Modeling the Influence of Thermal Phenomena in Inductors and Capacitors on the Characteristics of the SEPIC Converter

Abstract

The paper presents the results of measurements and calculations of the SEPIC converter characteristics, taking into account thermal phenomena in semiconductor devices and passive elements. Compact electrothermal models of the MOSFET transistor, diode, capacitor, and inductor are proposed. Parasitic phenomena are also included in these models. The form of the developed models and the method of determining the values of their parameters are presented. The correctness of the formulated models was verified experimentally. Calculations and measurements of the characteristics of SEPIC converters containing inductors with ferromagnetic cores made of different materials were carried out. The obtained results of the investigations are discussed, and the range of applicability of the formulated models is described. It was shown that, at the considered operating conditions at an ambient temperature equal to 22 °C, the temperature of capacitors can exceed 40 °C, whereas the temperatures of inductors can even reach 50 °C.

1. Introduction

DC-DC converters are the most important components of switching power supply systems, which are widely used in various industries [1,2,3]. These converters contain semiconductor devices, such as diodes and transistors operating as switches, as well as passive elements such as inductors and capacitors used to store electrical energy and filter the output voltage [3,4,5]. Many papers [6,7,8,9] describe selected constructions of DC-DC converters. DC-DC converters are divided into three groups: transformer, single-inductor, and dual-inductor converters [10]. The classical non-isolated DC-DC converters contain one inductor. Belonging to this group are, e.g., buck and boost converters [1,3,10]. The isolated DC-DC converters contain at least one transformer. Belonging to this group are, e.g., forward converters, flyback converters, and full-bridge converters [10]. In a dual-inductor converter, two inductors are used. According to the information given in [10], theoretically, infinitely many topologies of converters belonging to this group can be elaborated.

The SEPIC converter belongs to the most popular dual-inductor DC-DC converters. It can both step up and step down voltage [3,10,11]. For this reason, it is used in many applications. For example, in [12], a modified topology of the SEPIC converter dedicated to operating in LED systems used in electric vehicles was presented. In turn, in [13,14], a multi-input SEPIC converter topology was proposed, designed to operate in micro-networks cooperating with PV installations, energy storage systems, and railway systems [15,16].

The literature [3,10,17] describes properties of DC-DC converters containing ideal elements. Such elements are characterized by no energy losses and an unlimited, widely operating frequency range. In real systems, there are imperfections that can cause a decrease in the value of energy efficiency, an increase in the temperature of its individual components, and a change in the course of the output and transient characteristics of DC-DC converters [18,19,20]. Much attention is paid in the literature to the influence of imperfections of semiconductor devices on the characteristics of DC-DC converters [19,21,22], and only a few papers take into account the imperfections of passive elements [23,24,25,26]. Meanwhile, one of the factors influencing the limitation of the safe operation range of DC-DC converters is the selection of the magnetic material used for the construction of inductor or transformer cores [23,26].

The selection of the appropriate material from which the ferromagnetic core of the inductor or transformer is made determines the range of frequencies, voltages, and currents in which the considered converter can operate, as well as the necessary dimensions of the core [27]. The use of an inappropriate material may result in a significant reduction in the energy efficiency of DC-DC converters, among others, due to the increase in power losses in the core with an increase in frequency [23,27].

In DC-DC converters, apart from electrical phenomena, thermal phenomena also occur, which are the result of converting the energy loss in the electronic components into heat [28]. The mentioned thermal phenomena, such as self-heating, cause an increase in the temperature inside the converter components above the ambient temperature [26,29]. The increase in the temperature inside the components causes an increase in the parasitic resistance of these components, which results, among others, in a decrease in the energy efficiency of the considered systems [19,26]. Additionally, an increase in the junction temperature of the converter components causes a shortening of their lifetime [30,31,32], and a high value of this increase can cause damage to these components [31]. Therefore, it is very important to know the value of the junction temperature of each component during its operation.

