# Study of Partial Discharge Inception Voltage in Inverter Fed Electric Motor Insulation Systems
## 2 Literature Review
### Lusuardi Model
#### 2.2.4 Probability Model  
Both analytical and FEM-based models refer to a deterministic approach and, when certain conditions are met, PD will occur. However, this assumption is not always fulfilled. As mentioned several times in Section 2.1, the time when the applied voltage is maintained above the ignition is a key factor in the initial electrons' availability. If the applied voltage is maintained during too short a period, for example in overshoots, even though the applied voltage is higher than ignition, it is unlikely that initial electrons will be available. Therefore, probabilistic PDIV estimation models combining the probability associated with the initial electron availability and the inception voltage condition are developed considering the applied voltage evolution over time. Within these models, the volume-time theory deals with the air gap initial electrons' generation probability in consideration of temporal and spatial change in electric field distribution, to estimate discharge inception at bare electrodes. Based on this theory, [PERSON] et al. [31] proposed the extended volume-time theory, where, in addition to the number of electrons detached in the air gap, it also considers the number of electrons generated using electronic emission from the enamel coating of electrodes. The probability of initial electrons can be computed as described in Equation (18) and the PDIV is defined as when this probability is 50%. \[P(t)=1-e^{\int(R_{\rm V}(t)+R_{\rm S}(t))\mathrm{d}t} \tag{18}\]  
\[R_{\rm V}(t)=\int_{V_{\rm cr}}r_{\rm V}\Big{(}1-\frac{\eta}{\alpha}\Big{)} \mathrm{d}V \tag{19}\]  
\[R_{\rm S}(t)=\int_{S_{\rm cr}}r_{\rm S}\Big{(}1-\frac{\eta}{\alpha}\Big{)} \mathrm{d}S \tag{20}\]  
where \(R_{\rm V}(t)\) (Equation (19)) is the air gap initial electrons and \(R_{\rm S}(t)\) (Equation(20)) represents the initial electrons on the enamel surface. \(V_{\rm cr}\) and \(S_{\rm cr}\) stand for the critical air gap volume and enamel surface, respectively. \(r_{\rm V}\) and \(r_{\rm S}\) are the initial electron generation rates per unit time for the unit volume and unit surface. \(\alpha\) is the electron impact ionisation coefficient and \(\eta\) the electron attachment coefficient. On the one hand, \(V_{\rm cr}\) and \(S_{\rm cr}\) are calculated using electric field analysis using FEM, where [PERSON]'s Equation (Equation (4)) should be satisfied. Accordingly, \(V_{\rm cr}\) is defined as the volume comprising the air gap points where, for the specific electric field, the Schumann criteria is met. Similarly, \(S_{\rm cr}\) is defined for the enamel surface points. The authors do not specify the definition of the Schumann criteria constant, which is a key factor, as explained in Section 2.2.1. The critical surface and volume increase with the enhancement of the electric field. \(r_{\rm V}\) and \(r_{\rm S}\) depend on the instantaneous electric field strength, as defined in Equations (21) and (22), respectively. \[r_{\rm V}=k_{\rm d}n^{-} \tag{21}\]  
\[r_{\rm S}=aE^{2}e^{-\frac{b}{k}} \tag{22}\]  
where \(n^{-}\) is the O\({}_{2}^{-}\) density, a constant irrespective of time and space, and \(k_{\rm d}\) is the expected electron impact ionisation frequency, which depends on electric field strength. \(a\) and \(b\) are experimentally provided based on the Fowler-Nordheim Equation [77]. As shown by the author of [31], \(r_{\rm V}\) tends to stabilise with an increase in the electric field, whereas \(r_{\rm S}\) sharply increases even at high electric fields, which can generate initial electrons in the air gap with enamel-coated electrodes. Several observations were recorded by the authors in [31]. First, both \(R_{\rm V}(t)\) and \(R_{\rm S}(t)\) increased with the applied voltage. This behaviour was also noted in [43] based on the experimental results. On the other hand, the authors concluded that \(R_{\rm S}(t)\) can be more useful for PDIV estimation, as it has a higher increase rate and a higher value.