Effective versus standard Epstein loss figure in Fe-Si sheets

Abstract

The magnetic power losses have been measured at 50 Hz and different peak polarization values on different types of non-oriented and grain-oriented Fe-Si sheets using the Epstein frame, according to the current standards. The very same measurements have then been repeated by measuring polarization and tangential magnetic field by means of localized windings, centrally placed on the strips inside the Epstein frame windings, thereby retrieving the effective field and the true power loss figure. It is obtained that the ratio of the standard $P_{\textit{epst}}$ to the effective $P_{\textit{eff}}$ loss figure, which can be interpreted in terms of ratio of effective $l_{\textit{eff}}$ to conventional ( $l_{m}=$ 0.94 m) magnetic path length, evolves with the peak polarization $J_{p}$ , showing, in general, a monotonic increase with increasing $J_{p}$ . The deviation of $P_{\textit{epst}}$ from $P_{\textit{eff}}$ is observed to range from about $-$ 3% in the non-oriented alloys at low inductions to about $+$ 5% in the grain-oriented alloys at $J_{p}=$ 1.8 T. This behavior finds a rationale in the existence of a polarization profile $J_{p}(x)$ measured along the strip length and in the dependence of $P_{\textit{eff}}$ on $J_{p}$ , showing a power law $P_{\textit{eff}}(J_{p})\propto J_{p}^{n}$ , with $n>$ 1 and increasing with $J_{p}$ . The so calculated effective path length $l_{eff}=l_{m}\cdot P_{\textit{epst}}$ / $P_{\textit{eff}}$ consistently shows a monotonic increase with $J_{p}$ , which is more relevant in the GO alloys.

Keywords

Magnetic power losses Epstein frame magnetic steels

1. Introduction

The 25 cm Epstein test-frame with a defined magnetic path length $l_{m}=$ 0.94 m is a solidly assessed method for the characterization of the magnetic steel sheets. It ensures good measurement reproducibility and is widely adopted as an industry standard [1, 2, 3, 4]. It is also well established that the specific features of the employed magnetic circuit, with the double overlapping corners and the ensuing inhomogeneous flux distribution, make the value of the measured quantities, namely the magnetic losses, different from the value of the true quantities [5, 6]. True values could, however, be possibly accounted for by incorporating the complex response of the magnetic circuit into an effective magnetic path length $l_{\textit{eff}}$ , depending on the type of sheet, peak induction value, frequency, and type of excitation [7]. Should the effective (true) power loss figure $P_{\textit{eff}}$ be measured, one could express it in terms of standard power loss value $P_{\textit{epst}}$ according to

$P_{\textit{eff}}=P_{\textit{epst}}\cdot l_{m}/l_{\textit{eff}}.$ (1)

Measurements of $P_{\textit{eff}}$ using a single strip tester and an $H$ -coil were reported for non-oriented (NO) and grain-oriented (GO) sheets by Ahlers et al. [8]. They found at 50 Hz $P_{\textit{epst}}<P_{\textit{eff}}$ , that is $l_{\textit{eff}}<l_{m}$ in all materials, with maximum difference of the order of 8% in GO sheets at $J_{p}=$ 1.7 T. These authors justified their results in terms of effective length, expressed as $l_{\textit{eff}}=l_{0}+(\mu_{l}$ /2 $\mu_{c})\cdot l_{c}$ , the sum of the legs length $l_{0}$ and part of the corners length $l_{c}$ , depending on the ratio of the leg to corner permeabilities $\mu_{l}$ and $\mu_{c}$ . A number of literature experiments [5] actually show scattered outcomes, both in NO and GO materials, with $P_{\textit{epst}}$ either higher or lower than $P_{\textit{eff}}$ , and $P_{\textit{epst}}$ / $P_{\textit{eff}}$ generally decreasing with the peak polarization $J_{p}$ . Marketos et al. [7, 9], have determined $l_{\textit{eff}}$ by measuring GO and NO sheets with conventional 25 cm and reduced 17.5 cm Epstein frames. They find, assuming identical flux distribution in the corners of the two frames, that $l_{\textit{eff}}$ is always higher than $l_{m}$ (that is, $P_{\textit{epst}}$ / $P_{\textit{eff}}>1$ ) in GO high-permeability sheets and, in contrast with the results reported in [5], increasing with $J_{p}$ .

