**1. Introduction**

Nowadays, power electronic circuits often include DC–DC converters [1–5]. Designing these converters requires reliable methods of computer simulations [1,5,6]. They enable selection of optimal components and operating conditions of the designed converter. For more than 50 years, many scientists have been working on methods and models that allow fast computations of waveforms of currents and voltages as well as AC and DC characteristics of power converters [5–8]. To provide simulations of electronic circuits using the formulated models it is necessary to use software dedicated to electronic circuits simulation. SPICE (simulation program with integrated circuit emphasis) is commonly used for this purpose [5,6,8–10]. Its main advantage is easy implementation of any compact model of electronic components and devices.

Due to the switching mode of operation of DC–DC converters, their characteristics can be obtained with the sequential analysis of transient processes only. Such analysis requires, as a rule, a long time—and may cause convergence problems of computations [11]. In order to shorten the time of this analysis, the averaged models of DC–DC converters were proposed [5,8,12–15]. These models make it possible to compute characteristics of circuits at the steady state using a DC sweep (DC analysis) [5,8,16]. This is one of the types of the standard analyses realized in SPICE. Typically, a DC analysis needs a short computation time. In averaged models, the average values of voltages in nodes and currents flowing through branches of considered circuits are used instead of the instantaneous values of these quantities.

It is assumed that averaged DC–DC converters models should be formulated in accordance with two steps of their activation in "on" and "off" phases of a switch. In the first step, the transistor is turned on, and simultaneously the diode is turned <sup>o</sup>ff. In the second step, the transistor is turned off and simultaneously the diode is turned on. The considered states of operation are described using two subcircuits which depend on each other. For both the mentioned steps of operation, equations describing currents flowing through capacitors and voltages on inductors are formulated. Next, these equations are averaged across period T and compared to zero. Finally, equations describing dependences between average values of currents and voltage at the steady state in the considered DC–DC converter are obtained [5,8].

Averaged models of DC–DC converters have been described in literature for many years [5,8,12–21]. They can be presented in di fferent network forms and they can be used for: transient, AC and DC analysis [5,8]. They also enable obtaining small-signal frequency characteristics, steady-state voltage–current characteristics and waveforms of currents and voltages in considered converters.

As it can be seen, e.g., in [5,8,16], in all single-inductor DC–DC converters a diode–transistor switch can be distinguished. In such a switch a diode and one kind of transistors (BJT, JFET, MOSFET or IGBT) are included. The structure of such a switch, including insulated gate bipolar transistor (IGBT) and a diode, is shown in Figure 1.

**Figure 1.** Diagram of a diode–transistor switch with IGBT and the diode.

The considered switch contains 5 terminals: 3 terminals of IGBT and 2 terminals of the diode. During operation of such a switch in any DC–DC converter, a control signal in the form of a rectangular pulses train is connected to terminal number 5. The transistor and the diode are connected to other components of the considered DC–DC converter as shown in its diagram. Voltage between pairs of terminals 1, 2 (v1(t) voltage) and 3, 4 (v2(t) voltage) have the shape of rectangular pulses trains. At the steady state waveforms of voltages and currents are periodical.

In many papers, e.g., [5,8,16,17,20,21] averaged models of a diode–transistor switch of di fferent accuracy are described. In some of these models the diode and the transistor are described as ideal switches characterized by zero on-state resistance and infinite o ff-state resistance [5,8]. Another group of these models uses piecewise linear characteristics of the mentioned semiconductor devices [5,8,16,17,21]. Some of these models [21] take into account operation only in CCM (Continuous Current Mode), whereas models described in [5,8,16] make it possible to compute characteristics of these converters operating both in CCM and DCM (Discontinuous Current Mode).

During operation of DC–DC converters self-heating phenomena occur in semiconductor devices contained in these converters [11,16,17,22–24]. The result of these phenomena is an increase in internal temperature of these devices and changes in the course of characteristics of these converters. As it is indicated, among others in [11,17], thermal phenomena occurring in components of low-voltage converters significantly influence characteristics of these converters. Thermal phenomena also strongly affect reliability of such converters, as presented in [11,16,17]. A computer analysis can be useful in estimating thermal phenomena. In order to apply such a method, an electro-thermal model of electronic components comprising a DC–DC converter is necessary.

In literature, averaged models of a diode–transistor switch containing ideal switches [5,8,18,20] or MOSFETs [16,17,25] are described. The model of a diode–transistor switch containing IGBT and a diode presented in the study [21] has a significantly simplified form. In the considered model, characteristics of the diode and IGBT are modeled using piecewise linear functions consisting of only two pieces. Such a manner of modeling DC characteristics of IGBT and a diode can cause a significant error of computations, especially in the range of low currents. Moreover, this model was verified only in the CCM. As it is pointed out in many papers [5,8,16], omitting DCM can also constitute a cause of a significant error in computations. Finally, in the model presented in [21] thermal phenomena are not taken into account. The electrothermal averaged model of a diode–transistor switch including the power MOS transistor is described in [16,17]. Such models enable computations of voltages and currents in the considered converter and internal temperatures of semiconductor devices contained in the diode–transistor switch. Due to di fferent shapes of output characteristics of IGBT and MOSFET operating in the on–state, the cited models are not adequate to analyze characteristics of DC–DC converters including IGBT and a diode.

In this study, the electrothermal averaged model of a diode–transistor switch including the IGBT is proposed. It is dedicated to electrothermal analysis of DC–DC converters containing such a switch and it makes it possible to compute values of current, voltage and internal temperatures of semiconductor devices contained in the mentioned switch. In electrothermal analyses, both electrical phenomena occurring in the analyzed DC–DC converter as well as self-heating phenomena occurring in both semiconductor devices are taken into account. In Section 2, the form of the elaborated model is presented. In Section 3, a manner of estimating model parameters is described and some results of modeling DC characteristics of components of the diode–transistor switch are shown. In Section 4, the results of usefulness of the elaborated model for determining characteristics of the boost converter are presented and discussed.
