## 3 Analysis of the Switching Transients
### Turn-On Switching Transients  
_Stage 1 [(10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30; 31; 32; 33; 34; 35; 36; 37; 38; 39; 40; 41; 42; 43; 44; 45; 46;  
There are three independent state variables: the drain current \(i_{g}\), the gate-source voltage \(v_{gs}\) and the drain-source voltage \(v_{ds}\). The state equations derived from above equations are shown in Appendix A Equation (A1). As the power device is not activated, there is no switching loss \(P_{sw}\) during this period. _Stage 2 [11; 12; 13; 14; 15] current rise time:_ When the gate source voltage \(v_{gs}\) goes beyond the threshold voltage \(V_{th}\), the conductive pass (channel) enabling the current flow begins to form. Due to the fast speed of the SiC MOSFET power device, the drain current \(i_{d}\) surges, which generates the induced voltage across the parasitic inductances. This induced voltage deservers special attention, as it determines whether the drain current (id) will rise to its full value \(I_{DD}\) before or after the drain-source voltage \(v_{ds}\) drops to \(v_{gs}-V_{th}\) (the boundary condition between the saturation region and the ohmic region of MOSFET). If the power device works in the saturation region, the channel of the SiC MOSFET can be equivalent to a voltage-controlled current source (the relationship between the channel current \(i_{dh}\) and gate-source voltage \(v_{gs}\) is governed by (6)), otherwise it can be equivalent to a nonlinear resistance which is related to the junction temperature \(T_{j}\) and the drain current \(i_{d}\). Both of them are discussed as follows. \[i_{ch}=g_{f}(v_{gs}-V_{th}) \tag{6}\]  
where \(g_{f}\) is trans-conductance of the SiC MOSFET, which is also nonlinear. Figure 4: Equivalent circuits for the turn-on transients. (**a**) Stage 1. (**b**) Stage 2. (**c**) Stage 3. (**d**) Stage 4 and Stage 5. _Case I_: The MOSFET works in the saturation region. In this situation, as shown in Figure 4b, the drain current \(i_{d}\) consists of three parts: the channel current \(i_{ch}\) and the current flowing through the junction capacitor \(\text{C}_{gd}\) and \(\text{C}_{ds}\). The circuit equations can be expressed as:  
\[V_{GG}=R_{GG}i_{g}+L_{g}\frac{dig}{dt}+v_{gs}+L_{s}(\frac{di_{d}}{dt}+\frac{di _{g}}{dt}) \tag{7}\]  
\[i_{g}=(C_{gs}+C_{gd})\frac{dv_{gs}}{dt}-C_{gd}\frac{dv_{ds}}{dt} \tag{8}\]  
\[v_{ds}=V_{DD}-v_{f}-L_{p}\frac{di_{d}}{dt}-L_{s}\frac{di_{g}}{dt} \tag{9}\]  
\[i_{d}=g_{f}(v_{gs}-V_{th})+C_{ds}\frac{dv_{ds}}{dt}-C_{gd}\frac{d(v_{gs}-v_{ds} )}{dt} \tag{10}\]  
\[v_{f}=R_{f}(i_{d}-I_{DD})-V_{f} \tag{11}\]  
where \(L_{p}=L_{bus1}+L_{bus2}+L_{c}+L_{a}+L_{d}+L_{s}\), \(L_{c}=L_{c1}+L_{c2}\), \(L_{a}=L_{a1}+L_{a2}\), \(L_{d}=L_{d1}+L_{d2}\), \(L_{s}=L_{s1}+L_{s2}\). _Case II_: The MOSFET works in ohmic region directly. As discussed above, the channel of MOSFET can be represented by a nonlinear resistance in this scenario. Therefore, (10) is changed to (12), and the rest remains unchanged. \[i_{d}=\frac{v_{ds}}{R_{on}}+C_{ds}\frac{dv_{ds}}{dt}-C_{gd}\frac{d(v_{gs}-v_{ ds})}{dt} \tag{12}\]  
A new independent state variable \(i_{d}\) appears in this stage compared to the previous one, and the state equations derived from the above equations are shown in Appendix A Equation (A3). This stage ends when the drain current id reaches the bus current \(I_{DD}\). The switching loss \(E_{sw}\) expression in this period is as follows. \[E(\text{I})_{sw,g2}=\int_{t_{1}}^{t_{2}}i_{d}v_{ds}dt \tag{13}\]  
However, a more appropriate switching loss analysis is to replace (13) with (14), taking into account the assumption that the junction capacitances of MOSFET are only lossless energy buffers. The discrepancy between the two types of calculation methods will be further elucidated in Section 5 with the experimental verification.