spring 2006 dr. stuart longcourses.egr.uh.edu/ece/ece5318/longpwrptsch6-0221a71.pdf · spring 2006...

1

Chapter 6Chapter 6Part 1Part 1

Array BasicsArray Basics

ECE 5318/6352ECE 5318/6352Antenna EngineeringAntenna Engineering

Spring 2006Spring 2006Dr. Stuart LongDr. Stuart Long

2

Array RationaleArray Rationale

Single elementsSingle elements

Usually broad beamwidthRelatively low directivity

Often system requirements demand higher directivitiesOften system requirements demand higher directivities

Can increase electrical sizeor

form an “assembly” of elements in an electrical and geometrical

configuration called an

ARRAYARRAY

3

Assumptions for our coverageAssumptions for our coverage

1. All elements are identical2. No coupling between elements3. Total field is vector sum of individual

radiation patterns

Use array to control overall pattern shape by:Use array to control overall pattern shape by:

1. Geometrical configuration of overall array(linear, circular, rectangular, spherical, …)

2. Relative displacement between radiators3. Amplitude of excitement of each element4. Excitation phase of each element5. Individual element pattern

4

Two element arrayTwo element array

simplest casesimplest case

ψ

5

Two identical horizontal sources along the z axis, a distance “d ” apart, constant amplitudes; upper element leads lower one in phase by amount “β “

look at [ y - z ] plane (φ = 90°)only there

sin ψ = cos θ

Similar case of horizontal dipole above a ground plane, except 2nd. source not necessarily 180° out of phase – but similar analysis

Two element array Two element array (cont)(cont)

6

Total FieldTotal Field


22

)2

(

2 cos4

2

θπ

ηβ

reIkjE

rkj

o

−−

≅

2 cos2dr r θ≅ +

21 EEE +≅t

11

)2

(

1 cos4

1

θπ

ηβ

reIkjE

rkj

o

−−

≅

θcos21drr −≅ [6-2]

7

Total FieldTotal Field

[6-3]


⎥⎥⎦

⎤

⎢⎢⎣

⎡+≅

⎟⎠⎞

⎜⎝⎛ +

−⎟⎠⎞

⎜⎝⎛ +−

2cos

2cos

cos4

ˆβθβθ

θ θπ

η kdjkdjjkro

t eereIkjaE

1 22 2

1 21 2

ˆ cos cos4

j k r j k r

ot

k I e ejr r

β β

θη θ θ

π

⎛ ⎞ ⎛ ⎞− − − −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⎡ ⎤⎢ ⎥≅ +⎢ ⎥⎢ ⎥⎣ ⎦

E a

8

[6-3]

Element Factor Array Factor

[AF][Ee]

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ +

≅−

2coscos2cos

4ˆ βθθ

πη

θkd

reIkjE

rkjo

t a


9

θπ

η cos4

][EereIkj

rkjo

−

≅

⎟⎠⎞

⎜⎝⎛ +

≅2

coscos2[AF] βθkd

as function of kd and β⎟⎠⎞

⎜⎝⎛ +

≅2

coscos(AF) βθkdn

[6-4]


