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Antenna Array-VI: Dr. Yogesh Kumar Choukiker

The document summarizes the Dolph-Tschebyscheff array, which is an optimum antenna array design that achieves a good trade-off between beam width and side-lobe level. It discusses how the array factor can be expressed as a cosine series expansion that is related to Tschebyscheff polynomials. The design process involves determining the voltage ratio from the desired side-lobe level, finding the corresponding value of z, comparing the cosine expansion to the appropriate Tschebyscheff polynomial, and calculating the element excitation coefficients. Examples are given of designing 4-element and 7-element arrays with half-wavelength spacing to achieve specific side-lobe levels.

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111 views10 pages

Antenna Array-VI: Dr. Yogesh Kumar Choukiker

The document summarizes the Dolph-Tschebyscheff array, which is an optimum antenna array design that achieves a good trade-off between beam width and side-lobe level. It discusses how the array factor can be expressed as a cosine series expansion that is related to Tschebyscheff polynomials. The design process involves determining the voltage ratio from the desired side-lobe level, finding the corresponding value of z, comparing the cosine expansion to the appropriate Tschebyscheff polynomial, and calculating the element excitation coefficients. Examples are given of designing 4-element and 7-element arrays with half-wavelength spacing to achieve specific side-lobe levels.

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© © All Rights Reserved
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Lecture: 20

Antenna Array-VI

Dr. Yogesh Kumar Choukiker

School of Electronics Science Engineering


Microwave and Photonics Division
VIT University, Vellore, India

Dr. Yogesh Kumar Choukiker


Dolph - Tschebyscheff Array
The Dolph-Tschebyscheff (or Chebyshev) array is an optimum
array design to achieve a better trade-off between beam width
and side-lobe level
Let us now consider N Element array with equal spacing and phase
excitation (Broadside) but unequal amplitudes. The array factors are
M
AF 2 M an cos2n 1u
n 1
M 1 d
AF 2 M 1 an cos2n 1u Where u cos

n 1

The above array factors are nothing but a series expansion of cosine
terms, that are multiples (or harmonics) of a fundamental frequency
Hence the expansion of the above summation can be equated to
Tschebyscheff polynomials to obtain the excitation coefficients
Dr. Yogesh Kumar Choukiker
1
Dolph - Tschebyscheff Array
Expanding the cosine terms with the help of Eulers formula and
trigonometric identity sin2 = 1 - cos2

Dr. Yogesh Kumar Choukiker


2
Dolph - Tschebyscheff Array
By letting cosu = z, we can write the corresponding
Tschebyscheff polynomials as

These relations between cosine functions and Tschebyscheff


polynomials are valid in the range -1 z +1 since | cos(mu)| 1. If
the value of |z| > 1, then the polynomials are related to hyperbolic
Dr. Yogesh Kumar Choukiker
3 cosine functions
Dolph - Tschebyscheff Array
The recursive formula for Tschebyscheff polynomials is
Tm z 2 zTm 1 z Tm 2 z
The polynomials can also be computed from


Tm z cos m cos 1 z 1 z 1

Tm z cosh m cosh 1 z z 1, z 1
The other important factor is the voltage ratio of ratio of main lobe
voltage to side lobe voltage. This is also referred as side-lobe level
below main lobe maximum,
Main lobe voltage
R0
Side lobe voltage
From which z0 (max. value of z) is (where m is number of elements)
1 1/ P
P m 1
1/ P
z0 R0 R0 1 R0 R0 Dr.
2 2
1
YogeshKumarChoukiker
4 2
Dolph - Tschebyscheff Array

Axis that
determines the
design parameter
Increasing
order of the
polynomial

Dr. Yogesh Kumar Choukiker


5
Dolph - Tschebyscheff Array
Design Steps
Step 1 : First write down the array factor and expand the cosine terms in
terms of cosu
Step 2 : Find the dimensionless value of voltage ratio R0, from which
compute z0
Step 3 : Substitute the value of z/z0 in the place of cosu
Step 4 : Compare the cosine expansion to the Tschebyscheff polynomial
Tm(z), where m is one less than the number of elements
Step 5 : Find the coefficients a1, a2, a3 etc
Step 6 : Normalize the coefficients with end element values

Dr. Yogesh Kumar Choukiker


6
Dolph - Tschebyscheff Array

Voltage ratio
R0 Main lobe
level

Side lobe
level

Dr. Yogesh Kumar Choukiker


7 Maximum value of z
Dolph - Tschebyscheff Array
1. Rovr 10 R0 (in dB ) / 20

2. p No. of elements 1 N 1
1 1
3. z0 cosh cosh (20)
9
or
1

z0 20 400 1
2
20
1
9
400 1
1
9

Dr. Yogesh Kumar Choukiker


7
Problems
Problems

11. Design a four element broadside array of /2 spacing between


elements. The pattern is to be optimum with a side lobe level
19.1 dB down the main lobe maximum.

12. Design a seven element broadside array which has the optimum
pattern for a side lobe level of -20 dB. The spacing between
elements has to be /2.

Dr. Yogesh Kumar Choukiker


8

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