stttelkom.ac.id

Ch3. Introduction to Antenna
Chien-Wen Chiu(邱建文), Professor
Department of Electronic Engineering
National I-lan University
1, Sec. 1, Shern-Nong Rd., I-lan, Taiwan,
R.O.C.
e’mail address : [email protected]
March 16, 2010
1
Content
1. Antenna introduction and parameters
2. Field for the Short Dipole Antenna
3. Near field and Far field
4. Radiation Patterns and Beamwidth
5. Antenna Gain, Directivity, and Efficiency
6. Antenna Bandwidth
7. Polarization
8. Input impedance
9. Max. Power Transfer and Effective Aperture
2
1. Antenna Introduction
Example : Figure 13.1 (p. 634) by
Pazar’s microwave enginnering
Photograph of various millimeter wave
antennas. Clockwise from top: a highgain 38 GHz reflector antenna with
radome, a prime-focus parabolic
antenna, a corrugated conical horn
antenna, a 38 GHz planar microstrip
array, a pyramidal horn antenna with a
Gunn diode module, and a multibeam
reflector antenna.
3
1. Antenna Introduction
Basic operation of transmit and receive antennas.
Figure 13.2 (p. 635) Basic operation of transmit and receive antennas.
4
5

c

f
Frequency Band
 
c
f
 : wavelength
f : frequency
c: speed of light
6
Antenna - How it Works
The antenna converts radio frequency electrical energy fed to it (via the
transmission line) to an electromagnetic wave propagated into space.
The physical size of the radiating element is proportional to the
wavelength. The higher the frequency, the smaller the antenna size.
Assuming that the operating frequency in both cases is the same, the
antenna will perform identically in Transmit or Receive mode
[email protected]
7
Antenna reciprocity theorem
發射至空中
來自空中
發射機天線
RF能量
RF能量
發射機
接收機
(a)
(b)
天線的互易性定理(Antenna reciprocity theory): 任何天線，
若工作於相同之頻率，作用於發射端或接收端，當作發射天線或
接收天線都有相同之效率及特性，此種特性稱為天線之互易性。
8
Current distribution of
transmission line and
linear dipole
9
Sin(kl/2) ≒kl/2 as kl is very small
Significant interference
and cancelling will be noted
10
(900)
(1800)
11
Current variation of as a function of time for half-wavelength dipole
  E   j H
Radiation Mechanism
  H  j E
D  0  E
B  0  H
Why ?
How ?
What ?
12
(900)
(1800)
13
Radiation Mechanism for Dipole Antenna
(2)
(1)
(1)
(1)
(2)
(1)
(1)
(1)
傳播到空
中的電場
(a)
(b)
(c)
(d)
14
(900)
15
Maxwell Equations

 
  E   jH  M

 
  H  jE  J

D  

B  0
16
Types of antenna
1.Wire antennas (and loop antenna)
2. Aperture antennas
3. Microstrip antennas
4. Array antennas
5. Reflector antennas
6. Lens antennas
17
Antenna Parameters
Frequency band
Bandwidth
S11(dB) or VSWR@50 Ohm
Patterns
Gain
Efficiency
Polarization
Size
18
2. Field for the Short Dipole Antenna
I o L
1
j
 j r
cos  [ 2 
]
e
2
r
r 3
I o L
j
1
j
 j r
E 
sin [
 2 
]
e
2
r
r
r 3
Io L
j
1  j r
H 
sin [
 2 ]e
4
r
r
0
w

  , 

c
 0
0
Er 
H r  H   0, E  0
How to obtain these equations?
19
The elemental electric dipole
(詳細推導請參閱Cheng的電磁學第十章)
R=r
20
Derive Formula
Magnetic vector potential:
[ I ]
A
dlaˆ z
4 r
[ I ]  I o cos(t)  r ),  
R=r
2

 I o dl  j  r
Az  aˆ z
e
, (phasor form)
4 r
A  A( Ax , Ay , Az )  ( Ar , A , A )
For spherical coordinates:
Ar  Az cos  , A   Az sin  , A  0
H 
1

 A
I o dl
j
1
sin  [
 2 ]e  j  r
4
r
r
H r  H  0
 H 
21
E
1
j
Derive Formula
 H
 I o dl
1
j
 j r
cos  [ 2 
]
e
2
r
 r3
 I o dl
j
1
j
 j r
E 
sin  [
 2 
]
e
4
r
r
 r3
 Er 
E  0


w
(Conduction field, similar to that of dipole, as β =0)

