Chapter 2 :
Fundamental parameters of antennas
• From Radiation Pattern
–
–
–
–
–
–
Radiation intensity
Beamwidth
Directivity
Antenna efficiency
Gain
Polarization
• From Circuit viewpoint
– Input Impedance
1
Chapter 2 : Topics (2)
•
•
•
•
2
Antenna effective length and effective area
Friis transmission equation
Radar range equation
Antenna temperature
Definition of Radiation Pattern
• Once the electromagnetic (EM) energy
leaves the antenna, the radiation pattern
tells us how the energy propagates away
from the antenna.
• Definition :
Mathematical function or a graphical
representation of the radiation properties
of an antenna as a function of space
coordinates
3
Radiation Pattern Example
4
Radiation Pattern (1)
• Can be classified as:
– Isotropic, directional and omnidirectional
• Isotropic: Hypothetical antenna having equal
radiation in all directions
• Directional: having the property of
transmitting or receiving EM energy more
effectively in some directions than others
• Omnidirectional: having an essentially
nondirectional pattern in a given plane and a
directional pattern in any orthogonal plane
5
Radiation Pattern (2)
• Principal patterns (or planes):
– E-plane : the plane containing the electric field
vector and the direction of maximum radiation
– H-plane : the plane containing the magnetic
field vector and the direction of maximum
radiation
6
Radiation Pattern (3)
Eθ
Hφ
Omnidirectional
7
Field Regions
Far field region
(Fraunhofer zone)
R1
D
Reactive near
field region
D=largest antenna
dimension
8
R2
Radiating near field
region (Fresnel zone)
Reactive Near Field Region
• Region surrounding the antenna, wherein the
reactive field predominates
For D   : R  0.62
D3


For D   (small antenna) : R 
2
D3

 R  max[ ,0.62
]
2

Angular field distribution depends on distance from
antenna
9
Radiating Near Field Region
• Region between reactive near-field and farfield regions (Fresnel zone)
For D   :
2D 2

 R  0.62
D3

For D   (small antenna) : 3  R 

2
D3

 max[3 ,
]  R  max[ ,0.62
]

2

2D 2
Radiation fields predominate but angular field
distribution still depends on distance from antenna
10
Far Field Region
• Region where angular field distribution is
essentially independent of the distance from
antenna (Fraunhofer zone)
For D   : R 
2D 2

For D   (small antenna) : R  3
 R  max[3 ,
11
2D

2
]
Change of antenna amplitude
pattern shape
12
Spherical coordinate and
Solid Angle : Steradian
• Measure of solid
angle: 1 steradian =
solid angle with its
vertex at the center
of a sphere of radius
r that is subtended
by a surface of area
r2
dA  r 2 sin dd  r 2 d;
d  sin dd  solid angle
Quiz: What’s the solid angle subtended by a sphere?
13
Radiation Power Density
• Poynting vector = Power density
r
r
r
W E  H
r
W : instantaneous Poynting vector [W/m 2 ]
r
E : instantaneous electric field Intensity [V/m]
r
H : instantaneous magnetic field Intensity [A/m]
r
• Total power:
r
P  W  ds  W  nˆ da
S
S
P : instantaneous total power [W]
nˆ : unit vector normal to the surface
da : infinitesimal area of the closed surface [m 2 ]
14
Radiation Power Density (2)
• For time-harmonic
EMrfields
r
E ( x, y, z; t )  Re[E( x, y, z )e jt ]
r
r
H ( x, y, z; t )  Re[H ( x, y, z )e jt ]
• Poynting vector
r
r
r 1
r r* 1
r r j 2t
W E  H  Re[E  H ]  Re[E  He ]
2
2
• Time average Poynting vector (average power
density or radiation density)
r
r
r r*
1
Wav ( x, y, z )  [W ( x, y, z; t )]av  Re[E  H ]
2
1
2
15
r r
appears because E, H fields represent peak values
Radiation Power Density (3)
• Average power radiated power
Prad
r
r
r r*
1
 Pav   Wav  ds   Wav  nˆda   Re[E  H ]  ds
2 S
S
S
Example 2.1: The average power density is given by
r
sin 
Wav  rˆWr  rˆA0 2 [ W / m 2 ]
r
The total radiated power becomes
r
Prad   Wav  nˆda
S

2
0
16


0
(rˆA0
sin 
2
2
ˆ
)

r
r
sin

d

d



A0 [ W ]
2
r
Radiation Power Density (4)
• For an isotropic antenna
Prad
r
r
  W0  d s
S