In this paper we discuss measurements of power losses performed at 50 Hz on different types of NO and GO steel sheets, both according to the measuring standard (25 cm Epstein frame with $l_{m}=$ 0.94 m) and by detecting the tangential field and the induction derivative by collinear narrow $H$ - and $B$ -coils windings, placed directly upon the steel strips at the centre of the Epstein legs. In this way, the true power loss $P_{\textit{eff}}(J_{p}$ ) is obtained in comparison with the standard loss figure $P_{\textit{epst}}(J_{p}$ ). A ratio $P_{\textit{epst}}$ / $P_{\textit{eff}}$ monotonically increasing with $J_{p}$ is thus found in all materials. This can equivalently be expressed in terms of an effective path length similarly increasing with $J_{p}$ .

Figure 1.

Arrangement of the local sensing coils inside a leg of the Epstein frame. The flat multiturn $H$ -coil (thickness $\sim$ 1.5 mm) is placed in contact with the Epstein strip surface and the $B$ -coil is wound around it and the steel strips. The coils are about 17 mm wide and can slide along the whole length of the Epstein leg.

2. Experimental method

The magnetic measurements were performed at 50 Hz under sinusoidal induction waveform on the NO and GO alloys listed in Table 1. Both conventional (CGO) and high-permeability (HGO) grain-oriented sheets were investigated. A calibrated hysteresisgraph-wattmeter with digital control of the induction waveform was used, where signal acquisition and A/D conversion is made by means of a 12-bit 500 MHz HDO4054 LeCroy oscilloscope. The whole measuring process is performed within an Agilent VEE environment. The NO and GO alloys were tested in the polarization intervals 0.5 T–1.5 T and 1.0 T–1.8 T, respectively, with either eight or twelve strips inserted in the Epstein frame, depending on the sheet thickness. For any given material, a standard measurement was first made, followed by measurements with the centrally placed local windings. The arrangement of the 17 mm wide $H$ - and $B$ -coils, which are stuck together and placed inside the Epstein windings, is schematically shown in Fig. 1. The $H$ -coil (turn-area $N_{H}S_{H}=$ 2.11 $\cdot$ 10 ${}^{-2}$ m ${}^{2}$ , thickness $\sim$ 1.5 mm) is made of a few hundred turns (wire diameter 0.05 mm) wound on a rigid fibreglass plate and calibrated inside a field reference setup [10]. The 50-turn $B$ -coil enwraps the $H$ -coil and the strips under test. The air-flux contribution is usually negligible, but it is in any case compensated via software. Once the single standard Epstein measurement of the power loss $P_{\textit{epst}}(J_{0p}$ ) at a given polarization $J_{0p}$ is done, the local d $B$ /d $t$ and d $H_{\textit{eff}}$ /dt signals are simultaneously detected, under the identical exciting conditions, at a significant number of points, from corner to corner, along the length of the Epstein leg, amplified by calibrated low-noise amplifiers SR560, and integrated, These local measurements are then identically repeated on the other legs and the results are averaged. They provide the behaviours of $J(x)$ and $H_{\textit{eff}}(x)$ as a function of the distance $x$ from the centre of the leg for the Epstein measured polarization $J_{0p}$ . The distance $x$ ranges between $-$ 110 mm and $+$ 110 mm. Finally, the $H$ - and $B$ -coils are moved to the centre of the leg ( $x=$ 0) and the measurement of $P_{\textit{eff}}(J_{0p}$ ) is performed by imposing the local d $J$ /d $t$ sinusoidal with peak polarization value $J_{p}(0)=J_{0p}$ . It is remarked that across the 17 mm wide region occupied by the coils centred at $x=$ 0 the polarization is highly uniform. Given the low field levels involved, a certain background noise in the d $H_{\textit{eff}}$ /d $t$ signal is inevitable. This is dealt with by repeating the very same measurement a number of times, to make the random uncertainty contribution negligible. The process is further repeated on the other legs and the results are averaged.

Table 1
Physical parameters of the investigated non-oriented (NO), conventional (CGO) and high-permeability (HGO) grain-oriented steel sheets

Fe-Si alloy	Composition	Thickness (mm)	Density (kg/m ${}^{3})$	Resistivity ( $\Omega\cdot$ m)
NO-1	Fe-(3.2 wt%)Si	0.194	7650	52 $\cdot$ 10 ${}^{-8}$
NO-2	Fe-(3.5 wt%)Si	0.343	7600	56.4 $\cdot$ 10 ${}^{-8}$
CGO-1	Fe-(3 wt%)Si	0.255	7650	48 $\cdot$ 10 ${}^{-8}$
CGO-2	Fe-(3 wt%)Si	0.261	7650	48 $\cdot$ 10 ${}^{-8}$
HGO-1		0.257
HGO-2		0.295