, 


c
 I o dl
1
 j r
 Er 
cos

[1

]
e
2 r 2
j r
 I dl
1
1
 j r
E  j o  sin  [1 

]
e
4 r
j  r ( r )2

0
I o dl
1
H  j
 sin  [1 
]e  j  r
4 r
j r
22
(Conduction field, similar to that obtained by Biot-Sarvart theorm, as β0=0)
3. Near field and Far field for small dipole
( / 50  L   / 10)
1. Reactive near field region
2. Radiating near field region
(Fresnel region)
3. Radiating far field region
(Fraunhofer region)
(have Er)
23
Near field and far field for infinitesimal dipole
j
1
j
 2 
)
3
r
r
r

1
1
 2 
r
r
 r3
E  (
 r 1 r 

2

I . r 
(Near-field-region)
2
 I o dl
 j  j r
Er 
cosθ
e
3
2
r
(Conduction field, similar to that of dipole, as β =0)
 I o dl
 j  j r
E 
sinθ
e
3
2
r
0
H 
I o dl
1
sinθ 2 e  j  r
4
r
(Conduction field, similar to that obtained by Biot-Sarvart theorm, as β0=0)
24
Near field and far field for infinitesimal dipole
 Io L
1
j
 j r
cos  [ 2 
]
e
2
r
 r3
 Io L
j
1
j
 j r
E 
sin  [
 2 
]
e
2
r
r
 r3
Io L
j
1  j r
H 
sin  [
 2 ]e
4
r
r
Er 

II . r 
(Far-field-region)
2
 I o dl
j   j r
E 
sinθ
e
4
r
(same phase, orthogonal direction)
I o dl
j   j r
H 
sinθ
e
4
r
Er  0, E / H   
25
4. Antenna Radiation Pattern
I. Radiation Pattern(2D or 3D)
 A graphical representation of the intensity of the
radiation vs. the angle from the perpendicular.
 The graph is usually circular, the intensity
indicated by the distance from the center based
in the corresponding angle.
[email protected]
26
 I o dl
j   j r
E 
sinθ
e
 E  sin 
4
r
H-plane : the plane containing the magnetic-field vector and the direction of maximum radiation
E-plane : the plane containing the electric-field vector and the direction of maximum radiation
The pattern of the H-field is same as that of E-field
27
28
The type of system you are installing will help determine the type of
antenna used. Generally speaking, there are two „types‟ of antenna:
1.
Directional
- this type of antenna has a narrow beamwidth; with the power being more
directional, greater distances are usually achieved but area coverage is sacrificed
- Yagi, Panel, Sector and Parabolic antenna
- Some time we will use this type of antenna in both Point to Point and Point to
Multipoint communications
2. Omni-Directional
- this type of antenna has a wide beamwidth and radiates 3600; with the power being
more spread out, shorter distances are achieved but greater coverage attained
3. - Omni antenna
[email protected]
29
Radiation Pattern
(elevation)
main lobe
boresight
side lobe
Radiation Pattern
[email protected]
30
Side lobes
Upper Side Lobe
Suppression (dB)
[email protected]
31
Directive Radiation Pattern (Broadband)

Log periodic dipole array (LPDA)
Directional Radiation
Pattern
Dipoles
Transmission
line
- very wide BW, with constant SWR
- typical gain 10 dBi
main lobe
•
Reflector
Yagi antenna
Driven element (dipole)
Directors
back lobe
side lobe
- BW is smaller than LPDA
- typical gain 12 – 14 dB
[email protected]
main lobe
32
Antenna Radiation pattern
Directional Antenna Radiation Pattern
Horizontal plane
Vertical plane
Horizontal-plane and Vertical plane : based on the earth
[email protected]
33
Antenna Radiation pattern
Omni-directional Antenna Radiation Pattern
H-plane
E-plane
H-plane : the plane containing the magnetic-field vector and
the direction of maximum radiation
E-plane : the plane containing the electric-field vector and
the direction of maximum radiation
34
(Horizontal Plane)
(Horizontal Plane)
Typical Radiation Pattern for a Yagi
[email protected]
35
270 0
-3
-6
0
0
-15
-20
-15
-20
-30
-30
-10
dB
90 270 0
-3
-6
-10
dB
90
180
180
Typical Radiation Pattern for a Sector antenna
[email protected]
36
Three-dimensional Pattern
[email protected]
37
II. Power Patterns and Beamwidth
Power Density:
 *
1
Wav  Re[ E  H ] W/m 2
2
Power Pattern
Normalized Power Pattern
Beamwidth:
-3dB : half-power beamwidth
-- Half Power Beamwidth (HPF)
-- Beam Efficiency (BE)
2
1
0
2
0
0
0
U ( , ) sindd


