Influence of annealing temperature on nanocrystalline alloy’s excess loss parameters

Abstract

In this paper, both nanocrystalline alloy (Fe_73.5Cu₁Nb₃Si_15.5B₇) ribbon samples and toroidal samples (wound ribbon) are annealed at different temperatures in order to consider the influence of inner stress on the magnetization properties. Then the AC magnetization properties of these samples are measured. Combined with the measured results, the influence of inner stress on nanocrystalline alloy’s microstructure is analyzed quantitatively based on the loss separation principle and the statistical theory of loss. By comparing measured macroscopic magnetization characteristics and excess loss, the equivalent stress state of the toroidal sample is evaluated. Furthermore, two kinds of samples’ excess loss under different annealing temperatures are analyzed, and the effectiveness of stress relief at optimal annealing temperature is validated.

Keywords

Nanocrystalline alloy loss separation excess loss annealing stress relief

1. Introduction

In the high-frequency application, the traditional magnetic materials are gradually replaced by the amorphous alloy or nanocrystalline alloy, which are used to manufacture the iron cores of the energy conversion devices, such as the high-frequency transformer. On the one hand, the nanocrystalline alloy relies on the annealing treatment to separate out small enough grain 𝛼-Fe to acquire advanced magnetization characteristics. The 𝛼-Fe grain is obtained through annealing between the first and second crystallization temperature, and materials’ microstructure could be changed in this way [1]. On the other hand, similar to FeSi, the stress introduced during the cores’ manufacture process can further change the magnetization and loss properties of materials, while the annealing could reduce the effect of stress. Therefore, it is significant to reveal the correlation contributed from inner stress, the microstructure, and annealing temperature (T _A) of nanocrystalline alloy.

The effect of inner stress on magnetic materials has received much attention from materials-developed researchers and engineers. M.C. Ri et al. [2,3], proposed a re-winding method on amorphous rings to reveal the influence of inner stress on the macroscopic magnetization characteristics, and made contributions to improving the characteristics of Fe-based metallic glasses. They analyzed the effect of stress relief on the magnetic characteristics of amorphous under external stress through various annealing treatments. According to the loss separation principle, losses of ferromagnetic materials can be divided into the hysteresis loss, eddy current loss, and excess loss, which is related to the interactions of eddy currents and microstructure [4]. The domain wall pinning strength related closely to hysteresis loss is strongly dependent on T _A [5], so the hysteresis loss is inevitably influenced by T _A. A. Boglietti et al. compared differences of these loss components resulted from the punching process [6], then they applied annealing process to improve the magnetization characteristics, and obtain the high-permeability and low-loss. The initial permeability of the magnetic material was inclined to increase with temperature below the optimal T _A, while exceeding this temperature, the permeability would go down again [5,7]. Therefore different permeabilities of nanocrystalline alloy could be acquired by setting appropriate heat treatment conditions. The influence of T _A (500–600 °C) on macroscopic magnetic characteristics has been illustrated overall in [8], and 10–30 min were validated as the optimal heat treatment time. Furthermore, some scholars study the distinctive magnetization characteristics of nanocrystalline alloy annealed under transversal or longitudinal field [9–11]. Different anisotropic were obtained by various annealing treatments in [12], which pointed out that anisotropic would increase with the annealing field intensity.

In the previous researches, inner stress has a strong effect on materials’ macroscopic magnetization characteristics, while the relationship between magnetization characteristics and inner stress at the microscopic level has not been discussed quantitatively. Besides, the excess loss, which represents microscopic eddy current loss, was investigated under different anisotropic, while the influence of manufacture process on the excess loss was also analyzed, but the variation of this property along with T _A is still unknown [12,13]. Accordingly, for nanocrystalline alloy, a comprehensive analysis is significant on the differences resulted from inner stress introduced by the bending process, and on the characteristics improvement from appropriate annealing treatment.

In this paper, we do firstly annealing treatments on the nanocrystalline ribbon samples and toroidal samples under different temperatures. Then the AC magnetization characteristics are measured, and the results are analyzed based on the loss separation principle and the loss statistical theory. The excess loss parameters are identified under every experimental group, and the microstructure’s change related to T _A is also to be investigated based on these parameters. Furthermore, according to the different strains existed in two kinds of samples of every group, we compares their magnetization characteristics and analyzes the dominant equivalent stress state of the toroidal samples. Based on these results, the impact of stress on microstructure, and annealing on stress relief are assessed quantitatively.