3. Experimental results and discussion

The general outcome of the measurements performed on the six different types of soft magnetic steels described in Table 1, two NO sheets of thickness 0.194 mm and 0.343 mm, two conventional and two high-permeability GO sheets, of thickness ranging between 0.255 and 0.295 mm, is that the standard Epstein loss figure $P_{\textit{epst}}$ can either overestimate or underestimate the effective power loss $P_{\textit{eff}}$ , but the ratio $P_{\textit{epst}}$ / $P_{\textit{eff}}$ is always a monotonically increasing function of $J_{p}$ . The local $J_{p}=J_{p}$ (0) value involved in the measurement of $P_{\textit{eff}}$ is obviously made to coincide with the polarization value $J_{0p}$ previously determined through the whole Epstein secondary winding. While this result may appear partly at odds with previous literature outcomes [5, 8], we shall observe in the following how the behavior of $P_{\textit{epst}}$ / $P_{\textit{eff}}$ can be justified in terms of inhomogeneity of the induction along the Epstein legs and the power law dependence of $P_{\textit{eff}}$ on $J_{p}$ . Let us therefore observe in Fig. 2 the overall experimental behaviors of ( $P_{\textit{epst}}-P_{\textit{eff}})$ / $P_{\textit{eff}}$ versus $J_{p}$ and of the related effective magnetic path length $l_{\textit{eff}}=l_{m}\cdot(P_{\textit{epst}}$ / $P_{\textit{eff}}$ ). Similar trends versus $J_{p}$ are followed by the NO and GO materials, but the standard power loss $P_{\textit{epst}}$ becomes significantly higher, around 4 %–5 %, than the true loss $P_{\textit{eff}}$ at the highest $J_{p}$ values in the GO sheets. Table 2 provides a comparison of the measured power losses $P_{\textit{epst}}$ and $P_{\textit{eff}}$ .

Table 2
Power loss at 50 Hz obtained by the standard Epstein measurement ( $P_{\textit{epst}}$ ) and the localized measurement ( $P_{\textit{eff}}$ ) as a function of peak polarization on three different Fe-Si sheets

$f=$ 50 Hz	NO-1		CGO-2		HGO-1
$J_{p}$ (T)	$P_{\textit{epst}}$ (W/kg)	$P_{\textit{eff}}$ (W/kg)	$P_{\textit{epst}}$ (W/kg)	$P_{\textit{eff}}$ (W/kg)	$P_{\textit{epst}}$ (W/kg)	$P_{\textit{eff}}$ (W/kg)
0.50	0.241	0.248	–	–	–	–
0.75	0.480	0.489	–	–	–	–
1.0	0.784	0.755	0.345	0.343	0.303	0.299
1.2	1.125	1.135	0.499	0.498	0.431	0.426
1.4	1.656	1.665	–	–	–	–
1.5	2.070	2.050	0.811	0.787	0.681	0.664
1.7	–	–	1.198	1.154	0.914	0.874
1.8	–	–	1.587	1.514	1.123	1.063

Figure 2.

The experimental dependence on $J_{p}$ of the ratio of standard to true power losses $P_{\textit{epst}}$ / $P_{\textit{eff}}$ in NO and in conventional (CGO) and high-permeability (HGO) Fe-Si alloys (a) is paralleled by the behavior of the effective magnetic path length $l_{\textit{eff}}$ (b).

Figure 3.

Effective field $H_{\textit{eff}}$ measured versus the distance $x$ from the Epstein leg centre in the NO-2 sheet. It is $H_{\textit{eff}}=H_{\textit{sol}}-H_{d}$ , the difference between the field $H_{\textit{sol}}$ generated by the primary winding and the field $H_{d}$ originating from the free poles distributed along the strip length. It is worth to note the demagnetizing and magnetizing effect of $H_{d}$ beneath and outside the solenoid length. The horizontal dotted line shows the conventional field $H_{\textit{epst}}$ , calculated assuming the magnetic path length $l_{m}=$ 0.94 m.

