which is the final expression for the Jacobian matrix in the nonlinear TDFEM formulation if the scalar constitutive relation is used.
Nonlinear TDFEM Formulation with a Vector Constitutive Model
After the Newton-Raphson scheme of the TDFEM is constructed, the nonlinear equation can be solved iteratively.
Polarization Technique Based Newton-Raphson Scheme for TDFEM
Another approach of constructing the Newton-Raphson scheme for the TDFEM is to use the idea of Taylor series expansion Eqs.
The solution process of the nonlinear TDFEM is the same as that described in Section 3.3.3.
Since the excitation current has a majority of energy concentrating at very low frequencies or even at dc, the tree-cotree splitting (TCS) [18,19] has been applied to the TDFEM solver, in order to eliminate the low-frequency breakdown problem.
Since it is a scalar B-H relation, the nonlinear TDFEM formulation (22) is used for this example.
The comparison between the results obtained by the proposed nonlinear TDFEM solver and the measurement data  is shown in Fig.
From the nonlinear TDFEM formulations presented in the preceding section, it is clear that to model the magnetic hysteresis phenomenon, the accurate generation of [[bar.[upsilon]].sup.d.sub.r] and H is critical to the fast convergence of the Newton-Raphson method.
where [H.sup.n] and [B.sup.n] are the solutions from the previous time step n, and [B.sup.n+1.sub.k] is the estimated solution at the kth Newton iteration of the current time step n + 1, which is produced from the TDFEM solver.
In (2), however, the derivative of H is taken with respect to B, which is given by the estimated solution A from the TDFEM solver.