## 4 Investigations Results
### Verification of the Method  
In order to prove the correctness of the proposed computation method (new method), the results obtained with this method were compared with the results obtained with the classical method of transient analysis and the method using one non-physical thermal time  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **System A** & **System B** & **System C** \\ \hline \(R_{th}\) (K/W) & 42.5 & 3.2 & 0.7 \\ \hline \(a_{1}\) & 0.04 & 0.03 & 0.2 \\ \hline \(\tau_{th1}\) (s) & 0.8 & 0.0001 & 0.0004 \\ \hline \(a_{2}\) & 0.86 & 0.07 & 0.15 \\ \hline \(\tau_{th2}\) (s) & 42 & 0.01 & 0.0045 \\ \hline \(a_{3}\) & 0.1 & 0.5 & 0.65 \\ \hline \(\tau_{th3}\) (s) & 105 & 0.7 & 0.006 \\ \hline \(a_{4}\) & & 0.05 & \\ \hline \(\tau_{th4}\) (s) & & 8 & \\ \hline \(a_{5}\) & & 0.35 & \\ \hline \(\tau_{th5}\) (s) & & 540 & \\ \hline \hline \end{tabular}
\end{table}
Table 1: Values of parameters of the considered cooling systems.  
Figure 4: Transient thermal impedance waveforms normalized to thermal resistance for the considered cooling systems.  
constant (one tau method) [4]. In the one tau method, only one thermal time constant equal to 5 ms is used. The verification of the proposed method is performed for the switch operating with \(R_{0}=4.5~{}\Omega\) cooling system C and three values of frequency equal to 1 kHz, 10 kHz and 100 kHz, respectively. This cooling system is selected for such investigations because it is characterized by the shortest time needed to obtain the steady state. Due to such thermal properties of the tested transistor, the biggest changes in the junction temperature are expected. For other cooling systems, such changes should be smaller. The obtained results of the computations are given in Figure 5. In this figure the results obtained using the new method are marked with red, using the classical method of transient analysis, with blue, and using the one tau method, with black.  
As can be observed, the results obtained with the use the new method and the classical transient analysis are very close to each other. The differences between these results do not exceed 1 \({}^{\circ}\)C and the peak-to-peak values obtained with these methods are practically the same. In contrast, the results obtained with the one tau method are characterized by the same average value as the other considered methods, but the peak-to-peak values of the obtained waveforms are more than two times lower than for the other methods. It is also worth observing that an increase in the switching frequency from 1 to 100 kHz causes a visible decrease (over 20 times) in the peak-to-peak value of the considered temperature and an increase in the average value of this temperature. The presented results prove the usefulness of the proposed method. The calculation time for the new method is five times shorter than for the classical method at a frequency equal to 1 kHz. The computation time of the new method does not depend on the frequency and values of thermal time constants. In contrast, this time, while using the classical method, increases nearly linearly with an increase in the values of the longest thermal time constant and switching frequency.  
Figure 5: Computed waveforms of the junction temperature of the transistor operating as a switch with the cooling system C at frequency/ equal to 1 kHz (a), 10 kHz (b) and 100 kHz (c).  
In order to illustrate the usefulness of the proposed method for different values of load resistance, some computations were repeated for a load resistance equal to 10 \(\Omega\). The results of such computations obtained at a frequency equal to 1 kHz and 100 kHz are shown in Figure 6.  
As is visible, a good agreement between the computation results obtained using the new method and the classical transient analysis was also obtained for a higher value of load resistance. An increase in frequency causes an increase in the average value of the junction temperature and a decrease in the peak-to-peak value of this temperature.