Electrothermal Model of Coupled Inductors with Nanocrystalline Cores

: The paper proposes a new electrothermal model of a coupled inductor containing a nanocrystalline core dedicated to the analysis of electrical energy conversion systems. The formulated model has a form of a subcircuit for SPICE. The model takes into account the inﬂuence of direct current, frequency, skin effect, temperature, self-heating, and mutual thermal couplings on the parameters and characteristics of the coupled inductors. The form of the developed model and the method of model parameter estimation are presented. The applied measuring system is presented. The results of measurements and calculations made with the use of the proposed model are presented and commented on.


Introduction
The pursuit of the miniaturisation of electric energy conversion devices requires an increase in switching frequency [1,2]. The factor limiting this increase in frequency are dynamic properties of semiconductor and magnetic elements (including the inductor) contained in the mentioned devices [1,[3][4][5]. Additionally, the increasing application requirements in the field of designing devices characterised by high watt-hour efficiency and their failure-free operation force the manufacturers of electronic components and systems to design and produce systems characterised by very high reliability [6,7].
In order to obtain a long lifetime of electronic components, an effective cooling system is indispensable. Some forced cooling systems are described in [8,9], but for magnetic devices typically free cooling systems are used.
Computer software supporting the work of designers is used to analyse the properties of electronic components and systems. One of the popular kinds of software of this type is SPICE [10]. This software can use averaged models of switch-mode power supplies [11][12][13]. The mentioned models allow the values of currents and voltages at steady state to be computed using fast DC analysis, instead of time-consuming transient analysis. On the other hand, such models ignore, for example, the non-linearity and inertia of semiconductor devices and the non-linear dependence L(i) of magnetic components [11].
To take into account the influence of thermal phenomena (such as self-heating and mutual thermal couplings between components of an electronic component) occurring in electronic elements on their electrical characteristics, electrothermal models are used [13][14][15][16].
In the literature, electrothermal models of semiconductor and magnetic elements are described [17][18][19][20][21]. However, in the case of magnetic elements, the mentioned models are limited to classic inductors (single-winding inductors) or transformers [11,22], yet there are no such models for coupled inductors.
For example, in [23], an electrothermal model of an inductor with a ferromagnetic core based on the isothermal Jiles-Atherton model is proposed. Although this model takes into windings n. The new model additionally takes into account the skin effect, self-heating phenomena, and mutual thermal coupling between inductor components. All the mentioned phenomena were omitted in the presented model in paper [27]. Due to the simplification of the description and the improvement of its comprehensibility, the considerations are limited to the case where the coupled inductor contains three windings. The considerations were carried out for an inductor with a nanocrystalline core. The proposed model is in the form of a subcircuit for SPICE. The network representation of this model is shown in Figure 1.  The developed model consists of three blocks: the main circuit, the auxiliary circuit, and the thermal model. The structure of the proposed model is shown in Figure 2. This model takes into account the influence of frequency, skin effect, direct current, temperature, and thermal phenomena on the characteristics of the coupled inductors.
The main circuit describes the properties of individual windings of the modelled inductors connected between terminals Ai and Bi (i = 1… .n). The model of each winding consists of six series of connected elements: a voltage source with zero efficiencies VLSi; a linear inductor with inductance Li; controlled voltage sources ELSi, EMi, and ERSi; and a resistor RSi. The capacitor Ci, connected in parallel to the terminals Ai and Bi, represents parasitic capacitance of the winding.
The controlled voltage source ELSi represents the inductance of the i-th inductor winding, and its efficiency is described by the following formula [11]: where zi is the number of winding turns, SFe is the cross-section area of the core, Bsat is the saturation of the magnetic flux density, A is the field parameter, H is the magnetic force, lFe is the magnetic path length, lp is the air gap length, ai is the scaling factor, and VLi is the voltage on a linear inductor Li. The controlled voltage source EMi represents the mutual inductance between the i-th winding and the other windings, expressed by the following formula [27]: where kj is the magnetic coupling coefficient, VELSi is the voltage at the source ELSj, and n is the number of inductor windings. The developed model consists of three blocks: the main circuit, the auxiliary circuit, and the thermal model. The structure of the proposed model is shown in Figure 2. This model takes into account the influence of frequency, skin effect, direct current, temperature, and thermal phenomena on the characteristics of the coupled inductors.   The controlled voltage sources EW1, EW2, and EW3 included in the thermal model of the windings model the nonlinear thermal resistance of individual windings.
As per the literature [16,19,33,34] and the measurements made, thermal resistance is a non-linear function of temperature and the number of windings n, in which the power is dissipated.
In the considered case of coupled inductors, the efficiency of these sources is the same and described by the empirical dependence resistor R Si . The capacitor C i , connected in parallel to the terminals A i and B i , represents parasitic capacitance of the winding.
