Choosing a suitable core loss model and accurately predicting energy loss are crucial in designing magnetic devices with high efficiency based on soft magnetic composites (SMCs). In this work, FeSiAl and FeSi SMCs with a uniform insulating layer were fabricated by a phosphating process. The effects of excitation waveform and DC bias field on core loss have been investigated in depth. The results show that different remagnetization rates of SMC under ideal sinusoidal and square waves result in different core losses at the same frequency and flux density. The improved general Steinmetz equation and the modified Steinmetz equation were shown to be suitable for calculating core loss under square excitation waves without DC bias field. When the DC bias field is applied, the core loss of FeSiAl and FeSi SMCs was found to increase significantly due to the magnetization state gradually approaching saturation. Interestingly, the variation tendency of core loss can be accurately predicted using the waveform coefficient Steinmetz equation under square excitation conditions with a DC bias environment. This work not only provides deep insights into core loss under different excitation waveforms and DC bias fields but also determines the application scope of different core loss models.

In conventional power systems, the core loss of soft magnetic composites (SMCs) plays a crucial role in determining the efficiency of energy transfer.1–3 To improve efficiency and avoid overheating problems, the core loss of SMCs under specific operating conditions deserves more attention.4–6 It is worth noting that SMC-based magnetic devices usually work under complex high-frequency excitation waveforms, such as square voltage excitation with or without DC bias fields.2,7–9 Thus, it is necessary to accurately estimate the energy loss of magnetic components operating in nonsinusoidal waves and under DC bias conditions.10–13 

In detail, when sinusoidal excitation fields are applied to magnetic devices, the Steinmetz equation (SE) method can accurately predict core loss.14 However, when magnetic devices work in nonsinusoidal environments, the calculation error of core loss induced by the SE method increases significantly due to changes in excitation waveforms.15 To minimize the calculation error for nonsinusoidal conditions, the modified Steinmetz equation (MSE), the improved general Steinmetz equation (IGSE), and the waveform coefficient Steinmetz equation (WcSE) were developed.16–18 However, these methods have their own application scopes, and accuracy differs from one another under different operating conditions. What is worse is that SMC-based magnetic devices typically operate in a DC bias field, which would significantly affect permeability and thus the core loss of magnetic components.19–22 Overall, the influence of working conditions (nonsinusoidal waveforms with and without DC bias field) on core loss for SMCs should be thoroughly investigated, accompanied by the development of appropriate calculation methods with high computational accuracy.

In this work, core losses for FeSiAl and FeSi SMCs under sinusoidal and square excitation waveforms with and without DC bias fields are studied in detail. It is found that the remagnetization rate caused by ideal sinusoidal waveforms is different from that by square excitations, resulting in a difference in core loss at the same operating flux density (Bm) and frequency ( f). IGSE and MSE methods can accurately predict the variation of core loss under square waves without DC bias fields. When the DC bias field is superimposed, the core loss of FeSiAl and FeSi SMCs obviously increases, resulting from magnetization states closer to magnetic saturation induced by the DC bias field. In addition, the WcSE method is more suitable for predicating the core loss of SMC-based magnetic components under square waves with DC bias fields.

The raw gas-atomized FeSi (94.5 wt. %Fe, 5.5 wt. %Si) and FeSiAl (84 wt. %Fe, 9.6 wt. %Si, 5.4 wt. % Al) powders were purchased from Changsha Hualiu Powder Metallurgy Company. The orthophosphoric acid, ethanol, epoxy resin, and zinc stearate were supplied by Sinopharm Chemical Reagent Co., Ltd.

