Analytical study and corresponding experiments for iron loss inside laminated core under ac-dc hybrid excitation

Abstract

The modeling and reduction of the iron loss is increasingly concerned in both analysis and design of large electromagnetic devices under today’s extreme excitations. This paper aims to investigate the effective and practical approaches to exactly determine the iron loss inside laminated core under ac-dc hybrid excitation in the numerical analysis, and validate the proposals based on the product-level models.

Keywords

Ac-dc hybrid excitation magnetic property measurement and numerical analysis iron loss

1. Introduction

With the development of the HVDC transmission system, the ac power transformer near the HVDC system is often affected by the dc current, which will flow into the windings through earthed neutrals. A laminated iron core under ac-dc hybrid excitation generates a distorted hysteresis loop, resulting in a higher iron loss compared with that under sinusoidal excitation [1, 2, 3].

Under ac-dc hybrid excitation, the major difficulties of the iron loss modeling are: (1) coming from magnetic property: the magnetic property data, such as B-H properties, are quite different from those under “standard” sinusoidal excitation, which are varying with both the dc excitation and the ac excitation [1], and the existing property data of the magnetic material provided by manufacturers are usually not sufficient or none; (2) coming from loss calculation method: the iron loss is not only the function of the amplitude of the ac component of flux density ( $B_{m}$ ), but also the dc-biased flux density ( $\Delta B$ ) dependent, so the traditional calculation method of the iron loss, based on the specific loss data ( $B_{m}-W$ curve), is not feasible.

The purpose of this paper is to present a practical measurement method for the magnetic properties of laminated cores under ac-dc hybrid excitation, to investigate the effect of the variation of dc bias on iron loss, and to find an effective and practical numerical approach to confidently determine the iron loss inside GO silicon steel laminated core of large electromagnetic devices with ac-dc hybrid excitation, and validate their usefulness based on typical engineering-oriented models.

2. Modeling of magnetic property

In this paper, a novel measurement scheme for the magnetic properties of GO silicon steel lamination under ac-dc hybrid excitation is proposed. The newly developed system has the advantage that there is no interplay between ac and dc excitations, both ac and dc excitation applied to one and the same exciting winding, making the control easy, compared with the usual technique using a ring core or SST, having separate ac and dc exciting windings [4, 5].

Measurements of the magnetic properties under ac-dc hybrid excitation were carried out with the aid of a laminated core model (referred to as LCM), at the product level, with 45 ${}^{\circ}$ mitred step-lap joints, made of GO silicon steel 30Q140, as shown in Fig. 1.The detailed structural dimension and some individual design parameters for the LCM are shown in Fig. 1b and Table 1.

Table 1
Parameters of core and coils

Parameters	Unit	LCM	Remarks
Number of turns of exciting coil	turns	108/212/312	Coil turns can be changed
Number of turns of search coil	turns	108/212/312
Mean length of magnetic path	mm	1240	–
Length of silicon sheet	mm	420	–
Width of silicon sheet	mm	110	–
Net area of cross section	mm ${}^{2}$	2667.5	Packing factor: 0.975
Total weight of iron core	kg	26.3

Figure 1.

Laminated core model of product quality. (a) Photo of laminated core model. (b) Structural dimensions (mm).

The experimental scheme based on the LCM is set up as shown in Fig. 2. The dc current passing through the exciting winding of the LCM, supplied by the dc source, is in series with the ac current. $U_{1}$ and $U_{2}$ is the exciting voltage and induced voltage of the LCM, respectively. The power analyzer WT-3000, Yokogawa, is used for measuring the output from the exciting coils and search coils of the LCM.

Figure 2.

Experimental system of dc-biased magnetic property.

Figure 3a shows the example of hysteresis loop and the definitions of physical values under ac-dc hybrid excitation. Figure 3b and c show waveforms of flux density and magnetic field strength, respectively. $\Delta B$ is the dc-biased flux density and $H_{dc}$ is the dc-biased magnetic field strength. $H_{b}$ is the magnetic field strength at the instant when flux density reaches the maximum; when an average value of eddy current in an electrical steel sheet is zero. $B_{m}$ is the amplitude of the ac component of total flux density, and the waveform of the ac component of $b$ (total flux density) is controlled as a sinusoidal wave.

Figure 3.

