# Study of Partial Discharge Inception Voltage in Inverter Fed Electric Motor Insulation Systems
## 4 Analysis of PDIV and Modelling Tool Performance for Variable Temperature
### Performance of PDIV Models for Variable Temperature
#### 4.2.5 Proposed Model  
Within the previously analysed models (Dakin, parallel-plate, Lusuardi, and FEM and Paschen's law), [PERSON] and FEM and Paschen's law models were the ones capable to obtain good accuracy on the estimation of the PDIV at high temperatures. However, the Lusuardi model needed an experimentally measured PDIV value and did not consider the insulation permittivity, while the FEM and [PERSON]'s law model required a high computational load. Hence, in order to avoid the experimental tuning of the model and reduce the computational load, a new model was proposed. The proposed model was based on parallel-plate model (Equation (10)) [59]. In the parallel-plate model, [PERSON]'s law is used (Equation (2)). As explained in Section 2.2.1, this equation does not consider how the temperature changes the air properties. As a result, it is not accurate at high temperatures. However, it is possible to apply Peek (Equation (6)) or Dunbar (Equation (7)) corrections to solve this issue. Thus, the use of these temperature correction methods in a parallel-plate model was proposed. Firstly, it was necessary to adjust \(\gamma\). In Figure 15, a comparison of the experimentally measured data (sample group C) and parallel-plate model, evaluated for different \(\gamma\) values proposed in the literature at ambient temperature, is depicted. \(\gamma=0.01\) is a well-known value for metallic electrodes, which was also used in insulated twisted pairs [63; 69]. \(\gamma=0.001\) is the value used by [59] to develop the parallel-plate model (the one that best fits their experimental data). \(\gamma=0.0006\) was proposed by [62] for PAI overcoat samples, \(\gamma=0.00065\) was proposed for PI insulation by [73], and \(\gamma=0.0078\) was reported by [70] for epoxy resin. It can be seen that the \(\gamma\) values proposed for PAI and PI best fit the experimental results. The PDIV estimation in both cases does not differ, due to close \(\gamma\) values and the error for both being around 1%. Once \(\gamma\) was adjusted, the Peek and Dunbar corrections were applied to Equation (10). This way, the parallel-plate model was corrected as Equation (23) with Peek correction (based on Equations (2), (6), and (10)) and Equation (24) with Dunbar correction (based on Equations (2), (7) and (10)). \[\mathrm{PDIV}=min\left[\frac{\frac{P_{\mathrm{new}}}{P_{\mathrm{ref}}}\frac{T_ {\mathrm{ref}}}{T_{\mathrm{new}}}B\frac{P_{\mathrm{ref}}\ d_{\mathrm{air}}}{ln \left(\frac{\frac{T_{\mathrm{ref}}}{T_{\mathrm{ref}}}d_{\mathrm{air}}}{k} \right)}}{d_{\mathrm{air}}}\left(d_{\mathrm{air}}+\frac{t_{\mathrm{ins1}}}{ \varepsilon_{\mathrm{r1}}}+\dots\right)\right] \tag{23}\]  
Figure 15: Comparison of experimental measurements and parallel-plate model for different \(\gamma\) values at room temperature (sample group C). Crosses represent the error. \[\text{PDIV}=min\left[\frac{B\frac{P_{\text{ref}}\frac{T_{\text{ref}}}{T_{\text{ref}}} \frac{d_{\text{air}}}{T_{\text{ref}}}}{ln\left(\frac{A_{\text{ref}}\frac{T_{ \text{ref}}}{T_{\text{ref}}}\frac{d_{\text{air}}}{T_{\text{ref}}}}{k}\frac{d_{ \text{air}}}{k}\right)}{d_{\text{air}}}\left(d_{\text{air}}+\frac{t_{\text{ins1} }}{\varepsilon_{\text{r1}}}+\dots\right)\right] \tag{24}\]  
where \(A\) and \(B\) are experimental constants and depend on the gas composition (for air: 11.25 Pa\({}^{-1}\)m\({}^{-1}\) and 273.75 VPa\({}^{-1}\)m\({}^{-1}\), respectively, [58]). \(d_{\text{air}}\) is the straight distance between surfaces. \(k\) is defined in Equation (3). \(P_{\text{ref}}\) and \(T_{\text{ref}}\) are the pressure and temperature at room conditions, respectively, and \(P_{\text{new}}\) and \(T_{\text{new}}\) are the pressure and temperature at which the PDIV is estimated, respectively. \(t_{\text{ins1}}\) and \(\varepsilon_{\text{r}}\) are the thickness and the relative permittivity of each insulation layer, respectively. The results obtained using the correction of temperature using Equations (23) and (24) are illustrated in Figure 16 in comparison with the experimental measurements (sample group C).