A comparison of perturbation formulas for a square electromagnetic resonator

Several perturbation formulas are available for the resonant frequency of an electromagnetic cavity subjected to shape and material perturbations. Each is based on a mathematical identity from vector calculus, a choice of trial fields, and the manner in which the boundary conditions are met. The limits of usefulness of a given perturbation formula depend on these factors. Slater proposed a perturbation formula for the resonant frequency of cavities that maintained currency for a long time. However, what remained obscure was that Dombrowski also proposed a perturbation formula, making a correction to that of Slater. A critical performance comparison of various perturbation formulas for a fixed geometry was not undertaken during the time they were proposed, for want of an exact solution to validate their accuracy. They were assessed only in some landmark or limiting cases. Today, several efficient numerical packages are available, and the need for perturbation formulas is rather grim or even completely obviated. If there is any interest in revisiting these formulas, it is all the more a matter of curiosity until there comes a revival in this area. The aim of this tutorial is then to apply about five different perturbation formulas to the same geometry (in the present case, a square cavity perturbed in shape); to obtain a closed-form expression for each; and to compare the limits of their accuracy. The performance of these formulas is rated with the help of commercially available HFSS software, using the eigenmode solver, both visually by plotting, and also using a power series. Along with these formulas, a new heuristic perturbation formula is proposed in this paper, which ranks number one among all of these in terms of accuracy as far as the present geometry is concerned.