*4.3. Results of Measurements and Calculations*

In order to verify the usefulness of the thermal model of inductors proposed in Section 2, some measurements and computation were performed. In computations the nonlinear thermal model of the inductor was used. Results of these measurements and computations are shown in Figures 5–17.

In these figures, the results obtained for particular inductors were marked using the following markers and colour rules: an inductor with 14 × 8 mm dimensions of the cup core is marked as the small cup core (blue), an inductor with 18 × 11 mm dimensions of the cup core, the medium cup core (green), and an inductor with 26 × 16 mm dimensions of the cup core is marked as the big cup core (red). An inductor containing toroidal core with 16 mm diameter of the core is marked as a toroidal core

16 (violet), an inductor with 20 mm diameter of the core is marked toroidal core 20 (green), an inductor with 30 mm diameter of the core is marked toroidal core 30 (yellow) and an inductor with 40 mm diameter of the core is marked toroidal core 40 (blue). Additionally, it is worth remembering that the volume of toroidal core with 16 mm diameter corresponding to the volume of medium cup core. In all the figures, lines denote the results of calculations, whereas points refer to the results of measurements.

**Figure 6.** Measured and calculated waveforms of transient thermal impedance of the core for inductors with (**a**) cup and (**b**) toroidal cores of different dimensions.

**Figure 7.** Measured and calculated waveforms of transient thermal impedance of the winding for inductors with (**a**) cup and (**b**) toroidal cores.

**Figure 8.** Measured and calculated waveforms of transient thermal impedance of the medium cup core at selected values of dissipated power.

**Figure 9.** Measured and calculated waveforms of transient thermal impedances of the inductor containing the toroidal core 16.

**Figure 10.** Measured and calculated waveforms of transient thermal impedances of the inductor containing the medium cup core.

**Figure 11.** Measured and calculated waveforms of transient thermal impedances of the inductor containing the toroidal core 16.

**Figure 12.** Measured and calculated waveforms of transient thermal impedances of the core (**a**) and of the winding (**b**) for inductors containing each considered core.

**Figure 13.** Measured and calculated dependences of thermal resistance *RthC* of inductors with (**a**) cup cores and (**b**) toroidal core on dissipated power in the core.

**Figure 14.** Measured and calculated dependences of thermal resistance *RthW* of inductors winding with (**a**) cup cores and (**b**) toroidal core on dissipated power in the winding.

**Figure 15.** Measured and calculated dependences of thermal resistance *RthW* of inductors with (**a**) cup cores and (**b**) toroidal core on dissipated power in the winding.

At first, measured and calculated waveforms of transient thermal impedances occurring in the proposed thermal model of an inductor are presented. Next, dependences illustrating an influence of dissipated power in components of the tested inductors on thermal resistances occurring in the considered model are shown and discussed. Finally, an analytical description of the dependences of thermal resistances and capacitances on the effective volume of the core contained in the tested inductors are proposed and experimentally verified for these inductors.

**Figure 16.** Measured and calculated dependences of thermal resistances (**a**) *RthC*, (**b**) *RthW* and (**c**) *RthCW* of inductors with the medium cup core and the toroidal core on power dissipated in the core.

**Figure 17.** Measured and calculated dependences of thermal resistance of the core *RthC* (**a**) and thermal resistance of the winding *RthW* (**b**) on the effective volume of the core for inductors with cup cores and toroidal cores.

Figure 6 presents the calculated and measured waveforms of transient thermal impedance of the core of the considered inductors containing cup cores (Figure 6a) and toroidal core (Figure 6b) obtained at dissipation in the core the power of the amplitude equal to 2.5 W.

As can be seen, from the obtained waveforms of transient thermal impedance of the core, for the smallest volume of the core, the value of *ZthC*(*t*) at the steady state is more than twice higher than the value of *ZthC*(*t*) at the steady state for the biggest volume of the core (big cup core) and over 60% higher than the value *ZthC*(*t*) at the steady state for the core of medium volume—the medium cup core. It can be concluded from this relation that the ability to remove heat characterised by thermal resistance of the core *RthC* decreases with an increase in the core size. This is due to an increase in the surface area, at which convection heat transfer rate can occur. On the other hand, the time needed to reach the thermally steady state for the big core of the inductor is more than twice longer for the small inductor core. This means that thermal capacitance of the core increases with its size. In the case of the inductor with the toroidal core with the 16 mm diameter the value of *ZthC*(*t*) at the steady state is more than twice higher than the value of *ZthC*(*t*) at the steady state for the toroidal core with the 40 mm diameter and about 50% higher than the value of *ZthC*(*t*) at the steady state for the toroidal core with the 20 mm diameter. It is also worth noticing that the good agreemen<sup>t</sup> between the results of measurements and the results of calculations was obtained. For the toroidal core, the maximum error of calculations does not exceed 5% and for the cup core it is smaller than 8%.