The controlled voltage source E LSi represents the inductance of the i-th inductor winding, and its efficiency is described by the following formula [11]: where z i is the number of winding turns, S Fe is the cross-section area of the core, B sat is the saturation of the magnetic flux density, A is the field parameter, H is the magnetic force, l Fe is the magnetic path length, l p is the air gap length, a i is the scaling factor, and V Li is the voltage on a linear inductor L i . The controlled voltage source E Mi represents the mutual inductance between the i-th winding and the other windings, expressed by the following formula [27]: where k j is the magnetic coupling coefficient, V ELSj is the voltage at the source E LSj , and n is the number of inductor windings. The controlled voltage source E SRi represents a voltage drop in the series resistance of the inductor as a result of the skin effect and self-heating. Its efficiency is described by the following formula [11]: where V RSi denotes the voltage on the resistor R Si ; d d is the diameter of the winding wire; l d is the length of the winding; α p is the temperature coefficient of copper resistivity; T Wi , T a , and T 0 are the winding temperature, ambient temperature, and reference temperature, respectively; µ 0 is the magnetic permeability of the vacuum, ρ is the copper resistivity, and i Li is the current flowing through the i-th inductor winding. In turn, the resistance value of the resistor R Si corresponds to the series resistance of a single winding for the direct current and temperature T 0 . The value of this resistance is calculated from the classical formula The auxiliary circuit block consists of two controlled voltage sources, EB and EH, used to determine the magnetic flux density of the core B and the magnetic force, respectively. The efficiencies of these sources are described by the dependences [31,32] where i i is the current of the i-th winding. Due to the very narrow hysteresis loops of the considered inductor core in the model, the core losses are neglected.
In turn, the thermal model ( Figure 2) is used to determine the temperature of each winding T W1 , T W2 , and T W3 , and the temperature of the core T C of the coupled inductor. This model takes into account the phenomenon of self-heating in each winding and thermal coupling between each pair of windings and between each winding and the core.
This model consists of 10 subcircuits, nine of which are included in the thermal model of the winding and one in the thermal model of the core. For better readability of the model description, the thermal model of the winding was divided into four blocks. Block 1, contained in the thermal model of the winding, includes three subcircuits for determining the temperature of the first winding T W1 , the second winding T W2 , and the third winding T W3 , taking into account the self-heating in each of them and mutual thermal couplings. The subcircuits included in blocks 2, 3, and 4 are used to determine a temperature rise resulting from the mutual thermal coupling between the windings. In block 2, a temperature rise in the first winding caused by heat generation in the second (T W12 ) winding and a temperature rise in the first winding caused by heat generation in the third (T W13 ) winding are determined. The subcircuits in block 3 make it possible to determine a temperature rise in the second winding caused by heat generation in the first winding (T W21 ) and in the third winding (T W23 ). On the other hand, block 3 is used to determine temperature increases in the third winding caused by heat generation in the first (T W31 ) and second (T W32 ) windings. The proposed thermal winding model includes nine controlled current sources. Three of them are included in block 1, where they represent the power dissipated in each winding P W1 , P W2 , and P W3 . The controlled current sources G 1 , G 2 , G 3 , G 4 , G 5 , and G 6 included in blocks 2, 3, and 4 represent the power dissipated in the first winding P W1 (G 3 , G 5 ), the second winding P W2 (G 1 , G 6 ), and the third winding P W3 (G 2 , G 4 ). However, the thermal model of the core contains one controlled P CW current source, the efficiency of which corresponds to the sum of the power dissipated in each winding (P W1 + P W2 + P W3 ).
The controlled voltage sources E W1 , E W2 , and E W3 included in the thermal model of the windings model the nonlinear thermal resistance of individual windings.
As per the literature [16,19,33,34] and the measurements made, thermal resistance is a non-linear function of temperature and the number of windings n, in which the power is dissipated.
In the considered case of coupled inductors, the efficiency of these sources is the same and described by the empirical dependence where P Wi is the power dissipated in the i-th winding; T W is the temperature of the winding in which the power is dissipated; w 1 and α are the scaling coefficients, taking into account the number of windings in which the power is dissipated; R thW0 is the minimum value of thermal resistance of the winding; R thW1 is the maximum change value of the winding thermal resistance; T zW determines the slope of R thW (T W ); and n U is the number of windings in which the power is simultaneously dissipated. The controlled voltage source E C models the nonlinear thermal resistance between the winding and the core R thCW . The efficiency of this source is described by the empirical dependence where T C is the core temperature; w 2 and w 22 are the scaling coefficients, taking into account the influence of the number of windings in which the power is dissipated on mutual thermal resistance between the core and the winding; R thC0 is the minimum value of mutual thermal resistance between the core and the winding; R thC1 is the maximum change in the value of this resistance temperature; T zC determines the slope of R thC (T C ); and P CW is the power dissipated in the winding. In turn, the controlled voltage sources E W11 , E W12 , and E W13 model a temperature increase in individual windings resulting from thermal couplings between individual windings. The efficiency of these sources is described by the following empirical formula: where T W12 , T W13 , T W21 , T W23 , T W31 , and T W32 denote the temperature rises in individual windings caused by thermal couplings, which are determined in blocks 2, 3, and 4. The controlled voltage sources E WW12 , E WW13 , E WW21 , E WW23 , E WW31 , and E WW32 model mutual thermal resistance between the windings. The description of the efficiency of these sources for each pair of windings is the same and is described by the following empirical formula: where T W2 is the temperature of winding 2, where no power is dissipated; R thWW0 is the minimum value of mutual thermal resistance between the windings; R thW1 is the maximum change in the value of mutual thermal resistance between the windings; and i WWi is the current of the source G i . The voltage sources V 1a , V 1b , V 1c , V 2a , and V 4 model the ambient temperature T a .

