverhouding over de tijd uitgemiddeld.
Enkele middelen om de signaal-tot-ruisverhouding te verbeteren, zijn filteren, vooral handig als een gemoduleerd signaal wordt bestudeerd, of versterken, wat nodig is
om ervoor te zorgen dat het signaal het totale ingangsbereik van de analoog-naardigitaalomzetter bestrijkt. Versterkers kunnen niet alleen gebruikt worden om het
signaal van een sensor te versterken. Ze kunnen ook ingezet worden om met een
lusantenne een fout te injecteren in systemen. Deze techniek wordt gebruikt bij foutaanvallen, die verwant zijn met nevenkanaalsaanvallen omdat ook hier de implementatie wordt aangevallen en niet het wiskundige algoritme. Over de noodzaak van het
voorkomen van reflecties voor een goede signaal-tot-ruisverhouding doorheen de tijd
bestaat geen twijfel, maar het is niet eenduidig te zeggen of dit ook noodzakelijk is
voor nevenkanaalsanalyse. In de meeste gevallen wel, maar sommige omstandigheden
kunnen leiden tot een beter resultaat in geval van reflecties.
xliv
Nederlandse samenvatting
Besluit
Roosterantennes hebben vele toepassingen. Hier werden er twee besproken, die buiten het feit dat de elementen translaties zijn van een basiselement, en dat steeds
de driedeling elementen, signaalconditionering en -combinatie terugkomt, weinig gemeenschappelijk hebben. De verschillen springen meer in het oog. Bij een typische
telecommunicatietoepassing zijn de signalen smalbandig, bij nevenkanaalsanalyse algemeen gesproken niet. Ook zal een signaalbron zich typisch in het verre veld bevinden voor een telecommunicatietoepassing. De afstand tussen bron en sensorrooster
daarentegen is doorgaans klein. Bijgevolg kan in dit laatste geval de roosterbenadering niet gebruikt worden en kan de invloed van element en roostergeometrie op de
roostereigenschappen niet zondermeer gesplitst worden. Ook is de afstand tussen de
elementen in het sensorrooster veel kleiner, dit wederom omwille van het feit dat de
signaalbron zich niet in het verre veld bevindt. Tot slot wordt bij een telecommunicatietoepassing doorgaans naar een optimale signaal-tot-ruisverhouding doorheen de
tijd gestreefd, terwijl voor nevenkanaalsanalyse enkel de signaal-tot-ruisverhouding
op een welbepaald tijdstip van belang is.
Chapter 1
Introduction
1.1
Array Antennas
With the ever increasing importance of wireless communication, antennas have occupied a prominent position in everyday’s live. Though antennas are no more than
a transition between a guided wave on a transmission line system and a radiated
wave, many different ways of implementing this transition are in use. Indeed every
application has its specific demands regarding bandwidth, antenna size and radiation
directivity, resulting in a different antenna.
As a Fourier transform relates the currents on an antenna with the radiation pattern, Sect. 3.2, it can be understood that in order to obtain a radiation pattern that
concentrates nearly all power in one spatial direction, the current carrying antenna
area should be large. One example of such antennas, are parabola dish antennas.
Another way of enlarging the antenna area, is (periodically) placing distinct antennas
into some array configuration, filling a much larger space. This however does not
allow current flowing over the entire area, as the current can only flow in the antenna
conductors, not in the space between the separate antennas. The disadvantage is
that in this way less spatial resolution can be obtained. The array of antennas, or
array antenna, has a wider main beam in its radiation pattern than an antenna of
the same size that allows current to flow over its entire area. Array antennas however
allow varying the direction of the main beam in an electronic way, without mechanical
movements and its associated problems such as wear and slow reconfiguration.
From an antenna point of view, it is natural to explain increased spatial resolution
of a structure of separate antennas based on the larger antenna area. From a signal
processing point of view, array antennas use multiple elements to sample a signal with
a spatial diversity in order to increase signal quality. Both approaches are equivalent.
1
2
Chapter 1. Introduction
This work will only discuss the classical array antennas where each radiating element
is a translation of a base element, which is not the case for e.g. conformal arrays, and
where all array elements share the same signal, as opposed to the situation in true
Multiple Input Multiple Output (MIMO) systems.
Indeed, conformal arrays are generally spoken planar arrays that are bent and attached to a surface with a certain curvature such as the body of an airplane, or the
hull of a ship. Consequently, the array elements are no longer translations of the
base element, but rotations are necessary to obtain all elements from the base element. These rotations void the assumptions made in order to be able to split element
and location effects on the radiation properties of the array antenna, resulting in the
definition of the array factor. Hence classical array theory does not apply to conformal array antennas, but fortunately literature, e.g. [1], is available on analysis and
synthesis of this specific array type.
Though literally spoken, any system using multiple antennas on both receiver and
transmitter can be regarded as a MIMO system, true MIMO supposes the use of
multiple channels. More specific, the information that is to be transmitted is divided
into several streams that are then divided in some way over (a subgroup of the)
multiple antennas [2, 3]. MIMO counts on the orthogonality of the channels, due
to the scattering and fading in the environment, and to the design of the sets of
receiving and transmitting antennas, to improve link capacity. Hence the antennas
in receiver and transmitter array do not share the same signal and moreover should
preferably not be translations of a base element, but have different radiation patterns
or polarizations and should preferably be spaced far apart [4].
Still of the classical phased arrays, an abundance of examples can be found in every
day life. Without having the intention of being exhaustive, a small enumeration of
systems that were studied, to a lesser degree or greater extent, during this work, is
given below:
• Communication: Global System for Mobile communication (GSM) base station
antennas (Gamma Nu EDBDP-900F/1800-17-65), satellite antennas for mobile
communication (IRIDIUM Sect. 4.4.2)
• RAdio Detection And Ranging (RADAR): missile detection radars (Thales
SMART-L), earth observation radars (RADARSAT Sect. 4.4.2), air traffic control radars (Secondary RADAR in Bertem Sect. 3.3.2.3),
• Radio Astronomy: deep space probing (Square Kilometer Array (SKA) [5])
1.2. Outline of this Thesis
1.2
3
Outline of this Thesis
This thesis tackles array antennas used for yet two other selected, very different applications. The first application, satellite communication, explained in detail in
Chapter 4, is a typical communication application. Bandwidths are generally spoken
small and all standard telecommunication engineering methods are valid. Designing
for space, however, requires special attention due to the hostile environment. Consequently, in the design of a system for up link of in-situ collected data to an earth
observation satellite, much effort was spent on material and component selection.
Another interesting peculiarity of the design, was the application of an analog base
band implementation of a technique often used for digital beam forming.
Electromagnetic side channel analysis of cryptographic hardware, which is the second application and is discussed in depth in Chapter 7, requires an approach sometimes very different from standard telecommunication engineering methods. When
observing direct radiation of small currents performing cryptographic operations in
silicon hardware, the antennas are designed to be small and sensitive to magnetic
fields. Matching is not performed in order to assure power transfer, but to obtain a
high signal-to-noise ratio. The signal should be digitized with as less quantization error as possible to allow calculation of correlation with a hypothesis in post-processing.
Array antennas should perform beam forming on very wide band signals and preferably off-line to allow simultaneous monitoring of different active regions in the chip.
Covering two very distinct applications, allows to point out what aspects of array
theory and practice are application independent, and how applications will alter the
array design. Hence, besides the very technical treatments in this work, this leaves
some room for meta-discussions on the topic of array antennas in a more philosophical
way, which is an inevitable prerequisite for obtaining a Doctor of Philosophy (PhD)
degree.
In general, as explained at the beginning of Part I, the array antenna consists of:
• antenna elements,
• a signal shaping device for each antenna, to ensure constructive summing with
signals from other array elements,
• and the summing or combining network.
Obviously those three items will be discussed for both applications covered in this
work.
But firstly the general theory of array antennas is reviewed. At a sufficiently high
level of abstraction, some common possible ways of implementing beam forming will
be summed up. Then the mathematics of the array factor will demonstrate the
working principle of arrays and provide a basis for array topology design.
4
Chapter 1. Introduction
In a second part of this work, the application of array antennas for space communication is discussed. After a chapter of introduction to this application, with some
definitions, its relevance, an overview of the current state of the art and reference to
the general design approach for space and its pitfalls, following chapters will zoom in
on the three parts of an array antenna, mentioned above: the antenna element, the
signal shaping device and the combining network.
The second application, also chronologically, is treated in a third part of this work.
Again, a first chapter introduces the reader to the world behind this application: side
channel analysis of cryptographic hardware. Here, again, the three parts, namely
antenna element, signal conditioning for constructive interference and signal combination will make up the extensive treatment of the array for this application.
1.3
The Thesis at a Glance
Essentially, this work reviews much of the existing array theory and aims at indicating
what aspects of array theory and practice are application (in)dependent. This study is
performed by working out two applications in detail, namely satellite communication
and side channel analysis. Consequently, many problems and solutions at a smaller
scale, encountered in these two applications, are discussed throughout the work.
These smaller scale problems and solutions, in turn, are sometimes covered by existing
theory or common knowledge. When appropriate this existing material is reviewed.
Some new contributions, developed during the work reported here, are listed below,
to allow readers experienced in the field, to quickly browse to that parts that might
be of most interest to them.
• Sect. 4.1 describes a system design for in-situ data up link to an earth observation satellite. This system would allow, for the first time in earth observation
history, to retrieve data from locations other than those that are under observation with the imaging tool on board the satellite.
• Sect. 6.1 reports for the first time on an array that was designed and build by ir.
P. Delmotte to demonstrate an analog implementation of a technique commonly
used in digital beam forming.
• Sect. 6.2 adapts the design of Sect. 6.1 for usage on a satellite.
• Sect. 8.2 covers matching of several types of shielded loops.
• Sect. 8.3 generalizes the design of an RFID reader antenna, commonly described
in the literature for antennas that are small compared to the wavelength, to a
technique that is valid regardless of antenna size.
1.3. The Thesis at a Glance
5
• Sect. 8.4 calculates the maximal resolution of an inductive sensor based on
bandwidth and signal amplitude requirements.
• Appendix A elaborates on the possibility of compensating Doppler shift by
frequency scanning.
• Appendix D gives an improved signal processing technique to reconstruct an
image on a computer display after an interception of the radiation of the display
with an antenna.
The array content in Part III is very limited. While working on the side channel
analysis application, the lack of a good sensor element emerged. As the need for an
element, at the time of writing, was much stronger than the need for an array of
sensors, much effort was spent in studying and designing such element. As time is
however limited, this had its consequences on how much time could be invested in
studying arrays for this application.
invisible filling
Part I
Array Theory
7
9
Overview
In this part, the general theory, applicable to any classical phased array antenna,
where the elements are translations of a base element, is reviewed.
Array antennas, when used for reception, essentially sample an incoming wave at
distinct spatial points. By combining the slightly different signals captured at all
those points, the original signal can be reconstructed, while interferers coming from
other sources, at other locations, can be suppressed, resulting in a higher Signal to
Noise Ratio (SNR).
A straightforward example is an array antenna with all array elements equidistantly
spaced along a straight line (LESA as defined in 3.3.1.1.1) which is depicted in Fig. 1.1.
When a wave approaches the array, with its propagation vector under an angle1 θDOA
with the line of the array, the same signal will arrive at each of the array elements,
but with a time difference between two consecutive elements of:
d cos (θDOA )
∆t =
(1.1)
c
where d is the element spacing and c the speed of light.
d
d
d
d × cos (θDOA )
θDOA
d
Figure 1.1: The principle of beam steering.
If the direction the signal is coming from is known, this time delay can be compensated
for when recombining the signals of all separate array elements into one signal. It is
obvious that a signal s coming from any other direction, is partly cancelled by this
recombination, unless the signal has a large autocorrelation Rs (τ ) [6]:
Z T2
1
4
Rs (τ ) = lim
s(t) · s(t + τ )dt = hs(t) · s(t + τ )i
(1.2)
T →∞ T − T
2
for τ equal to ∆t of Eq. (1.1).
1 This
angle indicates the Direction of Arrival (DOA) of the signal.
10
When an array antenna is used for transmission, the direction of constructive interference, i.e. the direction in which most of the power will be radiated, can be found by
drawing spheres around each element, with different radius related to the time delay
applied to that specific element, see Fig. 1.2. The direction in which constructive interference takes place, is the direction in which (at infinity) all circles of the same set
(i.e. at the same point in the signal) intersect. The relation between time delay and
spatial direction of the maximum radiation is again expressed by Eq. (1.1). In fact,
any mathematical expression valid for an array when transmitting, is valid too in case
the array is used for reception, as long as the array contains no active components,
such as amplifiers.
ve
cti
u
r
st
on
fc
o
ion
ect
r
i
d
dir
er
erf
int
ce
en
ect
ion
of
con
str
uc
tiv
ei
nte
r fe
ren
ce
Figure 1.2: Spheres to determine main lobe for time delay beam forming.
Consequently, in this case too, the array properties, such as beam width and side
lobe level, not only depend on the array geometry and excitation, but also on the
signal that is transmitted or received, as they depend on the autocorrelation (or time
self-similarity) as defined in Eq. (1.2). Hence, derivation of these array properties
requires stochastic signals calculus. A priory knowledge of the stochastic2 signal,
such as bandwidth limitation, allows to limit the set of possible signals in the array,
such that the array properties for this set of possible signals can be derived.
2 Any communication signal will be stochastic, for it would not contain information if it were
deterministic.
11
If array properties are studied in the frequency domain, as in Chapter 3, which is
equivalent with a study in the time domain as both are linked via a Fourier transform,
time shifting translates into phase shifting. But here, again, all possible signals in the
array should be considered. It is only by limiting the bandwidth to zero, that the set
of possible signals is reduced in such a way that averaging over all possible signals
can be avoided.
Besides the possibility of studying array antenna properties in time or frequency domain, steering of the direction of maximum power of the array can also be obtained
with either a time or phase shift. Both are discussed, with advantages and disadvantages, in Chapter 2, where also some practical implementations of the techniques are
reviewed.
In Chapter 3, the influence of the number of array elements, their locations and the
way of combining them, on the array properties is analyzed in a mathematical way in
the frequency domain, for a single frequency signal, where time delay and phase shift
coincide.
When implementing array antennas in hardware, the array elements and the beam
forming can be designed independently. In turn, beam forming can again be split in
a signal shaping step and a signal combination. Fig. 1.3 illustrates this (arbitrary)
division.
signal combination
signal shaping
array elements
Figure 1.3: The three parts of a phased array antenna.
When discussing the two applications in Parts II and III, these three elements will
constitute the treatment.
invisible filling
Chapter 2
Beam Forming
Several techniques in order to shape the signals from the distinct array elements so
that they sum up constructively are in use. They can be categorized into three classes:
time delay (Sect. 2.2.1), phase shift (Sect. 2.2.2) and frequency change (Sect. 2.2.3).
Both analog and digital implementations are in use. Due to the advantage of high
flexibility and reconfigurability, the digital systems are gaining popularity. For some
applications, however, analog-to-digital conversion, that is indispensable prior to performing digital arithmetic in a processing unit, is not yet feasible due to the high
bandwidth or number of elements in the array. In such case, the analog systems, that
already have proven their capabilities for several decades, are to be used.
But before elaborating on these beam forming techniques, the choice of applying
the beam forming at the different stages in the receiving (or transmitting) chain,
namely Radio Frequencies (RF), Intermediate Frequencies (IF) or baseband (BB),
is discussed. Beam forming by frequency change is excluded from this discussion,
because of its limited relevance for practical systems. Beam forming by frequency
change is difficult to implement. Since it requires change of the mixing frequency
in both transmitter and receiver, the occupied bandwidth is large, the transmitters
and receivers are complex devices, and a separate channel is required to make sure
transmitter and receiver use the same1 carrier frequency [7]. Consequently, it is
merely applied in RADAR and sensing systems, where transmitter and receiver can be
combined into one device. Nevertheless is the study of the phenomenon of frequency
scanning important, as it unintentionally occurs in arrays using delay lines for signals
that have a non-zero bandwidth.
1 A bandpass communication signal can indeed only be received around the carrier frequency, hence
varying frequency for scanning destroys signal reception if synchronization between transmitter and
receiver is not assured in one way or another.
13
14
Chapter 2. Beam Forming
It will become clear that from the beam forming techniques covered in Sect. 2.1, the
only one that does not suffer from InterSymbol Interference (ISI), even for wide band
signals, is a time delay at RF. A phase shift of the carrier is always an approximation
for signals with small bandwidth, which is often the case in communication systems.
Besides those errors due to the non-zero bandwidth of signals, digital implementations
of beam forming introduce errors due to their discrete nature. The impact of those
approximation errors is evaluated in Sect. 2.3.
2.1
Beam Forming in the Receiving Chain
Essentially the signals that are received at the several elements of a phased array, are
time shifted versions of each other. Let us define ∆t as the time shift between the
signals of two adjacent elements, already calculated in Eq. (1.1), then the bandpass
signal at the two elements, can be written as:
R(t) cos (ωc t + θ(t))
(2.1)
R(t + ∆t) cos ωc (t + ∆t) + θ(t + ∆t)
where ωc = 2πfc is the carrier pulsation containing the carrier frequency fc . R and
θ contain the information in the form of an amplitude and phase deviation from the
carrier, similar to the split implicitly made when using complex envelope notation [6].
At Radio Frequencies (RF) it is obvious that by delaying the first signal with ∆t,
both signals get synchronized again such that they can be combined again. If the
bandwidth of the signal is small, this time delay can be approximated by a phase
shift δ̃ = ωc ∆t in the first signal of Eqs. (2.1):
R(t) cos (ωc t + ωc ∆t + θ(t))
(2.2)
R(t + ∆t) cos ωc (t + ∆t) + θ(t + ∆t)
This approximation is only valid when the information signal varies slowly:
R(t) ≈ R(t + ∆t) and θ(t) ≈ θ(t + ∆t)
(2.3)
as indeed only the carrier is rotated, not the information signal. If Eq. (2.3) does not
hold, inter symbol interference (ISI) will occur.
At intermediate frequencies (IF), the signals are already multiplied with a Local
Oscillator (LO) signal cos (ωLO t), so that after a low pass filter (to stop the (ωc +ωLO )
component), one gets2 :
R(t) cos (ωc − ωLO )t + θ(t)
(2.4)
R(t + ∆t) cos (ωc − ωLO )t + ωc ∆t + θ(t + ∆t)
2 The multiplication of two cosines is: 2 cos (x) cos (y) = cos (x + y) + cos (x − y), the LPF stops
the high frequency component cos (x + y). Filters behind mixers are not obligatory as the frequency
response of amplifiers acts as a filter for the ωc + ωLO component.
2.1. Beam Forming in the Receiving Chain
15
The phase shift in order to synchronize the signals is again ωc ∆t. If a time delay is
applied, this delay should be ωc ∆t/(ωc − ωLO ).
One thus sees that the phase shift stays the same (the angles of the cosines are simply
summed when expanding the product of two cosines), but the time delay is longer in
the IF than in the RF case. Therefore at IF, the recombination after a time delay is
not exact anymore; the approximation that is made here, is:
ωc
R t+
∆t ≈ R(t + ∆t)
(2.5)
ωc − ωLO
Generally, ωc /(ωc −ωLO ) > 1. If ωLO ≈ ωc , then ωc /(ωc −ωLO ) 1. If the bandwidth
of the signal R(t) is not small compared to (ωc −ωLO )/(ωLO ∆t), this will, again, result
in ISI.
For baseband (BB) the mixing signal cos (ωc t) has the same frequency as the RF
carrier, resulting in:
R(t) cos (θ(t))
(2.6)
R(t + ∆t) cos ωc ∆t + θ(t + ∆t)
so that the phase shift that should be applied to the first signal, is again ωc ∆t, as in
the RF case. Summing both signals again implies that Eq. (2.3) is valid. At BB a
carrier phase shift is not feasible with a time delay.
Besides the “correctness”, also the practical implementation issues of RF, IF and BB
phase shifting are to be considered. Time delay in IF results in longer lines and thus
more losses (though the substrates for IF are less lossy than for RF). Pure phase shift
at lower frequencies (IF and BB) requires more relative bandwidth of the components
but in general components at lower frequencies are much cheaper.
The topology of the signal chain with beam forming in IF, RF and BB can be found
in Fig. 2.1. In the RF and BB case, the IF stage is left out: instead of using the
heterodyne topology, a direct conversion scheme is depicted. This has the advantage
that no image rejection filters3 are needed, unless the signal is Single Side Band (SSB)
signal. If this is the case, beam forming at RF, still only needs one image rejection
filter for the entire array, namely after beam forming, before mixing down. For IF
beam forming, an image rejection filter is needed for every element prior to mixing
to IF, where beam forming takes place.
In case of signals with non-symmetrical side bands, such as e.g. SSB signals, the direct
conversion to BB has to be a quadrature mixing. Otherwise the upper and lower side
band will interfere after mixing, as Fig. 2.4 clearly shows.
3 Whenever mixing with an LO of pulsation ω
LO , not only the frequencies above the LO frequency,
but also those below are mixed to baseband. In case of IF mixing, where ωLO = ωc − ωIF , this causes
signals at the mirror frequencies (ωLO − (ωc − ωLO )) to interfere with the useful IF signal. Hence
prior to mixing, an image rejection filter should suppress those frequencies [8].
16
Chapter 2. Beam Forming
out
(a) The Signal Chain in Case of RF phase steering.
out
(b) The Signal Chain in Case of IF phase steering.
out
(c) The Signal Chain in Case of BB phase steering.
Figure 2.1: Phase steering at different stages in the receiver.
2.1. Beam Forming in the Receiving Chain
17
Phase shifting at RF has the advantage that only one mixer is needed, the phase
shifters will however be more expensive than those for BB phase shifting. Note that
mathematically spoken, it makes no difference whether the phase shift is applied to
the local oscillator signal or to the signal after mixing (see Fig. 2.2).
Figure 2.2: The phase shift can also be applied to the local oscillator signal.
Suppose that, in the BB case, the approximation made by shifting in phase is not
good enough, then still a time delay (∆t) can be applied, which results in an exact
recombination. Instead of solely shifting the first signal in phase by ωc ∆t and summing
both:
R(t) cos (ωc ∆t + θ(t)) + R(t + ∆t) cos (ωc ∆t + θ(t + ∆t))
(2.7)
again hoping that Eq. (2.3) applies, the first signal is shifted in phase by ωc ∆t and
then delayed by ∆t before summing: [9]
R(t + ∆t) cos (ωc ∆t + θ(t + ∆t)) + R(t + ∆t) cos (ωc ∆t + θ(t + ∆t))
Table 2.1 summarizes the discussion, and confirms the findings in [10].
Table 2.1: Comparison of phase steering at RF, IF and BB.
RF
IF
BB
∆t
exact
expensive switching components
approximation for narrow band
(worse than ∆φ for ωLO ≤ 12 ωc )
longer lines (losses)
mixer and image rejection filter
per element needed
not feasible
∆φ
approximation for narrow band
expensive shifter components
approximation for narrow band
mixer and image rejection filter
per element needed
approximation for narrow band
quadrature mixing required
mixer per element needed
cheap shifter components
(2.8)
18
2.2
Chapter 2. Beam Forming
Some Beam Forming Implementations
Neither with the intention of being exhaustive, nor of going into much technical
details, the basic principle of the three classes of beam forming, as introduced at the
beginning of the chapter, will be reviewed. References will point the reader to sources
where all technical details can be found.
Both analog and digital implementations are in use. A disadvantage of digital techniques is that the signal has to be sampled (or digitized) first, putting high demands
on the Analog to Digital Converter (ADC). If the signal has to be digitized at RF
frequencies, due to the Nyquist-Shannon sampling theorem [11, 12], the sample rate
has to be at least twice the carrier frequency.
Sampling at a rate equal to twice the bandwidth of the signal, is sufficient for reconstruction of the modulating signal at BB [13]. This however requires perfect filtering
prior to sampling (and inherently mixing down) to avoid aliasing from signals at any
other frequency. Recall, no time shifting at BB is possible, as already mentioned in
Sect. 2.1.
The huge advantage of digital beam forming techniques, is that heavy array processing
can be done (even off-line). This way, significant better performance can be obtained
[14]. And the entire half space as seen by the array can afterwards be analyzed and
received, simply by performing the digital calculations over and over again but with
other constants for other “directions of interest”. It is no surprise that many military
RADAR systems make use of this possibility [15].
2.2.1
Time Delaying
By making the feed line to a certain antenna element longer or shorter, a variable
time delay is introduced. This is mostly obtained by switching line sections in and
out of the feed line, similar to playing a trumpet by pushing the valves, using diodes
[16] or Micro Electro Mechanical System (MEMS) switches [17].
Digitally, this is even more straightforward to implement, using shift registers with
reconfigurable taps.
A more advanced approach is to delay the signal by modulating the propagation
velocity in the medium. This is mostly achieved by changing the permittivity = 0 r
(in ferroelectric materials) [18] or the permeability µ = µ0 µr (in ferrites) [19] of the
medium. This changes the propagation velocity:
c0
1
c= √ = √
µ
r µr
(2.9)
and thus the travelling time to the end of the line. The physical processes that cause
this change are way beyond the scope of this work.
2.2. Some Beam Forming Implementations
19
An advantage of the latter technique is that the signal path is not switched while
reconfiguring the array. For both analog and digital straightforward implementations
indeed a part of the signal is simply dropped, or repeated, when changing the time
delay, voiding that part of the received signal.
2.2.2
Phase Shifting
Eq. (1.1) explained the working principle of a phased array with time delay. For
narrow band applications, another way of explaining, is saying that the array in
receiving mode is sensitive to signals that arrive at each antenna element with a
phase which is the inverse of the phase shift imposed by the beam forming network.
In transmitting mode, the array will concentrate its power in locations (or Directions
of Departure (DOD) θDOD for far field approximation) where the signals departing
from each antenna element arrive in phase, which results in constructive interference.
This phase shift can easily be derived from the time delay of Eq. (1.1):
δ̃ = ωc ∆t = ωc ×
d cos (θDOD )
d × cos θ
= 2π
λfc
λ
(2.10)
Note that the beam can indeed be steered by applying the appropriate phase shift,
but that the antenna elements should be sensitive (for receiving mode) or be able to
radiate (for transmitting mode) in that direction as well to obtain a beam. A detailed
elaboration of this is delayed to Chapter 3.
The digital way to implement this, is by performing the matrix and vector multiplications of Sect. 2.2.2.1. The technique explained in that Sect. 2.2.2.1, referred to as
quadrature phase shifting, is also implementable in an analog way, by using analog
summing and multiplication devices. The block diagram is given in 2.6, a more detailed discussion is delayed to Sect. 6.1.1. Apart from this technique, many others are
in use. Some count on the phase change due to a reflection on a line end, terminated
with a complex impedance. Diodes are then used to switch capacitances in and out
in order to change the complex impedance [20]. Similar to this technique, a voltage
divider over a complex impedance can be used. If for this complex impedance a semiconductor is used, its capacitance can simply be changed by changing the bias and
hence the thickness of the depletion layer [21].
2.2.2.1
Quadrature Phase Shifting
A bandpass signal, as indeed in Eq. (2.1) too, can be written as the real part of a
complex envelope multiplied with a carrier phaser, see Fig. 2.3(a):
v(t) = <{R(t)ejθ(t) ejωc t } = R(t) cos (ωc t + θ(t))
(2.11)
20
Chapter 2. Beam Forming
In a quadrature demodulator this signal is mixed with the sine and cosine of the LO
frequency:
I = v(t) × cos (ωLO t)
(2.12)
Q = v(t) × sin (ωLO t)
so that it can easily be combined into the original RF signal with the same device:
v(t) = I × cos (ωLO t) + Q × sin (ωLO t) = v(t) × cos2 (ωLO t) + sin2 (ωLO t) (2.13)
If now the LO frequency is taken equal to the RF frequency and the 2ωc t component
is stopped by a filter, the I and Q components are indeed the in-phase (with the RF
carrier) and the orthogonal (to the RF carrier) component of the BB signal:
(
LP F
I = R(t) cos (ωc t + θ(t)) × cos (ωLO t) = R(t) cos (θ(t))
LP F
Q = R(t) cos (ωc t + θ(t)) × sin (ωLO t) = −R(t) sin (θ(t))
(2.14)
and the complex notation of the BB signal is:
R(t) cos θ(t) = <{R(t)ejθt } = <{I − jQ}
(2.15)
Suppose, in the most general case of Eqs. (2.14), that a linear combination of I and
Q leads to I 0 and Q0 [22]:
0
I
= a cos (∆θ) × I + a sin (∆θ) × Q
(2.16)
Q0 = −a sin (∆θ) × I + a cos (∆θ) × Q
then these I 0 and Q0 are the in-phase and quadrature components of the original
signal, shifted in phase by ∆θ and amplified by a (this can also be interpreted as the
LO delayed by ∆θ):
(
av(t) × cos (ωLO t − ∆θ)
av(t) × sin (ωLO t − ∆θ)
LP F
= I0
LP F
= Q0
(2.17)
so that after recombination (with cos (ωLO t) and sin (ωLO t)) the shifted version is
obtained. The block diagram is given in Fig. 2.3(b). If the I and Q signals are
sampled by an ADC, resulting in two vectors, I˜ and Q̃, the phase shift can also be
obtained by a simple matrix multiplication in digital logic. The vectors I˜0 and Q̃0 are
the in phase and quadrature vectors if the carrier in the original signal was rotated
over a phase ∆θ [22]:
I˜0
cos (∆θ)
=
− sin (∆θ)
Q̃0
This is similar to a coordinate rotation.
sin (∆θ)
I˜
·
cos (∆θ)
Q̃
(2.18)
2.2. Some Beam Forming Implementations
21
=
R(t)ejθ(t) ejωc t
Q
ejωc t
θ(t)
I
v(t)
<
(a) Complex Envelope Notation
0/90
0/90
(b) I-Q Phase Shifter (BB)
Figure 2.3: Working principle of the I-Q phase shifter.
If the LO shifts the signal to an IF instead of to BB, the implementation of the phase
shifter is even cheaper because only two instead of four variable gain amplifiers are
needed [23]. This is because the I and Q signal are not independent in the IF case
(but phase π/2 shifted version of each other), whereas in case of BB I and Q are both
needed to construct the original signal as they have only half the bandwidth of the
original signal, see Fig. 2.4, [13].
In some more detail for IF, the Q signal is thus the time delayed version of I (for BB
this is definitely not the case as I and Q are independent) so that:
(
LP F
I
= vbase (t) × cos (ωIF t)
(2.19)
LP F
Q
= vbase (t) × cos (ωIF t + π2 ) = vbase (t) × sin (ωIF t)
which gives us the possibility of obtaining a shifted version of I by simply adding the
two amplified signals.
vbase (t) × cos (ωIF t − ∆θ) = I 0 + Q0 = cos (∆θ) × I + sin (∆θ) × Q
(2.20)
A huge disadvantage of this technique is however that in order to obtain Q a time
delay is applied with possible ISI as a result.
22
Chapter 2. Beam Forming
<
=
ωLO
ω
ωLO
−ωLO
(a) I signal for BB mixing.
ω
−ωLO
(b) Q signal for BB mixing.
<
=
ωLO
ωLO
−ωLO
ω
(c) I signal for IF mixing.
ω
−ωLO
(d) Q signal for IF mixing.
Figure 2.4: Spectra for BB and IF. For IF Q is I after a π/2 phase shift.
Q
I 0 + Q0
0
Q
0/90
I
∆θ
I0
I 0 + Q0
I
(a) I-Q shifting in IF
(b) I-Q Phase Shifter (IF)
Figure 2.5: The I-Q phase shifter for IF.
The shifter of Fig. 2.3(b) takes an RF signal in and outputs an RF signal, rotated in
phase. The shifting itself is done in BB. When sending or receiving, one of the mixing
stages can be left out. We thus get the schemes of Fig. 2.6(a) and 2.6(b).
Any type of modulation can be sent or received by this scheme, but the appropriate
I and Q should be provided at the input for transmission. For a QPSK signal, these
are the two orthogonal bit streams, but for other schemes such as FM the generation
of I and Q is less straightforward.
2.2. Some Beam Forming Implementations
I
I0
I
I0
0/90
Q
23
0/90
Q0
0
Q
Q
(a) Transmission
(b) Reception
Figure 2.6: The shifter of Fig. 2.3(b) for transmission and reception.
2.2.3
Frequency Scanning
By designing the feed lines to each of the radiating elements all with a different length
(e.g. in a series-fed array such as a slotted waveguide), the phase shift to each element
will change as the frequency of the signal that is sent is changed, for the line lengths
are fixed but the wavelength changes. Hence this is time delay beam forming, but
with a deliberate modulation of the signal autocorrelation. This becomes very clear
in [24]. A more in depth explanation with example can be found in Appendix A.
As mentioned at the beginning of this chapter, practical problems arise with this
technique. When used in a receiver, a pilot channel to set the carrier frequency at
the transmitter based on the needs of the receiver is obligatory. Unless, of course this
technique is used to compensate for an unwanted frequency shift, such as a Doppler
shift. This idea is set out in Appendix A.
For use in RAdio Detection And Ranging (RADAR), this changing frequency is not
an issue. Hence many RADARs [25, 26, 27] and sensor arrays [28] use the technique.
The frequency shift is always applied by modulating an oscillator that mixes the signal.
Apart from that, no extra hardware is required, which makes this technique very cheap
and compact. Again, this frequency shift can be applied at BB, IF or RF. For the
accuracy, this does not make a difference, as was the case with phase shifting, due to
the summing of the frequencies in the cosine when multiplying. But as the frequency
shift stays the same at BB, IF and RF, the shift relative to the mixer frequency
becomes smaller for higher frequencies. Hence modulating the mixing frequency at
RF is the cheapest solution.
Let frequency scanning be a technique with limited usability for communication, all
array antennas using phase shifting do suffer from an unintended frequency scanning.
Looking at Eq. (2.10), one sees that decreasing the frequency and hence increasing
the wavelength λ results in an increase of cos θ and hence scans the beam further
away from broadside, as soon as δ̃ differs from zero.
24
Chapter 2. Beam Forming
This is a second order effect, but shows up as soon as the signal has a bandwidth in
% that is comparable to the beam width in ◦ , due to the aperture fill effect [29]. The
amplitude of this effect depends on the bandwidth of the signal and the scan angle
of the array. The more the array is scanned off broadside, the more pronounced this
effect becomes. The inter-element distance d is of no influence as δ̃ linearly depends
on d. This follows immediately from Eq. (2.10):
∆ cos θ =
2.3
θ
∆λ × 2πd×cos
∆λ × δ̃
∆λ
λ
=
=
× cos θ
2πd
2πd
λ
(2.21)
Beam Forming Approximation Effects
For any digital beam forming implementation, not every possible time, phase or
frequency shift can be selected. This introduces errors. Moreover, all analog implementations are controlled in a digital way, hence the system deals with the same
problem4 .
How this affects the array factor in case of phase shift beam forming, is explained in
Sect. 3.4.2. For time delay beam forming, the reader is referred to Sect. 9.4.
4 There is one way of circumventing the problem, namely in case the beam always has to perform
the same scanning movement, the controlling signals can be Low Pass Filtered (LPF), resulting in a
smooth transition with the beam maximum always pointing towards the receiver.
Chapter 3
Phased Array Design
In Sect. 2.2.2 was explained that for signals with sufficiently small bandwidth, the
beam of an array antenna can be steered by applying the appropriate phase shifts.
In this chapter a mathematical treatment of such phased arrays is given. For starters
is demonstrated in Sect. 3.1.1, that from Maxwell’s laws, the radiation properties, as
introduced in Sect. 3.1.2, of an array antenna, can be split into a part only depending
on the array element and another part only depending on the array geometry and
excitation.
The influence of array geometry and excitation can be studied via the properties of
the Fourier transform, as explained in Sect. 3.2. This is worked out in detail for
geometry in Sect. 3.3.1 and for phase and amplitude of the excitation in Sect. 3.3.2.1
and 3.3.2.2 respectively. The recommendations and techniques for designing arrays
resulting from the discussion, are summarized in Sect. 3.3.3.
All theory in this chapter can be found elsewhere. If appropriate, references are given.
No overview was available at the moment of writing, however, where all array theory
was linked with Fourier transform properties. Unless stated differently, all graphs
were obtained with numerical calculations in MATLABTM .
3.1
3.1.1
Definitions
Array Factor
An array antenna consists of N identical elements, translated from the origin by the
vectors ~bn (with n : 1 → N ) as on Fig. 3.1(a). The radiated field of the array in
linear, homogeneous media is found by summing the contributions of all elements.
25
26
Chapter 3. Phased Array Design
element 1
element 2
~b2
z
~ir
θ
~b1
|Rn |
r~0 n~ir
element n
reference element
y
~bn
φ
reference element
x
~ |R|
r~0 ir
~ir
(a) Array in Spherical Coordinate System.
~r
(b) Far Field Approximation.
Figure 3.1: The geometry of an array antenna.
~ for the array results in [30]:
Writing out the vector potential A
~ r) =
A(~
N
X
n=1
A~n (~r) =
Z
N
X
µ(r~0 n )
e−jβRn 0
J~n (r~0 n )
dvn
4π
Rn
0
vn
n=1
(3.1)
~ r) is
where Rn = |~r − r~0 n | is the distance between the observation point where A(~
calculated and a varying point r~0 n in the current carrying volume of the nth element
vn0 that is integrated. J~n (r~0 n ) is the current density in this varying point.
Neglecting the possible distortion on the current profile that mutual coupling between
array elements can cause, the current distribution on each element is supposed identical to the current distribution J~ref on the reference element, except for a complex excitation coefficient an , expressing the difference in amplitude and phase of the applied
signal with respect to the signal on the reference element. Thus J~n (r~0 n ) = an J~ref (r~0 )
where r~0 n = r~0 + ~bn (see Fig. 3.1(b)).
For large distances Rn → ∞ (i.e. far field approximation) ~ir , the vector of unit length
that points in the direction of ~r, see also Fig. 3.1(a):
~ir = (sin θ · cos φ, sin θ · sin φ, cos θ)
(3.2)
is the same for each point r~0 n . This results in an approximation for Rn :
Rn ≈ |~r − r~0 ·~ir − ~ir · ~bn |
(3.3)
For the value of Rn in the denominator of Eq. (3.1), this can even further be approximated by |~r|, as no phase information has to be preserved.
3.1. Definitions
27
Eq. (3.1) for the Rn → ∞ far field hence becomes:
Z
N
−jβ|~
r|
X
µ(r~0 )
~ ~
~0 ~
~ ff (~r) = e
~ ref,ff (~r) · F (~r)
A
an ejβ bn · ir = A
J~ref (r~0 )ejβ r · ir dv 0
4π|~r|
0
v
n=1
(3.4)
such that the vector potential of the array can be written as the product of the vector
~ ref,ff (~r) and the array factor F (~r):
potential of the reference element A
F (~r) = F (r → ∞, θ, φ) =
N
X
an ejβ bn · ir
~
~
(3.5)
n=1
Note that the array factor can be found as the values on the unit sphere after a Fourier
transform of a function that reflects the position and excitation of the elements. This
is explained in depth in Sect. 3.2
3.1.2
Array Radiation Properties
As antennas generally do not radiate their power equally into all directions, a quantity
is needed to express the amount of power radiated in each direction r → ∞, θ, φ. Such
quantity is the directivity Dant , which is the ratio of the power density that is emitted
by the antenna in a particular direction P (θ, φ) to the power density that would be
emitted in that direction if the radiated power Ps were radiated isotropically:
Dant (θ, φ) =
~ r) × H
~ ∗ (~r)i
P (θ, φ)
hE(~
=
Ps
Piso
4πr 2
(3.6)
~ and magnetic H
~ far fields of the array can
With Maxwell’s equations the electric E
be written as:
(
1
~ r) = H
~ ref,ff (~r) · F (~r)
~ ff (~r) =
H
µ ∇ × Aff (~
(3.7)
1
~
~
~ ref,ff (~r) · F (~r)
Eff (~r) = jω ∇ × Hff (~r) = E
Consequently Dant of the array, is no more than the directivity of the reference element
Dref multiplied with the square of the array factor, because1 :
Dant (θ, φ) =
~ ref (~r) · F × H
~ ∗ (~r) · F ∗ i
hE
ref
Ps
4πr 2
=
Pref
Dref (θ, φ) · |F |2
Ps
(3.8)
with Pref the power radiated by the reference element. This scaling only accounts for
the fact that mutual coupling alters the power needed to impose currents on the array
elements as compared to the case of imposing the same current on a single element
(without the fields of neighboring elements).
1 Note that the E
~ and H
~ field corresponding to any value for the power radiated by the reference
element can be taken because the directivity is only obtained after a division by the radiated power.
28
Chapter 3. Phased Array Design
Very much related with the directivity of an antenna, is the gain Gant , which is
the ratio of P (θ, φ) to the power density that would be emitted in that direction if
the total (radiated and dissipated) power delivered to the antenna Pin , were radiated
isotropically:
Gant (θ, φ) =
P (θ, φ)
Pin
4πr 2
=
Ps
Dant (θ, φ) = ηrad Dant (θ, φ)
Pin
(3.9)
Thus if the antenna efficiency ηrad = 1 (no dissipation losses), the directivity is equal
to the gain of the antenna.
From this gain, another quantity is derived, namely the normalized (with maximum
at 0 dB) radiation pattern p(θ, φ). This is the gain of the antenna normalized to the
maximum gain:
Gant (θ, φ)
p(θ, φ) =
(3.10)
max {Gant (θ, φ)}
The array factor is thus the only additional information that is needed to predict the
properties of an array if the radiating element is known.
3.1.3
Example: Subarray of the ATCRBS in Bertem
As an example the array factor and radiation properties of a subarray of the secondary RADAR in Bertem are calculated. This array is a part of the Air Traffic
Control (ATC) system of Belgocontrol. Besides Airport Surveillance Radars (ASR)
monitoring aircrafts landing and taking off and Airport Surface Detection Radars
(ASDR) mapping taxiing movements, also Air Route Surveillance Radars (ARSR)
are needed for enroute traffic. Such ARSR is located in Bertem, with on top of it an
Air Traffic Control Radar Beacon System (ATCRBS). The latter is strictly spoken
not a RADAR as it does not transmit pulses for detection and ranging, but instead
transmits queries to aircraft transponders to request their altitude and identification.
This way the two dimensional mapping of azimuth and range of the ARSR can be
extended to a three dimensional one. Hence the ARSR is often referred to as the
primary RADAR, and ATCRBS as the secondary [31].
The secondary RADAR consists of 35 subarrays. Each subarray is a linear array of
ten λ/2 dipoles2 , with element spacing d = 14.6 cm at a frequency3 of 1.06 GHz
(λ = 28, 3 cm), see Fig. 3.3.
For a λ/2 dipole the directivity is: [30]
!
#2
cos (β λ4 · cos θ) − cos β λ4
Dant = 1.64
sin θ
2 Actually
3 This
(3.11)
the dipoles measure 12 cm, slightly less than λ/2 = 14.6 cm.
frequency is chosen as the mean of the 1.03 GHz transmit and 1.09 GHz receiving frequency.
3.1. Definitions
29
This result is plotted, together with the array factor of Eq. (3.5) and the array directivity of Eq. (3.8) for this array.
|F (φ)|2 [dB]
10
Darray from MATLABTM
Darray from MAGMAS
|F (φ)|2
Ddipole
0
−10
−20
−30
−40
−50
0
15
30
45
60
75
90 105 120 135 150 165 180
φ
Figure 3.2: The directivity of an array Darray can be approximated (neglecting mutual coupling) as the product of the directivity of the reference element Ddipole and the
square of the array factor |F (φ)|2 .
The directivity can also be obtained from a MAGMAS4 gain simulation with zero
losses. As it is impossible to insert non-physical structures (such as an ideal dipole)
into MAGMAS, the dipole was drawn as two long and small rectangular patches, fed
with an active patch in between, as can be seen on Fig. 3.3.
We indeed see on Fig 3.2 that the calculation and the simulation give the same results,
mainly because mutual coupling was not taken into account in the simulation.
Also note that even if the array factor is not zero, the (far field) array radiation
pattern will be zero if the (far field) element pattern is zero. But still the near field
can differ from zero, causing mutual coupling and EMC problems often overlooked,
as e.g. in [33].
In our example all excitation coefficients were equal (an ≡ 1). But applying phase
differences between the excitations of the elements allows steering of the direction of
the beam. Applying amplitude differences allows shaping of the beam. This will be
explained in Sect. 3.3.2. A reprise of the secondary RADAR example can be found
in Sect. 3.3.2.3.
4 Model for the Analysis of General Multilayered Antenna Structures (MAGMAS) is an electromagnetic full wave integral equation solver developed at ESAT-TELEMIC, based on the method of
moments, see [32].
30
Chapter 3. Phased Array Design
0.05 cm
passive patch
active patch
12 cm
passive patch
array
element
Figure 3.3: The array of dipoles as inserted in MAGMAS
3.2
The Array Factor via a Fourier Transform
Eq. (3.5) reflects that the array factor can be found via the inverse Fourier transform of
a function that reflects the position and excitation of the elements. Define a function
f (x, y, z) that consists of Dirac impulses multiplied with the excitation coefficient on
the locations of the array elements and is zero elsewhere:
X
an δ 3 (~r − ~bn )
(3.12)
f (~r) =
N
The inverse Fourier transform of this function is:
F (~u)
= F (u, v, w) = F −1 {f (~r)}
yX
X
~
=
an δ 3 (~r − b~n )ej2π~r · ~u d~r =
an ej2πbn · ~u
N
(3.13)
N
This is a function over three variables (u, v, w), but only the values on the sphere
with a radius of one in this transformed domain have a physical meaning5 as these
dn ). The interpretation
are the only points where |~u| = 1 such that ~bn · ~u = |~bn | cos(ub
of u = sin θ · cos φ, v = sin θ · sin φ and w = cos θ results in the array factor of
Eq. (3.5). This is of course only true if the position of the elements as expressed in
~bn is measured in wavelengths.
5 The points that lie on the sphere with radius two have a physical meaning too as they represent
the array factor of the array when ~bn gives the element locations measured in 2λ and so on.
3.2. The Array Factor via a Fourier Transform
31
The properties of the Fourier transform hence apply. But be aware of the fact that the
transformed domain is not only the unit sphere on which the values with a physical
meaning lie. Consequently, care should be taken when studying the array using the
Fourier properties. E.g. when applying the theorem of Parceval [6]:
y
2
|f (~r)| d~r =
y
2
|F (~u)| d~u 6=
Zπ Z2π
0
0
|F (θ, φ)|2 sin θdθdφ.
(3.14)
The integration of the rightmost expression is over the unit sphere and gives the
power radiated by the array with isotropic array elements. The other two expressions
integrate over the entire (u, v, w) and (x, y, z) domain, which, because the integration
of a squared Dirac impulse is infinite, equals ∞.
Consequently, even two arrays of isotropic radiators, that share the same set of excitation coefficients an , but have different geometries, are not guaranteed to radiate
the same amount of power. This is a result of the fact that the elements are in the
presence of their mutual fields6 between array elements. Physically this is due to the
fact that the power is found as:


Z E
~ J~∗ 
P =<
dv 0
(3.15)


2
v0
If the geometry of the array is changed, the same currents are to be imposed in a
possibly different electric field, resulting in an other value for the consumed (radiated)
power. The shortcoming in treating an element as if it were not embedded in the
array is thus that it violates the conservation of energy. But still the approximation
to suppose all current distributions as identical to the distribution on the reference
element does not conflict with the conservation of energy.
A possible way to compare two arrays is calculating the power by integrating the
product of the square of the array factor and the element pattern of both arrays and
then scaling the set of excitation coefficient of one array until the power is equal
to the power of the other array. By doing this the supposition about the current
distribution is maintained but the fact that the elements are in their mutual electric
fields is reckoned by scaling the Zin of all elements resulting in the same power for
both arrays. This is not physically correct, as only in single mode infinite arrays
the elements experience the same environment and hence all have identical input
impedances.
6 The effect of this is actually twofold. Firstly, fields caused by neighboring elements can alter the
current profile of an element. This effect is bypassed by imposing currents (identical to the current on
the reference element except for an excitation coefficient an ). Note that in practice arrays are voltage
driven. Hence imposing current, limits the practical applicability of the results tremendously. The
second effect, is the changing input impedance of an element when in the presence of fields coming
from neighboring elements. This effect is discussed here.
32
Chapter 3. Phased Array Design
For finite arrays, the outer array elements have input impedances different from those
of the center array elements. Consequently, because most arrays are not current
driven, the (complex) current amplitudes will differ over the elements, resulting in
array properties different from those predicted when simply scaling the excitation
coefficients. Array factor calculations are a good starting point. Full wave simulation
of the array at a further design stage is ever sensible.
For a 1D array along the x-axis, the y and z values of ~bn are all zero such that the
value of F in Eq. (3.13) only depends on u. Thus for all (θ, φ) that give the same
result for u = sin θ · cos φ the array factor is the same. This results in a radiation
pattern that is rotation symmetric around the u axis, as on Fig. 3.4, because the
locus of the points where u is a constant and r = 1 is a circle around the u-axis
(u = r · sin θ · cos φ). For a 2D array in the x, y-plane the same derivation results in a
pattern that is mirrored by the u, v-plane (same array factor if u and v are the same
and r = 1). For a 3D array each point on the unit sphere can result in a different
value for the array factor.
Note that for the translation vectors ~bn any set of vectors can be chosen. Also arrays
with non-equally spaced radiators or even arrays with element locations on irregular
geometric figures have applications.
3.3
Array Design with Array Factor
3.3.1
Influence of the Array Geometry
To study the influence of the element locations of an array on the array factor, we
will first evaluate the equally spaced uniform linear array. This is an array where
all elements are fed uniformly (ai ≡ aj ) and are spaced equally along a line (as
for example on Fig. 3.3). Then linear arrays with non-equally spaced elements are
discussed. After that we look at non-linear arrays.
Throughout the discussion it will be clear that all results stem with the properties
of the Fourier transform. The emphasis in the text is on the physical interpretation,
related to array antennas.