10

ExampleExample

4λ

=d

242 πλλπ

==dk

( )AF cos cos4n

π θ⎡ ⎤= ⎢ ⎥⎣ ⎦

(a) 0=β

( ) θcosEe =n


11

ExampleExample

4λ

=d

242 πλλπ

==dk

( ) ( )AF cos cos 14n

π θ⎡ ⎤= +⎢ ⎥⎣ ⎦

(b) 2πβ =

( ) θcosE e =n


12

ExampleExample

( ) ( )AF cos cos 14n

π θ⎡ ⎤= −⎢ ⎥⎣ ⎦

(c)2πβ −=

( ) θcosEe =n


4λ

=d

242 πλλπ

==dk

13

General example for TwoGeneral example for Two--Element ArrayElement Array

( ) ⎟⎠⎞

⎜⎝⎛ +

=2

coscosAF βθkdn

nulls whennulls when

( ) πβθ ⎥⎦⎤

⎢⎣⎡ +

±=+2

12cos21 nkd n

0,1, 2, 3,n =

[ ]⎭⎬⎫

⎩⎨⎧

+±−= − πβπλθ )12(

2cos1 n

dn

for first null due to [AF] to exist

2λ

≥d

oror

for not phase shift

( )0=β

Defining Nulls

02

coscos =⎟⎠⎞

⎜⎝⎛ + βθkd

14

General example for TwoGeneral example for Two--Element ArrayElement Array

Defining Nulls

( ) nn θcosEe =

null at null at θθ nn = = 9090°°

15


NN--Element Uniform Element Uniform Linear ArrayLinear Array



16

NN--element uniformelement uniformlinear arraylinear array

βjnn eII 1−=

constant amplitudeconstant amplitudeconstant phase shiftconstant phase shiftelements along a lineelements along a line

(e.g. # 3 leads # 2 by β radians)

for convenience assume isotropic sources

AssumptionsAssumptions

[6-6]

17

[ ] ( )1

1 cos AF

Nj n

nkd e ψψ θ β −

=

= + ⇒ = ∑let

( ) ( ) ( )βθβθβθ +−+++++ ++++= cos)1(cos2cos1AF kdNjkdjkdj eee

[6-6]

NN--element uniform linear array element uniform linear array (cont)(cont)

( )[ ]∑−

+−+=N

n

kdnje1

cos1AF βθ

18

11[AF]

−−

= ψ

ψ

j

jN

ee

ψψψψ jNjjj eeee +++= 2[AF]

[6-7]


Multiply RHS by and subtract from [AF]e j ψ

( ) ψψ jNj ee −=− 11[AF]

19

ExampleExample

Look at N = 5

⇒

ψψ 41AF jj ee ++++=

ψψψψψ 542[AF] jjjjj eeeee ++++=

( ) ψψ 511[AF] jj ee −=−

11[AF]

5

−−

= ψ

ψ

j

j

ee


20


this term due to asymmetric locationof elements; shifting to center or taking magnitude eliminates it

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−

−=

−−

=−

−⎥⎦⎤

⎢⎣⎡ −

22

222

1

11[AF] ψψ

ψψψ

ψ

ψ

jj

NjNjNj

j

jN

ee

eeeee

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎥⎦⎤

⎢⎣⎡

⎥⎦⎤

⎢⎣⎡

=⎥⎦⎤

⎢⎣⎡ −

2sin

2sin

[AF] 21

ψ

ψψ

N

eNj

[6-10]

21


⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎥⎦⎤

⎢⎣⎡

⎥⎦⎤

⎢⎣⎡

=

2sin

2sin

[AF]ψ

ψN

or ‘Normalized’

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎥⎦⎤

⎢⎣⎡

⎥⎦⎤

⎢⎣⎡

=

2sin

2sin

1[AF]ψ

ψN

Nn

22


as ψ → 0

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡⎥⎦⎤

⎢⎣⎡

≅

2

2sin

[AF] ψ

ψN

or⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡⎥⎦⎤

⎢⎣⎡

=

2

2sin

1[AF] ψ

ψN

Nn

N→[AF] 1[AF] →nor

[Note: for small arguments ; thus,

for small x (but not necessarily small)]

xxx

=→

sinlim0

ψN

23

NullsNullssin x

x

.707

.63

1.0

1.39 π.217

π2

2π

3π2

x

nulls occur when

(where )πψ nN±=

2

xx

nsin[AF] ≅

02

sin =⎟⎠⎞

⎜⎝⎛ ψN

( )cos2 nN kd nθ β π⇒ + = ±

but not n = 0,3,2,1=n

⎭⎬⎫

⎩⎨⎧

⎥⎦⎤

⎢⎣⎡ ±−= − πβ

πλθ

Nn

dn2

2cos 1

[6-11]


