**5. Conclusions**

In the paper, a compact nonlinear thermal model of the inductor was proposed. This model makes it possible to calculate values of temperature of the core and the winding of the inductor taking into account occurrence of self-heating in every mentioned component of the inductor and mutual thermal couplings between the core and the winding. It also takes into account the influence of power dissipated in every component of the inductor on thermal resistance of the core and the winding and mutual thermal resistance between the core and the winding. A manner of calculating the value of parameters of this model was also proposed.

Correctness of the worked out model was verified for selected inductors containing ferrite cores made of the same ferrite material, but these cores were characterised by a di fferent shape or by a di fferent size. As a result of the comparison of the obtained results of calculations and measurements it was shown that the elaborated model is universal, i.e., it makes it possible to obtain the good agreemen<sup>t</sup> of these results over a wide range of changes of power dissipated in each component of the inductor at di fferent shapes and dimensions of the core.

Comparing the findings obtained for di fferent sizes of cup cores, it was observed that an increase in the dimensions of the core of the considered shape caused a decrease in the value of thermal resistance and extension of time indispensable to obtain the thermally steady state in the examined inductor. Taking into account the fact that a basic mechanism of removing heat generated in the core of the inductor is convection, it can be said that the value of thermal resistance of the core is a decreasing function of the surface of the cup core. Referring to the results of measurements shown in the paper [36] it can be stated that in the description of the considered dependence spatial orientation of the inductor should be also taken into account. In turn, the thermal capacitance of the core, deciding the time of settlement of the waveform of transient thermal impedance, depends on the volume of the ferrite core. Then, the thermal capacitances of the core can be described with an increasing function of the volume of the core.

The authors proposed analytical Equations describing dependences of thermal resistances and thermal capacitances of the core and the winding of the inductor on the volume of the core. Correctness of the formulated Equations was proved for both the considered shapes of cores and a good match between the results of measurements and calculations was obtained. The di fferences between the results of calculations and measurements do not exceed 15% maximum. The obtained results of calculations performed using the new thermal model of the inductor confirm usefulness of the formulated model.

A change in the shape of the core also influences waveforms of transient thermal impedances occurring in the new nonlinear compact thermal model of the inductor. At the similar volume of the core, greater even by 20% values of thermal resistance were obtained for the inductor with the cup core. The observed changes in the value of thermal resistances in the function of volume of the core are higher for inductors with cup cores than inductors with toroidal cores. From the thermal managemen<sup>t</sup> point of view, it is more profitable to use toroidal cores than cup cores.

The obtained results of investigations make it possible to model the thermal properties of inductors in a simple way. The proposed thermal model of inductors can be used in power electronics applications. In the mentioned applications, properties of magnetic elements strongly influence watt-hour e fficiency. Using the new model, the designers of power electronic circuits can calculate thermal parameters and temperature of every component of the designed inductor. They can also determine usefulness of selected inductors in the anticipated operating conditions of the designing step.

The results of investigations presented in this paper correspond to one ferromagnetic material only. In further investigations other ferromagnetic materials and other shapes of the cores will be analysed.

**Author Contributions:** Conceptualization, K.G.; methodology, K.G. and K.D.; measurements, K.D.; computations, K.G. and K.D.; resources, K.D.; writing—original draft preparation, K.G. and K.D.; writing—review and editing, K.G. and K.D.; visualization, K.G. and K.D.; supervision, K.G. All authors have read and agreed to the published version of the manuscript.

**Funding:** Project financed in the framework of the program by Ministry of Science and Higher Education called "Regionalna Inicjatywa Doskonało´sci" in the years 2019–2022, project number 006/RID/2018/19, the sum of financing 11,870,000 PLN.

**Conflicts of Interest:** The authors declare no conflict of interest.
