Magnetic Flux Paths in Single Sheet Tester as Modeled Numerically

International standards assume that dissipative losses of the single sheet tester (SST) are restricted to the 450 mm long free sample region, according to negligible induction and losses in yoke and pole face regions. This article reports non-linear magnetic equivalence circuit calculation (MACC)-simulation of induction distributions, with focus on sample/yoke contact regions, for grain-oriented (GO) material and non-oriented (NO) material, respectively. Neglecting stray flux along the free sample region, it is shown that the flux of a GO-sample tends to propagate toward the center of the pole pieces, in order to enter into the yokes in distributed ways. The propagation length depends strongly on the air gap length <inline-formula> <tex-math notation="LaTeX">${g}$ </tex-math></inline-formula> from coating and pole face finishing. NO-steel shows weaker penetration, the flux going “round the corner” for small <inline-formula> <tex-math notation="LaTeX">${g}$ </tex-math></inline-formula>. As a consequence, the conventional assumption of a constant path length of 45 cm cannot be corrected. Effective adaptation of the latter to the many involved impact factors is not realistic.


I. INTRODUCTION
M AGNETIC energy losses of electric steel show high relevance for the estimation of machine core performance, in particular with recent concerns of restricted energy resources.This justifies attempts to optimize standardized methods for an exact determination of losses P (in W/kg) for laminated core materials, as a function of peak value of induction.For electric steel, two IEC standards are applied in synchronous ways, [1] for the single sheet tester (SST) and [2] for the Epstein Tester (ET).In both cases, the total power consumption is measured for an inhomogeneous magnetic circuit that includes the sample of material.The losses are calculated by allocating a so-called effective, nominal path length L NOM to the sample.
Standards [1] sets L NOM as a constant of 450 mm.This value represents a nomination that obviously is not based on detailed investigations.The literature does not include any detailed experimental or numerical analysis.Papers like [3], [4], [5], [6], [7], and [8] concern other, less defined magnetic systems.Moreover, they do not consider here relevant instantaneous impact factors, as discussed in our earlier study [9], or in this work.
In [10], we presented for the first time our novel method-multi-directional non-linear magnetic equivalence circuit calculation (MACC).It allows consideration of multiple anisotropy functions by iterative modification of both local permeability values and local induction values until acceptable "harmonization" is attained for all considered locations of the considered system; detailed descriptions are given in [10].
In a recent paper [11], we modeled induction distributions of the ET by means of "Multidirectional non-linear MACC" (MACC, e.g., [10], [11], [12]).MACC revealed very significant anisotropy-caused differences for grain-oriented (GO) steels, in comparison to non-oriented (NO) ones.Here, we report about corresponding MACC-modeling of the SST.The aim of this study was to estimate flux distributions of SST in different sub-regions, as a basis of improved test design.This may include the formulation of an effective magnetic path length if applying the so-called current method [13] for loss measurements.On the other hand, it also concerns pre-conditions for the so-called field method that uses a tangential H-coil for direct detection of the magnetic field strength of the sample surface.

II. APPLIED MACC-MODEL
As described in detail in several references like [10], [11], and [12], MACC is a method of numerical modeling that renounces high spatial resolution as being typical for FEM, in order to avoid high processing times.Instead, the user takes advantage of simple possibilities to consider flux elements of pre-given geometry (2-D or 3-D) and non-linear, multi-directional permeability functions.This allows for the consideration of complex anisotropy conditions, as well as of meshing of multi-scale elements such as air gaps of bulk regions-all being relevant for the here given case.

