Numerical Determination of Cascaded LC Network Elements from Return Loss Coefficients

The data for describing the behavior of a lossless network and for synthesizing a network which exhibits that behavior can be presented in many ways. One of the lesser used descriptions is the Taylor series expansion in
s
of the return loss about a transmission zero. As is well known, a return loss is the natural logarithm of the reciprocal of the reflection coefficient as measured between a resistance termination and the remainder of the network. While it is realized that the return loss expansion is not so immediately useful a network function as, for example, the input impedance or the reflection coefficient, the analytical aspects are very interesting in themselves and may well find application in future work. If the low-pass LC ladder network of
n
elements starting with a series
L
is considered, the first
(2n - 1)
coefficients of a given return loss expansion about the transmission zero at infinity contain all the necessary information for finding numerical values of the ladder elements. It can be shown that the first coefficient depends on the first ladder element, the third coefficient depends on the first and second elements, etc. Formulas for finding up to four elements from the return loss expansion are available. However, a recursion form for extending the range of these formulas is not immediately evident from these available formulas. Two general equations, one for the series
L
's and one for the shunt
C
's are presented. The equations depend only on a knowledge of the Taylor coefficients for the particular type of ladder network under consideration. The method of finding the
L
's and
C
's is a straightforward algebraic approach and is novel only in that the elimination of redundant information leads to simple expressions for the
L
's and
C
's. Application of the equations leads to a recursion method for alternately finding an
L
, the succeeding
C
, the next
L
, etc. Accumulated results from one equation are used in finding the next equation.