W pracy przedstawiono zarys historii stosowania numerycznych metod analizy pola elektromagnetycznego w obliczeniach przetworników elektromechanicznych. Zwrócono uwagę na dwa główne nurty prac wykorzystujących te metody: (a) prace polegające na stosowaniu metod polowych do obliczania parametrów schematów zastępczych przetworników, (b) prace nad modelami polowo-obwodowymi i algorytmami, w których równania pola są rozwiązywane równocześnie z równaniami stanu przetwornika. Omówiono badania nad dwuwymiarowymi modelami polowo-obwodowymi. Przedstawiono ogólną postać równań modeli trójwymiarowych.
EN
The paper presents shortly the history of the applications of the numerical methods of electromagnetic field analysis in the electromechanical converter calculations. Attention is paid to the finite difference (FD) and finite element (FE) methods. It has been shown that there are two main lines of the application of these method: (a) the methods are applied in the calculations of integral parameters of the equivalent electric and magnetic circuits, and (b) the equations of FD or FE method are combined with the circuit equations of converter windings and with circuit equations of supply system (the circuit and field equations create the so called coupled field-circuit model of the converter). Summarily, the research works in relation to the 2D field-circuit models are discussed in the section 2. The 3D field-circuit models are presented in the section 3. In the considered models, the magnetic vector potential formulation is applied. First, the equations for the models with the edge element (EE) are presented. The finite elements are assumed to be edge elements in relation to magnetic vector potential A. The EE equations of the magnetic field are combined with the equations that describe the distribution of the electric field of conducting currents. In the case of the region with filamentary conductors, the EE equations are combined with the voltage equations which describe the winding currents i(c), see Eq. (1). These currents represent the edge values of current vector potential T. However, for the regions with solid conductors, the electric scalar potential V formulation is applied, see Eq. (3). Matrix equation (3) can be represented by the equations of the reluctance-conductance network. The network representation of tetrahedron is shown in Fig. 1 (reluctance network - Fig. 1a), conductance network - Fig. 1b). The edge element method has been compared with the classical nodal element method for the magnetic vector potential formulation. Finally, the unifed matrix of the equations for the 3D field models have been proposed, see Eq. (6). Equations (6a), (6b) are related to A-T formulation and A-V formulation, respectively. In the case of A-T formulation the finite elements are assumed to be nodal or edge elements in relation to A and T. However, for A-V formulation the finite elements are assumed be nodal in relation to V and are assumed to be nodal or edge elements in relation to A.
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W pracy przedstawiono algorytmy analizy dynamicznych stanów pracy przetworników elektromechanicznych na podstawie trójwymiarowych modeli polowych. Rozpatrzono układy z masywnymi elementami przewodzącymi o uzwojeniach wykonanych z cienkich przewodów. Szczególną uwagę poświęcono zagadnieniom związanym z symulacją ruchu na modelu o elementach skończonych. Przedstawiono wyniki obliczeń dla dwóch przykładów. Analizowano zjawiska w silniku magnetoelektrycznym o magnesach wykonanych z materiału przewodzącego oraz zjawiska w silniku liniowym o masywnej części wtórnej.
EN
The paper presents algorithms for dynamic analysis of electromechanical converters - on the basis of three dimensional field models. Systems containing solid conducting parts and filamentary windings have been considered. Special attention for movement finite element modelling has been paid. In this case, the equation of the mechanical motion must be taken into consideration. In order to solve this equation, the explicit or implicit time-stepping algorithm can be applied - expressions (2) and (3). The implicit procedure is unconditionally stable but it requires the iterative calculations. During the movement, the geometric structure of the considered region is changing. Therefore, the grid remeshing is required. Method of remeshing depends on the type of problem. Two main approaches are possible to applied: (a) the nodes within moving region are fixed and (b) the nodes move in accordance with this region. In case of infinitely long moving part with constant permeability, as for example in linear induction motor, the movement modelling is relatively simple task. During the dynamic process, the entirely constant mesh can be applied. The additional term in equation (1), proportional to velocity of the moving body may be omitted if the special procedure of estimation of derivative dA/dt - according to formulae (6) - is applied. Problem is still more complicated in case of short moving part. In such case, the polynomial or Fourier's interpolation of the FE matrices is proposed. The grid is entirely constant. On the basis of "grid positions", the Newton polynomial or Fourier's series is constructed. If the total displacement of the moving body is limited, then the distorted element method may be applied. However, the distortion of the elements causes additional formal changes of the magnetic stored energy and therefore causes additional parasitic components of the electromagnetic force or torque. As an illustration of the elaborated algorithms, the operation of the synchronous motor with conducting permanent magnet and start-up operation of the linear induction motor have been investigated.
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