BE   
  U ( , ) sindd
Beamwidth of half-wavelength dipole antenna is 78o
38
Beamwidth
10dB Beamwidth
3dB Beamwidth
Peak - 10dB
Peak - 3dB
60° (eg)
Peak
Peak - 3dB
[email protected]
120° (eg)
Peak
Peak - 10dB
39
III. Power Density and Radiation Intensity
A. Radiation Power Density
Poynting vector:
W  E  H * Watt / m 2 (Instantaneous Poynting vector)
P

S
W  dS 

S
ˆ (Instantaneous total power)
W  nda
Average Power Density: Wav ( x, y, z )  Wav ( x, y, z, t )  
av
Average Power: Prad  Pav 

S
Wav  dS 
1
2

S
1
Re[ E  H * ]
2
Re[ E  H * ]  dS
dS  r 2 sin  d d

S
m2
(Time average)
1
Radiation Power density: Re[ E  H * ]
2
B. Radian & Solid angle
solid angle: d  
W
dS
 sin  d d
r2
d     4
40
Power Density and Radiation Intensity
C. Radiation Intensity
U  r 2Wav
W
unit solid angle
: far zone parameter
2
r2
r2 
2
2

U ( ,  ) 
E (r , ,  ) 
E
(
r
,

,

)

E
(
r
,

,

)



2
2 
2
1
2
  U ( ,  )  U ( ,  )  , E (r )


r
Prad 


Ud   
2
0


0
U sin  d d
Example: Calculate the total radiated power
sin 
If an antenna has : Wav  aˆrWav  aˆr Ao 2
U=Aosin
r
2 
sin 
Prad   Wav  dS    aˆr Ao 2  aˆr r 2 sin  d d   2 A0 (W )
S
0
0
r
Prad 


Ud   
2
0


0
Ao sin  sin  d d   2 Ao (W )
D. Isotropic Source:
Prad 


U o d   4 U o  U o 
Prad
4
41
Power Density and Radiation Intensity
Example:
A hypothetical isotropic antenna is radiating in free space. At a
distance of 100 mm from the antenna , the total electric field E ( )
is measured to be 5 V/m. Find
(a) the power density (Wrad)
(b) the power radiated (Prad)
42
5. Antenna Directivity, Gain, and Efficiency
Directivity : Maximum radiation intensity/Average radiation intensity
U
Umax : MAX radiation intensity
D  rad
U0
, U0 : radiation intensity of isotropic source
U max
4 U max , P : radiation power from antenna
rad
=

Prad / 4
Prad
, (U0 = Prad /4 ) (lossless isotropic source)
Gain : 4 Maximum radiation intensity/Input Power
G
U max
4 U max

Pin / 4
Pin
,Prad : input power from the input port
Efficiency : Gain/ Directivity
ecd =Prad / Pin = G/D
* A short dipole antenna : D = 1.5 = 1.76 dBi (10logD)
A half-wavelength dipole antenna : D = 1.64= 2.15 dBi
Relative Gain or Directivity : Gr(dBd) =G(dBi)-2.15
43
Directivity and gain definition
D
=
U rad
U0
U max
4 U max

Prad / 4
Prad
4
44
Gain
Unless otherwise specified, the gain usually
refers to the direction of maximum radiation.
Gain of this direction
[email protected]
45
Gain Unit
Antenna gain is usually expressed in dBi or dBd
dBi
Gain relative to an isotropic antenna when the
reference antenna is an isotropic antenna.
dBd
Gain relative to a half-wave dipole when the
reference antenna is a half-wave dipole.
[email protected]
46
dBd and dBi
isotropic radiator
2.15dB
eg: 0dBd = 2.15dBi
half-wave dipole
[email protected]
47
Dipoles
Wavelength
1/4 Wavelength
1/2 Wavelength
1/4 Wavelength
1/2 Wavelength
Dipole
1900MHz ：78.95mm
800MHz ：187.5mm
[email protected]
48
Dipoles
One dipole
multiple dipoles
Received Power：1mW
Received Power ：4 mW
GAIN= 10log(4mW/1mW) = 6dBd
[email protected]
49
Sector antenna compared with the Dipoles
Antenna
(down look)
Omnidirectional array
Sector antenna
Received Power ：1mW
Received Power ：8mW
10log(8mW/1mW) = 9dBd
[email protected]
50
Measurement at anechoic chamber
GAUT = Gstandard + PAUT - Pstandard
GAUT : Gain of AUT (dB)
Gstandard : Gain of standard gain antenna (dB)
PAUT : Measured power of AUT (dBm)
Pstandard : Measured power of standard gain antenna (dBm)
Standard gain antenna : 1. BBHA 9120 LFA 700MHz-6GHz
2. TDK 9120D Horn antenna (900MHz-18GHz)
3. EMCO 3115 Double-Ridged Horn(1-18GHz)
4. Spectrum Technology : DRH-0118(1-18GHz)
(3D chamber)
51
Yagi
- better suited for shorter links
- lower dBi gain; usually between 7 and 15 dBi
[email protected]
52
Parabolic
- used in medium to long links
- gains of 18 to 28 dBi
- most common
[email protected]
53
Sector antenna(扇型天線)
-A sector antenna is a kind of directional antenna with a sectorshaped radiation pattern. In mobile communications, these antennas
are typically installed in base station sites for point-to-multipoint
connections
- directional in nature, but can be adjusted
anywhere from 450 to 1800
- typical gains vary from 10 to 19 dBi
0
-15
-20
-30
270 0
-3
-6
-10
dB
90
180
[email protected]
54
Omni
- used at the some communications for wide coverage
- typical gains of 3 to 10 dBi
55
Directivity
A. Directivity :
the radiation intensity in a given direction
the radiation intensity averaged over all directions
U Wav 4 U
D