2. Loss separation principle and the statistical theory of loss

According to the loss separation principle, total loss P _tot (W/m³) of ferromagnetic materials can be divided into hysteresis loss P _h, classical loss P _c, and excess loss P _e [4,14]. These loss components satisfied: $\begin{eqnarray}\displaystyle P_{\text{tot}}=P_{h}+P_{c}+P_{e}. & & \displaystyle\end{eqnarray}$ (1) Assuming magnetic flux is distributed uniformly in the cross section of the thin magnetic sheet, the classical loss would be: $\begin{eqnarray}\displaystyle P_{c}(t)={\displaystyle \frac{{\sigma}d^{2}}{12}}\left({\displaystyle \frac{\text{d}B}{\text{d}t}}\right)^{2} & & \displaystyle\end{eqnarray}$ (2) where σ is electrical conductivity, d is the magnetic sheet thickness, and B is magnetic flux density. For sinusoidal alternating flux, the classical loss can be expressed as: $\begin{eqnarray}\displaystyle P_{c}={\displaystyle \frac{1}{6}}{\sigma}{\pi}^{2}d^{2}B_{m}^{2}f^{2} & & \displaystyle\end{eqnarray}$ (3) where f represents the frequency, B _m is the peak of magnetic flux density. Bertotti model has been verified to be applicable in the mostly ferromagnetic materials [4], in which three parts of loss P (W/m³) could be transferred into: $\begin{eqnarray} \displaystyle {\displaystyle \frac{P_{\text{tot}}}{f}}-k_{c}B_{m}^{2}f=B_{m}^{n}k_{h}+k_{e}B_{m}^{{\frac{3}{2}}}f^{{\frac{1}{2}}} & & \displaystyle \end{eqnarray}$ (4) where k _c = (1∕6) ∙ σπ² d ². The first term at the right of formula represents the hysteresis loss, and k _e is the coefficient of excess loss. The excess loss is actually a linear function depending on f ^1∕2, and ((4)) can be expressed as the form of W (J/m³), which means the each cycle loss in unit volume: $\begin{eqnarray}\displaystyle W_{\text{tot}}-W_{c}=W_{h}+k_{e}\cdot B_{m}^{1.5}f^{0.5}. & & \displaystyle\end{eqnarray}$ (5) According to loss separation principle, the excess loss in every cycle is identified as following, $\begin{eqnarray}\displaystyle W_{e}(B_{m},f)=W_{\text{tot}}(B_{m},f)-W_{c}(B_{m},f)-W_{h}(B_{m}). & & \displaystyle\end{eqnarray}$ (6) Remarkably, this loss component is influenced by the frequency, but excess loss parameters are independent of the frequency. If knowing total losses W _tot (J/m³) at any two different frequencies without skin effect, the classical loss W _c corresponding the frequency f can be calculated by ((3)). As shown in Fig. 1, the hysteresis loss W _h and k _e at a certain flux density B ₀ could be respectively obtained by the intercept and slope of the linear function ((5)).

Fig. 1.

W _tot–W _c vs. f ^1∕2 (B _m = B ₀).

