## 5 Reliability Evaluation  
The switching devices used in power electronic converters are the most failure-prone components and they are easily damaged. As power modules are composed of different layers with various thicknesses, materials, and physical properties, the thermal expansion for each layer will show a different behavior over a given temperature. Thus, the power module is subject to failure due to this thermal stress caused within its structure. For this reason, methods such as thermal cycling are enormously useful in identifying degradation mechanisms that dominate the lifetime of power modules [52]. The thermal stress will gradually lead to fatigue and the useful life of the module will be the same as the number of thermal cycles, which reduces with increasing junction temperature's fluctuation. Power cycling is an efficient method for subjecting the bonded connections, soldered joints, and substrate lamination to thermal cycling [53]. During the power cycling test, the power semiconductors are repeatedly heated up by the power losses inside the module; then, they are cooled down by the cooling system. The cycle length is usually between several milliseconds to a few seconds [54].  
There are various analytical methods to link the failure mechanism to reliability performance, and by using these methods, an accurate prediction of the module lifetime would be expected. The most well-known lifetime model is the Coffin-Manson model, which is a strain-based model. The Coffin-Manson model is presented as follows:  
\[N_{f}=\alpha\big{(}\Delta T_{j}\big{)}^{-n} \tag{22}\]  
where \(\alpha\) and \(n\) are constant parameters that are empirically obtained. This simple equation simply indicates that the number of cycles to failure is directly related to the temperature change (\(\Delta T_{j}\)). The parameter of \(\Delta T_{j}\) shows the difference between the maximum and minimum temperatures for each cycle. In [55], it has been concluded that the average junction temperature also affects the lifetime and must be included in the aforesaid equation (Equation (22)). Thus, the Coffin-Manson model was modified by the Arrhenius approach as follows [55]:  
\[N_{f}=\alpha\big{(}\Delta T_{j}\big{)}^{-n}\exp(\frac{E_{a}}{k_{b}T_{jm}}) \tag{23}\]  
where \(T_{jm}\), \(E_{a}\), and \(k_{b}\) are the average junction temperature (in Kelvin), activation energy, and Boltzmann constant (8.617 \(\times\) 10\({}^{-5}\) eV/K), respectively. Another analytical lifetime model is known as Norris--Landzberg model, which also considers the frequency of temperature cycles. The equation related to the Norris-Landzberg model is given in the following equation [56]:  
\[N_{f}=\alpha\times f^{n_{2}}\times\big{(}\Delta T_{j}\big{)}^{-n_{1}}\exp( \frac{E_{a}}{k_{b}T_{jm}}) \tag{24}\]  
If the frequency of the temperature cycle is 1 Hz, this model will be the same as the improved Coffin-Manson model (Equation (23)). It should be noted that all parameters of a Coffin-Manson model must be characterized prior to employment, according to the structure of the desirable power module. To obtain the model parameters, the power cycling capability diagram provided in the datasheet should be used. Since the Fuji U  
Figure 8: Closed recursive loop for determining the semiconductors’ junction temperature.  
series power module is utilized in this paper, the power cycling diagram for this module is shown in Figure 9.  
Here, the curve fitting toolbox in MATLAB is used to extract the lifetime model's parameters. The obtained parameters for the examined power module are calculated as \(\alpha\) = 3.128 \(\times\) 10\({}^{12}\), \(n\) = 4.326, and \(E_{a}\) = 0.06606 eV. To neglect the effect of cycling frequency in the lifetime model, the input power would be in the form of a square pulse (heating 0.5 s and cooling 0.5 s) as shown in Figure 10.