Selected Problems in Measuring Magnetic Hysteresis Loops and Determining Losses per Unit of Volume for Ferrite and Nanocrystalline Cores

Abstract

This paper analyses a method for measuring the magnetization curves of ferrite and nanocrystalline materials and determining the loss of the ferromagnetic material used in inductor and transformer cores. Analogue and digital methods for determining B(H) characteristics and the corresponding measurement setup are described. Limitations of the analogue method are highlighted, and example magnetization characteristics obtained using both methods over a wide range of flux density amplitude and frequency changes are shown and discussed. Using the magnetization curves measured using both methods, the dependence of the loss of the tested cores on the frequency and amplitude of flux density was determined. It was demonstrated that the classical analogue hysteresis method produces distorted B(H) waveforms and significantly overestimated values of loss for the tested materials compared to the manufacturer’s data. An analysis was also conducted of the suitability of the classical Steinmetz model for modelling the dependence of the loss of the considered materials on the frequency and amplitude of magnetic flux density. It was demonstrated that the coefficients in this formula should be dependent on the above-mentioned quantities to achieve satisfactory accuracy between the model and the manufacturer’s data. Using a digital method for measuring the hysteresis loop of a ferromagnetic core and measuring its temperature using an optical method, the thermal resistance values of the tested cores were determined. Using the digital measurement method and the modified Steinmetz formula, a significant improvement in the accuracy of determining the losses of ferrite and nanocrystalline cores was achieved.

1. Introduction

Ferromagnetic cores are an important component of magnetic elements such as inductors and transformers [1,2,3]. They are commonly used in switch-mode power conversion systems, where they operate as energy storage elements or provide galvanic isolation between the input and output of the system [3,4,5,6]. The properties of these cores limit the maximum operating frequency of these systems and their energy efficiency.

During operation of the above-mentioned elements, their cores are periodically remagnetized, causing energy loss. The macroscopic effect of these losses is an increase in the core temperature during operation [7,8,9,10,11,12,13]. According to the classical Steinmetz formula [1,14,15,16], core loss increases with an increase in the frequency f and amplitude of magnetic flux density B_m.

One of the commonly used methods for determining the loss per volume P_V (lossiness) of ferromagnetic cores is to measure the area of their magnetization curve B(H) [11,17,18,19], representing the dependence of the flux density B on magnetic force H. During operation of the above-mentioned elements, their cores are periodically remagnetized, causing energy loss. The macroscopic effect of these losses is an increase in the core temperature during operation [7,8,9,10,11,12,13]. According to the formula given, among others, in [11,16,17,20], the loss are the product of the frequency and the integral of the magnetic flux density B with respect to the magnetic force H, calculated along the closed magnetization curve. This approach to determining the loss is called the hysteresis method. As shown in [21], for ferrite cores, the loss values obtained using the hysteresis method are overestimated compared to the calorimetric method proposed in the cited paper. The discrepancies between the obtained results can be several-fold.

Unfortunately, the method described in [21] is applicable only to ferromagnetic cores made of materials with low resistivity, which are not brittle and whose surface is not coated with an insulating substance. These limitations make it impossible to use this method for loss measurements of nanocrystalline materials, Ni-Zn ferrites, and powdered iron materials or its alloys.

The hysteresis method [17] is simple to implement in the low frequency range, but causes some problems in the high frequency range [22,23,24]. In this range, significant deformations of the magnetization curve B(H) can be observed. Therefore, a typical simplification is to measure the core loss for selected values of frequency f and the amplitude of magnetic flux density B_m, determine the parameters occurring in the Steinmetz model based on the obtained results, and extrapolate the loss values over a wide range of amplitudes of magnetic flux density and frequency values. For this reason, the P_V(B_m, f) dependences presented by ferromagnetic core manufacturers are straight lines on a log–log scale [25,26].

Papers [27,28,29,30,31] present various methods and measurement systems for characterizing the properties of magnetic materials based on B(H) hysteresis loop measurements. Fiorillo [27] highlights the problems with the repeatability and accuracy of measurement results. Csizmadia et al. [28] propose a measurement system for recording hysteresis curves of printed iron cores. Fathabad et al. [29] and Charubin [30] describe a computer-controlled, automated system for assessing the magnetic properties of thin, soft magnetic materials and measuring hysteresis loops. Tumański [31] presents modern methods for measuring the magnetic parameters of electrical sheets. In particular, a hysteresis loop measurement system is presented, which includes an RC integrator with an operational amplifier. Orosz et al. [32] describe the method of both the modelling and measurements of iron power loss in the rotating machines.