FeSi powders (100 g) were dispersed in 50 ml EtOH (50 ml) containing orthophosphoric acid (0.8 g). Afterward, FeSi mixtures were stirred for 30 min at room temperature before being dried at 80 °C for 1 h. The insulated FeSi powders were then mixed with 0.5 wt. % zinc stearate (lubricate) and 0.5 wt. % epoxy resin (binder), respectively. Subsequently, the powders were compressed into rings at a pressure of 1705 MPa. Finally, these rings were annealed at 730 °C for 40 min under an N2 atmosphere to obtain FeSi SMC. FeSiAl SMC was prepared by using the same method, except that FeSiAl powders and 1.5 g orthophosphoric acid were used. It is worth mentioning that FeSiAl and FeSi SMCs have similar effective permeabilities of 63.1 and 62.3, respectively (Fig. S1 in the supplementary material).

A field emission scanning electron microscope (FE-SEM, Hitachi SU8020) equipped with an Oxford Instrument INCA energy-dispersive spectrometer (EDS) was used to investigate morphology and elemental distribution. A laser particle size analyzer (Malvern Mastersizer 2000) was used to determine the particle size distribution of FeSiAl and FeSi powders. A B-H analyzer (Iwatsu SY-8218, tolerance less than 0.5%) equipped with a DC bias source (Iwatsu SY-961) and an AC blocker (Iwatsu SY-962) was used to measure core loss.

In this work, the two-winding method was chosen to measure the core loss of toroidal samples. The excitation current I1(t) is applied to the primary winding with turns N1 to evaluate the magnetic field strength H(t). The secondary winding with turns N2 is then used to measure the induced voltage V2(t) to calculate B(t), which can radically exclude winding loss to improve test accuracy. H(t) and B(t) can be evaluated using the following formulas:2 
Ht=N1I1(t)le,
(1)
Bt=1N2Ae0TV2(t)dt
(2)
where le and Ae represent the effective magnetic path length and effective core section of toroidal samples, respectively, and T is the cycle time of excitation waves. The core loss per unit volume can be expressed as
Pt=fHdB=N1fN2V0TV2tI1(t)dt,
(3)
where V is the sample volume, f is the frequency of excitation waves, and Pt is the core loss.

1. Steinmetz equation

The Steinmetz equation (SE) is commonly used to calculate core loss under ideal sinusoidal waveforms as follows:14 
Pt=kfαBmβ,
(4)
where Bm is the maximum flux density; f is the operating frequency; and k, α, and β are the Steinmetz coefficients, respectively.

2. Modified Steinmetz equation

Given that the flux density change rate has a vital impact on core loss, a modified Steinmetz equation (MSE) was proposed to calculate core loss under non-sinusoidal waves. The MSE is given as follows:16 
Pt=k(feqα1Bmβ)f,
(5)
feq=2ΔB2π20TdBdt2dt,
(6)
where Bm is the maximum flux density and ΔB is the peak-to-peak magnetic flux density. It should be noted that when a sample works with DC bias field, the ΔB is not equal to double Bm.

3. Improved generalized Steinmetz equation

In addition to the rate of flux density change, the instantaneous value of flux density should also be considered. Therefore, an improved generalized Steinmetz equation (IGSE) was proposed to describe core loss under non-sinusoidal excitation waves. The IGSE can be expressed as17 
Pt=kiT0TdBdtαΔBβαdt,
(7)
ki=k2βα(2π)α102πcosθαdθ.
(8)

4. Waveform-coefficient Steinmetz equation

The waveform-coefficient Steinmetz equation (WcSE) was developed to establish the correlation between any arbitrary waveform and sinusoidal waves having the same peak flux density and frequency. The WcSE can be calculated as18 
Pt=λkfaBmβ,
(9)
λ=4T0T4φi(t)dt4T0T4φsin(t)dt,
(10)
where φi(t) and φsin(t) are the instantaneous value of flux inside magnetic cores related to the arbitrary and sinusoidal excitation, respectively, and λ is the waveform coefficient.

SEM images of raw FeSiAl and FeSi powders are shown in Figs. 1(a) and 1(d), respectively. Both the uninsulated FeSiAl and FeSi powders have a scale-like surface, which may be generated during atomization. Figures 1(b) and 1(e) show SEM images of insulated FeSiAl and FeSi powders, respectively. Apparently, the scale-like surface is buried in a rough phosphate layer gained in the phosphating process. In addition, two powders share a similar particle size distribution as shown in Figs. 1(c) and 1(d). The D50 particle sizes of FeSiAl and FeSi powders are 29.140 and 32.934 µm, respectively.