At the design stage of electronic systems, computer analyses of their properties are carried out. Various computer programs are used for this purpose [22,33,34,35]. These programs allow for the selection of appropriate types or values of system components, ensuring their correct operation and obtaining high energy efficiency. These programs also enable the elimination of irregularities that may occur in the system at the design stage, which allows the costs of the design to be reduced and for the launch of the system to be achieved.

Popular programs of this type include SPICE [22,33,35,36]. This program contains built-in compact models of semiconductor and passive components [37]. These models include important physical phenomena which occur in the aforementioned components, as well as parasitic phenomena. This program is accompanied by libraries containing values of the model parameters for selected types of the considered components. However, these models do not take into account thermal phenomena occurring in electronic elements. In order to take these phenomena into account, it is necessary to formulate special electrothermal models. Such models are formulated for many semiconductor devices and described, among others, in [38,39,40,41,42,43]. There are also known electrothermal models of inductors and transformers [44,45,46,47].

The electrothermal models consist of two parts. There are an electrical model describing the relation between voltages and currents and a thermal model making it possible to calculate the device junction temperature. In the literature, many papers are devoted to the descriptions of such thermal models. Some information about selected models of this kind is given below.

The paper cited in [48] describes a model of an inductor operating in a quasi-saturation state in a boost converter. This model consists of three parts: magnetic, electrical, and thermal. The paper presents the structure of a thermal model in the form of a Cauer network to determine the core temperature of this component.

The paper cited in [49] presents a nonlinear compact thermal model of a smartphone taking into account all heat dissipation mechanisms. In order to take into account the nonlinearity of these phenomena, an iterative simulation procedure was proposed. The calculation results using the proposed model were compared with the commercial program Icepack. The obtained calculation results using the Icepack program, the proposed thermal model, and the measurements were compared, and it was shown that the proposed model can not only accurately estimate the steady-state temperature, but also effectively capture the transient thermal behavior.

In the paper cited in [50], it was pointed out that designing a high-efficiency electrical energy conversion system requires taking into account thermal phenomena occurring in these systems, which significantly influence power losses inside them. It was pointed out that appropriate models are often used for this purpose, but they are largely dependent on the topology and technology, which often results in simplified representations that underestimate thermal phenomena or complex sets of differential equations. Therefore, the cited paper proposes an automatic method for characterizing semiconductor power losses through the thermal dynamics of a converter. The proposed model uses an optimization program that identifies the optimal linear discrete-time model according to a set of power profiles in relation to temperature.

In the paper cited in [51], it was noted that existing thermal models of semiconductor devices often ignore the temperature distributions of the integrated circuit, and their applicability under high-temperature operating conditions is rarely discussed. Therefore, this paper proposes a new three-dimensional physical model of the RC network for an IGBT module to describe its thermal properties. A new method for determining parameters of this model is presented to monitor temperatures at critical locations of the IGBT module while taking into account the non-uniform power density distribution within it.

Although the models described in the literature have a complex form, which enables accurate modeling of the characteristics of these elements, they are too complex to be effectively used in the analysis of the characteristics of DC-DC converters. Therefore, there is a need for electrothermal compact models of DC-DC converter components that provide both the desired accuracy of calculations and an acceptable time for the calculations [33].

Some papers [19,22,33,38,44,45,52,53,54,55] describe the results of electrothermal analyses of DC-DC converters. For example, in the paper cited in [52], the problem of modeling a buck converter using behavioral electrothermal models of the IGBT and a p-i-n diode was considered. In all the mentioned papers, the thermal phenomena occurring in capacitors and inductors are omitted. That is why it is impossible to calculate the temperatures of inductors and capacitors.

The aim of this paper is to analyze the influence of thermal phenomena occurring in passive elements and semiconductor devices on the characteristics of the SEPIC converter. Electrothermal models of capacitors, inductors, diodes, and MOSFET transistors in the form of a subcircuit for SPICE are proposed. In contrast to the classical modeling method of a DC-DC converter, the thermal phenomena occurring in capacitors and inductors are taken into account. Additionally, both the DC losses and dynamic losses in inductors are taken into account. The method for determining the values of the parameters of these models is presented, and the results of calculations and measurements of non-isothermal characteristics of the investigated converter are presented. SEPIC converters containing inductors made of two magnetic materials, powdered iron and ferrite, are considered. The measurements and analyses are performed in a wide range of values of load resistance of the converter, what makes it possible to investigate the properties of this converter operating both in Continuous Conducting Mode (CCM) and Discontinuous Conducting Mode (DCM).