Parameter Estimation of the Electrothermal Model of Coupled Inductors
The use of the electrothermal model of the coupled inductor described in Section 2 requires the determination of the parameters of this model. The parameters of the considered electrothermal model of coupled inductors can be divided into three groups: a.
Geometrical parameters of the windings (l d , S d , d d , z); b.
Magnetic parameters of the core (k, B sat , A, l p , l Fe , S Fe ); c.
Values of the geometrical parameters are obtained directly from the construction of the inductor.
Values of individual parameters are determined on the basis of the catalogue data available on the manufacturer's website or the measured characteristics of the inductor. While determining the value of these parameters, the local estimation method described in [35] is used. The input data for this procedure are the measured characteristics of the considered inductor using the measuring setup shown in Section 4 and the catalogue data from the manufacturer's website [36].
The thermal parameters of the inductor were measured with the use of the measurement setup presented in Section 4. For different values of the power dissipated in one winding, the temperature of the core and all the windings of the inductor were recorded. Based on the obtained measurement results, thermal resistance of the winding, mutual thermal resistance between the core and the winding, and mutual thermal resistance between the windings were determined from the classical dependence [16,33].
In turn, to determine the magnetic parameters, the method described in [35] and the measurement setup presented in Section 4 were used. It is the most popular method for measuring the hysteresis loop described in the literature, which is based on the observation and recording of the time waveforms of the current in the primary winding and the voltage waveforms on the secondary winding of a transformer with a ferromagnetic core. Waveforms H(t) and B(t) are determined indirectly using the measured current and voltage waveforms.

