Upload
others
View
14
Download
0
Embed Size (px)
Citation preview
1
Chapter 2 :
Fundamental parameters of antennas
• From Radiation Pattern
– Radiation intensity
– Beamwidth
– Directivity
– Antenna efficiency
– Gain
– Polarization
• From Circuit viewpoint
– Input Impedance
2
Chapter 2 : Topics (2)
• Antenna effective length and effective area
• Friis transmission equation
• Radar range equation
• Antenna temperature
3
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
4
Radiation Pattern Example
5
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
6
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
7
Radiation Pattern (3)
Omnidirectional
Eθ
Hφ
8
Field Regions
DR1
R2
Reactive near
field region
Radiating near field
region (Fresnel zone)
Far field region
(Fraunhofer zone)
D=largest antenna
dimension
9
Reactive Near Field Region
• Region surrounding the antenna, wherein the
reactive field predominates
]62.0,2
max[
2 :antenna) (smallFor
62.0:For
3
3
DR
RD
DRD
Angular field distribution depends on distance from
antenna
10
Radiating Near Field Region
• Region between reactive near-field and far-
field regions (Fresnel zone)
]62.0,2
max[]2
,3max[
23 :antenna) (smallFor
62.02
:For
32
32
DR
D
RD
DR
DD
Radiation fields predominate but angular field
distribution still depends on distance from antenna
11
Far Field Region
• Region where angular field distribution is
essentially independent of the distance from
antenna (Fraunhofer zone)
]2
,3max[
3 :antenna) (smallFor
2:For
2
2
DR
RD
DRD
12
Change of antenna amplitude
pattern shape
13
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
Quiz: What’s the solid angle subtended by a sphere?
anglesolidsin
;sin 22
ddd
drddrdA
14
Radiation Power Density
• Poynting vector = Power density
• Total power:
HHHHEEEEWWWWrrr
[A/m]Intensity field magnetic ousinstantane :
[V/m]Intensity field electric ousinstantane :
][W/m vector Poynting ousinstantane : 2
HHHH
EEEE
WWWW
r
r
r
SS
dand ˆWWWWWWWWrr
sP
][m surface closed theof area malinfinitesi :
surface the tonormalr unit vecto :ˆ
[W]power totalousinstantane :
2da
n
P
15
Radiation Power Density (2)• For time-harmonic EM fields
• Poynting vector
• Time average Poynting vector (average power
density or radiation density)
]),,(Re[);,,( tjezyxtzyx
Err
EEEE
]),,(Re[);,,( tjezyxtzyx
Hrr
HHHH
]Re[2
1]Re[
2
1 2* tje
HEHErrrrrrr
HHHHEEEEWWWW
speak valuerepresent fields, because appears2
1
]Re[2
1)];,,([),,( *
HE
HEW
rr
rrrr avav tzyxzyx WWWW
16
Radiation Power Density (3)
• Average power radiated power
]W[sinˆ)sin
ˆ(
ˆ
0
22
0 0
2
20 Addrrr
Ar
danPS
avrad
Wr
SS
av
S
avavrad ddandPP sHEWW ]Re[2
1ˆ *
rrrrs
Example 2.1: The average power density is given by
The total radiated power becomes
]m/W[sin
ˆˆ 2
20r
ArWr rav
W
r
17
Radiation Power Density (4)
• For an isotropic antenna
The power density is then given by
]m/W[4
ˆˆ 2
200r
PrWr rad
W
r
]W[4sinˆ)](ˆ[ 0
22
0 0
2
0
0
WrddrrrWr
dPS
rad
sWrr
18
Radiation Intensity
• Definition : The power radiated from an
antenna per unit solid angle
2
0 0sin ddUUdPrad
Total power can be given by
radWrU2
][W/mdensity radiation :
angle] solid[W/unit intensity radiation :
2
radW
U
19
Radiation Intensity (2)
• Radiation intensity is related to the far-zone
electric field of antenna
]|),(||),([|2
1
]|),,(||),,([|2
|),,(|2
),(
22
222
22
ooEE
rErEr
rr
U
Er
space) freein 377 ( medium theof impedance intrinsic :
antenna theof components field-electric zone-far :,
),( antenna theofintensity field-electric zone-far :),,(
η
EE
r
er
jkro
EErr
20
Radiation Intensity (3)
000 4 UdUdUPrad
]W[sin
sin
0
22
0 0
2
0
2
0 0
AddA
ddUPrad
sin0
2AWrU rad
Example 2.2: The radiation intensity is given by
The total radiated power becomes
For an isotropic antenna
40
radPU
21
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
22
Beamwidth (2)
23
Beamwidth (3)
)20,0(cos)( 2 U
4)707.0(cos
707.0cos5.0cos|)(
1
2
h
hhU
Example 2.3: The normalized radiation intensity of an
antenna is represented by
The angle θh at which the function equal to half of its maximum can be found by
Since the pattern is symmetric with respect to the
maximum, HPBW = 2 θh = π/2
Likewise, FNBW = 2θn = π since2
)0(cos0|)( 1
nnU
24
Directivity
• Ratio of radiation intensity in a given direction
from the antenna to the average radiation intensity
less)-(dimension4),(
),(0 radP
U
U
UD
Note that the average radiation intensity equals to the
radiation intensity of an isotropic source.
