Physically consistent multi-frequency magnetic loss testing of silicon steel

For more than 100 years, magnetic energy losses of electric silicon steel sheets are determined by several versions of the Epstein Tester (ET).It roughly simulates the magnetic core of a single-phase transformer through about two dozens of sample strips of 28 cm lengths and 3 cm widths, according to IEC Standards.¹ For about 40 years, an alternative that is available is the Single Sheet Tester (SST). According to IEC Standards,² it requires a sample of 50 cm length. Usually, a square sample is investigated, with the advantage that tests of non-oriented steel can be performed for magnetization either in the rolling direction (RD) or in transverse direction (TD).

Though ET and SST are based on the same physical principle of a Watt-metric measurement, the results of testing tend to differ by up to about 10%.^3,4 For industrial applications, this presents a dilemma.³ There is a need for absolute accuracy, in particular for applications of grain-oriented steel for power transformers. More recently, it was expressed in Refs. 5 and 6 that significant reductions in difference would be attained by averaging the results of measurement. However, industry expresses the need for a single, accurate, easy test, to be used as the basis for new designs, e.g., to predict the building factor of a large machine.

Since the start of co-existence of the two standards,^1,2 attempts are made to bridge the practically very significant differences—however, with limited success. As discussed further down, we attribute this situation to a non-consistent physical basis of the Watt-metric principle of measurement. In recent studies,^7–10 we proposed novel concepts of testing that are based on generally acknowledged theories of Maxwell and Poynting. The present paper describes the main result of an extensive project that is aimed on a measurement system that satisfies physically consistent demands as a pre-requisite for a striven-for standardization.

As mentioned above, the standardized Single Sheet Tester (SST) is using the Watt-metric determination of losses, as defined as

where ρ is the density of tested material, T the magnetization period, H the magnetic field strength, B the induction, f the frequency and B′ the induction time change.

According to the SST standard,² B′ is determined by an induction coil that surrounds the free section of the 50 cm long sample. On the other hand, H is not determined by a direct method. Instead, it is assumed that it is linearly proportional to the magnetization current i, according to the equation

where N_M is the turn number of the magnetization coil and L_M is the so-called magnetic path length.

Already in 1988, the International Electro-Technical Commission (IEC) discussed possibilities of replacing the Watt-metric principle (the so-called Current Method, CM), by a direct detection of field through a tangential field coil (H-coil).¹¹ However, standardization of this so-called Field Method (FM) was halted due to concerns of potential drawbacks of H-coils, such as mechanical instability, time instability (aging), and low signal intensity. A major concern was also the problem of detecting the field H directly at the surface of the specimen. The use of two H-coils at different heights h above the sample surface was discussed in connection with extrapolation—an arrangement that however was expected to complicate the FM.

From the perspective of physics, the FM was also affected by the lack of consistency between H-detection and B-detection. The induction B was intended to be measured by means of a B-coil around the free section of almost the entire sample. On the other hand, the H-detection would be restricted to a rather small sample surface region that may not be representative for the material, due to inhomogeneity of its structure. For example, the latter may result from coarse grains locally, as well as from rolling imperfections in larger regions.

Due to such concerns, the focus was later placed on improvements of the already standardized apparatuses ET and SST, as both were based on the CM. Specific focus was placed on the problem of the magnetic path length L_M as the link between field H and current i, according to (2).

Contrary to the development described above, we intensified our own work on the establishment of concepts for the replacement of the CM by the FM. In the recent years, we have put forth these concepts for discussion in Refs. 7–10 (that contain large numbers of corresponding references, not to be repeated here).

As a second step, we designed a measurement concept that addresses all major concerns. It is the aim of the present paper to put forth this physically “Consistent Loss Testing” (CLT) for discussion. Prior to detailed descriptions, we summarize the state of the art in (B), as well as our own so far performed pre-studies in (C).

As a basis for CLT, we, here, want to summarize the respective state of the art, as attained by international research, since the first publication of standards of SST,² and ET,¹ i.e., 1992 and 1996, respectively. We address both testers with respect to the fact that both are based on the Current Method (CM).

Many attempts were made to develop improved designs of the two testers. Very few concepts were developed for an increase in the dimensions of the sample and tester. For the ET, large increases in size (up to 65 × 10 cm²) were made in our own work, as described further down. On the other hand, decreased sample size for the ET was reported, in particular for attempts to determine effective values of path length L_M, according to (2). As already mentioned, attempts were also made to avoid the definition of L_M. The authors of Refs. 12 and 13 reported tests with frames of different limb lengths, evaluating results from two testers, or even three testers, respectively. However, it is not likely that such expenditure would be widely accepted beyond basic research.

For the SST, increased sample size of 80 cm width and 60 cm length is reported in the product catalogue.¹⁴ Decreased sample size was introduced for a number of down-sized apparatus. Many authors^15–28 replaced the usual 50 × 50 cm² large quadratic sheet by a rectangular, smaller one, as summarized in Table I. Obviously, the aim was to attain more compact versions of both the sample and tester. With respect to the sample, this offers less expenditure for preparation, as well as more practical for archiving. For the tester, it reduces the mass of yoke, which is significant if we consider that the standard format corresponds to a yoke system mass that is close to 150 kg.

Interestingly enough, most so far developed test designs were based on the CM. But more than 20 years research on the problem of L_M yielded very limited success. More recently, we concentrated our own work on the thematic, with very disappointing results.²⁹ We found that L_M significantly depends on the type of material, due to modifications of flux distributions during the period of magnetization. As a new finding, we proved that L_M is a function of time itself. We defined a “path length function” L_M(t) that shows a maximum at instants of time that are characterized by maximum permeability μ, for instantaneous induction values close to 1 T. In principle, it would be possible to introduce the function L_M(t) into the evaluation of losses. However, corresponding attempts failed due to the finding that L_M(t) is not unique, due to the hysteresis of sample material.²⁹ This means that it has loop character itself.

The situation described above indicated that the use of the CM in connection with the path length should be abolished.¹⁰ As an alternative, we introduced the FM for both ET and SST. For the ET, a first corresponding approach was already made in 1978 by Pfützner, as reported in Refs. 30–32. For the SST, a corresponding discussion was made by the IEC in 1990, according to Ref. 11.

