Analytical time domain normal mode fields in a rectangular resonator

Abstract In this study, a direct and causal analytical Time Domain Normal Mode (TDNM) solution for a lossless electromagnetic rectangular resonator is presented in three-dimensional Cartesian coordinates. The TDNM method is based on a solution of an inhomogeneous wave equation for a magnetic vector potential in time domain. Using the separation of variables technique combined with a Sturm-Liouville technique, a space domain (spatial) differential equation and a time domain (evolutionary) differential equation are extracted separately. The link between these equations is constructed over eigenvalues of the former part. This link addresses a significant gap in the literature for time domain problems. Then, the latter part is solved by using a Laplace transform with initial conditions. Thus, an exact, explicit and analytical TDNM solution is obtained for a causal sine source and a unit-step source. The transient and steady-state parts are revealed in the field formulation. In terms of field distributions and temporal responses of the considered modes, all the TDNM results are verified by comparison with other analytical and numerical solutions of the same problem. Good agreement is observed between these methods. Specifically, it is shown that the fields within the resonator exhibit a non-sinusoidal time response under the unit-step pulse excitation. Methodological and practical novelties of the TDNM method lie in the use of the time domain magnetic vector potential and a physically positioned source, respectively. Furthermore, obtaining the explicit solution demonstrates evident computational efficiency. The findings provide valuable reference data in time domain for the systematic validation of numerical solutions and the precise calibration of experimental measurements.