**1. Introduction**

Inductors are important components of switch-mode power converters. These components are used to store electrical energy [1–5]. Properties of these components indeed depend on physical phenomena occurring in the winding and in the ferromagnetic core contained in the inductor [1,6–10]. At present, producers of ferromagnetic cores offer many types of cores made of different ferromagnetic materials. However, the most frequently used material to construct inductor cores are ferrites produced as a result of dwighting powdered metal oxides. Ferrites are characterised by high hardness, high resistivity, and low losses of eddy currents [5,9–14].

When an inductor operates in switched mode power converters an increase in power losses in this component is observed, which is the effect of current flowing through the winding of the inductor and remagnetisation of its core [1,5–16]. Power losses in the inductor are converted into heat. Heat generated in components of the inductor causes an increase in temperature of both the core and the winding above ambient temperature as a result of self-heating phenomena and mutual thermal couplings between the core and the winding of the inductor [8,11,15,17–19].

Temperature strongly influences the properties of electronic components, especially their reliability [20–22]. Therefore, it is essential to know the value of internal temperature of any element in the anticipated conditions of its operation. To calculate this temperature at well-known waveforms of power lost in the element, thermal models are used [23–26].

Thermal models presented in the literature have a character of microscopic models [6,7,27,28], dedicated to calculate distribution of temperature in the electronic component or macroscopic (compact) models [8,29–31], making it possible to calculate one value of internal temperature of the whole electronic component. Microscopic models, due to a high level of complexity, are not frequently used to analyse electronic circuits and they are used only to analyse thermal properties of single electronic components.

Typically compact thermal models are used. These models are often presented in the form of a network analogue [23,24,29,30,32]. Such an analogue usually consists of the current source, representing power dissipated in the modelled component and the RC network representing transient thermal impedance *Zth*(*t*), characterising the ability of a component to remove heat generated in this component [23–26]. Voltage on the current source corresponds to an excess of internal temperature of this component above ambient temperature [23–26].

The compact thermal model takes into account simultaneously all mechanisms of heat removal to the surroundings, i.e., conduction, convention, and radiation [24,30]. Transient thermal impedance occurring in the compact thermal model is typically described with the use of dependence of the form [24,29,30].

$$Z\_{tll}(t) = R\_{tll} \cdot \left[1 - \sum\_{i=1}^{N} a\_i \cdot \exp\left(-\frac{t}{\tau\_{tlli}}\right)\right] \tag{1}$$

where *Rth* is thermal resistance and *N* the number of thermal time constants τ*thi* corresponding to coefficients *ai*. At the steady-state the value of transient thermal impedance is equal to thermal resistance *Rth*.

In the literature many compact thermal models of inductors and transformers are described [6,8,15,17,33]. However, these models are highly simplified. Some of them [6,8] do not even take into account differences in temperature between the core and the winding. Thermal models of inductors taking into account nonlinearity of phenomena responsible for heat transfer are not known to the authors. Such nonlinearity is observed, among other things, in thermal models of semiconductor devices [30,34,35] or in the results of measurements of thermal properties of transformers shown among others in the papers [36,37]. Nowadays, there are no compact thermal models of inductors, which take into account influence of nonlinearity of thermal phenomena as well as the shape and the size of the core on thermal parameters of inductors.

The aim of this paper is to examine influence of the size and the shape of the ferromagnetic core and power losses on parameters of a thermal model of the inductor. In Section 2, a new nonlinear thermal model of the inductor is discussed. Section 3 describes a manner of estimation of parameters of the new model. The obtained results of calculations and measurements illustrating the usefulness of the proposed model are shown and discussed in Section 4.

## **2. New Nonlinear Thermal Model of Inductors**

A nonlinear thermal model of inductors presented in this section was developed by the authors. It belongs to a group of compact thermal models of electronic components. Let us assume that these electronic components contain more than one heat source. Such models were described in the literature for IGBT (insulated gate bipolar transistor) modules [23] and LED (light emitting diode) modules [38] in which self-heating phenomena and mutual thermal couplings also occur.

