This report will analyze Uniform Linear Arrays (ULA) using MATLAB to run simulations and create plots. More specifically, how the number of elements affect certain properties of antenna array factors. These properties include: side lobe levels (SLL), directivity, and beam width. Furthermore, the amount of phase shift will be varied as well to see what effect it has. The Dolph-Tschebyscheff method will be used to create a tapered distribution for an array factor containing 14 elements with SLL of at least -18 dB. This is to investigate what changing the side lobes does to antenna behavior. These analyses will grant a better understanding both for antenna theory and practical applications.
II. UNIFORM LINEAR ARRAY
The array factor is created by summing complex exponentials using equation (1) from Antenna Theory: Analysis and Design by Constantine Balanis. The spacing d of each element is half of a wave length. The absolute value was found for each element of the array factor and then normalized to provide a plot in decibels. The MATLAB function is able to do this for both odd and even element arrays. Specifically, array factors consisting of 6 and 14 elements were computed and to compare to results given in Balanis' book.
Fig 1. ULA with 6 isotropic elements, d = λ/2.
Fig 1. ULA with 14 isotropic elements, d = λ/2.
(5) AF=n=1Nanejn-1(kdcosγ+Β) (1)
Both of these graphs plot the array factor (dB) against the theta value (in degrees). The MATLAB program is able to calculate both the directivity and beam width. Here is a table comparing the results from equation (2) from the Balanis to the results from the program.
| 6 Element Program| 6 Element Book| 14 Element Program| 14 Element Book| Directivity(dB)| 10.87 | 7.78 | 16.64 | 11.46 |
3dB Beamwidth(degrees)| 17| 16.98| 7| 7.25|
Table 1. Directivity and Beamwidth Comparison.
(5) D0≅2N(dλ) (2)
It is seen from the table that the results from the book and from the MATLAB program are very similar. Both plots were analyzed again but using a d value of three-fourths of a wavelength. Increasing the spacing allows increases the directivity and number of side lobes but decreases the beam.
Fig 3. ULA with 6 isotropic elements, d = 3λ/4.
Fig 4. ULA with 14 isotropic elements, d = 3λ/4.
II. TAPERING AND ELEMENT SHIFTING
The Dolph-Tschebyscheff method as described in Chapter 6 was used in order to lower the side lobes below -18 dB for the a 14 element antenna array. Using the polynomial recursion relation a 7 by 7 matrix was developed to compute the coefficients for the cosine summation.
(5) Tmz= 2zTm-1z- Tm-2(z) (3)
Fig 4. ULA with tapered distribution for 14 isotropic elements, d = λ/2.
All the side lobes have been successfully reduced in this plot. The tapered distribution is not too different from original array factor. To understand how tapering process affects the antenna array, a table shows side by side data comparison.
| 6 Element | 14 Element | 14 Element Tapered|
Sidelobe Heights (dB)| -15.25,-12.43| -22.87,-22.44, -21.48, -19.91,-17.47,-13.13| -18.21,-18.22,-18.22, -18.21,-18.21,-18.26| 3dB Beamwidth(degrees)| 17| 7| 7|
Table 2. Side lobe and Beamwidth Comparison.