Compact Modeling of Thermal Properties of Selected Planar Transformers with Ferrite Cores

Abstract

This paper presents the results of modeling and measuring the thermal properties of selected planar transformer designs. A compact thermal model for the steady state of the planar transformer is presented. A method for measuring its parameters is discussed. The investigation results, illustrating the practical usefulness of the considered thermal model for describing the thermal properties of transformers containing ferrite cores made of different materials, are presented and discussed. The influence of the transformer’s operating frequency and load resistance on the temperatures of each component of the tested transformers is discussed. It is demonstrated that the proposed model enables the correct determination of these steady-state temperatures over a wide range of operating conditions for transformers with cores made of different ferrite materials.

1. Introduction

Planar transformers are becoming increasingly popular in power electronics [1,2]. The advantage of such transformer designs is their small size and the ability to integrate them with a printed circuit board (PCB) upon which the transformer windings are made [3,4]. The small size (volume between 0.25 and 80 cm³) [5] of the transformer means that its windings can contain only a few or a dozen turns in one layer [6]. Therefore, such solutions are dedicated to transformers operating at a low voltage amplitude or high frequency [4,7,8].

As is known from the literature [3,8,9], temperature strongly affects transformer components. Its increase causes an increase in winding resistance, due to the positive sign of the temperature coefficient of copper resistivity [3]. For ferrites, a decrease in the core saturation flux density is observed when its temperature increases, because the increasing thermal vibrations of the atoms disturb the magnetic ordering within the material [9,10]. Additionally, a change in the core’s magnetic permeability and a change in its losses are also observed when the temperature value changes [3,8,11]. Furthermore, an increase in temperature shortens the lifetime of electronic devices [12,13].

When transformers operate, energy losses occur both in the core and in the windings [14,15]. The result of these losses is an increase in the temperature of all transformer components, i.e., the core and each winding, due to the self-heating phenomenon and mutual thermal coupling between the transformer components [16].

Computer simulations are commonly used in the design and analysis of electronic components and systems [17,18]. Thermal analyses are a special type of such simulations [19,20]. They can be conducted using the finite element method (FEM) [20] or compact models [19]. FEM analysis allows for the determination of the temperature distribution in the analyzed element, but it requires greater computational effort. A compact model, on the other hand, allows for the determination of a single temperature value for the entire element [21] or for each of its components [16,19], but requires significantly less computational effort than FEM. Therefore, compact models are typically used in the analysis of electronic systems.

The compact thermal models of planar transformers described in the literature are very simplified. For example, in [22], a model for FEM analysis was proposed, which assumed a uniform power density generated throughout the transformer. In turn, in [4], it was assumed that power is only generated in the transformer windings.

Paper [23] presents an investigation of the magnetic and thermal properties of a planar toroidal transformer consisting of two planar toroidal coils. The electromagnetic properties are described using Maxwell’s equations, and the heat transfer using the Navier–Stokes equations. Calculations were performed using FEM and COMSOL Multiphysics software.

Paper [24] analyses a thermal modeling method for planar transformers, considering different winding configurations and distribution of power loss. The non-uniform temperature distribution in the tested transformer was observed. A simplified analytical model was proposed in the cited paper.

Paper [25] proposes simplified thermal models of a planar transformer, based on the concept of transient thermal impedance. Additionally, a simplified thermal model of this transformer is proposed based on the theory of thermal radiation. This model uses only the total power losses in the transformer to calculate the temperature rise in this component.

Paper [26] presents a literature review on thermal modeling of planar transformers. It was shown that compact models are widely used for early stage designs, due to their simplicity. Such models have a network form that is easy to use in classical software for analysis electronic circuits.

The aim of this paper is to verify the usefulness of a compact thermal model of a planar transformer for selected designs of this component. The considered model makes it possible to calculate the values of temperature for each winding and the core of the considered kind of transformers. A form of thermal model of such a transformer, designed for steady-state analysis, is described. A method for determining the model’s parameters is presented, and the results of model validation are presented for selected transformer designs operating over a wide range of frequency and load resistance variations. The differences between the values of temperature of the transformer components were analyzed and discussed.

Section 2 describes the compact thermal model of a planar transformer. Section 3 presents a method for determining the model’s parameters. Section 4 describes the tested transformers. Section 5 presents the results of the experimental verification of the model. Section 6 discusses the scope of the tested model’s applicability.

