## 2 Fatigue Characteristics of High-Power IGBT Model
### Analytical Fatigue Model  
Different BAFTs are based on different \(I_{1}\), as shown in Figure 15. The horizontal axis in Figure 15 is the number of cycle times, and the vertical axis is cumulative failure rate. Under \(I_{1}\) at 1200 A and 900 A, the mathematical model between cumulative failure rate \(F\) and cycle number \(x\) is as follows:  
\[F_{B1200}(x)=1-e^{-(\frac{x}{900.85})}{}^{8.01} \tag{21}\]  
\[F_{B900}(x)=1-e^{-(\frac{x}{917.254})}{}^{5.96} \tag{22}\]  
Obtained from Equation (12)  
\[F_{B1500}(x)=1-e^{-(\frac{x}{317.254})}{}^{12.98} \tag{23}\]  
Figure 14: Comparison of Weibull distribution fitted by GA and curves fitted by weighted LSM (least squares method).  
In Equations (21)-(23), \(F_{\rm B1500}(x)\), \(F_{\rm B1200}(x)\) and \(F_{\rm B900}(x)\) are the cumulative failure efficiencies of \(I_{\rm 1}\) at 1500A, 1200A and 900A, respectively, and their mean values are 304890, 479170 and 851580. The corresponding equipment service life \(N_{\rm f}\) can be expressed in terms of the mean value.  
In addition, the ambient temperature \(T_{\rm a}\) should also be considered [30; 31]. According to Equation (9), the \(N_{\rm f}\) expression considering \(T_{\rm a}\) is obtained.  
\[N_{\rm f}=f(I_{\rm eq})\times g(T_{\rm a}) \tag{24}\]  
In Equation (24), \(g\) is specific functions.  
Finally, after the fitting calculation, Equation (9) is expressed as:  
\[N_{\rm f}=(1.17I_{\rm 1}^{2}-3697.11I_{\rm 1}+3.23\times 10^{6})\times e^{k( \frac{1}{I_{\rm a}}-\frac{1}{25})} \tag{25}\]  
In Equation (25), \(k\) is an Arrhenius coefficient. According to [PERSON]'s law (a temperature increase of \(10\)\({}^{\circ}\)C means that the service life is shortened by half), \(k\) is calculated to be 60.65. The experimental and fitting curves are shown in Figure 16.  
Considering the RMS calculation scheme, the amplitude is \(\sqrt{2}\) times of the effective value, so:  
\[I_{\rm 1}=\sqrt{2}I_{\rm eq} \tag{26}\]  
From Equation (26):  
\[N_{\rm f}=(2.34I_{\rm eq}^{2}-5228.35I_{\rm eq}+3.23\times 10^{6})\times e^{60. 65(\frac{1}{I_{\rm a}}-\frac{1}{25})} \tag{27}\]  
The data obtained in the experiment and the curve that was obtained by mathematical fitting is shown in Figure 16. The vertical axis in Figure 16 is the cycle counts that a DUT could survive before failure. The horizontal axis is the amplitude value of the test current. As shown in Figure 16, the obtained curve conforms to the obtained data very well.  
By Equation (27), when \(I_{\rm eq}\) and \(T_{\rm a}\) are known, the service life or failure rate of the IGBT can be predicted. Equation (27) has been subjected to actual field data. The example we consider here is a metro line application in Shenzhen, a southern-China city, with the annual temperature average of 23 \({}^{\circ}\)C. The \(I_{\rm eq}\) has been recorded to be 489 A in a single year; therefore, \(N_{\rm f}\) is calculated to be 2,197,460. Considering that a train normally operates 204 cycles per day and normally 300 days per year, the train is capable of operating around 36 years under the same working conditions.  
Figure 15: Curves obtained under different BAFT aging currents.  
According to statistics offered by metro operating agency, the initial failure rate of a new module is 157 FIT. Given that they hold that the failure rate rise of 200% means that the switching devices in a batch have failed, the statistical service life of the applied power module in the specific metro line is around 47 years (offered by the agency). This means that the approach that was proposed has a prediction error of around 23%, which is within project tolerance (normally 20%-30%). It should be noted that the lifetime predicted by our model in Equation (27) is more likely to be true and more convincing, because a metro train is designed to operate for no more than 30 years, and the 47-year lifetime prediction does not take into consideration the acceleration of the aging process, as is shown by a bathtub curve.