U o Wave
Prad
If the directivity is not specified, it implies the direction of maximum radiation intensity.
Dmax 
D 
U
max
Uo

U max 4 U max

Uo
Prad
4 U 
4 U
, D 
( Prad )  ( Prad )
( Prad )  ( Prad )
Example: (linear dipole l<<) (Homework 3.1)
if : Wav  aˆr Ao sin 2  / r 2 (W / m 2 )
3
Please proof : Dmax  , D( ,  )  1.5sin 2 
2
56
Gain & Antenna Efficiency
Radiation intensity
U ( ,  ) U ( ,  )
G  4
 4

Total input power
Pin
( Pin / 4 )
Prad  ecd Pin  G ( ,  )  ecd D ( ,  ), (ecd obtained by measurement
G  G ( ,  ) max  ecd D( ,  ) max  ecd D
G (dB)  G (dB)  G (dB)
G (dB)  10 log10 (ecd D)
57
Gain & Antenna Efficiency
  Antenna Efficiency
er  (1   ) : Reflection (Mismatch) efficiency
2
ec : Conductivity efficiency
ed : Dielectric efficiency
ecd  ec ed Antenna radiation efficiency
eo  er ec ed  ecd (1   ) :
2
Total efficiency
58
Gain Example
Example: A lossless dipole antenna, with input impedance 73 Ohms, is to be connected to
a transmission line whose characteristic impedance is 50 Ohms. Assume that the
pattern of the antenna is given approximately by :
U ( , )  Bo sin 3 
Find the overall maximum gain of this antenna.
2 
U max  Bo  Prad 
  U ( ,  ) sin  d d
0 0

 Prad
D  4
3 2
 2 Bo  sin  d  Bo (
),
4
0
4
U max
16

 1.697
Prad
3
ecd  1  G  ecd D  1.697  2.297( dB )
73  50 2
)  0.965
73  50
 er  0.965  0.155 dB
er  (1   )  1  (
2
eo  ecd
G e
eo
cd D  0.965  1.697  2.142 (dB)
59
Gain (Homework 3.4)
A lossless resonant half-wavelength dipole antenna,
with input impedance of 100 ohms, is to be connected
to a transmission line whose characteristic impedance
is 25 ohm. Assuming that the pattern of the antenna is
given approximately by U  Bo sin3  find the overall
maximum gain of this antenna.
Answer : 2.997 (dB)
[email protected]
(Refer to p.61of Balanis‟s “antenna theory”)
60
Average Gain & Antenna Efficiency for portable device
  PeakGain :
Radiation intensity
U ( ,  ) U ( ,  )
G  4
 4