25
Directivity (2)
AdU
UD
dU
UD
4
')','(4
')','(
),(4),(
'
maxmax
'
'
')','( dUPrad
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
Since
max
' max
')','(
UPdU
UAradA
Dmax: maximum directivity
26
Directivity (3)
• If the direction is not
specified, it implies the
directivity of maximum
radiation intensity
(maximum directivity)
expressed as
less)-(dimension4 max
0
max0max
radP
U
U
UDD
27
Directivity (4)
2
0
2 sinAWrU r
]W/m[sin
ˆˆ 2
2
2
0r
ArWrW rav
2
0 00
2
03
8sinsin AddAUdPrad
Example 2.4: The radial component of the radiated power density of an infinitesimal linear dipole is given by
where A0 is the peak value of the power density. The radiation intensity is given by
The maximum radiation is directed along θ = π/2 and Umax = A0.
The total radiated power is given by
Thus,
22
0max
0 sin5.1sin and 2
34 DD
P
UD
rad
28
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
)||1( 2
0 cdrcd eeee
inalinput term at thet coefficien reflection voltage:
esefficienci dielectric ,conduction :,
efficiencyradiation antenna :
efficiency (mismatch) reflection :
efficiency total:0
dc
dccd
r
ee
eee
e
e
29
Gain
• It takes into account the efficiency of the antenna as well
as its directional properties. (Directivity only measures
directional properties.)
inP
UG
),(4),(
loss)power dielectric and Ohmic:(
)||1(
antenna; input toPower :
2
loss
lossradoin
in
P
PPPP
P
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
30
Gain (2)
• Using ecd, Prad= ecdPin and
),(),(
4),(
DeeP
UG cdcd
rad
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
source) isotropic (lossless
),(4),(
inP
UG
31
Gain (3)
• Absolute gain takes into account impedance mismatch
losses at the input terminals in addition to losses within
antenna
),()||1(),(
4
)||1(),(
4),(
4),(
0
2
2
DeeP
U
P
U
P
UG
cd
rad
ino
abs
32
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.
0, where)(
,)(Let
yoxo
jkzj
yoy
jkzj
xox
EEeeEzE
eeEzE
y
x
);(ˆ);(ˆ);( tzytzxtz yx EEEEEEEEEEEE r
)cos();(
)cos();(
yyoy
xxox
kztEtz
kztEtz
EEEE
EEEE
x
y
z
dependence timetje
Wave propagating in –z
direction
33
Polarization (2)
0or 0(i) yoxo EE
z
22
yoxo EE
A. Linear Polarization
Example
20,tan
,...2,1,0 where
(ii)
1
xo
yo
xy
E
E
n
n
δ, γ determine polarization state
or
34
Polarization (3)
4/)1(tan (i) 1 oyoxo EEE
B. Circular Polarization
,...2,1,0 where
CCW/LCP;2
12n
CW/RCP;2
12n
(ii)
n
xy
and
Note that the sense of rotation is observed along the direction of propagation.
