Teaching Scattering Matrix in Electrical Engineering—Misconceptions and Clarifications

Abstract

This paper presents a new approach to introducing scattering parameters for dispelling doubts and misconceptions. Normalization of voltage and current is first introduced for the subsequent descriptions of normalized matrices. The analogy with lossless transmission lines is used to define voltage waves in terms of circuit voltages and currents. Scattering parameters are interpreted as reflection and transmission coefficients for real normalizing impedance. Power considerations are made to interpret scattering parameters as power ratios. The condition for the lossless circuit is derived, and it is shown that the definitions of the voltage waves satisfy maximum power transfer conditions for real input impedance. Reciprocity conditions are derived to show that the normalized, not the unnormalized scattering matrix, is symmetric. The application of losslessness and reciprocity in filter design is clarified. Next, it is shown why definitions of forward and reverse voltage have to be changed for complex normalizing impedances. The relationships of matrices for complex normalizing impedance are shown to reduce to those for real normalizing impedance. The procedure for changing normalizing impedance is given, and its application for amplifier design is briefly described. The difference between circuit symmetry and physical symmetry is pointed out. Finally, it is shown how odd and even mode analysis arises from a mathematical consideration of symmetrical two-port networks. A straightforward discussion of Wilkinson power divider is given to illustrate the application of these modes.