**2. Proposed Model**

The formulated electrothermal model of a diode–transistor switch is based on the concept described in the study [16,17] for such a switch including the power MOS transistor. The network representation of the proposed electrothermal model of a diode–transistor switch (DTS) with IGBT and a diode is shown in Figure 2.

**Figure 2.** Network representation of the electrothermal averaged model of a diode–transistor switch (DTS) including IGBT and a diode.

Terminals of the presented model denoted as 1, 2, 3 and 4 correspond to terminals of the real diode–transistor switch in the diagram shown in Figure 1. This model can be connected to other parts of the analyzed DC–DC converters through the mentioned terminals in accordance with the diagram of these converters. The control signal is not shown in the model, but frequency f and duty cycle d of this signal are parameters of the presented model that are used in descriptions of some controlled voltage and current sources existing in this model. The voltages on terminals TjT and TjD correspond to internal temperatures of IGBT and the diode, respectively.

The presented model consists of 4 blocks. These are: main circuit, aided block, thermal model and CCM/DCM. Blocks and all components contained in them are described below.

The main circuit includes two controlled voltage sources *ER* and *ET* modeling average voltage between output terminals of IGBT and controlled current source GD modeling the average value of diode current. Voltage source V1 of zero value is used to monitor the average value of diode current. Values of currents *I*1*av* and *I*2*av*, as well as voltages *V*1*av* and *V*2*av* in the main circuit, are average values of diode and transistor voltages and currents. The connected in series controlled voltage sources *ER* and *ET* model a voltage drop between transistor output terminals. Controlled current source *GD* models diode current. The output values of these sources are described with elaborated formulas using the concept presented in the papers [16,17]. According to this concept, in each period of the control signal, current flows through the transistor for a time equal to d·T, and through the diode—for a time equal to (1-d)·T, wherein T is period of the control signal. Diode current–voltage characteristic and transistor output characteristic are described by piecewise linear functions. Parameters describing such piecewise linear functions depend on internal temperatures of the diode and IGBT. Values of these parameters are computed in the aided block, whereas internal temperatures of semiconductor devices are computed in the thermal model. The equivalent duty cycle *Veu*, adequate for operation of the DC–DC converter containing the modeled diode–transistor switch, is computed in CCM/DCM block shown in Figure 2. The value of *Veu* depends on duty cycle and frequency of the control signal, inductance of the inductor contained in the analyzed DC–DC converter and output current of this converter.

Controlled voltage and current sources existing in the main circuit are described as follows

$$E\_T = \frac{1 - V\_{eu}}{V\_{eu}} \cdot (V\_{2av} + V\_D) \tag{1}$$

$$E\_R = \frac{V\_{IGBT}}{V\_{cu}} + \frac{I\_{1av} \cdot R\_{IGBT}}{V\_{cu}} + \frac{I\_{1av} \cdot R\_D}{V\_{cu}^2} \cdot (1 - V\_{cu}) \tag{2}$$

$$G\_D = \frac{1 - V\_{cu}}{V\_{cu}} \cdot I\_{1av} \tag{3}$$

where current *I*1*av* and voltage *V*2*av* are denoted in Figure 2, *Veu* is voltage on controlled voltage source *Eu*, *VD* voltage and *RD* resistance are described by a piecewise linear characteristic of forward biased diode, whereas *VIGBT* voltage and resistance *RIGBT* are described by a piecewise linear output characteristic of switched-on IGBT. Parameters *VIGBT*, *VD*, *RD* and *RIGBT* depend on the device internal temperature (TjT for IGBT and TjD for the diode) and on current flowing through these devices. Values of the mentioned parameters correspond to voltages on controlled voltage sources EVIGBT, EVD, ERD and ERIGBT, respectively. These sources are included in the aided block. Descriptions of the mentioned parameters are as follows:

$$V\_D = \begin{cases} V\_{D1} \cdot \left(1 + \left(T\_{jD} - T\_0\right) \cdot \alpha\_{VD1}\right) \text{ if } I\_{2uv} < a\_1\\\ V\_{D2} \cdot \left(1 + \left(T\_{jD} - T\_0\right) \cdot \alpha\_{VD2}\right) \text{ if } a\_1 \le I\_{2uv} < a\_2\\\ V\_{D3} \cdot \left(1 + \left(T\_{jD} - T\_0\right) \cdot \alpha\_{VD3}\right) \text{ if } I\_{2uv} \ge a\_2 \end{cases} \tag{4}$$

