## 4 Electro-Thermal Modeling
### Thermal Modeling and Implementation  
MOSFETs are susceptible to the working temperature; in addition, thermal cycling is the major reason for various PE component failures and reduction in lifetime. MOSFET's temperature impacts the design of a power electronic converter; also, the cooling system has an influence on the size of a converter. The maximum operational temperature of a MOSFET indicates the cooling capacity of the device required and what kind of cooling is required, for example, liquid cooling or air cooling with a proper heatsink [35].  
In the operation of the semiconductor devices, for instance, in the course of turn-on and -off, conduction and the switching losses depend on the device's temperature (and vice versa). Therefore, careful design is essential for confirming accurate functionality with all the temperature-dependent conditions over the entire expected temperature range.  
The power losses have been injected into the fourth-order, Cauer-type, thermal network, as represented in Figure 15, and the thermal parameters have been extracted from the producer's MOSFET datasheet. Equivalent thermal impedance \(Z_{th}\) of MOSFET and body diode are calculated using Equation (55) and the time constant \(\tau\) (56).  
\[Z_{\text{th}}=R_{\text{th}}\Big{(}1-e^{-\frac{t}{\tau}}\Big{)} \tag{55}\]  
\[\tau=R_{\text{th}}C_{\text{th}} \tag{56}\]  
Figure 14: Lookup table to determine diode \(E_{rr}\).  
Figure 13: Lookup table to determine energy loss of MOSFET.  
Now, MOSFET and the diode junction temperature could be calculated by using our designed state-space thermal model. Such a model has been based on the one element section of the Cauer network, which would total a third-order equivalent thermal resistance \(R_{\text{th}}\) and equivalent thermal capacitance \(C_{\text{th}}\) network for MOSFET or the diode, about which the details have been given in the datasheet. Such a compact thermal model consists of two inputs: the total power loss (\(P_{total_{loss}}\) (W)) and case temperature (\(T_{c}\) (\({}^{\circ}\)C)) of an inverter, and its two outputs are heat flow (\(P_{c}\) (W)) and junction temperature (\(T_{j}\) (\({}^{\circ}\)C)) (Figure 16). The basic heatsink has been modeled in the MATLAB/Simulink according to thermal conductivity from a case to the sink and from a sink to the ambient (convective for air cooling). It is at that time incorporated with the thermal model that determines heat flow and an ambient temperature, the case, and sink temperatures, as presented in Figure 16. The state-space thermal model has been defined by (57) and (58).  
\[\frac{dx}{dt}=\left[-\frac{1}{R_{\text{th}}\,C_{\text{th}}}\right]x+\left[ \frac{1}{R_{\text{th}}\,C_{\text{th}}}\frac{1}{C_{\text{th}}}\right]\begin{bmatrix} T_{c}\\ P_{total_{loss}}\end{bmatrix} \tag{57}\]  
\[\begin{bmatrix}T_{j}\\ P_{c}\end{bmatrix}=\begin{bmatrix}1\\ \frac{1}{R_{\text{th}}}\end{bmatrix}x+\begin{bmatrix}0\,0\\ -\frac{1}{R_{\text{th}}}\,0\end{bmatrix}\begin{bmatrix}T_{c}\\ P_{total_{loss}}\end{bmatrix} \tag{58}\]  
The procedure mentioned above is a simple method for estimating the power module's junction temperatures; in addition to this, the transfer function of the thermal model is easily executed in the simulation. The state-space thermal model is implemented after discretization in the MATLAB simulation (with the integration of the non-linear loss model). In the lookup table block of Figure 16, the switching and conduction characteristics of MOSFET are modeled using \(I_{ds}\), \(V_{ds}\), and \(T_{j}\). Average total power loss is calculated by accumulating average switching and conduction power loss. The thermal state-space model is further applied to estimate the junction temperature and total heat flow. The heat  
Figure 16: Power losses and thermal model block diagram.  
Figure 15: Third-order Cauer-type thermal network.  
flow is used as the input of the heatsink model to estimate the sink temperature and case temperature with respect to the ambient temperature.