3.3.1.1
Linear Equally Spaced Arrays (LESA)
The array factor of Eq. (3.5) for the case of a uniform linear array along the x-axis,
is a complex periodic (in u = sin θ cos φ) function [30]:
N
X
d
d
F (θ, φ) =
cos 2πn sin θ cos φ + j
sin 2πn sin θ cos φ
λ
λ
n=1
n=1
N
X
(3.16)
3.3. Array Design with Array Factor
33
(c) 3D array
(a) 1D array
(b) 2D array
(d) Array layouts: Nx = Ny = Nz = 4, d = 0.9λ
Figure 3.4: Array factors of a 1D, 2D and 3D array.
3.3.1.1.1 Element Spacing d In Fig 3.5 the array factor of a uniform LESA
with N = 10 elements is plotted as a function of u. This is sufficient because the
array factor of a linear array only depends on u (see Sect. 3.2). As the array factor
is the inverse Fourier transform of a discrete function, it is a periodical function in u
(not in φ). The value for d determines the periodicity and thus the number of periods
mapped to the u = [−1, 1] visible interval.
As soon as d ≥ λ, we see that a second maximum appears on the edges of the plot,
because a direction exists in which the waves of each element have a difference in
path length of exactly one wavelength λ (in phase). Mathematically, more than one
period of the array factor function lies in the u = [−1, 1] or φ = [0, π] interval. These
additional maxima are called grating lobes and must be avoided, even if the far field
pattern of the reference element is zero, because they causes in-phase mutual coupling
between the elements in the near field.
The advantage of taking d as close to λ as possible is that the beam width is as small
as possible: with the same number of elements, a bigger d results in a longer antenna.
But one has to avoid that the array pattern raises too much at the edges of the [0, π]
interval, thus avoid that the grating lobe comes up higher than the side lobes.
If d is changed and hence another number of periods lies in the visible interval, also
the total power radiated by the array changes. The explanation7 for this was given
in Sect. 3.2.
7 Curves similar to those in Fig. 3.6 were assigned to surface waves in [34]. This is only partially
true.
34
Chapter 3. Phased Array Design
|F (φ)|2 [dB]
10
0
d=λ
d = 2λ
d = 0.2λ
−10
−20
0
15
30
45
60
75
90 105 120 135 150 165 180
φ
(a) Array Factor as Function of φ (Wider Beams at Edges).
|F (u)|2 [dB]
10
0
−10
−20
−1
−0.75 −0.50 −0.25
0.00
0.25
0.50
(b) Array Factor as Function of u = cos φ sin θ for θ =
0.75
π
2
u
(Periodic).
Figure 3.5: Array factors (Linear Equally Spaced Array, N = 10, varying d).
In case of an array with isotropic radiating elements, the radiated power equals the
integral of the square of the array factor:
P =
Zπ Z2π
0
0
2
|F (θ, φ)| sin θdθdφ =
Z
1
2π|F (u)|2 du
(3.17)
−1
where the last equality sign holds because for a linear array, F only depends on u.
This is plotted left in Fig. 3.6 for the N = 10 LESA. If now the maximum directivity
of the array is calculated as a function of d:
Dmax (d) =
~ ×H
~ ∗}
~ ref × H
~ ∗ )|F |2 }
max{E
max{(E
ref
=
Ps
Ps
(3.18)
then we see, again for the case of isotropic radiating elements, that a maximum
appears where Eq. (3.17) is at minimum. Choosing d such that the first grating lobe
lies just behind the edges of the visible interval results in maximum array gain8 .
8 For the case of uniform amplitude and linear tapering, that is. Theoretically it is possible to
achieve any desired gain value, when carefully selecting amplitude and phase of each array element [35]. This is referred to as superdirectivity [36].
3.3. Array Design with Array Factor
35
Darray
7
6
5
4
3
2
1
0
N =5
N =2
N =1
0
1
2
3
4
5
6
7
8
9
10
d [λ]
Figure 3.6: Darray of a uniform LESA varies with d.
As soon as less than one period lies in the interval, enhancing the gain even more is
possible by scanning the beam further than endfire [37]. This mathematical result
skips over the fact that practical problems, such as matching difficulties, might arise,
severely complicating the feeding of the array.
3.3.1.1.2 Number of Elements N The more elements in an array, the higher
the gain will be, even though the same amount of power P is divided among more
elements. The power fed to √
an element
is P/N . The amplitude of the field caused
√
by this element will be E ∝ P / N . The maximum field strength (in the point of
constructive interference)
for the complete √
array√is the sum of the electric fields of all
√ √
elements, ∝ N P / N or, in power, ∝ (N P / N )2 thus N times the power in the
case of one element (fed with the same power).
For a receiving antenna not the signal gain Gr but the signal-to-noise (e.g. from
interference signals in the side lobes) ratio should be maximized. MIMO even goes one
step further by maximizing the channel capacity C, which is essentially the ultimate
goal.
3.3.1.1.3 Length of Array As mentioned in previous examples, the beam width
of an antenna decreases if the antenna itself becomes larger. If the length of the array
is kept constant, we can increase the gain by increasing N if d does not become too
small. But the beam width stays the same. This can be seen on Fig. 3.7.
This stems with the properties of the Fourier transform. The function f (~r) can be
written, for a linear array, as a Dirac comb with inter spacing d that is windowed by
a rectangular function (Π) of size N d:
f (~r) =
∞
X
n=−∞
δ 3 (~r − (nd, 0, 0)) × Π(
x
)
Nd
(3.19)
36
Chapter 3. Phased Array Design
|F (u)|2 [dB]
d = 0.2λ, N = 101
d = 2λ, N = 11
d = 4λ, N = 6
40
30
20
10
0
−10
−20
−1
−0.75 −0.50 −0.25
0.00
0.25
0.50
0.75
u
Figure 3.7: Array factors (LESA, length 20λ, varying N).
The array factor is the convolution9 of a Dirac impulse train with the inverse Fourier
transform of the rectangular window (a sinc(x) = sin(πx)/(πx) function):
F (u) = F −1 {f (~r)} = N d
∞
sin(N dπu) O 1 X 3
n
δ (u − )
N dπu
d n=−∞
d
(3.20)
Form this equation we indeed get the conclusions:
• as the element spacing d increases, the spacings between the Dirac impulses in
the u domain is smaller (causing grating lobes);
• as more elements in the array make the window larger, the beam width will
become smaller for the frequency of the sine function is higher;
• as the length N d of the array is kept the same, the Fourier transform of the
window will remain the same; if then more elements are added, the spacing d
decreases such that the separation between the Dirac impulses in the u domain,
and hence the grating lobes, will be larger.
Obtaining a higher gain by adding elements (increasing N ) however has its limits. As
N becomes so high that d is smaller than d at the maximum directivity on Fig. 3.6,
adding more elements will on the contrary result in less gain.
9 x(t)
N
y(t) =
R +∞
−∞
x(t)y(u − t)du
3.3. Array Design with Array Factor
3.3.1.2
37
Linear Non-equally Spaced Arrays
Arrays with unequally spaced elements can be divided into three groups10 :
• Sparse Array: a LESA where elements are left out,
• Perturbed Array: a LESA (or sparse array) where elements are displaced
(randomly) from their original grid position,
• Fixed Array: an array where the placing of the elements is according to a
formula (e.g. geometric series).
Sparse arrays have less mutual coupling and need fewer radiating elements. This
however results in a lower gain, which depends on the number of elements. An example
is the multi-band RADAR of the Wide Band Sparse Elements (WISE) research project
at T.U.Delft, where several sparse arrays are put on the same substrate, resulting in
one antenna, containing arrays for multiple applications (as on Fig. 3.8). [38]
1111
0000
00 1
11
0
0 1
1
0
0000
1111
00
11
1
0
1
1
0000
11 0
00
0 1111
1
0 0
1
0
1
0000
1111
0000
1111
111
000
00
11
000
111
00
11
000
111
00
11
000
111
000
111
111111111
000000000
000000000
111111111
000000000
111111111
000000000
0
1
0111111111
1
0
1
000000000
111111111
1
0
0
1
1
0
000000000
0 1
1
0111111111
0
1
000000000
111111111
000000000
111111111
000000000
111111111
000000000
111111111
000000000
111111111
111
000
000
111
000
111
000
111
000
111
11111111
00000000
00000000
11111111
00000000
11111111
00000000
11111111
0
1
0
1
00000000
11111111
1
0
0
00000000
11111111
0 1
1
1
0
00000000
11111111
00000000
11111111
00000000
11111111
00000000
11111111
00000000
11111111
Figure 3.8: Several sparse arrays are combined into one antenna.
Another example of a sparse array, used e.g. in ultrasound medical imaging, is the
perfect array . This is an array that has one and just one pair of elements that are
separated by distance nd for every integer of the set n = 1 → N (N − 1)/2. Such
arrays have the smallest possible beam width (for a certain antenna length) and thus
the best possible resolution. But, as shown on Fig. 3.12, this results in very high side
lobes, which is not a disadvantage in a quiet measurement environment. No perfect
array with more than four elements exists [39].
The amplitude of the array factor function of any 11 array with excitation coefficients
all real (i.e. when not scanned), is symmetrical (in u). If f (x) is real12 , then F ∗ (u) =
F (−u) where F (u) = F −1 {f (x)} such that:
p
p
(3.21)
|F (u)| = F (u)F ∗ (u) = F ∗ (−u)F (−u) = |F (−u)|
10 Every (finite) fixed or perturbed array is also a sparse array for it can be obtained from a LESA
with d equal to the greatest common factor of the element spacings in the fixed or perturbed array.
11 Including sparse arrays, even though the array geometry itself is not symmetric.
12 If f (x) is real and even f (x) = f (−x), then:
Z ∞
Z ∞
f (x)e−j2πxy dx =
f (x)ej2πxy dx = F −1 {f (x)}.
F {f (x)} =
−∞
−∞
38
Chapter 3. Phased Array Design
But this can also intuitively be seen. Because only the far field radiation pattern is of
interest, the only path lengths that have an influence on the interference and thus the
radiation pattern are the path lengths from the radiating elements to the phase front
of the direction that we are looking at. These differences are equal in both cases,
except for a minus sign, resulting in the same amplitude after summing.
−10◦
+10◦
phase f
ront
phase to front: 6,5 and 2
or -1,-4 and -6
ront
phase f
phase to front: 1,4 and 6
or -6,-5 and -2
Figure 3.9: The radiation pattern of a sparse array is symmetrical.
Perturbed arrays, such as for example random arrays (where the elements are
displaced randomly) are mainly used to avoid grating lobes without having to lower
d. Moreover random arrays are very robust: even when a radiating element fails, the
performance of the array is hardly influenced [40]. The disadvantage of random arrays
is that even though the grating lobes can be lowered, the side lobes are higher than in
the equidistant case. A means to combine the advantages of equidistant arrays and
random arrays is the so called fractal array. Here the boundaries of the area that is
filled with elements are generated as fractals [41].
An example of a fixed array is the geometric array . Here the placement of
the elements is based on a geometric series. The distance between two consecutive
elements becomes smaller or bigger at the edges of the array, depending on whether
0 < a < 1 or a > 1: (xn = −x−n )
d|n|+1 = a × d|n| = a|n| × d1
(3.22)
The array factor of some geometric arrays with 10 elements and an array length of 6λ
with a > 1 is given in Fig. 3.10. We see that there is no use in taking the a > 1 for
this results in a wider beam and a higher side lobe level. On Fig. 3.11 some examples
of geometric arrays with a < 1 are given. In that case beam width can be traded off
against side lobe level.
As an overview, on Fig. 3.12 some linear arrays with N = 4 are given as well as their
array factor. We see indeed that the perfect array has the smallest beam width. The
perturbed array suppresses the raising grating lobe at the edges of the plot better
than the equidistant array. The geometric array trades off the beam width for a
higher (overall) side lobe level.
3.3. Array Design with Array Factor
39
a=2
a = 1.5
a = 1.2
a=1
(a) Array layouts.
|F (u)|2 [dB]
10
0
−10
−20
−1
−0.75 −0.50 −0.25
0.00
0.25
0.50
0.75
u
(b) Array factors.
Figure 3.10: Array factors of geometric arrays (a > 1, N = 10).
3.3.1.3
Non-linear Arrays
An example is an Nx × Ny grid with spacing dx and dy along the x- and y-direction
respectively.
This array can be seen as a linear array of perpendicularly oriented linear subarrays
Hence, the array factor can be calculated as the product of two array factors of linear
arrays (see Fig. 3.13).
Regular 2D arrays can, as well as the 1D arrays, be thinned or perturbed, resulting
in a sparse or random planar array. The latter, as mentioned before, is often used to
tackle grating lobes (that appear when d ≥ λ). This is shown on Fig. 3.14(d). But on
Fig. 3.14(h) we see that using a random array for the case where d < λ gives a larger
beam width and higher side lobe level than the equidistant array. Moreover, it is
nearly impossible to make a random array for small values of d when the dimensions
of the radiating elements are not infinitely small.
Another example that is used in practice is the circular array. This is an easy way to
obtain a two dimensional array (and thus an array pattern with a beam instead of a
rotational symmetric pattern) with very few elements. It has been applied on ESAs
MSG-I satellite [42]. But in [43] it is used to generate a rotation symmetric pattern
for the reception of GEOs in cars.
40
Chapter 3. Phased Array Design
a = 0.3
a = 0.5
a = 0.8
a=1
(a) Array layouts.
|F (u)|2 [dB]
10
0
−10
−20
−1
−0.75 −0.50 −0.25
0.00
0.25
0.50
0.75
u
(b) Array factors.
Figure 3.11: Array factors of geometric arrays (a < 1, N = 10).
3.3.2
Influence of Excitations
In previous examples all excitation coefficients were equal (same amplitude and phase).
But exciting each element in the array with a different phase allows steering of the
beam in a particular direction. Applying a particular amplitude distribution can
suppress side lobes. This again follows naturally from the properties of the Fourier
transform in Eq. (3.13), f (~r) is now rewritten as:
f (~r) =
∞
X
n=−∞
δ 3 (~r − ~bn ) × w(~r)
(3.23)
where w(~r) is a windowing function such that the excitation coefficient of the nth
element an = w(~bn ).
3.3.2.1
Phase
Applying the properties of the Fourier transform, we see that if the windowing function
w(x) of Eq. (3.23) is multiplied with e−j2πûx , this results in a shift in the transformed
domain F (u − û) On Fig. 3.15 we however see that this shift causes the beam to
broaden in φ due to the non-linearity of the φ = arccos (u) transformation.
3.3. Array Design with Array Factor
41
geometric array a = 1.5
perturbed array N = 4
LESA N = 4, d = 2λ
perfect array N = 4
(a) Array layouts.
|F (u)|2 [dB]
10
0
−10
−20
−1
−0.75 −0.50 −0.25
0.00
0.25
0.50
0.75
u
(b) Array factors.
Figure 3.12: Comparison of some linear non-equally spaced arrays.
(a) LESA Nx = 4, dx = 0.5λ.
(b) LESA Ny = 4, dy = 0.5λ.
(c) Four-by-four array.
Figure 3.13: Array factor of regular planar array.
42
Chapter 3. Phased Array Design
(a) Factor of Equidistant Array (d = 2λ).
(b) Factor of Random N=100 Array.
20
18
18
16
16
14
14
12
12
y
y
20
10
10
8
8
6
6
4
2
4
2
4
6
8
10
12
14
16
18
2
20
2
4
6
8
10
12
x
14
16
18
20
x
(c) Layout of Equidistant Array (d = 2λ).
(e) Factor of Equidistant Array (d =
(d) Layout of Random N=100 Array on same
area as c).
3
λ).
4
(f) Factor of Random N=100 Array.
7
7
6
6
5
5
4
4
3
3
2
2
1
1
1
2
3
4
5
6
(g) Layout of Equidistant Array (d =
7
3
λ).
4
1
2
3
4
5
6
7
(h) Layout of Random N=100 Array on same
area as g).
Figure 3.14: Random arrays have no grating lobes but should be sparse.
3.3. Array Design with Array Factor
43
visible interval
visible interval
|F (u)|2 [dB]
10
∆u = 0.3
0
−10
−20
−2
−1.5
−1.0
−0.5
0.0
0.5
1.0
1.5
u
(a) Beam Steering in u for LESA N = 10, d = λ/2.
2
|F (φ)| [dB]
17.46◦
10
δ̃ = 0◦
δ̃ = 54◦
0
−10
−20
0
15
30
45
60
75
90 105 120 135 150 165 180
φ
(b) Beam Steering in φ for LESA N = 10, d = λ/2.
Figure 3.15: Beam steering makes use of a Fourier property.
The excitation coefficients to steer the beam to (û, v̂, ŵ) can be found as:
an = ej δ̃n = e−j2π(û,v̂,ŵ) · bn
~
(3.24)
or expressed as a phase:
δ̃n = −
2π
(sin θ cos φ · xn + sin θ sin φ · yn + cos θ · zn )
λ
(3.25)
This indeed results in constructive interference:
∀n : an ejβ(ir · bn ) = An ej δ̃n ejβ(ir · bn ) ≡ 1
~
~
~
~
(3.26)
If d > λ/2, it is possible that by shifting the visible interval, more than one maximum
appears. We conclude that as soon as beam steering is required over the full range of
φ : 0 → π, the element spacing must be d ≤ λ/2.
44
3.3.2.2
Chapter 3. Phased Array Design
Amplitude
For the windowing function w(~r) any function can be chosen. Plenty of windowing
functions are possible. Some examples for linear arrays are given in [44]. The most
important ones are repeated in Table 3.1. Shading or tapering an array by choosing
a particular windowing function allows to tradeoff between beam width and side lobe
level Rsl .
The influence of a tapering function can be studied by comparing its inverse Fourier
transform to the inverse Fourier of the rectangular window:
#
!
O
X
−1
3
F −1 (w(~r))
(3.27)
F (u, v, w) = F
δ (~r − ~bn )
n
Table 3.1: Shading window functions and their properties, (after [44]). The side lobe fall is
expressed in dB/Oct., indicating the difference in dB between the height of the
side lobes at u and 2u.
Window Type
Rectangle
Triangle
Hamming
Blackmann
Dolph-Chebychev (Rsl,dB =50)
Dolph-Chebychev (Rsl,dB =80)
Side Lobe Level
-13 dB
-27 dB
-43 dB
-58 dB
-50 dB
-80 dB
Side Lobe Fall
-6 dB/Oct.
-12 dB/Oct.
-6 dB/Oct.
-18 dB/Oct.
0 dB/Oct.
0 dB/Oct.
Beam Width
∆u = 0.89/N d
∆u = 1.28/N d
∆u = 1.30/N d
∆u = 1.68/N d
∆u = 1.33/N d
∆u = 1.65/N d
One can conclude from Table 3.1 that the more a window decays from the central
element, the lower the side lobe level will be. This results however in a wider main
lobe.
Due to this Fourier relationship, one could also calculate the appropriate excitation
coefficients directly as the forward Fourier transform of the desired array factor function.
We will demonstrate this method for the example of a LESA with Dolph-Chebychev13
shading. The Chebychev polynomial is the solution of a differential equation [37]:
dTm
d2 Tm
−u
+ m2 T m = 0
2
du
du
One way of expressing the polynomial of order m is:
(1 − u2 )
Tm (u) = cos ((m) · arccos(u))
(3.28)
(3.29)
13 The Dolph-Chebychev excitation coefficients give us the best possible combination of low side
lobe level and small beam width, for the side lobes all have the same value.
3.3. Array Design with Array Factor
45
Tm=6 is plotted in Fig. 3.16. It is not a periodic function whereas the array factor
function is. Therefore the array factor function is a periodical repetition of an interval
of the Chebychev polynomial.
Suppose that the desired side lobe level Rsl,dB , for an array with N = 7, (hence
m = N − 1 = 6), is 33 dB, then the maximum of the period (see Fig. 3.16) should be
33 dB = 44.7 above the ripple of 0 dB = +1. This means that the period for F (u) is
the ] − u0 , u0 ] interval of the Chebychev polynomial, where u0 can be found from:
Rsl = 10
Rsl,dB
20
= cos (m arccos (u0 )) or u0 = cos
arccos (Rsl )
m
= 1.2937.
(3.30)
The array factor should be:
F (u) = TN −1 (u0 cos (uπ)) = cos ((N − 1) · arccos(u0 cos (uπ)))
(3.31)
N equidistant samples (over [0, 1[ in u) of this function give us, after Discrete Fourier
Transform (DFT), the samples that contain the excitation coefficient. See Table 3.2
for the numeric values of this example. Of course, the excitation coefficients do not
depend on d as not the visible interval, but one period of F (u) is taken for the
equidistant samples.
Table 3.2: Numeric values of the DFT pair for Chebychev tapering synthesis.
u
F (u)
an
0
44.7
16.43
1/7
15.1
39.63
2/7
-0.79
63.56
3/7
0.18
73.68
4/7
0.18
63.56
5/7
-0.79
39.63
6/7
15.1
16.43
|F (u)|
40
30
20
10
0
0.2
0.4
0.6
0.8
Figure 3.16: T6 (u) mapped to one interval of F (u) with Eq. (3.31)
1.0
u
46
Chapter 3. Phased Array Design
Note that using a DFT for array synthesis introduces errors intrinsic to DFT, such
as leakage. This is due to the fact that DFT supposes that the function is periodic
and discrete in both domains. This is certainly not the case.
A disadvantage of any shading is gain degradation. The uniform tapering provides
the highest possible gain due to the smallest main lobe, as can be seen on Fig. 3.17.
This figure also shows that the maximum D for the different taperings lies at other
values for d. This is because the beam width differs for different taperings, such that
the point where the grating lobe lies just behind the edges of the interval, differs. d
equal to any multiple of λ/2 always results in the same value for Dmax . Indeed, if
only one period or several periods of the periodic function are averaged out, makes
no difference. This value, moreover, is the limit for an infinite number of periods, or
stated differently, for d → ∞. In [45] is proven that this value Dmax = ηA × N with
P
( an )2
P
ηA =
(3.32)
N × a2n
the tapering efficiency of the array.
Darray
12
11
10
9
8
7
6
5
4
3
2
1
0
gabled
binomial
uniform
0
0.5
1.0
1.5
Chebychev Rsl = 50 dB
Chebychev Rsl = 80 dB
2.0
2.5
3.0
d [λ]
Figure 3.17: Any type of tapering results in a lower value for Dmax (N = 7).
3.3.2.3
Example: Subarray of the ATCRBS in Bertem
The reader will already have noticed that the radiation pattern of Fig. 3.2 in Sect. 3.1.3
is not practical for a secondary RADAR as half of the power in the main beam goes
into the ground. An airplane can only be expected under an elevation angle, between
0◦ and 90◦ . Eq. (3.25) can be used to elevate the beam above the horizon. But as
explained in Sect. 3.3.2.1, this will only shift the pattern. It is however recommended
to broaden the main beam to cover all elevation angles from 0◦ to 90◦ .
3.3. Array Design with Array Factor
47
The latter can be obtained by not applying a linear phase increment over the entire array, but by selecting appropriate (conflicting) phases for each element, so that
the maximum becomes lower, but constructive interference occurs in more directions.
Tapering broadens the beam even more. The phases and amplitudes used in the secondary RADAR in Bertem were measured with a near field probe (Texas Instruments
FA-9764/5) and are listed in Table 3.3.
Table 3.3: Excitation coefficients of the ATCRBS subarray.
n
phase [◦ ]
amplitude [dB]
1
23
-26
2
-9
-24
3
-27
-28
4
-51
-20
5
-103
-17
6
-167
-20
7
157
-27
8
-111
-32
9
-148
-24
10
160
-24
The resulting array directivity is depicted in Fig. 3.18. Not only D as calculated
with MAGMAS, implying the excitation coefficients of Table 3.3 is shown, also the
measured D is plotted. The difference is due to the inaccuracy of the measured phases
and amplitudes of Table 3.3 as well as due to the metal base that holds the dipoles
and was not taken into account when simulating.
Garray measured
Darray from MAGMAS
Garray [dB]
0
−10
−20
−30
−40
−50
0
15
30
45
60
75
90 105 120 135 150 165 180
φ
Figure 3.18: Directivity of the ATCRBS subarray.
3.3.3
Common Design Techniques
From the preceding sections, it is obvious that the most straight forward way of array pattern synthesis, is sampling the desired array factor and performing an Inverse
DFT (IDFT), as in the Chebychev example of Sect. 3.3.2.2. Essentially this is equivalent with the technique of mapping Taylor expansions or projections onto other sets
of basic functions of the array factor function, a common technique in times when
computer access was not as common as it is now.
48
Chapter 3. Phased Array Design
With the increase in computer power, the approach of random search and optimization
routines become popular in all kinds of fields. Even so in array synthesis. Many
examples of genetic algorithms can be found, all differing in the trade-off between
generality and calculation times, using more or less information about the topology
resulting in faster or slower convergence.
3.4
3.4.1
What if the Assumptions no longer Hold
Errors that can be solved by Calibrating
When implementing beam steering in a phased array in practice, sometimes the excitation coefficients in reality are not those intended by the designers. One cause
for this difference are the mutual electric fields (that even depend on the direction
in which the beam points) of the elements in the array. This will change the complex input impedance of the elements, resulting in a varying phase and amplitude
of the current. A second cause are hardware imperfections. Some examples will be
given in Sect. 6.1.3.1. There also a basic calibration technique will be explained that
compensates for systematic phase and amplitude errors.
3.4.2
Quantization Errors
Quantization of the phase by digital phase shifters, or digital control of analog phase
shifters, causes errors. This was already touched in Sect. 2.3, and now will be worked
out to some extent.
Suppose that a phase shifter can only be controlled by n bits, then the number of
possible states is limited, introducing a phase quantization error of at most:
∆δ̃ =
1 2π
.
2 2n
(3.33)
If the array consists of only two elements, this phase error can easily be related to the
beam direction error starting from Eq. (3.25):
∆δ̃ = −
2π
λ
∆u ⇔ ∆u = n+1 .
λ
2
d
(3.34)
If the array has more elements, the phase approximations not only obstruct the beam
from being steered under certain angles, but also causes broadening of the main beam
and increasing of the side lobe levels. This effect is caused by the fact that when
looking at pairs of array elements, these pairs do not all share the same direction of
constructive interference.
3.5. Conclusions
49
Hence the conflicting phases of the array elements, that were deliberately introduced
in Sect. 3.3.2.3 to obtain a good ATCRBS radiation pattern, are to be avoided in case
of directive antennas, as it is obvious that main beam broadening and side lobe level
rise, result in gain degradation.
Studying the beam width, side lobe level and gain degradation is a topic on its own
as it strongly depends on the array topology. Values from a case study on the array
antenna of Sect. 6.1.2 can be found in [46]. The conclusion there was that a 3 bit
phase shifter is sufficient, taking into account the effects of the other imperfections in
the system, such as the fact that grating lobes appear when the beam is scanned 48◦
off broadside.
3.4.3
Errors that Void the Theory
If the array elements are not all translated versions of a reference element, or the
current profile on all elements is not identical in shape, Eq. (3.1) can not be split into
an array and an element part, voiding all statements and formulas derived from it.
Fortunately theory on this matter is available, e.g. in [1].
3.5
Conclusions
In this chapter, the array factor was introduced, to describe the influence of geometry
and excitation in an array with phase shift beam forming. This chapter did not contain
any new contribution to science, but reviewed a lot of array theory and linked every
statement with Fourier properties. The introduced theory only holds for arrays where
every element is a translation of the base or reference element, hence not for conformal
or MIMO arrays. Moreover, the radiation properties (directivity) are only studied in a
far field approximation. This assumption holds for most communication applications,
as there indeed the intention is to transmit information over a considerable distance.
If the far field assumption does not hold, e.g. in case of the sensor application of
Part III, the approximations applied in the chapter are invalid.
Using Fourier transform properties, it can be understood that the larger the aperture
that is spanned by the array elements, the smaller the beam will be and the higher
the gain. The most basic illustration, is comparing a linear array with a conical beam
to a planar (2D) array with a pencil beam. Still following the same principle, the use
of tapering is disrecommended if the gain of the array has to be high.
50
Chapter 3. Phased Array Design
Tapering can however be very useful for side lobe level reduction to avoid interference
from communications in other directions. The trade off between beam width and side
lobe level can also be used at another extreme, as in sparse arrays: by allowing very
high side lobes but having a very narrow main beam, hence a very high resolution, in
applications with sensor arrays in a quiet measurement environment.
Adding more elements generally increases the gain, but only up to a certain level, if
this diminishes the element spacing d too much.
Scanning of an array is essentially using the shifting property of the Fourier transform.
Note that as the beam scans towards the edges of a planar or linear array, the beam
widens.
Part II
Application: Satellite
Communication
51
53
Overview
Array antennas are a logical choice for LEO satellite communication antennas. As
power on the satellite is scarce, and the distance between satellite and receiver on
earth is considerable, preferably high gain antennas should be used for the satelliteto-ground link. These high gain antennas should moreover be scanned when the
satellite flies by a ground based receiver at a speed of about 7 km/s, the typical LEO
satellite orbit speed.
Mechanical scanning of the antenna has its associated disadvantages such as wear and
vibrations. As in space reparation is out of the question, wear should be avoided. If
the LEO satellite is used for earth observation and thus carries an imaging sensor,
vibrations would dramatically limit the image resolution.
Consequently, for the system of in-situ data up link to a LEO earth observation
satellite, an array antenna was selected. Chapter 4 explains the full system layout,
justifies the choice for an array antenna, shows with a link budget calculation that
the link is feasible with the proposed antenna and explains that the hostile space
environment requires special attention when designing electronics for satellites.
Chapter 5 is dedicated to the design of a space qualified antenna element and the
determination of the array geometry, i.e. of the array antenna elements locations. As
high gain is especially needed when the satellite elevation is low, the array should
have its gain maximized under large scan angles. A technique to achieve this for
linear equidistant arrays is proposed, but unfortunately the technique is not easily
generalized for planar arrays.
The signal shaping and combination parts of the array are discussed in Chapter 6.
There the analog baseband implementation of a technique commonly used for digital
beam forming, designed by ir. P. Delmotte, is reported and evaluated. Afterwards
this implementation is adapted to the space environment for use in the system for
in-situ data up link, deployed in Chapter 4.
invisible filling
Chapter 4
Introduction to the
Application
As is often the case, earth observation was first used for military purposes. Soon governments started using it to obtain better maps. Later on, better imaging techniques
were developed and earth observation became a tool in monitoring and assessing
natural resources such as water and forest. With the unfortunate evolutions in climate, it is nowadays applied to monitor the health status of our planet (CryoSAT,
EarthCARE) [47].
Earth observation sensors not only “look” at the earth in many parts of the ElectroMagnetic (EM) spectrum (such as visible light, infrared and microwaves). Also
active systems, e.g. RADARSAT Sect. 4.4.2, that emit power and analyze the delay,
intensity and polarization of the reflections, are operational for already quite some
years.
If now this “image” (in a broad meaning) data could be combined with in-situ collected data, much more correlations could be investigated, increasing the knowledge
on the impact of factors such as rain, temperature and carbon dioxide concentrations
on crops, climate and so on. Moreover, if this in-situ data is available1 in the imaging
satellite, on board processing could reduce the amount of data that has to be down
linked to the ground station. That this down link is indeed a bottleneck, can easily be understood looking at the vast amount of data that imaging sensors produce.
RADARSAT-I fills up to 76 GB with only 15 minutes of imaging [49].
1 If the satellite does not use the in-situ data, there can still be reasons for up link. Simple
transport can be done way cheaper and easier over terrestrial networks, even for sensors at the poles
of the earth, using sensor networks that transport the data to one connection node of the sensor
network, e.g. ASTRonomisch Onderzoek in Nederland (ASTRON) LOw Frequency ARray (LOFAR)
AGRO project [48]. But sometimes performance (in terms of uptime or delay) of existing terrestrial
networks is insufficient and beyond control.
55
56
Chapter 4. Introduction to the Application
In this chapter, first a concept of a system for in-situ data up link will be investigated.
The link budget and a calculation of the loss of image resolution due to vibration,
motivate the use of electronic beam steering on earth observation satellites. Some
introduction into space design at the end of the chapter makes the reader aware
of the fact that designing electronics for use in space is very much different from
designing consumer electronics. The details on the implementation are postponed to
Chapters 5 and 6.
4.1
System Overview
In-situ data of about 2002 sensors is collected and stored by data loggers. When the
satellite flies by, it activates the ground station that sends the recorded data to the
satellite. There the data is ideally processed with the imaging data, but for the time
being, only stored and down linked together with the image data.
Except for the unconventional routing of the data packets, this system looks very similar to a standard Supervisory Control And Data Acquisition (SCADA) system [50].
Similar services are available from SpaceChecker [51]. That system uses a Geostationary Earth Orbit (GEO) satellite. But obviously up link of the data to an earth
observation satellite offers a huge advantage if the up linked data can be processed
and combined with the imaging data on board of the satellite.
All numbers given here, follow either from the link budget in Sect. 4.3.3 or from the
system design choices in Sect. 4.1.4.
4.1.1
In-Situ Data Collection
In the field, data of about 200 sensors, measuring e.g. temperature, humidity and pH,
is linked up per terminal. The sensors can be interfaced with RS-232, (Power over)
Ethernet (PoE), ISM 868 MHz, 802.11 (WLAN) or in most cases a simple electrical
wire. They should all be connected to a data logger that stores their value hourly for
about four days, the maximum revisit period of a Low Earth Orbit (LEO) satellite.
Hence the data logger should have at least 100 kB of data storage, which is certainly
the case in contemporary devices3 . The ELTEK SQ-1000 data logger, for example,
can store 250 k measurements. It can however only interface with 124 sensors, thus
for 200 sensors two data loggers are needed.
2 The
number of 200 sensors was agreed upon by the future users of the system.
is obvious that in the future smart data loggers will be designed to combine and extract
relevant information from far more than 200 sensors. As long as the amount of information from the
data logger corresponds to the equivalent of 200 sensors, it can still be linked up with the system
proposed.
3 It
4.1. System Overview
57
The data loggers should be connected with the ground station modem via an RS232 or Ethernet link, possibly wireless (up to about 7 km with the DECAGON RM1
Radio Receiver). The data rate is not an issue, as only few data is to be exchanged.
Even the standard RS-232 baud rate will do. The ELTEK SQ-1000 uses RS-232
(38400 baud).
The ELTEK SQ-1000 needs 9 − 14 V. A battery pack should deliver power to the
sensors, data loggers, and a terminal. The data loggers are only active once an hour.
The terminal needs, as a rule of thumb, about 5 W in receive mode and, from the
link budget in Sect. 4.3.3, about 15 W in transmit, for about 16 s/day. Indeed, to
send 100 kB of total data, at a data rate of 6400 bit/s, see Sect. 4.3.3, about 16 s
is needed. With the assumption that 6 W is sufficient for the entire ground segment
and that the night lasts for 14 h, a battery is needed that delivers 12 V and has a
capacity of at least 17 Ah. The PANASONIC LC-RD1217P lead acid battery can
serve these requirements, but batteries up to 33 Ah are available and if that is still
insufficient more than one can be used in parallel. To charge the batteries, solar
cells, with a charger to avoid damage, can be used. They are sold in large variety.
For demonstration purpose, the SOLARKING PT15-300 Powerfilm can be used.
To communicate with the satellite, a separate Transmit (Tx) and Receive (Rx) patch
antenna on spacers in air (r = 1), with two feeds for circular polarization is recommended, as this gives the highest attainable gain and least cross polar leakage. The
terminal (including antennas) should be able to withstand rain and ice (class4 IP67).
An overview of the ground segment, can be found in Fig. 4.1.
4.1.2
Space-Ground Communication Protocol
Software at the Telemetry, Tracking and Command (TT&C) ground station should
make up a time schedule for the satellite on when to query which ground terminals.
One record in this to-do list could be:
time stamp
ground terminal serial number
ground station GPS coordinates
This list will be forwarded to the satellite and be stored there. Once the time is equal
to the time stamp of querying a certain ground station, the activation message for
this ground station has to be broadcasted.
Another approach could be that the satellite sends out a request for serial numbers.
Any terminal hearing this request starts a random timer. When the timer counts down
to zero, it transmits its serial number. The satellite, that listens for serial numbers,
selects the first one, that comes in without collision, for up link and broadcasts an
activation message containing this serial number.
4 The International Protection Rating (IP) code, defined in norm IEC-60529 [52], specifies the
degree of protection against intrusion of objects and dust (first digit 0-6) and water (second digit
0-8) in electrical enclosures.
58
Chapter 4. Introduction to the Application
sensor data report
activation
test seq + sernr + end time stamp
Tx RHCP patch IP67 3 dBRx RHCP patch IP67 3 dB
5 W RF 2.0 GHz
RF 2.2 GHz
modem QPSK FEC sernr
data report
query
battery pack
solar cell
data logger (> 100 kB)
query
data
...
sensor 1
sensor 1
sensor 124
Figure 4.1: Subsystems of the ground segment.
This method looses precious communication time on hand shaking and requires a
more intelligent, hence more expensive, terminal. For a small number of terminals,
this is to be considered. In case of a large number of terminals, all intelligence should
be shifted to the space segment.
The ground station that receives an activation message should check its serial number against the one that was transmitted. If the XOR operation results in an all-zero
string, the ground station can up link the data it got from the data logger and stored
in a buffer for transmission until the time indicated in the activation message elapsed.
The up link will have, as stated above, a 6.4 kbit/s effective data rate. The actual bit
rate will be higher due to Forward Error Correction (FEC). To prevent the system
from being spammed by useless sensor data reports, the terminals should use a secret
key to sign the message.
The satellite will receive the up linked list with sensor data and process it in combination with the imaging data, or store it in a store & forward unit. If the satellite
passes over the ground station used to down link the image data, also the data in
the store & forward unit can be fetched and transmitted over an X-band (8-12 GHz)
down link to the ground station.
4.1. System Overview
59
satellite
(1)
(2) (2) (2)
terminal
TT&C ground station
(3)
(4)
terminal
terminal
data ground station
Figure 4.2: Steps of the communication protocol: (1) Sending up schedule, (2) Broadcasting
activation message, (3) Sensor data up link, (4) Sensor data down link.
4.1.3
Space Segment
The space modem in contrast to the modem on the ground, has to deal with I
and Q baseband signals from the array antenna that is described in more detail in
Sect. 6.2.1. It also takes care of FEC. It will output the received data and receive
activation messages over the satellite Controller Area Network (CAN) bus. To connect
to this CAN bus, a CAN interface controller is obligatory. This component will
wrap all data into packages according to the CAN standard, ISO-11898 [53]. An
overview of the space segment can be found in Fig. 4.3.
CAN ctrl ctrl software
modem
array antenna
Figure 4.3: Subsystems of the space segment.
60
4.1.4
Chapter 4. Introduction to the Application
System Design Choices
One of the most important choices, is the selection of the frequency. From a technological point of view, higher frequencies result in smaller antennas, which is advantageous as satellite surface is scarce. A higher frequency however, results in more
expensive components. But the main driving force in selecting a frequency, is the licensing. Up link at 1980 − 2010 MHz and down link at 2170 − 2200 MHz was selected,
keeping ease of licensing in mind, [54].
Although for frequencies around 2 GHz and higher, Faraday rotation in the ionosphere, which is a polarization rotation, is negligible, this effect is noticeable up to
5 GHz, [55]. It is hence better to use circular instead of linear polarization. Moreover
because in that case no attention should be payed to the relative position (rotation)
of the satellite and receiving antenna.
The bandwidth that is available, combined with the available transmit power and the
losses, determine the data rate. The link budget in Sect. 4.3.3 results in 6.4 kbit/s.
This is sufficient for the application. If the modem is implemented as a Software
Defined Radio (SDR), the data rate can still be changed afterwards.
4.2
Orbit
With the aid of Newton’s laws of motion, the speed of a satellite in its orbit can be
calculated. The force that keeps the satellite in its orbit is the gravitational force Fg
between the earth and the satellite:
Me m
Fg = −G 2
(4.1)
Rd
with Me = 6.0 × 1024 kg the mass of the earth, m the mass of the satellite, G =
6.67×10−11 Nm2 /kg2 the universal constant of gravity, and Rd is the distance between
the center of the earth and the satellite. From Fg the speed of the satellite in its
circular motion can be found combining Eq. (4.1) with Newton’s First Law:
r
Me
v2
⇔v= G
(4.2)
Fg = ma = m
Rd
Rd
v is independent of the mass of the satellite.
LEO satellites circle (according to all definitions) around the earth in a circular (or
elliptical) orbit less than 2000 km above the surface of the earth. In Table 4.1 the
velocity for some orbits is calculated5 . They were checked with the NASA Orbital
Velocity Calculator [57] and agree with the values in [58]. Orbits lower than 185 km
are unstable, because of too much air drag, and therefore never used.
5 In our calculation it is assumed that the earth is a perfect sphere and the influence of the moon
and the sun and air drag is neglected. In [56] it is explained how to take these effects into account.
4.2. Orbit
61
Table 4.1: Orbit characteristics for Low Earth Orbits. The LOS time is calculated with
MATLABTM , as explained in Appendix B for an orbit inclination of 0◦ and 180◦
and a ground station with latitude 0◦ . Down link starts as soon as α > 10◦ .
h [km]
200
800
1000
2000
v [km/s]
7.70
7.46
7.35
6.90
revolution
89 min
101 min
105 min
127 min
LOS [min]
6.5-7.4
14.3-16.4
16.4-19.0
26.2-31.3
down link
4 min
9 min
11 min
17 min
dα [km]
846
2366
2762
4435
loss [dB]
130
138
140
144
Because the speed of the satellite is so high, a user on earth only sees the satellite for
a few minutes. This can be calculated as follows. A user will only see that satellite if
it is above his horizon. On Fig. 4.4 one can see that the length of the arc of the orbit
above the horizon Larc is calculated as:
Larc = 2ψRo = 2ψ (Re + h)
(4.3)
with Ro the orbit radius, Re = 6371 km the earth radius [59] and h the height of the
satellite orbit. ψ can be found, using geometry, as:
Re
h
ψ = arccos
= arccos 1 −
(4.4)
Re + h
Re + h
Combining this length Larc with the speed that was found earlier, we obtain the time
a user has the satellite in Line of Sight (LOS). The results in the fourth column of
Table 4.1 however take the rotation of the earth into account. This is done for an
inclination of the orbit of 0◦ − 180◦ by simulating the (x, y, z) coordinates of satellite
and ground station (latitude of 0◦ ) as a function of time with the earth rotating at
ωe = 7.2925 × 10−5 rad/s (sidereal rotation, which means relative to the stars [60])
and calculating whether the satellite was above the tangent plane through the ground
station or below (see Appendix B). Using the same simulation method one can see
that an orbit height of 35789 km gives an infinite down link time, for the satellite has
a geostationary orbit (GEO).
In practice the time that a satellite can communicate with the ground station is even
shorter than the LOS time, because in most cases buildings or trees obstruct the
sight as long as the satellite has an elevation of less than 5◦ − 10◦ . This results in
the down link times (without earth rotation) of the fifth column in Table 4.1. Due to
the short communication window, for telecommunication purposes a constellation of
several LEO satellites with a hand-over procedure is obligatory. For earth observation
the high speed due to the low orbit is advantageous. This way the satellite revisit
time is shorter.
62
Chapter 4. Introduction to the Application
L
αε
Horizon
ψ
Earth Radius
Orbit Radius
Figure 4.4: Satellite is in LOS when above users horizon. For down link α ≥ 10◦ .
Suppose that the down link starts as soon as the satellite has an elevation of α = 10◦
above the horizon, then the total (satellite to ground station) path length is:
p
(4.5)
Rα =10◦ = Re cos (90◦ + α ) + Re2 cos2 (90◦ + α ) + (h2 + 2Re h)
This long path gives the worst case of the transmission losses for the link, the more
because this is also the case where the path length through the atmosphere is at
maximum. The length of this path as well as its free space path loss Gf s (thus
without influence of the atmosphere) that will be calculated in Sect. 4.3, are given in
the sixth and seventh column of Table 4.1 respectively.
As can readily be seen on Fig. 4.4 the distance between the satellite and the ground
station varies as the satellite flies over. This means that the satellite has a velocity
component towards the ground station. This causes a Doppler shift of the down link
signal at the receiver. In general this Doppler shift is small compared to the bandwidth
of the signal (see Appendix A), but still it can cause problems when demodulating.
4.3
4.3.1
Link Budget
Theory
If the satellite were a point source of radiation, the power density that reached the
earth could be calculated using the formula of the surface of a sphere with radius Rd .
4.3. Link Budget
63
This density, multiplied by the area A = λ2 /(4π) of an isotropic antenna results in
the power received by the isotropic antenna:
Pr =
Ps λ 2
= Gf s × Ps
4πRd2 4π
(4.6)
This attenuation factor Gf s is referred to as the free space path loss.
Because an antenna is not a point source and hence does not have an omni-directional
radiation pattern, the radiated power Ps is amplified by the gain of the transmit
antenna Gs . Similarly, adding the gain of the receiving antenna Gr and an attenuation
(Gatm < 1) due to atmospheric absorption [61]:
Pr = Gatm × Gr Gs
Ps λ 2
Ps
λ2
= Gatm Gs
Aeff with: Aeff = Gr
2
2
4πRd 4π
4πRd
4π
(4.7)
In literature this relation is known as the Friis transmission formula [62]. Aeff is the
effective area of the receiving antenna. Eq. (4.7) is graphically represented in Fig. 4.5.
The attenuation is largest when the path length is long. The path length as obtained
from Eq. (4.5) for an elevation of 10◦ above the horizon thus should be used when
calculating the link budget.
P [dB]
Ps
Gs
Gr
Gatm
Pr
Rd
Figure 4.5: Link budget.
Looking at Eq. (4.7), it is obvious that the received signal level depends on the product
of Gs and Gr . It is thus possible to trade off between both gains without losing overall
signal strength. The quality of a channel, however, depends on the Signal-to-Noise
Ratio (SNR) rather than on the signal level. Noise comes in at several stages of the
link. The total noise is found as an integral over the radiation pattern of the antenna,
but a simplified description based on main lobe and side lobe picked up noise, as
illustrated in Fig. 4.6, gives more insight:
1. When the antenna on earth looks at the satellite, the main lobe also picks up
some galactic noise with a temperature of 2.7 K and some sky noise with a
temperature of 10 K (for 2 GHz) [63].
64
Chapter 4. Introduction to the Application
2. Due to side lobes, the receiving antenna picks up noise from several directions
thus from several other communications or from noise sources at the earths
surface with a temperature of about 273 K. The higher the gain of the receiving
antenna, the lower the side lobe level will be and thus the more this noise will
be attenuated.
3. In case of rain, other communications are scattered on rain drops and are hence
picked up as noise by the receiving antenna main and side lobes. The heavier
the rain, the smaller Gatm (or the more negative, when expressed in dB) and
the more interfering signals will be received.
N12.7
Nscatter
K
Gatm
∼
N273
1
Gatm
K
∼
1
Gr
side lobe
transmitter
receiver
Gs
Gatm
Gr
main lobe
Figure 4.6: SNR in the link (without transmitter and receiver noise).
In order to minimize the noise energy that is picked up by the receiver, the receiving
antenna should be directive. The cosmic noise coming from the direction of the
satellite can however not be discarded this way. The signal of the satellite should
thus be large enough to detect it in a surrounding of 12.7 K noise. The performance
of the system will however not significantly increase when the signal level is increased
even further, which can be seen on Fig. 4.7, due to the logarithmic behavior of the
channel capacity Cc [64]:
Pr
(4.8)
Cc = B log2 1 +
Pn
where B is the bandwidth and Pn the noise power. A high capacity is however useful,
as it permits to exchange huge amounts of data in very short time. This is useful as
even a ground station near the poles only sees the satellite for about 60 min per day.
4.3.2
Space Ground Trade Off
Though in all cases, the best SNR is obtained by using an antenna with high gain on
the ground and a more modest antenna in space, sometimes the economical reality
imposes other rules to be followed. For a system with millions of users worldwide,
the hand held receiving equipment should be as light and as cheap as possible [65].
This implies that the antenna of the receiver will be of very poor quality, putting high
demands on the transmitting antenna on board of the satellite in order to obtain a
high Gr × Gs value. A parabola dish on the satellite seems to be the logic choice.
4.3. Link Budget
65
Cc [bit/s]
5
4
3
2
1
0
0
5
10
15
20
25
30
35
Pr /Pn
Figure 4.7: Channel capacity from Eq. (4.8) with B = 1 Hz as a function of SNR.
However, in case of time division multiple access (TDMA) in order to increase the
number of simultaneous calls, the pointing of a dish would be far too slow. Therefore
a large array antenna is chosen in most cases (e.g. IRIDIUM [66]). This results in a
lower gain than would be obtained with a dish, but the beams can be pointed more
flexibly.
But even if the antenna on the ground has a large gain, there is still an incentive to
use a directional antenna on the satellite. Because the (piggy back) launch has a cost
(at least $ 10.000,00 [67]) per kilogram, the weight of solar cells has to be kept at a
minimum, so any measure to lower the power consumption is worth considering.
The disadvantage of directional antennas is that they must be pointed accurately in
the direction of the receiving antenna, as the satellite points towards the center of
the earth. For TT&C antennas, this is even more important, as in case of failure
of pointing, or in case the satellite starts spinning, this could mean the loss of the
satellite. Hence in emergency cases, it should always be possible to switch to a more
omni-directional antenna.
4.3.3
Example: Link Budget for an Orbit at 600 km
In Table 4.2, numeric values are given for an example of a link budget. Again, for
worst case α = 10◦ . The gain of both ground terminal and satellite antenna is
chosen as 3 dB, a fair supposition for a patch antenna. If the array on the satellite is
in operation, Gr = 10 dB, so that if the modems are implemented as SDR, the data
rate can be increased.
For the noise power, the noise temperature is assumed to be Tn = 273 K.
Pn = kTn B = N0 B = 1.38 × 10−23 Tn 12.5 × 103
(4.9)
with k = 1.38 × 10−23 J/K Boltzmann’s constant and N0 the Power Spectral Density
(PSD) of the Noise. The allowable Bit Error Rate (BER) is taken to be 10−6 .
66
Chapter 4. Introduction to the Application
Table 4.2: Numeric example for link budget. Input left, output right.
input
h
α
fc
data rate
B
Ps
Gs
Gr
Gatm
Tn
BER
value
600 km
10◦
2 GHz
6400 bit/s
12.5 kHz
5W
3 dB
3 dB
-0.5 dB
273 K
10−6
output
Rα =10◦
Gf s
Pr
Pn
Eb /N0
margin (PLL)
margin (no PLL)
Eq.
