Practical Approximation of Sheet Losses Taking into Account the Guillotine and Laser Cutting Effect

Abstract

Reducing losses in electrical devices is essential for reducing global energy consumption. Losses in the core of electrical machines constitute a significant part of the overall losses—their share increases with the number of machines powered by PWM converters, especially for high-speed machines. Limiting core losses requires precise determination at the design stage of the device. Achieving this goal is possible thanks to numerical or analytical simulation. A necessary input for this process is the correct determination of the properties of the core material. The sheet loss, however, changes due to the machining process, primarily punching. The subject of the work is to develop a sufficiently accurate approximation of electrical steel sheet-specific loss, taking into account the effects of cutting and the width of a given machine element for a wide range of induction and frequency. The method also enables the extrapolation of losses for higher frequencies relevant from the point of view of generating losses in the machine. The developed loss approximation can be used in the finite element simulation and in applying analytical methods. The technique can be successfully used for many grades of non-oriented sheet metal, provided that the requirements specified in the work are met. The proposed approximation allows us to determine the loss of a sample of a certain width in a wide range of magnetic induction magnitude and frequency with an accuracy not worse than 4%.

1. Introduction

Electric motors and motor-driven systems account for around 70% of the total global electrical use of the industrial sector [1].

One should strive to reduce all power losses when designing electric motors with higher efficiency classes. In the case of motors operating at the supply voltage frequency of 50/60 Hz, losses in the windings dominate. Still, losses in the core can reach 20–30% of all losses, and their share increases with the increase of frequency, and for high-speed motors, they become dominant. Accurate calculation of power losses in the motor core requires considering the appropriate loss model and the manufacturing process, which may affect the deterioration of the core material properties. Punching is the most commonly used technology for producing electric motor sheets. Phenomena during punching are similar to when cutting a sheet with a guillotine. Guillotine cutting is the generally accepted method for sample preparation for Epstein and single-sheet tester measurements [2,3].

Laser cutting is often used in the case of unit production and the construction of prototypes of new construction solutions. The cutting process causes partial degradation of the core material and the resulting deterioration of the magnetization characteristics and increased sheet loss [4,5,6,7,8].

The impact of the punching process on the magnetic properties is complicated and depends on many factors, both related to the sheet metal production process, such as its thickness, content of silicon and other additives or grain size, and the parameters of the punching process, such as cutting clearance, wear state of punch and die and punching speed [9,10,11].

The study of the impact of punching requires the use of several advanced research methods, such as the magneto-optical Kerr effect, electron backscatter diffraction in a field emission scanning microscope, microhardness measured using the Vickers method and neutron grating interferometry [9,10,12]. Thanks to these methods in conjunction with the modeling of mechanical phenomena, it is possible to explain the impact of punching parameters on electromagnetic properties [9].

To simulate the distribution of the electromagnetic field and losses using the finite element method (FEM), dividing the punched sheet metal into layers and assigning each layer other parameters determined based on a model or experiment is advisable. However, experimental determination of such parameters is difficult due to the low voltages in the needle method or the need to drill holes in the material, which violates its structure [13,14,15,16,17]. For example, such a solution was proposed in [17], where a specific degradation profile, degradation factor at the cut edge, and degradation depth characterize the degradation. The degradation in permeability was represented by exponential or power functions of cut edge distance [8,18,19,20,21]. It should be noted that there is no consensus on the nature of the analytical function describing degradation that is the degradation profile. In [16,22], the hysteresis loss coefficient was assumed to be linearly dependent on the distance from the cut edge. In [23], the hysteresis and excess loss coefficients were assumed to be exponentially dependent on the distance from the cut edge. In [24] polynomial dependence was used. Paper [22] proposed a continuous loss model to include the cutting effect in FE simulation. In this approach, the magnetic properties varied at each element of the FE mesh according to the distance of each mesh element from the nearest cut edge. It should be emphasized that the described methods are very laborious because they require modification of the existing software to introduce the element’s properties in FEM, depending on its location in the model. At the same time, they are utterly useless in the case of analytical methods where area properties are needed.

An alternative to measuring local distributions is using measurements on samples of various widths and matching the obtained results to the assumed degradation function. This approach was used in the papers [23,25]. A similar approach can be found in [26,27,28,29], where the parameters are averaged based on the assumed width of the damaged zone.

