2.4.1. Mathematical Description of the FEM

In the FE models of this paper, a magnetoquasistatic formulation over the magnetic vector potential - *A* is considered. From the Maxwell equations the parabolic partial differential equation

$$\nabla \times (\nu \nabla \times \vec{A}) + \sigma \frac{\partial \vec{A}}{\partial t} = f\_s \tag{17}$$

the reluctance *ν*, the conductivity *σ*, and the injected stator current density -*J*<sup>s</sup> can be derived [14]. A discretization of the magnetoquasistatic formulation in (17) applying the weighted-residual method, which is necessary for the FEM, leads to the weak vector potential formulation

$$\int\_{V} \left( \nabla \times (\nu \nabla \times \vec{A}) + \sigma \frac{\partial \vec{A}}{\partial t} \right) \cdot \vec{w}\_{\text{i}} \, \text{d}V = \int\_{V} \vec{f\_{\text{s}}} \cdot \vec{w}\_{\text{i}} \, \text{d}V \tag{18a}$$

$$
\vec{A} = \sum\_{j} a\_{\vec{\jmath}} \vec{v}\_{\vec{\jmath}} \tag{18b}
$$

with *w*<sup>i</sup> being weighting functions. The magnetic vector potential is thereby divided into a finite sum of shape functions *v*j, and thus is described by its degrees of freedom *a*<sup>j</sup> [14]. Using Galerkin's method [14], which prescribes *v*<sup>j</sup> = *w* j, by transformations based on integral theorems and boundary conditions of the solution space, the system of equations and the resulting matrix notation can be expressed as [15]

$$\sum\_{j} (a\_{\parallel} \underbrace{\int\_{V} \boldsymbol{\nu} \nabla \times \vec{w}\_{\mathbf{i}} \cdot \nabla \times \vec{w}\_{\mathbf{j}} \, \mathrm{d}V}\_{K\_{\boldsymbol{\nu}, \vec{\boldsymbol{\mu}}}} \quad + \quad \underbrace{\frac{\partial a\_{\parallel}}{\partial t} \int\_{V} \boldsymbol{\sigma} \vec{w}\_{\mathbf{i}} \cdot \vec{w}\_{\mathbf{j}} \, \mathrm{d}V}\_{M\_{\boldsymbol{\nu}, \vec{\boldsymbol{\mu}}}} \quad = \underbrace{\int\_{V} \boldsymbol{\int}\_{\mathbf{s}} \cdot \vec{w}\_{\mathbf{i}} \, \mathrm{d}V}\_{\text{f}} \tag{19a}$$

$$\mathbf{K}\_{\nu}\vec{a} \qquad \qquad \qquad + \quad \mathbf{M}\_{\sigma}\frac{\partial \vec{a}}{\partial t} \qquad \qquad \qquad = \quad \vec{f}. \tag{19b}$$

Here **K***<sup>ν</sup>* is called stiffness matrix, **M***<sup>σ</sup>* is called mass matrix, and *f* is called load vector. From this system, the degrees of freedom of the magnetic vector potential and from it the flux density distribution in the machine can be determined.

#### 2.4.2. Time Harmonic Finite Element Model

In the TH-FEM, the time courses of physical quantities are simplified as sinusoidal and can therefore be described by complex phasors [14]

$$y(t) = \Re(\underbrace{\ddot{y}}e^{j\omega t}).\tag{20}$$

This has the advantage that the time derivative *<sup>∂</sup> <sup>∂</sup><sup>t</sup>* passes into a multiplication by j*ω*. The differential equation from (19b) thus transforms into the linear system of equations

$$(\mathbf{K}\_{\mathcal{V}} + \mathbf{j}\omega \mathbf{M}\_{\mathcal{F}})\vec{a} = \vec{f} \tag{21}$$

which can be solved with low computational effort. The disadvantage of the time-harmonic simulation is the underlying assumption that no time harmonics with an order greater than one exist, which is why in particular the accuracy of the calculated iron losses decreases compared with the T-FEM [16].

### Slip Transformation

Since the linear equation system (21) simulates the stator frequency *ω*<sup>S</sup> in the entire solution domain, the lower frequency *ω*<sup>R</sup> in the rotor is not considered. This can be accounted for by a slip transformation [17,18]. For this, the conductivity of the cage *σ*R,comp is scaled with the slip *s*, which changes the mass matrix **M***σ* for rotor nodes such that the lower rotor frequency is considered in the induction effect.
