Symbolic formulation of coefficients of the characteristic polynomial of a restricted class of RCL networks

Abstract

The stated objective is to describe a procedure for determining the symbolic coefficients of the characteristic polynomial of a restricted class of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">RLC</tex> networks through the eigenvalue approach of Bashkow's <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</tex> matrix. Theorem 2 is an algebraic method to determine each coefficient of the characteristic polynomial of an <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">LC</tex> network (called half-degenerate) which has no <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</tex> -only-circuits nor <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</tex> -only-cutsets. The method uses Wang algebra but does not have to enumerate trees. Even for a large scale of half-degenerate <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">LC</tex> network each coefficient can be obtained algebraically, as well as individually, from Wang algebra operations of its elements <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\{C_1,C_2,\cdots,C_a,L_1,L_2,\cdots,L_{n-a}\}</tex> . This implies that for its determination the new method requires less effort in computation over the existing tree enumeration methods based on Wang algebra. Theorem 1 is a method to determine the characteristic polynomial of an RLC network which is generated from a half-degenerate <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">LC</tex> network by inserting resistors in series with <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</tex> 's and in parallel with <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</tex> 's. The significance of Theorem 1 is that 1) the characteristic polynomials of the half-degenerate <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">LC</tex> subnetworks, which are used to express the characteristic polynomial of the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">RLC</tex> network, can be obtained from Theorem 2 in forms of power series of the complex frequencies variable 2, and then 2) the effect of insertion of loss parameters into a lossless network of order <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> is clear.