Advanced Electromagnetic Analysis of Passive and Active Planar Structures
With increasing operating frequencies of digital electronic devices, limitations are becoming evident in existing modelling techniques. This book acts as a bridge between the mathematical abilities of the pure EM theorist and those of the FET circuit modeller.
Inspec keywords: waveguides; electromagnetic wave propagation
Other keywords: multilayered substrate; 2D analysis; passive waveguides; active planar waveguides; active planar structures; passive planar structures; advanced electromagnetic analysis; closed waveguides; 3D analysis
Subjects: Filters and other networks; Waveguides and microwave transmission lines; Electromagnetic wave propagation; Transmission line links and equipment
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 aim of this chapter is to introduce the tools of electromagnetism that we shall use next. These are obtained from Maxwell equations and include the dyadic Green functions for stratified media with sources of finite dimensions. The derivation of some useful formulae is shown with a certain degree of detail, with a view to making the reader aware of the practical subtleties needed in their efficient utilisation, while leaving a more formalised and holistic description of the proposed approaches to the remainder of the book.
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.
The purpose of this section is to introduce some of the most widely used “full wave” techniques, namely techniques accounting for the hybrid nature of the modes supported by planar waveguides. This description is included not only for the sake of completeness, but also in order to provide a tool, whose results may be used as preliminary to those of a more recent technique. The latter is hence the central argument of the present chapter.
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.
The ultimate target of any analysis tool is the simulation of complex systems or subsystems involving filters, amplifiers, couplers and so on. In the previous chapter we have treated structures by assuming they were uniform along the propagation direction, namely, we dealt with 2D structures. The greater part of the information obtained may be directly used in the complete 3D design: once the characteristic impedance and the propagation constant of a transmission line are known, it is possible, to a first degree of approximation, to treat by a circuit approach a system involving cascaded lines or interconnected subsystems. Each element is treated by means of its matrix representation (whatever matrix has been selected: S, Z, transmission matrix...) and the matrix representation of the complete system may be computed by composing the matrix representation of each subpart. However, this way we are neglecting what happens when different structures are interconnected together, that is, whenever there is an abrupt change in a structure along the propagation direction of the wave. In this case, we have what is named a “discontinuity”, that is, a genuine 3D problem.
This appendix gives some helpful vector and differential identities.
In this appendix we prove identities (2.56) of Chapter 2. Let us consider a terminated transmission line to be excited by two different source distributions; let V(z) and I(z) be the voltage and current induced along the line by the source distribution v(z), i(z), while V'(z) and I'(z) are the voltage and the current induced by v'(z) and i'(z). Each set of values separately satisfies the transmission line equations (2.54).
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.
This book section presents the appendix of the Advanced Electromagnetic Analysis of Passive and Active Planar Structures.
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.
