A Fast Engineering Method for Estimating Iron Losses in Induction Type Motor Spindles Based on Equivalent Magnetic Circuit


 The iron losses in the motor of motorized spindles have a significant effect on the heat generation, working efficiency, and speed-torque characteristics of motorized spindles as well as on their thermal deformation and machining accuracy. The existing finite element and analytical methods based on Maxwell’s equations are too complicated to be suitable for engineering designers. A fast engineering method for estimating iron losses in the spindle motor is presented based on equivalent magnetic circuit (EMC) where the problem of solving a complex electromagnetic field inside the spindle motor is simplified into a simple magnetic circuit calculation by the assumption that the magnetic flux density distribution of any cross section along the magnetic flux direction in the spindle motor is uniform. The EMC is combined with the Boglietti’s model. They are integrated into a developed program by compiling source codes to achieve the analysis and prediction of iron losses in the spindle motor. The results obtained from the proposed method are compared with the prototype experiment data to verify its validity. With the purpose of ensuring accurate experiment results of iron losses, a method of no load running combined with a sudden loss of power supply is proposed to eliminate the braking torque and electromagnetic losses of the spindle motor, namely to achieve the separation of the mechanical loss from the total losses.


Introduction
Motor or Motorized spindles belong to a new kind of machine tool spindles, where middle transmission chains like belts or gears are cancelled, namely achieving zero transmission chain or direct drive. They have prominent advantages over traditional mechanical spindles, e.g., compact structure, small inertia, fast dynamic response, high speed (high productivity), high precision, wide speed range, low vibration and noise, and easy to realize automatic and precise control [1]- [3]. Motorized spindles have been increasingly applied to high speed precision machining such as high speed milling, high speed grinding, high speed turning, and high speed drilling [4]- [5].
Induction motors (IMs) are more popularly chosen as a drive motor of motorized spindles than permanent magnet synchronous motors (PMSMs) because they have simple and reliable rotor structure, no rotor demagnetization, and high performance-pri ce ratio. As a drive motor of motorized spindles, IMs are always fed by a voltage source inverter. The output voltage waveform of the inverter is not a sinusoidal wave but a series of rectangular pulse waves [6]- [7]. Numerous harmonic components from inverter supply are introduced into the stator winding current of the spindle motor. This makes the iron losses of the spindle motor in their physical origins more complicated. It is a challenge task to analyze and predict them accurat ely. The calculation results of motorized spindles such as heat generation, thermal deformation [8]- [9], and working efficiency are further affect ed when motorized spindles are developed and designed. Therefore, to find a fast and accurate prediction method of iron losses in the spindle motor is very important for reducing the development and design cost of motorized spindles, and improving their performance.
The ongoing study of iron losses in motors has been done because of their complex generation mechanisms [10]- [12 ]. The proper modeling of soft magnetic materials (SMMs) is the basis of performing the exact analysis and prediction of iron losses in motors. There are two basic types of loss models. One type is frequency domain model and the other type is time domain model.
Various frequency domain models have been developed [13]- [18 ]. Bertotti [14] proposed a three-term loss separation model based on the Steinmetz's equation [13], in which the additional power consumption during a magnetization process of SMMs due to domain wall effect is considered, and the total losses can be expressed as a three-t erm sum of the hysteresis loss, the classical eddy current loss, and the excess loss. However, the Steinmetz's equation and the Bertotti's model can achieve satis factory accuracies within a certain range of magnetic flux density levels and frequencies. Moreover, they are valid for only sinusoidal magnetic flux waveform excitation.
Boglietti et al [15]- [16] pres ented a general loss model according to the fact that any loss component contribution in it depends on the characteristics of supply voltage and the Bertotti's model [14]. The Boglietti's model is suitable for any voltage waveform supply. On the basis of the Boglietti's model, variable parameter models [17]- [18], where the hysteresis and eddy current loss coeffi cients are assumed as the functions of the magnitude and frequency of the magnetic flux density, were investigated for obtaining more accurate results over wide frequency and magnetic flux density ranges, but they need more experiment data to extract the model parameters before use.
Besides various frequency domain models, different time domain models have also been developed. Reinert et al [19] investigated a modifi ed Steinmetz's equation (MSE) in time domain to estimate the loss in SMMs under arbitrary flux waveforms. In time domain, Barbisio et al [20] developed a general loss model of SMMs with regard for minor hysteresis loops. However, the modeling [19]- [20] is to model the magnetic charact eristics of SMMs in the case of only an alternating magnetic field.
In order to describle the hysteresis phenomena and magnetic properties of SMMs under both rotating and alternating fields, a generalized vector hysteresis model [21]- [22] in time domain was established from a traditional Chua-type vector hysteresis model [23]. A comparison study between frequency and time domain models was made to predict the excess loss [24].
Time domain models [19]- [24] can gain better accuracy than frequency domain models [13]- [18] because of considering more realistic situations such as magnetic saturation, hysteresis properties, magnetic fi eld rotating and alternating, and distorted magnetic flux waveforms. However, they require extensively experimental tests to construct the databas e of the model parameters. In addition, they are always combined with the finite element to use. The combination of the finite element analysis (FEA) and the loss models of SMMs has become a fashionable method of cal culating iron losses in motors [8], [25]- [29]. The FEA is used along with the Bertotti's model to estimate iron losses in a slot-less PMSM with the stator cores made from two different types of SMMs [8], and together with the MSE to calculate the hysteresis loss in a surface-mounted PMSM [27]. The desired or accurate results can be obtained from the FEA, but it requires a long enough time for the preparation work including motor modelling, model meshing, and material definition. Also the computation is time-consuming. The analytical method based on Maxwell's equations was studied to improve computational efficiency [30]- [31]. However, both FEA and analytical methods bas ed on Maxwell's equations are too complicated to be suitable for engineering designers.
In this paper, a fast engineering method to estimate iron losses in the spindle motor is proposed based on equivalent magnetic circuit (EMC), where the problem of solving a complex electromagnetic fi eld inside the spindle motor is simplified into a simple magnetic circuit calculation by assuming that the magnetic flux density distribution of any cross section along the magnetic flux direction in the spindle motor is uniform. The EMC is used together with the Boglietti's model to realize the analysis and prediction of iron losses in the spindle motor. The validity of the proposed method is proved by prototype experiment.