Total input power
Pin
( Pin / 4 )
  Mean gain : (Average Gain)
calculted by averaging the measred gain at sufficient points
on a (typical spherical) surface around the handset. If the
antenna was lossless, the mean gain woud be 0dBi.
 3D Average gain  10 log10  (3-dimensional average gain)
3D Average gain=0 dB   =100%
  Two-dimensional average gain (Gavg ) for notebook computer:
N
Gavg  10 log10
 G ( )  G ( )
i 1
h
i
N
v
i
[from 0 to 3600 , Npoints]
61
6. Antenna Bandwidth
B  f H  f L
0
-5
1. GSM: 880-960 MHz
-10
2. DCS : 1710-1880 MHz
-15
-20
3. PCS : 1850-1990 MHz
4. WLAN/BlueTooth : 2400-2484 MHz
fH
fL
-25
-30
5. GPS : 1575.42 ±1.023 MHz
0.7
0.8
0.9
1.0
6. W-CDMA : 1.920 -2.170 GHz
Return Loss :
7. CDMA : 869-894 MHz(Qualcomm)
( RL )  20 log 
1.1
一般天線頻寬與反射係數的關係是取小S11=-10dB以下的頻率範圍當作頻
寬，當小於-10dB以下的頻寬時，其反射係數=1/3、駐波比VSWR=2:1，
表示此天線在此頻率範圍至少有90%以上的能量輻射出去。
Broadband antenna : eq. 10:1
Narrowband antenna : eq.10%
fraction bandwdith=
B
*100%
fc
62
VSWR＝
VSWR

Zin  Z 0 100  50

 1/ 3
Zin  Z 0 100  50
Vmax
Vmin

1 
1 
(1 )
forward: 10W
100
ohms
50 ohms
reverse: 1W
9W
Return Loss：-20log(1/3) ≒ 10 dB
VSWR (Voltage Standing Wave Ratio)
Vr2 / Z 0
Vr



2
Vin
Vin / Z 0
1 
VSWR＝
＝2
1 
Pr
1

 1/ 3
Pin
10
Usual Request：VSWR2.0
Reflection Coefficient：=(VSWR-1)/(VSWR+1)
Return Loss：RL=-20lg 
63
7. Polarization
An antenna polarization is relative to the E-field of antenna.
– If the E-field is horizontal, than the antenna is Horizontally
Polarized.
– If the E-field is vertical, than the antenna is Vertically Polarized.
No matter what polarity you choose, all antennas in the same RF
network must be polarized identically regardless of the antenna
type.
[email protected]
64
7. Polarization
Polarization:
-- Polarization of an antenna: the polarization of the wave transmitted by an antenna.
-- Polarization of radiated wave:
-- Electric-field :
Time varied.
A fixed point in space.
-- Linear: a function of time along a line.
Vertical polarization
Horizontal polarization
-- Circular: CW(clockwise), CCW(counterclockwise).
-- elliptical:
-- Co-polarization:
-- Cross polarization:
E = E e(jwt+1)
E = E e(jwt+2)
E-field direction :
Vertical Polarization
Horizontal Polarization
Circular Polarization
65
Polarization
Vertical
Horizontal
Vertical Polarization:
The electric field is vertical to the ground (In the maximum gain direction)
Horizontal Polarization:
The electric field is parallel to the ground (In the maximum gain direction)
[email protected]
66
Polarization
+ 45degree slant
- 45degree slant
[email protected]
67
Polarization
V/H (Vertical/Horizontal)
Slant (+/- 45° )
[email protected]
68
Polarization may deliberately be used to:
– Increase isolation from unwanted signal sources (Cross
Polarization Discrimination (x-pol) typically 25 dB)
– Reduce interference
– Help define a specific coverage area
Horizontal
Vertical
[email protected]
69
Polarization of plane wave
1. Linearly polarization : E ( z )  aˆ x E x ( z )
2. Consider the superposition of linearly polarized wave:
E ( z )  aˆ x E x ( z )  aˆ y E y ( z )  aˆ x E xo e  jkz  aˆ y E yo e  jkz  e  j / 2


E ( z , t )  Re [ aˆ x E x ( z )  aˆ y E y ( z )]e jwt  aˆ x E xo cos(t  kz )  aˆ y E yo cos(t  kz 

Set z  0
E (0, t )  aˆ x E x (0, t )  aˆ y E y (0, t )  aˆ x E xo cos(t )  aˆ y E yo sin(t )
As wt increases from 0 through  /2, the tip of the vector will have an locus.
E (0, t )
cos(t )  x
E xo
sin(t ) 
E y (0, t )
E yo
 E (0, t ) 
 1  cos 2 (t )  1   x

