2019 Volume 60 Issue 8 Pages 1577-1582
This paper enhances the Enokizono and Soda (E&S) two dimensional rotating magnetic properties model by integrating the impact of DC-biased field in the E&S model. An experimental set up is developed to apply the DC voltage excitation superimposed on a constant AC excitation, and to investigate their collective impact on the testing sample of electrical steel sheet. The magnetic properties B and H of the testing sample are measured under the influence of rotating and DC-biased magnetic fields. Based on the measurement results of B and H, the E&S model is enhanced by adding rotating reluctivity coefficient (vrk), rotating hysteresis coefficient (vhk), and DC reluctivity coefficient (v0k) to integrate the effect of DC-biased field with rotating magnetic field. Rotating and DC reluctivity coefficients in conjunction with rotating hysteresis coefficient are obtained by Fourier series expansion of experimental B and H waveforms. The accuracy of an enhanced model under DC-biased field is proved by the close agreement between experimental hysteresis loops of electrical steel sheet and the ones obtained by the proposed model.
Fig. 7 Integrated model coefficients extraction and verification procedure.
The wide spread application of power electronics converters causes DC-biased magnetization in the iron core of transformers and electrical motors. The iron loss, temperature and vibration of transformers and electrical motors are increased under the simultaneous AC and DC-biased excitations.1,2) The magnetic properties in the rolling direction of electrical steel sheet under DC-biased field are studied by applying the Epstein system and single-sheet tester.3–5) However, Refs. 3–5) don’t incorporate the effect of rotating magnetic field in motors stator core and the T joint of transformers core while investigating the impact of DC-biased excitation, which leads to inaccurate magnetic field analysis.6)
The flux density vector is not parallel to the magnetic field strength vector; moreover, the direction of the rotating flux varies in a magnetization cycle.7) Enokizono and Soda (E&S) proposed a two-dimensional hysteresis model for describing the properties of rotating magnetic field in the electrical steel sheet.8) However, the E&S model does not incorporate the effect of DC-biased field, which commonly occurs in the cores of motor stator and transformer caused by the power electronics converters.
This paper investigates the rotating magnetic properties under the influence of DC-biased field. Experimental set up has been developed and data is collected when the electrical steel sheet is magnetized by both the rotating magnetic and DC-biased fields. Based on the measured data, the E&S model is improved by introducing rotating reluctivity coefficient (vrk), rotating hysteresis coefficient (vhk), and DC reluctivity coefficient (v0k) to incorporate the effect of DC-biased field on the electrical steel sheet. These parameters are acquired using the Fourier series expansion of the experimental data. The accuracy of the proposed model is investigated by comparing the measured B-H locus with the one obtained by the enhanced model.
The paper is organized as follows: section 2 describes the DC-biased magnetizing conditions and experimental setup for measuring the two dimensional B and H waveforms under DC-biased field. Section 3 uses the measured B and H waveforms, which include the influence of DC-biased field and improves the E&S model of rotating magnetic properties. The accuracy of the enhanced model is investigated in section 4 by comparing the experimental B-H curve results with the ones obtained by the proposed model. Finally, concluding remarks along with the contributions made by this work are presented in section 5 of this paper.
The magnetizing condition of electrical steel sheet under DC-biased field is shown in Fig. 1. The orientation of steel sheet is defined as x, and the direction perpendicular to that is defined as y. Without DC-biased magnetization, the locus of rotating magnetic induction intensity over one magnetizing cycle is approximated by an elliptical shape.9) The locus is identified by two parameters: the maximum rotating magnetic induction intensity Bmax and the axis ratio α, which is the ratio between minimum and maximum rotating magnetic induction intensities.
DC-biased magnetization condition.
When the electrical steel sheet is under DC-biased field, DC excitation is superimposed on the AC excitation. The AC excitation is kept constant and DC-biased field is varied to analyze its impact.10) DC-biased field is defined by two parameters: the amplitude of DC magnetic induction intensity, Bdc, and the DC magnetizing angle, θdc, which is the angle between the direction of DC magnetic induction intensity and the x axis.