FIG. 1.

SEM images of (a) raw and (b) insulated FeSiAl powders. (c) Particle size distribution of the raw FeSiAl powder. SEM images of (d) raw and (e) insulated FeSi powders. (f) Particle size distribution of the raw FeSi powder.

FIG. 1.

SEM images of (a) raw and (b) insulated FeSiAl powders. (c) Particle size distribution of the raw FeSiAl powder. SEM images of (d) raw and (e) insulated FeSi powders. (f) Particle size distribution of the raw FeSi powder.

Close modal

EDS elemental distribution maps of the cross sections of FeSiAl and FeSi SMCs are shown in Figs. 2(a) and 2(b), respectively. Obviously, Fe (red), Si (green), and Al (magenta) elements are concentrated in the FeSiAl magnetic particle zones, while Fe (red) and Si (green) elements are clearly observed in the FeSi magnetic particle zones. In addition, P (yellow) and O (cyan) elements can be clearly identified around FeSiAl and FeSi particles. This indicates that the insulating layers remain stable and intact in the gaps between the magnetic particles. Consequently, the interparticle eddy current can be well suppressed, as shown in Fig. 2(c).

FIG. 2.

Typical cross-sectional SEM image and corresponding EDS elemental distribution maps of (a) FeSiAl and (b) FeSi SMCs. (c) Schematic illustration of eddy current in SMC.

FIG. 2.

Typical cross-sectional SEM image and corresponding EDS elemental distribution maps of (a) FeSiAl and (b) FeSi SMCs. (c) Schematic illustration of eddy current in SMC.

Close modal

The core loss of magnetic devices, which constitutes a major part of the total energy loss, is a fundamental criterion for assessing whether a power system meets the design requirements. Core losses of SMC under high frequency sinusoidal excitations based on the conventional loss separation model are shown in Fig. S2 in the supplementary material. The results show that the inter-particle eddy current loss is lower than hysteresis loss, intra-particle eddy current loss, and excess loss, indicating that the insulating layer has effectively blocked the conduction path between magnetic particles.23,24

Since magnetic components are usually used as storage cells in DC transformers, magnetic devices always work in non-sinusoidal excitation environments. In addition, the shape of excitation waves has a significant impact on the core loss of magnetic devices, which makes the design of decreasing temperature rise more complicated in light of the cooperation of other parameters, including size, structure, and thermal resistance.25–27  Figure 3 shows the dependence of core loss on f and Bm for FeSiAl and FeSi SMCs under square (Duty = 0.5) and sinusoidal voltage-excited waves. The variation tendency of core loss at square voltage excited waveforms is similar to that under ideal sinusoidal waves with increasing operating flux density and frequency. However, core losses of samples under square waves are always lower than those under sinusoidal excitation waves. Meanwhile, as the frequency increases, the gaps between core losses under two types of waves gradually increase. Therefore, the traditional SE method fails to precisely calculate the core loss of SMCs under nonsinusoidal excitations.

FIG. 3.

Core losses of (a) FeSiAl and (b) FeSi SMCs under sinusoidal and square (Duty = 0.5) waves vs f and Bm.

FIG. 3.

Core losses of (a) FeSiAl and (b) FeSi SMCs under sinusoidal and square (Duty = 0.5) waves vs f and Bm.