Section 2 presents the forms of the developed electrothermal models of the mentioned converter components. Section 3 contains a description of the investigated converter. Section 4 describes the method of determining the values of the parameters of the developed models and the results of their accuracy verification. Section 5 presents the obtained results of the measurements and calculations and discusses their compliance.

2. Electrothermal Models of Converter Components

The SEPIC converter component models presented in this section belong to the category of compact electrothermal models [19,47]. These models allow for the time waveform of voltages and currents to be determined on the external terminals of the modeled component and their internal temperature [43]. These models are designed to be used in the electrothermal analysis of transient states in the SPICE program. Therefore, they take the form of a circuit that is a circuit analog of such a model. The forms of compact electrothermal models of the MOSFET transistor, diode, inductor, and capacitor are presented later in this section.

A. 

Inductor model

An electrothermal model of an inductor has the form shown in Figure 1.

This model takes into account the nonlinear dependence of the inductor inductance on the current, the series resistance of the winding and its parasitic capacitance, as well as the self-heating phenomenon. The described model can be divided into three blocks: the main circuit, the thermal model, and the auxiliary block. The main circuit describes the current–voltage characteristic of the inductor, taking into account the influence of temperature on the properties of the inductor. The terminals of this circuit marked with numbers 1 and 2 correspond to the external terminals of this model. Inductor L₁ with constant inductance is used to determine the value of the derivative of the current of the modeled inductor. The controlled voltage source E_LS is used to describe an influence of current i flowing through the winding on the inductor inductance. The following formula describes the voltage of this source:

$${\mathrm{E}}_{\mathrm{LS}}={\mathrm{V}}_{\mathrm{L}1}\times \left(\mathrm{MIN}\left(\mathrm{A}\times \exp \left(-\mathrm{B}\times \left(\mathrm{i}-\mathrm{C}\right)\right)+{\mathrm{L}}_{\min },{\mathrm{L}}_{\max }\right)-{\mathrm{L}}_1\right){/\mathrm{L}}_1$$

(1)

where MIN(x,y) is a standard SPICE function that takes the value of the smaller of the arguments of this function and denotes the inductor current; V_L1 is a voltage drop across inductor L₁; L_max is the maximum value of the inductor inductance; and A, B, and C are model parameters describing the dependence of the inductor inductance on its current.

The voltage source V₁ is used to monitor the value of current i. Resistor R_S represents the inductor winding resistance, while capacitor C_L represents the parasitic capacitance of the winding.

In the auxiliary circuit, the peak-to-peak voltage value on the inductor is calculated. It is necessary to determine power losses in the inductor core. Diodes D₁ and D₂ are described using a built-in model in the SPICE program. In order to limit the influence of a voltage drop on these diodes on the determined peak-to-peak voltage value of the inductor, the output voltage of the controlled voltage source E₂ is equal to the product of 10 and the sum of the voltages on the source E_LS and on inductor L₁. The capacitance of capacitor C₂ should be selected in such a way that its impedance module at the frequency of the alternating voltage on the inductor is as small as possible. In turn, the time constant C₃R₃ should be significantly greater than the period of this voltage.

In the thermal model, C_thL denotes the thermal capacity of the inductor, and R_thL indicates its thermal resistance. The controlled current source G_PL represents the power lost in the inductor. The output current of this source is described by the formula:

$${\mathrm{G}}_{\mathrm{PL}}={\mathrm{V}}_{\mathrm{RL}}\times \mathrm{i}+{\mathrm{P}}_{\mathrm{C}0}\times {\mathrm{V}}_{\mathrm{Lpp}}\times \left(1+{\mathsf{\alpha }}_{\mathrm{f}}\times \left(\mathrm{f}-{\mathrm{f}}_0\right)\right)$$

(2)

where V_RL is the voltage across resistor R_S, f is frequency of the voltage across the inductor, V_Lpp is the peak-to-peak value of the voltage across the inductor, P_C0 is the proportionality coefficient determined at the reference frequency f₀, and α_f is the frequency loss coefficient in the inductor core.