Investigated Device and Measurement Setup
In the investigations, a coupled inductor with compensation type DK3H-3352-1011-NK made by Schurter ( Figure 3) was used. It contains a toroidal nanocrystalline core in the housing made of plastic of the UL 94V-0 flammability class. The dimensions of the considered coupled inductor are an external diameter of 38.3 mm, internal diameter of 25.3 mm, and height of 15.2 mm. Three windings made of enamelled copper wire with a diameter of 1.1 mm are wound on the core [36]. Each winding contains 11 turns.

Investigated Device and Measurement Setup
In the investigations, a coupled inductor with compensation type DK3H-3352-1011-NK made by Schurter ( Figure 3) was used. It contains a toroidal nanocrystalline core in the housing made of plastic of the UL 94V-0 flammability class. The dimensions of the considered coupled inductor are an external diameter of 38.3 mm, internal diameter of 25.3 mm, and height of 15.2 mm. Three windings made of enamelled copper wire with a diameter of 1.1 mm are wound on the core [36]. Each winding contains 11 turns. In order to verify the correctness of the developed model, the influence of temperature and the direct winding current on the parameters of the coupled inductor were investigated.
To determine the coupled inductor parameters, the measurement setup shown in Figure 4 was used. Figure 5 shows a photo of the measurement setup used.   In order to verify the correctness of the developed model, the influence of temperature and the direct winding current on the parameters of the coupled inductor were investigated.
To determine the coupled inductor parameters, the measurement setup shown in Figure 4 was used. Figure 5 shows a photo of the measurement setup used.

Investigated Device and Measurement Setup
In the investigations, a coupled inductor with compensation type DK3H-3352-1011-NK made by Schurter ( Figure 3) was used. It contains a toroidal nanocrystalline core in the housing made of plastic of the UL 94V-0 flammability class. The dimensions of the considered coupled inductor are an external diameter of 38.3 mm, internal diameter of 25.3 mm, and height of 15.2 mm. Three windings made of enamelled copper wire with a diameter of 1.1 mm are wound on the core [36]. Each winding contains 11 turns. In order to verify the correctness of the developed model, the influence of temperature and the direct winding current on the parameters of the coupled inductor were investigated.
To determine the coupled inductor parameters, the measurement setup shown in Figure 4 was used. Figure 5 shows a photo of the measurement setup used.

