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Computation of shift in oscillation frequency in the cavity resonator caused by anisotropicity and inhomogeneity of the permittivity

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Abstract

Shift in oscillation frequency in cavity resonator caused by anisotropicity and inhomogeneity of permittivity utilizing perturbation method is formulated in this paper. Permittivity of medium is a tensor quantity with anisotropicity only in xy plane assumed. For this problem, governing Maxwell equations are decomposed into linear combination of two equations, i.e., perpendicular (xy) plane and z direction. By cavity perturbation method applied to combination of linear equations, shift in oscillation frequency is computed. The same cavity resonator is simulated on HFSS software for vacuum and a simple anisotropic medium, and results are cross-verified. Error in oscillation frequency is calculated for small inclusion of anisotropicity in vacuum. Perturbations are of the order \(10^{-3}\) to \(10^{-6}\), thus satisfying the necessity for employing numerical perturbation method. Oscillation frequencies in media having permittivity tensor with continuous inhomogeneous functions are also evaluated, which is superior to the theory of finite element method. Permittivity tensor with complex diagonal entries is also tackled in this paper but cannot be consummated on HFSS (high-frequency simulation software).

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Acknowledgements

The authors would like to thank the very useful guidance, criticisms and contributions extended by the people behind the scene. It is with great pride and gratitude that I present this research paper. Every pain and effort that I have taken would not have been fruitful if I had not the support and guidance of those who stood by me.

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Correspondence to Pragya Shilpi.

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Appendix A

Appendix A

Perturbation method for numerical analysis

Rely on the assumption that a small change in the problem induces a small change in the solution. First, transform the problem rewriting it in terms of a small perturbation parameter (let \(\delta \)). Set \(\delta = 0\) and solve the resulting system for solution \(f_0\) which is exact known solution. Perturb the system by allowing \(\delta \) to be nonzero (perturbation parameter \(\delta \ll 0\)). Then, formulate the solution to the related, perturbed system as a perturbation power series.

$$\begin{aligned} f=f_0+\delta f_1+\delta ^2 f_2+\delta ^3 f_3+\cdots \end{aligned}$$
(23)

This method gives an approximate solution also known as asymptotic approximation.

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Shilpi, P., Upadhyay, D. & Parthasarathy, H. Computation of shift in oscillation frequency in the cavity resonator caused by anisotropicity and inhomogeneity of the permittivity. Eur. Phys. J. Plus 135, 59 (2020) doi:10.1140/epjp/s13360-019-00089-w

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