24

No matter how small ψ, always have max at ψ = 0 , (m = 0)

Absolute maximaAbsolute maxima

[ ]⎭⎬⎫

⎩⎨⎧

±−= − πβπλθ m

dm 22

cos 1

m = 0,1,2,3,…Absolute max. when πψ m±=2

⎥⎦

⎤⎢⎣

⎡= −

dm πλβθ

2cos 1

[6-12]

[6-13]


25

Half Power PointHalf Power Point

Half Power Point ⇒⎭⎬⎫

⎩⎨⎧

⎥⎦⎤

⎢⎣⎡ ±−= −

Ndh782.2

2cos 1 β

πλθ

for small ψ : (see appendix)x

xn

sin[AF] ≅

3 dB point ⇒ when 391.12

±=ψN707.0sin

→x

x

[6-14]


26

Secondary maxima ⇒ approx. when is max = ± 1(minor lobes)

sin N2

ψ

Secondary maximaSecondary maxima

( )2

122

πψ+±= sN

S = 1,2,3,…1 2 1cos2s

sd N

λθ β ππ

− ⎧ ⎫⎡ + ⎤⎡ ⎤= − ±⎨ ⎬⎢ ⎥⎢ ⎥⎣ ⎦⎣ ⎦⎩ ⎭ [6-15]


27

First minor lobe maximaFirst minor lobe maxima

⎭⎬⎫

⎩⎨⎧

⎥⎦⎤

⎢⎣⎡ ±−= −

Ndsπβ

πλθ 3

2cos 1

23

2πψ ±=

NMaxima value of first minor lobe occurs

at NOT at 2π

[6-16]


28

At this point the magnitude of the AF

(note this side lobe is independent of N )

First minor lobe maximaFirst minor lobe maxima

212.0

231

2

2sin

[AF]

23

≅=⎟⎠⎞

⎜⎝⎛

≅

==

πψ

ψ

πθθ s

N

N

n

[ ] ( ) [ ]dB5.13212.0log20dB −=⇒in

[6-17]


29


Broadside ArraysBroadside Arrays



30

BROADSIDE arraysBROADSIDE arrays

maximum radiation in direction normal to line of array maximum radiation in direction normal to line of array ((θθ = 90= 90°°))

⇒ Broadside radiation pattern when all elements are in phase( independent of separation dist “d” )

ββ = 0= 0

00cos90 =⇒=+⇒= ββθθ kdfor

0cos0 =+⇒= βθψ kd whenMax.[6-18]

31

BROADSIDE arrays BROADSIDE arrays (cont)(cont)

ββ = 0= 0

NullsNulls

MaximaMaxima

Half Power PointsHalf Power Points

Side Lobe MaximaSide Lobe Maxima

⎭⎬⎫

⎩⎨⎧

⎥⎦⎤

⎢⎣⎡ +

±≅

⎥⎦

⎤⎢⎣

⎡±≅

⎥⎦

⎤⎢⎣

⎡±=

⎥⎦

⎤⎢⎣

⎡±=

−

−

−

−

Ns

d

dN

dm

dNn

s

h

m

n

122

cos

391.1cos

cos

cos

1

1

1

1

λθ

πλθ

λθ

λθ ,3,2,,3,2,1

NNNnn

≠=

,3,2,1=s

,3,2,1,0=m

assuming 1<<λ

π d

assuming 1<<λ

π d

[Table 6.1]

32

ββ = 0= 0

ΘS

NOTE FIRST SIDE LOBE BEAMWIDTHNOTE FIRST SIDE LOBE BEAMWIDTH

(not the width of any beam)

First Null BeamwidthFirst Null Beamwidth

Half Power BeamwidthHalf Power Beamwidth⎭⎬⎫

⎩⎨⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛−=Θ −

dNh πλπ 391.1cos

22 1

⎭⎬⎫

⎩⎨⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛−=Θ −

dNnλπ 1cos

22

[Table 6.2]