2. Compact Thermal Model

The compact thermal model of the transformer considered in this paper allows for the determination of the steady-state core temperature T_C, primary winding temperature T_W1, and secondary winding temperature T_W2. This model accounts for changes in ambient temperature T_a and self-heating in each transformer component, as well as the thermal coupling between each pair of these components.

The method of formulation compact thermal modeling of electronic devices containing thermally coupled components is presented in [19]. Using this method, the formulas describing the temperatures of individual transformer components in a steady state were formulated. They have the following form:

$$T_C=T_a+R_{thC}\cdot p_C+R_{thW1C}\cdot p_{W1}+R_{thW2C}\cdot p_{W2}$$

(1)

$$T_{W1}=T_a+R_{thW1}\cdot p_{W1}+R_{thW1C}\cdot p_C+R_{thW1W2}\cdot p_{W2}$$

(2)

$$T_{W2}=T_a+R_{thW2}\cdot p_{W2}+R_{thW2C}\cdot p_C+R_{thW2W1}\cdot p_{W1}$$

(3)

In Formulas (1)–(3), R_thC, R_thW1, and R_thW2 denote the thermal resistances of the core, primary winding, and secondary winding, respectively. In turn, R_thW1C, R_thW2C, R_thCW1, R_thCW2, R_thW1W2, and R_thW2W1 denote the mutual thermal resistances between the primary winding and the core, the secondary winding and the core, the core and the primary winding, the core and the secondary winding, and between the windings, respectively. The average values of the power dissipated in the steady state in the core, primary winding, and secondary winding are designated by the symbols p_C, p_W1, and p_W2.

Because the heat transfer between the transformer components is equally intense in both directions, the individual pairs of mutual thermal resistances are equal. This means that the following equalities hold: R_thW1C = R_thCW1, R_thW2C = R_thCW2, and R_thW1W2 = R_thW2W1.

3. Model Parameters Estimation

As can be seen from the description of the compact thermal model of a planar transformer, its practical implementation requires determining the values of six parameters. These are the thermal resistances of each of the transformer components, R_thC, R_thW1, and R_thW2, as well as the mutual thermal resistances between each pair of the components R_thW1C, R_thW2C, and R_thW1W2.

The values of these parameters can be determined based on their definitions. The adopted concept of transformer thermal modeling assumes that each component has a different temperature, but the temperature distribution within each component is uniform. The thermal resistances of individual transformer components are defined as the steady-state temperature rise in that component above the ambient temperature T_a, divided by the power dissipated in that component. During this measurement, no power is generated in other transformer components. For example, the thermal resistance of the core, R_thC, is described by the formula

$$R_{thC}={\displaystyle \frac{T_C-T_a}{p_C}}$$

(4)

In the case of measuring R_thW1, the temperature T_C should be replaced by the temperature of the primary winding T_W1, and the power p_C by the power p_W1. In turn, when determining R_thW2, the temperature of the secondary winding T_W2 and the power dissipated in this winding p_W2 are used.

To determine the mutual thermal resistances, it is necessary to measure the temperature of one transformer component while dissipating power in another component. For example, the mutual thermal resistance between the core and the primary winding R_thCW1 is described by the formula

$$R_{thCW1}={\displaystyle \frac{T_{W1}-T_a}{p_C}}$$

(5)

During this measurement, no power is generated in either winding. To determine the mutual thermal resistance R_thCW2, we replace the temperature T_W1 with the temperature T_W2 in Formula (5). In turn, to determine R_thW1W2, it is necessary to measure the temperature T_W2 with the power dissipation p_W1 in the primary winding.

Formulas (4) and (5) define the self and mutual thermal resistances used in the presented model. They determine the method for measuring the above parameters. The temperature and power values in these formulas refer to the steady state. The power dissipated in individual transformer components, including the core, is determined by the external circuit supplying the transformer.

As can be seen from the presented definitions, determining each parameter of the compact thermal model of a transformer requires dissipating power in one component of the tested transformer until a steady state is achieved. Once a steady state is achieved, the power and temperature of the corresponding transformer component are measured. The temperatures of individual transformer components were measured using a pyrometer. By dissipating power in one transformer component and measuring the steady-state temperature of each component, the values of three parameters of the considered model can be determined in a single measurement process: one thermal resistance and two mutual thermal resistances.