As a drawback, the above designs suffered from the fact that rather small H-coils were applied, which means that the field H does not show physical consistency with the induction B. Unavoidable inhomogeneity of the magnetic circuit causes erroneous instantaneous pairings of H and B. This is a source of systematic errors of loss determination. For the first time, we had addressed this problem already in 1991,³³ introducing an extremely large H-coil into a commercial SST with 50 × 50 cm² sample size. The coil size was 50 cm width × 20 cm length. However, this was still too short to fulfill the demand for consistent detections of H and B.

Our following approaches concerned the ET. We designed a “large Epstein tester”³⁴ with a roughly doubled strip width of 6.25 cm, linked with a strip length of 50 cm. Consistent B/H-detection was verified with four H-coils of 26 cm length each. However, the width proved to be still too small to avoid errors from slitting effects. Thus, next we constructed a “Giant ET”³⁵ with a more than threefold strip width of 10 cm, linked with 65 cm length. This apparatus is equipped with four 3D-printed H-coils of 30 cm length, consistently linked with four 30 cm long B-coils. However, both micro-sensor tests and numerical modeling^36,37 proved that the detection region still shows a degree of magnetic inhomogeneity that does not promise loss results of high absolute accuracy.

Our next step was to give up the idea of ET and instead develop a kind of SST that fulfills all practical and physical demands. Applying the FM, we constructed a “Low Mass SST” (LM-SST = 50 cm-tester) with a sample size of 50 × 50 cm² that shows very good performance.³⁸ 

Despite this good performance, we finally doubted whether a large square sample of 50 cm size may be a broadly accepted solution, in particular with tendencies as reflected by Table I. This stimulated us to establish smaller versions of apparatus, still based on the common principle. In particular, a sample format of 50 × 17 cm²—according to a 17 cm-tester—showed extremely good performance. Therefore, we wish to put forth the general principle for discussion, through a detailed description in the following sections of this paper.

The development of CLT is based on the following seven targets:

• Full physical consistency with generally accepted, basic theories of Maxwell/Poynting, consequently avoiding compromises or conventions (as for L_M in Refs. 1 and 2).

• Apart from high reproducibility of measured losses, also possibilities for an assessment of absolute accuracy.

• Applicability for all types of silicon steel, including non-annealed samples of material.

• Applicability also for increased values of frequency, as a novel demand.

• Sample mass and size sufficient to be industrially representative for a given type of steel.

• Simple and rapid performance of testing.

• Test system of compactness and affordability, also for medium and small enterprises.

With these demands in mind, the application of the Epstein tester could be excluded a priori, just (iv) and (vii) speaking for it. Also, all options of SST could be excluded that are based on the Current Method, considering that the effective path length would be based on a mere convention^1,2—in contradiction to (ii). A priori, the focus was on the FM, in connection with a single sample.

The first crucial decision concerned the size of the sample sheet and thus also the geometry of the functional design of the tester device. In logical ways, the starting point of consideration was the standardized SST. Standards² define the length of sample as L = 50 cm. The corresponding data of more recent developments in Table I indicate a clear tendency toward a shorter length, with an average order of about 25 cm. Apart from preferences of compact apparatus, such a decrease may be justified by improved manufacturing of modern steel products. However, even most recent products show significant variations, in particular of thickness, as a need to average over a considerably long sample. Such considerations yielded the decision to keep the standardized length of L = 50 cm upright. This also considers that the magnetic detection region cannot comprise the whole length of the sample, due to—quite unavoidable—lowered induction in the sample end regions.

Recent scientific studies as listed in Table I, demonstrate an even clearer preference of decreased width of the sample—apart from the fact that the existing standards² do not demand the presently most usual width of 50 cm. In contrast to it, Table I indicates a tendency to an order of just 6 cm (or 10 cm), as an average.

With respect to our above listed targets, a value W = 10 cm would be compatible with (i), (iv), (vi), and (vii). On the other hand, specific tests proved that it would not be compatible with demand (iii), in particular for non-annealed steels. This is especially valid for steels with domain refinement that should not be tested after stress annealing. As is well known, high annealing temperature may reduce the effectiveness of treatments through stress coatings or laser scribing—a fact that led to the ban of testing through the Epstein tester.³⁹ 

Our above mentioned experiences with enlarged ETs^34,35 indicated that 10 cm width is still too low to avoid significant impact of slitting on the results of loss tests. In addition, specific studies in the literature, such as Refs. 40–43, indicate that effects do not cease below an order of about 15 cm. Finally this led to the decision of W = 17 cm. As a benefit, this also allows us to cut a square 50 cm-sample into three 16.7 cm wide samples, for a relation of results of loss tests of standard SST and a 17 cm-tester.

The last concern was that the strong reduction of width may cause results that differ from the SST results due to some other unknown reason. For a clarification, we manufactured and compared two versions of testers:

• 50 cm-tester (Low Mass SST³⁸)—the square format of 50 × 50 cm² allowing direct comparisons with SST.

• 17 cm-tester (VM-PLM)—the sample format of 17 × 50 cm² ranging between sheet and strip.

A very large number of comparison tests did not indicate systematic differences between results from 17 cm and 50 cm, respectively. Rather, irregular 1%-variations seem to be unavoidable, due to tolerances of sample cutting (geometry and inhomogeneity of material inner stress). Tests confirm W = 17 cm as an optimum compromise that meets all three requirements: a quite compact sample, sufficient representativeness and acceptable impact of sample preparation. With this in mind, we concentrate the following parts of the present paper on the 17 cm-tester.

A further crucial decision concerned the demand of standards—²whether the magnetization of the sample should be supported by a double yoke system. For demands (v and vi), we decided for a renouncement, as a major easement of system development. Finally—thanks to the field method—it was verified without drawbacks of accuracy, as discussed in depth in the following.

The above considerations yielded a test system as summarized in the block diagram of Fig. 1, for both the 17 cm-tester and the 50 cm-tester. According to it, the test apparatus consists of the sample unit, an analog/digital conversion card (A/D card), a computer, and a power amplifier.

Here, it should be stressed that the following descriptions are based on several versions of the test system, with modifications, in particular, with respect to the already discussed geometrical features. Focus is on features of design that indicated optimum performance for loss tests in the wide, industrially relevant frequency range 16 2/3 Hz to 10 kHz.

The sample of 50 cm length is located on the ground of a magnetization coil system that consists of a 30 cm long M-coil and two almost 10 cm long “booster M-coils.” This system of almost 50 cm length is complemented by a rather thin, U-shaped, soft magnetic yoke, contacted via ∼1 mm long air gaps.

Above the sample, two H-coils are located, with 30 cm length, as the so-called detection length L_DET. The width of the yoke corresponds to the sample width of W = 17 cm or 50 cm. Along L_DET, an induction coil (B-coil) for the direct measurement of B′, is arranged within the central M-coil.

