Practical Verification of the Approximation Method of the Influence of Guillotine and Laser Cutting and Core Dimensions on Losses and Magnetization of Induction Motor Cores

Abstract

Induction motors are a significant consumer of electricity. One of the crucial elements of losses in an induction motor is core losses. Core losses become dominant in motors powered by an inverter at higher frequencies. The core sheet loss depends on cutting the core using a die or laser. The article presents a practical method for approximating loss characteristics, enabling the determination of losses with high accuracy and the approximation of the magnetization characteristic. The method’s accuracy was verified using sheet metal samples cut with both a guillotine and a laser. The method is an area method; therefore, it is well-suited for use in analytical methods and calculations of power losses based on FEM (Finite Element Method) post-processing. Then, the developed approximations were used to calculate losses in sample induction motors of different powers. The results indicate that the developed method is accurate, making it a viable alternative to the approximate correction factor.

1. Introduction

Electric motors account for approximately 45 percent of the world’s electricity consumption. Most installed electrical machines are induction machines.

The number of installed electric motors is estimated to exceed 300 million and is growing rapidly [1]. With an assumed motor life of approximately 15 years, purchasing a motor typically accounts for only a few percent of the operating costs. For this reason, there has been a gradual shift in the focus of electric machine design, with a primary goal of reducing losses and thereby increasing energy efficiency. In induction motors operating at a mains frequency of 50 or 60 Hz, the dominant losses are those in the windings, which account for more than 50% of all losses. Reduction in these losses can only be achieved by increasing the usable area of the slots, which, while maintaining the overall dimensions of the machine, involves more significant use of the motor core and, as a result, increased losses in the core, both basic and additional, which in most motors amount to 30 to 40% of all losses.

Knowing the core material’s characteristics is necessary to calculate these losses at the motor design stage. These characteristics depend primarily on the type of electrical sheet metal, the induction value and frequency occurring in the individual elements of the motor core, the dimensions of the teeth and the motor yoke, as well as several additional factors, including sheet metal cutting technology. Punching causes changes in the domains, crystallographic structure, and macroscopic properties of soft ferromagnetic materials. Research on this issue began with Bozorth’s work in the 1940s and 1950s [2]. A broad review of the work on the problem can be found in [3,4,5,6]. Low- and medium-power machines, as well as those with fine-tooth geometries, are most susceptible to degradation. Determination of magnetization characteristics and losses of non-oriented electrical sheets for core elements of electrical machines of different widths is possible based on measurements on two samples: a wide one, for which we can assume that the cutting effect is negligible, and a narrow one, for which we can assume that the material has been subjected to the adverse impacts resulting from cutting over its entire width. Unfortunately, this possibility applies only to elements cut mechanically using a guillotine or a punch [7,8,9,10,11,12]. For laser cutting, the nature of sheet degradation is different, which makes it impossible to use such an approach [13,14,15].

The paper presents an extended method (preliminarily presented in [16]) for high-quality approximation of the magnetization characteristics and loss characteristics of two non-oriented electrical sheet metal types used in the construction of rotating machine cores, as a function of three variables: magnetic induction, frequency, and sample width. This allowed us to show the influence of laser and guillotine cutting on the material characteristics of various types of sheet metal [17]. The developed loss approximation method, which takes into account the influence of the cutting width and method, was incorporated into the analytical algorithm for modeling the induction motor and the proprietary STAT program [18]. Of course, the accuracy of loss modeling using the modified loss ratio is also determined by the core loss modeling algorithms used in this program, with particular emphasis on surface losses, which constitute a significant portion of the total losses in low- and medium-power machines. An accurate analytical model for estimating iron losses enables the improvement of motor parameters at the design stage, not only for induction motors [1,19,20,21].

Section 2 covers the basic elements of the work, including the approximation of the magnetization characteristic and the approximation of losses in the sheet metal. The approximation of the magnetization characteristic is as vital as the approximation of losses because it determines the accuracy of determining the magnetizing current, which, for low-power machines, is a significant portion of the motor current and influences losses in the windings. The approximation of losses utilizes the classical division of losses into hysteresis and eddy current losses. By dividing the losses into the tested frequency range and sample width, accurate approximation formulas were developed. These allow for the determination of losses depending on the material width in the tested machine area.

Section 3 presents the basics of calculating losses, including higher harmonic losses in the analytical model of the motor.

Section 4 contains a verification of the presented approximation methods and loss calculations, based on calculations for four motors of different power and frequency, comparing the obtained values with measurements.

2. Approximation of the Magnetization and Loss Characteristics of the Electrotechnical Sheet Metal

The main object of the analysis was an example of electrical sheet metal used for the cores of M470-50A induction motors with a thickness of 0.5 mm. For comparison, the results obtained for sheet metal with a thickness of 0.35 mm—M270-35A, which can be used for the cores of motors with increased efficiency, were also presented. As shown in [21,22,23,24,25], the magnetization characteristics and loss characteristics of each type of electrical sheet metal depend not only on the value of magnetic induction and frequency but also, especially in the range of low values of the magnetic field intensity, on the applied cutting technology and the width of the sheet metal strip.

