Absolute Calibration of the Small Black Spider Antenna for the

Transcription

Absolute Calibration
of the Small Black Spider Antenna
for the Pierre Auger Observatory
von
Oliver Seeger
Diplomarbeit in Physik
vorgelegt der
Fakultät für Mathematik, Informatik und Naturwissenschaften
der
Rheinisch-Westfälischen Technischen Hochschule Aachen
Oktober 2010
angefertigt am
III. Physikalischen Institut A
i
Erstgutachter und Betreuer
Zweitgutachter
Prof. Dr. Martin Erdmann
III. Physikalisches Institut A
RWTH Aachen
Prof. Dr. Thomas Hebbeker
III. Physikalisches Institut A
RWTH Aachen
ii
Contents
1 Introduction
1
2 Phenomenology of Ultra-High-Energy Cosmic Rays
3
2.1
Energy Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
2.2
Origin of Cosmic Rays . . . . . . . . . . . . . . . . . . . . . . . . . .
5
2.3
Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
2.4
Anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
3 Extensive Air Showers
11
3.1
Components of an EAS . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.2
Experimental Methods of EAS Observation . . . . . . . . . . . . . . . 12
3.3
3.2.1
Measurement of the Lateral Shower Profile . . . . . . . . . . . 13
3.2.2
Measurement of the Longitudinal Shower Profile . . . . . . . . 13
Emission of Radio Signals from EASs . . . . . . . . . . . . . . . . . . 15
3.3.1
Radio Emission Mechanisms . . . . . . . . . . . . . . . . . . . 15
3.3.2
Simulation of EASs . . . . . . . . . . . . . . . . . . . . . . . . 17
3.3.3
Measurements of Radio Signals from EASs . . . . . . . . . . . 18
4 The Pierre Auger Observatory
21
4.1
Surface Detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.2
Fluorescence Detector . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.3
Radio Detector - AERA . . . . . . . . . . . . . . . . . . . . . . . . . 24
5 Fundamental Characteristics of Antennas
27
5.1
Vector Effective Height . . . . . . . . . . . . . . . . . . . . . . . . . . 27
5.2
Matching Theory and Antenna Impedance . . . . . . . . . . . . . . . 29
5.2.1
Conjugate Matching . . . . . . . . . . . . . . . . . . . . . . . 30
iv
Contents
5.3
5.2.2
Real Matching
. . . . . . . . . . . . . . . . . . . . . . . . . . 31
5.2.3
Transformation Factor . . . . . . . . . . . . . . . . . . . . . . 31
5.2.4
Reflection Coefficient . . . . . . . . . . . . . . . . . . . . . . . 31
Absolute Gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
5.3.1
Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
5.3.2
The Friis Transmission Equation . . . . . . . . . . . . . . . . 33
5.4
Group Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
5.5
A Roadmap for Obtaining the Vector Effective Height . . . . . . . . . 34
6 The Small Black Spider LPDA
37
6.1
Logarithmic Periodic Dipole Antennas . . . . . . . . . . . . . . . . . 37
6.2
Properties of the Small Black Spider LPDA . . . . . . . . . . . . . . 39
6.3
Cross Talk between Antenna Planes . . . . . . . . . . . . . . . . . . . 40
6.4
Relative Phasing of Signals Received by both Antenna Planes . . . . 42
6.5
Reflection Characteristics
6.6
Antenna Impedance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
. . . . . . . . . . . . . . . . . . . . . . . . 44
7 Simulating Antennas with NEC-2
7.1
7.2
7.3
49
The Concept of NEC-2 . . . . . . . . . . . . . . . . . . . . . . . . . . 49
7.1.1
Fundamental Integral Field Equations . . . . . . . . . . . . . 49
7.1.2
Means of Determining Current Expansion . . . . . . . . . . . 50
7.1.3
Means of Determining the Electric Field . . . . . . . . . . . . 52
Simulation of Antenna Characteristics . . . . . . . . . . . . . . . . . 53
7.2.1
Simulation of a Setup with One Antenna . . . . . . . . . . . . 54
7.2.2
Simulation of a Transmission Measurement Setup . . . . . . . 55
Ground Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
7.3.1
Signal Reflections by a Ground Plane . . . . . . . . . . . . . . 56
7.3.2
Signal Propagation Towards the Horizon . . . . . . . . . . . . 60
7.3.3
Other Interference Effects . . . . . . . . . . . . . . . . . . . . 61
Contents
v
8 The BBAL 9136 Biconical Antenna
65
8.1
Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
8.2
Simulation of the BBAL 9136 in NEC-2
8.3
Calibration Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
8.4
Reflection Characteristics and Antenna Impedance . . . . . . . . . . 69
. . . . . . . . . . . . . . . . 66
9 Absolute Gain
9.1
9.2
71
Initial Near-Field Measurements . . . . . . . . . . . . . . . . . . . . . 71
9.1.1
Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . 71
9.1.2
Frequency Dependence . . . . . . . . . . . . . . . . . . . . . . 72
9.1.3
Near Field and Far Field . . . . . . . . . . . . . . . . . . . . . 73
Horizontal Absolute Gain . . . . . . . . . . . . . . . . . . . . . . . . 74
9.2.1
Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . 74
9.2.2
Frequency Dependence . . . . . . . . . . . . . . . . . . . . . . 76
9.2.3
Azimuth Dependence . . . . . . . . . . . . . . . . . . . . . . . 78
9.2.4
Zenith Dependence . . . . . . . . . . . . . . . . . . . . . . . . 80
9.3
Reconstruction of the Simulated Vertical Gain . . . . . . . . . . . . . 80
9.4
Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
9.4.1
Reproducability of Absolute Gain Measurements . . . . . . . . 83
9.4.2
Signal Reflections . . . . . . . . . . . . . . . . . . . . . . . . . 84
9.4.3
Topography and Weather Conditions . . . . . . . . . . . . . . 85
10 Group Delay and Effect of a Low-Noise Amplifier
87
10.1 Group Delay of the Small Black Spider LPDA . . . . . . . . . . . . . 87
10.2 Group Delay of the BBAL 9136 Biconical Antenna . . . . . . . . . . 88
10.3 Uncertainty of the Group Delay . . . . . . . . . . . . . . . . . . . . . 89
10.4 Directional Dependence of SBS Group Delay . . . . . . . . . . . . . . 90
10.5 Effect of a Low-Noise Amplifier . . . . . . . . . . . . . . . . . . . . . 92
11 Reconstruction of the Vector Effective Height
97
11.1 Horizontal Component . . . . . . . . . . . . . . . . . . . . . . . . . . 97
11.2 Horizontal Component with a Low-Noise Amplifier . . . . . . . . . . 102
vi
Contents
12 Summary and Outlook
103
A Appendix
105
A.1 List of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
A.2 NEC-2 Simulation Files
. . . . . . . . . . . . . . . . . . . . . . . . . 106
A.2.1 Small Black Spider Steering Card . . . . . . . . . . . . . . . . 106
A.2.2 BBAL 9136 Biconical Antenna Steering Card . . . . . . . . . 107
A.2.3 Steering Card for a Transmission Measurement . . . . . . . . 108
A.3 Biconical Antenna Calibration Certificate . . . . . . . . . . . . . . . . 109
A.4 FSH 4 Spectrum Analyzer . . . . . . . . . . . . . . . . . . . . . . . . 111
A.4.1 Calibration of the FSH 4 (Transmission Measurement) . . . . 111
A.4.2 Quality of Calibration for a Transmission Measurement . . . . 111
A.5 RG-213 Coaxial Cable . . . . . . . . . . . . . . . . . . . . . . . . . . 112
A.6 Influences of Aachen Radio Background . . . . . . . . . . . . . . . . . 112
A.7 Defective Antenna . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
References
119
Acknowledgements – Danksagungen
121
List of Figures
2.1
Energy spectrum of cosmic rays . . . . . . . . . . . . . . . . . . . . .
4
2.2
The Hillas plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
2.3
Energy loss of primary particles . . . . . . . . . . . . . . . . . . . . .
8
2.4
Aitoff-Hammer projection of the sky in galactic coordinates . . . . . . 10
3.1
Illustration of an air-shower cascade . . . . . . . . . . . . . . . . . . . 12
3.2
Lateral shower profile of an extensive air shower . . . . . . . . . . . . 14
3.3
Longitudinal shower profile of an extensive air shower . . . . . . . . . 15
3.4
REAS1 and REAS2 simulations of the frequency-dependent radiopulse amplitude for various distances to the core of an EAS . . . . . . 18
3.5
REAS1 simulation of the radio-pulse amplitude depending on the primary particle energy for various distances to the core of an EAS . . . 19
3.6
Pulse amplitude measured with the LOPES experiment depending on
the distance to the shower axis . . . . . . . . . . . . . . . . . . . . . 20
3.7
Measured pulse amplitude depending on the energy of the primary
particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4.1
Southern site of the Pierre Auger Observatory . . . . . . . . . . . . . 22
4.2
SD station . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4.3
Photo of an FD building . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.4
Sketch of an FD telescope . . . . . . . . . . . . . . . . . . . . . . . . 24
4.5
Illustration of the Auger Engineering Radio Array . . . . . . . . . . . 25
4.6
A Small Black Spider LPDA in the Argentinian Pampa . . . . . . . . 26
5.1
Reference coordinate system of an antenna . . . . . . . . . . . . . . . 28
5.2
Equivalent circuit diagram for antenna read-out . . . . . . . . . . . . 29
6.1
The Small Black Spider LPDA . . . . . . . . . . . . . . . . . . . . . . 37
6.2
Basic concept of an LPDA . . . . . . . . . . . . . . . . . . . . . . . . 38
viii
List of Figures
6.3
A set of Small Black Spider antennas folded together . . . . . . . . . 41
6.4
Cross talk between SBS antenna planes . . . . . . . . . . . . . . . . . 42
6.5
Illustration for the two possible antenna plane configurations of an SBS 43
6.6
Relative phasing of SBS planes . . . . . . . . . . . . . . . . . . . . . 43
6.7
SBS prepared for setup . . . . . . . . . . . . . . . . . . . . . . . . . . 44
6.8
SBS set up for measurements . . . . . . . . . . . . . . . . . . . . . . 45
6.9
Reflection curve of Small Black Spider LPDA . . . . . . . . . . . . . 46
6.10 Absolute value of the Small Black Spider’s antenna impedance . . . . 47
7.1
NEC-2’s method of determining current expansion on wires . . . . . . 52
7.2
Cylindrical coordinate system for a NEC-2 wire segment . . . . . . . 53
7.3
Simulated horizontal absolute gain of SBS . . . . . . . . . . . . . . . 54
7.4
Simulated transmission measurement between two SBS LPDAs . . . . 55
7.5
Simulated free-space gain pattern in 3D . . . . . . . . . . . . . . . . . 57
7.6
Simulated 3D gain pattern including a ground plane . . . . . . . . . . 58
7.7
Illustration of a signal reflection from a ground plane . . . . . . . . . 59
7.8
Sketch concerning destructive interference towards the horizon . . . . 59
7.9
Power law of radiation density towards horizon . . . . . . . . . . . . . 61
7.10 Power versus distance between transmitter and receiver . . . . . . . . 62
7.11 Power versus height of antenna over ground . . . . . . . . . . . . . . 63
7.12 Illustration for the radio path difference in a setup including a ground
plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
8.1
The BBAL 9136 biconical antenna . . . . . . . . . . . . . . . . . . . 65
8.2
Simulated BBAL 9136 biconical antenna . . . . . . . . . . . . . . . . 66
8.3
Simulated total free-space gain of the biconical antenna . . . . . . . . 67
8.4
Frequency dependence of BBAL 9136 absolute gain . . . . . . . . . . 68
8.5
Reflection curve of BBAL 9136 biconical antenna . . . . . . . . . . . 70
8.6
Antenna impedance of BBAL 9136 biconical antenna . . . . . . . . . 70
9.1
Frequency dependence of near-field horizontal gain . . . . . . . . . . . 72
9.2
Simulated convergence towards horizontal absolute gain . . . . . . . . 74
List of Figures
ix
9.3
Sketch of the setup for a horizontal absolute gain measurement . . . . 75
9.4
BBAL 9136 prepared for a transmission measurement . . . . . . . . . 75
9.5
BBAL 9136 readily set up for a transmission measurement . . . . . . 76
9.6
Frequency dependence of horizontal absolute gain . . . . . . . . . . . 77
9.7
Degree scale on the workshop construction for the SBS . . . . . . . . 79
9.8
Azimuth dependence of horizontal absolute gain . . . . . . . . . . . . 79
9.9
Zenith dependence of horizontal absolute gain . . . . . . . . . . . . . 80
9.10 Method to reconstruct the vertical absolute gain . . . . . . . . . . . . 81
9.11 Simulated vertical absolute gain . . . . . . . . . . . . . . . . . . . . . 82
9.12 Reproducability of absolute gain measurements . . . . . . . . . . . . 83
9.13 Frequency-dependent deviation of minimum gain in azimuth pattern . 84
9.14 SBS absolute gain simulation for various ground types . . . . . . . . . 86
9.15 Weather conditions during horizontal absolute gain measurement . . . 86
10.1 Group delay of the Small Black Spider LPDA . . . . . . . . . . . . . 88
10.2 Group delay of the BBAL 9136 biconical antenna . . . . . . . . . . . 89
10.3 Spread from two SBS group delay measurements . . . . . . . . . . . . 90
10.4 Azimuth dependence of SBS group delay . . . . . . . . . . . . . . . . 91
10.5 Zenith dependence of SBS group delay . . . . . . . . . . . . . . . . . 91
10.6 Signal propagation for different frequencies and zenith angles . . . . . 92
10.7 Power gain of a low-noise amplifier . . . . . . . . . . . . . . . . . . . 93
10.8 Horizontal absolute gain of SBS with an LNA . . . . . . . . . . . . . 94
10.9 Frequency-dependent group delay of an LNA . . . . . . . . . . . . . . 95
11.1 Frequency dependence of horizontal vector effective height . . . . . . 99
11.2 Azimuth dependence of horizontal vector effective height . . . . . . . 100
11.3 Zenith dependence of horizontal vector effective height . . . . . . . . 101
11.4 Horizontal component of vector effective height with an LNA . . . . . 102
A.1 SBS steering card . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
A.2 BBAL 9136 steering card . . . . . . . . . . . . . . . . . . . . . . . . . 107
A.3 Steering card for a transmission measurement . . . . . . . . . . . . . 108
x
List of Figures
A.4 Biconical antenna calibration certificate (1) . . . . . . . . . . . . . . . 109
A.5 Biconical antenna calibration certificate (2) . . . . . . . . . . . . . . . 110
A.6 Calibration of FSH 4 before and after measurement . . . . . . . . . . 111
A.7 Attenuation of RG-213 coaxial cable . . . . . . . . . . . . . . . . . . 112
A.8 Relative radio spectrum during transmission measurement . . . . . . 112
A.9 Frequency dependence of received power fraction in transmission measurement (A001) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
A.10 Frequency dependence of received power fraction in transmission measurement (A036) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
1. Introduction
One of the most interesting phenomena in astroparticle physics are ultra-high-energy
cosmic rays (UHECRs). These particles of not yet determined origin travel across
the vastness of the universe at extremely high energies up to several 1020 eV. When
a particle of such enormous energy hits the Earth’s atmosphere, a particle cascade
containing millions of secondary particles is formed and showers down upon the planet’s surface. We call this phenomenon an extensive air shower (EAS).
The Pierre Auger Observatory in Argentina has been constructed to observe UHECRinduced extensive air showers. Since the flux of particles at extremely high energies is
very low, huge detectors are needed to observe a noticeable amount of them and thus
gain appropriate statistics. Hence the Pierre Auger Observatory contains a grid of
surface particle detectors covering an area of approximately 3100 km2 . They detect
secondary particles from EASs that reach the surface of the Earth. Another detector
is present in the form of fluorescence telescopes, which directly observe fluorescence
light in the sky that is produced by the interaction between shower particles and
air molecules. The duty cycle for this detector is very low, though, since moonless
nights are required to be able to observe the weak fluorescence light.
Both detectors are completely independent from each other. In consequence the
Pierre Auger Observatory operates the largest hybrid detector for the detection of
UHECRs in the world.
It is also possible to measure electromagnetic signals emitted by EASs. Due to
the properties of the air showers coherent electromagnetic radiation is emitted in
the radio regime. Test measurements have shown that it can be measured with
antennas operating in a suitable frequency range. A radio detector features several
advantages, e.g. a longer duty cycle than the fluorescence telescopes.
With the Auger Engineering Radio Array (AERA) a radio detector covering an area
of 20 km2 is about to be set up. For the first stage of the setup 24 “Small Black Spider” logarithmic periodic dipole antennas (LPDAs) – especially designed for AERA
– have been shipped to and set up in Argentina.
To reconstruct the electric field of an incident signal emitted by an EAS and measured with a radio detector, it is necessary to understand the detector’s response to
the incoming signal. Therefore, an absolute calibration of the Small Black Spider
LPDA is performed in this thesis.
Various characteristics of the antenna – including its directional sensitivity, impedance and group delay – are studied. Measurements are compared to simulations,
2
Introduction
discussing the importance of environmental conditions in realistic experimental setups and the relevance of the ground below the antenna. The effect of a low-noise
amplifier is investigated.
With the obtained results the voltage measured with the Small Black Spider LPDA
can be related to the absolute electric field strength of the incident signal.
2. Phenomenology of
Ultra-High-Energy Cosmic Rays
Cosmic rays have been discovered at the beginning of the 20th century. Up to then
scientists thought that the only source for the ionization of molecules in air is radiation, originating from the decay of radioactive materials in the Earth’s crust. It
was postulated that the ionization level of the atmosphere would decrease for higher
altitudes.
On April 12th in 1912 the Austrian physicist Victor Hess proved that this is incorrect.
During a balloon flight he measured that the ionization of the atmosphere increases
when gaining altitude [1]. He concluded that the origin of a major fraction of the
radiation responsible for the ionization of air molecules is extraterrestrial. For this
discovery of cosmic rays he was granted the Nobel prize in 1936.
Two years later, in 1938, Pierre Auger discovered that an incoming cosmic ray
signal can be detected coincidently with more than one particle detector, even if the
detectors are separated by a great distance. This lead to the conclusion that a cosmic ray induces a particle cascade when interacting with the Earth’s atmosphere. It
produces many secondary particles during this process, which are distributed across
several square kilometres when arriving on the planet’s surface. The phenomenon is
referred to as extensive air shower. Pierre Auger estimated the energy of the shower
particles to be in the order of 1014 eV [2]. In consequence he concluded that the primary particles are situated at energies greater than 1015 eV. Today primary particle
energies greater than 1018 eV are observed. They are referred to as ultra-high-energy
cosmic rays (UHECRs).
This chapter will give a basic overview on UHECRs. Their origin, propagation
and energy spectrum will be discussed.
2.1
Energy Spectrum
Throughout numerous experiments cosmic rays have been observed over a wide
energy range. The energy spectrum ranges from around 108 eV to beyond 1020 eV,
spanning more than twelve orders of magnitude. While for energies below some GeV
the spectral behaviour is dominated by magnetic fields of stellar origin, for higher
4
Phenomenology of Ultra-High-Energy Cosmic Rays
Figure 2.1: Energy spectrum of cosmic rays [3, 4, 5, 6, 7, 8, 9, 10, 11, 12].
energies the differential flux of cosmic rays with respect to their energy E and per
steradian of solid angle Ω can be described by a power law of the form
d2 φ
∝ E −γ ,
dEdΩ
(2.1)
where γ is referred to as spectral index. The energy dependence of the flux is shown
in figure 2.1. For energies from about 1010 − 1015.5 eV the differential flux from equation (2.1) decreases with a spectral index of γ ≈ 2.7. For each order of magnitude
in energy the flux decreases by almost a factor of 1000. At the so called “knee” of
the spectrum, where the spectral index changes to γ ≈ 3.1, the flux is so small that
only one particle per square metre per year can be observed.
At energies above 1018 eV we speak of ultra-high-energy cosmic rays. Situated
around 1018.5 eV is the “ankle” of the cosmic-ray spectrum upon which the spectral index reverts back to γ ≈ 2.7. Cosmic rays of higher energies are believed to
be of extragalactic origin [13]. Typical observation rates for particles at these energies are one per square kilometre per year or, in case of the highest energies above
1020 eV, even per century.
2.2. Origin of Cosmic Rays
5
Events at the highest energies are extremely rare. We show great interest in them
since currently particles cannot be accelerated to such high energies by any artificial
means. Even the most recent Large Hadron Collider experiment at CERN is only
able to accelerate particles to a centre-of-mass energy (CME) in the TeV1 regime.
An UHECR at an energy greater than 1022 eV causes a CME in the order of 1015 eV
when impacting the Earth since the CME formula
p
√
s ≈ 2mT E
(2.2)
√
for a fixed target experiment holds in that case. s is the CME, mT the mass of
the collision target and E the energy of the UHECR.
With UHECRs many fundamental properties of the universe, which are related to
their origin and propagation, can be studied.
2.2
Origin of Cosmic Rays
The favoured model describing the origin of cosmic rays has been developed by
Enrico Fermi [14]. According to this model cosmic rays are particles that travel
through interstellar space and are accelerated by magnetic fields that occupy this
interstellar medium. The dimensions of these magnetic fields are situated around
several light years.
A charged particle drifting through the interstellar medium will be deflected due to
the presence of a magnetic field. For a homogeneous field it will gyrate on a circular
trajectory of the Larmor radius
rg ∝
E15
.
ZBµG
(2.3)
E15 is the energy of the particle in 1015 eV and Z its atomic number. BµG denotes
the strength of the magnetic field component perpendicular to the particle’s velocity
vector. BµG is given in microgauss2 .
