EE 543 Theory and Principles of Remote Sensing
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Transcript EE 543 Theory and Principles of Remote Sensing
EE 542
Antennas and Propagation for
Wireless Communications
Array Antennas
1
Array Antennas
• An antenna made up of an array of individual
antennas
• Motivations to use array antennas:
– High gain more directive pattern
– Steerability of the main beam
• Linear array: elements arranged on a line
• 2-D planar arrays: rectangular, square,
circular,…
• Conformal arrays: non-planar, conform to
surface such as aircraft
2
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Radiation Pattern for Arrays
Depends on:
• The type of the individual elements
• Their orientation
• Their position in space
• The amplitude and phase of the current
feeding them
• The total number of elements
3
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Array Factor
The pattern of an array by neglecting the
patterns of the individual elements; i.e.
assume individual elements are isotropic
4
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Linear Receive Array
j
A
+
Receiver
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5
Case 1:
Array Factor for Two Isotropic Sources with Identical
Amplitude and Phase (d = l/2)
P(x,y,z)
z
r1
r
r2
q
(2)
(1)
x
d
(I0,j0)
(I0,j0)
Isotropic sources are assumed for AF calculations. The radiated fields are
uniform over a sphere surrounding the source.
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6
Radiation from an Isotropic Source
E
e
jk r
r
r
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Case 1: Total E Field
E (q , j )
I
o
e
Ioe
jj o
jj o
e
jk r1
r1
Ioe
jj o
e
jk r2
r2
jk r
e jk r
e
r2
r1
1
2
where
r1 r1 ;
r2 r2 ;
r1 r
r2 r
O. Kilic EE 542
d
2
d
2
ˆ
x
ˆ
x
8
Case 1: Far Field Approximation
In the far field, r>>d or (d/r) <<1
r1 r1 r1
1
2
2
d
d
2
ˆ
r 2r x
2
4
1
2
2
ˆ d
r x
d
d
1 d
ˆ
r 1 2 2
r 1 rˆ x
2
r 2
4r
r
4 r
2
1
d
ˆ
r 1 rˆ x
2
r
r
d
2
co s q
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Case 1: Far Field Approximation
Similarly,
r2 r2 r2
1
2
2
d
d
2
ˆ
r 2r x
2
4
1
2
2
2
ˆ d
r x
d
d
1 d
ˆ
r 1 2 2
r 1 rˆ x
2
r 2
4r
r
4 r
1
d
ˆ
r 1 rˆ x
2
r
r
d
2
co s q
Thus, in the far field
r1
r2
r
r
d
2
d
2
co s q
co s q
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10
Case 1: Far Field Geometry
r1
r2
r
r
d
2
d
2
co s q
z
co s q
P(x,y,z)
r1
r
r2
q
(2)
(1)
x
d
If the observation point r is much larger than the separation d, the vectors r1, r
and r2 can be assumed to be approximately parallel. The path lengths from the
sources to the observation point are slightly different.
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11
Case 1: Total E in the Far Field
E (q , j ) I o e
e
jj o
d
jk r co s q
2
d
r 2 co s q
d
Ioe
Ioe
jj o
jj o
2 Ioe
e
jk r
e
jk r
r
jj o
e
d
r 2 cos q
d
jk
co s q
j k 2 co s q
2
e
e
r
r
d
d
jk
co s q
jk 2 c o s q
2
e
e
jk r
r
I oe
e
jj o
d
jk r co s q
2
d
co s k
co s q
2
12
The slight difference in path length can NOT be neglected for the exponential term!!
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Case 1: Total E for d=l/2
Note that d=l/2
E (q , j ) 2 I o e
2 Ioe
jj o
e
jk r
r
jj o
e
jk r
r
2 l
co s
co s q
l 4
co s co s q
2
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Case 1: Array Factor
The array factor is described as the magnitude of E at a
constant distance r from the antenna (i.e. unit V)
A F (q , j ) A F r E (q , j )
2 I o co s co s q
2
q
0
/2
3/2
AFn
0
1
0
1
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Normalized values
14
Case 1: Radiation Pattern
z
q
(2)
(1)
x
(I0,j0)
(I0,j0)
Notice how the two element array is more directive than the single element;
which is an isotropic source.