33

Example Example

NullsNulls

Absolute MaximaAbsolute Maxima

0;4

;10 === βλdN

3754cos

6652cos

52cos

1

1

1

=⎥⎦⎤

⎢⎣⎡±=

=⎥⎦⎤

⎢⎣⎡±=

⎥⎦⎤

⎢⎣⎡±=

−

−

−

n

n

nn

θ

θ

θ ,3,2,,3,2,1

NNNnn

≠=

⇒==5

2

410

nndN

nλλλ

,3,2,1,0=m⇒= md

m 4λ [ ] ( ) 900cos4cos 11 ==±= −− mmθ


ββ = 0= 0

34

Half Power BeamwidthHalf Power Beamwidth

Example Example

( ) 4.208.79902 =−=Θh


⇒⋅

⋅=

52391.1391.1

ππλdN

8.795

2391.1cos1 =⎥⎦⎤

⎢⎣⎡

⋅⋅

±= −

πθh


ββ = 0= 0

35

Side Lobes MaximaSide Lobes Maxima

01010cos

53106cos

10122cos

1

1

1

=⎥⎦⎤

⎢⎣⎡±=

=⎥⎦⎤

⎢⎣⎡±=

⎭⎬⎫

⎩⎨⎧

⎥⎦⎤

⎢⎣⎡ +

±=

−

−

−

s

s

ss

θ

θ

θ⇒⎥⎦⎤

⎢⎣⎡ +

=⎥⎦⎤

⎢⎣⎡ +

10122

1012

2ss

dλ ,3,2,1=s


ββ = 0= 0

Example

36

at θ = 53°

Array Factor MagnitudeArray Factor Magnitude

Example Example

10 2 5sin sin cos sin cos12 2 4 2[AF] 2.19

2 0.45sin sin cos sin cos2 4 2

n

N π λ πψ θ θλ

ψ π λ πθ θλ

⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠≅ = = = =⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠

[ ] ( ) [ ]1AF 2.19 0.219 20 log 0.219 13.2 dB10n = ⋅ = ⇒ = −


ββ = 0= 0

37

Array Factor MagnitudeArray Factor Magnitude

at θ = 0°

Example Example

5sin12[AF] 1.414

0.707sin4

π

π

⎛ ⎞⎜ ⎟⎝ ⎠≅ = =⎛ ⎞⎜ ⎟⎝ ⎠

[ ] ( ) [ ]1AF 1.414 0.1414 20 log 0.1414 17.0 dB10n = ⋅ = ⇒ = −


ββ = 0= 0

38

NullsNulls

20.4°

θ 53°

37°

79.8°

66°

MaximaMaxima

0.14140°

0.21953°

1.090°

[AF]nθm

Half PowerHalf Power

Θh = 2(90-79.8) = 20.4°θ h = 79.8°

20.4°

Summary Summary


ββ = 0= 0

at θ n = 66° & 37°

39


Ordinary Ordinary Endfire ArrayEndfire Array



40

ORDINARY ENDFIRE ARRAYSORDINARY ENDFIRE ARRAYSMaximum radiation in direction along line of array Maximum radiation in direction along line of array

ββ == ±± kdkd

for max. in for max. in θθ = 0= 0°°⇒⇒

kd cos θ + β = 0 ⇒

ψ = 0 for θ = 0° ⇒

β = - kd

for max. in for max. in θθ = 180= 180°° ⇒⇒

ψ = 0 for θ = 180° ⇒

kd cos θ + β = 0 ⇒

β = + kd

[6-20]

41

ORDINARY ENDFIRE arrays ORDINARY ENDFIRE arrays (cont)(cont)