However, while this approach is effective in the case of mechanical cutting, where the phenomena in the damaged zone do not depend (with a sufficiently broad sample) on the width of the sample, in the case of laser cutting, the width of the sample affects, probably due to the change in the cooling process of the cut area, the area damaged. It should be added, however, that similar problems can also arise with mechanical punching when high speeds are used due to thermal phenomena.

For this reason, also for use in FEM simulation, but mainly for analytical models, it is advisable to define area models corresponding to the actual width of the machine core elements, such as stator and rotor teeth, tooth heads, yokes, etc.

The aim set in the article by the authors was to obtain an area approximation with reasonable accuracy for mechanically cut, and laser cut sheets and to bring it to a form that would allow it to be easily used both in the FEM analysis and mainly in the analytical method.

2. Approximation of Losses in Electrical Sheets

In the literature, you can find many articles on how to describe the losses in electrical sheets. Generally, we can divide them into three groups: those derived from Steinmetz’s work [30].

$$p_{Fe}=k_{Se}{f}^{\alpha }\hspace{0.17em}{B}^{\beta }$$

(1)

where p_Fe is specific core loss, k_Se, α, and β are specific coefficients, f is the frequency of the magnetic field, and B m is the magnitude of flux density of the applied magnetic field; suggested by Jordan [31] where iron losses are separated into static hysteresis losses and dynamic eddy current losses,

$$p_{Fe}=k_{h}f\hspace{0.17em}{B}^2+k_e{f}^2\hspace{0.17em}{B}^2$$

(2)

and introduced by Bertoti [32,33,34] with the addition of excess losses

$$p_{Fe}=k_{h}f\hspace{0.17em}{B}^2+k_e{f}^2\hspace{0.17em}{B}^2+k_{exc}{f}^{1.5}\hspace{0.17em}{B}^{1.5}$$

(3)

It was then modified, among others, in [35] to a model known as the model with all the coefficients variable with four coefficients.

$$p_{Fe}=k_{h}f\hspace{0.17em}{B}^{\alpha }+k_e{f}^2\hspace{0.17em}{B}^2+k_{exc}{f}^{1.5}\hspace{0.17em}{B}^{1.5}$$

(4)

Various modifications have been made to all methods, an overview of which can be found in papers [36,37,38,39,40]. The last paper [40] introduced variable coefficients approximated by polynomials of orders 3 and 5.

$$p_{Fe}=k_e\left(f,B\right)f^2\hspace{0.17em}{B}^2+k_h\left(f,B\right)f\hspace{0.17em}{B}^2$$

(5)

Similarly, in [33], a polynomial of degree 10 and a logarithmic function were used. In [38] the Piecewise Variable Parameter Loss Model was developed to increase the accuracy

$$w_{Fe}=k_h{B}^{{\alpha }_1}f\left(k_{{}_1}{B}^{{\beta }_1}\right)+k_e{B}^2{f}^2\left(1+k_2{B}^{{\alpha }_2}\right)$$

(6)

where k₂ and β₂ are considered piecewise, varying with the magnitude and the frequency of flux density, instead of the constants α₁ and α₂.

It should be noted, however, that for higher frequencies, the dependence of eddy current losses on the square of induction will only occur for a linear material with constant magnetic permeability [41,42,43]. After considering the change in permeability along the thickness of the sheet resulting from the non-linearity of the magnetization characteristics and the non-uniformity of the field distribution due to the effect of field skin effect for higher frequencies, the relationship ceases to be square [44,45]. A detailed analysis of the phenomena, taking into account the field skin effect and the non-linearity of the sheet, can be found in [41]. The original approach of the authors using these considerations was to introduce a corrected expression to calculate the loss in the core of the electric machine in the form

$$p_{Fe}=k_{h}f\hspace{0.17em}{B}^{\alpha }+k_e{f}^2\hspace{0.17em}{B}^{\beta }$$

(7)

with four coefficients k_h, α, k_e and β. To obtain adequate accuracy, the frequency range being the subject of loss approximation was divided into sub-ranges.