 E xo 
2
2
 E y (0, t ) 
 E x (0, t ) 
 1

 
E
E


xo
yo


2.1 Circular polarization:Exo  E yo
2
2.2 Elliptical polarization:E xo  E yo
70
2
)
71
Electrical Field Representation

E ( z , t )  aˆ x E x ( z , t )  aˆ y E y ( z , t )
E x ( z , t )  Re[ E x e j (t  kz ) ]  Re[ E xo e j (t  kz  x ) ]  E xo cos(t  kz   x )
E y ( z , t )  Re[ E y e j (t  kz ) ]  Re[ E yo e
j (t  kz  y )
]  E yo cos(t  kz   y )
For linear polarizati on :
   y   x  n
For circular polarizati on :
E xo  E yo
When
   y   x   ( 2n  12 ) ,
n  0,1,2,  for CW
   y   x  ( 2n  12 ) ,
n  0,1,2,  for CCW
For elliptical polarizati on :
E xo  E yo
When
or
   y   x   ( 2n  12 ) ,
n  0,1,2,  for CW
   y   x  ( 2n  12 ) ,
n  0,1,2,  for CCW
   y   x  
n
 , n  0,1,2  ,  0, for CW
2
 0, for CCW
72
Derive Formula
E x  E xo cos(t  kz   x )  E xo cos(t   xo )
E y  E yo cos(t  kz   y )  E yo cos(t   yo )
Ex
 cos t cos  xo  sin t sin  xo , (1)
E xo
Ey
E yo
 cos t cos  yo  sin t sin  yo , (2)
(1) sin  yo  (2) sin  xo  sin  yo
Ey
Ex
 sin  xo
 cos t (cos  xo sin  yo  cos  yo sin  xo )
E xo
E yo
 cos t sin  ,
(1) cos  yo  (2) cos  xo  cos  yo
Ey
Ex
 cos  xo
 sin t (cos  xo sin  yo  cos  yo sin  xo )
E xo
E yo
 sin t sin  ,


1
sin 2 
1
sin 2 
[(sin  yo
[(
   yo   xo
   yo   xo
Ey 2
Ey 2
Ex
E
 sin  xo
)  (cos  yo x  cos  xo
) ] 1
E xo
E yo
E xo
E yo
Ex E y
Ey 2
Ex 2
)  2 cos 
(
) ] 1
E xo
E xo E yo
E yo
73
Derive Formula
Y
Y‟
x  x' cos   y ' sin  ; y  x' sin   y ' cos 
( A cos 2   B cos  sin   C sin 2  ) x'2  (2 A cos  sin   2C sin  cos 
 B (cos   sin  ) x' y '
2
2
X‟

x
 ( A sin 2   B cos  sin   C cos 2  ) y '2  k
1
B
 C  A sin 2  B cos 2  0    tan 1 (
)
2
AC
1
1
1
A 2 ,
B  2 cos 
, C 2
E xo
E xo E yo
E yo
 
2 E xo E yo
1
tan 1 ( 2
cos  )
2
2
E xo  E yo
Major axis :
2
4
2
OA  { 12 [ E xo2  E yo
 ( E xo4  E yo
 2 E xo2 E yo
cos 2 ) 2 ]}2
1
1
2
4
2
OB  { 12 [ E xo2  E yo
 ( E xo4  E yo
 2 E xo2 E yo
cos 2 ) 2 ]}2
1
Axial Ratio :
OA major axis

OB min or axis
1
,1  AR  
74
Polarization Loss Factor
 Incoming wave (electric field)

Ei  ˆ t Ei
 The receving antenna :

Er  ˆ r Er
Ei  aˆ x Eo ( x, y )e  jkz , Er  (aˆ x  aˆ y ) E ( x, y )e  jkz
ˆt  ?, ˆ r  ?, PLF  ?
Polarization Loss Factor :
PLF  ˆ t  ˆ r  cos
2
Example :
Linear polarization:
2
solve:
ˆ t  aˆ x , ˆ r 
1
(aˆ x  aˆ y )
2
2
Polarization efficiency :
 PLF (dB )  10 log10 ( ˆ t  ˆ r )  3dB
  2
le  Einc
Pe   2  2
 Receiving mode
le Einc

 Transmitting mode
le : Vector effective length of the antenna.

Einc : Incident electric field.
75
8. Input impedance of an Antenna
Z A  RA  jX A
,

 RA  Rr  RL
R = 2Pt/I2
: Input resistance
2 Prad
Rr  2
I0
: Radiation resistance
RL  Loss
(equivalent)
Resistance due to skin effect
or dielectric loss (heat)
76
Radiation Resistance for Infinitesimal Dipole
1
2
1
2
For Hertzian dipole: W  [ E  H *]  [aˆr E H *  aˆ Er H * ] (Poynting vector)
 I ol sin 2 
1
Wr 
[1  j
]
2
8 
r
(  r )3
2
 I ol cos  sin 
1
W  j
[1

]

16 2 r 3
( r )2
2
P

S
W  dS  
2
0


0
(aˆrWr aˆ W )  aˆr r 2 sin  d d
 I ol
1

[1  j
]
3 
(  r )3
2
 Prad  j 2 (Wm  We )
~
2We
1
Q

Prad
( kr) 3
2
2 Prad
2 l
Rr  2  80
,
Io
2
Example : (Infinitesimal dipole)
l  0.01 ,
l