The presented nonlinear thermal model of the inductor takes into account the fact that both the core temperature *TC* and the winding temperature *TW* depend on the value of ambient temperature *Ta* and a temperature excess, which is a result of a self-heating phenomenon in each component of the inductor and mutual thermal couplings between these components. The temperature excess of the inductor component caused by a self-heating phenomenon depends on the transient thermal impedance of the core *ZthC*(*t*) and transient thermal impedance of the winding *ZthW*(*t*). Thermal couplings between the core and the winding are characterised by mutual transient thermal impedance between the core and the winding *ZthCW*(*t*). Nonlinearity of phenomena responsible for transporting heat generated in the inductor to the surroundings is also taken into account.

The new nonlinear thermal model of the inductor is dedicated to the SPICE (simulation program with integrated circuits emphasis) program and has the network form shown in Figure 1.

**Figure 1.** Nonlinear compact thermal model of the inductor.

This model consists of four subcircuits. Two of them, visible on the left side of Figure 1, allow for the calculation of the winding temperature *Tw* and the core temperature *TC*. According to the rules of formulating a thermal model of electronic devices the Foster network is used [39]. In the mentioned network, current sources which describe the time dependence of power dissipated in the electronic element are used. Current sources *IW* and *IC* correspond to powers dissipated in the winding and in the core. Self-transient thermal impedances of the winding *ZthW*(*t*) and the core *ZthC*(*t*) are modelled using *CWi* and *CCi* capacitors and controlled current sources *GWi* and *GCi*. Voltages on these circuits correspond to a temperature excess caused by a self-heating phenomenon occurring in the winding and in the core. Influence of mutual thermal couplings between the core and the winding is modelled using the controlled voltage sources E1 and E2. Voltages on these sources correspond to the values of temperature excesses *TWC* and *TCW*. In contrast, voltage sources *V*1 and *V*2 model ambient temperature.

The other two subcircuits allow calculating excesses of the core temperature *TCW* and the winding temperature *TWC* caused by thermal coupling between the inductor components. In these subcircuits, current sources represent power dissipated in the winding *IWC* and in the core *ICW*. The networks connected to these sources model mutual transient thermal impedance between the core and the winding *ZthCW*(*t*). All transient thermal impedances occurring in the presented model are described by Equation (1).

As shown in the papers [23,35,36,40], waveforms of transient thermal impedance of electronic components depend on power or internal temperature of the considered electronic component. From the papers [23,36,38] and from measurements performed by the authors, it results that influence of power dissipated in the modelled component practically does not influence heat capacitances of this component, while influence of power dissipated in the considered component on thermal resistance could be significant. Therefore, in the presented nonlinear thermal model of the inductor, constant values of thermal capacitances are used. In contrast, thermal resistances occurring in the considered model depend on power generated in the inductor. Empirical dependence of thermal resistance *Rth* on the dissipated power p is proposed. This dependence is expressed by the empirical Equation:

$$R\_{th} = R\_{th0} + R\_{th1} \cdot \exp\left(\frac{-p}{b}\right) \tag{2}$$

where is power dissipated in a heating component (core or winding), *Rth*0, *Rth*1 and *b* are the model parameters. Of course, for each of the three transient thermal impedances (1), there is a different set of parameter values describing thermal capacitances and thermal resistances.

Changes of individual components of thermal resistance are modelled using the controlled current sources *Gi* according to the Equation [33]:

$$G\_i = \frac{V\_{Gi}}{a\_i \cdot R\_{tl}} \tag{3}$$

where *VGi* is voltage on source *Gi, Rth* thermal resistance calculated using the Equation (2), and *ai* is the coefficient in the Equation (1) which corresponds to i-th thermal time constants τ*thi*.

Values of parameters occurring in Equations (1) and (2) were determined using the concept of local estimation [41,42] based on the measured waveforms of transient thermal impedances occurring in the considered thermal model of the inductor.
