*2.5. Transient Finite Element Model*

The transient simulation solves the differential Equation (19b) for discrete time steps *t*<sup>k</sup> within a time interval Δ*T* considering the field solution of the previous time step and the motion of the rotor. Due to the discretization, the time derivative can be described as the difference quotient *a*k+1−*a*k <sup>Δ</sup>*<sup>t</sup>* with the time difference Δ*t* between two time steps. From this the linear equation system results in

$$(\mathbf{K}\_{\upsilon}\Delta t + \mathbf{M}\_{\sigma})\vec{a}\_{k+1} = \vec{f}(t\_{k+1})\Delta t + \mathbf{M}\_{\sigma}\vec{a}\_{k\prime} \tag{22}$$

which must be solved for each time step [14]. The corresponding rotor position Θ follows from the Newton equation of motion

$$J\frac{\partial^2 \Theta}{\partial t^2} = T(t) \tag{23}$$

with the moment of inertia *J* of the rotating body. This equation is also discretized in time, with the torque *T* updated for each time step based on the calculated field solution. By solving the differential equation in the time domain, temporal harmonics in particular are modeled, which allows a more detailed view of the machine behavior. Since the Shannon sampling theorem prescribes a sampling frequency *f* sample greater than or equal to 2 *f* for physical consideration of a given frequency *f* [19], the time difference between each time step must be small enough to consider the higher order harmonics, but the total time interval large enough for consideration of the low frequencies in the rotor. This, in combination with the required transient time of the vector potential solution, leads to a high calculation effort of the transient Finite Element (FE) simulation. Faster transient is made possible by a hybrid approach via estimation of a starting solution from an open-circuit simulation [20].

In Figure 6 the simulated magnetic flux density of an exemplary IM at an saturated operating point is shown. In Figure 6a the simulation was conducted using the TH-FEM and in Figure 6b using the T-FEM. The simulation results show quite similar field characteristics.

**Figure 6.** Simulated magnetic flux density and isolines for an exemplary IM.
