## 2 System Modeling
### Thermal Modeling of the Power Electronic Converter and Device Lifetime Estimation
#### 2.4.2 Life Consumption Estimation  
For silicon carbide (SiC) MOSFETs, cycles in \(T_{j}\) are one of the primary factors leading to device failure for failure modes such as bond wire liftoff, solder fatigue, and gate oxide breakdown [32; 33]. Analysis of the impacts the temperature profile has on the lifetime of the device is typically conducted using physics-of-failure (PoF) models, data-driven models, or some combination of the two. In general, PoF models are developed for a specific failure mode, while data-driven models may consider a variety of failure modes  
Figure 4: The simplified Cauer thermal network used to estimate junction temperature for each device. This figure only represents two of the six devices that comprise the two-level voltage source converter. The dots are representative of the other four devices feeding into the same node.  
depending on the dataset. In this work, a common PoF method, the Coffin-Manson model, is used to estimate the number of cycles to failure \(N_{f}\) for a specific device operating at a given thermal cycle amplitude \(\Delta T_{j}\) and minimum thermal cycle temperature \(T_{min}\), as shown in (13) [34]. The Coffin-Manson model is often used to estimate device failure caused by bond wire liftoff and solder fatigue [33]. The scaling terms in (13) are defined for IGBT technologies rather than SiC MOSFETs as there is a lack of publicly available data that provides sufficient information to derive the scaling terms in (13) for SiC technologies. Unfortunately, this does lead to general inaccuracy in the estimation of \(N_{f}\). The primary issue with utilizing scaling terms from other device technologies is, as the Coffin-Manson model looks at failures within and between the materials that make up the device, the scaling terms essentially represent the material properties of the specific device under test. Considering the differences between IGBTs and SiC MOSFETs, the material properties of SiC technologies are likely not encapsulated in the definition of the scaling terms in (13). Although the absolute values calculated for \(N_{f}\) are not determined for the specific device technology assumed in this work, these estimates provide a meaningful representation of how thermal stresses change under the proposed control method.  
\[N_{f}=1.017^{(20-T_{min})^{1.16}}\cdot 1.26\cdot 10^{13}\cdot\Delta T_{j}^{-4.51} \tag{13}\]  
Life consumption (\(LC\)) is a metric that assesses the cumulative damage a device has accumulated over a mission profile. The life consumption \(LC\) of the device during the mission profile can be estimated using the Palmgren-Miner rule [35], which is shown in (14). The Palmgren-Miner rule calculates the cumulative damage on a device considering the number of cycles \(n_{c,i}\) experienced at a specific stress level defined by a thermal cycle amplitude \(\Delta T_{j,i}\), minimum cycle temperature \(T_{min,i}\), and a corresponding estimated number of cycles to failure \(N_{f,i}\) calculated for the corresponding stress metrics. As the estimate for \(LC\) approaches 1, the device is expected to fail. The lifetime of the converter can be approximated as the reciprocal of \(LC\), assuming the converter repetitively experiences the same profile throughout its deployed life.  
\[LC=\sum_{i}\frac{n_{c,i}}{N_{f,i}} \tag{14}\]  
Pertinent thermal cycle data must be extracted from the mission profile to be used in (14). This can be achieved using various cycle counting methods, with rainflow counting being one of the most popular. MATLAB has a built-in rainflow counting function that returns values for \(n_{c,i}\), \(\Delta T_{j,i}\), and mean cycle temperature (which can be used to calculate the minimum cycle temperature) that will be used to calculate the cumulative \(LC\) over the entire mission profile [36]. More information on the rainflow counting algorithm can be found in [36].