## 3 Proposed Power Losses Model
### Proposed Switching Losses Model  
Figure 6 shows the response of the collector current and the collector-emitter voltage drop in one switching period. The red and black lines represent the real curves and it is difficult to calculate the power losses based on the real curves since these two curves are changing at the instant of turn-on and turn-off, especially for the determination of turning points which needs to be measured in every operating point. Thus, a piecewise linearization switching loss model is proposed in this paper and the piecewise functions are expressed as:  
\[I_{C}=\left\{ \begin{array}{ccc}0&t_{0}\leq t\leq t_{1}\\ k_{1}(t-t_{1})&t_{1}<t\leq t_{2}\\ I_{L}&t_{2}<t\leq t_{9}\\ k_{3}(t-t_{9})&t_{9}<t\leq t_{10}\\ 0&t_{10}<t\leq t_{11}\end{array} \right.V_{CE}=\left\{ \begin{array}{ccc}V_{L} &t_{0}\leq t\leq t_{2}\\ k_{2}(t-t_{2})+V_{L}&t_{2}<t\leq t_{4}\\ V_{T}&t_{4}<t\leq t_{7}\\ k_{4}(t-t_{7})+V_{T}&t_{7}<t\leq t_{9}\\ V_{L}&t_{9}<t\leq t_{11}\end{array} \right. \tag{34}\]  
where \(I_{L}\) represents the load current flow through the IGBT; \(V_{T}\) and \(V_{L}\) are the conduction voltage drop and dc-link voltage, respectively. \(k_{1}\) and \(k_{3}\) represent the slope of the collector current rise from zero to the conduction current, and the slope of the collector current drop from the conduction current to zero, respectively. \(k_{2}\) and \(k_{4}\) are the slope of the collector current drop from the dc-link voltage to the conduction voltage, and the slope of the collector current rise from the conduction voltage to the dc-link voltage, respectively. As shown in Figure 6, the time interval between \(t_{1}\) and \(t_{2}\) is defined as the current rise time \(t_{r\_on}\), which can be obtained from the datasheet; the time interval between \(t_{2}\) and \(t_{4}\) is defined as the voltage fall time, which needs to be measured. Once the drive circuit is determined, the voltage fall time is  
Figure 6: Waveform of the collector current and the collector–emitter voltage drop of IGBTs. a fixed value. According to the proposed piecewise linearization switching loss model, the turn-on energy is calculated as:  
\[E_{\text{ON}}\approx\int_{t_{1}}^{t_{2}}V_{L}(t-t_{1})k_{1}dt+\int_{t_{2}}^{t_{ 4}}I_{L}((t-t_{2})k_{2}+V_{L})dt \tag{35}\]  
Equation (35) can be simplified as:  
\[E_{\text{ON}}\approx\frac{1}{2}V_{L}k_{1}(t_{2}-t_{1})^{2}+\frac{1}{2}I_{L}k_{ 2}(t_{4}-t_{2})^{2}+V_{L}I_{L}(t_{4}-t_{2}) \tag{36}\]  
Considering the relationship \(V_{T}<<V_{L}\), Equation (36) can be further simplified as:  
\[E_{\text{ON}}\approx\frac{1}{2}V_{L}I_{L}(t_{4}-t_{1})=\frac{1}{2}(1+k_{on})V_{ L}I_{L}t_{r\_on} \tag{37}\]  
where \(k_{on}\) is defined as the turn-on coefficient and can be expressed as: \(t_{4}-t_{2}=k_{on}t_{r\_on}\). Similarly, the turn-off energy is calculated as:  
\[E_{\text{OFF}}\approx\int_{t_{7}}^{t_{9}}I_{L}(t-t_{7})k_{4}dt+\int_{t_{9}}^{t _{10}}V_{L}(t-t_{9})k_{3}dt \tag{38}\]  
Equation (38) can be simplified as:  
\[E_{\text{OFF}}\approx\frac{1}{2}V_{L}I_{L}(t_{10}-t_{7})=\frac{1}{2}\big{(}1+k _{off}\big{)}V_{L}I_{L}t_{f\_off} \tag{39}\]  
where \(k_{off}\) is defined as the turn-off coefficient and can be expressed as: \(t_{9}-t_{7}=k_{off}t_{r\_off}\); \(t_{f\_off}\) represents current fall time (\(t_{f\_off}=t_{10}-t_{9}\)). The switching losses in one switching period can be expressed as:  
\[E_{\text{SW}}=E_{\text{ON}}+E_{\text{OFF}}\approx\frac{1}{2}V_{L}I_{L}\big{(}( 1+k_{on})t_{r\_on}+\big{(}1+k_{off}\big{)}t_{r\_off}\big{)} \tag{40}\]  
Defining \(\tau_{on}=(1+k_{on})t_{r\_on}\) and \(\tau_{off}=\big{(}1+k_{off}\big{)}t_{r\_off}\), they are called the IGBT turn-on and turn-off time constants. Since the time constants are related to IGBT parameters and these parameters are affected by the IGBT junction temperature, the time constants are dependent on the junction temperature. Thus, the IGBT turn-on and turn-off switching energy dissipation can be represented as:  
\[