Download Fundamental Parameters of Antennas

Fundamental Parameters of Antennas
Ranga Rodrigo
June 23, 2010
Lecture notes are fully based on Balanis [1]. Some diagrams and text are directly
from the books.
Contents
1 Polarization
1
2 Input Impedance
6
3 Antenna Effective Length and Effective Areas
11
1 Polarization
Definition 1 (Polarization). Polarization is the curve traced by the end point of
the arrow (vector) representing the instantaneous electric field. The field must
be observed along the direction of propagation.
• Polarization is classified as linear, circular, or elliptical.
• If the vector that describes the electric field at a point in space as a function of time is always directed along a line, the field is said to be linearly
polarized.
• In general, the figure that the electric field traces is an ellipse, and the
field is said to be elliptically polarized.
1
z
y
Polarized
light
la
Po
d
roi
E
x
Polarization Types
• Linear polarization and circular polarization are special cases of elliptic
polarization.
• Polarization can be clockwise (CW, right-hand polarization), or counter
clockwise (CCW, left-hand polarization).
2
Linear, Circular and Elliptic Polarization
• The instantaneous electric field of a plane wave, traveling in the negative
z direction, can be written as
E (z; t ) = â x E x (z; t ) + â y E y (z; t ).
• By considering the complex counterpart of these instantaneous components, we can write
E x (z; t ) = E xo cos(ωt + kz + φx ),
E y (z; t ) = E yo cos(ωt + kz + φ y ).
where E xo and E yo are the maximum magnitudes of the x- and y-components.
• By defining ∆φ = φ y − φx , we can state these as
E x (z; t ) = E xo cos(ωt + kz),
E y (z; t ) = E yo cos(ωt + kz + ∆φ).
Linear polarization
∆φ = nπ,
n = 1, 2, . . .
y
E yo
E xo
x
Circular Polarization
|E § | = |E y | ⇒ E xo = E yo .
( ¡
¢
+ 2n + 21 π,
¡
¢
∆φ =
− 2n + 21 π,
y
E yo
n = 0, 1, 2, . . .
CW,
n = 0, 1, 2, . . .
CCW.
⊗ k
x
E xo
Elliptic Polarization
|E § | 6= |E y | ⇒ E xo 6= E yo .
3
µ
¶
1
∆φ = ± 2n + π,
2
OR
∆φ 6= ±n
y
E yo
π
2
n = 0, 1, 2, . . . .
n = 0, 1, 2, . . . .
⊗ k
E xo
x
Polarization Loss Factor and Efficiency
• In general, the polarization of the receiving antenna will not be the same
as the polarization of the incoming (incident) wave. This is commonly
stated as polarization mismatch.
• The amount of power extracted by the antenna from the incoming signal
will not be maximum because of the polarization loss.
• Assuming that the electric field of the incoming wave can be written as
E i = ρ̂ w E i ,
where ρ̂ w is the unit vector of the wave. The polarization of the electric
field of the receiving antenna can be expressed as
E a = ρ̂ a E a ,
where ρ̂ a is its unit vector.
• The polarization loss can be taken into account by introducing a polarization loss factor (PLF). It is defined, based on the polarization of the
antenna in its transmitting mode, as
¯
¯2 ¯
¯2
PLF = ¯ρ̂ w · ρ̂ a ¯ = ¯cos ψp ¯ .
where ψp is the angle between the two unit vectors.
• The PLF is dimensionless and corresponds to the polarization efficiency
ep .
4
ρ̂ w
ρ̂ a
ψp
Example 2. The electric field of a linearly polarized electromagnetic wave given
by
E i = â x E 0 (x, y)e − j kz
is incident upon a linearly polarized antenna whose electric-field polarization
is expressed as
E a = (â x + â x y)E (r, θ, φ).
Find the polarization loss factor (PLF).
PLF for Aperture Antennas
ψp
¯
¯2
PLF = ¯ρ̂ w · ρ̂ a ¯ = 1
ψp
¯
¯2 ¯
¯2
PLF = ¯ρ̂ w · ρ̂ a ¯ = ¯cos ψp ¯
¯
¯2
PLF = ¯ρ̂ w · ρ̂ a ¯ = 0
PLF for Linear Antennas
ψp
ψp
¯
¯2
PLF = ¯ρ̂ w · ρ̂ a ¯ = 1
¯
¯2 ¯
¯2
PLF = ¯ρ̂ w · ρ̂ a ¯ = ¯cos ψp ¯
5
¯
¯2
PLF = ¯ρ̂ w · ρ̂ a ¯ = 0
2 Input Impedance
Definition 3 (Input Impedance). The impedance presented by an antenna at
its terminals or the ratio of the voltage to current at a pair of terminals or the
ratio of the appropriate components of the electric to magnetic fields at a point.
Antenna in the Receiving Mode
ZA = RA + j X A .
• Z A = antenna impedance at terminals a − b (Ω).
• R A = antenna resistance at terminals a − b (Ω).
• X A = antenna reactance at terminals a − b (Ω).
