NUS/ECE
EE4101
Antenna Arrays
1 Introduction
Antenna arrays are becoming increasingly important in
wireless communications. Advantages of using antenna
arrays:
1. They can provide the capability of a steerable beam
(radiation direction change) as in smart antennas.
2. They can provide a high gain (array gain) by using
simple antenna elements.
3. They provide a diversity gain in multipath signal
reception.
4. They enable array signal processing.
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An important characteristic of an array is the change of its
radiation pattern in response to different excitations of its
antenna elements. Unlike a single antenna whose radiation
pattern is fixed, an antenna array’s radiation pattern, called
the array pattern, can be changed upon exciting its elements
with different currents (both current magnitudes and current
phases). This gives us a freedom to choose (or design) a
certain desired array pattern from an array, without changing
its physical dimensions. Furthermore, by manipulating the
received signals from the individual antenna elements in
different ways, we can achieve many signal processing
functions such as spatial filtering, interference suppression,
gain enhancement, target tracking, etc.
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2 Two Element Arrays
z

Far field
observation
point
r1
Dipole 2
I 2  Ie j
d
Dipole 1

r
I1  I
r1  r  d cos  , 0    
x
Two Hertzian dipoles of length dℓ separated by a distance
d and excited by currents with an equal amplitude I but a
phase difference  [0 ~ 2).
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E1 = far-zone electric field produced by antenna 1 =
E2 = far-zone electric field produced by antenna 2 =
aˆ  E1
aˆ  E2
 kI1d   e jkr    
 kd   e  jkr 
sin  θ    j
cos θ I1
E1  j




4  r  
2
4  r 
 kI 2 d   e  jkr    
 kd   e jkr 
sin  θ    j
cos θ I 2
E2  j




4  r1  
2
4  r1 
1
1
Use the following far-field approximations:
0
1 1

r1 r
e  jkr1  e  jk  r d cos 
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The total E field is:
E   E1  E2  aˆ 
 kd   e jkr 
jkd cos 


 aˆ  j
θ
I

I
e
cos
1
2




4  r 
 kI1d   e jkr 
j  jkd cos 


 aˆ  j
cos θ 1  e e


4  r 
 kId   e jkr 
 aˆ  j
cos θ AF


4  r 
where
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AF  Array Factor  1  e j e jkd cos 
j
1  e e
jkd cos 
  2e
j
1
   kd cos  
2
1

cos     kd cos  
2

The magnitude of the total E field is:

1

I1  I 2e jkd cos
I1

 jkr

 kId  e 
E  aˆ  j
cos θ AF


4  r 
 radiation pattern of a single Hertzian dipole  AF
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Hence we see the total far-field radiation pattern |E| of the
array (array pattern) consists of the original radiation pattern
of a single Hertzian dipole multiplying with the magnitude
of the array factor |AF|. This is a general property of
antenna arrays and is called the principle of pattern
multiplication.
When we plot the array pattern, we usually use the
normalized array factor which is:
1
1
1

AFn  AF  2cos     kd cos  


2

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7
where  is a
constant to
make the
maximum value
of |AFn| equal
to one.
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Examples of array patterns using pattern multiplication:
Array pattern of a two-element array of Hertzian dipoles ( = 0°, and d = /4)
1
1
1
1  




AFn  2cos     kd cos    2cos   cos  

2
 

2  2
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Array pattern of a two-element array of Hertzian dipoles ( = -90°, and d = /4)
1
1
1
1   



AFn  2cos     kd cos    2cos     cos  

2
 

2 2 2
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In many practical arrays, the element radiation pattern is
usually chosen to be non-directional, for example the -plane
pattern of a Hertzian dipole or a half-wave dipole. Then in
this case, the array radiation pattern will be totally
determined by the array factor AF alone, as shown in the
example below:

Element pattern
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
=
|F()| Array factor |A()| = Array pattern (normalized)
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3 N-Element Uniform Liner Arrays (ULAs)
Dipoles are
parallel to the z
direction
Far field
observation
point
y
r