A. System Regions
In the here-given case, we considered an SST according to IEC Standards [1], i.e., with double yokes of 25 mm thickness.Fig. 1 shows a schematic outline of the well-known overall design.Assuming approximate symmetry, modeling was restricted to the lower, right quarter of system, including the lower half of sample.
Our model comprised the following six system regions: 1) SF: the free sample region of 225 mm in length 2) SE: the embedded sample end of 25 mm in length 3) G: the gap of 10 µm between sample and yoke, estimated from 5 µm coating thickness, and assuming 5 µm surface roughness of pole face 4) YV: the vertical yoke region of 500 × 100 × 25 mm size 5) YC: the yoke corner region of 500 × 25 × 25 mm size 6) YH: the horizontal yoke region of 500 × 225 × 25 mm size Due to its specific ability to combine regions of very different geometries and non-linear permeability functions, MACC offered itself as the most promising tool for the here given task of modeling.In particular, this is true for gaps due to coating and pole face irregularities.On the other hand, we neglected stray fluxes between sample and horizontal yoke regions, as modeled by Nakata with FEM [14].This means that this study flux assumes constant induction in region SF.In practice, it tends to sink by several percent, toward the end of free sample region SF.

B. Permeability Functions
Fig. 2(a) depicts the corresponding measured permeability functions of highly GO (HGO) material, used for the simulation of the yoke in rolling direction (RD) and transverse direction (TD), respectively.The HGO-steel shows maximum RD-permeability µ RD ≈ 50 000 for the RD and µ TD ≈ 2500 for the TD.These high values are relevant for the sample, while-according to much lower permeability values-the yoke is weakly magnetized, even less than 10 mT (see Results).A numerical calculation of the permeability function below 10 mT would be possible based on Rayleigh low [15].However, this would not be justified due to the extremely low impact.In this work, we estimated the corresponding µ-functions by logarithmic extrapolation, since measurements below 10 mT are not possible in simple ways, with justifiable expenditure.It yielded orders below 500, i.e., close to initial µ-values.For the yoke corner, mean values were assumed, considering that laminations are stacked with cross-orientation [1].Being aware about low precision of extrapolation, we checked consequences of data variations, finding that they have very weak impact on the distribution of flux to the five flux paths A to E (see Fig. 3) -as the focus of this study.Fig. 2(b) shows the assumed permeability functions for a 500 µm thick NO material, based on typical data from the literature.It was also assumed that the NO material exhibits an isotropic behavior (µ RD = µ TD ).
As it is well known [1], the standard SST is magnetically excited by a magnetization coil around the entire free sample region SF.For simplification, we neglect coil designs (like effective height) and assume perfect performance without stray flux.As shown in Fig. 3, the reluctances for the simulation of the stray flux are neglected.We impress the defined sample induction value B (1.5 T for NO steel, 1.7 T for GO-steel) in the sample center x = 0 in a concentrated way, assuming that it remains constant to the border region with the pole face, according to x = 225 mm.In the axial direction, we assume five elements of sample steel.
Between sample and yoke material, we assume a coating layer of 5 µm thickness.We add 5 µm to simulate a further gap from rather unavoidable insufficiencies of finishing of pole faces.This yields a sum gap height of 10 µm, on top of the aforementioned yoke height of 100 mm.
In practice, the gap may exceed 10 µm due to technical imperfections.One is given by the more pronounced roughness We estimate consequences by increasing the gap length to g = 1 mm.We also estimate the case of no-gap by setting g = 0.1 µm.
We assume that the sample material is magnetized in its RD.Within the yoke we also consider the TD, according to extrapolated permeability functions µ RD (B RD ) and µ TD (B TD ), as shown in Fig. 2(a).
According to Standard [1], the yoke corner is X-wise laminated.This is considered by a medium permeability function µ YC (B YC ), analogous to quasi-isotropic sum behavior.On the other hand, the free horizontal yoke region is assumed to be anisotropic like the vertical yoke region.
This study was focused on flux paths through the yoke, starting in the 25 mm long sample end that is enclosed by yoke material.Thus, we set four elements to model this short region.In it, the axial sample flux is transferred into yoke flux, assuming five paths (see Fig. 3), path A as the most inner one, up to path E as the most peripheral one.A corresponding net for the vertical yoke region (YV) is modeled by 27 elements in RD and TD.The transfer flux into the five paths A to E is assumed to pass through the already discussed gap of length g, as modeled by five elements.
The vertical yoke region yields the yoke corner region (YC) of x-wise stacking.For it, we assume a total of 40 identical elements.There follows the horizontal yoke region (YH) with 5 × 5 elements in RD and 4 × 4 elements in TD.
The here applied MACC software does not consider hysteresis.This means that modeling for a given value of sample induction estimates quasi-static distributions, independent of other instants of time.