For an “induction-scan” of testing, the user inputs into the PC the characteristics of the sample, the wished frequency f of magnetization, and the range of peak induction values B_PEAK. For a given pairing of f and B_PEAK, the unit of PC + A/D card synthesizes an output voltage u(t) as a promising magnetization start voltage. The resulting induction change B′(t) is measured, and through a well-known hill-climbing method, the target course of time B(t) is approximated. After attainment of exact sinusoidal induction of Form Factor FF = 1.111, unique loss values are calculated according to (1) from induction change B′ and field H of the sample surface. H is extrapolated from the values H₁ and H₂ of the two tangential field coils. For losses P, the extrapolation can be renounced, assuming H = H₁.

Detailed descriptions of the components of test system are given in the following sections.

As a matter of fact, a great handicap of standardized SSTs is the need for a double yoke system that yields a very high system mass of typically several 100 kilograms. This results from the demand of CM that the performed measurement of the total of power consumption J (in W) comprises the through magnetic circuit of the sample and yoke system.² A pre-condition of so far standardized loss measurement is that the value J_S of the sample exceeds the J_Y of the yoke by at least two orders of magnitude.

Apart from the demand of negligible J_Y, the standard yoke should exhibit negligible magnetic resistance. For an effective determination of the field H of the sample, the field of the yoke should approach zero. This demand is also valid for the contact region between the sample and yoke, yielding needs for high cross section of pole faces, in connection with high expenditure for their finishing.

In the case of the here applied CLT, all the above demands are without relevance. Imperfections of yoke, and air gaps between yoke and sample have no impact on measured losses and field, provided that the evaluated detection region of the sample is magnetized in defined homogeneous ways. Air gaps prove to be even advantageous because of balancing the imperfections of contact regions in the transverse direction (especially for the wide 50 cm-tester).

In practice, it is not feasible to attain homogeneous conditions in RD, for the entire sample length of 50 cm. According to Fig. 1, this is considered by restricting the measurement to the central “detection region” of length L_DET = 30 cm.

Actually, the length of 30 cm represents a compromise that considers the following two contrary demands:

• L_DET should be as large as possible to guarantee sufficient representativeness for the given product of steel, for inhomogeneity of crystalline structure and uniformity (e.g., thickness variations from rolling).

• L_DET should be small, restricted to the quasi-homogeneously magnetized sample region. Very intensive analyses showed that this region is quite large for high values of instantaneous induction B (e.g., 1.7 T), due to approaching saturation. However, maximum instantaneous losses (see Sec. IV C) arise in instants around as little as 1 T (compared with Ref. 43), when the sample end regions are characterized by strong drops of magnetization.

Restricting the detection of induction and field to L_DET = 30 cm means that 10 cm long, critical sample end regions are excluded, remaining without any impact on the result of testing. This includes also the complexly magnetized contact regions between the sample and yoke. In order to stabilize the overall excitation demand, we arrange small air gaps of about 1 mm height at both sample ends. Variations prove to be without relevance.

As illustrated in Fig. 1, the yoke is manufactured with small cross section, with thickness values up to 10 mm, depending on the type of applied yoke material. Standards² of SST define a thickness of 25 mm, in connection with a pack of grain-oriented laminations, manufactured with precisely polished pole faces. On the other hand, CLT proves performance effectiveness in a wide frequency range with a thickness of just 10 mm of cheap, non-oriented SiFe, or about 5 mm with amorphous high-frequency bands, respectively.

As to be expected, neither the standardized nor the above, here-used arrangement can offer exactly homogeneous magnetization of the whole sample length of 50 cm. However, very satisfying degrees of homogeneity of the 30 cm long detection region are attained through the following two measures:

• Arrangement of additional “booster M-coils” along the sample end regions. Compared to the central M-coil, they show about threefold density of turns, thus restricting peripheral drops of magnetization along the 10 cm long end regions, as caused by stray fluxes of magnetization.

• The above-described yoke, the main advantage of which proves to be the significant reduction of power demand of the amplifier.

As mentioned above, we performed an intensive study to optimize the length of detection region—as finally to L_DET = 30 cm. For it, we arranged, e.g., five search coils along the sample, in order to determine approximate induction profiles B(x) along the sample axis. In fact, these attempts were handicapped through very strong variations, due to multiple impact factors, such as sample structure, internal mechanical stress, induction, and frequency (affecting permeability in complex ways).

Figure 2 illustrates the problem by earlier results of longitudinal field scanning of surface field H(x, B_PEAK) by an array of tangential Hall plates (from Ref. 30). From a first glance, profiles show distinct inhomogeneity, except for approaching saturation, i.e., 2 T. The depicted profiles concern a 28 cm long sample, where approximate homogeneity is restricted to the central 50% of sample length, the two 25% end regions exhibiting distinct decreases of H_x, and thus also of B_x. On the other hand, smaller values of B_PEAK—as being of practical relevance—are characterized by chaotic maxima and minima of local field that yield a masking of ordered tendencies.

We conclude from the above that an objective assessment of a “homogeneous sample region” is not justified by experiments that reveal rather chaotic behaviors of single sheets (a packaging of sheets—as given for ET—favoring more smoothed behaviors). Evidence exists on a large number of impact factors, such as grain structure, rolling, variations of stress, and slitting in addition to the role of the magnetization system.

As a conclusion, the definition of a detection region of about 60% of total sample length represents an acceptable compromise. So far experiments revealed that booster magnetization coils with tripled magnetization current density along the two 20% long end regions (Fig. 1) contribute to this compromise in satisfying ways.

As criteria for acceptable homogeneity, we assumed a maximum of 3% regional induction drop, for high B_PEAK, at the borders between the detection region and end region. Actually, this also proved to be attained for the further abovementioned air gap length L_GAP of ∼1 mm. In a rather unexpected way, increases of L_GAP proved to show very weak impact on measured losses P. As theoretically discussed in (D), gap increases up to 3 mm prove to affect P by less than 1%, positive or negative changes being possible, depending on the sample material, yoke material/thickness, induction, and frequency.

As a conclusion, the magnetic circuit exhibits very high robustness. This is also demonstrated by the fact that approximation of exact sinusoidal induction is attained with very precise Form Factor FF = 1.111 in wide ranges of B_PEAK and f, according to the undermentioned results of measurement.