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Case 2:
Array Factor for Two Isotropic Sources with Identical
Amplitude and Opposite Phase
P(x,y,z)
z
r1
r
r2
q
(2)
(1)
x
d
I1 I 2 I 0
(I1,j1)
(I2,j2)
j1 j 0
j2 j0
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Case 2 – Far Field Geometry
r1
r2
r
r
d
2
d
2
co s q
z
co s q
P(x,y,z)
r1
r
r2
q
(2)
(1)
x
d
I1 I 2 I 0
(I1,j1)
(I2,j2)
j1 j 0
j2 j0
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Case 2: Total E in the Far Field
E (q , j ) I o e
Ioe
Ioe
Ioe
jj o
e
jk r1
Ioe
r1
jj o
jj o
jj o
j j o
e
jk r2
r2
jk r
e jk r
e
j
e
r2
r1
1
2
jk r
e jk r
e
r
r
1
2
1
e
2 jI o e
jk r
r
jj o
2
d
d
jk
co s q
jk 2 co s q
2
e
e
e
jk r
r
d
sin k
co s q
2
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18
Case 2: Radiation Pattern
Note that d=l/2
E (q , j ) j 2 I o e
q AFn
0
1
/2 0
1
3/2 0
jj o
e
jk r
r
sin co s q
2
z
q
(2)
(1)
x
(I0,+j0)
(I0,j0)
Observe how the pattern is rotated compared to Case1 by simply changing the
phase of element 2
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19
Case 3:
Array Factor for Two Isotropic Sources with Identical
Amplitudes and 90o Phase Shift
Homework: Show that:
E (q , j ) 2 I o e
j jo
4
e
jk r
r
co s 1 co s q
4
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q
0
/2
3/2
Case 3
AFn
0
cos(/4)
1
cos(/4)
z
q
(2)
(1)
x
(I0,+j0)
(I0,j0)
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Generalization to N Equally Spaced
Elements
r
dcosq
dcosq
dcosq
q
d
0
d
1
d
2
3
N-1
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General Case for Linear Array
Total E field:
E (q , j ) I o e
e
jj o
jk r1
I1e
r
jj1
e
jk r2
r
I N 1e
jj N 1
e
jk rN 1
r
jk r
r
e
e
jk r
r
I o e
N 1
jj o
I ne
I1e
jj n
e
jj1
e
jk d co s q
I N 1e
jj N 1
e
j ( N 1 ) k d co s q
jn k d co s q
n0
Array Factor:
A F r E (q , j )
N 1
I ne
jj n
e
jn k d co s q
n0
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Special Case (A)
Equally Spaced Linear Array with Linear
Phase Progression
j n n
E (q , j )
N 1
I ne
jn k d co s q
n0
N 1
I ne
jn
n0
w h e re
k d co s q
Fourier series
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Some Observations
* A F ( ) A F ( 2 ) p e rio d ic in (p e rio d = 2 )
* A F is a fu n ctio n o f q o n ly , n o t . (R o t a tio n a l sy m m e try )
* * N o te th a t th e e le m e n t p a tte rn ca n b e a fu n ctio n o f .
* V isib le re g io n : q : 0 - 2
q : 0 - 2
kd kd
2kd
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Special Case (B)
Uniformly Excited, Equally Spaced Linear Array
with Linear Phase Progression
L e t I o I1
E (q , j )
e
jk r
r
In
N 1
Io e
jn
;
k d co s q
n 0
N 1
A F r E (q , j ) I o e
I o 1 e
jn
n 0
Io
1e
1e
jN
j
Io
e
e
j
N
2
j
2
N
j
e
N
2
e 2
e
j
j
2
e 2
e
j
A F Io
j
e
( N 1)
j
2
I e
o
N
s in
2
s in
2
O. Kilic EE 542
j2
j ( N 1)
N
sin
2
sin
2
26
Observations
A F Io
N
s in
2
s in
2
• AF similar to the sinc function (i.e. sinx/x) with a major
difference:
• Sidelobes do not die off for increasing values because
the denominator is a sine function, and does not
increase beyond a value of 1.
• AF is periodic with 2.
• Maximum value (=Io) occurs at 0, 2k.
