Simultaneous Realization of the Transfer and Reflection Factors of Two-Ports and n-Ports

Abstract

A unified theory of n -ports (n = 1, 2, 3, \cdots ) with RLC or R(L + r)(C + g) circuit elements (without ideal transformers) has been described. The essential parts are the theory of the M function F(\Lambda) = U(\omega)) + j V(\omega) ; the realization of M functions by ladder networks; the two-ports specified by the general characteristic equation (GCE) g(\Lambda)g(\Lambda) = f(\Lambda)f(\Lambda) + h_{1}(\Lambda)h_{1}(\Lambda) + j_{1}(\Lambda)j_{1}(\Lambda) , or the sum of transmitted, reflected, and dissipated powers equals the maximum delivered power; the introduction of pseudoattenuation poles; the deformation of the curve of an M function; the principles of rearrangement of the GCE; and the reduction of an n -port to n - 1 two-ports. Simultaneously prescribed transfer (apart from a constant) and reflection factors are realized 1) by a ladder network for attenuation poles (AP) in the closed left-half \Lambda plane, and 2) by parallel-connected networks for AP in the open right-half \Lambda plane. Without the restriction to a minimum number of circuit elements, perfect coupling of a transformer in case 1) can be avoided. Case 2) requires a superfluous number of circuit elements, but consists of no transformer and can be applied to case 1). Case 2) is valid for RC networks and case 1) may be extended to RC networks. In case 1), an equivalent circuit of a physical system can be absorbed in the network without disturbing the prescribed frequency characteristic, and the values of circuit elements of the ladder network are explicit functions in U(\omega), V(\omega)) , and their first derivatives with respect to \omega at AP. These facilitate the application of network theory to the field of distributed circuit elements. In case 2), the values of circuit elements for realizing a prescribed voltage transfer factor can be expressed as functions of the coefficients of g(\Lambda) and f(\Lambda) ; thus the computations are simplified. The possibility is mentioned of extensions of this theory, such as the ladder network without transformers, etc.