# Study of Partial Discharge Inception Voltage in Inverter Fed Electric Motor Insulation Systems
## 2 Literature Review
### Lusuardi Model  
Based on the experimental results from measuring the PDIV at different pressure levels using 50 Hz AC, [PERSON] et al. [26] proposed an analytical model for the estimation of the PDIV at different pressure and temperature levels. First, they defined a new empirical model using linear regression based on the PDIV measured values as a function of the logarithm of pressure (in bar). Then, since the ionisation process depends on the mean free path, which is inversely proportional to the number density (number of gas molecules per cubic meter), they reported the results as a function of the number density. The number density is dependent on the pressure and temperature, so Equation (11) can be deduced as normalised to reference values. \[\frac{\mathrm{PDIV}_{\mathrm{Pew},T_{\mathrm{new}}}}{\mathrm{PDIV}_{ \mathrm{ref}}}=1+0.299\mathrm{ln}\bigg{(}\frac{P_{\mathrm{new}}}{P_{\mathrm{ ref}}}\frac{T_{\mathrm{ref}}}{T_{\mathrm{new}}}\bigg{)}+0.0446\mathrm{ln} \bigg{(}\frac{P_{\mathrm{new}}}{P_{\mathrm{ref}}}\frac{T_{\mathrm{ref}}}{T_{ \mathrm{new}}}\bigg{)}^{2} \tag{11}\]  
where \(\mathrm{PDIV}_{\mathrm{Pew},T_{\mathrm{new}}}\) is the estimated PDIV value at the desired pressure and temperature, \(\mathrm{PDIV}_{\mathrm{ref}}\) is the experimentally measured PDIV value at the reference pressure and temperature, \(P_{\text{ref}}\) and \(T_{\text{ref}}\) are the pressure and temperature at which experimental PDIV measurement is collected, respectively, and \(P_{\text{new}}\) and \(T_{\text{new}}\) are the pressure and temperature at which the PDIV is estimated, respectively. The author's reasoning for the use of this model is that the Paschen's curve is not able to adequately reproduce the dependency of the measured PDIV on pressure. However, the author mentions that it is not valid if the relative permittivity has a considerable dependency on the temperature. The maximum temperature analysed is far from the glass transition temperature and the permittivity change is negligible within this temperature range for the used samples. Hence, this model might not be valid for insulations with glass transition temperatures close to the operating temperature. In this kind of insulation, the contribution of the permittivity increases at high temperatures, so the permittivity should be considered as a separate variable. Estimation Considering Rise Time and Pulse Width  
Based on the extended volume-time theory, explained later, [PERSON] et al. [43] proposed a model to estimate the PDIV for a single pulse based on the experimental results, where the rise time and pulse width are considered. Assuming that the initial electron generation probability (defined using the Extended Volume-Time theory) can be approximated as the accumulated PD probability, the authors stated that this probability can be obtained from the experimental PDIV value and time delay value (the time period since the applied voltage meets the ignition voltage until the PD is developed; see Figure 5). \[P(t)=1-e^{-A(t-t_{0})} \tag{12}\]  
\[P(t)=1-e^{-A\left(t-\frac{V_{\text{in}}\text{-}t}{0.08}\text{\emph{v}}_{ \text{apply}}\right)} \tag{13}\]  
where \(A\) is the equivalent initial electrons from the air volume and enamel layer surface and \(t_{0}\) is the time when the ionisation process starts. Under the same environmental conditions, \(t_{0}\) corresponds to the instant at which the applied voltage (\(V_{\text{apply}}\)) is equal to the inception voltage (\(V_{\text{inc}}\)). In this case, the authors considered that the ignition voltage is the measured PDIV in the test sample under a constant DC voltage. As the rise time (\(t_{\text{r}}\)) is defined between 10% and 90% of the applied voltage, a 0.8 factor is used. Therefore, Equation (12) can be rewritten as Equation (13). With the aim of simplifying the acquisition of the equivalent initial electrons, the authors propose Equation (14) for the quantification of \(A\), considering Equation (13) and experimental PDIV (defined when the probability is 50%) and delay time data.