*Energies* **2020**, *13*, 3033

$$R\_D = \begin{cases} R\_{D1} \cdot \left(1 + \left(T\_{jD} - T\_0\right) \cdot a\_{RD1}\right) \text{ if } I\_{2uv} < a\_1\\ R\_{D2} \cdot \left(1 + \left(T\_{jD} - T\_0\right) \cdot a\_{RD2}\right) \text{ if } a\_1 \le I\_{2uv} < a\_2\\ R\_{D3} \cdot \left(1 + \left(T\_{jD} - T\_0\right) \cdot a\_{RD3}\right) \text{ if } I\_{2uv} \ge a\_2 \end{cases} \tag{5}$$

$$V\_{IGBT} = \begin{cases} V\_{IGBT1} \cdot \left(1 + \left(T\_{jT} - T\_0\right) \cdot \alpha\_{VIGBT1}\right) \text{if } I\_{1av} < b\_1\\\ V\_{IGBT2} \cdot \left(1 + \left(T\_{jT} - T\_0\right) \cdot \alpha\_{VIGBT2}\right) \text{if } b\_1 \le I\_{1av} < b\_2\\\ V\_{IGBT3} \cdot \left(1 + \left(T\_{jT} - T\_0\right) \cdot \alpha\_{VIGBT3}\right) \text{if } I\_{1av} \ge b\_2 \end{cases} \tag{6}$$

$$R\_{IGBT} = \begin{cases} R\_{IGBT1} \cdot \left(1 + \left(T\_{jT} - T\_0\right) \cdot \alpha\_{RIIGHT1}\right) if \ I\_{1av} < b\_1\\ R\_{IGBT2} \cdot \left(1 + \left(T\_{jT} - T\_0\right) \cdot \alpha\_{RIIGHT2}\right) if \ b\_1 \le I\_{1av} < b\_2\\ R\_{IGBT3} \cdot \left(1 + \left(T\_{jT} - T\_0\right) \cdot \alpha\_{RIIGHT3}\right) if \ I\_{1av} \ge b\_2 \end{cases} \tag{7}$$

In Equations (5) and (6) symbols *VD*1, *VD*2, *VD*3, *RD*1, *RD*2, *RD*3, α*VD*1, α*VD*2, α*VD*3, α*RD*1, α*RD*2, α*RD*3, *a*1 and *a*2 denote parameters of a piecewise linear model of diode DC characteristic. In turn, in Equations (6) and (7) symbols *VIGBT*1, *VIGBT*2, *VIGBT*3, *RIGBT*1, *RIGBT*2, *RIGBT*3, α*VIGBT*1, α*VIGBT*2, α*VIGBT*3, α*RIGBT*1, α*RIGBT*2, α*RIGBT*3, *b*1 and *b*2 denote parameters of a piecewise linear model of IGBT output characteristics. Values of temperatures *TjD* and *TjT* are computed in the thermal model, whereas *T*0 represents reference temperature.

In the thermal model values of internal temperature of IGBT (*TjT*) and the diode (*TjD*) are computed with self-heating phenomena taken into account. The classical electrical analog of a DC compact thermal model described, e.g., in [11,16,17,22,26,27] is used. In this analog temperature corresponds to voltage in selected nodes of this analog, whereas dissipated power is represented by current sources. The ability to remove heat generated in the diode and in IGBT is characterized by thermal resistance. In the proposed model voltage source *VTa* represents ambient temperature, resistors *RthT* and *RthD* denote thermal resistance of IGBT and the diode, respectively. Average values of power dissipated in the considered semiconductor devices are represented by controlled current sources GTT and GTD. Currents flowing through these sources are described by the following formulas

$$\mathbf{G}\_{TD} = \left(V\_D + \frac{R\_D \cdot I\_{2uv}}{1 - V\_{cu}}\right) \cdot \frac{I\_{2uv}}{1 - V\_{cu}}\tag{8}$$

$$G\_{TT} = \left(V\_{IGBT} + \frac{R\_{IGBT} \cdot I\_{1av}}{V\_{cu}}\right) \cdot \frac{I\_{1av}}{V\_{cu}}\tag{9}$$

The proposed model can be used for computations of characteristics of DC–DC converters operating in CCM or DCM. In both mentioned modes of operation in each period of a control signal the transistor is turned on in time equal to the product of duty cycle d and period T. In turn, the diode is turned on in time equal to (1-d)·T in CCM and this time is shorter in DCM. In order to take into account influence of duty cycle d, frequency f of the control signal and inductance L of the inductor included in the tested DC–DC converter on voltage *Veu*, CCM/DCM block is included in the model. This block includes controlled current source *Ga*, voltage source *Va* of zero value, resistor *Ra* and controlled voltage sources *Eu* and *Ex*.

Voltage *Veu* on voltage source *Eu* is described by the formula

$$V\_{\rm cu} = \begin{cases} \; 0 \; if \; V\_x < 0\\ \; V\_x \; if \; 0 < V\_x \; : \; 1\\ \; 1 \; if \; V\_x > 1 \end{cases} \tag{10}$$

where voltage *Vx* on voltage source *Ex* is described as follows

$$V\_x = \text{LIMT}\left(\text{MAX}\left(d\_\prime \frac{d^2}{d^2 + 2 \cdot L \cdot f \cdot \frac{I\_{Va}}{V\_{2av} + V\_D}}\right), 0, 1\right) \tag{11}$$

In Equation (11) LIMIT (·) and MAX (·) are SPICE standard functions described, e.g., in the book [28] and *IVa* denotes current flowing through controlled current source *Ga* described with the formula of the form:

$$G\_{\rm d} = MAX(I\_{1av}, 0) \tag{12}$$

Resistor *Ra* must be included in this block due to formal rules of SPICE. Voltage source *Va* is used to monitor the value of current *IVa*.
