# Study of Partial Discharge Inception Voltage in Inverter Fed Electric Motor Insulation Systems
## 2 Literature Review
### PDIV Modelling
#### 2.2.1 Paschen's Law  
[PERSON]'s law describes the breakdown voltage for a given gas and electrode material under uniform electric field conditions, as a function of the product of gas pressure and gap length. This law is developed under constant pressure and temperature conditions [25]. The analytical expression derived for the breakdown voltage (\(V_{\text{B}}\)) is described in Equation (2):  
\[V_{\text{B}}=B\frac{Pd_{\text{air}}}{In\Big{(}\frac{APd_{\text{air}}}{k}\Big{)}} \tag{2}\]  
\[k=ln\bigg{(}1+\frac{1}{\gamma}\bigg{)} \tag{3}\]  
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]), \(P\) is the air pressure, \(d_{\text{air}}\) is the path followed by the electronic avalanche (the electric field lines), and \(\gamma\) is the Townsend's secondary electron emission coefficient, which describes the average released electrons from the cathode per incident positive ion [25]. It is important to note that this voltage is considered as peak to peak [59]. Despite [PERSON]'s law being widely used in the literature for the estimation of the PDIV [53; 59; 60; 61; 62; 63; 64; 65; 66; 67; 68], its application to the case of enamelled twisted pairs presents some limitations that must be taken into account. 1. In the general use of [PERSON]'s law (using parallel plate electrodes), for a given pressure, the first ionisation phenomena (\(\alpha^{\prime}\)) is considered constant in the distance \(d_{\text{air}}\), defining \(d_{\text{air}}\) as the perpendicular distance between electrode surfaces. Several authors have used the same definition to model the PDIV on twisted pairs with FEM simulations [60; 68; 69] or analytical models [59]. Indeed, the difference between the real and straight field length has been shown to be lower than 1% for small distances, while it is about 7% for distances larger than 50 \(\upmu\)m [69]. Addversely, [PERSON] et al. [70] stated that this simplification of assuming a constant effective first ionisation phenomenon is not correct for this specified geometry. In twisted pairs, the path defined by the perpendicular distance between wires does not match with the path followed by the electric field lines. Therefore, this approach is questioned. 2. [PERSON]'s law is based on [PERSON]'s ionisation mechanism. Although several authors used [PERSON]'s law for the definition of the PDIV [59; 60; 62; 63; 70], in [70], the author notes that some works have proven that the streamer inception mechanism is more effective. Additionally, [PERSON] et al. [71] argued that, as enamelled twisted pairs, electrodes are covered by an insulation film, so the emission of secondary electrons from the insulated cathode by positive ions is extremely unlikely. Accordingly, [PERSON]'s law can not be used to estimate the PDIV and should be replaced by the streamer criterion. The main concept of this criterion is the single-electron-avalanche mechanism, which defines the breakdown voltage of the air when the number of electrons reaches a critical number. The relationship between the number of electronsalong a field line and the critical electron number is defined by the [PERSON] criteria (Equation (4)):  
\[k_{\rm s}=\int_{0}^{d_{\rm air}}\alpha_{\rm eff}(x)\,\mathrm{d}x=\ln N_{\rm c} \tag{4}\]  
where \(\alpha_{\rm eff}\) is the effective ionisation coefficient of the anode, \(N_{\rm c}\) is the critical number of electrons for the breakdown, and \(k_{\rm s}\) is a dimensionless constant, generally taken as 18 for air. 3. Independent of the use of [PERSON]'s law or [PERSON]'s criteria, there is no consensus in the literature regarding the definition of the value \(\gamma\) for [PERSON]'s law and \(k_{\rm s}\) for the [PERSON] criteria. Concerning \(\gamma\), although it is well defined for metal electrodes, there is uncertainty about its value for different kinds of enamel materials [62; 70].