High-Speed Permanent Magnet Synchronous Motor Iron Loss Calculation Method Considering Multiphysics Factors

For the high-speed permanent magnet synchronous motors (HSPMSMs), their magnetic fields are more complicated with the rotating magnetization and harmonics considered. Furthermore, there are interaction effects on motor iron loss from high frequency, temperature, and compressive stress. In order to obtain the accurate HSPMSM iron loss calculation model, different iron loss models are proposed, in this article, considering these physical factors, and the iron loss model considering the interaction effect of the multiphysics factors is given. The magnetic field, temperature field, and stress field are analyzed for HSPMSM by finite-element method (FEM). Then, the proposed models are employed to calculate the iron loss of the silicon steel sheet and the prototype. The accuracy of the proposed iron loss model, which can consider the interaction effect of the multiphysics factors, is verified by the experimental test measurements on the prototype.


I. INTRODUCTION
H IGH-SPEED permanent magnet synchronous motors (HSPMSMs) are increasingly popular owing to their excellent performance including high power density, compact size, high efficiency, and the suitability for high-speed direct-drive applications without gearboxes, which are widely utilized in the industrial field, such as gas compressors, distributed power generation, electrical turbocharging, turbines and flywheel energy storage system [1]- [3]. However, there are some characteristic issues for HSPMSMs. High power density leads to high power loss density [4]. Due to the high frequency of the magnetic field alternating in the steel core lamination [5], the high temperature rise in total machine, and the stress of other parts in motor acting on the core, iron loss can be significantly higher than that in ordinary motor [6]. Therefore, the accurate iron loss calculation poses a particular challenge for HSPMSM. The iron loss in the silicon steel sheet is generated by the alternative magnetic field, and it is affected by the magnetization of the magnetic field. It is learnt from the literatures that both alternating magnetization and rotating magnetization exist simultaneously in the motor iron core, wherein the rotating one is the dominant one [7]. The iron loss taking into account of the rotating magnetization is closer to the measured one [8]. The magnetic field harmonic content of high-speed machines was analyzed by the Fourier decomposition method, whose effect cannot be ignored [9], [10]. Iron loss increases exponentially with frequency [11]. In addition, high frequency also causes the skin effect on the surface of the silicon steel sheet, which further increases the iron loss [12]. Thus, the influences of the abovementioned factors on motor iron loss need to be considered.
The losses of silicon steel sheets are affected by temperature, which decrease with temperature increasing. Andreas Krings and Oskar Wallmark studied the effects of motor operating temperatures on iron losses and magnetic properties [13]. The iron loss models that consider temperature effects are proposed based on a large number of experiments [14]. Some researchers have corrected the temperature coefficient through experiment measurements, whereas the correctness of the models is verified by silicon steel sheets and motor [15], [16]. Due to the difficulty in heat dissipation for HSPMSMs, over high temperature problems need to be avoided for motor during operation. Therefore, it is necessary to consider the influences of temperature in the iron loss calculation for HSPMSMs.
The loss characteristics of the silicon steel sheet vary with stress. Studies indicate that the loss of silicon steel sheets increases with compressive stress, and decreases with tensile stress [17]. When the compressive stress is increased from 0 to 100 MPa, the unit loss of a certain type of silicon steel sheet doubles [18], [19]. The effect from stress on the silicon steel sheets' loss is limited. When the stress reaches to a certain value, the loss then maintains at a stable value [20]. The motor iron loss can be affected by both compressive stress and electromagnetic stress, which are generated during motor assembly process and operation [21]. The influences of stress also need to be included in motor iron loss calculation for HSPMSMs. In this article, the improved iron loss models considering different physical factors are proposed. Then, an HSPMSM, whose maximum speed up to 30 000 r/min is taken as an example to analyze the magnetic field, temperature field, and stress field. Finally, the iron losses for the motor under different conditions are calculated by the proposed models, with their results compared to summary some conclusions. The accuracy of the proposed models is verified on the silicon steel sheet and the motor. The work done in this article provides a valuable reference for the research of motor iron loss accurate calculation.

