Successive variational approximations of impedance parameters in a coupled antenna system

In this paper, a new variational formulation for a single impedance parameter of an
m
-antenna system is presented. This formulation enables one to determine any self impedance
Z_{ii}
, one at a time, merely by exciting antenna
i
alone and leaving all the other antennas open circuited. For determining any mutual impedance
Z_{ij}
, only two independent excitations, one the same as that used for determining
Z_{ii}
and the other for determining
Z_{jj}
, are required. Thus, if all the
m(m+1)/2
impedance are required, only
m
independent excitation conditions are needed. In contrast to this, the formulation available in the literature is based on
m(m+1)/2
independent excitation conditions. Because of a reduced number of excitation conditions and the way they are assumed, the physical nature of the problem is made simpler and easier to comprehend. Such comprehension helps considerably in the choice of trial current distributions for a specific application. Two methods of evaluating the successive higher-order approximations are also given. One is based upon an orthogonalization process, and the other is based upon the successive inversion of matrices. In the evaluation of a certain order approximation, both methods have the advantage of utilizing all the work already done for the lower-order approximations; and at the same time, additional work required is considerably reduced. It is believed that the formulation, as well as the two methods of successive approximations, will also be useful in other problems.