A physically consistent determination of losses P is based on (1) that links the induction change B′ of the investigated sample region with the corresponding field H of the latter. On the other hand, it should be repeated that, for standardized testers,^1,2 H is not measured in direct ways, but replaced by the global magnetization current i according to (2).

For the standard SST, induction is determined for the free sample region of length about 45 cm. On the other hand, the current i represents the excitation of the whole magnetic circuit that exhibits strong 3D inhomogeneity along its closed magnetic path, i.e., sample, yoke and two sample/yoke contact regions. This means that the current i—as an assumed measure for the field—cannot show consistency to the induction, a priori.

As to be expected from the inhomogeneity of the standard SST’s magnetic circuit, the course of time i(t) deviates from that of actual H(t) throughout the cycle. As shown in Ref. 7, the deviations are seemingly weak. But they also are linked with phase shifts that tend to affect the measured value P, in most sensitive ways. This means that a direct measurement of field cannot be replaced by a current measurement, if correct loss determination is striven for.

In the case of CLT, the induction B represents the detection region of 30 cm length. For the directly consistent field, it would be needed to measure the field H along these 30 cm directly on the sample surface at height h = 0, which, however is not possible due to technical reasons. The usually applied tangential field coil (H-coil) tends to show an effective thickness of about 2 mm. This means that the field is detected at a (mean) height h of one mm above the surface. Specific studies reported increases of field with gradients of the order 3% for about 3 mm height, according to Fig. 3.^30,31,44

For CLT, we analyzed the field conditions in closer ways, through lifting a given coil, or by comparing the signals of two coils, as sketched in Fig. 1. Applying two coils yields the crucial challenge that their sensitivity values are known in very exact ways. Actually, in practice, it is quite impossible to manufacture two coils of identical sensitivity. Depending on the manufacturing technique (winding, printing, etching, etc.), deviations tend to arise of the order of one percent.

In order to determine the actual, effective sensitivity, a process of calibration is necessary, which is difficult in cases of wide coils, as given here with W = 17 cm, or even more with 50 cm. In acceptable ways, we calibrate in sets of solenoids of about 2 m length, in iterative ways. This means that measurements are started with narrow H-coils in a just about 7 cm thick solenoid that offers the ultimately defined field in its central region. Then in step-wise ways, the H-coils are arranged in thicker solenoids, until a degree of confidence is attained of several per thousand. We assess higher demands as academic, due to the experience that, e.g., mere temperature changes of testers may affect test results in comparable ways. This is also valid for changed direction and/or intensity of the geomagnetic field.

The practical application of calibrated coils revealed that the increase of field with rising distance from the sample surface depends on many parameters. We interpret the observed tendencies by physical mechanisms, as illustrated in Fig. 4 in schematic ways.

For a rough theoretical discussion, we assume that the overall arrangement of the sample, air gaps, and yoke generates in the sample a magnetizing, “driving” field H₀ that is almost constant in the samples in close vicinity. Now, we apply the theory of “magnetic charges” analogous to electrostatics (see, e.g., Refs. 30, 45, and 46). According to this theory, the sample shows a polarization M that “charges” the right end positively, the left one negatively, corresponding to a demagnetizing field H_DEM to the left side. Due to increasing path length, H_DEM decreases with rising h. Consequently, the sum vector field H = H₀ + H_DEM rises, with increasing h, as a reason for the detection problem. For its solution, we introduce the rather high air gap height of 1 mm, to neutralize variations.

For systematic, practical determinations of the field H(t) at the sample surface (h = 0), a sandwich of two H-coils of about 2 mm thickness was arranged, according to Fig. 1. A lower H-coil 1 is placed directly on the sample, with an effective height close to h₁ = 1.5 mm. On it, a second coil 2 detects at an effective (mean) height h₂ = 3 mm. For basic studies, linear extrapolation was applied to determine H(t) of the sample surface. However, a very high number of experiments revealed that a detection at 1.5 mm height can be assumed to yield practically exact losses P, without further measures. This is discussed in depth in Sec. II E in theoretical ways, and is also confirmed by Ref. 44. However, the extrapolation proves to be relevant for an exact determination of the peak value H_PEAK of the field.

Finally, some practical experiences with the H-coils should be added here, considering the objections of IEC, as mentioned in Sec. I A. They resulted from quite small H-coil dimensions of 8.5 cm × 17 cm × 1 mm, in combination with 850 turns of a just 160 µm thick wire. Obviously, in particular, the small thicknesses of the coil and wire yielded concerns about lacking stability.

In contrast to the above, the 17 cm-tester exhibits sevenfold volume with “giant” dimensions of about 30 cm × 17 cm × 2 mm, which allows for robust windings of 800 turns of 500 µm thick wire (the 50 cm-tester even exhibiting 20-fold volume).

With the above data, the H-coils offer high signal intensity, linked with mechanical robustness. Also, defined sensibility is favored by coil manufacturing by a multi-materials 3D-printer as described in detail in Ref. 37.

Here, it should be mentioned that the large detection area offers itself also for flat, multi-directional magnetostriction sensors. A representative detection is attained by extra-long strain gauges of ∼6.4 × 1 cm² size (HBM LY41⁴⁷). Locally, at an edge of the detection region (see Fig. 1), h increases by about 1 mm, which, however, raises the overall measured value P in negligible ways.

The induction is detected by an induction coil (B-coil) that encloses the sample. In the case of standardized SST, the coil encloses the entire free sample length of about 45 cm. As a drawback, this also includes parts of the strongly inhomogeneous end regions (Fig. 1). On the other hand, they are excluded in cases of CLT.

As is well known, the major problem of a B-coil is that its signal does not represent the mere flux φ of the sample, but also the air flux φ_AIR of the enclosed air volume. This flux tends to be compensated by an indirect air flux compensation (AFC) that is based on the field without sample.² However, this standardized method shows three sources of systematic error:

• Extremely large ratio air (+plastic) volume vs steel volume within the M-coil (sum slot height tending to be ∼25 mm for both ET and SST, i.e., the B-coil height is ∼100 times the thickness of GO steel).

• H_COMP related to the field configuration without sample of very restricted relevance, due to complex de-magnetizing field components.

• Homogeneous field in, above and below, the sample assumed. In fact, inhomogeneous components tend to rise with rising distance (Fig. 2).

Striving for high absolute accuracy, we replaced this indirect detection by a “true-field air flux compensation” (TF-AFC; Fig. 5). It reduces all three errors through the following:

• The sum slot height is reduced to about 2 mm below and 5 mm above the sample, according to 7 mm (as B-coil height), i.e., by at least 70%, compared to standardized testers.

