**2. The Model Form**

A block diagram of the electrothermal model (ETM) of SiC BJT proposed by the authors is demonstrated in Figure 1.

**Figure 1.** A block diagram of the electrothermal model of SiC bipolar junction transistor (BJT).

As presented, the ETM contains: the electrical model, the heat power model and the thermal model. The function of the electrical model is to determine the static characteristics of the transistor for a known temperature (Tj) of its interior. The heat power model is designed to determine the value of electrical power generated in the transistor based on the currents and clamping voltages values on the transistor leads. The thermal model calculates the actual value of transistor junction (inner) temperature based on the ambient temperature (Ta) and the thermal parameters associated with the transistor cooling method.

The form of the considered electrical model is based on the G–P model. However, in our model, a voltage-dependent collector resistance is used for modelling the quasi-saturation mode that starts to be visible in the higher temperature range in SiC BJTs' static output characteristics. The quasi-saturation region model is based on the Kull model and consists of the temperature-dependent intrinsic carrier concentration model dedicated to the devices made of silicon carbide. A network form of the proposed electrical model is demonstrated in Figure 2 [1,16,19].

**Figure 2.** Network form of the electrical model of SiC BJT.

Diodes D1–D4 model the generation and recombination phenomena in the layers of the collector/emitter-base junctions, whereas D1–D2 and D3–D4 represent the ideal and non-ideal component of these junctions, respectively. Values of the collector and the base current of the transistor are of the form [19]:

$$\mathbf{I}\_{\rm B} = \frac{\mathbf{I}\_{\rm BE1} + \mathbf{I}\_{\rm BC1}}{\beta} + \mathbf{I}\_{\rm BE2} + \mathbf{I}\_{\rm BC2} \tag{1}$$

$$\mathbf{I}\_{\rm C} = \frac{\mathbf{I}\_{\rm BE1}}{\mathbf{K}\_{\rm qb}} - \frac{\mathbf{I}\_{\rm BC1}}{\mathbf{K}\_{\rm qb}} - \frac{\mathbf{I}\_{\rm BC1}}{\beta} - \mathbf{I}\_{\rm BC2} \tag{2}$$

where the currents IBE1, IBE2, IBC1, IBC2 are defined as follows [19]:

$$\mathbf{I}\_{\rm BE1} = \text{IS} \cdot \left[ \exp\left(\frac{\mathbf{V}\_{\rm BE}}{\rm NF \cdot V\_{\rm t}}\right) - 1\right] \tag{3}$$

$$\mathbf{I}\_{\rm BE2} = \text{ISE} \cdot \left[ \exp\left(\frac{\mathbf{V}\_{\rm BE}}{\rm NE \cdot V\_{\rm t}}\right) - 1\right] \tag{4}$$

$$\mathbf{I}\_{\rm BC,1} = \text{IS} \cdot \left[ \exp\left(\frac{\mathbf{V}\_{\rm BC}}{\rm NR \cdot V\_{\rm t}}\right) - 1\right] \tag{5}$$

$$\mathbf{I}\_{\rm BC,2} = \text{ISC} \cdot \left[ \exp\left(\frac{\mathbf{V}\_{\rm BC}}{\rm NC \cdot V\_t}\right) - 1\right] \tag{6}$$

where β is the current gain factor, VBE is the intrinsic (B'–E') base-emitter voltage, VBC is the intrinsic (B'–C') base-collector voltage, IS, ISE and ISC are the saturation currents, NF and NR are the current emission coefficients, NE and NC are the leakage emission coefficients and coefficient Kqb is of the form [19]:

$$\mathbf{K}\_{\rm qb} = 0.5 \cdot (1 - \frac{\mathbf{V}\_{\rm BC}}{\mathbf{V}\_{\rm AF}} - \frac{\mathbf{V}\_{\rm BC}}{\mathbf{V}\_{\rm AF}})^{-1} \cdot \left[ 1 + \left( 1 + 4 \cdot \left( \frac{\mathbf{I}\_{\rm BE1}}{\rm IKF} + \frac{\mathbf{I}\_{\rm BC1}}{\rm IKR} \right) \right)^{\rm NK} \right] \tag{7}$$

where VAF, and VAR are early voltages, IKF and IKR are the knee current parameters, and NK is the high-current roll-off coefficient.

Efficiency of the controlled-current source GN from Figure 2 is of the form [19]:

$$\mathbf{I}\_{\rm N} = \frac{\mathbf{I}\_{\rm BE} - \mathbf{I}\_{\rm BC}}{\mathbf{K}\_{\rm qb}} \tag{8}$$

The coefficient β existing in Equations (1) and (2) are defined as [20]:

$$\beta = \frac{\wp\_0 \cdot \left(1 + \mathbf{a} \cdot \left(\mathbf{T}\_{\mathbf{j}} - \mathbf{T}\_0\right)\right)}{1 + \mathbf{b} \cdot \left(1 + \mathbf{c} \cdot \left(\mathbf{T}\_{\mathbf{j}} - \mathbf{T}\_0\right)\right) \cdot \sqrt{\mathbf{l}\_{\mathbf{C}}^2 + \mathbf{d}\_1}} \cdot \left[1 - \sum\_{\mathbf{i}} \gamma\_{\mathbf{i}} \cdot \exp\left(\mathbf{a}\_{\mathbf{i}} \cdot \sqrt{\mathbf{l}\_{\mathbf{C}}^2 + \mathbf{d}\_1}\right)\right] \tag{9}$$

where T0 is the nominal temperature, β0, αi, γi, a, b, c, and d1 are the model parameters.