Close modal

To understand the origin of the difference in core loss at different excitation waves, Fig. 4 compares waveforms of flux density B(t), magnetic field strength H(t), and varying rate of flux density (dB/dt) under sinusoidal and square voltage excitations. As shown in Fig. 4, when magnetic cores work under ideal sinusoidal waves, the shapes of B(t) wave and H(t) wave are consistent with excited voltage waves. When square voltage excited waves are applied to the samples, the waveforms of B(t) and H(t) are triangular, being different from the original square waveforms. In addition, the typical measured waveforms of the magnetic field H(t) and flux density B(t) are shown in Fig. S3 in the supplementary material. Clearly, the shapes of waveforms are close to ideal ones. It is accepted that square waves can be divided into many sinusoidal waves with different frequencies and amplitudes, indicating that the remagnetization rate is different from that of magnetic cores under ideal sinusoidal waves. Because the remagnetization rate is closely related to core loss, the difference in core loss between square and sinusoidal waveforms can be attributed to dB/dt determined by the remagnetization rate.16 Therefore, the traditional core loss calculation model, SE, based on ideal sinusoidal excitation, cannot accurately predict the variation of core loss under nonsinusoidal waveforms.28,29

FIG. 4.

Schematic diagram of the sinusoidal voltage and square voltage excitation waves.

FIG. 4.

Schematic diagram of the sinusoidal voltage and square voltage excitation waves.

Close modal

Based on the above-mentioned analysis, it is necessary to accurately predict the variation of core loss for SMCs under square waves. To find suitable loss models, IGSE, MSE, and WcSE methods are employed in this work to calculate core loss under nonsinusoidal excitation waves using only the parameters used in the SE model. In the first step, Figs. 5(a) and 5(b) show the fitted three-dimensional mesh surface of the core loss for FeSiAl and FeSi SMCs under ideal sinusoidal excitation waveforms based on the SE method, respectively. The fitted surfaces closely match the measured points, and the R-square values exceed 0.99, which means that the fitted results are reliable. The SE’s coefficients of FeSiAl and FeSi SMCs are shown in Figs. 5(c) and 5(d), which can be used by the IGSE, MSE and WcSE methods, respectively.

FIG. 5.

Fitted three-dimensional mesh surface of core loss vs f and Bm: (a) FeSiAl and (b) FeSi SMCs. Coefficients of SE as a function of Bm: (c) FeSiAl and (d) FeSi SMCs.

FIG. 5.

Fitted three-dimensional mesh surface of core loss vs f and Bm: (a) FeSiAl and (b) FeSi SMCs. Coefficients of SE as a function of Bm: (c) FeSiAl and (d) FeSi SMCs.

Close modal

Figure 6 shows the measured core loss and calculated values at different Bm values derived from IGSE, MSE, and WcSE methods. The corresponding error percentages are provided in the top left corner. The results show that the calculated values using IGSE and MSE methods can better match the measured points. In detail, the average percentage of errors using IGSE and MSE is less than 5%, while that of WcSE is closer to 13%. Given that SMC-based magnetic components are always placed in complex excitation environments with many harmonics waves, WcSE adopting the waveform-coefficient method ignores the impact of larger harmonics content from excitation current, resulting in a significant deviation of the calculated core loss from the measured points.29 In contrast, IGSE and MSE methods can better match the measured data because the rate of flux density change has been considered.

FIG. 6.

Measured and fitted core losses at different frequencies for (a)–(e) FeSiAl and (f)–( j) FeSi SMCs under square voltage waves at different Bm values. The insets show the corresponding percentage of errors.

FIG. 6.

Measured and fitted core losses at different frequencies for (a)–(e) FeSiAl and (f)–( j) FeSi SMCs under square voltage waves at different Bm values. The insets show the corresponding percentage of errors.