Distorted asymmetrical hysteresis loop under dc-biased magnetization. (a) Hysteresis loop under dc bias. (b) Total flux density waveform. (c) Magnetic field strength waveform.

The magnetic properties of the GO silicon steel sheet laminations are measured under dc-biased working condition, using the proposed experimental system. The ac exciting voltage $U_{1}$ is adjustable, allowing the ac working point ( $B_{m}$ , see Fig. 3) of the LCM to vary from 0.1 T to 1.7 T; the dc current applied to LCM is also adjustable, allowing the dc-biased magnetic field strength ( $H_{dc}$ , see Fig. 3) to vary from 25 A/m to 425 A/m (see Fig. 2).

Figure 4.

DC-biased B-H property (30Q140 silicon steel lamination, $B_{m}=$ 0.2 T, 50 Hz).

Figure 4 shows the effect of dc bias on the B-H property, demonstrating that the area of hysteresis loop increases when $H_{dc}$ is large, and resulting in an increase in the total iron loss when there is a large dc bias. Note that, the $B$ in Fig. 4, is only the ac component of the total magnetic flux, because of the dc component has not been captured by the search coil in the measurement.

Figure 5.

DC-biased specific total loss (30Q140 silicon steel lamination, 50 Hz).

Figure 5 shows the effect of dc bias on the iron loss at 50 Hz. Figure 6 shows the comparison of total iron losses at different $H_{dc}$ ( $B_{m}=$ 0.2 T, 0.7 T, 1.0 T and 1.7 T, 50 Hz).

For 30Q140 lamination, the total iron loss $W_{t}$ at $H_{dc}=$ 425 A/m is about 842% ( $B_{m}=$ 0.2 T), 180% ( $B_{m}=$ 0.7 T), 93% ( $B_{m}=$ 1.0 T) and 7% ( $B_{m}=$ 1.7 T) higher compared to the total loss at $H_{dc}=$ 0 A/m, respectively.

It is clear that the total iron loss increases more slowly at higher ac working points (e.g. $B_{m}=$ 1.7 T) when $H_{dc}$ is large ( $H_{dc}=$ 425 A/m) compared to lower ac working points (e.g. $B_{m}=$ 0.2 T). This is mainly because of the saturation effect of the lamination when both the $B_{m}$ and $H_{dc}$ are strong.

3. Analysis and validation of iron loss

In this paper, a practical approach for calculation of iron loss under ac-dc hybrid excitation is presented, and the numerical modeling results are verified by the measured data.

Effects of variation of dc-biased magnetic field intensity $H_{dc}$ and ac working magnetic flux density $B_{m}$ on the iron loss and flux are examined in detail. The exciting mode of LCM is listed in Table 2.

Table 2
Different excitation conditions

Cases	$B_{m}$ (T)/ $H_{dc}$ (A/m)	Cases	$B_{m}$ (T)/ $H_{dc}$ (A/m)
1	0.5/100	6	0.5/400
2	1.0/100	7	1.0/400
3	1.4/100	8	1.4/400
4	1.7/100	9	1.7/400
5	1.8/100	10	1.8/400

Figure 6.

Iron loss under dc-biased magnetization (50 Hz). (a) $B_{m}=$ 0.2 T. (b) $B_{m}=$ 0.7 T. (c) $B_{m}=$ 1.0 T. (d) $B_{m}=$ 1.7 T.

2-D magnetic field distribution in the LCM is examined, as shown in Fig. 7, where the different flux distributions caused by the different treatment of permeability, can be clearly seen.

Figure 7.

2-D magnetic field distribution. (a) Isotropic permeability. (b) Anisotropic permeability.