A. Flux Paths for GO Steel
For GO-steel, we concentrate on the induction B = 1.7 T, as an industrially relevant peak induction, e.g., for transformer cores.Fig. 4 shows corresponding induction distributions, as resulting from the model.
With the aforementioned assumption of an ideal magnetization coil without stray flux, the B-value of 1.7 T is kept upright to the end of the free sample region, i.e., x = 225 mm.With a negligible gap [Fig.4(a)], the sample flux is immediately transferred to yoke flux.That is, it passes in path A at the very beginning of the pole face.
Obviously, the latter case is generally assumed in Standard [1], corresponding to an over-all path length of L CONV = 450 mm-which would mean that the yoke is without any impact.However, the results of model indicate that induction values of more than 10 mT should be expected for considerably large yoke regions.This concerns the vertical yoke region where the flux spreads up from path A to path C.These quite concentrated fluxes hardly cannot be neglected for the over-all path length-also not with respect of losses.In the horizontal yoke region, homogeneous conditions arise, with some deload of the peripheral path E. Fig. 4(b) shows the case of g = 10 µm that can be assumed to be typical for SSTs of precise design.Here, the flux penetrates into the sample end region in a distributed way.Even in the middle of yoke thickness, the sample still exhibits B ≈ 0.6 T, a considerable flux portion entering the yoke path C. The vertical yoke region initiates a gradual homogenization that is completed after the yoke corner.The horizontal yoke region exhibits B = 10 mT, in a throughout way.
Fig. 4(c) illustrates the consequences of excess gap length g = 1 mm.This case may result from unprecise manufacturing of the SST.As well, it may be caused by imperfections of the sample of steel.With this high value, the sample flux enters into the yoke with almost even distribution.Thus, we find

B. Flux Paths for NO Steel
The above flux distributions for GO steel are clearly controlled by the high anisotropy of the test material.For a comparison, Fig. 5 depicts results for NO steel of low anisotropy.Fig. 5(a) concerns the low gap length of g = 10 µm, being valid also for the gap-less case, in approximation.In contrast to GO steel, the sample flux here turns "round the corner" into the yoke, right away into the innermost path A (and B).The corresponding induction starts out from the relatively high order of 20 µT.The weak anisotropy favors homogenization, except from flux round the yoke corner also here.
Fig. 5(b) concerns a pronounced gap of g = 1 mm.It yields some distribution of take-over of sample flux, into the yoke.However, the overall flux distribution equals that of GO steel with a normal gap length.
As the most significant difference, the penetration of sample flux into the embedded sheet end is much shorter.The corresponding elongation of the overall path length is less pronounced.Probably it can be neglected for the practical performance of testing, as closer discussed in Section IV.

