Lecture 33 Antenna Arrays and Phase Arrays · PDF file2 ECE 303 – Fall 2007 –...
Transcript of Lecture 33 Antenna Arrays and Phase Arrays · PDF file2 ECE 303 – Fall 2007 –...
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ECE 303 – Fall 2007 – Farhan Rana – Cornell University
Lecture 33
Antenna Arrays and Phase Arrays
In this lecture you will learn:
• Antenna arrays
• Gain and radiation pattern for antenna arrays
• Antenna beam steering with phase array antennas
ECE 303 – Fall 2007 – Farhan Rana – Cornell University
Two Hertzian Dipoles – A Two Element Array
y
z
( ) ( )03
00 ˆ hrdIzrJrrrr
−= δ
x
0hr
1hr0J
r1Jr
( ) ( )13
11 ˆ hrdIzrJrrrr
−= δ
( ) ( ) [ ]( )
( ) ( )φθφθ
θπ
ηθ
θπ
ηθ
,,,
sin4
ˆ
sin4
ˆ
10
10
.ˆ
0
1.ˆ
0
00
.ˆ1
.ˆ0
Fr
eIIe
IIe
rdIkj
eIeIerkdjrE
hrkjhrkjrkjo
hrkjhrkjrkjoff
Ε=
⎥⎦
⎤⎢⎣
⎡+=
+=
−
−
r
rr
rr
rr
One can write the E-field in the far-field as a superposition of the E-fields produced by all the elements in the array:
Consider first an array of just two Hertzian dipoles:
Element Factor Array Factor
Each antenna in the array is an “element” of the array
• The “element factor” is just the E-field produced by the first element if it were sitting at the origin
• The “array factor” captures all the interference effects
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ECE 303 – Fall 2007 – Farhan Rana – Cornell University
An N-Element Antenna Array - IConsider an N-element antenna array where the elements are:
- all identical (all loops, or all Hertzian dipoles, or all Half-wave dipoles, etc) - all oriented in the same way- but with possibly different current phasors
Let the current phasor of the m-th antenna be Im
Let the position vector of the m-th antenna be mhr
y
x
mhr
z
mI
0I1I
1−NIzOne can write the E-field in the far-field as a superposition of the E-fields produced by all the elements in the array:
( ) ( )
( ) ( )φθφθ
φθ
,,,
,, 1
0
.ˆ
0
Fr
eIIrrE
N
m
hrkjmff m
Ε=
∑Ε=−
=r
rrr r
ECE 303 – Fall 2007 – Farhan Rana – Cornell University
An N-Element Antenna Array - II
( ) ( )
( ) ( )φθφθ ,,,
1
0
.ˆ
0
Fr
eIIrrE
N
m
hrkjmff m
Ε=
∑Ε=−
=r
rrrr r
( )( )
( ) ( )222 ,, ofpart angular
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ˆ.,, φθφθ
πφθ Fr,
rP
rtrSG ⎥⎦
⎤⎢⎣⎡ Ε∝=
rrr
( ) ( )( )max,
,,φθ
φθφθG
Gp =
Gain:
Pattern:
y
x
mhr
z
mI
0I1I
1−NIz
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ECE 303 – Fall 2007 – Farhan Rana – Cornell University
An N-Element Hertzian Dipole Phase Array
x
z
y
N-element Hertzian dipole array
a
αjeα2je( )α1−Nje
• Current magnitude is the same for all the dipoles
• Current phase difference between successive dipoles is α
1
αj
m
m eI
I=+1
( ) ( ) ( )
( ) ( ) ( )( )21
0
cossin2
1
0
cossin.ˆ1
0
.ˆ
0
,
, 0
∑=⇒
∑=∑=
−
=
+
−
=
−
=
N
m
makj
N
m
makjmjhrkjN
m
hrkjm
eF
eeeeIIF m
αφθ
φθα
φθ
φθrr
( ) ( )( )
( ) ( )( )⎥⎦⎤
⎢⎣⎡ +
⎥⎦⎤
⎢⎣⎡ +
=φθα
φθα
cossin21sin
cossin2
sin
2
2
ka
kaN
Array Factor:
Element Factor:
( ) ( ) rkjo er
dIkjr −=Ε θπ
ηθφθ sin4
ˆ ,, 0r
ECE 303 – Fall 2007 – Farhan Rana – Cornell University
x
z
y
N-element Hertzian dipole phase array antennaa
αjeα2je( )α1−Nje
1
( ) ( )( ) ( )( )
( ) ( )( )⎥⎦⎤
⎢⎣⎡ +
⎥⎦⎤
⎢⎣⎡ +
=φθα
φθαθφθ
cossin21sin
cossin2
sinsin1,
2
22
2ka
kaN
Np
Pattern:
Coming from the element factor
array factor
( ) ( ) ( )222 ,sin1, φθθφθ F
Np =
An N-Element Hertzian Dipole Phase Array
NdIkP o 2012π
η=
( )( )
( ) ( )222 ,sin
23
4
ˆ.,, φθθ
πφθ F
NrP
rtrSG ==
rr
Coming from the element factor
array factor
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ECE 303 – Fall 2007 – Farhan Rana – Cornell University
An N-Element Hertzian Dipole Phase Array: Maxima
Lets look at radiation in the x-y plane:
φ
a
( )φcosa
y
x
Radiation going in the φ direction from adjacent elements will add in-phase, and one will have a big maximum in the radiation pattern, provided:
( ){ KK,3,2,1,0
2cos=
±=+n
nak παφ
( )φπθ ,2=p
φ
2
4
20
π
πα
=
−=
=
ka
N