35
36
Polarization (4)
2/,4/
)sin(
)2/cos();(
)cos();(
xo
xoy
xox
kztE
kztEtz
kztEtz
EEEE
EEEE
Example: RCP
z
37
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.
38
Polarization (6)
yoxo
xy
EEn
n
, FOR,...2,1,0 where
CCW/LEP;0
CW/REP;0
2 (ii)if
C. Elliptic Polarization
yoxo
xy
EEn
AND,...2,1,0 where
CCW/LEP;2
12n
CW/REP;2
12n
(i)if
39
Polarization (7)
AR1;AxisMinor
AxisMajor (AR) Ratio Axial
cos
2tan
2
1
2 22
1
yoxo
yoxo
EE
EE
40
Polarization Loss Factor (PLF)
• Electric field of incoming wave
• Electric field of receiving antennaiww EE
r
less)(dimension
|cos||ˆˆ|PLF 22
paw
aaa EEr
on vectorpolarizati:ˆ
wave theofr unit vecto:ˆ
where
a
w
a
w
41
PLF example
jkzj
oy
jkzj
ox eeEEeeEE yx ;
2
ˆˆˆwhere
2ˆ)ˆˆ()ˆˆ( 2/
yjx
eEeEyjxeeEyEx
w
jkz
ow
jkz
o
jkzj
oo
Er
*ˆˆwa
LCP wave: δ=-π/2,φx=0φy=-π/2
dB 012
ˆˆ
2
ˆˆPLF
2
yjxyjx
If the antenna is also LCP,
If the antenna is RCP,2
ˆˆˆ
yjxa
02
ˆˆ
2
ˆˆPLF
2
yjxyjx
42
Input Impedance
Generator
Antenna
Terminal
a
b
dohmicloss
loss
r
lossrA
RRR
R
R
RRR
)dielectric(ohmic, resistance Loss :
resistanceRadiation :
Transmitting case
Impedance presented by the antenna at its terminal
speak value are ,
)(,
gg
Agggggg
IV
ZZIVjXRZ
][)()()( AAA jXRZ
gAgA XXRR ,Maximum power delivered to the antenna occurs when conjugate
matched:
43
Input Impedance (2)
2
2
2
)(8
||||
2
1
2 lossr
rg
rgrad
A
g
gRR
RVRIP
R
VI
2
2
2
)(8
||||
2
1
lossr
lossg
lossglossRR
RVRIP
lossradg PPP
lossr
g
gggRR
VRIP
1
8
||||
2
12
2
When conjugate matched:
NOTE:
Radiated Power
Power loss to heat
Power loss in Rg
44
Input Impedance (3)
slossradin PPPP2
1
g
g
A
g
A
g
gggsR
V
R
V
R
VVIVP
4
||
4
||
22
1]
2
1Re[
22*
*
ing PP
lossr
r
in
radcd
RR
R
P
Pe
1,,00 If cdinradlossloss ePPPR
Power supplied by generator when conjugate matched:
antenna radiation
efficiency
sg PP2
1
45
Receiving Antenna
speak value are ,
8
||||
2
1
2
22
TT
T
TTTT
T
TT
IV
R
VRIP
R
VI
ATlossrAT XXRRRR ,
condition matched conjugateunder Receiving case
Impedance presented by the antenna at its terminal
TTTAAA jXRZjXRZ ,
2
22
)(8
||||
2
1
lossr
rTrTscatt
RR
RVRIP
Power delivered to
load
Power scattered or re-radiated
46
Receiving Antenna (2)
T
TTTc
R
VIVP
4
||]Re[
2
1 2*
2
22
)(8
||||
2
1
lossr
lossTlossTloss
RR
RVRIP
cT PP2
1
clossscatt PPP2
1
Tlossscattc PPPP
Power supplied by generator when conjugate matched:
captured/collected power
lossscattT PPP Power lost to heat
47
Antenna equivalent Area
• Used to describe the power capturing characteristics of
an antenna when a wave impinges on it
),(direction fromdensity flux power Incident
antenna receiving of lsat termina availablePower
),(direction in (aperture) area effective),(
eA
]m[2
|| 22
i
TT
i
Te
W
RI
W
PA
aveincident w ofdensity power :
load todeliveredpower :
i
TT
W
RP
48