Magnetic fields in the interstellar medium are not homogeneous, though. They
are established according to the laws of the interstellar matter’s hydrodynamics,
resulting in a turbulent magnetic field structure [15]. The turbulences are caused
by shock wave pulses e.g. originating from supernova explosions. When a gyrating
particle encounters an irregularity in the field, it can gain energy. The gain of energy
is of the form
∆E
∝ βη,
(2.4)
E
with η being a model-dependent parameter [16]. The field irregularities can be
abstracted as extremely heavy obstacles moving at an average velocity of β = vc
relative to the vacuum speed of light c. In reality this is the speed of a plasma-wave
1
2
1 TeV = 1012 eV.
1 µG = 10−6 G = 10−10 T.
6
shockfront. In consequence, a single particle can be accelerated to great amounts of
energy, forming cosmic and even ultra-high-energy cosmic rays in the process.
To what extent a cosmic ray can be accelerated depends on the properties of the
turbulent magnetic field. For an increasing energy the Larmor radius of the particle
also increases. Eventually the particle will have enough energy to leave the zone of
the turbulent field. Hence it will not be accelerated anymore.
The size L of the turbulent field is strongly related to its source. A. M. Hillas related
the energy that a cosmic ray can be accelerated to by a turbulent magnetic field to
the size L of that field [17]. The upper limit of the energy is given by
1
E15 < ZβB µG L pc ,
2
(2.5)
with L given in units of parsec3 . Taking this condition into account, Hillas showed
which sources are able to accelerate particles to ultra-high-energy cosmic rays of a
certain energy, depending on the strength and size of the turbulent magnetic field
that a source causes. Figure 2.2 shows these possible candidates.
Objects below the green line do not have the properties to accelerate an iron nucleus
to an energy of 1020 eV. Candidates for protons with an energy of approximately
1020 eV are neutron stars, gamma-ray bursts, radio-galaxy lobes and active galactic
nuclei (AGNs). White dwarfs and the galactic halo would be able to accelerate an
iron nucleus to that energy.
2.3
Propagation
Eventually an UHECR will escape the turbulent magnetic-field zone surrounding the
accelerator and propagate through the universe. On its way it is likely to interact
with the cosmic microwave background, which was discovered in 1965 and is present
throughout the entire universe[19].
In the rest frame of a proton-UHECR photons from the cosmic microwave background appear as high energetic gamma rays. Interaction leads to a ∆+ resonance
of the form
p + γ −→ ∆+ −→ p + π 0 ,
(2.6)
p + γ −→ ∆+ −→ n + π + ,
in which neutral and positively charged pions are produced. This effect was predicted
by Greisen, Zatsepin and Kuzmin [20, 21] and hence was named GZK cutoff. The
energy in the rest frame of the UHECR needs to be sufficiently high to generate the
mass of the pions.
During the pion-production process the proton looses energy. This significantly
reduces the flux of particles at the highest energies. Figure 2.3 shows the mean
energy loss of high-energetic protons as they propagate through space.
The cross section for the ∆+ resonance from equation (2.6) depends on the proton’s
energy. The resonance is more likely to occur for higher energies. This results in a
3
1 pc ≈ 3.26 ly.
2.3. Propagation
7
Figure 2.2: Possible accelerator candidates for UHECRs. If the accelerated particle
is a proton of 1020 eV, the accelerator must be situated on or above the dashed line
(adapted from [18]).
convergence of the proton energy for large propagation distances, meaning that protons with different initial energies will end up with the same energy. For UHECRs
of the highest energies this will happen after a propagation length of a few hundred
Mpc. In consequence most UHECRs observed on Earth are likely to originate from
the galactic neighbourhood in the vicinity of 100 Mpc.
It is important to point out that – depending on their energy – cosmic rays are
deflected to different extents by locally varying magnetic fields. For low energies
the propagation can be compared to a diffusion process, meaning that the cosmic
rays propagate through the universe on chaotic trajectories. For high energies the
propagation can be described as a sequence of scattering processes. Applying the
8
Figure 2.3: Energy loss of UHECR primary particles. The four curves represent
the mean energy loss of protons starting propagation with different initial energies
[22].
constraint from inequation (2.5) for field strengths in the microgauss regime yields
the result that cosmic rays at energies smaller than 1018 eV are confined to the Milky
Way. The galactic magnetic field causes them to gyrate within the halo of our galaxy. Thus, cosmic rays at these energies cannot be backtracked to their sources
when observed on Earth since the original directional information is lost during the
gyration process.
For energies greater than 1018 eV particles are able to escape the galactic confinement. They propagate through intergalactic space and are deflected by extragalactic
magnetic fields in the order of nanogauss4 . According to simulations UHECRs can
be backtracked to their source for energies greater than 4 · 1019 eV since enough
information about the original propagation direction is preserved in that case [23].
4
1 nG = 10−9 G = 10−13 T.
2.4. Anisotropy
2.4
9
Anisotropy
Cosmic rays at lower energies are randomized with respect to their arrival direction when they are observed on Earth since they have gyrated through the galactic
halo. We expect an isotropic distribution of arrival directions for these cosmic rays.
As mentioned before, UHECRs of extragalactic origin can be backtracked to their
source for high enough energies since they have not been deflected too much. In
consequence they should not be randomized with respect to their arrival directions
when observed on Earth. Because of this we expect to see an anisotropy in the distribution of UHECR arrival directions in regions around possible source candidates
across the sky.
A possible anisotropy of UHECR arrival directions has been observed by the Pierre
Auger Observatory [24]. To achieve this the correlation between cosmic rays at the
highest energies and AGNs in the vicinity of the Milky Way has been studied.
318 AGNs that lie within the Pierre Auger Observatory’s field of view with distances
up to DMax ≈ 75 Mpc from Earth have been taken into account. Only UHECRs with
an energy greater than EMin = 55 EeV5 were checked for correlation since for lower
energies the deflections by interstellar magnetic fields are expected to be too strong
and the cosmic rays do not necessarily point back to their sources anymore. Furthermore, an UHECR was counted as correlated to an AGN if the angular distance
◦
ψ between AGN and incoming direction was less than 3.1 from the direction in
which the AGN is located. The reasons for this are the limited angular resolution
of the detector and the fact that even at the highest energies UHECRs are slightly
deflected by cosmic magnetic fields.
Figure 2.4 shows an Aitoff projection of the sky observed at the Pierre Auger Observatory. For the given parameters DMax , EMin and ψ, 69 UHECR events have
been observed. 55 of them have been suitable candidates to examine correlation
with AGNs. It was found that 13 out of 55 events are correlated to nearby AGNs.
Due to the choice of parameters a maximum amount of 11.5 out of 55 events would
have been correlated to AGNs on average in the case of an isotropic distribution
of UHECR arrival directions. Hence the observation of 13 out of 55 event correlations indicates a directional anisotropy of incoming UHECRs. However, more data
is needed since the confidence level is very low.
5
1 EeV = 1018 eV.
10
Figure 2.4: Aitoff-Hammer projection of the sky in galactic coordinates. The
area below the solid black line is the fraction of the sky that can be observed from
the southern Pierre Auger Observatory. The blue circles mark the positions of 318
◦
AGNs. Their radius corresponds to an angular distance of ψ = 3.1 and a darker
blue resembles a higher relative exposure. The black dots mark 69 UHECR events
that meet the criteria for DMax and EMin . If an event lies inside the circle belonging
to an AGN, the two are correlated to each other [24].
3. Extensive Air Showers
When an UHECR primary particle arrives in the Earth’s atmosphere it will interact
with an air molecule at typical altitudes of approximately 20 km, creating secondary
particles in the process. A chain of multiple interactions will form a particle cascade
that is called extensive air shower (EAS). If the primary particle is a nucleus heavier
than a proton with an atomic mass number A, the shower can be described as a
superposition of A proton showers.
The secondary particles propagate through the planet’s atmosphere at almost the
speed of light and form a particle front that in first order can be approximated as a
flat disk. The disk of an EAS contains millions of particles. In case of UHECRs the
shower disk’s radius may span several kilometres while its thickness is only situated
in the range of a few metres.
The properties of extensive air showers as well as their emission of electromagnetic
radiation in the radio regime will be discussed in this chapter.
3.1
Components of an EAS
Figure 3.1 shows an illustration of the particle cascade that forms an EAS. The first
interaction induces a dominantly hadronic cascade, in which 90 % pions and 10 %
kaons are produced. These particles decay in the form of
π + −→ µ+ νµ ,
π − −→ µ− ν̄µ ,
K + −→ µ+ νµ ,
(3.1)
resulting in a muonic component of the shower and neutrinos. Neutral pions π 0
decay into photons by
π 0 −→ γγ
(3.2)
and compose the electromagnetic component of the shower. The photons generate
electron-positron pairs that cause even more photons due to Bremsstrahlung. The
overall electromagnetic component of an extensive air shower forms about 90 % of
its particle content.
12
Extensive Air Showers
Figure 3.1: Illustration of an air-shower cascade induced by a proton. The shower
includes a muonic, hadronic and electromagnetic component as well as neutrinos
(adapted from [25]).
3.2
Experimental Methods of EAS Observation
The properties of an air shower can be measured with suitable detectors. In case of
surface detectors a large area has to be occupied by many detector units to observe
a noticeable amount of UHECR-induced events at the highest energies. This is due
to the low flux of UHECRs discussed in chapter 2.
Since an UHECR-induced air-shower disk extends over an area of several kilometres
the spacing between individual detector units can be greater than 1 km. Noting the
arrival times of a shower at different detector stations enables the reconstruction of
the shower axis and its inclination.
Another method to obtain information about the properties of an extensive air shower is to observe the fluorescence light that is created as the shower propagates
3.2. Experimental Methods of EAS Observation
13
through the atmosphere. Nitrogen atoms in the air are excited by shower particles
and isotropically emit fluorescence light during deexcitation.
This weak light can be observed directly by telescopes containing mirrors and photo
multipliers. This is only possible during moonless nights, though, resulting in a
low duty cycle of approximately 10 % for a fluorescence detector. Recording a time
sequence or observing the shower with more than one telescope enables the reconstruction of the shower axis.
3.2.1
Measurement of the Lateral Shower Profile
With a grid of surface detector stations the lateral profile of an extensive air shower
can be measured. The lateral distribution of the particle density in the disk can be
described by the Nishimura-Kamata-Greisen function [26]
α−η
−α r
r
1+
.
(3.3)
n(r) = κ
r0
r0
r is the distance from the shower axis and r0 the Molière unit1 . α and η are model
parameters while κ is proportional to the size of the EAS. Measuring n(r) yields the
lateral shower profile, from which the position of the shower core can be reconstructed
and the energy of the primary particle can be estimated. The total number N of
particles that reach the ground can be obtained from the lateral distribution of the
particle density at ground level by
Z
N = n(r) dr.
(3.4)
An example for a lateral shower profile can be seen in figure 3.2.
3.2.2
Measurement of the Longitudinal Shower Profile
Fluorescence telescopes allow us to observe the fluorescence light that is emitted by
an EAS and follow its longitudinal evolution. The number of electrons and positrons
contained within the shower disk at a certain slant depth2 X can be parameterized
by the Gaisser-Hillas function [27]
Ne (X) = Ne,Max
X − X0
XMax − X0
XMax−X
0
λ
e
XMax −X
λ
,
(3.5)
where NMax is the maximal number of particles during the development of the airshower cascade at the corresponding slant depth XMax . X0 and λ are referred to as
shape parameters [28]. They represent the slant depth of the first interaction and a
1
The Molière unit denotes the radius of a cylinder around the shower axis in which 90 % of the
shower’s energy is deposited.
2
The slant depth is the integrated column density that a particle has traversed through since
g
entering the atmosphere. It is given in units of cm
2 . Since the shower propagates at approximately
the speed of light the quantity X also provides information about the shower’s age, meaning the
time since the first interaction of the primary particle.
14
Figure 3.2: Lateral shower profile of an extensive air shower measured at the Pierre
Auger Observatory. We see the amount of muons hitting a surface detector station
vertically depending on the distance to the shower core. The circles denote surface
detector stations that have seen a signal while the triangles denote stations that
have not. The red mark represents a station that has been triggered accidentally.
The green line represents a fit of the lateral distribution model from equation (3.3)
with the fit uncertainties seen in grey.
particle’s absorption length.
For small slant depths the number of electrons and positrons in the shower disk will
increase due to the increasing electromagnetic component of the air-shower cascade.
Eventually the maximum amount of NMax electrons and positrons will be present
at the slant depth XMax . For slant depths greater than XMax the latter term of
equation (3.5) will dominate and Ne decreases again due to absorbtion and decay of
the particles. This means that for strongly inclined showers only a small fraction of
electrons and positrons will reach the ground.
The energy EPrimary of the primary particle can also be estimated from the longitudinal shower profile since the relation
Z
EPrimary ∝ Ne (X) dX
(3.6)
holds.
An example for a longitudinal shower profile measured with fluorescence telescopes
at the Pierre Auger Observatory can be seen in figure 3.3.
3.3. Emission of Radio Signals from EASs
15
Figure 3.3: Longitudinal shower profile of an extensive air shower measured at the
Pierre Auger Observatory [28]. We see the energy deposit of the shower particles
depending on the slant depth of the shower disk. The energy of the primary particle
was approximately 6 · 1019 eV.
3.3
Emission of Radio Signals from EASs
As mentioned before, an extensive air shower propagating through the atmosphere
forms a flat disk with a thickness of a few metres. Within the shower disk processes
take place that lead to the emission of electromagnetic radiation in the MHz regime.
Various model approaches have been made to explain this phenomenon.
3.3.1
Radio Emission Mechanisms
Askaryan was the first to propose that an extensive air shower emits electromagnetic
pulses in the radio regime [29]. According to his understanding a radio pulse originates from Cherenkov-light emission within the shower. The particles that form the
shower disk propagate through the atmosphere at speeds very close to the vacuum
speed of light c. Their speed is greater than the speed of light in air, meaning that
the particles are able to produce Cherenkov light. The thickness of the shower disk
ranges within a few metres, resulting in coherent emission of electromagnetic radiation at these wavelengths. The corresponding frequency range is the radio regime
(MHz).
This model requires a charge excess in the shower front since otherwise the radiation
from positive and negative charges would cancel each other out. A negative charge
excess is present at the shower front due to the fact that the positrons created by
pair production in the electromagnetic component of an EAS are likely to be absorbed faster by the atmosphere than the electrons.
16
Measurements have shown that the amplitude of radio pulses emitted by EASs is
influenced by Earth’s magnetic field [30]. This cannot be explained by Askaryan’s
model. Therefore, the currently most favoured models consider radio pulses from
EASs to be geo-magnetic radiation.
The principle of the geo-magnetic radiation models is that the Lorentz force separates electrons and positrons in an EAS. A transverse current forms in the process
and emits coherent electromagnetic radiation in the MHz regime. The first approach
based on this principle was made by Kahn and Lerche [31]. Recent models are based
on a macroscopic description of coherent geo-magnetic radiation from cosmic rays
[32] and the phenomenon of synchrotron radiation [33].
One great advantage of the geosynchrotron model is that the theory of synchrotron
radiation is well understood. The power PS of the synchrotron radiation generated
by a single particle is
q2c 4 4
β γ .
(3.7)
PS =
6π0 r2
q denotes the particle’s charge while r is the radius of its gyration trajectory. β = vc⊥
is the particle’s velocity perpendicular to its rotation axis and relative to the speed
of light while γ is the Lorentz factor3 . 0 denotes the vacuum permittivity [34].
In case of an air shower a particle with the mass m is forced on a circular orbit of
the radius
γmv⊥ c
(3.8)
r=
qB
by the Lorentz force in Earth’s magnetic field B. The total radiated power can then
be expressed as
2 2
q 4 v⊥
B
PS =
E 2.
(3.9)
6π0 m4 c5
Because the radiated power of a synchrotron pulse is proportional to m−4 the strongest pulses are emitted by electrons and positrons. Since the electromagnetic component is dominant in EASs pulses generated by other particles such as muons can be
neglected. PS describes the power radiated by one particle in an EAS with m = me
and q = ±e as the electron mass and the elementary charge. The total power of a
pulse is calculated by superimposing N particles with a total mass of m = N · me , a
total charge of q = ±(N · e) and a total energy of ETot = N · E. It is related to PS
by
PTot = N 2 · PS ,
(3.10)
increasing with the squared number of particles in the shower disk. This yields a relation between the electric field strength E of an emitted synchrotron pulse and the
2
total energy ETot of the N particles generating it since PTot ∝ |E |2 and PTot ∝ ETot
.
By integrating over the longitudinal and lateral shower profiles and thus following
the evolution of the shower the energy of the primary particle can be reconstructed
from measured synchrotron pulses.
Due to the properties of the shower disk coherent synchrotron pulses are also expected in the radio regime in this model. Furthermore, the model takes the influence
3
γ=√
1
1−β 2
≈
E
m.
E is a particle’s total energy.
17
of the geo-magnetic field into account as the pulse formation in an EAS has been
related to the Lorentz force, which is generated by the velocity vector of shower particles and Earth’s magnetic field. Hence the pulse evolution depends on the angle
between the velocity vector and the magnetic field vector. Thus, the behaviour of
the emitted radio pulses is related to the incoming direction of an EAS.
3.3.2
Simulation of EASs
Basic radio signal properties can be studied with simulations such as REAS4 [35].
The simulation uses the modelling for the lateral and longitudinal shower profiles of
an EAS given in equations (3.3) and (3.5) to simulate the properties of the shower.
Radio pulses are then calculated by superimposing geosynchrotron pulses emitted
by individual electron-positron pairs. The latest version REAS3 considers additional
effects such as charge variation within a shower [36]. Since REAS2 the code is also
coupled to the EAS simulation tool CORSIKA [37] to provide realistic information
concerning the particle content of air showers. With REAS interesting properties of
radio pulses have been predicted.
According to simulations the radiation emitted from EASs suffers loss of coherence
for high frequencies. The effect also depends on the distance to the shower core
and is shown in figure 3.4. Within the frequency range of coherent radiation an
exponential law of the form
−
|E (r, ν)| ∝ e
ν−10 MHz
ν0 (r)
(3.11)
can be fitted to the amplitude of the electric field vector for different distances r to
the shower core. The parameter ν0 increases along with the distance to the shower
core. This shifts the loss of coherence to smaller frequencies.
The original REAS1 simulation also predicts that the amplitude of a pulse’s electric field is related to the energy EPrimary of the primary particle that induced the
corresponding EAS by
|E (r, EPrimary )| = E0 ·
with
µV
mMHz
E0 =

 0.265 µV
mMHz


 10.18
κ(r)
EPrimary
1017 eV
κ(r)
,
at
r = 20 m
at
r = 500 m,
(3.12)
(3.13)
close to one.
The almost linear relation is visualized in figure 3.5. We see again that increasing
the distance to the shower core leads to the radio signal becoming incoherent. This
4
Radio Emission from extensive Air Showers
18
Figure 3.4: Simulation of the frequency-dependent radio pulse amplitude for a
shower induced by a primary particle of EPrimary = 1017 eV. A loss of coherence is
predicted towards high frequencies. The differently coloured lines represent various
distances to the shower core. The thin lines represent the results from the original
simulation REAS1. The thick lines show the results for REAS2, which considers the
properties of EASs in a more detailed way [38].
effect slightly increases with the energy of the primary particle.
The REAS simulation is a useful tool to understand the process of radio emission from UHECR-induced extensive air showers based on the currently favoured
geo-magnetic and geosynchrotron radiation models.
3.3.3
Measurements of Radio Signals from EASs
The coherent radio signals emitted by extensive air showers can be detected with
suitable antennas in the radio regime. Just as in case of other detectors the shower
axis can be reconstructed if the radio signal is measured with an array of antennas
and the time information is stored.
H.R. Allan [40] found out that the pulse amplitude increases linear with the primary
particle’s energy. He modelled the pulse’s amplitude per unit bandwidth with respect
to various parameters according to
−r
µV
EPrimary
sin (α) cos (θ) exp
.
(3.14)
f = 20
1017 eV
r0 (f, θ)
mMHz
f denotes the centre frequency of the used receiver. Due to the normalization to
a receiving antenna’s bandwidth equation (3.14) can be used to compare different
19
Figure 3.5: Electric field amplitude emitted by a vertically incident EAS at a
frequency of 10 MHz and depending on the energy of the primary particle. From top
to bottom the different colours denote the distances 20 m, 100 m, 180 m, 300 m and
500 m to the shower core [39].
experiments to each other. α is the angle between Earth’s magnetic field vector
and the shower axis while r represents the distance to the shower axis. The factor
cos(θ) in equation (3.14) considers that showers inclined at a zenith angle θ have
a reduced pulse amplitude due to them propagating through the atmosphere for a
longer time. r0 is a model parameter obtained from measured data. Figure 3.6 shows
the dependence of the measured pulse amplitude on the distance to the shower axis
that was measured with the LOPES experiment.
While Allan found the model parameter r0 to be 110 m, the LOPES experiment
yielded a value of r0 = (230 ± 51) m. Figure 3.7 shows the dependence of the pulse
amplitude on the primary particle’s energy, which was also measured with LOPES.
Detailed models concerning the properties of UHECR-induced extensive air showers
and their radio emission already exist and a lot of research is being conducted in the
field of radio detection. Measurements have shown that the energy of an UHECR
primary particle can be related to the radio emission of an extensive air shower.
In contrary to fluorescence telescopes radio detection arrays have no limitations with
respect to their duty cycle. Much effort is being put into constructing self-triggered
radio-detection arrays to further investigate radio emission from EASs. One of them
is AERA – the Auger Engineering Radio Array – located at the southern site of the
Pierre Auger Observatory in Argentina.