B. 

Capacitor model

The electrothermal model of a capacitor has the circuit representation shown in Figure 2.

The described model of a capacitor takes into account the parasitic resistance of the capacitor and the parasitic inductance of this element, as well as the self-heating phenomenon. The presented model consists of the main circuit describing the current–voltage characteristics of the capacitor. Terminals 1 and 2 represent the external terminals of the modeled capacitor. The presented model should be connected to the analyzed system using these terminals.

The main circuit consists of three elements connected in series. Capacitor C_C represents the capacitance of the modeled capacitor. Inductor L_C represents the parasitic inductance of the modeled element, and resistor R_C represents its resistance in the series equivalent circuit. In turn, the thermal model is used to calculate an increase in the temperature of the capacitor’s interior above the ambient temperature due to the self-heating phenomenon. The controlled current source G_PC models the power lost in the capacitor. It is equal to the product of the current and voltage on resistor R_C. R_thC is the thermal resistance of the capacitor, and C_thC is its thermal capacitance.

C. 

Model of the MOS transistor

The electrothermal model of the power MOS transistor belongs to the group of the hybrid electrothermal model [19]. Its circuit representation is shown in Figure 3.

The presented model takes into account the influence of the transistor interior temperature changes on the resistance of the switched-on channel and the self-heating phenomenon. The main circuit includes transistor M₁, described using the MOS transistor model built into the SPICE program (Level = 3) [37], and the controlled voltage source E_M. The output voltage of this source is described by the formula:

$${\mathrm{E}}_{\mathrm{M}}={\mathrm{R}}_{\mathrm{O}\mathrm{N}}\times {\mathrm{i}}_{\mathrm{D}}\times {\mathsf{\alpha }}_{\mathrm{R}\mathrm{O}\mathrm{N}}\times \left({\mathrm{T}}_{\mathrm{T}}-{\mathrm{T}}_o\right)$$

(3)

where R_ON is the resistance of the switched-on channel at the reference temperature T₀, i_D is the drain current of the transistor, R_ON is the temperature coefficient of resistance R_ON, and T_T is the interior temperature of the transistor determined in the thermal model. In the thermal model, the controlled current source G_PM models the power dissipated in the transistor. This power is equal to the product of the voltage between the drain and source terminals and current i_D. R_thM is the thermal resistance of the transistor, and C_thM is its thermal capacitance.

D. 

Diode model

The electrothermal model of the diode also belongs to the category of so-called hybrid electrothermal models [19]. Its circuit representation is shown in Figure 4.

The presented model takes into account the influence of the temperature changes inside the diode on its series resistance and forward voltage, as well as the self-heating phenomenon. The main circuit includes diode D₁, described using the model built into the SPICE program, and the controlled voltage source E_D. The output voltage of this source is described by the formula:

$${\mathrm{E}}_{\mathrm{D}}=\left({\mathsf{\alpha }}_{\mathrm{V}}+{\mathrm{R}}_{\mathrm{D}}\times {\mathrm{i}}_1\times {\mathsf{\alpha }}_{\mathrm{RD}}\right)\times \left({\mathrm{T}}_{\mathrm{D}}-{\mathrm{T}}_{\mathrm{o}}\right)$$

(4)

where R_D is the series resistance of the diode at the reference temperature T₀, i₁ is the diode current, R_D is the temperature coefficient of resistance R_D, α_V is the temperature coefficient of the forward voltage of the diode, and T_D is the temperature inside the diode determined in the thermal model.

3. Investigated Converter

The subject of the investigations is a two-inductor SEPIC converter, the diagram of which is shown in Figure 5.