2.1. Magnetization Characteristics

In the analytical method used to calculate the parameters and characteristics of the induction motor, the B-H magnetization characteristic is necessary to calculate the magnetizing current of the motor. Figure 1 shows the magnetization characteristics of the M470-50A sheet (Figure 1a) and the M270–35A sheet (Figure 1b), measured at a supply voltage frequency of 50 Hz, averaged for different cutting angles to the rolling direction, for several samples of various widths, cut using both guillotine and laser cutting technologies. Measurements for different cutting angles enable consideration of the magnetic anisotropy that occurs. For each of the seven sample widths, measurements were taken for five cutting directions [23].

As can be seen from the curves presented in Figure 1, the differences for these two types of sheet metal are minor. However, the differences in the magnetization characteristics for samples cut using different technologies in the induction range below 1.4 T are considerable for small sample widths. In contrast, for higher values of magnetic induction (above 1.4 T), the characteristics of samples cut with a laser exceed those of samples cut with a guillotine. As the induction increases further, they become closer to each other.

Magnetization characteristics for any sample width cut with a guillotine can also be determined theoretically using the method described in [23]. For laser-cut samples, a 5 mm wide sample is regarded as completely damaged in the entire range of magnetic induction; however, in this case, a correction of the bilateral width of the inactive zone should be introduced, depending on the sample width and the value of the magnetic field intensity in the sample. For laser-cut samples with a width of less than 5 mm, the magnetization characteristics can be determined only by measurement. In the STAT analytical calculation program, the magnetization characteristics are approximated using 9th, 11th, and 13th-order polynomials in the following form [18]:

$$H_m=a_{13}{B_m}^{13}+a_{11}{B_m}^{11}+a_9{B_m}^9+a_1{B}_m$$

(1)

where H_m—is the maximum value of the magnetic field intensity, and B_m—is the maximum value of the magnetic induction. The degrees of the polynomial approximating the magnetization characteristic were chosen based on empirical criteria, due to the nature of this characteristic. Fitting this curve using lower-order polynomials produces unacceptable errors compared to the measured characteristic.

The coefficients a₁, a₉, a₁₁ and a₁₃ appearing in Equation (1) were determined as a function of the sample width x and for the M470-50A sheet metal they are listed in

Table 1. Coefficients approximating the magnetization characteristics of M470-50A sheet metal at 50 Hz, as a function of sample width, for guillotine-cut samples.

	x < 10 mm	Sample Width-x 10 mm ≤ x < 30 mm	x ≥ 30 mm
a1	0.00530579x2 − 0.10468422x + 0.66216722	0.00020397x2 − 0.00999001x + 0.22456257	−0.00022896x + 0.11500792
a9	0.00305011x2 − 0.09882070x + 0.74171217	0.00054375x2 − 0.02565346x + 0.25082739	0.00034587x − 0.04197109
a11	−0.00103136x2 + 0.05376345x − 0.44505727	−0.00034513x2 + 0.0157832x − 0.12466469	−0.00060647x + 0.05851273
a13	0.00002318x2 − 0.00708952x + 0.07130214	0.00005455x2−0.00241898x+ 0.01955544	0.00015443x − 0.00899416

(for the guillotine-cut samples) and

Table 2. Coefficients approximating the magnetization characteristics of M470-50A sheet metal at 50 Hz, as a function of sample width, for laser-cut samples.

	x < 10 mm	Sample Width-x 10 mm ≤ x < 30 mm	x ≥ 30 mm
a1	0.00456309x2 − 0.13203596x + 1.29456589	0.00075238x2 − 0.04232056x + 0.77703872	−0.00192148x + 0.24180396
a9	0.01293691x2 − 0.21691801x + 0.88129844	0.00003138x2 + 0.00069892x − 0.00836756	−0.00058466x + 0.05819845
a11	−0.00952104x2 + 0.16197559x − 0.66266359	0.00002706x2 − 0.00332134x + 0.03946426	0.00043548x − 0.04844816
a13	0.00171481x2 − 0.02959676x + 0.12739373	0.00001410x2 + 0.00105769x − 0.00693274	−0.00007705x + 0.01436672

(for the laser-cut samples), while for the M270-35A sheet metal they are listed in

Table 3. Coefficients approximating the magnetization characteristics of M270-35A sheet metal at 50 Hz, as a function of sample width, for guillotine-cut samples.

	x < 10 mm	Sample Width-x 10 mm ≤ x < 30 mm	x ≥ 30 mm
a1	0.00331095x2 − 0.07264214x + 0.51856131	0.00009547x2 − 0.00533485x + 0.16473088	−0.00014719x + 0.09443036
a9	0.00864965x2 − 0.17804632x + 0.93836887	0.00024135x2 − 0.01184702x + 0.11433901	0.00055313x − 0.04111274
a11	−0.00685244x2 + 0.13726040x − 0.65793720	−0.00014073x2 + 0.00601744x − 0.01473377	−0.00081288x + 0.06394361
a13	0.00128762x2 − 0.02533188x + 0.11865067	0.00002036x2 − 0.00072455x − 0.00102191	0.00024061x − 0.01171574

(for the guillotine-cut samples) and

Table 4. Coefficients approximating the magnetization characteristics of M270-35A sheet metal at 50 Hz, as a function of sample width, for laser-cut samples.

	x < 10 mm	Sample Width-x 10 mm ≤ x < 30 mm	x ≥ 30mm
a1	0.01016331x2 − 0.22758058x + 1.63954634	0.00067168x2 − 0.03644296x + 0.67684692	−0.00226978x + 0.25556552
a9	0.01231831x2 − 0.21019253x + 0.83897053	−0.00016388x2 + 0.00701705x − 0.09634859	−0.00005479x − 0.03493911
a11	−0.00976185x2 + 0.16618097x − 0.65662005	0.00003194x2 − 0.00119065x + 0.04802653	0.00010525x + 0.04086413
a13	0.00188548x2 − 0.03194502x + 0.13110823	0.00000742x2 − 0.00036562x + 0.00092435	−0.00002596x − 0.00322681

(for the laser-cut samples).

