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

The stated objective is to describe a procedure for determining the symbolic coefficients of the characteristic polynomial of a restricted class of
RLC
networks through the eigenvalue approach of Bashkow's
A
matrix. Theorem 2 is an algebraic method to determine each coefficient of the characteristic polynomial of an
LC
network (called half-degenerate) which has no
C
-only-circuits nor
L
-only-cutsets. The method uses Wang algebra but does not have to enumerate trees. Even for a large scale of half-degenerate
LC
network 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-degenerate
LC
network by inserting resistors in series with
L
's and in parallel with
C
's. The significance of Theorem 1 is that 1) the characteristic polynomials of the half-degenerate
LC
subnetworks, which are used to express the characteristic polynomial of the
RLC
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
n
is clear.