R A = R rad + R L .
• R rad = radiation resistance of the antenna.
• R L = loss resistance of the antenna.
Power
• The current developed within the loop which is I g .
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• The power delivered to the antenna for radiation is given by
¯ ¯2 ·
¸
¯Vg ¯
R rad
1 ¯¯ ¯¯2
P rad = I g R rad =
.
2
2
(R rad + R L + R g )2 + (X A + X g )2
• The power delivered to the antenna dissipated as heat is
¯ ¯2 ·
¸
¯Vg ¯
RL
1 ¯¯ ¯¯2
P rad = I g R L =
.
2
2
(R rad + R L + R g )2 + (X A + X g )2
• The power dissipated as heat on the internal resistance R g of the generator,
¯ ¯2 ·
¸
¯Vg ¯
Rg
1 ¯¯ ¯¯2
P rad = I g R g =
.
2
2
(R rad + R L + R g )2 + (X A + X g )2
• The maximum power delivered to the antenna occurs when we have conjugate matching:
R rad + R L = R g ,
X A = −X g .
• For this case
¯ ¯2 ·
¯Vg ¯
¸ ¯¯ ¯¯2 ·
¸
Vg
R rad
R rad
P rad =
=
.
2
4(R rad + R L )2
8
(R rad + R L )2
¯ ¯2 ·
¸
¯Vg ¯
RL
PL =
.
8
(R rad + R L )2
¯ ¯2 ·
¸ ¯¯ ¯¯2
¸ ¯¯ ¯¯2 ·
¯Vg ¯
Rg
Vg
Vg
1
=
.
Pg =
=
2
8
(R rad + R L )
8
R rad + R L
8R g
P g = P rad + P L
• Power supplied by the source during conjugate matching is
"
#
·
¸
∗
V
|Vg |2
1
1
1
g
∗
P S = Vg I g = Vg
=
.
2
2
2(R rad + R L )
4
R rad + R L
7
• Of the power that is provided by the generator, half is dissipated as heat in
the internal resistance (R g ) of the generator and the other half is delivered
to the antenna. This only happens when we have conjugate matching.
• Of the power that is delivered to the antenna, part is radiated through
the mechanism provided by the radiation resistance and the other is dissipated as heat which influences part of the overall efficiency of the antenna.
• If the antenna is lossless and matched to the transmission line (e o = 1),
then half of the total power supplied by the generator is radiated by the
antenna during conjugate matching, and the other half is dissipated as
heat in the generator.
Antenna in the Receiving Mode
• The incident wave impinges upon the antenna, and it induces a voltage
VT which is analogous to Vg of the transmitting mode.
• The power P rad delivered to R rad is referred to as scattered (or reradiated)
power.
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• Power delivered to R T , R rad , and R L , are, respectively,
·
¸
·
¸
|VT |2
RT
1
|VT |2
|VT |2
PT =
=
=
,
8
(R rad + R L )2
8
R rad + R L
8R T
·
¸
·
¸
R rad
|VT |2
R rad
|VT |2
=
,
P rad =
2
4(R rad + R L )2
8
(R rad + R L )2
·
¸
|VT |2
RL
PL =
.
8
(R rad + R L )2
• The induced (collected or captured) power is
·
¸
·
¸
VT∗
1
1
|VT |2
1
∗
P C = VT I T = VT
=
.
2
2
2(R rad + R L )
4
R rad + R L
• Under conjugate matching of the total power collected or captured half
is delivered to the load R T and the other half is scattered or reradiated
through R rad and dissipated as heat through R L .
• If the losses are zero (R L = 0), then half of the captured power is delivered
to the load and the other half is scattered.
• This indicates that in order to deliver half of the power to the load you
must scatter the other half.
• The most that can be delivered to the load is only half of that captured and
that is only under conjugate matching and lossless transmission line.
Antenna Radiation Efficiency
• The antenna efficiency takes into account the reflection, conduction, and
dielectric losses.
• The conduction and dielectric losses are difficult to compute and in most
cases they are measured.
• Even with measurements, they are difficult to separate and they are usually lumped together to form the e cd efficiency.
• The resistance R L is used to represent the conduction-dielectric losses.
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Definition 4 (Conduction-Dielectric Efficiency). The conduction-dielectric efficiency e cd is defined as the ratio of the power delivered to the radiation resistance R rad to the power delivered to R rad and R L .
e cd =
R rad
.
R rad + R L
• DC resistance of a rod
R DC =
1 l
.
σA
• RF resistance of a rod
R RF
R RF
1 l
, where δ is the skin depth δ =
=
σ 2πbδ
r
l
ωµo
.
=
2πb
2σ
s
2
,
ωµ0 σ
Example 5. A resonant half-wavelength dipole is made out of copper (σ = 5.7 ×
107 S/m) wire. Determine the conduction-dielectric (radiation) efficiency of the
dipole antenna at f = 100 MHz if the radius of the wire b is 3 × 10−4 λ, and the
radiation resistance of theλ/2 dipole is 73 Ω.
