## 5 MTTF Analysis  
In this paper, the single-phase, two-phase and three-phase interleaved boost converters are evaluated from the MTTF point of view. Also, the redundant-switch configurations are applied to the single-phase, two-phase and three-phase interleaved dc-dc converters. Four steps for the MTTF analysis are considered as follow:  
* Step 1: The MTTF curves for the parallel-switch-based and standby-switch-based single-phase converter are extracted. The obtained curves are illustrated as triangular fuzzy memberships. * Step 2: The fuzzy curves related to the parallel-switch-based and standby-switch-based two-phase converter are traced based on the obtained fuzzy-MTTF functions. * Step 3: With respect to the results obtained from the two previous steps and the three-phase operation condition, fuzzy-based MTTF curves are extracted. * Step 4: The fuzzy results are defuzzied to real values. This step provides a fair comparison to analyse the reliability of the considered converters in different modes. The Markov reliability modes of a three-phase converter with parallel-switch and standby-switch configurations are shown in Figure 3. As shown in these figures, four states can be identified as follows:  
* State I: All the three-phases are healthy; this state is as shown in Figure 3(a) (for parallel-switch) and Figure 3(d) (for standby-switch). * State II: One phase fails. Then, it is detected and isolated, and thus the system proceeds to operate as a two-phase converter; this state is shown in Figure 3(b) (for parallel-switch) and Figure 3(e) (for standby-switch). * State III: Another phase fails. Again, it is detected and isolated and thus the system proceeds as a single-phase converter (the conventional dc-dc boost converter); this state is shown in Figure 3(c) (for parallel-switch) and Figure 3(f) (for standby-switch). * State IV: All the phases fail (absorbing state). Figure 3: _Cont._The overall Markov model of the considered converter is shown in Figure 4. In the shown Markov model, the failure rate of a \(k\)-phase converter is \(\lambda^{(k)}\), which is a constant value. The probability of successful operation of the fault-managing system is considered \(P_{C}\). For example, as shown in Figure 4, the transition rate from three-phase-converter state to two-phase-converter state is \(3P_{c}\lambda^{(3)}\). If the fault-managing system fails, the probability of the transition from three-phase-converter state to the absorbing state is \(3(1-P_{c})\lambda^{(3)}\). The MTTF of the single-phase and two-phase converters can be evaluated by referring to Figure 4. The Markov model of the two-phase operation can be obtained by assuming \(\lambda^{(3)}=0\). As the same, the Markov model of the single-phase converter is obtained by considering \(\lambda^{(3)}=0\), and \(\lambda^{(2)}=0\). With respect to Equations (2)-(4), the junction temperatures and temperature factors of the switches  
Figure 4: Overall Markov model of the considered converter with redundant switch configuration. Figure 3: Markov chain and reliability curves of the three-phase converter: (**a**) healthy mode of the three-phase converter with parallel-switch configuration; (**b**) two-phase converter with parallel-switch configuration; (**c**) single-phase converter with parallel-switch configuration; (**d**) healthy mode of the three-phase converter with standby-switch configuration; (**e**) two-phase converter with standby-switch configuration; (**f**) single-phase converter with standby-switch configuration. in the studied converters are listed in Table 1. By using these values, the MTTFs of the phases can be easily calculated.