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.

Inspec keywords: Helmholtz equations; Rayleigh-Ritz methods; ridge waveguides; finite element analysis; boundary-value problems; eigenvalues and eigenfunctions

Other keywords: finite element method; eigenvalues; functionals; wave equation; eigenvectors; Rayleigh-Ritz procedure; scalar homogeneous Helmholtz differential equation; magnetic boundary condition; variational calculus; ridge waveguide

Subjects: Waveguides and microwave transmission lines