Antenna Theory EC 544 Lecture#5

Chapter 5 Dolph – Tchebycheff Design Equal Side Lobes Equal Side Lobes

Properties of Tschebyscheff Polynomial Number of Zeroes = Order of Polynomial (m) Number of Zeroes = Order of Polynomial (m) Number of Lobes = m – 1 Number of Lobes = m – 1 For - 1 ≤ Z ≤ 1 | T m (z)| ≤ 1 For - 1 ≤ Z ≤ 1 | T m (z)| ≤ 1 For | z | ≥ 1 | T m (z)| ≥ 1 For | z | ≥ 1 | T m (z)| ≥ 1 All polynomials, of any order, pass through the point (1,1) All polynomials, of any order, pass through the point (1,1) All roots occur within - 1 ≤ Z ≤ 1, and all maxima and minima have values of +1 and - 1, respectively. All roots occur within - 1 ≤ Z ≤ 1, and all maxima and minima have values of +1 and - 1, respectively. Number of Zeroes = Order of Polynomial (m) Number of Zeroes = Order of Polynomial (m) Number of Lobes = m – 1 Number of Lobes = m – 1 For - 1 ≤ Z ≤ 1 | T m (z)| ≤ 1 For - 1 ≤ Z ≤ 1 | T m (z)| ≤ 1 For | z | ≥ 1 | T m (z)| ≥ 1 For | z | ≥ 1 | T m (z)| ≥ 1 All polynomials, of any order, pass through the point (1,1) All polynomials, of any order, pass through the point (1,1) All roots occur within - 1 ≤ Z ≤ 1, and all maxima and minima have values of +1 and - 1, respectively. All roots occur within - 1 ≤ Z ≤ 1, and all maxima and minima have values of +1 and - 1, respectively.

The Expansion of Cos (mμ) m = 0 cos(mμ) = 1 m = 0 cos(mμ) = 1 m =1 cos(mμ) = cos(μ) m =1 cos(mμ) = cos(μ) m = 2 cos(mμ) = cos(2μ) = 2 cos 2 (μ) - 1 m = 2 cos(mμ) = cos(2μ) = 2 cos 2 (μ) - 1 m = 3 cos(mμ) = cos(3μ) = 4 cos 3 (μ) - 3 cos(μ) m = 3 cos(mμ) = cos(3μ) = 4 cos 3 (μ) - 3 cos(μ) m = 4 cos(mμ) = cos(4μ) = 8 cos 4 (μ) - 8 cos 2 (μ) + 1 m = 4 cos(mμ) = cos(4μ) = 8 cos 4 (μ) - 8 cos 2 (μ) + 1 m = 5 cos(mμ) = cos(5μ) = 16 cos 5 (μ) - 20 cos 3 (μ) + 5 cos(μ) m = 5 cos(mμ) = cos(5μ) = 16 cos 5 (μ) - 20 cos 3 (μ) + 5 cos(μ) m = 0 cos(mμ) = 1 m = 0 cos(mμ) = 1 m =1 cos(mμ) = cos(μ) m =1 cos(mμ) = cos(μ) m = 2 cos(mμ) = cos(2μ) = 2 cos 2 (μ) - 1 m = 2 cos(mμ) = cos(2μ) = 2 cos 2 (μ) - 1 m = 3 cos(mμ) = cos(3μ) = 4 cos 3 (μ) - 3 cos(μ) m = 3 cos(mμ) = cos(3μ) = 4 cos 3 (μ) - 3 cos(μ) m = 4 cos(mμ) = cos(4μ) = 8 cos 4 (μ) - 8 cos 2 (μ) + 1 m = 4 cos(mμ) = cos(4μ) = 8 cos 4 (μ) - 8 cos 2 (μ) + 1 m = 5 cos(mμ) = cos(5μ) = 16 cos 5 (μ) - 20 cos 3 (μ) + 5 cos(μ) m = 5 cos(mμ) = cos(5μ) = 16 cos 5 (μ) - 20 cos 3 (μ) + 5 cos(μ)