Figure 2 shows example curves of coefficients approximating the sheet’s magnetization characteristics as a function of the sample width at a supply voltage frequency of 50 Hz, using guillotine cutting technology for the M470-50A sheet and the M270-35A sheet.

Figure 3 compares the measured and approximated magnetization characteristics at a frequency of 50 Hz for several samples of different widths cut with a guillotine for sheet metal M470-50A (Figure 3a) and M270-35A (Figure 3b). Figure 4 shows the averaged magnetization characteristics for samples of sheet metal M270-35A of different widths cut using guillotine cutting technology (Figure 4a) and laser cutting (Figure 4b) at different supply voltage frequencies.

As can be seen from Figure 4, the influence of the supply voltage frequency on the averaged sheet magnetization characteristics primarily occurs in the range of low magnetic induction values (below 1.2 T), which is attributed to the different shapes of the magnetic hysteresis loop at various frequencies. This influence decreases as the sample width decreases. This influence is much smaller when cutting samples with a laser than when cutting with a guillotine.

Magnetization characteristics are used to determine the magnetic voltages in individual elements of the engine core (teeth and yokes of the stator and rotor) caused by the fundamental harmonic of magnetic induction, and as a result, the magnetizing current of the motor. The vast majority of motors are supplied with a voltage at a mains frequency of 50–60 Hz. Additionally, the magnetic induction in the motor core is usually greater than 1.2 T, and it is most often in the range of 1.6 to 1.8 T. The magnetization characteristics determined at different voltage values depend only on the sample width for such induction values. Therefore, in practice, for the calculation of the magnetizing current in induction motors, both when using the guillotine and laser sheet metal cutting technology, the magnetization characteristics determined for different sample widths at a frequency of 50 Hz can be used, regardless of the frequency of the voltage supplying the motor.

2.2. Approximation of Loss Characteristics

The starting point is the assumption that losses in electrical sheet metal can be presented in the classical approach as the sum of hysteresis losses, proportional to frequency, and eddy current losses, proportional to the square of the frequency. For such an approach, the quotient of losses and frequency as a function of frequency for constant induction is a straight line. Figure 5 shows the relationship between the quotient of losses and frequency for 0.5 mm sheet metal, covering two frequency ranges: from 5 Hz to 50 Hz and from 2000 Hz to 4000 Hz (4 mm wide samples cut with a guillotine). The coefficient of determination value for the linear approximation, R², is given for each curve.

As can be seen, the R² coefficient is greater than 0.95, which proves that the assumption of linearity of the tested dependence is correct. Dividing the frequency and induction ranges into sub-ranges made it possible to obtain the required accuracy. Similar tests were previously carried out for 0.35 mm sheet metal and 0.2 mm thick NO20 sheet metal with an equally positive effect [18,26].

The result of loss distribution is the relationship

$$p_{Fe}=c_{h}f+c_e{f}^2$$

(2)

The simple loss-frequency ratio c_h and c_e coefficients are then approximated by a power-law relationship. The author’s original approach presents the loss relationship in the form

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

(3)

with four coefficients: k_h, α, k_e, and β [16,18].

The values of these coefficients for samples of different widths cut with a guillotine and a laser, in the full range of magnetic induction and frequency, for M270-35A sheet metal with a thickness of 0.35 mm, were given in [16].

Figure 6 shows sample approximation results for guillotine cutting in the width range from 4 to 10 mm and the frequency range from 2000 Hz to 4000 Hz, and for laser cutting in the width range from 10 to 60 mm and the frequency range from 5 to 50 Hz for M470-50A sheet metal with a thickness of 0.5 mm. Here, too, the R² coefficients are greater than 0.95, indicating the correct choice of the approximation form.

The final step is to approximate the obtained coefficients as a function of the sample width using a second-degree polynomial. Thus, the approximation requires determining 12 coefficients for each of the ranges. Figure 7 shows example approximations for the 5–50 Hz range, guillotine cutting, induction above 1.2 T, and the width range of 10–60 mm.

Figure 8, Figure 9 and Figure 10 compare the loss characteristics of both sheet metal grades calculated using the developed approximations with the measurement results.

The measuring systems and methods for measuring the material characteristics of electrical sheets were presented in [16,23].

3. Method of Calculation of Induction Motor Core Losses

The calculation of basic losses in the core was calculated for the fundamental harmonic of the magnetic field, using the loss characteristics of the sheets approximated for the actual tooth widths and the height of the stator yoke.

The higher harmonics of the air gap magnetic fields were considered in the analytical calculations of the additional core losses.