In this circuit, a unipolar transistor M₁ of the IRF540 type is used, controlled from the rectangular signal source V_G via a resistor R_G = 47 Ω, a Schottky diode D₁ of the S6D02065A type, two electrolytic capacitors with capacities C₁ = C₂ = 220 µF, and two inductors containing toroidal cores made of different ferromagnetic materials, i.e., iron powder (RTP) and ferrite (RTF). The circuit is powered by a constant voltage source V_in, and a resistor R_L is its load. The time waveform of the transistor output voltage is recorded using an oscilloscope (OSC), to which a current probe (CP) is also connected, which allows the inductor current L₁ to be recorded. The inductance of each inductor at a zero value of the constant current component is about 40 µH. The value of this inductance depends on the value of the DC component of the current, the frequency, and the material used to construct the core of this element [22].

The tested SEPIC converter and the utilized measurement set-up are shown in Figure 6.

The applied magnetic materials are characterized by various properties expressed by the values of the parameters listed in

Table 1. Values of the parameters of the ferromagnetic materials used.

Type of Core	Bsat [T]	TCurie [°C]	PV [kW/m3]	µi
RTP	1.39	1023	103	75
RTF	0.47 (25 °C) 0.38 (100 °C)	220	80 @(100 °C, 25 kHz) 450 @(100 °C, 100 kHz)	2300 @(25 °C, 10 kHz)

[56,57]. These parameters are: saturation of magnetic flux density B_sat, Curie temperature T_Curie, losses per unit of volume P_V, and initial permeability μ_i.

It can be seen that the values of these parameters for both the cores differ significantly. In particular, the significantly higher value of the magnetic permeability of the RTF core means that, for the same value of the inductor inductance at a zero current value for the RTP inductor, it was necessary to wind 25 turns of wire of a diameter equal to 0.8 mm, while, for the RTF inductor, only 4 turns of wires of the same diameter were needed.

4. Method of Determining the Values of Models Parameters

In order to make practical use of the models presented in Section 2, it is necessary to determine the values of their parameters for specific types of components that will be used in the analyzed circuit. Thermal parameters were measured using the classical optical method described, among others, in [58]. The parameters of the built-in transistor and diode models were taken from the SPICE program libraries [19,59]. The values of the temperature coefficients α_V, α_RD, and α_RON were determined based on the information provided in the catalog data of these semiconductor devices [60,61]. The values of the parameters of passive elements were determined based on the approximation of the frequency characteristics of the inductors and capacitors measured using an RLC bridge manufactured by GWINSTEK. In the case of capacitors, the investigated elements were connected directly to the terminals of this measuring device. In the case of inductors, the measuring set-up shown in Figure 7 was used.

The presented measurement set-up uses a supply voltage source V_DC and resistor R = 0.5 Ω, limiting the bias current from this source. Capacitors C₁ and C₂ of capacitances 4.7 µF are used to protect the bridge from the DC component of the current. Inductor L is used to suppress the AC component of the current flowing through the voltage source V_DC. It has a much higher inductance (1.3 mH) than the inductance of the tested inductor (DUT).

The measurement of the inductor inductance was performed using an automatic RLC bridge, GWINSTEK. During the measurements, the bridge operated in the mode of inductance in a series equivalent circuit. The voltage amplitude generated by the bridge was 1 V, and its frequency was 200 kHz. Additionally, in order to examine the effect of the constant component on the core and the winding temperature of the inductor, a pyrometer connected to the data acquisition card (DAQ) connected to a computer was used, as well as an infrared camera (TC) to observe the temperature distribution on the surface of the investigated elements. The measurements were performed for the inductors with annular cores of approximately 25 mm in diameter made of powdered iron (−26), designated in a later part of the paper as RTP, and 3C90 ferrite, designated in a later part of the paper as RTF.

As a result of the measurements and necessary calculations, the following values of the parameters of the converter component models described in Section 3 were determined and are presented below. The parameters of the transistor were taken from [19], whereas the parameters of the diode were taken from [59].