The excess loss is closely related to the materials’ microscopic structure, and the excess field intensity could be expressed as follows [14], $\begin{eqnarray}\displaystyle H_{e}(t)={\displaystyle \frac{{\sigma}GS}{n(t)}}{\displaystyle \frac{\text{d}B}{\text{d}t}} & & \displaystyle\end{eqnarray}$ (7) where G = 0.1356 is a dimensionless constant, and S represents the materials’ cross section. Magnetic object (MO) is the whole domain structure restricted by single grain in the cross section, in which the material’s local coercivity field is distributed homogeneously and n represents the number of MO. Once applied field exceeds the excess field intensity H _e, MO is in active state. The more number of active MO indicate the magnetization process is smoother, and the lower is the excess field H _e [15]. Besides, the number n is changeable with time, so it can be expressed as the function of the excess field H _e(t): $\begin{eqnarray}\displaystyle n(H_{e})=n_{0}+{\displaystyle \frac{H_{e}}{V_{0}}} & & \displaystyle\end{eqnarray}$ (8) where V ₀ is a statistical parameter reflecting the average minimum separation between different local coercivity field. n ₀ represents the number of MO located at the material’s cross section at demagnetization status. The formula (7) can be transferred into: $\begin{eqnarray}\displaystyle n(t)={\displaystyle \frac{{\sigma}GS\left({\displaystyle \frac{\text{d}B}{\text{d}t}}\right)^{2}}{H_{e}(t){\displaystyle \frac{\text{d}B}{\text{d}t}}}}. & & \displaystyle\end{eqnarray}$ (9) Under sinusoidal excitation, the time average value of the number of MO satisfied: $\begin{eqnarray}\displaystyle {\tilde{n}}={\displaystyle \frac{{\displaystyle \frac{1}{T}}\int _{0}^{T}{\sigma}GS4{\pi}^{2}f^{2}B_{m}^{2}\cos ^{2}({\omega}t)∼\text{d}t}{{\displaystyle \frac{1}{T}}\int _{0}^{T}H_{e}(t){\displaystyle \frac{\text{d}B}{\text{d}t}}∼\text{d}t}}={\displaystyle \frac{2{\pi}^{2}{\sigma}GSB_{m}^{2}f^{2}}{P_{e}}}. & & \displaystyle\end{eqnarray}$ (10) Combining (8) and (9), a quadratic equation about the excess field H _e(t) is acquired and can be solved as: $\begin{eqnarray}\displaystyle H_{e}={\displaystyle \frac{n_{0}V_{0}}{2}}\left(\sqrt{1+{\displaystyle \frac{4{\sigma}GS}{n_{0}^{2}V_{0}}}{\displaystyle \frac{\text{d}B}{\text{d}t}}}-1\right). & & \displaystyle\end{eqnarray}$ (11) The time average excess magnetic field intensity can be expressed as: $\begin{eqnarray}\displaystyle \tilde{H}_{e}={\displaystyle \frac{P_{e}}{4B_{m}f}} & & \displaystyle\end{eqnarray}$ (12) while excess loss is calculated by (6), the average value of n and excess magnetic field intensity can be acquired from (10), (12) respectively. Then n, H _e are substituted into (8), which is a linear function about the excess field H _e. In the next, the fitting results obtained from at least two frequencies at every B _m can lead to the determination of the microscopic parameters n ₀ and V ₀.

The inner stress introduced in the toroidal sample inevitably caused the variation of materials’ microstructure, which could be observed from the variation of excess parameters n, V ₀. In the next section, the influence of inner stress on nanocrystalline alloy’s microstructure would be discussed based on the statistical theory of loss, and distinctions under different T _A are also to be compared.

3. Experimental results and analysis

3.1. Measurement setup and methods

The investigated nanocrystalline alloy in this paper is 1K107B (Fe_73.5Cu₁Nb₃Si_15.5B₇), which is the most popular commercial content in the market, and adopted by transformer manufacturers and electrical machine designers. According to the datum provided by the manufacturer, crystallization temperature and curie temperature of 1K107B are 530 °C and 570 °C, respectively. The annealing treatment might result in the shift of curie temperature, and the magnetization state around crystallization phenomenon is also interesting. Therefore, annealing temperature T _A is set as 500 °C, 520 °C, 540 °C, 560 °C, 580 °C and 600 °C in this paper. The annealing time is set as 10 min referring to the optimal time [8]. The samples are annealed in a vacuum environment without annealing magnetic field so that the anisotropic could be negligible. After annealing treatment, materials’ alternating magnetic characteristics are measured by different instruments according to their geometrical shapes. Ribbon samples are measured by single sheet tester TD8160, as shown in Fig. 2a, which uses induction voltage combined with Faraday’s Law to calculate flux density, and adopts the H-coil method to acquire field intensity. Meanwhile, the TD8120 based on the traditional two-winding method is used to measure the magnetization characteristics of toroidal samples. Since the magnetic circuit length l of this sample is easily accessible, the field intensity could be resolved based on the Ampere’s Law, as shown in Fig. 2b. At frequency f = 50 Hz, the macroscopic magnetic characteristics of both samples, which are annealed at optimal T _A = 560 °C, are tested and listed in Table 1. Tested toroidal sample (40 × 36 × 15 mm) are limited by the accuracy of instruments at low induction, where acquired signals are extremely weak, so that only the magnetization characteristics above B _m = 0.4 T are measured. However, compared to ribbon samples, it is still obvious that toroidal samples have a higher permeability, lower coercivity, and lower magnetic losses. These differences are attributed to the inner stress existed in the toroidal samples, and could result in distinctions in microstructure. In the next, the statistical theory of losses is applied to both samples, and variations in the microstructure are to be analyzed quantitatively.