IV. DISCUSSION
As already stressed in Section I, the Standard [1] of SST is based on the assumption, that a massive double yoke of as much as 25 mm thickness yields negligible induction intensities in all SST regions, except the free sample region.
This assumption would justify the generally applied "nominal path length" of 450 mm.
On the other hand, the results of the here reported numerical modeling indicate that sample end regions cannot be neglected.As well, contributions to the overall path length come from specific regions of the yoke.However, all these contributions depend on several impact factors.One is the steel type, anisotropy favoring a "penetration" of sample flux into the sample ends.The high permeability in RD favors straightforward flux against branch-off through the alwaysexistent gap of coating plus air gap.
A very significant factor is the unavoidable gap between the sample and the yoke.A minimum (basic) gap results from the coating that is relevant at least for GO steel.The stronger impact is indicated for the case that the finishing of pole faces is not perfect, or that small tilts arise between the sample surface and pole face.Then elongated path length is given, even for NO steel.
Of course, the above tendencies have an impact also on the measured losses.However, numerical estimations are very difficult.One reason is that loss data are not available here given the low intensities of yoke induction.On the other hand, the in-sample intensities of penetration are much higher.But corresponding loss values again are not available, since strong distortion is to be expected at all locations.Obviously, this corresponds to strongly increased loss values, compared to well-known data for sinusoidal magnetization.
Here, it should be stressed that the above results are in clear analogy with those of our earlier analogous MACC modeling of ET [11].Also there, the low anisotropy of NO steel proved to yield a concentration of flux in inner corner regions of the Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.As a global conclusion, the above means that both standardized test methods should be evaluated with increased path length values, in particular for GO steel.However, both cases are complicated by the indication that the configuration of flux is not constant, apart from additional impact factors, as we have discussed earlier in [9].For the case of SST, this study indicates that the flux paths depend mainly on: 1) the type of steel; 2) its anisotropy; 3) the coating thickness; and 4) on the precision of contact between the sample and yoke.All the aforementioned factors cause errors of SSTs that are based on the so-called current method [13], in connection with a so-called path length.An effective adaptation of the latter to the many involved impact factors is not realistic.
The only effective alternative is given by the application of a field coil for the direct determination of the magnetic field strength.Earlier, field coils were characterized by several drawbacks like mechanical and thermal instability (including problems of aging), restricted sensitivity due to small dimension, or detection errors due to high distance from sample surface.Today, all these drawbacks are avoided through measures of defined, fully automatic manufacturing of high precision, linked with shielded encapsulation in the SSTs coil system.Finally, full-area "giant" H-coils offer high signal levels as well as physical consistence with the Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
As closer described in [16], a novel multi-frequency SST (MF-SST) is not affected by instability any longer.On the contrary, a mechanically shielded H-coil system provides very low levels of repeatability errors, below 0.2%.A more recent study by our group, to be published soon, indicates reproducibility errors of loss measurements below 1%, which is superior to current SSTs.Finally, the absolute uncertainty is estimated to be below 3% which is superior as well.

V. CONCLUSION
This study yields the following main conclusions: 1) While experimental analyses of flux distributions in SST are inhibited by a lack of small sensors and FEM analyses by problems to combine non-linearity with gaplike inclusions, MACC proves to be an effective tool to model individual sub-regions with respect to distributed flux paths.2) Contrary to standards, the sample´s magnetization proves to be not restricted to the free sample region, but to penetrate into the sample ends that are enclosed by the yokes.3) Due to its high anisotropy, a GO sample shows strong penetration to the central region of pole face, linked with distributed flux through the yoke system.4) A NO sample shows restricted penetration, linked with flux round the corner, the latter, however, being restricted in the case of residual air gaps between sample coating and pole faces.5) As a consequence, it is not feasible to express the field strength through a standardized, constant path length, since the latter is a material-dependent function of time of air gap length and of yoke characteristics.6) As a main conclusion, it is not feasible to attain substantial improvements in the accuracy of the measurements of the current-based SST.7) The only promising alternative is the SST with field coil, that at present avoids earlier drawbacks with respect to stability, sensitivity, and robustness, through fully automatic manufacturing and shielded encapsulation into the SSTs coil system.

Fig. 2 .
Fig. 2. Permeability functions.µ RD (B RD ) for flux paths in RD, µ TD (B TD ) for flux paths in TD, including extrapolations.(a) For the GO steel type, as assumed for the yoke and for the GO sample.(b) For the non-oriented (NO) steel sample.

Fig. 4 .
Fig. 4. Results of modeling for GO-steel for an induction B = 1.7 T, averaged over the sample cross section.(a) Assuming negligible gap length (g = 0.1 µm).(b) Assuming g = 10 µm, as a practically ideal case.(c) Assuming g = 1000 µm, corresponding to construction lacks.Notice: All data is given in T, mT, or µT, respectively.

Fig. 5 .
Fig. 5. Results of modeling for NO-steel for an induction B = 1.5 T, averaged over the sample cross section.(a) Assuming g = 10 µm, as a practically ideal case.(b) Assuming g = 1000 µm, corresponding to construction lacks.Notice: All data are given in T, mT, or µT, respectively.