ANTENNA ARRAYS - GUC · no minor lobes for the arrays with spacing of λ/4 and λ/2 between the...
Transcript of ANTENNA ARRAYS - GUC · no minor lobes for the arrays with spacing of λ/4 and λ/2 between the...
ADAPTIVE ANTENNAS. PROF. A.M.ALLAM
1
ANTENNA ARRAYS
ADAPTIVE
ANTENNAS
ADAPTIVE ANTENNAS. PROF. A.M.ALLAM
2 2/3/2015 LECTURES 2
Why antenna arrays?-1
1.The element pattern
2.The amplitude excitation of the individual element
3.The phase excitation of the individual element
4.The relative displacement between elements
5.The geometrical distribution of the array, linear,
rectangular, circular, spherical, …
The array has 5 degrees of freedom which help in
configuring some parameters
ADAPTIVE ANTENNAS. PROF. A.M.ALLAM
3 2/3/2015 3
N Elements linear arrays: Uniform amplitude and spacing
An array of identical elements fed with identical magnitude and a progressive
phase. The array factor can be obtained by considering the elements to be point
sources then multiplied times the element pattern
cos
:
.....1
1
)1(
1
cos)1(
cos)1(cos2cos
kdwhere
eAF
aswrtittenbecanwhich
eAF
eeeAF
N
n
nj
N
n
kdnj
kdNjkdjkdj
ADAPTIVE ANTENNAS. PROF. A.M.ALLAM
4 2/3/2015 LECTURES 4
2
1sin
2sin
1
1
)1()1(
);2()1(
2
1
2
1
2
1
222
1
N
e
ee
eee
e
eAF
eeAF
gSubtractin
Nj
jj
Nj
Nj
Nj
j
jN
jNj
If the reference point is the physical center of the array, the
array factor reduces to:
2
1sin
2sin
N
AF
)2(.....
;
)1(.....1
)1(32
)1(2
jNNjjjjj
j
Njjj
eeeeeAFe
ebysidesbothgMultiplyin
eeeAF
ADAPTIVE ANTENNAS. PROF. A.M.ALLAM
5 5
2
1
2sin
N
AF
The maximum value of the array factor is equal to (N), to normalize the array
factor, so that the maximum value of each is equal to unity,
For small value of ψ it
is approximated by:
For small values of ψ , the above expression can be approximated by:
2
2sin
N
N
AFn
2
1sin
2sin
N
N
AFn
coskd
ADAPTIVE ANTENNAS. PROF. A.M.ALLAM
6
Nulls
The maximum values occur when:
Maximum
2
12
22
..3,2,12
12
2sin
2
..3,2,12
12
2cos
2
12cos
2
12
sin
1
1
s
d
dfor
ss
dor
ss
d
skd
Nor
N
s
s
s
s
Maxima of minor lobes
They occur approximately when:
The null values of the array occur when:
m=0 for main lobe
m=1,2,… for grating lobes
...2,1,022
cos
cos2
1
2
1
mmd
mkd
m
n
...3,2,
...3,2,12
2cos
2
02
sin
1
NNNn
n
N
n
d
nN
N
n
n
ADAPTIVE ANTENNAS. PROF. A.M.ALLAM
7 LECTURES 7
The maximum radiation of any array is directed normal to the axis of the
array (broadside θo=90o) which occurs when:
valueanyofbecand
putkd o
oo
;0
900cos
(a) Broadside array
dnLL
Do )1(...............2
2
1sin
2sin
N
N
AFn
coskd
The total phase difference ψ:
1- current excitation phase β=0
2-path difference phase (kdcosθ)=0,
ψ =0, i.e.,
constructive normal to the array axis θ=90o
ADAPTIVE ANTENNAS. PROF. A.M.ALLAM
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N=10 N=10 Only the maximum in
broadside direction
One more
maximum appears
( grating lobes) due
to large spacing
If the spacing between the elements is chosen between λ < d < 2λ, then the
maximum toward θ0 = 0◦ shifts toward the angular region 0◦ < θ < 90◦
while the maximum toward θ0 = 180◦ shifts toward 90◦ < θ < 180◦
GRATING LOBES d > λ/2
NO GRATING LOBES d ≤ λ/2
The separation is an important constraint you have to
mind to avoid the onset of the grating lobes in
broadside arrays
d=λ/4 d=λ
If d = 2λ, there are maxima toward 0◦, 60◦, 90◦, 120◦ and 180◦
ADAPTIVE ANTENNAS. PROF. A.M.ALLAM
9 9
(b) Endfire array
The maximum radiation of any array is directed along to the axis of the array
(θo=0o or θo=180o) which occurs when:
dnLL
Do )1(...............4
2
1sin