Investigated Device and Measurement Setup
In the investigations, a coupled inductor with compensation type DK3H-3352-1011-NK made by Schurter ( Figure 3) was used. It contains a toroidal nanocrystalline core in the housing made of plastic of the UL 94V-0 flammability class. The dimensions of the considered coupled inductor are an external diameter of 38.3 mm, internal diameter of 25.3 mm, and height of 15.2 mm. Three windings made of enamelled copper wire with a diameter of 1.1 mm are wound on the core [36]. Each winding contains 11 turns. In order to verify the correctness of the developed model, the influence of temperature and the direct winding current on the parameters of the coupled inductor were investigated.
To determine the coupled inductor parameters, the measurement setup shown in Figure 4 was used. Figure 5 shows a photo of the measurement setup used.    In the setup presented in Figure 4, the windings of the investigated element were marked with the symbols L 1 , L 2 , and L 3 . Depending on the position of switches S 1 and S 2 , the dependence of the inductor inductance on its current was measured when the current flowed through one winding L 1 , two windings L 1 and L 2 , or three windings L 1 , L 2 , and L 3 . The resistor R 1 of the resistance equal to 1 Ω limited the current of the DC voltage source V DC . If the value of this current ranged from 0 to 10 A, it was measured with the multimeter UNIT-T 804. In turn, if this current exceeded 10 A it was measured using the oscilloscope (OSC) of the type RIGOL DS 1052E with the current probe (CP) of the type PINTEK PA-677.
The capacitor C 1 , with a capacity of 6.7 µF, protected the RLC bridge, making a flow of the direct current through this bridge impossible. In turn, the inductor L B , with an inductance of 1.45 mH, reduced the value of the alternating current of the voltage source V DC . The automatic bridge GWINSTEK LCR-8110G [37] was used to measure the inductance of the investigated element. During the measurements, the LCR bridge worked in the mode of measurement inductance with series equivalent resistance. Additionally, by using the FLIR ETS-320 (INFRARED CAMERA) [38] infrared camera connected to a computer, the temperature distribution was recorded on all the components of the investigated inductor.
In order to ensure a reliable measurement result, it was necessary to keep the relation of L i >> L B . The L B contained the powdered iron core (type 102 material) with 20 turns of copper wire in enamel with a diameter 2 mm. The geometric dimensions of the inductor were as follows: outer diameter of 10.3 cm, inner diameter of 5.6 cm, and height of 6.5 cm. Figure 6 shows the measured dependence L B (I DC ) of the blocking inductor. The measurements were made in the circuit shown in Figure 4. As a blocking inductor, a series connection of five identical inductors connected in series was used.
Energies 2022, 15, 224 8 of 20 In the setup presented in Figure 4, the windings of the investigated element were marked with the symbols L1, L2, and L3. Depending on the position of switches S1 and S2, the dependence of the inductor inductance on its current was measured when the current flowed through one winding L1, two windings L1 and L2, or three windings L1, L2, and L3 The resistor R1 of the resistance equal to 1 Ω limited the current of the DC voltage source VDC. If the value of this current ranged from 0 to 10 A, it was measured with the multimeter UNIT-T 804. In turn, if this current exceeded 10 A it was measured using the oscilloscope (OSC) of the type RIGOL DS 1052E with the current probe (CP) of the type PINTEK PA-677.
The capacitor C1, with a capacity of 6.7 μF, protected the RLC bridge, making a flow of the direct current through this bridge impossible. In turn, the inductor LB, with an inductance of 1.45 mH, reduced the value of the alternating current of the voltage source VDC. The automatic bridge GWINSTEK LCR-8110G [37] was used to measure the inductance of the investigated element. During the measurements, the LCR bridge worked in the mode of measurement inductance with series equivalent resistance. Additionally, by using the FLIR ETS-320 (INFRARED CAMERA) [38] infrared camera connected to a computer, the temperature distribution was recorded on all the components of the investigated inductor.
In order to ensure a reliable measurement result, it was necessary to keep the relation of Li >> LB. The LB contained the powdered iron core (type 102 material) with 20 turns of copper wire in enamel with a diameter 2 mm. The geometric dimensions of the inductor were as follows: outer diameter of 10.3 cm, inner diameter of 5.6 cm, and height of 6.5 cm. Figure 6 shows the measured dependence LB(IDC) of the blocking inductor. The measurements were made in the circuit shown in Figure 4. As a blocking inductor, a series connection of five identical inductors connected in series was used. As can be seen in Figure 6, the dependence of inductance LB of the blocking inductor on the direct current IDC in the range of 0 to 5 A was almost constant. For IDC > 5 A the inductance of the inductor decreased to the value of approx. 800 μH at IDC = 19 A.
In turn, in order to characterise parameters of the nanocrystalline core of the considered coupled inductor and to determine its saturation of magnetic flux density Bsat, the hysteresis loop was measured using the classic measurement system shown in Figure 7 [31,32].
In the measurement setup presented in Figure 7, the JV5603P function generator was used as the source of the sinusoidal input signal Vsin [39]. To amplify the input signal, the AETECHRON 7228 power amplifier (PA) was used [40]. The resistor R1 was used to limit the current flowing from the source Vsin, and the resistor R2 and the capacitor C2 represented the integrator. By recording with the RIGOL1102D [41] oscilloscope (OSC), the in-  As can be seen in Figure 6, the dependence of inductance L B of the blocking inductor on the direct current I DC in the range of 0 to 5 A was almost constant. For I DC > 5 A the inductance of the inductor decreased to the value of approx. 800 µH at I DC = 19 A.
In turn, in order to characterise parameters of the nanocrystalline core of the considered coupled inductor and to determine its saturation of magnetic flux density B sat , the hysteresis loop was measured using the classic measurement system shown in Figure 7 [31,32].
In the measurement setup presented in Figure 7, the JV5603P function generator was used as the source of the sinusoidal input signal V sin [39]. To amplify the input signal, the AETECHRON 7228 power amplifier (PA) was used [40]. The resistor R 1 was used to limit the current flowing from the source V sin , and the resistor R 2 and the capacitor C 2 represented the integrator. By recording with the RIGOL1102D [41] oscilloscope (OSC), the input current waveforms using the current probe (CP) PINTEK PA / 677 [42], and the voltage time waveform on the capacitor C 2 , the time waveform of the magnetic force from the dependence (6) was determined, and the magnetic flux density was determined from the dependence: Energies 2022, 15, 224 9 of 20 put current waveforms using the current probe (CP) PINTEK PA / 677 [42], and the voltage time waveform on the capacitor C2, the time waveform of the magnetic force from the dependence (6) was determined, and the magnetic flux density was determined from the dependence:  Figure 8 shows the hysteresis loop of the investigated nanocrystalline core obtained with the use of the measurement setup shown in Figure 7. The measurements were carried out at the input signal frequency f = 2.6 kHz. The resistance values of the resistor R1 and R2 were 0.5 Ω and 100 Ω, respectively, whereas the capacitance of the C2 capacitor was equal to 3.19 μF. As can be seen, the investigated core was characterised by a low value of the saturation of magnetic flux density Bsat equal to 110 mT and low power losses presented by a very narrow hysteresis loop.
Based on the estimation procedure described in Section 3, the values of the parameters of the electrothermal model of the investigated inductor were determined and are collected in Tables 1-3.   Figure 8 shows the hysteresis loop of the investigated nanocrystalline core obtained with the use of the measurement setup shown in Figure 7. The measurements were carried out at the input signal frequency f = 2.6 kHz. The resistance values of the resistor R 1 and R 2 were 0.5 Ω and 100 Ω, respectively, whereas the capacitance of the C 2 capacitor was equal to 3.19 µF.
Energies 2022, 15, 224 9 of 20 put current waveforms using the current probe (CP) PINTEK PA / 677 [42], and the voltage time waveform on the capacitor C2, the time waveform of the magnetic force from the dependence (6) was determined, and the magnetic flux density was determined from the dependence:  Figure 8 shows the hysteresis loop of the investigated nanocrystalline core obtained with the use of the measurement setup shown in Figure 7. The measurements were carried out at the input signal frequency f = 2.6 kHz. The resistance values of the resistor R1 and R2 were 0.5 Ω and 100 Ω, respectively, whereas the capacitance of the C2 capacitor was equal to 3.19 μF. As can be seen, the investigated core was characterised by a low value of the saturation of magnetic flux density Bsat equal to 110 mT and low power losses presented by a very narrow hysteresis loop.
Based on the estimation procedure described in Section 3, the values of the parameters of the electrothermal model of the investigated inductor were determined and are collected in Tables 1-3.  As can be seen, the investigated core was characterised by a low value of the saturation of magnetic flux density B sat equal to 110 mT and low power losses presented by a very narrow hysteresis loop.
Based on the estimation procedure described in Section 3, the values of the parameters of the electrothermal model of the investigated inductor were determined and are collected in Tables 1-3.   Tables 1-3 summarise the values of the geometric parameters, the magnetic parameters, and the thermal parameters of the electrothermal model of the coupled inductor. The model assumed that all the windings had the same value for all the parameters.