2.1 Experimental setup under DC-biased magnetizingFigure 2 shows the overall setup designed for data collection and simultaneous excitation of electrical steel sheet with rotating magnetic and DC-biased fields. The electrical steel sheet sample is placed at the center of the cross-shaped testing setup as shown in Fig. 2(a). The excitation windings are placed in the direction of x axis and y axis around the sample. The yokes around the windings provide a closed path for the magnetic field to suppress the leakage flux. Two coils on the x and y axis are connected separately in series. The coils along x and y axis are simultaneously connected with AC and DC excitation using a power amplifier and DC power supply respectively, which are shown in Fig. 2(b). An excitation signal from a multifunction I/O device is given to the power amplifier to apply AC excitation to the coils. The DC power supply excites the coils along x and y axis with DC-biased field. Two B coils are wound in the middle of the electrical steel sheet sample for measuring magnetic induction intensity along the x and y axis separately. Also, one H coil is located close to each side of the testing sample for measuring magnetic field strength, as shown in Fig. 2(a). The induced voltages from B coils and H coils are filtered and are fed back to the multifunction I/O device after amplification through a preamplifier. The induced voltages are then processed in the computer to calculate the rotating components and DC components of the magnetic induction intensity and magnetic field strength.
(a) The electrical steel sheet sample test setup (b) AC and DC supply and measurement system.
First, the sample and yokes are demagnetized. Next, the offset voltage is measured and removed from the system. Then, the windings are excited with AC supplies. Without DC-biased field, rotating magnetic properties are obtained through the analogue feedback system. In this work, Bmax and α are fixed at 1.0 T and 0.5 respectively. Finally, DC excitation with a controllable slope rate and rise time is added while AC excitation remains unchanged. The angle θdc is varied between 0 and 90 degrees at 15 degree intervals, and Bdc is changed between 0 and 1.5 T with a step of 0.1 T. Multifunction I/O device obtains DC and AC components of B and H separately. The drift in the measurement is reduced by a signal processing.12) Rotating magnetic properties under DC-biased field of different Bdc and θdc are measured. Using the induced voltages obtained from B and H coils, the rotating components of B and H can be calculated by eqs. (1) and (2):
| \begin{equation} B_{ack} = \frac{1}{K_{Bk}}\int v_{Back}dt \end{equation} | (1) |
| \begin{equation} H_{ack} = \frac{1}{K_{Hk}}\int v_{Hack}dt \end{equation} | (2) |
The DC excitation is added with a controllable rising rate and induced voltages from B and H coils are collected. The DC components of magnetic induction intensity and magnetic field strength are calculated by integrating the induced voltages over multiple rotating cycles11) using eqs. (3) and (4):
| \begin{equation} B_{dck} = \frac{1}{K_{Bk}}\int_{0}^{nT}v_{Bk}dt \end{equation} | (3) |
| \begin{equation} H_{dck} = \frac{1}{K_{Hk}}\int_{0}^{nT}v_{Hk}dt \end{equation} | (4) |
This section details the enhanced integrated model which incorporates the effects of rotating magnetic and DC-biased fields on magnetic properties of the electrical steel sheet. Subsection 3.1 analyzes the experimental data acquired under the influence of both the rotating magnetic field and DC-biased field. The enhanced model, based on the experimental data analysis, is proposed in subsection 3.2.
3.1 Analysis of the rotating magnetic properties under DC-biased fieldFigure 3 shows the loci of two-dimensional magnetic quantities Bx-By and Hx-Hy under different DC-biased magnetizing conditions. This figure shows sample experimental results while Bdc varies from 0 to 1.0 T with steps of 0.2 T and Bmax and α are kept constant. The Bx-By and Hx-Hy of 0, 45, and 90 degree magnetizing angles are shown in Fig. 3(a)–(b), (c)–(d), and (e)–(f) respectively. It can be observed that the loci begin to distort under DC-biased field. Also, offset and asymmetry of Bx-By and Hx-Hy loci occur when DC excitation is applied. Larger DC excitation results in more serious distortions of the Bx-By and Hx-Hy loci. The amplitude of magnetic field strength rises with the increase of Bdc. Due to anisotropy of the steel sheet, the distortions of Bx-By and Hx-Hy loci are least obvious when θdc equals to zero. In this case, an increase in the magnetic field strength amplitude is the least evident.
The loci of Bx-By and Hx-Hy under different directions and amplitudes of DC induction intensity (a) Bx-By under 0 degree magnetizing angle (b) Hx-Hy under 0 degree magnetizing angle (c) Bx-By under 45 degrees magnetizing angle (d) Hx-Hy under 45 degrees magnetizing angle (e) Bx-By under 90 degrees magnetizing angle (f) Hx-Hy under 90 degrees magnetizing angle.
