QUICK FINITE ELEMENT QUASI–STATIC ANALYSIS OF COPLANAR WAVEGUIDE DISCONTINUITIES Krzysztof Nyka * , Micha l Mrozowski * Abstract In this paper we present a procedure of quasi-static analysis of coplanar waveguide discontinuities based on their equivalent schematic representation. Capacitances and inductances are calculated from scalar potentials. For magnetostatic formulation the boundary conditions for scalar potential are introduced by means of partitioning surfaces. The use of adaptive finite element method and multigrid method provides required flexibility with respect to analyzed geometry, optimal discretization and good efficiency. A quasi–static model is compared with the results of full wave simulation. 1 Introduction In the commercial CAD software of microwave circuits an analyzed structure is usually treated as a network of transmission lines connected by lumped elements regarded as discontinuities. These elements are usually represented by equivalent circuits, which are characterized based on the quasi-static analysis, however the typical libraries of such building blocks comprise only the pre designed shapes. The emerging trend in the CAD of microwave structures is to give the designer the freedom to input and modify arbitrary topology of a circuit. As this approach requires repetitive solution of field equation in the 3D region of each separate discontinuity, it is particularly important to use numerical methods of field analysis that offer significant reduction of computer resources. The quasi-static analysis of arbitrarily shaped elements is a reasonable trade off between the accurate but time consuming full-wave simulation and the use of pre calculated libraries of approximated models. It has been shown [1] that this strategy applied to the monolithic and hybrid MIC structures provides good results up to millimeter wave range. In this contribution we concentrate on the coplanar waveguides (CPW), because they have recently been gaining a growing interest owing to its advantages over microstrip techniques such as: less dispersion, ease of shunt connections compared to via-holes, better isolation between adjacent lines and capability of wafer probe. In spite of this the CPW elements are relatively poorly represented in the available CAD software. In the present analysis of the CPW discontinuities we have introduced several enhancements to the methods used in [1] that provide better efficiency and flexibility of quasi-static solvers, by using the following techniques [2]: (a) discretization of wave equation by means of finite element (FE) instead of finite difference (FD) method, (b) scalar potential formulation of the magnetostatic problem in general case of structures without symmetry [3, 4], (c) adaptive refinement of the discretization mesh [5], (d) multigrid method of the solution to the system of linear equations resulting from the problem discretization [6, 7], (e) extrapolation techniques to improve discretization accuracy [2]. 2 Evaluation of the equivalent circuits The capacitances and the inductances of the lumped element representation of discontinuities are derived from a static solution of Laplace equation for electric and magnetic potential in 3D. The formulation for the scalar magnetic potential in general asymmetric structures is possible due to the special technique similar to the concept of potential partitioning surface [3]. This method will be described in the next section. In order to provide flexibility in shaping the structure geometry the FE method is used to transform continuous problem into sparse matrix system of linear equations. The overall speed of the solution to the given boundary problem depends on two basic factors: the size of the system matrix and the efficiency of the solution to the system of equations. For a given discretization accuracy, the minimum number of the system unknowns is achieved by the use of adaptive FE mesh refinement. This leads to the optimal discretization that conforms not only to the boundary singularities but also to the solution being currently computed. The regions of more intensive field changes are covered by finer mesh. To solve the system of equation we use multigrid methods that are known [6] to provide the highest convergence rate at nearly optimal computational load (time and memory demand) of O(N ). Besides, the multigrid methods can be naturally combined with the adaptive mesh refinement [5]. * Department of Electronics, Telecommunications and Informatics, Technical University of Gda´ nsk, 80–952 Gda´ nsk, Poland. E-mail: nyx@pg.gda.pl,mim@pg.gda.pl