In order to find a rationale for the $J_{p}$ dependent relationship between $P_{\textit{epst}}$ and $P_{\textit{eff}}$ , it is useful to analyze the distribution of field and polarization along the magnetic circuit, as retrieved by the previously described local measurements. To start with, we provide an example in Fig. 3, concerning the NO-2 sheet, of field decomposition along a leg of the frame for a standard Epstein measurement at a given polarization value $J_{p}$ . The solenoid surrounding each leg has length 2 $L=$ 195 mm. By subtracting the effective field $H_{\textit{eff}}(x)$ , measured by sliding the $H$ -coil along the leg, from the field $H_{\textit{sol}}(x)$ applied by the primary solenoid, we obtain the behavior of the magnetostatic field $H_{d}(x)$ . This exerts a demagnetizing action towards the strip portion of length 2 $L$ underlying the Epstein winding, adding instead to $H_{\textit{sol}}$ towards the corners, to eventually impose a magnetic path length $l_{\textit{eff}}$ longer than the solenoid length. The effect of $H_{d}$ is less important at high inductions, where the permeability is lower and the free poles are more localized around the solenoid edges. The distribution of the polarization $J_{p}(x)$ along 2 $L$ is however moderately affected, as shown by the examples regarding the samples NO-2 and HGO-1 shown in Fig. 4. These curves are representative of the flux distribution found in all materials and bring to light the fact that, because of the strong non-linear dependence of the power loss on $J_{p}$ , the true loss value cannot be recovered by a standard Epstein measurement, adjusted through a simple constant (in this case the conventional magnetic path length $l_{m})$ . By denoting the polarization measured through the secondary Epstein winding $J_{0}(t)=J_{0{p}}\sin(\omega t)$ , where $J_{{0p}}=\frac{1}{2L}\int_{-L}^{L}J{}_{p}(x)\,dx$ , we can write the power loss per unit volume measured with the standard method

$P_{\textit{epst}}(J_{\textit{0p}})=\frac{1}{T}\int\limits_{0}^{T}{H_{\textit{% epst}}}(t)\cdot J_{\textit{0p}}\omega\cos(\omega t)dt,$ (2)

where $H_{\textit{epst}}(t)=N_{H}i_{H}(t)$ / $l_{m}$ , $N_{H}$ is the number of turns of the primary winding and $i_{H}(t)$ is the magnetizing current. The true power loss corresponding to the condition met with the standard Epstein measurement is therefore given by the average of the local $P_{\textit{eff}}(x)$ across the length 2 $L$

$P_{\textit{eff}}(J_{\textit{0p}})=\frac{1}{2L}\int\limits_{-L}^{L}{dx}\,\;% \frac{1}{T}\int\limits_{0}^{T}{H_{\textit{eff}}}(t,x)\cdot J_{\textit{p}}(x)% \cdot\omega\cos(\omega t)dt=\frac{1}{2L}\int\limits_{-L}^{L}{P_{\textit{eff}}}% (J_{\textit{p}}(x))\;dx.$ (3)

We thus obtain the effective magnetic path length corresponding to such condition

$l_{\textit{eff}}(J_{\textit{0p}})=l_{\textit{m}}\cdot\frac{P_{\textit{epst}}(J% _{\textit{0p}})}{P_{\textit{eff}}(J_{\textit{0p}})}.$ (4)

Figure 4.

Examples of measured distribution of the reduced polarization $J(x)$ / $J$ (0) upon the portion of strip length underlying the Epstein secondary winding in the NO-2 and HGO-1 sheets.

Figure 5.

The measured effective power loss $P_{\textit{eff}}(J_{p}$ ) increases with the peak polarization $J_{p}$ according to a power law $P_{\textit{eff}}\propto J_{p}^{n}$ , with $n$ an increasing function of $J_{p}$ .

