4.2.1. Electromagnetic Machine Models

For the electromagnetic simulation of the IM, the machine models described in [27] are used. These are the Fundamental Wave Model (FWM), the Harmonic Wave Model (HWM) and the Extended Harmonic Wave Model (E-HWM), three analytical models, and the Time Harmonic Finite Element Model (TH-FEM) and the Transient Finite Element Model (T-FEM), two numerical models. The models differ in their range of values and level of detail. The following is a brief description of the models.

The fundamental wave model is based on a single-phase ECD, also presented in [39]. For the calculation of the elements of the ECD, which can be derived exclusively from the machine geometry and constant parameters, reference is made to the literature [40].

The HWM and the E-HWM are based on the harmonic wave theory of the IM presented in [41–43]. The HWM is the implementation under the assumption of an infinite permeability of the stator and rotor iron. The E-HWM is an extension of the HWM, where the influence of saturation is modeled by multiplying the flux densities by an air gap conductance function. The circumferential location dependent air gap conductance function [40] is a description of the effective air gap of the machine. The air gap is is increased on average by a saturation factor *kh* > 1 as a result of the main field saturation. In the region of large iron saturation, i.e., in the maximum of the air gap flux density *B*, the air gap is further enlarged by an additional saturation factor *kh*1, and reduced at zero crossings. The air gap conductance function defined in this way moves synchronously with the fundamental wave field. The air gap flux density flattened by the saturation follows from multiplying the air gap flux density calculated with the HWM by the air gap conductance function.

The TH-FEM is based on the state of the art FEM for time-harmonic simulation of electromagnetic components including a slip transformation [44–46]. To consider for the nonlinear material behavior of the stator and rotor laminations, an iterative procedure is used in the field solution. For this purpose, the successive substitution approach or the Newton method can be used.

For the computation of the T-FEM, an in-house state-of-the-art Finite Element (FE) solver called *i*MOOSE/*py*MOOSE [47] is used. To reduce the computational effort of the T-FEM, a hybrid simulation approach presented in [48] is applied.

The FWM, HWM, E-HWM, and TH-FEM are implemented in Matlab®, whereas the T-FEM is implemented in pythonTM and C++. An operating map simulation needs 15 min using the TH-FEM on an PC with an i7 3.6 GHz Processor and 16 GB RAM and 12 h using the T-FEM on the compute cluster of the RWTH Aachen University.