The papers [20,33,34] describe standard methods for determining the lesses of soft metallic and powder ferromagnetic materials. They are based on the observation that the loss is proportional to the area of the hysteresis loop. These methods utilize sinusoidal excitation with an adjustable amplitude, and the values of the magnetic force H, flux density B, and loss P_V are determined based on the measured values of the current in primary winding, the voltage in secondary winding, and the geometric dimensions of the tested toroidal core. The scope of applicability of the described method is limited to frequencies ranging from 20 Hz to 100 kHz. To determine the loss value, it is necessary to integrate the product of the voltage in secondary winding and the current in primary winding. Neither the cited standard [20] nor the cited papers above specify how to perform this operation. The cited papers do not consider the suitability of the described method for ferrite or nanocrystalline cores. The method of measuring the magnetization curve B(H) is also not considered in the cited standard, meanwhile many technical parameters can be obtained analysing this curve.

In the book [2], a classic RC integrator was used to determine the time course of B(t). The output voltage of this circuit was proportional to the flux density in core B. This method of determining flux density was hereinafter called the analogue method. A characteristic feature of the analogue method is the use of an analogue RC network to integrate the voltage on transformer secondary winding, and the output voltage of this network is used to determine the waveform of flux density B(t). As mentioned in cited papers [2,30,31,33,34], the problem of measuring magnetic hysteresis loops and determining loss per unit of volume of the magnetic core, especially for frequencies higher than 100 kHz, is very difficult and problematic.

In the case of recording the waveform of the voltage on the secondary winding of a transformer, the B(t) course can be determined using numerical integration. This method is called the digital method. A characteristic feature of the digital method is the use of the waveform of the voltage on the secondary winding as the input data for the computer algorithm of numerical integration. The result of such an algorithm is the waveform of flux density B(t).

As mentioned earlier, although there are many papers devoted to methods for measuring the hysteresis loop, despite the standardization of this procedure, many contemporary papers still draw attention to the problem of measuring this curve and determining power loss in ferromagnetic cores [28,29,35]. Both analogue and digital methods are reported in the literature; however, a direct comparison between these methods, as well as a discussion of the advantages and disadvantages of each of them, has not been provided for different types of ferromagnetic materials.

The aim of this paper was to compare the usefulness of the analogue and digital measurement methods to obtain magnetization characteristics of ferromagnetic cores and to analyze the influence of the accuracy of the measurements of B(H) characteristics on the accuracy of determining the loss of ferromagnetic cores made of selected soft magnetic materials. Investigations were performed for two kinds of ferromagnetic cores made of ferrite or nanocrystals. Particular attention should be paid to the fact that this paper analyzes ferrite cores, which are commonly used in power electronic systems, as well as nanocrystalline cores, whose importance is increasing and which are attracting growing interest due to their high magnetic permeability. Unfortunately, the information in the datasheet for these cores is limited and often insufficient for the proper design of inductors or transformers used in the mentioned systems [36,37]. Measurements illustrating the influence of frequency and flux density on the accuracy of the considered measurement methods were performed, and their results are shown and discussed. Based on the data provided by the producers, power loss in both types of cores operating at different values of the flux density amplitude B_m and frequency f were determined and compared with the results of calculations using the classical Steinmetz model. Analytical relationships taking into account the effect of core operating conditions on the Steinmetz model coefficients were proposed and experimentally verified. The accuracy of measurements of power dissipated in the tested ferromagnetic core, performed using hysteresis loops measured with analogue and digital methods and calculated using the modified Steinmetz model, was analyzed using manufacturer data.

The presented results prove that the analogue method can cause large distortions of the measured B(H) characteristics, especially in the high frequency range. Using the area values of such measured characteristics in the calculation of power dissipated in the ferromagnetic core may result in a large overestimation of this power. The use of the digital method allows one to eliminate the above problems.

2. Measurement Method and Setup

The concept of the hysteresis measurement method requires determining the magnetization characteristics B(H) of the tested ferromagnetic cores at different values of the amplitude of magnetic flux density B_m and frequency f. The measurement of this characteristic was performed using the setup shown in Figure 1.

In the considered measurement setup, the tested core (DUT) was contained within a transformer. The transformer’s primary winding was powered by the power amplifier W, which was excited by a sinusoidal voltage generator V_sin. The primary winding current I₁ was limited by the resistor R₁. The waveform of this current was recorded by an oscilloscope (OSC) via a current probe (CP). The transformer’s secondary winding load is represented by an integrating circuit containing resistor R₂ and capacitor C₁. The waveforms of voltage V_C across the capacitor and the voltage V₂ across the secondary winding were also recorded using the oscilloscope. The results recorded by the oscilloscope were transmitted via the USB port to a computer, which performed the appropriate calculations. Components R₂ and C₁ (marked in Figure 1 with red) were used only in the analogue method.

The measurement setup used a Textronix, Beaverton, OR, USA, TCPS 300 current probe (CP) with a cut-off frequency of 100 MHz [38]. A GWINSTEK, New Taipei City, Taiwan, GDS2104A oscilloscope with a 100 MHz bandwidth [39] was used to record the voltage and current waveforms.

