Synthesis of transfer functions with poles restricted to the negative real axis into two parallel R-C ladders and an ideal transformer

This paper presents a method of synthesizing a circuit to obtain a transfer function whose poles lie on the negative real axis and whose zeros lie anywhere in the complex frequency plane including the positive real axis by means of two parallel resistance-capacitance (R-C) ladders and an ideal transformer. The polynomial in the numerator of the given transfer function is first resolved into the difference of two component polynomials whose zeros are all negative real numbers. The method of resolution is based on conditions for collinearity of points that represent nth degree polynomials (with real coefficients) in n-dimensional space. The two component polynomial fractions for the given transfer function are then synthesized into two R-C ladders and the resulting network is the parallel combination of these two ladders through an ideal transformer and terminated by another R-C ladder. The method applies to more generalized cases and usually gives a network that contains less resistive and capacitive elements than any of the other methods available. If the resolution of the numerator polynomial is as a sum of polynomials instead of a difference, more than two components may be required. An expression for the upper bound to the least number of such components is also included in the paper.