The power dissipated in the windings was determined as being the product of the winding current and the voltage across the winding when the winding was supplied from a DC source. The power dissipated in the core was determined by supplying the primary winding with a sinusoidal signal with amplitude V_m and frequency f. During this measurement, the secondary winding was loaded with a high-resistance resistor, ensuring a low current flowing through both windings. Based on the data provided by the core manufacturer regarding its cross-sectional area S_Fe (given in m²) and the number of turns z₁ on the primary winding, the flux density amplitude in the core (given in T) was determined using the following formula [22]:

$$B_m={\displaystyle \frac{V_m}{2\cdot \pi \cdot z_1\cdot S_{Fe}\cdot f}}$$

(6)

Of course, the value of B_m has to be lower than saturation flux density B_S, because the core cannot operate in the saturation range. The average value of the power lost in the core is determined based on the Steinmetz formula [3,27], using the following relationship

$$p_C=V_e\cdot P_{V0}\cdot f^{\alpha }\cdot B_m^{\beta }$$

(7)

In Formula (7), V_e denotes the core volume provided by the manufacturer and given in m³, and P_V0, α, and β are the Steinmetz model parameters. The values of these parameters are determined based on the approximation of the P_V(f) and P_V(B_m) characteristics provided by the manufacturer of the core material. Parameters α and β do not have any units, whereas P_V0 is given in W/m³s^α/T^β.

4. Tested Transformers

The tests were carried out for six planar transformers. Their construction was based on cores containing two ELP 22/6/16 profiles made of various manganese–zinc ferrites. Each profile has the shape of the letter E. The numbers occurring in the symbol of this profile denote that it is 21.8 mm long, 15.8 mm wide, and 5.7 mm high. The profile thickness is 2.5 mm. The side columns are 2.5 mm wide, and the central column is 5 mm wide. The cores under consideration have the following geometric parameters: magnetic path length l_Fe = 32.5 mm, core cross-sectional area S_Fe = 78.3 mm², and core volume V_e = 2540 mm³ [28].

For individual transformers, four ferrite materials were used, designated by the manufacturer as N49, N87, N92, and N97.

Table 1. Parameter values of ferrite materials N49, N87, N92 and N97 [29,30,31,32].

Material	N49	N87	N92	N97
Parameter
µi @ TC = 25 °C	1500	2200	1500	2300
Bs @ TC = 25 °C [mT]	490	490	500	510
Bs @ TC = 100 °C [mT]	400	390	440	410
fmin–fmax [kHz]	300–1000	25–500	25–500	25–500
Pv @ f = 25 kHz, Bm = 200 mT, TC = 100 °C [kW/m3]	-	57	70	45
Pv @ f = 100 kHz, Bm = 200 mT, TC = 100 °C [kW/m3]	-	375	410	300
Pv @ f = 300 kHz, Bm = 100 mT, TC = 100 °C [kW/m3]	330	390	410	340
Pv @ f = 500 kHz, Bm = 50 mT, TC = 100 °C [kW/m3]	80	215	230	205
Pv @ f = 1 MHz, Bm = 50 mT, TC = 100 °C [kW/m3]	475	-	-	-
TCurie [°C]	>240	>210	>280	>230
ρ [Ωm]	17	10	8	8

summarizes the values of the selected parameters of these materials provided by the manufacturer [29,30,31,32]. They include the initial permeability μ_i, saturation induction B_S, recommended operating frequency range f_min–f_max, losses per unit of volume P_V at selected frequency values f, flux density amplitudes B_m, core temperatures T_C, the Curie temperature T_Curie and resistivity ρ.

As can be seen, the µ_i values for individual materials differ by up to 50%. B_S values at 25 °C fall within the range of 500 ± 10 mT, while at 100 °C, this range is wider: 415 ± 25 mT. The recommended operating frequency range is from 25 to 500 kHz for the N87, N92, and N97 materials, while for the N49 material, the optimal operating frequency range is from 300 kHz to 1 MHz. The N49 material has the lowest loss. The T_Curie temperature for each of the materials considered exceeds 210 °C. The resistivity ranges from 8 to 17 Ωm.

The transformer windings were made on a 1.6 mm-thick FR-4 laminate substrate covered with 35 µm copper foil. Two types of oval-shaped windings were used. Windings designated W_Z1 contain 6 turns in a 0.5 mm wide track, while windings designated W_Z2 contain 13 turns in a 0.15 mm wide track. The distance between adjacent turns in each of the considered windings is 0.2 mm. The ends of each winding are led out to solder pads, to which wires connecting the planar transformer windings to the measuring system are soldered. A view of the manufactured boards with windings is shown in Figure 1.