3.1 Treatment of anisotropy

The well established ${\bm{T}}$ - $\Omega$ method is used to solve for the magnetic field, governing equation involving anisotropic and nonlinear materials in eddy current region is given by Eq. (1) [6],

$\nabla\times\left(\frac{1}{c_{p}}[\sigma]^{\;-1}\nabla\times{\bm{T}}\right)+[% \mu]\frac{\partial({\bm{T}}-\nabla\Omega)}{\partial t}=0$ (1)

The anisotropic and nonlinear permeability [ $\mu$ ] of Eq. (1) can be represented by Eq. (2),

$[\mu]=\left[{{\begin{array}[]{*{20}c}{\mu_{0}/(1-c_{p})}\hfill&\hfil&\hfil\\ \hfil&{c_{p}\mu_{y}}\hfill&\hfil\\ \hfil&\hfil&{c_{p}\mu_{z}}\hfill\\ \end{array}}}\right]$ (2)

Note that in the numerical implementation the vector magnetic property of the material can be easily taken into account. The anisotropic conductivity [ $\sigma$ ] of the sheets can be dealt with as

$[\sigma]=\left[{{\begin{array}[]{*{20}c}{c_{p}\sigma_{x}}\hfill&\hfil&\hfil\\ \hfil&{c_{p}\sigma_{y}}\hfill&\hfil\\ \hfil&\hfil&{c_{p}\sigma_{z}}\hfill\\ \end{array}}}\right]$ (3)

where $\sigma_{y}=\sigma_{z}=\sigma_{yz}$ in sheets, $c_{p}$ is the packing factor of the laminations [7].

To effectively reduce the computational cost, only the first few laminations are modeled individually as shown in Fig. 8a.

An air region is used to analyze the leakage flux in the surface layers. The inner laminated sheets are modeled as bulk and the electric conductivity ( $\sigma$ ) is modeled as anisotropic. The tangential conductivity (y-z plane) is isotropic, but the normal conductivity (x-axis) is zero (or near to zero due to non-zero $\sigma$ limitation for [ $\sigma$ ]-1 in T- $\Omega$ ), as shown in Fig. 8b.

Figure 8.

3-D eddy current simulation model. (a) Surface layer. (b) Inner bulk.

A zoned analysis method is employed in the GO silicon steel sheet’s region, i.e., a very thin mesh used in the surface layer and a coarse mesh in the remaining inner bulk region of the finite element model [7, 8, 9].

To further reduce the cost of the electromagnetic field computation in the bulk region, especially for a large-scale laminated transformer core, all the eddy currents induced in the bulk domain are neglected. So the induced eddy currents in its surface layer and then the additional eddy current loss caused by normal leakage flux into the surface layer of the laminated core have to be taken into account. The zoned conductivity [ $\sigma$ ] inside the surface layer and the inner bulk of the lamination is applied, as shown in Eq. (4),

$\displaystyle[\sigma]=\left\{\begin{array}[]{ll}\left[{{\begin{array}[]{ccc}0% \hfill&\hfil&\hfil\\ \hfil&{\sigma_{y}}\hfill&\hfil\\ \hfil&\hfil&{\sigma_{z}}\hfill\\ \end{array}}}\right]\\ \left[0\right]\\ \end{array}\right.\;{\begin{array}[]{*{20}c}\hfil\\ {\textit{for surface layer}}\hfill\\ \hfil\\ \textit{for inner bulk}\hfill\\ \end{array}}$ (4)

The magnetic property of the GO silicon steel 30Q140 is certainly anisotropic, however, the applied field to LCM is almost along the rolling direction. Therefore, the working property of the LCM is, in fact, weakly anisotropic; so the measured specific total iron loss curves $W_{t}-B_{m}$ and B-H property, see Figs 4 and 5, are sufficient.

3.2 Modeling of iron loss

The eddy current can be neglected in the electromagnetic analysis of the inner bulk because the sheet is very thin (0.3 mm thick). Of course, $B_{m}$ obtained in this way will be somewhat different from the real flux density $B_{m}$ . Fortunately, the measured specific total iron loss of the LCM already includes the eddy current loss induced by the planar flux(x-y plane, x-z plane, and y-z plane, see Fig. 8).

When measuring dc-biased magnetic properties using the proposed measuring system, only the tangential excitation is applied; the flux sometimes enters the laminated sheets perpendicularly and that leads to an extra eddy current loss which is referred to as an additional iron loss; the eddy current reaction from such excitation condition is however much lower and can be neglected.