• Relation to the actually true instantaneous field configuration, averaged in x-direction over the detection region.

• Field increases with rising distance are interpolated and extrapolated and averaged in the z-direction from H₁(t) and H₂(t) of the H-coils.

The above enable tests at high levels of induction, close to saturation. As an additional advantage versus the CM-testers, the TF-AFC does not require any further hardware components—the compensation is based on the signals of the two H-coils and the B-coil. For the volume of the sample and its close surrounding, H₁(t) is assumed. For more peripheral regions (air volume of sample insert slot + plastic coil former), H₂(t) can be assumed. Details of field inhomogeneity are neglected, considering that a correction problem of 2nd order is given.

As a matter of fact, the large range of frequency f corresponds to an unacceptably high range of the B-coil signal. As a most effective option, this problem was solved by arranging two separate coils for low f < 1 kHz and for high f > 1 kHz, respectively. This meets the demand that the intensity of B-coil signals should exceed that of H-coil signals as weakly as possible, in order to reduce phase shifts of the corresponding signal amplifiers. This is relevant to avoid phase errors of the loss calculation procedure,^30,31 as discussed in Sec. II F. For hardware, we applied low turn numbers of N_B = 30 and N_B = 10, respectively, evenly distributed along the detection region length L_DET = 30 cm.

Finally, standard loss testers tend to exhibit some weak induction harmonics, despite feedback application. Instead, CLT uses a simple hill-climbing algorithm for an exact synthetization of sinusoidal induction. A Form Factor of 1.111 is established for the full ranges of induction and frequency. This complete lack of harmonics yields uniqueness of loss results. Furthermore, it guarantees that (principally very strong) harmonics of field have no impact on the result of loss determination.

In the case of standardized testers, losses P (in W/kg) are determined by the Watt-metric principle, according to (2), by a time-averaged multiplication of the induction voltage and the magnetization current. This yields the power consumption J (in W) of the total of magnetic circuit. In the case of SST, the latter comprises the sample (including its end regions of reduced induction), the contacts between the sample and yoke (with 3D inhomogeneity of flux, planar eddy currents, and stray fields), and the yoke that is assumed as showing J = 0. The losses P are assumed to be represented by the sample's free region of 45 cm (“effective” path) length, in exact ways. However, from the standpoint of physics, a priori, this concept cannot be expected to yield accurate results.

The CLT is based on (2), i.e., on the link between the two quantities B′ and H. Determined with the earlier described considerations, this promises a high degree of accuracy, a priori.

As illustrated in Fig. 6, an analog/digital card (A/D-card; e.g., of National Instruments) receives three dynamic signals, i.e., u_B′(t) from a B-coil (LF for low, technical frequencies, HF for higher frequencies, respectively), as well as u_H1(t) and u_H2(t) from a lower and an upper H-coil, respectively. Despite low turn numbers of B-coils, the order of u_B′(t) may exceed that of u_H1(t) and u_H2(t) in strong ways. This impedes the performing of time multiplexing within a given period of magnetization. Instead, we take advantage of steady-state conditions and process the three signals in three subsequent periods, thus avoiding well-known phenomena of electric charging. The mathematical link is performed as a post-task of mere digital nature.

The above concept promises a high degree of absolute accuracy. However, in practice, one crucial pre-requisite is that air flux compensation (AFC) is without error—as favored by the further earlier described “true field AFC.” A second prerequisite is that the field is detected in sufficient vicinity to the sample. With a distance h below 2 mm, some rest-error still exists, according to Fig. 3. However, in Ref. 30, it was demonstrated that it tends to affect losses P in very restricted ways. Applying the model of Fig. 4, this is due to the fact that for soft magnetic material, the polarization μ₀ M can be assumed to be identical to the induction B, as assumed also by standards.^1,2 This means that H_DEM (Fig. 4) is proportional to B, and the curl integral over the product M · dB results as zero.³⁰ Practical experiments indicated that a 2% error in H corresponds to less than 1% in P.

Figure 6 depicts a second, upper H-coil, for extrapolation to the sample surface (h = 0). However, practical experience showed significant problems that result from the well-known spike-character of H(t). Even high sample rate of field detection (e.g., 250 kHz) does not guarantee a unique detection of the field’s peak value H_PEAK. Thus—contrary to theory—it is possible that h₂ yields lower field intensity than h₁. From this experience, we conclude that it is preferable to renounce extrapolation, provided h₁ < 2 mm (as also confirmed by Ref. 44). This offers increased reproducibility of results, linked with increased simplicity of both tester and test procedure.

A second, already mentioned, possible source of error is given by different phase shifts φ of analog amplifiers of A/D-card for field and induction (see Fig. 6). To reduce differences, the order of corresponding signal differences should be kept to a minimum. Respective problems result for high f that yields high voltage levels of B-coil. For a reduction, Fig. 6 indicates the use of a low-turn “B-coil HF” for f > 800 Hz. To check actual consequences, test measurements were performed with channel commutation (see Refs. 30 and 31). They did not indicate significant errors for the here performed restriction to test frequencies up to 10 kHz, at least for the here used rather high sampling rate of 250 kHz.

A further potential source of error results from harmonics of induction, beyond those of the field. The concept of CLT avoids errors by exact approximation of sinusoidal induction. The corresponding Form Factor FF is consistently kept to 1.111. With this value, the multiplier (Fig. 6) takes over the role of a lock-in amplifier, reducing effective harmonics of the field to the fundamental one. This means that the usually high peaks of H(t)have no impact on the evaluated losses P. On the other hand, the peaks, of course, contribute to the evaluated peak value H_PEAK and to the RMS value H_RMS of the field.

According to Standards,² the Single Sheet Tester (SST) is dedicated to low technical frequency, in particular 50 and 60 Hz. A priori, the development of CLT was aimed at a much wider range of test frequency f. In the following, we make a distinction of three (so far non-standardized) ranges:

• LF: Low (Technical) Frequency, 16^2/3 Hz up to 100 Hz

• MF: Middle Frequency, up to 1 kHz

• HF: High Frequency, up to 10 kHz

Apart from different fields of application, the three ranges show different technical features, as well as different demands for test accuracy. Thus we discuss them in separate ways. The following results were attained from the following two samples:

Sample S1—GO, scribed steel, thickness nom. 270 µm/tested as 263 µm, corresponding to 27ZDKH95.¹⁴ 

Sample S2—GO, thickness nom. 300 µm/tested as 286 µm, roughly corresponding to 30Z120.¹⁴ 

Note that almost all further given results were derived with the 17 cm-tester, equipped with a single H-coil, i.e., without extrapolation to the sample surface. This considers the experience of negligible effects on measured loss values for distances of the here-given order of just about 2 mm.

