Among the very large number of guiding structures proposed and used in microwave applications, planar waveguides have proved to be the most successful solution to the problem of the interconnections in microwave and millimetric wave circuits. There are many reasons for this success; at first, it is apparent that planar waveguides are a “natural” solution in microwave monolithic integrated circuits (MMICs), being the direct evolution of classical interconnecting wires; hence microstrips as well as coplanar waveguides are completely compatible with the existing technological procedures for the metallisation and passivation of monolithic integrated circuits. Moreover a wide range of characteristic impedances are easily obtained by varying the line cross section: shape and dimensions of the waveguide are fully controlled by masking in the lithographic process; an additional opportunity is provided by strips on semiconductor substrates, behaving as Schottky junctions at the same time as electrically tunable transmission lines.

The very nature of this chapter is theoretical and investigative, mainly consisting, in the first part, of a study of the consequences of the Lorentz' reciprocity theorem as applied to closed waveguides. This part, extending up to Section 3.2, requires a bit of “mathematical technicality” - as for the entire Chapter 2 - while remaining quite abstract. Nevertheless, as the final step of an electromagnetic analysis normally involves the computer solution of some kind of equation, physical insights in the expected solution are really helpful in avoiding numerical trouble and in recognising so called “spurious solutions”. Spurious solutions, as we will see, are solutions of the numerical problem holding no physical meaning. Hence, we will derive general constraints linking propagation constant, power flow, dissipation and storage in closed, uniform waveguides loaded with linear, isotropic media. Some constraints are also derived on the frequency variation of the above quantities. In the particular case of a lossless waveguide, critical points in the characteristics and complex waves are discussed. The complex wave phenomenon in lossless media still posing some intriguing questions, is carefully investigated leading to the statement of a fundamental theorem.

In spite of the very large literature covering the subject of ideal planar structures, only a few and quite recent works address the problem of real planar structures and the concepts involved. As a further proof of the analytical challenge that losses pose, there is also a nearly complete lack of available experimental data in the literature. The main purpose of the this chapter is to explore this kind of structures by means of a generalised transverse resonance-diffraction method.

This appendix gives some helpful vector and differential identities.

In this appendix are reported the close formulae for the complex propagation constants of the FET two line model (grounded source) described in previous chapters.

The purpose of this appendix is the derivation of the DGF, linking electric fields and electric sources, for a 3-dimensional boxed dielectric slab, as required in Chapter 6.