20
Figure 3.6: Pulse amplitude measured with the LOPES experiment, depending on
the distance to the shower axis. The distance was reconstructed from data obtained
with the KASCADE-Grande surface detectors that observed the events coincidentally with the LOPES antennas [41].
Figure 3.7: Pulse amplitude measured with the antennas from the LOPES experiment. The amplitude depends on the energy of the primary particle that induces
an EAS. The energy has been reconstructed by coincidentally observing the air showers with the KASCADE-Grande experiment. The data has been corrected from
the dependence on the distance to the shower axis [41].
4. The Pierre Auger Observatory
The Pierre Auger Observatory constitutes the largest experiment in the world that
is able to detect ultra-high-energy cosmic rays at energies greater than 1018 eV.
The southern site of the observatory is located in Argentina, near the town Malargüe in the province of Mendoza. Extensive air showers are observed with a grid of
surface detectors and with fluorescence telescopes that surround the surface detector grid. An overview of the site is given in figure 4.1. The two detection methods
are completely independent from each other. Thus, the southern site of the Pierre
Auger Observatory operates a hybrid detector, enabling cross calibration between
the detectors and high-precision measurements.
A site in the northern hemisphere is currently being planned. In combination with
the southern site it will allow the observation of the entire sky.
In this chapter the detectors operated at the southern site of the observatory are
introduced. The principle of the surface and fluorescence detectors is explained and
the largest extension of the Pierre Auger Observatory – the Auger Engineering Radio
Array – is presented.
4.1
Surface Detector
The surface detector (SD) of the Pierre Auger Observatory consists of 1600 individual tanks that are filled with extremely pure destilled water. Each tank contains
12 m3 of water. The tanks are arranged in a hexagonal grid with a spacing of 1.5 km.
When a particle from an EAS impacts a tank and traverses through the water
inside, it causes the emission of Cherenkov light due to its speed being greater than
the speed of light in water. Diffusers ensure that the emitted photons will be detected by photomultipliers located at the ceiling of an SD tank. A photo of an SD tank
can be seen in figure 4.2.
If three or more neighbouring stations detect a signal within an adequate time frame,
the data is read out and sent by wireless communication to central data acquisition.
The surface detector is able to measure the arrival direction of UHECRs with an
◦
angular resolution of 1.1 or better. The energy of the primary particle can be
reconstructed with an accuracy of at least 12 %. The resolution for the arrival direction and the energy reconstruction becomes even better for events at the highest
energies since a greater number of detector stations is hit in that case.
22
The Pierre Auger Observatory
Figure 4.1: Bird’s eye view of the southern Pierre Auger Observatory. The grid of
surface detector stations is distributed over the entire area belonging to the observatory and thus covers approximately 3100 km2 . The red dots mark individual surface
detector stations. The blue lines denote the observation field of the fluorescence
telescopes. Each line corresponds to an individual telescope. They are situated in
groups of six at the four positions labelled in blue. The location of some extensions
to the observatory – such as AERA – can also be seen [42].
Figure 4.2: A water tank belonging to the surface detector grid of the Pierre
Auger Observatory. Solar panels ensure that the station is supplied with electricity
autonomously. The station communicates measured data via cell phone technology
[43].
4.2. Fluorescence Detector
23
Figure 4.3: The FD building at Los Leones. The shutters protecting the telescopes
are open (adapted from [44]).
4.2
Fluorescence Detector
The fluorescence detector (FD) consists of 24 individual telescopes that observe the
fluorescence light directly emitted by an EAS. They are grouped in six and are
located at the four sites Coihueco, Loma Amarilla, Los Leones and Los Morados,
which are labelled in figure 4.1. Figure 4.3 shows a photo of an FD building at one
of the sites. The buildings are positioned at suitable locations in order to oversee
the whole Pierre Auger Observatory.
Each telescope consists of a 3.5 m × 3.5 m spherical mirror that reflects incoming
fluorescence light onto a camera with 440 photomultipliers. The telescopes have an
◦
◦
opening azimuth angle of 30 and an opening zenith angle of 28.6 . A sketch of an
individual telescope can be seen in figure 4.4.
The fluorescence detector is able to detect the energy that a shower radiates as
fluorescence light with a resolution better than 5 %. A limiting factor to the accuracy for the energy reconstruction of the primary particle, however, is the uncertainty
on the fluorescence yield1 . It used to be in the order of 20 %, but with the recently
performed AirLight experiment [45] it has been improved to about 15 %.
The duty cycle of the fluorescence detector is limited to 10 % since good observation
conditions (i.e. moonless nights) are required to detect the weak fluorescence light
caused by extensive air showers.
1
The fluorescence yield is defined as the ratio of excited atoms that emit a fluorescence photon
to the total number of excited atoms.
24
Figure 4.4: Sketch of an FD telescope (adapted from [43]).
4.3
Radio Detector - AERA
As discussed in chapter 3, extensive air showers emit electromagnetic radiation in
the MHz regime. It is possible to relate the amplitude of the electric field vector
E to properties of the primary particle, which makes radio detection of UHECRs
possible. Since 2006 radio test setups have been investigated at the Balloon Launch
Station (BLS, see figure 4.1) and the Central Laser Facility (CLF). They confirmed
that the detection of EAS-induced radio signals is possible at the Pierre Auger Observatory, which led to the idea of a self-triggered radio detector.
The Auger Engineering Radio Array (AERA) is currently being set up to exploit the
radio technique for UHECR detection. It is being constructed in three subsequent
stages, of which the first one is almost completed. The array will consist of approximately 160 individual antennas as detector stations that will span an area of about
20 km2 . The spacing between the antennas will increase in each stage, starting from
150 m for stage one, followed by 250 m for stage two and finally extending to 380 m
in stage 3. An illustration of AERA can be seen in figure 4.5.
4.3. Radio Detector - AERA
25
Figure 4.5: Illustration of the Auger Engineering Radio Array [46]. The antennas
that are already set up for stage one of AERA are marked in red. The blue antennas
indicate the setup planned for stage two while the green antennas will be set up in
stage three. The location of AERA can also be seen in figure 4.1.
A set of 24 logarithmic periodic dipole antennas (LPDAs) has already been set up
for the first stage of AERA. They are the Small Black Spider (SBS) LPDAs that
were developed and mass-produced by our institute’s workshop at RWTH Aachen
University. Their operation bandwidth ranges from 30 − 80 MHz and the antennas
feature a good sensitivity towards almost the entire sky. Figure 4.6 shows a photo
of an SBS in the Argentinian Pampa.
An antenna station communicates its received data to a central radio station (seen
in figure 4.5) by wireless signal transmission. For this purpose an additional small
antenna is attached to every station. The communication antennas operate in the
GHz regime, meaning that no interference with measured signals in the radio regime
26
Figure 4.6: A Small Black Spider LPDA in the Argentinian Pampa [46]. A special
low-noise amplifier that amplifies incoming signals within the antenna’s operation
bandwidth and filters signals outside of it has been designed for the SBS [47].
is to be expected. As in case of the surface detector stations, the individual antenna
stations are supplied with electricity by solar panels.
The antennas for AERA are able to measure two polarization directions of an incoming pulse independently. By measuring the same pulse with many detector stations
it is possible to reconstruct the propagation direction of the incoming pulse, gaining
a third polarization direction of the electric field vector in the process. Thus, the
absolute electric field vector of the incident signal can be reconstructed, if the radio
detector is calibrated. An absolute calibration of the Small Black Spider LPDA is
carried out in this thesis.
AERA is a suitable candidate to realize an increased hybrid detector duty cycle in
combination with the surface detector since the duty cycle of AERA will be close to
100 % . During moonless nights the fluorescence detector is operational, too, which
means that the Pierre Auger Observatory will be able to operate a super-hybrid
detector.
5. Fundamental Characteristics of
Antennas
This chapter will explain fundamental parameters of antennas that are associated
with the process of measuring radio pulses from extensive air showers. It focuses
on the response of an antenna to the incident electric field and will outline why
exactly an absolute calibration is needed for a radio detection array. The antenna
theory presented in this chapter is based on an excellent book from Balanis [48] and
is enhanced at topics which are important for our purpose.
5.1
Vector Effective Height
The vector effective height of an antenna is of central importance when setting up a
radio detection array with that antenna. It fully describes the structure’s response
towards an incoming pulse, including signal amplifications, dispersion effects and
reflections. The quantity is frequency and direction-dependent and relates the complex voltage U , obtained by reading out the antenna, to the complex electric field
vector E applied to the structure by
U (f, φ, θ) = H (f, φ, θ) · E (f, φ, θ) = Hφ Eφ + Hθ Eθ .
(5.1)
The indices φ and θ denote vector components in the azimuth and zenith direction of
a spherical coordinate system. The reference point of this coordinate system usually
is the antenna footpoint1 . However, for a radio detection setup close to the ground
it is suitable to define the system antenna plus ground as an effective antenna, of
which the response pattern contains all additional ground effects. The origin of the
reference coordinate system is by convention then located at ground level below the
centre of the antenna structure. An illustration of the coordinate sytem can be seen
in figure 5.1. All quantities in equation (5.1) and the following equations in this
chapter refer to the electromagnetic far field, which means that the condition
r≥
2D2
λ
(5.2)
is fulfilled for the distance r between a receiving antenna and the source of the signal.
D is the largest dimension of the radiating antenna. Hence emitted signals can be
treated as spherical or for very large distances even as plane waves. Therefore, the
1
The footpoint of an antenna is the point where the signal received by the structure is read out.
28
Fundamental Characteristics of Antennas
Figure 5.1: Spherical reference coordinate system of an antenna. If the antenna
is located above a ground plane the origin of the coordinate system is located at
ground level below the centre of the structure.
electric field vector only has a component in the e φ and e θ direction of the reference
coordinate system while the signal propagates in the e r direction. With respect to
the reference coordinate system in figure 5.1 it is possible that an incoming radio
signal is polarized in e φ or e θ direction. In case of a polarization in e φ the electromagnetic wave is called horizontally polarized and in case of a polarization in e θ one
speaks of a vertically polarized wave.
An absolute calibration for an arbitrary antenna is performed by measuring the
values for the components of H . To relate the vector effective height to other fundamental antenna parameters one can make use of equations valid in the far-field
regime. The magnitude of the Poynting vector of an incoming radio signal polarized
in either e φ or e θ direction can be written as
|Ek |2
,
Sk =
Z0
(5.3)
where k = φ, θ denotes the polarization direction and Z0 ≈ 120π Ω is the characteristic impedance of free space (for vacuum and in good approximation also for air).
The same quantity can also be expressed in terms of the receiving antenna as
Sk =
PR,k
4πPR,k f 2
= 2
,
Aeff
c GR,k
(5.4)
with PR,k as power received by the antenna and Aeff as its effective area. The effective
area of the receiving antenna is then expressed in terms of the signal frequency f ,
the (vacuum) speed of light c and its absolute gain GR,k . By equating the equations
(5.3) and (5.4) and solving for the electric field component we obtain
|Ek |2 =
4πZ0 PR,k f 2
.
c2 GR,k
(5.5)
5.2. Matching Theory and Antenna Impedance
29
Figure 5.2: Equivalent circuit diagram for an antenna plus read-out coaxial cable.
The measured power PZ depends on the properties of the read-out impedance Z,
which influences the system additionally to the antenna impedance ZA and the
antenna footpoint voltage U0 .
By taking the absolute value of equation (5.1), squaring and plugging into equation
(5.5) we end up with
4πZ0 PR,k |Hk |2 f 2
2
,
(5.6)
|Uk | =
c2 GR,k
if we assume that the incoming signal is either polarized in e φ or e θ direction.
In the calibration experiments presented in this thesis power magnitudes PR,k are
measured. Hence we need to find a way to express the magnitude of the voltage |Uk |
induced to our read-out system in terms of the measured power PR,k . To achieve
this it is necessary to understand the basic principles of matching theory.
5.2
Matching Theory and Antenna Impedance
The impedance of an antenna is a fundamental quantity that is needed to understand the signal transmission and reception of a specific antenna structure. It is
strongly related to an antenna’s ability of power radiation and reception at a certain
frequency. Hence it is a frequency-dependent quantity. As soon as one interacts
with the antenna system, for example, by reading out a signal with a coaxial cable
during a measurement, the antenna impedance becomes a relevant factor since its
relation to the read-out system determines whether mismatch losses occur in the
setup or not and how strong they are throughout the operation bandwidth of the
antenna. The smaller the mismatch losses are, the better the antenna is matched to
the read-out system.
To understand the matching concept and the consequences for measurements, we
can prescind the system antenna plus load impedance over which the system is read
out to an equivalent circuit diagram for an arbitrary but fixed frequency f . An
illustration can be seen in figure 5.2. In the most general case the load is a complex
30
impedance Z that has the form
Z = R + iX.
(5.7)
The real part R of equation (5.7) is called resistance while the imaginary part X is
named reactance. The measured power PZ is then given by the relation
PZ = Re (U · I ∗ ) ,
(5.8)
where U is the voltage read out over the impedance Z and I ∗ is the complex conjugate
of the current I within the system. By expressing I in terms of U and Z and applying
the mesh rule we can write
Z
U = U0 ·
(5.9)
ZA + Z
and, after plugging into equation (5.8) and simplifying, end up with
2
PZ = |U0 | · Re
Z
2
|ZA | + ZA · Z ∗ + ZA∗ · Z + |Z|2
.
(5.10)
One already sees that the read-out impedance Z is crucial and determines what
amount of power a measurement will yield. To further simplify equation (5.10), Z
needs to be specified. Two very popular matching techniques will be pointed out in
the following subsections.
5.2.1
Conjugate Matching
We speak of conjugate matching if the read-out impedance is selected to be the
complex conjugate of the antenna impedance for every frequency f , meaning the
equation
Z = ZA∗
(5.11)
must hold. Let
ZA = RA + iXA
(5.12)
be the general form of the antenna impedance. In consequence the quantities defined
in equation (5.7) become
R = RA ,
X = −XA .
(5.13)
(5.14)
After applying this specification to equation (5.10) the measured power becomes
PZ,CM
|U0 |2
=
,
4RA
where CM stands for conjugate matching.
(5.15)
5.2. Matching Theory and Antenna Impedance
5.2.2
31
Real Matching
When performing real matching we only specify X = 0 in equation (5.7), resulting
in
Z = Z ∗ = R,
(5.16)
with R being an arbitrary resistance. For coaxial cables often used in measurements
and at the AERA site in Argentina this means R = 50 Ω. Applying real matching
to equation (5.10) yields
PZ,RM = |U0 |2
R
,
(R + RA )2 + XA2
(5.17)
where RM stands for real matching. It is important to note that the power calculated
from equations (5.15) and (5.17) can differ vastly, meaning that one needs to know
the exact read-out impedance properties of an antenna calibration setup in order to
interpret the measurement results correctly.
5.2.3
Transformation Factor
It is easy to express the magnitude of the antenna footpoint voltage |U0 | in terms of
the measured power PZ,CM in a system with conjugate matching. Since all calibration
measurements for this thesis are performed with 50 Ω real matching, it is useful to
define a transformation factor between the power depiction in both systems. Relating
equations (5.15) and (5.17) to each other in the form of
PZ,RM = F · PZ,CM
(5.18)
leads to a frequency-dependent transformation factor of
F =
4RA R
(R + RA )2 + XA2
(5.19)
that contains the antenna impedance and the cable resistance as parameters.
5.2.4
Reflection Coefficient
After introducing the principles of matching theory it is now useful to define the
complex amplitude reflection coefficient as
ρ=
ZA − Z
.
ZA + Z
(5.20)
When feeding a signal into the antenna the squared absolute value of equation (5.20)
represents the fraction of power that is reflected by the antenna structure. Thus,
equation (5.20) provides a relation between the signal power reflected by an antenna
and the antenna impedance.
32
5.3
Absolute Gain
The absolute gain of an antenna contains information about its directional and
frequential sensitivity towards an incoming radio signal. In contrary to the vector
effective height defined in equation (5.1), it is a scalar quantity applied in the power
domain. Thus, it only provides a simplified description of the antenna response.
Nevertheless, it gives crucial calibration information and can be used to obtain the
vector effective height.
5.3.1
Definition
When using an antenna as receiver, the gain can be defined as the frequency- and
direction-dependent ratio
PR,k (f, φ, θ)
PR,k,Iso (f )
Gk (f, φ, θ) =
(5.21)
of the power PR,k that is received from a k-polarized incoming signal with a fixed
direction and the power PR,k,Iso that a lossless perfect isotropic structure would
receive from the same signal. For the case of a transmitting antenna we find
Gk (f, φ, θ) =
Sk (f, φ, θ)
,
Sk,Iso (f )
(5.22)
Sk being the radiation density at distance r of the transmitter and Sk,Iso the corresponding radiation density caused by a lossless perfect isotropic radiator. Only
referring to the antenna’s far-field leads to the equivalence of the definitions presented in the equations (5.21) and (5.22). If one talks about absolute gain, additional
effects caused by interaction between the antenna structure and the read-out system – described in the previous section – are included within the quantity. Another
important issue is the polarization of the incoming or radiated signal2 . For a e φ polarized pulse we speak of the horizontal gain Gφ while for a e θ -polarized pulse we
speak of the vertical gain Gθ . A total gain can then be defined as sum of the two
partial gains by
GTot (f, φ, θ) = Gφ (f, φ, θ) + Gθ (f, φ, θ) .
(5.23)
For a given frequency f the directional pattern G (f = const., φ, θ) provides a three
dimensional deformed spheroid. In case of a lossless perfect isotropic radiator the
pattern is a sphere with the gain factor G = 1 as radius. The radiation density at
distance r from this structure in free space is given by
SIso =
2
PT,Iso
,
4πr2
(5.24)
Careful: equation (5.21) only yields the correct gain for a fixed polarization k = φ, θ of the
incoming signal. The reason for this is that while the radiation density of an incoming signal is a
linear quantity the power measured by the receiver is not (i.e. Sφ + Sθ = STot , but PR,φ + PR,θ 6=
PR,Tot ).
5.3. Absolute Gain
33
where PT,Iso is the total power radiated by the structure. Any deviation from these
conditions results in a deformation of the sphere and the radiation density becoming
G (φ, θ) PT,Iso
.
(5.25)
4πr2
However, due to the laws of energy conservation the total amount of radiated power
PT,Iso must not change when deforming the gain sphere. This can be expressed in
the relation
Zπ Z2π
G (φ, θ) sin (θ) dφ dθ = 4π.
(5.26)
S (φ, θ) =
0
0
One can see that the gain of an antenna can only be increased in one direction at
cost of other directions. Extreme cases are high-gain directional antennas that can
be used for directive measurements and low-gain omnidirectional antennas.
5.3.2
The Friis Transmission Equation
Transmission measurements are a possible method to obtain the absolute gain of an
antenna. Such a setup contains a transmitting and a receiving antenna, which are
separated by a sufficient distance r to ensure far-field conditions. After aligning the
transmitter to a fixed polarization k = φ, θ and in the way that its maximum gain
direction faces the receiver, the radiation density at the location of the receiver can
be expressed in terms of the transmitting antenna by
S (f ) =
GT (f ) PT (f )
.
4πr2
(5.27)
GT is the gain of the transmitter in the direction of maximal sensitivity and thus
is only frequency-dependent, PT depicts the input power fed into the transmitter
(corresponding to the total radiated power of a lossless perfect isotropic radiator
PT,Iso ) and r is the distance between the two antennas. In addition to equation
(5.27) one can also express the radiation density in terms of the receiving antenna
according to equation (5.4), which results in
S (f ) =
4πPR (f, φ, θ) f 2
,
c2 GR (f, φ, θ)
(5.28)
with GR being the gain of the receiver and PR the received power. By equating the
two depictions and solving for GR we obtain the Friis Transmission Equation
GR (f, φ, θ) =
16π 2 r2 2 PR (f, φ, θ)
f
.
c2
GT (f ) PT (f )
(5.29)
It contains the characteristics of the transmitting and receiving antennas as well as
the loss due to free space propagation of the signal. Thus, the gain GR of the receiving antenna can be measured in a transmission measurement, if the transmitter
gain GT is already known.
34
For measurements close to the ground a gain pattern is deformed due to ground
reflections of electromagnetic waves. The deformed pattern GR can be associated
with an effective antenna defined as receiver plus ground. This convention makes
sense for calibration measurements when preparing a radio detection array for the
detection of extensive air showers since the antennas are also close to the ground
in the final setup. Hence the deformed gain pattern including the ground effects is
what we are really interested in.
5.4
Group Delay
The group delay of an antenna describes the signal dispersion throughout the structure. Therefore, it is a frequency-dependent quantity that is defined as
τg = −
dψ
.
dω
(5.30)
ψ is the phase of an electromagnetic signal passing through the antenna and ω = 2πf
denotes the angular frequency of the signal. As the word “delay” implies, the group
delay has the dimension of time. It can be interpreted as the time that an incoming
signal needs to pass through the antenna up to its footpoint. Depending on the
internal dispersion and geometry of the antenna structure this time is different for
different signal frequencies and even can depend on the direction of the incoming
signal. In addition to the possible variation of the group delay over the frequency
bandwidth of an antenna one will also encounter a delay offset when measuring the
group delay with a transmission measurement setup. The reason for this is that an
electromagnetic signal needs to propagate from the transmitter to the receiver in
such a setup. Even in the fastest case this can only happen at the vacuum speed of
light c, resulting in the constant signal delay
τOff =
r
c
(5.31)
for all frequencies with r as the distance between transmitter and receiver.
