Causal oscillations in an electromagnetic resonator

Abstract

In this study, an analytical time domain solution for electromagnetic waves generated in a rectangular resonator is given. A point causal cosine source is used to excite the resonator. All of the resonator walls are assumed to be perfect electric conductor. Considering three-dimensional inhomogeneous (with a given source) wave equation for a vector potential, initial boundary value problem is solved analytically in time domain. Using separation of variable method together with Sturm-Liouville technique, time domain and space domain differential equations are completely separated from the inhomogeneous wave equation. Applying the boundary conditions, the ordinary space domain differential equation is solved in a classical manner of eigenvector-eigenvalue concept. Taking into account the initial conditions, the ordinary time domain differential equation is solved by using Laplace transformation. Therefore, the obtained analytical solution is causal. Specially, it is worth noting that the time domain differential equation has the similar form of a displacement equation for a harmonic oscillator in quantum mechanics. The link between the space and time domain differential equations is eigenvalues that correspond to the resonance frequencies of the resonator modes. At resonance, combining space and time solutions, an explicit expression for the vector potential is found in time domain. The resonance behaviors in time domain and in frequency domain are clearly revealed in graphical results.

Full text

Causal Oscillations in an Electromagnetic Resonator
Osman Said Bişkin, Talha Saydam, Serkan Aksoy
Electronics Engineering Department
Gebze Technical University
Gebze, Kocaeli
TÜRKİYE
Abstract: In this study, an analytical time domain solution for electromagnetic waves generated in a rectangular
resonator is given. A point causal cosine source is used to excite the resonator. All of the resonator walls are
assumed to be perfect electric conductor. Considering three-dimensional inhomogeneous (with a given source)
wave equation for a vector potential, initial boundary value problem is solved analytically in time domain. Using
separation of variable method together with Sturm-Liouville technique, time domain and space domain
differential equations are completely separated from the inhomogeneous wave equation. Applying the boundary
conditions, the ordinary space domain differential equation is solved in a classical manner of eigenvector-
eigenvalue concept. Taking into account the initial conditions, the ordinary time domain differential equation is
solved by using Laplace transformation. Therefore, the obtained analytical solution is causal. Specially, it is worth
noting that the time domain differential equation has the similar form of a displacement equation for a harmonic
oscillator in quantum mechanics. The link between the space and time domain differential equations is
eigenvalues that correspond to the resonance frequencies of the resonator modes. At resonance, combining space
and time solutions, an explicit expression for the vector potential is found in time domain. The resonance
behaviors in time domain and in frequency domain are clearly revealed in graphical results.
Key-Words: Analytical time domain solutions, causal oscillations, rectangular resonators.