Using different printed circuit boards for the primary and secondary windings and different core materials, planar transformers were constructed for testing. Six transformers were constructed, designated TR1-TR6. Their design details are provided in

Table 2. Design data of the tested planar transformers.

Transformer	Core Material	Primary Winding	Secondary Winding
TR1	N49	WZ2	WZ2
TR2	N49	WZ2	WZ1
TR3	N49	WZ1	WZ2
TR4	N87	WZ2	WZ2
TR5	N92	WZ2	WZ2
TR6	N97	WZ2	WZ2

, and a view of the assembled transformer is shown in Figure 2. The transformer windings were connected to the measuring system, using 2.5 mm² copper wires. These wires were soldered to the winding leads using standard solder.

As you can see, transformers TR1, TR4, TR5, and TR6 have a turns ratio of one. Transformer TR2 steps down the voltage, and transformer TR3 steps up the voltage. Three transformers contain an N49 core (TR1–TR3), while the remaining transformers contain cores made of other materials.

Based on data provided by the manufacturer, the values of the parameters appearing in the Steinmetz model were determined for each of the ferromagnetic materials used, approximating the catalog dependencies P_V(B_m) and P_V(f), using the Steinmetz formula. The values of these parameters are summarized in

Table 3. The values of the Steinmetz model parameters for the considered ferrite materials.

Material	PV0 [kW/m3sα/Tβ]	α	β
N49	0.096	1.115	2.5448
N87	0.0181	1.23	2.33
N92	1.02	0.944	2.28
N97	0.0308	1.176	2.276

.

Each tested transformer contains four 1.6 mm-thick PCBs. The total thickness of the planar transformer laminate stack is therefore 6.4 mm. The boards outside this stack contain the primary and secondary windings, while the boards inside the stack contain no copper layer. This results in weaker thermal coupling between the primary and secondary windings than between each winding and the core.

Using the model parameter estimation method described in Section 3 and the measured temperatures of the components of the tested transformers given in [7], the values of thermal resistances and mutual thermal resistances occurring in the planar transformer model for the considered components were determined. These values are summarized in

Table 4. The values of the thermal model parameters for the tested transformers.

Transformer	RthW1 [K/W]	RthW2 [K/W]	RthC [K/W]	RthW1W2 [K/W]	RthW1C [K/W]	RthW2C [K/W]
TR1	17.1	21.05	18.1	14.1	15.8	15.9
TR2	19.2	21.1	24	14.2	15.9	15.9
TR3	21.1	18.9	24.1	13.9	15.1	15.2
TR4	25.5	25.6	23.9	19.9	22.1	22.2
TR5	18.1	18.9	18.2	11.1	12.9	12.9
TR6	16.5	17.5	18.1	11.5	12.8	12.9

.

During the measurements, the ambient temperature was 21 °C. The power dissipated in the heat source component was selected so that the steady-state temperature of this component was between 100 and 120 °C. For example, for transformers with a W_Z1 winding with fewer turns and wider paths, the measured value of power dissipated in this winding was approximately 5 W, while for transformers with a W_Z2 winding, it was only 3 W.

It is worth noting that the secondary winding has a higher thermal resistance than the primary winding. The highest thermal resistance values were obtained for the transformer with an N87 core, up to 30% higher than for the other cores. Mutual thermal resistances are always lower (up to 30%) than the thermal resistances of the individual transformer components. The thermal resistance of the core is similar to the thermal resistance of the windings.

5. Measurement and Calculation Results

The thermal properties of the tested transformers were measured and simulated, operating at different supply voltage amplitudes V_m and frequencies f. In each case, a load resistor R_L was connected to the secondary winding. Various values of this resistance were considered.

