The previous section assumed that every element in the array radiated with the same element pattern. In fact an element in an array does not radiate the same pattern as if it were radiating in an isolated environment, nor does it present the same impedance or support the same current or field distribution. These effects are all evidence of a phenomenon called mutual coupling, depicted in Fig. 9(a), by which every element of the array interacts or couples with every other element. In the general case even the shape of the current distribution on each element changes with scan angle, and one must use a higher-order current approximation to evaluate the total radiation for each angle in space. Between these two limits, one assuming no interaction and one assuming fine scale changes with the scan angle, is the case where one can speak of single-mode elements and single-mode mutual coupling.

The solution of the general array mutual coupling problem involves a complex integral equation formulation and numerical solution. The solution is expressed in terms of a series of basis functions (possibly the harmonics in a Fourier series distribution) used to approximate the total current or aperture field. For the purpose of this discussion, it is convenient to think of these basis functions as modes and to consider the case where a single mode is a good representation for the current distribution on each antenna. This is often a good approximation because most array elements are small compared to a wavelength, and all element currents or fields are often nearly the same.

This one-mode assumption makes it simpler to explain the two complementary views of array mutual coupling. We will call these the mutual impedance viewpoint and the element pattern viewpoint.

From the mutual impedance perspective, we assume the single mode radiates with a pattern f(u, v), which might be a vector quantity. Each transmission line excites an element that radiates into all other transmission lines through their elements, as indicated in Fig. 9(a). For an N element array on some nth element, the radiating field or current, here called In, is related to an input voltage matrix for the whole array by the square N x N impedance matrix Z (V = ZI). The In are unknown, so to compute the

-90 -72 -54 -36 -18 0 18 36 54 72 90 |

Angle 9 (deg)

Figure 9. Array mutual coupling. (a) Coupling between array elements. (b) Element pattern P(e) and reflection coefficient magnitude R of center element of unloaded waveguide array (B/X = a/X = 0.4) after Wu (3).

array radiation, one needs to invert the impedance matrix that relates the applied signals Vm to the produced In:

In summary, from this perspective one can find the array radiation from the applied sources by solving for the actual currents (or fields) that result. It turns out that the common problem of synthesizing a desired radiation pattern is handled by solving for the desired current and then using the impedance matrix to find the necessary applied sources.

The alternative point of view is focused on the array “element patterns” that radiate when each element is excited separately, with all other elements terminated in a matched load. When only one transmission line is excited, the total pattern is generated as the sum of contributions from all the element radiation. Consider a small aperture element that supports a single mode of field with radiation pattern f(u, v). If that nth element alone were to radiate when excited by an incident signal an from the nth transmission line, then the radiation pattern of that element would be ane(u, v)e-jk• xxn. The exponential term is due to the location of the element in the array. However, as indicated in Fig. 9(a), that element scatters its radiation into every other element, inducing a field on any m’th element that is given mn’th term of the scattering matrix S. The total radiation from the array with one element excited is

thus: |

ARRAY SYNTHESIS |

where

iV

■jk xixm — x„)

This expression shows the radiation to consist of a primary radiation from the excited element, plus a scattered term given by Snn times the primary term, plus terms Smn times the primary term but radiating from the location of the other elements (xm). From this perspective each element radiates with a different element pattern fn(u, i>)because each element pattern contains radiation from every element of the array. The total radiation is the sum of all these element patterns weighted by the incident transmission line signals. Figure 9(b) shows an example (3) of the element pattern of an isolated antenna element and the same element in an array. This figure shows the element pattern of the center element of an array of N waveguide elements for N = 5, 9, 11 and an infinite array. The figure shows the effect of mutual coupling on element pattern as resulting in periodic ripples with higher periodicity for longer arrays and that the end-fire gain (00 = ±90 degrees) reduces because of coupling until it is zero for the infinite array case. One can show that the optimum gain varies like cos 0 for the infinite case.

Many useful pattern synthesis techniques for planar or linear antenna arrays follow directly from existing methods developed for aperture or continuous one-dimensional antennas. This is so for several reasons. First, as long as the elements are closely spaced and grating lobes well out of the radiating region, the array periodicity does not significantly alter the pattern structure. Second, the distinctions that do exist come from the mutual coupling and are evident in array edge effects, or equivalently from the observed different element patterns across the array. As long as the elements support only the single-mode fields, these issues do not alter the synthesis procedure, since one can synthesize in terms ofthe currents and aperture fields that create radiation, or in terms of the measured or computed array element patterns, and then include mutual coupling to evaluate the necessary applied excitation.

The basis for most aperture synthesis is the Fourier transform relationship between aperture field and far field for a continuous aperture. If the arrays are large and the elements closely spaced, this procedure is not sensitive to the discretization or edge effects, and the method is quite accurate. The transform method is also especially convenient because of its application to arrays that are not periodic and for arrays conformal to gently curved geometries.

Arrays periodic in one or two dimensions have far-field patterns describable by discrete Fourier transform pairs. In one dimension the array factor at wavelength X is written

kndx |

iV

(21) |

F(u) =