5.5
A Roadmap for Obtaining the Vector Effective Height
After introducing all relevant quantities equation (5.6) can be rewritten by expressing
|Uk | in terms of the antenna’s footpoint voltage U0,k , using equation (5.9). For a load
impedance specified to Z = R = 50 Ω equation (5.6) becomes
R2
4πZ0 PR,k |Hk |2 f 2
|U0,k | ·
=
.
c2 GR,k
(R + RA )2 + XA2
2
(5.32)
One can now use equation (5.17) to write the left side of equation (5.32) in terms of
the measured power. PR,k then cancels out on both sides and we obtain
R=
4πZ0 |Hk |2 f 2
.
c2 GR,k
(5.33)
5.5. A Roadmap for Obtaining the Vector Effective Height
35
Solving equation (5.33) for |Hk | yields
r
|Hk (f, φ, θ)| =
R
c
GR,k · ,
4πZ0
f
(5.34)
the magnitude of the desired component of the vector effective height. Of course, in
the most general case Hk is a complex quantity of the form
Hk (f, φ, θ) = |Hk (f, φ, θ)| · eiψH .
(5.35)
The phase ψH takes the phase shift of the signal into account as it passes through
the antenna structure. Equations (5.34) and (5.35) show what quantities need to be
measured to obtain the vector effective height and thus perform an antenna absolute
calibration.
One needs to obtain the absolute gain and take matching theory into account. It is
therefore helpful to know the antenna impedance. The phase of the vector effective
height can be obtained from the signal phase in a transmission measurement that
is performed with a vector network analyzer. It is strongly related to the group delay.
Many of the quantities to be measured depend on the direction and frequency of an
incoming radio signal. In consequence, this is also the case for the vector effective
height.
36
6. The Small Black Spider LPDA
The Small Black Spider (SBS) LPDA was especially designed for stage one of the
AERA setup in Argentina. It is designed to feature ultra-broadband sensitivity in
the 30 − 80 MHz radio regime and the ability to measure two polarization directions
of an incoming radio pulse independently. The antenna is easy to handle and has
proven to be resistive to various stress conditions. This chapter will illuminate the
principle the Small Black Spider antenna is based on – that of logarithmic periodic
dipole antennas – and will point out its additional features. A picture of a set up
Small Black Spider LPDA can be seen in figure 6.1.
6.1
Logarithmic Periodic Dipole Antennas
The concept of a logarithmic periodic dipole antenna was first established by D. E.
Isbell in 1960 [49]. The idea is to gain a large frequency bandwidth by assembling
Figure 6.1: The Small Black Spider LPDA on the roof of our institute at RWTH
Aachen University.
38
The Small Black Spider LPDA
Figure 6.2: Technical sketch of an LPDA. A specific relation between the distance
of each dipole to the antenna’s virtual array vertex and the resonance frequency
of that dipole needs to be fulfilled to obtain a large frequency bandwidth (adapted
from [50], based on [51]).
a sequence of half-wave dipoles of lengths li to a central waveguide, which routes
the signals received by the individual dipoles to the antenna’s footpoint, where the
antenna is read out. The basic concept of an LPDA can be seen in figure 6.2. An
LPDA is periodic with reference to the logarithm of an individual dipole’s resonance
wavelength. The equation
λi
= const.
Ri
(6.1)
must hold, in which λi stands for a dipole’s resonance wavelength and Ri for the
distance between the centre of that dipole and the antenna’s virtual array vertex.
Since the index i points to an arbitrary dipole belonging to the structure one can
apply equation (6.1) to two dipoles situated next to each other, take the logarithm
and write
log (λi ) = log (Ri ) − log (Ri−1 ) + log (λi−1 ) .
(6.2)
Equation (6.2) shows the periodicity in the logarithm of the wavelenghts λi . Furthermore, the resonance wavelength λi of a dipole is proportional to its actual length
li . Using this relation and equation (6.1) for sequenced dipoles leads to
Ri
λi
li
=
=
≡ τ,
Ri−1
λi−1
li−1
(6.3)
6.2. Properties of the Small Black Spider LPDA
39
defining an LPDA’s constant geometric ratio τ in the process. This constant can be
used to express the spacing between the dipoles of an LPDA as
1
Si = Ri−1 − Ri = Ri ·
−1
(6.4)
τ
and to relate all dipoles contained in the structure to the parameters of the longest
dipole by
li = τ i−1 l1 ,
Ri = τ
i−1
(6.5)
R1 .
(6.6)
These relations define the periodicity of a logarithmic periodic dipole antenna with
respect to the wavelength. A good signal reception is possible almost across the entire resonance frequency range that is spanned by the dipoles belonging to the LPDA.
Each dipole has a frequential resonance peak that is not discrete, but has a resonance width instead. It is important that the resonance widths of the individual
dipoles overlap to a certain extent to ensure a mostly constant sensitivity throughout
the frequency bandwidth. The operation bandwidth is also influenced by the presence of the central waveguide belonging to the LPDA structure. The boundaries of
the frequency range that a designed LPDA will operate at are approximately given
by
c
,
2 · lmax
c
,
≈
3 · lmin
λmax ≈ 2 · lmax
=⇒
fmin ≈
(6.7)
λmin ≈ 3 · lmin
=⇒
fmax
(6.8)
with lmax referring to the longest and lmin to the shortest dipole in figure 6.2 [51].
By assembling many dipoles with suitable proportions ultra-broadband sensitivity
can be achieved.
6.2
Properties of the Small Black Spider LPDA
The Small Black Spider LPDA operates in a frequency bandwidth from 30 − 80 MHz
and is sensitive towards almost all sky directions. Therefore, the antenna is well
suited to measure radio signals caused by extensive air showers. It contains two individual antenna planes labeled “North-South” and “East-West”, which are completely
independent from each other in terms of electronics and can be read out separately. The planes are orientated perpendicular to each other and enable polarizationsensitive measurements of incoming radio pulses.
Each antenna plane contains nine dipoles, eight of which are made of copper wire
while the longest dipole is made of an aluminium tube that helps to stabilize the
entire structure. All dipoles of a plane are connected to a central waveguide – a Lecher wire made of messing – that guides the signals to the footpoint located at the
top of the antenna. The Lecher wires are protected by a massive aluminium tube,
40
which is the central structure element of the Small Black Spider LPDA and can be
seen in figure 6.1. The central aluminium tube also contains two coaxial cables that
lead from the footpoint of the antenna to the connection ports at the bottom of the
structure, where the signals received by the antenna planes can be read out.
It is important to mention that the footpoint impedance of the Small Black Spider LPDA is situated around 200 Ω. When the signal transcends from the Lecher
wire to the coaxial cable, mismatch losses are to be expected since the characteristic
impedance of the coaxial cable is 50 Ω. Because of this a 4 : 1 balun is integrated in
the SBS antenna that transforms the structure’s impedance from around 200 Ω to
about 50 Ω within the operation bandwidth [52]. Hence the antenna is well matched
to a 50 Ω read-out system.
Additional ropes and GFK1 -pipe extensions provide a stable frame for the Small
Black Spider LPDA. The ropes are connected to the wire dipoles by springs and
help to maintain tension throughout the whole antenna structure. Despite the Small
Black Spider LPDA being quite a large object with dimensions of approximately
4.3 m × 4.3 m × 3.9 m, the use of aluminium and copper wires leads to an overall
weight of merely 18 kg. A folding mechanism is integrated to the antenna and by
folding both ends of the lowest dipoles on both planes upwards towards the central structure one can significantly reduce the dimensions of the Small Black Spider
LPDA. Figure 6.3 shows a set of folded SBS antennas.
The design of the SBS results in an extremely high wind resistivity up to gusts
. This was tested in January 2010, when the storm “Daisy” took
of at least 130 km
h
place in and around Aachen while a SBS antenna was installed on the roof of the
physics centre at RWTH Aachen University.
Furthermore, extensive stress testing was performed at the connection points between the wire dipoles of the LPDA and the central aluminium tube. Wires were
constantly oscillated at a frequency of 2 Hz over a time period of one year to ensure that the final antenna design would be able to handle equivalent environmental
conditions at the AERA site.
6.3
Cross Talk between Antenna Planes
As already mentioned the Small Black Spider LPDA consists of two individual antenna planes that are orientated perpendicular to each other. It is engineered to
minimize a possible cross talk between the antenna planes. This cross talk between
the antenna planes can be tested by performing a transmission measurement, using
one plane as transmitter and the other as receiver.
A FSH 4 spectrum analyzer from the company Rohde & Schwarz (please refer to
section A.4 in the appendix) is used to provide the input signal to one plane and
measure the power fraction received by the other. The device is operated as a vector
network analyzer, which means it measures the magnitude of the transmitted power
1
“Glasfaser verstärkte Kunststoffe” – fiber glass reinforced synthetics
6.3. Cross Talk between Antenna Planes
41
Figure 6.3: 15 Small Black Spider LPDAs folded together and ready for shipping
to Argentina. Due to light weight and compact dimensions the folded antennas are
easy to transport.
fraction and the change in the phase of the voltage amplitude.
Figure 6.4 shows the cross talk between the two antenna planes of the Small Black
Spider LPDA across its operation bandwidth. Even in the worst case the power
fraction received by one plane is less than 0.06 % of the signal power radiated by the
other.
Due to the cross talk signal residing in the same order of magnitude as other radio
signals present in Aachen, it is necessary to also measure the input from external
radio sources that disturbs the measurement (please refer to section A.6 in the appendix for a radio background measurement). The radio background has already
been eliminated from the data in figure 6.4. As one can see, the cross talk effect can
be completely neglected when measuring radio pulses with the Small Black Spider
LPDA.
42
Figure 6.4: Cross talk between antenna planes. The figure shows the fraction of
power a SBS plane receives if the other plane radiates a signal. NS and EW refer to
the North-South and East-West antenna planes.
6.4
Relative Phasing of Signals Received by both
Antenna Planes
One interesting aspect about the Small Black Spider LPDA is the relative phasing of
signals received by the North-South and East-West plane of the antenna. Depending
on the polarity that the dipoles belonging to an individual plane are connected with
to the central waveguide, there is a possibility to end up with a “North-South-EastWest” antenna or a “North-South-West-East” antenna (names by convention). An
illustration can be seen in figure 6.5.
To find out what configuration is present in case of the Small Black Spider LPDA,
another transmission measurement needs to be performed. This time, however, the
receiver is one of the SBS’s planes while the transmitter is a biconical antenna (please
refer to chapter 8 for details concerning that antenna). Starting with the receiving
SBS plane aligned parallel to the incident electric field vector E inc , transmission
measurements are performed for an azimuth angle range from 0 ◦ to 360 ◦ by rotating the SBS. This is done for both antenna planes. At first the signal pattern is
measured with the North-South plane. Then the antenna is rotated 90 ◦ clockwise
and the same measurement is repeated for the East-West plane.
The quantity we are interested in during this measurement is the phase of the incoming signal that is measured with the FSH 4 network analyzer. Figure 6.6 shows
the azimuth dependence of the signal phase for a given frequency f for both Small
Black Spider planes.
6.4. Relative Phasing of Signals Received by both Antenna Planes
43
Figure 6.5: Configuration of antenna planes. In the upper figure we see the polarity
configuration for a NS-EW antenna and in the lower figure the configuration for a
NS-WE antenna. E Inc is an incident electric field vector. When receiving the same
signal with the NS plane at first and then, after rotating 90 ◦ clockwise, the EW
plane, there should be an offset of 180 ◦ in the measured signal phase ψ for a NSEW configuration and no phase offset for a NS-WE configuration.
Figure 6.6: Azimuth angle dependence of the phase of an incoming signal at a
given frequency f , measured with both planes of the Small Black Spider LPDA.
The angle θ refers to the zenith direction of the incoming signal.
44
We see a constant phase offset of
∆ψ = 180 ◦
(6.9)
between the two curves. Comparing to the sketches for the two possible antenna
configurations in figure 6.5 leads to the conclusion that the Small Black Spider LPDA
is a North-South-East-West antenna according to our name convention.
6.5
Reflection Characteristics
The reflection characterstics of the Small Black Spider LPDA can be obtained by
performing a reflection measurement. Figure 6.7 shows the SBS during the setup
process and figure 6.8 shows the complete setup for this measurement. The antenna
is connected to the FSH 4 network analyzer with a coaxial cable. We use RG213 coaxial cables in our setups since this type of cable is characterized by small
attenuation factors within the operating frequency bandwidth from 30 − 80 MHz
(please refer to section A.5 in the appendix)[53].
Figure 6.7: The Small Black Spider LPDA during setup. The antenna is already
fully unfolded and attached to an extra aluminium pole.
A signal of constant power magnitude P0 is fed into the Small Black Spider LPDA
from the spectrum analyzer through the coaxial cable connection. If one assumes
that ohmic losses inside the SBS’s structure are neglectable, fractions of the power
can only be either reflected or radiated by the antenna.
Reflection losses occur because the coaxial cable has a constant impedance of Z =
R = 50 Ω for all frequencies while the impedance ZA of the Small Black Spider
LPDA varies for different frequencies. If antenna and read-out impedance are close
to each other, we speak of an antenna that is well matched. For the SBS this is the
case within the operation bandwidth from 30 − 80 MHz. However, there still remain
reflections that can be quantified by ascertaining the absolute value of the complex
6.5. Reflection Characteristics
45
Figure 6.8: The Small Black Spider LPDA attached to a pole especially designed
for calibration measurements. The construction has been designed by the mechanical
workshop of our institute at RWTH Aachen University. The lowest dipole is located
at an approximate height of 3 m above the ground, resembling the setup at the
AERA site in Argentina.
amplitude reflection coefficient defined in equation (5.20).
In a reflection measurement the ratio
|ρ|2 =
PRef
P0
(6.10)
of reflected power PRef to fed in power P0 yields the squared absolute value of the
reflection coefficient. The frequency-dependent quantity can be seen in figure 6.9.
The reflection curve of the SBS oscillates within the operation bandwidth. The
reason for this is the nature of an LPDA. The structure contains many dipoles
with individual resonance frequencies that influence each other. The position of
the resonance peaks is also influenced by other parts of the antenna structure such
as the Lecher wire that functions as a central waveguide. Although the antenna
features a mostly constant sensitivity throughout the whole operation bandwidth,
signal radiation and reception is even better close to the centre of a resonance width.
We see this as dips in the reflection curve in figure 6.9.
46
Figure 6.9: Reflection curve of the Small Black Spider LPDA. EW refers to the
East-West and NS to the North-South plane of the antenna. Within the bandwidth
ranging from 30 − 80 MHz less than 20 % of the fed in signal power is reflected.
Uncertainties are only shown for the East-West plane but are the same for the
North-South plane.
Furthermore, the reflection curve is identical for both SBS antenna planes. This
underlines the functionality of the tested SBS unit and proves the reproducability of
an antenna’s reflection characteristics. It shows that both antenna planes have been
engineered to have equal properties. More details concerning the reproducability of
SBS reflection curves can be found in [54].
6.6
Antenna Impedance
Solving equation (5.20) for the antenna impedance ZA and specifying the read-out
system to Z = R = 50 Ω yields the relation
ZA = R ·
1+ρ
1−ρ
(6.11)
for the antenna impedance. Since the antenna impedance is a complex quantity the
reflection coefficient ρ is needed in its complex form to obtain it. From the reflection
curve one only obtains the absolute value |ρ| of the quantity. However, the reflection
measurement is performed using the FSH 4 as a vector network analyzer, meaning
that the device does not only measure the magnitude of power fractions but also the
corresponding signal phase. Hence one is able to reconstruct the complex reflection
coefficient ρ and compute the antenna impedance ZA . Figure 6.10 shows the results
for the Small Black Spider LPDA.
6.6. Antenna Impedance
47
Figure 6.10: Magnitude of the SBS antenna impedance. Within the bandwidth
the impedance magnitude oscillates around 50 Ω, meaning that the antenna is well
matched to a 50 Ω read-out system in this frequency region. The data points are
black while the uncertainties are drawn in red.
The uncertainty of the measured value for the reflected power fraction is σ|ρ|2 =
0.3 dB [46]. No information has been available concerning the uncertainty of the
signal phase. Using error propagation after transforming σ|ρ|2 to a linear scale yields
an asymmetric uncertainty for the antenna impedance. However, figure 6.10 shows
that the uncertainties are very small compared to the large scale over that the
antenna impedance varies.
48
7. Simulating Antennas with
NEC-2
It is important that one is able to cross check quantities measured for the Small
Black Spider LPDA not only with additional measurements but also appropriate
simulation tools. The NEC-2 simulation software1 is a tool that can be used to
simulate reflection and even transmission measurement setups [55]. Fundamental
antenna parameters can be computed by the programme directly. This chapter
will explain the principle of the NEC-2 software and illuminate its usefulness for
cross checking antenna calibration results. Furthermore, it will point out interesting
phenomena caused by the presence of a ground plane below an antenna.
7.1
The Concept of NEC-2
The Numerical Electromagnetics Code has been developed at the Lawrence Livermore Laboratory in California. The project was sponsored by the Naval Ocean
Systems Center and the Air Force Weapons Laboratory. For our simulations concerning radio antennas we use the version NEC-2, which is publicly available.
7.1.1
Fundamental Integral Field Equations
NEC-2 computes the response of an arbitrary metal structure to an incident electromagnetic field or excitation. This is done by numerically solving two fundamental
integral equations that are derived from Maxwell’s equations [56]. The more important one to us is the Electric Field Integral Equation
Z
−i
J (r 0 ) · G(r , r 0 ) dV 0
(7.1)
E (r ) =
4πω0
V
that relates the electric field vector E to the volume current distribution vector J .
ω is the angular frequency, 0 the vacuum permittivity and G the dyadic Green’s
function given by
0
Gmn (r , r ) = ∂xm ∂xn
1
e−ik|r −r 0 |
+ k δmn
,
|r − r 0 |
2
NEC – Numerical Electromagnetics Code
m, n = 1, 2, 3
(7.2)
50
Simulating Antennas with NEC-2
with k = 2π
as the wave number and r and r 0 as position vectors. The convention
λ
for time evolution of the electric field is
E (r , t) = E (r ) eiωt .
(7.3)
The other fundamental equation is the Magnetic Field Integral Equation
−ik|r 0 −r 0 | Z
1
e
1
I
0
0
e n (r 0 ) × J S (r ) × ∇
dA0 ,
−e n (r 0 ) × H (r 0 ) = − J S (r 0 ) +
0
2
4π
|r − r 0 |
S
(7.4)
where e n (r 0 ) is the outward directed normal vector at the surface point r 0 of the
metal structure. H I is the incident magnetizing field without the structure and
J S the volume current distribution on the structure’s surface. The surface integral
included in equation (7.4) is also performed over the surface of the metal structure.
NEC-2 uses equation (7.1) to simulate structures consisting of thin wires and equation (7.4) to simulate structures made of surfaces. For a more complex structure
containing wires and surfaces a hybrid equation that combines the equations (7.1)
and (7.4) is used.
If the radius of a wire is not small enough in comparison to the length of the wire
and the signal wavelength, the current in the structure cannot be assumed to be
represented by a filament on the wire. In case of this the computation process is
more complex and an extended thin-wire approximation is used in NEC-2 to solve
the problem [56].
7.1.2
Means of Determining Current Expansion
To obtain the current expansion in a specified problem, NEC-2 uses the numerical
method of moments [57] to solve the field equations presented in the previous subsection. Generally, this method can be applied to any linear operator equation of
the form
Lζ = ξ.
(7.5)
ζ is an unknown response function (e.g. the volume current distribution), ξ a known
excitation function (e.g. the applied electric field) and L a linear operator. In
our case the linear operator is an integral operator. The response function can be
expanded into a sum of basis functions
ζ=
N
X
αl ζl
(7.6)
l=1
with the coefficients αl . These coefficients can be derived by taking the inner product
of equation (7.6) with a selected set of weighting functions wk and making use of
the operator linearity in equation (7.5), obtaining
N
X
l=1
αl < wk , Lζl > =< wk , ξ >,
k = 1, ... , N
(7.7)
7.1. The Concept of NEC-2
51
in the process. This is a set of linear integral equations that can be solved for the
coefficients αl . In case of the field equations to be solved in NEC-2 the inner product
is defined as
Z
(7.8)
< f, g >= f (r ) · g(r ) dA.
S
The programme integrates over the structure surface by using the weighting functions
wk (r ) = δ(r − r k ),
(7.9)
with r k denoting position vectors of individual points on the structure’s surface.
This point sampling of the original integral equations (7.1) and (7.4) is known as
collocation method of solution.
A wire is divided into a certain amount of straight segments with a sample point
at the centre of each segment. Surfaces are divided into flat patches with a sample
point at the centre of each patch. The choice of basis functions ζl is crucial in terms
of calculation time and accuracy of the point sampling.
For the current expansion on wires NEC-2 allocates a current of the form
∆l
(7.10)
2
to the lth segment of the structure. As before k is the wave number while s denotes
the position on the segment and sl is the coordinate of its centre. ∆l is the segment
length. This depiction of the current distribution on a wire was first used by Yeh
and Mei [58].
By allocating equation (7.10) to every segment and considering local conditions, two
of the constants Al , Bl and Cl can be eliminated for each segment. The third constant
yields a relation to the current’s amplitude and can be obtained by solving equation
(7.7). Using this method, the complete current expansion on a wire structure can
be determined. An illustration for a wire containing four segments can be seen in
figure 7.1.
Il (s) = Al + Bl sin (k(s − sl )) + Cl cos (k(s − sl )) ,
|s − sl | <
If surfaces have to be considered in the calculation, they are divided into small
patches with a sample point at the centre. The surface current density of the structure can then be written as
J S (r ) =
NP
X
(J1j e 1j + J2j e 2j ) Vj (r ).
(7.11)
j=1
Np denotes the number of patches, e 1j and e 2j are directional unit vectors defining
an individual flat surface patch. The parameters J1j and J2j represent the average
surface current densities for those directions on patch j and can be obtained from
a linear system of equations when applying the equations (7.7) and (7.11) to the
Magnetic Field Integral Equation (7.4). The weighting factor Vj (r ) is introduced as

for r on patch j

1
Vj (r ) =
(7.12)

0
otherwise.
