# Analysis of Permanent Magnet Motors in High Frequency--A Review
## 3 Analysis Tools
### Analytic Methods
#### 3.2.3 Parasitic Capacitances  
When the working frequency increases, the impedance of parasitic capacitances decreases, leading to new current flow paths inside the motor which affect its DM and CM impedances. The analytical calculation of capacitances is spread out in the literature. * **Stator-Winding-to-Rotor Capacitance:** In [41; 43] each slot is considered to be a plate capacitor. This capacitor consists of the airgap and the height of the slot opening with the permittivity of the air and the slot wedge and the upper slot insulation with the insulation material relative permittivity. This approach shows an approximate error of 15 % due to geometrical approximations. The method of image charges for the solution of Laplace equation is used by [PERSON] and [PERSON] to calculate winding-to-rotor capacitance. This method replaces the elements in the geometry with imaginary electric charges, replicating the boundary conditions of the problem. It uses line charges for the slot portion and ring charges for the end-winding portion. The results show small deviations with FEM results enabling the prediction of bearing voltage. Moreover, the end-winding contribution to the winding-to-rotor capacitance must be calculated. In [58], authors state that the end windings typically have a rectangular structure. Thus, the capacitance is estimated using the equation for a cylindrical capacitor (7), where \(\varepsilon_{r}\) is the relative permittivity, \(\varepsilon_{0}\) is the permittivity of the vacuum, \(l_{ew}\) is the length of the end winding and \(d_{r}\) and \(d_{air}\) are the rotor diameter and the airgap, respectively. The obtained result is added to the rest of the winding-to-rotor capacitance, validating it with FEM results, with some deviations. \[C_{ur,ew}=\frac{2\pi\varepsilon_{r}\varepsilon_{0}l_{ew}}{3}\ln\left(\frac{d_ {r}+2d_{air}}{d_{r}}\right)\] (7)
* **Rotor-to-Stator Capacitance:** The stator and rotor are considered to be cylindrical capacitors. To account for the influence of the slot openings, a coefficient is added similar to the one used to for the reduction of the airgap flux density in magneto-static calculations, called Carter's coefficient. The calculated values show a difference of 15% from measurements, but the author states that the values are in the safe side, as the bearing currents are overestimated by this method [41]. * **Bearing Capacitance:** The bearing acts as a capacitor if it rotates at a sufficient speed for an electrically insulating lubricating film to build up and if the voltage applied is lower than the electrical break-down threshold. The value is calculated taking into account the minimum thickness of the lubrication film and the Hertzian contact area [41]. The proposed model is validated with bearing voltage measurements. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Case**} & **Frequency** & **Iron** & **Proximity** & **Capacitive** & **Validation** \\  & **Range** & **Core** & **Effect** & **Coupling to Ground** & **V** \\ \hline
[40] & 0–80 k & Eddy Currents & & Parallel plates & Single coil \\
[37] & 0–1 M & Dumping resistance & Dumping resistance & Bulk winding & DM CM (Not adjusted resonances) \\
[47] & 0–200 k & Bertotti & & Lumped C network & \\
[50] & 0–1.5 k & & Laplacian Equation & & Joule Losses in PMSM (Error \textless{} 10\%) \\
[38] & 0–10 M & (Shielded cable) & Bessel function & & R \& X with FEM (Error \(\approx\) 12\%) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Analytic simulation comparison. * **Turn-to-turn Capacitance:** In [42], an analytical model of a winding is presented to predict the turn-to-turn capacitance. The capacitance for nonadjacent turns is not taken into account as it is assumed that the flux lines are only directed towards the next immediate conductor. The turn-to-turn capacitance is the series of the airgap and coating capacitances. In other words, the global accuracy of the analytical results compared to FEM results is questionable looking the relative error.