In the calculations of surface and pulsating losses in the stator and rotor core, both the stator ν_s and rotor ν_r winding space harmonics and the stator ν_sq and rotor ν_rq slot harmonics generated by the fundamental harmonic of the magnetic field in the air gap were taken into account. The relations define the orders of these harmonics (about the entire machine circuit) [27]

$$\begin{array}{l}{\nu }_s=p\left(k_{s}2m_{sph}+1\right){\nu }_{sq}=p\left(j_s{\displaystyle \frac{Q_s}{p}}+1\right)\\ {\nu }_r=p\left(k_r{\displaystyle \frac{Q_r}{p}}+1\right){\nu }_{rq}=p\left(j_r{\displaystyle \frac{Q_r}{p}}+1\right)\end{array}$$

(4)

where k_s(r) = ±1, ±2, ±3…, j_s(r) = ±1, ±2, ±3…

The frequency of the harmonics of the stator field on the rotor f_νs and the rotor field on the stator f_νr is, respectively, as follows:

$$f_{\nu s}=f_s\left|1-{\displaystyle \frac{{\nu }_s}{p}}\left(1-s\right)\right|f_{\nu r}=f_s\left|1+j_r{\displaystyle \frac{Q_r}{p}}\left(1-s\right)\right|$$

(5)

where Q_s—number of stator slots; Q_r—number of rotor slots; p—number of pole pairs; f_s—frequency of the fundamental harmonic of the field; m_sph—number of stator winding phases; s—rotor slip.

The individual winding space harmonics of the stator magnetic induction were calculated based on the following relationship [27]:

$$B_{f{\nu }_s}={\displaystyle \frac{\sqrt{2}{m}_{sph}{\mu }_0{N}_s{k}_{sw{\nu }_s}{k}_{s{\nu }_s}}{\pi \left|{\nu }_s\right|\delta k_{ns}{k}_{c{\nu }_s}}}{I}_s{k}_{s{\nu }_s}={\displaystyle \frac{\sin \left({\nu }_s\frac{b_{s1}}{D_{si}}\right)}{{\nu }_s\frac{b_{s1}}{D_{si}}}}$$

(6)

where μ₀- magnetic permeability of vacuum; N_s—number of series turns of the stator winding; $k_{sw{\nu }_s}$—resultant stator winding factor for the harmonic of the order ν_s; $k_{s{\nu }_s}$—stator slot gap factor for the harmonic of the order ν_s; b_s1—stator slot gap width; D_si—inner diameter of the stator core; δ—air gap thickness; k_ns—stator core saturation factor for the harmonic of the order ν_s; $k_{c{\nu }_s}$—Carter factor for the harmonic of the order ν_s; I_s—current in the phase winding of the stator.

The individual slot harmonics of the stator magnetic induction were calculated based on the following relationship:

$$\begin{array}{l}{B}_{q{\nu }_{sq}}=\pm {\displaystyle \frac{{\beta }_s}{2}}{k}_{c{\nu }_{sq}}{F}_{sq}{B}_p{F}_{sq}={\displaystyle \frac{2\sin \left(1.6⌊j_s⌋{\gamma }_s\pi \right)}{\pi ⌊j_s⌋\left[1-{\left(1.6j_s{\gamma }_s\right)}^2\right]}}\\ {\gamma }_s={\displaystyle \frac{b_{s1}}{t_s}}{t}_s={\displaystyle \frac{\pi D_{si}}{Q_s}}{\beta }_s=0.43\left(1-e^{-m_s}\right)m_s={\displaystyle \frac{\frac{b_{s1}}{\delta }-0.7}{3.3}}\end{array}$$

(7)

If the denominator in formula for F_sq is equal to zero, then $F_{sq}={\displaystyle \frac{1}{⌊j_s⌋}}$

The resultant harmonic induction for the harmonic of order ν_s is calculated from the following relationship:

$$B_{{\nu }_s}=\sqrt{B_{f\nu s}^2+B_{q{\nu }_{sq}}^2+2B_{f{\nu }_s}{B}_{q{\nu }_{sq}}\left(1-{\cos }^2\phi \right)}$$

(8)

where t_s—stator slot pitch; $k_{c{\nu }_{sq}}$—Carter factor for the harmonic of order ν_sq; cosφ—power factor; B_p—fundamental harmonic of induction in the air gap.

Similarly, the individual excitation harmonics of the rotor magnetic induction are calculated by inserting into formula (6) $k_{sk{\nu }_r},k_{r{\nu }_r},k_{nr},k_{c{\nu }_r},{\nu }_r$ respectively instead of $k_{sw{\nu }_s},k_{s{\nu }_s},k_{ns},k_{c{\nu }_s},{\nu }_s$ and b_r1, D_re respectively instead of b_s1, D_si, where $k_{sk{\nu }_r}$—rotor slot skew factor for the harmonic of the order ${\nu }_r$, while the phase current in the stator winding I_s is replaced by the current I_r1 in the rotor winding referred to the stator winding, calculated from the following relationship:

$$I_{r1}=\sqrt{I_s^2+I_0^2+2I_s{I}_0\left(1-{\cos }^2\phi \right)}$$

(9)

where I₀—no-load current.

The individual slot harmonics of the rotor magnetic induction were calculated based on formulas (7), substituting $k_{c{\nu }_{sq}},j_s,{\nu }_s,t_s,Q_s$ for $k_{c{\nu }_{rq}},j_r,{\nu }_r,t_r,Q_r$, respectively, where $k_{c{\nu }_{rq}}$ is the Carter coefficient for the ν_rq order harmonic.

