# 1.3: Chapter Outline

- Page ID
- 41001

The chapter begins by considering distributed circuits in Section 1.4 and identifying the types of structures to be considered in this book. Then Section 1.5 revisits Maxwell’s equations from a perspective different from the way it was (almost certainly) introduced to the reader. It is assumed that the reader has a knowledge of Maxwell’s equations which are nearly always introduced following a treatment of the static field laws: Biot–Savart law, Ampere’s circuital law, Gauss’s law, Gauss’s law for magnetism, and Faraday’s law. The classic treatment follows the historical time line and then Maxwell’s equations are introduced. It was almost as though Maxwell’s equations were derived from the static field laws but that was never stated because they were not. Maxwell’s equations represent a separate physical insight. Maxwell’s equations cannot be derived from the static field laws, however the static field laws can be derived from Maxwell’s equations. Maxwell had remarkable insight and interpreted the physical world in an amazing way and this places him among the greatest physicists of all time. He touched on relativity and quantum field theory in imagining that the electric and magnetic fields, and light, formed a two-component field where the spatial derivative of one component was related to the time derivative of the other. He imagined that this relationship imposed a cosmic speed limit. The components of Maxwell’s field are the electric and magnetic fields but the insight was much more fundamental than that.

The next section, Section 1.6, demonstrates that all of the early static field laws can be derived from Maxwell’s equation. Maxwell unified what seemed to be a number of unconnected observations but at the time many suspected that there was an underlying reality. The next few sections derive some important short-hand techniques that are useful in working with distributed circuits and in particular transmission lines. Section 1.7 describes how the EM fields interact with lossless materials. It is no secret that Maxwell’s equations are fiendishly difficult to work with unless great simplifications are made. While tremendous computational EM analysis tools are available the design engineer needs insight. One of the critical pieces of insight that guides microwave engineers is to imagine a magnetic wall analogous to a conductor which is an electric wall. A magnetic wall is approximated at the interface of two dielectrics. The concepts of electric and magnetic walls analyzed in Section 1.8 provide an insight that is particularly useful to the microwave engineer is developed. These results will be used throughout the chapters in this book. Actual dielectrics and conductors are lossy so the effect of loss is considered in Section 1.9. Examples of how insight and fortuitous use of geometry, symmetries, and physical insight can be used in EM calculations are given in Section 1.9. Generally only particular structures with the requisite symmetries and simple geometry are used in design because if a structure is too difficult to analyze then the important intuitive insight is hard to acquire.

The final part of this chapter is an appendix on mathematical foundations with the required essentials of trigonometric expansions, mathematical identities, complex arithmetic, Butterworth and Chebyshev polynomials used in matching network and filter design, the properties of circles on a complex plane are used in dealing with complex numbers plotted on a polar plot, Kron’s method used in network condensation and thus in simplifying designs, and the mathematics of random processes which are used in working with digitally modulated signals and with noise. Everything in this appendix is used somewhere in this book series.