(4.5)
(4.6)
(4.6)
(4.9)
(4.10) and (4.9)
14.35-10.5=
14.35-13.5=
value
1962 km
164 dB
-145 dB
-163 dB
14.35 dB
3.85 dB
0.85 dB
For Quadrature Phase Shift Keying (QPSK), this results in a desired energy per bit
Eb over N0 of Eb /N0 = 10.5 dB for coherent6 QPSK, from BER curves in e.g. [6]. To
calculate Eb out of Pr , the length of one bit interval is taken into account:
Eb =
4.4
Pr
data rate
(4.10)
Motivation of Electronic Beam Steering
That there sometimes is a reason for an antenna with higher gain on the satellite,
became clear in Sect. 4.3.2. But this can either be a mechanically steerable or electronically steerable one. The advantages of mechanically steerable antennas, are that
they have a higher gain for the same area and less losses in the feeding network. The
advantages of electronic beam steering are that they are lighter, cheaper, low profile,
have a longer lifetime (no friction abrasion as no parts move with respect to each
other) and do not require immediate counter-action of the satellite attitude control
system nor do they consume energy for scanning. But the main reason for choosing
electronic beam steering is that scanning is done extremely fast compared to mechanical scanning and that no vibrations occur when scanning. For any mission carrying
optical instruments, the use of dishes is hence ruled out as the drop in resolution due
to vibrations when scanning is dramatic, as will be explained in Sect. 4.4.1.
Reflector antennas are an evident choice on GEO satellites, because of the higher gain
and the fact that the antennas never have to scan.
6 The value of E /N = 10.5 dB is only true if a noise free version of the carrier frequency is
0
b
available, e.g. via a Phase Locked Loop (PLL) in the receiver. Else Eb /N0 = 13.5 dB.
4.4. Motivation of Electronic Beam Steering
67
For LEOs, depending on the mission, a large (for mobile communication with preferably low cost terminals) or small (for scientific data down link to only a few ground
stations) array antenna is preferred. This thought is confirmed by real life examples
in Sect. 4.4.2.
4.4.1
Note on Vibration
As mentioned earlier in the text, as a parabolic dish is rotated in order to lock on a
ground station, the satellite will rotate as well as it tries to preserve its angular momentum. A back-of-the-envelope calculation gives an idea of the order of magnitude
of this effect.
2l
ωbody
√
3
2 lM
ωant
Figure 4.8: Model of the satellite.
Assume the satellite is a cube with mass M and sides of length
√ 2l and the dish is
modeled by a triangular prism of mass MM with sides of length 3lM /2 as on Fig. 4.8.
In order to preserve the angular momentum, the rotation of the reflector antenna
shall be compensated by a rotation of the body of the satellite so that:
ωant LM = ωbody L
(4.11)
with ωant the angular velocity of the antenna, ωbody the angular velocity of the satellite
body, LM the moment of inertia of the prism and L the moment of inertia of the
cube. With the formulas in [68], the angular momentum around a perpendicular axis
through the joint point of the cube and the prism writes as:
#
!
√
MM
M
3 2
2
2
ωant
3(
lM ) + MM lM = ωbody
2(2l )2 + M l
(4.12)
36
2
12
and the rotation of the satellite, for M = 100 kg, MM = 1 kg, l = 0.25 m and
lM = 0.29 m, is found as:
ωbody = ωant
2
51 × MM lM
= 8.6 × 10−3 × ωant
2
80 × M l
(4.13)
68
Chapter 4. Introduction to the Application
For an orbit height of 600 km, ωant = 0.72◦ /s when in zenith above the ground
station. If the rotation of the bus would not be compensated by the attitude control
system, for 1/30 s which is a standard shutter time, this would give an angular
rotation of the bus of 2 × 10−4 ◦ or 3.4 × 10−6 rad, corresponding to a movement of
600 × 103 × 3.4 × 10−6 = 2.2 m on the ground.
Knowing that the resolution of the Japanese Earth Resources Satellite (JERS-1) is
18 m × 18 m and of the French SPOT-4 is 10 m × 10 m, we see that these vibrations
indeed lower the spatial resolution considerably. Even for RADAR sensing, vibrations
lower the resolution, because of displacement of the array elements, causing the main
lobe of the antenna to become wider. The same problem arises for RADARS mounted
on the wings of airplanes.
4.4.2
Examples of Phased Arrays on Satellites
R satellites [66]. Each of the 66 satellites carries
A first example are the IRIDIUM
three phased array panels. These allow each satellite to produce 48 fixed down link
beams. The disadvantage is that when the receiver is at the edge of a certain fixed
beam, the signal level will be lower. The beams are formed by use of Butler matrices:
16 RF signals (for one panel) are fed into a power divider, which sends each input
signal to the appropriate ones of 80 outputs. Each one of these 80 outputs is an
input to the Butler Matrices, which have 80 output beams that fill the scan volume
of the array, depicted in Fig. 4.9. Besides, each patch antenna has a 5-bit phase
shifter for temperature compensation, even though the panels are kept at a constant
temperature by heat pipes.
Figure 4.9: The 80 beams of the IRIDIUM antenna (from [66]).
4.4. Motivation of Electronic Beam Steering
69
IRIDIUM has a large array, as it is a satellite for personal communication. Scientific
satellites need a smaller array for data down link, e.g. the commercially available
Alcatel high-rate X-band antenna [69], especially designed for down link of earth
observation images to a ground station. Again, to obtain a high data rate at a low
BER, the received signal must be high. This can be achieved using an antenna with
high gain.
As can be seen on Fig. 4.10, the antenna consists of a truncated cone with 24 arrays
of stacked patch antennas mounted on it. Note that for this antenna the radiating
elements are not translations of a basic element. This means that the antenna is not
an array antenna, but a conformal array antenna. The advantage is that the beam
does not loose gain when it is pointed under an angle larger than 60◦ , which is a
problem with planar arrays of patch radiators.
The phase shifting is done by 5-bit phase shifters. Butler matrices are used in the
antenna as well, not for the scanning, but to allow (by means of destructive interference) to not excite the subarrays that are at the back of the antenna with respect
to the scanned beam. This could also be obtained by simply switching off certain
subarrays, but then the amplifiers after the phase shifters are not always loaded in
the same way. The structure weighs about 8 kg.
Figure 4.10: The Alcatel X-band antenna for LEO satellites [69].
Phased arrays not only serve as communication antennas. On RADARSAT [70] a 15
by 1.5 m slotted waveguide phased array is used as a steerable active remote sensing
RADAR. The azimuth beam pattern is fixed but the elevation angle of the main
beam can be varied (to obtain different incidence angles) by the use of 32 variable
phase shifters. Because these ferrite phase shifters are not reciprocal, they have to
switch back and forth each time the antenna is switched between transmit and receive
mode.
70
Chapter 4. Introduction to the Application
4.5
Designing Space Instruments
Due to the cost of a launch per kg, the power consumption and mass of any design
should be minimal. As reparation or upgrading in space is impossible, it has to resist
the severe space conditions. This makes going into space expensive.
4.5.1
Space Conditions
One could say that the conditions in space are hostile. But the advantage over the
circumstances on earth is that it is a more predictable situation. No fingers will poke
and no users will do unexpected things. [71]
Apart from space conditions, the structure has to endure severe shocks (with accelerations up to several times g = 9, 81 m/s2 ) and mechanical (with accelerations of
about 0.5g) as well as acoustical (up to 140 dB) vibrations during launch.
4.5.1.1
Pressure
Due to the absence of gas particles, the pressure in space is very low. A slight pressure
increase is caused by sun particles, resulting in (10−2 − 10−7 Pa) [72]. But this still
low pressure causes some problems:
• no convection: Because of the absence of gases, there is no convection. The
only way of heat dissipation is radiation. When the satellite is in sunlight, it
might become a problem to cool the electronics.
• outgassing: Some materials become volatile once the spacecraft is in orbit.
This of course alters the properties of the devices and can disturb the functioning. Moreover, the gases that are set free can condense elsewhere, causing a lot
of problems.
The advantages of low pressure are:
• less corrosion: Because the pressure is lower, there is less oxygen to oxidate
the metals. Note however that the oxygen that is present, is atomic oxygen
which is a lot more corrosive.
• perfect insulation: The breakdown voltage in vacuum is the highest possible.
However, due to outgassing, corona discharge and eventually an arc can occur.
A complete overview is given in Fig. 4.11.
4.5. Designing Space Instruments
Lubrication Problems
71
Conductivity Problems
Change in Operational Properties
(Mechanically and Electronically)
Vacuum
Outgassing
Modification Optical Properties
Gas Cloud
Condensation
Modification Radiation Properties
Cooling Problem
Modification Electrical Properties
Perturbation of
Measurements
Corona
Arc
Noise
Damage
Figure 4.11: Overview of the problems due to low pressure. (after [72])
4.5.1.2
Temperature
Temperature T is linked with the kinetic energy of moving particles (atoms or molecules)
of a body:
1
mv 2 = kT
(4.14)
2
The square of the speed of the atoms is averaged v 2 . m is the mass of one particle.
Applying Eq. (4.14) in a LEO environment gives a temperature of 700 − 1700 K
depending on the activity. Due to the lack of atoms, this heat is not transferred to
the satellite.
Because heat is only exchanged via radiation, the temperature at equilibrium can be
found using the black body radiation formula:
Pin = Pout = T AσT 4
(4.15)
with Stefan-Boltzmann’s constant equal to σ = 5.67 × 10−8 W/m2 K4 . T is the
thermal emissivity coefficient. A is the total outer surface of the object.
When the satellite is sunlit, the radiation that is captured comes from the sun and
from the earth. In eclipse only radiation from the earth is received. The sun radiates
4.18 × 1026 W. The distance between the earth and the sun is 1.49597 × 1011 m.
72
Chapter 4. Introduction to the Application
So the earth captures:
Pin,e = αT ×
4.18 × 1026
× πRe2 = 1.14 × 1017 W
4π(1.49597 × 1011 )2
(4.16)
where αT = 0.6 is the thermal absorption coefficient of the earth. With Eq. (4.15)
the temperature of the earth is: (T = 0.9) [73]
s
r
Pin,e
1.14 × 1017
4
Te =
= 4
= 257 K
(4.17)
T Ae σ
T σ4πRe2
Applying the same formulas (with αT /T = 1) to the case of the satellite in sunlight
as well as in eclipse, leads to the results in Table 4.3. The higher the orbit above
the earth’s surface, the colder the satellite. Note that the choice of material can
influence αT and T allowing to control the temperature. Two types of satellites were
calculated, based on the ratio:
fA =
Asat
Asat,proj
=
area that emits thermal radiation
area that captures thermal radiation
(4.18)
with Asat,proj the projected surface, namely satellites with fA = 2 such as satellites
with large solar panels or spheric objects, and with fA = 6 such as cubical satellites.
The formula used to obtain the values for eclipse in Table 4.3 is:
s
s
αT Pin,sat,m2
αT Pout,e
4
Tsat,eclipse =
= 4
(4.19)
T fA σ
T fA σ4π(Re + h)2
where the power density at the satellite Pin,sat,m2 is calculated from the power emitted
(or captured, as both are equal according to Eq. (4.15)) by the earth Pout,e that was
calculated in Eq. (4.16). When sunlit, the first equality of Eq. (4.19) can still be used,
with:
4.18 × 1026
(4.20)
Pin,sat,m2 =
4π(1.49597 × 1011 )2
The results in Table 4.3 show that the satellite undergoes a cyclic variation of temperature with amplitude of about 55 K.
Table 4.3: Temperature cycles for LEO satellites (αT /T = 1).
position
(fA = 2)
(fA = 2)
(fA = 6)
(fA = 6)
sunlight
eclipse
sunlight
eclipse
200 km
350 K
207 K
266 K
158 K
500 km
349 K
203 K
265 K
154 K
800 km
348 K
198 K
264 K
151 K
1000 km
347 K
195 K
264 K
148 K
2000 km
345 K
183 K
262 K
140 K
4.5. Designing Space Instruments
73
This cycle is experienced more than ten times a day as can be seen from the revolution
times in Table 4.1. Note that heat from amplifiers etc. is not taken into account
and that the steady state situation is calculated. In practice the total temperature
variation for a cubical satellite with passive thermal regulation (heat pipes) is about
20 K inside and 100 K on its surface. Of course locally the temperature variation can
be bigger (−100 ◦ C to 100 ◦ C [72]).
The main problem caused by this temperature variation is fracture due to mechanical
stress. The problem with the high temperatures can be outgassing and damaging
of electronics due to problems with cooling. The problem with low temperatures is
condensing of the outgassed components. An overview is given in Fig. 4.12.
Material Degradation
Increased Outgassing
Cooling Problem
High
Temperature
Loss of Protective Coating
Cycling
Mechanical Fatigue
Cracks / Fracture
Low
Condensation
Modification Electric Properties
Figure 4.12: Overview of the problems related to temperature. (after [72])
4.5.1.3
Radiation
In space, four types of high energetic charged particle radiation occur: [74]
• Galactic Cosmic Radiation (GCR): The source of these high energetic
protons and fully charged heavy ions must probably be found outside our Solar
System. An energy of up to hundreds of GeV can be carried per nucleus.
• Anomalous Component of GCR (ACR): A certain component of the GCR
diffuses in the outer spheres of the sun. There it interacts with the solar winds
and gets singly ionized by the UltraViolet (UV) radiation. Then it is released
back into space as radiation with an energy of up to 100 MeV per nucleus.
• Coronal Mass Ejection (CME): Comparable to the solar wind, which is a
constant mass ejection, sometimes large quantities of protons and intermediately
charged ions are ejected from the corona due to a sudden change in the magnetic
fields of the sun. Energies up to 430 MeV occur.
74
Chapter 4. Introduction to the Application
• Cosmic Ray Albedo Neutron Decay (CRAND): Cosmic rays interact
with atoms in the upper atmosphere of the earth, creating neutrons. These
neutrons decay after 630 s into a proton and an electron. The energy for a
proton can be as high as 300 MeV, for an electron only up to 7 MeV.
Indeed no X-rays (electromagnetic rays with a frequency about 1019 Hz) are mentioned, though they occur frequently as a result of solar flares, because their energy is
very low compared to the radiation listed above. Using Planck’s formula to calculate
the energy content of an electromagnetic wave this can easily be seen:
E = hp ν = 6.61 × 10−34 × 1019 ≈ 10−14 J ≈ 100 keV
(4.21)
The conversion is 1 eV = 1.6 × 10−19 J. hp is Planck’s constant.
Because all these high energetic particles are electrically charged, their movement
is influenced by the earth’s magnetic field. This gives raise to the phenomenon of
aurora borealis (also called as northern light) or, when the particles are trapped in
the magnetic field and thus not guided into the earth’s atmosphere, causes the Van
Allen radiation belt, a toroid around the earth at 700 − 10.000 km above the earth’s
surface, as shown in Fig. 4.13.
Magnetic Axis
Mirror Point: Particle is Trapped
Motion is a Helix due to:
F=v x B
Trapped Particles: Radiation Belt
(outer)
(inner, Van Allen)
Charged Particle
Magnetic Field Lines
Electrons
Protons
Aurora
Charged Particle
Belts
Movements
Figure 4.13: Some charged particles cause Aurora, others get trapped in Radiation Belts.
So the magnetic field of the earth forms a natural shielding against the violent space
radiation. But still LEO satellites can get hit by the radiation because: [75]
• in polar orbits (inclination > 50◦ ) they are exposed to unattenuated fluxes of
cosmic rays and solar energetic particles and encounter the outer electron belt
at auroral latitudes, and
• off the coast of Brazil the Van Allen Belts start at 400 km due to a dip in the
magnetic field (called the South Atlantic Anomaly SAA).
4.5. Designing Space Instruments
75
The harmful effects of the radiation, summarized in Fig. 4.14 are:
• displacement damage: (or material degradation or bulk damage) The radiation damages the lattice structure of the material, producing recombination
centers, which degrades the performance of semi-conductors.
• ionization damage: The charged particles that are trapped in the material,
introduce an electric charge. This can offset the bias of a transistor.
• Single Event Effects (SEE): (mainly caused by protons) Due to radiation a
bit flips (e.g. 0 → 1). This can result in wrong data, but can also cause fatal
errors if the bit in a system command was flipped.
The first two effects are cumulative, this means that during time more and more
radiation degrades the component until it stops functioning. Mathematically this
comes down to integrating the flux of radiation (that is stopped and absorbed) over
the life time of the component. This total dose is expressed in gray or rad where
1 Gy = 1 J/kg = 8.6207 × 1018 eV/kg = 100 rad.
Also UV radiation occurs. This causes embrittlement (the structure of e.g. polymers
changes) and gives raise to the photo-electric effect (electrons are excited out of the
material). The latter results in electrostatic charges (which can also be caused by
the electrons in the plasma surrounding the spacecraft) and eventually ElectroStatic
Discharge (ESD) problems.
Bias Offset
Arc Discharge
Surface Charging
Radiation
Lattice Damage
Aging
Single Event Upsets
Bit Flip
Latch Up
Figure 4.14: Overview of the problems related to radiation.
76
4.5.1.4
Chapter 4. Introduction to the Application
Space Debris
It is not unlikely that debris hits the satellite. On the one hand this can be natural
debris such as (dust of) a comet (with ice) or an asteroid (without), for many of them
have orbits that cross the earth’s orbit [76]. On the other hand also lots of wrecks
from former spacecrafts still orbit around the earth. A collision can cause severe
damage and can even leave a crater on the surface that was hit (triggering ESD).
4.5.2
Product Assurance
The science on designing instruments for the space environment is called product
assurance (PA). The design should be such that the chance that an error occurs is
as small as possible and if an error occurs the damage should be as small as possible
[56].
PA works on all levels of designing a satellite: materials, components, subsystems and
the entire satellite. Some materials are prohibited in space as they outgass (e.g. fluid
glue) or form whiskers (e.g. Zinc). All tested and space qualified components appear
on the Space Qualified Parts List (QPL) or the Preferred Parts List (PPL), available
from [77]. If Commercial Off-The-Shelf (COTS) or newly developed components
are considered for usage, they should be tested first, in order to asses the risk of
breakdown, and to find out what the failure modes are.
Based on the failure modes of the components, the subsystems are tested on their
behavior in case of failure. Some ways to enhance performance are shielding, redundancy and derating. Derating means that several components function in parallel so
that they are only partially loaded, which longens the lifetime. In case of redundancy,
the components also function in parallel but are fully loaded, so that if one fails,
another can be made active (standby redundancy) or a decision is taken by majority
vote over all parallel components (active redundancy).
In the end the satellite has to be tested. In former a prototype, called the Qualification Engineering Model (QEM) was built and tested to destruction. Nowadays a
protoflight model is used, which means that the QEM will become the Flight Model
(FM) which will fly if all tests succeeded. The tests are based on expected failures
that are related to system level failures, which are induced. In the end a figure for
the probability that the satellite will accomplish the mission is obtained.
Often the concept of the satellite is tested on a Bread Board Model (BBM). This is
done in an early phase of the design, even before the optimization phase.
4.6. Conclusions
4.6
77
Conclusions
In this chapter, a system for in-situ collected data up link to an earth observation
satellite was deployed. This system would bring the advantageous possibility of combining the data from the imaging sensor on board the earth observation satellite with
in-situ data. This way, preprocessing on the image data could be carried out on board
the satellite, drastically reducing the demands on the imaging data down link, which
is obviously a bottleneck for earth observation satellites. Due to the high orbit speed,
calculated to be typically 7 km/s in this chapter, the communication window with
the ground station is indeed limited.
Although the data that must be exchanged for the in-situ data up link system is very
modest, compared to the huge amount of image data, still the use of array antennas
for that system is advisable. Array antennas make sure that the power, that is scarce
on the satellite, is not wasted by radiating it into unwanted directions, and do not
suffer from wear or give raise to vibrations, which is the problem when mechanically
scanning high gain antennas such as parabola dishes. Moreover, the link budget
calculation and the examples of existing LEO satellite array antennas, given in this
chapter, indicate that the communication link for the in-situ data up link system is
feasible with the proposed antenna.
The orbit calculations in this chapter point out that the array antenna should have
its highest gain under large scan angles, used when the satellite elevation above the
ground station horizon is low. This will be the starting point for the array optimization
in Chapter 5. The same orbit calculation also shows that Doppler shift will be a
problem for the communication link. This issue will be dealt with in Chapter 6.
Another issue is the hostile space environment. In this chapter, the possible failures
associated with the LEO space environment were discussed, as well as the general
design rules to avoid or minimize damage. These guidelines at a high level of abstraction, will be translated into practical engineering rules, when designing antenna and
control circuitry in Chapters 5 and 6.
invisible filling
Chapter 5
Array Elements
In this chapter, first the element specifications that follow from the design choices
made in Sect. 4.1.4 and from the general rules for space design in Sect. 4.5, will be
summarized. Then, in Sect. 5.2, an appropriate substrate will be chosen and a patch
element will be designed in Sect. 5.3. With this element an array geometry can be
evaluated. A first step is taken in Sect. 5.4.
5.1
Element Specifications
In Sect. 4.1.4 the working frequency was fixed to 1980 − 2010 MHz and 2170 −
2200 MHz. This is a huge bandwidth for patch antennas, hence stacked patches
are preferred to obtain a dual band operation. In the same Sect. 4.1.4, circular polarization was selected. This can be obtained by a single feed and a modified patch
(e.g. a corner cut away). But a dual feed topology results in better RF performance.
The feeds should be implemented mechanically as robust as possible. Probe feeding
is preferred as it has no backward radiation, as opposed to aperture feeding, and is
compact, in contrast to line feeding. As long as a substrate is used, no force acts on
the probe which means that there is no danger of breaking the probes during launch.
Aperture feeding is not recommended as multi layer substrate needs gluing. Moreover,
as the power is low, Passive Inter Modulation (PIM), the prime reason why probes,
that need soldering, are avoided on satellites, will not be a problem.
To prevent the antenna substrate from being charged by radiation, a passivation
coating should be applied. This is closed space specific know how, that lies way
beyond the scope of this text. Related to passivation coating, a plastic cover should
be placed over the antenna to prevent debris impacts causing craters and ESD.
79
80
Chapter 5. Array Elements
The antenna should not degrade the performance of other systems on the satellite.
Conducting shielding walls are appropriate to avoid ElectroMagnetic Compatibility
(EMC) problems with other components, but can cause secondary radiation, and add
weight to the structure.
5.2
Antenna Substrates for Space Application
Choosing a substrate for an antenna or circuit design deserves special attention. In
this section we will first explain what the characteristics of substrates are, which
substrates are in use and what would be the best choice to use in space.
5.2.1
Tolerances
Substrates are only available for a limited number of values for . Indeed is it less
important to have a certain value for , than to have tight bounds on the variation
of over the entire substrate. A circuit or antenna after redesign with the correct
(measured) value for on a substrate with different from the originally intended,
overperform an implementation on a substrate with the right (mean) but with
a strong deviation depending on the place on the substrate. In other words, the
tolerances are much more important than the value. The ideal substrate has: [78]
• a well defined and constant dielectric constant
• a well defined thickness and a smooth surface
• a high purity with high uniformity and low losses and anisotropy
• a high resistivity
• a high thermal conductivity
5.2.2
Properties
5.2.2.1
Dielectric Constant The relative magnetic permeability for most substrates is µr = 1, but the electric
permeability differs considerably from substrate to substrate. All common substrates
have r between 2.10 and 10.2. Some examples can be found in Table 5.1.
5.2. Antenna Substrates for Space Application
81
Table 5.1: Some examples of the available r for RF substrates.
type
Arlon CuClad 217
Arlon DiClad 880
Rogers UltraLam 1217
Rogers RT/Duroid 5880
Rogers RO4003
Rogers RO6006
2.17
2.17
2.17
2.20
3.38
10.2
r
±
±
±
±
±
±
0.02
0.02
0.02
0.02
0.05
0.25
material
Crossplied Woven glass PTFE
Unidirected Woven glass PTFE
Woven glass PTFE
Random glass PTFE
Woven glass Polymer Ceramic
PTFE Ceramic
The value of r , at a certain frequency, of a substrate can be determined by measuring
the capacitance of a double cladded substrate panel:
C=
r A
Q
=
V
dt
(5.1)
where A is the area and dt the thickness of the substrate.
The higher the value for r , the lower the speed of light in the material due to:
c0
c= √
r
(5.2)
and thus the shorter the wavelength. For this reason, a high r is a good choice for
circuit substrates or for antenna miniaturization. For high gain antennas, r should
be small, to have a large antenna resulting in a higher gain. This also results in a
higher bandwidth.
The value of the characteristic impedance Zc for a micro strip line (of width b) also
depends on r : [79]
1
4b
1
Zc = √ 60 ln
∼√
(5.3)
r
d
r
5.2.2.2
Dissipation Factor tan δ
A double cladded substrate is a capacitor. Hence, to quantify the resistive losses in a
substrate, the dissipation factor or loss tangent [79]
tan δ =
dissipated energy
stored energy
(5.4)
can be measured, which is the ratio of the resistive part to the capacitive part of the
impedance of the lossy capacitor.
82
Chapter 5. Array Elements
5.2.2.3
Dielectric Thickness dt
The thickness or height of the substrate dt is the thickness of the bare substrate, thus
without metal. This is often expressed in mil 1 e.g. 50 mil is 0.050” or 1.27 mm.
A higher dt gives more bandwidth for micro strip antennas as it longens the resonator
at the sides. But due to a longer path length in the material, the losses increase, so
that the efficiency and gain decrease [80].
A constant thickness and very few pits are important. This is expressed in tolerance
grades from grade A (best) to C for pits and class 1 (best) to 4 for thickness variation.
5.2.3
Substrate Materials
5.2.3.1
Teflon (r : 2.10 → 10.2)
Teflon or PolyTetraFluoroEthyleen (PTFE) has a dielectric constant of 2.10. By
adding impurities a higher r can be obtained:
• ceramics result in high dielectric constants (r : 6 → 10).
• glass generally results in lower dielectric constants (r : 2.17 → 4).
The glass in the PTFE can either be a woven matrix or randomly distributed fibers.
The first results in a more rigid structure that is however more lossy and anisotrope.
Due to capillarity the substrate will also absorb more water than the random glass
counterpart. Both are though better on this issue than the Teflon with ceramic type.
PTFE with ceramic also has a lower peel strength and deforms easier than PTFE with
glass. Therefore often a metal backing is used. An overview is given in Table 5.2.
The more impurity added, the higher the r will be. But with more glass or ceramic
added, also the losses and anisotropy will increase. The peel strength fortunately also
increases with more impurities as this improves the adhesion.
Table 5.2: Overview of the different PTFE based substrates.
type
woven glass
random glass
ceramics
1 One
r
2.17 → 4
2.17 → 4
6 → 10
mil is 1/1000 of an inch.
strength
best
chips sink in substrate
use a metal backing
H2 O abs.
±0.02%
±0.01%
±0.05%
peel
++
+
-
5.2. Antenna Substrates for Space Application
5.2.3.2
83
Alumina (r : 8 → 10)
Alumina or Al2 O3 is a brittle material that thus has some difficulties with drilling, but
that is more rigid than Teflon. It has lower losses than its Teflon high r counterpart.
5.2.3.3
Thermoset Polymer (r : 3 → 10)
The most used thermoset substrate is FR4. This low cost material consists of glass
with r = 3 and a polymer resin with r = 6. The overall r ≈ 4.5, but the tolerances are not tight. Many polymer substrates were invented to combine the good
performance of the PTFE substrates and the low cost of FR4.
5.2.3.4
Sapphire (r ≈ 11) and Beryllia (r ≈ 6)
These materials are only used in applications where the substrate should have a
high thermal conductivity. Both materials are brittle and thus difficult to process.
Moreover, beryllia is toxic. Avoid these materials if possible.
5.2.3.5
PolyPhenyleenOxide PPO (r = 2.55)
This material disappeared from the laminate scene as it became sticky after etching
and cracked after drilling.
5.2.4
Metalization
Most substrates have metal on one or both sides. This can be etched to form the
strips of the circuit. There are two ways to put metal on substrates:
• sputtering gas ions collide with a negatively biased target, setting metal atoms
free that will settle on the positively charged substrate.
• cladding with the aid of heath and pressure the metal and the substrate are
stuck together.
The metal layer for the latter process can be either Electro Deposited (ED) or rolled
metal. ED is the standard method where the metal atoms are electrically joined to
form a sheet of the desired thickness. For a rolled metal, the thickness is obtained
by pushing a thick metal sheet through a pair of rollers. The rolled metal thus has
a better conductivity and a higher uniformity. This makes it suited for more critical
applications.
84
Chapter 5. Array Elements
Several metals are in use. Silver has the highest conductivity, but copper has become
standard due to its lower cost and easy soldering. Gold is the most corrosion resistant,
but does not adhere to ceramic material as it soaks into the substrate. For this reason
an adhesive layer of e.g. chromium can be used. An overview is given in Table 5.3.
Table 5.3: Overview of some commonly used metals.
metal
silver (Ag)
copper (Cu)
gold (Au)
5.2.4.1
σ [Ω−1 /m]
6.30 × 107
5.85 × 107
4.25 × 107
machining
soft
hard
very soft
soldering
good
corrosion
resistant
sensitive
very resistant
price
+
--
Thickness
The thickness of the metalization is indicated by the weight of a square foot of the
metal sheet in ounces. Standard values are 2 oz, 1 oz, 1/2 oz and 1/4 oz. Table 5.4
relates this weight with the height of the metalization layer. The larger the thickness,
the more problems with under-etching in case of small lines. In the worst case, the
line can even lift from the laminate. A small thickness avoids this problem, but results
in more conductive losses in case of high power. 1/2 oz is the standard for low-power
(mW to W) applications. But if the frequency is lower than 22 MHz, the thickness
should be more than 1/2 oz. To include 98% of the current, the metal should be at
least four skin depths high [78]. The skin depth δ being found from:
r
2
δ=
(5.5)
µ0 ωσ
with σ the electrical conductivity of the metal, e.g. 58 MS/m for copper.
Table 5.4: Weights and corresponding heights for copper. [79]
weight [oz]
1/2
1
2
5.2.4.2
thickness [inch]
0.0007
0.0014
0.0028
thickness [µm]
17.78
35.56
71.12
Peel Strength
The force that is needed to pull the metal sheet off the substrate is designated the
peel strength. Values range from 4 lbs/in to over 20 lbs/in. 4 lbs/in is very low but
this allows to scrape the metal off with a knife which is useful for prototyping. This
is not intended for finished circuits.
5.2. Antenna Substrates for Space Application
5.2.5
85
Space Environment
In this section selection criteria for substrates intended for space use will be given.
5.2.5.1
Pressure
Outgassing is measured with three quantities, all as percentage of the initial sample
mass:
• Total Mass Loss (TML) which is the difference in weight before and after an
outgassing test
• Collected Volatile Condensible Material (CVCM) or the mass of the condensed
material during an outgassing test
• Water Vapor Regain (WVR) or the mass of water that is re-absorbed after the
outgassing test during 24 hours at 25◦ C with a relative humidity of 50%
National Aeronautics and Space Administration (NASA) specifies [81] that materials
with T M L > 1.0% and/or CV CM > 0.1% should be avoided in space applications.
European Space Agency (ESA) [82] allows more TML as long as the mass loss is
mainly due to water, so that the Recovered Mass Loss (RML=TML-WVR) should be
less than 1.0%. Some substrates that meet the specs are listed in Table 5.5.
5.2.5.2
Temperature
The mean working temperature of the substrate in space will be lower than on earth,
but the issue of importance is the temperature cycling, rather than the extreme
temperatures. Depending on the orbit height, 12 to 16 cycles a day with an amplitude
of more than 50 K are experienced. The use of a thermal control paint could have a
positive effect on the amplitude of the temperature cycles. For this purpose a Flexible
Optical Solar Reflector (FOSR) was used on IRIDIUM [66].
During these cycles the properties of the substrate change constantly. The change of
is expressed with the Thermal Coefficient of r (TCr ):
r,T = r,T0 × (1 + TCr (T − T0 )).
(5.6)
with T0 an arbitrary chosen reference temperature. The change in dimensions in xyz
direction is expressed with the Coefficient of Thermal Expansion (CTE):
lxyz,T = lxyz,T0 × (1 + CTExyz (T − T0 )).
Some values can be found in Table 5.5.
(5.7)
Chapter 5. Array Elements
86
r
1.04
2.17, 2.20 ± 0.02
2.17, 2.20 ± 0.02
2.17, 2.20 ± 0.02
2.2 ± 0.02
2.33 ± 0.02
2.33 ± 0.02
2.33 ± 0.02
2.33 ± 0.02
2.40 − 2.60 ± 0.05
2.40 − 2.60 ± 0.05
2.40 − 2.60 ± 0.04
2.40 − 2.60 ± 0.04
2.40 − 2.60 ± 0.04
2.94 ± 0.04
3.27 ± 0.032
3.38 ± 0.03
3.38 ± 0.05
3.58 ± 0.03
6.00∗
9.2 ± 0.230
10.00∗
10.2, 10.5, 10.8 ± 0.25
tan δ
0.0017
0.0009
0.0009
0.0009
0.0009
0.0013
0.0013
0.0014
0.0012
0.001
0.001
0.0022
0.0022
0.0019
0.0012
0.0020
0.0025
0.0027
0.0035
0.0035
0.0022
0.0030
0.0023
TML
3.774
0.01
0.01
0.02
0.03
0.01
0.02
0.03
0.05
0.01
0.02
0.01
0.02
0.03
0.02
0.04
0.17
0.06
0.24
0.02
0.06
0.02
0.03
[%]
CVCM
0.071
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.02
0.01
0.00
0.01
0.00
0.00
0.00
0.00
0.00
0.00
WVR
2.356
0.00
0.00
0.02
0.02
0.00
0.00
0.02
0.04
0.00
0.00
0.00
0.00
0.02
0.01
0.03
0.02
0.02
0.07
0.00
0.04
0.00
0.02
-151
-160
-157
-125
-171
-161
-132
-115
-170
-153
-170
-153
-100
+12
+39
-87
+40
+50
-325
-43
-233
-425
TCr
29
25
46
31
23
17
31
22
18
14
18
14
15
16
16
15
11
16
12
16
14
24
28
34
47
48
24
29
35
28
19
21
19
21
15
16
16
15
14
18
14
16
16
24
[ppm/◦ C]
CTEx CTEy
246
525
236
237
194
217
203
173
177
173
177
173
200
24
20
52
46
59
62
20
37
24
CTEz
Table 5.5: Some substrates that meet the specs concerning outgassing (except for ROHACELL, a product of Röhm GmbH [83]). Values
taken from [84],[85] and [86]. (∗) The tolerance depends on the thickness. () No tolerance could be found. Arlon never
specifies tolerances on in their data sheets, the values here are indicative and taken from [87]. On request Arlon ships the
test results with a substrate.
manufacturer - type
Röhm
ROHACELL 31 HF
Arlon
CuClad 217
Arlon
DiClad 880
Arlon
IsoClad 917
Rogers
RT/duroid 5880
Arlon
CuClad 233LX
Arlon
DiClad 870
Arlon
IsoClad 933
Rogers
RT/duroid 5870
Arlon
CuClad 250GT
Arlon
DiClad 522
Arlon
CuClad 250GX
Arlon
DiClad 527
Rogers
Ultralam 2000
Rogers
RT/duroid 6002
Rogers
TMM 3
Arlon
25N
Rogers
RO4003
Arlon
25FR
Arlon
AR 600
Rogers
TMM 10
Arlon
AR 1000
Rogers
RT/duroid 6010
5.2. Antenna Substrates for Space Application
87
For the example of a patch antenna with a CTE of 16, a temperature difference of
100 K lengthens the patch with 0.16% and hence shifts the resonant frequency down
by 0.16% (for the supposition that r remains constant).
Under supposition that the length remains constant, a TCr of 12 changes the permittivity with 0.12% for√a 100 K difference and hence shifts the resonant frequency of
a patch down by 1 − 1/ 1 + 0.0012 = 0.06%. Most materials are however non-linear
due to phase transitions [88]. The values in Table 5.5 are mean values over the range
0◦ C to 100◦ C. Measurements in [89] revealed that most manufacturers provide values
for TCr that are way too low. A summary is given in Table 5.6.
Table 5.6: TCr from data sheets compared with measurements in [89].
type
RT/Duroid 5880
Ultralam 2000
CuClad 250
RT/Duroid 6010
TCr [ppm/◦ C]
-125
-100
-170
-425
measured
-606
-460
-513
-429
One tends to think that ideally a substrate has CTE ≡ 0 and TCr ≡ 0. Because,
however, the metalization on the substrate will expand and shrink during the temperature cycles, as do the component leads or the metalization of the through holes,
the CTExyz should be 16.6 ppm/◦ C in case of copper or 24.3 ppm/◦ C in case of
aluminum [79].
5.2.5.3
Radiation
One major problem in space is plasma charging. Free charged particles, e.g. in the
Van Allen belts or at the poles, can accumulate at the surface of the substrate and
lead to an arc discharge. A means to prevent this, can be metal shortening vias
to ground the substrate surface in some critical places. In [90] also a plasma jet to
compensate for charges and currents is mentioned as a possible, but more expensive,
solution.
To avoid differential surface charging, parts of the antenna that shadow others should
be avoided. Sunlit surfaces charge positively, due to the photo-electric effect that
excites electrons out of the material, whereas shadowed surfaces charge negatively,
because more collisions occur with electrons due to their lower mass, thus higher
velocity. This differential potential could cause Electro-Static Discharges (ESD).
Another problem is oxidation with atomic oxygen, causing mass loss and a change of
surface properties. Fortunately this problem can be reduced drastically with protective coatings (e.g. Germanium-Kapton passivation), that moreover have the advantage
to be conductive enough to prevent potential buildup from plasma charging [91].
88
Chapter 5. Array Elements
5.2.5.4
Debris
Impact of debris can also damage the antenna. The debris bumpers as discussed in
[90] are still rather expensive. Most antennas in current use are not protected, or
have a superstrate radome, so that the metal surface can not be hit by debris. In this
way at least no crater is formed on the metallic surface which could trigger ESD.
A passive way of protection is minimizing the probability of an impact by avoiding
to put the antenna on the side of the satellite that looks forward with respect to the
satellite movement.
5.2.6
Material Selection
5.2.6.1
Choice of Substrate
The r of the material should be as low as possible to give high gain. The lowest possible value is r = 1, which means vacuum as substrate. This introduces mechanical
challenges. ROHACELL, with an r slightly larger than 1 could be a good solution.
This is a foam, which means that the TML is unacceptable for space flight, unless
the material undergoes a special treatment before launch. The RF performance is
however very acceptable (see Table 5.1).
When the values for TCr , TML, CVCM and WVR are taken into consideration, the
preferred material for a satellite antenna is Rogers’ RT/DUROID 6002. The value for
r is rather high, so that 30% of the gain is lost compared to the ideal case of vacuum
substrate (see Fig. 5.1).
Gant [dB]
9
8
7
6
5
4
0
1
2
3
4
5
6
7
8
9 10 11
r
Figure 5.1: Gain as a function of r for the topology of Fig. 5.3 with dt of 120 mil.
dt should be high, as this results in a higher bandwidth and more mechanical strength.
Unfortunately this has a negative effect on the gain, see Fig. 5.2 and [80].
5.3. Element Design
89
Gant [dB]
8
7
6
5
0
50
100
150
dt [mil]
Figure 5.2: Gain as a function of dt for the topology of Fig. 5.3 on RT/DUROID 6002.
5.2.6.2
Choice of Metalization
As the chosen substrate is Rogers’ RT/DUROID 6002, the cladding material should
be copper as it matches CTExy , and probes or component pins should be made
of aluminum to match CTEz . As mentioned in Sect. 5.2.4.1, the cladding thickness
should be at least four skin depths δ. For a 2 GHz signal, Eq. (5.5) gives δ = 2.608 µm.
The standard 1/2 oz cladding with thickness 17.78 µm suffices.
5.3
Element Design
For the array element, a patch antenna can be used. Fig. 5.3 depicts a design for a
dual feed circularly polarized patch antenna, for the lower band, 1980 − 2010 MHz.
The patch is put on Rogers’ RT/DUROID 6002 with a thickness of 3.048 mm or 120
mil. By feeding the two probes with a phase difference of 90◦ , circular polarization is
obtained. Shifting probes more to the center, results in a smaller input impedance.
The limiting case of a probe in the center results in zero input impedance, a probe
at the side result in a high input impedance. This as the result of standing waves
on the patch. The probe position is optimized for a 50 Ω input impedance over the
1980 − 2010 MHz band. The scattering parameter depicted in Fig. 5.4 indeed stays
below −10 dB. The gain is 6.34 dB, which is in accordance with Fig. 5.1 and 5.2. The
design was simulated with MAGMAS, on an infinite ground plane and substrate. If
transmit and receive antenna should be integrated, e.g. to save space which is scarce
at the satellite, see Sect. 6.2.1, a stacked patch topology can be used with the upper
patch slightly smaller to add a second resonance at the 2170 − 2200 MHz band.
5.4
Selecting Array Geometry
In Sect. 4.2 was stated that a link budget should be calculated for α = 10◦ as
this case gives the worst signal attenuation. Here, a design method is introduced to
enhance the gain of the array under that angle, based on selecting the appropriate d,
see Fig. 3.6.
90
Chapter 5. Array Elements
4.235 cm
0.73 cm
4.235 cm
0.73 cm
TOP VIEW
SIDE VIEW
Figure 5.3: Scaled layout of the dual feed patch antenna.
1.95
0
1.96
1.97
1.98
1.99
2.00
2.01
2.02
2.03
2.04
f [MHz]
−10
−20
−30
|S11 |
Figure 5.4: Scattering parameter of the dual feed patch antenna.
This will result in the optimal array in the sense that it is the array with the highest
possible gain in the direction appropriate for this low elevation constellation, without
having grating lobes that could interfere with neighboring communications.
First the technique is explained on a one dimensional array, then generalized to a two
dimensional one.
5.4.1
Linear Array
5.4.1.1
Theoretical Background
Let the linear array be an N -element LESA along the x-axis with inter element
spacing d, as in Fig. 5.5. As explained in Sect. 3.3.1.1.1, the radiated power of the
LESA depends on d. One should choose d so that the first grating lobe lies just behind
the edges of the visible interval. In this case, the main beam is the smallest possible
without the appearance of grating lobes, so that the total power is low and thus the
gain high, without risk of interfering with neighboring communications.
5.4. Selecting Array Geometry
91
z
θ=
y
π
2
plane
θ
φ
x
Figure 5.5: The array topology, with dipoles as array elements.
If N → ∞, the beam width of the main lobe is zero. In this case, the optimal element
spacing is the inverse of the period in the u = sin (θ) cos (φ) = cos (φ) domain causing
the first grating lobe to lie just behind the edge of the visible interval. For an array
that scans the beam to φmax this would result in:
d=
λ
λ
=
= 0.50383λ
∆u
1 + cos (φmax )
(5.8)
For N 6= ∞ the beam width is not infinitely small. After calculating a beam width
related value ∆ue for d = λ, resulting in a value which can be tabulated for later use,
dopt can be found from Fig. 5.6. For d 6= λ, the beam width becomes ∆ue λ/d and:
∆ue λ
λ
= cos (φmax ) + 1 +
dopt
dopt
(5.9)
solves dopt as:
dopt =
1 − ∆ue
λ
1 + cos (φmax )
(5.10)
λ/d
visible interval
side lobe level
−1 −
∆ue λ
d
-1
0
cos (φmax )1
Figure 5.6: Supergaining on an array factor function graph.
u
92
Chapter 5. Array Elements
Example: LESA Scanned to φmax = 10◦
5.4.1.2
To illustrate the method, an example of a LESA with uniform tapering scanned to
φmax = 10◦ is optimized. The method works for any tapering type, but as tapering
causes gain degradation, it is better to start optimizing with a uniformly tapered
array.
The array factor function for a uniform LESA was given in Eq. (3.16). If d = λ is
taken, the value for ∆ue can be calculated as the value for u closest to u = 0 (the
maximum) where the array factor function equals the side lobe level. The results for
N = 2 → 18 for a uniform tapering, thus where the grating lobe is allowed to come up
to −13 dB, are summarized in Table 5.7. From these intermediate results the values
for dopt are easily obtained after applying Eq. (5.10) with φmax = 10◦ . The result is
plotted in Fig. 5.7. The dashed line gives the upper limit i.e. the element spacing for
N = ∞ obtained with Eq. (5.8).
Table 5.7: ∆ue for linear arrays with uniform tapering.
N
∆ue
N
∆ue
N
∆ue
2
0.428140
8
0.101330
14
0.057760
3
0.276260
9
0.090000
15
0.053910
4
0.204860
10
0.080960
16
0.050530
5
0.163040
11
0.073570
17
0.047550
6
0.135480
12
0.067420
18
0.044910
7
0.115930
13
0.062220
d [λ]
0.50
0.45
0.40
0.35
0.30
0.25
upper limit Eq. (5.8)
dopt for isotropic elements
dopt for dipoles
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 N
Figure 5.7: dopt for LESA with uniform tapering, scanned to φ = 10◦ .
What is gained in terms of directivity, can be seen on Fig. 5.8. The line labeled
approximation shows the results in case the gain of an array with N elements were
equal to N , which is an assumption often made, but only valid if d = λ/2, [45]. The
gain is thus higher than one would expect from the approximation. This is caused by
the fact that the main lobe lies partly beyond the visible interval due to the extreme
scan angle, so that less power than expected is actually sent.
5.4. Selecting Array Geometry
D [dB]
14
12
10
8
6
4
2
0
93
approximation N
isotropic elements
dipoles
1
2
3
4
5
6
7
8
N
9 10 11 12 13 14 15 16 17 18
Figure 5.8: Gain of the optimal LESA scanned to φ = 10◦ as a function of N .
5.4.1.3
Evaluation of the Method
To verify that the method indeed found the optimal array, the gain of several arrays
with N elements scanned to φ = 10◦ , is compared in Fig. 5.9. From φ = 90◦ down
to φ = 40◦ , the gain is independent of the scan angle. When the beam is scanned
further, the main lobe is partly scanned behind the edge of the visible interval. Arrays
with a higher gain at φ = 10◦ , have a lower gain at broadside. This is however not a
problem as it is still the φ < 40◦ region that is important when designing. This can
be understood when looking at the uncompensated part of the free space path loss
Gf s in Fig. 5.10.
D [dB]
13
12
11
10
9
8
7
6
d = 0.41
d = 0.43
d = 0.45
d = 0.47
d = 0.49
0
10
20
30
40
50
60
70
80
90
φ
Figure 5.9: Directivity of several arrays as a function of scan angle (N = 8).
To avoid interference coming in, the grating lobe is only allowed to come up as high
as the maximum side lobe level. This keeps the algorithm from finding the array
with the highest possible gain. Indeed, the directivity of several LESAs scanned to
φmax = 26◦ is given in Table 5.8. Comparing dopt found by Eq. (5.10) with dite
found by an iterative optimization search for maximum gain, reveals that all dopt are
smaller, to keep the grating lobe low. Table 5.8 moreover shows that for arrays with
N ≥ 11, allowing grating lobes, maximizes the gain even more.
94
Chapter 5. Array Elements
G [dB]
−120
−125
−130
−135
−140
−145
uncompensated part
array gain
d = 0.4528, N = 8
1/(4πRd2 )
0
10
20
30
40
50
60
70
80
90
φ
Figure 5.10: Free space path loss and array gain as a function of elevation angle.
Table 5.8: dopt of Eq. (5.10) compared to dite and ddip of an iterative optimization search,
for isotropic elements and dipoles, respectively. dite1 and ddip1 are the first local
maxima. dite is the global maximum in [0, 2]. φmax = 26◦ . The gain in case of an
array of dipoles is higher due to the gain of the dipoles.
N
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
dopt [λ]
0.3012
0.3812
0.4188
0.4408
0.4553
0.4656
0.4733
0.4793
0.4840
0.4879
0.4911
0.4939
0.4962
0.4983
0.5000
0.5016
0.5030
5.4.1.4
Gopt
2.1379
3.5382
4.7450
5.8199
6.8036
7.7276
8.6163
9.4893
10.3613
11.2438
12.1448
13.0696
14.0206
14.9981
16.0002
17.0228
18.0608
dite1 [λ]
0.3764
0.4143
0.4368
0.4517
0.4625
0.4706
0.4769
0.4820
0.4862
0.4897
0.4926
0.4951
0.4972
0.4991
0.5007
0.5021
0.5033
Gite1
2.3611
3.6366
4.7867
5.8299
6.7936
7.7034
8.5813
9.4451
10.3090
11.1837
12.0772
12.9946
13.9383
14.9085
15.9032
16.9187
17.9499
dite [λ]
0.3764
0.4143
0.4368
0.4517
0.4625
0.4706
0.4769
0.4820
0.4862
0.4897
1.5466
1.5488
1.5507
1.5524
1.5539
1.5553
1.5565
Gite
2.3611
3.6366
0.4368
5.8299
6.7936
7.7034
8.5813
9.4451
10.3090
11.1837
12.2132
13.2533
14.2915
15.3222
16.3472
17.3734
18.4057
ddip1 [λ]
0.3493
0.4027
0.4305
0.4480
0.4601
0.4689
0.4757
0.4812
0.4855
0.4892
0.4922
0.4947
0.4969
0.4988
0.5004
0.5019
0.5032
Gdip1
3.6115
5.1033
6.3695
7.4906
8.5186
9.4911
10.4365
11.3763
12.3268
13.3000
14.3042
15.3444
16.4227
17.5381
18.6871
19.8643
21.0621
Applicability to Non Isotropic Radiators
For non isotropic element types, the approach of only allowing the grating lobe as
high as the highest side lobe level, to maximize the gain, will definitely not work. This
is because although the array factor is scaled with varying d, the element radiation
pattern is not scaled.
5.4. Selecting Array Geometry
95
Moreover, the main lobe can be made even smaller by allowing grating lobes in the
visible interval that are suppressed by zeros in the radiation pattern of the elements
(as in [33]). It is unsure whether this is a wise approach, for this might cause huge
losses or be extremely narrow band. In such case the use of an iterative optimization
in combination with an electromagnetic solver is advisable.