Therefore, the total iron loss $W_{\textit{lamination}}$ inside the GO silicon steel sheet laminations at the specified $H_{dc}$ (or $\Delta B$ ) can be calculated by Eq. (5),

$W_{\textit{lamination}}=\sum\limits_{e=1}^{NE}{\left[{\,W_{t}^{(e)}(B_{m}^{(e)% })\,}\right]}\;V^{(e)}$ (5)

where $B_{m}^{(e)}$ and $V^{(e)}$ are the peak value of the flux density along the rolling direction inside the LCM and the volume per element, respectively. $W_{t}^{(e)}$ is the total iron loss per element. It is a function of $B_{m}$ and $H_{dc}$ (or $\Delta B$ ) due to the dc-biased magnetization. A post-processing method for establishing the relationship between $W_{t}$ and $B_{m}$ (without $\Delta B$ ) is proposed, as shown in Eq. (6), producing the measured $W_{t}-B_{m}$ curve, mentioned above.

$\left\{{\begin{array}[]{l}B=(B_{\max}-B_{\min})/2=B_{m}\\ B_{\max}=B_{m}+\Delta B\\ B_{\min}=\Delta B-B_{m}\\ \end{array}}\right.$ (6)

where, the relationship between $B_{\max}$ , $B_{\min}$ and $B_{m}$ , as shown in Fig. 9.

Note that, the measured $W_{t}-B_{m}$ curve at the specified $H_{dc}$ , in Fig. 5, is used in the field computation; i.e., the effect of dc-biased magnetic field intensity $H_{dc}$ on the total iron loss is already included in Eq. (5). Finally, the total iron loss $W_{\textit{lamination}}$ of Eq. (5) could turn out to be a function of $B_{m}$ and $H_{dc}$ .

Table 3

Total iron loss under different exciting cases

Iron loss (W/kg)
Cases	Measured	Calculated	Cases	Measured	Calculated
1	0.2654	0.2437	6	0.4034	0.3696
2	0.6122	0.5715	7	0.7555	0.7202
3	0.9787	0.9186	8	1.1038	1.0749
4	1.4388	1.3665	9	1.5034	1.4392
5	1.7152	1.6483	10	1.7487	1.6962

Table 4

Measured and calculated average flux

Flux inside iron core (mWb)
Cases	Measured	Calculated	Cases	Measured	Calculated
1	4.706	4.738	6	5.200	5.234
2	4.891	4.903	7	5.302	5.337
3	4.957	4.971	8	5.375	5.392
4	5.051	5.082	9	5.409	5.418
5	5.144	5.179	10	5.425	5.431

Figure 9.

Definition of $B_{\max}$ , $B_{\min}$ and $B_{m}$ under dc bias.

3.3 Modeling of flux inside lamination

The flux inside the entire lamination structure for each test case is examined. The magnetic flux $\phi$ is indirectly determined from the measured emf $e(t)$ at several time steps as follows:

$\phi=\frac{1}{n}\sum\limits_{i=1}^{N_{i}}{e(t_{i})}\Delta t$ (7)

where $n$ ( $=$ 312) is the number of turns of the search coil (as shown in Fig. 1), $N_{t}$ , the number of time step in one period, $e(t_{i})$ , the emf at $t_{i}$ , and $\Delta t$ , the time step.

Note that, the average flux determined by Eq. (7) is only the ac component of flux; however, an efficient approach to calculating the dc flux in the laminated core has been proposed and presented in [10].

3.4 Results and discussion

The measured and calculated results of total iron loss and average magnetic flux inside dc-biased LCM are shown in Tables 3 and 4 respectively. Tables 3 and 4 show good agreement between the measured and calculated results for each exciting case.

4. Conclusion

A new measuring system for the magnetic properties of Grain Oriented silicon steel lamination under dc-biased magnetization is proposed. The proposed method has an advantage in making the control easy since there is no interaction between ac and dc excitations.

The effect of the different dc-biased excitation patterns on both the iron loss and flux inside the laminated silicon steel sheets is investigated in detail. All the calculated and measured results of both iron loss and flux for different test cases are in good agreement. This proves that the proposed practical approach is effective in dealing with the dc-biased lamination configuration.

Footnotes

Acknowledgments

This project was supported in part by the NSFC (51107026/51677052/51237005).

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Analytical study and corresponding experiments for iron loss inside laminated core under ac-dc hybrid excitation

Abstract

Keywords

1. Introduction

2. Modeling of magnetic property

Table 1 Parameters of core and coils

Table 2 Different excitation conditions

4. Conclusion

Footnotes

Acknowledgments

References

Table 1
Parameters of core and coils

Table 2
Different excitation conditions