## 2 Materials and Methods
### Thermal Measurements  
The thermal impedance \(Z_{th}(t)\) and thermal resistance \(R_{th}\) of the samples were measured. For this purpose, the measuring set-up, shown in Figure 4, was used [16]. An indirect electrical method was used based on gate-emitter voltage V\({}_{\text{GE}}\) as a thermo-sensitive parameter [17]. During the measurements, the FR4 substrate with soldered IGBT was fixed to a liquid-cooled heat exchanger [18] to increase the cooling efficiency of the FR4 substrate [19]. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Soldering** & **Type of Heating** & **Temperature (\({}^{\circ}\)C)** & **Time (s)** & **Atmosphere** \\ \hline \multirow{3}{*}{Batch oven} & \multirow{2}{*}{Infrared radiation} & pre-heating 150–180 & 140 & \\  & & reflowing 210–254 & 200 & Air \\  & & cooling 254–170 & 180 & \\ \hline \multirow{3}{*}{3-zones tunnel oven} & \multirow{3}{*}{Convection} & pre-heating 160–180 & 60 & \\  & & reflowing 213–247 & 40 & Air \\  & & cooling 247–150 & 70 & \\ \hline \multirow{3}{*}{4-zones tunnel oven} & \multirow{3}{*}{Convection} & pre-heating 140–175 & 50 & \\  & & reflowing 210–240 & 50 & Air \\  & & cooling 240–140 & 30 & \\ \hline \multirow{3}{*}{Vapour phase oven} & \multirow{3}{*}{Heat conduction} & pre-heating 125–187 & 50 & \\  & & reflowing 206–227 & 60 & Inert \\  & & cooling 227–150 & 130 & \\ \hline \multirow{3}{*}{Hot air soldering} & \multirow{3}{*}{Convection} & pre-heating 120–195 & 15 & \\  & & reflowing 195–221 & 10 & Air \\  & & cooling 221–175 & 50 & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Soldering ovens. Figure 3: Soldering thermal profiles. In the measuring set-up, the thermal impedance \(Z_{th}(t)\) was determined in three steps. 1. Calibration of the thermometric characteristic. The characteristic describes the dependence of a thermo-sensitive parameter on temperature. The calibration was performed at a closed switch (S1). The measurement current (I\({}_{\text{M}}\)) of 1.3 mA flowed through the transistor. During calibration, the temperature of the transistor was regulated by a thermostat, in which the device being tested was situated. The slope of the thermometric characteristic was calculated on the basis of this characteristic;
2. Heating. In the second step of measurements, the heating power was dissipated in the transistor operating within the active range, and its internal temperature increased as a result of self-heating. In this stage, switch (S1) was opened. The current I\({}_{\text{H}}\) flowed through the self-heating device. This step ended after a thermally steady-state was obtained. At this point, junction temperature of the transistor should be around 100 \({}^{\circ}\)C. At the steady-state, the value of the voltage drop (V\({}_{\text{CE}}\)) in the self-heating device was measured. The value of heating power (\(P_{H}\)) was equal to the product of the voltage drop of the self-heating device and current I\({}_{\text{H}}\). In the case of the performed measurements, it was equal to 4.9 W;
3. Cooling. This step began after the change in the state of the switch S1 at time t = 0 and meant the measurement of the waveform of the thermo-sensitive parameter until the steady-state was obtained. Then, the current flowed through the transistor, which was equal to the calibration value. V\({}_{\text{CE}}\) voltage was measured at defined time intervals with an analogue-to-digital (A/D) converter and recorded by a PC. The waveform of junction temperature (\(T_{j}\)) of the IGBT were calculated according to the use of the thermometric characteristic measured during calibrations and the waveform of voltage V\({}_{\text{GE}}\)(t) measured in the third step of measurements. Finally, the transient thermal impedance was calculated using the following equation: \[Z_{th}(t)=\frac{T_{j}(t)-T_{a}}{P_{H}}\] (1) where \(T_{j}\) is junction temperature, \(T_{a}\) is ambient temperature and \(P_{H}\) is heating power. Thermal resistance \(R_{th}\) corresponds to the values of thermal impedance \(Z_{th}(t)\) at the steady-state.