For elements with a rotation symmetric radiation pattern, such as dipoles, a plane can
be found where the array radiation pattern contains only the array factor. Moreover,
this is the plane of interest as here lies the maximum of the element pattern. To
validate the applicability of the proposed algorithm to non isotropic radiators with a
rotation symmetric radiation pattern, dopt are now compared to the optimal element
spacings ddip in arrays of dipoles, as found with an iterative optimization search. See
Table 5.8 for the numerical results. The routine maximizes the directivity function of
Eq. (3.18) yielding ddip,ff from:
D(d) =
Ddipole (θ = 90◦ , φ = 10◦ )|F (90◦ , 10◦ , d)|2
Rπ 2π
R
Ddipole (θ, φ)|F (θ, φ, d)|2 sin θdθdφ
(5.11)
0 0
where Ddipole was defined in Eq. (3.11). No conditions on the grating lobes were
imposed. The integral used 1◦ × 1◦ discretizations. An analytical expression for
Eq. (5.11) can be found in [92].
The optimum search with near field calculation of Ps as in Eq. (3.15) gave, as expected
because both are theoretically identical, very similar values dnear , see Table 5.9. The
difference is due to numerical approximations. The current on the dipoles was discretized using 100 segments with a constant, uniform current profile. For the segment
to segment coupling the formulas for hertzian dipoles were used.
Table 5.9: Optimal d in dipole arrays from near and far field iterative search.
N
3
4
5
6
7
8
9
10
diso
0.3646
0.4006
0.4217
0.4356
0.4454
0.4528
0.4585
0.4630
dnear
0.3882
0.4137
0.4295
0.4405
0.4485
0.4547
0.4596
0.4636
dff
0.3861
0.4124
0.4287
0.4399
0.4481
0.4544
0.4593
0.4634
N
11
12
13
14
15
16
17
18
diso
0.4668
0.4699
0.4725
0.4747
0.4767
0.4784
0.4799
0.4812
dnear
0.4669
0.4697
0.4721
0.4742
0.4759
0.4776
0.4790
0.4803
dff
0.4667
0.4696
0.4720
0.4740
0.4759
0.4775
0.4789
0.4802
In Fig. 5.11 can be seen that for a N = 8 dipole array, scanned to φ = 10◦ , the global
maximum of the directivity appears for an inter-element distance that gives rise to
a grating lobe. The local maximum around d = λ/2 corresponds quite well with the
distance found by the algorithm explained.
96
Chapter 5. Array Elements
Not only for N = 8 does the optimal inter-element spacing for dipoles correspond
quite well with the one found by our algorithm. Only for smaller N a considerable
deviation from the result of Eq. (5.10) is observed, see Fig. 5.7. The difference in
gain is unnoticeable. For N > 11 the approximation yields a dopt that is slightly
larger than the exact solution. This results in a poor array performance due to the
sharp edge in Fig. 5.11. A slight overestimation of the optimal inter element spacing
results in a steep drop of gain. Hence for a broadband signal, the upper frequency
should be used when determining the optimal d, as for this frequency the highest d/λ
is obtained.
D
14
12
10
8
6
dipoles
4
isotropic elements
2
d [λ]
0
0.5
1.0
1.5
2.0
2.5
Figure 5.11: Directivity as a function of d for N = 8 LESA with dipoles and isotropic
elements.
The dipole gain of 2.15 dB is lost at higher N . Even with the optimal spacing in a
dipole array, the gain of a dipole can not be added to the gain of an optimal array
with isotropic elements, as the dipole array already is better at itself and less can be
gained by shifting the lobe outside the interval.
5.4.1.5
Applicability to Real Arrays
In any practical array, instead of imposing a current as was done in the theoretical
treatment, a power wave is divided by a network to all array antenna elements. When
the input impedance of all antenna elements is equal to the characteristic impedance
of the transmission lines used in the power divider network, all power travelling on
the network is absorbed by the antenna elements. If however, due to mutual fields,
the input impedance of the elements is altered, mismatch between the transmission
lines and antenna elements will cause reflections.
The amount of reflected power will depend on the amount of mismatch and hence on
the amount of change of the input impedance by the mutual fields. Consequently,
the actual (complex) current amplitude on the antenna elements will differ from what
was intended with the power divider, and the difference will depend on the scan angle
of the array. This was already touched in Sect. 3.4.1.
The proposed method is hence not suited for real arrays. Gain optimization under
low elevation angles requires full wave simulation, with the mutual fields.
5.4. Selecting Array Geometry
97
As an illustration, the curve of gain variation of a dipole array as obtained from a
simulation with imposed current from Fig. 5.11, is compared in Fig. 5.12 with a full
wave simulation with NEC of the same array fed with a power divider network. It
is clear that for small values of d the results are worse, mainly due to the change
in reactive input impedance that destroys the phase relations between the elements
needed for beam forming.
The curve has relatively high values for closely spaced dipoles. This is misleading as
the plot only shows the directivity. But as the input impedance of the dipoles for
that case is nearly entirely imaginary, it is practically impossible to feed power to the
dipoles.
D
imposed current
power divider
14
12
10
8
6
4
d [λ]
0
0.5
1.0
1.5
2.0
2.5
Figure 5.12: Directivity as a function of d for N = 8 dipole LESA with current imposed and
fed with a power divider network.
5.4.2
Planar Array
R
αmax
10◦
h
Earth Radius 6378 km
Figure 5.13: Satellite passing over a ground station.
For planar ground station antennas, maximizing the gain at a scan angle of θ = 80◦
is the design goal. For satellite antennas, θ depends on the orbit height. Some values
are given in Table 5.10. The situation of a zenith pass is depicted in Fig. 5.13. The
values of Table 5.10 apply to any over pass, not only when the satellite passes over
the zenith of the ground station.
98
Chapter 5. Array Elements
Table 5.10: Orbit heights h and their corresponding maximal deviation from broadside θmax .
h [km]
200
300
400
500
θmax [◦ ]
73
70
68
66
h [km]
600
700
800
900
θmax [◦ ]
64
63
61
60
The study of the linear array was only a first step in designing a 2D array that can
be used on satellites or ground stations. Generalizing the one dimensional to a two
dimensional array can be done in several ways, amongst others:
• a N elements LESA can be used as an element in an M elements LESA, obtaining an N × M regular grid array,
• or a N elements LESA can be repeated M times after a rotation by mπ/M for
m : 0 → M − 1 obtaining an array of N concentric circles.
Both possibilities, depicted in Fig. 5.14, are now compared, first in case of a broadside
beam, then for the scan angle θmax = 64◦ . In the end the best one for our purpose is
chosen.
5.4.2.1
Broadside
5.4.2.1.1 Concentric Circles For odd N , the center element is placed only once.
All elements, even the center element, should be excited with the same amplitude, as
tapering causes gain degradation.
For M > 1, the radiation pattern can not be derived from the radiation pattern of
the linear array. Especially not for odd N , where even the array factor does not equal
the sum of the array factors of the rotated arrays, as the center element is only placed
once. This means that mathematically, the conclusion on the optimal inter-element
spacing in a linear array will no longer hold for M > 1. We will now investigate how
good the approximation with Eq.(5.10) is for arrays of concentric circles.
The higher M , the less side lobes suffer from beam broadening [37] at the edges of
the scan interval. Then, when side lobes appear or disappear at the ends of the scan
interval, they cause a sudden change in gain, as opposed to in the case of a linear
array where the beam broadening compensates partly for a lobe shifting out of or in
to the visible interval, with a smoother curve in Fig. 5.15 as a result. The curves for
an array of concentric circles have a shape similar to the curve for a linear array. Also
the sharp edges behind the maxima are present, which means that in this case as well
overestimation of d, lowers the gain drastically.
5.4. Selecting Array Geometry
99
(a) Concentric circles array.
(b) Regular grid array.
Figure 5.14: Two types of planar arrays obtained from linear arrays.
G
M
M
M
M
40
=1
=2
=3
=4
30
20
10
0
0
0.5
1.0
1.5
2.0
2.5
d [λ]
Figure 5.15: Gain as a function of d for N = 7 concentric circles array of isotropic radiators,
scanned to θmax = 0◦ . Hence the first maximum lies around d ≈ 1.
Table 5.11 displays the results of an optimum search for the array of concentric circles.
In conclusion one could say that the approximation of Eq. (5.10) is not useful anymore.
5.4.2.1.2 Regular Grid As a regular grid array is a linear array that has a linear
array as array element, the array factor for a regular grid array is the product of the
array factor of the two linear arrays.
Unfortunately, again as the power consumed by the elements alters when in the present
of mutual fields, the total power of the regular grid array will not be equal to the
product of the power of both linear arrays, even with the same currents applied. In
this case again, Eq. (5.10) is not valid. As the shape is maintained, using the optimal
inter element spacing of a linear array as an approximation, is better than in the case
of concentric circles, as can be seen in Table 5.12.
100
Chapter 5. Array Elements
Table 5.11: Optimal d found with an iterative search for maximum gain of arrays of concentric circles scanned to broadside for M : 1 → 3. dopt found with Eq. (5.10)
differs largely from the actual values. The difference between dopt and dM =1 is
due to the condition of low side lobe level incorporated in Eq. (5.10).
N
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
dopt [λ]
0.5719
0.7237
0.7951
0.8370
0.8645
0.8841
0.8987
0.9100
0.9190
0.9264
0.9326
0.9378
0.9422
0.9461
0.9495
0.9524
0.9551
dM =1 [λ]
0.7143
0.7860
0.8284
0.8565
0.8765
0.8916
0.9033
0.9126
0.9203
0.9267
0.9322
0.9369
0.9409
0.9445
0.9476
0.9043
0.9530
GM =1
2.55
4.24
5.97
7.74
9.51
11.30
13.10
14.90
16.70
18.51
20.32
22.13
23.94
25.75
27.56
29.08
31.19
dM =2 [λ]
0.9021
0.5686
0.8939
0.7227
0.9256
0.8340
0.7583
0.8510
0.8013
0.8864
0.8366
0.8982
0.9572
0.9085
0.7616
0.9215
0.8861
GM =2
7.70
8.32
13.22
14.27
19.78
21.66
22.71
28.82
29.84
35.79
37.35
43.17
46.90
50.01
45.26
57.16
58.99
dM =3 [λ]
0.9360
0.8210
1.5000
0.9104
0.7533
0.6669
0.8535
0.7646
0.9118
0.8233
0.7696
0.8608
0.8194
0.7538
0.9551
0.9063
0.8975
GM =3
8.49
16.37
14.58
27.81
25.44
30.84
37.58
43.59
50.01
55.79
51.89
66.95
64.79
68.28
78.91
85.66
91.06
Table 5.12: The optimal inter element spacings for N × N regular grid arrays scanned to
broadside, compared with the dopt of linear arrays scanned to broadside.
N
2
3
4
5
6
7
8
9
10
dN ×1 [λ]
0.7143
0.7860
0.8284
0.8565
0.8765
0.8916
0.9033
0.9126
0.9203
GN ×1
2.55
4.24
5.97
7.74
9.51
11.30
13.10
14.90
16.70
dN ×N [λ]
0.6373
0.7327
0.7906
0.8274
0.8536
0.8731
0.8879
0.8997
0.9093
GN ×N
7.70
22.40
45.62
77.08
116.15
163.01
216.93
277.89
345.67
5.4.2.1.3 Comparison For a broadside beam, the regular grid array is preferable
to the array with concentric circles, as the gain of the regular grid array is by far higher
for the same area used on e.g. a satellite. This is seen from the Tables 5.12 and 5.11,
but is not surprising keeping the theory of sparse arrays in mind.
5.5. Conclusions
101
Sparse arrays can be used to obtain a small beam and thus a high resolving capability,
but have in general a lower gain due to higher side lobes [39], especially when the
array elements are concentrated in the middle of the array and the array becomes more
sparse on the outer side. This is indeed the case for an array of concentric circles or
its one dimensional analogy of a geometric linear array with geometric factor larger
than 1.
5.4.2.2
Scanned to θmax = 64◦
5.4.2.2.1 Concentric Circles Apart from the fact that Eq. (5.10) is no longer
applicable, now also the optimal distance changes with changing scan angle φ. If M
is however high enough, the symmetry of the structure is so high, that the geometry
is nearly the same, no matter under which φ angle the array is scanned. In this case,
the gain (and the optimal inter element spacing) is nearly independent of the scan
angle φ, as can be seen in Fig. 5.16(a).
5.4.2.2.2 Regular Grid For the regular grid, the relative difference in gain also
decreases with more elements used, as in Fig. 5.16(b).
5.4.2.2.3 Comparison For the purpose of an array antenna on board an earth
observation satellite, where the attitude of the satellite is dictated by the part of the
earth’s surface under observation, the array of concentric circles is the best choice, as
for this array, when M is chosen high enough, the gain does not depend on the scan
angle φ.
5.5
Conclusions
In this chapter, the array element was designed, taking into account the system specifications such as bandwidth and gain, as well as the general guidelines for space
design. Consequently, much effort was spent on the antenna substrate selection.
Once the element is designed, the array geometry can be optimized. For the application of the in-situ data up link to an earth observation satellite, the antenna should
have high gain for large scan angles, to partially compensate for the larger satellite
to ground station distance, and hence free space path loss, when the satellite has a
low elevation above the ground station’s horizon. A new technique for linear arrays
was introduced. Unfortunately, this technique is not easily generalized to planar arrays. Moreover, as the technique supposes current imposed on the array elements, the
practical relevance of the results is questionable, for most practical array antennas
are voltage driven.
102
Chapter 5. Array Elements
∆G [dB]
0.0
−0.1
−0.2
−0.3
−0.4
−0.5
M
M
M
M
M
M
−0.6
−0.7
−0.8
0
=1
=2
=3
=4
=5
=6
15
30
45
60
75
φ
(a) Gain variation over φ for N = 7, d = 0.4656 concentric circles array scanned to θ = 64◦ .
∆G [dB]
0.0
−0.1
−0.2
M
M
M
M
−0.3
=N
=N
=N
=N
=4
=5
=6
=7
−0.4
−0.5
−0.6
0
15
30
45
60
75
φ
(b) Gain variation over φ for dx = dy = 0.4656 regular grid array scanned to θ = 64◦ .
Figure 5.16: Gain variation over φ for concentric circle and regular grid array.
Chapter 6
Signal Modification and
Combination
After having designed the array element and the array geometry in Chapter 5, for the
system for in-situ data up link to an earth observation satellite that was described
in Chapter 4, this chapter will deal with the signal shaping and combination. Signal
combination is obtained by a BB summing opamp. For the signal shaping, an analog
BB implementation of a technique commonly used for digital BB beam forming,
touched in Sect. 2.2.2.1, will be used. The implementation was developed already
back in 2003 by ir. P. Delmotte [93]. Sect. 6.1.1 describes the implementation.
In recent years, the capabilities of Analog-to-Digital Converters (ADCs) have tremendously increased, both regarding speed and resolution. This evolution offers lots of
possibilities for digital beam forming and makes the implementation of Sect. 6.1.1 less
attractive. In some cases however, implementing the digital beam forming technique
in an analog way is useful. Indeed, when a large bandwidth is needed, the demands
on sampling speed might be beyond ADC specs. If the array has a large number
of antenna elements, the demands on processing power to combine the output of all
ADCs (used in parallel) might be unrealistic.
Ir. P. Delmotte built an array to demonstrate his analog BB phase shifter at 2.4 GHz,
in order to allow testing and demonstrational use of the array in the unrestricted
Industrial, Scientific and Medical (ISM) band. The demonstrated technique, however,
can be implemented at any frequency and with minor changes for a transmitting
array as well. This antenna architecture as well as its control and calibration will be
discussed in Sect. 6.1.2 as, unfortunately, no references on this work were available at
the time of writing. Afterwards, new work, namely the adaptation of the technique
for space operation is discussed.
103
104
Chapter 6. Signal Modification and Combination
6.1
Analog Quadrature BB Phase Shifter
6.1.1
Analog Implementation of Digital Technique
An analog way to implement additions, is using summing opamps. Multiplication can
be done by mixing. The disadvantage of using mixers, is that the phase of the output
signal changes with the amplitude of the input signals. Therefore multiplication was
implemented using Variable Gain Amplifiers (VGAs). Hence Eq. (2.16), restated here
in matrix notation:
I0
cos (∆θ)
=
Q0
− sin (∆θ)
sin (∆θ)
I
·
cos (∆θ)
Q
(6.1)
is implemented by the scheme depicted in Fig. 6.1, when the gain of the VGAs is set
to the values of the corresponding elements in the equation.
cos (∆θ)
I
I0
− sin (∆θ)
sin (∆θ)
Q0
Q
cos (∆θ)
Figure 6.1: Analog implementation of Eq. (6.1).
This implementation of a phase shifter is very similar to an implementation of a
Quadrature Amplitude Modulator (QAM) [94]. The difference is the lower baud rate
as the reconfiguration rate for scanning is expected to be much lower than the symbol
rate of a standard communication application.
If tapering of the array is desired, e.g. to lower the side lobe level, the signal of each
element n has to be amplified by the appropriate factor Gn . A natural way of doing
this, is by setting the gain of the VGAs to the corresponding element of Eq. (6.1)
after multiplication with Gn . Hence no extra amplifier per element is needed, but
at the expense of resolution. As the gain of the amplifiers is controlled in a digital
way, as explained in Sect. 6.1.2.3, the resolution would be proportional to Gn of the
antenna elements. Hence in this design, the tapering of the elements is taken care of
by setting the gain of the Low Noise Amplifier (LNA) in the mixing chip at the back
of the antenna.
6.1. Analog Quadrature BB Phase Shifter
6.1.2
105
Architecture of the Demonstrator Array Antenna
As any scanning phased array, this antenna too consists of radiating elements, described in Sect. 6.1.2.1, phase shifters and a power combination network, described
in Sect. 6.1.2.2, and hardware to control the phase shifters and apply the tapering,
detailed in Sect. 6.1.2.3. A photo of the array, taken inside the anechoic chamber, can
be found in Fig. 6.2.
Figure 6.2: Photograph of the array antenna inside the anechoic chamber.
6.1.2.1
Patch Antenna Element
The antenna element is a probe fed patch with a superstrate. The ground plane of the
patch also serves as the ground plane of the mixing circuitry, discussed in Sect. 6.1.2.2,
that sits behind the antenna element. FR4 was used for both patch superstrate and
circuitry board. For the probe, solid 1.5 mm2 copper wire was used. The geometrical
parameters are summarized in Table 6.1 and a photograph is shown in Fig. 6.3
The finiteness of the ground plane, and especially of the superstrate have strong
repercussions on the gain of the antenna element, This choice was made to obtain a
modular system. In order to illustrate the effect, the antenna element was simulated
in IE3D in [95]. The radiation pattern of the three different situations, i.e. infinite
superstrate and ground plane, finite ground plane and infinite superstrate and finite
ground plane and finite superstrate, are given in Fig. 6.4(a),6.4(b) and 6.4(c).
106
Chapter 6. Signal Modification and Combination
Table 6.1: Geometrical and electrical parameters of the patch.
feature
probe height
probe offset
patch width, length
ground plane width, length
superstrate thickness
superstrate dielectric constant
superstrate loss tangent
horizontal inter element spacing
vertical inter element spacing
dimension [mm]
5
10
40, 47
50, 50
1.575
±4.5
±0.0012
75
75
Figure 6.3: Photograph of the antenna element.
6.1.2.2
Signal Path
As soon as the signal is picked up by the antenna, it is amplified with an LNA
with controllable gain, so that tapering can be applied, as touched in Sect. 6.1.1.
Then it is mixed down by a quadrature mixer, and fed into the baseband phase
shifter that was introduced in Sect. 6.1.1. The mixer (MAX2701) is fed with an LO
that is distributed through a dedicated Printed Circuit Board (PCB) to all elements
over cables of (intentionally) identical length and amplified by a transistor per four
antennas. As a consequence the phase of the LO arriving at all mixers, can not
be expected to have exactly the same phase. This error will be corrected when
calibrating. To avoid problems with LO cross talk as mentioned in [8], the LO was
distributed at half the carrier frequency. This frequency is then doubled prior to
mixing, inside the mixer IC, at the back of each antenna element.
6.1. Analog Quadrature BB Phase Shifter
107
(a) ∞ ground plane and ∞ superstrate.
(b) Finite ground plane and ∞ superstrate.
(c) Finite ground plane and finite superstrate.
Figure 6.4: Influence of the finiteness of ground plane and superstrate on the array element
radiation pattern. The dimensions of the finite ground plane and superstrate
can be found in Table. 6.1
108
Chapter 6. Signal Modification and Combination
.. .. ..
...
Four analog BB shifters are grouped on one PCB. The summing of the output of the
VGAs (four per MLT04) is not done per element, as depicted in Fig. 6.1. Instead,
the summing (with an LM6172) is performed over the output of all phase shifters on
the PCB, in order to obtain the overall I 0 and Q0 of the four antennas connected,
as shown in Fig. 6.5. This way of implementing saves eight summing operational
amplifiers (opamps), but is less flexible, as only modular at the four elements subarray
level. The summing over all PCBs containing the phase shifters, is delayed to another
PCB. Yet another PCB contains the Bayonet Neill-Concelman (BNC) connectors for
outputting the Itot and Qtot of the array.
I2
Q0
RF2
+
-
Q2
I1
I0
RF1
+
-
Q1
Figure 6.5: Schematic of four phase shifters on one controller PCB.
6.1.2.3
Control Hardware
As all 64 elements can individually be addressed and configured with an amplitude
for tapering and four VGA settings for phase shifting, the control hardware takes a
considerable area on the PCBs. All control commands are passed to the array over
RS-232 serial communication. A command typically consists of a controller board
number, a Digital to Analog Converter (DAC) number, a channel number and the
new value. The controller board number makes sure the command is only passed
through by the micro controller on the appropriate phase shifter board. The DAC
number indicates for which of the four antennas connected to the board the new
settings apply and the channel number indicates to which VGA. A fifth DAC on
each phase shifting board allows controlling the gain of the LNAs in the mixer ICs
(MAX2701) at the back of the four elements. The new setting is thus a digital number.
The DAC (MAX5841) translates this into an analog voltage that, after amplification
(×2) and a DC offset (−2.5V) with a TL084C, can be applied to a VGA pin to control
its amplification. The control chain and the type numbers can be found in Fig. 6.7
6.1. Analog Quadrature BB Phase Shifter
109
Figure 6.6: Photograph of PCB with four phase shifters.
6.1.3
Measurement Results
Before using the array and hence before measuring, too, a calibration should be performed. Why this is necessary and how this is done is explained in Sect. 6.1.3.1. Afterwards the measurement setup and the results are given in Sect. 6.1.3.2 and 6.1.3.3.
As the antenna puts out a separate I and Q signal, all measurements are carried out
for both the I and Q output of the antenna. Hence calibration and measurement are
performed twice.
6.1.3.1
Calibration
Several causes make a calibration inevitable:
• the phase of the LO at all mixers will not be exactly identical
• the amplification characteristics of the VGAs differ mutually
• the I and Q signals of the mixers are not perfectly orthogonal
110
Chapter 6. Signal Modification and Combination
I’2 I’3 I’4
LM6172
MLT04
Itot
I
I’1
UTP
UTP
Q’1
Q
Qtot
TL084
Q’2 Q’3 Q’4
AT89C4051
RS-232
MAX5841
Figure 6.7: Implementation of the array control.
If calibration is needed on the fly, e.g. in adaptive “smart” antennas, additional ADCs
and a DSP processor will be needed, but these can operate at a lower data rate and
periodically instead of continuously, eliminating the need for fast or unobtainable
digital components [96]. The demonstration model discussed here uses a feed-forward
system, calibrated and controlled off-line using a network analyzer and a PC running
MATLABTM . The calibration discussed here is very limited as it only serves to obtain
reliable measurements. Much more advanced techniques are available in literature [97,
98].
For calibration, the array antenna is placed in an incident field of constant amplitude and frequency coming from the broadside direction of the array. Then all but
one antenna elements of the antenna are shut down, sequentially both VGAs that
contribute to the I 0 signal of the active element are set to an arbitrary value e.g.
half of the maximum, and the amplitude and phase of the down converted I 0 signal,
are measured. This measurement is repeated for all array elements, resulting in the
0∗
0∗
for n = 1 → 64. All measured amplitudes should
complex amplitudes In,1
and In,2
0∗
0∗
be equal and the phase ∠In,1 = 0 and ∠In,2
= π2 . But apparently this is not the case.
6.1. Analog Quadrature BB Phase Shifter
111
Next, one VGA, e.g. the first one, is chosen as the reference. All VGAs of all elements
0∗
0∗
will be adapted to line up with the reference. Suppose that the In,1
and In,2
signals
0∗
of the first and second VGA of the nth antenna divided by the reference I1,1
do not
equal 1 and j, as would be expected for an ideal array, but:
0∗ 0∗ I1,1
In,1
1
a b
0∗
=
C
·
· I1,1
=
·
(6.2)
I,n
0∗
0∗
jI1,1
j
In,2
c d
where CI,n is introduced as the calibration matrix for the VGAs of the nthe element
that contribute to the antenna Itot signal. With the aid of the calibration matrix,
erroneous deviations are compensated. This is proven by the fact that the calibrated
0∗
0∗
0∗
I 0 signals, denoted as Ical,n,1
and Ical,n,2
indeed equal I1,1
and its π2 shifted version:
0∗ 0∗
Ical,n,1
In,1
1 0∗
−1
·
=
I
(6.3)
=
C
0∗
0∗
I,n
In,2
j 1,1
Ical,n,2
Similarly, a calibration matrix CQ,n for each antenna element can be measured by
monitoring the output of each of the VGAs contributing to the overall Qtot signal of
the array.
0∗ 0∗ I1,1
Qn,1
1
e f
0∗
· I1,1
(6.4)
·
=
C
·
=
Q,n
0∗
j
g
h
jI
Q0∗
1,1
n,2
Inserting both calibration matrices into Eq. (6.1) results in:
I˜0
cos (∆θ) sin (∆θ)
0
0
=
0
0
− sin (∆θ) cos (∆θ)
Q̃0
−1 ˜
CI,n
I
·
·
−1
CQ,n
Q̃
(6.5)
This equation is sufficient for operating the array under any tapering and scan angle
and can be obtained by only measuring the responses of all VGAs contributing to
both Itot and Qtot of the array once.
This way of calibrating and compensating errors, however, supposes that the VGAs
do not drift with time and temperature, nor accounts for non-linearities in the gain
setting. If calibration is critical, it should hence be performed on a regular basis.
Moreover, though calibrating once should suffice for any steering angle and tapering,
it is advisable, again in case calibration is critical, to first apply the tapering and only
then perform a calibration. Of course the tapering should then be taken into account
so that the amplitude of the nth element An should be added to Eq. (6.2):
0∗ 0∗ An a b
An
In,1
I1,1
1 0∗
=
C
·
I
·
(6.6)
=
I
0∗
0∗
In,2
jI1,1
j 1,1
A1 c d
A1
and the equations derived should be adapted accordingly.
112
Chapter 6. Signal Modification and Combination
For the antenna described here, the calibration did not seem to be critical. With
the very basic calibration as explained above, the performance of the array is still
“acceptable”.
6.1.3.2
Measurement Setup
The array antenna was measured in the anechoic room. An RF source at 2.4001 GHz
was connected to an EMCO Type 3115 broadband horn that was used as transmit
antenna. Care was taken that the entire array was uniformly illuminated. The array
antenna was supplied with the 1.2 GHz LO signal. The I output signal of 100 kHz
was connected to the test input port of the receiver of an HP8510C Vector Network
Analyzer (VNA). The signal obtained by mixing the 2.4001 GHz RF and 1.2 GHz
LO signal, which also contains a component at 100 kHz, is connected to the reference
port of the receiver.
Usually, RF signals are sampled inside a test set of the VNA, down converted to
20 MHz and transmitted through an IF cable to the microwave receiver. Inside the
receiver, the 20 MHz signals are further filtered, amplified and down converted to
100 kHz [99, 100]. When feeding 100 kHz signals with a sufficient amplitude at the
20 MHz IF ports, enough signal gets through the filters, amplifiers and mixers to
produce repeatable measurement results.
Then the antenna is calibrated with the procedure explained in Sect. 6.1.3.1. Next
the antenna is configured for a certain tapering and scan angle, with the aid of a
computer running MATLABTM as explained in Sect. 6.1.2.3. Now the antenna can
be rotated, measuring the result of the complex division of test and reference signal
for all rotation angles. These measurements allow to plot the radiation patterns given
and discussed in Sect. 6.1.3.3.
6.1.3.3
Measurement Results
Firstly, the response of a phase shifter with calibration is measured. Secondly, the
functioning of the system is demonstrated with the measurement of radiation patterns
for some taperings and scanning angles after calibration.
6.1.3.3.1 Phase Shifter Constellation Plot First the phase shifter of one antenna element was measured. All other elements of the array were shut down and then
the phase shifter was configured for some selected angles and amplifications. Fig. 6.8
shows the constellation plot, i.e. the points in the complex plane that indicate the
ratio of I 0 over I for the configured shifter settings. The plot clearly shows that calibration is inevitable. The amplifier in the Q channel amplifies too much compared
to the one in the I channel. After calibration this is compensated.
6.1. Analog Quadrature BB Phase Shifter
113
Calibration was performed for the inner circle and gets less effective for the larger
circles, due to the fact that the errors are not linear. From the plot it is also obvious
that applying a tapering by lowering the gain of the VGAs deteriorates the accuracy
of the shifter, as already mentioned in Sect. 6.1.1. Indeed, the same absolute error, as
the digital control of the gain is linear, results in a higher relative phase and amplitude
deviation.
2
1.5
1
0.5
0
−0.5
−1
−1.5
−2
−2.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
Figure 6.8: Constellation plot of the phase shifter for some selected angles, with (o) and
without (+) calibration.
6.1.3.3.2 Radiation Patterns Fig. 6.9 and 6.10 show some array radiation patterns after calibration, to demonstrate the functioning of the array. The measurements indicate that if the array is scanned beyond 48◦ , grating lobes occur, because
dx = dy = 0.6λ. For the Chebychev taperings, the beam widening for lower side lobe
levels is clearly noticeable. The side lobes still come up higher than expected, due
to the approximation made by the digital control of the tapering. Only 4 effective
bits were used because if the gain was varied more, the amplifiers in some shifters
saturated. See Sect. 3.4.2 for a discussion on the effect of phase approximation on the
radiation pattern of a phased array.
114
Chapter 6. Signal Modification and Combination
p(θ) [dB]
0
−5
−10
0◦
15◦
30◦
45◦
−15
−20
−25
−30
−35
−40
−100
−50
0
50
θ [◦ ]
(a) Uniform tapering, φ = 0◦ , θ = 0◦ , 15◦ , 30◦ , 45◦ .
p(θ) [dB]
0
60◦
75◦
90◦
−5
−10
−15
−20
−25
−30
−35
−40
−100
−50
0
50
θ [◦ ]
(b) Uniform tapering, φ = 0◦ , θ = 60◦ , 75◦ and90◦ .
Figure 6.9: Copolar H-plane radiation patterns of the calibrated uniform eight-by-eight array.
6.1. Analog Quadrature BB Phase Shifter
115
p(θ) [dB]
0
0◦
15◦
30◦
−5
−10
−15
−20
−25
−30
−35
−40
−100
−50
0
50
θ [◦ ]
(a) Chebychev tapering, Rsl = −20 dB, φ = 0◦ , θ = 0◦ , 15◦ , 30◦ .
p(θ) [dB]
0
Rsl = −15dB
Rsl = −20dB
Rsl = −30dB
−5
−10
−15
−20
−25
−30
−35
−40
−100
−50
0
50
θ [◦ ]
(b) Chebychev tapering, Rsl = −15 dB, −20 dB, −30dB, φ = 0◦ , θ = 0◦ .
Figure 6.10: Copolar H-plane radiation patterns of the calibrated Chebychev eight-by-eight
array.
116
6.2
Chapter 6. Signal Modification and Combination
Space Qualified Phase Shifter
This section contains the results of a feasibility study on how to modify the architecture of the shifter in Sect. 6.1.1 to obtain an implementation suitable for space
application.
One approach could have been to simply copy the implementation, in COTS and
test for space environment. However, the tests are costly and this involves a risk of
increasing the number of design cycles. Using space graded versions of every component is expensive but excludes risk at component level. For this implementation, a
compromise is suggested. For components sitting at the back of the antenna, space
graded components should be used. For others, COTS components can be chosen,
as long as their working temperature ranges was from -55 ◦ C to +125 ◦ C and they
are not implemented in CMOS technology. If a space graded version of a component
is available from one of the main vendors (Analog Devices, Maxim Dallas, Radial,
National Semiconductor), this is preferred.
As no suitable quadrature VGAs could be found, the option of using mixers was
reconsidered again. Now in the form of multipliers, which are essentially mixers
implemented in an IC that compensates for the non-linearity.
For the driving of the DAC, one microcontroller with direct addressing was chosen.
All DACs are connected to the same data output of the microcontroller. Which DAC
is updated depends on which output pin of the microcontroller is high. Hence as many
output pins of the microcontroller are needed as there are DACs. As no more than
seven elements are used in the array, this is a defendable choice. For more elements, a
microcontroller with binary addressing has to be used, reducing the number of output
lines to a number smaller than the number of elements. One microcontroller per DAC
is another solution. This is undoubtedly the option with the lowest reconfiguration
time. But the most expensive one, too.
As malfunctioning of the array must not lead to the loss of the mission, a switch in
order to disable all but the backup element, can be inserted in the signal path. This
way, in case of error, the system falls back on using a single element.
In order to compensate for the Doppler shift at the receiving ground station, the
transmit frequency can be varied. This can be obtained by using a Voltage Controlled Oscillator (VCO) for the Tx LO, with an analog input voltage controlling the
frequency. Experiments performed by ir. Aliakbarian showed that to make sure that
the quadrature mixing to baseband is perfectly in phase (needed for easy QPSK detection) the use of a PLL is indispensable. This PLL should be narrow band, not to
follow the phase jumps in the QPSK signal. Consequently its pull-in region1 will not
be large enough to track the Doppler shift. Yet another control signal is needed, to
set the frequency of the voltage controlled LO for Rx to the appropriate value.
1 The
pull-in region is the frequency range in which the PLL is able to lock onto a carrier, [6].
6.2. Space Qualified Phase Shifter
117
Again, as was stated earlier in this chapter, a digital implementation of beam forming,
and even of the PLL, might bring tremendous advantages. That would, e.g. make
it possible to lock the PLL digitally on a varying carrier frequency, by calculating
the fourth power of the QPSK signal, which is a sine wave because 4 × π/2 = 2π, as
explained in [101].
6.2.1
Space Segment System Overview
The array with space qualified phase shifter is actually intended to serve as a part of
the system described in Sect. 4.1.3. Hence, Fig. 6.12 that gives an overview of the
system described below, is a more detailed reprise of Fig. 4.3.
The array antenna as proposed, consists of seven identical elements. The number of
seven allows to optimally use all outputs of the micro controller. Moreover, this allows
an array topology of concentric circles (with one center element), as in Sect. 5.4.2 with
N = 3 and M = 3, depicted in Fig. 6.11. This way a planar array antenna is obtained,
hence with a beam instead of a rotation symmetric radiation pattern, with a limited
number of elements and hence a manageable complexity.
Figure 6.11: 7 elements concentric circles array (M = N = 3).
One of the seven elements will also be used for Tx. It will broadcast the activation
message. And it will be used in the array for Rx, as the one element without phase
shifting and tapering. This way, this element can be seen as a backup element in case
the beam steering fails. Instead of this Tx/backup element, a separate Tx element
could be used. This would simplify the element design tremendously, as in that case
no elements should be dual band. As area on the satellite is however scarce and hence
expensive, space saving solutions must be applied whenever possible.
The beam forming network as proposed is slightly different from the one in
Sect. 6.1.2.2. The I and Q signals from the six Rx elements are phase shifted and
combined to one overall I and Q signal. Adding the I and Q from the backup element
is delayed to a next summing opamp to allow an emergency switch to disconnect
the array signal in case of a failure.
The signals to control the emergency switch, to drive the VCO of the LO with the
correct voltage, and to set the gain of the VGAs in the BFN to the correct value,
come from a microcontroller that receives the necessary input from software on the
main CPU over the CAN bus. The software routines are summarized in Sect. 6.2.4
and given in pseudo code in Appendix C. The signal path and controller are now
worked out into some more detail. The components numbers are listed in Table 6.2.
118
Chapter 6. Signal Modification and Combination
array
Tx/backup
RF 2.0 GHz
5 W RF 2.2 GHz
7 × RF 2.0 GHz
LO
QMix bank
7 × BB I/Q
BFN
BB I/Q
circulator
PLL
RF 2.0 GHz
5 W RF 2.2 GHz
array ctrl
QMix
QMix
emerg switch
BB I/Q
LO
sum opamp
BB I/Q
QPSK BB modem
activate msg
data 6 kbit/s
power bus
CAN bus
CAN interface
Figure 6.12: Subsystems of the space segment.
6.2.2
Signal Path
At the back of each antenna element the RF signal is mixed down in an I/Q demodulator. Then, inside the satellite, to keep the electronics shielded (at least to
some extend) from temperature cycles and radiation, I and Q are rotated with four
multipliers that are set to the correct amplification by the array controller board signals Camp,x,y where x denotes the number of the antenna element and y indicates
the number of the multiplier. The four multiplied signals then pass a switch so that
in case of malfunctioning the element can be shut down i.e. disconnected from the
summing opamp that sums all rotated I 0 and Q0 and passes its output to the modem.
The transmitting part is much less complicated as this is a single element. Hence the
signals from the modem have to be mixed up and combined into an RF signal that
can be amplified and via a circulator put on the antenna.
6.2. Space Qualified Phase Shifter
119
Cdop,Rx
I/Q demodulator
Cx,4
Cx,3
VCO
Cx,2
circulator
Cx,1
RF amplifier
Cdis,x
mixer
0/90
opamp
Cdop,T x
modem
Figure 6.13: Implementation of the signal path of the array hardware.
6.2.3
Control Hardware
The controller mainly gets digital settings for the control voltages Camp,x,y for the
VGAs, Cdop,T x and Cdop,Rx for the VCOs and Cdis,x for the disable switches. It has
to pass this digital settings to the appropriate DAC and update the DAC output. The
settings can be put on the four parallel output ports, available at the microcontroller.
The remaining pins can be used to address the appropriate DAC and give the update
trigger.
Each DACs has four analog outputs, which is equal to the number of VGAs per
antenna element. Hence N − 1 DACs are used to control the VGAs of the N − 1
phase shifters. A second group, of d((N − 1) + 2)/4e DACs is used to enable or
disable N − 1 elements and control the two VCOs.
6.2.4
Software Overview
A lot of routines have to be performed on board the satellite to steer the array,
compose the activation message and compensate the Doppler shift. The pseudo code
can be found in Appendix C. To get an idea on the complexity and the variables that
the code has to receive from the satellite on board CPU, Fig. 6.15 shows the relations
and dependencies of the routines.
Chapter 6. Signal Modification and Combination
µcontroller
clk
D0-D11
DS0-DS1
LS8 LS7CS
12
2
Camp
Camp
Camp
Camp
Camp
DAC
DAC
DAC
DAC
DAC
DAC
DAC
LS6 . . . LS2 LS1CS
Camp
DAC
120
Cdis
Cdis/dop
clk
Figure 6.14: Implementation of the control of the array hardware. DS0-DS1 is used to select
one out of four DACs in each package. D0-D11 is the 12 bit setting of a DAC.
CS (chip select) updates the output of the DACs with what is in the registers.
Hence one CS for all DACs controlling the VGAs is used. A second one controls
the other DACs. LS (latch select) is different for each DAC, to make sure that
only the register of the corresponding DAC is update and not any other.
Table 6.2: Components in signal path and controller. The number of elements N will determine the number of components needed. The type numbers used in Sect. 6.1.2
are given as a reference. SQ indicates whether or not the considered component
is space qualified.
component
I/Q demodulator
VCO Tx
VCO Rx
VGA
switch
opamp
mixer
RF amplifier
circulator
clock
microcontroller
DAC
#
N
1
1
4 × (N − 1)
N −1
2
2
1
1
1
m
l 1
N + (N −1)+2
4
Sect. 6.1.2
MAX2701
MLT04
LM6172
AT89C4051
considered
IR0104LC1Q
ROS-2432-119+
ROS-2432-119+
AD534SH/883B ($ 86)
ADG201HSTQ ($ 13)
AD844SQ/883B ($ 30)
MO205W8
TGA8310-SCC
F2588
9920
MRAM8051 ($ 5000)
SQ
y
n
n
y
y
y
y
y
n
y
y
MAX5841
AD664TD-BIP ($ 360)
y
6.3. Conclusions
121
TT&C
...
CAN
time stamp next
ground station
satellite
velocity & position & attitude position & serial numberground station query
compose
calculate
Doppler shift compensation
DAC
activation message
calculate
settings VGA phase shifter
invert & scale
BFN
VCO Rx VCO Tx
Figure 6.15: Overview of the Routines that are needed on the Satellite.
6.3
Conclusions
In this chapter, the design of an array antenna for a satellite communication application to up link in-situ collected data to an earth observation satellite, was finished.
After the antenna element design and determination of the array geometry, both performed in Chapter 5, this chapter zoomed in on the signal shaping and combination.
The combination was implemented in BB using a summing opamp. The beam forming
technique was an analog BB implementation of a technique, often used in digital
beam forming, that rotates the phase of the carrier. Back in 2003 already, ir. P.
Delmotte developed the analog implementation. His implementation was evaluated
in this chapter and adapted to a space qualified design.
At the time of writing the tests on this space qualified prototype were still ongoing.
Consequently no performance data could be given on the new design.
invisible filling
Part III
Application: EM Side
Channel Analysis
123
125
Overview
The second application for which array antennas are applied in this work, is electromagnetic side channel analysis. This is a field of study related to cryptography.
Similar to cryptology, that seeks for weaknesses in mathematical algorithms and cryptographic protocols, side channel analysis looks for weaknesses in implementation.
Such weaknesses might lead to observable variations in current consumption, execution time or electromagnetic radiation, that correlate with a secret parameter, inside
the cryptographic device, which should not be revealed.
Chapter 7 explains the underlying physical principles that cause information on the
logic operations that are performed in a cryptographic device to leak via e.g. electromagnetic radiation. This chapter also indicates that no practical system can ever be
100% secure. No matter what countermeasures are applied, the only system that does
not radiate any secret information, is a system that does not do anything. With an
unlimited amount of time and equipment, any system can be broken. Designers can
only make sure that the effort needed to break a system is unfavorably high compared
to the benefit of breaking the system. Where hackers try to break systems for their
own benefit, the side channel community tries to evaluate the effort needed to break a
system and seeks to improve security by detecting weaknesses and suggesting counter
measures.
The application of electromagnetic side channel analysis would benefit a lot from
array antennas as this would increase the signal-to-noise ratio and would allow (simultaneous if off-line beam forming is applied) monitoring of different regions in the
cryptographic device. For an FPGA it is obvious that this would reveal a lot of extra information, as there is a direct link between active regions in the FPGA and
operations performed. For a microcontroller it would allow to monitor activity in
controller, memory, input/output blocks and so on. Apart from some minor remarks
on how these arrays should best be implemented and used, the array content in this
part is limited. While working on the side channel analysis application, the lack of a
good sensor element emerged. As the need for an element, at the time of writing, was
much stronger than the need for an array of sensors, much effort was spent in studying
and designing such element. As time is however limited, this had its repercussions on
the time that could be invested in studying arrays for this application.
Chapter 8 provides a profound theory for designing inductive sensors for electromagnetic side channel analysis. Much of the contemporary work is carried out in a
haphazard way, with little or no understanding of radio frequency engineering. This
chapter studies the sensors in a systematic way. First the specifications for a good
sensor are defined. Next some well known inductive antenna types are studied and the
maximum practical resolution limit for small coils is calculated as a function of the
bandwidth and signal amplitude required and the input impedance of the measurement device. This calculation also provides a design method for maximal resolution
loop sensors.
126
As a side step of the study of existing loop antennas, the design of an RFID reader
antenna, commonly described in the literature for antennas that are small compared
to the wavelength, was generalized to a technique that is valid regardless of antenna
size, in Sect. 8.3.
At the end of Chapter 8, in Sect. 8.5, some hints and suggestions are given on how to
implement an array of sensors for the application. At the same time, it is indicated
that still many problems need to be solved in order to come to an implementation of
an array of sensors.
Just as there was little or no theory nor good practice available on sensors for cryptographic analysis, no guidelines on the measurement setup were at hand. Chapter 9,
again, tries to study this in a systematic way, and compares some possible setups regarding signal-to-noise ratio. As the theory developed here, is useful for a setup with
a single sensor, also arrays of sensors benefit from application of the insights presented
here. Apart from a note suggesting off-line digital time shifting as the preferred beam
forming technique, in Sect. 9.4, again no specific array aspects are discussed.
Chapter 7
Introduction to the
Application
With the shift towards electronic implementation of almost all everyday actions, the
applicability area of cryptography has broadened from purely strategic (military,
diplomacy and intelligence) to practically any aspect of modern life [102]. Consequently anyone uses cryptography on a daily basis, some probably even without
knowing. Obvious examples are online banking and electronic signatures with an
eID. But even entertainment, with e.g. water marking and copy protection, has become a field of application.
Essentially, cryptography tries to hide a secret by transforming a plaintext into a
ciphertext. The intended reader can transform in the opposite direction, by using the
key that was agreed with the transmitter (symmetric key cryptography) or by using
the personal secret key (public key cryptography). An unintended reader can not get
to the secret information as he or she lacks the key. Hence the problem of keeping a
secret is shifted to protecting a key.
Shannon [103] showed that in the trivial case were the secret and the key are of the
same length (Vernam scheme) the encryption is perfectly safe1 , as he shows that a
key with a data rate equal to or greater than the data rate of the message leads to
a system with perfect secrecy. In any other practical situation where the key length
is smaller than the message length, the system can be broken with an appropriate
number of trials. In short: any cryptographic system is breakable, designers have
to make sure that the effort to break a system is unfavorably high compared to the
benefit of breaking the system.
1 Under
the assumption that the key is never reused: one-time pad.
127
128
Chapter 7. Introduction to the Application
Moreover, even if a system were perfectly safe, the implementation of the transformation mentioned adds another link to the chain that can be used to attack the system.
Only limiting the discussion to electronic semiconductor hardware, the electrons that
move when performing logic operations will cause a radiation that carries away some
secret information on the transformation, the secret information or even on the secret
key. Such alternative information channels that can be used in attempts to break the
system are referred to as side channels [104].
The work of this part is merely developing sensor systems and measurement setups
for evaluating vulnerability to side channel analysis. Hence no extensive treatment on
cryptography as such can be expected. As an example, one cryptographic primitive2
is reviewed. This should give the reader an idea of the logic functions that are carried
out by the cryptographic hardware that is under test when evaluating side channel
leakage.
7.1
Example of A Block Cipher
To illustrate how a cryptographic operation can be implemented in Silicon hardware,
an example of a block cipher is given. The example, the KeeLoq block cipher [105],
should preferably not be used in practice. Both the mathematics behind the cipher
and the only commercially available implementation have been proven to be insecure,
by a sliding attack in [106] and a power analysis in [107].
A block cipher, as opposed to a stream cipher, divides the message or plaintext input
stream into blocks of a certain size, e.g. 32 bits. Those 32 bit blocks are encrypted with
the aid of a secret key that is only known by the two parties communicating. After
encryption, a 32 bit block of ciphertext is obtained, that can ideally only be decrypted
by the intended addressee. After decryption of all blocks by the addressee, the original
message, apart from some padding if the message length was not a multiple of 32 bit,
is obtained by concatenation of all blocks.
Hence, first the message block is shifted into the 32 bit shift register of Fig. 7.1.
Encryption of a block is done by executing 528 rounds. In one round the Non Linear
Function (NLF) is applied to five bits that are tapped from the shift register at
positions 1,9,20,26 and 31 as indicated on Fig. 7.1. This result is combined with two
more bits tapped from the shift register at positions 0 and 16, and the bit at position
0 of the 64 bit shift register used to cycle the key. The resulting value is shifted into
the 32 bit shift register and the 64 bit shift register is cycled, ready for the next round.
2 A cryptographic primitive is a low level cryptographic algorithm such as a stream cipher or a
hash function.
7.2. Cryptographic Hardware
31
129
26
20 16
9
10
NLF
+
63
0
Figure 7.1: Schematic of the KeeLoq block cipher.
The NLF can be implemented as a LookUp Table (LUT), where the five input bits
of the NLF indicate the address or bit position in the 32 bit 0x3A5C742E string:
31
0
30
0
29
1
28
1
...
101001011100011101000010
3
1
2
1
1
1
0
0
An eXclusive OR (XOR, ⊕) implementation of the NLF is possible too:
N LF (x4 , x3 , x2 , x1 , x0 ) = x4 x3 x2 ⊕ x4 x3 x1 ⊕ x4 x2 x0 ⊕ x4 x1 x0 ⊕
x4 x2 ⊕ x4 x0 ⊕ x3 x2 ⊕ x3 x0 ⊕ x2 x1 ⊕ x1 x0 ⊕
x1 ⊕ x0
(7.1)
In Appendix E, the transceiver is described that was used to implement and test the
attack on KeeLoq as described in [106].
7.2
Cryptographic Hardware
Cryptographic primitives can be implemented on any electronic digital device. They
can be run on general purpose microprocessors, such as a computer Central Processing
Unit (CPU), or on dedicated hardware such as an Application Specific Integrated
Circuit (ASIC).
130
Chapter 7. Introduction to the Application
Due to the possible information leakage via side channels mentioned in Sect. 7.3,
cryptographic primitives should be preferably implemented on dedicated hardware,
as this allows to implement countermeasures against this leakage also at gate level3 .
But again this is a cost versus security trade-off. In many consumer electronic devices,
such as mobile phones, a microcontroller (µC) or Field Programmable Gate Array
(FPGA) is used to perform cryptographic transformations. Even in some smart cards,
µC without any special precautions regarding information leakage are used. Industry
and academia are however continuously improving designs, e.g. [108], and trying to
reduce the cost.