To correctly apply the above formula required checking whether the quotient of losses and frequency has a linear course as a function of frequency for a constant value of magnetic flux density magnitude. Figure 1 shows the exemplary curve of the quotient of specific losses and frequency for the tested sample of the electrical sheet with a width of 4 mm cut by guillotine for the frequency range from 5 Hz to 50 Hz for constant magnetic flux density magnitude values from 0.2 T to 1.5 T. The results obtained in this part of the work were obtained for the M270-35A non-oriented sheet metal with a thickness of 0.35 mm.

All curves were approximated using linear functions. The coefficients of the linear function were estimated by the least squares method, and the intercept is present in the model. To study the linearity of the characteristics, both the coefficient of determination for the linear approximation R² and the value of the relative mean square error for the approximation with a linear function were calculated. Graphs of these values as a function of magnetic induction are shown in the following Figure 2.

As can be seen, the coefficient of determination for the linear approximation R² is greater than 0.95, which proves a perfect fit of the curves obtained to the linear distribution. Also, the relative mean square error value for the approximation with a linear function for individual induction values does not exceed 3%.

The following Figure 3 shows the values of the relative mean square error for the approximation by a linear function for individual induction values, for a 5 mm wide sheet sample in the frequency range of 50–200 Hz, for a 7 mm wide sheet in the 200–500 Hz frequency range and 10 mm wide in the frequency range 1000–4000 Hz.

The graphs presented in Figure 1, Figure 2 and Figure 3 were made for sheets cut with a guillotine. Figure 4 shows analogous graphs for samples cut with a laser.

From the analysis of the presented results, it can be concluded that for the tested nonoriented electrical steel sheets, the division of losses into the hysteresis part proportional to the frequency and the eddy current part depending on the square of the frequency is correct, and the errors are within acceptable limits. This conclusion applies to both samples cut with a guillotine and a laser. The authors’ previous research shows that this approach is also correct for other types of nonoriented sheets, such as those tested in [46] M600-50A with a thickness of 0.5 mm, NO20 with a thickness of 0.2 mm, NO12 with a thickness of 0.12 mm, and also amorphous steel Alloy 2605SA1 with the thickness 0.00254 mm. Similar results were obtained in [47].

As a result of the division of losses, we obtain the relation

$$p_{Fe}=c_{h}f\hspace{0.17em}+c_e{f}^2$$

(8)

Then, the c_h and c_e coefficients obtained for the available induction range are approximated using the power function to obtain dependence (7). Figure 5 and Figure 6 show example approximations of c_h and c_e coefficients in the range of 200–500 Hz for samples with a width of 10 mm cut with a guillotine.

The four coefficients k_h, α, k_e, and β appearing in Equation (7) are determined based on the approximation. Dividing the frequency and induction ranges into sub-ranges allows for good-quality approximation. On this basis, we obtain sets of coefficients for different sample widths. For example,

Table 1. Coefficients k_h, k_e, α, and β approximating the specific loss characteristics of electrical steel sheets M270-35A as a function of the width of the sample cut with a guillotine for the frequency range 5–50 Hz and the induction range up to 0.6 T.

Sample Widths (mm)	kh (10−2)	α	ke (10−4)	β
4	2.64	1.82	1.96	2.00
5	2.17	1.70	3.06	2.12
7	1.47	1.52	4.60	2.28
10	1.09	1.40	5.28	2.31

shows coefficients for a guillotine punched sheets with a width of 4 mm to 10 mm for the frequency range 5–50 Hz and the induction range up to 0.6 T,

Table 2. Coefficients k_h, k_e, α, and β approximating the specific loss characteristics of electrical steel sheets M270-35A as a function of the width of the sample cut with a laser for the frequency range 5–50 Hz and the induction range up to 0.6 T.

Sample Widths (mm)	kh (10−2)	α	ke (10−4)	β
4	2.71	1.06	2.43	2.00
5	2.67	1.06	2.38	1.89
7	2.59	1.09	2.26	1.77
10	2.41	1.16	2.01	1.79

for a laser,

Table 3. Coefficients k_h, k_e, α, and β approximating the specific loss characteristics of electrical steel sheets M270-35A as a function of the width of the sample cut with a guillotine for the frequency ≥4000 Hz and the induction range up to 0.3 T.