( )2,2 φπθ =F
2
4
20
π
πα
=
−=
=
ka
N
(degrees) φ
nulls
smallmaxima(side lobes)
bigmaxima(main lobes)
main lobes
side lobes
ECE 303 – Fall 2007 – Farhan Rana – Cornell University
An N-Element Hertzian Dipole Phase Array: Maxima
φ
a
( )φcosa
y
x
Condition for a big maximum in the x-yplane:
( ){ KK,3,2,1,0
2cos=
±=+n
nak παφ
At a big maximum the value of the array factor is:
( )( )( )
( )( )
2
2
22max
cos21sin
cos2
sin,2
N
ka
kaN
F
=
⎥⎦⎤
⎢⎣⎡ +
⎥⎦⎤
⎢⎣⎡ +
==φα
φαφπθ
And the value of the antenna gain is:
( )2,2 φπθ =F
2
4
20
π
πα
=
−=
=
ka
N
(degrees) φ
nulls
smallmaxima(side lobes)
bigmaxima(main lobes)
( ) ( ) ( )
NG
FN
G
23,
2
,sin23,
max
22
=⎟⎠⎞
⎜⎝⎛ =⇒
=
φπθ
φθθφθ
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ECE 303 – Fall 2007 – Farhan Rana – Cornell University
An N-Element Hertzian Dipole Phase Array: Nulls
φ
a
y
x
Radiation in the x-y plane:
( )( )( )
( )( )⎥⎦⎤
⎢⎣⎡ +
⎥⎦⎤
⎢⎣⎡ +
==φα
φαφπθ
cos21sin
cos2
sin,2
2
22
ka
kaN
F
The array factor will give a null in the radiation pattern provided:
( )φcosa
( )
{ KK,3,2,,0
2cos
NNNnN
nak
≠
±=+παφ
( )2,2 φπθ =F
2
4
20
π
πα
=
−=
=
ka
N
(degrees) φ
nulls
smallmaxima(side lobes)
bigmaxima(main lobes)
ECE 303 – Fall 2007 – Farhan Rana – Cornell University
An N-Element Hertzian Dipole Phase Array: Width of Big Maxima
( )2,2 φπθ =F
2
4
20
π
πα
=
−=
=
ka
N
φ
φ
a
y
x
φ∆Angular width of a lobe associated with a big maxima:
For a big maxima: ( ) {
,3,2,1,022cos KK±±±===+ nN
nNnak ππαφ
At the nulls nearest to the above maxima we must have:
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cosN
nNak ±=+⎟
⎠⎞
⎜⎝⎛ ∆
+ παφφ
Which can be solved for ∆φ – and for N large (N >> 1) one gets approximately: ( )
sin14
φπφ
kaN≈∆
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ECE 303 – Fall 2007 – Farhan Rana – Cornell University
x
z
y
aka = π
αjeα2je( )α1−Nje
Phase difference between successive elements is α
N = 2 N = 10
( )φθ ,p ( )φθ ,p
1
N-element Hertzian dipole array
An N-Element Hertzian Dipole Phase Array: Examples
ECE 303 – Fall 2007 – Farhan Rana – Cornell University
x
z
y
aka = π / 2
αjeα2je( )α1−Nje
Phase difference between successive elements is α
N = 2 N = 10
( )φθ ,p ( )φθ ,p
1
N-element Hertzian dipole array
An N-Element Hertzian Dipole Phase Array: Examples
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ECE 303 – Fall 2007 – Farhan Rana – Cornell University
An (N+1)-Element Hertzian Dipole Binomial Array
x
z
y
(N+1)-element Hertzian dipole array
a • Current phase is the same for all the dipoles
• Currents magnitudes in the dipoles follow a binomial distribution
( )!!!
0 mNmN
IIm
−=
( ) ( )( ) ( )
( ) ( ) ( ) ( ) ( )⎥⎦⎤
⎢⎣⎡=+=⇒
∑−
=∑===
φθφθ
φθ
φθ
φθ
cossin2
cos21,
!!!,
222cossin2
0
cossin.ˆ
0
.ˆ
00
akeF
emNm
NeeIIF
NNNakj
N
m
makjhrkjN
m
hrkjm mrr
Array Factor:
Element Factor:
( ) ( ) rkjo er
dIkjr −=Ε θπ
ηθφθ sin4
ˆ ,, 0r
π
λ
=⇒
=
ka
a2
ECE 303 – Fall 2007 – Farhan Rana – Cornell University
Lets look at radiation in the x-y plane:
φ
a
( )φcosay
x
Radiation going in the φ direction from adjacent elements will add in-phase, and one will have a big maximum in the radiation pattern, provided:
( ){
( ) n
nnak
πφπ
πφ
±=⇒
=±=
cos2
,3,2,1,0 2cos
KK
An (N+1)-Element Hertzian Dipole Binomial Array: Maxima and Nulls
( ) ( )⎥⎦⎤
⎢⎣⎡== φπφπθ cos2
cos2,2 222 NNF
( ) ( )⎥⎦⎤
⎢⎣⎡== φπφπθ cos2
cos,2 2Np
N+1=2
N+1=40
N+1=2N+1=40
( )φπθ ,2=p ( )φπθ ,2=p
Note:
• No side lobes• Two nulls at φ = 0,π
(degrees) φ
(degrees) φ
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ECE 303 – Fall 2007 – Farhan Rana – Cornell University
Phased Array TRack to Intercept Of Target
Alaska
Applications of Antenna Arrays - I
ECE 303 – Fall 2007 – Farhan Rana – Cornell University
Communication satellite with two phase array antenna panels MRI system with phase array
RF receivers
Applications of Antenna Arrays - II
Jicamarca facility (Peru) with an array of half-wave dipoles
SETI