**3. Method of Model Parameters Estimation**

Values of parameters of the thermal model described in Section 2 can be determined as a result of realisation of a series of measurements and calculations. These measurements are performed in the measuring set-up shown in Figure 2.

**Figure 2.** Set-up to measure thermal parameters of the inductor.

The considered set-up contains a voltage source E, a resistor R limiting the value of current, an ammeter, a voltmeter, the examined inductor L, a pyrometer, the acquisition data system DAQ, and a PC. To measure the value of DC current and DC voltage multimeters of the type UNIT UT-803 were used. The uncertainty of the measurements of DC voltage is ±0.025%, of the DC current ±0.1%, and of temperature measured by the pyrometer PT-3S is equal to ±3% [43].

In this set-up transient thermal impedances of the core *ZthC*(*t*) and the winding *ZthW*(*t*), and also mutual transient thermal impedance between the core and the winding *ZthCW*(*t*) are measured. These measurements are realised with the use of the indirect method and the following definitional Equations.

$$Z\_{\rm thC}(t) = \frac{T\_{\rm C}(t) - T\_a}{p\_{\rm C}} \tag{4}$$

$$Z\_{thW}(t) = \frac{T\_W(t) - T\_a}{pw} \tag{5}$$

$$Z\_{\rm thWC}(t) = \frac{T\_{\rm C}(t) - T\_a}{pw} \tag{6}$$

where *pC* denotes power dissipated in the inductor core, whereas *pW* refers to power dissipated in the winding. In Equations (4)–(6), the temperatures of the core and the winding and powers dissipated in the core and in the winding also appear.

In the considered measuring set-up power in the shape of a jump is dissipated during the flow of current through one of two components of the inductor depending on the position of switch S. In the position 1 of the switch, power is dissipated in the core, and in the position 2 of this switch power is dissipated in the winding. The value of current flowing through components of the inductor is regulated by means of voltage source E and resistor R. The temperature of the core is registered by means of the pyrometer PT-3S [43] configured to work in the continuous operation, as well as the card

of data acquisition and a computer. In turn, temperature of the winding is measured indirectly on the basis of measurements of the winding resistance.

The value of voltage and current flowing through the core or the winding is regulated over a wide range of the measured waveforms *ZthC*(*t*), *ZthW*(*t*) and *ZthCW*(*t*) for different values of power *pC* and *pW.* Basing on the registered waveforms of the mentioned transient thermal impedances of the inductor, values of parameters *Rth, ai,* τ*thi* occurring in the Equation (1) for every applied value of power *pW* and *pC* are estimated using the program ESTYM [42]. For every transient thermal impedance of the modelled inductor at the highest applied values of the dissipated power, average values of parameters *ai* and thermal capacitances occurring in the proposed thermal model are calculated on the basis of the Equation:

$$C\_i = \frac{\tau\_{thi}}{a\_i \cdot R\_{th}} \tag{7}$$

Based on the measured dependences of thermal resistance on power *RthC*(*pC*), *RthW*(*pW*)*,* and *RthCW*(*pW*), values of parameters *Rth*<sup>0</sup>*, Rth*1, and *b* occurring in Equation (2) are estimated with the method of local estimation [39,40] for every considered dependence separately.