Sample Widths (mm)	kh (10−2)	α	ke (10−4)	β
4	8.18	1.85	0.413	1.992
5	7.29	1.83	0.406	1.966
7	6.15	1.86	0.381	1.932
10	6.03	1.81	0.319	1.927
20	5.34	1.79	0.298	1.921
60	4.93	1.78	0.286	1.918

for a guillotine for the frequency ≥4000 Hz and the induction range up to 0.3 T and

Table 4. Coefficients k_h, k_e, α, and β approximating the specific loss characteristics of electrical steel sheets M270-35A as a function of the width of the sample cut with a laser for the frequency ≥4000 Hz and the induction range up to 0.3T.

Sample Widths (mm)	kh (10−2)	α	ke (10−4)	β
4	11.36	1.69	0.651	2.049
5	10.21	1.68	0.593	1.997
7	8.46	1.64	0.499	1.951
10	7.20	1.49	0.415	2.027
20	6.34	1.58	0.354	1.985
60	5.32	1.70	0.299	1.929

similar for a laser. It should be noted that the α coefficient is significantly different from 2 and, the β coefficient for most cases is also different from 2.

To check the sensitivity of the coefficients, the courses of coefficients k_h, k_e, α, and β were additionally examined as a function of frequency for individual widths of the M270-35A sheet samples cut with a guillotine, shown in the Figure 7 and cut with a laser (Figure 8).

Based on Figure 7 and Figure 8, for samples cut with a guillotine and a laser above the frequency of 1000 Hz, the approximation coefficients for all sample widths are practically constant, which allows a limit of the range of measurements.

The four coefficients k_h, α, k_e, and β are then approximated by a second-order polynomial that best represents their variation as a function of sample width (Figure 9).

Sets of 12 obtained coefficients can be the basis for calculating losses for individual elements of the motor core.

3. Method and Range of Measurements of Loss Characteristics of Electrotechnical Steel Sheets

The primary measuring tool used in work was a specialized measuring device MAG8.1, an automated single sheet tester made following the IEC standard [3], which allowed testing single sheets with dimensions: length of 300 mm, width of 60 mm. For the measurement, samples were made of one or more metal strips consisting of a rectangle with dimensions close to 300 × 60 mm with the following configurations (number of strips × nominal width of a single strip): 1 × 60 mm, 2 × 30 mm, 3 × 20 mm, 4 × 15 mm, 6 × 10 mm, 8 × 7 mm, 12 × 5 mm, 15 × 4 mm, 9 × 6 mm. Individual samples were characterized by a cut angle measured about the rolling direction −0 degrees means a consistent direction, and 90 degrees means a perpendicular direction. The samples were cut at the following angles: 0°, 30°, 45°, 60° and 90°. Exemplary samples are shown in Figure 10.

Using samples cut at different angles to the rolling direction results from sheet anisotropy (obviously much smaller than oriented sheets). Since the field in the motor has a diverse and changing direction about the direction of punching, the characteristics averaged for all tested directions were used for approximation. We can obtain magnetization characteristics, losses, hysteresis loops, and waveforms of all field quantities using a measuring device. An example of the measured hysteresis loop family is shown in Figure 11.

The averaged loss characteristic was compared with the results for the toroidal sample to check adequacy. The toroidal sample better reflects the phenomena occurring in the motor’s magnetic circuit. The problem, however, is cutting such a sample because each shape would require the construction of an expensive die. For this reason, the measurements were compared for laser-cut samples. The toroidal sample had dimensions of 420/400 mm (external diameter/internal diameter), which allowed us to obtain field homogeneity similar to that found when testing a rectangular sample. A detailed description of the measurement system has been presented in [48,49]. Figure 12 shows the toroidal wound sample with the measuring system.

A comparison of loss characteristics in a wide frequency range for a laser-cut rectangular sample with a width of 10 mm, measured using the MAG8.1 device and measured in the system shown in Figure 12 of a laser-cut toroidal sample of the same width, is shown in Figure 13.

It indicates high compliance of the characteristics measured on the toroidal sample with the averaged values calculated for the rectangular samples.

