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

Chapter 6Chapter 6Part 1Part 1

Array BasicsArray Basics

ECE 5318/6352ECE 5318/6352Antenna EngineeringAntenna Engineering

Spring 2006Spring 2006Dr. Stuart LongDr. Stuart Long

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

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

Two element arrayTwo element array

simplest casesimplest case

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)

Total FieldTotal Field

2 cos4

reIkjE

−−

2 cos2dr r θ≅ +

21 EEE +≅t

1 cos4

reIkjE

−−

θcos21drr −≅ [6-2]

Total FieldTotal Field

⎥⎥⎦

⎢⎢⎣

⎡+≅

⎟⎠⎞

⎜⎝⎛ +

−⎟⎠⎞

⎜⎝⎛ +−

ˆβθβθ

θ θπ

η kdjkdjjkro

t eereIkjaE

1 22 2

1 21 2

ˆ cos cos4

j k r j k r

k I e ejr r

θη θ θ

⎛ ⎞ ⎛ ⎞− − − −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⎡ ⎤⎢ ⎥≅ +⎢ ⎥⎢ ⎥⎣ ⎦

Element Factor Array Factor

[AF][Ee]

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ +

≅−

2coscos2cos

4ˆ βθθ

reIkjE

η cos4

][EereIkj

⎟⎠⎞

⎜⎝⎛ +

coscos2[AF] βθkd

as function of kd and β⎟⎠⎞

⎜⎝⎛ +

coscos(AF) βθkdn

ExampleExample

242 πλλπ

( )AF cos cos4n

π θ⎡ ⎤= ⎢ ⎥⎣ ⎦

(a) 0=β

( ) θcosEe =n

ExampleExample

242 πλλπ

( ) ( )AF cos cos 14n

π θ⎡ ⎤= +⎢ ⎥⎣ ⎦

(b) 2πβ =

( ) θcosE e =n

ExampleExample

( ) ( )AF cos cos 14n

π θ⎡ ⎤= −⎢ ⎥⎣ ⎦

(c)2πβ −=

( ) θcosEe =n

242 πλλπ

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

( ) ⎟⎠⎞

⎜⎝⎛ +

coscosAF βθkdn

nulls whennulls when

( ) πβθ ⎥⎦⎤

⎢⎣⎡ +

12cos21 nkd n

0,1, 2, 3,n =

[ ]⎭⎬⎫

⎩⎨⎧

+±−= − πβπλθ )12(

2cos1 n

for first null due to [AF] to exist

for not phase shift

( )0=β

Defining Nulls

coscos =⎟⎠⎞

⎜⎝⎛ + βθkd

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

Defining Nulls

( ) nn θcosEe =

null at null at θθ nn = = 9090°°

NN--Element Uniform Element Uniform Linear ArrayLinear Array

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

[ ] ( )1

1 cos AF

nkd e ψψ θ β −

= + ⇒ = ∑let

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

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

( )[ ]∑−

+−+=N

kdnje1

cos1AF βθ

11[AF]

−−

ψψψψ jNjjj eeee +++= 2[AF]

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

( ) ψψ jNj ee −=− 11[AF]

ExampleExample

Look at N = 5

ψψ 41AF jj ee ++++=

ψψψψψ 542[AF] jjjjj eeeee ++++=

( ) ψψ 511[AF] jj ee −=−

11[AF]

−−

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

⎥⎥⎥

⎢⎢⎢

−−

−⎥⎦⎤

⎢⎣⎡ −

11[AF] ψψ

ψψψ

NjNjNj

⎥⎥⎥⎥

⎢⎢⎢⎢

⎥⎦⎤

⎢⎣⎡

⎥⎦⎤

⎢⎣⎡

=⎥⎦⎤

⎢⎣⎡ −

[AF] 21

[6-10]

⎥⎥⎥⎥

⎢⎢⎢⎢

⎥⎦⎤

⎢⎣⎡

⎥⎦⎤

⎢⎣⎡

[AF]ψ

or ‘Normalized’

⎥⎥⎥⎥

⎢⎢⎢⎢

⎥⎦⎤

⎢⎣⎡

⎥⎦⎤

⎢⎣⎡

1[AF]ψ

as ψ → 0

⎥⎥⎥⎥

⎢⎢⎢⎢

⎡⎥⎦⎤

⎢⎣⎡

[AF] ψ

or⎥⎥⎥⎥

⎢⎢⎢⎢

⎡⎥⎦⎤

⎢⎣⎡

1[AF] ψ

N→[AF] 1[AF] →nor

[Note: for small arguments ; thus,

for small x (but not necessarily small)]

sinlim0

NullsNullssin x

1.39 π.217

nulls occur when

(where )πψ nN±=

nsin[AF] ≅

sin =⎟⎠⎞

⎜⎝⎛ ψN

( )cos2 nN kd nθ β π⇒ + = ±

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

⎭⎬⎫

⎩⎨⎧

⎥⎦⎤

⎢⎣⎡ ±−= − πβ

πλθ

2cos 1

[6-11]

