**3. Results of Measurements and Simulations**

To test the accuracy of the presented electrothermal model, static isothermal and non-isothermal characteristics of the BT1206-AC SiC BJT were measured and compared to the results of SPICE calculations. The model was implemented in SPICE in the form of the subcircuit. A set of electrical parameter values of the ETM was used in the simulations. The values of the model parameters were acquired from the measured characteristics with the use of the proper parameters estimation procedure described in [20,23]. The values of parameters occurring in Equation (15) were acquired from measurements of the transistor transient thermal impedance.

Isothermal characteristics of the selected transistor were measured by means of Keithley Source Measure Unit type 2602 in order to minimize the impact of the self-heating phenomenon on the measured characteristics of the tested device. During the measurements, a signal with a duty cycle of less than 0.0001 and a pulse duration of 50 μs was used. The non-isothermal characteristics were performed by the "point by point" measurement method for the transistor operating without any cooling system under the thermal steady-state. During the measurements, the case temperature TC of the transistor was measured by a TM-2000 precision thermometer comprising a platinum Pt-100 sensor. The temperature sensor was glued on the transistor case by a heat conductive AG Termoglue, 10 G.

The measured and simulated isothermal static output characteristics of the BT1206-AC transistor are presented in Figure 4, where points represent the results of measurements and lines represent the results of simulations using the proposed ETM.

As seen in Figure 4, good agreemen<sup>t</sup> between the results of the isothermal measurements in comparison with the simulations was obtained even in the quasi-saturation region at both values of the ambient temperature, which expresses a percentage relative error (absolute error quotient and exact value) of less than 8%. The increase of the ambient temperature causes the collector current value drop in the bipolar silicon carbide transistors, but it is commonly known from the literature that in the case of the bipolar transistors made of silicon, this dependence is reversed [24,25].

**Figure 4.** Isothermal output characteristics of the BT1206-AC transistor.

Measured and calculated non-isothermal I-V characteristics of the considered SiC BJT transistor are shown in Figure 5. The investigations were performed at four fixed values of the control current in the range of 10–150 mA. As shown in Figures 4 and 5, the differences between the results of isothermal and non-isothermal measurements of the static output characteristics iC(vCE) reach over 50% for a control current equal to iB = 100 mA and a voltage vCE of 4 V at room temperature.

**Figure 5.** Non-isothermal output characteristics of the BT1206-AC transistor.

As seen, the satisfying agreemen<sup>t</sup> between the results of simulations and measurements was expressed by a relative error value of usually less than 10%. The measurements were carried out to obtain the transistor junction temperature (Tj) close to the maximum value of this parameter given in the catalogue by the manufacturer. The junction temperature was estimated on the basis of case temperature (TC) measurements and catalog value of the thermal resistance Rthjc between the transistor junction and case.

The simulated output characteristic at the base current equal to 150 mA has a specific shape. The negative slope of the collector current with increasing collector-emitter voltage is caused by the strong self-heating phenomenon. In the literature, this type of characteristic is called an N-shape characteristic [8].

The important parameter in the description of the bipolar transistors is the common-emitter current gain factor (β), defined as the ratio of the output and control current (β = IC/IB). It is known from the literature that the value of β depends on the operating point. In catalogs, usually only one value of this parameter is given (under set operating conditions). Manufacturers deliberately choose a specific transistor operating point to present the highest value of the current gain factor β. Figure 6

shows the authors' measured and calculated values of β factor for the considered transistor in the isothermal conditions at a fixed value of the collector-emitter voltage equal to 3 V.

**Figure 6.** Isothermal dependence of the current gain factor at fixed values of an ambient temperature.

As seen, in the range of collector current values up to about 6 A, the β(iC) dependence is a monotonically increasing function. In addition, the dependence of the current gain factor β(T) at a fixed value of a collector current is a decreasing function. This is a characteristic feature of bipolar power transistors made with silicon carbide technology [8,25–27]. The results of simulations using the proposed β model has a good agreemen<sup>t</sup> with the results of the measurements of the β(iC) characteristics in all ranges of temperature, which indicates that the transistor's current gain properties deteriorate with the temperature increase.

Figure 7 presents the isothermal and non-isothermal dependence of the current gain factor β of the considered transistor on the ambient temperature at a fixed value of the collector-emitter voltage and the control current equal to 3 V and 50 mA, respectively. The points represent the results of measurements and lines represent the results of calculations using the proposed ETM.

**Figure 7.** Isothermal and non-isothermal dependence of the current gain factor.

As seen, the non-isothermal characteristic of SiC BJT lies under the isothermal counterpart. Moreover, at room temperature the differences between isothermal and non-isothermal measurements reach over 25%. This phenomenon is due to the fact that when the device lattice temperature and phonon scattering in SiC BJTs increase, they cause both the carrier mobility and the current gain to drop [28].