Results of Measurements and Calculations
In order to verify the correctness of the developed electrothermal model of coupled inductors described in Section 2, the influence of temperature, frequency, and direct current IDC on the parameters of the investigated element was examined. The measurement setup presented in Figure 4 was used to investigate the coupled inductors. The obtained measurement results were compared to the results of the calculations obtained with the use of the mentioned model. In all figures presented in this section, the principle that the points denote the measurement results and the lines illustrate the results of the calculations made with the use of the model described in Section 2 was adopted.
Additionally, in order to investigate the influence of the ambient temperature Ta on parameters of the investigated inductor, a thermostat with the considered inductor was applied to the setup shown in Figure 4. The obtained results show that the ambient temperature did not influence the inductor parameters; hence, the mentioned results are omitted in this section. Figure 9 shows the dependence of inductance L 1 of a single winding, inductances of a series connected by two windings L 12 , and inductances of a series connected by three windings L 123 of the coupled inductors on the direct current I DC flowing through these coils at a frequency of the exciting signal equal to f = 200 kHz. The mentioned windings were wound on the same core.

Characteristics of Coupled Inductor
The dependences L(I DC ) presented in Figure 9 show that the results of the computations performed using the proposed model fit the results of the measurements very well in all the considered connections of the windings. It is also visible that in the case when two windings were connected in series, their inductance L 12 was over two-fold higher than inductance L 1 of the single winding. If three windings were connected in series, their inductance L 123 was over three-fold higher than inductance L 1 . All the presented dependences L(I DC ) are monotonically decreasing functions. Figure 10 presents thermograms illustrating the temperature distribution in the investigated inductors when the current flowed through one winding (Figure 10a), two windings connected in series (Figure 10b), and three windings connected in series (Figure 10c). In each of the considered cases, the power dissipated in one winding, two series-connected windings, and three series-connected windings was equal to 2.5 W. The dependences L(IDC) presented in Figure 9 show that the results of the computations performed using the proposed model fit the results of the measurements very well in all the considered connections of the windings. It is also visible that in the case when two windings were connected in series, their inductance L12 was over two-fold higher than inductance L1 of the single winding. If three windings were connected in series, their inductance L123 was over three-fold higher than inductance L1. All the presented dependences L(IDC) are monotonically decreasing functions. Figure 10 presents thermograms illustrating the temperature distribution in the investigated inductors when the current flowed through one winding (Figure 10a), two windings connected in series (Figure 10b), and three windings connected in series ( Figure  10c). In each of the considered cases, the power dissipated in one winding, two seriesconnected windings, and three series-connected windings was equal to 2.5 W. It can be seen (Figure 10) that the temperature distribution of the core and the winding was uneven and depended on the number of windings through which the current flowed. It can be noticed that when the current flowed through only one winding, the temperature TW1 of that winding reached up to 78 °C, and the temperature of the TC core, resulting from thermal couplings between the core and the winding, reached approx. 45 °C and was almost 10 °C higher than the temperature of the other windings through which no current flowed. This proves that thermal coupling between the core and the winding was stronger than between the windings. On the other hand, when the current flowed through two or three series-connected windings, the temperature of such windings TW12 and TW123 was equal to 55 and 58 °C, respectively. It was also observed that the tempera-  The dependences L(IDC) presented in Figure 9 show that the results of the computations performed using the proposed model fit the results of the measurements very well in all the considered connections of the windings. It is also visible that in the case when two windings were connected in series, their inductance L12 was over two-fold higher than inductance L1 of the single winding. If three windings were connected in series, their inductance L123 was over three-fold higher than inductance L1. All the presented dependences L(IDC) are monotonically decreasing functions. Figure 10 presents thermograms illustrating the temperature distribution in the investigated inductors when the current flowed through one winding (Figure 10a), two windings connected in series (Figure 10b), and three windings connected in series ( Figure  10c). In each of the considered cases, the power dissipated in one winding, two seriesconnected windings, and three series-connected windings was equal to 2.5 W. It can be seen ( Figure 10) that the temperature distribution of the core and the winding was uneven and depended on the number of windings through which the current flowed. It can be noticed that when the current flowed through only one winding, the temperature TW1 of that winding reached up to 78 °C, and the temperature of the TC core, resulting from thermal couplings between the core and the winding, reached approx. 45 °C and was almost 10 °C higher than the temperature of the other windings through which no current flowed. This proves that thermal coupling between the core and the winding was stronger than between the windings. On the other hand, when the current flowed through two or three series-connected windings, the temperature of such windings TW12 and TW123 was equal to 55 and 58 °C, respectively. It was also observed that the temperature of each winding through which the current flowed was the same. Additionally, it was It can be seen (Figure 10) that the temperature distribution of the core and the winding was uneven and depended on the number of windings through which the current flowed. It can be noticed that when the current flowed through only one winding, the temperature TW1 of that winding reached up to 78 • C, and the temperature of the T C core, resulting from thermal couplings between the core and the winding, reached approx. 45 • C and was almost 10 • C higher than the temperature of the other windings through which no current flowed. This proves that thermal coupling between the core and the winding was stronger than between the windings. On the other hand, when the current flowed through two or three series-connected windings, the temperature of such windings T W12 and T W123 was equal to 55 and 58 • C, respectively. It was also observed that the temperature of each winding through which the current flowed was the same. Additionally, it was observed that when the current flowed through two series-connected windings, the core temperature between the windings through which the current flowed was equal to 46 • C and was almost 5 • C higher than the temperature of the core near the winding through which the current did not flow. On the other hand, when the current flowed through three series-connected windings, the core temperature between all the windings was similar and was equal to 44 • C. Figure 11 shows the dependence of thermal resistance of a single winding R thW , thermal resistance between the core and winding R thCW , and the mutual thermal resistance between the windings R thWW on the temperature of this winding T W1 . which the current did not flow. On the other hand, when the current flowed through three series-connected windings, the core temperature between all the windings was similar and was equal to 44 °C. Figure 11 shows the dependence of thermal resistance of a single winding RthW, thermal resistance between the core and winding RthCW, and the mutual thermal resistance between the windings RthWW on the temperature of this winding TW1. Figure 11. Dependence of thermal resistances of the inductor components on the temperature of the winding in which the power is dissipated.
As can be seen, thermal resistance of the winding RthW, thermal resistance between the core and the winding RthCW, and mutual thermal resistance between the windings RthWW decreased with an increase in the first winding TW1 temperature, in which the power was dissipated. In each considered case, an increase in the winding temperature, in which the power was dissipated, of about 46°C caused a decrease in the values of the mentioned thermal resistances by approx. 5 K/W. Figure 12 presents the dependence of mutual thermal resistances between the core and one RthCW1, two RthCW12, or three RthCW123 windings on the winding, in which the power was dissipated.    Figure 11. Dependence of thermal resistances of the inductor components on the temperature of the winding in which the power is dissipated.
As can be seen, thermal resistance of the winding R thW , thermal resistance between the core and the winding R thCW , and mutual thermal resistance between the windings R thWW decreased with an increase in the first winding T W1 temperature, in which the power was dissipated. In each considered case, an increase in the winding temperature, in which the power was dissipated, of about 46 • C caused a decrease in the values of the mentioned thermal resistances by approx. 5 K/W. Figure 12 presents the dependence of mutual thermal resistances between the core and one R thCW1 , two R thCW12 , or three R thCW123 windings on the winding, in which the power was dissipated.