A. Iron Loss Separation Model
The iron loss separation model was proposed by Bertotti, which can be divided into the hysteresis loss, eddy current loss, and additional loss [22]. Its specific expression is where p h is the hysteresis loss, p c is the eddy current loss, p e is the additional loss, k h and α are hysteretic loss coefficients, respectively, k e is the additional loss coefficient, d is the thickness of silicon steel sheet, δ is the density, and ρ is the equivalent resistivity.
When excitation is sinusoidal, (1) can be simplified as The additional loss is a small percentage of the motor iron loss. Therefore, in order to simplify the calculation of the iron loss, the additional loss term is not considered in this article.

B. Iron Loss Model With Magnetic-Effect
Calculating iron loss ideally, the flux density waveform is sinusoidal and the magnetization is alternating magnetization. In fact, due to the influence of stator slotting and rotor structure, the flux density waveform is the superposition of fundamental wave and harmonics. And the induced current in the three-phase winding is incompletely symmetrical, resulting in the rotating magnetization.
The flux density waveform can be decomposed into a fundamental wave and many harmonics. Therefore, the rotating magnetization can be equivalent by two mutually orthogonal alternating magnetizations in the calculation process [23]. Hence, the expression for the iron loss model with magnetic effect is where B mkmax and B mkmin are the magnitudes of the flux density represented by the long and short axes in the elliptical rotating magnetization trajectory, respectively. The loss characteristics of silicon steel sheet can be affected by high frequency significantly. The skin effect results in uneven distribution of eddy current on the laminates. It can be represented by a frequency coefficient [24] k where D = d √ πμσ; μ is the average magnetic permeability of silicon steel sheet; and σ is the conductivity of silicon steel sheet.

C. Iron Loss Model With Temperature-Effect
The iron loss of silicon steel sheet decreases with temperature rising. The temperature effect on total loss can be reflected in hysteresis loss and eddy current loss, respectively. k h and k c both vary with temperature. The original equation improved with temperature effect is [14] p Due to the influence of equivalent resistivity by temperature, the temperature coefficient k t is proposed for equivalent resistivity in the iron loss model. Assuming that equivalent resistivity of nonoriented silicon steel sheet increases linearly with temperature if k t takes a positive value Eddy current coefficient is where ρ 0 is the base value of equivalent resistivity, Δt is a temperature variation, and k t is the temperature coefficient, which is obtained by fitting iron loss curve under different temperature.

D. Iron Loss Model With Stress-Effect
Assuming that the direction of compressive stress acting on the silicon steel sheet is positive, iron loss of silicon steel sheet increases with compressive stress. The microstructure of the silicon steel sheet is changed by compressive stress. The internal magnetic domain is newly arranged consuming more energy, which leads to loss increasing [17]. Therefore, the loss coefficient as a function of compressive stress is proposed. The iron loss model with stress effect can be expressed as where k h (σ) and k c (σ) are the loss coefficient affected by the stress, which is obtained by fitting iron loss curve of stress using the least mean square method.

E. Iron Loss Model With Multiphysics Factors
To analyze the relationship between each physical factor and the loss coefficient, the correlation coefficient is applied. Let (X, Y) be a binary random variable, the correlation coefficient is defined as (9). The closer the correlation coefficient is to 1, the higher the correlation between X and Y The correlation analysis between physical factors and loss coefficient is shown in Fig. 1. The correlation coefficient between stress and hysteresis loss coefficient is close to 1. The correlation coefficient between temperature and eddy current loss coefficient is close to 1. Therefore, the stress is the main factor affecting the hysteresis loss coefficient under different frequencies. And the temperature is the main factor affecting the eddy current loss coefficient under different frequencies.  Based on the iron loss models considering different physical factors and the correlation analysis, the physical factors with the highest correlation are considered in the hysteresis loss and eddy current loss, respectively. An iron loss model with multiphysics factors is proposed, shown as follows: where k h (σ), k c0 (f), and k t are all loss coefficients that are obtained by fitting iron loss curves.

A. Structure and Main Paraments of HSPMSM
The structure of HSPMSM analyzed in this article is shown in Fig. 2. It adopts a two-pole and 24-slot structure.
The windings are connected in a double-layer short-distance distributed form, as shown in Fig. 3. Three-phase windings A, B, and C differ by 120°electric angle in turn.
The rated parameters of HSPMSM are shown in Table I.

