A waveguide used in many broadband microwave equipments is the ridge geometry. An important feature of this sort of waveguide compared to the conventional rectangular waveguide is the wider separation between the cut-off numbers of its dominant and first higher order mode. Another is the fact that its impedance is bracketed between that of the regular rect angular waveguide (377 Ω) and those of coaxial and stripline structures (50 Ω). The original ridge waveguide consisted of a regular rectangular waveguide with one or two ridge inserts. Most passive components that may be realised in conventional rectangular waveguides are also available in ridge geometry. This chapter includes some typical arrangements by way of introduction.

Important quantities in the description of any waveguide are the definition of its propagation constant and the voltage-current, power-voltage and power-current definitions of its characteristic impedance. One purpose of this chapter is to summarise some closed form descriptions of propagation, power flow and impedance in the ridge waveguide based on the transverse resonance method (TRM). Since the different notations introduced in its characterisation are on occasion difficult to follow readily, another purpose of this chapter is to reproduce the existing literature in a single nomenclature. Still another is to summarise graphically its voltage-current, power voltage and power-current definitions of impedance in a unified way.

One means of calculating the cut-off space and impedance of a ridge wave guide which avoids the need to describe the fringing fields at the edges of the ridges is the finite element method (FEM). One way of obtaining any of the definitions of impedance at any frequency in such a waveguide is to make use of the relationship between that at finite and infinite frequencies. The impedance at infinite frequency is readily deduced by recognising that the electric field distribution is invariant at cut-off and at infinite frequency and that the magnetic field at infinite frequency is simply related to the electric field there by the impedance of free space. This suggests that a knowledge of the electric field at infinite frequency or at the cut-off space is in practice sufficient for calculation.

The finite element method (FEM) has been widely utilised in the analysis of the cut-off space and propagation constant of dielectric loaded waveguides containing isotropic and anisotropic media.The purpose of this chapter is to give some calculations on the cut-off space, the propagation constant and the impedance of the ridge waveguide with a dielectric filler between its ridges based on a hybrid finite element calculation.

The ridge waveguide is not in practice restricted to the rectangular waveguide with one or two ridges. One or more ridges have by now been introduced into circular, square and triangular waveguides. A property of the round or square waveguide symmetrically loaded by four ridges is that its dominant mode can be decomposed into counter-rotating circular polarised waves on its axis. This sort of ridge waveguide supports Faraday rotation provided it is perturbed by a gyromagnetic material along its axis. Since the dominant mode solution of this structure has two-fold symmetry it is sufficient, insofar as it is concerned, to investigate the problem regions revealed by introducing suitable orthogonal magnetic and electric walls in all combinations. Its mode nomenclature coincides with that of the round or square waveguide obtained by removing the ridges. The nodal finite element method (FEM) again provides one means of investigating this sort of isotropic waveguide. The effect of depositing dielectric tiles on the ridges is also given some attention.

The transition between any two different waveguides is of some interest in the design of filters and other waveguide components. One canonical representation of such a discontinuity is a lumped element susceptance in cascade with an ideal transformer. The purpose of this chapter is to summarise some experimental data on a transition between an ordinary waveguide and a single ridge one. This sort of data may be experimentally extracted by making separate measurements on the external quality factor and resonant frequency of a half-wave long prototype. The network variables of the over all arrangement are separately established by using the ABCD notation. The chapter includes a careful characterisation of this problem region for each possible definition of impedance in the waveguide. The work outlined here omits, in keeping with some prior art, the effect of the lumped element susceptance of the discontinuity on the description of the external quality factor but, in keeping with some previous work, retains it in that of the midband frequency.

The design of 2-port microwave filters relies either on an exact synthesis procedure in the t-plane involving one kind of element separated by UEs or on an approximate s-plane technique involving one kind of element separated by immitance inverters. This sort of topology enters in the realisation of directly coupled bandpass filters using half-wave long cavities connected by metal or inductive posts. The circuit is not canonical since the impedance inverters do not contribute to the overall amplitude response of the filter but only provide a practical layout of the circuit elements. The design is completed by physically realising the impedance inverters. The original immittance inverter took the form of a simple quarter-wave impedance transformer. Such filters are known as quarter-wave coupled. The modern version, in which the inverter is realised by a step discontinuity, is known as a directly coupled arrangement. Since the notion of the immittance inverter is central to the design it is given special attention notwithstanding that it is a classic topic in the literature.

One important class of ridge waveguide hardware is the nonreciprocal 2-port one. Two classic devices are the nonreciprocal phase shifter and the resonance isolator. Each device relies on the two different interactions between a spinning electron with a clockwise or anticlockwise circularly polarised alternating radio frequency magnetic field in a suitably magnetised ferrite medium. If the alternating magnetic field rotates in the same direction as that of the electron spin then the medium exhibits a scalar permeability with one distinct value; if, however, it rotates in the oppposite direction it again behaves as a scalar medium but with a different value. This property of a magnetised ferrite medium is the basis of a number of important non reciprocal devices. One means of realising such components in a ridge wave guide is to recognise that such polarisation always exists at the interface between any two different semi-infinite dielectric regions and everywhere outside and that its hand is determined by the direction of propagation. Another significant property of this sort of arrangement is that the senses of rotations are in opposite directions on either side of a dielectric rib. Such dielectric inserts are therefore suitable for the construction of non reciprocal ferrite devices such as phase shifters and isolators. It is assumed, for simplicity, throughout, that the introduction of thin H or E-plane ferrite or garnet tiles in the vicinity of the dielectric wall does not in the first instance disturb the polarisation in its neighbourhood. While the double ridge waveguide does not have natural planes of circular polarisation in its trough regions it does have such planes at its electric symmetry wall. An H-plane bifurcation produces one possible carrier for the design of nonreciprocal phase shifters and isolators.

The 3-port circulator is a unique nonreciprocal symmetrical junction having one typical input port, one output port and one decoupled port. The fundamental definition of the junction circulator has its origin in energy conservation. It states that the only matched symmetrical 3-port junction corresponds to the definition of the circulator. One possible model of a circulator is a magnetised ferrite or garnet gyromagnetic resonator having three-fold symmetry connected or coupled to three transmission lines or waveguides.

A number of the calculations on the ridge waveguide described in this book amount to obtaining the eigenvalues and eigenvectors of the related planar problem region by using the finite element method. It is therefore appropriate to include a chapter on this technique. A property of a typical eigen solution of any problem region is that it must satisfy both the wave equation and the boundary conditions of the region. Solutions of the wave equation based on a separation of variables technique, however, only exist for regular geometries such as rectangular and circular structures. For irregular structures, a variational approach based on the fact that the stationary values of the energy functional of the problem region also satisfy the related scalar homogeneous Helmholtz differential equation must be employed. The stored energy of the circuit when integrated over the problem region is known as the functional of the problem region. If the region consists of top and bottom electrical walls then this quantity automatically satisfies a magnetic boundary condition on the side-wall of the problem region. If the problem region has top and bottom magnetic walls then it automatically satisfies an electric wall at its side-walls. If the former structure contains an electric wall or segments of such walls then these have to be separately catered for. A dual statement applies to the latter problem region. The process of obtaining a solution to a Helmholtz differential equation by extremising a functional is called a variational method. The use of the word 'functional' in this context serves as a reminder that it is not in itself a function but rather a function of functions. The stationary values obtained in this way satisfy, as will be demonstrated, the homogeneous Helmholtz differential equation.