Characterization of Microwave Circuits: S ‐Parameters

Abstract

The purpose of this article is to demonstrate that the conventional S‐parameters cannot be applied to devices having complex characteristic impedances as in some cases the reflection coefficient can be greater than 1 even when terminated by a passive conjugate load for maximum power transfer. This is particularly pertinent for antenna problems. Also, the expression for the transfer impedance becomes quite complex and not suitable for optimization for maximum power transfer. These artifacts do not exist when the power wave scattering parameters are used. Thus, the first objective of this article is to introduce two different types of S‐parameters generally used to characterize microwave circuits with lossy characteristic impedance. The first one 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 1, even for a passive load impedance with a conjugately matched transmission line. Also, the power balance cannot be obtained simply from the powers associated with the incident and reflected waves. Hence, they cannot be applied for broadband characterization of antennas. The second type of S‐parameters is called the power wave scattering parameters . They are useful when one is interested in the power relation 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 relations 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 multiport) can be defined using complex reference impedances at each of the ports without any transmission line being present, so that the concept of characteristic impedance 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 the definition of 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 the pseudo‐wave and the power wave scattering parameters provide the same results. Finally, a general methodology with examples is presented to illustrate how the S‐parameters can be computed for an arbitrary network without any a priori knowledge of its characteristic impedance.