which the current did not flow. On the other hand, when the current flowed through three series-connected windings, the core temperature between all the windings was similar and was equal to 44 °C. Figure 11 shows the dependence of thermal resistance of a single winding RthW, thermal resistance between the core and winding RthCW, and the mutual thermal resistance between the windings RthWW on the temperature of this winding TW1. As can be seen, thermal resistance of the winding RthW, thermal resistance between the core and the winding RthCW, and mutual thermal resistance between the windings RthWW decreased with an increase in the first winding TW1 temperature, in which the power was dissipated. In each considered case, an increase in the winding temperature, in which the power was dissipated, of about 46°C caused a decrease in the values of the mentioned thermal resistances by approx. 5 K/W. Figure 12 presents the dependence of mutual thermal resistances between the core and one RthCW1, two RthCW12, or three RthCW123 windings on the winding, in which the power was dissipated. Figure 12. Dependence of mutual thermal resistance between the inductor core and the winding on the winding temperature for one, two, and three series-connected windings.   Figure 12. Dependence of mutual thermal resistance between the inductor core and the winding on the winding temperature for one, two, and three series-connected windings.
As can be seen, the number of windings in which the power was dissipated significantly influenced thermal resistance of the inductor core. An increase in the number of windings in which the power was dissipated from one to three caused a decrease in the value of thermal resistance by up to 25%. Figure 13 shows the dependence of thermal resistance of one R thW1 , two R thW12 , and three R thW123 series-connected windings in which the power was dissipated on the temperature of these windings. cantly influenced thermal resistance of the inductor core. An increase in the number of windings in which the power was dissipated from one to three caused a decrease in the value of thermal resistance by up to 25%. Figure 13 shows the dependence of thermal resistance of one RthW1, two RthW12, and three RthW123 series-connected windings in which the power was dissipated on the temperature of these windings. As in the case of thermal resistance of the core, the number of windings in which the power was dissipated significantly influenced thermal resistance of the inductor winding. An increase in the number of the mentioned windings from one to three caused a decrease in the value of thermal resistance RthW123 of the windings by up to 70% compared to RthW1. Figure 14 shows the temperature dependence of all the inductor components on the direct current IDC obtained when power was dissipated in one winding (a) and in three windings connected in series (b).
As can be seen in Figure 14a, an increase in the direct current flowing through one winding L1 caused an increase in the temperature of all the components of the considered inductor as a result of mutual thermal interactions between the windings. An increase in the temperature of windings TW2 and TW3 at the direct current IDC = 14 A was equal to 25 °C, whereas an increase in the core temperature TC at IDC = 14 A was equal to 33 °C. On the other hand, from the dependence shown in Figure 14b, it can be seen that in the case of the power dissipated in three series-connected windings, the core and the winding temperature for IDC < 6 A were the same. For IDC > 6 A, the temperature of the winding was higher than the core temperature by about 10 °C. As in the case of thermal resistance of the core, the number of windings in which the power was dissipated significantly influenced thermal resistance of the inductor winding. An increase in the number of the mentioned windings from one to three caused a decrease in the value of thermal resistance R thW123 of the windings by up to 70% compared to R thW1 . Figure 14 shows the temperature dependence of all the inductor components on the direct current I DC obtained when power was dissipated in one winding (a) and in three windings connected in series (b). This shows a strong thermal coupling between the components with increasing numbers of windings in which the power was dissipated. It should also be noted that the results of the calculations are in good agreement with the results of the measurements and the difference between the results of the calculations and measurements did not exceed 4%. Figure 15 shows the dependence of the inductor core temperature on the direct current IDC when the current flowed through one TC1 winding, two series-connected TC12 windings, and three series-connected TC123 windings. As can be seen in Figure 14a, an increase in the direct current flowing through one winding L1 caused an increase in the temperature of all the components of the considered inductor as a result of mutual thermal interactions between the windings. An increase in the temperature of windings T W2 and T W3 at the direct current I DC = 14 A was equal to 25 • C, whereas an increase in the core temperature T C at I DC = 14 A was equal to 33 • C. On the other hand, from the dependence shown in Figure 14b, it can be seen that in the case of the power dissipated in three series-connected windings, the core and the winding temperature for I DC < 6 A were the same. For I DC > 6 A, the temperature of the winding was higher than the core temperature by about 10 • C.
This shows a strong thermal coupling between the components with increasing numbers of windings in which the power was dissipated. It should also be noted that the results of the calculations are in good agreement with the results of the measurements and the difference between the results of the calculations and measurements did not exceed 4%. Figure 15 shows the dependence of the inductor core temperature on the direct current I DC when the current flowed through one T C1 winding, two series-connected T C12 windings, and three series-connected T C123 windings. Figure 14. Dependence of the temperature of the inductor components on the direct current (a) when power is dissipated in one winding, (b) when the power is dissipated in three series-connected windings.
This shows a strong thermal coupling between the components with increasing numbers of windings in which the power was dissipated. It should also be noted that the results of the calculations are in good agreement with the results of the measurements and the difference between the results of the calculations and measurements did not exceed 4%. Figure 15 shows the dependence of the inductor core temperature on the direct current IDC when the current flowed through one TC1 winding, two series-connected TC1 windings, and three series-connected TC123 windings. Figure 15. Dependence of the inductor core temperature on the direct current IDC when the curren flows through one, two, or three windings connected in series.
As expected, an increase in the number of windings in which power was dissipated caused a rise in the core temperature. In the case when the power was dissipated in only one winding, the temperature of the core TC1 at IDC = 14 A was equal to 59 °C. An increase in the number of windings in which the power was dissipated from one to three caused an increase in the temperature of TC123 to almost 100 °C. As can be seen, the increase in the temperature of the inductor core with an increase in the number windings in which the power was dissipated was non-linear, which was shown in Section 2. Additionally, it is worth noting that good agreement was obtained between the results of the calculations and measurements, and the difference between them did not exceed 5%. As expected, an increase in the number of windings in which power was dissipated caused a rise in the core temperature. In the case when the power was dissipated in only one winding, the temperature of the core T C1 at I DC = 14 A was equal to 59 • C. An increase in the number of windings in which the power was dissipated from one to three caused an increase in the temperature of T C123 to almost 100 • C. As can be seen, the increase in the temperature of the inductor core with an increase in the number windings in which the power was dissipated was non-linear, which was shown in Section 2. Additionally, it is worth noting that good agreement was obtained between the results of the calculations and measurements, and the difference between them did not exceed 5%. Figure 16 shows the dependence of the inductor winding temperature on the direct current I DC obtained when the current flowed through one T W1 , two series-connected windings T W12 , and three series-connected windings T W123.
As can be seen, an increase in the number of windings in which the power was dissipated caused increases in the temperature of the inductor winding. It is observed that changes in the winding temperature with a change in the number of windings in which the power was dissipated had a non-linear character. If the current flowed through only one winding, the winding temperature at I DC = 14 A was equal to 78 • C. When the current flowed through three series-connected windings, the temperature of the winding T W123 exceeded 100 • C. Additionally, it is worth noting that the results of the measurements and calculations are in good agreement and the difference between them did not exceed 5%.