The choice between a µC or an FPGA is nowadays mainly an economical choice. Both
FPGAs and µCs have some overhead4 Generally µCs require less space on the PCB as
they have less pins, as intended for sequential program execution. When parallelism
is indispensable, the fully hardware programmable FPGA is a better choice. For mass
production, in the supposition that no further reconfiguration of the FPGA is needed,
the logic of the FPGA can be implemented in a (non-reconfigurable) ASIC.
Recently, many microcontroller cores were made available as blocks to load into an
FPGA. Consequently, it is not inconceivable that µC will disappear in favor of FPGAs. Moreover, nowadays FPGAs that contain Random Access Memory (RAM) are
available on the market.
7.2.1
Field Programmable Gate Array (FPGA)
An FPGA consists of a huge number of logical blocks, all implemented as LUTs, e.g.
with four inputs and one output, that can be interconnected at will by the designer
by (re)configuring switching matrices at the intersections of the communication lines
connecting all blocks. A schematic representation of a general FPGA can be found
in Fig. 7.2.
Programming an FPGA generally consists of writing VHSIC (Very High Speed Integrated Circuit) Hardware Description Language (VHDL) code and compiling it into a
bit file. This bit file essentially contains the settings (conducting or non-conducting)
of the transistors in the switching matrices in order to obtain the circuit that implements the logic and functionality described in the VHDL code. Programming the
device then means, putting the transistors in the appropriate state, corresponding to
the value in the bit file. This can be, depending on the technology used, done in a
reversible (SRAM), or irreversible (antifuse) way [109].
3 Other
levels of design abstraction that can be secured, are e.g. system, protocol and algorithm.
FPGAs the number of gates needed depends on the configuration. Some gates of the FPGA
will idle, as it is very unlikely that the number of gates needed exactly matches the number of gates
available on the device. For µCs a certain instruction set is provided. It is very unlikely that every
instruction of the set is used.
4 In
7.2. Cryptographic Hardware
131
I/O
I/O
I/O
I/O
I/O SM
SM
SM
SM I/O
SM
LUT
I/O SM
SM
LUT
SM
LUT
I/O SM
SM
SM
SM
SM
LUT
SM
LUT
SM
LUT
SM
SM
SM I/O
LUT
SM
LUT
SM
SM
SM
SM I/O
LUT
SM
I/O SM
SM
SM
SM I/O
I/O
I/O
I/O
I/O
Figure 7.2: Schematic of a typical FPGA.
It is perfectly possible that the same VHDL file after compilation will result in a
different bit file, resulting in the operations taking place in other parts of the device.
Generally spoken, intermediate results are set on relatively long lines for processing
in other logical blocks.
7.2.2
Microcontroller (µC)
A µC is essentially a computer on a chip: it contains a controller, memory, storage of
the program and some input/output (I/O) blocks. This device can be programmed
by writing a program in assembly. This assembly code is then transferred to the
on-chip Read Only Memory (ROM) or RAM. The CPU can then read operands and
instructions from this ROM and execute the program. Intermediate results can be
written to the RAM. Definite results can be transferred to the I/O blocks.
Opposite to what happens in FPGAs, all operations are performed in one area and
all intermediate results written to another area in the chip. As an example, the block
diagram of the AT89C4051, a µC of the 8051 family 5 , used in Sect. 6.1.2.3, is given
in Fig. 7.3.
5 The 8051 family of µCs was developed by Intel, under the name MCS-51. These products have
been discontinued by Intel. Other vendors, such as Atmel, still manufacture 8051s.
132
Chapter 7. Introduction to the Application
Besides ROM to save the program code, RAM to store intermediate results, a timer
and interrupt block, also some communication blocks are provided: four parallel 8 bit
I/O ports, and a Universal Asynchronous Receiver/Transmitter (UART) controller
to facilitate interfacing with RS-232.
ROM
I/O
UART
INTERRUPT
RAM
TIMERS
PU
Figure 7.3: Block diagram of a typical 8051 family µC.
7.3
Side Channels
Instead of using mathematical techniques to break cryptographic ciphers, physical
properties of the device on which an implementation of the cipher is running can be
used to break the cipher. It is commonly agreed that in the mid nineties, Kocher [110]
was the first to publish a side channel in a practical attack. Since then, a lot of research
effort is spent on side channel analysis, and it is regarded as a separate cryptographic
field of study.
Military were, however, and again as usual, aware of the problem for more than fifty
years in advance. According to [111] Herbert Yardley in 1918 can be regarded as one
of the first to notice and exploit emanation of electronic devices processing secure
information. Later on, in 1943 during World War II, a spike at an oscilloscope screen
caused by an encryption machine started an entire TEMPEST research program on
electromagnetic emanation of sensitive information by the National Security Agency
(NSA) [112].
Apart from Appendix D on compromising emanation of computer displays, this work
focuses on side channels of cryptographic semiconductor hardware. Some of the possible side channels in that case are: timing [110] (Sect. 7.3.1), power consumption [113]
(Sect. 7.3.2), electromagnetic radiation [114] (Sect. 7.3.3) and sound [115, 116].
7.3. Side Channels
133
Somehow related to side channel analysis, knowledge of the physical implementation
of a cryptographic primitive can be (ab)used to deliberately induce errors during
execution, e.g. by pulsing laser light or RF power into the device. Observing the
output of the device after an induced error, and possibly comparing this with the
normal error free behavior of the device might again reveal some information that is
intended to be kept secret. This way of studying vulnerability of systems is generally
referred to as fault attacks [117, 118].
Yet another way of inducing errors in semiconductor devices, also mentioned in [117],
is bombing a device with an ion beam. The consequences this can have, were already
discussed in a completely different context of designing for space in Sect. 4.5.1. The
protective measures (shielding and redundancy) suggested there remain valid. Such
measures have a cost however, so that only for critical devices this protection against
(possibly) hypothetical threads such as Electro Magnetic Pulse (EMP) [119] can be
justified.
7.3.1
Timing
In many cases the time it takes for some cryptographic hardware to process some (partial) message and (partial) key inputs into a (partial) ciphertext output will slightly
differ. Reason for the deviation can be: condition branches, cache hits and misses,
difference in execution time of multiplications and divisions, and so one. Careful observation of the time duration of processes might hence disclose sensitive information.
A way to avoid information leakage via this mechanism is to make sure all operations
have the same time duration. This often requires bypassing of the cross compiler
optimizations and is even harder, if not impossible, if caching is involved. Another
approach could be blinding or masking of the inputs prior to processing. This however
merely adds another step and hence an additional effort for an observer, but does
not solve the problem profoundly as the masking step can not be supposed to be
perfectly secret [120]. A better approach might be to always calculate the results
for all branches in a conditional tree and afterwards select the appropriate value to
output. This at the expense of computational power.
7.3.2
Power Consumption
With each clock rise, transistors in (parts of) the cryptographic device have to switch
to charge or discharge registers (or capacitors). Depending on the value of the capacitance, either the capacitor (dis)charge current or the short current when the two
complementary transistors conduct together when switching, dominates. In the first
case, observation of the current consumption reveals the difference in hamming weight
between the old and new state of a register. In the second case, the transition count
(number of registers that update, not necessary to a new value) is disclosed.
134
Chapter 7. Introduction to the Application
This current I (or power P = V × I as V is the constant voltage of the power supply)
can be measured [121] by a current probe or by measuring the voltage drop over a
small resistor (e.g. 1 Ω) inserted in the power line6 of the voltage source.
The best approach to suppress this information leakage, is to use secure logic, such
as dual rail logic, where always one register holds the logic results and another one
its complement to balance the power consumption to a value independent of the
logic result. This technique can be implemented on FPGAs [122] or by using dual
rail logic cells in the design. In the latter case, non-standard routing algorithms
must be used as well, as now the number of signal lines doubles. Regardless of the
implementation, care should be taken that the capacitance of both the signal and
complementary register are exactly the same. This can never be perfectly achieved,
hence the problem of information leakage is not solved in a fundamental way, at best,
again, additional effort is required from an observer.
7.3.3
Electromagnetic Radiation
As the current mentioned in Sect. 7.3.2 not only draws power from the source, but
also causes EM radiation, observing the fields around the chip, might also be a way
to get to certain sensitive information. The radiation caused directly by the currents
that perform the actual logic operation, is, generally spoken, rather small and difficult
to observe in the presence of radiation of clock lines and the like. Due to typical EMC
issues, such as common ground impedance or cross-talk, the nearly unobservable signal
in the logic often modulates a much stronger signal, e.g. the clock signal. Hence the
information can be captured as the modulation7 of a strong carrier, improving its
observability drastically [123].
It is commonly agreed in literature that the EM side channel contains more information than just the power side channel. An intuitive way of explaining this, is that in
the case of power analysis only one signal is at the disposal of an adversary, whereas
the number of signals that can be obtained from the EM channel is virtually unlimited. By positioning the EM sensor slightly differently, a signal with another weighting
from all radiating source in the chip is obtained. Indeed if the chip is modeled as a
set of elementary dipoles that all radiate a different signal, the relative weight (and
phase) of the different signals in the sum signal at a certain location depends on the
distance between source and observation point.
6 This resistor can be put between the positive output of the power supply and the power rail
of the circuit, but then the voltage must be measured differentially in order not to short the power
supply by the ground of the oscilloscope probe. A resistor between the ground of the power source
and the ground of the circuit makes the latter floating. Especially when the circuit is fed with
multiple power supplies (at different voltages) this is not recommended.
7 It can be expected that (some of) these modulations are present too in the power consumption
signal.
7.4. Frequency Spectrum
135
Hence many publications e.g. [124, 125] report on measurements obtained after randomly shifting around the sensor until a position was reached were statistical attacks
similar to those used for power traces succeeded (even for hardware that was resistant
against power analysis). Inspection of the signal picked up by the sensor on an oscilloscope or spectrum analyzer while varying the position of the sensor is then often used
as an ad hoc method to avoid running the attack for every possible spatial position
of the sensor. That the scientific nature of this method is questionable, will be clear.
The method is not that surprisingly, either: an attacker, in general, does not have the
design plans of the device under test; nor are standard methods at hand to position
the sensor.
By collecting signals at different spatial points simultaneously, with EM sensors arranged in some array structure, it should theoretically be possible to reconstruct the
separate original signals to some extent. The question remains whether this is useful
as statistical methods might even be able to distract the information from this set
right away, i.e. without reconstructing the original signals. Nevertheless it is obvious
that array antennas are very promising in this application, as several of these differently weighted sum signals will, after appropriate combination, allow to improve
the SNR of the system. This could hence be a way to overcome the tedious shifting
around of the sensor to find the optimum location or, at least, facilitate the process.
It is expected that moving the sensor away for a considerable distance from the
radiating device, the weighting of all sources will be equal, resulting in a signal equal
to the power consumption signal.
Countermeasures to prevent leakage via the EM channel include all common EMC
techniques, such as: shielding, bypass capacitors and cross coupling aware design.
Besides that, again in this case dual rail logic or additional noise sources could be
considered.
7.4
Frequency Spectrum
The clock frequency of a device is selected such that the (chain of) executions that
should be performed during a clock cycle are finished well in time before the following
rising clock edge. Consequently, the spectrum of a power or EM signal from these
executions will go up to several tens of the clock frequency, depending on the time
constants of the (gate) capacitances in the logic device.
Thus, a wide band is needed to reconstruct the original signal. As it is, however, only
the information in the signal that is of interest, it might already be sufficient to only
use a fraction of the bandwidth and still be able to extract sensitive information. This
will depend on the hardware used (µC or FPGA) and on the code that is executed.
Unfortunately, at the time of writing, no founded theory was available on this matter.
136
Chapter 7. Introduction to the Application
Contemporary analysis hence always involves scanning of the frequency spectrum. It
is obvious that interesting and challenging research on this subject can and should be
performed.
Instead of working on the time signals, and looking for correlation between measurements and key bits, it is possible to do all calculations in the frequency domain. As
the Fourier transform is a one-to-one transformation, this is not surprising. What
is however surprising, is that when leaving out the phase information, and working
with the Power Spectral Density (PSD) instead of with the Fourier transform, side
channel analysis still seems to work [126]. Here, the PSD was calculated from the
time signal. But a spectrum analyzer, or even better, real time spectrum analyzer,
can directly measure (instantaneous) PSD. The latter even without missing events,
which is a problem with the first one, due to the non-zero sweep time.
Even if a PSD should not be sufficient for a side channel analysis, it will already give
an indication of the amount of information leakage. If a PSD as calculated on the
time signal recorded with e.g. an oscilloscope is compared with an instantaneous PSD
of a spectrum analyzer, deviation of both will signal a deviation from the general
behavior in a short time interval. This reveals data or instruction dependency of the
signal.
7.5
Conclusions
In this chapter the underlying physical principles were explained that cause information on the logic operations that are performed in a cryptographic device to leak via
e.g. electromagnetic radiation. This chapter also indicated that no practical system
can ever be 100% secure. With an unlimited amount of time and equipment, any
system can be broken. Designers can only make sure that the effort needed to break
a system is unfavorably high compared to the benefit of breaking the system.
After a short review of two hardware platforms often used for cryptographic operations, namely the FPGA and µC, some side channels such as power consumption,
execution time and electromagnetic radiation were discussed. A note on the lack of
knowledge about the optimal frequencies for EM side channel analysis concluded the
chapter.
Chapter 8
Array Elements
In this chapter, first the specifications for the sensor are summarized in Sect. 8.1.
Next the sensor can be designed. Three designs are reviewed: a shielded loop in
Sect. 8.2, a loop suitable for RFID in Sect. 8.3 and a tiny loop in Sect. 8.4. With
this small element, again an array can be implemented. Some details on this can be
found in Sect. 8.5.
8.1
Element Specifications
As already mentioned in Sect. 7.3.3, several approaches to retrieving information
via EM radiation, or ElectroMagnetic Analysis (EMA) can be followed: either the
information radiated directly by the (small) current performing the operations can
be captured, or the information can be extracted from a much stronger field that
is modulated by the same current via capacitive, inductive or resistive coupling. A
clear example is when the small current performing the logical operation couples to a
clock line running nearby, slightly altering the amplitude of the clock pulses. In such
case information might be extracted more easily by looking at the variations of the
field radiated by the clock lines, than when looking for the radiation of the current
performing the logical operation itself.
In the case of direct radiation by the current performing the logical operation, the
fields will be very small and nearly unnoticeable in the presence of the radiation of all
other parts in the chip. In such case the sensor should be small and put very close to
the radiating part of the device. Possibly the chip must be (partially) unpackaged1
to further reduce the distance and obtain distinguishable signals.
1 Note that the removal of the plastic around the silicon device puts the semiconductor in a
corrosive environment, reducing the lifetime of the chip dramatically.
137
138
Chapter 8. Array Elements
In the case of logical currents modulating a stronger signal, positioning of the sensor
is of less importance. Some other specifications, such as resonance frequency, become
more important, though.
Also mentioned in Sect. 7.3.3, was the fact that it is possible with EM probes to perform a power analysis without the need for physically contacting the device, allowing
a certain distance between the adversary and the device under test. This will again
result in different specifications.
All three approaches are compared in Table 8.1, regarding specifications. Following
subsections clarify the specifications. Only when eavesdropping on direct radiating
current, the sensor must be sensitive to magnetic fields. When a modulated carrier
or the integrated flux (proportional to the power consumption) is to be captured,
the sensor could also be an electric field probe. If the classical techniques of power
analysis are to be used on the signal recorded via the EM side channel, the sensor
should preferably have a flat frequency response, to preserve the signal in the time
domain as much as possible. If an integration is performed first, this issue is solved
and the sensor should preferably follow Eq. (8.7). Whether matching and a balun is
needed, depends on the rest of the signal chain, described in Chapter 9. Hence those
properties are not mentioned in the table.
Table 8.1: Overview of the relevant probe specs, depending on the approach to EMA. All
types need a rigid implementation, see Sect. 8.1.5.
type
Sect.
power
direct
modulated
8.1.1
field
8.1.1
E or H
H
E or H
resolution
8.1.2
low
high
medium
sensitivity
8.1.3
low
high
medium
bandwidth
8.1.4
large, flat response
large
around carrier
distance
large
small
medium
Capacitive and Inductive Sensors
In EMC problems, the sensors used, either pick up magnetic or electric fields. In
side channel analysis, it is not relevant to know the value of electric nor magnetic
field at a certain point. The only thing needed, is a signal that can be analyzed with
the aid of statistics and might reveal information on a secret key. This means that
a sensor for side channel attacks not necessarily has to discriminate between electric
and magnetic fields. Nevertheless in many cases the magnetic and electric fields will
differ in magnitude.
8.1. Element Specifications
139
10000
Electric dipole
|E|/|H|
1000
100
Magnetic dipole
10
0.01
0.1
r [λ]
1
10
Figure 8.1: |E|/|H| of a dipole as a function of distance.
If a source can be seen as a magnetic dipole, e.g. when current I flows in a loop
with radius rl that is small compared to the sensor and the wavelength, the magnetic
component will dominate the near field, as the field of a magnetic dipole [30]:
jβIπrl2 −jβr
1
~r =
~
e
1
+
(8.1)
H
2πr 2
jβr cos θ × ir
2
2
~θ = − β Iπrl e−jβr 1 + 1 − 21 2 sin θ × i~θ
(8.2)
H
4πr
jβr
β r
2
jβIπr
1
E~φ =
−ωµ 4πr l e−jβr 1 + jβr
sin θ × i~φ
(8.3)
β = 2π/λ is the wave number. The spherical coordinate system was illustrated on
Fig. 3.1(a). If the distance between source and observation rβ 1, indeed Hθ terms
dominate. In the far field, the ratio Z0 = |E|/|H| ≈ 120π is the free space wave
impedance. The relative amplitude of both components is depicted on Fig. 8.1.
If on the contrary the current I flows in a loop that is large compared to the sensor,
the source can be approximated as a set of electric dipoles with momentum Il, with
a mainly electric component in near field [127]:
p µ Il −jβr 1
~
E~r =
e
1
+
(8.4)
2
2πr
jβr cos θ × ir
p µ jβIl −jβr
1
− β 21r2 sin θ × i~θ
(8.5)
1 + jβr
E~θ =
4πr e
jβIl −jβr
1
sin θ × i~φ
(8.6)
H~φ =
1 + jβr
4πr e
Due to the similarity, except for some constants, between both sets, the results in
Fig. 8.1 are each others dual, apart from a constant Z0 .
140
Chapter 8. Array Elements
A capacitive sensor can be useful when attacking power lines running to the cryptographic device, or package pins. This capacitive sensor, e.g. a monopole, can be held
either along the wire or perpendicular to the wire. The latter measures E~r , whereas
the first position measures E~θ , but in practice the measured signal strength is the
same.
8.1.2
Resolution
The resolution to aim for, is in the order of magnitude of 100 µm or less. The area to
cover (or scan) is about 3 cm by 3 cm. The ITRS roadmap [128] indicates for 2009 a
maximum die size of 723 mm2 , which is about 2.6 cm by 2.6 cm. The ratio of package
size over die size can be less than 1.2, for so called Chip Scale Packages (CSP), but is
usually higher. In that case, bonding wires, which typically have a diameter of 20 µm
and a pitch of 35 µm, will be longer. A typical FPGA, the Xilinx Virtex II XC2V1000
that came out in 2002, has a die of approximately 1 cm2 [129]. The array of LUTs is
40 by 32 elements [130]. Hence one block in 2002 was 250 µm. The 8051 core in the
SCARD [131] test chip has a die size of about 1 mm2 for about ten thousand gates.
The resolution that can be obtained, depends on the design and on the implementation. Shielded loops will, due to their topology always require more physical space
than just coiled conductors. But shielded loops from PCB traces manage to achieve
considerably smaller implementations than can be obtained with coaxial cables. Masuda [132] reports an aperture of 20 µm × 1 mm, compared to a diameter of at least
a centimeter with RG-58 coax.
Using a loop sensor to detect varying magnetic fields is based on the Faraday-Lenz
law, that indicates that a varying magnetic flux through a surface will cause the line
integral of the electric field along the line enclosing the surface to be non-zero2 :
H
I
~ A
~
d A Bd
dφ
~ ~l = − B = −
V = Ed
(8.7)
dt
dt
s
φB is the magnetic flux. The integral is taken over the line s or surface A closed by
the loop. It is clear that location sensitivity necessitates a small loop. Otherwise, flux
from several sources is combined into one signal. Unfortunately, again from Eq. (8.7)
smaller loops result in a smaller induced voltage. This problem can be solved by
using more than one turn. This however limits the bandwidth of the sensor, as will
be explained in Sect. 8.1.4.
If the loop is intended for a contact less power analysis, a large loop can be used, as
now the entire flux of the device should be integrated.
2 This voids Kirchhoff’s second law which states that the sum of all directed voltages in a closed
circuit is zero. Introducing an extra voltage source with voltage equal to Eq. (8.7) is a workaround
that allows to keep using Kirchhoff’s laws, and apply circuit theory, though invalid in this case.
8.1. Element Specifications
141
Note that if the device that is eavesdropped is placed right in the middle of a large
shielded loop, a net flux of zero will be measured no matter what the power consumption is. The best location for the device under investigation is at the side of the loop,
as will become clear in Fig. 8.38.
8.1.3
Sensitivity
What sensitivity is desired, heavily depends on the rest of the signal capturing chain.
Hence a more in-depth discussion with order of magnitudes is delayed to Sect. 9.3.4.
There it will also be pointed out that not only the amplitude of the signal is important,
but that it is ultimately the SNR that determines the system quality.
Sensitivity will always be traded-off either versus spatial resolution, as explained in
Sect. 8.1.2, or versus bandwidth, see Sect. 8.4.1 and 8.4.2.
8.1.4
Bandwidth
From the discussion in Sect. 7.4, it will be clear that the sensor should have an upper
working frequency fH of at least several times the clock frequency of the device under
test. Preferably fH should be as high as possible to allow detection of transition
currents3 and other typical short time semiconductor phenomena. As the bandwidth
of a signal chain is limited by the lowest and highest working frequency of the separate
links in the chain, there is not much use in imposing a fH for the sensor that is much
higher than the upper frequency of e.g. the oscilloscope bandwidth.
Common clock frequencies for digital logic range from a few kHz to a few MHz. The
maximum working frequency, determined by the delays in the critical path of the logic,
results in the least calculation or operation time, at the expense of power consumption.
Indeed, the average dynamic power consumption of an inverter is linearly proportional
to the clock frequency fclk :
2
Pdyn,avg = CL Vdd
fclk P0→1
(8.8)
CL is the load capacitance, Vdd the power supply voltage, and P0→1 the average
number of transitions from 0 to 1 [133]. With the scaling of the transistors following
Moore’s law in combination with the aggressive scaling of clock frequencies, an ever
increasing amount of power must be dissipated in an ever decreasing volume. By now
the limits are more or less met (or the thermal wall is hit) and it is expected that
clock frequencies, nowadays commonly about 3 GHz for CPUs, will not increase a
great deal any more [134].
3 In Complementary Metal Oxide Semiconductor (CMOS), which is the most commonly used logic
style nowadays, short circuit current flows for a short period of time during the switching of a gate
when both complementary transistors are conducting.
142
Chapter 8. Array Elements
The lowest operating frequency for a loop fL will never be 0 Hz. From Eq. (8.7) it
is understood that a DC flux will never induce a voltage over the loop. Moreover,
the induced voltage is linearly proportional to the frequency. Hence the frequency
response of a loop will never be flat, or translated to the time domain: the shape
of the signal will not be preserved. Some techniques can, however, result in a flat
frequency response for a limited frequency band, but again, at the cost of sensitivity.
Examples will be given in Sect. 8.4.2.
As soon as the perimeter of the loop is considerable compared to the wavelength,
wave phenomena come into play and the insertion of a voltage source with value of
Eq. (8.7), in order to keep using simple circuit theory, becomes inaccurate.
8.1.5
Rigidity
It will be clear from Eq. (8.7) that the loop should not deform as this alters the flux
through the surface and hence the measured signal. Deformation during a measurement would obstruct interpretation of the results. Deformation when applying or
removing would inhibit comparison between different measurements. Therefore all
loops should be implemented rigidly.
8.1.6
Matching
As soon as the sensor becomes a part of a signal chain, the sensor will be interfaced:
either with cables or with measurement devices or amplifiers. This introduces extra
boundary conditions for the sensor design. It might be necessary to match the sensor
to the cable, commonly4 50 Ω, to avoid reflections. This will again become much
clearer when worked out in detail in Sect. 9.3.2. Some generalities can however already
be discussed here.
In general antenna design, whenever transmission lines, such as coaxial cables, are
used, telecommunication applications require that the loads at both ends of the line
are matched to the characteristic impedance of the line, in order to assure a maximum
power transfer. Moreover a minimization of the reflections back into the transmitter
lenghtens the lifetime of the transmitter. However, when measuring magnetic fields,
a transfer of power is not the issue, as not necessarily analog circuitry is driven, but
the voltage signal might be sampled and digitally stored. In that case, the amplitude
of the measured signal, or even more correct, the signal-to-noise ratio, is what is to be
maximized. Therefore, matching as performed in the measurement setup of Chapter 9
differs from the standard approach in RF engineering.
4 The value of 50 Ω was the compromise between 30 Ω, for maximum power carrying capability,
60 Ω, for highest breakdown voltage, and 77 Ω for minimum attenuation, according to tests by Lloyd
Espenschied and Herman Affel at Bell Labs in 1929.
8.1. Element Specifications
143
Matching of a single port device (or one port 5 ), such as an antenna, is characterized
by measuring the scattering parameter at the port of the device under test:
S11 =
V1−
V1+
(8.9)
which is the ratio of the complex amplitude of the reflected voltage wave V1− from port
1 over the complex amplitude of the incoming voltage wave V1+ in port 1. Squaring
gives the ratio in reflected power to incoming power, as the characteristic impedance
is the same for both voltage waves.
Matching can be measured with a VNA, such as an HP 8510C or HP 8753C, depending
on the frequency range of interest, or with a scope and function generator as explained
in Appendix F.
8.1.7
Baluns
Transmission lines, and more generally systems, can be grouped into balanced and
unbalanced ones. Many ways to define these terms can be found. Ultimately, regardless of the definition chosen, the grouping is identical. However, not all definitions are
as rigorous as one would want them to be.
Defining a balanced transmission line as a system of two conductors that are electrically identical and can hence be interchanged, is about the only way of seeing it
without introducing an arbitrary ground or reference point, complicating things more
than necessary.
The IEEE6 dictionary [136] defines a balanced line as: A transmission line consisting
of two . . . conductors capable of being operated in such a way that when the voltage of
the two . . . conductors at any transverse plane are equal in magnitude and opposite
in polarity with respect to ground, the total currents along the two . . . conductors are
equal in magnitude and opposite in direction.
According to [137], transmission lines with two conductors are said to be: balanced if
the impedance between each conductor and ground is equal or unbalanced if one of the
conductors is connected to the ground. Although the terminology suggests that each
system should either be balanced or unbalanced, this way of defining, hence allows
a third more general case, where both impedances are non-zero and not equal. This
situation is depicted in Fig. 8.2
5 A device with two terminals has exactly one port to connect signals as for signal transfer at least
two conductors are required in order to close the current loop.
6 The IEEE name was originally an acronym for the Institute of Electrical and Electronics Engineers, Inc. Today, the organization’s scope of interest has expanded into so many related fields, that
it is simply referred to by the letters IEEE, [135].
144
Chapter 8. Array Elements
Z1−2
conductor 1
Z1
Z
2
conductor 2
ground
Figure 8.2: This system is not balanced nor unbalanced if Z1 6= Z2 .
Both, balanced and unbalanced transmission lines, can (and should) be excited in
such a way, that the current in both conductors is balanced, canceling out any fields
at larger distance7 . Otherwise the transmission line would radiate and hence loose
power instead of simply transferring a signal from one place to another with as few
losses as possible. If however balanced and unbalanced systems are interconnected,
modes of transmission are generated, due to different boundary conditions, that cause
currents to flow in an unbalanced way, resulting in emitted radiation propagating and
carrying away signal power.
Dipole and loop antennas are examples of a balanced system. Coaxial cables are
unbalanced. Again, when excited in a proper way, no radiation from the transmission
line will occur, as the current will only run in the center conductor and at the inner
side of the coaxial conductor. This gives the coaxial cable its good shielding property.
If now a balanced antenna is connected to this unbalanced cable, current will run at
the outer side of the outer conductor of the unbalanced coaxial cable near the point
of connection to the balanced antenna, just as a result of boundary conditions. Due
to the alternative path to the ground, via outer side of the cable sleeve, possibly with
less impedance, now the current is not forced to go into the antenna anymore. In case
the loop should be very location sensitive, this should be avoided as it deteriorates
the sensor resolution.
A balun, or balanced-unbalanced transition, can be used to avoid that current can
flow on the outer side of the conductor, by adding a λ/4 transmission line around
the outer conductor that is shortened at the end. The transmission line transforms
this short into an open so that no signal can enter the transmission line. An outline
of the balun is drawn in Fig. 8.3. A common implementation of the same principle
is depicted in Fig. 8.4. At λ/4 from the end of the coaxial cable, an open stub is
connected to the outer conductor. Usually a short is placed between O and B and
the antenna is connected to B and C, as opposed to the scheme from Fig. 8.3 where
the antenna is simply connected to C and O. This implementation shows several
correspondences with shielded loops, see Sect. 8.2.
7 At closer distance there will always be field observable, allowing for e.g. current consumption
monitoring, as a field around the conductors is indispensable for propagation
8.1. Element Specifications
145
B
B
λ
4
C: center conductor
O: outer conductor
B: balun sleeve
O C O
Figure 8.3: Schematic drawing of a sleeve balun with dipole antenna.
λ
4
C: center conductor
B
B
O: outer conductor
B: outer conductor of coax for balun
O C O
Figure 8.4: Schematic drawing of an implementation of a sleeve balun with dipole antenna.
Another balun type, avoiding bandwidth limiting resonance stubs, that moreover allows to transform the load to a value needed for matching, is a transformer balun [138].
A practical implementation is described in [139] and schematically drawn in Fig. 8.5.
By winding wires over a magnetic core, selecting the appropriate number of turns for
primary and secondary side, and by grounding different points in the primary and
secondary coils, this can be obtained. The use of a magnetic core however excludes
usage for higher frequencies.
Vunbal
Vbal
Figure 8.5: Working principle of a transformer balun.
Unfortunately, baluns are not broad band devices. For the stub balun because the
physical length of the stub has to be λ/4. For the current transformer balun due to
the resonance of the inter turn capacitance and the inductance of the coils.
146
Chapter 8. Array Elements
Another way of dealing with the problem, is adding lossy material, such as ferrite
beads, which is a very common technique in measurement of small antennas, such as
Ultra Wide Band (UWB) and handset antennas [140].
Balun problems with symmetric antennas can be avoided by using two coaxial cables,
both connected to the scoop or one terminated with a 50 Ω load, or by connecting the
symmetric antenna directly to the measurement device, without cables. This relates
to the measurement setup, which is discussed in Chapter 9.
8.2
Shielded Loop
8.2.1
Loop Types
Although a vast amount of shielded loop magnetic field sensors are in use, only four
implementations, depicted in Fig. 8.6, namely an unshielded loop, a symmetrical loop,
a balanced loop and a mœbius loop will be discussed and compared. First, the types
will however be introduced briefly, with remarks when necessary. A schematic for
each type can be found in Fig. 8.7.
Figure 8.6: Photograph of the shielded loops. The upper one is the EMCO loop. Below
from left to right are the unshielded, symmetrical, balanced and mœbius with
and without short.
The naming of the loops is in accordance with the naming in [141], but might be
confusing as interfering with the naming in [142]. The latter interpretation of a
shielded loop is depicted in Fig. 8.8, but will not be discussed in this work, as it
requires biax8 cable, which can not be interfaced as is with a standard oscilloscope.
Moreover, its characteristics are similar to those of the balanced shielded loop.
8A
biaxial cable is a signal cable with a sleeve and two inner conductors.
8.2. Shielded Loop
147
50 Ω
(optional)
(a) Non-Shielded Loop
(b) Symmetrical Shielded Loop
(c) Balanced Shielded Loop
(d) Mœbius Type Balanced Shielded Loop
Figure 8.7: Schematic drawings of the four loop types.
Figure 8.8: A shielded loop according to the definition of [142], with biax cable
All types should be isolated, to prevent contact with e.g. pins of components in the
measured circuit. Care should be taken that the loops do not deform when used, as
this alters the signal. Therefore the loops were attached to cardboard with waxed
cords. A solution like embedding the coax in injection molded plastic, used by EMCO
for the near field probe set model 7405 [143], is better. A PCB implementation of a
shielded loop will also sidestep this issue.
148
Chapter 8. Array Elements
Table 8.2 lists the DC resistance between inner and outer conductor of the sensors.
This allows quick verification of the contacts before using the sensor and reveals the
matching at low frequencies: the non shielded, symmetrical and mœbius shorted
loop will have an S11 = 1 (0 dB), whereas the balanced and mœbius without short
are matched to 50 Ω so that S11 = 0 (−∞ dB). In Table 8.3 the advantages and
disadvantages of each type, now explained in detail, are summarized.
8.2.1.1
Non-Shielded Loop
A straightforward way to implement a loop sensor, is to simply bend a wire and solder
one end to the center conductor and the other end to the outer conductor of a coaxial
cable.
This system needs a balun and will, without balun, suffer from the antenna effect,
i.e. current will run at the outer side of the outer conductor of the coax and will
hence pick up signals and influence the output signal of the sensor, as was explained
in Sect. 8.1.7.
8.2.1.2
Symmetrical Shielded Loop
A loop is formed by the connection of center and outer conductor at the end of the
coaxial cable to the outer conductor of the coaxial cable at the beginning of the loop.
In this way, a line integral for Eq. (8.7) similar to the one of the non-shielded loop is
obtained. A piece of outer conductor is cut away, however, obstructing current from
flowing on the outer side of the outer conductor of the coax. Without this gap the
current flowing would cancel the magnetic field so that the measured signal would be
zero.
Table 8.2: Measured DC resistance values (rounded) between connectors of the four loop
types. i: inner conductor, o: outer, sa: same end, op: opposite end. 50 Ω will
only be measured if one port of the sensor is loaded with 50 Ω. If left open, ∞
will be measured.
type
non-shielded
symmetrical
EMCO φ = 6 cm
balanced
mœbius
mœbius with short
o-i sa
0Ω
0Ω
0Ω
50 Ω
50 Ω
0Ω
o-i op
50 Ω
0Ω
0Ω
o-o op
0Ω
50 Ω
0Ω
i-i op
0Ω
50 Ω
0Ω
8.2. Shielded Loop
149
Table 8.3: Advantages and disadvantages of the four loop types.
type
non-shielded
symmetrical
balanced
mœbius
local pickup
-+
+
+
antenna effect
+
+
imped match
+(LF)/-(HF)
-(LF)/+(HF)
amplitude
1
1
1/2
1
The short at the end of the coaxial cable will cause large reflections. Inserting a
50 Ω resistor, as in Fig. 8.7(b) can solve this problem, at the cost of loosing signal
strength and generating extra noise. Indeed, following the expression of JohnsonNyquist noise [144], the root mean square (rms) value of the noise generated in a
resistance R is proportional to R:
p
hv 2 i =
√
4kT BR.
(8.10)
The loop antennas of the EMCO near field probe set model 7405 are of this type. In
fact, this type is widely used in EMC diagnostics as it allows to measure values for
the magnetic field strength without errors due to the presence of electric fields. This
antenna is indeed less sensitive to the electric field, compared to the non-shielded
loop, due to the shielding of the outer conductor.
8.2.1.3
Balanced Shielded Loop
If a coaxial cable is taken and the two ends are connected to a scope (or one to a
scope and one to a 50 Ω load), then the reflection problem is avoided. Moreover, also
the issue with current flowing on the outer side of the sleeve is avoided, as now the
interconnection of an unbalanced cable and a balanced loop is avoided by using two
(unbalanced) cables, balancing each other.
A loop is obtained by bending the coaxial cable. Where the coaxial cable does not have
to form a loop, some insulation is removed and the outer conductor of the adjacent
coaxial cables is soldered together. Again a piece of outer conductor in the loop is
cut away as otherwise the measured signal would be zero.
The signal that is picked up will have the same magnitude if the loop is about the
same size compared to the previous types. The signal however now has to be measured
between the two ports of the scope, so that an oscilloscope with math function comes
in handy.
150
Chapter 8. Array Elements
8.2.1.4
Mœbius Type Balanced Shielded Loop
Instead of cutting away a piece of outer conductor, the coaxial cable is cut in two
and the inner conductor of both parts is connected to the outer conductor of the
other part. Doing so, creates a double loop, hence the name, so that the amplitude
of the signal picked up will be twice as large. This is explained in more detail in
Sect. 8.2.3. The shorting of the two outer conductors outside the loop was omitted
in one prototype and applied to a second.
8.2.2
Matching
As RG-58 cable was used for the loops, only matching to 50 Ω is relevant. This is
advantageous as most equipment to measure scattering parameters also works with
50 Ω as the reference.
The scattering parameter S11 of the loops was measured with a HP 8510C VNA. As
this device is only specified for frequencies higher than 45 MHz due to an IF stage in
the machine of 20 MHz, obtained by the signal (or one of its harmonics) of a local
oscillator between 65 MHz − 300 MHz [100], the measurements between 22 MHz and
45 MHz are not fully accurate. They nevertheless nicely align with the measurements
between 1 kHz and 20 MHz obtained with a scope and function generator as explained
in Appendix F, and can hence be expected not to be totally unreliable.
0
100
200
300
400
500
600
700
800
0
−5
−10
−15
−20
−25
−30
−35
S11 [dB]
unshielded
symmetrical
EMCO symm.
balanced
mœbius
mœbius with short
Figure 8.9: S11 of the loops for 22 MHz − 1 GHz
900
f [MHz]
8.2. Shielded Loop
0
5
151
10
15
20
25
30
35
40
0
−5
−10
−15
45
f [MHz]
unshielded
symmetrical
balanced
mœbius
mœbius with short
−20
S11 [dB]
Figure 8.10: S11 of the loops for 1 kHz − 50 MHz
The measured values for S11 are displayed in Figs. 8.9 and 8.10. For lower frequencies,
all probe types with shorting of inner and outer conductor have a scattering parameter
S11 = 1. The mœbius without short and balanced type are matched to 50 Ω at lower
frequencies. For higher frequencies however, it is less evident for a current to take
the short between the inner and outer conductor if this path would have a large
inductance. Hence the mœbius with and without short behave the same for higher
frequencies.
The periodic peaks for both mœbius types are due to the non-perfect characteristic
impedance Zc of the coaxial cable. The cable used was of the RG-58 type, defined
in [145], but canceled in [146] and hence invalid as a standard. The manufacturer
specified Zc as 50 ± 2 Ω [147]. At 200 MHz, a value of 52 Ω was measured. This
value was obtained as the geometric mean of an impedance measurement with a short
(Zin,0 ) and an open (Zin,∞ ). The validity of this technique is easily checked, by writing
out Zin in terms of scattering parameters and then filling in 0 and ∞ for ZL :
v
u
L −Zc
L −Zc
u 1 + ej2kl Z
p
1 + ej2kl Z
ZL +Zc
ZL +Zc
×
Zin,0 × Zin,∞ = Zc t
L −Zc
L −Zc
1 − ej2kl Z
1 − ej2kl Z
ZL +Zc
ZL +Zc
s
1 + ej2kl
1 − ej2kl
= Zc
×
= Zc
(8.11)
1 + ej2kl
1 − ej2kl
At higher frequencies Zc becomes complex, due to losses. Being e.g. 52 + 8i Ω at
500 MHz, the cables transform the load on circles around the center of a smith chart
referenced to 52 + 8i Ω. On a smith chart referenced to 50 Ω, those circles become
egg shaped curves around the point corresponding to 52 + 8i Ω, i.e. a little up to the
right from the center point.
152
Chapter 8. Array Elements
Figure 8.11: S11 of the mœbius with short on a Smith Chart.
The curve for the mœbius without short comes up as high as −2 dB around 100 MHz.
This peak is due to a short when the capacitance between the two adjacent outer
conductors before and behind the loop starts to conduct signals. The capacitance
between two cables of radius rw with distance d between the wire centers, is [148]:
C =l×c=l×
log
d
2r
π
q
d 2
+ ( 2r
) −1
(8.12)
This expression is not exact in the case of hollow coax outer conductors due to the
skin effect, but still, it can be concluded that the capacitance between the cables
is influenced by both the length l of the cables, and the distance d between them.
Hence this peak can be avoided by moving the 50 Ω termination at the end of the
second cable to just behind the loop, so that the second cable has zero length. A
measurement, shown in Fig. 8.12, indeed reveals that this action solves the matching
issue. This sensor is indeed matched sufficiently over the entire frequency range.
Such sensor would however only partially pick up signals as the voltage induced by the
magnetic flux stands partly between the two outer conductors and does not contribute
to the signal that arrives at the scope.
This actually means that none of the four probe types is matched over the entire
frequency range of interest. If, for some application, only frequencies below e.g.
50 MHz are to be used, the balanced loop can be used without matching problems.
If, for another application a band will be used around a frequency of some hundreds of
MHz, that will be mixed down in a receiver, the mœbius loop with short can be used.
8.2. Shielded Loop
0
100
153
200
300
400
500
600
0
−5
−10
700
800
900
f [MHz]
mœbius large C
mœbius small C
−15
−20
−25
−30
−35
S11 [dB]
Figure 8.12: S11 of a mœbius without short with different capacitances between the two
adjacent cables.
Only if for some reason the entire band from (nearly) DC to 1 GHz is important, a
combination of the balanced and mœbius with short can be the solution. A schematic
layout of this solution is given in Fig. 8.13. Capacitances and an inductor are used
to obtain the connections for a balanced loop at low frequencies and the connections
for a mœbius loop at high frequencies.
Figure 8.13: Layout of a loop that combines the balanced and mœbius loop.
154
8.2.3
Chapter 8. Array Elements
Sensitivity
Generally, it is assumed that shielded loops are only sensitive to magnetic fields. A
small remark should be made here. Indeed will only the magnetic field through the
loop have an electric field associated9 with it that lies along the metal of the loop.
This way, as in the case of an electric dipole in a linearly polarized electric field,
electrons in the metal will rearrange to cancel out the electric field in the conductor,
resulting in a current or induced voltage, in case the loop is not terminated with a
load.
If the electric field on the loop antenna is symmetric with respect to the feed axis
of the loop, no voltage will be measured over the loop. This is because the voltage
induced in one half of the loop is exactly compensated by the voltage over the other
half of the loop. Hence when using a loop as an electric field sensor, two half loops
can be used and the difference current at two opposite measuring points should be
observed, as in [149].
The true value of the shield is hence not to shield the loop from electric fields, but
to assure that metal in the surroundings does not unbalance the loop and shield the
loop from local perturbations, as explained in [142]. If however, as is the case for
the symmetrical loop, the signal can not continuously use the inner side of the outer
conductor, the loop will, despite the shield, still suffer from the antenna effect.
How the voltage is found with Eq. (8.7) in case of a non-shielded loop, is straightforward. In case of a shielded loop, this is less obvious. Actually Eq. (8.7) should be
applied to the shield, in this case. This results in a voltage over the gap, which then
excites a wave in the coax, as explained on Fig. 8.14. Indeed, as the square formed
by the arrows Vc,L , Vc,R and Vgap and the inner conductor can be supposed to be
infinitesimally small, the flux through this square can be said to be zero, hence with
Eq. (8.7):
Vgap = Vc,L + Vc,R .
(8.13)
Generally spoken, if the load at both sides of the gap is equal, Vc,L = Vc,R = Vgap /2.
The balanced loop will hence have a signal amplitude that is only halve of the amplitude for the non-shielded loop. The symmetric loop, however, shorts Vc,R , so that
Vc,L = Vgap , if the loop can be assumed to be small compared to the wavelength.
Otherwise the short at the end is transformed by the piece of coax between the gap
and short. Hence, it is natural to have the gap right next to the short, at the neck as
in [141]. For the mœbius with short, the line integral of Eq. (8.7) again leads directly
to the same voltage as in case of the non-shielded loop. Now, however, the line of the
line integral runs partially over the outer conductor.
9 −jωµ H
~
0
~ see also Eq. (8.7), one of Maxwell’s equations.
= ∇ × E,
8.3. RFID Loop
155
Vc,L
Vc,R
Vgap
Figure 8.14: Working principle of a shielded loop.
8.3
RFID Loop
Following part was reported in [150]. The designed loop is not exactly a sensor for
EMA, but a loop for a reader that extends the reading range of an ISO-14443A RFID
system. Although Radio-frequency identification (RFID) is beyond the scope of this
text, the design of the loop is relevant, as it fits in analyzing existing loop design.
Indeed many aspect studied here will turn out to be very relevant when designing a
small sensor in Sect. 8.4.
RFID was developed during the second world war by the British, to identify friendly
from other airplanes, Identify Friend or Foe (IFF). It took however several years until
the technique was used in industry. With the ever decreasing cost of silicon logic, it
is not unlogical that sooner or later, RFID will replace bar codes. RFID offers some
advantages over bar codes. Besides the obvious application of tracking goods, it can
be used for counterfeit control and is writable. More reasons to use RFID can be
found in [151]. A drawback of RFID is that it requires special attention concerning
privacy. Access to sensitive information should be denied to unauthorized parties.
RFID can be implemented in several ways: active or passive, inductive or capacitive
and so on. Appendix G gives an overview and explains one inductive standard, ISO14443A for 13.56 MHz in more detail. This is the standard that is also used in
the Belgian passports. Often remarks on privacy and security issues are countered
by the small read out distance of the system, often assumed to be about 10 cm.
Kirschenbaum and Wool [152] however designed a low-cost, extended range RFID
skimmer. With the RFID loop designed here and the amplifier of Sect. 9.3.3.1, an
even larger reading distance can be bridged.
8.3.1
Loop Design
The reader antenna has to provide the tag with a field that is sufficient to power up
the hardware in the tag. Hence the antenna will be designed in such a way that the
magnetic field at a certain read out distance rd is large enough. In second order, the
antenna should also be suited to receive the answer from the tag. This condition is not
considered here, as an adversary is presumed to have access to a separate receiving
antenna if needed.
156
Chapter 8. Array Elements
The parameters of the loop that can be chosen are shape, size, number of turns and
wire diameter. These are discussed separately below.
In order to feed the loop in a balanced way, a push-pull amplifier is preferred to a
single ended amplifier. Else, a balun, discussed in Sect. 8.1.7, should be used.
8.3.1.1
Shape
Of all antennas that can be used to excite a magnetic field, a circular loop is clearly
the best choice in case the current on the entire antenna is in phase. As the distance
from the point where the tag is located to all current carrying parts of the antenna is
equal in this case, the contributions of all parts of the antenna arrive in phase at the
tag, resulting in constructive interference.
For a larger loop, where the current over the loop can not be supposed constant, it is
less obvious. A spiral can slightly compensate for the phase difference over the loop
by a difference in propagation distance to the tag. Moreover, a spiral has a lower
inductance L than a circular coil with the same number of turns Nt , resulting in an
excellent coupling factor k for a system with two spiral coils [153]:
k=√
M
,
L1 L2
with M the mutual inductance between the two coils:
H
H
~ 2 dA
~
~ 1 dA
~
Nt,1 A1 µH
Nt,2 A2 µH
M=
=
Il1
Il2
(8.14)
(8.15)
which is the ratio of the flux in one coil to the current in the other. Where this
may be interesting from a power transfer point of view, it is rather irrelevant for the
application envisaged here. When simply looking at maximum attainable magnetic
field strength starting from a certain loop current Il , the circular loop is still the better
choice. Indeed, only then will all current carrying parts be at the optimal distance
under the optimal angle, that will be derived in Sect. 8.3.1.2. Moreover the circular
loop outperforms the spiral in case of lateral misalignment [154], which is very likely
to occur in case of an adversary secretly reading out tags. As a consequence, the
circular loop is the best choice here too.
8.3.1.2
Size
The larger the circular reader antenna loop is, the more current carrying parts add
a contribution to the magnetic field. If the loop becomes too large, however, these
contributions are very weak due to the large distance from the current carrying part
to the tag.
8.3. RFID Loop
157
Hence, there will be an optimal loop diameter and this diameter is ruled by the read
out distance. Suppose that the circular loop has a radius rl and the read out distance
is rd , then rl should be chosen so that the magnetic field at a distance rd from the
center is maximal.
y
rl
I
x
r
rd
~
dH
α
z
Figure 8.15: Reader loop geometry.
When phase differences10 along the total wire length of the loop (2Nt πrl ) are taken
~ z | the amplitude of the magnetic field in the direction perpendicular
into account, |H
to the tag, at a distance rd becomes (see Fig. 8.15 for conventions):
Z
2πrl Nt I exp (j 2πl ) cos α l
λ
~ z| = dl
|H
0
4π(rd2 + rl2 )
q
−1
2
t
Il rl λ 2(1 − cos ( 4πλ rl )) NX
4π 2 nrl
j
λ
p
=
(8.16)
e
8π 2 (rl2 + rd2 )3
n=0
with Nt the number of turns of the loop, Il the amplitude of the loop current and α
the angle indicated in Fig. 8.15.
Finding the optimal value for rl thus boils down to (numerically) finding the root of
the derivative of Eq. (8.16):
~z |
d|H
rl
= 0 with ρ = .
dρ
rd
(8.17)
Writing Eq. (8.16) as the product of two functions f (ρ)g(ρ) and leaving out all constants, because of no importance for the derivative, results in:
s
2
4π ρrd
ρ
1
−
cos
f (ρ) =
(8.18)
3
λ
(1 + ρ2 ) 2
N −1
!N
#
t
t
X
X
4π 2 nrl
4π 2 ρrd
j
λ
− Nt
g(ρ) = (Nt − n) cos n
e
=2×
λ
n=0
n=0
10 The phase due to the distance between the source current and the tag location is discarded as
this distance is the same for all current carrying parts of the circular loop.