The instantaneous value of the magnetic field H was determined using the following formula:

$$H={\displaystyle \frac{z_1\cdot I_1}{l_{Fe}}}$$

(1)

where z₁ is the number of turns on the primary winding of the transformer, I₁ is the current of this winding, and l_Fe is the length of the magnetic path in the core.

In turn, in the analogue method, the value of the magnetic flux density B is described by the following formula [2]:

$$B={\displaystyle \frac{V_C\cdot R_2\cdot C_1}{S_{Fe}\cdot z_2}}$$

(2)

where z₂ is the number of turns on the secondary winding of the transformer, S_Fe is the active cross-sectional area of the core, and V_C, R₂ and C₁ are marked in Figure 1.

The condition for obtaining the correct waveform B(t) based on Formula (2) is that the resistance R₂ is much higher than the impedance modulus of capacitor C₁. This means that the variable component of voltage V_C should be much smaller than voltage V₂ on the secondary winding of the tested transformer. This results in a small amplitude of voltage V_C and the requirement for a high-resolution voltage measurement using an oscilloscope. To eliminate the problem described above, the digital method was used, in which the time waveform of voltage V₂ on the secondary winding of the transformer was measured and the corresponding waveform of magnetic flux density B(t) was calculated in Excel software using the following formula:

$$B\left(t\right)={\displaystyle \frac{1}{S_{Fe}\cdot z_2}}\cdot {\displaystyle \int V_2\left(t\right)dt}$$

(3)

The measurement setup uses an AETECHRON, Elkhart, IN, USA, 7228 (W) power amplifier, a Tektronix TP-300 (CP) current probe, and a GWINSTEK GDS2104A (OSC) oscilloscope. The values of the passive components are R₁ = 47 Ω, C₁ = 3.3 µF.

While measuring the hysteresis loop after achieving a steady state, the temperature T_C was measured using an Optex, Shiga, Japan, PT-3S pyrometer. This measurement was performed only for the ferrite core. For the nanocrystalline core, such a measurement is invalid because the core is placed inside a plastic container, the temperature of which can be significantly lower than the core temperature.

3. Tested Cores

The tests were carried out for toroidal ferrite and nanocrystalline cores. Both the tested cores had similar dimensions (outer diameter: 25 mm, inner diameter: 15 mm, height: 10 mm). A core made of 3F3 ferrite [40] and a nanocrystalline core made of M-070 material [41] were considered. The medium-frequency power material, 3F3, is suitable for use in power and general-purpose transformers at frequencies of 0.2–0.5 MHz. This material belongs to soft ferrites Mn-Zn. In turn, M-070 is a nanocrystalline material made of the alloy Fe_73.5 Cu₁ Nb₃ Si_15.5 B₇. This alloy contains iron, copper, niobium, silicon and boron. It is made of a tape with a thickness of approximately 20 μm. The values of the basic material parameters of the tested cores are summarized in

Table 1. Values of the basic material parameters of the tested cores [40,41].

Parameter	3F3	M-070
μi (TC = 25 °C, f = 10 kHz, Bm = 0.25 mT)	2000	60,000
Bsat (TC = 25 °C) [T]	0.44	1.2
Bsat (TC = 100 °C) [T]	0.37	1.1
PV (TC = 100 °C, f = 100 kHz) [kW/m3]	80 @ Bm = 0.1 T	<735 @ Bm = 0.3 T
PV (TC = 100 °C, f = 400 kHz, Bm = 50 mT) [kW/m3]	150	-
TCurie [oC]	200	600
ρ(f = 0, TC = 25 °C) [Ω × m]	2	0.0115

.

The initial magnetic permittivity μ_i ranges from 2000 for the 3F3 core to 60,000 for the M-070 core. The saturation of magnetic flux density B_sat for the ferrite core is about three times lower than for the nanocrystalline core. P_V for the nanocrystalline core is nine times higher than for the ferrite core. The Curie temperature T_Curie is 200 °C for the 3F3 core and 600 °C for the M-070 core. The resistivity ρ is 200 times higher for the ferrite core than for the nanocrystalline core.

The tested transformers contain windings with a diameter of 1 mm, made of enameled copper wire. Two windings are located on opposite sides of the core. During testing, the transformers were placed horizontally on a table.

4. Modelling Power Loss in Ferromagnetic Cores

According to the classical dependence [17,20], the core loss per unit of volume P_V can be determined based on the hysteresis loop area using the following formula:

$$P_V=f\cdot {\displaystyle \oint BdH}$$

(4)

The P_V(f, B_m) dependence is often approximated using the classical Steinmetz model of the following form [1,9]:

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

(5)

where P_V0, α, and β are model parameters.

