Objectives In this course you will learn the following Array specified by only its nulls. Radiation pattern of a general array. Array synthesis. Criterion for choosing number of elements in synthesized array.
In practice we need to design a variety of radiation patterns with a very tight control over the nulls and the side lobe amplitudes. A user specifies the radiation pattern and the antenna engineer is asked to design an array which can give the radiation pattern as close to the specified pattern as possible. In general we have to control the complex currents of the individual elements of an antenna array to get the desired radiation pattern. The array synthesis problem is not a very straight forward problem as there may not be unique array for a given radiation pattern. Here we discuss only the synthesis of linear arrays. Array specified by its nulls In an interference prone environment the arrays are to be designed to suppress the effect of the interfering signals. The interfering signals can be suppressed by placing nulls of the array factor in the directions of their arrival. Let there be an array of elements with complex currents respectively, where (i.e. currents are normalized with respect to ) The array phase is. Writing, the array factor of the array is --------------- (A) The is a polynomial of of degree. The polynomial can be factorized in binomials as --------------- (B) where are roots of the polynomial. The nulls in the radiation pattern occur for i.e. when. The array currents can be obtained from the knowledge or the nulls of the radiation pattern i.e. an array can be synthesized for given nulls in the radiation pattern. Array Synthesis Procedure (1) Given directions of nulls.
(2) Select appropriate inter-element spacing,. Generally is chosen to be. (3) Find corresponding to nulls : where (4) Find roots of the array factor (5) Substitute in equation (B). Expand the polynomial. (6) Coefficients of the polynomial give the array currents. Note : In this case there is no control over the direction of the main beam of the array and the array is completely defined by its nulls. Radiation Pattern of a General Array Let there be an array of (2N+1) elements with conjugate excitation as shown in Fig. The excitation currents are complex in general. The center element of the array is the reference element and its phase is zero by definition. The elements around the center have conjugate symmetry, i.e. equal amplitudes but opposite phases. Let us define the array phase is, and. The array factor then can be written as The conjugate terms can be combined to get the array factor as Writing the complex current, and, we get The RHS is the Fourier series expansion of the array factor in with having period. The array synthesis problem therefore reduces to the Fourier expansion of the AF. The Fourier coefficients give the
complex currents of the array. Note: To represent a periodic function, in general the Fourier series has infinite terms. However, the AF is expanded with a truncated series of N terms. Consequently, the synthesized radiation pattern is an approximation to the prescribed pattern. Larger the array (i.e. more elements in the array) better is the approximation to the desired pattern.
The synthesized pattern is the least square fit to the desired radiation pattern. The Fourier synthesis needs the radiation pattern specified in the specified as a function of. -domain, whereas in practice the radiation pattern is Conversion from to needs the knowledge of the inter-element spacing, d. For the Fourier synthesis, the array designer has to select a-prior the inter-element spacing. For the visible radiation pattern the angle, i.e.. This is called the visible range of. The visible range of is a function of the inter-element spacing, d, whereas for the Fourier expansion the period is always. For a unique synthesis, therefore the inter-element spacing should be equal to. The array synthesis steps are as follows: (1) Specify the radiation pattern for. (2) Select the inter-element spacing, d and the number of elements N in the array. (3) Get radiation pattern as a function of. If, there is uncovered range of within to. Chose some arbitrary function over the uncovered range of. (4) Find the Fourier series for the function and retain only (N+1) terms including DC. (5) The excitation currents are the Fourier coefficients. Some times the number of elements are not decided a-priori but the rms error between the synthesized and the desired pattern is specified. In this case the rms deviation between the two patterns is calculated for different values of N and satisfactory choice of N is made.
Recap In this course you have learnt the following Array specified by only its nulls. Radiation pattern of a general array. Array synthesis. Criterion for choosing number of elements in synthesized array.