2
 Rr  73
(l  
)
50 (  120 )
1 2
Rr  80 2 (
)  0.079 Ohm
100
77
Radiation Resistance for a Small Dipole
Rrad : Radiation resistance
RL : Loss due to material and metal
G  ecd D,
ecd 
Rrad
Rrad  RL
-- From the circuit point of view:1. Power loss --- R 2. Stored energy --- L,C
For Small Dipole:( /50<l   /10)
1
1
Wav  Re[ E  H * ]  Re[aˆr E H * ] (Poynting vector)
2
2
j  I ol
1
2

E aˆr , E =
sin  e- j  r ,   120
2
8 r
Prad
1
  Wav  dS 
S
2
2

0
0
 
I o  2  2l 2
2
2
sin


r
sin  d d
2 2
64 r
2
I o  l 2
l
2

( )  10 2 ( ) 2 I o
12


1 2
2
2 l 2
 I rm
R

I
R

R

20

( ) ,
radiation resistance
s r
o r
r
2

2
Prad
78
Radiation Resistance of a Small Dipole
Example :
A dipole antenna for a paging system operates at f=930 MHz. If the dipole
is 30 mm long (l) with a radius a of 0.1mm, find the input impedance of
the dipole, and its radiation efficient. Assume a copper conductor .
(  5.7 107 S/m)
We know :
Rs l
2 f o
l
60  l

Rrad  20 2 ( )2 , X 
ln(
)

1

,
R


,
R

loss
s


 l  a
6 a
2
(short dipole)(λ/50<L<λ/10)
Solve : Rs  7.96 103 , Rloss  0.13, Rrad  1.70, X  967, and erad  93%
79
(Refer to p.133 of Pozar‟s “microwave and RF design of wireless systems)
Antenna Radiation Efficiency
Example :
7
(


5.7

10
S/m)
A resonant half-wavelength dipole is made out of cooper
wire. Determine the conduction-dielectric (radiation) efficiency of the
dipole antenna at f=100 MHz if the radius of the wire is 3 104  ,
And the radiation resistance of the half-wavelength dipole is 73 ohm.
Solve :
RL 
1
Rhf (see p.78, “Antenna theory “ by C. A. Balanis)
2
wo
Rhf 
2 b 2
 Rrad 
ecd  
  99.52%
 RL  Rrad 
80
8. Input Impedance (Transmitter)
(a)Transmitter:
ZA: Input impedance
ZA=RA+jXA, RA=Rr+RL
Rr: Radiation resistance of the antenna
RL: Loss resistance of the antenna
Zg=Rg+jXg
Zg: Source impedance(Generator Internal impedance)
a
Generator
Zg
Radiated wave
b
2
Vg 

1 2
Rr
Pr  I g Rr 


2
2  ( Rr  RL  Rg ) 2  ( X A  X g ) 2 
a
Maximum power delivered to the antenna:
Vg
Ig
 RA  Rr  RL  Rg
, Conjugate match

X


X
 A
g
2
Rg
Pg 
Rr
2
Vg 
Vg 


Rr
RL
 Pr 
,
P

L



,
2  4( Rr  RL ) 2 
2  4( Rr  RL ) 2 
2
RL
2
Xg
b
2
Vg 
Vg 
Rg
 Vg  Rr  RL 

1


P

P

r
L




2
2
2  4( Rr  RL ) 
8  ( Rr  RL ) 
8  ( Rr  RL ) 
XA
Ps: power supplied by the generator(conjugate matching)=Pg+Pr+PL
Ig 
Vg
( Rr  RL  Rg )  j ( X A  X g )
, Ig 
( R
r
Vg
 RL  Rg ) 2  ( X A  X g ) 2
81

1/ 2
8. Input Impedance (Receiver)
1. The power delivered to RT:
(b) Receiver:
RL : antenna loss
2
2
V
VT
RT
1
PT  T

( Rr  RL  RT , X A   X T )
2
8 ( Rr  RL )
8 Rr  RL
RT : load resistance
X T : load reactive
2.The power delivered to Rr:
a
2
Load
ZT
Incident wave
2
V
VT
Rr
Rr
Pr  T

2 4( Rr  RL ) 2
8 ( Rr  RL ) 2
b
3.The power delivered to RL:
2
a
RT
RL
VT
IT
V
RL
PL  T
, PL  Pr  PT
2
8 ( Rr  RL )
IT 
VT
( Rr  RL  RT )  j ( X A  X T )
4.Induced power (collected or capture):
XT
Rr
b
XA
1
1
PC  VT IT *  VT
2
2
2
 VT *
 VT
1
, PC  Pr  PL  PT