Figure 4 shows DC component of magnetic field strength under different DC-biased conditions. Bdc is increased from 0 to 1.5 T and θdc is varied from 0 to 90 degrees. It can be observed from Fig. 4 that DC component of magnetic field strength below 0.8 T is quite small because of the small value of DC excitation. The difference in the field strength between various θdc becomes more obvious when Bdc gets higher than 0.8 T, and DC component of magnetic field strength increases rapidly. The increase of Hdc between 0.8 to 1.5 T is relatively more obvious in the direction of 60 degree when compared to the rest of the angles. This is because 60 degree is the hard magnetization direction of electrical steel sheet.
DC component of magnetic field strength under different Bdc and θdc.
Since the electrical steel sheet is hard to be magnetized at 60 degree, DC-biased field has a more obvious impact on rotating magnetic properties at this angle. The harmonic components of rotating magnetic induction intensity and rotating magnetic field strength at 60 degrees are larger compared to the other angles. Therefore, the magnitude of harmonic components for 60 degree θdc is analyzed and the results are shown in Fig. 5. Figure 5(a) to (d) show that fundamental component of both B and H is the most dominant one, while the magnitude of the rest of the harmonic components decreases as their order increases. The magnitude of 7th order harmonics is very small. Therefore, higher than 7th order harmonics are neglected. Figure 5(a) and (b) show that the fundamental component of rotating magnetic induction intensity along x and y axis has relatively a smaller variation. However, the fundamental component of rotating magnetic field strength rises significantly as Bdc increases, as shown in Fig. 5(c) and (d).
The harmonic components magnitude of rotating magnetic properties when θdc equals to 60 degrees (a) Bacx (b) Bacy (c) Hacx (d) Hacy.
The E&S model has been proposed for describing the two-dimensional magnetic properties of rotating magnetic field. However, this model does not include the effect of DC-biased field on the rotating magnetic quantities. Therefore, an improvement in the E&S model for describing rotating magnetic properties under DC-biased field can be suggested as:
| \begin{equation} H_{k}(\tau) = v_{rk}B_{ack}(\tau) + v_{hk}\int B_{ack}(\tau)\mathrm{d} \tau + v_{0k}B_{dck} \end{equation} | (5) |
The measured magnetic induction intensity and magnetic field strength can be expressed by eqs. (6) and (7) using the Fourier series expansion:
| \begin{equation} B_{k} = \sum_{n = 1}^{N}[R_{nB_{k}}\sin n\tau] + \sum_{n = 1}^{N}[I_{nB_{k}}\cos n\tau] + B_{dck} \end{equation} | (6) |
| \begin{equation} H_{k} = \sum_{n = 1}^{N}[R_{nH_{k}}\sin n\tau] + \sum_{n = 1}^{N}[I_{nH_{k}}\cos n\tau] + H_{dck} \end{equation} | (7) |
| \begin{equation} v_{rk} = \frac{ \begin{array}{l} \displaystyle\sum_{n = 1}^{N}[R_{nH_{k}}\cos n\tau] \cdot \sum_{n = 1}^{N}\left[\dfrac{1}{n}R_{nB_{k}}\sin n\tau \right]\\ \quad + \displaystyle\sum_{n = 1}^{N}[I_{nH_{k}}\sin n\tau] \cdot \sum_{n = 1}^{N}\left[\dfrac{1}{n}I_{nB_{k}}\cos n\tau \right] \end{array} }{ \begin{array}{l} \displaystyle\sum_{n = 1}^{N}\left[\dfrac{1}{n}R_{nB_{k}}\sin n\tau \right] \cdot \sum_{n = 1}^{N}[R_{nB_{k}}\cos n\tau]\\ \quad + \displaystyle\sum_{n = 1}^{N}\left[\dfrac{1}{n}I_{nB_{k}}\cos n\tau \right] \cdot \sum_{n = 1}^{N}[I_{nB_{k}}\sin n\tau] \end{array} } \end{equation} | (8) |
| \begin{equation} v_{hk} = \frac{ \begin{array}{l} \displaystyle\sum_{n = 1}^{N}[I_{nH_{k}}\sin n\tau] \cdot \sum_{n = 1}^{N}[R_{nB_{k}}\cos n\tau]\\ \quad - \displaystyle\sum_{n = 1}^{N}[R_{nH_{k}}\cos n\tau] \cdot \sum_{n = 1}^{N}[I_{nB_{k}}\sin n\tau] \end{array} }{ \begin{array}{l} \displaystyle\sum_{n = 1}^{N}\left[\dfrac{1}{n}R_{nB_{k}}\sin n\tau \right] \cdot \sum_{n = 1}^{N}[R_{nB_{k}}\cos n\tau]\\ \quad + \displaystyle\sum_{n = 1}^{N}\left[\dfrac{1}{n}I_{nB_{k}}\cos n\tau \right] \cdot \sum_{n = 1}^{N}[I_{nB_{k}}\sin n\tau] \end{array} } \end{equation} | (9) |
| \begin{equation} v_{0k} = \frac{H_{dck}}{B_{dck}} \end{equation} | (10) |