Dipole 1
rN-1
x
Dipole N
d
An N-element uniform antenna array with an element separation d
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The principle of pattern multiplication can be extended to
N-element arrays with identical antenna elements and
equal inter-element separation (ULAs). If the excitation
currents have the same amplitude but the phase difference
between adjacent elements is  (the progressive phase
difference), the array factor for this array is:
AF  1  e j ( kd cos    )  e j 2( kd cos    )    e j ( N 1)( kd cos    )


sin  N 

N

1
j
N


 2 e
2
  e j ( n1) 


n 1
sin  
2
where   kd cos    and 0   ,   2
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The normalized array factor is:


sin  N 
1
 2
AFn   
 sin  
 
2
where  is a constant to
make the largest value of
|AFn| equal to one. Note
that  is not necessarily
equal to N.
The relation between |AFn|, , d, and  is shown graphically
on next page. Note that |AFn| is a period function of ,
which is in turn a function of . The angle  is in the real
space and its range is 0 to 2. However,  is not in the real
space and its range can be greater than or smaller than 0 to
2, leading to the problem of grating lobes or not achieving
the maximum values of the |AFn| expression.
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|AFn( )|
 = kd cos +
kd


kdcos
The relation between AFn,, d, and 
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Properties of the normalized array factor AFn:
1. |AFn| is a periodic function of , with a period of 2.
This is because
|AFn( + 2)| = |AFn()|.
2. As cos() = cos(-), |AFn| is symmetric about the line of
the array, i.e.,  = 0 & . Hence it is enough to know
|AFn| for 0    .
3. The maximum values of |AFn| occur when (see
Supplementary Notes):
 1
 (kd cos    )   m , m  0,1,2,
2 2
1  
(   2m ) 
 max  main beam directions  cos 
 2 d

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Note that there may be more than one angles max
corresponding to the same value of m because cos-1(x) is
a multi-value function. If there are more than one
maximum angles max, the second and the subsequent
maximum angles give rise to the phenomenon of grating
lobes. The condition for grating lobes to occur is that d 
 (disregarding the value of ) as shown below:
2nd grating lobe
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1st grating lobe
Main lobe
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1st grating lobe
2nd grating lobe
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2nd grating lobe 1st grating lobe
1st grating lobe
Main lobe
2nd grating lobe
|AFn( )|

(1) When d  0.5, no
grating lobes can be
formed for whatever
value of . (2) When d
 , grating lobe(s) is
(are)
formed
for
whatever value of .
(3) When 0.5 <d< ,
formation of grating
lobes depends on .
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Visible region
kd

=kdcos
17
General conditions to
avoid grating lobes
with  ϵ [0,2] and d ϵ
[0.5,]:
1.For 0   < , the
requirement is:
kd +   2
2. For    < 2, the
requirement is:
kd -   0
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4. There are other angles corresponding to the maximum
values for the minor lobes (minor beams) but these angles
cannot be found from the formula in no. 3 above.
5. When  and d are fixed, it is possible that  can never be
equal to 2m. In that case, the maximum values of |AFn|
cannot be determined by the formula in no. 3.
6. The main beam directions max are not related to N. They
are functions of  and d only.
7. The nulls of |AFn| occur when:
n

n  1,2,3,
 ,

N
2
n  N ,2 N ,3 N ,
2n 
1  
 null  null directions  cos 
(    ) 
N 
 2 d
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Note that there may be more than one angles null
corresponding to a single value of n because cos-1(x) is a
multi-value function.
8. The null directions null are dependent on N.
9. The larger the number N, the closer is the first null (n = 1)
to the first maximum (m = 0). This means a narrower
main beam and an increase in the directivity or gain of the
array.
10.The angle for the main beam direction (m = 0) can be
controlled by varying  or d.
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Example 1
A uniform linear array consists of 10 half-wave dipoles with
an inter-element separation d = /4 and equal current
amplitude. Find the excitation current phase difference 
such that the main beam direction is at 60 (max = 60).
Solutions d = /4, max = 60, N = 10
main beam dirction  max


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
2





m
2

 60  cos  



 2 d

1
   m2   cos  60   0.5

    m2  45  360  315,
4
20
when m  1
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Other values of  corresponding to other values of m are
outside the range of 0    2 and are not included.
 