The quantity $P_{\textit{eff}}(J_{0p})$ can be measured, according to Eq. (3), by integrating the previously discussed local measurements, which are represented as a function of $J_{p}$ in Fig. 5, over the $J_{p}(x)$ distribution shown in Fig. 4. This is easily done through knowledge of the measured dependence of $P_{\textit{eff}}(J_{p})$ on $J_{p}$ , which, as shown in Fig. 5, follows a power law $P_{\textit{eff}}(J_{p})\propto J_{p}^{n}$ , with $n$ an increasing function of $J_{p}$ . By introducing the $P_{\textit{eff}}(J_{0p})$ calculated by Eq. (3), we obtain the behavior of $l_{\textit{eff}}(J_{0p}$ ) in the different materials shown in Fig. 6. On the other hand, the previously defined $P_{\textit{eff}}(J_{p})$ (Fig. 2) is the true loss measured at $x=$ 0 when $J_{p}$ (0) $=$ $J_{p}$ , which is compared with the Epstein power loss when the secondary winding provides the same polarization value $J_{p}$ . $l_{\textit{eff}}(J_{p}$ ) is correspondingly defined through Eq. (1). Consequently, $l_{\textit{eff}}(J_{0p}$ ) and $l_{\textit{eff}}(J_{p}$ ) do not usually coincide. $l_{\textit{eff}}(J_{0p}$ ) is, in any case, the magnetic path length to be applied in substitution of $l_{m}$ when making the standard Epstein testing at 50 Hz at the specific measured polarization level $J_{0p}$ . It takes into account the fact that the peak polarization $J_{0p}$ measured by the secondary Epstein winding is the average of $J_{p}(x)$ between $\pm\ L$ , according to the behaviors of $J_{p}(x)$ shown in Fig. 4. It will coincide with $l_{\textit{eff}}(J_{p})$ for homogeneous $J_{p}(x)$ across the solenoid length 2 $L$ , a limiting unattainable condition. The calculated $l_{\textit{eff}}(J_{0p}$ ) and the measured $l_{\textit{eff}}(J_{p}$ ) nevertheless display quite similar increasing trends vs. $J_{p}$ , both in NO and GO alloys, as illustrated by their behaviors shown in Fig. 6. It is appreciated the fact that, as shown in Figs 2 and 6, the effective magnetic path length in the GO alloys tends to be higher than in the NO sheets, besides being larger than the conventional Epstein value $l_{m}=$ 0.94 m. This implies that $P_{\textit{epst}}$ / $P_{\textit{eff}}$ is similarly higher. The present results actually show that $J_{p}(x)$ is more homogeneous in the GO strips (see the example shown in Fig. 4), thereby better approaching the ideal condition of perfectly homogeneous magnetization over the whole 1 m long Epstein circuit.

Figure 6.

Effective magnetic path lengths $l_{\textit{eff}}(J_{0p})$ and $l_{\textit{eff}}(J_{p})$ versus peak polarization in the investigated NO and GO steel sheets. $l_{\textit{eff}}(J_{0p})$ is calculated through Eqs (2)–(4). It permits one to retrieve the true power loss value from the standard loss figure for peak polarization $J_{0p}$ measured with the secondary Epstein winding. $l_{\textit{eff}}(J_{p})$ is the same quantity obtained for $J_{p}=J_{p}$ (0), where $J_{p}$ (0) is the polarization measured at the centre of the Epstein leg.

4. Conclusions

Measurements of the true power losses in different types of non-oriented and grain-oriented materials using localized $H$ - and $B$ -coils placed inside the legs of a standard 25 cm Epstein frame have been compared, upon a range of peak polarization values, with the loss figures obtained according to the usual procedure prescribed by the measuring standards. It is found that true $P_{\textit{eff}}$ and standard $P_{\textit{epst}}$ power loss figures are in a relationship dependent on the imposed peak polarization value $J_{p}$ , with the ratio $P_{\textit{epst}}$ / $P_{\textit{eff}}$ exhibiting a monotonic increase with $J_{p}$ in all materials in the investigated polarization range 0.5 T $\leqslant J_{p}\leqslant$ 1.8 T. This behavior, which can be interpreted in terms of an effective magnetic path length $l_{\textit{eff}}$ , to be used as a substitute for the conventional fixed length $l_{m}=$ 0.94 m in the expression for the applied field, is justified in terms of non-uniform profile of the polarization $J_{p}(x)$ along the strip portion underlying the Epstein secondary winding and non-linear increase of $P_{\textit{eff}}$ with $J_{p}$ . The effective path length $l_{\textit{eff}}(J_{p}$ ) can then be calculated, by which true and Epstein power losses are reconciled.

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Effective versus standard Epstein loss figure in Fe-Si sheets

Abstract

Keywords

1. Introduction

Table 1 Physical parameters of the investigated non-oriented (NO), conventional (CGO) and high-permeability (HGO) grain-oriented steel sheets

Table 2 Power loss at 50 Hz obtained by the standard Epstein measurement ( P 𝑒𝑝𝑠𝑡 ) and the localized measurement ( P 𝑒𝑓𝑓 ) as a function of peak polarization on three different Fe-Si sheets

References

Table 1
Physical parameters of the investigated non-oriented (NO), conventional (CGO) and high-permeability (HGO) grain-oriented steel sheets

Table 2
Power loss at 50 Hz obtained by the standard Epstein measurement ( $P_{\textit{epst}}$ ) and the localized measurement ( $P_{\textit{eff}}$ ) as a function of peak polarization on three different Fe-Si sheets