Rotating magnetic reluctivity coefficient vrk and hysteresis coefficient vhk are functions of τ. The numerator and denominator of vrk and vhk should be calculated separately to avoid the extreme points where the denominator is equal to zero along with changing τ. Thus, vrk and vhk can be rewritten as vik/vsk and vmk/vsk. Through some algebraic manipulation, the coefficient vik, vmk, and vsk can be expressed by eq. (11):
| \begin{equation} v_{ik},v_{mk},v_{sk} = \sum_{n = 1}^{N}C_{n}^{ik,mk,sk}\sin n\tau \end{equation} | (11) |
The coefficients under different DC magnetizing angles and amplitudes (a) $C_{1}^{ix}$ (b) $C_{1}^{iy}$ (c) $C_{1}^{mx}$ (d) $C_{1}^{my}$ (e) $C_{1}^{sx}$ (f) $C_{1}^{sy}$.
Figure 7 details the step-by-step procedure for experimental data collection, extracting and interpolating the rotating reluctivity coefficient (vrk), rotating hysteresis coefficient (vhk), and DC reluctivity coefficient (v0k) to predict the behavior of B-H loci for any given magnetizing conditions. First, vrk, vhk, and v0k are calculated based on the experimental data using eqs. (6)–(10). Then, coefficients $C_{n}^{ik,mk,sk}$ are calculated using eq. (11). The cubic spline interpolation is applied in the interpolation block of Fig. 7 to acquire the $C_{n}^{ik,mk,sk}$ coefficients and v0k for any DC-biased operating condition.
Integrated model coefficients extraction and verification procedure.
The magnetic field strength for any operating condition can be calculated using the interpolated values of $C_{n}^{ik,mk,sk}$ coefficients and v0k in the enhanced model to verify the accuracy of the proposed model. For example, in this work the enhanced model is tested for two arbitrarily magnetizing conditions: Bdc = 0.62 T, θdc = 22° and Bdc = 1.26 T, and θdc = 67°. The corresponding coefficients $C_{n}^{ik,mk,sk}$ and v0k are extracted from the interpolation results as shown in Fig. 7. Next, vrk and vhk for these operating conditions are calculated using eq. (11). Finally, the B-H loci for these two test cases are constructed based on eq. (5). Also, the B-H loci for the same operating conditions is acquired from the experimental setup. The B-H loci obtained from the enhanced model and the ones that are experimentally obtained are plotted together in Fig. 8(a)–(d) under the aforementioned magnetization conditions. The distortion in the B-H locus under DC-biased field shows that the improved model is able to detect the magnetic saturation. The lag angle difference between H and B shows that the enhanced model can describe magnetic hysteresis of electrical steel sheet. The close agreement between the B-H loci calculated using the proposed model and experimentally acquired loci proves the accuracy of the integrated model for incorporating the effects of DC-biased field on rotating magnetic properties.
The calculated and measured B-H loci (a) Bx-Hx locus when Bdc = 0.62 T, θdc = 22 degrees (b) By-Hy locus when Bdc = 0.62 T, θdc = 22 degrees (c) Bx-Hx locus when Bdc = 1.26 T, θdc = 67 degrees (d) By-Hy locus when Bdc = 1.26 T, θdc = 67 degrees.
This work successfully enhanced the E&S model of two dimensional rotating magnetic properties by integrating the impact of DC-biased field in the model. The experimental setup for applying the rotating magnetic field and DC-biased field is developed for testing the magnetic properties of electrical steel sheet. Rotating magnetic properties under DC-biased field have been measured, and an enhanced model based on the acquired data is developed. Finally, the enhanced model is tested for calculating the magnetic properties under two operating conditions. The magnetic properties acquired experimentally and the ones obtained using the proposed integrated model are compared. The close agreement between the experimental results and the ones obtained by the model prove that the proposed model can effectively and accurately incorporate the impact of DC-biased field on the rotating magnetic properties.
This paper is supported by National Key R&D Program of China (No. 2016YFC0800100).
