The far-field pattern of an array of N elements in the generalized geometry of Fig. 6 can be written in terms of the incident transmission line voltage signals an and the element locations r„ as
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for
rn =XXn +2Zn k —xkx +ykv +£k.
where kx = ku, ky
kv, kz = k cos в, and k = 2n/X.
The parameters u = sin в cos ф; v = sin в sin ф are called direction cosines of the angular location (в, ф). The use of the direction cosine parameters u and v greatly simplifies the evaluation of beamwidth, bandwidth, and scanning parameters, so they are routinely used in array analysis.
In these expressions, k is the vector wave number or propagation constant of a plane wave radiating in the direction of (в, ф) and k = 2n/X at wavelength X. fn (в, ф) is the far-field pattern of that nth element in the array environment.
The multiplying factor exp(-jkR0)/R0 is common to all expressions, and it will be suppressed hereafter. In this expression we have assumed each element to have a single input port, but an obvious generalization would include additional ports by including additional an fn terms. Finally, the above summation applies for generalized array of any dimensionality and conforming to any surface.
In general, the element patterns fn (в, ф) are different for each array element even though all the elements are the same. These differences occur because each element pattern includes the scattered radiation from all other elements, from the array edges, and any other scattering from the mounting structure. Element pattern differences due to element interaction will be discussed later. In the special case of array elements mounted conformal to some nonpla — nar body, like a rocket or aircraft nose radome, the elements do not even point in the same direction, so the element patterns are fundamentally different in this case.
Before discussing the general cases, let us consider a linear array of elements located in a plane at xn with yn = zn =0 but otherwise arbitrarily spaced in the x direction:
If all of the element patterns fn (в, ф) are the same [and given by f(e, ф)], then the expression becomes
When this separation can be made, the pattern is expressed as the product of an element pattern /(в, ф) and an array factor (the indicated summation).