• Transistor IRF 540: R_ON = 60 mΩ, α_RON = 8 × 10⁻³ 1/K, R_thT = 50 K/W;
• Diode S2D20065A: α_V = −2 mV/K, R_D = 0.15 Ω, R_thD = 70 K/W, α_RD =3 × 10⁻³ 1/K;
• Capacitors: C_C = 200 μF, R_C = 0.15 Ω, L_C = 130 nH, R_thC = 32 K/W;
• Values of parameters describing models of both the inductors are collected in

  Table 2. Values of parameters describing models of both the inductors with different cores.

  Parameter	A [μH]	B [1/A]	C [A]	Lmin [μH]	Lmax [μH]	RL [Ω]	CL [pF]	C2 [μF]	C3 [mF]	R2 [Ω]	RthL [K/W]	PC0 [nJ]	f0 [kHz]	αf [ms]
  Core RTP	21	−0.25	1	21	45	0.5	3	100	1	100	10	14.5	50	0.1
  Core RTF	33	−1.35	0.01	10	45	0.4	40	100	1	100	10	55	50	0.1

  .

Figure 8, Figure 9 and Figure 10 show the calculated (lines) and measured (points) characteristics of both the inductors and the capacitor. Figure 8 shows the calculated and measured dependences of the impedance modulus of the inductors containing iron powder cores (RTP) and ferrite cores (RTF).

As can be seen, decreasing dependences of Z(I_DC) were obtained for both the inductors in a wide range of current values. For the inductor with the RTP core, the impedance module value was constant for current I_DC < 1.5 A, while for the inductor with the RTF core, a strong decrease in the Z value was visible for current I_DC > 10 mA with an increase in I_DC. For the inductor with the RTF core, for current I_DC > 4 A, the impedance modulus value was 40 times lower than for I_DC = 0. For the inductor with the RTP core, the minimum impedance modulus value was only twice lower than its value at I_DC = 0. Despite such large differences in the Z(I_DC) curves for both the inductors, good agreement between the calculation and measurements results was obtained for each of them.

Figure 9 illustrates the influence of frequency on the impedance modulus of the investigated inductors.

As can be seen, for the inductor with the RTF core, the maximum of the Z(f) dependence occurred at a frequency of about 1.5 MHz, and the minimum at f = 9 MHz. For the inductor with the RTP core, the maximum occurred at a frequency exceeding 10 MHz. Based on the measured characteristics, the values of the parasitic capacitances of both the inductors were determined. In the range of frequency below 1 MHz, very good accuracy of the proposed model was achieved. In this range, the differences between the results of modeling and measurements did not exceed a few percentage points. These differences were the highest for the inductor with the RTF core in the range between 1 and 2 MHz, and they even reached 50%. These results show that, for very high values of frequency, additional parasitic phenomena should be taken into account.

Figure 10 shows the calculated and measured dependences of the modulus (Figure 10a) and phase (Figure 10b) of the impedance of the modeled electrolytic capacitors on the frequency.

As can be seen, the Z(f) dependence had a minimum at f equal to about 20 kHz. The phase(f) dependence graph shows that the impedance was capacitive at f < 1 kHz and inductive at f > 1 MHz. It is clearly visible that the applied capacitor model enabled good agreement to be obtained between the results of calculations and measurements in the entire considered range of frequency changes. The differences between the measured and calculated values of the impedance module did not exceed 2%. Higher differences could be observed for the curve phase (f) for frequency between 10 and 20 kHz, which could even reach 20°. Outside this range, these differences did not exceed 3°.

5. Investigations Results

In order to analyze the influence of thermal phenomena occurring in passive elements and semiconductor devices on the properties of the SEPIC converter, calculations and measurements of the characteristics of this converter operating under different power supply, load, and control conditions were performed. The example results are presented in Figure 11, Figure 12, Figure 13, Figure 14, Figure 15, Figure 16, Figure 17, Figure 18, Figure 19 and Figure 20. In these figures, the results of the calculations are marked with lines, and the results of the measurements with points. The blue color refers to the converter containing the inductors with RTF cores, and the red color indicates the inductors with RTP cores. In all the presented cases, the input voltage was equal to 6 V, and the control signal frequency was 50 kHz. In each case, inductors L₁ and L₂ were the same, i.e., both of them contained RTP or RTF cores.