Neglecting the rotor induction harmonics generated by the stator excitation and slot harmonic fields, whose values are relatively small, the average value of the resultant harmonic induction amplitude of the order ν_r of the rotor magnetic field on the stator surface is defined by the following relationship:

$$B_{{\nu }_r}=\sqrt{B_{f{\nu }_r}^2+B_{q{\nu }_{rq}}^2}$$

(10)

Additional surface losses in the stator and rotor teeth are calculated from the relationship proposed by the authors.

$$\begin{array}{l}{P}_{s\left(r\right){\nu }_{r\left(s\right)}}^{pow}=w_{Fe}\left(B_{{\nu }_{r\left(s\right)}},f_{{\nu }_{r\left(s\right)}},b_{s1\left(r1\right)}\right){\rho }_{Fe}{S}_{s\left(r\right)}{\lambda }_{s\left(r\right){\nu }_{r\left(s\right)}}\\ S_{s\left(r\right)}=\pi D_{si\left(re\right)}{k}_{t_{s\left(r\right)}}{L}_{s\left(r\right)}{k}_{Fe}\\ k_{t_{s\left(r\right)}}={\displaystyle \frac{t_{s\left(r\right)}-b_{s1\left(r1\right)}}{t_{s\left(r\right)}}}{\lambda }_{s\left(r\right){\nu }_{r\left(s\right)}}={\displaystyle \frac{\pi D_{si\left(re\right)}}{2\left|{\nu }_{r\left(s\right)}\right|}}{P}_{s\left(r\right)}^{pow}={\displaystyle \sum _{{\nu }_{r\left(s\right)}}{P}_{s\left(r\right){\nu }_{r\left(s\right)}}^{pow}}\end{array}$$

(11)

where w_Fe—specific electrical sheet loss of at the frequency f_{ν r(s)}, and magnetic flux density B_r(s).νr(s), approximated based on measured loss characteristics as a function of magnetic flux density, frequency, stator (rotor) tooth head dimensions, and the applied core cutting technology; λ_s(r).νr(s)—depth of penetration of individual harmonics of the rotor (stator) magnetic field into the stator (rotor) teeth [17]; ρ_Fe—specific mass of the core material; L_s(r)—stator (rotor) core length; k_Fe—core filling factor.

Additional pulsation losses in the stator teeth $P_s^{puls}$ caused by the rotor field harmonics were calculated using the formula in the form given in [6].

$$P_{s{\nu }_r}^{puls}=w_{Fe}^{puls}\left(B_{puls{\nu }_r},f_{{\nu }_r},b_{sdavr}\right)m_{sd}{P}_s^{puls}={\displaystyle \sum _{{\nu }_r}{P}_{s{\nu }_r}^{puls}}$$

(12)

where $w_{Fe}^{puls}$—part of the electrical sheet loss caused by eddy currents, approximated for the frequency f_νr, magnetic induction B_pulsνr, for the average stator tooth width and the core cutting technology used; m_sd—stator tooth mass.

The individual harmonics of magnetic induction were calculated based on the following relationship:

$$\begin{array}{l}{B}_{sd2{\nu }_r}={\displaystyle \frac{t_s{k}_{ts}{\eta }_s{B}_{{\nu }_r}}{k_{Fe}{b}_{sd2}{k}_{n1}}}{\eta }_{s1}={\displaystyle \frac{\pi k_{ts}{\nu }_r}{Q_s}}{\eta }_s={\displaystyle \frac{\sin {\eta }_{s1}}{{\eta }_{s1}}}\\ B_{sd3{\nu }_r}=B_{sd2{\nu }_r}{\displaystyle \frac{b_{sd2}}{b_{sd3}}}{b}_{sdavr}={\displaystyle \frac{b_{sd2}+b_{sd3}}{2}}\\ B_{puls{\nu }_r}=\sqrt{{\displaystyle \frac{B_{sd2{\nu }_r}^2+B_{sd2{\nu }_r}{B}_{sd3{\nu }_r}+B_{sd3{\nu }_r}^2}{3}}}\end{array}$$

(13)

where b_sd2 and b_sd3—stator tooth widths at approximately 1/3 and 2/3 of the tooth height (for drip slots—at the height of the diameter of the upper and lower semicircle of the slot); b_sdavr—average stator tooth width.

Additional power losses in the core can therefore be calculated from the following relationship:

$$P_{dd}=P_s^{pow}+P_s^{puls}+P_r^{pow}=P_{dds}+P_{ddr}$$

(14)

Total core losses are calculated as the sum of basic losses in the core P_c and additional losses.

$$P_{Fe}=P_c+P_{dd}$$

(15)

When the magnetic induction in the stator yoke exceeds 1.6 T, additional power losses in the magnetic casing adjacent to the core P_cas must be considered, and their value may be comparable to the additional losses in the core P_dd. The share of additional power losses caused by the leakage flux around the winding end connections P_cc, the rotor slot skew P_sk, and, in motors with cast rotor cages, occurring on the slotted surface of the rotor P_Al and resulting from the lack of insulation between the cage bars and the core P_cocb, is very small and generally does not exceed 1% of the total losses in the motor.