Antenna Equivalent Area (2)
22
2
)()(2
||
TATlossr
T
i
Te
XXRRR
R
W
VA
2
2
)(8
||
lossr
r
i
Ts
RR
R
W
VA
lossri
T
Ti
Tem
RRW
V
RW
VA
1
8
||
8
|| 22
Under conjugate matched condition:
power radiated-reor scattered the toequal is density,
powerincident by multipliedn which whearea, scattering:
load todeliveredpower toequal is density,power
incident by the multipliedn which whearea, effective :e
s
T
A
R
A
Maximum effective area
49
Antenna Equivalent Area (3)
2
22
)(8
||
4
||
lossr
Tlossr
i
T
Ti
Tc
RR
RRR
W
V
RW
VA
2
2
)(8
||
lossr
loss
i
Tloss
RR
R
W
VA
antennaby capturedpower total the toequal is density,
powerincident by multipliedn which whearea, Captured:
load todeliveredpower toequal is density,
powerincident by the multipliedn which whearea, loss :loss
c
loss
A
R
A
Under conjugate matched condition:
lossse AAA
losssec AAAA
50
Antenna Equivalent Area (4)
ri
Tem
RW
VA
8
|| 2
2
2E
Wi
ElVT
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
Since the dipole is very short, the induced current can be
assumed to be constant and of uniform phase. The induced
voltage is
22
2222
2
119.08
3
)/80)(2/(8
)(
lE
ElAem
For a uniform plane wave, the incident power density is given by
thus
51
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 a far-field quantity.
ZA
b
a
VT
[m]),(ˆ),(ˆ),( llle r
ltagecircuit vo-open :
field electricincident :
ocT
i
VV
E
r
e
i
oc lEVrr
Example 2.6 : The electric field of a short dipole is given by
sin8
ˆr
lekIj
jkr
ina
Er
sin2
ˆ lle r
e
jkrina le
r
kIjEE
rr
4ˆˆE
52
Effective area & Directivity
)1(4or
4
2
2
RP
PAD
R
ADPAWP
t
rrt
rttrtr
204 R
PW t
If antenna #1 were isotropic, its radiated power density at a
distance R would be
where Pt is the total radiated power. Because of the directivity,
the actual power density becomes
The power collected by the antenna would be
204 R
DPDWW tt
tt
53
Effective area & Directivity(2)
rm
r
tm
t
A
D
A
D 00
)2(4 2R
P
PAD
t
rtr
r
r
t
t
A
D
A
D
If antenna #2 is used as a transmitter, 1 as a receiver, and the
medium is linear, passive and isotropic, one obtains
From (1) and (2),
Increasing the directivity of an antenna increases its effective
area:
r
rmtm
D
AA
0
If antenna #1 is isotropic,
54
Effective area & Directivity(3)
0
2
4DAem
43
2
8
3 22
0
r
rmtm
D
AA
4
2
00 rtmrrm DADA
For example, if antenna #2 is a short dipole, whose effective
area is 3λ2/8π and directivity is 1.5, one obtains
and
In general, maximum effective area of any antenna is related to its maximum directivity by
If there’re conduction-
dielectric loss, polarization
loss and mismatch:
22
0
2
|ˆˆ|)||1(4
awcdem eDA
55
Friis Transmission Equation• The Friis transmission equation relates the power
received to the power transmitted between two antennas separated by a distance R > 2D2/λλλλ.
