An Exposition on the Choice of the Proper S Parameters in Characterizing Devices Including Transmission Lines with Complex Reference Impedances and a General Methodology for Computing Them

The purpose of this paper is to demonstrate that the recently published paper [1] dealing with little known facts and some new results on transmission lines is due to an incomplete interpretation of the nonphysical artifacts resulting from a particular mathematical model for the S parameters. These artifacts are not real and do not exist when a different form of the S parameters is used. The first objective of this paper is thus to introduce the two different types of S parameters generally used to characterize microwave circuits with lossy characteristic impedance. The first type is called the pseudo wave, an extension of the conventional traveling-wave concepts, and is useful when it is necessary to discuss the properties of a microwave network junction, irrespective of the impedances connected to the terminals. However, one has to be extremely careful in providing a physical interpretation of the mathematical expressions, as in this case the reflection coefficient can be greater than one, even for a passive load impedance with a conjugately matched transmission line. The power balance also cannot be simply obtained from the powers associated with the incident and reflected waves. Hence, this cannot be applied for broadband characterization of antennas. The second type of S parameter is called the power-wave scattering parameter. This is useful when one is interested in the power relationship between microwave circuits connected through a junction. In this case, the magnitude of the reflection coefficient cannot exceed unity, and the power delivered to the load is directly given by the difference between the powers associated with the incident and the reflected waves. Since this methodology deals with the reciprocal relationships between powers from various devices, this may be quite suitable for dealing with a pair of transmitting and receiving antennas where power reciprocity holds. This methodology is also applicable in network theory, where the scattering matrix of a two-port (or a multi-port) can be defined using complex reference impedances at each of the ports without any transmission line being present, so that the concept of characteristic impedances becomes irrelevant. Such a situation is typical in small-signal microwave transistor amplifiers, where the analysis necessitates the use of complex reference impedances in order to study simultaneous matching and stability. However, for both definitions for the S parameters, when the characteristic impedance or the reference impedance is complex, the scattering matrix does not need to be symmetric, even if the network in question is reciprocal. The second objective is to illustrate that when the characteristic impedance of the line or the reference impedances in question is real and positive, then both of the pseudo-wave and the power-wave scattering parameters provide the same results. Finally, a general methodology is presented with examples to illustrate how the S parameters can be computed for an arbitrary network without any a priori knowledge of its characteristic impedance.