All the presented results of the investigations refer to the thermally steady state. Non-isothermal characteristics of the investigated converter were calculated using the transient analysis in the SPICE program with one non-physical thermal time constant [19]. This method of analysis was chosen in order to obtain an acceptably long calculation time. The value of all the thermal capacitances occurring in the models was 50 μJ/K.

Figure 11 shows the calculated and measured dependences of the output voltage of the investigated converter on the duty cycle of the control signal at load resistance R_L = 4.7 Ω. At this value of R_L, the considered converter operated in the CCM (continuous conducting mode).

As can be seen, the considered dependence had a maximum at d = 0.7. The use of the inductors with the RTF cores allowed a higher value of the output voltage to be obtained at low values of d. A decrease in the V_out value at high d values resulted from power losses in the converter components.

Figure 12 illustrates the dependence of the energy efficiency of the tested converter on the value of the duty cycle of the control signal d.

For the converters containing the inductors with both types of ferromagnetic cores, the maximum of the η(d) characteristic was visible at d equal to approximately 0.5. It is worth noting that, when using the inductors with RTP cores, a much higher energy efficiency value was obtained. This efficiency was low due to a low value of the input voltage V_in of the converter, at which the voltage drops in the turned-on semiconductor devices were at the level of several percent of V_in. The differences between the measured and calculated values of energy efficiency did not exceed 5%.

Figure 13 illustrates the dependence of the transistor’s internal temperature on the control signal duty cycle.

The T_T(d) dependence obtained from the calculations and measurement was a monotonically increasing function. An increase in temperature T_T to 140 °C at d = 0.7 was a factor limiting the permissible range of d changes at a fixed value of load resistance. The inductor core material had a low impact on the T_T temperature value.

Figure 14 shows the calculated and measured dependences of the temperature of the inductor L₁ (Figure 14a) and capacitor C₁ (Figure 14b) on the duty cycle d.

The T_L1(d) dependence is a monotonically increasing function, while the T_C1(d) dependence had a maximum at d equal to approximately 0.75. The values of both these temperatures took on higher values for the converter containing the inductors with RTF cores. It is worth noting that the temperatures of the considered passive elements exceeded the ambient temperature by up to 25 °C. In particular, an increase in the temperature of electrolytic capacitors had a very unfavorable impact on their lifetime.

Figure 15, Figure 16, Figure 17, Figure 18, Figure 19 and Figure 20 show the calculated (lines) and measured (points) dependences of the output voltage, energy efficiency, and temperatures of the SEPIC converter components on the load resistance at the duty cycle of this signal (d = 0.5).

Figure 15 presents the dependence of the converter output voltage on the load resistance.

As expected, the dependence presented in Figure 15 shows that, with an increase in load resistance, the value of the converter output voltage increased. For a load resistance above 10 Ω, the converter switched to the DCM mode. It can also be observed that the calculation results obtained using the proposed model remained in good qualitative agreement with the measurement results. The greatest differences between the results of measurements and calculations could be observed for load resistance R_L > 100 Ω, when the converter operated in the DCM mode and the output voltage value was determined by the properties of the inductor. However, it should be noted that a better match of the calculation and measurement results could be observed for the converter including the inductor with the RTP core.

Figure 16 shows the dependence of the converter energy efficiency on the load resistance.

As can be seen, both for the converter containing the inductors with the ferrite cores and for the converter containing the inductors with the iron powder cores, the η(R_L) dependence had a local maximum for load resistance of R_L = 50 Ω. It can also be observed that the energy efficiency values were about 10% higher for the converter containing the iron powder inductors.

Figure 17 shows the dependence of the temperature of inductor L₁ contained in the considered converter on the load resistance.

With an increase in load resistance, the temperature T_L1 of the inductor increased. It can be observed that, for the inductor with the RTF core, higher values of the core temperature of this element were obtained by approx. 6 °C for R_L < 100 Ω.

Figure 18 shows the dependence of the temperature of capacitor C₁ contained in the considered converter on the load resistance.