Model under Sinusoidal Flux Excitation
It is well known that a classical model is the Bertotti's three-t erm loss separation model [14], in which the total losses are divided into three components: the hysteresis loss, the classical eddy current loss, and the excess loss. The model can be expressed by a mathematical formula as h ec e P P P P    Where P denotes the total losses. h P , ec P and e P are the respective hysteresis, classical eddy current, and excess losses.
The power consumption or hysteresis loss inside electrical steel sheets (ESSs) is generated by a change in the direction of a magnetic field applied to them. T he hysteresis loss can be formulated as hp x P afB  Where a represents the hysteresis loss coeffi cient related to magnetic material properties, f is a magnetic field frequency, p B is the peak value of the magnetic flux density, and x denotes the Steinmetz coefficient.
The classical eddy current loss is resulted from that ESSs are incised by a rotating magnetic field to induce an electromotive force (EMF). The eddy currents inside ESSs are formed with an induced EMF and the subsequent Joule heat is produced. T he classical eddy current loss can be formulated as Where b repres ents the classical eddy current loss coefficient and it is formulated as Where σ , d and ρ are the electrical conductivity, the thickness, and the density of soft magnetic materials (SMMs). The additional power consumption or excess loss is caused by domain wall effect during a magnetizing process of SMMs. The excess loss can be characterized by a formula as Where e denotes the excess loss coefficient. If the term of the excess loss in (1) is ignored then (1) can be reduced to h ec P P P  The ignored term of the excess loss is actually included in the other two terms: the hysteresis loss and the classical eddy current loss [15]- [16]. Equations (2) and (3) are substituted into (6), then Formula (7) is valid for only sinusoidal flux waveform excitation. The unit of the results calculated from (7) is expressed in W/kg.

Model with Inverter Supply
The Boglietti's model is a represent ative loss model of SMMs with inverter supply [15]. In the model, minor hysteresis loops are usually neglected because of high inverter switching frequency and the additional harmonic eddy current losses due to inverter supply are taken into consideration by a model correction factor ( ) is the MCF,  denotes the ratio between the root mean square value of the total voltage from inverter supply and that of its fundamental voltage. On the right side of (8), the sum of the first two terms stands for the total fundamental losses and the last term represents the total harmonic losses.