B. Magnetic Field Analysis
Due to the same trend of radial flux density waveform and the tangential one at the same position of the integer pitch of motor stator pitch, the feature points are taken only on one stator pitch. The distribution of feature points is shown in Fig. 4. The selection principle considering the top, middle, and bottom of stator teeth and yoke are uniformly followed.
There are mainly two magnetization types that include alternating magnetization and rotating magnetization. Due to the limitation of this article length, only two representative points, which are shown in Fig. 5, are selected to illustrate the complex magnetization characteristics of the stator.
The flux density of point c varies in both radial and tangential directions, indicating that its magnetization mode is alternating magnetization and rotating magnetization. The flux density of point i changes only in the radial direction, indicating that its magnetization mode is only alternating magnetization. Hence, the alternating and rotating magnetization exist simultaneously in the stator.
Because of the existence of harmonics, the waveform of the magnetization trajectory is distorted. The harmonic decomposed graphs of the feature points' magnetization track are shown in Fig. 6.
The harmonic content of each feature point is different. However, the third harmonic is the main harmonic component of all feature points, whose content can reach 48% of fundamental wave content. Hence, harmonic effects cannot be ignored in the iron loss calculation.

C. Temperature Field Analysis
Assuming that there is no heat exchange between the motor stator and rotor, temperature rise calculation can be carried out by the two-heat-source heat circuit method [25]. The temperature distribution on motor stator can be observed by finite element method (FEM) method directly.
Under the natural heat dissipation condition, the boundary condition is set as follows.
1) All circumferential symmetrical surfaces on both sides of the solution domain model are set as wall boundary conditions, and the heat transfer option is set as adiabatic.
2) The contact surface between air gap and sheath is set as the boundary condition of rotating wall, and the angular velocity is set to 3140 rad/s (30 000 r/min) to simulate the actual rotation of the rotor. The internal temperature field of the motor is simulated and analyzed. The temperature distribution on the stator is shown in Fig. 7. Fig. 7 shows the stator temperature distribution of HSPMSM. And the maximum temperature can reach 126.5°C. Hence, temperature effects cannot be ignored in the iron loss calculation.

D. Stress Analysis
The stator is mainly subjected to mechanical compressive stress of the frame during assembly. The static mechanical compressive stress between the stator core and the frame is determined by the static interference [26]. According to the theory of thick-walled cylinders of elastic mechanics, the equation for calculating the static interference is [27] δ s = r so − r fi .
(11)  The interference between the stator and the frame is set as 0.19 mm. The FEM simulation result is shown in Fig. 8.
The maximum value occurs at the bottom of the stator slot, which is 114.32 MPa. The minimum value occurs in the middle of the stator yoke, which is 4.5452 MPa.

A. Measurement System
In order to measure the magnetic properties of silicon steel sheets under different conditions, the monolithic measurement method is adopted. The schematic diagram and physical map are shown in Fig. 9.  Variables for measuring conditions include frequency, temperature, and stress. The measured frequency range is 50 Hz-1 kHz. The measured temperature range is 20-200°C. The measured stress range is 0-100 MPa. And the example specimen of silicon steel sheet is B20AT1500.

B. Calculation and Analysis of Loss Coefficient
Based on the test data of the silicon steel sheets loss, the loss coefficients are fitted by a combination method of the Levenberg-Marquardt and the general global optimization method. The loss coefficients obtained by fitting at different frequencies (T = 20°C, σ = 0 MPa) are shown in Table II. It can be seen that the hysteresis loss coefficient fluctuates in a small range and the eddy current loss decreases with the increase of frequency.
The loss coefficients obtained by fitting at different temperatures (f = 500 Hz, σ = 0 MPa) are shown in Table III. The hysteresis loss coefficient fluctuates in a small range and the  II  LOSS COEFFICIENTS   TABLE III  LOSS COEFFICIENTS   TABLE IV  LOSS COEFFICIENTS temperature coefficient decreases with the increase of temperature. Due to the frequency remains the same, k c0 is constant.
According to the measured loss curves from 0 to 100 MPa (f = 500 Hz, T = 20°C), the loss coefficients with the compressive stress are obtained in Table IV. The eddy current loss coefficient hardly varies with compressive stress (see Table III), which indicates the independence between eddy current loss and stress. The hysteresis loss coefficient increases with compressive stress less than 40 MPa. With the compressive stress more than 40 MPa, the hysteresis loss coefficient starts to increase slowly, indicating that the hysteresis loss coefficient is affected by the compressive stress weakly.

