Symbolic formulation of coefficients of the characteristic polynomial of a restricted class of RCL networks
Mamoru Tanaka, C. Hishinuma, Susumu Mori · IEEE Transactions on Circuits and Systems · 1975
The stated objective is to describe a procedure for determining the symbolic coefficients of the characteristic polynomial of a restricted class ofRLCnetworks through the eigenvalue approach of Bashkow'sAmatrix. Theorem 2 is an algebraic method to determine each coefficient of the characteristic polynomial of anLCnetwork (called half-degenerate) which has noC-only-circuits norL-only-cutsets. The method uses Wang algebra but does not have to enumerate trees. Even for a large scale of half-degenerateLCnetwork each coefficient can be obtained algebraically, as well as individually, from Wang algebra operations of its elements\{C_1,C_2,\cdots,C_a,L_1,L_2,\cdots,L_{n-a}\}. 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-degenerateLCnetwork by inserting resistors in series withL's and in parallel withC's. The significance of Theorem 1 is that 1) the characteristic polynomials of the half-degenerateLCsubnetworks, which are used to express the characteristic polynomial of theRLCnetwork, 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 ordernis clear.