52
Figure 7.1: Current expansion on a wire containing four segments. We see the
current basis function for each segment and the superposition as the resulting overall
current distribution on the structure [56].
A more complex approach is needed when taking structures into account that include wires connected to surfaces (please refer to [56] for details on that approach).
However, the structures used to simulate the calibration measurements for the Small
Black Spider LPDA do not contain wire-surface transitions.
7.1.3
Means of Determining the Electric Field
In a wire structure the currents of the individual wire segments are of the form given
in equation (7.10). For each segment a corresponding electric field is computed. The
total electric field is then gained by superposing the results for all segments.
A wire should be divided into segments of a length smaller than 0.1 λ. It has been
shown that, if NEC-2 uses the thin-wire and the extended thin-wire approximations
for an adequate segment length, the numerical errors of the field integral equation
solutions become less than one percent [56].
In the thin-wire kernel NEC-2 assumes the current distribution for each segment
to be a filament on the segment axis. The point where the current is observed at,
however, lies on the surface of the segment. Hence one can define a cylindrical coordinate system around the segment axis, with the segment’s surface being separated
by a constant cylinder radius a from the cylinder axis. An illustration can be seen
in figure 7.2.
The electric field E j (r ) for a segment j can then be derived from the known current
distribution I j (r ) for that segment since the corresponding field integral equation
can be solved numerically. Superposition of all segments then yields the total electric
field E (r ).
In case of the extended thin-wire kernel, which holds if a wire is too thick to use the
regular thin-wire approximation, the current distribution is not represented by a filament on the z-axis of a cylindrical coordinate system. Instead, one can approximate
the surface-current density on the segment’s surface by
J(z) =
I(z)
,
2πa
(7.13)
7.2. Simulation of Antenna Characteristics
53
Figure 7.2: A wire segment on the z-axis of a cylindrical coordinate system. The
value for the electric field E j (r ) of segment j is derived from the current distribution
I j (r ) located on the z-axis and observed from the wire surface at radius a [56].
which means that the current filament is distributed across the segment’s surface.
For wires made of high-conductivity materials this is a good approximation due to
the nature of the skin effect. The skin effect predicts that in case of a perfectly
conducting wire the distribution of the current density will be limited to the surface
of that wire [59].
7.2
Simulation of Antenna Characteristics
To simulate the properties of an antenna with the NEC-2 software, the structure
needs to be specified in the programme. This is done with simulation input files
known as steering cards. Steering cards define material properties, transmission
lines and the geometric layout of an antenna.
Furthermore, it can be chosen to either excite the antenna structure by a voltage
source or simulate an incident electromagnetic field at an arbitrary frequency. It has
to be specified whether the thin-wire or extended thin-wire kernel should be used in
the simulation process. It is possible to introduce a ground plane to the setup that
can be characterized in terms of its permittivity and conductivity.
Many more special features and options are available and can be included in a NEC-2
steering card, but not all of them are necessarily relevant for simulations concerning
antenna calibration. Please refer to section A.2 in the appendix for some sample
steering cards.
A steering card is processed by NEC-2 and an output file containing the simulation results in ASCII format is produced. The output file can be parsed for various
parameters such as the current, impedance and voltage values at the centre of each
antenna segment.
54
Figure 7.3: Directional horizontal absolute gain pattern of the Small Black Spider
LPDA at f = 63.6 MHz. A perfectly conducting ground plane is included in the
simulation. The visualization is achieved by using 4NEC-2X [60].
7.2.1
Simulation of a Setup with One Antenna
In case of an incident electric field, it is interesting to observe the footpoint voltage
U0 of the antenna. It yields fundamental information about the antenna’s response
to the incoming wave. Figure 7.3 shows a visualization of the directional absolute
gain pattern for the Small Black Spider LPDA.
One can see that, for an incoming wave polarized in e φ direction, NEC-2 predicts
the Small Black Spider LPDA to be sensitive for incoming directions perpendicular
to the antenna’s dipoles and blind for signals that propagate in the plane containing
the antenna dipoles.
When using NEC-2 to simulate the excitation of a structure by a voltage source, i. e.
when simulating an antenna in transmitting mode, observing the footpoint segment
yields crucial parameters such as the antenna impedance ZA . The total power PT ,
radiated by the structure, and even the radiation efficiency can also be observed.
yields information about ohmic losses within the antenna.
7.2. Simulation of Antenna Characteristics
55
Figure 7.4: Simulated transmission measurement between two Small Black Spider
LPDAs in NEC-2. One antenna is excited and the response of the other antenna
can be observed.
7.2.2
Simulation of a Transmission Measurement Setup
NEC-2 can be used to simulate a transmission measurement by placing two antennas
into one NEC-2 steering card. One antenna is then excited and the response of the
other antenna to the radiated electromagnetic signal is observed. A visualization of
a simulated transmission measurement can be seen in figure 7.4.
Using the Friis Transmission Equation (5.29), the absolute gain of the receiving
antenna can be reproduced by reading out the relevant quantities and expressing
the footpoint voltage U0 in terms of the received power PR . This, of course, should
yield the same result for the absolute gain as observing it directly from the output
file does. This is the case if the received power is calculated as
PR =
|U0 |2
,
4RA
(7.14)
just as in equation (5.15), meaning that the NEC-2 simulation assumes conjugate
matching at the footpoint of an antenna.
Since the footpoint voltage U0 is provided in its complex form by the NEC-2 simulation one automatically obtains the signal phase when observing that quantity.
Hence the group delay τg defined in equation (5.30) can also be computed from a
simulated transmission measurement.
We conclude that all important calibration quantities associated with the vector
effective height of the Small Black Spider LPDA, namely the antenna impedance
ZA , the absolute gain G and the group delay τg , are accessible with the NEC-2 software.
The simulation is limited by the accuracy of the antenna structure properties specified in the NEC-2 steering card and the lack of possibility to simulate a completely
realistic environment, which considers ground topography and environmental conditions.
56
7.3
Ground Effects
It is necessary to include a ground plane into the simulation when comparing results
to a measurement since its presence influences quantities such as the absolute gain.
Signal reflections from the ground give rise to numerous interesting phenomena that
are discussed in this section.
7.3.1
Signal Reflections by a Ground Plane
NEC-2 allows to simulate an antenna structure such as the Small Black Spider LPDA
in a free space environment. Consequently, all simulated parameters refer to the characteristics of the antenna structure only. An example for a free space gain pattern
can be seen in figure 7.5.
When introducing a ground plane to the setup, signal reflections occur at the ground.
NEC-2 considers these reflections and computes the current and electric field contributions of the ground by introducing mirror current distributions [56]. The overall
behaviour of the structure is different to the case of free space. The absolute gain
pattern is deformed, as can be seen in figure 7.6. The deformed effective gain pattern is one of the antenna calibration quantities we are interested in since at the
AERA site the Small Black Spider LPDAs are also set up only three metres above
the ground (referring to the longest dipole).
When a ground plane is present the gain rapidly decreases for zenith angles close to
◦
90 , meaning that the antenna is blind with respect to signals coming in from those
directions. The reason for this is that the incoming signal is a superposition of two
signals: one wave propagating directly from the source to the receiving antenna and
one wave that reaches the receiver after being reflected by the ground.
For example, the electric field E hR caused by the reflection of a wave polarized in e φ
direction is given by
E hR = Rφ · E h .
(7.15)
E h is the electric field from the wave propagating towards the ground before the
reflection and Rφ is the reflection coefficient
p
ZR cos (θ) − 1 − ZR2 sin2 (θ)
p
.
(7.16)
Rφ =
ZR cos (θ) + 1 − ZR2 sin2 (θ)
ZR contains information about the ground and is given by
r
0
ZR =
,
1 − i σω1
(7.17)
with 1 and σ1 being the permittivity and conductivity of the ground [56]. An
illustration of a signal reflection from the ground can be seen in figure 7.7.
7.3. Ground Effects
57
Figure 7.5: Simulated total absolute gain of SBS at f = 50 MHz in free space. The
upper figure shows a 3D-visualization of the directional gain pattern. The lower
figure shows the zenith angle dependence of the gain for a given azimuth angle of
◦
φ = 0 . For directions perpendicular to the SBS’s dipoles the antenna is clearly
sensitive towards all zenith angle directions.
58
Figure 7.6: Simulated total absolute gain of SBS at f = 50 MHz including a
ground plane. The upper figure shows a 3D-visualization of the directional gain
pattern while the lower figure shows the zenith angle dependence of the gain for a
◦
given azimuth angle of φ = 0 . Since antenna and ground form an effective antenna
the gain sphere is deformed and the structure is only sensitive towards the sky.
7.3. Ground Effects
59
Figure 7.7: Reflection of a horizontally polarized signal from a ground plane (adapted from [56]). Due to the phase shift occuring during the reflection process the
reflection coefficient from equation (7.16) can be negative.
The total incident electric field E hInc reaching the receiving antenna is then given by
E hInc = E hD + E hR = E hD + Rφ · E h .
(7.18)
E hD is the electric field vector belonging to the wave propagating directly from the
◦
transmitter to the receiver. For a zenith angle of θ = 90 the reflection coefficient
becomes Rφ = −1. As the distance r between transmitter and receiver is increased
(figure 7.8), the vector E h converges in the form of
lim E h = lim ◦ E h = E hD .
r→∞
(7.19)
θ→90
Applying these conditions to equation (7.18) yields that the total incident field E hInc
completely vanishes. By convention we consider this ground effect in the gain pattern
of the receiving antenna. By doing so the gain decreases to zero for a zenith angle
◦
converging to θ = 90 .
Figure 7.8: Sketch concerning the limit in equation (7.19).
60
7.3.2
Signal Propagation Towards the Horizon
The consequence of the previously described signal reflections is an interesting behaviour of the radiation density caused by a radiation source above a ground plane.
Though this is not an exclusive property to the Small Black Spider LPDA, it is
important to note that for an incoming signal from directions towards the horizon
(large zenith angles) the radiation density does not hold to
S=
P0
,
4πr2
(7.20)
where P0 is the power radiated by a source at a fixed frequency f and r is the
distance between the signal source and the receiver. NEC-2 simulations have shown
that the relation can be generalized to
S=
P0 q
r
4π
(7.21)
to correctly describe the radiation density for signals propagating close and almost
parallel to the ground.
This finding has motivated a cross check measurement on the rooftop of the physics
centre at RWTH Aachen University. In this transmission measurement, a BBAL
9136 biconical antenna (chapter 8) is used as a transmitter and one plane of the
Small Black Spider LPDA as a receiver. Signal generation and readout is done using
the FSH 4 network analyzer. The dipole axes of both antennas are aligned parallel to each other for optimal signal transmission of a horizontally polarized signal.
Measurements are then performed for various distances r between the centre of the
transmitter and the centre of the lowest dipole of the receiving SBS LPDA. The
results can be seen in figure 7.9.
In the simulations the behaviour of the radiation density shows significant deviations from the expected power law with the exponent qexp = −2. Though not
exactly in agreement with the simulation, the measurements confirm this behaviour.
The extent of the deviation depends on the frequency of the transmitted signal and
on the selected measurement distances. For the same frequency the power law is
different in the range from 3 − 8 m than in the range from 9 − 15 m.
One reason for this may be near field effects. Although the far-field condition from
inequation (5.2) is fulfilled at all times with D = 1.94 m being the largest dimension
of the biconical antenna, for small distances the Small Black Spider LPDA responds
differently to the incoming signal than for large distances. This is because for large
distances the curvature of the radiated spherical electric far field can be neglected in
comparison to the dimensions of the receiving antenna. In consequence the incoming
signal behaves like a plane wave.
However, a more dominant effect that causes the deviation of the exponent in the power law is the reflection of electromagnetic waves due to the presence of the ground.
The origin of the receiver based coordinate system is not located at the centre of
the lowest SBS dipole. Instead, it is located at ground level below the centre of
7.3. Ground Effects
61
Figure 7.9: Distance-dependent power fraction received by SBS during a transmission measurement with the BBAL 9136 biconical antenna. The left graph shows the
results for distances from 3 − 8 m and the right graph for distances from 9 − 15 m at
fixed frequencies f . q denotes the measured exponent of the power law in equation
(7.21) while qS stands for the value obtained from a corresponding NEC-2 simulation. The transmitter and the lowest dipole of the SBS are situated at h = 1.9 m
above the ground.
the antenna. When increasing the distance between transmitter and receiver while
keeping the transmitter at a constant height of h = 1.9 m above the ground, from
the Small Black Spider’s point of view the zenith angle θ of the incoming signal
converges to 90 ◦ .
As discussed in the previous subsection the total incident electric field gets very
weak for large zenith angles until it vanishes completely for θ = 90 ◦ . We see this as
a deviation from the power law given in equation (7.20) and as deformation of the
antenna’s gain pattern (figure 7.6). The deviation is caused by the superposition
of the wave propagating directly from transmitter to receiver and a wave reflected
from the ground. It increases for larger distances between transmitter and receiver.
An independent confirmation of the power law deviation can be found in [61]. In our
case the measurement was motivated by the predictions of the NEC-2 simulation.
The phenomenon is one example for the tremendous effect the presence of a ground
plane has on an antenna setup and that the most convenient approach is to define
receiving antenna plus ground as an effective antenna.
7.3.3
Other Interference Effects
Electromagnetic waves in the radio regime behave very similar to those in the optical
regime in terms of reflection and interference. The NEC-2 simulation can be used
to exploit those effects.
Imagine a setup of two identical simple dipoles, separated by a distance r, at the
same height over a metal ground plane. One dipole is used as a transmitter and the
other one as a receiver.
62
Figure 7.10: Received power versus distance in a simulated transmission measurement between two simple dipoles of l = 4.25 m length above a ground plane. Perfect
ground stands for a perfectly conducting mirror that does not absorb a fraction of
the signal. The dashed curve shows the same simulation for free space, where the
power law P (r) ∝ r−2 holds with P (r) as the received power at a given distance r.
Once again we observe the behaviour of the radiation density by increasing the
distance between the two dipoles but leaving the height at h = 22.5 m. Figure 7.10
shows that in this case not even equation (7.21) holds anymore. For certain distances
– in this case for r = 60 m – the signal vanishes completely. At r = 81.5 m on the
contrary, we locally get a maximum signal.
Let’s examine the same effect from a different point of view. The distance r between
transmitter and receiver is now fixed to r = 60 m. Instead, we now vary both dipoles’
height above the ground plane, but leave it equal for transmitter and receiver at all
times. Figure 7.11 shows that at presence of a ground plane the received signal
oscillates during height variation. Again, we see minima and maxima, just as in
figure 7.10. In free space, varying the z component of both antennas in the coordinate
system at the same time has no effect at all, since their relative positioning remains
unchanged.
To understand the minima and maxima seen in figures 7.10 and 7.11 one needs
to apply the same principles as in geometrical optics. Analog to the optical path
difference we can define the radio path difference between the signal propagating
directly from transmitter to receiver and the signal being reflected from the ground
and then travelling to the receiver. The radio path difference is
r
r2
λ
λ
−r+ ,
(7.22)
∆s = s2 − s1 + = 2 h2 +
2
4
2
7.3. Ground Effects
63
Figure 7.11: Height-dependent power fraction received during a simulated transmission measurement between two simple dipoles of l = 4.25 m length.
Figure 7.12: The radio path is different for the signal directly propagating to the
receiver and the reflected signal.
with the geometric properties of the setup as seen in figure 7.12 and a phase shift
corresponding to λ2 . The phase shift occurs because the ground is a denser medium
than vacuum (respectively denser than air in reality). At the minima and maxima
in figures 7.10 and 7.11 we can now check if a relation of the form
∆s = m ·
λ
2
(7.23)
holds with m being an integer value. For odd m we expect the signal to vanish and
for even m we expect it to become maximal.
Checking at r = 60 m, h = 22.5 m and λ = 7.5 m yields
∆s = 18.75 m = 5 ·
7.5 m
.
2
(7.24)
64
We have destructive interference at this point, which is why the signal vanishes in
figures 7.10 and 7.11. Checking at r = 81.5 m, h = 22.5 m and λ = 7.5 m on the
contrary yields
7.5 m
∆s = 15.35 m ≈ 4 ·
.
(7.25)
2
In this case we have constructive interference, which is why we see a maximum in
figure 7.10.
We have seen that one can nicely relate some of the power law oddities to interference effects. All phenomena discussed in this section show how important it is
to appropriately consider the ground plane when analyzing measurement data and
comparing it to simulations. Reflections from the ground cannot be neglected and
crucially influence many antenna related quantities.
8. The BBAL 9136 Biconical
Antenna
The BBAL 9136 antenna is a biconical antenna from the German manufacturer
Schwarzbeck Mess-Elektronik [62]. A photo of the antenna can be seen in figure 8.1.
Since the antenna is pre-calibrated we know its absolute gain for a 50 Ω read-out
system. This makes the biconical antenna suitable for transmission measurements
with the Small Black Spider LPDA, from which the absolute gain of the latter
antenna can be obtained.
8.1
Basic Properties
The BBAL 9136 biconical antenna is basically a fat dipole antenna with a dipole
length of lBic = 1.94 m. The maximum vertical diameter of each cone is dBic = 57 cm
and the operation bandwidth of the biconical antenna ranges from 30 − 300 MHz.
A VHBB 9124 4:1 balun is integrated in the central structure of the antenna. Just
as in case of the Small Black Spider LPDA, the device transforms the impedance at
the antenna’s footpoint from around 200 Ω to about 50 Ω. This reduces mismatch
losses when reading out or feeding in a signal to the antenna through a 50 Ω system.
Figure 8.1: The BBAL 9136 biconical antenna. The type index BBAL 9136 actually refers to the two aluminium cones that can separately be assembled to the
central structure containing a VHBB 9124 4:1 balun.
66
The BBAL 9136 Biconical Antenna
From the footpoint of the biconical antenna, located between the two cone elements,
a received signal is guided through the central structure to the connection port at
the bottom.
The BBAL 9136 features a maximum directional gain in the plane perpendicular
to the dipole axis. Absolute gain values for this plane are provided by the manufacturer for various frequencies.
8.2
Simulation of the BBAL 9136 in NEC-2
To be able to cross check measurements including the BBAL 9136 biconical antenna
as transmitter it is necessary to simulate the antenna structure in NEC-2. Figure
8.2 shows a sketch of the simulated antenna structure.
The simulated cones consist of wires with a diameter of dBic,w = 6 mm and during
the simulation process the extended thin wire kernel is applied. Furthermore, the
1
conductivity σAl = 37.7 µmΩ
of aluminium is considered to provide a corresponding
load impedance for the simulated structure. The structure contains 117 individual
segments to simulate the antenna absolute gain at an appropriate accuracy. When
using the simulated antenna as a transmitter, it is excited at the footpoint segment
at the centre of the structure.
Figure 8.2: Simulated BBAL 9136 biconical antenna in NEC-2. For the visualization of the structure the software xnecview has been used [63].
The absolute gain of the simulated antenna, seen in figure 8.3, shows the known
behaviour of a dipole antenna. The antenna is blind and does not radiate any power
in the direction of the dipole axis. Perpendicular to that axis the antenna has a
maximum sensitivity for all zenith directions.
8.2. Simulation of the BBAL 9136 in NEC-2
67
Figure 8.3: Total free-space gain of the BBAL 9136 biconical antenna for a frequency of f = 50 MHz. The upper figure shows a 3D-visualization of the directional
gain pattern and the lower figure shows the gain versus the zenith angle for a given
◦
azimuth angle of φ = 0 . Due to symmetry of the antenna structure the maximum
directional gain has no zenith angle dependence.
68
8.3
Calibration Data
Using the programmed NEC-2 simulation we can now cross check the absolute gain
calibration data of the biconical antenna that was provided by the manufacturer.
The frequency-dependent values of the gain for the directional plane perpendicular
to the biconical antenna’s dipole axis can be seen in figure 8.4. The included uncertainties of 0.7 dBi are given in the calibration certificate (please refer to section A.3
in the appendix).
Figure 8.4: Comparison of simulated and measured total gain of the BBAL 9136 biconical antenna. Measured data was provided by the German manufacturer Schwarzbeck Mess-Elektronik for a 50 Ω read-out system, including a calibration certificate.
Schwarzbeck Mess-Elektronik executed an advanced version of the so called two antenna method to obtain the absolute gain of the BBAL 9136 biconical antenna. They
have performed transmission measurements with two identical BBAL 9136 antenna
units, using one as transmitter and one as receiver.
It is interesting to note that the measurements have been performed for small distances between transmitting and receiving antenna, which could result in significant
errors due to near field effects. However, due to the dipole length of the BBAL 9136
being lBic = 1.94 m, the condition from inequation (5.2) is fulfilled even at distances
only a few metres away from the transmitter. Because of the equal properties of
the receiver and the zenith angle symmetry of the biconical antenna a transmission
measurement at a distance of r = 4 m already yields the desired far field gain pattern.
The provided gain values are supposed to be for free space, meaning that all possible
ground effects have been eliminated from the measurement data. The gain values
refer to the BBAL 9136 structure exclusively. These values are needed when performing a transmission measurement with a receiving Small Black Spider LPDA since
due to our convention only the gain pattern of the latter antenna is affected by the
ground and we need the free-space pattern for the transmitter.
8.4. Reflection Characteristics and Antenna Impedance
69
To cross check the calibration data we use the simulated biconical antenna to reproduce an ideal transmission measurement setup in free space. To ensure far field
conditions, two BBAL 9136 units are separated several kilometres from each other
in a NEC-2 steering card. The footpoint voltage U0 of the receiving antenna, the
radiated power PT and the antenna impedance ZA is read out from the output file.