The remaining additional losses (previously assumed as a fraction of a percent of the rated power) are, therefore, as follows:

$$P_d=P_{cas}+P_{cc}+P_{sk}+P_{Al}+P_{cocb}$$

(16)

The total losses in the motor and the electrical input power are calculated from the following relationship:

$${\displaystyle \sum P}=P_{Fe}+P_{ws}+P_{wr}+P_d+P_m{P}_{in}=P_{out}+{\displaystyle \sum P}$$

(17)

where P_ws(r)—stator (rotor) winding losses; P_m—mechanical losses; P_in(out)—input (output) power.

4. Operating Parameters of Selected Induction Motors Made of Different Types of Electrotechnical Sheet Metal and Cutting Technology

The presented method of determining core losses was adopted for calculating the parameters and operating characteristics of four selected induction motors of different designs, with a wide power range, supplied with different voltage values and frequencies. Motors 1 and 2 are high-power motors supplied with mains-frequency voltage, where motor 1 is a double-cage motor with die-cast aluminum rotor cage, and motor 2 is a single-cage motor with welded bar-type copper cage rotor. Motors 3 and 4 are low-power model motors with die-cast aluminum rotor cages, supplied from an inverter, operating in a wide frequency range. The rated data of these motors and selected design parameters are listed in

Table 5. Basic nominal data and construction dimensions of the tested machines.

Data	M 1	M 2	M 3	M 4
Rated power [kW]	45	300	1.4	0.20
Rated voltage [V]	480	1140	230	230
Frequency [Hz]	50	50	350	300
Rated current [A]	71	180	5.82	1.56
Number of poles	4	4	4	2
Rated power factor	0.83	0.85	0.65	0.69
Rated efficiency [%]	92.0	94.5	73.0	68.5
Outer diameter of stator core [mm]	327	440	118	87.4
Shaft axis height [mm]	200	250	71	56
Length of stator core [mm]	234	550	120	32
Number of stator slots	48	48	36	18
Number of rotor slots	40	40	32	11
Number of series turns/phase	62	46	114	96
Average stator tooth width [mm]	7.07	9.15	3.25	3.86
Stator yoke height [mm]	28.85	38.20	7.52	12.52
Average rotor tooth width	8.16	12.87	3.06	5.61
Rotor yoke height [mm]	30.58	41.94	12.62	4.7

.

Motors 1, 2, and 3 have cores made of 0.5 mm-thick M470-50A sheet metal, while motors 4 have cores made of both 0.5 mm-thick M470-50A sheet metal and 0.35 mm-thick M270-35A sheet metal. Measurement and calculation results for motor 3, with a core cut from M270-50A sheet metal using a guillotine and laser, were presented in [6]. In addition, motor 4 was made in two versions, i.e., with a core cut with a die and a core cut with a laser.

For all motors, calculations of parameters and losses were made using the proprietary STAT program. When calculating basic and additional losses in the core, both the loss characteristics determined as a function of induction and frequency for a toroidal sample with a constant width of 10 mm and the characteristics determined for rectangular samples and approximated as a function of 3 variables w_Fe(B,f,x) were used, i.e., induction, frequency and dimensions of the core magnetic circuit element, as well as taking into account the core cutting technology.

The calculation results are presented in

Table 6. Calculation results for motor 1 with a core made of M470-50A sheet metal, for measurement conditions (columns: A—M470-50A, calculated loss from toroidal sample; B—M470-50A loss from die-cut rectangular samples; C—M470-50A loss from laser-cut rectangular samples; D, E and F like A, B and C but for M270-35A), f_s = 50 Hz, P_out = 45.2 kW, U_s = 479 V, P_m = 106 W.

	Meas.	A	B	C	D	E	F
Is [A]	69.90	73.12	73.20	73.30	73.43	73.25	73.89
I0 [A]	31.3	31.20	30.87	33.79	32.0	31.33	32.17
Bsd [T]	-	1.617	1.606	1.604	1.602	1.602	1.597
Bsy [T]	-	1.635	1.632	1.634	1.625	1.639	1.624
Brd [T]	-	1.653	1.643	1.647	1.638	1.639	1.633
Bry [T]	-	0.988	0.991	0.993	0.989	0.992	0.991
PFe [W]	1518	776	1413	1614	731	1094	1383
Pd [W]	218	210	181	193	204	216	196
Pws [W]	1526	1670	1671	1672	1683	1674	1704
Pwr [W]	1055	1063	1036	1054	1055	1040	1050
∑P [W]	4423	3825	4407	4639	3779	4130	4439
η [%]	91.09	92.20	91.12	90.69	92.28	91.62	91.06
cos φ [-]	0.816	0.816	0.818	0.808	0.810	0.815	0.813

,

Table 7. Calculation results for motor 2 with a core made of M470-50A sheet metal, for measurement conditions (columns: A—M470-50A, calculated loss from toroidal sample; B—M470-50A loss from die-cut rectangular samples; C—M470-50A loss from laser-cut rectangular samples; D, E and F like A, B and C but for M270-35A), f_s = 50 Hz, P_out = 300 kW, U_s =1138 V, P_m= 1320 W.