2
0


0
[rˆW0 (r )]  rˆr 2 sin dd  4r 2W0 [ W ]
The power density is then given by
r
Prad
2
W0  rˆW0  rˆ
[
W
/
m
]
2
4r
17
Radiation Intensity
• Definition : The power radiated from an
antenna per unit solid angle
U  r Wrad
2
U : radiation intensity [W/unit solid angle]
Wrad : radiation density [W/m 2 ]
Total power can be given by
Prad  Ud  

18
2
0


0
U sin dd
Radiation Intensity (2)
• Radiation intensity is related to the far-zone
electric field of antenna
2
r2 r
r
U ( ,  ) 
| E( r ,  ,  ) | 2 
[| E (r ,  ,  ) |2  | E (r ,  ,  ) |2 ]
2
2
1

[| Eo ( ,  ) |2  | Eo ( ,  ) |2 ]
2
r
ro
e  jkr
E(r , ,  ) : far - zone electric - field intensity of the antenna  E ( ,  )
r
E , E : far - zone electric - field components of the antenna
η : intrinsic impedance of the medium ( 377  in free space)
19
Radiation Intensity (3)
Example 2.2: The radiation intensity is given by
U  r 2Wrad  A0 sin 
The total radiated power becomes
Prad  
2
0
 A0 
2
0


0


0
U sin dd
sin 2 dd   2 A0 [ W ]
For an isotropic antenna
Prad  U 0 d  U 0  d  4U 0


Prad
 U0 
4
20
Beamwidth
• Beamwidth is the angular separation
between two identical points on opposite
site of the pattern maximum
• Half-power beamwidth (HPBW): in a
plane containing the direction of the
maximum of a beam, the angle between
the two directions in which the radiation
intensity is one-half value of the beam
• First-Null beamwidth (FNBW): angular
separation between the first nulls of the
pattern
21
Beamwidth (2)
22
Beamwidth (3)
Example 2.3: The normalized radiation intensity of an
antenna is represented by
U ( )  cos 2  (0     ,0    2 )
The angle θh at which the function equal to half of its
maximum can be found by
U ( ) |  h  cos 2   0.5  cos  h  0.707
1
 h  cos (0.707) 

4
Since the pattern is symmetric with respect to the
maximum, HPBW = 2 θh = π/2
1
Likewise, FNBW = 2θn = π since U ( ) |   0   n  cos (0) 
n
23

2
Directivity
• Ratio of radiation intensity in a given direction
from the antenna to the average radiation
intensity
U ( ,  ) 4U

(dimension - less)
D ( ,  ) 
U0
Prad
Note that the average radiation intensity equals to the
radiation intensity of an isotropic source.
24
Directivity (2)
Since
Prad  U ( ' ,  ' )d'
'
D ( ,  )  4
U ( ,  )
U ( ' ,  ' )d'
'
Dmax
4
 4

U ( ' ,  ' )d'  A
U max
Dmax: maximum directivity
'
ΩA is called beam solid angle, and is defined as “solid
angle through which all the power of the antenna would
flow if its radiation intensity were constant and equal to
Umax for all angles within ΩA
U ( ' ,  ' )
 A  
d'  Prad  AU max
U max
'
25
Directivity (3)
• If the direction is not
specified, it implies the
directivity of maximum
radiation intensity
(maximum directivity)
expressed as
Dmax
26
U max 4U max
 D0 

(dimension - less)
U0
Prad
Directivity (4)
Example 2.4: The radial component of the radiated power
density of an infinitesimal linear dipole is given by
sin 2 
2
Wav  rˆWr  rˆA0
[
W/m
]
2
r
where A0 is the peak value of the power density. The radiation
intensity is given by
U  r 2Wr  A0 sin 2 
The maximum radiation is directed along θ = π/2 and Umax = A0.
The total radiated power is given by
Prad   Ud A0 
Thus,
27