Applications of LF comprise drives of electrical trains with 16^2/3 Hz, all kinds of conventional machines with 50 or 60 Hz, and specific—e.g., frequency controlled—drives, e.g., with 80 Hz. Highest demands for absolute accuracy exist, in particular, for power transformers, as built up from advanced highly grain-oriented (GO) steel types. For a full exploitation of materials, loss characteristics should be determinable up to 1.8 T, according to Standards.² Industrial product catalogs^14,48 for GO steel tend to also include 1.9 T, which, however, is highly error prone^30,31 and thus is not considered in the following.

Figure 7 shows examples of results of losses P as a function of peak induction B_PEAK, as determined at sample S1. The measured functions P(B_PEAK) show non-linear increases as reported in the wide, well-known corresponding literature. We see analogous increases for the lower frequency values of 16^2/3 and 25 Hz, as well as for the higher values 80 and 100 Hz frequency ranges, where the relevance of eddy current losses arises in continuous ways.

In comparison to results of the 17 cm-tester of Fig. 7, those of the 50 cm-tester tend to deviate by the order of 2% in irregular ways. Clear tendencies—such as weaker slitting effects—cannot be identified. Obviously, they are mantled by different features of the larger sample with respect to inhomogeneity of steel, as well as differences of slitting and possible internal stress.

For the steel type S1, detailed comparison data are available from catalog¹⁴ (p. 163), for both 50 Hz and 60 Hz, derived from an SST sample of 80 cm width and 60 cm length. According to Table II, the correspondences are within 3% for 50 and 60 Hz. Considering different charge, almost sixfold sample size, and different slitting method, we assume that the correspondences can be assessed as very satisfying.

According to the further above definitions, we speak about MF for the frequency range between 100 and 1000 Hz. This range has high relevance for transformers of air planes and—in rapidly increasing ways—for drives of electric vehicles. On the other hand, reports on losses are rather rare. Obviously due to technical reasons, applications of SST are restricted to very small SST samples, such as 7 × 3 cm² in Ref. 27. More frequent is the application of Epstein Tester (ET), as reported in Ref. 49. Wider data are given in material catalogs,^14,48 also listing results from ET in connection with a Hay bridge, for several steel types.

Figure 8 shows test results of the 17 cm-tester for the sample S1. With increasing peak induction B_PEAK, losses P show non-linear increases very similar to the range of LF. However, the slopes are stronger, obviously due to higher impact of eddy currents that become dominant. Total losses rise up to 100 W/kg, with instantaneous maxima close to 200 W/kg—set as the upper limits of CLT. For an illustration of corresponding thermal conditions, 10 kg of steel would allow to warm up a sleeping room. As well, the temperature rise starts to affect the intensity of losses P.⁵⁰ On the other hand, short-time loss increases with high values of temperature are accepted for electric drives.

For the given sample type S1, data for direct comparison are not available. However, catalog¹⁴ includes Hay Bridge (HB) data for somewhat thicker, non-scribed steel, very close to sample S2. Table III shows some examples for corresponding tests. Considering these differences, the deviations are surprisingly weak. The negative sign may be due to annealing, without experimental conformation.

According to the earlier above definitions, we speak about HF in cases where the frequency exceeds the value f = 1 kHz. Corresponding applications of silicon steel gained industrial relevance in recent time through electric drives, such as those of cars, with up to several kHz.⁴⁹ Due to highest densities of eddy currents, the materials tend to be magnetized with rather low induction values, of the order of 0.1 T. This is also reflected by data catalogs of leading steel producers, such as by Refs. 14 and 48. Test results are also included in academic reports, however, in particular for non-oriented steels (e.g., Refs. 51 and 52, for up to 10 kHz).

According to our already mentioned earlier study,⁵⁰ the to-be-expected corresponding rise of temperature Θ may yield a decrease of conductivity, as a reason of restricted rise of P. On the other hand, rising Θ tends to deteriorate domain refinement of GO steels through stress coating or laser scribing. This may counter-balance the restriction, at least for lower frequency levels.

For HF, also the use of CLT is restricted to rather low induction. One reason is to prevent damage of apparatus (in particular H-coils). A further reason is that it becomes difficult to attain exactly sinusoidal magnetization for highly dissipative excitations. Values of Form Factor FF tend to deviate from 1.111, with undefined flux distortions impeding effective comparisons of results from loss tests. Reduced accuracy of testing may also result from crucial phase angles, as discussed in Refs. 30 and 53.

For the scribed sample S1, results of testing are given in Fig. 9. Losses P prove to rise with increasing B_PEAK in an enhanced non-linear way, due to strongly rising eddy currents. The order of 100 W/kg is already attained with 0.9 T for 2 kHz, and even for 0.2 T for 10 kHz.

As in the case of MF, also for HF, no data for sample S1 are given by the corresponding product catalog.¹⁴ For quantitative comparison of results, we thus again refer to some results for the sample S2, as listed in Table IV. It offers direct comparisons with a somewhat thicker steel type as documented in the catalog.¹⁴ For certain frequency values, it reveals zero-deviations, obviously by accident. For 10 kHz, the deviation is 6.7%, still not dramatic, if we consider differences of both, the tested sample and applied test method.

Even for weak induction values, the table lists very high loss values. We assume that these values have restricted practical relevance, due to both energetic and technical reasons, since the high values of P are linked with very small values of permeability μ, as a constructional drawback. As an example, for 50 Hz with 1.5 T, S2 shows a loss value of just 0.65 W/kg combined with μ = 37 000. On the other hand, according to Table V, for 1 kHz, the combination is 117 W/kg with 5290. Finally, for 0.4 T and 5 kHz we measure 110 W/kg with just 2430. At least from a theoretical view, it is doubtful if such a performance may be attractive in practice.

As a conclusion for high frequency (HF), the concept of CLT proves to be applicable up to at least f = 10 kHz. Tests prove to be possible for a considerably high induction range that obviously is not accessible by any other test apparatus for representatively large samples of material.