NullsNulls

MaximaMaxima


Side Lobe MaximaSide Lobe Maxima

ββ == ±± kdkd

⎭⎬⎫

⎩⎨⎧

⎥⎦⎤

⎢⎣⎡ +

−≅

⎥⎦

⎤⎢⎣

⎡−≅

⎥⎦

⎤⎢⎣

⎡−=

⎥⎦

⎤⎢⎣

⎡−=

−

−

−

−

Ns

d

dN

dm

dNn

s

h

m

n

122

1cos

391.11cos

1cos

1cos

1

1

1

1

λθ

πλθ

λθ

λθ,3,2,

,3,2,1NNNn

n≠=

,3,2,1=s

,3,2,1,0=m

assuming 1<<λ

π d

π dλ

assuming << 1

[Table 6.3]

42

First Null BeamwidthFirst Null Beamwidth

Half Power BeamwidthHalf Power Beamwidth

ββ == ±± kdkd

ExampleExample

kddN −=== βλ ;4

;10

⎭⎬⎫

⎩⎨⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛−≅Θ −

dNh πλ391.11cos2 1

⎭⎬⎫

⎩⎨⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛−≅Θ −

dNnλ1cos2 1

hh θ2=ΘHPBW HPBW ⇒⇒

[Table 6.4]

ORDINARY ENDFIRE arrays ORDINARY ENDFIRE arrays (cont)(cont)

43


Scanning Scanning Phased ArrayPhased Array



44

SCANNING PHASED ArraysSCANNING PHASED Arrays

Can Can ““scanscan”” beam between endfire and broadsidebeam between endfire and broadsideby allowing phase shift by allowing phase shift ββ to beto be

-- kd kd << ββ <<kdkd

0 ≤ β ≤ kd -kd ≤ β ≤ 0or

θθo o

45

for for maximiummaximium radiation at radiation at θθ == θθoo ⇒⇒

-- kd kd << ββ << kdkd

kd cos θo + β = 0 ⇒

ψ = 0 for θ = θo ⇒

β = - kd cos θo [6-21] θθo o

PHASED Arrays PHASED Arrays (cont)(cont)

46



(use general expressions for nulls, max, sidelobes)

For HPBW the main beam is no longer symmetric about θ = 90° or θ = 0° ;

subtracting the two θ h (+ and – sign)

⎟⎟⎠

⎞⎜⎜⎝

⎛+−⎟⎟

⎠

⎞⎜⎜⎝

⎛−=Θ −−

dkNdkN ooh782.2coscos782.2coscos 11 θθ

47



L••

•

••

•

L = (N-1)d

L + d = Ndcan also write Θh as

⎟⎟⎠

⎞⎜⎜⎝

⎛+

+−⎟⎟⎠

⎞⎜⎜⎝

⎛+

−=Θ −−

dLdL oohλθλθ 443.0coscos443.0coscos 11

[6-22]

48

Example Example

60;4

;10 === odN θλ

785.04

60cos2

60cos4

2cos −=−=−=−=−=ππλ

λπθβ okd

7.234.472.715.2

443.021cos

5.2443.0

21cos 11 =−=⎟

⎠⎞

⎜⎝⎛ +−⎟

⎠⎞

⎜⎝⎛ −=Θ −−

h


49

Example Example

-12.6°

+11.2°

71.2°

47.4°

NOTE: MAIN BEAM IS NOTNOTE: MAIN BEAM IS NOTSYMMETRIC ABOUT ITSSYMMETRIC ABOUT ITSMAX ATMAX AT θθ = 60= 60°°

ALSO NOTE ROTATIONALALSO NOTE ROTATIONALSYMMETRY GIVES 3SYMMETRY GIVES 3--DDPATTERN OF CONICAL BEAMPATTERN OF CONICAL BEAM


50


DirectivityDirectivity



51

DIRECTIVITYDIRECTIVITY““the ratio of the radiation intensity in a given direction from tthe ratio of the radiation intensity in a given direction from the he antenna to the radiation intensity averaged over all directionsantenna to the radiation intensity averaged over all directions””

radoo P

UUUD max4π==

Fig. 2.13 3-D directivity pattern of a λ/2 dipole

52

For BROADSIDE For BROADSIDE and small spacingand small spacing

( ( d d << << λλ ))