D0 
2
0


0
8
sin  sin dd  A0
3
2
4U max 3
 and D  D0 sin 2   1.5 sin 2 
Prad
2
Antenna Efficiency
• The overall antenna efficiency take into the
following losses:
– Reflections because of the mismatch between the
transmission line and the antenna
– Conduction and dielectric losses
e0  ecd er  ecd (1 |  |2 )
e0 : total efficiency
er : reflection (mismatch) efficiency
ecd : antenna radiation efficiency  ec ed
ec , ed : conduction, dielectric efficiencies
 : voltage reflection coefficient at the input terminal
28
Gain
• It takes into account the efficiency of the antenna as well
as its directional properties. (Directivity only measures
directional properties.)
U ( ,  )
G ( ,  )  4
Pin
Pin : Power input to antenna;
Pin  (1 |  |2 ) Po  Prad  Ploss
( Ploss : Ohmic and dielectric power loss)
Gain : ratio of radiation intensity in a given direction to the
average radiation intensity that would be obtained if all the
power input to the antenna were radiated isotropically
29
Gain (2)
• Using ecd, Prad= ecdPin and
U ( ,  )
G ( ,  )  4
ecd  ecd D( ,  )
Prad
Relative gain: ratio of power gain in a given direction to the
power gain of a reference antenna in the same direction. The
power input must be the same for both antennas. If the
reference antenna is a lossless isotropic source, then
U ( ,  )
G ( ,  )  4
Pin (lossless isotropic source)
30
Gain (3)
• Absolute gain takes into account impedance mismatch
losses at the input terminals in addition to losses within
antenna
U ( ,  )
U ( ,  )
2
Gabs ( ,  )  4
 4
(1 |  | )
Po
Pin
U ( ,  )
 4
(1 |  |2 )ecd  e0 D( ,  )
Prad
31
Polarization
• Property of an EM wave describing the time varying
direction and relative magnitude of the electric field.
The figure traced as a function of time by the tip of
the electric field and the sense in which is traced, as
observed along direction of propagation.
r
Wave propagating in –z
direction
E ( z; t )  xˆEx ( z; t )  yˆ Ey ( z; t )
Let E x ( z )  E xo e j x e jkz ,
y
j y
E y ( z )  E yo e e jkz where E xo , E yo  0
ẑ
x
Ex ( z; t )  E xo cos(t  kz   x )
e jt time dependence Ey ( z; t )  E yo cos(t  kz   y )
32
Polarization (2)
A. Linear Polarization
(i) E xo  0 or E yo  0
or
(ii)   y   x  n
Example
where n  0,1,2,...
 E yo 

, 0   
  tan 
2
 E xo 
2
2
E xo
 E yo
1
δ, γ determine polarization state
33
ẑ
Polarization (3)
B. Circular Polarization
1
(i) E xo  E yo  Eo    tan (1)   / 4
and

1
 2n  2  ; CW/RCP


(ii)   y   x  
  2n  1  ; CCW/LCP
 
2
where n  0,1,2,...
Note that the sense of rotation is observed along the
direction of propagation.
34
35
Polarization (4)
Example: RCP
   / 4,    / 2
ẑ
Ex ( z; t )  Eo cos(t  kz   x )
Ey ( z; t )  Eo cos(t  kz   x   / 2)
  Eo sin(t  kz   x )
36
Polarization (5)
C. Elliptic Polarization
A wave is elliptically polarized if it is not linearly or
circularly polarized.
Linear and circular polarization are special cases of
elliptic polarization.
To have elliptic polarization:
1. Field must have two orthogonal linear components.
2. The two components can be of the same or different
magnitude.
37
Polarization (6)
C. Elliptic Polarization

1
 2n  2  ; CW/REP


(i)if    y   x  
  2n  1  ; CCW/LEP
 
2
where n  0,1,2,... AND E xo  E yo
n
(ii)if    y   x  
2
 0; CW/REP

 0; CCW/LEP
where n  0,1,2,... FOR E xo , E yo
38
Polarization (7)
Major Axis
; 1  AR  
Axial Ratio (AR) 
Minor Axis
 2 E xo E yo

1
1 

   tan  2
cos

2

2 2
E

E
yo
 xo


39
Polarization Loss Factor (PLF)
• Electric field of incoming wave
r
E w  ̂ w Ei
• Electric field of receiving antenna
r
E a  ̂ a Ea
where
ˆ w : unit vector of the wave
ˆ a : polarization vector
PLF | ˆ w  ˆ a |2 | cos p |2
(dimensionless)
40
̂ w
̂ a
PLF example
LCP wave: δ=-π/2,φx=0φy=-π/2
j x
j y
E x  Eo e e ; E y  Eo e e jkz
jkz
r
E  ( xˆEo  yˆEo e  j / 2 )e jkz  ( xˆ  jyˆ ) Eo e jkz  ˆ w 2 Eo e jkz
xˆ  jyˆ
where ˆ w 
2
*
If the antenna is also LCP,
ˆ a  ˆ w
2
xˆ  jyˆ xˆ  jyˆ
PLF 