	Meas.	A	B	C	D	E	F
Is [A]	185.1	189.4	187.7	188.9	190.5	187.7	188.8
I0 [A]	59.4	56.3	53.01	54.24	58.37	53.94	55.19
Bsd [T]	-	1.701	1.696	1.701	1.692	1.690	1.686
Bsy [T]	-	1.702	1.701	1.704	1.695	1.697	1.693
Brd [T]	-	1.640	1.638	1.639	1.629	1.631	1.627
Bry [T]	-	0.904	0.908	0.907	0.904	0.908	0.908
PFe [W]	4200	4059	4111	4800	3816	3021	3966
Pd [W]	1535	1601	1452	1108	1254	1338	1429
Pws [W]	5613	5877	5772	5849	5846	5769	5839
Pwr [W]	6853	6852	6795	6823	6865	6824	6870
∑P [W]	19,521	19,709	19,650	19,900	19,101	18,272	19,424
η [%]	93.90	93.84	93.91	93.78	94.01	94.26	93.92
cos φ [-]	0.876	0.856	0.863	0.868	0.851	0.861	0.859

,

Table 8. Calculation results for motor 3 with a core made of M470-50A sheet metal cut with both a die and a laser, for measurement conditions f_s = 350 Hz.

	Die Cut			Laser Cut
	Measured	Toroidal	Rectangular	Measured	Toroidal	Rectangular
Pout [kW]	1.375	1.375	1.375	1.063	1.063	1.063
Us [V]	221.4	221.4 Y	221.4	219.9	219.9 Y	219.9
Is [A]	5.59	5.55	5.60	4.94	4.63	4.90
I0 [A]	-	0.561	0.577	-	0.624	0.776
Bsd [T]	-	0.329	0.327	-	0.331	0.329
Bsy [T]	-	0.420	0.421	-	0.675	0.421
Brd [T]	-	0.375	0.363	-	0.389	0.371
Bry [T]	-	0.182	0.155	-	0.186	0.159
PFe +Pd	170.9	147.7	164.0	347.7	154.0	342.4
Pws [W]	88.5	87.2	92.0	66.9	58.8	65.8
Pwr [W]	56.9	59.5	59.9	37.1	32.2	40.1
Pm [W]	81.4	81.4	81.4	82.2	82.2	82.2
∑P [W]	397.7	375.8	397.3	533.9	327.2	530.5
η [%]	77.57	78.53	77.58	66.56	76.46	66.70
cos φ [-]	0.841	0.849	0.824	0.863	0.872	0.868

and

Table 9. Calculation results for motor 4 with a core made of both M470-50A and M270-50A sheet metal cut with a die, for measurement conditions f_s =300 Hz.

	M470-50A			M270-35A
	Measured	Toroidal	Rectangular	Measured	Toroidal	Rectangular
Pout [kW]	0.184	0.184	0.184	0.184	0.184	0.184
Us [V]	221.9	221.9 Y	221.9	226.0	226.0 Y	226.0
Is [A]	1.01	1.00	1.01	0.99	0.98	0.99
I0 [A]	-	0.738	0.696	-	0.743	0.702
Bsd [T]	-	0.856	0.852	-	0.872	0.868
Bsy [T]	-	0.764	0.762	-	0.778	0.777
Brd [T]	-	0.979	0.975	-	0.997	0.994
Bry [T]	-	0.826	0.856	-	0.828	0.873
PFe +Pd	18.21	15.22	18.58	15.13	12.68	14.61
Pws [W]	8.36	8.32	8.26	7.96	7.72	8.09
Pwr [W]	4.72	4.58	4.58	4.06	3.57	4.72
Pm [W]	18.68	18.68	18.68	16.74	16.74	16.74
∑P [W]	49.96	46.73	50.10	43.89	40.71	44.16
η [%]	78.66	79.76	78.62	80.76	81.91	80.67
cos φ [-]	0.520	0.546	0.554	0.597	0.581	0.591

. The following symbols are used in the tables: U_s—supply voltage; B_sd, B_rd—inductions in the stator and rotor teeth; B_sy, B_ry—inductions in the stator and rotor yoke; η—motor efficiency.

Table 6 and Table 7 additionally present the calculation results for replacing the M470-50A sheet with the M270-35A sheet and using different core punching technologies.

The manufacturer of these motors provided the measurement results for motors 1 and 2, while motors 3 and 4 were measured in laboratory conditions.

As can be seen from Table 6 and Table 7, in the case of high or medium power motors supplied with a voltage of 50 Hz, the share of the sum of core losses and additional losses in the motor does not exceed 40% of all losses; however, despite this, the accuracy of calculating these losses has an impact on the motor efficiency. The use of material characteristics approximated for a 10 mm wide laser-cut toroidal sample in calculations yields efficiency results similar to those of core motors with die-cut cores, differing by approximately 1% from the measurement results. This difference depends on the extent to which the tooth dimensions vary from 10 mm. Yoke widths in these motors have less influence because the loss characteristics for widths greater than 10 mm are similar. The use of loss characteristics determined as a function of induction, frequency, and the dimensions of the core magnetic circuit element in calculations yields results that are somewhat more similar to the measurement results. Still, the differences between these calculations are minimal and do not exceed 1%. This results from the fact that in high-power motors with shaft axis heights of 200 mm and greater, the dimensions of the teeth and motor yoke are close to 10 mm or larger, so, as shown in Figure 8, the loss characteristics determined for samples of such width are also similar. The change in core cutting technology and sheet metal type also has a relatively small effect. However, using M270-35A sheet metal with lower losses than M470-50A sheet metal allows for an increase in efficiency of about 0.5%. Figure 11 shows a graphical summary of losses for motor 1.