The model described by Formula (5) does not provide satisfactory agreement with the measurement results for modern ferromagnetic materials. Therefore, a modified form of this model described by following formula was proposed:

$$P_V=P_{V0}\cdot f^{\alpha }\cdot B_m^{\beta }\cdot \left(1+d\cdot {\left(T_C-T_m\right)}^2\right)$$

(6)

where d and T_m are model parameters, and T_C denotes the core temperature. Formula (6) is a modified version of the classical Steinmetz model, and the values of the parameters α, β, and T_m depend on the frequency f and the amplitude of magnetic flux density B_m.

The dependence of P_V on temperature T_C introduced in Formula (6) is described by a quadratic function. Its form results from the approximation of the P_V(T_C) dependence given in the datasheets of ferrite cores, e.g., [42]. In these dependences, the minimum of the considered function for fixed values of B_m and f occurs at a fixed value of temperature T_m.

The manufacturer’s data presents graphs illustrating the influence of the amplitude of magnetic flux density B_m, frequency f, and temperature T_C on the loss of the materials used in the tested ferromagnetic cores. The P_V(B_m) and P_V(f) graphs are straight lines on a log–log scale. Based on the manufacturer’s data, the values of the parameters of the loss model, described by Formula (5), were determined for the considered cores. When determining the values of these parameters, the coordinates of five points located on these characteristics corresponding to a fixed temperature value, two B_m values at a constant f value and two f values at a constant B_m value were used. The values of these parameters are summarized in

Table 2. Values of the loss model parameters for considered cores.

Parameter	β	α	Pv0 [Wsα/m3/Tβ]
3F3 core	2.568	1.45	2
M-070 core	1.75	1.54	0.05

.

As can be seen in Table 2, the values of α parameter determined for considered cores differ only slightly (less than 10%), whereas the values of β parameter are much more diverse—they differ by as much as 40%. The P_V0 coefficient changes its value up to forty-fold.

Figure 2, Figure 3 and Figure 4 show the dependences of the loss of the tested cores on the frequency f, temperature T_C and amplitude of magnetic flux density B_m, given by the manufacturer [40,41] (points) and as calculated (lines). In these figures, the dashed lines denote the results of the calculations based on Formula (5) and the values of the parameters given in Table 2. The solid lines correspond to the results obtained from Formula (6), which takes into account the empirical dependences of the parameters α, β and T_m on f, and B_m.

Figure 2 shows the dependence of P_V in the 3F3 core on B_m at selected frequency values, f.

As can be seen, the P_V(B_m) dependence in a log–log scale is a straight line. It can be seen that the catalogue data and the calculation results using Formula (5) are close to each other only at f = 200 kHz. At other frequency values, the calculated P_V values differ (up to two-fold) from the values declared by the manufacturer. In contrast, calculations using Formula (6) model the manufacturer’s dependences very well.

The influence of temperature on 3F3 core loss is illustrated in Figure 3 for selected values of f and B_m.

Figure 3 shows that the loss model given by Formula (6) allows for good agreement between the calculation and measurement results for all considered operating conditions. In contrast, the classic Formula (5) does not take into account the influence of temperature. Therefore, the dashed line in this figure indicates the results obtained from Formula (6), with fixed values of all parameters appearing in this formula. It is clear that omitting the influence of f on the model parameter values results in significant discrepancies between the calculation results and the manufacturer’s data. At f = 50 kHz and B_m = 200 mT, these discrepancies reach up to 100%.

The dependences of parameters α and β used in the calculations shown in Figure 2 and Figure 3 are as follows:

$$\alpha =\left\{\begin{array}{l}{k}_1\cdot f^2+k_2\cdot f+k_3\hspace{1em}if\hspace{1em}f<75\mathrm{kHz}\\ k_1\cdot f^2+k_4\cdot f+k_5\hspace{1em}if\hspace{1em}f\ge 75\mathrm{kHz}\end{array}\right.$$

(7)

$$T_m=\mathrm{MAX}\left(k_6\cdot f^2+k_7\cdot f+k_8,80°\mathrm{C}\right)$$

(8)

Formulas (7) and (8) were obtained by approximating the dependences α(f) and T_m(f) using a second-degree polynomial. The coordinates of individual points on the approximated characteristics were obtained by selecting the values of the parameters α and T_m that ensure the best fit of the calculated and measured P_V(B_m) and P_V(T_C) graphs for individual values of frequency f.

The MAX(x,y) function in Formula (8) takes the value of the larger of its two arguments. The values of the parameters in Formulas (6)–(8) for the 3F3 core are summarized in

Table 3. Values of the loss model parameters for the 3F3 core, described by Formulas (6)–(8).

Parameter	β	d [K−2]	Pv0 [Wsα/m3/Tβ]	k1 [s2]
Value	2.568	1.37 × 10−4	2	8.4 × 10−14
Parameter	k2 [s]	k3	k4 [s]	k5
Value	3.485 × 10−6	1.557	6.6 × 10−8	1.428
Parameter	k6 [°C·s2]	k7 [°C·s]	k8 [°C]
Value	−7.4 × 10−10	2.6 × 10−4	90

.