The absolute gain of the biconical antenna can then be computed from the Friis
Transmission Equation (5.29) after specifying
GR = GT = G
(8.1)
for this setup of two identical antennas. We need to add the effect of the VHBB
9124 4:1 balun into the simulation by modifying the antenna impedance and the
footpoint voltage to be
ZA,B =
ZA
,
4
U0,B =
U0
.
2
(8.2)
The received power PR,CM in a read-out system with conjugate matching is not
affected by that and given by equation (5.15) in both cases. However, this power
needs to be transformed to a 50 Ω read-out system using equation (5.19) with the
antenna impedance ZA,B . The simulation then yields the free-space absolute gain
of the biconical antenna in a 50 Ω system, just as the provided calibration data
does. Figure 8.4 also shows the frequency dependence of the simulated absolute
gain. Simulation and measured data are in good agreement with each other, which
is a good confirmation for the calibration data and also points out the quality of the
NEC-2 simulation software.
8.4
Reflection Characteristics and Antenna Impedance
Figure 8.5 shows the fraction of power from equation (6.10) that is reflected by the
BBAL 9136 biconical antenna during a reflection measurement.
Although the operation bandwidth of the BBAL 9136 is said to range from 30 −
300 MHz, the antenna is not matched very well to a 50 Ω read-out system for frequencies around 20 − 80 MHz, which are relevant for the Small Black Spider LPDA.
The reflection losses are included in the calibration data provided by Schwarzbeck
Mess-Elektronik since the absolute gain values are given for a 50 Ω read-out system,
as mentioned in section 8.3.
Figure 8.6 shows the magnitude of the antenna impedance of the biconical antenna
that has been obtained by using equation (6.11).
70
Figure 8.5: Reflection curve of the BBAL 9136 biconical antenna. Accordning to
the manufacturer the antenna has an operation bandwidth from 30 − 300 MHz. At
the lower end of the operation bandwidth, however, the antenna is not matched very
well to a 50 Ω read-out system.
Figure 8.6: Magnitude of the BBAL 9136 antenna impedance. Within its operation
bandwidth the impedance magnitude mostly resides in the vicinity of 50 Ω. However,
this is not the case for the lower frequency range that overlaps with the SBS’s
operation bandwidth.
9. Absolute Gain
Measuring the absolute gain of the Small Black Spider LPDA is required to learn
about the directional and frequency-dependent sensitivity of the antenna towards an
incoming signal. The gain is obtained from a transmission measurement between a
transmitting BBAL 9136 biconical antenna (chapter 8) and a receiving SBS (chapter
6). The frequency-dependent power ratio
ζ=
PR
PT
(9.1)
of received power to fed-in power is measured with the FSH 4 vector network analyzer. The distance between receiver and transmitter as well as the absolute gain of
the transmitter in a 50 Ω read-out system (including mismatch losses) are known.
Hence the quantities can be used to compute the absolute gain of the Small Black
Spider LPDA with the Friis Transmission Equation (5.29).
In this chapter the measurements performed will be evaluated. We discuss the
properties of the measurement setup with respect to near-field effects, polarization
issues and uncertainties caused by environmental conditions.
9.1
Initial Near-Field Measurements
To test the validity of the Friis Transmission Equation given in equation (5.29),
a transmission measurement is performed on the rooftop of the physics centre at
RWTH Aachen University. The goal is to verify the method of obtaining the absolute
gain in a simple setup since a complex setup is required for measuring the full
directional gain pattern of the Small Black Spider LPDA.
9.1.1
Experimental Setup
In the test setup the Small Black Spider is attached to a pole on the edge of the roof.
Its lowest dipole is located at h = 1.9 m height above the rooftop, which resembles the
ground in this setup. The BBAL 9136 biconical antenna is situated at approximately
the same height above the rooftop for this test measurement. Transmitter and
receiver are separated by a distance of d = 11 m, meaning the distance between the
centre of the SBS’s lowest dipole and the centre of the biconical antenna. Hence
the distance between biconical antenna and origin of the SBS reference coordinate
system is
√
(9.2)
r = d2 + h2 ≈ 11.2 m.
72
Absolute Gain
Figure 9.1: Near-field horizontal gain versus frequency. The blue and red curve
shows the measured data while the black curve shows the simulation of an equivalent
setup above a perfect ground plane in NEC-2. θ and φ denote the direction of the
incoming radio signal while r denotes the distance to the source.
Both antennas are aligned for a maximal signal transmission, meaning that the SBS’s
dipoles are orientated parallel to the dipole axis of the biconical antenna. That way
the radiated electric field is polarized in e φ direction when observed from the SBS’s
coordinate system, which means the horizontal gain of the antenna will be measured.
The antennas are connected to the FSH 4 network analyzer through RG-58U coaxial
cables. It provides the feed-in signal for the transmitter and measures the power
received by the Small Black Spider LPDA.
9.1.2
Frequency Dependence
The frequency-dependent gain calculated from the Friis Transmission Equation (5.29)
can be seen in figure 9.1. It constantly increases throughout the operation bandwidth
of the SBS and reaches a maximum value around 80 MHz. For higher frequencies it
decreases again due to mismatch losses.
The uncertainty of the gain values results from uncertainties on the measured power
fraction, the absolute gain of the transmitter and the measured distance between
transmitter and receiver. The individual uncertainties are
σζ = 0.3 dB,
σGT = 0.7 dBi,
σr = 0.1 m.
(9.3)
The NEC-2 simulation corresponding to the test setup yields gain values for a conjugate matched read-out system. For comparison with the measured data the simulation results have been transformed to a 50 Ω read-out system using equation (5.19),
9.1. Initial Near-Field Measurements
73
after including the effect of a 4:1 balun by applying equation (8.2). If not mentioned
elsewise, this is done for all simulations from now on.
Simulation and measurement yield very similar results. The ground plane included in the simulation is a perfectly conducting mirror and it seems that this is also a
good approximation for the flat rooftop where the measurement has been performed.
Due to the small distance between biconical antenna and Small Black Spider there
are still near-field effects present in this transmission measurement. Therefore, the
measured gain is by convention named horizontal near-field gain.
9.1.3
Near Field and Far Field
Near-field effects are present in a transmission measurement, if the distance between
transmitter and receiver is not selected to be large enough. In case of the Small
Black Spider LPDA, the shape of the electromagnetic field, radiated by the biconical
antenna, is of crucial importance. For a distance of r = 11.2 m the far-field condition
2
2 · lBic
≈ 0.8 − 2 m
(9.4)
λ
is fulfilled within the SBS operation bandwidth from 30 − 80 MHz. The largest
antenna dimension of the biconical antenna is lBic = 1.94 m. According to this
condition far-field approximations are valid in the test setup.
However, this only means that we can assume the radiated electric field to propagate
like a spherical wave. The distance between SBS and biconical antenna is still not
large enough to neglect the curvature of this spherical wave in comparison to the
dimensions of the SBS. Hence it can not be approximated as a plane wave. This
causes a different response of the Small Black Spider LPDA to the incoming signal
and hence leads to different values for the gain.
The absolute gain of an antenna is defined as a far-field quantity in the sense that
the incoming signal is a plane wave. This gain is not obtainable from a transmission
measurement in which the receiver is located too close to the transmitter due to the
curvature of the radiated field. Hence the term near-field gain makes sense, even
though the conventional far-field condition is fulfilled at all times.
r≥
Figure 9.2 shows the gain obtained from a simulated transmission measurement
for various distances between the BBAL 9136 and the SBS. The direction of the
incoming signal is the same as in the measurement shown in figure 9.1. For increasing distances r the curvature of the radiated electric field becomes more and more
insignificant and the gain calculated with the Friis Transmision Equation converges
towards the absolute gain that can be obtained directly from the simulation.
From the test measurements and the simulations we conclude that it is possible
to obtain the absolute gain of the Small Black Spider LPDA using the Friis Transmission Equation. However, the distance between transmitter and receiver should
be as large as possible in the setup to minimize near-field effects.
74
Absolute Gain
Figure 9.2: Frequency dependence of the simulated horizontal gain for various
distances between biconical antenna and SBS. Due to near-field effects the absolute
gain can only be obtained from a transmission measurement for large distances
between transmitter and receiver. In the radio regime the required distances may
range well into a few hundred metres, depending on the direction of the incoming
signal. This makes it impossible to completely avoid near-field effects in any of our
setups.
9.2
Horizontal Absolute Gain
The quantity that describes the SBS’s sensitivity towards an incoming plane wave
polarized in e φ direction is the horizontal absolute gain. When observing radio
signals caused by extensive air showers this is the quantity one is interested in since
the incoming radio pulses can be approximated as plane waves in the operation
bandwidth of the SBS [64]. It is obtained by a transmission measurement with
a sufficiently large distance r between the BBAL 9136 biconical antenna and the
Small Black Spider LPDA. A complex setup is necessary to measure the frequency,
azimuth angle and zenith angle dependence of the horizontal absolute gain.
9.2.1
Experimental Setup
The measurements are performed at the physics centre of RWTH Aachen University.
A sketch of the setup can be seen in figure 9.3. The SBS is set up on the ground, as
seen in figure 6.8. The biconical antenna is positioned next to a corner of the roof of
the physics centre (figures 9.4 and 9.5). Due to the positioning at the corner of the
rooftop and the dipole axis of the BBAL 9136 being orientated perpendicular to the
wall of the physics centre, signal reflections from the wall into the SBS are avoided.
The antennas are connected to the FSH 4 network analyzer by RG-213 coaxial
cables with a total length of lCable = 100 m. The biconical antenna is orientated in
e φ direction with respect to the SBS’s reference coordinate system.
9.2. Horizontal Absolute Gain
75
Figure 9.3: Sketch of the setup for a horizontal absolute gain measurement.
Figure 9.4: Metal construction to hold the BBAL 9136 biconical antenna in position for a transmission measurement. Due to its weight the construction can easily
counter balance the crank caused by the biconical antenna several meters away from
the construction’s center. GFK-pipes were chosen to hold the antenna to minimize
additional metal structures close to it.
76
Absolute Gain
Figure 9.5: The BBAL 9136 biconical antenna as transmitter in a horizontal absolute gain measurement for the SBS LPDA. Attached to a construction especially
designed for this measurement, the antenna is located a few meters away from a
rooftop corner of our institute. Reflection effects are minimized and a distance of
approximately 30 − 40 m to the SBS is guaranteed in this setup. A brazing solder
leads from the antenna to the point on the ground below it. It is needed to measure
the dimensions of the setup, from which the distance r between BBAL 9136 and
SBS is ascertained.
With this setup all dependencies of the horizontal absolute gain can be measured.
The network analyzer performs a frequency sweep. The construction seen in figure
6.8 and designed by our mechanical workshop is rotatable in e φ direction of the SBS
reference coordinate system, enabling the variation of the azimuth angle. By moving
the entire construction across the ground the zenith angle can be selected.
9.2.2
Frequency Dependence
As described in section 9.1 the fraction of received power ζ given in equation (9.1)
is measured with the network analyzer.
The distance between the BBAL 9136 biconical antenna and the Small Black Spider
LPDA is acquired from the the distance d = 31.4 m between the SBS and the lower
end of the brazing solder and the height h = 21.5 m of the physics centre. Due to the
ground not being completely flat throughout the entire setup, there is also a height
offset h0 = 2.3 m between the ground below the BBAL 9136 and the ground below
the SBS.
The distance r between the antennas is hence given by
r=
q
d2 + (h − h0 )2 ≈ 36.8 m.
(9.5)
77
Figure 9.6: Frequency dependence of the horizontal absolute gain. The blue and
red curve shows the measured data while the solid black line shows the results for a
simulation of the equivalent setup in NEC-2. The dashed line represents the absolute
gain obtained from the NEC-2 simulation output file of a Small Black Spider LPDA
over a ground plane. The green curve shows the equivalent output for a free space
simulation. θ and φ denote the direction of the incoming radio signal in the SBS’s
reference coordinate system while r depicts the distance between transmitter and
receiver.
From this we can also calculate the zenith angle as
d
◦
≈ 58 .
θ = arctan
h − h0
(9.6)
The uncertainties σζ and σGT are the same as in section 9.1. The uncertainties of
the measured lenghts and the zenith angle are
σr = 0.6 m
σd = σh = σh0 = 0.5 m
=⇒
(9.7)
◦
σθ = 1 .
σr and σθ are calculated by applying error propagation to equations (9.5) and (9.6).
With all known quantities the horizontal absolute gain can be computed using the
Friis Transmission Equation (5.29). We see the result in figure 9.6.
As from 25 MHz the absolute gain steeply increases along with the frequency. The
reason for this is that mismatch losses abruptly decrease when entering the operation
bandwidth of the SBS. In the range from 55 − 70 MHz we see a dip in the curve.
Comparing to the simulations for free space and for a ground plane we can conclude
that this dip is caused by the presence of a ground in the setup. The direct signal
78
Absolute Gain
interferes destructively with the signal reflection from the ground in that frequency
range. Past the upper end of the operation bandwidth the gain decreases again due
to mismatch losses becoming relevant again.
Over a large intervall of the SBS’s operation bandwidth the difference between simulation and measurement does not exceed 2 dBi. Significant deviations are only
seen in the regions around the lower and the upper end of the operation bandwidth.
The measured gain is always smaller than the simulated gain. One possible reason could be a misalignment of the biconical antenna, meaning that its maximum
gain direction would not face the SBS. The gain GT of the transmitter would then
have to be calculated as
GT = GT,0 cos2 (φOff ) ,
(9.8)
where GT,0 is the gain provided by the manufacturer and φOff is the offset angle
quantifying the misalignment. To explain a difference of more than 2 dBi, though,
◦
φOff has to be greater than 35 . This has not been the case in the measurements
since the antennas were aligned very carefully.
Generally, deviations around 2 − 4 dBi are to be expected since the simulation assumes a flat and infinite ground plane and in no way resembles the complexity of
the environment around the actual setup.
9.2.3
Azimuth Dependence
The azimuth dependence of the horizontal absolute gain can be measured with the
transmission measurement setup by rotating the aluminium pole of the construction
that the Small Black Spider LPDA is attached to. The azimuth angle can be adjusted
on the degree scale on the aluminium panel of the construction. A photo can be
seen in figure 9.7. The result of the measurements is seen in figure 9.8.
When the dipoles of the SBS are aligned parallel to the dipole axis of the transmitting
biconical antenna, the SBS’s sensitivity becomes maximal for a fixed frequency f .
If the dipoles are aligned perpendicular to the dipole axis of the transmitter, the
◦
◦
gain decreases to zero. This is the case for φ = 90 and φ = 270 and should be
completely independent of the signal frequency.
There is a slight difference between the minima of the two curves in figure 9.8. It
actually is caused by reflections from walls present in the setup environment. Due
to convenient positioning of the antennas, however, the effect is hardly visible in the
data shown in figure 9.8. This has not been the case in all measurements (please
refer to subsection 9.4.2 for details).
79
Figure 9.7: The degree scale on the workshop construction for the Small Black
Spider LPDA. By rotating the pole holding the SBS one is able to adjust the azimuth angle in the SBS’s reference coordinate system. The uncertainty on the value
◦
approximately is σφ ≈ 2 [65].
Figure 9.8: Azimuth dependence of the horizontal absolute gain for different frequencies f . The gain is shown as a linear factor in this plot, not to be mistaken for
the dBi scale used in the other plots. θ denotes the zenith direction of the incoming
signal and r the distance between the BBAL 9136 and the SBS.
80
Absolute Gain
Figure 9.9: Zenith dependence of horizontal absolute gain for different frequencies
f . φ denotes the azimuth direction of the incoming signal.
9.2.4
Zenith Dependence
By varying the position of the Small Black Spider LPDA along with the entire
workshop construction from figure 6.8, the zenith-angle direction of the incoming
radio signal can be changed. For this purpose a set of wheels has been attached to
the construction.
For each position the distance d to the brazing solder above the ground needs to
be measured to obtain the distance r between the antennas with equation (9.5)
and the zenith angle θ by using equation (9.6). Due to an uneven topography
the positions suitable for measurements are limited at the physics centre of RWTH
Aachen University. Hence the gain dependence is only measured for a small intervall
of zenith angles. The result is shown in figure 9.9.
Depending on the frequency f , the directional gain sphere takes on a different shape
as number and position of the side lobes – seen in figures 7.3 and 7.6 in chapter 7
and caused by the presence of a ground plane – change. This leads to the different
slope values for the curves seen in figure 9.9.
9.3
Reconstruction of the Simulated Vertical Gain
It is more difficult to obtain the vertical absolute gain of the Small Black Spider
LPDA. In a transmission measurement the transmitter then needs to be orientated
in e θ direction in the SBS reference coordinate system. This is difficult to realize in
an experiment.
Another approach, however, is to perform two measurements for given angle values
θ and φ. In one measurement the dipole axis of the BBAL 9136 is aligned to the
9.3. Reconstruction of the Simulated Vertical Gain
81
Figure 9.10: The left sketch shows the transmitter orientated in e θ direction for a
measurement of the vertical absolute gain. This is hard to realize in a setup. The
right sketch shows the transmitter orientated parallel to the y and z-axes of the
SBS’s reference coordinate system. These alignments are much easier to attain in a
setup.
y-axis of the SBS reference coordinate system and in the other it is aligned to the
z-axis. An illustration can be seen in figure 9.10.
One can express the power PR,θ that the SBS would receive from a wave radiated
by the transmitter and polarized in e θ direction as
PR,θ =
|Uθ |2
.
R
(9.9)
To obtain equation (9.9) equations (5.9) and (5.17) were used with Z = R = 50 Ω.
By using equation (5.1) and assuming that the transmitter behaves like a dipole, we
can write
|Hθ Eθ |2
|Hθ (Ey cos (θ) + Ez sin (θ))|2
PR,θ =
=
R
R
(9.10)
|Uy cos (θ) + Uz sin (θ)|2
=
.
R
Ey and Ez are the projections of Eθ to the y and z-axes of the SBS reference coordinate system. Consequently, Uy and Uz are the voltages measured in transmission
measurements with a corresponding alignment of the biconical antenna.
The only difference between the two transmission measurements is the orientation of
the biconical antenna. Hence Uy and Uz are complex voltages with the same signal
phase ψ. Thus, we can write equation (9.10) as
PR,θ
|Uy |2 cos2 (θ) + |Uz |2 sin2 (θ) + 2 |Uy | |Uz | cos (θ) sin (θ)
=
R
= PR,y cos2 (θ) + PR,z sin2 (θ) + 2
p
PR,y PR,z cos (θ) sin (θ) .
(9.11)
82
Absolute Gain
Figure 9.11: Frequency dependence of simulated vertical absolute gain. The black
curve shows the output from a NEC-2 simulation file while the red curve was obtained from two simulated transmission measurements, using the reconstruction from
equation (9.11). θ and φ denote the direction of the incoming signal in the SBS
reference coordinate system.
PR,y and PR,z are the power fractions measured in the individual transmission measurements and PR,θ is the power one would measure if the transmitter was orientated
in e θ direction. The reconstructed value for PR,θ can be plugged in to the Friis
Transmission Equation to obtain the vertical absolute gain.
The reconstruction of the gain by using equation (9.11) has been verified in NEC-2.
Figure 9.11 shows the reconstruction for simulated transmission measurements.
Due to the polarization of the signal in e θ direction the vertical gain is maximal for
◦
an azimuth angle of φ = 90 . The reconstruction from equation (9.11) yields the
vertical absolute gain in good approximation. Hence the presented method is a good
approach for future vertical gain measurements. Assuming a simple dipole behaviour
for the transmitter results in slighty inaccurate values for the reconstruction.
9.4. Uncertainties
83
Figure 9.12: Reproducability of absolute gain measurements. The histogram shows
the difference of the measured gain values to the mean value for a given direction
◦
◦
of the incoming signal. The azimuth angle varies from 0 − 375 in steps of 15 and
each measurement is performed 10 times. This results in an overall amount of 250
entries. θ denotes the zenith direction of the incoming signal and r is the distance
between the BBAL 9136 biconical antenna and the SBS LPDA.
9.4
Uncertainties
Apart from the uncertainties mentioned in the previous sections, differences between
simulations and measurements occur due to environmental conditions at the setup.
This subsection will quantify the reproducability of absolute gain measurements and
give details on effects that disturb the measurements.
9.4.1
Reproducability of Absolute Gain Measurements
Apart from the systematic error σζ = 0.3 dB, given by the manufacturer of the FSH 4
network analyzer for the measured power fraction ζ in a transmission measurement,
one can test the reproducability of a measurement and e.g. repeat it several times
within a short time frame. This has been done for the test setup on the roof of the
physics centre, which was discussed in section 9.1. The spread of the absolute gain
values obtained from the Friis Transmission Equation can be seen in figure 9.12.
The spread of the measured gain values is completely insignificant, only being
σGR ,Stat = 0.008.
(9.12)
It is translated to the logarithmic dBi-scale by
σgR ,Stat,+ = 10.0 ( log10 (GR + σGR ,Stat ) − log10 (GR )) ,
σgR ,Stat,− = 10.0 ( log10 (GR ) − log10 (GR − σGR ,Stat )) .
(9.13)
84
Absolute Gain
Figure 9.13: Frequency-dependent deviation of the minimum gain in an azimuth
pattern. In case of the blue and red curves, a minimum gain does not necessarily
occur when the dipoles of the SBS are aligned perpendicular to the BBAL 9136.
This is due to reflected signals from walls close to the setup. The black curve shows
the equivalent deviation in a setup with the walls being further away. NS and EW
refer to the North-South and East-West planes of the SBS.
For absolute gain values around 2 − 8 dBi equation (9.13) yields uncertainties in
the order of 0.01 − 0.02 dBi in the logarithmic scale. Other effects such as signal
reflections cause differences between measurements at a higher order of magnitude.
