Hilbert space methods in microwave engineering.

摘要

In this dissertation electromagnetic boundary value problems (BVPs) of the elliptic type are investigated, with an emphasis on the mathematical framework required for well-posed and efficient solution methods.; First, we shall examine, clarify and simplify the existence and uniqueness theory of the classical solutions to the elliptic BVPs. Then, using the conditions for the existence of the solutions to the Helmholtz BVPs, we solve a long standing problem regarding the theoretical representation of non-unique solutions for a lossless resonant system found in the Helmholtz formalism of the BVPs. A novel method capable of suppressing virtually all resonant modes is presented for power/ground plane structures.; Second, we summarize and clarify the variational methods, discuss common errors in application of these methods and establish a guideline for employing valid and efficient numerical methods related to the generalized solutions for solving the BVPs. We apply the Hilbert space theory to microwave engineering, and develop a new error estimate for the numerical analysis based on the energy-norm. This new estimate, constructed from the S-parameters used in analysis of microwave networks, provides a rigorous posterior error estimate and correlates commonly measured engineering quantities with both the underlying physics and a rigorous mathematical analysis.; Third, we illustrate an intrinsic difficulty in obtaining accurate solutions of BVPs with material discontinuities and singularities, and suggest approaches to resolve the difficulty.; Finally, using the conditions assuring the uniqueness of solutions to the Helmholtz BVPs, we develop a new boundary element (BE) algorithm, which accepts piecewise constant coefficients in the BVPs, and calculates the relative error in the BE solutions. Having successfully implemented the new BE formulation, we apply this technique to analyzing a simplified biological structure and a power/ground distribution structure of the type found in printed circuit boards and multichip modules. These numerical examples demonstrate that the effects of material non-uniformities on the RF signals in the power/ground structure and in the biological structure can be extracted from the surface data with an appropriate simulation scheme.