In the case of the M-070 core, no impact of temperature on the loss value was observed; however, obtaining consistency of the calculation results with the manufacturer’s data required taking into account the dependence of α and β coefficients on B_m. These dependences take the following forms:

$$\alpha =k_{11}\cdot B_m^2+k_{12}\cdot B_m+k_{13}$$

(9)

$$\beta =k_{14}\cdot B_m^2+k_{15}\cdot B_m+k_{16}$$

(10)

where k₁₁, k₁₂, k₁₃, k₁₄, k₁₅ and k₁₆ are model parameters. The values of coefficients occurring in Formulas (5), (9) and (10) for core M-070 are summarized in

Table 4. Values of the loss model parameters for the M-070 core described by Formulas (5), (9) and (10).

Parameter	Pv0 [Wsα/m3/Tβ]	k11 [T−2]	k12 [T−1]	k13
Value	0.047	53.5	−16.7	3.17
Parameter	k14 [T−2]	k15 [T−1]	k16
Value	1.926	−0.2	1.577

.

Formulas (9) and (10) were obtained by approximating the dependences α(B_m) and β(B_m) using a second-degree polynomial. The coordinates of individual points on the approximated characteristics were obtained by selecting the values of parameters α and β that ensured the best fit of the calculated and measured P_V(f) graphs for each frequency, B_m.

Figure 4 presents the calculated and catalogue dependence of P_V on B_m for a nanocrystalline core.

Figure 4 shows that the Steinmetz model with fixed values of α and β parameters enables accurate determination of the loss value in a wide range of frequency changes only at lower values of B_m.

For the highest of the considered B_m values, the slope of the P_V(f) dependence increases significantly, and the literature model does not take this phenomenon into account. In turn, the proposed modification of the classical model allows for obtaining good agreement between the calculation results and the manufacturer’s data in all considered operating conditions.

5. Measurements of B(H) Characteristics

This section analyses the problem of determining the hysteresis loop B(H) of selected ferromagnetic cores using the measurement setup shown in Figure 1. The results presented in this section were obtained for two values of resistance R₂ equal to 47 Ω and 10 kΩ, respectively, using the analogue and digital measurement methods.

Magnetization curves were measured for transformers containing cores made from the materials described in Section 3. The measurements were performed over a wide range of f and B_m values. The range of B_m values up to the B_sat values for the tested cores was considered. The frequencies at which the measurements were performed are in the range from 50 Hz to 100 kHz, which corresponds to almost the entire range expected by the standard [20]. The results are shown in Section 5.1 for the 3F3 core and in Section 5.2 for the M-070 core. The results obtained using the analogue and digital methods were compared in each case.

5.1. 3F3 Core

Figure 5 and Figure 6 present the B(H) curves measured at B_m = 0.2 T (Figure 5) and B_m = 0.3 T (Figure 6). During the measurements, the resistance of resistor R₂ was 47 Ω, and 12 turns were wound on both windings of the transformer.

As can be seen, an increase in the frequency value, at a fixed value of B_m, results in an increase in the size of the hysteresis loop, and, in particular in, its area. However, it is worth noting in Figure 5 that the changes in the hysteresis loop area are smaller than the changes in frequency. For example, a change in frequency from 1 kHz to 10 kHz only caused a doubling of the hysteresis loop area. In turn, an increase in frequency from 10 kHz to 25 kHz resulted in more than a doubling of the hysteresis loop area.

Comparing the B(H) curves shown in Figure 5a,b, significant differences in the shapes of these curves can be seen. The values of B_m obtained using both methods also differ significantly. The analogue method achieved a B_m of 0.2 T, whereas the digital method achieved a B_m of 0.32 T. This means a discrepancy between the measurement results of up to 60%.

In turn, Figure 6 shows that the shape of the hysteresis loop and its dimensions significantly depend on the used measurement method. Differences were observed for the highest of the considered frequency values. The hysteresis loops presented in Figure 6a deviate from the shape of such relationships presented in the literature. This is particularly true for the curve corresponding to the frequency f = 25 kHz.

In turn, B(H) curves visible in Figure 6b have a shape similar to that presented in the manufacturer’s materials. In particular, the saturation range of the magnetization curve at B_m = 0.44 T is clearly visible.

The discrepancies between the magnetization curve measurement results obtained using both the considered methods, presented in Figure 5 and Figure 6, are the result of using a low-resistance resistor R₂ in the measuring setup. For the 3.3 μF capacitor C₁, the integrator’s time constant was only 155.1 μs. This means that this setup can correctly integrate the voltage V_C and ensure the validity of Formula (2) only for frequencies significantly higher than 1 kHz. At the same time, the voltage amplitude V_C should be much smaller than the voltage amplitude V₂, which limits the accuracy of measuring these voltages. Therefore, in further studies, a circuit containing a 10 kΩ resistor R₂ was used. In this case, with the same capacitance C₁ value as in the previous examples, a time constant of 33 ms was obtained. This means that the circuit will correctly integrate signals with frequencies significantly higher than 5 Hz.

