# Study of Partial Discharge Inception Voltage in Inverter Fed Electric Motor Insulation Systems
## 2 Literature Review
### Lusuardi Model
#### 2.2.3 FEM Simulations  
Two different approximations have been identified in the literature for the estimation of the PDIV based on the FEM simulations. In all of them, the FEM simulations are developed in order to determine the electric field or potential difference across two insulated wires [38; 60; 62; 63; 69; 70]. The geometry used is defined as in Figure 6. Then, the results are evaluated by applying different methods to define the PDIV value (Paschen's law or Schumann criteria). The FEM model attempts to simulate the most critical situation of turn-to-turn insulation, where one conductor is subjected to high voltage (HV, defined as peak-to-peak voltage) and the other to 0 V (GND) and there is no impregnation. \(d_{\text{in}}\) is the conductor nominal diameter, \(t_{\text{ins}}\) is the insulation thickness, \(d_{0}\) is the minimum distance between wires, \(d\) is the horizontal distance between the wires, and \(d_{\text{e}}\) is the length of the electric field line. Using this model, the voltage distribution of the air gap is computed in the range of the gap distance (defined as \(d\) or \(d_{\text{e}}\)). In order to define the PDIV value using a combination of FEM and Paschen's law, FEM simulations for different voltage level inputs are developed. The voltage distribution of the air gap in the range of gap distance is calculated using FEM and subsequently compared to [PERSON]'s law (Figure 7). If the FEM curve crosses the Paschen's curve, there is PD activity. If the FEM curve does not cross the Paschen's curve, there is no PD activity. Thus, the PDIV is defined as the point where the FEM curve is tangential to the Paschen's curve, as shown in Figure 7.  
The main idea behind this method is included by different authors. Nevertheless, as described in Section 2.2.1, there is no agreement in the literature as to whether \(d_{\text{air}}\) is defined as the horizontal distance between the wire surfaces (defined as \(d\) in Figure 6) or the path followed by the electric field lines must be used (defined as \(d_{\text{e}}\) in Figure 6) to define the distance followed by the electrons (\(d_{\text{air}}\) in Equation (2)). Figure 6: FEM model geometry. The straight-line distance between the two surfaces is used to define the electric field in [38; 60; 62; 69]. Indeed, when using a straight distance, an error of around 2% [69], less than 5% [60], and less than 3% [38] is reported in the literature in the estimation of the PDIV. Additionally, of the cited studies, [38] obtained PDIV values for different temperatures by considering the permittivity dependence on the temperature and the Paschen's curve temperature correction. In the case of [62], this method was used to obtain the \(\gamma\) value as described in Section 2.2.1. However, other authors [63; 70] used constant electric field lines to estimate the PDIV and defined the \(\gamma\) value based on the reasons stated in Section 2.2.1. For that, the electric field in the air gap between the wire surfaces is first computed using FEM. Then, constant electric field value points are identified and a function based on a second-order polynomial is defined. Finally, the \(d_{\mathrm{e}}\) is calculated using the integral of polynomial between two points. FEM and Schumann Criteria  
As explained in Section 2.2.1, [PERSON] et al. [71] reported that, in twisted pairs, there are nonuniform electric fields. Moreover, the emission of secondary electrons from the insulated cathode is unlikely, so [PERSON]'s law cannot be used to predict the PDIV and must be replaced by the streamer inception criterion. With this criterion, the PDIV is defined as when the number of electrons of an electric field line fulfils the Schumann criteria. The method proposed by [PERSON] et al. [71] uses FEM simulation and Schumann criteria to define the PDIV, considering the effect of the temperature. The Schumann constant is defined as 5.98, as explained in Section 2.2.1. The method consists of the following steps:  
1.