Figure 7 presents the calculated and measured waveforms of transient thermal impedance of the winding of the considered inductors containing cup cores (Figure 7a) and toroidal cores (Figure 7b) obtained at power dissipated in the core equal to 2 W.

As can be seen, from the obtained waveforms of transient thermal impedance of the winding of the inductor with cup cores (Figure 7a), for the medium volume of the core, the value of *ZthW*(*t*) at the steady state is the same as the value of *ZthW*(*t*) at the steady state for the big cup core. In the case of the inductor with the toroidal core with the 16 mm diameter the value of *ZthW*(*t*) at the steady state for the considered inductor is five times higher than for the same inductor with the core with the 40 mm diameter. Additionally, di fferences between the results of calculations and measurements do not exceed 11% for the inductor with the cup core and 13% for the inductor with the toroidal core.

Figures 8 and 9 show the waveforms of thermal transient impedance of the medium cup core (Figure 8) and of the toroidal core (Figure 9) for selected values of power dissipated in the core.

As can be seen in Figure 8, due to changes in power dissipated in the cup core, values of *ZthC*(*t*) at the steady state change by not more than 20%. An increase in the value of power causes a decrease of *ZthC*(*t*) value. It is observed that power does not influence time indispensable to achieve the steady state of *ZthC*(*t*) waveform. This means that thermal capacitance is practically independent of power dissipated in the core.

However, an increase in the value of power dissipated in the toroidal core (Figure 9) causes a decrease in the value of *ZthC*(*t*) at the steady state. These changes reach almost 15%. At the same time, it can be seen that the value of power dissipated in the core practically does not influence time, in which the waveform of *ZthC*(*t*) achieves the steady state. As can be seen, di fferences between the results of calculations and measurements do not exceed 3%.

Figures 10 and 11 present the measured and calculated waveforms of transient thermal impedances *ZthW*(*t*), *ZthC*(*t*)*,* and *ZthCW*(*t*) for the inductor containing the medium cup core (Figure 10) at power dissipated in the core and in the winding equal to 2.5 W, and for the toroidal core, 16 (Figure 11) at power dissipated in the core *pC* = 1.7 W.

It can be seen that, at the steady state, values of *ZthW*(*t*) are up to 50% higher than *ZthC*(*t*). Time necessary to obtain the steady state is the shortest for waveform *ZthW*(*t*) and the longest for *ZthC*(*t*). Di fferences in the values of these times reach 20%, and are a result, among others, of di fferences in the mass of the core and windings. Also, the good agreemen<sup>t</sup> between the results of measurements and calculations was obtained. The maximum deviation does not exceed 8%.

Similarly to the cup core, the highest values are obtained for *ZthW*(*t*) (Figure 11). They are even 30% higher than the value of *ZthC*(*t*). Waveforms of *ZthC*(*t*) and *ZthCW*(*t*) di ffer from each other by not more than 5%. These di fferences are due to construction of the inductor, which causes that the winding is directly cooled by the air surrounding the inductor, and due also to the fact that the core is surrounded by the winding, in contrast to the cup core, which is not. Also the good agreemen<sup>t</sup> between the results of measurements and calculations is achieved and the di fferences between them do not exceed 3%.

Figure 12 presents a comparison of waveforms of transient thermal impedances of the core *ZthC*(*t*) and the winding *ZthW*(*t*) for the inductor containing the toroidal core and the cup core at power dissipated equal to 1.7 W.

The presented comparison shows that values of transient thermal impedances *ZthC*(*t*) and *ZthW*(*t*) are about 30% higher for the inductor with the toroidal core. The setting time for *ZthC*(*t*) waveform is longer for the inductor with the cup core, while the setting time for *ZthW*(*t*) waveform is practically the same. In this case, the error of calculations does not exceed 3%.

Figures 13–15 shows the calculated with the use of the Equation (2) and measured dependences of thermal resistances *RthC* (Figure 13), *RthW* (Figure 14), *RthWC* (Figure 15) occurring in the thermal model of the considered inductors with cup cores and inductors with toroidal cores on power dissipated in these components of the inductor.

As can be seen in Figure 13, dependence *RthC*(*pC*) is a decreasing function for both the inductors with the cup cores (Figure 13a) and with the toroidal core (Figure 13b). It is also visible that as the core size increases, thermal resistance values decrease. The biggest di fferences in the values of this parameter for the considered cores can be seen in the range of low values of power *pC*. Di fferences between the calculation results and the measurement results do not exceed a dozen percent for the inductor with the cup core and do not exceed 8% for the inductor with the toroidal core.

Figure 14 presents the calculated and measured dependences of thermal resistance *RthW* of the winding of the considered inductors with cup cores (Figure 14a) and inductors with toroidal cores (Figure 14b) on power dissipated in these cores. Due to the limited size of the small core, it was impossible to wind eight turns of copper wire in enamel with a diameter of 1 mm on inductor cores, so in the following results, a comparison between the big and medium cup cores only are presented in Figure 13a.