The Expansion of Tschebyscheff Polynomial T m (z) m = 0 T m (z) = T 0 (z) = 1 m = 0 T m (z) = T 0 (z) = 1 m = 1 T m (z) = T 1 (z) = z m = 1 T m (z) = T 1 (z) = z m = 2 T m (z) = T 2 (z) = 2 z 2 – 1 m = 2 T m (z) = T 2 (z) = 2 z 2 – 1 m = 3 T m (z) = T 3 (z) = 4 z 3 – 3 z m = 3 T m (z) = T 3 (z) = 4 z 3 – 3 z m = 4 T m (z) = T 4 (z) = 8 z 3 – 8 z 2 +1 m = 4 T m (z) = T 4 (z) = 8 z 3 – 8 z 2 +1 m = 5 T m (z) = T 5 (z) = 16z 5 – 20z 3 +5z m = 5 T m (z) = T 5 (z) = 16z 5 – 20z 3 +5z m = 0 T m (z) = T 0 (z) = 1 m = 0 T m (z) = T 0 (z) = 1 m = 1 T m (z) = T 1 (z) = z m = 1 T m (z) = T 1 (z) = z m = 2 T m (z) = T 2 (z) = 2 z 2 – 1 m = 2 T m (z) = T 2 (z) = 2 z 2 – 1 m = 3 T m (z) = T 3 (z) = 4 z 3 – 3 z m = 3 T m (z) = T 3 (z) = 4 z 3 – 3 z m = 4 T m (z) = T 4 (z) = 8 z 3 – 8 z 2 +1 m = 4 T m (z) = T 4 (z) = 8 z 3 – 8 z 2 +1 m = 5 T m (z) = T 5 (z) = 16z 5 – 20z 3 +5z m = 5 T m (z) = T 5 (z) = 16z 5 – 20z 3 +5z

If we let z = cosμ in the Tschebyscheff polynomial, it can be find that If we let z = cosμ in the Tschebyscheff polynomial, it can be find that m=0 cos(mμ) = 1 = T 0 (z) m=0 cos(mμ) = 1 = T 0 (z) m =1 cos(mμ) = cos(μ) = z = T 1 (z) m =1 cos(mμ) = cos(μ) = z = T 1 (z) m = 2 cos(mμ) = cos(2μ) = 2cos 2 (μ) – 1= 2z 2 – 1= T 2 (z) m = 2 cos(mμ) = cos(2μ) = 2cos 2 (μ) – 1= 2z 2 – 1= T 2 (z) m = 3 cos(mμ) = cos(3μ) = 4cos 3 (μ) – 3cos(μ)= 4z 3 – 3z = T 3 (z) m = 3 cos(mμ) = cos(3μ) = 4cos 3 (μ) – 3cos(μ)= 4z 3 – 3z = T 3 (z) So for So for If we let z = cosμ in the Tschebyscheff polynomial, it can be find that If we let z = cosμ in the Tschebyscheff polynomial, it can be find that m=0 cos(mμ) = 1 = T 0 (z) m=0 cos(mμ) = 1 = T 0 (z) m =1 cos(mμ) = cos(μ) = z = T 1 (z) m =1 cos(mμ) = cos(μ) = z = T 1 (z) m = 2 cos(mμ) = cos(2μ) = 2cos 2 (μ) – 1= 2z 2 – 1= T 2 (z) m = 2 cos(mμ) = cos(2μ) = 2cos 2 (μ) – 1= 2z 2 – 1= T 2 (z) m = 3 cos(mμ) = cos(3μ) = 4cos 3 (μ) – 3cos(μ)= 4z 3 – 3z = T 3 (z) m = 3 cos(mμ) = cos(3μ) = 4cos 3 (μ) – 3cos(μ)= 4z 3 – 3z = T 3 (z) So for So for Relation Between cos(mμ) and T m (z) - 1 ≤ Z ≤ 1 z = cos(μ) z = cos(μ) cos(mμ) = T m (z) - 1 ≤ Z ≤ 1 z = cos(μ) z = cos(μ) cos(mμ) = T m (z)

| z | ≥ 1 | z | ≥ 1 z = cosh(μ) z = cosh(μ) cosh(mμ) = T m (z) cosh(mμ) = T m (z) The recursion formula for Tschebyscheff polynomial is It can be used to find one Tschebyscheff polynomial if the polynomials of the previous two orders are known. Each polynomial can also be computed using But for But for T m (z) = cos[m cos -1 (z)] - 1 ≤ Z ≤1 T m (z) = cos[m cos -1 (z)] - 1 ≤ Z ≤1 Tm(z) = cosh[m cosh-1(z)] | z | ≥ 1 Tm(z) = cosh[m cosh-1(z)] | z | ≥ 1 T m (z) = 2z T m-1 (z) – T m-2 (z)

Determination of R o and Z o

R o is defined as the ratio of the major -to- minor lobe intensity. R o is defined as the ratio of the major -to- minor lobe intensity. z o is defined as the point (z) at which T m (z o ) = R o. The order m of the Tschebyscheff polynomial is always one less than the total number of elements N. z o is defined as the point (z) at which T m (z o ) = R o. The order m of the Tschebyscheff polynomial is always one less than the total number of elements N. R o is defined as the ratio of the major -to- minor lobe intensity. R o is defined as the ratio of the major -to- minor lobe intensity. z o is defined as the point (z) at which T m (z o ) = R o. The order m of the Tschebyscheff polynomial is always one less than the total number of elements N. z o is defined as the point (z) at which T m (z o ) = R o. The order m of the Tschebyscheff polynomial is always one less than the total number of elements N.

