## 5 MTTF Analysis
In general, three failure rates, which are seen in Figure 3, are described as follows:  
\(\lambda_{H}^{(k)}\): Failure rate of a switch when the converter operates in \(k\)-phase mode, and the two parallel-switches conduct the phase current in parallel state;  
\(\lambda_{F}^{(k)}\): Failure rate of a switch when the converter operates in \(k\)-phase mode and only one of the standby-switches conducts the phase current;  
\(P^{\text{C}}\): Probability of successful operation of the fault-managing system. As shown in Figure 5, the basic failure rate is considered as a fuzzy function The triangular memberships depend on three constant parameters a, b, and c, (a = 0.01, b = 0.012, and c = 0.014 for the basic failure curve and a = 0.6, b = 0.7 and c = 0.8 for the probability of successful operation of the fault managing system curve), according to Table 1 and the basic failure rate, the failure rates of power switches in different redundant configuration can be obtained. As seen in Figure 3, to calculate MTTF of a single-phase, two-phase and three-phase converter, \(\lambda_{H}^{(3)}\), \(\lambda_{F}^{(3)}\), \(\lambda_{H}^{(2)}\), \(\lambda_{F}^{(2)}\), \(\lambda_{H}^{(1)}\), and \(\lambda_{F}^{(1)}\) must be determined. In these parameters, superscript 1,2,3 show the operation mode of the converter, i.e., single-phase, two-phase and three-phase states. Subscript \(F\) and \(H\) show the states of the switch that conducts either half the current of a phase or the whole of it, respectively. These parameters are fuzzy and can be extracted as follows:  
\[\widetilde{\lambda}_{H}^{(k)} =\pi_{\text{Q}}\pi_{A}\pi_{E}\times\pi_{TH}^{(k)}\times \widetilde{\lambda}_{b} \tag{9}\] \[\widetilde{\lambda}_{F}^{(k)} =\pi_{\text{Q}}\pi_{A}\pi_{E}\times\pi_{TF}^{(k)}\times \widetilde{\lambda}_{b} \tag{10}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Converter Type**} & **Redundant-Switch** & **Junction Temperature (T\({}_{j}\))** & **Temperature Factor (\(\pi_{t}\))** \\  & **Configuration** & **(Ambient Temperature = 25 °C)** & **25 °C** \\ \hline \multirow{2}{*}{Single-phase} & Parallel & 87 °C & 3.065 \\  & Standby & 130.2 °C & 5.422 \\ \hline \multirow{2}{*}{Two-phase} & Parallel & 59.3 °C & 1.951 \\  & Standby & 74.2 °C & 2.497 \\ \hline \multirow{2}{*}{Three-phase} & Parallel & 52.3 °C & 1.736 \\  & Standby & 61 °C & 2.023 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Junction temperature and temperature factor of power switch in parallel-switch and standby-switch configurations. Figure 5: Fuzzy membership: (**a**) Basic failure, (**b**) Probability of successful operation of the fault managing system. In order to calculate the MTTF of the single-phase, two-phase and three-phase converter, referring to Figure 3, the probability of successful operation of the fault-managing system is defined in Table 2 where the differential equations for reliability analysis are given as follow:  
\[\frac{d}{dt}\left[ \begin{array}{c}P_{3}(t)\\ P_{2}(t)\\ P_{1}(t)\\ P_{fault}(t)\end{array} \right]=\left[ \begin{array}{cccc}-3\widetilde{\lambda }^{(3)}&0&0&0\\ 3\widetilde{P}_{c}\widetilde{\lambda}^{(3)}&-2\widetilde{\lambda}^{(2)}&0&0\\ 0&2\widetilde{P}_{c}\widetilde{\lambda}^{(2)}&-\widetilde{\lambda}^{(1)}&0\\ 3(1-\widetilde{P}_{c})\lambda_{Th}&2(1-\widetilde{P}_{c})\widetilde{\lambda} ^{(2)}&\widetilde{\lambda}^{(1)}&0\end{array} \right]\left[ \begin{array}{c}P_ {3}(t)\\ P_{2}(t)\\ P_{1}(t)\\ P_{fault}(t)\end{array} \right] \tag{11}\]  
The reliability functions of the three-phase converters can be obtained as follows:  
\[1-P_{fault}(t)=R(t) \tag{12}\]  
Regarding Equation (5), (11), and (12), the MTTF of the three-phase converter is given by:  
\[MTTF^{(3)}=\frac{1}{3\widetilde{\lambda}^{(3)}}+\frac{\widetilde{P}_{c}}{ \widetilde{\lambda}^{(2)}}+\frac{\widetilde{P}_{c}^{2}}{\widetilde{\lambda}^ {(1)}} \tag{13}\]  
Similarly, the MTTF functions of the two-phase and single-phase converters are extracted by Equation (14) and (15), respectively.