Close modal

Magnetic devices are a major component in switching-mode power supplies (SMPS), resulting in SMC-based magnetic components being subjected to DC bias fields.8,27,30 Therefore, the effect of DC bias field on core loss should be considered. Figures 7(a) and 7(b) show the core loss vs ΔBm and DC bias field at 50 kHz. As DC bias field and ΔBm gradually increase, the energy loss presents the obvious rising tendency, suggesting that the DC bias field has significant influences on core loss.31  Figure 7(c) shows the core losses normalized to the energy loss without DC bias condition at changing DC bias fields under sinusoidal waves excitation ( f = 50 kHz and ΔBm = 100 mT). In addition, the core loss vs the DC bias field is presented in the top left corner. Apparently, as the DC bias field increases, the normalized core loss of FeSiAl and FeSi SMCs gradually increases, and when the DC bias field reaches 200 Oe, the core loss of FeSi SMC is close to FeSiAl SMC, which differs from the core loss performance without DC bias field. In addition, the normalized core loss of FeSiAl SMC is much larger than that of FeSi SMC, suggesting that the influence of the DC bias field on the core loss of FeSiAl SMC is greater than FeSi SMC. The ratio of Bm to their corresponding Bs (Bm/Bs), which represents the degree of magnetization state away from magnetic saturation, can be used to clarify the relationship between core loss and DC bias field [Fig. 7(d)]. As the DC bias field gradually increases, the magnetization state of FeSiAl and FeSi SMCs gradually tends to the magnetic saturation point. This means that resistance to domain wall motion and magnetization rotation enhances, which further leads to an increase in core loss. Furthermore, the rate of magnetization state variation in FeSi SMC is obviously slower than that of FeSiAl SMC. It can be inferred that when samples are subjected to the same DC bias condition, the magnetization state of FeSiAl SMC approaches magnetic saturation more closely, leading to a sharp increase in core loss under high DC bias fields. Therefore, FeSi SMC exhibits a similar core loss to FeSiAl SMC under higher DC bias fields.

FIG. 7.

Core loss at different ΔBm and DC bias fields for (a) FeSiAl and (b) FeSi SMCs under sinusoidal waves at 50 kHz. (c) Normalized core loss for FeSiAl and FeSi SMCs (f = 50 kHz; ΔBm = 100 mT), and core loss vs DC bias field is shown in the top left corner. (d) Dependence of magnetization state (Bm/Bs) on DC bias fields.

FIG. 7.

Core loss at different ΔBm and DC bias fields for (a) FeSiAl and (b) FeSi SMCs under sinusoidal waves at 50 kHz. (c) Normalized core loss for FeSiAl and FeSi SMCs (f = 50 kHz; ΔBm = 100 mT), and core loss vs DC bias field is shown in the top left corner. (d) Dependence of magnetization state (Bm/Bs) on DC bias fields.

Close modal

When samples are subjected to nonsinusoidal excitations under DC bias fields, the core loss can still be calculated using the SE method.17, Figures 8(a) and 8(b) show the dependence of SE’s coefficients on different DC bias fields. It should be noted that SE parameters for different magnetization states need to be adjusted in comparison with zero bias conditions. In addition, the R-square exceeds 0.99, confirming a high level of accuracy in predicting core loss. However, since operating frequency is constant, the coefficient of α keeps at the fixed value within DC bias field range of 0–200 Oe.17 As a result, the core loss with square waveforms at the same ΔBm and f under different DC bias fields can be derived from IGSE, MSE, and WcSE methods using parameters of SE. Consequently, k and β should be adjusted to describe additional core losses brought by the DC bias field. When β remains relatively constant, k increases monotonically as the DC bias field increases. When β presents slight variations, it can lead to a drastic change in k. Figures 8(c) and 8(d) show the measured and fitted core loss for FeSiAl and FeSi SMCs under square voltage-excitation conditions (Duty = 0.5; ΔBm = 100 mT; f = 50 kHz) as a function of DC bias fields. Corresponding percentage errors for FeSiAl and FeSi SMC are shown in Figs. 8(e) and 8(f), respectively. The average error percentage for FeSiAl and FeSi SMCs using the WcSE method is nearly 12% and 4%, respectively. While the average error percentage for FeSiAl and FeSi SMCs using IGSE and MSE methods is close to 40% and 20%, respectively. We should point out that this result is contrary to the fitting results without the DC bias field. As discussed above, a larger DC bias field brings the magnetization state closer to magnetic saturation, which may be the dominant factor for increased core loss. In addition, considering the reduced core loss for square voltage-excitation conditions compared to sinusoidal voltage-excitation conditions, the influence of magnetization state on core loss is more significant than harmonics content. Because the WcSE method is based on the average value of the operating magnetic flux density, it can preferably reflect the variation in the magnetization state.18,29

FIG. 8.

