Abstract: The purpose of this project is to create a computer tool for the different Antenna Synthesis Method. Using Matlab, a graphical interface is created where the user will have the possibility to choose between the different methods available. For each one of the methods, matlab mfiles are generated with different input parameters. In this report, the Antenna Synthesis Methods are presented theoretically and an example of the different matlab output is presented. The following figure is the graphical interface for the Antenna Synthesis Method, with a scroll to choose between the different methods and the different input parameters needed to generate the results of the method.
Figure1: Graphical Interface
- 2/10 -
• Schelkunoff’s Polynomial Method: Given: - Spacing d - Phase Difference β - Position of nulls z1, z2, z3…zn Array Factor Calculation: AF = (z- z1) (z- z2)… (z- zn) Z= ejΨ and Ψ= kd cos(θ+β) Î plot AF vs θ Matlab file: Schelkunofftab.m
Figure2: Array Factor for 1 null at 90º ; d = 0.5λ and d = λ
- 3/10 -
•
Fourier Transform Method:
i) Continuous Line Source Distribution: Given SF(θ)d for a range of θ1≤ θ ≤ θ2 Assume kz=0 = phase distribution along length. ξ= kdcos(θ)- kz Calculate range for ξ corresponding to range for θ Current distribution is derived with: ξ 1 2 I a ( z' ) = SF (ξ )e − jz 'ξ dξ Where SF (ξ ) = SF (θ ) d 2π ξ∫1 Integrate and obtain I(z’) for −
Calculate SF (θ ) a =
l/2
∫ I ( z ' )e
l l ≤ z' ≤ 2 2
− jz 'ξ
dz '
−l / 2
Matlab file: FourierLineSource.m Example with the desired radiation pattern symmetrical about θ = π/2 given by: SF(θ) = 1 for π/4< θ > [cnorm,N] = FourierLinearArray(10,0.5); N = 21 >> cnorm' ans = -0.0100 0.0455 -0.0496 0.0101 0.0518 -0.0895 0.0578 0.0558 -0.2170 0.3582 1.0000 0.3582 -0.2170 0.0558 0.0578 -0.0895 0.0518 0.0101 -0.0496 0.0455 -0.0100 >>
- 6/10 -
•
Woodward Lawson:
i) Continuous Line Source Distribution: Given: Desired SF(θ)d - Length l Choose M ≤ l/λ The total number of sampling is: 2M+1 With - Sampling separation: Δ = λ/l - Sampling angle θm = cos-1(m Δ) - Coefficient bm=[ SF(θ= θm)] Calculation of the approximated SF: M
SF(θ) a =
∑ S (θ )
m=− M
m
⎧ ⎡ kl ⎤⎫ ⎪⎪ sin ⎢ 2 (cosθ − cosθ m ⎥ ⎪⎪ ⎣ ⎦ S m = ⎨bm ⎬ kl ⎪ (cosθ − cosθ m ⎪ ⎪⎭ ⎪⎩ 2 Î Plot SF(θ) a vs θ and SF(θ) d vs θ Matlab file: WoodLineSource.m
Figure6: Space Factor Plot vs θ – Woodward Method for Line Source
- 7/10 -
ii) Linear Array: Given: -
-
Desired AF(θ)d Length l Spacing d
The number of elements in the array is then: N = (l/d) +1 The method is the same as for the line source; Δ, θm, and bm are calculated. M
AF(θ) a =
∑ f (θ )
m=− M
m
⎧ ⎡ kl ⎤⎫ ⎪⎪ sin ⎢ 2 (cosθ − cosθ m ⎥ ⎪⎪ ⎣ ⎦ f m = ⎨bm ⎬ kl ⎪ (cosθ − cosθ m ⎪ ⎪⎭ ⎪⎩ 2 Î Plot AF(θ) a vs θ and AF(θ) d vs θ Matlab function: WoodArray.m
b) SF (θ )desired = cos 3 (θ )
a) SF (θ )desired = sin 3 (θ )
Figure7: Space Factor Plot vs θ – Woodward Method for Linear Array for different desired Pattern
- 8/10 -
• Taylor Line Source (one parameter): Given: - length = l - Ro = 1st Side Lobe level in dB Î Calculate R0 = 10^(R0indB/20) sinh(πB) and B with R0 = 4.603 * πB
The space Factor can be calculated with the following formula: ⎡ sinh (πB ) 2 − u 2 SF (θ ) = ⎢ ⎢⎣ (πB ) 2 − u 2
⎡ sin u 2 − (πB) 2 ⎢ ⎢⎣ u 2 − (πB) 2
With u =
⎤ ⎥ for u 2 ≤ (πB) 2 ⎥⎦
⎤ ⎥ for u 2 ≥ (πB) 2 ⎥⎦
πl cos(θ ) λ
Matlab file: TaylorOneParameter.m
Figure8: Taylor Line Source plot for different side Lobe level
- 9/10 -
Analysis of Results: The output results give significant results, and the plots are correct. This tool still needs some improvements: - The Taylor Line Source with Tschebyscheff error has not been implemented yet. - The schelkunoff’s Method does not accept multiple nulls as input parameters. (The .m function does accept separately one table as a parameter but has not been implemented in the graphical interface). This tool is still a nice example for the different Antenna Synthesis Method, and the modifications shouldn’t be too difficult to make.