For HF, an assessment of accuracy of testing is difficult due to the absence of reference methods. The reproducibility of results proves to be very good, order of about 0.2%. However, absolute errors can be assumed to increase with rising f. Tests with commutation of A/D channels indicate loss differences of about 2%, starting at 5 kHz. For even higher f, capacitive effects of phase shifts have to be expected, as well as deviations of induction approximations, despite the high sample rate of 250 kHz.

Golden standards for loss measurements do not exist. This disfavors the assessment of absolute accuracy of loss measurement. Standards² do not address accuracy at all. A restriction is given to reproducibility, claiming “1% for GO steel” and “2% for NO steel” (point 4.3.4).

From our experience, an assessment of absolute accuracy is complicated. The key parameter is given by the peak induction B_PEAK. For it, Ref. 2 claims the use of a digital voltmeter having an accuracy of 0.2% (point 4.2.1.1). However, in reality, the absolute accuracy of B_PEAK is mainly affected by the actual characteristics of the sample and, in particular, by the afore-described crucial air flux compensation. For CLT, we estimate that errors of B_PEAK affect those of losses by up to 1% for LF, and more for HF.

Completely different conditions concern the magnetic field strength. For the determination of relative permeability μ, the peak value H_PEAK is needed. For it, point 5.5 of Ref. 2 claims a reproducibility of “3%.” This order proves to be typical also for CLT. However, impact factors such as geomagnetic fields, demagnetizing field components, as well as inner stress or torsion of sample may cause specific variations. We think that corresponding improvements would not be justified due to large effort. Analogous problems—in milder ways—concern RMS-values H_RMS of the field.

With regard to time-averaged losses P, as the most significant value, the reproducibility of CLT proves to be very good. For an illustration, Fig. 10 shows successively displayed loss values after up/down turns of sample. For an a priori flat sample, it indicates deviations as weak as about 0.1% from a loss mean value of about 0.817 W/kg. Plastically pre-bent samples show stronger differences, due to interior elastic stress, as a result of flattening. However, acceptable global results are attainable from averaging over two tests with positions up and down, respectively.

Finally, there lacks the assessment of the absolute accuracy of CLT that cannot take advantage from a golden standard. We base a careful estimation on systematic errors of the following three impact factors:

• The detection of field H – H(t) is detected very close to the sample surface, averaged over the whole detection region. Under the condition of FF = 1.111 of induction B(t), harmonics of H(t) have no impact on P. Demagnetizing field components affect P in mild ways. Thus, we estimate field-caused errors to be below 1%.

• The detection of induction B—for high H, it may be affected by residual errors of air flux compensation. Although the latter is based on “true-field compensation,” induction-caused errors up to 1% cannot be excluded for high induction.

• Linkage between H and B, according to (1)—taking advantage of advanced components of signal processing, we assume errors to be negligible, at least for LF and MF.

From the above, we estimate the total uncertainty for CLT to be within 2.5% for LF and MF. For HF, errors due to minor phase shifts cannot be excluded between the basic harmonics of field and induction, as earlier discussed in Ref. 30.

The above descriptions and results are focused on the quite compact 17 cm-tester, due to the experience that it may present a most attractive solution. This is valid for both, the design of apparatus and its practical application.

On the other hand, the 50 cm-tester—with many results reported in Ref. 54—shows the advantage that its sample size is identical with that of most standard SSTs, thus allowing for direct comparison tests at LF. Typical deviations for 50 Hz prove to be of the order of 3%, which is acceptable if derived from different samples, including data from product catalogs. On the other hand, the high sample width of 50 cm proves to disfavor the needed power for magnetization with high frequency (results in Ref. 54 being restricted to 500 Hz). It also complicates homogeneous induction along the transverse direction, with respect to constant flux take-over by the yoke. Finally, the preference for small samples, as expressed by Table I, speaks against the 50 cm-tester.

The above considerations indicate that a 17 cm-tester may represent an optimum compromise that satisfies most practical demands of testing in acceptable ways. Here, it also should be pointed to the rather small dimensions of steel components of drive motors that can be assumed to be best represented by analogously small dimensions of test material.

A good compromise may exist also with respect to the design of the 17 cm-tester, including its low mass, with respect to the very large range of frequency. The corresponding results of Chapter III indicate an applicability range that starts with 16^2/3 Hz—as being relevant for train drives—and ends with 10 kHz. Still, we point to experiments that indicate that the concept of CLT can be used even for higher f, an objective assessment however being impeded by the lack of comparison data.

The results of Chapter III are restricted to grain-oriented (GO) steels. However, as reported elsewhere,⁵⁵ very good results prove to be attained also for non-oriented (NO) steels. The corresponding induction range is significantly restricted—due to the already mentioned thermal problematic, but mainly due to the involved textures. Table VI lists an example for loss data for 1000 Hz, which indicates that the critical value of 100 W/kg is already reached for an induction of 1.3 T. Beyond that, increased temperature may affect the results of testing.

Of course, for NO steel, the 50 cm-tester is attractive, since allowing for tests in both RD and TD at one and the same sample. On the other hand, as discussed in Ref. 55, the 17 cm-tester offers specific information from a compact set of samples cut in RD and TD, respectively. This favors the assessment of the frequency-dependent “effective anisotropy” that proves to decrease with increasing f,⁵⁵ as a possible benefit for electric drives.

As a global—subjective—assessment, we claim that CLT offers a first methodology that allows for comparative loss tests for the whole range of frequencies that comprise three orders of magnitude. A specific benefit is that all tests can be taken at one-and-the-same single sample that is large enough to be representative with its (standardized) length of 50 cm. On the other hand, a width of 50 cm yields the drawback of high demands of amplification power. An optimal compromise can be assumed to be given by the 17 cm-tester, which links the acceptable mass of apparatus with a compact format of sample.

A significant feature of the 17 cm-tester is that the impact of sample cutting effects is sufficiently weak. For the first time, this offers versatile possibilities to perform comparative tests for all types of steel—including those with domain refinement by stress coating and/or laser scribing.

Finally, it should be mentioned that CLT allows also measurement of instantaneous magnetization power H B'/ρ—as a novel field of basic research. For corresponding results and discussions, we point to our recent publications.^54,56 As an example for grain-oriented steel, Fig. 11 shows a magnetization power function p(t) for a 300 µm thick steel type that indicates maximum losses P_MAX that exceed time-averaged losses P in strong ways.

It should be stressed (see Refs. 54 and 56) that P_MAX does not arise in moments of B_MAX (where instantaneous losses are zero, due to dB/dt = 0). Rather it arises in the early region of the period, for instantaneous induction values close to 1 T. This means that testers should exhibit optimum performance here—and not beyond 1.5 T, as frequently assumed. For example, in Fig. 11, P_MAX arises at the time of about 1 ms, i.e., shortly after the start of the sinusoidal period of magnetization.

