# Study of Partial Discharge Inception Voltage in Inverter Fed Electric Motor Insulation Systems
## 2 Literature Review
### PDIV Modelling
#### 2.2.1 Paschen's Law
Regarding the Schumann constant \(k_{\rm s}\), although it is generally defined as a constant value (18 for air), the prediction of the PDIV using this value seems not to be very accurate [71]. 4. It is well known that the breakdown voltage of the air is affected by atmospheric conditions such as temperature, pressure, and humidity. Several authors [26; 30; 38; 53; 65; 72] stated that [PERSON]'s law is not valid for certain atmospheric conditions and proposed corrections for Equation (2). Different kinds of solutions are proposed in the literature to overcome the issues related to the definition of [PERSON]'s secondary ionisation coefficient (\(\gamma\)) and the [PERSON]'s criteria constant (\(k_{\rm s}\) ), as well as to consider the influence of the temperature on the [PERSON]'s law. These solutions are collected in the following lines. [PERSON]'s Secondary Ionisation Coefficient (\(\gamma\)) and [PERSON]'s Criteria Constant (\(k_{\rm s}\)) Correction  
As mentioned before, researchers do not completely agree about the value of the \(\gamma\) coefficient. \(\gamma=0.01\) is a well-known value for metallic electrodes. Several researchers have obtained good accuracy in the estimation of PDIV by applying this value to polymeric enamelled twisted pairs [63; 69]. On the other hand, others have adjusted the \(\gamma\) value in order to achieve good accuracy in the estimation of the PDIV compared to the experimental results. For instance, [PERSON] et al. [59] adjusted \(\gamma=0.001\), which best fits their experimental measurements using a PEI (Polyester-imide) and PAI enamel combination. Bearing this in mind, [PERSON] et al. [62] proposed a method to predict the \(\gamma\) value by iterations, based on a mixture of numerical and FEM calculations and comparing with the experimental results. Firstly, a FEM model is built by using a precise enamelled profile with the help of a microscope and representing the true experimental configuration. Then, the 2D electrostatic problem is solved using FEM, with the experimental PDIV value being the input of a high voltage conductor (see Section 2.2.3), to calculate the voltage drops along straight field lines between the wires. The computed voltage drops are compared to the [PERSON]'s curve and the \(\gamma\) is iterated until the [PERSON]'s curve intersects with the voltage drops. Using this method, \(\gamma=0.0006\) is proposed by [62] for PAI overcoat samples. Moreover, negligible differences are observed for different diameters, meaning the same \(\gamma\) value can be used. A similar methodology was proposed by [PERSON] et al. [70] but by considering constant electric field lines to define \(d_{\rm air}\). In this case, the \(\gamma\) value is defined as 0.0078 based on the arithmetic mean of the value estimated for different diameters and relative permittivity values (the materials have different epoxy resins, not specified). It is shown that using this mean value, the maximum error achieved in the estimation of PDIV for different wire diameters and permittivity values at room temperature is about 5%. The same process for the estimation of \(\gamma\) is developed at 100 and 150 \({}^{\circ}\)C, showing that \(\gamma\) tends to decrease with an increase in the temperature. Nonetheless, after analysing the errors and variations in the estimation, the author concludes that the changes in the material type and manufacturer are more important than those in the temperature. Finally, [PERSON] et al. [73] proposed a different method for \(\gamma\) estimation. In a similar way to [70], the electric field lines and scalar potential were computed using FEM. Then, the results were integrated into self-developed software to predict the PDIV using [PERSON]'s theory in a range of \(\gamma\) from \(10^{-6}\) to \(10^{-2}\). The \(\gamma\) was interpolated for the defined experimentally PDIV. This way, \(\gamma=0.00065\) was proposed for the PI (Poly-imide).