Steady-State Algorithms for Nonlinear Time-Periodic Magnetic Diffusion Problems Using Diagonally Implicit Runge–Kutta Methods

A general framework is presented for the formulation of steady-state simulation algorithms for magnetically nonlinear eddy-current problems using implicit Runge-Kutta (RK) methods. A close analogy is drawn between equations discretized using the backward-Euler method and fully implicit RK methods. Detailed formulations of the time-periodic finite-element method (TPFEM) and the shooting-Newton method (SNM) are derived using the popular family of diagonally implicit RK (DIRK) methods. Both algorithms employ the generalized minimum residual method to solve the linear equations arising at each Newton iteration. The benefits of higher-order DIRK methods are demonstrated by simulating a surface mount permanent magnet synchronous machine. The effects of using a solid versus a laminated rotor back iron on the simulation time are examined. Simulation results indicate that the performance of TPFEM and SNM is quite similar and much faster than transient analysis.