## 2 Fatigue Characteristics of High-Power IGBT Model
### Analysis of the Results of Conventional Single-Directional Aging Experiments
In Figure 2, G1 is auxiliary GTO, \(I_{1}\) is a fatigue current and \(I_{2}\) is used to calculate \(T_{\rm j}\). R and C form an absorption branch. If G1 is turned on, \(I_{1}\) flows through the IGBT part. If G1 is turned off, G2 provides the freewheeling route of \(I_{1}\). C1 and C2 produce \(I_{1}\) and \(I_{2}\), respectively. During the experiment, aging current \(I_{1}\) only flows through IGBT. So, it can be called SAFT. During AFT, the ambient temperature is approximately 25 \({}^{\circ}\)C. \(I_{2}\) is 1.5 A, which is no more than 0.1% of the DUT rated current (1500 A) as a rule of thumb. The cycle time is 12 s with a duty cycle of 48%. During the AFT, DUT is placed in the actual TC power module and is then connected to the test platform. In AFT, \(F\) is calculated from the observed \(U_{\rm CES}\)[7], as shown in Equation (3). \[F=\frac{U_{\rm CESat}-U_{\rm CESat0}}{U_{\rm CESat0}} \tag{3}\]  
In Equation (3), \(U_{\rm CES0}\) is the initial saturation voltage of a new DUT, and \(U_{\rm CES}\) is the current value. When \(U_{\rm CES}\) rises to 1.2 times that of \(U_{\rm CES0}\), \(F\) is defined to be 1. With the data from SAFT, the corresponding \(F\) is calculated. Four different distribution models are fitted by the LSM method. As for the normal distribution, it gives:  
\[F(x)=\frac{1}{35500\sqrt{2\pi}}\int_{-\infty}^{x}e^{-\frac{(t-200254)^{2}}{2 \times 3500^{2}}}dt,\mu=309254,\sigma=35500 \tag{4}\]  
As for the lognormal distribution, it gives:  
\[F(x)=\frac{1}{0.127\sqrt{2\pi}}\int_{-\infty}^{\ln x}e^{-\frac{(t-22.665)^{2}} {2\times 0.127^{2}}}dt,\mu=12.665,\sigma=0.127 \tag{5}\]  
As for the Weibull distribution, it gives:  
\[F(x)=1-e^{-(\frac{x}{317254})12.98},\lambda=317254,\ k=12.98 \tag{6}\]  
As for the Logarithmic Weibull distribution, it gives:  
\[F(x)=1-e^{-(\frac{\ln x}{\ln 6})12.667},\lambda=165,\ k=12.667 \tag{7}\]  
In order to select the most suitable one of the four distributions, we plot the curve of the experimental data and four distributions, and the density functions are also plotted. The results are shown in Figures 3 and 4. It should be noted that in Figures 3 and 4, the curve of the Weibull distribution coincides with the curve of the Logarithmic Weibull distribution. In Figures 3 and 4, the horizontal axis is the number of cycle times, and the vertical axis is cumulative failure rate, which is calculated according to DUT saturation voltage drop [13; 17]. Figure 2: Schematic diagram of single-directional accelerated fatigue test. From Figure 4, it seems that the Weibull distribution is closer to the practical fatigue process (hence the fault transition process) compared with the normal distribution, because the latter one shows a larger error. What is more, the lognormal distribution is basically consistent with normal distribution; hence the same fitting accuracy is obtained. However, the transient component of failure density in the case of the lognormal distribution function is too high; therefore, it is not suitable to describe the failure density of high-power IGBTs. The logarithmic Weibull distribution has the same  
Figure 4: Comparison of the density function of Weibull distribution, Logarithmic Weibull distribution and normal distribution, lognormal distribution under the single-direction 1500 A fatigue test. Figure 3: Comparison between data curves of single-direction 1500 A fatigue test and four distribution fitting results. characterization accuracy as the Weibull distribution, however its calculation process is much more complicated. Finally, the Weibull distribution is the most suitable one.