Characteristics of Coupled Inductor
The usefulness of the proposed electrothermal model of the coupled inductor in the analysis of switching systems was verified by calculating the influence of the load resistance on the output voltage. Additionally, the influence of the load resistance on the temperature of individual components of the coupled inductor operating in the boost converter was investigated. The analysed circuit is shown in Figure 17. changes in the winding temperature with a change in the number of windings in which the power was dissipated had a non-linear character. If the current flowed through only one winding, the winding temperature at IDC = 14 A was equal to 78 °C. When the curren flowed through three series-connected windings, the temperature of the winding TW12 exceeded 100 °C. Additionally, it is worth noting that the results of the measurements and calculations are in good agreement and the difference between them did not exceed 5%. Figure 16. Dependence of the inductor winding temperature on the direct current IDC flowing through one TW1, two TW12, or three TW123 windings connected in series.

Characteristics of Coupled Inductor
The usefulness of the proposed electrothermal model of the coupled inductor in the analysis of switching systems was verified by calculating the influence of the load re sistance on the output voltage. Additionally, the influence of the load resistance on the temperature of individual components of the coupled inductor operating in the boost con verter was investigated. The analysed circuit is shown in Figure 17.

Characteristics of Coupled Inductor
The usefulness of the proposed electrothermal model of the coupled inductor in the analysis of switching systems was verified by calculating the influence of the load resistance on the output voltage. Additionally, the influence of the load resistance on the temperature of individual components of the coupled inductor operating in the boost converter was investigated. The analysed circuit is shown in Figure 17. Figure 17. Analysed circuit of a boost converter.
In the considered circuit, a MOSFET transistor IRF540 (M1), a Schottky diode 1N5822 (D1), and passive elements R2 = 30 Ω and C1 = 50 μF were used. The resistor RL is the load In the considered circuit, a MOSFET transistor IRF540 (M 1 ), a Schottky diode 1N5822 (D 1 ), and passive elements R 2 = 30 Ω and C 1 = 50 µF were used. The resistor R L is the load of the converter, and the voltage source V in is the power source. The signal V p is a square wave generator with voltage levels equal to 0 and 10 V. Figure 18 shows the calculated dependences of the output voltage of the considered converter on load resistance at the control signal frequency f = 50 kHz, the control signal duty cycle d = 0.5, and the value of the input voltage V in = 24 V. The case of the current flowing through one winding L 1 , two L 12 windings connected in series, and three L 123 windings connected in series was considered.
As can be seen, the dependence V out (R L ) had a local extreme at R L = 20 Ω when the current flowed through only one winding. In the case of the increasing number of windings in which the power was dissipated, this extreme shifted to the left side of the considered characteristic. It is also observed that an increase in load resistance R L from 0.1 to 10 Ω caused an almost seven-fold increase in the value of the output voltage V out of the boost converter. On the other hand, at the load resistance R L > 100 Ω, the output voltage of the boost converter was constant. Additionally, an increase in the number of windings from one to three in which the power was dissipated reduced the value of the output voltage of the considered converter by over 10%. The maximum observed at R L equal to about 10 Ω is a result of the nonlinearity of the dependence L(i) of the inductor used. This dependence influenced the location of the border between the CCM and DCM modes.
wave generator with voltage levels equal to 0 and 10 V. Figure 18 shows the calculated dependences of the output voltage of the considered converter on load resistance at the control signal frequency f = 50 kHz, the control signal duty cycle d = 0.5, and the value of the input voltage Vin = 24 V. The case of the current flowing through one winding L1, two L12 windings connected in series, and three L123 windings connected in series was considered. Figure 18. Dependence of the output voltage on the load resistance of the boost converter when the current flows through one coil, two coils, or three coils connected in series.
As can be seen, the dependence Vout (RL) had a local extreme at RL = 20 Ω when the current flowed through only one winding. In the case of the increasing number of windings in which the power was dissipated, this extreme shifted to the left side of the considered characteristic. It is also observed that an increase in load resistance RL from 0.1 to 10 Ω caused an almost seven-fold increase in the value of the output voltage Vout of the boost converter. On the other hand, at the load resistance RL > 100 Ω, the output voltage of the boost converter was constant. Additionally, an increase in the number of windings from one to three in which the power was dissipated reduced the value of the output voltage of the considered converter by over 10%. The maximum observed at RL equal to about 10 Ω is a result of the nonlinearity of the dependence L(i) of the inductor used. This dependence influenced the location of the border between the CCM and DCM modes. Figure 19 shows the dependence of the core temperature TC and the winding temperature TW1, TW2, and TW3 of the coupled inductor operating in the boost converter on the load resistance. In this case, switches S1 and S2 were closed and the current flowed through only one coil L1. Figure 19 shows that an increase in the load resistance of the boost converter caused a decrease in the temperature of the core and all the windings of the coupled inductor. An increase was observed of the core temperature to TC = 51 °C at RL = 0.1 Ω and the temperature of the windings in which no power was dissipated at the same load resistance to TW1 = TW2 = 53 °C. This is the result of the strong thermal couplings between the windings and between the winding and the inductor core.  Figure 19 shows the dependence of the core temperature T C and the winding temperature T W1 , T W2 , and T W3 of the coupled inductor operating in the boost converter on the load resistance. In this case, switches S 1 and S 2 were closed and the current flowed through only one coil L 1 .
Energies 2022, 15,224 17 of 20 Figure 19. Dependence of the core and winding temperature on the load resistance of the boost converter when the current flows through one coil.

Conclusions
The paper proposes an electrothermal model of a coupled inductor with a nanocrystalline core dedicated to the SPICE software. The formulated model takes into account electrical phenomena occurring in the windings, magnetic phenomena in the core, and thermal phenomena. In contrast to the model proposed in the literature, in the proposed model, the temperature of the core and each winding are computed. The thermal phenomena include self-heating in each of the windings and mutual thermal couplings between each pair of windings and between each winding and the inductor core.
In Table 4 some properties of the proposed model are compared with select models of the inductors presented in the paper [24,27].   Figure 19 shows that an increase in the load resistance of the boost converter caused a decrease in the temperature of the core and all the windings of the coupled inductor. An increase was observed of the core temperature to T C = 51 • C at R L = 0.1 Ω and the temperature of the windings in which no power was dissipated at the same load resistance to T W1 = T W2 = 53 • C. This is the result of the strong thermal couplings between the windings and between the winding and the inductor core.

Conclusions
The paper proposes an electrothermal model of a coupled inductor with a nanocrystalline core dedicated to the SPICE software. The formulated model takes into account electrical phenomena occurring in the windings, magnetic phenomena in the core, and thermal phenomena. In contrast to the model proposed in the literature, in the proposed model, the temperature of the core and each winding are computed. The thermal phenomena include self-heating in each of the windings and mutual thermal couplings between each pair of windings and between each winding and the inductor core.
In Table 4 some properties of the proposed model are compared with select models of the inductors presented in the paper [24,27]. Table 4. The phenomena included in the models described in the current paper and in papers [24,27].

Phenomena
New Model Model From [24] Model From [27] Coupling factor YES NO YES The correctness of the developed model was verified experimentally for a coupled inductor containing three identical windings wound around a nanocrystalline core. The obtained results of the calculations and the measurements confirm that the developed model enabled calculations of the dependence L(i) and the temperature of the windings and the core under various conditions of supplying these windings. The differences between the obtained results of the calculations and measurements are small and did not exceed a few percent.
It has been proven that in the investigated inductor, there can be a strong variation between the temperatures of the core and the individual windings. These differences exceed as much as 20 • C. The proposed model enables the correct determination of the value of the temperature of the mentioned components of the coupled inductor, taking into account the non-uniformity of the power density generated in the investigated element and the non-linearity of the phenomena responsible for heat dissipation. The practical usefulness of the developed model for the analysis of properties of DC-DC converters was demonstrated with the example of a boost converter. The performed calculations allowed both the electrical characteristics of the considered system and the dependence of the temperature of individual components of the investigated inductor on the load resistance of the considered converter to be determined.
The formulated model can be used for the analysis of select power conversion systems. The use of this model will allow a more detailed analysis of the influence of the designed inductor to be performed on properties of this class system.