Coupling Matrix Synthesis Using Groebner Basis

Abstract

This paper presents a new analytical approach for a problem of coupling matrix (CM) synthesis. The approach is based on reformulation of well-known equations representing filter scattering parameters S <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">11</inf> and S <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">21</inf> for a given coupling matrix. With the use of relation connecting inverse matrix its determinant and its adjugate matrix the system of equations is formed. The equations are reformulated in a way allowing direct comparison of polynomials coefficients for numerator and denominator of S <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">11</inf> and numerator of S <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">21</inf> . This enables to form the system of polynomial equations, where unknowns are all entries of searched coupling matrix. In the proposed approach no conversion from S-parameters to Y-parameters is needed like in most of published techniques. SINGULAR software (GPL license) using Groebner basis for solution of such a system of polynomial equations was used. Two numerical experiments have been presented. First one CM synthesis for 4th order (one cross-coupling) filter. Second one is CM synthesis for 8th order (two cross-couplings) filter. Both experiments showed excellent results obtained with the proposed method. The method works for S-parameters of filters with arbitrary topologies.