3482IEEE TRANSACTIONS ON MAGNETICS, VOL. 40, NO. 6, NOVEMBER 2004Characterization of Noise and Vibration Sources in Interior Permanent-Magnet Brushless DC MotorsHong-Seok Ko and Kwang-Joon KimAbstract—This paper characterizes electromagnetic excitation forces in interior permanent-magnet (IPM) brushless direct current (BLDC) motors and investigates their effects on noise and vibration. First, the electromagnetic excitations are classified into three sources: 1) so-called cogging torque, for which we propose an efficient technique of computation that takes into account saturation effects as a function of rotor position; 2) ripples of mutual and reluctance torque, for which we develop an equation to characterize the combination of space harmonics of inductances and flux linkages related to permanent magnets and time harmonics of current; and 3) fluctuation of attractive forces in the radial direction between the stator and rotor, for which we analyze contributions of electric currents as well as permanent magnets by the finite-element method. Then, the paper reports on an experimental investigation of influences of structural dynamic characteristics such as natural frequencies and mode shapes, as well as electromagnetic excitation forces, on noise and vibration in an IPM motor used in washing machines. Index Terms—Brushless machines, electromagnetic forces, noise, permanent magnet, vibrations.Fig. 1.Cross sections of BLDC motors.I. INTRODUCTIONCONVENTIONAL direct current commutator motors with permanent magnets are easy to control and require few semiconductor devices. Yet, they have serious operational problems in association with brushes. For examples, the brushes require regular maintenance and induce noise by friction with the commutators. A solution for these problems is brushless direct current (BLDC) motors. BLDC motors can be classified into two types, as shown in Fig. 1 according to the geometric shape and location of permanent magnets. Compared with surface mounted permanent-magnet (SPM) motors, interior permanent-magnet (IPM) motors have several advantages. One advantage comes from the position of magnets. Because permanent magnets are embedded in the rotor, the IPM motors can be used at higher speeds without debonding of the permanent magnets from the rotor due to the centrifugal forces. Another obvious advantage of the IPM motors is higher efficiency. That is, in addition to the mutual torque from the permanent magnets, the IPM motors utilize the reluctance torque generated by the rotor saliency [1].Manuscript received June 28, 2002; revised June 7, 2004. H.-S. Ko was with the Mechanical Engineering Department, Korea Advanced Institute of Science and Technology (KAIST), Daejon 305-701, Korea. He is now with Samsung Electronics Company Ltd., Suwon 443-742, Korea (e-mail: hskatom@yahoo.co.kr). K.-J. Kim is with the Mechanical Engineering Department, KAIST, Daejon 305-701, Korea (e-mail: kjkim@mail.kaist.ac.kr). Digital Object Identifier 10.1109/TMAG.2004.832991Regarding the noise and vibration, the IPM motors have more sources than the SPM motors. Furthermore, analysis of magnetic field in the IPM motors is more difficult due to the magnetic saturations, especially in the rotors. In an IPM motor, the electromagnetic excitation sources can be classified into three parts: cogging torque, ripples of mutual and reluctance torque, and fluctuations of radial attractive force between the rotor and stator. In an SPM motor, only the mutual torque is generally considered and an analytical method can be used [2], [3]. For the IPM motors, however, the finite-element method (FEM) is used to account for the magnetic saturation at the rotor core and, besides the mutual torque, the reluctance torque needs to be considered. In addition, although only the permanent magnet may be considered to calculate the radial attractive forces between the rotor and stator in the IPM motors [4], the electromagnetic field due to the currents may become significant depending on the loading and generate serious excitation forces. In this paper, a technique that can efficiently calculate the cogging torque as a function of rotor position by including saturation effects is proposed. Then, a torque equation for characterizing the space and time harmonics with respect to the mutual and reluctance torque ripples is used to extract their fluctuating components. The radial attractive forces due to the electric currents in the stator as well as the permanent magnets in the rotor are calculated by the FEM and its effects on noise and vibration are investigated. The noise and vibration in the motors are mostly generated by the electromagnetic sources and subsequently can be amplified by the dynamic characteristics of the motor structure. Therefore, influences of natural frequencies and mode shapes of the structures are experimentally investigated for the noise and vibration of an IPM motor under study. II. ELECTROMAGNETIC EXCITATION SOURCES Electromagnetic excitations in electric motors are caused by variation of both circumferential and radial forces acting between the stator and the rotor with respect to the time and space.0018-9464/04$20.00 © 2004 IEEEKO AND KIM: CHARACTERIZATION OF NOISE AND VIBRATION SOURCES IN IPM BLDC MOTORS3483Torque ripples in an IPM motor, the result of dynamic circumferential forces multiplied by an appropriate radius, are composed of two sources; cogging torque and ripples of mutual and reluctance torque. The cogging torque is due to physical geometry of the stator teeth and the rotor magnets. The ripples of mutual and reluctance torque are produced by harmonics of the flux linkages related to magnets, inductances, and currents. In addition, fluctuation of attractive forces in the radial direction between the rotor and stator works as excitation sources. The cogging torque, ripples of mutual and reluctance torque, and fluctuation of the radial attractive forces will be discussed next in more detail.Fig. 2. Geometric configuration of IPM motor.A. Cogging Torque The cogging torque is defined as a torque produced by magnetic forces in the circumferential direction between the stator teeth and the magnets of rotor. Because it is superposed on the mean output torque as a fluctuating component, it can be an important performance index of noise and vibration as well as smoothness