Equivalent Magnetic Circuit
In order to simplify the problem of solving a complex electromagnetic fi eld inside motors into a simple magnetic circuit calculation, it is assumed that the magnetic flux density distribution of any cross section along the magnetic flux direction in motors is uniform. A typical magnetic circuit of induction motors (IMs) per pole is shown in Figure 1. According to Ampere's loop law, the integral result is independent of the path. A magnetic circuit which passes through the center line between two adjacent or a pair of magnetic poles is usually chosen as one for calculation, as illustrated in Figure 1. The magnetic circuit consists of five parts: airgap, stator tooth, stator yoke, rotor tooth, and rotor yoke segments. Only one half of the magnetic circuit is calculated because of its symmetry.
The magnetic flux of IMs per pole can be cal culated from the following formula [32] dp Φ 2 22 Where E is an induced winding back electromotive force (BEMF) per phase, f is a power supply frequency, N  denotes the number of stator winding series conductors per phase, and dp K is a stator winding coefficient which depends on the number of slots per pole per phas e, the electrical angular degree between two adjacent slots, and the number of slots spanned by two effective sides of a winding coil.
The peak value of the magnetic flux density in airgap can be obtained from a mathematical expression [32] as Where s F is the waveform amplitude coeffici ent of the magnetic flux density in airgap (the ratio of the peak value of the magnetic flux density in airgap to its average value) and it is a function of magnetic saturation occurring in stator and rotor core teeth,  is the pole pitch expressed in that the circumference of a circular inner hole of the stator core is divided by the number of poles, and ef l denotes an effective core length.
The respective calculation formulas of the magnetic flux densities of stator and rotor teeth can be expressed as K is a laminated coeffi cient, 1 l and 2 l are the respective lengths of the laminated stator and rotor cores.
The magnetic flux densities of different yoke cross sections along the magnetic flux direction are not same. When a yoke cross section passes through the center line between two adjacent poles, the magnetic flux density of the section reaches its maximum value, but when a yoke cross section passes through the center line of a pole, the magnetic flux density of the section is exactly equal to zero, as shown in Figure 1. Thus, the total magnetic flux of yoke parts is only one half of the magnetic flux per pole. The maximum magnetic flux densities of stator and rotor yoke parts can be formulated respectively as sj,p sj Where sj A and rj A are the respective flux areas of stator and rotor yoke parts, sj sj s = A h l  and rj rj r = A h l  . Where sj h and rj h are the respective calculation heights of stator and rotor yokes, and they are commonly determined by their geom etrical yoke heights together with slot types.

Calculation of Iron Losses in S pindle Induction Motors
The equivalent magnetic circuit (EMC) is combined with (8) to estimate iron losses in induction motors (IMs) of motorized spindles. The spindle motor is supplied with a voltage source inverter. The EMC is used to calculate the magnetic flux densities of the spindle motor stator and rotor cores and the corresponding loss densities are calculated by (8). The total iron losses in the spindle motor can be calculated from the following formula . Where  is the density of ESSs, st h and rt h are the stator and rotor tooth calculation lengths used for magnetic circuit calculating and they are related to slot types but the notch heights of slots are usually ignored in them, 1 D and i2 D denote the outer and inner diameters of stator and rotor cores.
In (15), both fundament al and harmonic components from inverter supply are significant contributors to iron losses in the stator core. However, there is almost no contribution of the fundamental component from inverter supply to iron losses in the rotor core owing to a very small rotor slip. The iron losses in the rotor core are mainly caused by the harmonic components from invert er supply. These may explain why the loss densities of stator and rotor cores are different.
The proposed calculation method of iron losses in the spindle motor is integrated into a developed program [33] by compiling source codes. The detailed calcul ation processes of the developed program are shown in Figure 2.  Figure 2 Calculation of iron losses in the spindle motor STEP1: The data of the spindle motor design parameters, e.g., voltage, frequency, shaft power, the number of pole pairs, and geometrical dimensions are input to the developed program before its run.
STEP2: The magnetic flux density levels of airgap, stator and rotor cores can be calculated from the EMC and the known values of the design parameters.
STEP3: The loss densities of the tooth and yoke parts of stator and rotor cores are obtained from (8), the magnetic flux density levels calculat ed from STEP 2, and a known inverter supply frequency. There is need for determining the loss coeffici ents in (8) from the physical characteristic and experimental data of ESSs before model use.
STEP4: The total masses of the tooth and yoke parts of stator and rotor cores are calculated from their structures, and the corresponding geometrical dimensions and material densities.
STEP5: The total iron losses of the spindle motor are divided into two parts: static and dynamic iron losses. The static iron losses in the spindle motor are produced due to magnetic field alternating. They can be expressed as a sum of multiplying the total masses of the tooth and yoke parts of stator and rotor cores by the corresponding loss densities.
STEP6: The dynamic iron losses in the spindle motor are caused by rotor rotating motion combined with tooth and slot effect. They are commonly taken into account based on the static iron losses by introducing empirical coefficients (whose values are always more than 1) because of their complex generation mechanisms. STEP7: The total iron losses can be calcul ated and obtained according to (14), the results of the total masses of the tooth and yoke parts of stator and rotor cores and the corresponding loss density values calculat ed from STEPs 3 and 4 STEP8: The calculation will be not stopped until the perform ance requirem ents of the spindle motor are satis fied. The performance of the spindle motor can be optimized by changing design parameters.