 1  0 dB
2
2
xˆ  jyˆ
If the antenna is RCP, ˆ a 
2
2
xˆ  jyˆ xˆ  jyˆ
PLF 

0
2
2
41
Input Impedance
Impedance presented by the antenna at its terminal
Transmitting case
a
[ ]
RA  Rr  Rloss
Rr : Radiation resistance
Generator
Antenna
Terminal
Z A ( )  RA ( )  jX A ( )
b
Rloss : Loss resistance (ohmic, dielectric)
Rloss  Rohmic  Rd
Z g  Rg  jX g , Vg  I g ( Z g  Z A )
Vg , I g are peak values
Maximum power delivered to the antenna occurs when conjugate
matched:
R  R , X  X
A
42
g
A
g
Input Impedance (2)
When conjugate matched:
Ig 
Vg
2 RA
2
 Prad
| Vg |
1
Rr
2
 | I g | Rr 
2
8 ( Rr  Rloss ) 2
2
Ploss
| Vg |
Rloss
1
2
 | I g | Rloss 
2
8 ( Rr  Rloss ) 2 Power loss to heat
1
Pg  | I g |2 Rg 
2
NOTE:
43
Radiated Power
| Vg | 2
8
1
Rr  Rloss
Pg  Prad  Ploss
Power loss in Rg
Input Impedance (3)
Power supplied by generator when conjugate matched:
*
2
2
V
|
V
|
|
V
|
1
1
g
g
g


Ps  Re[ Vg I g* ]  Vg
2
2 2 RA 4 RA
4 Rg
1
Pg  Ps
2
1
Pin  Prad  Ploss  Ps
2
Prad
Rr
ecd 

Pin
Rr  Rloss
Pg  Pin
antenna radiation
efficiency
If Rloss  0  Ploss  0, Prad  Pin , ecd  1
44
Receiving Antenna
Impedance presented by the antenna at its terminal
Z A  RA  jX A , Z T  RT  jX T
Receiving case
under conjugate matched condition
RT  RA  Rr  Rloss , X T   X A
VT
1
| VT |2
2
IT 
 PT  | I T | RT 
2 RT
8 RT
2
Power delivered to
VT , I T are peak values
load
Pscatt
2
1
|
V
|
Rr
2
T
 | I T | Rr 
2
8 ( Rr  Rloss ) 2
Power scattered or re-radiated
45
Receiving Antenna (2)
Power supplied by generator when conjugate matched:
Ploss
2
Rloss
1
|
V
|
2
T
 | I T | Rloss 
2
8 ( Rr  Rloss ) 2
Power lost to heat
PT  Pscatt  Ploss
2
1
|
V
|
Pc  Re[VT I T* ]  T
2
4 RT
captured/collected power
1
PT  Pc
2
Pscatt  Ploss
46
1
 Pc
2
Pc  Pscatt  Ploss  PT
Antenna equivalent Area
• Used to describe the power capturing characteristics of
an antenna when a wave impinges on it
Ae ( ,  )  effective area (aperture) in direction ( ,  )
Power available at terminals of receiving antenna

Incident power flux density from direction ( ,  )
PT | I T |2 RT
Ae 

Wi
2 Wi
[m 2 ]
PT : power delivered to load RT
Wi : power density of incident wave
47
Antenna Equivalent Area (2)
| VT |2
Ae 
2Wi


RT

2
2
(
R

R

R
)

(
X

X
)
loss
T
A
T
 r

Under conjugate matched condition:
| VT |2 | VT |2
1
Aem 

8Wi RT
8Wi Rr  Rloss
Maximum effective area
| VT |2
Rr
As 
8Wi ( Rr  Rloss ) 2
Ae : effective area, which when multiplied by the incident
power density, is equal to power delivered to load RT
As : scattering area, which when multiplied by incident power
density, is equal to the scattered or re - radiated power
48
Antenna Equivalent Area (3)
Under conjugate matched condition:
Aloss
| VT |2
Rloss

8Wi ( Rr  Rloss ) 2
| VT |2 | VT |2 Rr  Rloss  RT

Ac 
4Wi RT
8Wi ( Rr  Rloss ) 2
Ae  As  Aloss
Ac  Ae  As  Aloss
Aloss : loss area, which when multiplied by the incident power
density, is equal to power delivered to load Rloss
Ac : Captured area, which when multiplied by incident power
density, is equal to the total power captured by antenna
49
Antenna Equivalent Area (4)
Example 2.5: a uniform plane wave is incident upon very
short dipole, whose radiation resistance is Rr=80(πl/λ)2.
Assume that Rloss = 0, the maximum effective area reduces to
| VT |2
Aem 
8Wi Rr
Since the dipole is very short, the induced current can be
assumed to be constant and of uniform phase. The induced
voltage is
VT  El
For a uniform plane wave, the incident power density is given by
E2
Wi 
2
( El ) 2
32
2
Aem 


