8/13/2019 L14_Arrays2

1/14

Nikolova 2012 1

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

8/13/2019 L14_Arrays2

2/14

Nikolova 2012 2

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

2.94kd

N

for a maximum at 0 , (14.2)

2.94kd

N

for a maximum at 180 . (14.3)

Conditions (14.2)(14.3) are known as the HansenWoodyard conditions for

end-fire radiation. They follow from a procedure for maximizing the directivity,

which we outline below.The normalized patternAFnof a uniform linear array is

sin cos2

cos2

n

Nkd

AFN

kd

(14.4)

if the argument coskd is sufficiently small (see previous lecture). We

are looking for an optimal , which results in maximum directivity. Letpd , (14.5)

where d is the array spacing andp 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)

orsin

n

ZAF

Z , where ( cos )Z q k p .

8/13/2019 L14_Arrays2

3/14

Nikolova 2012 3

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 zkq 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)

8/13/2019 L14_Arrays2

4/14

Nikolova 2012 4

( ) 1.472

Ndk p .

1.47, where2 2

Ndk Ndpdp

( ) 1.472

Ndk

2.94 2.94kd kd

N 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 minimum beamwidth(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

2.942.94

2.942

Nkd

Nkd

N

(14.18)

Since we want to have a minimum of the pattern in the 180 direction, wemust ensure that

8/13/2019 L14_Arrays2

5/14

Nikolova 2012 5

180| | . (14.19)

It is easier to remember the Hansen-Woodyard conditions for maximum

directivity in the 0 direction as

1800

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 mustensure 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 that

the complementary requirement | | at the opposite direction can be met ifthe following relation is observed:

1

4

Nd

N

. (14.24)

If N is large, / 4d . Thus, for a large uniform array, Hansen-Woodyardcondition can yield improved directivity only if the spacing between the array

elements is approximately / 4 .

8/13/2019 L14_Arrays2

6/14

Nikolova 2012 6

ARRAY FACTORS OF A 10-ELEMENT UNIFORM-AMPLITUDE HWEFA

Solid line: / 4d

Dash line: / 2d N = 10

kdN

Fig. 6.12, p. 273, Balanis

8/13/2019 L14_Arrays2

7/14

Nikolova 2012 7

2. Directivity of a linear array2.1. Directivity of a BSA

2

22

sin cossin2

( )cos

2

n

Nkd

ZU AF N Z

kd

(14.25)

0 00 4

rad av

U UD

P U , (14.26)

where / (4 )av rad U P . The radiation intensity in the direction of maximumradiation / 2 in terms of nAF is unity:

0 max ( / 2) 1U U U ,1

0 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

cos

2

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

Z

U dZN kd Z

, (14.29)

22

2

1 sin

Nkd

av

Nkd

ZU dZ

Nkd Z

. (14.30)

8/13/2019 L14_Arrays2

8/14

Nikolova 2012 8

The function 1 2( sin )Z Z is a relatively fast decaying function as Zincreases.

That is why, for large arrays, where / 2Nkd is big enough 20 , the integral(14.30) can be approximated by

21 sin

av

Z

U dZNkd 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 EFAConsider an EFA with maximum radiation at 0 , i.e., kd .

2

22

sin cos 1sin2

( )

cos 12

n

NkdZ

U AFN Z

kd

, (14.35)

where (cos 1)2

NZ kd . The averaged radiation intensity is

2 22

0 0 0

1 sin 1 sinsin sin4 4 2rad

av P Z ZU d d d Z Z

.

Since

(cos 1)2

NZ kd and sin

2

NdZ kd d , (14.36)

8/13/2019 L14_Arrays2

9/14

Nikolova 2012 9

it follows that

2/2

0

1 2 sin

2

Nkd

av

ZU dZ

Nkd Z

,

2/2

0

1 sinNkd

avZU 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)

Comparing (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 D0in terms of the array

lengthL =(N1)d:

0 4 1

L d

D d

. (14.40)

For large arrays, the following approximation holds:

0 4 / if D L L d . (14.41)

2.3. Directivity of HW EFAIf the radiation has its maximum at 0 , then the minimum of avU is

obtained as in (14.13):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 .

8/13/2019 L14_Arrays2

10/14

Nikolova 2012 10

2

min1 2 0.878

1.85152 2

avUNkd Nkd

. (14.43)

The directivity is then

0min1 1.789 4

0.878avNkd 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 lengthL of the array as

0 1.789 4 1 1.789 4L d L

Dd

. (14.45)

Example: Given a linear uniform array of Nisotropic 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 dB

dD N

b)Ordinary EFA 0 4 10 10 dB

dD N

c)HW EFA 0 1.789 4 17.89 12.53 dB

dD N

8/13/2019 L14_Arrays2

11/14

Nikolova 2012 11

3. Pattern characteristics of linear uniform arrays - recapitulationA. Broad-side array

NULLS ( 0nAF

):

arccosnn

N d

, where 1,2,3,4,...n and ,2 ,3 ,...n N N N

MAXIMA ( 1nAF ):

arccosnm

d

, where 0,1,2,3,...m

HALF-POWER POINTS:1.391

arccoshNd

, where

d1

HALF-POWER BEAMWIDTH:

1.3912 arccos , 1

2h

d

Nd

MINOR LOBE MAXIMA:

2 1arccos

2s

s

d N

, where 1,2,3,...s and

d1

FIRST-NULL BEAMWIDTH (FNBW):

2 arccos2

nNd

FIRST SIDE LOBE BEAMWIDH (FSLBW):

32 arccos , 1

2 2s

d

Nd

8/13/2019 L14_Arrays2

12/14

Nikolova 2012 12

B. Ordinary end-fire arrayNULLS ( 0nAF ):

arccos 1nn

N d

, where 1,2,3,...n and ,2 ,3 ,...n N N N

MAXIMA ( 1nAF ):

arccos 1nm

d

, where 0,1,2,3,...m

HALF-POWER POINTS:

1.391

arccos 1h Nd

, whered

1

HALF-POWER BEAMWIDTH:

1.3912arccos 1 , 1h

d

Nd

MINOR LOBE MAXIMA:

2 1arccos 1 2s

s

Nd

, where 1,2,3,...s and 1d

FIRST-NULL BEAMWIDTH:

2arccos 1nNd

FIRST SIDE LOBE BEAMWIDH:

3

2arccos 1 , 12sd

Nd

8/13/2019 L14_Arrays2

13/14

Nikolova 2012 13

C.Hansen-Woodyard end-fire arrayNULLS:

arccos 1 1 2 2n n Nd

, where 1,2,...n and , 2 ,...n N N

MINOR LOBE MAXIMA:

arccos 1ss

Nd

, where 1,2,3,...s and 1

d

SECONDARY MAXIMA:

arccos 1 [1 (2 1)] 2m m Nd

, 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

8/13/2019 L14_Arrays2

14/14

Nikolova 2012 14

4. 3-D characteristics of a linear arrayIn the previous considerations, it was always assumed that the linear-array

elements are located along the z-axis, thus, creating a problem, which is

symmetrical around thez-axis. If the array axis has an arbitrary orientation, thearray 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 between the array axis and the position vector to

the observation point. Thus, if the array axis is along the unit vector a ,

sin cos sin sin cosa a a a a a x y z (14.47)

and the position vector to the observation point is

sin cos sin sin cos r x y z (14.48)

the angle can be found as

cos sin cos sin cos sin sin sin sin cos cos ,a a a a a a r x y z

cos sin sin cos( ) cos cosa a a . (14.49)

If ( 0 )a a z , then cos cos , .