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N=4 Case
AF
AF (N=4)
Period: 2
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
/2
0
0.5
1
1.5
3/2
nulls
2
2.5
3
3.5
4
4.5
5
5.5
6
6.5
(rad)
AF
sin(y/2)
sin(Ny/2)
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More Observations
A F Io
N
s in
2
s in
2
For all k values except
when y/2 becomes an
integer multiple of
• Zeroes (Nulls) @ N/2 = k ok=2k/N, k=0,1,2, …
• This implies that as N increases there are more sidelobes (i.e. more
secondary null points) in one period.
• Sidelobe widths are 2/N.
• First null at o1=2/N.
• Within one period, N null points N-2 sidelobes (Because we
discard k = N case, which corresponds to the second peak. Also 2
nulls create one sidelobe.)
• This implies that as N increases, the main beam narrows.
• Main lobe width is 2*2/N, twice the width of sidelobes.
• Max value ( = NIo) @ =2k, k=0,1,2, …
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Effect of Increasing N
AF
AF for Different N
12
10
8
6
4
2
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
6.5
y (rad)
N=3
HW: Regenerate this plot.
N=5
N = 10
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Construction of Polar Plot from AF(y)
• The angle is not a physical quantity.
• We are more interested in observing the
AF as a function of angles in real space;
i.e. q, j.
• Since linear arrays are rotationally
symmetric wrt j, we are concerned with q
only.
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Case 1: Construction of Polar Plot
N = 2, d = l/2, = 0 (uniform phase)
Using the general representation from Page 24
A F Io
N
sin
sin
2
Io
2 I o co s
2
sin
sin
2
2
k d co s q ;
0, d l / 2
co s q
co s q
A F 2 I o co s
2
Compare to page 62
z
r
q
Io, j =0
Io, j =0
l/2
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x
32
Normalized AF for Case1
f ( )
A F ( )
A Fm a x
co s
2
co s q
co s
2
Normalized AF, N = 2
Period = 2
2.5
f(Y)
2
1.5
1
0.5
0
-7
-6
2
-5
-4
-3
-2
-1
0
y (rad)
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1
2
3
4
5
6
7
2
33
Normalized AF for Case1 – Polar Plot
Visible range: q: [0-] : [-kd,kd]
= kdcosq = cosq
Visually relate q to
Circle of radius kd
q1
q2
1
2
; x
kd
Normalized AF, N = 2
2.5
f(Y)
f()
2
1.5
f1
1
f2
0.5
0
-7
-6
-5
-4
-3
-2
-1
0
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1
1
2
2
3
34
4
5
6
7
Constructing the Polar Plot
Circle of radius kd
1
2
; x
kd
Normalized AF, N = 2
2.5
f(Y)
f()
2
1.5
f1
1
f2
0.5
0
-7
-6
-5
-4
-3
-2
-1
0
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1
1
2
2
3
35
4
5
6
7
Case 2
N = 2, d = l/2, =
Note: AF() same for all N=2
A F Io
N
sin
2
2 I o co s
2
sin
2
k d co s q
Value of different, depends on , d
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36
Case 2: Polar Format
= kdcosq +
Circle of radius kd
q1
Shifted by
1
kd
; x
Normalized AF, N = 2
2.5
f()
2
1.5
f1
1
0.5
0
-7
-6
-5
-4
-3 -2
-1
00
1
2
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3
4
51
6 2 7
37
Normalized AF for Case 2 – Polar Plot
Circle of radius kd
2
1
0
; x
Normalized AF, N = 2
f(Y)
f()
2.5
2
f2
1.5
f1
1
0.5
0
-7
-6
-5
-4
-3
-2
-1
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0
EE 542
0
1
2
3
4
1
5
2
6
38
7
Shift by
k d co s q
q : [0, ]
: [ k d , k d ]
kd
kd
kd
q1
1
kdcosq1
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Generalize to Arbitrary N
A F Io
N
s in
2
s in
2
kd cos q
Visible Range:
Shift by
: [ k d , k d ]
2kd
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General Rule
• AF plot with respect to is identical for all
cases with identical N.
• The polar plot is determined by shifting the
unit circle by , the linear phase
progression amount.
• Visible range is always the 2kd range
centered around that point.
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Shift and construct
Observe the dependence of main beam direction on , the phase progression.