0
.
119

thus
8( E 2 / 2 )(80 2l 2 / 2 ) 8
50
Vector Effective Length
• Vector effective length (or height) is a quantity used to
determine the voltage induced on the open-circuit
terminals of an antenna when a wave impinges on it. It is
r
a far-field quantity.
a
ZA
VT
ri
E : incident electric field
VT  Voc : open - circuit voltage
b
le ( ,  )  ˆl ( ,  )  ˆl ( ,  ) [m]
ri r
Voc  E  le
r
r
kI

jkr
in
E a  ˆE  ˆE    j
e le
4r
Example 2.6 : The electric field of a short dipole is given by
 jkr
r
kI
le
E a  ˆj in
sin 
8r
51
r
l
ˆ
le   sin 
2
Effective area & Directivity
If antenna #1 were isotropic, its radiated power density at a
distance R would be
Pt
W0 
4R 2
where Pt is the total radiated power. Because of the directivity,
the actual power density becomes
Pt Dt
Wt  W0 Dt 
4R 2
The power collected by the antenna would be
Pt Dt Ar
Pr  Wt Ar 
4R 2
Pr
or Dt Ar  4R 2 (1)
Pt
52
Effective area & Directivity(2)
If antenna #2 is used as a transmitter, 1 as a receiver, and the
medium is linear, passive and isotropic, one obtains
Pr
Dr At  4R 2 (2)
Pt
Dt Dr
From (1) and (2),

At
Ar
Increasing the directivity of an antenna increases its effective
area:
D0t D0 r

Atm Arm
If antenna #1 is isotropic,
53
Arm
Atm 
D0 r
Effective area & Directivity(3)
For example, if antenna #2 is a short dipole, whose effective
area is 3λ2/8π and directivity is 1.5, one obtains
Arm 32 2 2
Atm 


D0 r 8 3 4
2

and
Arm  D0 r Atm  D0 r
4
In general, maximum effective area of any antenna is related
to its maximum directivity by
2
Aem 
D0
4
If there’re conductiondielectric loss, polarization
loss and mismatch:
54
2
Aem 
D0 ecd (1 |  |2 ) | ˆ w  ˆ a |2
4
Friis Transmission Equation
• The Friis transmission equation relates the power
received to the power transmitted between two antennas
separated by a distance R > 2D2/λ.
Power density at distance R from the transmitting antenna:
Pt Gt ( t , t ) et Pt Dt ( t , t )
Wt 

2
4R
4R 2
( r , r )
( Pt , Gt , Dt , ecdt , t , ˆ t )
( t , t )
( Pr , Gr , Dr , ecdr , r , ˆ r )
The effective area of the
receiving antenna:
2
Ar  er Dr ( r , r )
4
The amount of power collected by the receiving antenna:
2
2 Dt ( t , t ) Dr ( r , r ) Pt
2
ˆ
ˆ
Pr  er Dr ( r , r ) Wt  et er
|



|
t
r
4
(4R) 2
55
Friis Transmission Equation(2)
The ratio of the received to the input power:
Pr
2 Dt ( t , t ) Dr ( r , r )
2
ˆ
ˆ
 et er
|



|
t
r
Pt
(4R ) 2
or
2
Pr
2
2  Dt ( t , t ) Dr ( r , r )
2
ˆ
ˆ
 ecdt ecdr (1 | t | )(1 | r | )
|



|
t
r
Pt
(4R ) 2
For reflection and polarization-matched antennas aligned for
maximum directional radiation and reception:
2
Pr   

 G0t G0 r
Pt  4R 
56
Radar Cross Section
• Radar cross section or echo area (σ) of a target is
defined as the area intercepting that amount of power
which, when scattered isotropically, produces at the
receiver a density which is equal to that scattered by the
actual target.
rs 2
rs 2