In contrast, core laser cutting slightly worsens it. In motor 2 (Table 7), the stator tooth width is somewhat smaller than 10 mm, while the rotor tooth width is slightly larger, and these differences are minimal, so the calculation results obtained using a 10 mm wide toroidal sample and samples of different widths are very similar. For both motors, the magnetizing current values are practically independent of the sample width, as the magnetization characteristics determined for samples of different widths coincide for inductions greater than 1.6 T.

Table 8 presents the results of calculations and measurements for model motor 3, powered by a voltage of 350 Hz, with a core made of M470-50A sheet metal. The cores were cut in motors with a cutting die and a laser. These motors were tested without housing. In motors supplied with voltages significantly higher than the mains voltage, the share of the sum of the core losses and additional losses in the motor when cutting the core with a die exceeds 40% of all losses, while when cutting the core with a laser, it is greater than 60%. In this case, using material characteristics approximated for a 10 mm wide laser-cut toroidal sample for calculations yields higher efficiency results than measurements for motors with a core cut using a die, because the widths of the teeth are significantly smaller than 10 mm. However, because for these motors the average values of magnetic induction in the teeth and yoke of the stator and rotor do not exceed 0.5 T, the differences in efficiency do not exceed 1%, because in this range of induction, the loss characteristics measured for samples of different widths differ little from each other (Figure 9 and Figure 10). In the case of laser-cut cores, it is necessary to use the loss characteristics determined for samples cut using this technology for calculations, because calculations performed using the characteristics specified for a toroidal sample yield values of losses and magnetizing current that are significantly underestimated compared to the measurement results.

Table 9 compares the measurement and calculation results for motor 4, which has a core made from M470-50A and M270-35A sheet metal. In these motors, the average values of magnetic induction in the teeth and yoke of the stator and rotor range from 0.7 to 1 T. In this range of induction, the loss values measured for samples with a width of approximately or less than 5 mm are greater than the loss values determined for a sample with a width of 10 mm (Figure 9). For this reason, the results of motor efficiency calculations made using material characteristics that take into account the core dimensions are closer to the measurement results than the results of calculations made using characteristics determined for a toroidal sample.

As can be seen from Table 8 and Table 9, using sheet metal characteristics approximated for the actual dimensions of the stator teeth and yoke in the calculations yields results for losses, efficiency, and magnetizing current, and consequently, the current in the machine windings, that are more similar to the measured results, especially in motors where the tooth width or stator yoke height is significantly less than 10 mm. This is particularly evident in motors with laser-cut cores, which operate at higher supply voltage frequencies (Table 8). Figure 12 shows a graphical summary of losses for motor 3 and Figure 13 for motor 4.

5. Conclusions

The article presents the application of the method of approximating losses in the core of an induction motor, taking into account the increase in losses caused by mechanical cutting using a punch and die or cutting using a laser for non-oriented electrical sheet metal of thickness 0.5 mm and 0.35 mm, and its influence on the operating parameters of induction motors of different power supplied with varying values of voltage and frequencies.

The conducted tests confirm that the method can be successfully applied to most non-oriented sheets. A key element determining the method’s applicability is the ability to describe losses using the classical division into eddy current and hysteresis losses. The first hysteresis component varies linearly with frequency, while the eddy current component varies with the square of the frequency. This approach eliminates the need to introduce anomalous losses proportional to the frequency to the power of 1.5. The introduction of anomalous losses significantly complicates the loss approximation process due to the appearance of a third factor. The hypothesis that anomalous losses can be omitted is verified by examining the linearity of the loss-to-frequency ratio as a function of frequency.

The proposed method, verified by comparing approximated and measured losses, enables high accuracy in determining core losses due to the division of frequency and induction ranges into sub-ranges during approximation. Of course, a crucial element, es-pecially for low- and medium-power machines, is considering the influence of the width of the machine’s magnetic circuit elements, such as the teeth and yokes, on losses and magnetization characteristics, as well as the core cutting technology, which until now has most often been considered in the form of empirical correction factors resulting from dis-crepancies between calculation results and measurements on the built machine. The developed method, which is the result of several years of work in determining electromagnetic parameters, especially losses in induction motors, enables the correct calculation of machine losses and efficiency with sufficient accuracy at the design stage, without the use of any correction factors. Since the only way to verify the method’s validity is to compare the core losses of the actual machine with those calculated using approximate loss characteristics, an essential element of the work was to perform such a comparison for several selected machines of varying power, powered both by the mains and an inverter. This comparison demonstrates the very good accuracy of the methods used.

The developed method was adapted to the proprietary STAT program, but it can also be adapted to other programs that utilize material characteristics of electrical sheets. For example, it can be applied in any FEM package during postprocessing. The authors have also successfully used it in conjunction with the Opera 2D package from Dassault Systèmes 3DS. The proposed model can be applied to any sheet metal and only requires performing appropriate measurements on samples of this sheet metal of various widths. It is beneficial for calculating losses in low-power or multi-pole motors with small tooth or stator yoke widths (especially less than 10 mm), as well as in motors powered by inverters operating at high frequencies.