## 3 Proposed Power Losses Model
### Proposed Conduction Losses Model
#### 3.1.4 The Proposed Conduction Losses Model  
Through the simplification of Equation (21), Equation (19) can be expressed as:  
\[D(\alpha)=\frac{1}{2}\bigg{(}1+n\bigg{(}\cos(\alpha)-\frac{3\sqrt{3}}{8\pi} \cos(3\alpha)\bigg{)}\bigg{)} \tag{23}\]  
Substitute Equation (23) into Equation (18):  
\[P_{igbt\_cur\_cond}(\theta)=\frac{1}{2\pi}\int_{\frac{\pi}{2}}^{\frac{\pi}{2}} \big{(}V_{c\alpha\alpha}+R_{ce}I_{p}\cos\theta\big{)}I_{p}\cos\theta\times \frac{1}{2}\bigg{(}1+m\bigg{(}\cos(\theta+\varphi)-\frac{3\sqrt{3}}{8\pi}\cos 3 (\theta+\varphi)\bigg{)}\bigg{)}d\theta \tag{24}\]  
Figure 3: The waveform of \(d(\alpha)\), \(d_{3}(\alpha)\) and \(d_{5}(\alpha)\). (**a**) Comparison of the waveform for \(d(\alpha)\) and \(d_{3}(\alpha)\). (**b**) Comparison of the waveform for \(d(\alpha)\) and \(d_{5}(\alpha)\). (**c**) Error comparison. By solving the integral in Equation (24), the proposed conduction losses model of IGBT based on current cycle under SVPWM can be calculated. \[P_{igbt\_cur\_cond}=\frac{R_{ce}{I_{p}}^{2}}{8}(1+\frac{8\times m\cos\varphi}{3 \pi}-\frac{8\sqrt{3}\times m\cos(3\varphi)}{40\pi^{2}})+\frac{V_{\alpha\alpha} I_{p}}{2\pi}(1+\frac{2\pi\times m\cos\varphi}{8}) \tag{25}\]  
Similarly, the proposed conduction losses model of FWD can also be calculated, but the difference is that the duty ratio of FWD is \(1-D(\alpha)\). \[P_{f\_fund\_cur\_cond}=\frac{R_{f}{I_{p}}^{2}}{8}(1-\frac{8\times m\cos\varphi} {3\pi}+\frac{8\sqrt{3}\times m\cos(3\varphi)}{40\pi^{2}})+\frac{V_{f}I_{p}}{2 \pi}(1-\frac{2\pi\times m\cos\varphi}{8}) \tag{26}\]  
where \(R_{f}\) and \(V_{f}\) are the FWD temperature-dependent parameters. Since the parameters (\(V_{\alpha\alpha}\),\(r_{\alpha\
u}\),\(V_{f}\), \(r_{f}\)) of IGBTs and FWDs are temperature-dependent, they are measured based on the double-pulse experiment and the Infineon datasheet under the different temperatures. The linear functions of the junction temperature are obtained based on the reference values at temperature 25 \({}^{\circ}\)C and 150 \({}^{\circ}\)C shown in Equations (27) and (28), and the parameters can be calculated according to the real-time junction temperature. The fitting relationships of IGBT parameters and temperature are depicted in Figure 4.  
\[V_{\alpha\alpha}(T)=\frac{V_{\alpha\alpha}(150)-V_{\alpha\alpha}(25)}{150-25}( T-150)+V_{\alpha\alpha}(150) \tag{27}\]  
\[r_{\alpha}(T)=\frac{r_{\alpha}(150)-r_{\alpha}(25)}{150-25}(T-150)+r_{\alpha \alpha}(150) \tag{28}\]  
Therefore, the proposed conduction losses model can be expressed as:  
\[P_{igbt\_cur\_cond}=\frac{R_{\alpha\epsilon}(T){I_{p}}^{2}}{8}(1+\frac{8\times m \cos\varphi}{3\pi}-\frac{8\sqrt{3}\times m\cos(3\varphi)}{40\pi^{2}})+\frac{V_ {\alpha\alpha}(T)I_{p}}{2\pi}(1+\frac{2\pi\times m\cos\varphi}{8}) \tag{29}\]  
\[P_{f\_fund\_cur\_cond}=\frac{R_{f}(T){I_{p}}^{2}}{8}(1-\frac{8\times m\cos \varphi}{3\pi}+\frac{8\sqrt{3}\times m\cos(3\varphi)}{40\pi^{2}})+\frac{V_{f}( T)I_{p}}{2\pi}(1-\frac{2\pi\times m\cos\varphi}{8}) \tag{30}\]  
The principle diagram of the proposed conduction losses model is shown in \[P_{igbt\_cur\_cond}(\theta)=\frac{1}{2\pi}\int_{\frac{\pi}{2}}^{\frac{\pi}{2}} \big{(}V_{c\alpha\alpha}+R_{ce}I_{p}\cos\theta\big{)}I_{p}\cos\theta\times \frac{1}{2}\bigg{(}1+m\bigg{(}\cos(\theta+\varphi)-\frac{3\sqrt{3}}{8\pi}\cos 3 (\theta+\varphi)\bigg{)}\bigg{)}d\theta \tag{24}\]0 It can be seen from Equations (29) and (30) that there are still two unknown variables: the voltage modulation ratio \(m\) and the power factor angle \(\varphi\). \[P_{igbt\_cur\_cond}(\theta)=\frac{1}{2\pi}\int_{\frac{\pi}{2}}^{\frac{\pi}{2}} \big{(}V_{c\alpha\alpha}+R_{ce}I_{p}\cos\theta\big{)}I_{p}\cos\theta\times \frac{1}{2}\bigg{(}1+m\bigg{(}\cos(\theta+\varphi)-\frac{3\sqrt{3}}{8\pi}\cos 3 (\theta+\varphi)\bigg{)}\bigg{)}d\theta \tag{24}\]1 The fitting relationship of IGBT parameters and the temperature. According to the definition \(m=\frac{2V_{r}}{V_{L}}\), \(m\) is related to the reference voltage vector and dc-link voltage.