## 2 Fatigue Characteristics of High-Power IGBT Model  
### Analysis of the Results of Conventional Single-Directional Aging Experiments  
The analytic model of the IGBT is the analytical relationship between the sensitive factor in the IGBT and the number of failures (i.e., the lifetime). The aging data is fitted correspondingly, and the expression is intuitive. In order to describe the relationship between the cumulative failure rate \(F\) and the number of experiments (i.e., the number of cycles), it is necessary to select an appropriate life distribution. If the distribution model is properly selected, the \(F\) under various working conditions could be obtained with more accuracy [15]. Common life distribution models include normal distribution, lognormal distribution, Weibull distribution and log-Weibull distribution. The choosing of distribution model is the key to switching device fault reconfiguration. The normal distribution, also known as the Gauss distribution, is the most common and widely used distribution of the probability distribution of all random phenomena. It can be used to describe many natural phenomena and various physical properties. In practical applications, there are many experimental data that can be fitted with a normal distribution as its approximate distribution [16]. The lognormal distribution is a typical model that is derived from a normal distribution. When the logarithm of the expiration time \"\(In\) (\(t\))\" obeys the normal distribution, \(t\) obeys logarithmic normal distribution. Logarithmic transformation can reduce the larger number to a smaller number. Lognormal distribution is widely used in fatigue life research [17]. The Weibull distribution is a continuous distribution that is widely used in reliability, and its advantage lies in its strong adaptability to various types of test data. It is widely used in reliability engineering, especially for the distribution of wear and tear failure of electromechanical products. Since it can easily infer its distribution parameters using probability values, it is widely used in data processing of various life tests [15]. The log-Weibull distribution is a special correlation distribution of the Weibull distribution which is obtained by taking the logarithm of \(x\) and obeying the Weibull distribution. The effect is similar to the lognormal distribution, and the larger number can be reduced by logarithmic transformation [18]. Table 1 is the equation for the distribution function and density function of the four distributions. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Name** & **Distribution Function** & **Density Function** \\ \hline Normal distribution & \(F(x)=\frac{1}{e\sqrt{2\pi}}\int_{-\infty}^{x}e^{-\frac{(y-\mu)^{2}}{2\mu^{2}}}dt\) & \(f(x,\mu,\sigma)=\left\{\begin{array}{ll}\frac{1}{e^{\sqrt{2\pi}}}e^{-\frac{(y- \mu)^{2}}{2\sigma}}&,x>0\\ 0&,x\leq 0\end{array}\right.\) \\ Lognormal distribution & \(F(x)=\frac{1}{e\sqrt{2\pi}}\int_{-\infty}^{\ln x}e^{-\frac{(y-\mu)^{2}}{2\sigma }}dt\) & \(f(x,\mu,\sigma)=\left\{\begin{array}{ll}\frac{1}{e\sqrt{2\pi}}e^{-\frac{(y- \mu)^{2}}{2\sigma}}&,x>0\\ 0&,x\leq 0\end{array}\right.\) \\ \hline Weibull distribution & \(F(x)=\left\{\begin{array}{ll}1-e^{-(\frac{x}{x})^{x}}&,x>0\\ 0&,x\leq 0\end{array}\right.\) & \(f(x,\Bbbk,\lambda)=\left\{\begin{array}{ll}\frac{k}{\lambda}(\frac{x}{ \lambda})^{k-1}e^{-(\frac{x}{x})}&,x>0\\ 0&,x\leq 0\end{array}\right.\) \\ Logarithmic Weibull distribution & \(F(x)=\left\{\begin{array}{ll}1-e^{-(\frac{x}{x})^{x}}&,x>0\\ 0&,x\leq 0\end{array}\right.\) & \(f(x,\Bbbk,\lambda)=\left\{\begin{array}{ll}\frac{k}{\lambda}(\frac{x}{ \lambda})^{k-1}e^{-(\frac{x}{x})^{x}}&,x>0\\ 0&,x\leq 0\end{array}\right.\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Distribution functions and density functions of the four distributions. Because the design life of IGBTs is generally more than 20 years, aging data is generally obtained by means of accelerated fatigue test (AFT) [19].