Synthesis of Some Classes of Multivariable Cascaded Transmission-Line Networks

Since the concept of multivariable positive real functions was introduced, the synthesis of these functions and matrices, concerning passive networks composed of lumped elements and of lumped and transmission-line elements, has been one of the most interesting problems in the field of circuit theory. This paper considers the synthesis of cascaded transmission-line networks of noncommensurate unit elements with the following structures: 1) a cascade connection ofmtransmission linesNᵢ (i= 1, 2, ⋯ ,m)in a prescribed order, eachNᵢbeing a transmission line with open-ended stubs, composed of unit elements of a single variablepᵢ; 2) a transmission line with open-ended stubs composed of unit elements ofP₂, connected in cascade between two transmission lines composed of unit elements ofp₁; 3) a cascade connection ofmtransmission linesNᵢ (i = 1, 2, ⋯ ,m)in an arbitrary order, eachNᵢbeing a transmission line composed of unit elements ofpᵢ; and 4) a transmission line composed of commensurate unit elements with open-ended stubs, partitioned from a multivariable network.