2sin
N
N
AFn
kdkd cos
The total phase difference ψ:
1- current excitation phase β=-kd
2-path difference phase (kdcosθ)=kd,
ψ =0, i.e.,
constructive along the array axis θ =0
kdkdkd
kdkdkd
o
o
0cos
0cos
180
0
ADAPTIVE ANTENNAS. PROF. A.M.ALLAM
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N=10 d=λ/4
β = -kd=-π/2 θo= 0o β = kd =π/2 θo=180o
θo=0o
θo=180o
ADAPTIVE ANTENNAS. PROF. A.M.ALLAM
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(c) Phased (scanning) array
The maximum radiation of the array is required to be oriented at an angle
θ0(0◦ ≤ θ0 ≤ 180◦). To accomplish this, the phase excitation β between the
elements must be adjusted so that:
oo kdkdkdo
coscoscos
Thus by controlling the progressive phase difference between the elements,
the maximum radiation can be squinted in any desired direction to form a
scanning array
2
1sin
2sin
N
N
AFn okdkd coscos
nNN
n
2..............0
2sin
mkd
n
cos2
1
2
m=0,1,2,…
Maxima
Nulls
n=1,2,…
n≠N,2N,…
ADAPTIVE ANTENNAS. PROF. A.M.ALLAM
The total phase difference ψ:
1- current excitation phase β=-kdcosθo
2-path difference phase (kdcosθ)= kdcosθo
ψ =0, i.e.,
constructive along the direction of scanning to maximum at θ = θo
It is clear that the general array can be scanned to any direction within the visible region
(-kd to kd) . The control parameter is the phase progressive of the excitation current; β
0 β kd -kd βo
The visible region
To avoid the onset of the grating lobe while scanning :
βo-2π
-kd > βo- 2π cosθo>1 - λ/d for an array along z-axis
The
main
lobe
The
grating
lobe
ADAPTIVE ANTENNAS. PROF. A.M.ALLAM
13 2/3/2015 LECTURES 13
(d) Different array orientations
•Linear array oriented along z-axis
2
1sin
2sin
N
N
AFn
coskd
ADAPTIVE ANTENNAS. PROF. A.M.ALLAM
14 2/3/2015 LECTURES 14
•Linear array oriented along x-axis
2
1sin
2sin
N
N
AFn coskd
cossinkd
ADAPTIVE ANTENNAS. PROF. A.M.ALLAM
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•Linear array oriented along y-axis
2
1sin
2sin
N
N
AFn coskd
sinsinkd
ADAPTIVE ANTENNAS. PROF. A.M.ALLAM
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Odd Even
N Elements linear arrays: Nonuniform amplitude and uniform spacing
Broadside arrays β=0 , Array along z-axis
ADAPTIVE ANTENNAS. PROF. A.M.ALLAM
17 2/3/2015 LECTURES 17
•Array factor for even number
ADAPTIVE ANTENNAS. PROF. A.M.ALLAM
18 2/3/2015 LECTURES 18
•Array factor for odd number
ADAPTIVE ANTENNAS. PROF. A.M.ALLAM
19 19
Binomial broadside array
Binomial expansion
s triangle’Pascal
5Example N=
10Example N=
β = 0
ADAPTIVE ANTENNAS. PROF. A.M.ALLAM
20 2/3/2015 LECTURES 20
•It’s observed that there are
no minor lobes for the arrays
with spacing of λ/4 and λ/2
between the elements
•Binomial arrays have very
low level minor lobes nearly
zeros, but they exhibit large
beamwidth
•A major disadvantage of
binomial arrays is the wider
variation between the
amplitudes excitation of the
different elements of an array
•Example
N=10
One more maximum appears
( grating lobes) due to large
spacing d=3λ/4 , d=λ
ADAPTIVE ANTENNAS. PROF. A.M.ALLAM
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Dolph Tschebyscheff broadside array
The order of the polynomial should be one less than the total number of elements
of the array ( array of 10 elements means Chebyshev polynomial of order 9)
-Since the array factor of an even or odd number of elements is a summation of cosine terms, whose
form is a polynomial as the Dolph-Chebyscheff polynomials, then the unknown coefficients of the