4 – Select the appropriate array factor

5. Expand the array factor. Replace each cos(mμ) function (m = o, 1, 2, 3,…..) by its appropriate series expansion in terms of cos(μ). 6. Determine the point z = z o such that T m (z o ) = R o (voltage ratio). The design procedure requires that the Tschebyscheff polynomial in the -1≤ z ≤ z 1, where z 1 is the null nearest to z = +1, be used to represent the minor lobes of the array. The major lobe of the pattern is formed from the remaining part of the polynomial up to point z o ( z 1 ≤ z ≤ z o ). 5. Expand the array factor. Replace each cos(mμ) function (m = o, 1, 2, 3,…..) by its appropriate series expansion in terms of cos(μ). 6. Determine the point z = z o such that T m (z o ) = R o (voltage ratio). The design procedure requires that the Tschebyscheff polynomial in the -1≤ z ≤ z 1, where z 1 is the null nearest to z = +1, be used to represent the minor lobes of the array. The major lobe of the pattern is formed from the remaining part of the polynomial up to point z o ( z 1 ≤ z ≤ z o ).

7. Substitute cos(μ) = z / z o cos(μ) = z / z o in the expansion of the array factor. The cos(μ) is replaced by z / z o, and not by z, so that cos(μ) = z / z o would be valid for | z | ≤ | z o |, and it attains its maximum value of unity at z = z o. in the expansion of the array factor. The cos(μ) is replaced by z / z o, and not by z, so that cos(μ) = z / z o would be valid for | z | ≤ | z o |, and it attains its maximum value of unity at z = z o. 8. Equate the array factor from step (6), after substitution of step (7), to a Tm(z), where, (m) is equal to (N – 1). This will allow the determination of the excitation coefficients an, s. 9. Write the array factor of step (4) using the coefficients found in step (8). 7. Substitute cos(μ) = z / z o cos(μ) = z / z o in the expansion of the array factor. The cos(μ) is replaced by z / z o, and not by z, so that cos(μ) = z / z o would be valid for | z | ≤ | z o |, and it attains its maximum value of unity at z = z o. in the expansion of the array factor. The cos(μ) is replaced by z / z o, and not by z, so that cos(μ) = z / z o would be valid for | z | ≤ | z o |, and it attains its maximum value of unity at z = z o. 8. Equate the array factor from step (6), after substitution of step (7), to a Tm(z), where, (m) is equal to (N – 1). This will allow the determination of the excitation coefficients an, s. 9. Write the array factor of step (4) using the coefficients found in step (8). continued

An Alternative Technique for Determining the Excitation Coefficients The excitation coefficients of a Dolph- Tschebyscheff array can be derived using various documented techniques. One method, whose results are suitable for computer calculations, is that by Barbiere. The coefficients using this method can be obtained using.. The excitation coefficients of a Dolph- Tschebyscheff array can be derived using various documented techniques. One method, whose results are suitable for computer calculations, is that by Barbiere. The coefficients using this method can be obtained using..

An Alternative Technique for Determining the Excitation Coefficients

The half-power beamwidth of a Dolph- Tschebyscheff array can be determined by: The half-power beamwidth of a Dolph- Tschebyscheff array can be determined by: 1 – Calculating the beamwidth of a uniform array (of the same number of elements and spacing) as shown previously. 2 – Multiplying the beamwidth of part (1) by the appropriate beam broadening factor f. The half-power beamwidth of a Dolph- Tschebyscheff array can be determined by: The half-power beamwidth of a Dolph- Tschebyscheff array can be determined by: 1 – Calculating the beamwidth of a uniform array (of the same number of elements and spacing) as shown previously. 2 – Multiplying the beamwidth of part (1) by the appropriate beam broadening factor f.

From the previous figure, it can be concluded that: 1 – The directivity of a Dolph-Tschebyscheff array, with agiven side lobe level, increases as the array size or number of elements increases. 2 – For a given array length, or a given number of elements in the array, the directivity does not necessarily increase as the side lobe level decreases. As a matter of fact, a -15 dB side lobe array has smaller directivity than a -20 dB side lobe array. This may not be the case for all other side lobe levels.