Published in IET Microwaves, Antennas & Propagation Received on 20th February 2009 Revised on 17th July 2009 doi: 10.1049/iet-map.2009.0059 ISSN 1751-8725 Parametric history analysis of resonance problems via step-by-step eigenvalue perturbation technique S. Gu ¨ nel E.Y. Zoral Department of Electrical and Electronics Engineering, Dokuz Eylu ¨l University, Tinaztepe Kampusu, 35160 Kaynaklar, Buca, I ˙ zmir, Turkey E-mail: serkan.gunel@eee.deu.edu.tr Abstract: The Galerkin procedure to solve Maxwell’s equations associated with a perturbed system approximately, yields a generalised eigenvalue perturbation problem. Instead of solving the generalised eigenvalue problem, perturbed eigenvalues and eigenvectors can be approximated in terms of unperturbed ones. Although solving the perturbed eigenvalue problem can be computationally attractive, the small perturbation requirement may be quite restrictive. This restriction can be relaxed using iterative perturbation techniques in which the problem is divided into small perturbation steps, and then each subsequent problem is solved depending on the solutions of the previous step. Besides, the step-by-step iterative solution also provides a parametric history of the system behaviour. In this study, the parametric history analysis of electromagnetic resonant structures has been accomplished using the step-by-step eigenvalue perturbation method. To illustrate the proposed method, the reanalysis of perturbation of a cylindrical cavity with a dielectric sample has been examined. The results obtained using the parametric history analysis are compared with the theoretical, the experimental and the results of the classical perturbation approach. 1 Introduction The perturbation techniques involve approximating the solutions of a slightly modified system in terms of the solutions of the unmodified system. Essentially, such techniques require the perturbed parameter to be small compared to the original parameters of the unperturbed system. These techniques are widely used in electromagnetic engineering problems [1, 2]. Typically, the deviation of the resonant frequency of a modified resonant structure is examined. Consider two electromagnetic resonant structures that have slightly different properties. If the fields are designated as E i , D i , H i and B i in corresponding structures with n i  E i ¼ 0 on surface S i and the material properties as 1 i , m i (i ¼ 1, 2), it can be shown that the resonant frequencies of these structures are related with the following formula Dv v 1 ¼ Ð V (E 2  D 1  E 1  D 2 )  (H 2  B 1  H 1  B 2 )dv Ð V (E 1  D 2  H 1  B 2 )dv (1) where V is the union of the volumes enclosed by the boundaries S i : Dv ¼ v 2  v 1 , and v i ’s are the resonant frequencies of corresponding resonant structures [3]. A typical application is to perturb a structure with a material of unknown properties. The electrical properties of the sample can then be derived using (1) by measuring the frequency shift Dv [4–7]. An alternative approach is to consider the problem as an eigenvalue problem and to approximate the perturbed eigenvalues and eigenvectors in terms of the unperturbed ones. It is known that, given a regular pencil A  lB, the 466 IET Microw. Antennas Propag., 2010, Vol. 4, Iss. 4, pp. 466–476 & The Institution of Engineering and Technology 2010 doi: 10.1049/iet-map.2009.0059 www.ietdl.org