Parameters of SE at different DC bias fields: (a) FeSiAl and (b) FeSi SMCs. Measured and fitted core loss vs DC bias field under square waveform excitations: (c) FeSiAl and (d) FeSi SMCs. The error percentages for (e) FeSiAl and (f) FeSi SMCs.

FIG. 8.

Parameters of SE at different DC bias fields: (a) FeSiAl and (b) FeSi SMCs. Measured and fitted core loss vs DC bias field under square waveform excitations: (c) FeSiAl and (d) FeSi SMCs. The error percentages for (e) FeSiAl and (f) FeSi SMCs.

Close modal

Overall, both excitation waveforms and DC bias field have significant effects on core loss. For waveforms, when ideal sinusoidal waveforms are subjected to SMC, the changing trend of core loss can be accurately calculated using the SE method. When square excitations are applied to SMC-based magnetic components, the core loss is lower than that under sinusoidal waves at the same f and Bm. IGSE and MSE methods have been shown to be more suitable for predicting core loss under square waves. Given that the core loss under DC bias fields is closely related to the magnetization state, the WcSE method can better describe the changing trend of core loss with square wave under DC bias fields.

In summary, well-insulated FeSiAl and FeSi SMCs were prepared using a phosphating technique. The effects of excitation waveforms and DC bias field on core loss have been systematically studied. Due to the difference in remagnetization rate, the core loss of both FeSiAl and FeSi SMCs under sinusoidal and square excitation waveforms exhibits different trends of variation, and lower core loss can be achieved under square waves. In addition, when SMC-based magnetic components operate under ideal sinusoidal fields, the core loss can be accurately predicted by using the SE method; while square voltage excitation is applied to samples, the MSE and IGSE methods are more suitable for calculating the core loss. On the other hand, as the DC bias field increases from 0 to 200 Oe, the core loss of FeSiAl and FeSi SMCs gradually increases. It indicates that the DC bias field can urge the magnetization state to approach magnetic saturation, resulting in an increase in core loss. Moreover, the magnetization state of FeSiAl SMC is closer to magnetic saturation than FeSi SMC at the same DC bias field, resulting in a sharp increase in core loss. Therefore, the core loss of FeSiAl and FeSi SMCs shows a similar value at 200 Oe DC bias field. In addition, when square voltage excitation is applied to SMC-based magnetic components under DC bias fields, the calculated value obtained by the WcSE method can match better with the measured points than IGSE and MSE methods. It can be deduced that magnetization state, rather than harmonics content, plays a more crucial role in determining core loss under DC bias fields.

See the supplementary material for effective permeabilities, core loss separation results, and typical waveforms of H(t) and B(t) for FeSiAl and FeSi SMCs.

This work was supported by the Anhui Provincial Natural Science Foundation (Grant No. 2308085QE133), the Major Project of Science and Technology Innovation 2025 in Ningbo City, China (Grant No. 2020Z062), the Science and Technology Project of State Grid Corporation of China (Grant No. 5500-202118252A-0-0-00), and the Huaian Key Research and Development Plan (Grant No. HAG 202114).

The authors have no conflicts to disclose.

Chao Mei: Formal analysis (lead); Investigation (lead); Writing – original draft (lead); Writing – review & editing (lead). Kun Wan: Formal analysis (equal); Investigation (equal). Bowei Zhang: Formal analysis (equal); Investigation (equal). Xu Zhu: Formal analysis (equal). Feng Hu: Investigation (equal). Wei Liu: Conceptualization (equal); Funding acquisition (equal); Writing – review & editing (supporting). Zhongqiu Zou: Conceptualization (supporting); Funding acquisition (equal). Hailin Su: Conceptualization (lead); Funding acquisition (lead); Project administration (lead); Resources (lead); Writing – review & editing (supporting).

The data that support the findings of this study are available from the corresponding authors upon reasonable request.

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