This paper describes the procedure for analyzing nonlinear transient electromagnetic phenomena in electrical machines and devices with the finite element method. Two time integration methods are used which are based on (1) an implicit forward difference scheme and (2) the Crank-Nicholson method. The quasilinearization of the nonlinear matrix equation is handled by a simple chord iteration method in the former and the Newton-Raphson scheme in the latter. The associated variational expressions for the time-dependent diffusion equation are obtained in terms of energy-related functionals. The paper includes illustrative examples of application of the methods to one-and two-dimensional time-dependent eddy current problems in a conducting slab, a rotating machine under asynchronous operation, and a three- phase bus-bar enclosure.

Nonlinear Transient Finite Element Field Computation for Electrical Machines and Devices

A 2-dimensional section of a synchronous machine is modelled using 1st-order triangular finite elements. Equations are formulated by the use of Ampere's law. The transient performance of the machine is obtained by determination of the vector potential distribution at successive instants in time, the effect of closed circuits being taken into account in the formulation of the equations by the introduction of conductivity matrices. Variable permeability of steel and the effect of eddy currents are taken into account. The method is applied to calculate the performance during a field-decrement test.

Numerical analysis of transient field problems in electrical machines

This paper details the use of computerized magnetic field solutions, based on magnetic vector potential, to develop a simplified approach to the calculation of saturated values of steady state reactances of turbogenerators under various load conditions. When magnetic nonlinearity of the stator and rotor materials is taken into account, the turbogenerator reactances become dependent on the saturation conditions in the machine, which are a function of the operating conditions (terminal voltage, current and power factor). Quadrature axis synchronous reactances are reduced by as much as 40% from their classical unsaturated values under rated load. Torque angles deduced from the model at rated load are lower by as much as 20% than values obtained by standard methods. The method is applied to two typical 4-pole, 659 MVA and 2-pole, 733 MVA turbogenerators of the gas inner-coaled type. Supportive shop floor test results are provided.

A simplified approach to determination of saturated synchronous reactances of large turbogenerators under load

Finite difference and finite element methods are investigated as applied to Laplace and Poisson linear equations in two dimensions. The comparative analysis concentrates on first-order square, rectangular, triangular and polar algorithms which are characteristic to each method. The results are found to be independent of the method used to establish the matrix equations but dependent upon the choice of grid systems employed to discretize the problem. It is found that for special cases the solutions are also independent of the type and structure of the grid systems used.

Comparison of First-Order Finite Difference and Finite Element Algorithms for the Analysis of Magnetic Fields. Part I: Theoretical Analysis

In this paper a new method for computing non-linear magnetostatic fields is introduced, which allows the simultaneous coupling of a finite element structure with a magnetic network. Combining the advantages of both methods while avoiding their drawbacks, this coupling yields both an accurate and time-efficient computation.

The traditional method of the unknown mesh fluxes is applied for the solution of the magnetic network. The finite element solution, on the other hand, is based on a classical first-order interpolation of the unknown vector potential. The coupling is established by a proper organization of the unknowns on the boundary common to the finite element and network regions. In this way, a single system of non-linear equations is obtained.

Moreover, it is shown that the coupled system of equations is equivalent to a single finite element system if generalized base functions are allowed. Consequently, various results from finite element theory may be applied. For instance, the matrix governing the iteratively linearized system of equations can directly be shown symmetrical and positive definite.

Finally, the field inside a permanent magnet motor is calculated with the coupled method. Although the number of unknowns is dramatically reduced compared to a full FE calculation, the same level of accuracy is achieved. Hereby, the benefit of the coupled method is clearly proved.

Coupling finite elements and magnetic networks in magnetostatics

The performance of electrical machines is largely dictated by the action of current and flux in the core length. The field in a cross-section obeys Poisson's equation and approximate solutions have been obtained by finite difference and element methods. The finite difference method requires a large number of nodes and is slow to converge as permeability is variable. The finite element method is more flexible being more readily fitted to iron-air boundaries and has better convergence. However, it is difficult to formulate a legitimate variational formulation for transient conditions in the presence of dissipation. Here, discrete equations are formed by applying Ampere's circuital law around each node. Careful choice of contour lines give a current distribution superior to that obtained with finite elements. Fast convergence is obtained and the method is applicable under transient conditions.

A. Hannalla

Papers

A nodal method for the numerical solution of transient field problems in electrical machines