4. Verification of the Proposed Approximation Method for Samples Cut with the Use of a Guillotine

In the case of guillotine cutting, which corresponds to die cutting, the problem of necessary data for approximation is less complicated than in the case of laser cutting. As shown in the papers [49,50], it can be assumed with some approximation that the two-sided width of the damaged zone is equal to about 3.38 mm and is constant regardless of the width of the sample (for the tested sheet). This allowed the development of a relatively simple method of calculating the characteristics for a sample of any width based on the results of loss measurements obtained for a 60 mm broad sample, in which the impact of the damaged zone is practically negligible, and for a 4 mm wide sample, which can be approximately treated as completely damaged [49]. However, in this comparison, only the data obtained based on measurements were used as more reliable.

It should be emphasized that, following the conclusions in the paper [46], both with the use of FEM and the analytical method developed and verified by the authors, calculations of losses necessitated extrapolation of the specific loss characteristic of the material for frequencies above 4000 Hz. This extrapolation used the measured values in the 1000–4000 Hz range and was a given dependence (7). The method of determining the coefficients k_h, α, k_e, and β in specific ranges of magnetic induction and frequency, approximated by polynomials as a function of the sample width w, is discussed in Chapter 2.

Examples of measured and approximated loss characteristics of electrical sheets M270-35A for different sample widths at frequencies of 5 Hz, 10 Hz, and 20 Hz are shown in Figure 14.

Examples of measured and approximated loss characteristics of electrical sheets M270-35A for different sample widths at frequencies of 50 Hz and 100 Hz are shown in Figure 15.

Examples of measured and approximated loss characteristics of electrical sheets M270-35A for different sample widths at frequencies of 200 Hz, 500 Hz, and 750 Hz are shown in Figure 16.

Examples of measured and approximated loss characteristics of electrical sheets M270-35A for different sample widths at frequencies of 1000 Hz, and 1400 Hz are shown in Figure 17.

Examples of measured and approximated loss characteristics of electrical sheets M270-35A for different sample widths at frequencies of 2000 Hz and 3000 Hz are shown in Figure 18.

Examples of measured and approximated loss characteristics of M270-35A electrical sheets for different sample widths at a frequency of 4000 Hz are shown in Figure 19a.

The coefficients k_h, k_e, α, and β approximating the loss characteristics of electrical sheets M270-35A as a function of the sample width, determined for frequencies up to 4000 Hz, were used to determine the loss characteristics for frequencies higher than 4000 Hz (up to 14,400 Hz) in the magnetic induction range up to 0.2 T. (Figure 19b), for samples of different widths.

The specific losses characteristics of the electrical sheet, determined in a wide range of frequencies, can be used to calculate the basic and additional losses caused by higher harmonics of magnetic induction, taking into account the actual width of individual elements of the magnetic circuit of an induction motor with a core punched out.

5. Verification of the Proposed Approximation Method for Samples Cut with the Use of the Laser

Figure 20 shows the average for different cutting angles about the direction of sheet rolling, measured at the frequency of the supply voltage of 50 Hz, and the loss characteristics of the M270-35A sheet for samples of different widths cut using the laser punching technology.

The method for determining substitute loss characteristics for samples cut with a guillotine of various widths, proposed in [49], is based on the results of loss measurements obtained for a 60 mm wide sample, in which the influence of the damaged zone is practically negligible, and for a small width sample, which can be treated as completely damaged. For samples cut with a laser, even taking into account the variable width of the inactive zone depending on the width of the sample and the strength of the magnetic field in the sample, this approach leads to significant errors. Therefore, to determine the loss of a sample of any width, it is necessary to measure the loss of a larger number of samples (e.g., 8–10) cut with a laser in the width range corresponding to the dimensions of the magnetic core of the electric machine, and then approximate it according to the relation (4), determining the quantities k_h, k_e, α, and β in the specified ranges of magnetic induction and frequency in the form of polynomials as a function of any sample width w from the tested range.

Examples of measured and approximated loss characteristics of electrical sheets M270-35A for different widths of the laser-cut sample at frequencies of 10 Hz and 20 Hz are shown in Figure 21.

Examples of measured and approximated loss characteristics of M270-35A electrical sheets for different widths of the laser-cut sample at 50 Hz, 100 Hz, and 200 Hz are shown in Figure 22.

Examples of measured and approximated loss characteristics of electrical sheets M270-35A for different widths of laser-cut samples at frequencies of 300 Hz, 500 Hz, and 750 Hz are shown in Figure 23.