158
Chapter 8. Array Elements
The derivative of those functions is:
r
2 1 − cos 4π λρrd
f 0 (ρ) =
(1 − 2ρ2 ) +
5
(1 + ρ2 ) 2
g 0 (ρ)
= −2 ×
Nt
X
n=0
n(Nt − n)
2 2π 2 ρrd sin 4π λρrd
r
2 (8.19)
3
λ(1 + ρ2 ) 2 1 − cos 4π λρrd
4π 2 ρrd
4π 2 rd
sin n
λ
λ
(8.20)
Knowing that:
(f g)0 = f 0 g + f g 0
(8.21)
the equation to be solved in order to find the optimal ρ, is:
2 2 1 − cos 4π λρrd
λ(1 − 2ρ2 ) + 2π 2 ρrd (1 + ρ2 ) sin 4π λρrd
P
2 Nt
4π 2 ρrd
4π ρrd
2
(N
−
n)
cos
n
× 2
−
N
−
2ρ(1
+
ρ
)
1
−
cos
t
t
n=0
λ
λ
P
2
Nt
4π
ρr
2
d
=0
(8.22)
n=0 n(Nt − n)4π rd sin n
λ
After a division by λ, an equation is obtained that leads to the solutions for rd /λ.
Hence solving this equation once leads to design curves valid for any frequency. For
the case of a single turn (Nt = 1), the optimal value for ρ and hence rl is found as
the solution of:
2
sin ( 4π λρrd )
(2ρ2 − 1)
rd
−
2π 2 (1 + ρ2 ) = 0.
4π 2 ρrd
ρ
λ
1 − cos ( λ )
(8.23)
The result, as well as the solutions for Nt = 2, 3, 5, 10, plotted in Fig. 8.16, show
that as rd increases, rl also increases, even up to rd /λ = 1/2π ≈ 0.16. But the ratio
between rl and rd decreases as rd increases. For any number of turns, the limit for
rd /λ → 0:
√
rl
lim ρ = rlim
(8.24)
= 2.
rd
d
r
λ →0
λ →0 d
This can be expected as in this limiting case, also rl /λ → 0. The assumption of√a
constant current over the entire wire length of the loop, leading to the ratio rl /rd = 2
as published in a.o. [155] and [156] in such case surely holds.
8.3.1.2.1 Total wire length of loop small compared to wavelength Under
this precondition, the current can be assumed to be constant over the loop. Eq. (8.16)
then simplifies to,
Z 2πrl
2
N
I
r
N
I
cos
α
t
l
t
l
l
~z | = = p
(8.25)
|H
dl
2 (r2 + r2 )3 ,
4πr2
0
l
d
which can e.g. also be found in [157].
8.3. RFID Loop
159
rl /rd
Nt
Nt
Nt
Nt
Nt
1.0
=1
=2
=3
=5
= 10
0.5
0
0
0.05
0.10
0.15
rd /λ
Figure 8.16: rl /rd as a function of rd /λ.
The optimal value for rl can be found with Eq. (8.24), as assuming that 2πrl /λ → 0
is equivalent with assuming that rd /λ → 0, because both rl and rd are of the same
order of magnitude.
The ISO-14443 standard [158] specifies the minimum (rms) magnetic field strength
for the cards to operate11 as Hmin = 1.5 A/m. Combining Eq. (8.25) with (8.24)
yields:
N I r2 N I Nt Il rl2
t l d t l ~
.
(8.26)
|Hz | = p 2
= p
= √
2 (rl + rd2 )3 (3rd2 )3 27rd The minimum current (rms) Imin needed in the reader antenna (with rd in m) is:
√
Imin × Nt = 27 × Hmin × rd = 7.8 × rd .
(8.27)
Assume, e.g. Nt = 1, rd = 0.1 m and L = 1 µH. The current with Eq. (8.27) will be
0.78 A and the voltage:
√
(8.28)
|V | = |ωL 27Hmin rd | ≈ 66 V.
It is obvious that applying this voltage directly to the loop is not practical. Hence
current enhancement techniques, covered in Sect. 8.3.2, should be used.
8.3.1.2.2 Total wire length of loop comparable to wavelength As soon as
the total wire length of the loop is considerable, say λ/10 as a rule of thumb, the
magnetic field of the loop will be smaller than what would be expected when using
Eq. (8.25), which is indeed invalid in this case. Due to the phase differences over
the loop, the contributions of all parts of the loop at the location of the tag will not
be in phase and partial cancellation will occur. In that sense it might even be more
advantageous to use a slightly smaller loop with more current, as Fig. 8.16 indeed
indicates.
11 That
is to build up a voltage high enough to power up the hardware in the tag.
160
Chapter 8. Array Elements
8.3.1.2.3 Total wire length of loop multiple of half wavelength If the loop
perimeter equals half a wavelength or a multiple, standing waves will occur. The
loop itself will then resonate and the reactance on the Smith chart crosses the real
axis, going from inductive to capacitive impedance. This dependence of the reactance
on the frequency has its effect on the usage of the coil in an RLC circuit to enhance
current. If e.g. Eq. (8.35), the resonance condition for a combined series-parallel RLC,
has, due to frequency dependence, multiple solutions for different ω, the RLC system
will resonate at multiple frequencies and the energy will be divided amongst them.
Fig. 8.22 shows multiple crossings of the curves for the resonance condition and the
imaginary part of the input impedance.
8.3.1.3
Number of Turns Nt
Looking at Eq. (8.25), which is only valid for small loops and in the case the coil
is fed with a current source, it is tempting to think that a high Nt will result in a
high magnetic field H at the tag location. The considerations below, however, argue
against a value of Nt > 1.
8.3.1.3.1 Phase degradation The more turns are used, the longer the total wire
length of the loop and hence the more pronounced the effect described in Sect. 8.3.1.2.2
becomes. For a small loop, fed by a current source, the magnetic field can be boosted
by taking more turns. But as soon as the total wire length of the loop can not be
regarded small compared to the wavelength, it is advantageous to take Nt smaller.
Fig. 8.17 depicts the magnetic field as a function of rd in the case of Nt = 1, 2, 3, 5, 10
when the optimal rl as calculated in Sect. 8.3.1.2 is used. The conclusion that using
more turns is only advantageous for smaller rd , is clear.
~ z | [dB]
|H
Nt
Nt
Nt
Nt
Nt
10
0
=1
=2
=3
=5
= 10
−10
−20
−30
0
1
2
3
4
rd [m]
~ z | at rd when optimal rl for that value of rd is used. Il = 1 A and fc =
Figure 8.17: |H
13.56 MHz.
8.3. RFID Loop
161
8.3.1.3.2 Loop impedance For a loop, fed by a voltage source, discarding
the small ohmic and radiation resistance with respect to ωL, following Schrank and
Mahony [159]:
|Nt Il | = Nt
|Vl |
|Vl |
|Vl |
= Nt
∝
,
ωL
ωNt2 L1,Nt =1
Nt L1,Nt =1
(8.29)
with L1,Nt =1 the inductance of a loop of the same size with only one turn. Thus
~ z | in Eq. (8.25), means Nt = 1 and L1,N =1
maximizing Nt Il , in order to maximize |H
t
small.
8.3.1.3.3 Bandwidth When the loop antenna is placed in an RLC chain and
used for a communication link, raising Nt is not without a price. As L is proportional
to the quality factor, defined later on in Eq. (8.34), it can only be increased up to a
certain level as otherwise the bandwidth of the system would become too small. This
is not in contradiction with the results of Yates et al. [160], namely that the power
transfer ratio is proportional to Nt2 , but they do not consider bandwidth issues.
8.3.1.3.4 Self-resonance frequency When the loop perimeter is about half a
wavelength, the loop will resonate. But if the loop has multiple turns, resonance at
lower frequency occurs due to the parallel resonance of the inductance of the loop and
the capacitance between its turns. Above its resonance frequency, the loop starts to
behave as a capacitance. Therefore it is important to know the resonance frequency
fres of the loop.
The capacitance between two turns Ctt of bare wire can be calculated with Eq. (8.12).
If the wire has an insulation with a relative permittivity r different from 1, in
Grandi [161] the formula becomes:
Ctt = 2πrl ×
r 0 π
,
log ( 2rdw )
(8.30)
where the insulation of both wires is supposed to touch, resulting in d = 2rw + 2t
with t the insulation thickness. This formula only holds for a radial electric field in
the insulation, which is surely not the case for r → 1. Hence substituting r = 1 into
Eq. (8.30) does not result in Eq. (8.12) due to approximations used in the model that
led to Eq. (8.30).
If an inductor with multiple turns is used, the equivalent capacitance is found as the
series circuit of all turn-to-turn capacitances. This is a simplification and it assumes
that the capacitance between non-adjacent turns can be neglected. In Fig. 8.18 the
equivalent circuit of an inductor with all capacitances is drawn. The capacitances
that are neglected are drawn with dashed lines.
162
Chapter 8. Array Elements
LN
CN
...
...
L3
C3
L2
C2
L1
C1
...
Figure 8.18: Equivalent circuit of an inductor.
Note that discarding the capacitance between non-adjacent√turns implies that the
self-resonance frequency only shifts downwards by a factor Nt when adding more
turns. Indeed, the resonance frequency is found as:
fres
1
1
√
p
=
=
=
2π L1 Ctt
2π Nt L1,Nt =1 Ctt
r
2
fres,Nt =2 ,
Nt
(8.31)
where L1,Nt and Ctt indicate the inductance (in the presence of the other turns), resp.
capacitance of a single turn. L1,Nt = Nt × L1,Nt =1 still depends on the number of
turns, L1,Nt =1 is the inductance of a single turn in the absence of all other turns.
Table 8.4 shows the result obtained from Eq. (8.31).
Table 8.4: Self-resonance frequency of loops with rw = 1 mm with and without insulation of
0.2 mm with r = 4 for varying loop radius. The last column gives the resonance
frequency in case of a single turn.
rl [m]
0.01
0.1
1
Ctt [pF]
2.016
20.16
201.6
Lt [µH]
0.07729
1.352
19.3
fres,Nt =2 (0 )
403.2 MHz
30.49 MHz
2.551 MHz
fres,Nt =2 (r = 4)
125.6 MHz
9.498 MHz
0.7947 MHz
fres,Nt =1 (0 )
4775 MHz
477.5 MHz
47.75 MHz
8.3. RFID Loop
163
Taking the inter-turn capacitance into account, the impedance of the coil equals:
Zcoil =
jωNt2 L1,Nt =1
.
1 − ω 2 Nt L1,Nt =1 Ctt
(8.32)
which is again frequency dependent, possibly causing resonances at multiple frequencies, similar to those mentioned for Nt = 1 in Sect. 8.3.1.2.3.
8.3.1.3.5 Resistance of Parallel Wires Another disadvantage of multiturn
loops, is their increased ohmic resistance. Due to the capacitive inter-turn coupling,
the current in the loop wire is even more confined than should be expected due to the
skin effect alone. Smith [162] provides formulas to calculate this effect in the case of
a loop that is small compared to the wavelength.
The argumentation above leads to the conclusion that Nt should be taken small unless
only power transfer is considered or the loop antenna is fed by a current source and
rd is rather small.
8.3.1.4
Wire Diameter
Sect. 8.3.1.3 revealed that when a voltage source is used, or the loop is placed in an
RLC chain and used for data communication, its inductance L should be small. Using
a wire with large diameter reduces L.
8.3.2
Power source and current enhancement
As is mentioned before and expressed in Eq. (8.27), a minimum amount of current is
needed to activate the RFID tag. This current can be directly drawn from an external
power source, but sometimes it can be more convenient to enhance the current if for
example a powerful power source is not available. This can be done either passively
by means of an RLC circuit or actively with the aid of an amplifier. The choice of
power source and enhancement technique affects the design of the loop so these need
to be taken into consideration. Some passive enhancement techniques are treated
below. For active enhancement, with the aid of amplifiers, the reader is referred to
Sect. 9.3.3.
When adding a capacitor and a resistor to the loop, in order to obtain an RLC circuit,
many combinations with a capacitor, inductor and resistor can be made. Only the
ones with L, the inductance of the loop antenna, and R in series are considered for
this application, because the loop resistance is inherently in series with the inductance
of the antenna [163]. If an external resistor has to be added, it is preferably added in
series with the loop, for the same reason. The internal resistance of the capacitors is
smaller and will be neglected. An overview of the possible circuits is given in Fig. 8.19.
164
Chapter 8. Array Elements
In this figure the top left configuration is the best choice if a source can deliver an
unlimited amount of current, the top right setup is optimal when a source can provide
the circuit with high voltage. On the other hand, the bottom configurations ease the
need for high voltage or current. This is explained in following sections.
Cs
R
R
Cp
L
L
Series
Parallel
Cs
Cp
Cs
R
Cp
L
Series-Parallel
R
L
Parallel-Series
Figure 8.19: Schematic of (top left) Series, (bottom left) Combined Series-Parallel, (top
right) Parallel and (bottom right) Combined Parallel-Series RLC resonance
circuit, four possibilities of passive current enhancement.
8.3.2.1
Series RLC chain
In this case, the current in the loop will be maximal when Cs = 1/(ω 2 L), the
impedance Zseries = R is minimal. The voltage over the loop is:
Vl = j
ωL
ωL
Vsource = jQRLC Vsource , with QRLC =
,
R
R
(8.33)
where QRLC is the quality factor of the circuit. The higher QRLC , the larger the
current in, and the voltage over, the loop, but the lower the bandwidth B of the
circuit. This follows from the definitions [30]:
QRLC =
stored energy
f0
and QRLC = ,
dissipated energy per cycle
B
(8.34)
with f0 the resonance frequency. As a minimum bandwidth is needed for data transfer,
the value for QRLC is upper bounded by the data rate.
8.3. RFID Loop
8.3.2.2
165
Parallel RLC chain
Here, the current in the loop only depends on the voltage applied to the RLC
chain, but the current drawn from the source will be minimum if the condition
Cp = L/(ω 2 L2 + R2 ) or Zparallel = R + ω 2 L2 /R is met. In this case the impedance of
the chain is maximal as is the current amplification.
8.3.2.3
Combined series/parallel RLC chain
The loop current is maximized when
ω 2 Cs =
1 − 2ω 2 LCp + ω 2 Cp2 (ω 2 L2 + R2 )
.
L − Cp (ω 2 L2 + R2 )
(8.35)
Due to the second degree of freedom, Cp , any value for the impedance can be obtained:
ZcombinedSP =
1−
2ω 2 LCp
R
.
+ ω 2 Cp2 (ω 2 L2 + R2 )
(8.36)
Hence this is the best choice for the resonance circuit, as it allows to match the
internal resistance of any source, to ensure maximum power transfer to the load. If
the loop antenna is located at a distance from the reader, a transmission line has to
be used to connect both and the use of the combined chain is obligatory: of the four
circuits, only this one can match the characteristic impedance of any line.
8.3.2.4
Combined parallel/series RLC chain
The loop current is maximized when Cs = 1/(ω 2 L). This is identical to the resonance
condition of the series resonance chain. The second degree of freedom, Cp , can again
be used to choose the input impedance of the chain:
ZcombinedPS =
R − jωR2 Cp
,
1 + ω 2 R2 Cp2
(8.37)
but to a lesser extent than was the case in the series/parallel chain as ZcombinedPS will
always be smaller than R.
In order to use the RLC equations mentioned above, the inductance L of the antenna
should be known. Appendix H learns that this is not straightforward. Even more,
the value for L also depends on the surroundings, especially for larger loops. The
presence of metal is one of the reasons for this alteration. Consequently, the value for
the capacitance C needed after installation can differ slightly from the value calculated
with the design equations given above.
166
Chapter 8. Array Elements
This problem is easily solved by tuning12 the resonance circuit, using trimming capacitors. Automatic tuning compensates on the fly, but, at the cost of increased
complexity. One example [164] uses a control circuit to set the DC bias in a ferrite
core to change the inductance of a coil.
8.3.2.5
Impact of R on Cs and Cp
The last parameter to determine is the resistance R. The total resistance R of the
chain will be the internal resistance of the loop Rl and an external resistor that is deliberately added. The equations above show that the value of R can also influence the
resonance frequency of the chain. This deviation can also be corrected by tuning the
capacitors, so that the value for R can be determined only based on the requirements
for the bandwidth, mathematically condensed in QRLC .
8.3.3
Validation
Three antennas were made to validate the formulas and statements: two solid copper
wire loops, one with Nt = 1, another one with Nt = 2 and one copper tube loop, see
Fig. 8.20. Their geometrical parameters are summarized in Table 8.5.
The MIFARE Pegoda MF RD 700 is used as a reader. Its RF output is a voltage source
with internal resistance. For such a source, as explained in Sect. 8.3.2, the best choice
is the series-parallel circuit. Tuning of this circuit is necessary because the frequency
response of the circuit is very sensitive to L, Cp and Cs . The capacitors bought had
a tolerance of 10 %. Moreover, neither the measured Lmeas nor the calculated Lcalc
are exact due to the problems mentioned in Appendix H. The calculated inductances
Lcalc were verified against values derived from measurements. The values obtained
can be found in Table 8.5. Due to balun problems the measurements were totally
unreliable for the copper tube loop. Instead, the inductance of the loop was measured
in an indirect way: the loop was taken out of the circuit and replaced by a lumped
element inductance with a certain value L. This was repeated for lumped elements
with different values of L, until the resonance frequency matched the original one.
The resonance circuit is a variation on the series-parallel circuit: it consists of an
upright and a mirrored version of it, see Fig. 8.21, to feed the antenna in a balanced
way. If this is not done, problems as those shown in Fig. 8.22 can arise because of
a transition from a balanced loop to an unbalanced vectorial network analyzer. The
simulated curve is obtained from a NEC simulation with 400 divisions along the circle
perimeter.
12 Try to find the values for the capacitors that result in the largest current in the loop. Monitoring
of the current in the loop can be done by 1) using a current probe, but this adds another inductance,
2) measuring the voltage over the loop, but voltage probes always form a small loop and pick up
fields or 3) using a field probe. The last method is preferred.
8.3. RFID Loop
167
A
rl
B
d
(a) Layout. A = top view, B = side
view. Nt = 1. The same schematic holds
for Nt > 1, only more turns are stacked.
(b) Photograph. Largest = copper
tube loop, Smallest = two turn solid
copper wire loop
Figure 8.20: Layout drawing and pictures of the RFID reader loops designed and made to
validate the formulas and statements. The values for rl and d of the different
antennas can be found in Table 8.5.
Cs,1
Cp,1
R
Cp,2
Cs,2
L
Figure 8.21: Resonance circuit used to determine the reading range.
Furthermore, Table 8.5 also lists some electrical parameters: the magnetic field H at
the origin of the loop and the maximum reading distance with a MIFARE card. For
the latter measurement no other alterations were done to the setup but tuning the
resonance circuit. For the first measurement an EM CO − 902, 3 cm magnetic field
probe is utilized [143]. Although Table 8.5 indicates that the loop with two turns
generates a higher magnetic field compared to the loop with one turn, this does not
result in a larger reading distance because of a QRLC which is too high, so that even
at a very small distance no communication can take place in this setup. Adding an
external resistor can solve this, but this is beyond the scope of validating the formulas
and statements. When holding a card very close to the copper tube, this card could
be read by the reader, but the reader could not supply enough current to obtain a
functioning system with a card at the origin or further along the z-axis.
168
Chapter 8. Array Elements
Another difference between the two solid wire loops is the remarkably lower fres of
the two turn loop. This confirms the effect of Ctt , calculated with Eq. (8.30) for
r = 4 [165], on fres .
simulation
measurement
Eq. (8.35)
L[µH]
10
5
f [MHz]
0
10 20 30 40 50 60 70 80 90 100
−5
Figure 8.22: Frequency dependent value of L for the copper tube loop as obtained from simulation and measurement. A balanced to unbalanced system transition causes
this value to fluctuate heavily around 13 MHz so that the loop is useless unless
fed in a balanced way.
Table 8.5: Characteristics overview of the RFID reader loops: A-B) Solid copper wire loops,
C) Copper tube loop, (*) reading distance not measurable because of QRLC too
high, (**) reading distance not measurable because the reader could not supply
enough current, this requires an amplifier.
Nt
A
B
C
8.4
1
2
1
rl
[cm]
8.25
8.25
56.5
d
[cm]
0.1
0.1
1.5
Rl
[Ω]
0.5
2.09
200
coil
Lcalc Lmeas
[µH]
[µH]
0.537 0.569
2.15
2.27
3.1
3.45
fres,calc
[MHz]
579
48
85
fres,meas
[MHz]
583
41
27
RLC circuit
H(000)
rd
[A/m] [cm]
4.16
10
5.35
(*)
0.12
(**)
Maximal Resolution Sensor
As the sensor should have the largest possible area, resulting in a higher amplitude,
for the shortest possible perimeter, resulting in a larger working frequency band, the
loop shape should be circular. Nt and rl are determined by trade-off as explained
in Sect. 8.4.1. There it will also become clear that the wire diameter, which should
be as large as possible for low resistance, is determined by material selection and
fabrication technique as soon as rl is determined. Sect. 8.4.1 makes abstraction of
the fact that a considerable amount of turns can not be wound circularly without
stacking turns, distancing from the source. Sect. 8.4.3 points out that in such case it
is more appropriate to fabricate spirals.
8.4. Maximal Resolution Sensor
8.4.1
169
Geometrical Design
Sect. 8.2 zoomed in on shielded loops. Those loops were quite large in comparison
with a micro-controller, ASIC or even FPGA and were consequently not suited for
localized measurements. Smaller implementations of the same concept do manage to
achieve smaller resolutions. Masuda et al. [132] reports an aperture of 20 µm × 1 mm.
One of the commercially available high resolution sensors is the NEC CP-2S with a
resolution of ≈ 250µm. This is a shielded loop printed on glass ceramic multilayer with
thin film technology, after [166]. Even shielded loops on chips have been fabricated,
as in [167].
Non-shielded loop sensors can inherently be made smaller than the shielded loops.
The commercially available types, such as the ICR near field microprobes, sold by
Langer EMV-Technik, are sold under specification of 80 µm resolution.
Sect. 8.4.1.1 discusses the maximal spatial resolution of non-shielded loop sensors.
Examples in Sect. 8.4.1.2 give numerical results. A straightforward design methodology can not be derived due to the complexity of the equations. Numerical search
methods are to be applied. The practical limits on the resolution are however at
hand. For additional calculations, Sect. 8.4.1.3, which was left out in [168] due to the
page limit, contains details on the numerical implementation of the displayed theory.
8.4.1.1
Theory
In this section the minimum achievable dimension of a circular inductive sensor for
usage in a frequency interval [fL , fH ] is evaluated and values for the number of turns
Nt and the loop radius rl are derived starting from the value of the magnetic field
strength B and a minimum amplitude Vmin that should be generated over a load Z,
the input impedance of the measurement device, in parallel with the loop. This value
Vmin is determined by the measurement equipment and relates to the minimum voltage
that can be measured by an oscilloscope or the minimum signal amplitude that has
to be fed to an amplifier connected to the loop to obtain a reasonable signal-to-noise
ratio or the like.
8.4.1.1.1
quency:
General Case: Arbitrary Z Rewriting Eq. (8.7) for a single fre
dφB = ωNt AB.
|V | = (8.38)
dt The rightmost equality sign implies that the loop is positioned orthogonally to the
magnetic field.
170
Chapter 8. Array Elements
If a load Z is attached to the terminals of the loop sensor, current will flow, resulting
in a voltage over Z equal to:
Z
,
|V | = ωNt AB (8.39)
jωL + R + Z with the loop inductance L [169]:
L=
Nt2 µ0 rl
ln
8rl
rw
−2
(8.40)
and the loop resistance R [163]:
R=
2πrl Nt
− (rw − δ)2 )
(8.41)
2
σπ (rw
with δ the skin depth, defined in Eq. (5.5) and σ the electrical conductivity of the
metal, see Table 5.3. If δ > rw , δ should be replaced by rw in the formula, for then
not the skin depth, but the wire diameter is the limiting factor.
The contour lines of Eq. (8.39) for rl /rw = 16, f = 250 kHz and Z = 50 Ω are drawn
in Fig. 8.23. Designing a loop sensor with maximal resolution for signal amplitude
Vmin boils down to finding the pair (Nt , rl ) on the |V | = Vmin contour where rl is
minimum. These loci are connected by the solid black line on Fig. 8.23.
1
0.9
0.8
0.7
rl [mm]
0.6
0.5
0.4
0.3
0.2
0.1
100
200
300
400
500
600
700
800
900
1000
Nt
Figure 8.23: The contour lines of Eq. (8.39) as a function of rl and Nt . The solid black
line connects the minima for rl on the different contour lines. rl /rw = 16,
f = 250 kHz and Z = 50 Ω.
8.4. Maximal Resolution Sensor
171
The optimum value for Nt is a trade off between increasing Nt to increase the induced
voltage of Eq. (8.38), and decreasing Nt to lower L ∝ Nt2 and R ∝ Nt , avoiding that
|jωL + R| |Z| in the denominator of Eq. (8.39).
Still, not all (Nt , rl ) pairs found as the minimum on the appropriate contour are
valid. Eq. (8.39) implies that the total wire length of the loop is small to avoid signal
cancellation due to phase differences over the loop:
Nt 2πrl <
λH
c
λH
or Nt rl <
=
10
20π
10ωH
(8.42)
with c the speed of light. Due to the inverse proportionality with ω, this inequality
condition only has to be validated for fH , the upper bound of the intended frequency
band.
Moreover, for similar reasons, explained in more detail in Sect. 8.4.2, the resonance
frequency of the system fres , consisting of loop sensor and measuring device, should
be higher than ten times the highest frequency in the working frequency band:
fres =
1
=
2π Ltot Ctot
√
2π
r
1
Nt2 L1,Nt =1
Ctt
Nt
+ CZ
> 10fH
(8.43)
with CZ the capacitive part of Z. The capacitance between two turns Ctt can be
calculated with Eq. (8.12). The inductance of one turn (in the absence of the other
turns) is given in Eq. (H.2). Filling in Eq. (8.12) and (H.2) into Eq. (8.43) results in:
v u
u ln 8rl − 2 2π 2 rl + Nt CZ ln (α)
p
t
rw
0
λ
> 2π Nt rl
(8.44)
10
ln (α)
s
2
d
d
with α =
+
−1
2rw
2rw
In conclusion, in the general case, for arbitrary values of Z, the minima for rl on
the contour lines of Eq. (8.39) must be sought for: e.g. by a minimum search, in the
(Nt , rl ) domain bound by the conditions Eq. (8.42) and (8.44), for all frequencies in
the [fL , fH ] interval. Sect. 8.4.1.3 provides additional details.
8.4.1.1.2 Ideal Case: Z = ∞ In the ideal13 case that no load is attached to the
loop sensor, the voltage between its terminals simply equals the voltage induced by
the varying magnetic field, see Eq. (8.38).
13 This case is ideal in the sense that the voltage measured over the loop terminals is maximal.
Any load between the terminals would cause a current through the loop, resulting in a smaller loop
voltage.
172
Chapter 8. Array Elements
Combining Eq. (8.42) with (8.38), only to be checked for fL , the lower bound of
the intended frequency band, due to the proportionality of |V | ∝ ω, results in the
maximum amplitude that can be obtained with the best sensor, still obeying the
condition imposed on the total wire length:
|V | ≤
πc2 ωL B
2 N = Vmax .
100ωH
t
(8.45)
This leads to the obvious conclusion that, in case Z = ∞, for maximum amplitude,
Nt = 1. rl , related to the choice for Nt via Eq. (8.42), will then be as large as possible
and A maximal.
If the voltage that is needed, Vmin < Vmax , then Nt ≥ 1. In this case a trade-off
between good resolution, meaning small A, and large frequency band, meaning small
Nt , can be made, still resulting in the same value for Nt A. As soon as Nt > 1,
however, Eq. (8.44) again bounds the solution space. Eq. (8.44) can be rewritten, in
case Z = ∞ and CZ = 0, as:
r
Nt
c
Nt rl <
(8.46)
10ωH Nswitch
with Nswitch the value where both Eq. (8.42) and (8.44) are equivalent:
l
−2
ln 8r
rw
2
!
#.
Nswitch = 2π
r
ln
d
2rw
+
d
2rw
2
(8.47)
−1
If Nt > Nswitch , only Eq. (8.42) should be checked, and the (integer) number of turns
for the loop with minimal dimension or maximal resolution is found with Eq. (8.45)
as:
πωL c2 B
Nmax =
,
(8.48)
2 V
100ωH
min
else, only Eq. (8.46) should be checked. Eq. (8.46) combined with Eq. (8.38):
r
r
c
Nt Vmin
Nt
<
(8.49)
Nt rl =
πωL B
10ωH Nswitch
reveals that this condition is independent of the value for Nt . Stated otherwise,
Eq. (8.46) for any value of Nt is equivalent with Eq. (8.42) for Nt = Nswitch .
Consequently, the design of an ideal inductive loop sensor with optimal resolution
consists of: calculating Nmax with Eq. (8.48) and Nswitch with Eq. (8.47). If Nmax ≥
Nswitch , then Nt = Nmax , else Nt = 1. Once Nt is determined, rl follows from
Eq. (8.38), again only to be evaluated for the lower working frequency, due to |V | ∝ ω:
r
Vmin
.
(8.50)
rl =
ωL BπNt
8.4. Maximal Resolution Sensor
8.4.1.2
173
Results - Maximal Resolution
To give some realistic numerical values, this section evaluates the formulas in Sect. 8.4.1.1
for a circular inductive sensor to measure a magnetic field of B = 2µT that should
deliver at least Vmin = 1 mV.
The values for rw and d are set to:
rw
d
= rl /16
(8.51)
=
(8.52)
2.4rw
corresponding to the rules of thumb of bending radii of wires in [170] and breakdown
voltage between conductors.
The rl calculated below are of the order of magnitude of 10 µm. Loops of such
small diameter, with conductors of even smaller dimensions can be produced, as is
illustrated in e.g. Seidermann and Büttgenbach [171].
8.4.1.2.1 Ideal Case: Z = ∞ Evaluating Eq. (8.47) with the values in Eq. (8.51)
and (8.52) results in Nswitch = 91. Calculating Nmax with Eq. (8.48) and evaluating
Eq. (8.50) with the appropriate Nt as explained in Sect. 8.4.1.1.2, for zero bandwidth,
meaning fL = fH , results in the radii depicted by the solid line in Fig. 8.25. This is
the practical resolution limit for Vmin = 1 mV. The sudden discontinuity in the curve
as f = 10 GHz is due to the jump from Nt = 91 → 1, as indicated on Fig. 8.24. Also
note that the curve stops at f = 900 GHz, as above this frequency, no sensor can be
designed to deliver V ≥ Vmin due to Eq. (8.45).
100k
Nt
10k
Z=∞
Z = 1 MΩ
1k
100
10
1
1MHz
10MHz
100MHz
1GHz
10GHz
100GHz
fL = fH
Figure 8.24: Nt as function of fL = fH for Z = ∞ and Z = 1 MΩ.
174
Chapter 8. Array Elements
6
x 10
5
Z=∞
Z = 1 MΩ
rl [µm]
4
3
2
1
0
1MHz
10MHz
100MHz
1GHz
10GHz
100GHz
fL = fH
Figure 8.25: Minimum rl as a function of fL = fH for Z = ∞ and Z = 1 MΩ.
Fig. 8.26 depicts the loop radius in case fL is fixed and fH is varied from 1MHz →
10 GHz. This figure nicely illustrates the trade-off between resolution and working
frequency band. At a certain value for fH , no sensor can be designed to deliver
V ≥ Vmin due to Eq. (8.45) and the curve goes to zero. The curve has no meaning for
values of fH < fL and is hence set to zero. The flat part in the curves corresponds
with Nt = 1. For zero bandwidth, Fig. 8.25, the radius decreased again with increasing
frequency after the steep rise, due to the proportionality of V with ω in Eq. (8.38).
For a non-zero bandwidth, fL limits the resolution, resulting in the flat part of the
curve.
Effects of the parameters on the optimal resolution Eq. (8.47) reveals that
Nswitch depends slightly on rw /rl and heavily on d/2rw (especially for d/2rw ≈ 1,
which is often the case when winding a conductor). Indeed, for d ≈ rw the capacitance
Ctt is much larger, hence Eq. (8.42) will only overrule Eq. (8.44) for a higher Nt .
Fig. 8.27 plots this dependency in the interval of interest for rw /rl and d/2rw If Nswitch
drops, a higher frequency upper bound can be achieved with the sensor, although this
implies that if the same resolution has to be kept, the lower frequency bound has to
go up. Fig. 8.28 shows the effect of varying ratio d/2rw on the resolution.
8.4.1.2.2 General Case: Arbitrary Z As soon as a load is attached to the
sensor, the resolution is equal to or worse than in the ideal case of no load over the
loop. This is due to the division in Eq. (8.39).
8.4. Maximal Resolution Sensor
6
175
x 10 −4
fL = 100 MHz
fL = 1 GHz
5
rl [m]
4
3
2
1
0
100MHz
1GHz
10GHz
fH
Figure 8.26: Minimum rl for two loop sensors with varying working frequency band.
400
350
Nswitch
300
250
200
150
100
50
0
1
1.2
1.4
1.6
1.8
d/2rw
2
100
80
60
40
20
0
rl /rw
Figure 8.27: Variation of Nswitch as function of rl /rw and d/2rw .
For Z = 1 MΩ, the difference in resolution is negligibly small, except for smaller
frequencies. The dashed line in Fig. 8.25 indeed deviates from the solid line below
300 MHz. This is due to the difference in Nt . In case of no load Z = ∞, Nt should
be taken as high as possible with Eq. (8.48). In case of a finite load, however, an
excessive14 value for Nt results in a smaller loop voltage as |jωL + R| |Z| in
Eq. (8.39).
14 From a practical point of view, N = 104 can be regarded as excessive too. This treatment is
t
however purely mathematical as a starting point.
176
Chapter 8. Array Elements
x 10 −5
6
d/2rw = 1.2
d/2rw = 1.5
d/2rw = 2
5
rl [m]
4
3
2
1
0
1MHz
10MHz
100MHz
1GHz
10GHz
100GHz
fL = fH
Figure 8.28: rl as function of fL = fH for unloaded loops with different d/2rw .
The cases of Sect. 8.4.1.2.1 are reviewed here, for Z = 50 Ω and Z = 1 MΩ k 13 pF,
which are typical oscilloscope input impedances. For the high impedance and zero
bandwidth case, in Fig. 8.29, the curve for rl shows several spikes. Those abrupt
changes in resolution are due to a a decrease by one of Nt (which has to be an
integer), similar to the spike in the ideal case for the transition of Nt : Nswitch → 1.
The results in the non-zero bandwidth case are depicted in Fig. 8.30.
10
x 0.1
9
8
Z = 1 MΩ k 13 pF
Z = 50
7
rl [mm]
6
5
4
3
2
1
0
1MHz
10MHz
100MHz
1GHz
10GHz
100GHz
fL = fH
Figure 8.29: Minimum rl as function of fL = fH for Z = 1 MΩ k 13 pF and Z = 50.
8.4. Maximal Resolution Sensor
177
16
fL
fL
fL
fL
14
12
= 0.1 MHz, Z = 1 MΩ k 13 pF
= 1 MHz, Z = 1 MΩ k 13 pF
= 0.1 MHz, Z = 50 Ω
= 1 MHz, Z = 50 Ω
rl [mm]
10
8
6
4
2
0
1MHz
10MHz
100MHz
1GHz
10GHz
100GHz
fH
Figure 8.30: Minimum rl for Z = 1 MΩ k 13 pF with varying working frequency band.
As the number of turns in e.g. Fig. 8.24 is impractically high for some values of
fL = fH , the effect of limiting Nt ≤ 30 is illustrated for the case of Z = 50 Ω in
Fig. 8.31. This deteriorates the resolution for lower values of fL .
10
x 0.1
9
8
Nt ≤ 30
Nt is unbound
7
rl [mm]
6
5
4
3
2
1
0
1MHz
10MHz
100MHz
1GHz
10GHz
100GHz
fL = fH
Figure 8.31: Minimum rl as function of fL = fH for Z = 50 Ω with and without restriction
on Nt .
178
Chapter 8. Array Elements
Effects of the parameters on the optimal resolution d/2rw now no longer has
any effect. The curves for d/rw = 1.5 and 2 coincide with the curve for d/rw = 1.2
(and Z = 1 MΩ k 13 pF) on Fig. 8.29. Fig. 8.32 shows the effect of varying ratio
rw /rl on the resolution in case of Z = 1 MΩ k 13 pF.
8
x 10 −4
7
6
rl [m]
5
4
3
rl /rw = 16
rl /rw = 20
2
1
0
1MHz
10MHz
100MHz
1GHz
fL = fH
Figure 8.32: rl as function of fL = fH for Z = 1 MΩ k 13 pF and several rw /rl .
8.4.1.3
Numerical Implementation of the Optimum Search Routine
The value of minimum rl was found by first determining the corresponding minimum
value of rl for all values of Nt in the way described below. Afterwards the minimum
rl out of this set for all values of Nt was selected.
To find the minimum value of rl for a fixed Nt , first the region in the (rl , ω) domain
is determined were all conditions are satisfied. Fig. 8.33 depicts an imaginary case to
illustrate the procedure. The minimum and maximum frequency, resp. ωL and ωH
bound the possible solution area with two horizontal lines, the resonance condition
Eq. (8.42) bounds the r-dimension between 0 and rres .
If Nt > 1 the resonance frequency changes and also Eq. (8.44) should be satisfied.
This condition, which is most restrictive for ω = ωH , bounds the r-dimension further with two vertical lines r = ri1 and r = ri2 . If rres > min (ri1 , ri2 ), the solution area is bounded by the rectangle formed by the lines ω = ωL , ω = ωH , r =
max (min (ri1 , ri2 ), 0) and r = min (max (ri1 , ri2 ), rres ), in case rres < min (ri1 , ri2 )
there is no solution.
In this solution area, the minimum rl needs to be found such that ∀ω ∈ [ωL , ωH ] the
voltage from Eq. (8.39) is at least Vmin .
8.4. Maximal Resolution Sensor
179
ω
ωH
W1
W2
∆
K
ωL
ri3
rresri2
r
Figure 8.33: Graphical representation of the conditions in the ω,rl plane for the minimum
rl search for a fixed Nt
As the remark on δ > rw below Eq. (8.41) indicates, the formula to calculate this
voltage differs left and right of the curve δ = rw . With Eq. (8.51) this curve ∆
can be determined. In Fig. 8.33, the contour lines for V = Vmin are drawn for the
case where δ ≥ rw and δ < rw . Left from the line ∆ the area between W1 and
W2 is the valid area. Right from the ∆-line the grey colored area bounded by K
is part of the possible solution area. The shape of the contour lines is constructed
to explain the next steps in the search algorithm. In most cases the contour lines
however have a much less complex shape. Consequently not all cases implemented in
the algorithm will occur in practice, but being complete in coding avoids undefined
states and premature program termination.
Inside this area the minimum rl can only be either 1) an intersection of the W or K
curve with the lines w = wL and w = wH or 2) points of inflection of the W or K
curve. For each of this points it should be checked whether the vertical line through
the point lies completely in the feasible interval. Of all the points to which this final
condition applies, the minimum one should chosen. This is the minimum rl for this
specific Nt .
To allow manipulation of the contour lines in MAGMA [172]. First, the function:
2
Z
2
2
− Vmin
=0
(8.53)
|V |2 − Vmin
= ωNt πrl2 B
jωL + R + Z was rewritten in polynomial form,
2
v 4 Nt rl2 ZR δ(2rl v − aδ)
X 2 + v4 Y 2 −
Vmin
πB
2
X 2 − v4 Y 2
with:
X
Y
α
2
=0
= ZR δ(2rl v − aδ) + 2v 2 arl Nt /σ − v 4 ZR CZ Nt2 µrl αδ(2rl v − aδ)
= Nt2 µrl αδ(2rl v − aδ) + v 2 ZR CZ 2arl Nt /σ
r
√
rl
8
2
−2 ; a=
; v = ω and δ =
= log
a
rw
σµ
(8.54)
180
8.4.2
Chapter 8. Array Elements
Enhancement Design
In Sect. 8.4.1, the conditions imposed by Eq. (8.42) and (8.43) made sure that the
system is used way below resonance. The reason for this is twofold. Firstly, in case
of resonance, either by inter-turn capacitance or standing waves on a multiple of half
a wavelength, the impedance of the loop does not equal jωL, voiding Eq. (8.39). E.g.
in case of resonance due to inter-turn capacitance, Eq. (8.39) can be replaced by:
Z
.
(8.55)
|V | = ωNt AB Z − ω 2 LNt Ctt Z + jω(L + RNt Ctt Z) Fig. 8.34 illustrates the difference between Eq. (8.39) and (8.55) for several real values
of Z. For f < fres /10, the deviation is less than 1.01%. If Z has a capacitive part,
it can be rewritten as Z = RZ k CZ . CZ can be summed with Nt Ctt , as both are in
parallel, as can be seen on Fig. 8.35, which depicts the equivalent circuit of the loop
sensor connected to a load, and is identical to the model used in [149].
From Fig. 8.34(a) it can be concluded that allowing resonance could lead to even
higher signal amplitudes, for high values of RZ , that is, but again traded off against
bandwidth. Hence, e.g. in case of a sensor for cryptographic analysis on a signal
modulated by current performing the logical operation, it might be advantageous to
design a sensor and deliberately add capacitance parallel to the loop in order to tune
the resonance frequency of the system to the frequency of the signal analyzed, and
hereby boost the signal, as in [149] for field magnitude measurement sensors or as in
Sect. 8.3.2 for transmitting antennas. The proper bandwidth can then be selected by
adding resistance in parallel to broaden bandwidth at the cost of signal amplitude. A
mathematical means that gives the relation between the resistance and bandwidth in
resonating systems, is the quality factor Q that was defined in Sect. 8.3.2.1. Besides
requiring the amplitude to be more or less constant in the frequency band of interest,
also the phase should vary linear with frequency around the center frequency, to
preserve the signal in the time domain. This can indeed also be regulated by selecting
the appropriate value for RZ , as can be seen on Fig. 8.34(b).
Fig. 8.34(b) also illustrates the second reason why in Sect. 8.4.1 resonance was ruled
out by imposing strict conditions on the sensor geometry. When using a sensor
starting from (nearly) BB, in order to preserve the signal in the time domain, the
phase should vary linearly with frequency. Around resonance, however, the phase
varies linearly with a much larger derivative. Consequently the phase curve has a
bend somewhere between BB and resonance. Starting from BB, the sensor can only
be used up to this bend.
8.4.3
Practically Implementing a Large Nt
From the order of magnitude of the results in Sect. 8.4.1.2, it is clear that the loops
should preferably be implemented on-chip.
8.4. Maximal Resolution Sensor
181
100
10
|V |
1
0.1
0.01
Eq. (8.55) RZ = 0.1
Eq. (8.55) RZ = 1
Eq. (8.55) RZ = 100
0.001
0.01
0.1
Eq. (8.39) RZ = 0.1
Eq. (8.39) RZ = 1
Eq. (8.39) RZ = 100
1
ω/ωres
10
100
10
100
(a) Amplitude
0
-0.5
-1
∠V
-1.5
-2
-2.5
-3
0.01
Eq.
Eq.
Eq.
Eq.
Eq.
Eq.
(8.39)
(8.39)
(8.39)
(8.55)
(8.55)
(8.55)
0.1
RZ
RZ
RZ
RZ
RZ
RZ
= 0.1
=1
= 100
= 0.1
=1
= 100
1
ω/ωres
(b) Phase
Figure 8.34: The difference between Eq. (8.39) and (8.55) for several real values of Z.
Designing loops on a lossy and heterogeneous material as silicon is a topic on its own,
way beyond the scope of this work. Moreover, as the number of layers is not infinite,
it can be expected that the number of turns will be very limited, when implementing
a loop on-chip. The best solution to this problem, is to use a spiral instead of solenoid
loop.
182
Chapter 8. Array Elements
R
Nt Ctt + CZ
L
RZ
V
Figure 8.35: Equivalent circuit of the loop sensor connected to a load.
This way, L of the loop is smaller, resulting in a smaller denominator in Eq. (8.39).
Moreover, stacking turns enlarges the distance from the source, resulting in a decrease
of the magnetic field proportional to r3 , from Eq. (8.1). In a spiral the area enclosed
by the turns only decreases proportional to r2 with each turn added.
8.4.4
Twins
In order to improve resolution, one immediately thinks of tiny loops. In large loops,
however, it is possible too to extract information on the location, but only if the
signal amplitude is known (or two measurements are recorded with different sensor
locations), and only one signal source is present. Hence this is not useful in case of
side channel analysis on chips. It is however relevant to choose the sensor location of
a large (e.g. shielded) loop with respect to the hardware, and it points out that there
is an ambiguity or crosstalk between two dipole sources at opposite sides of a loop.
Indeed, consider an electric dipole at the origin along the y axis. Moving a loop
sensor along the x axis, results in a voltage amplitude depending on the loop location
as depicted in Fig. 8.36. Moving the sensor around until the zero is found, is the
only way to localize the current dipole. Indeed, due to the symmetry, except for the
difference in sign, no distinction can be made between a dipole at the left or at the
same distance at the right of the loop. Moreover, the minus sign still does not solve
the ambiguity between an electric dipole at the left of the loop and the case of a
dipole with opposite phase at the right of the loop. Fig. 8.37 makes this clear and
also suggest two loops in a twin configuration to solve the problem.
In case of a magnetic dipole, which is a better building block for modeling a cryptographic device, the curve for V is entirely symmetrical, without a reverse of sign,
as can be seen on Fig 8.38. Here, again, two sensor loops are advisable as they give
more location information than a single loop.
8.4. Maximal Resolution Sensor
183
V
1.0
0.5
−0.02
−0.01
0.01
0.02
x [m]
−0.5
−1.0
Figure 8.36: Analytically calculated V of a loop sensor as function of x position with respect
to dipole along y axis at origin. Maximum scaled to one. Loop at 1 mm above
x, y plane.
≡
(a) Flux caused by current at left or opposite current at right is equal
(b) A twin configuration of loops can solve this duality
Figure 8.37: A twin loop configuration solves location ambiguity of a dipole source.
From the same graph it is observed that the loop size should be of the order of
magnitude of the source loop. A smaller loop will give a very low signal amplitude,
again due to the small area in Eq. (8.7). A loop that is too large, will however enclose
flux in both positive and negative directions, canceling out. As a consequence, for
a loop with certain diameter, it is advantageous to position the loop such that the
source lies at the loop perimeter.
A last advantage of twin loops is that they can suppress noise as an unwanted flux will
most probably be caused by a source at a larger distance from the loops. The signal
picked up from this unwanted source will hence be identical for both loops. Hence
distracting the signal of both loops will suppress the noise and enhance the measured
signal. This is of course not necessary if the measurements are performed in a shielded
environment. Be aware, however, that shielding is costly. All power or signal lines that
interface between the shielded environment and the data recording devices, should be
properly filtered and will even then import unwanted signals into the measurement
setup. The only way to avoid this problem, is including the data recording devices
into the shielded environment, and powering everything with batteries.
184
Chapter 8. Array Elements
V
1.0
sensor = 0.1source
sensor = 1source
0.5
sensor = 2source
sensor = 5source
sensor = 10source
−0.02
−0.01
0.01
0.02
x [m]
−0.5
Figure 8.38: Analytically calculated V of differently sized loops as function of x position
with respect to magnetic dipole at origin. Maximum scaled to one. Sensor
loop, with diameter sensor at 1 mm above x, y plane. Source magnetic dipole
has a diameter of source = 1 mm
clock line
(a) A clock line runs between both loops
clock line
(b) A clock line runs above both loops
(c) Combining both loops enhances the signal (d) Combining both loops suppresses the noise
Figure 8.39: Twin loops can be used to suppress noise.
8.5
Array Design
The essential difference with classical application of arrays, is the fact that in this
application the array elements are put as close as possible to the source, as otherwise
the signal is too weak. Hence the far field approximation that allowed to approximate
Rn as in Eq. (3.3) and ultimately led to the introduction of the array factor, is invalid
in this case. Indeed, now one source is seen under different angles by the different
array elements. Hence not only the distance, that in this case causes a huge amplitude
difference in the field strength, proportional to r3 , is different. Also the orientation
of the sensor, that determines the signal strength proportional to the cosine of the
angle under which the source is seen, differs.
8.6. Conclusions
185
Consequently, definitely not all elements of the array will pick up a signal from one
location in the chip with a sufficient amplitude to allow beam forming and enhance
the location sensitivity. Only the sensors in the vicinity of the source will be useful
in shaping the signal. In other words, the sensor right above the source location and
all its twins (in the meaning of Sect. 8.4.4) will form the active region of the array for
a certain signal source. Fortunately, digital off-line beam forming allows to monitor
the entire chip that is observed, as the recorded date of all sensors in the array can
afterwards be processed active region per active region.
As opposed to in the case of classical arrays, the array topology design is straightforward. The array should be a planar honeycomb. A close packing of the sensors has
to make sure that every part of the chip is covered by a sensor. This implies that
the inter element spacing is generally much smaller than in case of an array antenna
for telecommunication purposes. Indeed, now the entire half space should be covered.
Implementation limitations and certainly the connection that leads away the signal
picked up by the sensors for further processing, will however complicate things, turning the implementation of the straightforward topology into a research topic on its
own.
Indeed, with Eq. (5.3) the width of a micro strip line for Zc = 50 Ω on FR4 of
dt = 1 mm is 1.5 mm, which is even more than the pitch for a Small Outline Integrated Circuit with 8 pins (SOIC8), which is ±1.27 mm. Consequently it might be
advantageous to integrate at least some signal processing with the sensor array in
order to reduce the number of channels. Care should be taken that no information
crucial for the side channel analysis is dropped this way. It is however not clear at
the moment of writing how to determine what information is crucial and what is not.
One advantage of the array of sensors will be that different parts of the chip can be
monitored simultaneously. Currently, most work on EM side channel analysis is done
with a single probe. A first step then consists of choosing the sensor location. For
this, often a two dimensional map of the radiation from the chip is made by moving
around the probe above the chip, and looking at the signal amplitude or spectrum
at that location. This is commonly referred to as EM cartography. One of the first
works showing such map is [124]. The method is still popular, [173]. An array would
bypass this time consuming step, that moreover is not guaranteed to lead to the best
location, as it is uncertain what signal properties correlate with the degree of success
of an attack.
8.6
Conclusions
In this chapter first the specifications for a good sensor for side channel analysis were
defined. If direct radiation of the current performing the cryptographic operation is
to be observed, the sensor should be a small, rigid loop sensor, with a large bandwidth
and large signal amplitude.
