Synthesis of a network containing a cascade of commensurate transmission lines and lumped elements

Simple necessary and sufficient conditions are presented under which a 2-variable admittance function may be realised as a cascade of commensurate lossless transmission lines with several lumped shunt capacitors located at specified positions at the junctions of adjacent lines and terminated in a positive resistance. Two types of networks are considered. The first consists of equal-delay transmission lines, the second contains not necessarily equal-delay but commensurate-delay transmission lines. The main results are obtained on the basis of Trudi's theory of bigradient arrays.