Main beam
qpeak
cos(qpeak) = /kd
q1
- kd
Normalized AF, N = 2
1
f()
6
+ kd
5
4
3
2
f1
1
0
-6
-5
-4
-3
-2
-1
0
1 O. Kilic2 EE 5423
4
5
6
42
Shift and construct
Observe the dependence of main beam direction on , the phase progression.
Main beam
- kd
Normalized AF, N = 2
f()
6
+ kd
1
5
4
3
2
1
0
-6
-5
-4
-3
-2
-1
0
1 O. Kilic2 EE 5423
4
5
6
43
Array Pattern vs kd
• If kd > 2; i.e. d>l/2 multiple peaks can
occur in the visible range. These are
known as grating lobes, and are often
undesirable.
• Why??
– Will cause reduced directivity as power will be
shared among all peaks
– Likely to cause interference
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Grating Lobes
Three main beams.
, x
kd
-kd
N=5
5
4
f(Y)
3
2
1
0
-15
-10
-5
0
5
10
15
, x
Y (rad)
45
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Pattern Multiplication
• So far only isotropic elements were considered.
• Actual arrays are made up of nearly identical
antennas
• AF still plays a major role in the pattern
F (q , j ) e(q , j )f (q , j )
Normalized
Array Pattern
Normalized
element
pattern
Normalized Array factor
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Validation with Dipoles
• Consider the case of an ideal dipole array
as below.
r
dcosq
dcosq
q
d
I0
d
I1
d
I2
I3
(N-1)d
47
0
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Sum of the E fields
For the center dipole, assuming z << l
z / 2
ˆI
A z
z / 2
e
jk R
4 R
d z ' z Az I
e
jk r
4 r
z
E E q qˆ jw sin q A z
E sin q A z e q
sin q
Normalized pattern
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Vector Potentials for Each Dipole
e
Az I0
4 r
0
A z I1
e
1
Az
m
jk r
Im
z
jk r
4 r
e
e
jk d co s q
z
jk r
4 r
e
jm k d co s q
z
w h e re
Rm
r m d co s q
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Total Vector Potential
Az
N 1
Az
m 0
e
jk r
4 r
e
m
z I 0 I 1 e
jk r
4 r
z
N 1
Ime
jk d co s q
I N 1e
jk ( N 1 ) d co s q
jk m d co s q
m 0
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Total E field
Eq
jw sin q A z
jw sin q
E q sin q
N 1
e
jk r
4 r
Ime
z
N 1
Ime
jk m d co s q
m 0
jk m d co s q
m 0
Array factor
Array pattern
Normalized element pattern
F (q , j ) e(q , j )f (q , j )
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51
Directivity of Linear Arrays
D (q , j )
U (q , j )
Uo
4 U
Pra d
where
U (q , j ) r
Pra d
2
S av
S av . d s
4
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Radiation Intensity
1
E (q , j ) E o f (q , j )e (q , j )
S a v rˆ
rˆ
U (q , j ) r
1
2
1
2
2
E (q , j )
Eo
2
S av
2
f (q , j )
1
2
r
Eo
2
2
e (q , j )
f (q , j )
2
2
1
r
2
e (q , j )
2
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Total Radiated Power
Pra d
S av . d s
4
2
Eo
2 r
Eo
2
f (q , j )
2
2
e (q , j ) r rˆ.rˆ sin q d q d j
2
4
2
2
f (q , j )
2
2
e (q , j ) sin q d q d j
4
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Directivity
D (q , j )
4 f (q , j )
f (q , j )
2
e (q , j )
2
2
2
e (q , j ) sin q d q d j
4
A
4 f (q , j )
2
e (q , j )
2
A
f (q , j )
2
2
e (q , j ) sin q d q d j
4
Dm ax
4
A
4
f (q , j )
2
2
e (q , j ) sin q d q d j
4
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Directivity for Arrays with Isotropic
Elements
• Easier to calculate
• Represents an approximate solution for elements
with broad patterns
• Uniform amplitude and equal spacing will be
assumed.