186
Chapter 8. Array Elements
Before however coming to the design of such maximum resolution loop sensor, some
existing loop antenna types, the shielded loops and loops used for RFID, were studied. The design of the maximum resolution sensor is essentially a trade-off between
bandwidth, signal amplitude and sensor size. Indeed, to obtain a reasonable signal
amplitude in case of a small sensor, multiple turns should be used. Due to inter-turn
capacitance this however limits the bandwidth. For an infinite input impedance of
the measurement device, the design of the maximum resolution sensor can be done
analytically. For any practical value, an iterative search as proposed in this chapter
should be used.
Instead of using a single sensor, it is advised to use a twin configuration. This solves
the location ambiguity inherently connected to magnetic dipole sources, and allows
common mode noise suppression. This is also a first step in the direction of arrays.
Indeed, with a regular grid of sensors, each loop sensor could be used in conjunction
with all its neighboring twins. Off-line beam forming would then allow to use these
active subregions of the array to monitor distinct parts of a cryptographic device
simultaneously and with an improved signal-to-noise ratio.
Chapter 9
Signal Modification and
Combination
With all elements in the antenna or sensor array of Chapter 8 intercepting magnetic
fields, the voltages from all elements should be combined into one signal for further
processing.
For the application in Part II, essentially, apart from the trend of digitizing as much as
possible for cost and flexibility reasons, and the recent advances in the field of digital
beam forming, the signal could be analog up to the demodulator, where the bit values
are decided based on the received signal. For the application that is discussed in this
part, the information extraction does not simply follow from the digitization of the
analog signal. Sect. 9.1 will explain how recorded signals are correlated with certain
values of a secret, let it be (a part of) the key or plaintext, in order to extract
information from a measurement.
Consequently, the digitization should ideally be performed using an infinite amount
of quantization levels, to limit the quantization noise. Moreover, it should generally
be implemented at an early stage in the signal chain, in order to obtain a high SNR.
Sect. 9.2 elaborates on the sources of noise in the measurement setup. Techniques
to improve SNR, such as filtering and amplification, should be applied with care, as
they inevitably add noise to the signal as well. Sect. 9.3 will go more into detail on
this topic.
Improving SNR is ever a wise thing to do, for a single sensor as well as for an array.
Sect. 9.4 narrows the discussion again to sensor arrays. In that section it is argued
that combining the signals from the distinct elements in the array should be performed
in a digital way. The limitations on sample frequency and processing power will be
translated into the spectral and spatial resolution of the system.
187
188
9.1
Chapter 9. Signal Modification and Combination
Digitization
In this section, one approach for side channel analysis is given. Many other implementations or variations on the same idea are used. The only intention of this section
is to indicate that digitization of the measured side channel signal is compulsory.
Suppose an attacker wants to obtain a secret key inside a device. One possible approach to retrieve the key, is obtaining another identical device where the key can be
programmed. By programming any possible key and measuring the EM emanation of
the device for every key, an attacker can make up a list of template signals, hence its
name template attack, [174]. If now the EM emanation of the device with unknown
key is measured and compared to all templates of the list, the key can be found as
the key corresponding to the template signal that matches the signal of the device
with unknown key.
In the presence of noise, it is obvious that the template signal for a certain key should
be an average of several measured signals for that key. The more noise, the more
measurements should be averaged. Eventually, besides the average signal, also the
standard deviation can be stored, in order to use probability intervals instead of simply
the Euclidean distance when matching a signal to a template. In that case, maximum
likelihood decoding is performed. The probability that the measured signal occurs,
under the assumption that they has a certain value, is calculated for all possible key
values. The key that results in the highest probability is supposed to be the key
residing in the device.
The technique as explained above is essentially a brute force attack as all possibilities
for the key are to be checked. Fortunately, or unfortunately, the amount of work
needed, can be reduced by not working on the entire key, but parts (or single bits)
of the key. Then, not only different templates for different key bits are needed, but
for the same key bit, different templates correspond with different values for registers
with intermediate results. Fortunately, from the plaintext, the starting state of the
registers is known. Moreover, during execution, the next register state can be derived
from the previous one for the possible values of the key bit. This limits the number
of possibilities and moreover provides a means of error correction. An example can
be found in [175].
In case an attacker does not have a programmable identical device at its disposal, still
other ways are possible to retrieve the secret key inside the device.
For template attacks as explained above, as well as for any other technique, distance
measures between signals, and correlations between a secret and signals are calculated.
Consequently, the measured signals must be available in a digital way to allow further
mathematical or statistical processing. An ADC is hence inevitable in a side channel
analysis measurement setup. This will in most cases be an oscilloscope due to its
easy interfacing with computers, where the computations to extract information can
be performed in mathematical software such as MATLABTM .
9.2. Measurement Setup
189
amplifier
cable
oscilloscope
sensor
radiation
crypto device
Figure 9.1: Typical measurement setup for a EM side channel analysis.
9.2
Measurement Setup
The typical measurement setup for a cryptographic analysis can be found in Fig. 9.1.
EM radiation from a device is picked up by a probe and led to a digitizing device, such
as an oscilloscope by a cable. Eventually an amplifier can be added in front of the
oscilloscope or behind the sensor. For details on cryptographic hardware, the reader
is referred to Sect. 7.2. For details on magnetic field probes, see Sect. 8.2 and 8.4.
Details on amplifiers can be found in Sect. 9.3.3.
The coax cable is typically a Zc = 50 Ω coax cable, as most oscilloscopes only offer
50 Ω input impedance besides the high impedance state (usually 1 MΩ k ±10 pF).
Still 50 Ω coax is available in large variety. Different dielectric materials are used,
resulting in a different signal speed in the cable vcab . Typically vcab = 2/3 × c for
a cable filled with the dielectric material polyethylene (PE). The value for vcab for
some cables is listed in Table I.1. Another important cable characteristic, is the signal
attenuation, α, which is the real part of the complex propagation constant γ = α+jβ.
α is expressed in Np/m. The attenuation depends on the losses of the dielectric
material inside the cable as well as the losses of the inner and outer conductor. This
value is frequency dependent. For a Sucoflex 102E cable from Huber & Suhner, the
attenuation at 450 MHz is 0.25 dB/m, resulting in α = 0.25/8.686 = 0.0288Np/m.
Apart from the ambient fields, that can be regarded as unwanted signals, the noise
in this setup comes from the amplification (in the analog oscilloscope front-end or
external amplifiers) or quantization.
9.2.1
Noise Contributions
Due to the digitization of the signal, information is lost. This is mathematically
expressed by introducing quantizing noise eq :
e2q =
Z
∞
−∞
e2q f (eq )deq =
Z
δADC
2
−
δADC
2
e2q
1
δADC
deq =
2
δADC
,
12
(9.1)
with δADC the step size of the quantizer and f (eq ) the Probability Distribution Function (PDF) of eq . The rightmost equality only holds in case the quantizing noise is
uniformly distributed in a discretization interval. From Eq. (9.1) it is clear that δADC
should be as small as possible. Hence, ideally the (peak-to-peak) signal perfectly
matches the input range of the ADC (of the oscilloscope) used.
190
Chapter 9. Signal Modification and Combination
The second source of noise is the oscilloscope noise which exists out of two components. The first component is fixed Nf and caused by the oscilloscope’s front-end
attenuator and amplifier. Its rms value is generally around nf,rms = 200 − 300 µV.
The relative importance of this component depends on the amplitude setting. It is
significant for the most sensitive setting (lower V/div) and almost negligible for the
higher V/div settings. The second component is variable Nv and finds its origin in
the ADC. The rms value of this part nv,rms equals a few percent of the oscilloscope’s
V/div setting. As such it is important for higher V/div settings but overruled by
nf,rms for the most sensitive settings. Table 9.1 displays the values that were measured on a Tektronix DPO7254 with full bandwidth. Indeed, limiting the bandwidth
limits the noise power, as will be explained in Sect. 9.3.1.
Table 9.1: Measured rms values of fixed and variable noise for a Tektronix DPO7254.
mV/div
< 100
> 100
< 100
> 100
Zs
50 Ω
50 Ω
1 MΩ
1 MΩ
nf [mV rms ]
0.13
0.9
0.27
0.3
nv [% of mV/div]
1.1
1.0
1.5
1.5
The rms values can be converted into power by:
x2
Px = rms =
Zs
p
hx2 (t)i
Zs
2
(9.2)
where Zs is the oscilloscope impedance.
Obviously, an amplifier amplifies the noise at its input by a factor Gamp = Vout /Vin ,
the amplification. Besides, it adds noise Nextra . The noise figure N F relates the noise
at the output of the amplifier to the noise at the input:
N FdB = 10 log
Nextra + Nin G
Nin G
(9.3)
Amplifier noise consists of a fixed part and a part that is relative to the input noise
power, too. For normal use, the fixed part will be negligible, compared to the noise
that is proportional to the input noise power. Hence N F in most data sheets only
reports on the variable part of the amplifier noise. The EMCO 7405 amplifier, for
example, is specified with G = 30 dB and N F = 3.5 dB. Moreover, N F specified on
most datasheets is only valid in matched systems. Hence, a Northon/Thevenin equivalent for current and voltage noise is the only correct way of modeling the noise, [176].
In general N F is proportional to the gain and bandwidth of the amplifier.
9.3. Improving Signal-to-Noise Ratio
191
Each amplifier has its minimum noise source impedance, [176]. A Low Noise Amplifier
(LNA) specified for 50 Ω systems can be hoped to have this minimum N F in case of
a source impedance of 50 Ω. When in doubt, the minimum noise figure impedance
should be measured. A trade-off can be made between matching the sensor for minimum noise and for maximum signal amplitude. Some more considerations on this
can be found in Sect. 9.3.4.3.
9.3
9.3.1
Improving Signal-to-Noise Ratio
Filtering
As noise is present at every frequency, it is advantageous to suppress frequencies where
the PSD of the signal is too low compared to the PSD of the noise, as in such case the
overall SNR can be improved by not including that frequency interval. An example
could be a signal with the PSD of Fig. D.1 in the presence of white noise, which has
a PSD that is constant over all frequencies.
This is especially important in case not the direct radiation, but a modulation onto
a carrier is observed, see Sect. 7.3.3. In that case, either a heterodyne receiver or a
direct down conversion scheme can be applied, see Fig. 2.1. Here, again, as explained
in Sect. 2.1 an IF stage requires appropriate filters to suppress the mirror frequencies.
A mixer-less approach, if the modulation is plain AM modulation, which is often the
case, could be to use a simple diode detector. This circuit, depicted on Fig. 9.2, uses
the non-linearity of a diode to implicitly mix down all frequencies to BB. In that
sense it is of uttermost importance to have a band pass filter in front of the diode
detector. Without it, noise of all frequencies is superimposed at BB. A LPF behind
the diode detector becomes obsolete, as the RC system acts as a LPF with 3 dB upper
frequency:
f3
dB,up
=
1
2πRC
(9.4)
Filtering the input prior to down mixing is less important in case of mixing, again
except for the image rejection filters. A LPF at BB prior to digitization will be
sufficient. Therefore, the cost that is saved by using a diode detector might be lost
again when designing the custom band pass filter needed in front of the diode detector.
Besides improving the SNR by suppressing noise, in many cases also quantization
noise is decreased. This is because the peak-to-peak amplitude of the signal will
generally be smaller if the bandwidth of the signal is limited.
192
Chapter 9. Signal Modification and Combination
D
Vin
R
C
Vout
Figure 9.2: Circuit of diode detector.
9.3.2
Avoiding Standing Waves
Although it might be possible, mathematically, to reconstruct1 the original signal
from a measurement with reflections, these reflections will enlarge the peak-to-peak
amplitude and hence should be avoided as they increase the quantization noise.
A detailed discussion on standing waves, reflections and oscillations in cables can
be found in Appendix I. In short, in order to avoid reflections or standing waves,
at least one side of the cable between sensor and oscilloscope should be terminated
with a load that is matched to Zc of the cable. Fig. 9.3 indeed shows that only
in case of mismatch at both sides, standing waves are observed. It however makes
some difference, whether both sides (smallest amplitude), the sensor side (largest
amplitude) or the oscilloscope side is matched. This can be understood by looking at
the voltage division in Fig. 8.35 and filling in the values for R of the sensor and Z of
the oscilloscope (after transformation over a cable with Zc and length l).
9.3.3
Amplification
Essentially two types of amplifier are available on the market: those that are matched
to 50 Ω impedances and those that have generally higher input impedance, typically
about a few MΩ. The first ones require no further action and can simply be connected
to an oscilloscope and to cables. The latter ones generally require adding resistors to
determine the amplification.
An example of a 50 Ω matched amplifier is the EMCO 7405 amplifier. A way to build
a similar amplifier is to start from a matched amplifier IC, such as the minicircuits
MAR-6+. They can be cascaded to increase the gain. But as the output port of
the MAR-6+ is also used as the DC bias port, DC blocking capacitors are needed
between consecutive stages. This results in a circuit as in Fig. 9.4. Eventually the
MAR-6+ can be replaced by a MAR-3+ in the last stage(s) of the amplifier. This
component can give more output power, at the cost of loosing gain. An example of
the other category is the class E amplifier, discussed in Sect. 9.3.3.1.
1 It is expected that with common signal processing techniques it is possible to extract information
and hence signal power from the reflections. A deconvolution with the system response might be a
good starting point. Further details are irrelevant for this work.
9.3. Improving Signal-to-Noise Ratio
193
V
1
t
0
(a) Source current
V
0.04
0.02
0
−0.02
−0.04
t
(b) Scoop 1 MΩ and loop 50 Ω
V
0.04
0.02
0
−0.02
−0.04
t
(c) Scoop 50 Ω and loop 50 Ω
V
0.04
0.02
0
−0.02
−0.04
t
(d) Scoop 1 MΩ and loop shorted
V
0.04
0.02
0
−0.02
−0.04
t
(e) Scoop 50 Ω and loop shorted
Figure 9.3: The importance of matching
9.3.3.1
Class E Power Amplifier for ISO-14443A
The Class E topology is not directly suited to amplify a signal picked up by a loop. It
is however the ideal amplifier to drive a loop. Hence, this part, reported extensively
in [177], is relevant for fault injections.
194
Chapter 9. Signal Modification and Combination
Vcc
Rbias
Rbias
Lbias
Cblock
in
Lbias
Cblock
A06
Rbias
Lbias
Cblock
A06
A06
out
Figure 9.4: Possible circuit for a matched amplifier with a MAR-6+.
The application envisaged here, was amplifying the signal between an ISO-14443A
reader and the antenna, see Fig. G.6 in Appendix G. To power up an RFID card with
inductive coupling over great distances, very large primary coil currents are needed,
which build up high coil voltages. A class E amplifier has a parallel-series tuned load
network, see also Fig. 8.19, that following the discussion in Sect. 8.3.2.4 is essentially
a series RLC resonance chain with a parallel capacitor to tune the impedance. The
series capacitor then acts as a DC decoupling capacitor which relieves the voltage
stress on the transistor. The parallel capacitor adjusts the (phase of the) impedance
of the load and bypasses the transistor to guide most of the current.
9.3.3.1.1 Class E Amplifier Design Formulas As stated in [178], the class E
amplifier obtains a high power efficiency by minimizing the time in which current and
voltage exist simultaneously in the transistor. In a class D amplifier, this is obtained
by driving a parallel RLC chain with a square voltage wave. The dual of this, namely
driving a series RLC with a square current wave, is exactly what happens in a class E
amplifier.
Obviously, this requires a current source, which is in this case obtained by using a
choke inductor Lchoke . Fig. 9.5 shows the basic class E topology. As opposed to e.g. in
case of the class A amplifier, where the transistor is used as a variable resistor, here the
transistor is used as a switch S. Consequently the transistor is driven by a square wave
switching the transistor between off state and saturation. The series resonance chain
of a capacitor Cser , inductor L, and resistor R is connected in parallel with capacitor
Cpar to compensate for the phase so that the current is zero, with derivative of zero,
when switching the transistor into saturation.
9.3. Improving Signal-to-Noise Ratio
195
Several sets of formulas can be found in the literature. The analysis of Raab [179]
coincides with simple formulas in [180] in case of a class E amplifier and a duty cycle
of 50%:
R
=
Cpar
=
Lchoke
≥
L =
Cser
=
0.577
(Vcc − Vsat )2
PS
(9.5)
0.2
ωR
10
ω 2 Cpar
QRLC R
ω
1
ω 2 L − 1.1525ωR
(9.6)
(9.7)
(9.8)
(9.9)
with PS the output power, Vcc the supply voltage and Vsat the saturation voltage of the
transistor. As these formulas model the diodes and transistor as ideal components,
the value for Cpar should be the sum of the external capacitor and the intrinsic
capacitance of transistor, diodes and any other components added in parallel to the
transistor.
Lchoke
S
Cser
R
Cpar
L
Figure 9.5: Schematics of an ideal class E amplifier
9.3.3.1.2 Push-pull topology The single ended class E amplifier treated in
Sect. 9.3.3.1.1, can be converted into a push-pull configuration by simply doubling
the class E amplifier and connecting the load between the two output ports. By doing
this, the topology of Fig. 9.6 is obtained.
Taking Cser,1 = Cser,2 equal to 2 × Cser , and R1 = R2 equal to R/2 of the single ended
class E, and closing switch S1 when S2 is open and vice versa, this circuit works just
as the single ended class E amplifier. Indeed, if S1 is closed (or S2 ), the circuit reduces
to the circuit of a single class E amplifier.
This push-pull configuration consumes more power, as now during the entire cycle
current runs through a transistor drain-to-source-resistance RDS(ON ) , as opposed to
half a cycle in the single ended case.
196
Chapter 9. Signal Modification and Combination
Vcc
Lchoke,1
S1
R1
Cpar,1
Cser,1
L
Cdist
Cser,2
R2
Cpar,2
Lchoke,2
S2
Figure 9.6: The topology of a push-pull class E amplifier.
The current drawn from the source also increases, requiring the switch transistors
to conduct more current. The transistor currents can be partially reduced by redistributing Cpar by adding a Cdist drawn in dashed on Fig. 9.6 as explained in [181].
Fortunately the current through the loop increases too, as now during the entire cycle,
one end node of the series chain is excited.
9.3.3.1.3 Modulation The class E amplifier of Sect. 9.3.3.1.1 was able to amplify a continuous sine wave. For the ISO-14443A communication, data has to be
modulated onto this sine carrier with ASK, see Appendix G. This can be done by
modulating the voltage on the power supply [182]. Changing the amplitude of the
square wave on the transistor gate is not an option, because shutting down the class E
transistor causes the continuous current delivered by the choke inductor to charge Cpar
to a value that breaks the transistor.
The power can be modulated by adding a switch transistor Smod between the power
source and the power line of the class E amplifier. The gate of this transistor is driven
by a gate driver controlled by the modulation signal (envelope). In case the envelope
is high, the switch must close, otherwise the switch must be open.
If the use of a gate driver is inappropriate, e.g. because no additional power supply
voltage is available, the driver can be implemented by the circuit in Fig. 9.7, where
the envelope signal is applied to a transistor Senv that switches the voltage at the
gate of Smod between Vcc and Vcc Rb /(Ra + Rb ).
As the current in the choke inductor of the class E amplifier only varies slowly, the
current will not immediately drop when the modulation circuit opens the transistor in
the power line. This delay adds up to the influence of the quality factor QRLC of the
load of the class E amplifier and should be kept low in order to be able to silence the
carrier when needed. Therefore it is advised to keep the value of the choke inductor
Lchoke low.
9.3. Improving Signal-to-Noise Ratio
197
Smod
Vmod
Vcc
Rb
Ra
Xmod
D1
Senv
Venv
Figure 9.7: Circuit schematic of the modulation circuit.
In [182] it is reported that the design values for the class E amplifier will not differ
more than 15% from the ones obtained with the formulas of Sect. 9.3.3.1.1, as long
as:
2
π
Rl
Rl
Lchoke >
+1
≈ 3.5 .
(9.10)
4
fc
fc
If a lower value is required, other design formulas apply due to a non-constant current
in the choke inductor that was assumed in the derivation of the formulas.
In [182], also a formula to calculate Lchoke based on the maximum modulation frequency fmod is suggested. This fmod will generally spoken be much lower than the
carrier frequency fc of 13.56 MHz. Hence the class E amplifier can be modelled with
a low frequency equivalent resistance as in Raab [179], or derived from Eq. (9.5) and
knowing that the power consumption PS = Vcc2 /RDC :
RDC ≈
1
R = 1.7337R
0.577
(9.11)
Lchoke can now be determined by imposing requirements on the transfer function
K(fmod ) of Vmod to the voltage over RDC , as depicted in Fig. 9.8:
|K(fmod )| = p
RDC
(RDC + Rmod )2 + (2πfmod Lchoke )2
(9.12)
where Rmod is the resistance of the power supply, well approximated by RDS(ON ) of
Smod and in most cases negligible compared to RDC .
The data rate of ISO-14443A is given in [183] as fmod = fc /128. Compared to
Eq. (9.10), |K| = 0.995. Compared to Eq. (9.7), |K| = 0.975. As it is best to pass
the modulation signal as good as possible, and to keep Lchoke as low as possible, as
indicated earlier in this section, Eq. (9.10) is best used to determine the value for
Lchoke .
198
Chapter 9. Signal Modification and Combination
Rmod
Vmod
Lchoke
∼
RDC
Figure 9.8: Equivalent low frequency scheme of the class E amplifier to determine Lchoke
When opening the switch in the power line of the class E amplifier, the current through
the choke inductor can not change immediately, causing the voltage over the inductor
to change abruptly. To prevent break down of Smod due to an excessive VDS , a diode
can be added. This component keeps Vmod equal to zero, at least until the choke
current is zero. After the choke current becomes zero, the voltage at both sides of the
choke is equal and Vmod will follow the voltage of the drain of the class E transistor,
as can be seen on Fig. 9.9. If Smod , an IRF9530 in our design, is not modelled as
a switch but with its spice model as supplied by the vendor, its capacitances cause
a resonance in series with the choke inductor in the absence of a diode. With the
ES1D diode, Vmod again stays zero until the choke current becomes zero and then
again Vmod starts oscillating due to the series resonance of the choke inductor and the
capacitances of the IRF9530 and the diode.
120
2.5
VDS IRF510
Vmod
Ichoke
100
2
1.5
60
1
40
Ichoke [A]
VDS , Vmod [V]
80
0.5
20
0
0
−20
1.896
1.898
1.9
1.902
1.904
Time [s]
1.906
1.908
1.91
−0.5
1.912
−4
x 10
Figure 9.9: Ichoke and drain voltages of the IRF510 and Smod .
9.3. Improving Signal-to-Noise Ratio
199
100
80
With Diode
Without Diode
VDS of IRF510
VDS , Vmod [V]
60
40
20
0
−20
−40
−60
−80
1.896
1.898
1.9
1.902
1.904
1.906
Time [s]
1.908
1.91
1.912
−4
x 10
Figure 9.10: The drain voltage of the IRF9530 with and without the diode.
Adding another transistor switch between Vmod and the ground that closes when the
IRF9530 opens and vice versa, instead of using a diode, is a better solution as this
achieves the same without the risk of oscillation of Vmod .
9.3.3.1.4 Determining Cser and Cpar The two turn antenna of [150] or Table 8.5 was used to load the amplifier. Hence L = 2.2 µH and Rl = 2 Ω. The
formulas in Sect. 9.3.3.1.1 combined with the remarks in Sect. 9.3.3.1.2 then lead to
Cser = 130 pF and Cpar = 439 pF. The actual values differ from these values as
indicated in Sect. 9.3.3.1.1: the intrinsic output capacitance of the transistor adds
to the effective Cpar of the circuit. This output capacitance Coss can be retrieved
from the data sheet [184], but will vary with the voltage VDS , as does the junction
capacitance Cj of the diodes with varying bias Vbias [185]:
Cj = Cj0
1−
Vbias
Vj
m
(9.13)
with Cj0 the junction capacitance when no voltage is applied, Vj the intrinsic junction
voltage depending on the doping and m a constant indicating the type of junction.
Hence the effective output capacitance for a voltage swing can be obtained e.g. via the
technique explained in [186] where essentially the transistor is biased and connected
in series with a resistor, allowing the derivation of drain-to-source capacitance CDS
by measuring the RC time constant.
200
Chapter 9. Signal Modification and Combination
Printed Circuit Board (PCB) traces add up to the series capacitance. Formulas
from [187] indicate about 1pF/cm. As a result both Cser and Cpar are smaller in the
actual design than obtained with the formulas of Sect. 9.3.3.1.1. Trimming capacitors
were used to select the appropriate values for both capacitors.
9.3.3.1.5 Simulations The circuit was simulated with eldo 2 . First the class E
amplifier as designed above was simulated, with an extra resistor of Rext = 3.5 Ω
added to the loop inductance to meet the ISO-14443A requirements for QRLC . The
envelope of the decay, shown in Fig. 9.11, reveals that the ISO-14443A specifications
are met. As it seemed not straightforward to obtain a resistor able to dissipate
several watts without adding too much inductance, the measurements were carried
out without the extra resistor, as this is inherent to the antenna design and not
considered here. To limit the current, however, resistors were added to the choke
inductances. This is equivalent to feeding the amplifier at a lower supply voltage.
The simulation results for this case are also added on Fig. 9.11. In this case the decay
is too slow, according to the explanation above.
100
90
80
without Rext
with Rext
70
Il [%]
60
50
40
30
20
10
0
2
2.01
2.02
2.03
2.04
2.05
time [s]
2.06
2.07
2.08
2.09
2.1
−4
x 10
Figure 9.11: Simulation of loop current with and without Rext .
Fig. 9.12 shows that the amplifier was operating in the class E working point. The
non-zero voltage when the transistor is conducting is due to the finite RDS(ON ) of the
class E transistor.
2 Eldo
(by Mentor Graphics) is simulation software for analog electronics circuits.
9.3. Improving Signal-to-Noise Ratio
201
80
5
VG
VDS
IDS
70
4
60
3
IDS [A]
VG , VDS [V ]
50
40
2
30
20
1
10
0
0
−10
2.002
2.0022
2.0024
2.0026
time [s]
2.0028
−1
2.0032
2.003
−4
x 10
Figure 9.12: Waveforms of a class E amplifier with the complete SPICE model.
9.3.3.1.6 Component Selection The components were selected to withstand
the high currents and voltages. In case of alternatives, the ones with the smallest
capacitance were selected in order to preserve the bandwidth as much as possible.
The parts list can be found in Table 9.2. To drive the gates of the two class E
switches, a gate driver was used. For controlling the switch on the power line, both a
gate driver (Xenv ) and the circuit of Fig. 9.7 were tested.
The class E amplifier works at a frequency of 13.56 MHz and can provide the inductive
load a current of at maximum 8 A. This is a practical limit, as a higher current will
break the components. The amplifier works with a 30 V power supply.
Table 9.2: Components selected for the class E amplifier
SEMICONDUCTOR DEVICES
Smod
IRF9530
Senv
BS170
S1,2
IRF510
D1
ES1D
Xmod
HA-5002
Xenv
IR2821
X1,2
LM5111
PASSIVE DEVICES
Lchoke
2µH
Ra
5kΩ
Rb
10kΩ
L
2.2µH
Rl + Rext 2Ω + 3.5Ω
Cser
85pF
Cpar
10pF
202
Chapter 9. Signal Modification and Combination
9.3.3.1.7 PCB Design In order to provide a good ground, a four layer PCB with
power and ground plane was used. The signal traces were kept as short as possible
and decoupling capacitors were added. A heat sink of proper size was added to all
power MOSFETs and an extra driver for the class E transistors and switch in the
power line were added for redundancy. A picture of the hardware and the PCB layout
can be seen in Fig. 9.13 and 9.14.
Figure 9.13: Picture of the class E amplifier.
Figure 9.14: PCB layout of the class E amplifier.
9.3.3.1.8 Measurements To illustrate the class E behavior of the amplifier,
Fig. 9.15 depicts the voltages at the gate and drain of the class E transistors. Apart
from the voltage drop due to RDS(ON ) , the voltage over the transistor stays zero when
the transistor is conducting. In the remaining half of the cycle, the voltage builds up
and decays to zero, with, under perfect conditions, a derivative of zero at turn on.
9.3. Improving Signal-to-Noise Ratio
203
35
30
VG , VDS [V]
25
20
VG
VDS
15
10
5
0
−5
4
5
6
7
8
9
time [s]
10
11
12
13
−8
x 10
Figure 9.15: Gate and drain voltage of the class E transistors.
The signal picked up with a commercial magnetic field probe [143] kept parallel at
a distance of 30 cm above the center point of the transmitting loop, is represented
in Fig. 9.17. The measured peak-to-peak voltage of approximately 100 mV corresponds to a magnetic field strength of 0.63 A/m. To build up this field, the loop
carries 2.8 A. A maximum read out distance of 21.79 cm can be inferred from this.
Fig. 9.16 illustrates that the decay does not completely comply with the ISO-14443
recommendations. This is due to the lack of an external resistor, a choice that was
made in Sect. 9.3.3.1.5. To obtain a modulated signal, given in Fig. 9.16, an envelope
was applied to the gate driver Xenv of the Smod transistor.
Performing these measurements was a very tedious experience as things tended to
break continuously. Indeed, when one component fails, this causes an over current
in all connected components, breaking nearly the entire circuit. Eventually, some
protection was included by adding current limiting resistors between the gate drivers
and the class E transistors. Addition of thermal circuit breakers, or control systems
based on Current Sensing Resistors (CSR) is unavoidable if any further action with
the circuit is intended. The use of a current mirror can be considered as well, this
however consumes twice as much power as in the controlling branch the same current
runs as in the controlled branch. Hence it is not recommendable for a circuit that
consumes 8 A. With a current mirror, the modulation circuit of Fig. 9.7 is replaced
by the one in Fig. 9.18.
204
Chapter 9. Signal Modification and Combination
0.05
0.04
0.03
0.02
Vl [V]
0.01
0
−0.01
−0.02
−0.03
−0.04
−0.05
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
time [s]
−5
x 10
Figure 9.16: Field transmitted by the loop antenna driven by the class E amplifier.
0.05
0.04
0.03
0.02
Vl [V]
0.01
0
−0.01
−0.02
−0.03
−0.04
−0.05
0
0.5
1
1.5
2
2.5
time [s]
3
3.5
4
4.5
5
−7
x 10
Figure 9.17: Zoom of the signal as received with a magnetic probe at 30 cm.
9.3.4
Comparison of some Setups
In order to compare different setups, to be able to select high or low oscilloscope
input impedance, or to use or not to use an amplifier, the SNR of the different
setups is compared. The question remains whether this is a useful value for side
channel analysis, as in some exceptional situations a reflection might even enhance a
correlation sought for. Anyhow, in general a higher SNR will result in less traces (or
measurements) needed to extract secret information from a measurement.
9.3. Improving Signal-to-Noise Ratio
205
Vcc
Vmod
R=
Vcc
Imax
Venv
Figure 9.18: Circuit schematic of the modulation circuit with current mirror.
Starting from a signal and noise, picked up by the sensor, and passing through the
general setup of Fig. 8.35, the signal entering the cable depends on the impedance
of the measurement setup seen by the sensor at its end of the coax and on the
sensor impedance itself. To make abstraction of the sensor, different measurement
setups are compared starting from the same signal amplitude entering the chain sin .
SN Rin = Sin /Nin at the input of the measurement chain, leads to an overall SN Ros ,
with the os subscript denoting the oscilloscope side, equal to:
SN Ros =
Nos,in +
Sos,in
.
δ2
12 + Nf + Nv
(9.14)
Sos,in as well as Nos,in can be derived from Sin and Nin , the signal and noise entering
measurement chain, when the cable properties and G and N F of the amplifiers are
known.
To allow comparison between having the amplifier at the sensor and at the oscilloscope
side, a general setup with two amplifiers is investigated. If one or both amplifiers are
not present, G and N F of the amplifier concerned are set to 1.
Zin,meas
sensor
Γss
Γos
Zc = 50 Ω
oscilloscope
Figure 9.19: General schematic of the measurement chain.
206
Chapter 9. Signal Modification and Combination
9.3.4.1
Oscilloscope Side at 50 Ω Impedance
The unknowns in Eq. (9.14) become, including both amplifiers of Fig. 9.19:
Nos,in
Sos,in = e−2αl Glin,1 Glin,2 Sin
NF
−1
= e−2αl N Flin,1 + Glin,2
Glin,1 Glin,2 Nin
lin,1
(9.15)
(9.16)
with subscript lin denoting linear quantities as opposed to dB.
9.3.4.2
Oscilloscope Side at 1 MΩ input impedance
When the oscilloscope (with amplifier in front) has an input impedance of 1 MΩ, a
reflection due to mismatch between oscilloscope and characteristic impedance of the
cable occurs. If then the sensor side impedance neither matches Zc of the cable, an
infinite number of delayed version of the signal arrive at the oscilloscope. This is
explained in detail in Appendix I. Depending on whether these delayed reflections
are considered as useful signal or as noise, this leads to a different value for the Sos,in
and Nos,in .
9.3.4.3
Simulation Results and Conclusions
SN Ros of Eq. (9.14) was calculated and simulated (with random time signals) in
MATLABTM for the setup of Fig. 9.19 for different cases. As an example, the results
for a cable of l = 0.3 m are depicted in Fig. 9.20. In all cases the oscilloscope used
was the Tektronix DPO7254, with noise contributions as in Table 9.1. The amplifier
used was the EMCO 7405, discussed in Sect. 9.2.1. The setups are:
• case 1: oscilloscope at 50 Ω input impedance, no amplifier
• case 2: oscilloscope at 50 Ω input impedance, with amplifier
• case 3: oscilloscope at 1 MΩ input impedance, reflections regarded as signal
• case 4: oscilloscope at 1 MΩ input impedance, reflections regarded as noise
For small peak-to-peak signal amplitudes Vpp , an amplifier is advantageous, as amplification allows to use the entire range of the ADC of the oscilloscope at the most
sensitive setting. For large signals, even without amplifier the signal spans the range
of the ADC. In such case, an amplifier should not be used as it will inevitably add
extra noise. Introducing a mismatch and hence causing reflections is advantageous at
lower values for Vpp as then again the range of the ADC can be spanned because of
amplitude increase. From a certain value of Vpp onwards, however, a signal increase
will not result in a better SNR.
9.4. Digital time shifting
207
SN Ros,lin
2
1.5
case 2
1
case 3
0.5
case 4
0
−5
case 1
0
5
SN Rin,dB
0
1
2
3
4
5
Vpp [mV]
Figure 9.20: SN Ros for four different setups.
In this comparison, the signal as entering the signal chain was supposed to be independent of the input impedance of the measurement setup. When taking the voltage
division of sensor and measurement setup input impedance into account, also the
dependence of amplifier noise figure on the source impedance should be taken into
account. Indeed, not the signal amplitude, but SNR should be maximized.
9.4
Digital time shifting
Where the other topics discussed in this chapter are applicable to single sensor setups
too, this section narrows the discussion to setups of sensor arrays. As mentioned in
Sect. 9.1 the signal picked up by the sensor should be digitized. Sect. 8.5 learned
that the signals of the different sensors should be digitized separately to allow off-line
beam forming. This way multiple beams can be extracted from the measured sensor
signals and hence multiple parts of the cryptographic hardware can be monitored
simultaneously.
As the signal can be supposed to be wide band when observing the direct radiation
of the currents performing the cryptographic operations, the signal shaping method
should preserve wide band signals. As the signal is digitized per element, digital time
shifting is the most obvious way of (off-line) beam forming.
208
Chapter 9. Signal Modification and Combination
When time delay beam forming is performed in real time, e.g. by switching in and
out delay lines of different lengths, the digital nature of the technique introduces
errors and approximations similar to those mentioned in Sect. 3.4.2. If however,
signal recombination is done off-line with computers, the sampling frequency of the
ADC only limits the bandwidth of the system. The spatial resolution is unaffected
by the digitization. Indeed, by simply interpolating the signals, any time delay can
be applied, resulting in a spatial distance between two source points that is only
limited by computational power. Adding interpolation points obviously increases
both processing and memory demands. SNR and the spatial sensitivity of the sensor
element will have a much larger impact on the spatial resolution of the sensor array.
9.5
Conclusions
The aim of this chapter was to provide a systematic way of selecting the appropriate devices and parameter settings in a measurement setup for side channel analysis. Throughout the chapter, the only figure of merit was the SNR that should be
maximized. It is unclear whether a maximum SNR at a certain point in time or a
maximum mean SNR over time leads to the side channel analysis with the fewest
number of measurements needed. Further research is needed to clarify this.
In the chapter, all noise contributions, from digitization device, amplifier and oscilloscope were analyzed and expressed in numbers. This allowed comparison of different
setups. This comparison made clear that if the signal is small, an amplifier should be
used to make sure that the entire range of the ADC is spanned. Reflections should
in general be avoided, even in case of small signals, as they will only improve SNR
in a haphazard way. Consequently, at least one side of the cable between sensor and
oscilloscope must be matched.
If an amplifier is used that is matched to 50 Ω at input and output port, the amplifier can be placed at the sensor side, at the oscilloscope side or even halfway the
cable without any difference in performance. If the amplifier has a high input and
output impedance, it should be inserted at the oscilloscope side and the sensor should
be matched to 50 Ω. The ideal setup however has an amplifier with a high input impedance connected to the sensor, as this results in the highest resolution, see
Sect. 8.4, and an output impedance of 50 Ω driving the signal on the cable to the
oscilloscope that is set to an input impedance of 50 Ω as this was measured in this
chapter to give the smallest variable oscilloscope noise.
Although the comparison of measurement setups was useful for systems with a single
sensor too, it is a necessary step to take in array design as well. Sensor arrays for
side channel analysis will use off-line digital time shifting as beam forming technique
so that in that case too the signals from the individual sensor elements are to be
transported, in a way that preserves SNR as much as possible, to the digitizing device.
Part IV
Conclusions
209
211
In this work, the classical array antenna theory was reviewed and applied to two
different applications. The general array theory was derived from the properties of
Fourier transform of the source function that consisted of three dimensional Dirac
impulses with a complex amplitude that reflects the amplitude and phase of the
excitation. This way effect of topology, amplitude tapering and phase excitation were
coherently restated in terms of Fourier transform properties. The different ways of
beam forming, time delay, phase shift and frequency change, as well as the different
ways of implementing in the receiving chain, namely at radio frequencies, intermediate
frequencies and baseband, were compared regarding performance and cost.
The first application, satellite communication, was introduced and the benefit of electronic beam steering over mechanical pointing was clarified. Next some peculiarities
of designing for space were reviewed. Then the three parts of an array antenna, being
antenna elements, phase shifters and combining network were discussed. From this
work can be concluded that the challenges when applying array antennas to satellite communication are not scientific questions but rather design, development and
engineering issues. What requires special attention, is designing antenna and beam
steering hardware to withstand the hostile space environment. Another difficulty of
this application is that gain should be maximized for low elevation angle as in this
case, the satellite-to-ground station distance is largest and hence the free space path
loss at maximum.
For the second application, electromagnetic side channel analysis, the link between
currents performing cryptographic calculations and the leakage of information over the
side channel of electromagnetic radiation from the chip was established. Specifications
for the ideal sensor were made up, for monitoring direct radiation from the currents as
well as for analysis via a modulated signal. Much work was done on designing sensors.
Some existing types, such as shielded loops and radio frequency identification coils
were investigated. The design of small loops, for obtaining a high spatial resolution,
was investigated as well. Besides the sensor, the other components of the measurement
chain, being cable, amplifier and digitizing device, were analyzed as well, in order to
characterize all noise sources in the measurement setup.
The second application, electromagnetic side channel analysis, differs much from the
first one, satellite communication, regarding the application of array antennas. Firstly
because the distance between signal source and array is not large enough to justify the
approximations made when deriving classical array antenna theory. This also results
in a much smaller inter element spacing in the case of side channel analysis. Secondly,
this application would benefit from off-line beam forming, which allows multi beam
configurations, or, stated differently, simultaneously monitoring of the different active
parts in the cryptographic hardware. In order to achieve this, a way has to be found
to bring out and process multiple signals. Eventually, signals could be combined in
signal processing hardware integrated with the sensor array. Care should however be
taken that this way no information crucial to the process of side channel analysis is
dropped.
212
Yet a third difference with classical array antennas, is that in the sensor array it is
not useful to combine the signals of all array elements into one signal. Ideally, for the
off-line processing, the signal of all elements should be digitized separately. And even
in the off-line recombinations will only signals from parts of the array be combined
into one signal. This is somehow related to MIMO systems.
As spatial resolution is of main importance, it can be expected that the array of
sensors is best implemented in a silicon device. This will not be straightforward
as silicon is definitely not the best antenna substrate, neither regarding losses nor
regarding predictability of electrical parameters.
As it is obvious that still some years of research are needed to make this sensor array
on chip available, meanwhile, side channel analysis on modulated signals should not
be discarded. Contemporary work is much like looking for a needle in a haystack.
Currently it is unclear whether the amount of information in a signal can be evaluated without actually performing a side channel analysis. It is unlikely that the
sensor should be positioned relative to the cryptographic device where the amplitude
is largest. It might sound more logical to look for sensor positions where spectral
components appear that do not appear elsewhere. A study of the spectra emitted by
cryptographic hardware might lead to a more scientific standard methodology.
In this text, the measurement setup was studied thoroughly which led to formulas and
curves to compare the signal-to-noise ratio in different setups. From this comparison
it can be concluded that for small signals, it is advantageous to use an amplifier and
reflections should be avoided. However, from assessment of orders of magnitude of
radiation, it would be possible to derive specifications for amplifiers and digitizing
devices, ideally in relation with the number of measurements needed for a successful
side channel analysis.
Much of the material in this work is a review of work that can be found elsewhere,
but from a different point of view or applied in a case study. A few topics, however,
can be regarded as new contributions to science. For the satellite communication
application, the idea of Doppler shift compensation by frequency scanning, and the
method to design a linear equally spaced array to maximize gain under low elevation
angle, both resulted in a publication. For the side channel analysis, not much work on
array antennas was carried out, due to the lack of theoretical background and a good
sensor element. Hence these gaps were filled first. A review of shielded loops and
their matching, and a mathematical study on the spatial resolution limit of magnetic
sensors were published.
In this work, whenever appropriate, the developed theory was materialized in designs. A radio frequency identification reader loop antenna, a class E amplifier and
a KeeLoq transceiver where built and tested. Measurements on a modular array
antenna illustrated the array theory.
213
Future Work
When given time, many more interesting things, both enjoyable and useful, could
have been investigated. A more in depth mathematical study of the effect of phase
approximations on array radiation patterns, would be very useful to decide on how well
performing digital phase shifters, and their control chain, have to be, in order not to be
the limiting factor. The further development of the combined balanced-mœbius sensor
seems to be very promising. A study of spectra emitted by cryptographic devices
resulting in a standard methodology for side channel analysis on modulated signals,
will be inevitable to make up standard tests for fast screening of vulnerability level
of cryptographic devices to electromagnetic side channel analysis. The translation of
signal-to-noise ratio in measurement setups to the number of measurements needed
for side channel analysis, will make clear whether or not signal-to-noise ratio is a
useful measure to compare measurement setups.
It can be hoped, that it is only a matter of years until side channel analysis on
modulated signals will not reveal any secret information. With the advances in secure
logic styles and the awareness of the need for applying best practice design methods
from electromagnetic compatibility theory, this can indeed be expected. Hence for the
long term, the development of an on chip sensor array is imperative. The road ahead
will be bumpy. Implementing loop sensors on silicon will not be straightforward.
Neither will interfacing the sensor array with the digitization devices over decent
transmission lines. Eventually integrated signal processing can be considered
Once this sensor array will be available, it will still be a challenge to apply the
appropriate weight functions to combine the signals of parts of the array into signals
useful for side channel analysis. Modeling of the cryptographic hardware is a first step
in understanding and predicting radiation spectra and magnitudes that are necessarily
know when deciding on the details of the sensor array.
invisible filling
Appendices
215
Appendix A
Doppler Shift Compensation
by Frequency Scanning
This appendix is [188] and evaluates Doppler shift compensation by varying the frequency in such way that it could also be used for frequency scanning.
A.1
Introduction
As a spacecraft might travel at high speeds, the Doppler shift of a down link signal
is considerable compared to the carrier frequency. As such this Doppler shift should
be taken into account when down converting for demodulation. A common technique
is to use a PLL to lock onto the signal and slowly follow the frequency shift, as
in [189]. For Spread Spectrum techniques, this is not feasible and hence many signal
processing algorithms are used to determine the actual frequency after an FFT e.g.
[190], [191]. Compensating the shift at the transmitter instead of at the receiving side
is also considered, a.o. in [192].
In this paper it is mathematically shown that the Doppler shift frequency caused by
the movement of a vehicle coincides with the frequency shift needed for frequency
scanning, if the ratio of the element spacing, expressed in wavelengths in free space,
over the additional line length, in wavelengths in the substrate medium, equals the
ratio of the vehicle speed over the speed of light. By choosing a high value for the
permittivity of the substrate, this additional line length can still be reduced by e.g.
a factor of ten, but this is insufficient for practical contemporary applications. For
LEO satellites for example, this technique is inappropriate as it would require several
hundreds of meters of additional line lengths per array element, but for future space
vehicles traveling at nearly the speed of light, this technique has obvious advantages.
217
218
Appendix A. Doppler Shift Compensation by Frequency Scanning
The drawback of the method is that with only one degree of freedom, namely the
varying frequency, the beam can indeed only be steered with one degree of freedom.
This is sufficient for a linear array, but a planar or any other array with dimension
more than one needs an additional steering mechanism.
A.2
Mathematical Description
In this section the mathematical proof will be given that frequency shift needed for
Doppler shift compensation can be used to perform frequency scanning of a linear
array. A discussion on the practical implications is delayed to Sect. A.3.
A.2.1
Doppler Shift
Suppose that a spacecraft, traveling at a constant speed of v0 on a trajectory that
could have any curvature, tries to communicate with a ground station it passes by.
The Doppler shift ∆fc of the carrier fc , as received by the ground station then depends
on the velocity of the spacecraft towards this ground station:
vt = vo × cos (φ)
(A.1)
with φ the angle between the tangent of the trajectory and a line connecting the
spacecraft and ground station. Once the velocity of the source in the direction of the
receiver is established, the Doppler shift is calculated as (e.g. in [193]):
∆fc = fc
vt
vo
= fc
cos (φ)
c − vt
c − vo × cos (φ)
(A.2)
where c stands for the speed of light (≈ 3 × 108 m/s).
A.2.2
Frequency Scanning
To point the beam of a directional equidistant linear array in a certain direction (angle
φ), each radiating element of the antenna array should be driven with the appropriate
phase shifted version of the signal. The phase difference δ̃ between two consecutive
elements was already calculated earlier in Eq. (2.10), and restated here: (see Fig. A.1)
δ̃ =
2πd
cos (φ)
λ0
(A.3)
where d is the inter-element spacing and λ0 is the wavelength of the signal in free
space.
A.2. Mathematical Description
219
This phase shift δ̃ can be obtained in various ways. One of the techniques is frequency
scanning. For this technique each radiating element is fed with a line of a certain
length: the length of a line to an element is equal to the length of the line to the
previous element in the array plus a certain length L (see also Fig. A.1). This is
commonly referred to as the series-fed array.
Length L is chosen so that at the center frequency fc , the phase difference between
consecutive elements is zero (modulo 2π) and the beam points at broadside. Therefore
√
L should be an integer multiple of the wavelength in the substrate λc = λc,0 / . If
the frequency is varied, the fixed length L is no longer an integer multiple of the
wavelength, giving raise to a phase difference δ̃:
δ̃ =
L
λc
2π
∆fc
fc
(A.4)
Combination of Eqs. (2.10) and (A.4) leads to the frequency shift ∆fc to be applied
in order to steer the beam in direction φ:
∆fc = fc
1−
d
√
L d
√
cos (φ)
L cos (φ)
(A.5)
phase= 0
RF
d
L
phase= δ
d
L
φ
phase= 2δ
0
L
phase= 3δ
δ
phase front
d
2δ
d.cosφ = ∆δ.λ
2π
Figure A.1: The principle of frequency scanning.
220
A.2.3
Appendix A. Doppler Shift Compensation by Frequency Scanning
Doppler Shift Compensation
Comparison of Eqs. (A.2) and (A.5), with the remark that indeed the angle φ between
the velocity vector and the line connecting spacecraft and ground station is the angle
under which a linear array should be steered, reveals that simply by choosing:
d
v
√ = o
c
L (A.6)
the Doppler shift is exactly compensated by the frequency shift to scan the beam (in
order to lock on a fixed point).
A.3
Applicability
In this section, the technique as explained in previous section is reviewed with respect
to applicability in two situations: a LEO satellite and a futuristic spacecraft passing
by a futuristic base station in space.
A.3.1
LEO satellite
In the case of a LEO satellite around the earth, vo depends entirely on the height h
of the circular orbit above the earth’s surface:
r
Me
vo = G
(A.7)
Re + h
with Me = 6.0 × 1024 kg the mass of the earth, G = 6.67 × 10−11 Nm2 /kg2 the
universal constant of gravity, and Re = 6371 km the earth’s radius. Filling in this
value and the value for φ, which can be calculated for every elevation angle α from
(see Fig. A.2):
sin ( π2 − φ)
sin ( π2 + α )
cos (φ)
(A.8)
=
=
Re
Re
Re + h
into Eq. (A.2), results in plots as in Fig. A.3(a), where ∆fc for a LEO satellite with
h = 800 km and several fc are plotted, or Fig. A.3(b), for fc = 4 GHz and varying h.
The rotation of the earth was neglected in previous calculations. When taken into
account, the orbit of the satellite is not a circle but a curved line in a coordinate
system that is fixed to the earth (see Fig. B.1 in Appendix B). This results in a
varying magnitude of the orbit speed vo , so that ∆fc will vary with the position of
the satellite. MATLABTM simulations of the speed as a function of time, with the
orbit simulator of Appendix B, revealed that the difference with the non-rotating
earth is negligible (see Fig. B.2).
A.4. Conclusion
221
vo
vt
φ
αε
horizon (plane)
h
Re
orbit (circle)
earth (sphere)
Figure A.2: The geometry of a LEO satellite that passes over a ground station.
The value of the Doppler shift in the case of a LEO satellite, is hence ±0.002 % of
fc . Referring to Eq. (A.6), for practical arrays, where d varies between λ2 and λ, L
should be about 105 × λc , thus for fc in the GHz region, at least a kilometer, which
is infeasible in practice.