For ORDINARY ENDFIRE For ORDINARY ENDFIRE and small spacingand small spacing

( ( d d << << λλ ))ψ

ψ

2

2sin

[AF] N

N

n

⎟⎠⎞

⎜⎝⎛

≅

ZZ

n

sin[AF] ≅

θ

θ

cos2

cos2

sin[AF]

kdN

kdN

n

⎟⎠⎞

⎜⎝⎛

≅

θcos2

kdNZ =where

1cos2

1cos2

sin[AF]

−

⎟⎠⎞

⎜⎝⎛ −

≅θ

θ

kdN

kdN

n

1cos2

−= θkdNZwhere

[6-38] [6-45]

DIRECTIVITY DIRECTIVITY (cont)(cont)

53



( ( d d << << λλ ))


( ( d d << << λλ ))( )[ ]

22 sinAF)( ⎟

⎠⎞

⎜⎝⎛==

ZZU nθ

90at1max == θU 180or0at1max == θU

rado P

UD max4π=

→2

dkNwith approx. that large; can calculate Prad

54

∫∫∫∫∫∫ ⎟⎠⎞

⎜⎝⎛==Ω=

Ω

ππππθθφθθφ

0

22

00

2

0sinsinsin d

ZZddUddUPrad

θθππ

dZ

Z sinsin22

0∫ ⎟⎠⎞

⎜⎝⎛=

[2-13], [6-39]


55


( ( d d << << λλ ))


( ( d d << << λλ ))

θcos2

kdNZ =

θθ dkdNZd sin2

−=

ZdZ

ZdkNP

Nkd

Nkdrad ∫ ⎟⎠⎞

⎜⎝⎛−=

2/

2/

2sin

2

2π

)1(cos2

−= θkdNZ

θθ dkdNZd sin2

−=

ZdZ

ZdkNP

Nkd

rad ∫ ⎟⎠⎞

⎜⎝⎛−=

0

2sin

2

2π

[6-40] [6-47]


56

with approx. that Ν k d

2→ large


( ( d d << << λλ ))


( ( d d << << λλ ))

ZdZ

ZdkN

Prad ∫∞+

∞⎟⎠⎞

⎜⎝⎛=

sin4π

[6-42]

ππdkN

Prad4

=

λπ dN

PUD

rado

24 max ==

ZdZ

ZdkN

Prad ∫∞+

⎟⎠⎞

⎜⎝⎛=

0

2sin4π

[6-49]

ππ 24dkN

Prad =

λπ dN

PUD

rado

44 max ==


57


( ( d d << << λλ ))


( ( d d << << λλ ))

for large array

[6-42]λ

π dNP

UDrad

o24 max ==

NdLdL =⇒>>

λLDo 2=

for large array

[6-49]λ

π dNP

UDrad

o44 max ==

NdLdL =⇒>>

λLDo 4=

[6-44] [6-49]


58


( ( d d << << λλ ))


( ( d d << << λλ ))

ExampleExampleExampleExample

ExampleExample Note:Note:

The directivity for the ENDFIRE case is exactly twice that for the broadside case since it is unidirectional instead of bidirectional

L2 4.5oDλ

=

2 5.0oNdDλ

=

4;10 λ

== dN

2;20 λ

== dN

19 or 20oD

5HPBW =

3.20Pozar =⇒ oD

2;20 λ

== dN

40or38=oD

32HPBW =

40Pozar =⇒ oD


59


Fig. 6.12 HalfFig. 6.12 Half--Power Beamwidth for Broadside, OrdinaryPower Beamwidth for Broadside, OrdinaryEndEnd-- Fire, and Scanning Uniform Linear ArraysFire, and Scanning Uniform Linear Arrays