sin 5  cos     
1
2



AFn 

1  


sin   cos     

2 2
  10
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4 Mutual Coupling in Transmitting Antenna Arrays
What we studies before about antenna arrays has assumed
that the antenna elements operate independently. In
reality, antennas placed in close proximity to each other
interact strongly. This interaction is called mutual
coupling effect and it will distort the array characteristics,
such as the array pattern, from those predicted based on
the pattern multiplication principle. We need to
consider the mutual coupling effect in order to apply the
pattern multiplication principle.
We study an example of a two-element dipole array. We
characterize the mutual coupling effect using the mutual
impedance.
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Consider two transmitting antennas as shown on next
page. They are separated by a distance of d and the
excitation voltage sources, Vs1 and Vs2, have a phase
difference of  but an equal magnitude. Hence if there is
no mutual coupling effect, the excitation currents also
differ by a phase difference of  and have an equal
magnitude. When the mutual coupling effect is taken
into account, the two coupled antennas can be modelled
as two equivalent circuits. Now because of the mutual
coupling effect, there is another excitation source (the
controlled voltage source) in the equivalent circuit. This
controlled voltage source is to model the coupled voltage
from the other antenna.
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Dipoles are
parallel to the z
direction
Far field
observation
point, r
y

Dipole 1
x
d
Dipole 2
Two coupled dipoles
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Antenna 2
Antenna 1
Vs1
Terminal
current
I1
Zg1
Excitation
voltage
source
Coupled
voltage
I1 a
1

b1
I2 a2
Vs2 
V12

Antenna
Self-impedance
25
V21
Z22
Zg2
Z11
Zg1
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a2
b2
d
Vs1 
Source
internal
Impedance
Vs2 
Zg2
a1
b1

I2
b2
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Z12  mutual impedance with antenna 2 excited
V12

I2
coupled voltage across antenna 1's open-circuit terminal

excitation current at antenna 2's shorted terminal
Voc12

I 2 I1 0,Vs1 0
Voc12  a
I2
b
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c
d
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Z 21  mutual impedance with antenna 1 excited
V21

I1
coupled voltage across antenna 2's open-circuit terminal

excitation current at antenna 1's shorted terminal
Voc 21

I1 I 2 0,Vs 2 0
I1
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a
b
Voc21  c
d
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Note that for
passive antennas,
Z12 = Z21
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Using the mutual impedance, the coupled voltages V12
and V21 can be expressed as follows:
V12  Z12 I 2
V21  Z21I1
I1 and I2 are the actual terminal currents at the
antennas when there is mutual coupling effect. From
the antenna equivalent circuits,
Vs1  V12
I1 
Z g 1  Z11
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Vs 2  V21
I2 
Z g 2  Z 22
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Is1 and Is2 are the terminal currents at the antennas when
there is no mutual coupling effect.
Vs1
I s1 
Z g1  Z11
Vs 2
I s2 
Z g 2  Z11
Our aim is to express I1 and I2 in terms of Is1 and Is2.
Vs 2  V21
I2 
Z g 2  Z 22
Vs1  V12
I1 
Z g 1  Z11
I1Z 21
 I s2 
Z g 2  Z 22
I 2 Z12
 I s1 
Z g1  Z11
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From these two relations, we can find:
I1 
Z12
I s1 
I s2
Z11  Z g1


Z12 Z 21
1 

  Z11  Z g1  Z 22  Z g 2  
, I2 
Z 21
I s2 
I s1
Z 22  Z g 2


Z12 Z 21
1 

  Z11  Z g1  Z 22  Z g 2  
That is:
1
I1   I s1  Z12 I s 2 
D
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1
 I s1 
I 2   I s 2  Z 21
D
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where
EE4101
Z12 Z 21
D  1
 Z11  Z g1  Z 22  Z g 2 
Z12
Z12 
Z11  Z g1
Z 21
 
Z 21
Z 22  Z g 2
Now if we want to find the array pattern E on the
horizontal plane (=/2) with mutual coupling effect, then
E is just equal to the array factor (see pages 10 and 6).
1
jkd cos 
Vector