Formulas (1)–(3) were used to calculate the temperature values of individual components. The values of thermal resistances and mutual thermal resistances are given in Table 4. The average values of power dissipated in individual windings were determined based on the measured values of the current amplitude of winding, I_W1 or I_W2, and their resistance, R_W1 or R_W2. These resistance values were measured when individual windings were supplied from a DC voltage source. The resistance of the W_Z2 windings at a temperature of 25 °C is 3.85 Ω, and for the W_Z1 windings it is 0.52 Ω. The temperature coefficient of change in the winding resistance, equal to 4.45 × 10⁻³ °C⁻¹ (estimated using data given in [33]), was also taken into account. In turn, the average value of power dissipated in the core is described by the formula

$$p_C=V_e\cdot P_{V0}\cdot f^{\alpha -\beta }\cdot {\left({\displaystyle \frac{V_{m1}}{2\cdot \pi \cdot z_1\cdot S_{Fe}}}\right)}^{\beta }\cdot \left(1+d\cdot {\left(T_C-T_m\right)}^2\right)$$

(8)

where V_m1 is the voltage amplitude on the primary winding, d is the squared temperature coefficient of dissipation, and T_m is the temperature at which the core dissipation reaches a minimum. The values of these parameters were determined based on the P_V(T_C) characteristics provided by the manufacturer.

The form of Equation (8) results from the modified Steinmetz model for ferrite cores given in [3]. This equation includes an empirical factor describing the effect of temperature T_C on the ferrite core loss. The flux density amplitude B_m is expressed in terms of the amplitude of the voltage V_m1 appearing on the transformer’s primary winding, the frequency f, and the transformer’s design parameters, in accordance with the relationship from [3].

During the measurements, the transformer’s primary winding was supplied with a sinusoidal voltage from the output of an AE Techron 7228 power amplifier [34] via a 4.7 Ω resistor. This amplifier was driven by a sinusoidal signal generator. The voltage waveforms in the system were measured using a Gwinstek GDS-2104A oscilloscope [35]. Optex PT-3S pyrometers [36] were used to measure the windings and core temperatures.

Figure 3 presents the measured (solid bars) and calculated (empty bars) temperature values of the components of the tested transformers when their primary windings are powered from a DC source. In this figure, the primary winding temperature measurements are marked in blue, the secondary winding temperature in black, and the core temperature in red.

In the operating mode under consideration, the only heat source is the primary winding, which is subject to self-heating. The supply current values for individual transformers were selected to ensure that the primary winding temperature exceeded 100 °C. Under these operating conditions, the primary winding resistance increased by over 35% compared to the ambient temperature of T_a = 20 °C. Figure 3 shows that for each of the tested transformers, the highest temperature was achieved for the primary winding (T_W1) and the lowest for the secondary winding (T_W2). The differences between them reach up to 50 °C. The transformer cores have an intermediate temperature between the winding temperatures. Deviations between the measurement and calculation results do not exceed 1.5 °C. The increase in the core and secondary winding temperature above the ambient temperature is the result of mutual thermal coupling between the transformer components. The highest temperature values for all the components were obtained for the transformer TR3. This is the result of the fact that the primary winding of this transformer dissipates more power than the other transformers.

The following research results concern the classic operation of transformers. Their primary winding is supplied by a sinusoidal voltage source, with amplitude V_m and frequency f. The secondary winding is loaded with a resistor with resistance R_L. In these figures, the points represent the measurement results, and the lines represent the calculation results.

Figure 4 shows the effect of the supply voltage frequency on the temperatures of the TR1 transformer components (Figure 4a) and the power dissipated in the individual transformer components (Figure 4b). The amplitude of this voltage is V_m = 30 V, and the resistance is R_L = 15 Ω.

Figure 4a shows that the temperature dependencies of all the transformer components on the frequency are monotonically decreasing functions. The highest values are achieved by temperature T_W2, and the lowest by temperature T_W1. The differences between them are greatest at the lowest frequency values, reaching 12 °C. Within the considered frequency range, the transformer component temperatures decrease from 130 °C to only 45 °C. This is the result of an increase in the primary winding impedance modulus with increasing frequency, which causes a decrease in the winding current.

Figure 4b shows that the power in the windings decreases with increasing frequency. For the operating conditions considered, the power dissipated in the core p_C is much lower than the power in each of the windings. Despite this, the core temperature T_C is close to the temperature T_W1, due to mutual thermal coupling between the core and the windings.

Figure 5a shows the effect of the load resistance of the TR1 transformer on the temperatures of its components, and Figure 5b shows the effect on the power dissipated in these components. The tests were carried out at a frequency f = 100 kHz and a supply voltage amplitude V_m = 30 V.