22 4
),(
4
),(
R
DPe
R
GPW ttttttttt
t
4
),(2
rrrrr DeA
2
2
22
|ˆˆ|)4(
),(),(
4),( rt
trrrtttrttrrrrr
R
PDDeeWDeP
Power density at distance R from the transmitting antenna:
)ˆ,,,,,( ttcdtttt eDGP )ˆ,,,,,( rrcdrrrr eDGP
),( rr ),( tt
The effective area of the
receiving antenna:
The amount of power collected by the receiving antenna:
56
Friis Transmission Equation(2)
2
2
222 |ˆˆ|
)4(
),(),()||1)(||1( rt
rrrtttrtcdrcdt
t
r
R
DDee
P
P
2
2
2
|ˆˆ|)4(
),(),(rt
rrrtttrt
t
r
R
DDee
P
P
The ratio of the received to the input power:
or
rt
t
r GGRP
P00
2
4
For reflection and polarization-matched antennas aligned for
maximum directional radiation and reception:
57
Radar Cross Section
• Radar cross section or echo area (σσσσ) 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.
2
22
2
222
||
||4lim
||
||4lim4lim
i
s
Ri
s
Ri
s
RRR
W
WR
H
H
E
Er
r
r
r
[A/m] field magnetic scattered incident, :,
[V/m] field electric scattered incident, :,
][W/mdensity power scattered incident, :,
[m] targefrom distancen observatio :
][msection crossradar :
2
2
si
si
si WW
R
HH
EErr
rr
58
59
Radar Range Equation
Transmitting
antenna
)ˆ,,,,,( ttcdtttt eDGP
)ˆ,,,,,( rrcdrrrr eDGP
w
Receiving
antenna
Target R1
Incident wave
R2
Scattered wave
2
21
2
2 )4(
),(
4 RR
DPe
R
PW tttt
tc
s
• 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 σσσσ.
2
1
2
1 4
),(
4
),(
R
DPe
R
GPWP ttttttttt
tc
The amount of captured power at
the target with the distance R1 from
the transmitting antenna:
The scattered power density:
60
Radar Range Equation(2)
2
2144
),(),(
RR
DDPeeWAP rrrtttt
rtsrr
2
2144
),(),(
RR
DDeeWA
P
P rrrtttrtsr
t
r
2
2
21
22 |ˆˆ|44
),(),()||1)(||1( rw
rrrtttrtcdrcdtsr
t
r
RR
DDeeWA
P
P
The amount of power delivered to the load:
2
21
00
44
RR
GG
P
P rt
t
r
Thus,
With polarization loss:
For reflection and polarization-matched antennas aligned for
maximum directional radiation and reception:
Brightness Temperature
• Energy radiated by an object can be
represented by an equivalent temperature
known as Brightness Temperature TB (K).
where
εεεε : emissivity (dimensionless) 0 ≤ εεεε ≤ 1
Tm : molecular (physical) temperature (K)
ΓΓΓΓ(θθθθ,,,,φφφφ) : reflection coefficient of the surface for the
polarization of wave
Example Ground 300 K
61
mmB TTT2
1),(),(
Antenna Temperature• Energy radiated by various sources appears
at antenna terminal as antenna temperature,
given by
where
TA : antenna temperature (effective noise
temperature of the antenna radiation resistance ; K)
G(θθθθ,,,,φφφφ) : gain (power) pattern of the antenna
62
2
0 0
2
0 0
sin),(
sin),(),(
ddG
ddGTT
B
A
Noise Power
• Assuming no losses or other contributions
between antenna & receiver, noise power
transferred to receiver:
where
Pr : antenna noise power (W)
k : Boltzmann’s constant (1.38×10-23 J/K)
TA : antenna temperature (K)
∆∆∆∆f : bandwidth (Hz)63
fkTP Ar
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:
where
Ta : antenna temperature at receiver terminals (K)
TA : antenna noise temperature at antenna terminals (K)
TAP : antenna temperature at antenna terminals due to
physical temperature (K)
αααα : 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
)1( 2
0
22 ll
AP
l
Aa eTeTeTT
PAAP TeT )1( 1
System Noise Power
• noise power transferred to receiver:
• If there’s thermal noise in receiver:
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
fkTP ar
fkTfTTkP sras )(
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.
67
03.3300)199(.)1( 11 PAAP TeT
149.00.0149,dB/m)/8.68(Np/m)( l
K904.190
)1(30003.3150
)1(
)149(.2)149(.2)149(.2
2
0
22
eee
eTeTeTTll
AP
l
Aa