E =AF   I1  I 2e
magnitude, not
I1
absolute value
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1
E   I1  I 2e jkd cos  
I1
1
 I s1  e jkd cos  
 I s1  Z12 I s 2    I s 2  Z 21

I1D
1
 I s1  I s 2e jkd cos    Z12  I s 2  I s1e jkd cos   

I1D
 with Z12  Z 21 

I s1
Is2
j  jkd cos 
j
jkd cos 
j 
1  e e
  with

 Z12  e  e
e 


I1D
I s1


I s1
j  kd cos    
j  kd cos    
j





 Z12e 1  e
1 e



I1D 



original pattern
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additional pattern
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We see that the array pattern now consists of two parts:
the original array pattern plus an additional pattern:
Z12 e j 1  e j kd cos     
which has a reverse current phase - and a modified
amplitude with a multiplication of a complex number
Z’12ej. Note that all parameters in the above formula
can be calculated except I1 which will be removed after
normalization. Normalization of the above formula can
only be done when its maximum value is known, for
example by numerical calculation.
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Absolute value
Example 2
Find the normalized array pattern |En| on the horizontal plane
(=/2) of a two-monopole array with the following
parameters with mutual coupling taken into account:
I s1  1, I s 2  e j , I s1  I s 2  1,   150
d   4,    4
Z12  Z 21  21.8 - j 21.9 Ω
Z11  Z 22  47.3  j 22.3 Ω
Z g1  Z g 2  50 Ω
d

I s1
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I s2
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Solution
I s1  1,
Is2  e
I s1  I s 2  1
I s1  0,
j
As the required array pattern |En| is on
the horizontal plane, it is equal to the
normalized array factor |AFn|.
I s 2    150  2.62 rad
2  
kd 

 4 2
Z12
Z 21
Z12 

 0.16  j 0.26
Z11  Z g 1 Z 22  Z g 2
Z12 Z 21
D  1
 1.042  j 0.09
 Z11  Z g1  Z 22  Z g 2 
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

I s1
1  e j kd cos       Z12 e j 1  e j kd cos     
E  AF 
I1D
0.95  j 0.08
1  e j 2.62e j 2  cos 

I1


  0.16  j 0.26  e j 2.62 1  e  j 2.62e j 2 cos  
0.94  j 0.37
j  2  cos 

1   1.14  j 0.40  e
I1
The pattern of f  1   1.14  j 0.40  e j 2  cos 
next page.
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is shown on
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f  1   1.14  j 0.40  e j 2  cos 
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Normalization
The pattern of f attains the maximum value when  = 180°.
When  = 180°,
E  180
0.94  j 0.37
j  2  cos 

1   1.14  j 0.40  e
 180
I1
1.83

I1
Hence we normalize |E| by this factor (1.83/|I1|) to get:
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0.94  j 0.37
1   1.14  j 0.40  e j 2 cos  
I1
En 
1.83
I1
 0.52 1   1.14  j 0.40  e
j  2  cos 
The polar plot of |En| is shown on next page.
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En  0.52 1   1.14  j 0.40  e
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j  2  cos 
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The case when there is no mutual coupling is shown below
for comparison.
1
En no mutual coupling effect  1  e j e jkd cos 

where  is a constant to make
the largest value of |AFn|
equal to one ( = 1.73).
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References:
1. C. A. Balanis, Antenna Theory, Analysis and Design, John Wiley
& Sons, Inc., New Jersey, 2005.
2. W. L. Stutzman and G. A. Thiele, Antenna Theory and Design,
Wiley, New York, 1998.
3. David K. Cheng, Field and Wave Electromagnetic, AddisonWesley Pub. Co., New York, 1989.
4. John D. Kraus, Antennas, McGraw-Hill, New York, 1988.
5. Fawwaz T. Ulaby, Applied Electromagnetics, Prentice-Hall, Inc.,
New Jersey, 2007.
6. Joseph A. Edminister, Schaum’s Outline of Theory and Problems
of Electromagnetics, McGraw-Hill, Singapore, 1993.
7. Yung-kuo Lim (Editor), Problems and solutions on
electromagnetism, World Scientific, Singapore, 1993.
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Antenna Arrays