lecture 14: linear array theory - part ii linear arrays
TRANSCRIPT
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LECTURE 14: LINEAR ARRAY THEORY - PART II
(Linear arrays: Hansen-Woodyard end-fire array, directivity of a linear array,
linear array pattern characteristics – recapitulation; 3-D characteristics of an
N-element linear array.)
1. Hansen-Woodyard End-fire Array (HWEFA)
The end-fire arrays (EFA) have relatively large HPBW as compared to
broadside arrays.
[Fig. 6-11, p. 270, Balanis]
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To enhance the directivity of an end-fire array, Hansen and Woodyard
proposed that the phase shift of an ordinary EFA
kd = (14.1)
be increased as
2.94
kdN
= − +
for a maximum at 0 = , (14.2)
2.94
kdN
= + +
for a maximum at 180 = . (14.3)
Conditions (14.2)–(14.3) are known as the Hansen–Woodyard conditions for
end-fire radiation. They follow from a procedure for maximizing the directivity,
which we outline below.
The normalized pattern AFn of a uniform linear array is
( )
( )
sin cos2
cos2
n
Nkd
AFN
kd
+
+
(14.4)
if coskd = + is sufficiently small (see previous lecture). We are looking
for an optimal , which results in maximum directivity. Let
pd = − , (14.5)
where d is the array spacing and p is the optimization parameter. Then,
( )
( )
sin cos2
cos2
n
Ndk p
AFNd
k p
−
=
−
.
For brevity, use the notation / 2Nd q= . Then,
sin ( cos )
( cos )n
q k pAF
q k p
−=
−, (14.6)
or sin
n
ZAF
Z= , where ( cos )Z q k p= − .
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The radiation intensity becomes
22
2
sin( ) n
ZU AF
Z = = , (14.7)
2sin ( )
( 0)( )
q k pU
q k p
−= =
− , (14.8)
2( ) sin
( )( 0) sin
n
U z ZU
U z Z
= =
= , (14.9)
where
( )z q k p= − ,
( cos )Z q k p= − , and
( )nU is normalized power pattern with respect to 0 = .
The directivity at 0 = is
0
4 ( 0)
rad
UD
P
== (14.10)
where ( )rad nP U d
= . To maximize the directivity, 0 / 4radU P = is
minimized.
22
0
0 0
1 sinsin
4 sin
z ZU d d
z Z
=
, (14.11)
22
0
0
sin ( cos )1sin
2 sin ( cos )
q k pzU d
z q k p
− =
− , (14.12)
2
0
1 cos2 1 1Si(2 ) ( )
2 sin 2 2 2
z zU z g z
kq z z kq
− = + + =
. (14.13)
Here, ( )0
Si (sin / )z
z t t dt= . The minimum of ( )g z occurs when
( ) 1.47z q k p= − − , (14.14)
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( ) 1.472
Ndk p − − .
1.47, where2 2
Ndk Ndpdp − − = −
( ) 1.472
Ndk + −
2.94 2.94
kd kdN N
− − = − +
. (14.15)
Equation (14.15) gives the Hansen-Woodyard condition for improved directivity
along 0 = . Similarly, for 180 = ,
2.94
kdN
= + +
. (14.16)
Usually, conditions (14.15) and (14.16) are approximated by
kdN
+
, (14.17)
which is easier to remember and gives almost identical results since the curve
( )g z at its minimum is fairly flat.
Conditions (14.15)-(14.16), or (14.17), ensure maximum directivity in the
end-fire direction. There is, however, a trade-off in the side-lobe level, which is
higher than that of the ordinary EFA. Besides, conditions (14.15)-(14.16) have
to be complemented by additional requirements, which would ensure low level
of the radiation in the direction opposite to the main lobe.
(a) Maximum at 0 = [reminder: coskd = + ]
0
0180
2.94
2.94
2.942 .
Nkd
Nkd
N
=
= =
= −
= − +
= − −
(14.18)
Since we want to have a minimum of the pattern in the 180 = direction, we
must ensure that
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180| | = . (14.19)
It is easier to remember the Hansen-Woodyard conditions for maximum
directivity in the 0 = direction as
0 180
2.94| | , | |
N N
= = = . (14.20)
(b) Maximum at 180 =
180
1800
2.94
2.94
2.942 .