Examples of measured and approximated loss characteristics of electrical sheets M270-35A for different widths of laser-cut samples at frequencies of 1000 Hz and 1400 Hz are shown in Figure 24.

Examples of measured and approximated loss characteristics of electrical sheets M270-35A for different widths of the laser-cut sample at frequencies of 2000 Hz and 3000 Hz are shown in Figure 25.

Examples of measured and approximated specific loss characteristics of electrical sheets M270-35A for different widths of a laser-cut sample at a frequency of 4000 Hz are shown in Figure 26a. As for samples cut with a guillotine, and also for samples cut with a laser, coefficients k_h, k_e, α, and β approximating loss characteristics of electrical sheets M270-35A, as a function of sample width for a frequency up to 4000 Hz, were used to determine loss characteristics for frequencies higher than 4000 Hz (up to 14,400 Hz) in the range of magnetic induction up to 0.3 T. Figure 26b shows the approximated specific losses characteristics for several selected frequencies in the range up to 14,400 Hz, for a rectangular sample cut with a laser with a width of 10 mm and compared with the characteristics measured in a wide frequency range (up to 14,400 Hz) for a ring sample cut with a laser with diameters of 400/420 mm.

For numerical illustration,

Table 5. Exemplary the differences between experimental and simulation results for the M270-35A sheet cut with a guillotine.

Frequency [Hz]	Flux Density [T]	Specific Losses for Sample Widths = 10 mm [W/kg]			Specific Losses for Sample Widths = 60 mm [W/kg]
		Measured	Approximated	Error [%]	Measured	Approximated	Error [%]
10	1.4	0.3635	0.3564	1.95	0.3144	0.3019	3.95
50	1.4	2.4953	2.4692	1.05	2.0838	2.0943	−0.50
100	1.4	5.8647	6.0438	−3.05	4.9967	4.9704	0.53
200	1.2	11.3984	11.1024	2.60	9.4816	9.3090	1.82
500	1.2	44.5892	43.7880	1.80	38.4084	37.5700	2.18
1000	0.6	34.6830	35.8368	−3.46	29.5318	29.2622	0.91
2000	0.5	70.9134	71.0834	−0.24	61.2496	61.3573	−0.18
4000	0.3	76.9264	77.3911	−0.60	68.1426	68.5557	−0.61

and

Table 6. Exemplary the differences between experimental and simulation results for the M270-35A sheet cut with a laser.

Frequency [Hz]	Flux Density [T]	Specific Losses for Sample Widths = 10 mm [W/kg]			Specific Losses for Sample Widths = 60 mm [W/kg]
		Measured	Approximated	Error [%]	Measured	Approximated	Error [%]
10	1.4	0.4221	0.4175	1.09	0.3309	0.3282	0.82%
50	1.4	2.9287	2.9365	−0.27	2.2125	2.2384	−1.17
100	1.4	6.9318	6.9457	−0.20	5.2760	5.2426	0.63
200	1.2	13.8982	13.6054	2.11	10.1718	10.0556	1.14
500	1.2	50.1598	50.2049	−0.09	39.9924	39.5264	1.17
1000	0.6	47.2180	48.3642	−2.43	32.5832	33.5303	−2.91
2000	0.5	95.7118	92.0008	3.88	65.802	64.2697	2.33
4000	0.3	105.7260	105.7494	−0.02	73.9746	74.5607	0.79

compare measured and simulated losses using the introduced dependencies. The source of differences between the experiment and the simulation are primary errors resulting from the limited accuracy of measurements and secondary errors resulting from the limited accuracy of the approximation used and its adequacy to describe the physical phenomena occurring in the models. However, it can be stated that the proposed approximation allows us to determine the loss of a sample of a certain width in a wide range of magnetic induction modulus and frequency with an accuracy not worse than 4%.

As in the case of cutting loss characteristics of the electrical sheet using a die, determined in a wide range of frequencies, characteristics of the electrical sheet cutting by laser can be used to calculate losses in an induction motor. This includes both basic and additional losses caused by higher harmonics of magnetic induction, taking into account the actual width of individual elements of the magnetic circuit of an induction motor.

