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On the synthesis of two-element-kind multiport networks Stein, Richard Adolph

Abstract

Procedures for the synthesis of a class of two-element-kind multiport networks are developed and illustrated by examples. In the RC case, the networks consist of the series connection of an R network and an RC network. The latter contains at least one capacitor tree so that its open-circuit impedance matrix vanishes at infinite frequency. It is shown that for a network within the class, the open-circuit impedance matrix and the normal form state model are equivalent in that, given one, the other can be written immediately. The synthesis problem then takes the form of the determination of a normal coordinate transformation such that the transformed state variables may be identified as capacitor voltage variables in a passive RC network. Two procedures are described for determining a transformation (modal) matrix which yields an irreducible realization of a given kxk, nͭʰ degree impedance matrix. There are ½(n-k) (n-k-l) degrees of freedom in the modal matrix. General analytical solutions are possible when n≤k+2, one greater than in existing methods. The main procedure yields a network with, in general, n + ½k(k+1) capacitors. A set of necessary conditions, easily applied to the given impedance matrix, is derived. Necessary and sufficient conditions are given for the special case k=2, n=3- An alternative procedure yields a network with n capacitors. Using either procedure, it is possible to simultaneously minimize both the number of elements and the total capacitance in the network. By introducing additional equations into the main procedure, numerical solutions for the modal matrix may be determined for any value of n. With k=2, the procedure yields a new class of minimal, grounded two-port networks consisting of one π-section and n-2 T-sections connected in parallel. The severity of the realizability conditions is approximately proportional to n. A given 2x2 impedance matrix may be realized exactly, or one driving-point impedance function may be realized, exactly and the transfer impedance function with a desired gain factor (within the limits of realizability). A computational procedure is given which minimizes the total capacitance and optimizes the voltage gain factor.