Nkd
Nkd
N
=
= =
=
= +
= +
(14.21)
In order to have a minimum of the pattern in the 0 = direction, we must ensure
that
0| | = . (14.22)
We can now summarize the Hansen-Woodyard conditions for maximum
directivity in the 180 = direction as
180 0
2.94| | , | |
N N
= = = . (14.23)
If (14.19) and (14.22) are not observed, the radiation in the opposite of the
desired direction might even exceed the main beam level. It is easy to show (use
the relation 2 /kd N + ) that the complementary requirement | | = at the
opposite direction can be met if the following relation is observed:
1
4
Nd
N
−
. (14.24)
If N is large, / 4d . Thus, for a large uniform array, Hansen-Woodyard
condition can yield improved directivity only if the spacing between the array
elements is approximately / 4 .
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ARRAY FACTORS OF A 10-ELEMENT UNIFORM-AMPLITUDE HW EFA
Solid line: / 4d =
Dash line: / 2d =
N = 10
kdN
= − +
Fig. 6.12, p. 273, Balanis
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2. Directivity of a Linear Array
2.1. Directivity of a BSA
Using the approximate expression for the AF, the normalized radiation intensity
is obtained as
2
22
sin cossin2
( )
cos2
n
Nkd
ZU AF
N Zkd
= = =
(14.25)
0 0
0 4rad av
U UD
P U= = , (14.26)
where / (4 )av radU P = . The radiation intensity in the direction of maximum
radiation / 2 = in terms of nAF is unity:
0 max ( / 2) 1U U U = = = = ,
10 avD U − = . (14.27)
The radiation intensity averaged over all directions is calculated as
2
2 2
20 0 0
sin cos1 sin 1 2
sin sin4 2
cos2
av
Nkd
ZU d d d
NZkd
= = .
Change variable:
cos sin2 2
N NZ kd dZ kd d = = − . (14.28)
Then,
22
2
1 2 sin
2
Nkd
av
Nkd
ZU dZ
N kd Z
−
= −
, (14.29)
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22
2
1 sin
Nkd
av
Nkd
ZU dZ
Nkd Z−
=
. (14.30)
The function 1 2( sin )Z Z− is a relatively fast decaying function as Z increases.
That is why, for large arrays, where / 2Nkd is big enough ( )20 , the integral
(14.30) can be approximated by
21 sin
av
ZU dZ
Nkd Z Nkd
−
=
, (14.31)
0
12
av
Nkd dD N
U
= =
. (14.32)
Substituting the length of the array ( )1L N d= − in (14.32) yields
0 2 1
N
L dD
d
+
. (14.33)
For a large array ( )L d ,
0 2 /D L . (14.34)
2.2. Directivity of ordinary EFA
Consider an EFA with maximum radiation at 0 = °, i.e., kd = − .
( )
( )
2
22
sin cos 1sin2
( )
cos 12
n
Nkd
ZU AF
N Zkd
−
= = = −
, (14.35)
where (cos 1)2
NZ kd = − . The averaged radiation intensity is
2 22
0 0 0
1 sin 1 sinsin sin
4 4 2
radav
P Z ZU d d d
Z Z
= = =
.
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Since
(cos 1)2
NZ kd = − and sin
2
NdZ kd d = − , (14.36)
it follows that
2/2
0
1 2 sin
2
Nkd
av
ZU dZ
Nkd Z
−
= −
,
2/2
0
1 sinNkd
av
ZU dZ
Nkd Z
=
. (14.37)
If ( Nkd ) is sufficiently large, the above integral can be approximated as
2
0
1 sin 1
2av
ZU dZ
Nkd Z Nkd
= . (14.38)
The directivity then becomes
0
1 24
av
Nkd dD N
U
= =
. (14.39)
The comparison of (14.39) and (14.32) shows that the directivity of an EFA is
approximately twice as large as the directivity of the BSA.
Another (equivalent) expression can be derived for D0 of the EFA in terms of
the array length L = (N−1)d:
0 4 1L d
Dd
= +
. (14.40)
For large arrays, the following approximation holds:
0 4 / ifD L L d= . (14.41)
2.3. Directivity of HW EFA
If the radiation has its maximum at 0 = , then the minimum of avU is
obtained as in (14.13):
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2
min minminmin
min min
1 2 cos(2 ) 1Si(2 )
2 sin 2 2av
Z ZU Z
k Nd Z Z
− = + +
, (14.42)
where min 1.47 / 2Z = − − .
