# Analysis of Permanent Magnet Motors in High Frequency--A Review
## 3 Analysis Tools
### Finite Element Methods (FEM)
#### 3.1.3 Parasitic Capacitances  
As mentioned in Section 2.3, some additional current paths appear at high frequency, due to the existing parasitic capacitances between the different parts of the machine. These paths must be calculated thoroughly as they might have a significant impact on the motor's high-frequency behavior. To analyze these capacitive couplings electrostatic FEM simulations must be conducted [28; 52; 87]. There are different methods to calculate the capacitances between floating conductors, as explained in [88]. In the electrical machine, those conductors can be the winding conductors as well as the stator and rotor cores, because they behave equally as coupling paths. These are the most used methods [65]:  
* Minimum Energy Method: It consist of performing \(N(N+1)/2\) simulations, where N is the number of regions to compute. Then, using the electric field value, the stored energy will be calculated. The simplest way to find the self-capacitance coefficients \(C_{ii}\), is to perform N simulations in which all but one conductor are defined with a boundary condition \(V_{j}=0\) for \(j\
eq i\). Then, the expression (2) is solved, where \(W_{i}\) is the stored energy and \(V_{i}\) is the applied voltage. To calculate the mutual capacitance between two regions \(C_{ij}\), the boundary conditions \(V_{j}\
eq V_{i}\
eq 0\) are defined, where the rest of the regions are set to zero. \[C_{ii}=2\frac{W_{i}}{V_{i}^{2}}\quad C_{ij}=C_{ji}=2\frac{W_{ij}-0.5(C_{ii}V_{i }^{2}+C_{jj}V_{j}^{2})}{V_{i}V_{j}}\] (2)
* Gauss Law method: It consists of performing N simulations in which the conductors are defined with a boundary condition \(V_{k}=0\) for \(k\
eq j\). Then, after solving the electric field distribution in each simulation, the expression (3) is applied, where \(Q_{ij}\) is the charge of each conductor due to the voltage excitation. \((N+1)/2\) simulations are enough, so that the number of required simulations is notably reduced. Hence, this method can be considered more suitable than the minimum energy method for electrical motors, which normally use coils comprising rather high number of conductors. \[C_{ij}=2\frac{Q_{ij}}{V_{j}}\] (3)  
With respect to the model geometry, the capacitances of the active part of the machine can be calculated by 2-D FEM electrostatic simulations, whereas the end-winding capacitances must be calculated by 3-D FEM simulations [73; 85]. It must be remarked that the capacitive coupling between the active part of the stator winding and the stator core is significantly bigger than the rest of the capacitive couplings, so that it is the predominant CM current path. Hence, in some cases 2-D FEM simulations are accurate enough to analyze this main CM current path [58]. However, the end winding represents up to 40% of the total winding-to-rotor capacitance and it influences the phase-to-phase capacitance, so it must be considered to obtain accurate results on bearing currents [10; 54]. Moreover, the inter-turn and turn-to-stator capacitances result in an NxN symmetric matrix, where N is the number of turns, as shown in (4), where \(C_{ij}\) is the capacitance between the \(i\)th and the \(j\)th turn and the \(C_{ii}\) it the \(i\)th turn-to-core capacitance [39; 42; 51; 56; 62; 70]. In [52; 55], phase-to-phase capacitance is also defined for multi-layer windings. \[[C]=\left( \begin{array}{ccc}\ddots&-C_{ij}&\ldots\\ \cdots&C_{io}+\sum_{j=1}^{N_{C}}C_{ij}&\ldots\\ \ldots&-C_{ji}&\ddots\end{array} \right) \tag{4}\]  
In [65] the importance of the mesh between conductors is stated, concluding that at least 2 mesh layers must be defined between conductors to achieve accurate results. Theauthors state that the computed matrix shows that the strongest couplings are found for capacitances between neighboring turns and between turns facing the stator core. They stated that the matrix can be simplified neglecting low values of distant conductors, as well as in [51]. In Table 2 the analyzed models are summarized including the purpose of the models and the considered capacitive couplings.