The determination of the electromagnetic fields within any region is dependent upon one's ability to solve explicitly the Maxwell field equations in a coordinate system appropriate to the region. Complete solutions of the field equations, or equivalently of the wave equation, are known for only relatively few types of regions. Such regions may be classified as either uniform or nonuniform. Uniform regions are characterized by the fact that cross sections transverse to a given symmetry, or propagation, direction are almost everywhere identical with one another in both size and shape. Nonuniform regions are likewise characterized by a symmetry, or propagation, direction but the transverse cross sections are similar to rather than identical with one another.

Waveguide structures are composite regions containing not only uniform or nonuniform waveguide regions but also discontinuity regions. The latter are regions wherein there exist discontinuities in cross-sectional shape; these discontinuities may occur within or at the junction of waveguide regions. The equivalent circuits representative of the discontinuities together with the transmission lines representative of the associated waveguides comprise a microwave network that serves to describe the fields almost everywhere within a general waveguide structure. The present chapter is principally concerned with the general nature and properties of the parameters that characterize such microwave networks.

A structure that contains a geometrical discontinuity is designated as a four-terminal, or two-terminal-pair, waveguide structure if it comprises an input and an output region each in the form of a waveguide propagating only a single mode. The over-all description of the propagating modes is effected by representation of the input and output waveguides as transmission lines and by representation of the discontinuity as a four terminal lumped-constant circuit. The transmission lines together with the lumped-constant circuit form a four-terminal network that determines the reflection, transmission, standing-wave, etc., properties of the over-all structure. The quantitative description of the transmission lines requires the indication of their characteristic impedance and propagation wavelength; the description of the four-terminal circuit requires, in general, the specification of three circuit parameters and the locations of the input and output terminal planes.

An arbitrary junction of four accessible waveguides, each propagating only a single mode, is termed an eight-terminal or four-terminal-pair waveguide structure. So likewise is termed an arbitrary junction of three accessible waveguides in one of which two modes can be propagated while in the remaining two of which only one mode can be propagated. The over-all description of the propagating modes in such structures is obtained by representation of the waveguide regions as transmission lines and by representation of the junction region as an eight-terminal lumped-constant equivalent circuit. The four transmission lines and lumped-constant circuit thereby required together comprise an eight terminal network indicative of the transmission, reflection, standing wave, etc., properties of the over-all structure. The quantitative description of the transmission lines requires the indication of their characteristic impedances and propagation wavelengths. For the description of the eight-terminal lumped-constant circuit it is, in general, necessary to specify ten circuit parameters and the location of the four associated terminal planes. This number of required circuit parameters may be reduced considerably if the structure possesses geometrical symmetries.