## The Wave Concept in Electromagnetism and Circuits

1

General Principles of the Wave Concept Iterative Process

1.1. Introduction

The iterative method, which uses a wave network, is an integrated method and is not based upon electric and magnetic fields, as are, for example, Electrical Field Integral Equation (EFIE), Magnetic Field Integral Equation (MFIE), or more generally the method of moments or a combination of both fields. These are likened to the amplitudes of transverse waves, both diffracting around obstacles and those in space, termed "free space", owing to the presence of evanescent fields. However, while the method of moments appeals to so-called admittance or impedance operators, within the wave iterative method (Wave Concept Iterative Process (WCIP)), the diffraction operators are restricted, thus leading to the convergence of all iterative processes based upon this particular formalism [BAU 99].

It may be noted that, with the method of moments, the solution to the problem often entails using a restriction in the given field so as to define trial functions that constitute the basis for given solutions. This often leads to both analytical and numerical problems. In the WCIP method, field conditions are simply described on the basis of pixels which make up the entire sphere.

Moreover, the iterative process has a significant resemblance to that used within harmonic equilibrium [KER 75]. Within this latter process the nonlinear component behaves in a way that is described in relation to time, while the rest of the circuit is described within the frequency sphere. The operator thus functions diagonally at given frequencies. With each iteration, we therefore proceed with a Fourier transform (using a time-frequency basis) so as to approach the detailed composition of boundary conditions at the shutdown level. Moreover, when writing equations in terms of components studied over time, an inverse Fourier transform (based upon frequency-time) is used.

The WCIP approach is closely related. By simply replacing time by a coordinate and the frequency by a "spatial frequency", the operation reverts to one within the spectral sphere. Outside of the Transverse Lines Matrix (TLM) method, which also necessitates the wave concept [KRU 94], the WCIP is based upon the systematic iteration between both incident and reflected waves. The approach used in the paragraphs below is as follows: select a wave definition which is consistent with pre-existing cases, in particular within waveguides, and ensure that it has a fundamental physical significance. The iterative process will then be described in the context of several types of problems, in particular quasi-periodic structures.

The objectives of this chapter are to first set out the WCIP, showing its potential for circuit modeling, antennae and quasi-optical devices within stratified environments [BAU 99, AZI 95, AZI 96, WAN 05, RAV 04, TIT 09]. There are two advantages to this method. Firstly, the iterative process is always convergent (excepting the frequency resonating from a mechanism such as that one which is also relevant to other digital methods). Secondly, by the description of all surfaces through the use of pixels, it is not necessary to use a network describing the part of the surface corresponding to a metallic coating (or indeed to the dielectric dual), as falls within the sphere of the method of moments.

In the second part of this chapter, the WCIP is outlined. The principles of the WCIP are adhered to. Through the use of combined equations, one is expressed in the spatial sphere and the other in the spectral sphere (also called the modal sphere). The solution is obtained by achieving equilibrium between these two spheres. The description of a given mechanism is not set by rectangular pixels but by cells restricted by periodic barriers, each containing periodic non-configured sources. The sources are described within the