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3 Antenna Effective Length and Effective Areas
• An antenna in the receiving mode, whether it is in the form of a wire,
horn, aperture, array, dielectric rod, etc., is used to capture (collect) electromagnetic waves and to extract power from them.
• For each antenna, an equivalent length and a number of equivalent areas
can then be defined.
Vector Effective Length
• This is a far-field quantity that is used to determine the voltage induced
on the open-circuit terminals of the antenna when a wave impinges upon
it.
`e (θ, φ) = â θ l (θ, φ) + â φ l (θ, φ).
• In transmitting mode
E a = â θ E θ + â φ E φ = − j η
• In receiving mode
Voc = E i · `e .
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k I in
`e e − j kr .
4πr
• Voc = open circuit voltage at antenna terminals.
• E i = incident electric field.
• E a = far-field electric field radiated by the antenna with current I in in its
terminals..
• `e = vector effective length.
Example 6. The far-zone field radiated by a small dipole of length l < λ/10 and
with a triangular current distribution id given by
k I in
`e e − j kr sin θ.
8πr
Determine the vector effective length of the antenna.
E a = â θ j η
Antenna Equivalent Areas: Effective Area A e
• With each antenna, we can associate a number of equivalent areas. These
are used to describe the power capturing characteristics of the antenna
when a wave impinges on it.
• Effective area (aperture)
Ae =
P T |I T |2 R T /2
=
.
Wi
Wi
where
A e = effective area (effective aperture) m2
P T = power delivered to the load W
A e = power density of incident wave W/m2
12
Maximum Effective Area A em
• Using the equivalent circuit to compute the current
·
¸
|VT |
RT
Ae =
.
2Wi (R rad + R L + R T )2 + (X A + X T )2
• Under conditions of maximum power transfer (R rad +R L = R T , X A = −X T )
·
¸
·
¸
|VT |
RT
|VT |
1
A em =
=
.
8Wi (R rad + R L )2
8Wi R rad + R L
• All of the power that is intercepted, collected, or captured by an antenna
is not delivered to the load.
• Capture area = Effective area + Scattering area + Loss area.
• Under conjugate matching, these are
Capture A c
h
i
|VT |
8Wi
R T +R rad +R L
(R rad +R L )2
Effective A em
h
i
|VT |
8Wi
RT
(R rad +R L )2
Scattering A s
h
i
|VT |
8Wi
R rad
(R rad +R L )2
|VT |
8Wi
Loss A L
h
RL
(R rad +R L )2
i
Aperture Efficiency
•
²ap =
A em Maximum effective area
=
.
Ap
Physical area
Example 7. A uniform plane wave is incident upon a very short lossless dipole
(l ¿ λ). Find the maximum effective area assuming that the radiation resis³ ´2
tance of the dipole is R rad = 80 πl
λ , and the incident field is linearly polarized
along the axis of the dipole.
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•
Maximum Directivity and Maximum Effective Area
• In general , the maximum effective aperture (A em ) of any antenna is related to its maximum directivity (D 0 ) by
A em =
λ2
D0 .
4π
• This assumes that there are no conduction-dielectric losses (radiation efficiency e cd = 1 ), the antenna is matched to the load (reflection efficiency
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e r = 1), and the polarization loss factor PLF and polarization efficiency p e
are unity.
• If the losses are included
µ
A em = e o
¶
µ 2¶
λ2
2
2 λ
D 0 |ρ̂ w · ρ̂ a | = e cd (1 − |Γ| )
D 0 |ρ̂ w · ρ̂ a |2 .
4π
4π
Friis Transmission Equation
µ
¶
¡
¢¡
¢ λ 2
Pr
2
2
= e cd t e cd r 1 − |Γt | 1 − |Γr |
D t (θt , φt )D r (θr , φr )|ρ̂ t · ρ̂ r |2
Pt
4πR
For reflection and polarization-matched antennas aligned for maximum directional radiation and reception
µ
¶
λ 2
Pr
=
G 0t G 0r
Pt
4πR
Radar Range Equation
µ
¶2
¡
¢¡
¢ D t (θt , φt )D r (θr , φr )
Pr
λ
2
2
= e cd t e cd r 1 − |Γt | 1 − |Γr | σ
|ρ̂ w · ρ̂ r |2
Pt
4π
4πR 1 R 2
where σ = radar cross section or echo area (m2 ), ρ̂ w = polarization unit vector
of the scattered waves, ρ̂ r = polarization unit vector of the receiving antenna.
For polarization-matched antennas aligned for maximum directional radiation and reception
µ
¶2
Pr
G 0t G0r
λ
=σ
Pt
4π
4πR 1 R 2
15
Antenna Temperature
• Read on your own, please.
References
[1]
Constantine A. Balanis. Antenna Throry: Analysis and Design. John Wiley & Sons, Inc., 2nd edition, 1997.
[2]
John D. Kraus, Ronaled J. Marhefka, and Ahmad S. Khan. Antennas for All Applications. Tata-McGraw-Hill, 3rd edition, 2006.
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