Receiving Properties of Antennas
Open-circuit voltage of short dipole antennas
For d << λ, quasistatic limit
Note that equipotentials (a) and (b) intercept the dipole at the
midpoints for rwire → 0, and are perpendicular.
-
-
-
ẑ
-
-
-
-
-
-
-
+
deff
+ +
(a)
vo.c.
+
(b)
-
+
+
+
plane
⇐ wave
-
E
+
-
E
+
+ +
-
+
θ
v o.c. = −E • zˆ deff = −E sin θdeff
↑ “open circuit”
+
+
+
+
+
+
equivalent charges
at infinity
L1
Equivalent circuit for short dipole antennas
-jX
+
v Th
2
Rr
jX =
1
here
jωCeq
Rr
+
v Th = v o.c. = −E sin θ deff
matched load for
maximum power received
antenna Thevenin
equivalent
L2
Available power from a short dipole antenna
⎛ V Th ⎞
Prec = ⎜
⎟
⎝ 2 ⎠
2
2Rr =
2 2
E deff
8Rr
deff = deff ,
r
t
sin2 θ
2
λ2 3 2
• sin θ
=
•
2ηo 4π 2
E
S(Wm-2)
incident
thin ­ wire,
short - dipole limit
-jX
jX
Rr
Rr
-
2
λ
A e (θ, φ ) =
G(θ, φ )
4π
“effective
area” (m2)
short
dipole
vTh
+
Rr = (2πηo 3 )(deff λ )2
V Th = Edeff sin θ
L3
Proof that A = Gλ2/4π for all reciprocal antennas
I2
+
I1
+
v1 -
N-port
-
v2
V = ZI , where Z ↔ S
I
-+ n
Impedance matrix for imbedded N-port
vn
t
t
If reciprocity applies : Z = Z , S = S (scattering matrix)
(Reference: Electromagnetic Waves, Staelin, Morgenthaler, and Kong, p. 459)
t
t
t
Reciprocity applies if u = u , ε = ε , σ = σ
[Excludes ferrites, magnetized plasmas, etc.]
(Reference: Op. Cit., p.454)
L4
Proof that A = Gλ2/4π for all reciprocal antennas
2
+
V1 -
1
antenna
(θ,φ)
I1 +
Rr
1
-
+
-
r
A1,G1
I1
A2,G2
I2
V2
z
Z=Z
+
V Th = Z12I2
1
-
Z21I1 = V Th
2
+
-
+
t
I2
Rr
2
-
Power received by antennas 1 and 2:
Pr = Z12 I2
1
2
Pr = Z 21I1
2
2
8Rr = Pt
1
2
8Rr = Pt
2
1
G2
2
• A1
2
• A2
4πr
G1
4πr
L5
Proof that A = Gλ2/4π for all reciprocal antennas
Power received by antennas 1 and 2:
G2
2
Pr = Z12 I2 8Rr = Pt
• A1
2
1
1
2 4 πr
G1
2
• A2
Pr = Z 21I1 8Rr = Pt
2
2
2
1 4 πr
G1A 2Pt
A1 A 2 Pt1 Pr1
1
Thus Pr Pr =
Therefore
=
•
•
2
1
G2 A1Pt
G1 G2 Pt Pr
2
2
2
But
Pr
1
Pr
2
A1
Therefore
=
G1
2
=
Z12 I2 Rr
2
2
Z 21I1 Rr
1
=
Pt
2
Pt
if Z12 = Z 21
1
A 2 λ2
t
t
for
all
antennas
if
µ
=
µ
,
ε
=ε
=
G2 4 π
L6
Example: Aeff for short dipole
2
λ2 3λ2 ~ ⎛ λ ⎞
= ⎜ ⎟ ≠ f (deff )
≤
A =G
4π 8π ⎝ 3 ⎠
≤ 3/2
λ/3
max
if matched
λ/3
e.g. λ = 300 m @ 1 MHz, yet d ≅ 1 m on car
e.g. cell phone @ 900 MHz → λ ≅ 30 cm, d ≅ 15 cm
Note: Aeff can be much larger than physical antenna when
the load is roughly impedance matched, but this match
may provide excessively narrow bandwidth ∆ω ≅ 1 Rr Ceq
Rr
Ceq
+
VTh
L7
Multi-conductor wire antennas
Short dipoles scatter.
How much?
