Eigenvalue Problems in Surface Acoustic Wave Filter Simulations

Abstract

Surface acoustic wave filters are widely used for frequency filtering in telecommunications. These devices mainly consist of a piezoelectric substrate with periodically arranged electrodes on the surface. The periodic structure of the electrodes subdivides the frequency domain into stop-bands and pass-bands. This means only piezoelectric waves excited at frequencies belonging to the pass-band-region can pass the devices undamped. The goal of the work presented is the numerical calculation of so-called "dispersion diagrams", the relation between excitation frequency and a complex propagation parameter. The latter describes damping factor and phase shift per electrode. The mathematical model is governed by two main issues, the underlying periodic structure and the indefinite coupled field problem due to piezoelectric material equations. Applying Bloch-Floquet theory for infinite periodic geometries yields a unitcell problem with quasi-periodic boundary conditions. We present two formulations for a frequency-dependent eigenvalue problem describing the dispersion relation. Reducing the unit-cell problem only to unknowns on the periodic boundary results in a small-sized quadratic eigenvalue problem which is solved by QZ-methods. The second method leads to a large-scaled generalized non-Hermitian eigenvalue problem which is solved by Arnoldi methods. The effect of periodic perturbations in the underlying geometry is confirmed by numerical experiments. Moreover, we present simulations of high frequency SAW-filter structures as used in TV-sets and mobile phones.

Structured backward error for palindromic polynomial eigenvalue problems

“…It was first raised in the study of the vibration in the structural analysis for fast trains [5,6], and then of the behavior of periodic surface acoustic wave (SAW) filters [18]. PQEP with ⋆ = H was raised in the computation of the Crawford number by the bisection and level set methods in [4].…”

Smith forms of palindromic matrix polynomials

“…Coefficients of such polynomials have symmetry under certain involutions: A i → εA T k−i in the T -palindromic case, or A i → εA * k−i in the * -palindromic case, where ε = ±1. T -palindromic polynomials arise in the vibrational analysis of railroad tracks excited by high speed trains [11,20], and in the modelling and numerical simulation of periodic surface acoustic wave (SAW) filters [12,28]. Complex * -palindromic polynomials occur when the Crawford number of two Hermitian matrices is computed [10], as well as in the solution of discrete-time linear-quadratic optimal control problems via structured eigenvalue problems [2].…”

Polynomial Eigenvalue Problems: Theory, Computation, and Structure

“…The analysis of rail noise caused by high speed trains also leads to a quadratic eigenproblem (QEP), but one with a complex T -palindromic matrix polynomial. Real and complex T -palindromic QEPs also arise in the numerical simulation of the behavior of periodic surface acoustic wave (SAW) filters [43,85]. Quadratic eigenproblems with T -alternating polynomials arise in the study of corner singularities in anisotropic elastic materials [7,8,70].…”