Figure 5a clearly shows that the temperatures of all transformer components decrease with increasing resistance R_L. This is related to the decreasing dependence of the power dissipated in the windings on the resistance R_L, which is visible in Figure 5b. Under the considered transformer operating conditions, the temperature T_W2 decreases from almost 120 °C to only 27 °C. The temperature values of the transformer components are determined by the power dissipated in the windings. For resistance R_L < 30 Ω, the power dissipated in the secondary winding dominates, while for R_L > 30 Ω, the power in the primary winding dominates. This means that for R_L > 30 Ω, the magnetizing current is a significant component of the primary winding current.

Figure 6 illustrates the effect of the frequency on the temperature of the components of the TR2 transformer (Figure 6a) and on the power dissipated in the components of this transformer (Figure 6b). During the measurements, the resistance R_L = 15 Ω and the supply voltage amplitude V_m = 45 V.

The dependencies visible in Figure 6a are strongly decreasing functions for f < 70 kHz and with a further increase in frequency, they have a practically constant course. Figure 6b shows that the power p_W2 is practically constant over the entire frequency range. The dominant component of the power dissipated in the transformer is p_W1, which is almost constant for f > 70 kHz. The low resistance of the secondary winding causes the power in the primary winding to be higher than in the secondary winding. It is worth noting that for f > 70 kHz, the temperature distribution in the transformer is almost uniform.

Figure 7 illustrates the effect of resistance R_L on the temperatures of the windings and core of the TR2 transformer (Figure 7a) and the power dissipated in its components (Figure 7b). The tests were carried out at f = 24 kHz and V_m = 45 V.

Of course, the planar transformers are typically used in a high-frequency range. On the other hand, it was shown that at a fixed value of the amplitude of the voltage on the primary winding, the power dissipated in the core increases with a decrease in frequency. In order to illustrate the influence of load resistance on the temperature of transformer components when the power of high value is dissipated in the core, the lower frequency value was selected in the example presented in Figure 7.

In Figure 7a, changes in temperature T_W1, T_W2, and T_C, due to changes in R_L, are visible for R_L < 20 Ω. For R_L > 20 Ω, the considered temperatures practically do not depend on the value of this resistance. The highest values are reached by temperature T_C, and the lowest by temperature T_W2. The differences between them reach up to 10 °C. Figure 7b shows that the power p_W2 decreases quickly with increasing R_L, while the values of p_W1 and p_C are almost constant for R_L > 20 Ω. Despite the fact that the power p_W1 is greater than p_C, the temperature T_C is higher than T_W1 due to the fact that the thermal resistance of the core is greater than the thermal resistance of the primary winding.

Figure 8 presents the dependence of the temperatures of the TR3 transformer components (Figure 8a) and the power dissipated in these components (Figure 8b) on the frequency. The tests were carried out at a resistance of R_L = 15 Ω and a supply voltage amplitude of V_m = 27.5 V.

Figure 8a shows that all considered temperatures are decreasing functions of frequency. The highest values were obtained for the primary winding temperature, and the lowest for the secondary winding. The differences between them are greatest at the lowest frequency values, reaching as much as 15 °C. Figure 8b shows that the power dissipated in the secondary winding is up to twice that in the primary winding, but due to the differences in the thermal resistance of both windings and the strong thermal coupling between them, the highest temperature is observed in the component that does not dissipate the highest power. The power dissipated in each transformer component decreases monotonically as a function of frequency.

Figure 9 illustrates the effect of resistance R_L on the temperatures of the windings and core of transformer TR3 (Figure 9a) and the power dissipated in its components (Figure 9b). The tests were carried out at f = 100 kHz and V_m = 27.5 V.

The decreasing temperature dependencies of each component on the resistance RL were obtained. Under the operating conditions considered, the highest power value was obtained for the secondary winding. The power dissipated in the core is much lower than in either winding. In Figure 9a, very small temperature differences between the windings and the core can be observed. The differences between these temperatures do not exceed 5 °C.

Figure 10 presents the dependence of the temperatures of the core and windings of the TR4 transformer (Figure 10a) and the power in these windings and core (Figure 10b) on the frequency. The tests were carried out at a resistance of R_L = 15 Ω and a supply voltage amplitude of V_m = 25 V.

An almost linearly decreasing dependence of each of the considered temperatures on the frequency was obtained. The highest temperature value was obtained for the primary winding, and the lowest for the core. The maximum difference between them reaches 12 °C. Analyzing the waveforms shown in Figure 10a,b, it can be concluded that the cooling efficiency of the primary winding is significantly worse than that of the secondary winding. It is worth noting that the power dissipated in the windings of transformer TR4 is up to two times lower than for transformer TR3 operating under identical supply and load conditions.