9.4.2
Signal Reflections
A signal emitted by the BBAL 9136 biconical antenna during a transmission measurement may be reflected from a wall in the vicinity of the setup and then received
by the SBS. This effect is minimal for the measurements presented in section 9.2.
Measurements with a setup in which the SBS is located closer to the walls of the
physics centre include observable reflection effects.
When observing the azimuth dependence of the horizontal absolute gain as in figure 9.8, the position of the minima should be independent of the signal frequency
◦
◦
and always be located at φ = 90 and φ = 270 . This is not the case if there
are walls close to the setup. In that case the position of the minima depends on
the signal frequency f , as the interference between direct and reflected signal also
depends on the frequency. Figure 9.13 shows the deviation φ0 of the minimum gain
◦
◦
from φ = 90 and φ = 270 .
For a setup in which walls are far away from the SBS, the minimum of the gain
◦
only deviates up to about 10 from the expected position. In a setup with walls
9.4. Uncertainties
85
◦
nearby, the minimum is almost 35 away from the expected azimuth angle for some
frequencies.
This leads to the conclusion that the gain pattern of the SBS is significantly deformed
by the presence of walls in the setup. Hence the measurements do not longer yield
the correct absolute gain for the system antenna plus ground.
9.4.3
Topography and Weather Conditions
The local topography around the setup also has a significant influence on the measured absolute gain.
In the test setup discussed in section 9.1, the simulated and the real transmission measurement are in good agreement with each other. This makes sense since
the measurements were performed on the perfectly flat roof of the physics centre at
RWTH Aachen University.
In the measurement for the horizontal absolute gain, however, the SBS was positioned in the grasslands behind the physics centre. Though the position was chosen
carefully, we had to ponder between a locally flat ground and being far away from
the walls of the physics centre. It was impossible to avoid a change of topography
between the position of the SBS and the BBAL 9136 biconical antenna.
Topography effects are not included in the simulated transmission measurements
and hence differences between simulations and measurements are to be expected.
Weather effects cannot be quantified easily since it takes a lot of time to prepare the
setup for an absolute gain measurement and we did not perform outdoor measurements during heavy rain.
It is expected that, for a wet ground with a high conductivity, approximating the
ground with a perfectly conducting mirror in the simulation is more accurate than
for a dry ground. The simulations that can be seen in figure 9.14 show that the gain
difference between setups with a mirror and setups with realistic grounds is in the
order of 1 − 2 dBi.
The measurements in this chapter have been compared to simulations including a
perfect mirror as ground since the ground properties in the setups were unknown.
This, of course, promotes deviations between simulations and measurements.
For grounds such as polar ice, e.g at the south pole, which are weakly conductive and have a permittivity close to 0 , the gain will behave as if the antenna were
located in free space.
The reason for this is that the parameter ZR from equation (7.17), which contains
ground specifications, converges in the form of
lim ZR = lim
1 → 0
1 → 0
σ1 → 0
σ1 → 0
r
0
= 1.
1 − i σω1
(9.14)
86
Absolute Gain
Figure 9.14: SBS absolute gain simulation for various ground types. θ and φ
denote the position on the gain sphere. The different colours represent different
ground types.
Figure 9.15: Weather conditions during the absolute gain measurement. On the
left we see the development of the air humidity while on the right we see the development of the ground temperature.
In consequence the reflection coefficient from equation (7.16) vanishes since
p
ZR cos (θ) − 1 − ZR2 sin2 (θ)
p
lim Rφ = lim
= 0.
(9.15)
ZR →1
ZR →1 Z cos (θ) +
1 − ZR2 sin2 (θ)
R
Figure 9.15 shows the time development of the air humidity and the ground temperature during the absolute gain measurements. The air humidity increases towards the
evening and the temperature of the ground slightly decreases. The overall weather
conditions remain stable.
10. Group Delay and Effect of a
Low-Noise Amplifier
The group delay τg of an antenna quantifies the frequency-dependent time a signal
needs to pass through the antenna structure. It has been defined in equation (5.30).
It is the derivative of a signal phase with respect to the angular frequency and hence
is related to the phase of the vector effective height from equations (5.34) and (5.35).
In this chapter group delay measurements are presented. The group delay of the
Small Black Spider LPDA is measured for one particular direction of an incoming
radio pulse. Using this we show that the BBAL 9136 biconical antenna has no
frequency-dependent group delay. The cross calibration is then used to investigate
the directional dependence of the SBS’s group delay.
Furthermore, the effect of a low-noise amplifier (LNA) on the horizontal absolute
gain of the SBS is studied.
10.1
Group Delay of the Small Black Spider LPDA
A transmission measurement between two Small Black Spider LPDAs needs to be
performed to obtain information on the SBS’s group delay. For this purpose, two
SBS LPDAs are set up as seen in figure 7.4 in chapter 7 and separated by a distance
of r = 44 m. They are connected to the FSH 4 network analyzer with RG-213 coaxial
cables. The transmitting and receiving antenna planes are aligned parallel to each
other for a maximal signal transmission.
The quantity of interest in this setup is the relative signal phase ψ between the fed
in and received signal. The frequency-dependent signal phase ψ is given as a set of
data points that can be differentiated numerically to obtain the overall signal delay
of the measurement. This is done by fitting a linear slope of the form
ψi (ωi ) = −τg,i · ωi + bi
(10.1)
to an intervall of 30 frequency bins. The FSH 4 records the signal phase data of
the observed frequency bandwidth in 631 frequency bins. ψi is the signal phase
and ωi = 2πfi the angular frequency in the ith bin. The size of each bin depends
on the observed bandwidth. For the group delay measurements a bandwidth from
0−100 MHz was chosen for the frequency f , resulting in a bin size of ∆f ≈ 0.16 MHz.
The paramater τg,i in equation (10.1) depicts the negative numerical derivative of the
signal phase with respect to the angular frequency in the ith frequency bin. Hence
it is the value of the group delay averaged over 30 frequency bins. The derivative is
88
Group Delay and Effect of a Low-Noise Amplifier
Figure 10.1: Group delay of the Small Black Spider LPDA for a horizontally
incoming signal. NS stands for the North-South and EW for the East-West plane of
the SBS LPDA. Both planes were used as a transmitter to test the reproducability
of the measurement. The black curve shows the results obtained from an equivalent
simulated setup in NEC-2. θ and φ denote the direction of the incoming signal in
the SBS’s reference coordinate system. r is the distance between the antennas.
calculated for every bin from iMin = 16 to iMax = 616. The obtained total signal delay
contains the frequency-dependent group delay of both SBS antennas and a constant
delay offset. The constant offset is always included in a transmission measurement
since according to equation (5.31) the electromagnetic wave needs to propagate from
the transmitting to the receiving antenna. Dividing the overall delay by two yields
the frequency dependence of the SBS’s group delay. The result can be seen in figure
10.1.
Two independent measurements have been carried out and yield the same result for
the group delay. The simulation is in acceptable agreement with the measured data.
Orignially, the simulated group delay was on average about 15 ns smaller than the
measured group delay. The reason for this is that the approximately 4 m of coaxial
cable inside the Small Black Spider LPDA are not included in the NEC-2 simulation
of the antenna. This offset in the simulated data has been corrected in figure 10.1.
10.2
Group Delay of the BBAL 9136 Biconical
Antenna
By exchanging one of the two SBS antennas from the setup described in section
10.1, e.g. the transmitter, with the BBAL 9136 biconical antenna, one can obtain
the latter antenna’s group delay by repeating the measurement.
After computing the overall signal delay in the measurement the group delay of the
10.3. Uncertainty of the Group Delay
89
Figure 10.2: Group delay of the BBAL 9136 biconical antenna. BIC refers to
the transmitting biconical antenna and NS to the receiving North-South plane of
the Small Black Spider LPDA. The black curve shows an equivalent simulation in
NEC-2. θ and φ denote the direction of the incoming signal in the SBS’s reference
coordinate system. r is the distance between the antennas.
SBS needs to be subtracted from the data. The group delay of the biconical antenna
remains and can be seen in figure 10.2.
Because of its fixed phase centre the group delay of the biconical antenna is not
frequency-dependent. This has also been confirmed by the manufacturer Schwarzbeck Mess-Elektronik [66]. The NEC-2 simulation of the antenna’s group delay is
not very accurate since only the antenna cones are included in the simulation. Hence
the electric properties contained in the central structure are not taken into account.
10.3
Uncertainty of the Group Delay
Figure 10.1 shows two independent measurements of the SBS’s group delay under
equal setup conditions. By obtaining the RMS1 of the spread between corresponding
group delay values the uncertainty of the group delay measurements can be estimated. Figure 10.3 shows the spread of the group delay. The estimated uncertainty is
στg = 1 ns.
1
Root Mean Square
(10.2)
90
Figure 10.3: Spread from two SBS group delay measurements. The RMS is a good
estimator for the uncertainty of the group delay.
10.4
Directional Dependence of SBS Group Delay
With the group delay from the biconical antenna it is now possible to analyze the
phase data obtained in the transmission measurements from chapter 9. Measurements have been performed for various azimuth and zenith angles. Hence the directional dependence of the Small Black Spider’s group delay can be studied. Figures
10.4 and 10.5 show the results.
The propagation offset from equation (5.31) has been corrected. It can be seen that
the group delay of the Small Black Spider LPDA is not significantly influenced by the
variation of the azimuth angle φ of the incoming signal. However, the fluctuations
are greater than the group delay uncertainty στg . Especially for azimuth values close
◦
to 90 the group delay deviates from the other curves because the SBS’s dipoles
are orientated nearly perpendicular to the dipole axis of the transmitter for those
angles. Only a small fraction of the radiated signal is received in that case. The
signal phase values are then dominated by other signals from external radio sources
in Aachen and hence are not reliable anymore.
In case of the zenith-dependent group delay, the propagation offset has also been
corrected. It has been taken into account that r varies along with the zenith angle
θ.
An increasing group delay can be seen for an increasing zenith angle θ. Within
the operation bandwidth, the effect is smaller for higher frequencies than for lower
frequencies. Referring to figure 10.6, it can be explained as follows: as mentioned
before, a radiated signal in a transmission measurement has a minimum delay since
it needs time to travel to the receiving antenna. This time is given by equation (5.31)
with r as the distance between the transmitter and the origin of the Small Black
Spider’s reference coordinate system. In this form, the propagation time does not
10.4. Directional Dependence of SBS Group Delay
91
Figure 10.4: Azimuth dependence of the Small Black Spider’s group delay. θ is
the zenith direction of the incoming signal and r the distance between the biconical
antenna and the SBS LPDA. The group delay is shown for various azimuth angles
φ.
Figure 10.5: Zenith dependence of Small Black Spider’s group delay. φ is the
azimuth direction of the incoming signal and r the distance between the biconical
antenna and the SBS LPDA. The group delay is shown for various zenith angles θ.
92
Figure 10.6: Signal propagation for different frequencies and zenith angles. The
true propagation distance rT from a transmitter to the SBS depends on the zenith
angle θ and the signal frequency f .
depend on the direction of the incoming signal. However, the SBS LPDA does not
have a localized phase centre, which means that for higher frequencies the shorter
dipoles will receive the signal and for lower frequencies the longer dipoles will.
Hence the true propagation distance rT of the radiated signal slightly changes with
the frequency and because of the selected reference coordinate system it also changes
with the zenith angle. Only correcting for the propagation distance r between transmitter and the origin of the SBS reference coordinate system in equation (5.31),
therefore, yields a frequency and zenith-angle-dependent group delay.
10.5
Effect of a Low-Noise Amplifier
Low-noise amplifiers will be used at AERA to amplify measured radio pulses within
the frequency operation bandwidth [47]. The influence of a low-noise amplifier on
the horizontal absolute gain of the Small Black Spider LPDA and its group delay
has been studied. A prototype LNA was used in the measurement since no final
version was available. It is attached to a read-out channel of the SBS and powered
by an external voltage source. Figure 10.7 shows the power gain of the LNA with
included filter elements.
The power gain of this prototype LNA is slightly different to that of the final series
production. The final version was especially designed to amplify incoming signals
between 30 MHz and 80 MHz. However, the prototype LNA is sufficient to study
10.5. Effect of a Low-Noise Amplifier
93
Figure 10.7: Power gain of the low-noise amplifier used for the SBS. In a frequency
bandwidth ranging from approximately 20−100 MHz an incoming signal is amplified
by the device.
the qualitative influence of a low-noise amplifier on the absolute gain and the group
delay of the Small Black Spider LPDA.
Figure 10.8 shows the measured horizontal gain of the SBS with and without an
LNA.
Noise caused by external radio stations is clearly visible in the measurement for
frequencies above 85 MHz. In the transmission measurements presented in chapter
9, this noise was situated orders of magnitude below the signal fed into the biconical antenna from the FSH 4 network analyzer and thus could be neglected. This
time, the feed-in signal was attenuated by 40 dB to protect the network analyzer
from receiving too strong signals because of the amplification by the LNA. Hence
the signal radiated by the biconical antenna is much weaker. In consequence radio
stations significantly disturb the measurement for frequencies above 85 MHz. The
strong relative signal received by the radio stations shows that a lesser attenuation
could have been chosen in the setup. However, below 85 MHz the characteristics of
the antenna are still visible.
The absolute gain of the SBS is significantly increased when including an LNA to
the setup. Since the gain is proportional to the power PR received by the SBS, the
power gain of the LNA and the antenna gain add up, resulting in a very high overall gain of the system antenna plus LNA within the LNA’s amplification bandwidth.
Figure 10.9 shows the group delay of the LNA. The data above 85 MHz is dominated by the influence of the external radio sources already seen in figure 10.8. For
those frequencies the signal phase data and thus the group delay is not reliable.
Within the SBS’s operation bandwidth there is a variation of about 10 ns. Compared
to the frequency-dependent group delay of the antenna structure itself the delay of
the LNA is very small. However, it will slightly influence the phase of the antenna’s
94
Figure 10.8: Horizontal absolute gain of the Small Black Spider LPDA. The black
and green curves represent measurements performed without an LNA while the blue
and red curves represent measurements in which an LNA was used. EW and NS
refer to the East-West and North-South planes of the SBS while θ and φ denote the
incoming signal’s direction. r is the distance between the biconical antenna and the
origin of the SBS’s reference coordinate system.
vector effective height. Thus a system of SBS and LNA will respond differently to
an incoming signal than the SBS by itself.
The measurements presented in this section qualitatively describe the influence of a
prototype LNA on the SBS’s response towards an incoming signal. The results show
that – for a specific LNA – it is advisable to compare antenna calibration results
obtained in a setup with and without the LNA.
10.5. Effect of a Low-Noise Amplifier
Figure 10.9: Frequency-dependent group delay of the tested LNA.
95
96
11. Reconstruction of the Vector
Effective Height
According to equations (5.34) and (5.35) the vector effective height of an antenna can
be reconstructed from absolute gain and signal phase measurements. This chapter
summarizes the reconstruction results for the directional and frequency dependence
of the Small Black Spider’s vector effective height. The effect of a low-noise amplifier
on the vector effective height is also studied.
11.1
Horizontal Component
Since the incoming signals in the transmission measurements from chapters 9 and
10 were always polarized in e φ direction, the reconstruction will yield the horizontal
component Hφ of the SBS’s vector effective height.
By using equation (5.34), the magnitude of Hφ can be calculated from the horizontal
absolute gain and the load impedance of the antenna read-out system.
The phase ψH of the vector effective height is obtained from the signal phase data
of the gain measurements. The measured phase ψ is the relative phase between the
feed-in signal and the signal that has returned to the network analyzer. It is of the
form
ψ = ψSBS + ψPropa + ψBic .
(11.1)
The propagation distance r between the antennas as well as their properties contribute to ψ. The propagation phase can be calculated as
ψPropa =
2πf r
.
c
(11.2)
The phase offset ψBic , caused by the biconical antenna, only depends linear on the
frequency f since the biconical antenna has no frequency-dependent group delay
(section 10.2). By applying equation (11.1) to the measurements from chapter 10
we obtain
ψS − ψR
,
2
= ψB − ψR − ψSBS,0 .
ψS = 2ψSBS,0 + ψR
=⇒
ψSBS,0 =
ψB = ψSBS,0 + ψR + ψBic
=⇒
ψBic
(11.3)
ψS denotes the phase value measured with the network analyzer in the setup with
two SBS antennas (section 10.1) and ψB is the value from the setup including the
98
Reconstruction of the Vector Effective Height
biconical antenna (section 10.2). ψR is the propagation offset due to the distance of
R = 44 m in those measurements. ψSBS,0 is the contribution of the SBS to the phase
in the measurements from chapter 10. Plugging the upper expression in equation
(11.3) into the lower one yields
ψBic = ψB −
1
(ψS + ψR ) .
2
(11.4)
Hence the phase of the Small Black Spider LPDA’s vector effective height is given
by
ψH ≡ ψSBS = ψ − ψPropa − ψBic .
(11.5)
The frequency dependence of Hφ is shown in figure 11.1. The directional dependence
can be seen in figures 11.2 and 11.3.
The magnitude of Hφ is maximal at frequencies around 45 MHz. In contrary to the
absolute gain it does not have another maximum for higher frequencies (compare
figure 11.1 to figure 9.6 from chapter 9). This is because of the factor f1 in equation
(5.34) that suppresses |Hφ | for high frequencies.
In case of the azimuth dependence, this effect causes the curves from figure 9.8 in
chapter 9 to switch places in figure 11.2. The reason for the different behaviour is the
division by the frequency in (5.34). One also sees that the sign of the vector effective
◦
◦
height is included in its phase. At azimuth angles around 90 and 270 there is a
◦
phase shift of 180 , which means a change in the sign of the vector component.
Due to the reflection effects discussed in section 9.4 the phase shift does not necessa◦
◦
rily occur at 90 and 270 , depending on the frequency. To obtain the phase of the
vector effective height, the data from the setup far away from walls was evaluated
to minimize reflection effects.
Variation of the zenith angle causes only small changes for the values of |Hφ |. The
extent depends on the position of sidelobes on the gain sphere (figure 7.6 in chapter 7) and hence on the frequency. The slope of the zenith-dependent phase is not
linear, confirming the zenith dependence of the SBS’s group delay discussed in section 10.4. For different frequencies we see different slopes, which corresponds to the
frequency-dependence of the SBS’s group delay.
All dependencies of Hφ have been reconstructed and shown in figures 11.1, 11.2 and
11.3. For a given direction the complex quantity can be calculated from the data as
Hφ (f, φ, θ) = |Hφ (f, φ, θ)| eiψH (f,φ,θ) .
(11.6)
For an incoming signal polarized in e φ direction in the SBS reference coordinate
system it is now possible to apply the relation
Eφ =
Uφ
.
Hφ
(11.7)
Uφ is the voltage measured with the antenna in a 50 Ω system. By using the measured
vector effective height Hφ the absolute electric field strength Eφ of the incident signal
can be calculated.
11.1. Horizontal Component
99
Figure 11.1: Frequency dependence of horizontal vector effective height. The
magnitude of Hφ is shown in the upper figure while the phase of the quantity – still
including the offset ψBic from the biconical antenna – is seen below. θ and φ denote
the direction of the incoming signal in the SBS’s reference coordinate system while
r is the distance between biconical antenna and SBS.
100
Figure 11.2: Azimuth dependence of the horizontal vector effective height for
different frequencies f . The magnitude of Hφ is shown in the upper figure while
the phase of the quantity is seen below with the offset ψBic eliminated from the
data. θ is the zenith direction of the incoming signal while r is the distance between
transmitter and receiver. The offset ψBic has been corrected.
11.1. Horizontal Component
101
Figure 11.3: Zenith dependence of the horizontal vector effective height for different frequencies f . The magnitude of Hφ is shown in the upper figure while the
phase of the quantity is seen below with the offset ψBic eliminated from the data.
φ is the azimuth direction of the incoming signal while r is the distance between
transmitter and receiver.
102
Figure 11.4: Horizontal component of the Small Black Spider’s vector effective
height. The blue and red curve shows the reconstruction from section 11.1 without
an LNA. The green and red curve shows the reconstruction for the system antenna
plus LNA. θ and φ denote the direction of the incoming radio signal while r is the
distance between transmitter and r eceiver. θ, φ and r refer to the green curve. The
deviation from the quantities used in section 11.1 is very small, thus enabling us to
compare the two curves to each other.
11.2
Horizontal Component with a Low-Noise Amplifier
The effect of a low-noise amplifier on the frequency dependence of |Hφ | is visualized
in figure 11.4.
Within the Small Black Spider LPDA’s operation bandwidth we see that the presence of an LNA significantly increases the length of the vector. This is due to the
LNA’s power gain that leads to a very high absolute gain of the system antenna plus
LNA. The data above 85 MHz is dominated by external radio sources due to the
setup properties discussed in section 10.5 and hence does not represent the antenna
characteristics.
The prototype LNA used in the measurements amplifies signals up to more than
100 MHz and thus increases the length of the vector effective height for those frequencies. The final LNA design for the first stage of AERA filters signals below
30 MHz and above 80 MHz. It is expected that the length of the vector effective
height will steeply decrease outside of that amplification bandwidth. To obtain
quantitative results, additional calibration measurements combining the SBS LPDA
and the final LNA design will have to be carried out.
12. Summary and Outlook
With AERA a very promising setup for the detection of radio pulses emitted from
UHECR-induced extensive air showers is being realized. The radio array features
a much longer duty cycle than the fluorescence telescopes. The Small Black Spider
(SBS) LPDA designed for AERA is well understood and for the first stage of the
setup 24 SBS units of excellent production quality have been shipped to and set up
at the Pierre Auger Observatory in Argentina.