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

Absolute maximaAbsolute maxima

[ ]⎭⎬⎫

⎩⎨⎧

±−= − πβπλθ m

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

⎥⎦

⎤⎢⎣

⎡= −

dm πλβθ

2cos 1

[6-12]

[6-13]

Half Power PointHalf Power Point

Half Power Point ⇒⎭⎬⎫

⎩⎨⎧

⎥⎦⎤

⎢⎣⎡ ±−= −

Ndh782.2

2cos 1 β

πλθ

for small ψ : (see appendix)x

sin[AF] ≅

3 dB point ⇒ when 391.12

±=ψN707.0sin

[6-14]

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

sin N2

Secondary maximaSecondary maxima

πψ+±= sN

S = 1,2,3,…1 2 1cos2s

λθ β ππ

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

First minor lobe maximaFirst minor lobe maxima

⎭⎬⎫

⎩⎨⎧

⎥⎦⎤

⎢⎣⎡ ±−= −

Ndsπβ

πλθ 3

2cos 1

2πψ ±=

NMaxima value of first minor lobe occurs

at NOT at 2π

[6-16]

At this point the magnitude of the AF

(note this side lobe is independent of N )

First minor lobe maximaFirst minor lobe maxima

≅=⎟⎠⎞

⎜⎝⎛

πθθ s

[ ] ( ) [ ]dB5.13212.0log20dB −=⇒in

[6-17]

Broadside ArraysBroadside Arrays

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]

BROADSIDE arrays BROADSIDE arrays (cont)(cont)

ββ = 0= 0

NullsNulls

MaximaMaxima

Half Power PointsHalf Power Points

Side Lobe MaximaSide Lobe Maxima

⎭⎬⎫

⎩⎨⎧

⎥⎦⎤

⎢⎣⎡ +

⎥⎦

⎤⎢⎣

⎡±≅

⎥⎦

⎤⎢⎣

⎡±=

⎥⎦

⎤⎢⎣

⎡±=

391.1cos

πλθ

λθ ,3,2,,3,2,1

,3,2,1=s

,3,2,1,0=m

assuming 1<<λ

[Table 6.1]

ββ = 0= 0

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

⎭⎬⎫

⎩⎨⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛−=Θ −

dNnλπ 1cos

[Table 6.2]

Example Example

NullsNulls

Absolute MaximaAbsolute Maxima

;10 === βλdN

3754cos

6652cos

=⎥⎦⎤

⎢⎣⎡±=

=⎥⎦⎤

⎢⎣⎡±=

⎥⎦⎤

⎢⎣⎡±=

θ ,3,2,,3,2,1

⇒==5

nλλλ

,3,2,1,0=m⇒= md

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

ββ = 0= 0

Half Power BeamwidthHalf Power Beamwidth

Example Example

( ) 4.208.79902 =−=Θh

⇒⋅

52391.1391.1

ππλdN

2391.1cos1 =⎥⎦⎤

⎢⎣⎡

⋅⋅

±= −

ββ = 0= 0

Side Lobes MaximaSide Lobes Maxima

01010cos

53106cos

10122cos

=⎥⎦⎤

⎢⎣⎡±=

=⎥⎦⎤

⎢⎣⎡±=

⎭⎬⎫

⎩⎨⎧

⎥⎦⎤

⎢⎣⎡ +

θ⇒⎥⎦⎤

⎢⎣⎡ +

=⎥⎦⎤

⎢⎣⎡ +

dλ ,3,2,1=s

ββ = 0= 0

Example

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 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

Array Factor MagnitudeArray Factor Magnitude

at θ = 0°

Example Example

5sin12[AF] 1.414

0.707sin4

⎛ ⎞⎜ ⎟⎝ ⎠≅ = =⎛ ⎞⎜ ⎟⎝ ⎠

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

ββ = 0= 0

NullsNulls

20.4°

θ 53°

79.8°

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°

Ordinary Ordinary Endfire ArrayEndfire Array

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]

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

NullsNulls

MaximaMaxima

Side Lobe MaximaSide Lobe Maxima

ββ == ±± kdkd

⎭⎬⎫

⎩⎨⎧

⎥⎦⎤

⎢⎣⎡ +

−≅

⎥⎦

⎤⎢⎣

⎡−≅

⎥⎦

⎤⎢⎣

⎡−=

⎥⎦

⎤⎢⎣

⎡−=

391.11cos

πλθ

λθ,3,2,

,3,2,1NNNn

,3,2,1=s

,3,2,1,0=m

assuming 1<<λ

π dλ

assuming << 1

[Table 6.3]