C. Calculation and Analysis of Silicon Steel Sheets Loss
In order to verify the accuracy of the iron loss model, the calculated and measured values of the silicon steel sheet loss are first compared. The average absolute error is used as one of the criteria for measuring the accuracy of the iron loss model.
The average absolute error is defined as where P cal i and P mea i are the calculated and measured values of the loss, respectively. M denotes the total counts of measurement.  The losses of silicon steel sheet from 50 Hz to 1 kHz are calculated by the iron loss model considering high frequency, as shown in Fig. 10. The error between the calculated value and the measured one is less than 3.6%.
The measured iron loss and predicted one are exhibited in Fig. 11. Since the loss curves between different temperatures are relatively close, the two temperatures (20°C and 150°C) are selected for illustration. The error between the calculated value and the measured one is less than 2.7% in the range from 20°C to 200°C.
The comparison between the measured iron loss and predicted values is shown in Fig. 12. The iron loss gradually increases with  the increase of the stress, but the growth rate is obviously slowed down from 60 to 100 MPa. The error between the calculated value and the measured one is less than 4.1% in the range from 0 to 100 MPa.

D. Comparison and Analysis of Iron Loss in HSPMSM
In order to study the influence of single variable on iron loss of HSPMSM, the motor iron loss considering single physical factor is calculated. Applying the iron loss model with magnetic-effect, temperature-effect, and stress-effect, respectively, the motor iron loss is calculated. The motor iron loss increases with the increase of frequency, as shown in Fig. 13. The motor iron loss decreases with the increase of temperature, as shown in Fig. 14. The motor iron loss increases first and then remains stable with the increase of stress, as shown in Fig. 15.
Based on the FEM analysis of HSPMSM, the average temperature on the stator is about 84°C and the average stress is about 60 MPa under no-load condition. The results of motor iron loss with the models considering different factors are shown in Table V.
The percentage variation of iron loss is used to express the effect for three factors on motor iron loss. The rotating magnetization, frequency, and harmonics of magnetic field, and the stress all result in the iron loss increasing, whereas the temperature result in the one decreasing. The higher percentage variation    Fig. 16, indicates that the magnetic field is the main influencing factor of iron loss. And the effect of temperature field and stress field should not be ignored in motor iron calculation. The purpose of the abovementioned analysis is to figure out the influencing factors of iron loss, and analyze the severity of the impact of each factor, which is helpful to take corresponding measures to reduce iron loss.
In order to verify the accuracy of the iron loss model considering multiphysics factors, the HSPMSM prototype shown in Fig. 17 is taken as the test object. The iron loss of no-load motor can be obtained by the loss separation method. The measured iron losses at different voltages are compared with the calculated iron losses as shown in Fig. 18.   Based on the experimental results, it can be learnt that the calculated iron losses for the model with multiphysics factors are closer to the measured iron losses than the result of the iron loss separation model.

V. CONCLUSION
This article analyzed the magnetic field, temperature field, and stress field distribution characteristics of HSPMSM. The improved models of HSPMSM iron loss considering multiphysics factors were proposed. The following conclusions are obtained.
1) The iron losses calculated by the model considering multiphysics factors are closer to the measured iron loss of prototype.
2) The loss coefficients are affected by the frequency, temperature, and stress. The hysteresis loss coefficient is mainly affected by stress, and the eddy current loss coefficient is mainly affected by frequency and temperature.
3) The iron loss models considering different physical factors are validated by silicon steel sheet. The errors between calculated and measured values are less than 5% in all three cases. 4) In HSPMSM, the calculated values of the iron loss model considering multiphysical factors are closer to the measured values, which proves that the accuracy of the model is higher than that of the iron loss separation model.