f (q , j )
2
e (q , j )
2
sin ( N / 2)
2
N sin ( / 2)
1
N
2
N
2
N 1
Using
sin( a b ) sin a cos b cos a sin b
( N m ) co s( m )
m 1
1
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Directivity:
Isotropic Elements, Linear Phase Progression,
Uniform Spacing, Uniform Amplitude
2
A
0
A
kd
2
d j f (q ) sin q d q 2
2
kd
0
kd
kd
2
1
f ( )
d
kd
2
f ( ) d
kd
2 1
k d N
2
kd
d
kd
N
2
N 1
N
m 1
m
kd
kd
co s m d
Dm ax
4
N
4
A
4
N
2
N 1
N m
m 1
m kd
2 co s m sin m k d
1
1
N
2
N
2
N 1
N m
m 1
m kd
c o s m sin m k d
O. Kilic EE 542
57
Non-Uniformly Excited Linear
Arrays
• We have seen the effects of phase shifting
on the beam direction.
• We can also shape the beam and control
the level of sidelobes by adjusting the
amplitude of the currents in an array.
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O. Kilic EE 542
Array Factor for Non-Uniform
Excitation
AF
N 1
Ime
jm
;
k d co s q
m 0
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Can we eliminate the sidelobes???
• Yes!
• First consider the 1x2 element array as in
case 1 we studied.
• Recall that the AF did not have any
sidelobes
AF = |1+ejj| = 1Z
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Binomial Series Coefficients
• If the amplitudes are equal to the coefficients of
the binomial series, no sidelobes.
• Consider the array factor, which is the square of
Case 1:
• AF = (1+Z)(1+Z)=1 + 2Z + Z2
• This corresponds to a three element array with
current amplitudes in the ratio of 1:2:1
• Since this array factor is simply the square of an
array factor with no sidelobes there are no
sidelobes.
61
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2-Dimensional Arrays
• The elements lie on a plane instead of a
line.
• Many geometric shapes are possible;
circle, square, rectangle, hexagon, etc.
• Will consider rectangular arrays
62
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Rectangular Array Geometry
z
r
q
rmn
y(n)
rmn
dx
j
dy
x(m)
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Individual Fields
E m 0 ,n 0 q , e q , I 00 e
E m n q , e q , I m ne
jj 0 0
e
jk r
r
jj m n
e
ik rm n
rm n
rm n r r m n
rm n r r m n
r
2
2 r .r m n r m n
2
r ˆ.
r r mn
E m n q , e q , I m ne
jj m n
e
ik r
r
e
jk . r m n
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O. Kilic EE 542
Total Field
E q ,
M
N
E m n q ,
m M nN
e q ,
e
ik r
r
M
N
I m ne
jj m n
e
jk . r m n
m M nN
Array factor
Element pattern
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O. Kilic EE 542
Array Factor
Sa
M
N
I m ne
jj m n
e
jk . r m n
m M nN
ˆ kyy
ˆ kzz
ˆ
k kxx
ˆ ndyy
ˆ
r mn m d x x
Sa
M
N
I m ne
jj m n
e
jm k x d x
e
jn k y d y
m M nN
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O. Kilic EE 542
Array Factor:
Linear Phase, Uniform Amplitude
j m n m x n y
Imn I0
S a I0
M
N
e
j m x n y
e
jm k x d x x
e
jm k x d x
e
jm k y d y
m M nN
I0
M
N
e
jn k y d y y
e
jn k y d y y
m M nN
I0
M
m M
e
jm k x d x x
N
nN
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O. Kilic EE 542
Factors of Planar AF
Sa SxSy
Sx
N
e
jn k x d x x
e
jn k y d y y
nN
Sy
N
nN
k x k sin q co s j
k y k sin q sin j
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O. Kilic EE 542
Homework, Problem 1
Show that the Array Factor for two isotropic sources
with identical amplitudes and 90o phase shift is given by
E (q , j ) 2 I o e
j jo
4
e
jk r
r
co s 1 co s q
4
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O. Kilic EE 542
HW Problem 2
• Construct by hand after plotting the AF for N=4, =
/2, d = l/2
• Hint: The AF vs should look like this:
AF
AF (N=4)
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
6.5
(rad)
AF
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O. Kilic EE 542
References
• Stutzman, et. al. “Antenna Theory”
provides an excellent discussion on array
antennas!!!
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O. Kilic EE 542