Figure 11 illustrates the effect of resistance R_L on the temperatures of the windings and core of the TR4 transformer (Figure 11a) and the power dissipated in its components (Figure 11b). The tests were carried out at f = 100 kHz and V_m = 25 V.

Figure 11a shows that practically identical values were obtained for all components of the TR4 transformer over a wide range of frequency changes. The power dissipated in both windings is also very close to each other. They are significantly greater than the power dissipated in the core over the entire range of the R_L resistance changes. The relationships between the power and temperatures of the windings and the core demonstrate that the heat dissipation efficiency of both windings is the same.

Figure 12 presents the dependence of the temperatures of the core and windings of the TR5 transformer (Figure 12a) and the power in these windings and core (Figure 12b) on the frequency. The tests were carried out at a resistance of R_L = 15 Ω and a supply voltage amplitude of V_m = 30 V.

For the transformer under consideration, Figure 12a shows the strongly decreasing dependencies of the temperatures of each component of the transformer as a function of frequency. It is worth noting that in Figure 12b, at a frequency f < 30 kHz, the power dissipated in the core p_C reaches almost 40% of the power dissipated in the primary winding. This power component is responsible for the significant temperature increase in all transformer components in the low-frequency range. A significant temperature difference between individual transformer components is visible, reaching up to 15 °C. The highest temperature is observed in the secondary winding, where the highest power is dissipated.

Figure 13 illustrates the effect of the resistance R_L on the temperatures of the windings and core of the TR5 transformer (Figure 13a) and the power dissipated in its components (Figure 13b). The tests were carried out at f = 100 kHz and V_m = 30 V.

In Figure 13a, it can be seen that the temperature of the secondary winding is higher than that of the primary winding, and the difference between them reaches 10 °C at resistance R_L = 12 Ω. The core temperature is only slightly lower than the T_W1 temperature. Figure 13b shows that the highest power is dissipated in the secondary winding. The powers in the windings are decreasing functions of resistance R_L, while the power dissipated in the core is an increasing function of this resistance. At R_L > 80 Ω, the powers dissipated in the core and in each winding equalize.

Figure 14 presents the dependence of the temperatures of the core and windings of the TR6 transformer (Figure 14a) and the power in these windings and core (Figure 14b) on the frequency. The tests were carried out at a resistance of R_L = 15 Ω and a supply voltage amplitude of V_m = 30 V.

The dependencies of the temperatures of individual components of the TR6 transformer on the frequency (Figure 14a) and the dependencies of the power lost in these components on the frequency (Figure 14b) have similar shapes to the dependencies presented in Figure 12 for the TR5 transformer. Figure 14b shows that at the lowest of the considered frequency values, the powers lost in both windings have similar values.

Figure 15 illustrates the effect of resistance R_L on the temperatures of the windings and core of the TR6 transformer (Figure 15a) and the power dissipated in its components (Figure 15b). The tests were carried out at f = 100 kHz and V_m = 30 V.

Figure 15a shows a weak differentiation between the temperatures of the transformer components, and Figure 15b shows that the powers dissipated in both windings have similar values. This relationship between the powers indicates that the currents in both windings have similar values.

6. Discussion

The presented calculation and measurement results indicate that the winding parameters and core material have a significant impact on the thermal properties of the transformers under consideration. The highest thermal resistance values were obtained for the transformer with a core made of the N49 or N87 materials and windings containing the same number of turns. Despite the identical mechanical construction of all tested transformers, significant differences are visible in the mutual thermal resistance values between the transformer components. The mutual thermal resistances between the core and each winding are the same, while the mutual thermal resistance between the windings is lower than the thermal resistance between the core and the windings.

The power dissipated in the transformer windings is a decreasing function of frequency and load resistance. In turn, the power dissipated in the core is a decreasing function of frequency and an increasing function of load resistance.

For transformers with an N49 core and a primary winding with a larger number of turns, the power dissipated in the core is significantly less than the power dissipated in each winding. In contrast, for a transformer with the same core and a primary winding with fewer turns, the power dissipated in the core is comparable to the power dissipated in the primary winding. Only when the transformer operates at low frequencies and with high load resistances is the highest temperature observed in the transformer core.

For each transformer, good agreement was achieved between the calculation and measurement results for all operating conditions considered.

Table 5. The values of the root mean square error of the temperature modeling of the components of the tested transformers presented in the figures in Section 5.

