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5 to account for the field coupling between elements and use transmission line relationships to account for the second kind of coupling as done by Agrawal (1969). To facilitate the application of reaction-matching, we assume that the antenna is perfectly conducting and that each loop, including its image, is divided into eight segments as shown in Fig. 37. The currents at the junctions are denoted by I0 1 , X42 , • • • , X424 . + ~ 1023 1021 Z 02! FIG. 37. Junction currents and their reference directions for three-element TEM-line antenna (courtesy of P.

134b) and then the solutions for these systems are respectively (x = —3 , y = 4),(X = 2, y = 4) and x = 3 , y = - 2). Here we see that the relative changes in the solutions are of the order of hundreds of per cent while the coefficient changes were only a fraction of 1 per cent. Each of these three ill-conditioned systems of equations has a nearly singular coefficient matrix which is the reason for the sensitivity of the solutions to the coefficient values. If one normalizes the coefficients in each equation, it will become obvious that the value of the coefficient determinant is nearly zero, being much less than unity in each case.

Z '* a 2M NM ... Zá11 zá12 z';21 ' za11 Zá 2 2 ... ,z aN2 .. Zá1M z 21 a11 Z~* 11 * ... 100) where the superscript * denotes the complex conjugate. The product [Z~]* [Z ] , where denotes the matrix transpose, yields an M x M complex symmetric matrix. The solution of eqn. 100) provides a least mean square "match" by M modes of the boundary conditions at N points on the wire. Only M simultaneous equations need be solved, where M, the number of modes, may be less than 10, while the number of match points, N, may be greater than 100, for example.

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