2 Ws 
2 |E | 
2 |H | 
  lim 4R
4R r i 2   Rlim
4R r i 2 
  Rlim
R 




Wi 
|E | 
|H | 



 : radar cross section [m 2 ]
R : observation distance from targe [m]
Wi ,Ws : incident, scattered power density [W/m 2 ]
ri rs
E , E : incident, scattered electric field [V/m]
ri rs
H , H : incident, scattered magnetic field [A/m]
57
58
Radar Range Equation
• The radar range equation relates the power delivered to
the receiver load to the power transmitted by an antenna,
after it has been scattered by a target with a radar cross
section of σ.
The amount of captured power at
the target with the distance R1 from
Target
R
the transmitting antenna:
Incident wave
1
Transmitting
Scattered wave
̂ w
antenna
( Pt , Gt , Dt , ecdt , t , ˆ t )
R2
Pt Gt ( t , t ) et Pt Dt ( t , t )
Pc  Wt  

2
4R1
4R12
The scattered power density:
Receiving
antenna
( Pr , Gr , Dr , ecdr , r , ˆ r )
59
Pc
Pt Dt (t ,t )
Ws  2 et
2
4R2
(4R1R2)
Radar Range Equation(2)
The amount of power delivered to the load:
Thus,
Pt Dt ( t , t ) Dr ( r , r )   


Pr  ArWs  et er
4
 4R1 R2 
Pr
Dt ( t , t ) Dr ( r , r )   


 ArWs  et er
Pt
4
 4R1 R2 
2
2
With polarization loss:
2
D ( ,  ) D ( ,  )   
Pr
 | ˆ w  ˆ r |2
 ArWs  ecdt ecdr (1 | t |2 )(1 | r |2 ) t t t r r r 
Pt
4
 4R1 R2 
For reflection and polarization-matched antennas aligned for
maximum directional radiation and reception:
G0t G0 r
Pr

Pt
4
60
  


 4R1 R2 
2
Brightness Temperature
• Energy radiated by an object can be
represented by an equivalent temperature
known as Brightness Temperature TB (K).

2

TB ( ,  )   ( ,  )Tm  1   Tm
where
ε : emissivity (dimensionless) 0 ≤ ε ≤ 1
Tm : molecular (physical) temperature (K)
Γ(θ,φ) : reflection coefficient of the surface for the
polarization of wave
Example Ground 300 K
61
Antenna Temperature
• Energy radiated by various sources appears
at antenna terminal as antenna temperature,
given by
TA
2

0
0
T ( ,  )G ( ,  ) sin dd



 
  G( ,  ) sin dd
B
2
where
0
0
TA : antenna temperature (effective noise
temperature of the antenna radiation resistance ; K)
G(θ,φ) : gain (power) pattern of the antenna
62
Noise Power
• Assuming no losses or other contributions
between antenna & receiver, noise power
transferred to receiver:
Pr  kTA f
where
Pr : antenna noise power (W)
k : Boltzmann’s constant (1.38×10-23 J/K)
TA : antenna temperature (K)
∆f : bandwidth (Hz)
63
System Noise Power Model
• Assume antenna & transmission line are
maintained at certain temperature, and
transmission line is lossy, then the model
below can be used to include all
contributions.
64
Antenna Temperature (2)
• Effective antenna temperature at receiver
terminals:
2l
2l
2l
Ta  TAe  TAP e  T0 (1  e )
where
Ta : antenna temperature at receiver terminals (K)
TA : antenna noise temperature at antenna terminals (K)
TAP : antenna temperature at antenna terminals due to
1
physical temperature (K)
TAP  (e A  1)TP
α : attenuation constant of transmission line (Np/m)
eA : thermal efficiency of antenna (dimensionless)
l : length of transmission line (m)
T0 : physical temperature of transmission line (K)
65
System Noise Power
• noise power transferred to receiver: Pr  kTa f
• If there’s thermal noise in receiver:
Ps  k (Ta  Tr )f  kTs f
where
Ps : system noise power (W)
Ta : antenna noise temperature at receiver terminals
Tr : receiver noise temperature at receiver terminals
Ts =Ta + Tr : effective system noise temperature at
receiver terminals
66
Ex 2.16 Effective antenna temp = 150 K.
Antenna is maintained at 300 K and has
thermal efficiency 99%. It is connected to a
receiver through 10-m waveguide (loss = 0.13
dB/m, temp = 300 K) Find effective antenna
temperature at receiver terminals.
1
A
1
TAP  (e  1)TP  (.99  1)300  3.03
 ( Np/m)   (dB/m)/8.68  0.0149,l  0.149
Ta  TAe
2l
 TAP e
2l
 T0 (1  e
2l
)
 150e  2 (.149 )  3.03e  2 (.149 )  300(1  e  2 (.149 ) )
 190.904 K
67