Resistors RE and RB represent series resistances of the emitter and base areas according to Equation [19]:

$$\mathbf{R\_{E}(T)} = \mathbf{R\_{E}} \cdot \left(\mathbf{1} + \text{TRE1} \cdot \left(\mathbf{T\_{j}} - \mathbf{T\_{0}}\right) + \text{TRE2} \cdot \left(\mathbf{T\_{j}} - \mathbf{T\_{0}}\right)^{2}\right) \tag{10}$$

$$R\_{\rm B}(T) = \,^1R\_{\rm B} \cdot \left(1 + \text{TRB} \mathbf{1} \cdot \left(\mathbf{T}\_{\rm j} - \mathbf{T}\_0\right) + \text{TRB} \mathbf{2} \cdot \left(\mathbf{T}\_{\rm j} - \mathbf{T}\_0\right)^2\right) \tag{11}$$

where TRE1, TRB1, TRE1 and TRE2 represent linear and quadratic temperature coefficients of the emitter and the base resistances, respectively.

The voltage-dependent collector resistance (RC) dedicated for the quasi-saturation region modeling is described as [1,16]:

$$\mathbf{R\_{C}} = \mathbf{R\_{C0}} \cdot \left(\frac{1}{2} + \frac{1}{4} \cdot \sqrt{1 + \frac{4 \cdot \mathbf{n\_{i}^{2}}}{\mathbf{N\_{epi}}^{2}}} \cdot \exp\left(\frac{\mathbf{q\_{i}^{\mathrm{V}} \mathbf{V\_{BC}}}{\mathbf{k \cdot T\_{\parallel}}}\right)} + \frac{1}{4} \cdot \sqrt{1 + \frac{4 \cdot \mathbf{n\_{i}^{2}}}{\mathbf{N\_{epi}}^{2}}} \cdot \exp\left(\frac{\mathbf{q\_{i}^{\mathrm{V}} \mathbf{V\_{BC}}}}{\mathbf{k \cdot T\_{\parallel}}}\right)\right)^{-1} \tag{12}$$

where Nepi is an epi-layer collector doping concentration, VB'C' and VB'C are voltages in the selected end of the collector resistance, q is the elementary charge, and ni is the intrinsic carrier concentration defined as [1,16]:

$$\mathbf{m}\_{\mathbf{i}} = \mathbf{A} \cdot \mathbf{T}\_{\mathbf{j}}^{\frac{2}{2}} \cdot \exp\left(-\frac{\mathbf{E}\_{\mathbf{g}0}}{2 \cdot \mathbf{k} \cdot \mathbf{T}\_{\mathbf{j}}}\right) \tag{13}$$

where Eg0 is SiC bandgap at zero temperature, k is Boltzmann's constant, A is the material constant independent of temperature.

The zero-bias collector resistance RC0 existing in Equation (12) is of the form [1,16]:

$$\mathbf{R}\_{\rm C0} = 2.143 \cdot 10^{-21} \cdot \exp\left(0.1302 \cdot \mathbf{T}\_{\rm j}\right) \tag{14}$$

A detailed specification of the mathematical description of the considered model is achievable in the SPICE user manual [19] and other papers [1,16,20].

In the thermal model, the heat interactions between the device interior and the ambient temperature were taken into account. As a rule, the compact thermal model of a semiconductor device is represented by the form of the RC network. Based on this form from other works [21,22], the non-linear compact thermal model of the SiC BJT was demonstrated. Thermal capacitances hardly depend on the temperature in contrast to the thermal resistance that changes with the temperature. In the presented model controlled voltage sources were used only instead of R elements. The network representation of the considered model is shown in Figure 3.

**Figure 3.** Network form of the non-linear thermal model of the SiC BJT.

The controlled-current source GP expresses the power dissipated in the device, controlled voltage sources ER1, ER2, ... , ERn express temperature changes between every part of the thermal dissipation path. The analytical form of these voltage source e fficiencies takes into consideration changes in the thermal resistance value expressing the thermal dissipation between the specific construction parts. Capacitors C1–Cn describe thermal capacitances of every construction part. The voltage at nodes Tj and Ta represents the transistor junction temperature and the ambient temperature respectively.

Efficiencies of controlled-current source GP and controlled-voltage sources are of the form [21,22]:

$$\mathbf{E}\_{\rm Rn} = \left( \mathbf{R}\_{\rm th1} \cdot \exp\left(\frac{\mathbf{i}\_n}{\mathbf{p}\_0}\right) + \mathbf{R}\_{\rm th0} \right) \cdot \mathbf{i}\_n \cdot \mathbf{d}\_{\rm i} \tag{15}$$

$$\mathbf{G\_{P}} = \mathbf{p\_{th}} = \mathbf{I\_{B}} \cdot \mathbf{V\_{BE}} + \mathbf{I\_{C}} \cdot \mathbf{V\_{CE}} \tag{16}$$

where di is the quotient of Rn thermal resistance and the transistor thermal resistance Rth, Rth0, Rth1, p0 are the model parameters, and in is the current of the source ERn.