The characteristics presented in Figure 14 have a similar shape as the characteristics presented in Figure 13. An increase in the core volume causes a decrease in thermal resistance of the winding. The di fferences between the results of measurements and calculations for inductors with the cup core do not exceed 10% for all the considered inductors with cup and toroidal cores. It is also worth noticing that thermal resistance of the inductor containing the toroidal core with the 16 mm diameter is higher by even 10 K/W than thermal resistance of the inductor containing the medium cup core. Additionally, the di fferences between the results of measurements and calculations do not exceed 11% for the inductor with the cup core and 15.5% for the inductor with the toroidal core.

Figure 15 presents the calculated and measured dependences of mutual thermal resistance between winding and core *RthWC* of the inductors with cup cores (Figure 15a) and inductors with toroidal cores (Figure 15b) on power dissipated in these cores.

As can be seen, an increase of the core size causes a decrease of thermal resistance of the considered inductors with cup and toroidal cores. In the case of the inductor with the toroidal core an increase in diameter from 16 to 40 mm causes a more than double decrease of thermal resistance, whereas an increase of the diameter from 18 to 26 mm of the cup core causes a decrease in thermal resistance by about 15%. Di fferences between the calculations and measurements results do not exceed a dozen per cent for the inductor with the cup core and they do not exceed 13% for the inductor with the toroidal core.

Figure 16 presents the measured and calculated dependences of thermal resistances *RthC*, *RthW* and *RthCW* occurring in the nonlinear thermal model of the inductor on power dissipated in the core (Figure 16a) and in the winding (Figure 16b) for inductors with the medium cup core and the toroidal core 16. As mentioned in Section 3, both the considered cores have similar volume.

As can be seen, dependences of all the thermal resistances on power dissipated in the core are decreasing functions. Values of the considered parameters for the toroidal core are higher than the values obtained for the cup core. The highest values were obtained for thermal resistance of the winding *RthW*, and the lowest values for mutual thermal resistance between the core and the winding *RthCW.* Values of these parameters differ between each other even twice. Due to the influence of changes in power values in the considered range, changes in individual thermal resistances up to 20% are observed. Also, the good agreemen<sup>t</sup> between the results of calculations and measurements was obtained. The differences between the results of calculations and measurements are less than 12%.

Analysing results of investigation presented above we formulated the analytic Equation describing an influence of core volume on thermal resistances existing in the thermal model of the inductor. The form of this Equation is as follows:

$$R\_{th} = R\_{th\text{A}} \cdot \left(1 + k\_1 \cdot \exp\left(-\frac{V\_{\varepsilon}}{m\_1}\right)\right) \tag{8}$$

where *RthA* denotes border value of the thermal resistance at the volume of core tending to infinity, *Ve* is equivalent core volume, whereas *m*1 and *k*1 are model parameters characterising the slope of the dependence *Rth*(*Ve*).

In the same way, an analytical description of the dependence of thermal capacitance on core volume was formulated. This dependence is given by following Equation:

$$\mathbb{C}\_{th} = \mathbb{C}\_{thA} \cdot \left(1 + k\_3 \cdot V\_{\mathfrak{e}}\right) \tag{9}$$

where *CthA* denotes border value of thermal capacitance corresponding to zero value of volume *Ve*, whereas *k*3 is volume coefficient of thermal capacitance.

Figure 17 presents the measured (points) and calculated (lines) using Equation (8) dependences of thermal resistance of the core *RthC*, occurring in the nonlinear thermal model of the inductor on effective volume of the core for inductors with the cup cores (red colour) and the toroidal cores (green colours). Measurements and computations were performed at power dissipated in the core equal to 2 W.

As it is visible, for both the considered shapes of the core the dependence *Rth*(*Ve*) is a decreasing function. Values of both thermal resistances for cup cores are smaller than for toroidal core in the range of low values of core volume, whereas in the range of high values of core volume these relation is opposite. It is worth noticing that in the considered range of change the core volume values of thermal resistance decreases over twice. For both the shapes of core, a good accuracy of modelling considered dependences are obtained.

Figure 18 illustrates an influence of core volume on selected thermal capacitances occurring in thermal model of tested inductors.

**Figure 18.** Measured and calculated dependences of selected thermal capacitance of the core *CthC*1 (**a**) and *CthC*2 (**b**) on the effective volume of the core for inductors with cup cores and toroidal cores.

As can be observed, the considered dependences are increasing functions. It is worth noticing that the thermal capacitance *CthC*1 of the tested inductors is smaller for inductors including toroidal cores, whereas the thermal capacitance *CthC*2 is smaller for inductors including cup cores. Also, the good agreemen<sup>t</sup> between the results of measurements and calculations was obtained. The di fferences between these results do not exceed 12% for both considered shapes of cores.