Spindle Motor
The spindle motor taken as a study case is a three-phase inverter-fed induction motor. The performance param eter values of the spindle motor are calculated and obtained from the developed program [33], as seen in Table 1.  Figure 3 shows the laminated sheets of the stator and rotor cores of the spindle motor. The respective slot types of stator and rotor laminated sheets are designed as pear and circle slots. The laminated sheets of stator and rotor cores have 48 and 38 slots, respectively. All the slots are uniformly distributed in their own circumferential directions.
A slot number combination of 48 and 38 is the combination in which there is an obvious difference between stator and rotor slot numbers to reduce electromagnetic vibration and noise. The slot number of the stator laminated sheet is designed as 48. It belongs to a more slot design to obtain a smaller stator outer diameter and more compact stator structure, and to suppress slot harmonic losses and asynchronous additional torque.
The bars in a rotor cage of the spindle motor are designed as copper ones to reduce rotor heat generation and slip, and to improve efficiency as illustrated in Figure  4 b. 35W300 ESSs are chosen to manufacture and fabricate the stator and rotor cores of the spindle motor to suppress eddy currents and reduce magnetizing current because of their thin thicknesses and high permeability, as shown in Figure 4 a and b. The properties of 35W300ESSs are given in Table 2.

Model Parameter Extraction
The fundam ental parameters in (8)  coeffi cient x is assumed as a constant value of 2 because the desired results of iron losses in motors can be achieved [34]- [35]. The value of the parameter b is calculat ed and obtained from (4) and the known physical characteristic data of 35W300 ESSs (Table 2). Base on the measured data of 35W300 ESSs under 50 Hz sinusoidal excitation, (8) is fitted by the least square to obtain the value of the parameter a. As a result, the values of the fundamental parameters in (8) are given in Table 3.

Experiment
The mechanical loss separation and measurement, and iron loss test are performed on a developed and manufactured motor spindle prototype ( Figure 5) which is supported by hydrostatic bearings. The rated power and rotational speed of the prototype are 35 kW and 6, 000 r/min, respectively. The specifications of the prototype motor are exactly the same as those of the spindle motor in Table 1. The prototype works under the condition of inverter power supply.