Fig. 2.

The measuring setup (a) TD8160 (for ribbon sample) (b) TD8120 (for toroidal sample).

Table 1

Macroscopic magnetic characteristic comparison (f = 50 Hz)

	Ribbon			Toroidal
B _m (T)	μ × 10³ (H/m)	H _c (A/m)	P _tot (W/kg)	μ × 10³ (H/m)	H _c (A/m)	P _tot (W/kg)
0.1	111	0.45	0.0009	—	—	—
0.2	147	0.68	0.0029	—	—	—
0.3	159	0.88	0.0057	—	—	—
0.4	151	1.06	0.0092	—	—	—
0.5	133	1.24	0.0135	324	0.81	0.0086
0.6	112	1.41	0.0188	304	0.91	0.0119
0.7	88	1.61	0.0252	246	1.03	0.016
0.8	68	1.76	0.0321	165	1.17	0.0216
0.9	47	1.94	0.0405	62	1.26	0.0279
1	21	2.42	0.0548	—	—	—

3.2. Quantitative assessment of microstructure parameters

The thickness of the magnetic sheets is 19.5 μm, with the conductivity σ = 8.33 × 10⁵ S/m. According to (3), in terms of the extremely thin sheets, the change of σ caused by different T _A could be neglected during calculating the classical loss, which is also far smaller than total losses. Therefore, the total losses at any two frequency points are firstly measured in abscence of the skin effect, then corresponding calculated classical losses are subtracted from the total losses. Combined with (5), the hysteresis losses and the excess loss parameters k _e are obtained. Then the excess losses are acquired by (6). Substituting the excess losses into (10), (12), the linear function n (H _e) is acquired, and n ₀, V ₀ equals to intercept and slope of this linear function respectively.

The parameters n (H _e) of toroidal samples and ribbon samples annealed at different T _A are shown in Figs 3–4. For this material, n ₀ = 0, and the law of ((8)) is changed into n = H _e∕V ₀. This phenomenon indicates that the number of MO is zero when the material is in a demagnetized state. Compared with the toroidal sample, the larger excess field intensity is required for MO to be active in the ribbon sample, and the number of MO in the latter one is far less than that in the former. Obviously, there is a minimum in the toroidal sample at T _A = 560 °C, and this phenomenon is initially inferred to be influenced by inner stress and annealing temperature comprehensively.

Fig. 3.

Identified n (H _e) for ribbon samples at different T _A. (a) 500 °C (b) 520 °C (c) 540 °C (d) 560 °C (e) 580 °C (f) 600 °C.

Fig. 4.

Identified n (H _e) for toroidal samples at different T _A. (a) 500 °C (b) 520 °C (c) 540 °C (d) 560 °C (e) 580 °C (f) 600 °C.

On the other hand, the parameter V ₀, which reflects the minimal separation between different local coercive field values, is pointed out to be influenced by many factors, such as the applied field and the composition of materials [4,14]. In this paper, as shown in Figs 5–6, the parameter V ₀ is not only impacted by T _A, but also related to the inner stress introduced by bending process. For ribbon samples, the variation of the parameter V ₀ along with annealing temperature T _A is relatively stable. Below crystallization temperature, the value of V ₀ has not a certain rule about B _m, while exceeding the curie temperature, it approximates to a constant, which indicates the parameter V ₀ is not sensitive to the annealing temperature T _A. Only when the annealing temperature T _A = 560 °C, V ₀(B _m) is a monotonically decreasing function, and this characteristic is similar to the amorphous METGLAS 2605 SC ribbon mentioned in [4], but is rare in other cases. In the investigated toroidal sample, similar to ribbon sample, the parameter V ₀ is found out to be independent of the applied field below the crystallization temperature. As T _A is higher than crystallization temperature, V ₀ is going to increase with applied field as shown in Fig. 6. Remarkably, the parameter V ₀ is much higher in the ribbon sample, or the toroidal sample annealed at optimal annealing temperature T _A, where the distinction of V ₀ between two samples is the smallest. These indicate a larger difference among local coercive field value, and higher excess loss is obtained from ribbon or optimal T _A. Obviously, both in the ribbon and toroidal sample, V ₀ at T _A = 560 °C is far larger than that at other annealing temperature T _A. The reason for this phenomenon is that the difference between the local coercive field is larger and could cause the magnetization process is not so smooth [15]. Although the lowest total loss at low frequency and lowest hysteresis loss obtained from this annealing temperature made itself as optimal T _A, with applied frequency increased, the excess loss would replace the hysteresis loss and be dominant in the total loss [12,16], especially for the application frequency of high-frequency transformer used in DC-DC converter. So it is necessary to pay more attention to the annealing temperature T _A of magnetic devices used for different frequency ranges.