Figure 7 shows the magnetization curves measured by both methods at selected frequency values.

As can be seen, despite the proper selection of the RC time constant in the measurement setup, there are clear differences between the measurement results obtained using both measurement methods. In particular, the use of the analogue measurement method causes significant distortions of the B(H) curve. Such distortions are not visible in Figure 7b, which shows the results obtained using the digital method. The differences between the results obtained for both methods increase with increasing frequency. In addition to the differences in shape, it is worth noting the differences in the areas of the determined hysteresis loops and the values of the determined amplitude of magnetic flux density.

The changes in the shape of the hysteresis loop with changing frequency, visible in Figure 5, Figure 6 and Figure 7, are the result of increased energy losses with increasing frequency. They are caused by more frequent remagnetization of the core and the associated friction of the Bloch walls of adjacent ferromagnetic domains. In turn, an increase in the flux density amplitude results in an increase in the number of ferromagnetic domains that require remagnetization during each period of the excitation signal.

Figure 8, Figure 9, Figure 10, Figure 11 and Figure 12 illustrate the influence of the amplitude of magnetic flux density on the magnetization curve for selected frequency values. These are 1 kHz (Figure 8), 10 kHz (Figure 9), 25 kHz (Figure 10), 50 kHz (Figure 11) and 100 kHz (Figure 12), respectively.

Observing the B(H) characteristics shown in Figure 8, one can see a strong influence of B_m on the hysteresis loop area. When this amplitude changes from 0.15 T to 0.46 T, the area of the B(H) curve increases as much as 12 times. It is worth noting that the results obtained using both methods are similar, but the hysteresis loop area obtained using the digital method is slightly larger than that obtained using the analogue method.

Similarly, the sizes of the magnetization curves shown in Figure 9 change, with the coercive field value for the B(H) curves obtained at f = 10 kHz being smaller when using the digital method. In Figure 8a and Figure 9a, small loops are visible at high values of H. This is the result of the low resolution of measurements performed using the analogue method.

B(H) curves determined using the classical method at frequencies of 25 kHz or higher (Figure 10a, Figure 11a and Figure 12a) significantly deviate from their typical shape. This defect is not visible in the B(H) curves determined using the digital method (Figure 10b, Figure 11b and Figure 12b).

Particularly large differences in the B(H) curves determined using both methods are visible for the highest values of B_m. For B(H) curves determined using the classical method and corresponding to the above-mentioned frequency values, at the highest of the considered B_m values, a decrease in the magnetic flux density value B was observed with an increase in the magnetic force H. Under these conditions, the magnetization curve surface increases with increasing B_m much more than observed at lower frequency values. It is also worth noting that an increase in frequency causes a decrease in the amplitude of magnetic flux density, at which saturation of the characteristic is observed.

It is worth paying attention to the B(H) curves shown in Figure 11a and Figure 12a. Both their shape (close to an ellipse) and B_m values (higher than the saturation of magnetic flux density) indicate an unphysical course of these dependences. On the other hand, the digital method allows for obtaining typical B(H) curves for each of the considered frequency values.

Figure 8, Figure 9, Figure 10, Figure 11 and Figure 12 show that only at the highest saturation flux density values does the tested core enter the saturation range, and its operating point forms a large hysteresis loop. At lower B_m values, one of the small hysteresis loops is observed, the size of which decreases with decreasing B_m values. Each of them exhibits central symmetry with respect to the origin.

5.2. M-070 Core

Figure 13, Figure 14, Figure 15, Figure 16, Figure 17 and Figure 18 present the results of measurements carried out for a nanocrystalline core made of M-070 material. All measurements, the results of which are presented in this section, were carried out with the resistance R₂ = 10 kΩ, and 18 turns were wound on each transformer winding.

Figure 13 illustrates the influence of frequency on the magnetization curves of the considered core at B_m = 1.2 T.

It is worth noting that the presented hysteresis loops are much narrower than the B(H) curves for ferrite cores presented in the previous section. The M-070 core enters saturation at H values of just a few A/m. Increasing the frequency causes an increase in the width of these loops. The typical shape of the B(H) curve was obtained only for a frequency of f = 1 kHz. For the considered case, the results obtained using the classical and digital methods were very similar.

Figure 14, Figure 15, Figure 16, Figure 17 and Figure 18 illustrate the influence of the amplitude of magnetic flux density on the course of the magnetization curves measured at different frequencies. Figure 14 concerns f = 50 kHz, Figure 15 concerns f = 25 kHz, Figure 16 concerns f = 10 kHz, Figure 17 concerns f = 1 kHz, and Figure 18 concerns f = 50 Hz. For most of the performed measurements, almost identical results were obtained using the classical and digital methods. Visible differences occurred only for the lowest of the considered frequencies. Therefore, Figure 14, Figure 15, Figure 16, Figure 17 and Figure 18 present only the measurement results obtained using the digital method.