6. Summary of the Impact of the Punching Method on the Power Loss of Electrical Sheet

Based on the measurements and approximation of the loss characteristics of electrical sheet samples, an analysis of the impact of the applied punching technology on the sheet loss values in a wide range of magnetic induction and frequency was carried out. The results obtained for samples of several selected widths cut with a guillotine and a laser were compared. In addition, the results of loss measurement for a water-cut sample 60 mm wide, in which material degradation practically does not occur (Figure 27 and Figure 28), are presented.

Table 7. Comparison of the relative (referred to measured loss values of water-cut sample 60 mm wide) loss values of M270-35A sheet metal samples cut with a guillotine and a laser.

Frequency [Hz]	Flux Density [T]	Sample Width = 4 mm		Sample Width = 10 mm		Sample Width = 60 mm
		Guillotine Cutting	Laser Cutting	Guillotine Cutting	Laser Cutting	Guillotine Cutting	Laser Cutting
50	1.0	1.68	2.02	1.23	1.75	1.01	1.15
300	1.0	1.50	1.87	1.21	1.60	1.03	1.12
1000	0.5	1.54	2.35	1.21	1.73	1.04	1.16
4000	0.25	1.34	2.16	1.17	1.64	1.04	1.13

presents a comparison of the relative (referred to measured loss values of water-cut sample 60 mm wide) loss values of M270-35A sheet metal samples cut with a guillotine and a laser, at several frequencies of the supply voltage, for a selected, constant value of magnetic induction.

As can be seen from Table 7, for small-width samples cut with a guillotine, the loss rate increases by about 50%, while for samples cut with a laser, it is more than twice as high as the loss of a 60 mm wide sample with a practically undamaged structure, i.e., water-cut. As the width of the sample increases, these differences decrease. For samples with a width of 60 mm, when cutting with a guillotine, the increase in loss is of the order of several percent, while when cutting with a laser, it does not exceed 15% of the loss of an undamaged sample.

7. Conclusions

The paper presents a method of approximating the loss of a non-oriented electrical sheet, considering the effect of cutting machine elements with a guillotine (corresponding to cutting with a punch) and with a laser.

The method can be used for various types of non-oriented sheets, provided that the loss characteristics meet the condition specified in Chapter 2, allowing for the division of losses into hysteresis losses proportional to frequency and eddy current losses proportional to the square of the frequency.

The method, as demonstrated in Chapters 3 and 4, allows approximation of electrical sheet loss depending on the width of the sample for a wide range of inductions and frequencies.

An essential advantage of the method is the correct approximation in the range of measured frequencies and the possibility of extrapolating losses beyond the measured range. The ability to determine losses for high frequencies is essential due to their significant share in total losses, especially for machines powered by a PWM inverter and high-speed machines.

For mechanical cutting (guillotine and punch), it is possible to obtain the necessary data only based on measurements for a 60 mm broad sample, in which the impact of the damaged zone is practically negligible, and for a 4 mm wide sample.

For laser cutting, it is necessary to take measurements for different widths of the sample. In both cases, the method presented in Chapter 2 allows for the approximation of loss characteristics for the actual width of individual elements of the magnetic circuit of the electric machine (yokes and teeth of the stator and rotor).

As seen from Chapter 5, the punching method’s effect is critically essential for correctly determining the increase of losses for the electrical sheet due to mechanical cutting, especially in laser cutting.

Because the proposed model operates in a wide range of frequencies, it enables the determination of not only basic losses but also dominant pulsating losses, especially in high-speed machines.

The condition of the tools significantly affects the cutting result. In the series production of electrical machines, the punching tool is replaced after specific punch cycles when it is worn out. Although it was proved that the worn tool degrades the electrical steel, the inclusion of this effect in the design tool has not been attempted yet. A degradation profile for the punch tool can be derived from successive measurements based on different stages of the tool’s life [51].

If we want to be able to reliably determine the efficiency of the motor at the design stage, we must make measurements for each type of sheet metal, also realizing that even for different batches of sheet metal of the same type, there may be several percent differences in loss, which may also depend on the current batch of sheets and for the present the condition of the punching tool or the parameters of the laser used. So far, when calculating core losses, the punching process has been considered with additional, most often empirical, correction factors. The proposed approach makes it possible to abandon these coefficients and use different loss characteristics in the calculation methods, considering the sheet metal processing process’s influence, depending on the magnetic circuit elements’ actual dimensions.