2
min1 2 0.878
1.85152 2
avUNkd Nkd
= + − =
. (14.43)
The directivity is then
0min
11.789 4
0.878av
Nkd dD N
U
= = =
. (14.44)
From (14.44), we can see that using the HW conditions leads to improvement of
the directivity of the EFA with a factor of 1.789. Equation (14.44) can be
expressed via the length L of the array as
0 1.789 4 1 1.789 4L d L
Dd
= + =
. (14.45)
Example: Given a linear uniform array of N isotropic elements (N = 10), find the
directivity 0D if:
a) 0 = (BSA)
b) kd = − (ordinary EFA)
c) /kd N = − − (Hansen-Woodyard EFA)
In all cases, / 4d = .
a) BSA
( )0 2 5 6.999 dBd
D N
=
b) Ordinary EFA
( )0 4 10 10 dBd
D N
=
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c) HW EFA
( )0 1.789 4 17.89 12.53 dBd
D N
=
3. Pattern Characteristics of Linear Uniform Arrays – Recapitulation
A. Broad-side array
NULLS ( 0nAF = ):
arccosn
n
N d
=
, where 1,2,3,4,...n = and ,2 ,3 ,...n N N N
MAXIMA ( 1nAF = ):
arccosn
m
d
=
, where 0,1,2,3,...m =
HALF-POWER POINTS:
1.391
arccoshNd
, where d
1
HALF-POWER BEAMWIDTH:
1.391
2 arccos , 12
h
d
Nd
= −
MINOR LOBE MAXIMA:
2 1
arccos2
s
s
d N
+
, where 1,2,3,...s = and
d1
FIRST-NULL BEAMWIDTH (FNBW):
2 arccos2
nNd
= −
FIRST SIDE LOBE BEAMWIDH (FSLBW):
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3
2 arccos , 12 2
s
d
Nd
= −
B. Ordinary end-fire array
NULLS ( 0nAF = ):
arccos 1n
n
N d
= −
, where 1,2,3,...n = and ,2 ,3 ,...n N N N
MAXIMA ( 1nAF = ):
arccos 1n
m
d
= −
, where 0,1,2,3,...m =
HALF-POWER POINTS:
1.391
arccos 1hNd
= −
, where
d1
HALF-POWER BEAMWIDTH:
1.391
2arccos 1 , 1h
d
Nd
= −
MINOR LOBE MAXIMA:
( )2 1
arccos 12
s
s
Nd
+ = −
, where 1,2,3,...s = and 1
d
FIRST-NULL BEAMWIDTH:
2arccos 1nNd
= −
FIRST SIDE LOBE BEAMWIDH:
3
2arccos 1 , 12
s
d
Nd
= −
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C. Hansen-Woodyard end-fire array
NULLS:
( )arccos 1 1 22
n nNd
= + −
, where 1,2,...n = and ,2 ,...n N N
MINOR LOBE MAXIMA:
arccos 1s
s
Nd
= −
, where 1,2,3,...s = and 1
d
SECONDARY MAXIMA:
arccos 1 [1 (2 1)]2
m mNd
= + − +
, where 1,2,...m = and 1
d
HALF-POWER POINTS:
arccos 1 0.1398hNd
= −
, where 1, -large
dN
HALF-POWER BEAMWIDTH:
2arccos 1 0.1398hNd
= −
, where 1, -Large
dN
FIRST-NULL BEAMWIDTH:
2arccos 12
nNd
= −
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4. 3-D Characteristics of a Linear Array
In the previous considerations, it was always assumed that the linear-array
elements are located along the z-axis, which is convenient to analyze in spherical
coordinate system. If the array axis has an arbitrary orientation, the array factor
can be expressed as
( )( ) ( )1 cos 1
1 1
N Nj n kd j n
n n
n n
AF a e a e − + −
= =
= = , (14.46)
where na is the excitation amplitude and coskd = + .
The angle is subtended by the array axis and the position vector to the
observation point. Thus, if the array axis is along the unit vector a ,
ˆ ˆ ˆ ˆsin cos sin sin cosa a a a a = + +a x y z (14.47)
and the position vector to the observation point is
ˆ ˆ ˆ ˆsin cos sin sin cos = + +r x y z (14.48)
the angle can be found as
ˆ ˆcos sin cos sin cos sin sin sin sin cos cos ,a a a a a = = + +a r
cos sin sin cos( ) cos cosa a a = − + . (14.49)
If ˆ ˆ ( 0 )a= = a z , then cos cos , = = .