A.3.2
Future spacecraft and base station
Imagine a spacecraft cruising at a constant speed of about 0.1×c along a straight line,
passing a base station. The same exercise over again, results in exact compensation
of the Doppler shift by frequency scanning if L ≈ 5 × λ. For fc = 10 GHz this would
be 30 cm, which is feasible.
A.4
Conclusion
Although Doppler shift and frequency scanning shift mathematically collide, in case
the linear array antenna is always aligned with the instantaneous velocity vector and
the magnitude of the velocity is constant, it is impractical to compensate for Doppler
shift by frequency scanning. The practical problems when implementing the suggested
compensation, can be solved if a component that magnifies phase shifts is found, or
vanish for vehicles cruising with approximately the speed of light.
222
Appendix A. Doppler Shift Compensation by Frequency Scanning
Doppler Shift of a h=800 km LEO Satellite Downlink Signal
(varying frequency)
300
2 GHz
4 GHz
8 GHz
12 GHz
200
doppler shift [kHz]
100
0
−100
−200
−300
0
20
40
60
80
100
elevation angle [deg.]
120
140
160
180
(a) h = 800 km, several fc
Doppler Shift of a LEO Satellite Downlink Signal at 4 GHz
(varying orbit height)
100
200 km
600 km
1000 km
2000 km
80
60
doppler shift [kHz]
40
20
0
−20
−40
−60
−80
−100
0
20
40
60
80
100
elevation angle [deg.]
120
140
160
180
(b) fc = 4 GHz, several h
Figure A.3: Doppler shift as a function of α for some LEO satellites.
Appendix B
The Orbit Simulator
In Appendix A some plots on the Doppler shift of a down link signal from a satellite
to a ground station were depicted. The LOS time and the Doppler shift are, however,
difficult to calculate with the effect of the rotation of the earth taken into account.
Also the latitude coordinate of the ground station and the inclination angle of the
orbit are of influence. Therefore it is easier to simulate the overpass. This appendix
gives some details on the simulation, which also takes the rotation of the earth into
account.
B.1
The Orbit
The orbit of a satellite is characterized by its height h and inclination angle αi .
Assume that the center of the earth is the center of a translating (not rotating with
respect to the stars) Cartesian coordinate system, then the place of the satellite as a
function of time can be written as:

 xsat (t) = (Re + h) cos (ωo t)
ysat (t) = (Re + h) sin (ωo t) cos (αi )
(B.1)

zsat (t) = (Re + h) sin (ωo t) sin (αi )
The orbit pulsation ωo depends on the height as is explained in Sect. 4.2. Because
the earth is rotating around the z-axis in the translating coordinate system, the
coordinates of the ground station will be:

 xgnd (t) = Re cos (αl ) cos (ωe t)
ygnd (t) = Re cos (αl ) sin (ωe t)
(B.2)

zgnd (t) = Re sin (αl )
where αl stands for the latitude coordinate of the ground station and ωe is the sidereal
rotation of the earth.
223
224
Appendix B. The Orbit Simulator
But to calculate the horizon (a tangent plane through the ground station) it is more
appropriate to consider a coordinate system fixed to the earth (thus rotating along
with the earth). For the z-axis this makes no difference. The new x0 and y 0 coordinates
can be found using:
 0
 x = cos (ωe t) × x + sin (ωe t) × y
y 0 = − sin (ωe t) × x + cos (ωe t) × y
(B.3)
 0
z = z
This leads to:
 0
 xsat =
y0 =
 sat
0
zsat
=
 0
 xgnd =
y0 =
 gnd
0
zgnd
=
(Re + h) (cos (ωo ) cos (ωe t) + sin (ωo ) cos (αi ) sin (ωe t))
(Re + h) (− sin (ωo ) cos (αi ) sin (ωe t) + cos (ωo ) cos (ωe t))
(Re + h) sin (ωo ) sin (αi )
(B.4)
Re cos (αl )
0
Re sin (αl )
(B.5)
An illustration of both coordinate systems is found in Fig. B.1.
(a) Translating coordinate system
(b) Translating rotating coordinate system
Figure B.1: Satellite orbit and ground station in two coordinate systems.
B.2
The Simulation
A perfect (zenithal) overpass occurs when the satellite is in position:1
 0
 xsat = (Re + h) cos (αl )
y0 = 0
 sat
0
zsat
= (Re + h) sin (αl )
(B.6)
If the time can be solved from the combination of Eqs. (B.4) and (B.6), then a time
interval around this zenithal position can be simulated. But the set of equations does
not always have a solution. In case sin (αl ) > sin (αi ) this is clear.
1 This
comes down to making the angle between ~
rsat and ~
rgnd equal to 0 as in [58].
B.2. The Simulation
225
Furthermore each equation of the set gives an infinite number of solutions (if any) so
that an infinite number of values must be checked in the other equations. Therefore
it is easier not to look for the solution of the perfect pass, but to look for a tsat when
the satellite is above a place with αl :
zsat = (Re + h) sin (ωo tsat ) sin (αi ) = (Re + h) sin (αl )
(B.7)
and a tgnd 6= tsat when the ground station is exactly beneath the satellite:
xgnd = Re cos (αl ) cos (ωe tgnd ) =
Re
xsat
Re + h
(B.8)
This gives:
tgnd
sin (αl )
1
arcsin
tsat =
ωo
sin (αi )
1
cos (ωo tsat )
=
arccos
ωe
cos (αl )
(B.9)
(B.10)
If the numerical calculation of tgnd is not accurate enough (not resulting in a pass
that is good enough) an optimization can be done by exhaustively searching a time
∆tgnd where the distance between the satellite and the ground station is nearly equal
to the orbit height:
q
(xsat − xgnd
)2
+ (ysat − ygnd
)2
+ (zsat − zgnd
)2
∆h
=h 1+
h
(B.11)
The more times that are checked, the smaller ∆h will become and thus the better the
overpass. A near overpass occurs every 30 days (∆h/h < 0.001).
When good values for tsat and tgnd are found, both times can be varied in an interval
around the calculated values, in order to simulate the position of satellite and ground
station over this time interval. This makes it possible to calculate the Doppler shift
as the speed of the satellite towards the ground station is the derivative to time of the
distance between both. Note that the values do not differ much from those obtained
neglecting the rotation of the earth, as can be seen on Fig. B.2. Here the worst case
is taken (ground station at 0◦ latitude and orbit inclination equal to 180◦ ).
But also the down link time can be calculated. Knowing that the expression for the
tangent plane through the ground station is:
0
0
0
0
x0gnd (x0 − x0gnd ) + ygnd
(y 0 − ygnd
) + zgnd
(z 0 − zgnd
)=0
(B.12)
the moment the satellite appears above the horizon will be found as the time where
the left hand side of Eq. (B.12), with the coordinates of the satellite filled in, reverses
sign.
226
Appendix B. The Orbit Simulator
Doppler Shift for satellite with h=800 km,
α =180 and groundstation α =0 (12 GHz)
i
l
300
without earth rotation
with earth rotation
doppler shift [kHz]
200
100
0
−100
−200
−300
−20
0
20
40
60
80
100
elevation angle [deg.]
120
140
160
180
Figure B.2: Doppler shift with and without earth rotation.
Also the down link time (α > 10◦ ) can be calculated because for every time t the
angle of elevation can be found as:


0
0
0
0
0
0
0
0
0
 ((xsat − xgnd ) · xgnd ) + ((ysat − ygnd ) · ygnd ) + ((zsat − zgnd ) · zgnd ) 
q
α = arccos  q

02
02
02
0
0
0 − y0
2
2
(x0sat − x0gnd )2 + (ysat
xgnd
+ ygnd
+ zgnd
gnd ) + (zsat − zgnd )
(B.13)
Appendix C
Pseudo Code of the Array
Control Routines
• Beam Steering From the attitude and GPS coordinates of the satellite and the
GPS coordinates of the groundstation, the beam direction and the excitation
phases can be calculated.
# ifndef PI
# DEFINE PI 3.1415;
# endif
void BeamSteer(float[] GroPoss, float[] SatPoss, float[] SatAtt) {
/* Calculate the settings for the VGAs in the array
based on
GroPoss: xyz vector with GPS coordinates of the groundstation
SatPoss: xyz vector with GPS coordinates of the satellite
SatAtt: phi,theta vector with attitude of the satellite
*/
float arrLoc[7][2]=. . .; //
//
float Poss[3];
//
//
float RotMat[3][3];
//
float BeamAtt[2];
//
xcoff[7];
//
float vgaSet[7][4];
//
int counter;
translation vectors of the seven
elements in the array (in wavelength)
direction vector between satellite
and groundstation
satellite rotation matrix
pair of steering angles of the array
excitation phase of the array elements
VGA settings for I and Q of all elements
227
228
Appendix C. Pseudo Code of the Array Control Routines
// Make up satellite rotation matrix
// Euler rotation with alpha=phi, beta=theta, phy=0
RotMat[1][1]=cos(SatAtt[1]);
RotMat[1][2]=-sin(SatAtt[1])*cos(SatAtt[2]);
RotMat[1][3]=sin(SatAtt[1])*sin(SatAtt[2]);
RotMat[2][1]=sin(SatAtt[1]);
RotMat[2][2]=cos(SatAtt[1])*cos(SatAtt[2]);
RotMat[2][3]=-cos(SatAtt[1])*sin(SatAtt[2]);
RotMat[3][1]=0;
RotMat[3][2]=sin(SatAtt[2]);
RotMat[3][3]=cos(SatAtt[2]);
// Find angles for the beam
for (counter=1;counter<4;counter++) {
Poss[counter]=SatPoss[counter]-GroPoss[counter];
}
for (counter=1;counter<4;counter++) {
Poss[counter]=Poss[1]*RotMat[counter][1]+
Poss[2]*RotMat[counter][2]+
Poss[3]*RotMat[counter][3];
}
for (counter=1;counter<4;counter++) {
Poss[counter]=Poss[counter]
/sqrt((Poss[1])^2+(Poss[2])^2+(Poss[3])^2);
}
// Calculation of phase excitations
for (counter=1;counter<8;counter++) {
xcoff[counter]=Poss[1]*arrLoc[counter][1]
+Poss[2]*arrLoc[counter][2];
vgaSet[counter][1]=cos(2*PI*xcoff[counter])
vgaSet[counter][2]=sin(2*PI*xcoff[counter]);
vgaSet[counter][3]=-sin(2*PI*xcoff[counter]);
vgaSet[counter][4]=cos(2*PI*xcoff[counter]);
}
// Convert to integers for VGA settings
}
229
• Doppler Shift Compensation From the GPS coordinates of the satellite and
the GPS coordinates of the groundstation, the Doppler shift can be calculated.
# ifndef PI
# DEFINE PI 3.1415;
# endif
# ifndef SOL // speed of light
# DEFINE SOL=3e8;
# endif
void DopplerComp(float[] SatPoss, float[] SatVeloc, float[] GroPoss) {
/* calculate the voltage offset for the VCO
in order to compensate Doppler shift
based on
SatPoss: xyz vector with GPS coordinates of the satellite
SatVeloc: xyz velocity vector of the satellite
GroPoss: xyz vector with GPS coordinates of the groundstation
*/
int offSet;
//
//
float vtowa;
//
float fc=2e9; //
float Poss[3]; //
float fd;
//
int counter;
offset for the Rx VCO (Tx VCO has opposite sign
and is scaled)
speed towards groundstation
carrier frequency
direction vector between satellite and groundstation
deviation frequency
// Find angle between velocity vector and direction vector
for (counter=1;counter<4;counter++) {
Poss[counter]=SatPoss[counter]-GroPoss[counter];
}
vtowa=(SatVeloc[1]*Poss[1]+SatVeloc[2]*Poss[2]
+SatVeloc[3]*Poss[3])/sqrt((Poss[1])^2+(Poss[2])^2+(Poss[3])^2)
// Calculation of Doppler shift
fd=fc*(1+vtowa/SOL);
// Convert to voltage offset of VCO
. . .
}
230
Appendix C. Pseudo Code of the Array Control Routines
• Activation Message
void ActMes(time_t NextTermTime, GroStaSerNr) {
/* compose up the activation message
with the information from the querying schedule
received from the TT&C groundstation
namely
NextTermTime: timestamp when to query next terminal
GroStaSerNr: serial number of ground station terminal to activate
*/
char TestSeq[]={. . .}; // test sequence
char ActMess[];
//activation message
time_t GuardTime=5;
// guard time to avoid overlap
time_t EndTime;
// stop time for this terminal communication
// Composition of Activation Message:
EndTime=NextTermTime-GuardTime;
sprintf(ActMess,‘‘%s%d%d’’,TestSeq,GroStaSerNr,EndTime);
}
Appendix D
Eavesdropping on Computer
Displays
This appendix elaborates on one example of information leakage through EM radiation, namely on what is referred to as Van Eck phreaking.
Back in 1985 [194] was the first to deal with the subject in the open literature. Again,
military and security agencies are supposed to apply the techniques much longer.
But [194] suggested that with a very modest investment, the technique was available
for anyone. This can be questioned, referring to the expensive receiving equipment
used in [195]. But tests reported in [196] clearly indicate that reconstruction of
information displayed on screens from a signal captured with an antenna is possible.
The equipment used in [196] is considerably cheaper than the military receiver used
in [195]. But if an FPGA could be used instead of an oscilloscope, cost would decrease
even more.
This appendix focusses on implementing the reconstruction of a screen image on
FPGAs, which is an ongoing master thesis project at the moment of writing. Much
general background information is omitted, as can be extensively found in [196].
D.1
Source of Radiation
Images are written on computer displays in a way similar to the way an image is
written on a TeleVision (TV). An electron beam is scanned over the entire screen,
line after line, hitting the phosphor dots that glow green, red or blue when hit by an
electron. By modulating the intensity of the electron beam, the intensity of a pixel
can be varied.
231
232
Appendix D. Eavesdropping on Computer Displays
The signal to control the intensity of the electron beam, as well as the signals to
control the blanking and fly back of the electron beam at the end of a line (HSync)
or screen (VSync) are send to the screen by the video card of the computer, over a
cable, using e.g. the Super Video Graphics Array (SVGA) standard for a 600 by 800
pixels resolution.
Both the ion beam and the cable radiate EM fields containing the video information.
As this is a digital signal, the use of a pixel clock, and the fact that the intensity
signal keeps its value over one pixel clock cycle, results in a Power Spectral Density
(PSD) of the video signal:
Wp (f ) = fp
sin (πf /fp ) X
W (f − nfp )
πf
n
(D.1)
calculated with the formulas for PSD of a standard flat-top sampled information
signal, as e.g. in [6]. fp is the pixel clock frequency. This PSD indicates that the
baseband information is repeated around the harmonics of the pixel clock frequency.
Without the low frequency component, however, as there the sinc function becomes
zero. The PSD is plotted in Fig. D.1. The impact on the image when using a repetition
around a harmonic instead of the BB signal, is illustrated in Fig. D.2. One sees that
the lack of low frequencies complicates interpretation of the image by the human eye,
mainly because only the transitions (and not the constant parts) on the horizontal
lines are visible. Hence all horizontal parts of the letters disappear. Using a routine
that checks for deviations of the mean value of the screen results in Fig. D.2(c), a
good starting point for pattern recognition, where all transitions are black. Here the
pixel values equal:
P
vSV GA 255 × vSV GA − 600×800
(D.2)
vp =
P
vSV GA max vSV GA − 600×800
with vp the resulting pixel value in the reconstructed image and vSV GA the original
SVGA pixel value.
Wp (f )
−3
−2
−1
1
2
Figure D.1: Schematic representation of a PSD of a video signal.
3
f /fp
D.2. Screen Reconstruction
233
(a) Image reconstructed from SVGA signal at BB.
(b) Image reconstructed from SVGA signal around 7th harmonic.
(c) Image reconstructed from SVGA signal around 7th harmonic with deviation from mean.
(d) Image reconstructed from antenna signal around 7th harmonic with deviation from mean.
Figure D.2: Effect of using a repetition instead of BB version of the video signal.
D.2
Screen Reconstruction
In order to intercept the signal radiated by the cable and screen, any antenna can
be used. The working frequency range will determine around which harmonics the
repeated signal can be captured. As Fig. D.1 clearly shows, the signal around the
first1 harmonics is strongest. But low frequencies require large antennas to obtain a
reasonable antenna gain. That, together with the level of ambient EM radiation, will
determine which antenna suits the most in a specific case.
For the example of Fig. D.2(d), an Anritsu MS2721A spectrum analyzer was used
to determine the pixel clock at 40.08 MHz by looking at the nulls of the PSD of the
SVGA signal. The 7th harmonic of this clock lies in the frequency band of a folded
dipole determined in [196] to be 88 − 108 MHz and 260 − 360 MHz.
The antenna signal of only one frame2 might not be sufficient to reconstruct an image
in case of ambient noise. To improve the quality of the reconstruction, multiple
frames can be captured and combined. This combination will eventually average out
the noise and keep the image, as can be seen on Fig. D.3. This combination is done
in a digital way, but still many ways of doing so are possible. An overview is given in
Table D.1
A first possibility is to combine the signals at RF, as in [196]. The antenna signal is
digitized by an oscilloscope. If desired, multiple recordings from different frames can
be combined. Then the digital signal is mixed down and filtered in a digital way and
converted into a screen image. The disadvantages of this technique are the bandwidth
requirements on the ADC and the wide band noise coming in, limiting the accuracy
of the digitization.
1 In fact the signal around 0 Hz is even stronger. But with ω = 0 no propagation takes place in
the Maxwell equations. Hence this signal will never reach an antenna.
2 One scan of the electron beam over the entire screen.
234
Appendix D. Eavesdropping on Computer Displays
(a) Image reconstructed from 1 frame in the antenna signal.
(b) Image reconstructed from 5 frames in the antenna signal.
(c) Image reconstructed from 25 frame in the antenna signal.
Figure D.3: Effect of averaging multiple frames on the image quality.
An analog filter solves this problem, but if another harmonic is selected due to changing ambient noise, the filter has to be redesigned. A more standard way is mixing the
antenna signal to BB and applying a LPF prior to digitization. This way, the same
LPF can be used, regardless of the chosen harmonic. An ADC with a bandwidth as
small as 50 MHz is sufficient in this case. Moreover the processing effort for averaging
and image reconstruction decreases tremendously. This however requires that the
phase between LO signal used for mixing and harmonic of the clock in antenna signal is constant over all measurements. Otherwise, the random phase of the different
frames would cancel out the signal by destructive summing. Consequently the LO
should be derived from the pixel clock (or HSync) by using a PLL.
An easy way to circumvent this problem, is using an envelope detector as built-in in
the receiver in [195], such as a diode detector see Fig. 9.2, at BB to at least prevent the
signal from becoming negative and canceling out. An equivalent, but more expensive
way regarding processing cost, is not combining the digital signal, but the image,
after applying Eq. (D.2). Implementing this technique in MATLABTM resulted in
the reconstructions of Fig. D.3. The same technique was implemented on FPGA for
a master thesis, ongoing at the time of writing, and gave similar results.
The measurement setup is depicted in Fig. D.4. An antenna is connected to a mixer,
fed by a sweeper generating the LO, followed by a LPF and ADC. An FPGA extracts
a screen image from each frame, averages out several screen images and writes the
result to an attached computer display.
Table D.1: Comparison of screen reconstruction methods.
combining method
signals at RF
images from RF
signals at BB
images from BB
ADC Cost
high
high
low
low
Noise
high
high
low
low
Processing Power
moderate
high
low
moderate
LO generation
phase with VSync
standard
D.2. Screen Reconstruction
235
FPGA
screen
Figure D.4: Measurement setup for eavesdropping on computer displays.
invisible filling
Appendix E
A KeeLoq Transceiver
KeeLoq, a block cipher designed in the ’80 of last century, used for wireless authentication such as car immobilizer, is lately proven to be insecure by several parties.
A power analysis revealed the secret key with considerable ease in [107]. A mathematical combination of slide and meet-in-the-middle attack in [106] also succeeded in
extracting the key from 216 plaintext-ciphertext (or challenge-response) pairs. In this
appendix, the transceiver is described that was used to collect 216 plaintext-ciphertext
pairs to implement and test the attack on KeeLoq as described in [106]. A photograph
can be found in Fig. E.1
Although, the attack requires 216 known plaintext-ciphertext pairs, the implementation allows to have 216 chosen plaintext and the corresponding ciphertext pairs. A
Xilinx virtex II FPGA was used to cycle through 216 plaintexts of 32 bit length, starting with the all zero sequence, 0x00000000 and increasing by one until 0x0000FFFF.
From the 32 bit challenge, the envelope of the RF ASK signal was derived, as explained in [197], namely three or five time intervals1 (TE = 200 µs) high, depending
on the bit being 0 or 1, followed by one interval low. Hence transmitting a challenge
requires at most2 38.4 ms.
Next the envelope is inverted and amplified by an opamp, MC33174PG, to obtain
a signal suitable to control a transistor, a PNP BJT 2N2904, in the power line of a
class E amplifier that can modulate the power level. The square wave needed by the
class E transistor itself, a N-MOSFET ZVN2106A, is also generated by the FPGA,
by toggling a signal between high and low at each rising clock edge. Consequently,
the FPGA is clocked at 250 kHz, to obtain a 125 kHz square wave. As was the case
for the envelope, this square wave has to be amplified by an opamp too. Indeed, logic
FPGA outputs always are buffered as the FPGA is not designed to deliver current.
1 The system can also be used in another mode with T = 100 µs, by simply flipping a bit in the
E
transponder configuration.
2 With the challenge being 0xFFFFFFFF.
237
238
Appendix E. A KeeLoq Transceiver
Figure E.1: Photograph of the KeeLoq transceiver.
The class E amplifier was again designed according to the formulas given in Sect. 9.3.3.1.1.
Starting form QRLC < 28.2 from [197], and the transmitting loop resistance Rl = 2 Ω
and inductance L = 44 µH, as measured at 125 kHz with a HP4275A LCR meter,
this results in Cser = 39.5 nF, Cpar = 127 nF and Lchoke = 120 µH. An external
resistance was not needed, as Rl = 2 Ω > 1.22 Ω = ωL/QRLC . The circuit is drawn
in Fig. E.3. The voltage over the loop, when transmitting the opcode, is given in
Fig. E.2.
V
7.0
3.5
0
0
1
2
3
4
5
6
7
8
t [ms]
Figure E.2: Envelope of voltage over transmitting loop sending opcode.
Indeed, as can be seen on Fig. E.1, the transmitting loop has Nt = 18 multiple turns,
which seems in contradiction with the discussion in Sect. 8.3.1.3. However, as in this
case fc = 125 kHz, and rl = 7 cm, a total wire length of λ/10 requires more than
1000 turns. Moreover, as ωL is so small, it is even negligible compared to Rl . Still
Nt = 18 does not require an external resistance to keep QRLC < 28.2. Hence the
conclusions of Sect. 8.3.1.3 are not valid for frequencies as low as fc = 125 kHz, and
loop radii as small as rl = 7 cm.
239
100 kΩ
100 kΩ
Vcc = 5 V
−
+
envelope
120 µH
100 kΩ
39.5 nF
100 kΩ
to Rx
−
+
127 nF
44 µH
square
Figure E.3: Circuit of the Tx part in the KeeLoq transceiver.
The KeeLoq transponder, or key, sends back the response (or ciphertext) by means of
load modulation. Hence an amplitude variation can be observed at the transmitting
loop, due to inductive coupling between transponder and reader coil. Now the amplitude is higher for TE , followed by a lower amplitude for one or two time intervals
TE , depending on whether a zero or one is transmitted. The envelope of the voltage over the transmitting loop is depicted in Fig. E.4. Consequently, transmitting
a response takes at most 19.2 ms. Taking guard time, start pause of 2TE , opcode
(which is 0b10001 for IFF challenge) and challenge-response into account, collecting
one pair requires at most 94.8 ms. Collecting 216 pairs is finished in less than 91 min,
in TE = 200 µs mode.
V
7
6
t [ms]
0
1
2
3
Figure E.4: Envelope of voltage over transmitting loop receiving response.
In order to decode the ciphertext from the transmitted response, an oscilloscope
probe can be connected over the transmitting loop. Down mixing or demodulation
and decoding result in the binary sequence.
240
Appendix E. A KeeLoq Transceiver
In order to lower the sample frequency needed from twice the carrier frequency of
125 kHz to twice 1/TE = 5 kHz, a simple diode detector can be used. Adding a
comparator, LM339N, after the diode detector even lowers the sample frequency to
once every TE , and moreover supplies FPGA compatible logic levels. Consequently, a
diode detector and comparators render the oscilloscope superfluous, and the response
can be decoded by the same FPGA used for transmitting the challenge.
One comparator was used for the challenge, one for the response. Essentially, these
are data slicers, converting an ASK signal into bits. The first one requires a reference
signal that is slightly (e.g. the forward voltage drop of a diode) less than the unmodulated carrier voltage, and compares with the rectified antenna signal. The second one
can use the same reference signal, when dividing the rectified antenna signal slightly,
to be below or above the reference depending on the value of the load in the transponder or key. To allow for adjustments if necessary, a tunable resistor was used for this
voltage division. The output of the comparators can then be applied to the gate
of a transistor, a BC337, pulling an FPGA input pin down when appropriate. The
circuit to obtain reference and both comparator input signals, is shown in Fig. E.5.
The comparators and the FPGA pin drivers are drawn on Fig. E.6. Obviously, the
challenge is known by the FPGA and the circuitry for decoding the challenge can be
discarded.
D
to Tx
6.8 nF
100 kΩ
resp
10 kΩ
10 kΩ
50 kΩ
6.8 nF
+
−
D
∞
chal
ref erence
1 MΩ
22 µF
Figure E.5: Circuit of the Rx part in the KeeLoq transceiver.
241
to FPGA in
Vcc = 5 V
resp
100 Ω
1 kΩ
+
−
ref erence
Vcc = 5 V
to FPGA in
100 Ω
1 kΩ
chal
+
−
Figure E.6: Circuit to interface the Rx part with the FPGA in the KeeLoq transceiver.
invisible filling
Appendix F
A Low Cost VNA
This Appendix, part of [198], explains how to obtain a low cost VNA from an oscilloscope and a function generator. The technique turned out to be relatively accurate.
The idea for this implementation is from ir. P. Delmotte.
To measure S11 , a sine wave of a certain frequency generated by an Agilent 33220A
function generator, was applied to a RG-58 50 Ω coax cable. At the end of the cable,
the loop and a scope were connected with a BNC-T connector to the end of the cable.
If the input impedance of the Tektronix TDS2024 scope, namely 1 MΩ, is regarded
as infinite, the signal measured with the scope is the actual voltage at the end of the
coaxial cable:
(F.1)
Vscope = V1+ + V1− = V2+ + V2−
The setup and symbol conventions can be found on figure F.1.
Now S11 is found as:
S11 =
V1−
Vscope − V1+
=
V1+
V1+
(F.2)
where Vscope can be read from the scope and V1+ from the function generator. All
voltages are complex values, as the phase is also of importance. Suppose that V +
and V − have the same magnitude, but are 120◦ out of phase, so that |Vscope | = |V1+ |,
then S11 = 1 whereas a scalar sum would give S11 = 0.
In order to obtain a complex measurement of the voltage Vscope , the signal V1+ should
also be connected to the scope. The best way is to use another signal source, synchronised with the first one, that generates exactly the same sine.
243
244
Appendix F. A Low Cost VNA
If a splitter is used, the reflections at the second port of the scope, used for reference,
might reach the first port and disturb the measurement. An attenuator of 10 dB at
each cable was used to suppress1 the reflection 20 dB more than the useful signal,
avoiding the more expensive setup of two sources.
scope
−10 dB
R T
function generator
V2−
−10 dB
V1+
V1−
V2+
loop
Figure F.1: Setup for the S11 measurements below 45MHz.
1 Adding attenuators to suppress reflections is a very common technique in antenna gain measurements [199]. Reflections have to pass the attenuator at least twice, as opposed to the signal that is
only attenuated once.
Appendix G
RFID Basics
Radio-frequency identification (RFID) is a wireless technique to identify and track
items. The items that should be identified or tracked are all tagged with a chip
that responds to the fields of an RFID reader by sending the requested information.
Depending on the environment and the specific application, other technological implementations are choosen. Hence the variety of active and passive tags, inductive
and capacitive tags and LF, HF, UHF or microwave tags.
The distinction as defined in [200], specifying an active tag as a tag that is able to
produce a radio signal as opposed to a passive tag which reflects and modulates a
carrier signal received from an interrogator, as by far the most convenient one. Having
a battery is not sufficient for a tag to be active. Besides sending a radio signal, a
battery in an RFID tag can have many other purposes: power up sensors, logging
data or maintaining volatile memory. A tag with battery that is not used to send a
radio signal is sometimes called a Battery Assisted Passive tag (BAP).
G.1
Different Transmission Systems
Depending on the frequency, several different transmission methods are used to provide a communication channel between reader and tag. The frequency and the fact
wether the tag is active or passive, rules in many case out some techniques and
definitely discourages others. Nevertheless there is no one-to-one relation between
frequency, active or passive and the system used for the link.
245
246
Appendix G. RFID Basics
The first two techniques, inductive and capacitive coupling, suppose that tag and
reader are in each others vicinity, as otherwise the modeling of the mutual inductance
or capacitance between tag and reader as lumped element, does not make sense. The
third method, back scattering, supposes the tag to be in the far field of the reader
antenna. Last method is supposed to work for small and large distance between tag
and reader.
G.1.1
Inductive Coupling
Just as coils in a voltage transformer can couple magnetically and pass power from
the primary to secondary windings, a reader with a coil or loop antenna will be able
to couple power and data to a tag with a coil or loop. Then the alternating current in
the transmitter coil will generate a magnetic field that is picked up by the receiver coil
and will induce a voltage in the receiving chain. Let ZT x and ZRx be the impedance
of the transmitter and receiver and LT x , LRx , RT x and RRx the inductances and
resistances of the reader and tag loop. The setup and its equivalent circuit is then
depicted in Fig. G.1. By changing the impedance of the receiver (e.g. by switching in
and out some extra resistor or capacitance), the voltage over the reader coil can be
altered and information can be sent back to the reader. This technique is commonly
referred to as load modulation.
Energy
PICC
Data
PCD
(a) setup
ZT x
Vreader
tag
RT xRRx
ZRx
LT xLRx
Vinduced,tag
(b) equivalent circuit
Figure G.1: Schematic representation of inductive coupling.
G.1. Different Transmission Systems
G.1.2
247
Capacitive Coupling
Similar to inductive cross talk being used for communication, also capacitive cross
talk can be used. In this case essentially a capacitor at the output of the transmitter
is used that has a higher capacitance to the ground of the reader circuit, via the two
metal plates in the tag, than directly to the ground of the reader circuitry. Let ZT x
and ZRx be the impedance of the transmitter and receiver and CT x , CRx1 and CRx2
the capacitance at the output of the reader and between reader and tag, and tag to
reader ground respectively. The setup and its equivalent circuit is then depicted in
Fig. G.2 By changing the load between the two tag capacitors, the voltage over the
reader output capacitor changes, allowing to pass information back to the reader.
Energy
PICC
Data
PCD
(a) setup
ZT x
Vreader
CRx1
CT x
tag
ZRx
CRx2
(b) equivalent circuit
Figure G.2: Schematic representation of capacitive coupling.
G.1.3
Back Scattering
As in any metal object current will flow, when excited with a magnetic field, any object
will scatter electromagnetic fields. The scattered power depends on the magnitude of
the currents and the size of the metal object. The setup is depicted in Fig. G.3 Hence
by changing the current amplitude on a metal plate in the tag, it becomes possible
to pass information back to the reader. In the reader, a circulator is used to split the
back scattered signal from the transmitted signal and allow reception.
248
Appendix G. RFID Basics
Energy
PICC
Data
PCD
Figure G.3: Schematic representation of back scattering.
G.1.4
Radio Transmission
If the tag holds a system that can emmit radiation on itself, the use of a standard
radio link becomes possible. For this system to work, separation either in time or
frequency domain, or other access division multiplexing is obligatory. The setup is
depicted in Fig. G.4
Data
battery
PICC
Data
PCD
Figure G.4: Schematic representation of radio transmission.
G.2
Link Budget
A distinction is made between whether the receiving antenna is in the near or far field
region of the transmitting antenna. Essentially the division of the space into these
regions depends on the antenna size. For small antennas1 , the transition between
(reactive) near field and far field region lies where kr < 1 or r < λ/(2π) as there the
transition between 1/r3 and 1/r domination in the expressions for elementary dipoles
in Sect. 8.1.1 occurs.
In the near field, depending on whether the transmitting antenna is electrical or
magnetical in nature, the electrical or magnetical field will be much stronger. In the
far field region, the Electric field |E| ≈ 120π|H| the magnetic field strength. As a
consequence, no distinction between electric and magnetic far field are made.
1 For larger antennas, the transition lies where r < 2D 2 /λ as there the wavefronts will become
reasonably planar. In this definition part of the near field is radiating. The near field formulas given
here only apply in the reactive near field.
G.2. Link Budget
G.2.1
249
Electric Far Fields
Starting from the power transmitted by the reader PT X , the reader and tag antenna
gain GT X and GRX respectively, the received power can be calculated based on Friis
formula:
GT X GRX
λ2
PRX = PT X ×
×
(G.1)
4πR2
4π
One tends to conclude from this equation that the received power will be larger if lower
frequencies and thus longer wavelengths are used. This is however a misperception
as it becomes easier to build antennas with a higher gain at higher frequencies. If a
parabola dish is taken as an example, this can be illustrated by looking at the formula
of the gain G of the parabola dish:
G=
π 2 D2
λ2
(G.2)
with D the diameter of the dish. At low frequencies, it becomes impractical to build
a dish that is reasonable large compared to the wavelength. If a fixed physical size is
used, λ falls out of the equation.
If attenuation due to trees and other obstacles, or even raindrops and gasses when
the frequency is above 10 GHz, are taken into account, in general, the received power
is inverse proportional to the wavelength.
In the case of an active tag, equation (G.1) is valid for the tag to reader communication
as well, but in that case the tag becomes the transmitter (Tx) and the reader the
receiver (Rx). In the case of a passive tag, the power that is received by the reader,
can be calculated from the power transmitted by the reader, again starting from
equation (G.1), but with addition of the scattering term [201]:
PRX = PT X ×
GT X GRX λ2 4Rrad
(4πR)2 |Za + Zc |2
(G.3)
where Rrad is the radiation resistance of the tag antenna, Za and Zc the impedance
of the antenna, respectively the IC in the tag.
G.2.2
Electric Near Fields
With the reasonable assumptions that the tag is kept parallel to the reader and that
both plates of the tag and reader have the same shape and size, CRX1 ≡ CRX2 =
2 × CRX . Hence solving the circuit of Fig. G.2(b):
PRX = PT X
RT X ω 2 ((CRX
2
RRX ω 2 CRX
2 Z2 ) + R
2
2 2
+ CT X )2 + ω 2 CT X CRX
RX ω CRX
RX
(G.4)
250
Appendix G. RFID Basics
Matching would require ZT X to be the complex conjugate of the load, but as CRX
changes with reading range, the load changes depending on the read out distance.
Moreover the mismatch in the load is used to transfer data back.
Indeed, the voltage over CT X contains the information that modulates ZRX :
V CT X = V T X
G.2.3
1 + jωCRX ZRX
1 − ω 2 CT X CRX ZRX ZT X + jω (CRX ZRX + ZT X (CRX + CT X ))
(G.5)
Magnetic Near Fields
Looking at the circuit in Fig. G.1(b), the power transfer formula becomes:
PRX = PT X
ω 2 M 2 RRX
2RRX (RRX + jωLRX + ZRX )2
(G.6)
In this case, again, matching is impossible due to the change of M with the read out
distance. Assuming matching in the unloaded case, i.e. ZT X = RT X − jωLT X and
ZRX = RRX − jωLRX , (G.6) becomes:
PRX = PT X
ω2 M 2
8RRX RT X
(G.7)
which is in accordance with [160].
Again the mismatch in the tag is modulated to pass data back to the reader. This
data can be read out from the voltage over LT X :
VLT X = VT X
G.3
(ZT X
jωLT X (jωLRX + RRX + ZRX ) + ω 2 M 2
+ RT X + jωLT X )(ZRX + RRX + jωLRX ) − ω 2 M 2
(G.8)
ISO-14443A RFID Standard
ISO-14443A 13.56 MHz RFID systems, defined in [183, 158, 202, 203] use inductive
magnetic coupling and a battery-less tag, as explained in Sect. G.1.1. The reader
or proximity coupling device (PCD) transmits a query to the tag or proximity IC
card (PICC) by a 100 % amplitude modulated (AM) magnetic field. A pauze at the
beginning of a bit interval indicates a zero. For a one the silence starts at half the
interval. The specifications for the pauze are delineated in Fig. G.5. With 2 µs <
t1 < 3 µs, t2 > 0.7 µs, t3 < 1.5 µs and t4 < 0.4 µs. The tag modulates the load with
Manchester encoding. A higher voltage at the reader coil during the first half bit
interval indicates a one, otherwise a zero was sent.
G.3. ISO-14443A RFID Standard
251
110%
90%
60%
5%
5%
t
60%
90%
110%
t4
t1
t2
t3
Figure G.5: Specifications of the pause in the PCD signal as defined in ISO14443
To provide enough energy to the PICC to power up, the magnetic field strength at
the tag should be at least 1.5A/m. The bit rate is 106 kbit/s.
The read out distance can not be infinite, because inductive coupling from the tag
to the reader implies that the tag must be in the near field of the reader. Indeed,
if the tag receives a traveling wave instead of a quasi-static field, the modifications
to the field due to the tag will never travel back to the reader. This means that the
tag must be located in the reactive near field [160] around the antenna, in this case
rd λ/(2π) ≈ 4 m.
Most COTS readers are designed for a reading range of 10 cm. Kfir and Wool [204]
and Kirschenbaum and Wool [152] have designed a system for relay attacks and a
low-cost, extended range RFID skimmer respectively. In these papers, little focus is
put on the actual design of the antenna and the relevant theory to achieve larger read
out distances with it. [150] filled the gap. [177] focusses on a powerful amplifier to
boost the signal in between the two building blocks, reader and antenna, as shown in
Fig. G.6. With an extended reading range and the fact that most implementations
are Reader Talks First (RTF), it is possible that tags can be read without the owner
being aware of it.
RFID logic
Class E
Antenna
Figure G.6: Schematic of the building blocks of the altered RFID system
invisible filling
Appendix H
Determining the Inductance
of a Coil
When modeling a coil or loop (antenna), the major parameter is the inductance of the
loop L. To find this value is not straightforward, especially not if a reasonable accuracy
is desired.If only an order of magnitude for L is sufficient, simply approximate L by
1 µH per meter of wire used.
Many formulas can be found. They give a good approximation, but are only valid
for specific types of coils, and moreover only for low frequencies i.e. only in case the
perimeter of the loop is small compared to the wavelength. It is possible to calculate
the value exactly with an integral, but this becomes rapidely time consuming for
larger loops as the area in the loop must be discretized. Moreover these formulas
assume a certain geometry e.g. circle or square, whereas the manufactured loop has
imperfections.
For these reasons, it is even better to measure the value for L. Several techniques can
be used, but measuring S11 and deducing Zin is the most accurate one.
H.1
Calculation
The way to calculate L, is to apply a current Il to a coil and calculate the magnitic
flux φB that is enclosed by the coil. For a circular loop with radius rl made of round
wire with radius rw , this leads to a numerical integration:
L=
φB
µ0 rl
=
Il
2
Z
0
rl −rw
x
Z
2π
0
rl − x cos α
3
(x2 + rl2 − 2xrl cos α) 2
253
dαdx
(H.1)
254
Appendix H. Determining the Inductance of a Coil
In figure H.1 is explained how the area of the loop is integrated.
dI
dα
α
rl
x
x
dx
Figure H.1: The integration over the area of the loop.
In literature an approximation for equation (H.1) is [169]:
8rl
−2
L = µ0 rl ln
rw
(H.2)
To illustrate the validity of the approximation, the result of formulas (H.1) and (H.2)
are plotted on figure H.2. The numerical integral of (H.1) gives a slightly higher result
due to the discretization. In this case 10000 division along the loop radius and 8 × 360
angular divisions were used. With each increase the result still converged more to the
result of (H.2).
L [µH]
4
formula (H.1)
formula (H.2)
3
2
1
0
0
0.1
0.2
0.3
0.4
rl [m]
Figure H.2: L as a function of rl (rw = 0.5 mm) calculted with formulas (H.1) and (H.2).
For a solenoide, the formula:
L=
rl2 N 2
9rl + 10l
with l the length of the solenoide, stays valid as long as l > 0.8rl [205].
(H.3)
H.2. Measurement
H.2
255
Measurement
The most straightforward and cheapest way to measure L, is to measure the current
through and the voltage over the inductor. Dividing both gives the impedance and
hence L:
Vl
(H.4)
ZL = jωL =
Il
This is the technique used in LCR meters, e.g. [206]. If no such device (for the
appropriate frequencies) is available, an oscilloscope can be used. Unfortunately the
voltage probe will also form a loop that picks up magnetic fields. This will induce a
voltage on the probe that voids the result. For lumped element inductors that do not
radiate, this technique is applicable. For loop antennas however, the use of voltage
probes should be avoided.
A technique that allows to avoid using a voltage probe, is putting the inductance in a
series RLC chain. Measuring the current drawn from the source with a current probe
when varying the frequency, will reveal the resonance frequency. This current probe
adds another inductance to the series chain and should hence be subtrackted again.
L = Lres − Lprobe =
1
2 C
ωres
− Lprobe
(H.5)
Again this technique is perfectly suited for lumped element inductors, but a loop
antenna does not have a constant input impedance over the entire frequency range.
Hence this technique reveals L at a certain frequency fres that is not necessarily the
frequency of interrest.
The most accurate, but most expensive way, is measuring the scattering parameters
of the antenna at the frequency of interrest. From this quantity, the input impedance
and hence L can be derived:
={Zin }
50 Ω
1 + S11
L=
=
=
(H.6)
jω
jω
1 − S11
This technique has the disadvantage that though the accuracy on the scattering parameter measurement might be good, the accuracy on L will be bad if S11 ≈ 1, which
can indeed be expected for an inductance.
Moreover all measurement techniques connect one side of the loop to the ground.
If the loop is of reasonable length so that wave phenomena occur, this results in
balanced-unbalanced incompatibility problems.
invisible filling
Appendix I
Standing Waves in
Measurement Setup
When the oscilloscope is put to an input impedance of 1 MΩ, oscillations and reflections occur. They are caused by the inductance of the sensor and the capacitive part
of the oscilloscope input impedance, or by mismatch of both sensor and oscilloscope
to the characteristic impedance of the cable. This appendix calculates the frequency
of the standing waves in a setup of a sensor, cable and oscilloscope as a function of
sensor inductance, cable parameters and oscilloscope input impedance.
I.1
Oscillations in a parallel RLC circuit
In the case an oscilloscope is used at high input impedance, Zs = RZ + jωCZ ≈
1 MΩ k ±10 pF. Directly connecting an inductive sensor to the oscilloscope will
add an inductance Lss to this parallel RC network, possibly causing a parallel RLC
resonance that spoils the signal to be measured. The resonance frequency is calculated
with Eq. (8.31).
I.2
Reflections
If an inductive sensor is connected to an oscilloscope at an input impedance different
from the characteristic impedance Zc of the connection cable, typically 50 Ω, the
impedance mismatch between cable and oscilloscope will cause the signal on the cable
to reflect back to the sensor.
257
258
Appendix I. Standing Waves in Measurement Setup
If the input impedance of the sensor neither matches the characteristic impedance of
the cable, an infinite number of reflections will take place. Let sss (t) be the signal
picked up by the sensor, the signal seen at the oscilloscope can be written as:
sos (t) = (1 + Γos )
∞
X
e−(2n+1)lα Γnos Γnss sss
n=0
(2n + 1)l
t−
vcab
(I.1)
with α the attenuation of the cable and Γ the reflection coefficient at the oscilloscope
(os) and sensor (ss) side.
Remark 1. Eq. (I.1) is only valid in case of real impedances. Complex impedances
give rise to a different reflection coefficient Γ for each frequency component and requires a Fourier transform to calculate the resulting waveform.
The spectrum of this signal at the oscilloscope is:
S̃os (f ) = e−lα S̃ss (f )(1 + Γos )
∞
X
−jω
(Γtot )n e
(2n+1)l
vcab
(I.2)
n=0
with S̃ss (f ) = F{sss (t)} the Fourier transform of the sensor signal and Γtot the resulting scaling factor between two consecutive arrivals of the signal at oscilloscope,
e.g. without amplifier:
Zss − Zc
RZ − Zc − jωZc CZ RZ
−2lα
−2lα
Γtot = e
Γss Γos = e
(I.3)
Zss + Zc
RZ + Zc + jωZc CZ RZ
This spectrum is dominated by the frequencies fref for which the additions for all n
are in phase. In this case standing waves will occur. In the following equation, the
left hand side is the phase of the component at fref of the (n + 1)th -arrival of the
signal at the oscilloscope. The right hand side is the phase of the first arrival at fref
when no reflection has occurred yet.
(2n + 1)l
∀n ∈ N : n × ∠Γtot (ωref ) − ωref
vcab
2l
∀n ∈ N : n × ∠Γtot (ωref ) − ωref
vcab
= −ωref
= k2π
l
+ k2π
vcab
(I.4)
where ∠Γtot is the phase of the scaling factor and k ∈ Z− . From this equation fref
can be solved in an iterative way and this has always a solution.
I.2. Reflections
259
Theorem 1. Eq. (I.4) has always a solution for ωres > 0 and k ∈ Z− .
Proof.
1. A solution for Eq. (I.4) where n = 1 is a solution for any value
n, because
2l
2l
if ∠Γtot (ωref ) − ωref vcab = k2π then n × ∠Γtot (ωref ) − ωref vcab = n × k2π =
k 0 2π with k 0 ∈ Z−
2. For ω = 0, the left hand side equals π, for ω = ∞, the left hand side equals
−∞.
3. 2ωl/vcab is continuous in the half open interval [0, ∞[, ∠Γtot as defined in
Eq. (I.3) can be shown to be continuous in the same interval by showing continuity of the real and imaginary part.
4. Point 2 and 3 state that the left hand side will take on every value between π
and −∞ including 2kπ for k ∈ Z− .
Fig. I.1 plots the lowest frequency fref as a function of the cable length in case CZ =
13pF and RZ = 1 MΩ for different Lss . This graph allows to determine the cable
length that avoids standing waves within the bandwidth of the oscilloscope if the
value Lss of the sensor is known.
9
10
Lss
Lss
Lss
Lss
8
= 2 nH
= 20 nH
= 0.2 µH
= 2 µH
fref [Hz]
10
7
10
6
10
0
1
2
3
4
5
6
7
8
9
10
l [m]
Figure I.1: Lowest frequency of the standing wave as a function of the cable length for
different Lss with CZ = 13pF and RZ = 1 MΩ.
260
Appendix I. Standing Waves in Measurement Setup
Fig. I.2 shows the frequency as a function of the cable length in case CZ = 13pF,
Lss = 0.2µH and RZ = 1 MΩ for different k in Eq. (I.4).
10
10
k
k
k
k
k
9
fref [Hz]
10
=0
=1
=2
=3
=4
8
10
7
10
6
10
0
1
2
3
4
5
6
7
8
9
10
l [m]
Figure I.2: Frequency of the standing waves as a function of the cable length for different k
in Eq. (I.4) with Lss = 0.2µH, CZ = 13pF and RZ = 1 MΩ.
The modulus of Γtot will be smaller than one for any practical situation. After a
certain number of reflections, the signal will be indistinguishable from the noise.
Remark 2. For k = 0 and n = 1 in Eq. (I.4), the solution can also be found based
on the equations in Sect. I.1. Adding a cable in between oscilloscope and sensor, will
transform the inductive load of the sensor to a value depending on the length l and
characteristic impedance Zc of the cable ([148]) at the oscilloscope side ( os):
Zos = Zc
ω
Zss + jZc tan ( vcab
l)
ω
Zc + jZss tan ( vcab
l)
,
(I.5)
with vcab the signal velocity in the cable. This results in an inductance Los :
Los =
Zos
.
jω
(I.6)
The combination of Los and the oscilloscope’s capacitance CZ cause a resonance at a
frequency fres or pulsation ωres = 2πfres :
r
1
2
.
(I.7)
ωres =
Los CZ
I.3. Validation
261
This frequency can only be calculated in an iterative way as Eq. (I.5) also depends
on fres . Eq. (I.7) can be rewritten with the aid of Eq. (I.6), (I.5) and, approximating
Zss ≈ jωLss , in:
res
l)
Zc − ωres Lss tan( ωvcab
,
(I.8)
ωres =
ωres
CZ Zc (ωres Lss + Zc tan( vcab l))
and can be shown to be equivalent with Eq. (I.4).
I.3
Validation
The theory above was validated with measurements. These are partially displayed
in Fig. I.3. Three different loops with different Lss from the EMCO 7405 near-field
probe set were used: the 1cm (nr. 903), 3cm (nr. 902) and 6cm (nr. 901) loop. The
values of the inductances of the loops were measured under the same conditions as
the actual standing wave frequencies. The influence of different vcab is quantified with
the use of three different cables with different dielectric material. All properties are
summarized in Table I.1.
Table I.1: The properties of the different cables.
Brand
Belden
Huber and Suhner
Fabbrica
Milanese
Conduttori
Type
9907 Ethernet coax
Sucoflex 102E
RG58C/U
l [m]
3.6
5
0.5
1
3.5
Zc [Ω]
50
50
vcab [m/s]
0.8c
0.77c
50
0.66c
9
10
k = 0, vcab = 66%c, Lss = 28.9 nH
k = 1, vcab = 77%c, Lss = 97 nH
k = 1, vcab = 80%c, Lss = 220 nH
8
fref [Hz]
10
7
10
6
10
0
1
2
3
4
5
6
7
8
9
10
l [m]
Figure I.3: Experimental validation of the theory on standing waves in Sect. I.2. CZ = 13pF
and RZ = 1 MΩ, Zc = 50 Ω. The crosses denote measured points.
invisible filling
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