60

GRATING LOBESGRATING LOBES

When visible range (VR) includes When visible range (VR) includes ψψ= = --22ππ oror ψψ = 2= 2ππ, a second major , a second major lobe of same magnitude N is produced lobe of same magnitude N is produced

-- This is called a This is called a ““grating lobegrating lobe”” --

61

⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

=

2sin

2sin

)(ψ

ψ

ψ

N

AF

62

GRATING LOBES GRATING LOBES (cont)(cont)

π≥kd

⎟⎠⎞

⎜⎝⎛ −≤

Nkd ππ 22

⎟⎠⎞

⎜⎝⎛ −≤

Nkd

2ππ

π2≥kd

⎟⎠⎞

⎜⎝⎛ −≤

Nkd ππ2 ⎟

⎠⎞

⎜⎝⎛ −−≤ βππ

Nkd 2

( )βπ −≥ 2kd

Broadside Broadside Other phases Other phases

For no part of grating lobe

For complete grating lobe

But to keep second lobe from being grater than side lobe need

Endfire Endfire

⎟⎠⎞

⎜⎝⎛ −≤

Nkd ππ ⎟

⎠⎞

⎜⎝⎛ −−≤ βππ

Nkd 22

63

GRAPHICAL SOLUTION FOR ARRAYS GRAPHICAL SOLUTION FOR ARRAYS

ζζ )(sin)( =nAF

ExampleExample

Will see several functions like

)cos()( δγζ += cff

kdN2

=c)cos(2

βθζ += kdN βδ2N

=where

64

Often easier to plot in rectangular form and then convert to polar plot for actual function of real angle θ

)(ζf

GRAPHICAL SOLUTION FOR ARRAYSGRAPHICAL SOLUTION FOR ARRAYS(cont)(cont)

65

1.First, draw rectangular plot of


)(ψAF ψvs.

2.Then, draw a semicircle of radius underneath,

which is offset by an amount

kdβ

3.Draw vertical lines to intersect semicircle

5.Mark corresponding magnitudes on radial lines

4.Draw radial lines to points of intersection

6.Connect points

ProcedureProcedure

66

ExampleExample


5–element Uniform Linear array

⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

=

2sin

2sin

)(ψ

ψ

ψ

N

AF

for N = 5

kd = 1.2π

β = - 0.2π

βθψ += coskd

25

n πψ = ±

Prin. Max. at ψ = 0 , mag. = 5

First null at ψ = ± 0.4π

Nulls at

(2n 1)5

πψ += ±Secondary Max. at

23.1)3.0(sin

1==

π1st. Side Lobe mag.

(±0.8π, ±1.2π, ±1.6π)

(±0.6π, ±π, ±1.4π)

67


(First, draw rectangular plot of ))(ψAF ψvs.

68

1.23

Semicircular radius kd = 1.2πOffset by β = -0.2π

Rectangular Rectangular

to Polarto Polar

69

Rectangular Rectangular

to Polarto Polar

Fig. 6.16 Rectangular to polar plot graphical solution

70

Broadside ⇒ β = 0 (no offset)Max. always along θ = 90º

Smaller kd gives broader beamwidth

If kd approaches 2πadditional main lobescan appear – called

“GRATING LOBES”

VISIBLE RANGE:values of ψ whichcorrespond to real

angles θ

NOTESNOTES

Larger kd gives narrower beamwidth

–kd ≤ ψ ≤ kd

GRAPHICAL REPRESENTATION GRAPHICAL REPRESENTATION FOR BROADSIDEFOR BROADSIDE

kdkd

kd

θ

θ

θ

71

GRAPHICAL GRAPHICAL REPRESENTATION REPRESENTATION

FOR ENDFIREFOR ENDFIRE

kd ≤ (2π - 2π/N)

Example for no part of grating lobe

β = - kd

β = kd

kd

kd

spring 2006 dr. stuart longcourses.egr.uh.edu/ece/ece5318/longpwrptsch6-0221a71.pdf · spring 2006...

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