2(
R

R
)
4
(
R

R
)

r
L 
r
L
Pr : represent the power is scattered or reradiated through R r
Set RL = 0, in order to deliver half of the power to the load, you must scatter the other half power.
82
9. Max. Power Transfer and Effective Aperture
to describe the power capturing characteristics of an antenna when a wave impinges on it
Za = Ra+jXa
Zl = Rl+jXl
Za = Zl *
Purpose :to prove
G
4
Aem

Effective Length or Aperture : Ae = P/ Wi
Power absorbed by load/ Power density of incident plane wave
→ Received power : Pr=Ae*Wi
2
83
9. Max. Power Transfer and Effective Aperture
84
Effective Aperture
IT 
VT
( Rr  RL  RT )  j ( X A  X T )
Antenna Equivalent Area:
2
Aperture Efficiency :  ap
I RT / 2
P
A e T  T
Wi
Wi
2
A e :Effective Area( m )
 ap 
Aem Maximum effective area

Ap
Physical area
P T : Power delivered to the load(W )
Wi : Power density of incident wave(W / m 2 )
2
V
P
1
Aem  T  T
: (Maximum effective aperture)
Wi 8Wi Rr  RL
(Under maximum power transfer)
2
V
P
Rr
AS  r  T
; (Scatter area)
Wi 8Wi ( Rr  RL ) 2
2
V
P
RL
AL  L  T
; (Loss area)
2
Wi 8Wi ( Rr  RL )
2
V RT  Rr  RL
P
AC  C  T
; (Capature area)(Total power capture)
Wi
8Wi ( Rr  RL ) 2
Capature area  Effective Area  Scatter area  Loss area
85
Example . Effective Aperture
A uniform plane wave is incident upon a very short lossless dipole. Find the maximum effective
area assuming that the radiation resistance of the dipole is Rr  80 2 (l /  ) 2
, and the incident
field is linearly polarized along the axis of the dipole
For a very short dipole(Hertzian dipole):
VT   E  dl  El ,
l
Wi : incident power density of plane wave
2
Aem
VT
1

( RL  0)
8Wi Rr  RL
( El ) 2 /(8 Rr )  l 2
E2
Wi 
 Aem 

2
2
E /(2 )
4 Rr
Rr  80 2 (l /  ) 2
 Aem
120 l 2
3 2
2



0.119

4[80 2 (l /  ) 2 ]
8
l

2
 Aem 
2
2
D0 
1.643  0.132
4
4
86
Max. Directivity and Max. Effective Aperture
Relation between Directivity and effective area :
Wo : radiated p ower density of isotrop ic antenna
Wo 
Pt
4r 2
Pt Dt
4r 2
PD
 Pr  Wt Aer  t 2t Aer
4r
P
 Dt Aer  r 4r 2
Pt
Wt  Wo Dt 
If the intervening media is linear, passive and isotropic, then
(介於中間的)
Dr Aet 

Pr
4r 2
Pt
Dt
A
 et
Dr
Aer
D0t
D
 0r
Atm
Arm
87
Max. Directivity and Max. Effective Aperture
If antenna 1 is isotrop ic, then D0t  1
Atm 
Arm
D0 r
(The max. effective area of an isotropic source )
The max. effective area of isotrop ic source
Atm
Arm
0.1192
2



(the other antenna is a very short dipole)
D0 r
1. 5
4
2
(The max. effective area of any antenna)
 Arm  D0 r Atm  D0 r
4
2
 Aem 
D0 (D0: maximum directivity, no loss approximation,)
4
2
2


Practically, Aem  ecd
D0  G
(Consider loss)
4
4
4
 G  2 Aem

2

2
(If reflection and polarization A  e (
ˆ
ˆ
)
D



em
t
o
w
a
4
losses are also included)
2
2
2
 ecd (1   )
Do ˆ w  ˆ a
4
l

2
 Aem 
2
2
D0 
1.643  0.132
4
4
88
Reference :
(1) Ch1-2, C. A. Balanis, Title: Antenna Theory, Analysis and
Design,1997. John Wiley & Sons, Inc.
(2) D. M. Pozar, “Microwave and RF design of wireless
systems”,2001, John Wiley & Sons, Inc.
(3) D.K. Cheng, Ch.10, “Fundamentals of Engineering
Electromagnetics”,1993, Addison-Wesley Publishing
Company, Inc..
89