Fig. 5.

The variation of V ₀ along with B _m for ribbon samples at different T _A. (a) 500 °C (b) 520 °C (c) 540 °C (d) 560 °C (e) 580 °C (f) 600 °C.

Fig. 6.

The variation of V ₀ along with B _m for toroidal samples at different T _A. (a) 500 °C (b) 520 °C (c) 540 °C (d) 560 °C (e) 580 °C (f) 600 °C.

Fig. 7.

Excess loss vs. annealing temperature at B _m = 0.8 T, f = 50 Hz.

3.3. Stress relief at different T _A

In order to illustrate the influence of inner stress on nanocrystalline alloy’s excess loss, and validate the effectiveness of annealing treatment on this material’s stress relief, Fig. 7 is plotted. In accordance with [17], the larger number of the parameter n leads to a higher excess loss. Although the outside of toroidal samples suffered from tensile stress while the inside of them suffered from compression stress, the equivalent stress state of the toroidal sample is unknown. Due to lower excess loss in the toroidal samples, combined with macroscopic magnetization characteristics given by Table 1, it could be inferred that the toroidal sample is comprehensively affected by tensile stress [18]. Remarkably, this consequence does not mean that all toroidal samples and power electronic components are suffered from tensile stress, this might be dependent on the rolling process and geometric properties, and it isn’t the highlight of this paper.

It is easy to notice that the excess loss P _e of the toroidal samples fluctuates with the annealing temperature T _A dramatically, and at the same time the excess loss of the ribbon sample is relatively stable. After approach to the crystallization temperature, the excess loss P _e is dropping apparently. However, at optimal annealing temperature T _A, annealing treatment could remove the effect of stress more apparently, so the difference in the excess loss P _e of two kinds of samples would be the minimum in this case. As for the high P _e at T _A = 500 °C, it can be attributed to that nanocrystalline alloy doesn’t possess excellent characteristics, and the inhomogeneous local coercive field value existed in this case.

4. Conclusion

The loss separation principle and the statistical theory of loss are applied to the magnetization characteristics analysis of nanocrystalline ribbon and toroidal samples which have been annealed at the different temperatures 500 °C, 520 °C, 540 °C, 560 °C, 580 °C, and 600 °C. As excess loss parameters n, V ₀ are acquired, the influence of inner stress on this material’ microstructure could be reflected quantitatively. Some conclusions are summarized as following:

Based on the number of MO investigated in this paper, it is revealed that, the whole magnetic domain restricted by the cross section of individual grain in the toroidal sample is much more than that of ribbon samples.

For the ribbon sample, the parameter V ₀ is relatively high and independent of the applied field below the crystallization temperature owing to the unsatisfactory magnetic property. Once 𝛼-Fe grains are separated out after reaching crystallization temperature, the parameter V ₀(B _m) is a monotonically increasing function except at optimal annealing temperature T _A. In terms of toroidal samples, there are two differences in this microscopic characteristic. Firstly, V ₀ is still increased with B _m at optimal T _A. The other is that the value of V ₀ is much smaller than that of ribbon samples, which indicate that a larger distinction existed in the ribbon samples’ local coercive value.

The parameter V ₀ approaches to the maximum value again at the optimal T _A, which reflects the inhomogeneous property of local coercive field, and is contrary to the trend of pinning strength. Thus the excess loss of the nanocrystalline alloy is not in a good condition at this annealing temperature and is different from the characteristic of hysteresis loss. Considering the significance of excess loss along with increasing frequency, a tradeoff among loss components are supposed to be paid attention during manufacturing the materials.

Compared with nanocrystalline ribbons annealed at the same condition, toroidal samples in this work comprehensively suffered from the tensile stress. So the reduction of excess loss P _e occurred to the toroidal samples is observed. At the optimal annealing temperature T _A, the impact of stress could be reduced effectively, so that the difference between these two kinds of samples at this point becomes the lowest.

Footnotes

Acknowledgements

This work was supported by the National Key Research and Development Program of China (2017YFB0903902), the National Natural Science Foundation of China (Grant No. 51677064).

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