Mechanical Loss Separation and Measurement
The mechanical loss is a major loss of the prototype. Whether it can be correctly separated from the total losses of the prototype by experiment or not will affect the credibility of the experimental results of iron losses.
The mechanical loss of the prototype resulted from bearing friction and rotor wind drag is assumed as a function of its rotational speed square Where m P denotes the mechanical loss of the prototype running at any speed, c is the rotor rotating friction factor, and m ω is the mechanical angular velocity of the prototype rotor.
The no load running is combined with sudden power cut to separate the mechanical loss of the prototype from the total losses. The inverter-fed prototype is started under no load and its speed rises to a rated speed of 6 k r/min by changing inverter supply frequency. Under no load, the prototype is kept running at the rated speed for a long enough time so that the temperature rise of the prototype is not changed any more. When the temperature rise is stable, the power supply of the prototype is suddenly cut off to eliminate the braking torque and electromagnetic losses of the prototype to obtain accurate test results. In other word, only the mechanical loss is left. Aft er a sudden loss of power supply, the speed of the prototype will not decreas e to zero immediately but show a drop process due to frictional forces. The mechanical loss of the prototype can be measured by measuring the rotor rotating friction factor of the prototype and its mechanical rotational speed, as seen in (16).
It is known from the conservation law of energy that under the condition of suddenly losing power supply, the loss of kinetic energy of the prototype in any time interval is exactly equal to the work done to overcome frictional forces in this time interval. The conservation equation of energy is written as Where 1 t and 2 t denote two different rotating motion moments of the prototype after sudden power cut and they can be measured by a stopwatch, The measured data of a rotational speed change in the prototype after sudden power cut are listed in Table 4. Based on the measured data in Table 4, (18) is combined with (16) and (17) to solve the value of the rotor rotating friction factor, namely = 0.0 080 c -1 N m s rad    .

Iron Loss Test
A simpler method is used to measure iron losses in the prototype than the traditional method, where there is no need for connecting the prototype under test to a synchronous motor with the same pole pairs. The no load active power of the prototype rotating at different speeds is measured by a power analyzer. The current of the no load operating prototype at any speed exceeds a power analyzer current limit value of 5 A. Thus, current sensors are required in the test. The current sensors with a trans former ratio of 1:40 can be chosen from a power capacity of the prototype. The iron losses in the prototype can be indirectly obtained from the following formula 2 ir no-load m s s P P P mR I    Where ir P denotes the total iron losses, no-load P is the measured no load active power, m P is the measured mechanical loss, m is the number of phase, s R and s I are the measured respective winding resistance and current per phase.
In (19), both stray and rotor resistance loss terms are neglected, but they are included in the total iron losses. The term of the stray loss at no load is very small so that it can be ignored. The term of the rotor resistance loss can also be ignored because of a very low rotor slip at no load. The good measurement accuracy can be still guarant eed even without considering them. Figure 6 shows the measured results of various losses of the no load running prototype at different speeds.  It can be found from Figure 6 that in no exceeding the rated speed range, the total iron losses in the prototype increas e with an increas e in its rotational speed. The reasons are summarized in two aspects. First, a V/f control strategy is adopted to keep the magnetic flux of the prototype constant within a rat ed speed range, but the static iron losses in the prototype due to magnetic field alternating are increas ed with a higher rotational speed or magnetic field frequency. Second, the dynamic iron losses in the prototype increase with an increas e in the rotational speed because of rotor rotating motion combined with tooth and slot effect.

Results and Discussion
It can be known from Figure 7 that the results obtained from the proposed method agree well with the prototype experiment data. The biggest error between the estimated and measured results is no more than 10 %. The validity of the proposed method is confirmed by experiment.
The proposed method is simpler and more suitable for engineering designers than the FEA and analytical methods based on Maxwell's equations. In the proposed method, there is no need to solve a complex electromagnetic field inside motors, and to perform 2D or 3D eddy current analysis and the corresponding post-processing.

Conclusions
(1) A fast Engineering method for estimating iron losses in the inverter-fed induction motor of motorized spindles is presented bas ed on equivalent magnetic circuit (EMC), in which the EMC is combined with the Boglietti's model and they are integrated into a developed program by compiling source codes to realize the analysis and prediction of iron losses in the spindle motor. (2) The proposed method is simpler and more suitable for engineering designers than the FEA and analytical methods based on Maxwell's equations because it does not need to solve a complex electromagnetic field inside motors. (3) The results obtained from the proposed method are compared with the prototype experiment data to verify its validity. It is demonstrated that the biggest error between the calculated and measured results is no more than 10%. The proposed method is valid for analyzing and predicting iron losses in the spindle motor and it can provide technical support for the analyses of the heat generation, working effi ciency, and speed -torque characteristics of motorized spindles as well as those of their thermal deformation and rotating error when motorized spindles are developed and designed. (4) A method of no load running combined with a sudden loss of power supply to separate and measure the mechanical loss of induction type motor spindles is proposed. The proposed method can eliminate the braking torque and electromagnetic losses of the spindle motor, and can achieve the separation of the mechanical loss from the total losses.

Availability of data and materials
The datasets supporting the conclusions of this article are included within the article.