## 1 Introduction  
Power semiconductor devices, particularly power transistors, are commonly used in switching circuits [1; 2; 3]. In these systems, the power \(p\) dissipated in transistors changes strongly as a function of time, and the junction temperature \(T_{j}\) can significantly exceed the value of the ambient temperature \(T_{a}\) as a result of the self-heating phenomenon [4; 5; 6]. In order to correctly compute the waveforms of the power \(p(t)\) and junction temperature \(T_{j}(t)\), it is necessary to perform electrothermal analyses using electrothermal models [7; 8; 9]. The models mentioned above take into account the influence of mutual interactions between the temperature \(T_{j}\) and the voltages and currents of the transistor [10; 11; 12]. In electrothermal models, compact thermal models describing the dependence of the junction temperature on the dissipated power \(p\) are often used. Transient thermal impedance \(Z_{th}(t)\) is the parameter characterizing the heating process of a transistor in a specific cooling system [13; 14; 15]. In this study transient thermal impedance, junction-ambient, is considered. The course of \(Z_{th}(t)\) is typically described using the sum of the exponential functions [7; 16; 17]\[Z_{th}(t)=R_{th}\cdot\left[1-\sum_{i=1}^{N}a_{i}\cdot\exp\left(-\frac{t}{\tau_{thi}} \right)\right] \tag{1}\]  
where \(R_{th}\) is thermal resistance, \(a_{i}\) is the weight coefficients related to thermal time constants \(\tau_{thi}\) and \(N\) is the number of thermal time constants. In Formula (1), there are thermal time constants \(\tau_{thi}\) characterizing individual components of the heat flow path, and their values differ from each other by several orders of magnitude [17; 18]. In the electrothermal analysis of switching circuits, there is a problem of how to solve the system of differential-algebraic equations, in which there are significant differences between the values of the time constants characterizing the electrical and thermal properties of the analyzed circuit. The differences between these time constants can reach nine orders of magnitude [4; 19], which causes an unacceptable increase in the duration time of the calculations necessary to obtain the steady state. Therefore, many methods have been developed that make it possible to shorten this time by applying various simplifications. Many of these methods are characterized in the paper [4]. For example, the use of the so-called electrothermal averaged models [20] allows the performance of fast analyses with the omission of time changes in the power dissipated in semiconductor devices. In the analyses carried out with these models, it is not possible to accurately determine the peak-to-peak value of ripples of the junction temperature \(T_{j}\); only its average value can be determined. When using the accelerated transients analysis [4], the influence of the shortest and longest thermal time constants on the determined waveforms \(T_{j}(t)\) is ignored. Therefore, the results of such calculations are not precise. In the paper [19] an iterative method of electrothermal transient analysis of DC-DC converters is proposed, in which the calculations are performed iteratively with the use of SPICE and a program extrapolating the values of voltages, currents and junction temperatures with the use of memoryless convolution algorithms [21]. This method does not require simplifications, but its practical application requires the preparation of a special program. Modern constructions of power transistors are characterized by a method of mounting a semiconductor die to the metal base, which causes the occurrence of thermal time constants in the transient thermal impedance of the transistor of a much lower value, even up to a dozen microseconds [22]. This may cause the occurrence of thermal cycles, which adversely affects the reliability of the considered class of transistors [23; 24].