From the dependence shown in Figure 18, it can be seen that, with an increase in the load resistance, the temperature of capacitor C₁ decreased. It can be seen that the type of the material used to build the inductor core contained in the considered converter had no significant impact on the dependence T_C1(R_L) for R_L > 10 Ω.

Figure 19 presents the dependence of the temperature of the diode contained in the considered converter on the load resistance.

From the T_D(R_L) dependence, as expected, it can be seen that, with an increase in load resistance, the diode temperature decreased. It can be seen that the type of material used to build the core of the inductors contained in the considered converter, similarly to the capacitor temperature, had no significant impact on the dependence T_D(R_L) at R_L > 5 Ω. It can also be seen that a good agreement was obtained between the results of the calculations obtained using the proposed model and the measurements.

In turn, Figure 20 shows the dependence of the transistor temperature on the load resistance of the considered converter.

As can be seen, the T_T(R_L) dependence has the minimum in both of the considered cases for R_L = 10 Ω. It can be observed that higher values of the transistor temperature were obtained when the converter contained the inductors with ferrite cores RTF. For R_L > 10 Ω, the temperature of the transistor operating in the converter containing the inductors with ferrite cores was 50 °C higher than the temperature of the same transistor operating in the converter containing the inductors with cores made of powdered iron (RTP). It is worth noting that, in the range of high values of resistance R_L, significant differences between measured and calculated values of the transistor temperature could be observed. In this range of resistance R_L values, the tested converter operated in the DCM mode, and the observed differences were a result of the use of the total power instead of thermal power in the thermal model of this device. This problem is discussed in the paper cited in [62]. Unfortunately, for the hybrid electrothermal model of a power MOSFET, it was impossible to calculate the thermal power dissipated in this transistor.

6. Conclusions

This paper presents electrothermal compact models of a MOS transistor, a diode, an inductor, and a capacitor. Both the form of these models and the method of determining their parameter values are presented. The correctness of these models in a wide range of currents and frequencies is experimentally demonstrated.

Using the proposed models, the non-isothermal characteristics of the SEPIC converter containing inductors with cores made of ferrite or powdered iron were determined. It was shown that the formulated models allowed us to obtain good agreement between the results of calculations and measurements for changes in load resistance and the control signal duty cycle in a wide range.

It is worth emphasizing that, in addition to the possibility of determining the values of electrical quantities in the modeled system, the proposed models provide a unique possibility of determining the value of the temperature of each of the semiconductor devices and passive elements. It was shown that the increases in the temperature of the listed components above the ambient temperature can be significant, and that they can even exceed 120 °C. The values of these temperatures calculated using the proposed models are in good agreement with the values obtained from the measurements. This confirms the practical usefulness of the formulated models. The main advantage of the elaborated models in comparison with the models described in the literature is the possibility to take into account the influence of non-idealities of passive components on the properties of the SEPIC converter.

Comparing the characteristics determined for the converter with the inductors containing cores made of RTF and RTP materials, it can be noted that the selection of the core material significantly influenced the value of the converter’s energy efficiency. The use of the RTP material allowed us to obtain a significantly higher (even by 20%) value of this parameter. In the case of operation of the converter at a high load resistance value, significant increases in the temperature of the transistor T_T and inductors were visible, especially for the converter containing the inductors with RTF cores. In turn, the temperature of the capacitors was highest at low load resistance values. In the considered range of changes in the duty cycle and load resistance, the temperatures of the capacitors were higher than the ambient temperature by 20 °C. For the inductors, the temperatures were higher than the ambient temperature by 30 °C.

In the range of very high load resistances, significant discrepancies between the results of calculations and measurements were observed. These discrepancies were probably caused by inaccurate modeling of the impulse properties of MOS transistors in the high-voltage range in the model of this device used in the SPICE program. In further research, the authors will make an attempt to improve this model by changing the description of the internal capacitances of the power MOSFET. This will make it possible to more accurately calculate the thermal power dissipated in this device and its junction temperature, which will enable more accurate modeling of this operating range.

The modeling method proposed in our article can also be used in modeling other power converters, especially multi-level DC/AC converters. The use of this method can help to improve the accuracy of modeling properties of the mentioned class of power converters.