SUPERPOSITION OF PHASORS; ANTENNA ARRAYS/APERTURES
Superposition of Waves, Applications:
Multiple waves launched by antenna
(Antennas designed to yield desired pattern)
+
-
+
-
+
-
Multiple reflections intercepted by receiver
(Multipath)
Multiple points of interception in receiver antenna
(Same as transmitting antenna, time reversed)
A
B
cos ωt + cos (ωt + φ) = A cos(ωt + θ) C
[= 2 cos (ωt + φ/2) cos (φ/2)]
Note: fields add, powers do not
A+B
t
A+C
D
A+D
L6-1
SUPERPOSITION OF PHASORS
Graphical Method:
A(t) = Re {Ae jωt } = Re {Re {A} + jIm {A} [cos ωt + jsin ωt ]}
= Re {A}cos ωt − Im {A}sin ωt
Im{A}
Superposition of Phasors:
Im{A}
j
Sin
Cos
ωt
Sin
Cos
ejωt
Re{A}
2 + j = (12 + 22)0.5 ejβ ⇒
2cos ωt -sin ωt = (12 + 22)0.5cos(ωt + β)
(“phase lead” of β)
β
Re{A}
Im{A}
i
B = ∑ i A i e j φi
0
φi
1 • e jπ 2 → A i = 1 ,
φi = π 2
Re{A}
L6-2
ANTENNA ARRAYS
Superposition of Waves:
E(r, θ, φ) = ∑ ai Eie− jkri
+
-
i
+
-
= E  ∑ aie− jkri  = (element factor E ) (array factor )


 i

E(r,θ,φ) can be factored if elements have the same orientation and pattern
(so Ei = E), but different locations ri, and amplitudes and phases ai, where ai
characterizes the currents driving each radiating element
Example, horizontal arrays of vertical dipoles:
In phase:
z
x
y At 30° relative power = 2
(90° out of phase)
1
λ/2 0
Array factor 1
λ/4
30°
x
Relative power in
x direction = 4
y
x
180° out of phase:
90° out of phase,
y relative power =2
Element λ/2
factor
Relative power in
y direction = 4
30°
λ/4
x
L6-3
TWO-ELEMENT ARRAYS
In-Phase, 2λ
λ Separation:
90° phase, λ/4 separation
y
null
y
3λ/2
λ/4
null
θ = cos-1(3/4)
x
λ/2
1
x
j (phase lead 90°)
2λ
180° phase, λ/2 separation
unequal sources
y
Nulls at:
θ = cos-1([λ/2]/2λ)
θ = cos-1(3λ/2]/2λ)
Peaks at:
θ =0
θ = cos-1(λ/2λ)
= π/3
λ/2
a
b
x
2 unequal sources
cannot produce a null
L6-4
LINEAR ANTENNA ARRAYS
D
Linear uniformly excited array:
θnull = sin λ/2L = ~λ/D
x
L
Phasor for Uniform 8-Element Array:
Im{A}
λ/2
Im{A}
ωt
Sin
null
Peak power ∝ 82
0
Re{A}
Cos
45o
Im{A}
45o
First null
D
Second null
First null
Im{A}
y
x
λ/2 ⇒ φ=1800
Peak ∝ 82
Re{A}
x
0
0
First null
Re{A}
θ
0
90o
Re{A}
G(θ) ∝ E(r, θ, φ)
2
z
θfirst null = sin-1[(λ/2)/(4D/7)]
First sidelobe
Second null
L6-5
FREQUENCY REUSE
Cell phones:
Frequency allocations are limited
Therefore multiple narrowband signals or other
orthogonal signals are used (e.g., TDM)
Base station antennas enable frequency reuse
All frequencies, α = 0, V i = V o e jmα
All frequencies, α = π
Element factor E(θ)
Typical 4-antenna face
Most clients can access non-faded frequencies
Assume 3λ
L6-6
MULTIPATH EFFECTS
Fading, scintillation: Causes of fading:
aEe− jφ
N paths
L
E
Moving reflectors (trucks, trees) vary φ
Moving sources and receivers
Variations in c due to temperature, humidity
Polarization rotation or variation
Examples:
FM radios in moving cars click during nulls below FM threshold
Snowy TV broadcast stations waiver as planes fly overhead or trucks pass
Strength of radio stations varies with the weather (also due to refraction)
Ionosphere (faraday) rotates linear polarization ~
< 3 GHz, causing fades
Doppler shift ∆f:
If the path L to the source increases at v = ∂L/∂t, we lose
fD = (∂L/∂t)/λ cycles per second = v/λ = fov/c Hz, so fD = fo(1 - v/c) Hz
Remedies:
Doppler shift – retune receiver
Fading – high sensitivity and dynamic range; error-correction codes;
space, frequency, polarization diversity
L6-7
DIFFRACTION
Uniformly Illuminated Aperture:
B=
x
jφ
jrˆ • x
dx
ˆ ∫ A(x)e
∑i Ai e i →
x
45o
First null
x
D
E(r, θ, φ)
θ
2
z
θfirst null = sin-1[(λ/2)/(4D/7)]
0
λ/2 ⇒ φ=1800
x'
Planar Aperture:
x
θfirst null = sin-1(λ/D)
y'
θ
r'
r
z'
θB
z
D
λ/2 ⇒ φ=180°
θB(eye) ≅ sin-1(λ/D) ≅ 0.5×10-6/10-3 ≅ 1 arc min
Uniformly Illuminated Aperture A:
(
)
5×10-4 radians
Eff ≅ θˆ jke− jkr 2πr cos θ ∫ E x (x ',y ')e jkrˆ •r ' da'
A
(~Fourier transform)(r is distance from origin to ff)
L6-8