array factor can be determined by equating the series representing the cosine terms of the array
factor to the appropriate Chebyscheff polynomial
Tm(z) = cos[mcos-1(z)] for -1≤z≤1 represents the region of the side lobes
Tm(z) = cosh[mcosh-1(z)] for z<-1, z>1 represents the region of the main beam
Tm(z)= 2zTm-1(z)- Tm-2(z) is the recursive formula for Chebyscheff polynomial
1yln(y)cosh 21- yNote:
ADAPTIVE ANTENNAS. PROF. A.M.ALLAM
1- All polynomial pass through the point (1,1)
2- For -1≤z≤1, Tm(z) have values within -1 to +1
3- All roots occur within -1≤z≤1 , all maxima and minima have values +1 and
-1 respectively
Tm(z)
z
ADAPTIVE ANTENNAS. PROF. A.M.ALLAM
2/3/2015 LECTURES 23
1- Select the appropriate array factor (even or odd)
2- Expand the array factor {cosmu=…..}
3- Determine the point zo, such that Tm(zo)=Ro
* The order (m) of the Chebyscheff polynomial is always one less than
the total number of the element
4- Substitute cos u=z/zo
5- Equate the array factor to Tm(z)
6- Determine an
•Design procedures
ADAPTIVE ANTENNAS. PROF. A.M.ALLAM
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Example:
Design a broadside Tschebyscheff array of 10 elements with spacing (d) between
the elements. The major to minor lobe level is 26 dB. Find the excitation
coefficients and form the array factor.
Solution:
9
1
29
1
2
1
1
9
5432110
5
1
2
112
1
0851.120cosh9
1cosh
cosh9cosh20)(
20log20263
exp9cos,7cos,5cos,3cos,cos
9cos7cos5cos3coscos2
cos,12cos1
ooooo
o
oo
ooo
n
nM
RRRRzor
z
zzT
RRdBR
ansionseriesbyuuuuuforsubstitute
uauauauauaAF
duunaAF
ADAPTIVE ANTENNAS. PROF. A.M.ALLAM
2/3/2015 LECTURES 25
uuuuuAFthen
aaaaa
or
aaaaa
formnormalizedin
aaaaa
zzzzz
az
zaa
z
zaaa
z
z
aaaaz
zaaaaa
z
zAF
zTwithstepCompare
stepAFtheinz
z
zuSubistute
ooo
oo
o
9cos7cos357.15cos974.13cos496.2cos798.2
189.0706.0485.0357.0
798.2496.2974.1357.11
:
8377.52073.51184.48308.2086.2
2565764321209
256576443211216
120552049753
)()4(5
)2(0851.1
cos4
10
12345
12345
12345
9753
5
9
54
7
543
5
5432
3
5432110
9
ADAPTIVE ANTENNAS. PROF. A.M.ALLAM
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MXN Elements planner arrays: Nonuniform amplitude and uniform spacing
where
The array factor for this planar array with its maximum along θ0, φ0 , for an even
number of elements in each direction can be written as
wmn is the amplitude excitation of the
individual element
For separable distributions wmn = wmwn
Number of excitation values need to
be computed are M+N
For nonseparable distributions wmn ≠ wmwn
Number of excitation values need to be
computed are MxN
i.e., wmn is a 1 x N vector for a
linear array and a M x N matrix
for a planar array
ADAPTIVE ANTENNAS. PROF. A.M.ALLAM
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•The amplitude coefficients an or wn control primarily the shape of the
pattern and the major-to-minor lobe level
•Tapered amplitude distributions exhibit wider beamwidth but lower
sidelobe
•The phase excitations control primarily the scanning capabilities of the
array
Therefore, an antenna designer can choose different amplitude
distributions and different phase excitation to conform to the
application specifications
In smart antenna systems the objectives of a DSP are to estimate:
1-The direction of arrival (DOA) of all impinging signals using DOA
algorithms, and
2. The appropriate weights using the adaptive beam forming algorithms to
ideally steer the maximum radiation of the antenna pattern toward the
SOI and to place nulls toward the SNOI
ADAPTIVE ANTENNAS. PROF. A.M.ALLAM
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