In Figure 14, Figure 15, Figure 16, Figure 17 and Figure 18, at higher frequencies than those considered, it is not possible to obtain an amplitude of magnetic flux density corresponding to the core saturation in the applied measurement setup. This is because the maximum amplitude of the output voltage of the power amplifier that supplies the primary winding of the tested transformer is limited. Increasing frequency means that the same voltage value in this winding can be obtained with increasingly lower core amplitudes of magnetic flux density. It is also worth noting that increasing frequency causes an increase in the coercive field value and an increase in the hysteresis loop area. At low frequencies, a broadening of the portion of the B(H) characteristic corresponding to the saturation range is observed.

Comparing the hysteresis loop dimensions obtained for different B_m values, changing the value of this parameter results in a change in the hysteresis loop area proportional to the square of the change in B_m. It is worth emphasizing that the magnetization curves determined using the analogue and digital methods differ only slightly when the frequency is not lower than 1 kHz. However, these differences are significant, especially in the saturation range at the lowest frequency considered (f = 50 Hz). This is the result of too small a difference between the period of the measured signal and the time constant R₂C₁ of the integrator. The oscillations visible in the measured hysteresis loops are the result of the low resolution of the analog-to-digital converters contained in the oscilloscope used for the measurements.

5.3. Discussion

Comparing the results obtained for both cores, using the analogue method, correct B(H) curves can be obtained when the integrator’s time constant is much greater than the supply signal period. This condition is not sufficient for measuring ferrite core characteristics. Therefore, using the digital method is beneficial for both core types and allows for reliable B(H) curve measurement results.

It is worth noting that small oscillations are visible in the measured hysteresis loop waveforms. These are related to the limited resolution of the analogue-to-digital converters in the oscilloscope used. Typically, for very high sampling frequency, the resolution of analogue-to-digital converters does not exceed 10 bits. It corresponds to the minimum allowable relative error equal to 0.1%. In turn, the steps visible in some of the presented B(H) waveforms result from ferromagnetic proprties related to the remagnetization of ferromagnetic domains [2].

6. Estimation of Power Losses in the Core

Based on the measured magnetization curves, the loss values of each tested core were determined and compared with the calculation results obtained using the modified Steinmetz model. The results of the core temperature measurements and calculations are presented. Section 6.1 concerns the 3F3 core, whereas Section 6.2 concerns the M-070 core.

6.1. 3F3 Core

Figure 19 shows the dependence of the power losses density P_V for a 3F3 core on B_m at selected values of frequency f. In this figure, the dashed lines indicate the results of the calculations using the modified Steinmetz model (Equation (6)). The solid lines represent the results of calculations obtained using Formula (4) and the digital hysteresis loop measurement method. The dotted lines represent the results of calculations obtained using Formula (4) and the analogue hysteresis loop measurement method.

The monotonically increasing dependencies of loss and flux density amplitude obtained for all considered operating conditions are related to the hysteresis loop B(H) curves presented in Section 4, whose area is an increasing function of f and B_m. The shape of this dependency indicates that an increase in the value of f with an imposed maximum value of permissible losses P_V results in a limitation of the maximum permissible value of B_m. This phenomenon is one of the limitations on increasing the maximum switching frequency in switched-mode power supplies.

In Figure 19, it is worth noting that the loss values determined by the hysteresis method using the analogue and digital B(H) curve measurement methods are close to each other only for the lowest frequency values considered. The discrepancies between these results increase with increasing frequency. At f = 100 kHz, these values differ by a factor of ten. It is also worth noting that the results obtained by the hysteresis method using the digital B(H) curve measurement method are close to the values calculated from Formula (5) using the model parameter values given in Section 3.

Figure 20 shows the measured and calculated dependences of the core temperature T_C on B_m for selected frequency values. In this Figure, the points represent the results of measurements made using a pyrometer in a thermally steady state, and the lines represent the results of calculations based on a compact thermal model of the core and the core loss values determined using the hysteresis method. The solid lines refer to the analogue hysteresis loop measurement method, and the dashed lines to the digital method.

The used compact thermal model is described by the following formula:

$$T_C=T_a+R_{th}\cdot V_e\cdot P_V$$

(11)

where T_a is the ambient temperature, V_e is the core volume, and R_th is the core’s thermal resistance. The R_th value characterizes the efficiency of heat dissipation generated in the core to the environment.

Good agreement between the calculation and measurement results was obtained at R_th = 35 K/W for f not exceeding 25 kHz. For higher frequencies, it was necessary to reduce the R_th value to 25 K/W (at f = 50 kHz) and 12 K/W (at f = 100 kHz). This indicates a problem with reliably determining the core loss power at higher frequencies using the hysteresis method. This may be due to delays in the recording of the measured signals, which translate into changes in the shape of the hysteresis loop B(H) and its area or the presence of loss components other than hysteresis losses in the tested core.

