**Rigorous Approach to Analysis of Two-Dimensional Potential Problems, Wave Propagation and Scattering for Multi-conductor Systems**

Galyna Safonova and Elena Vinogradova

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/61287

### **Abstract**

The research described in this chapter analyses two-dimensional potential problems for the multi-body systems, transverse electromagnetic wave propagation along multi-conductor transmission lines and two-dimensional plane wave scattering by various arrays. All conductors may be of arbitrary cross-sections; the only restric‐ tion on the system geometry is a smooth parameterization. These problems are mathematically modelled by Dirichlet boundary value problems for either the Lap‐ lace or the Helmholtz equation, with the classical integral representation of the sol‐ utions in the form of single-layer potential. The analytical-numerical algorithm presented here is based on the method of analytical regularization. The key idea be‐ hind this technique is an analytical transformation of the initial ill-posed integral equations to a well-conditioned Fredholm second kind matrix equation. The result‐ ing system of infinite linear algebraic equations is effectively solved using the trun‐ cation method: the solution of the truncated system converges to the solution of the infinite system with the guaranteed accuracy that only depends on the truncation number and thus may be pre-specified. The solution obtained is applied to the accu‐ rate analysis of 2-D electrostatic- and electrodynamic-field problems for multi-con‐ ductor systems with arbitrary profiled conductors. Examples of some conceptual shielded transmission lines incorporating various configurations of conductors and scattering problems for the arrays of thick strips establish the utility of our method and its reliability in various situations

**Keywords:** Scattering, propagation, analytical regularization, Laplace equation, Helmholtz equation

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