In this thesis the absolute calibration of the Small Black Spider LPDA has been
performed. By relating the voltage read out from the antenna to the incident electric field the SBS’s vector effective height has been measured for an incoming signal
polarized in e φ direction in the antenna’s reference coordinate system. This quantity
is mandatory to accurately reconstruct the waveform of radio pulses from extensive
air showers that are received by the antenna. It provides the SBS’s sensitivity including the signal dispersion within the structure. It was demonstrated how the vector
effective height depends on the frequency and the direction of an incoming radio
signal.
The cross talk between the North-South and East-West polarization plane of a Small
Black Spider LPDA was measured. It was found to be neglectable for actual measurements within the antenna’s operation bandwidth. The frequency-dependent antenna
impedance of the SBS was obtained from the measured reflection characteristics. By
taking matching theory into account, it has been shown that the influence of the load
impedance of an antenna-read-out system is of crucial importance when interpreting
measurement results.
Ground effects in an antenna setup were studied in measurements and compared to
simulations with NEC-2. The deformation of an antenna’s gain pattern due to the
presence of a ground plane has been related to reflected signals from the ground.
It was shown that the laws of geometrical optics can be applied to understand the
behaviour of the radiation density in setups including a ground plane. The behaviour of the radiation density has been measured and was shown to deviate from the
power law for a spherical wave in free space.
A calibrated transmitter can be used to obtain the absolute gain of the SBS with the
Friis Transmission Equation. For these measurements a biconical antenna from the
company Schwarzbeck Mess-Elektronik was used. The structure has been simulated
in NEC-2 and the simulated gain values were shown to be in agreement with the
measured values provided by the manufacturer.
104
Summary and Outlook
By performing transmission measurements the absolute gain of the Small Black Spider LPDA was measured for an incoming signal polarized in e φ direction. The
frequency dependence in the operation bandwidth as well as the full azimuth dependence of this horizontal gain was measured. Due to limitations of the setup the
zenith dependence of the gain was measured for an angle interval of approximately
◦
10 . It was shown that simulation and measurements yield a similar behaviour of
the absolute gain. However, measurements are significantly influenced by environmental conditions such as reflections from nearby walls and the ground topography.
The absolute gain of the SBS for an incoming signal polarized in e θ direction was
studied in simulations. A method was developed to reconstruct this vertical gain
from two transmission measurements, in which the transmitter is aligned parallel to
the y and z-axes of the SBS reference coordinate system. In a setup this is much
easier to realize than orientating the transmitter in e θ direction.
The group delay of the Small Black Spider LPDA has been obtained for a fixed
direction of the incoming signal. This was done with a transmission measurement
between two SBS antennas. By exchanging one SBS with the biconical antenna used
for the gain measurements, the group delay of the SBS was used to show that the
biconical antenna’s group delay does not depend on the frequency. Knowing this,
the directional dependence of the SBS’s group delay has been obtained.
The effect of a low-noise amplifier on the frequency dependence of the horizontal
absolute gain has been investigated. The results show that the gain of the antenna
and the power gain of the LNA combine, yielding an effective gain for the overall
system.
From the performed measurements the φ component of the vector effective height
has been reconstructed for a setup with and without LNA. For an incoming signal
polarized in e φ direction and received by the Small Black Spider LPDA it is hence
possible to calculate the electric field strength from the voltage measured with the
antenna.
Construction of the first stage of AERA has recently been completed and data
taking has started. First radio pulses from extensive air showers are expected to be
observed soon. By measuring the φ component of the Small Black Spider LPDA’s
vector effective height an important step has been made towards a precise reconstruction of the electric field of the incoming radio pulse. This calibrated reconstruction
will allow us to reveal important properties of the air shower and the nature of the
associated cosmic ray.
A. Appendix
A.1
List of Abbreviations
AERA
ASCII
BLS
CERN
CLF
CME
EAS
EW
FD
GFK
GZK
LNA
LPDA
NEC
NS
RG-58U
RMS
SBS
SD
UHECR
Auger Engineering Radio Array
American Standard Code for Information Interchange
Balloon Launch Station
Conseil Européen pour la Recherche Nucléaire
Central Laser Facility
Centre-of-Mass Energy
Extensive Air Shower
East-West
Fluorescence Detector
Glasfaser verstärkte Kunststoffe
Greisen Zatsepin Kuzmin
Low-Noise Amplifier
Logarithmic Periodic Dipole Antenna
Numerical Electromagnetics Code
North-South
Radio Guide-58 Universal
Root Mean Square
Small Black Spider
Surface Detector
Ultra-High-Energy Cosmic Ray
106
A.2
Appendix
NEC-2 Simulation Files
The following three subsections contain examples for NEC-2 steering cards.
A.2.1
Small Black Spider Steering Card
Figure A.1: Steering card for an SBS above a ground plane.
A.2. NEC-2 Simulation Files
A.2.2
107
BBAL 9136 Biconical Antenna Steering Card
Figure A.2: Steering card for the BBAL 9136 biconical antenna in free space.
108
A.2.3
Appendix
Steering Card for a Transmission Measurement
Figure A.3: Steering card for a transmission measurement in a setup including a
ground plane.
A.3. Biconical Antenna Calibration Certificate
A.3
Biconical Antenna Calibration Certificate
Figure A.4: Biconical antenna calibration certificate (page 1 of 2).
109
110
Appendix
Figure A.5: Biconical antenna calibration certificate (page 2 of 2).
A.4. FSH 4 Spectrum Analyzer
A.4
A.4.1
111
FSH 4 Spectrum Analyzer
Calibration of the FSH 4 (Transmission Measurement)
It is advisable to activate the device at least thirty minutes before data taking to
avoid thermal noise effects. Although a rechargable battery is included in the FSH 4,
it is better to provide electricity from the grid to ensure signal provision of constant
power to the transmitter [67].
All coaxial cables that will be included in the measurement need to be connected
to the FSH 4. The device contains two ports – one input and one output – and by
connecting these with the coaxial cables one creates a “through” connection. The
FSH 4 will then measure the frequency-dependent signal loss due to cable attenuation
and will eliminate this effect from any data measured after the calibration. After
that a test impedance needs to be loaded to the transmitting port. The FSH 4 is
then ready for data taking.
A.4.2
Quality of Calibration for a Transmission Measurement
The calibration will become unstable after some time, which could be a problem for
time-consuming measurements. However, figure A.6 shows that the deviations due
to calibration loss are very small.
Figure A.6: Frequency-dependent calibration offset before and after transmission
measurements spanning several hours. The offset on the measured power does not
exceed 0.2 dB within the SBS’s bandwidth and hence can be neglected.
112
A.5
Appendix
RG-213 Coaxial Cable
Figure A.7: Signal attenuation versus frequency [68].
A.6
Influences of Aachen Radio Background
Figure A.8: The Aachen radio background relative to the signal fed in to the
BBAL 9136 for the transmission measurements. In the 100 MHz regime various
radio stations can be seen. Within the SBS’s bandwidth the radio background is
situated orders of magnitude below the signal used in the transmission measurements
and hence can be completely neglected. As a consequence changes of the radio
background in time are also insignificant.
A.7. Defective Antenna
A.7
113
Defective Antenna
A.9 shows the received power fraction ζ measured with a receiving Small Black Spider LPDA (serial number 001) during a transmission measurement. The frequency
dependence of ζ is measured for both antenna planes of the Small Black Spider
LPDA.
The same measurement was also performed with another Small Black Spider unit
(serial number 036). The result is shown in figure A.10. Further testing of this SBS
unit showed that the longest and third longest dipole of the East-West plane had a
defective contact. Hence all further measurements were either performed with the
SBS 001 or the North-South plane of SBS 036.
Figure A.9: Power fraction received by the Small Black Spider LPDA with serial
number 001. θ and φ denote the direction of the incoming signal while r is the
distance between the biconical antenna and the SBS. Under equal setup conditions
both antenna planes receive almost exactly the same amount of power at a given
frequency.
114
Appendix
Figure A.10: Power fraction received by the Small Black Spider LPDA with serial
number 036. θ and φ denote the direction of the incoming signal while r is the
distance between BBAL 9136 and SBS. In the lower half of the bandwidth the EastWest plane receives significantly less power than the North-South plane.
References
[1] V. F. Hess, Über Beobachtungen der durchdringenden Strahlung bei sieben
Freiballonfahrten, Physik. Zeitschr., 13 (1912), pp. 1084–1091.
[2] P. Auger et al., Extensive cosmic-ray showers, Rev. Mod. Phys., 11 (1939),
pp. 288–291.
[3] S. Fliescher and T. Winchen, 2010. Private Communication.
[4] Seo et al., The Astrophysical Journal, 378 (1991), p. 763. scaled by factor 2.
[5] M. Nagano et al., J. Phys. G. Nucl. Part. Phys., 18 (1992), pp. 423–442.
[6] A. Haungs et al., Nuclear Physics B - Proc. Sup. 151, (2006), pp. 167–174.
[7] A. Haungs et al., in Proceedings of the 31st International Cosmic Ray Conference, 2009.
[8] Afanasiev et al., in Proceedings of the Int. Symposium of Extremely High
Energy Cosmic Rays, 1996, p. 32.
[9] M. A. Lawrence et al., J. Phys. G. Nucl. Part. Phys., 1991 (17), pp. 733–757.
[10] D. J. Bird et al., The Astrophysical Journal, 424 (1994), pp. 491–502.
[11] R. U. Abbas et al., Phys. Rev. Lett., 100 (2008).
[12] F. Schüssler for the Pierre Auger Collaboration, in Proceedings of
the 31st International Cosmic Ray Conference, 2009.
[13] P. L. Biermann and G. Sigl, Physics and Astrophysics of Ultra-High-Energy
Cosmic Rays, edited by M. Lemoine and G. Sigl, Springer, Berlin, Germany,
2001.
[14] E. Fermi, On the Origin of the Cosmic Radiation, Phys. Rev., 75 (1949),
pp. 1169–1174.
[15] H. Alfvén and C. G. Falthammar, Cosmical Electrodynamics, vol. 2, Clarendon Press, Oxford, United Kingdom, 1963.
[16] R. J. Protheroe, Acceleration and Interaction of Ultra High Energy Cosmic
Rays, arXiv:astro-ph/9812055v1, (1999).
[17] A. M. Hillas, The Origin of Ultra-High-Energy Cosmic Rays, Ann. Rev. Astron. Astrophys., 22 (1984), pp. 425–444.
116
References
[18] L. Anchordoqui et al., Ultrahigh Energy Cosmic Rays: The state of the art
before the Auger Observatory, arXiv:hep-ph/0206072v3, (2002).
[19] A. A. Penzias and R. W. Wilson, A measurement of excess antenna tem, Astrophysical Journal, 142 (1965), pp. 419–421.
perature at 4080 Mc
s
[20] K. Greisen, End to the cosmic-ray spectrum?, Phys. Rev. Lett., 16 (1966),
pp. 748–750.
[21] G. T. Zatsepin and V. A. Kuzmin, Upper limit of the spectrum of cosmic
rays, JETP Lett. (USSR), 4 (1966), pp. 78–80.
[22] C. Genreith and T. Winchen, 2010. Private Communication.
[23] K. Dolag et al., Constrained simulations of the magnetic field in the local universe and the propagation of ultrahigh energy cosmic rays, J. Cosmol. Astropart.
Phys., 1 (2005).
[24] The Pierre Auger Collaboration, Update on the correlation of the highest energy cosmic rays with nearby extragalactic matter, Accepted for publication in Astroparticle Physics on 31 August 2010.
[25] H. Dembinski, Aufbau einer Detektorstation aus Szintillatoren zum Nachweis
von kosmischen Teilchenschauern, Simulation und Messung, Diploma Thesis,
2005.
[26] M. Nagano and A. A. Watson, Observation and implications of the
ultrahigh-energy cosmic rays, Reviews of Modern Physics, 72 (2000), pp. 689–
732.
[27] T. K. Gaisser and A. M. Hillas, Reliability of the method of constant
intensity cuts for reconstructing the average development of vertical showers, in
Proceedings of the 15th International Cosmic Ray Conference, vol. 8, Plovdiv,
Bulgaria, 1977, pp. 353–357.
[28] M. Unger, Shower profile reconstruction from fluorescence and Cherenkov
light, GAP-2006-10, (2006).
[29] G. A. Askaryan, Soviet Phys. JETP Lett. (USSR), 14 (1962), p. 441.
[30] D. Ardouin et al., Geomagnetic origin of the radio emission from cosmic ray
induced air showers observed by CODALEMA, arXiv:0901.4502.
[31] F. D. Kahn and I. Lerche, Radiation from cosmic ray air showers, in Proceedings of the Royal Society of London, vol. 289, 1966, pp. 206–213.
[32] O. Scholten, K. Werner, and F. Rusydi, A Macroscopic Description of
Coherent Geo-Magnetic Radiation from Cosmic Ray Air Showers, Astropart.
Phys., 29 (2008), pp. 94–103.
References
117
[33] T. Huege and H. Falcke, Radio Emission from Cosmic Ray Air Showers:
Coherent Geosynchrotron Radiation, Astron. Astrophys., 412 (2003), pp. 19–34.
[34] J. D. Jackson, Classical Electrodynamics, vol. 2, John Wiley & Sons, Inc.,
1975.
[35] T. Huege, Radio emission from cosmic ray air showers., PhD thesis, Rheinische Friedrich-Wilhelms Universität Bonn, 2004.
[36] T. Huege et al., A detailed comparison of REAS3 and MGRM radio emission simulations, in Nuclear Instruments and Methods A (2011), to appear in
Proceedings of the 4th Int. Workshop on the Acoustic and Radio EeV Neutrino
Detection Activities (ARENA 2010), Nantes, France, 2010.
[37] M. Risse et al., EAS Simulations at Auger Energies with CORSIKA , in Proceedings of the 27th International Cosmic Ray Conference, Hamburg, Germany,
2001.
[38] T. Huege, R. Ulrich, and R. Engel, REAS2: CORSIKA-based Monte
Carlo simulations of geosynchrotron radio emission, arXiv:0707.3763v1, (2007).
[39] T. Huege and H. Falcke, Radio emission from cosmic ray air showers:
simulation results and parametrization, Astropart. Phys., 24 (2005), pp. 116–
136.
[40] H. R. Allan, R. W. Clay, and J. K. Jones, Radio pulses from extensive
air showers, Nature, 227 (1970), pp. 1116–1118.
[41] The LOPES Collaboration, Progress in air shower radio measurements:
Detection of distant events, Astropart. Phys., 26 (2006), pp. 332–340.
[42] The Pierre Auger Collaboration, Properties and performance of the
prototype instrument for the pierre auger observatory, in Nuclear Instruments
and Methods in Physics Research Section A, vol. 518(1-2), 2004, pp. 172–176.
[43] The
Pierre
Auger
Collaboration,
The
Pierre
ger
Observatory
Design
Report:
Second
Edition,
http://www.auger.org/admin/DesignReport/index.html.
Au1997.
[44] http://www.auger.org/observatory/gallery2005.html.
[45] T. Waldenmaier et al., Measurement of the Air Fluorescence Yield with
the AirLight Experiment, in Proceedings of the 29th International Cosmic Ray
Conference, Pune, India, 2007.
[46] S. Fliescher, 2010. Private communication.
[47] M. Stephan, Design and Test of a Low Noise Amplifier for the Auger Radio
Detector, Diploma Thesis, 2009.
118
References
[48] C. A. Balanis, Antenna Theory: Analysis and Design, vol. 3, John Wiley &
Sons, Inc., Hoboken, New Jersey, United States of America, 2005.
[49] D. E. Isbell, Log Periodic Dipole Arrays, IRE Transactions on Antennas and
Propagation, 8 (1960), pp. 260–267.
[50] S. Fliescher, Radio Detection and Detector Simulation for Extensive Air Showers at the Pierre Auger Observatory, Diploma Thesis, 2008.
[51] K. Rothammel, A. Krischke, Rothammels Antennenbuch, vol. 12, DARC
Verlag, Baunatal, Germany, 2001.
[52] K. Weidenhaupt, LPDA-Antennas for Large Scale Radio Detection of Cosmic Rays at the Pierre-Auger-Observatory, Diploma Thesis, 2009.
[53] G. Hilgers, 2010. Private communication.
[54] O. Seeger for the Pierre Auger Collaboration, Logarithmic Periodic Dipole Antennas for the Auger Engineering Radio Array, in Nuclear Instruments and Methods A (2011), to appear in Proceedings of the 4th Int. Workshop
on the Acoustic and Radio EeV Neutrino Detection Activities (ARENA 2010),
Nantes, France, 2010.
[55] G. J. Burke, A. J. Poggio, Naval Ocean Systems Center Report, NEC-1
(1977), NEC-2 (1981), NEC-3 (1983).
[56] G. J. Burke, A. J. Poggio, Numerical Electromagnetics Code (NEC): Description Theory, 1981.
[57] R. F. Harrington, Field Computation by Moment Methods, MacMillan, New
York, United States of America, 1968.
[58] Y. S. Yeh, K. K. Mei, Theory of Conical Equiangular Spiral Antennas, Part
I – Numerical Techniques, IEEE Trans. Ant. and Prop., (1967), p. 634.
[59] T. Fließbach, Elektrodynamik: Lehrbuch zur Theoretischen Physik II, Spektrum, Akademischer Verlag, Siegen, Germany, 2005.
[60] A. Voors. 4NEC-2, http://home.ict.nl/˜arivoors/.
[61] F. Schröder, H. Bozdog, O. Krömer, Test Measurements for the Time
Calibration of AERA with a Beacon, (2009).
[62] Schwarzbeck Mess-Elektronik, http://www.schwarzbeck.de.
[63] xnecview, http://wwwhome.cs.utwente.nl/˜ptdeboer/ham/xnecview/.
[64] A. Nigl et al., Direction Identification in Radio Images of Cosmic-Ray Air
Showers Detected with LOPES and KASCADE, Astron. Astrophys., 487 (2008),
pp. 781–788.
[65] B. Philipps, 2010. Private communication.
References
119
[66] D. Schwarzbeck, 2010. Private communication.
[67] M. Stephan, 2010. Private communication.
[68] T. Winchen, Measurement of the Continuous Radio Background and Comparison with Simulated Radio Signals from Cosmic Ray Air Showers at the Pierre
Auger Observatory, Diploma Thesis, 2007.
120
References
Acknowledgements –
Danksagungen
An dieser Stelle möchte ich mich gerne bei den Menschen bedanken, die zur Verwirklichung dieser Diplomarbeit maßgeblich beigetragen haben.
Zunächst bedanke ich mich bei meinem Betreuer Prof. Dr. Martin Erdmann, der
mir das spannende in dieser Arbeit behandelte Thema angeboten hat und mir dadurch einen einzigartigen und unvergesslichen Einblick in die aktuelle Forschung auf
dem Gebiet der Astroteilchenphysik ermöglicht hat.
Darüber hinaus weiß ich es sehr zu schätzen, dass es mir möglich gewesen ist internationale Konferenzen zu besuchen und dort Vorträge zu halten. Dadurch habe ich
wertvolle Erfahrungen gesammelt.
Weiterhin möchte ich mich bei Prof. Dr. Thomas Hebbeker bedanken, der sich
freundlicherweise als Zweitgutachter für meine Diplomarbeit zur Verfügung gestellt
hat.
Für die zahlreichen interessanten Diskussionen und Anregungen danke ich Stefan
Fliescher, auf dessen Erfahrung im Bereich der Radioastronomie ich oftmals gerne
zurückgegriffen habe. Auch Maurice Stephan und Klaus Weidenhaupt gebührt besonderer Dank, da ich bei Ihnen immer auf ein offenes Ohr gestoßen bin, wenn ich
Hilfe bei einer zeitaufwendigen und schwierig zu realisierenden Messung benötigt
habe.
Nicht vergessen möchte ich den Rest der Aachener Auger Gruppe: Dr. Christine Meurer, Christoph Genreith, Marius Grigat, Gero Müller, Matthias Plum, Nils
Scharf, Peter Schiffer, Stephan Schulte, David Walz und Tobias Winchen.
An dieser Stelle soll auch die unverzichtbare Hilfe von Barthel Philipps und der
mechanischen Werkstatt erwähnt werden, denn ohne die dort fertiggestellte Konstruktion wären die Messungen zur Kalibrierung der Small Black Spider LPDA nicht
in dieser Form möglich gewesen. Auch Günter Hilgers aus der elektronischen Werkstatt gebührt Dank für die Teilhabe an seinem reichhaltigen Erfahrungsschatz.
Ganz besonders bedanken möchte ich mich bei meinen Eltern und Großeltern. Sie
haben mir das Physik-Studium durch ihre Unterstützung überhaupt erst in dieser
Form ermöglicht und ich bin mir sicher, dass ich diese Zeilen hier ohne sie nicht unter
eine fertige Diplomarbeit schreiben würde. Der Rückhalt, den mir meine Familie die
ganze Zeit über gegeben hat, ist in vielerlei Hinsicht unbezahlbar.
122
Acknowledgements – Danksagungen
Abschließend möchte ich sagen, dass es mir eine Freude war meine Diplomarbeit am
III. Physikalischen Institut A der RWTH Aachen zu schreiben. Das letzte Jahr war
eine tolle Erfahrung und ich werde in Zukunft noch oft daran zurück denken.
Erklärung
Hiermit versichere ich, dass ich diese Arbeit einschließlich beigefügter Zeichnungen,
Darstellungen und Tabellen selbstständig angefertigt und keine anderen als die angegebenen Hilfsmittel und Quellen verwendet habe. Alle Stellen, die dem Wortlaut oder
dem Sinn nach anderen Werken entnommen sind, habe ich in jedem einzelnen Fall
unter genauer Angabe der Quelle deutlich als Entlehnung kenntlich gemacht.
Aachen, den 5. Oktober 2010
Oliver Seeger

Absolute Calibration of the Small Black Spider Antenna for the

Transcription

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