As can be seen in Figure 20, the core temperature values measured with the pyrometer and those calculated using Formula (11) with the digital hysteresis method show very good agreement. In turn, the results obtained using the classical hysteresis method differ significantly from the optical measurements, with the results being underestimated at low frequencies and overestimated by several times at frequencies above 25 kHz.

6.2. M-070 Core

Based on the hysteresis loops shown in Section 5, the losses of the considered nanocrystalline core were determined as a function of B_m at selected frequency values. The obtained test results are shown in Figure 21.

In this Figure, the measurement results provided by the manufacturer are marked with points, and the results of calculations performed using Formula (5) are marked with dashed lines.

Figure 21 shows that Formula (5) allows a good fit of the calculation results to the manufacturer’s data over a wide range of frequencies and magnetic flux density amplitudes.

Figure 22 compares the dependence of M-070 core losses on B_m obtained by the hysteresis method using the analogue (dashed lines) and digital (solid lines) hysteresis loop measurement methods with the results of calculations based on Formula (5) (dotted lines and points).

In Figure 22, it can be seen that the measurement results obtained using the analogue and digital methods have similar values. The differences are visible only at the lowest considered frequency. The calculation results obtained using Formula (4) and the values of the parameters describing the characteristics shown in Figure 21 are significantly overestimated compared to the hysteresis measurement results. The values obtained from this formula are up to three times greater than the values determined by the hysteresis method. The differences between the results obtained using both methods are particularly visible in the high-frequency range.

As frequency increased, the values of losses were measured over an increasingly narrow range of amplitude of magnetic flux density. This is due to the limited range of B_m values resulting from the allowable power losses in the core and related to its cooling efficiency. In the case of nanocrystalline cores, cooling is further complicated by the core being enclosed within a plastic housing, which protects the core from mechanical damage but also hinders convection on the core surface. For this reason, the core temperature was not measured. Therefore, for the nanocrystalline core, the accuracy of the digital loss measurement method was not verified by comparing the calculated and measured core temperature values. The only comparison was made to the loss values obtained from the digital measurement method and the Steinmetz formula. The agreement between these results confirms the practical usefulness of the digital measurement method.

7. Conclusions

This paper considers the problem of measuring the hysteresis loop B(H) and determining core losses. The measurement method and setup used are described. The magnetization curves of ferromagnetic cores made of ferrites or nanocrystalline material were determined using analogue and digital methods. The causes of incorrect results obtained when measuring magnetization curves using the analogue method were identified. One of these causes is an incorrectly chosen value of the RC circuit time constant. The second reason is the low accuracy of the voltage measurement at the output of the integrator. It is demonstrated that even with the correct selection of the integrator time constant, significantly deformation of the magnetization curves B(H) can occur when using the analogue measurement method. The digital method allows for the elimination of such distortion.

The core operating conditions corresponding to significant distortions of the B(H) characteristics were identified. These distortions become more pronounced the higher the frequency of the excitation signal.

The main results of this research are as follows:

(a)

Validations of the digital and analogue methods for measuring the magnetization characteristic B(H) of two types of ferromagnetic cores made of ferrite or nanocrystals;

(b)

Analysis of the influence of the frequency and flux density on the accuracy of the considered measurement methods;

(c)

Comparison and analysis of the accuracy of power loss measurements in the ferromagnetic core using hysteresis loops obtained using the analogue and digital methods and calculated using the modified Steinmetz model.

It has been shown that the classical Steinmetz’s formula for calculating core losses is applicable within a limited range of B_m and f changes, where no significant distortions of the hysteresis loop B(H) are observed. In particular, it has been shown that the analogue method exhibits significant errors at low frequencies and at B_m values close to the saturation of magnetic flux density. For ferrite and nanocrystalline cores, it is necessary to consider the dependence of α and β coefficients on the frequency and amplitude of magnetic flux density in order to achieve consistency between the P_V(f, B_m, T) formulas calculated from Steinmetz’s model and the manufacturer’s specifications.

Steady-state measurements of the ferrite core temperature T_C for selected B_m and f values confirmed that the loss factor determined using the digital measurement method allows for the correct calculation of the core temperature increase resulting from self-heating. However, achieving good agreement between the calculation and measurement results requires considering the decreasing dependence of thermal resistance on frequency or introducing an appropriate correction factor to the dependences describing the effect of frequency on the loss factor.

The presented results of this research can be used in the characterization of ferromagnetic cores. The proposed method allows determination of loss values over a wide range of magnetic flux density amplitudes and frequencies up to 100 kHz. References [43,44,45] indicate that in the higher frequency range, there are problems related to the observed phase shift between B and H. In the cited papers, special methods for calibrating measurement probes were proposed, enabling measurements in the higher frequency range, even exceeding several hundred kilohertz. The development of these methods will be the subject of our future research.