Rr
λ
d <<
2π
+
Short ~
≡ VT
circuit
Short
circuit
-
N1
Multi-conductor wire antennas
Rr
+
λ
d <<
2π
Short ~
≡ VT
circuit
Short
circuit
-
Recall:
⎛ E 2 d2
⎞
⎜
2
eff sin2 θ ⎟
Prec = V Th 8Rr ≅ Sinc • A matched = ⎜
⎟
⎟
⎜ 8Rr
⎝
⎠
2
2
Pscat = V Th 2Rr ≅ Sinc • A scat
Sinc = E 2ηo Wm ­ 2
(
)
Therefore A scat ≅ 4 A matched ≤ 3λ2 2π (for short dipoles )
~0.7λ
d << λ
Ascat
~0.7λ
N2
Scattering from a half-wave dipole
Rr ≅ 73Ω, G ≤ 1.64, X ≅ 0 because We ≅ Wm
Most EM energy (WT = We + Wm ≅ 2Wm) is stored
within a few wire radii
2
ωo WT
Resonance Q =
= ω ∆ω ≅ 10
Pd
where ωo = 2πc λ , Pd ≡ Pr
Sinc
I
∆ω
0
ωo
ω
Orbiting λ/2 needles
for passive satellite
communications link
(artificial ionosphere)
N3
Scattering from parasitic antenna elements
image current
λ/2
~
≤ λ/4
λ/2
~λ/2, resonant
≡
λ/4
λ/4
z
λ/4
reflector
radiator
pattern
N4
Phase control in isolated wires
i(t)
+
R
V = IZ = I(R + Ls + 1 Cs ) = I Z e jφ z
L
f Hz
s = jω here
C
ωo = 1 LC
Z ≅ R + jXo (ω − ωo ) for ω ≅ ωo
φz
π/2
ω
0
~100°
ωo
Q=
∆ω
-π/2
∆ω
ωo
Control phase φ in wire by:
reducing ωo → 0° < φ < 90° (lengthen wire)
increasing ωo → -90° < φ < 0° (shortening wire)
Increase Q and ∂φ/∂ω by increasing WT (thinning the wire)
N5
Directivity of parasitic wire antennas
I1
I2
short circuit λ/4 away, so I2 ≅ − I1 and radiated fields cancel
λ/2
∆ << λ/2π
open circuit
Reflectors:
PB
back
I1
PF (forward power)
I2 = I1
~reality
I2
(
forward
PF Wm −2
)
0 λ/4 λ/2
“reflector” is parasitic resonant dipole
λ/2 long (note: reflects at all D)
D
λ
D
PB (D = λ 4 ) ≅ 0
Directors:
I1
I1
PB
d
D
If d ≠ λ/2, then φ ≠ 0
If D,φ ∋ PB ≅ 0 and PF ≠ 0, then parasitic element is “director”
PF
N6
Multiple parasitic wires, Yagi antenna
Choice of di, Di, (i = 1, …N) originally was an art. Now
computers can optimize chosen specifications (e.g.
bandwidth, reactance, directivity)
driven element ≈ λ/2
i=N
di
Di
main beam
directors
(length < λ/2)
reflectors
(length > λ/2)
pattern
P1
Half-wave folded dipole antenna
IA
Io
IB
Zin ≅ ∞ for λ 4
+
-
Therefore ITEM ≅ 0 at all z,
and IA ≅ IB
λ 4
TEM parallel-wire line
Equivalent to:
D≅0
I(z )
+
V (z )
TEM line
I(z )
0
λ/2
V (z )
z
Pt = Io2Rro 2 (single dipole )
Pt = (2Io )2 Rro 2 = 2Io2Rr (folded dipole )
Io
IB
IB ≅ IA
Therefore Rr = 4Rro ≅ 300Ω
IA
“Half-wave folded dipole”
Half - wave dipole Rro ≅ 73Ω
P2
Half-wave folded dipole antenna
Cross-section of TEM “twin lead” line:
“TEM” mode
Common mode
ε ≅ 5εo (say )
(E sees less ε )
E
E
Cu wire
c TEM < c common mode and v ≅ 1 µε , so λ common > λ TEM
Therefore
λ/2 for common mode
to radiate
λ/2 for TEM mode
to force IA = IB
P3
A Balun couples balanced to unbalanced systems
e.g., this is okay
Suppose we want:
mirror
solder joint
mirror
C
A
Io
coax
λ/4
Solution:
λ/4
But current will flow down the
outside of C instead of into B
mirror
D
B
A
B
C
Conductors C and D form λ/4 TEM line shorted at the mirror,
yielding an open circuit at coax end, forcing current into B
P4
Helical antenna
L
e.g.
coax
D=2R
d
mirror
center conductor
If L >> D, standing wave at end is
small because of radiation losses.
Assume ~ TEM propagation
Waves add in phase in
the forward direction if
(2πr )2 + d2 − d = nλ
(If r = d, d = nλ
one 360°
turn of wire
hypotenuse
)
4π2 + 1 − 1
2π r
circumference
d
P5
Helical antenna
L
e.g.
coax
D=2R
d
mirror
center conductor
f-Hz dipole
rotation
I(t)
+ +
≅
-
+
-
-
I(t)
I(t)
I( t )
Long helices have weaker standing waves (less current at end)
P6
Log-periodic antennas
resonant at fo
active part of antenna at fo(d ≅ λ/2)
(moves with frequency)
fo input
long elements
not excited,
due to
radiation
losses
A
Too short to matter, has
a reactive effect.
Pattern, impedance
≠ f(f) (approximately)
Log spiral, radiates
circular polarization
B
P7