# Efficient PEEC-based solver for complex electromagnetic problems in power electronics

Sammanfattning: The research presented in this thesis discusses an electromagnetic (EM) analysis tool which is based on the partial element equivalent circuit (PEEC) method and is appropriate for combined EM and circuit simulations especially power electronics applications. EM analysis is important to ensure that a system will not affect the correct operation of other devices nor cause interference between various electrical systems. In power electronic applications, the increased switching speed can cause voltage overshoots, unbalanced current share between semiconductor modules, and unwanted resonances. Therefore, EM analysis should be carried out to perform design optimizations in order to minimize unwanted effects of high frequencies. The solver developed in this work is an appropriate solution to address the needs of EM analysis in general and power electronics in particular. The conducted research consists of performance acceleration and implementation of the solver, and verification of the simulation results by means of measurements. This work was done in two major phases. In the first phase, the solver was accelerated to optimize its performance when quasi-static (R,Lp,C)PEEC as well as full-wave (R,Lp,C,tau)PEEC simulations were carried out. The main optimizations were based on exploiting parallelism and high performance computing to solve very large problems and non-uniform mesh, which was helpful in simulating skin- and proximity effects while keeping the problem size to a minimum. The presented results and comparisons with the measurements confirmed that non-uniform mesh helped in accurately simulating large bus bar models and correctly predicting system resonances when the size of the problem was minimized. On-the-fly calculation was also developed to reduce memory usage, while increasing solution time. The second phase consists of methods to increase the performance of the solver while including some levels of approximations. In this phase sparsification techniques were used to convert a dense PEEC system into a sparse system. The sparsification was done by calculating the reluctance matrix, which can be sparsified by maintaining the accuracy at the desired level, because of the locality and the shielding effect of the reluctance matrix. Efficient algorithms were developed to perform sparse matrix-matrix multiplication and assemble the sparse coefficient matrix in a row-by-row manner to reduce the peak memory usage. The sparse system was then solved using both sparse direct and iterative solvers with proper preconditioning. The acquired results from the sparse direct solution confirmed that the memory consumption and solution time were reduced by orders of magnitude and by a factor 3 to 5. Moreover, the Schur complement was used together with the iterative approach, making it possible to solve large problems within a few iterations by preconditioning the system, and using less memory and lower computational complexity. Bus bars used in two types of power frequency converters manufactured by ABB were modelled and analysed with the developed PEEC-based solver in this research, and the simulations and measurements agreed very well. Results of simulations also led to improvement in the physical design of the bars, which reduced the inductance of the layout. With the accelerated solver, it is now possible to solve very large and complex problems on conventional computer systems, which was not possible before. This provides new possibilities to study real-world problems which are typically large in size and have complex structures.

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