Transformer	Figure	ΔTW1 [°C]	ΔTW2 [°C]	ΔTC [°C]
TR1	Figure 4a	0.774	1.46	1.002
TR1	Figure 5a	1.072	0.878	0.848
TR2	Figure 6a	1.145	1.495	1.031
TR2	Figure 7a	1.410	2.031	1.126
TR3	Figure 8a	0.941	0.689	0.512
TR3	Figure 9a	1.196	1.570	2.095
TR4	Figure 10a	0.829	1.612	0.977
TR4	Figure 11a	1.085	1.284	0.756
TR5	Figure 12a	1.691	1.359	1.325
TR5	Figure 13a	0.931	0.955	0.968
TR6	Figure 14a	0.844	1.259	1.170
TR6	Figure 15a	1.283	1.061	1.102

provides the root mean square error values for the temperature modeling of each component of the tested transformers. In most cases, the error considered slightly exceeds 1 °C.

Table 6. The maximum values of temperature and dissipated power in the components of tested transformers presented in the figures in Section 5.

Transformer	Figure	TW1max [°C]	TW2max [°C]	TCmax [°C]	pW1max [W]	pW2max [W]	pCmax [W]
TR1	Figure 4	126	125	125	3.93	2.98	0.34
TR1	Figure 5	94	111	102	2.18	3.12	0.02
TR2	Figure 6	125	89	118	2.91	0.46	0.89
TR2	Figure 7	101	92	103	3.27	2.1	0.64
TR3	Figure 8	132	115	125	2.31	4.9	0.5
TR3	Figure 9	115	127	120	1.7	3.94	0.03
TR4	Figure 10	103	91	90.5	1.94	2.5	0.26
TR4	Figure 11	119	121	111	2.16	2.41	0.02
TR5	Figure 12	134	127	123	3.66	4.18	1.67
TR5	Figure 13	108	118	104	2.66	3.26	0.07
TR6	Figure 14	120	120	115	3.48	3.72	1
TR6	Figure 15	113	123	111	3.53	4.11	0.07

illustrates the non-uniformity of the distribution of temperature and power dissipated in the components of the tested transformers. In this table, the maximum values of the considered temperature and power dissipated values obtained using data presented in Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11, Figure 12, Figure 13, Figure 14 and Figure 15 are collected.

As can be seen, the temperature differences between components in the tested transformers can reach up to 36 °C, but there are also cases in which this difference is as little as 1 °C. Typically, during testing, the power dissipated in the windings was significantly greater than the power dissipated in the core. Nevertheless, due to mutual thermal coupling, the difference between the temperatures of the transformer components was not as large. Only in a few cases did the power dissipated in the core reach half of the value of the power dissipated in the winding. The presented summary demonstrates the validity of using the described transformer thermal model that takes into account the temperature differences in its components, the differences in their thermal resistances, and the existence of mutual thermal coupling.

7. Conclusions

This paper considers the problem of modeling the thermal properties of planar transformers. A lumped DC model of such a transformer is proposed, which allows for the determination of the temperature of each winding and core during power dissipation in individual transformer components. This model accounts for both the self-heating phenomenon in each component and the mutual thermal coupling between each pair of these components.

The conducted tests demonstrate that the developed model provides good agreement between the calculation and measurement results over a wide range of frequencies and load resistances. The model’s accuracy was verified for six transformers containing cores made of various ferromagnetic materials and windings with varying numbers of turns and track widths.

Calculations and measurements have shown that, at a fixed supply voltage amplitude, the power dissipated in each winding decreases as a function of frequency and load resistance. The core losses, on the other hand, are greatest at low operating frequencies and high load resistances.

In the tested transformers, strong thermal coupling occurred between the transformer components. As a result of this coupling, the temperatures of individual transformer components differ significantly, with the maximum recorded temperature differences reaching 15 °C, which corresponds to approximately a 15% temperature difference above the ambient temperature. These differences are greatest at low frequencies and load resistances.

The presented research results may be useful for designers of planar transformers and power electronics systems incorporating such transformers. Further research will consider the effect of thermal inertia on the thermal properties of such transformers.

Further research will develop a dynamic compact thermal model of a planar transformer. This model will account for heat transfer delays between transformer components. A compact electrothermal model of a planar transformer will also be developed, which describes the interactions of electrical, magnetic, and thermal phenomena, taking into account the time delays between these phenomena.