First Null BeamwidthFirst Null Beamwidth

Half Power BeamwidthHalf Power Beamwidth

ββ == ±± kdkd

ExampleExample

kddN −=== βλ ;4

⎭⎬⎫

⎩⎨⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛−≅Θ −

dNh πλ391.11cos2 1

⎭⎬⎫

⎩⎨⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛−≅Θ −

dNnλ1cos2 1

hh θ2=ΘHPBW HPBW ⇒⇒

[Table 6.4]

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

Scanning Scanning Phased ArrayPhased Array

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

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)

(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

L••

••

L = (N-1)d

L + d = Ndcan also write Θh as

⎟⎟⎠

⎞⎜⎜⎝

+−⎟⎟⎠

⎞⎜⎜⎝

−=Θ −−

dLdL oohλθλθ 443.0coscos443.0coscos 11

[6-22]

Example Example

;10 === odN θλ

785.04

60cos2

60cos4

2cos −=−=−=−=−=ππλ

λπθβ okd

7.234.472.715.2

443.021cos

5.2443.0

21cos 11 =−=⎟

⎠⎞

⎜⎝⎛ +−⎟

⎠⎞

⎜⎝⎛ −=Θ −−

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

DirectivityDirectivity

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

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

[AF] N

⎟⎠⎞

⎜⎝⎛

sin[AF] ≅

sin[AF]

⎟⎠⎞

⎜⎝⎛

θcos2

kdNZ =where

sin[AF]

⎟⎠⎞

⎜⎝⎛ −

−= θkdNZwhere

[6-38] [6-45]

DIRECTIVITY DIRECTIVITY (cont)(cont)

( ( d d << << λλ ))

( ( d d << << λλ ))( )[ ]

22 sinAF)( ⎟

⎠⎞

⎜⎝⎛==

ZZU nθ

90at1max == θU 180or0at1max == θU

rado P

UD max4π=

dkNwith approx. that large; can calculate Prad

∫∫∫∫∫∫ ⎟⎠⎞

⎜⎝⎛==Ω=

ππππθθφθθφ

0sinsinsin d

ZZddUddUPrad

θθππ

Z sinsin22

0∫ ⎟⎠⎞

⎜⎝⎛=

[2-13], [6-39]

( ( d d << << λλ ))

θcos2

kdNZ =

θθ dkdNZd sin2

Nkdrad ∫ ⎟⎠⎞

⎜⎝⎛−=

)1(cos2

−= θkdNZ

θθ dkdNZd sin2

rad ∫ ⎟⎠⎞

⎜⎝⎛−=

[6-40] [6-47]

with approx. that Ν k d

2→ large

( ( d d << << λλ ))

Prad ∫∞+

∞⎟⎠⎞

⎜⎝⎛=

sin4π

[6-42]

ππdkN

λπ dN

24 max ==

Prad ∫∞+

⎟⎠⎞

⎜⎝⎛=

2sin4π

[6-49]

ππ 24dkN

Prad =

λπ dN

44 max ==

( ( d d << << λλ ))

for large array

[6-42]λ

π dNP

o24 max ==

NdLdL =⇒>>

λLDo 2=

for large array

[6-49]λ

π dNP

o44 max ==

NdLdL =⇒>>

λLDo 4=

[6-44] [6-49]

( ( 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

2;20 λ

19 or 20oD

5HPBW =

3.20Pozar =⇒ oD

2;20 λ

40or38=oD

32HPBW =

40Pozar =⇒ oD

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

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”” --

⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

GRATING LOBES GRATING LOBES (cont)(cont)

π≥kd

⎟⎠⎞

⎜⎝⎛ −≤

Nkd ππ 22

⎟⎠⎞

⎜⎝⎛ −≤

π2≥kd

⎟⎠⎞

⎜⎝⎛ −≤

Nkd ππ2 ⎟

⎠⎞

⎜⎝⎛ −−≤ βππ

( )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

GRAPHICAL SOLUTION FOR ARRAYS GRAPHICAL SOLUTION FOR ARRAYS

ζζ )(sin)( =nAF

ExampleExample

Will see several functions like

)cos()( δγζ += cff

=c)cos(2

βθζ += kdN βδ2N

=where

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

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

1.First, draw rectangular plot of

)(ψAF ψvs.

2.Then, draw a semicircle of radius underneath,

which is offset by an amount

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

ExampleExample

5–element Uniform Linear array

⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

for N = 5

kd = 1.2π

β = - 0.2π

βθψ += coskd

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

π1st. Side Lobe mag.

(±0.8π, ±1.2π, ±1.6π)

(±0.6π, ±π, ±1.4π)

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

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

Rectangular Rectangular

to Polarto Polar

Rectangular Rectangular

to Polarto Polar

Fig. 6.16 Rectangular to polar plot graphical solution

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

GRAPHICAL GRAPHICAL REPRESENTATION REPRESENTATION

FOR ENDFIREFOR ENDFIRE

kd ≤ (2π - 2π/N)

Example for no part of grating lobe

β = - kd

β = kd

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

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