Cascade synthesis of transmission lines and lossless lumped networks

Abstract

Considerable interest in mixed lumped/distributed networks has arisen in recent years. The use of semiconductor elements, which have essentially lumped equivalent circuits, in microwave networks is an obvious example, and the need to meet even more exacting filter specifications at ever higher frequencies has resulted in many problems which are essentially mixed lumped/distributed. In most applications, the cascade type of structure is required, and it has been claimed that the sufficient condition for a 2-variable bounded real reflection coefficient to be realised by a resistively terminated cascade of lossless transmission lines and lumped 2-ports is that the transmission zeros are the product of the transmission zeros of the basic sections. Primarily by a particular numerical example, it is shown that this result is not generally valid for a canonic cascade realisation. This result is because of the possibility of ‘dummy’ sections existing within the network without violating the bounded-real condition at the terminals. Consideration of the existence of ‘dummy’ sections leads to the conclusion that an additional n(n−1)/2 conditions are required, where n is the number of transmission matrices for both terminated and unterminated networks. Additional conditions for reciprocity etc. are given in the realisability conditions, which are entirely algebraic in nature.