Up to 1.7 T, power functions such as that of Fig. 11, exhibit a single, clearly pronounced positive maximum, close to doubled time-averaged losses (Table VII). Furthermore, peak-like extrema are absent. This confirms that the involved sample material exhibits a highly grain-oriented crystalline structure, strong magnetization in rolling direction, not involving significant orientation processes, in the course of atomic spin orientation (compare⁵⁶). Processes of alignment start at about 1.8 T, as clearly indicated by the pair of positive and negative spikes, which, however, are of non-dissipative nature.⁵⁶ Analyses of such mechanisms are a further benefit of the here applied CLT concept.

With the above features, we claim that CLT represents a concept that combines physical consistency with versatile practical applicability. In particular, the latter results from the possibility to test a single, rather compact sample within a very large range of frequency. At present, this is not offered by any other test system. The sample length of 50 cm may be assessed as being high in connection with the high frequency. But as an advantageous compromise, it allows for a link to standardized SST tests for 50 Hz, without the need of specific treatments, such as annealing.

All the above is focused on grain oriented steel. In the already mentioned separate paper, Ref. 55, we discuss applications of CLT on non-oriented steels, comparing magnetization in both the rolling direction and transverse direction, with high relevance of–rather complex—instantaneous power functions.

Considering the increasing interest for high frequency applications, we propose the test system for a corresponding international procedure of standardization. The target is to complement the SST for full-format testing with LF by CLT, for a more compact system for versatile extensions of the frequency range.

International standards² do not address the absolute accuracy of test results. Only the reproducibility is considered for “a relative standard deviation of 1% for grain oriented steel sheet and 2% for non-oriented steel sheet” (Sec. IV C 4). As a benefit for industry, this offers effective comparisons of different steels. However, after 100 years of loss testing, absolute determinations of losses are not possible.

Standards² provide the accuracy of SST only for certain components, e.g., ±0.5% for power measurement. These good levels of accuracy favor the comparability of testers, but not the absolute accuracy of test results. The latter is determined through the convention of a constant path length, which is the critical deficiency of the standard.

After 100 years of testing, without having a consistent way of determining absolute loss values, attempts should be made to aim for standards of a test principle that exhibits a clear physical basis for an exact determination of losses. As a starting point, we propose the here described principle of multi-frequency “consistent loss testing” (CLT) that promises assessable absolute accuracy.

With regard to the expectable accuracy of testing, the so far experiences indicate that the degree of errors is not determined by components of the tester, but rather by the state of the sample. Deviations result mainly from thickness variations of the tested steel type, as well as from its magneto–elastic state, e.g., internal stress due to flattening. In fact, the impact of variations of sample size proves to be comparable to that of tolerances of magnetic sensor performances. This is also valid for the relevance of demagnetization—as requested by standards.² 

We performed extensive experiments, which indicate that the effort of demagnetization is not justified for routine tests, due to many impact factors, such as the frequency.⁵⁷ However, it is also dependent on the orientation of the tester in the geomagnetic field, as earlier reported by Nakata in Ref. 58. Theoretically, it would be possible to correct for these factors, but this would complicate and increase the cost of the equipment in a way that is difficult to justify for routine tests.

The present study yields the following main conclusions:

1. The standardized Single Sheet Tester (SST)² is based on a constant path length that is fixed by mere convention for LF, which is a systematic source of error.

2. The concept of Consistent Loss Testing (CLT) is based on globally accepted theories of electro-dynamics, thus providing an assessable accuracy in a large range of frequency, up to 10 kHz.

3. Out of two dimensions of test apparatus studied, the 17 cm-tester provides a good compromise between compactness and low mass on one hand and negligible impact of sample preparation on the other. This is applicable for both non-annealed and annealed samples.

4. For a sample of 50 cm length, in the 30 cm long center part as a “detection region,” the induction B and magnetic field strength H yield consistent physical information.

5. Optimum detection of H is attained by a “giant” tangential field coil within just 1 mm effective distance from the sample surface.

6. The presented design also allows for additional measurement of magnetostriction, with negligible increase of measured loss values.

7. Optimum detection of B is attained with “true-field” air flux compensation.

8. Accurate evaluation of losses results from completely suppressed harmonics in the induction, also supported by a high sampling frequency. With aforementioned (5) and (7), this yields absolute loss accuracy levels estimated as 2% and 2.5%, for low and medium frequencies, respectively.

9. Test results for low frequency around 50 Hz indicate good correspondence with catalog data.

10. For medium and high frequency, direct reference data are not available, but single comparisons confirm consistent quantitative tendencies.

11. Determination of instantaneous losses offers effective physical insights, in particular for steel production.

12. The new CLT concept covers the whole frequency range relevant to industry, it provides simple and extremely rapid data collection and is using compact and yet representative samples.

13. The CLT concept is suggested for international standardization, as a method for multi-frequency tests.

The authors acknowledge the support from the Austrian Science Fund FWF (Project No. “Energy Losses” P 31596).

The authors have no conflicts to disclose.

H. Pfützner: Conceptualization (lead); Formal analysis (lead); Funding acquisition (equal); Methodology (lead); Investigation (equal); Writing – original draft (lead); Visualization (supporter). G. Shilyashki: Conceptualization (supporter); Formal analysis (supporter); Funding acquisition (equal); Methodology (supporter); Investigation (equal); Project administration (lead); Visualization (lead); Validation (supporter). C. Bengtsson: Conceptualization (supporter); Investigation (supporter); Project administration (supporter); Validation (lead); Visualization (supporter); Methodology (supporter).

Data available on reasonable request from the authors.

B (T)

induction

B′ (T/s)

induction change (= dB/dt)

f (Hz)

frequency

H (A/m)

magnetic field strength

h (m)

height (above the sample)

J (W)

power consumption

L_M (m)

magnetic path length

P (W/kg)

losses

p(t) (W/kg)

magnetization power function

T (s)

duration of period

Θ (°C)

temperature of sample

ρ (kg/m³)

density (of sample material)

AFC

air flux compensation

CLT

consistent loss testing

CM

current method

ET

Epstein tester

FM

field method

HF

high frequency (>1 kHz)

LF

low (technical) frequency (<100 Hz)

MF

medium frequency (100 Hz–1 kHz)

RD

rolling direction

SST

single sheet tester

TD

transverse direction