in rotations of the rotor. In order to calculate the resultant torque for a given position of the rotor relative to the stator by taking the magnetic saturation in the rotor core and the complex geometric shapes of the stator teeth and rotor magnets into account, it is inevitable to employ numerical methods such as the FEM. Since this torque is rotor-position dependent, the numerical calculation must be repeated for every position of the rotor, which should be very time consuming and, hence, may not be a good tool at the phase of parametric study [5]. In this section, an efficient technique that can be useful in the initial design and modification stages is suggested. The technique is composed of the following steps. The first step is to calculate the flux density through the magnet, rotor core, air gap, slotless stator, rotor core, and the magnet by employing the FEM, just once to deal with the saturation problems in the rotor core. The second step is to obtain the boundary conditions in the slotted air gap by employing the concept of relative permeance [6]. The third is to compute the flux density in the slotted air gap as a series solution of the magnetic potential equation with the boundary conditions obtained from the second step. Finally, the cogging torque is derived from the Maxwell stress formula with movement of the rotor with respect to the stator. The flux density and field intensity in the air gap can be related as given in the following equation by assuming magnetic saturation does not occur in the air-gap region: (1) where is the permeability of air. Since the field intensity can be represented in terms of a magnetic scalar potential defined as (2) governing equation of the magnetic potential in the air-gap region is given by (3) where is the number of slots. Therefore, the second boundary condition is written as The circumferential coordinate denotes the angular displacement of the stator-fixed coordinate and the circumferential coordinate denotes the angular displacement of the rotor-fixed coordinate as shown in Fig. 2. The coordinate is related to the , where the coordinate by the rotor movement is the rotational displacement of the rotor with coordinate respect to the stator and given by the rotation frequency multiplied by the time, i.e., is equal to . Therefore, the flux density on the inner surface of the stator in the radial direction and the one on the outer surface of the rotor in the circumferential direction in the slotless air gap can be respectively represented by Fourier series as (4) (5) where is the number of pole pairs. The flux density in the slotted air gap can be obtained by solving the governing equation (3) with two boundary conditions. One comes from the fact that the slotting effect on the circumferential flux distribution on the outer surface of rotor can be neglected. Therefore, the circumferential flux density along the outer surface of the rotor can be represented by (5). The other comes from the fact that the radial flux density on the inner surface of the stator in slotted air gap can be calculated by the product of the radial flux density in the slotless air gap and the relative permeance in (6) (6)(7)3484IEEE TRANSACTIONS ON MAGNETICS, VOL. 40, NO. 6, NOVEMBER 2004The general solution of (5) in the slotted air gap may be proposed as follows:(8) Hence, the flux density in the slotted air gap can be written asFig. 3.Cogging torque profile with rotor positions. TABLE I PARAMETERS OF IPM MOTOR UNDER STUDY(9)Fig. 4.Harmonic components of cogging torque.The circumferential stress in air gap is calculated by the Maxwell stress tensor as (11) Therefore, the cogging torque can be calculated as (12) where is an arbitrary circle in the air gap with the radius from the center of the rotor , and is the axial length of the rotor. Fig. 3 shows estimations of the cogging torque by the proposed technique together with those by measurements and the conventional FEM, where the flux density is computed with parameters as shown in Table I. The results of the proposed technique show good agreement with those of FEM and measurement both in magnitude and waveform. Fig. 4 shows the components of the cogging torque harmonics. Therefore, the(10)KO AND KIM: CHARACTERIZATION OF NOISE AND VIBRATION SOURCES IN IPM BLDC MOTORS3485cogging torque can generate the noise and vibration at the frequency of the rotor rotation multiplied by 24 and its higher harmonics. B. Mutual and Reluctance Torque Ripples As explained in the introduction, the output torque of an IPM motor is given by sum of the mutual torque and the reluctance torque, each of which can be expressed by using the following energy method [7]: (13)By substituting (15) and (17) into (13), the mutual torque can be rewritten as follows:(18) harmonics, and where is the order of the flux linkage is the order of current harmonics. When is zero, the mutual torque is constant, i.e., completely static. When and are multiples of three, the mutual torque should have harmonics at the source frequency multiplied by such multiples of three. By substituting (16) and (17) into (14), the reluctance torque can be rewritten as follows:(14) In the above equations, the coordinate is the electrical angle and given by the mechanical angle multiplied by the number , , , and the currents of pole pairs , i.e., is equal to in the coils of phase , , and , respectively, the inductance between the phase and the phase , and the flux by the permanent magnets linking the phase . Ripples of the mutual and reluctance torque, defined as fluctuating components of the output torque, are governed by several factors such as the shape of currents with respect to time, variations of inductances, and with respect to rotor movement, which are flux linkages further discussed in the following. in (13) can be represented by Fourier The flux linkage series as (15) where , , and are 0, , and , respectively. The inductance matrix can be formulated as follows [8]:(19) where and stand for the order of current harmonics and the order of inductance harmonics. It can be seen from (19) that or the reluctance torque become static only when is zero and, when , or are multiples of three, it should have harmonics at the source frequency times multiples of three. Equation (18) and (19) are very useful for characterizing and, hence, reducing the mutual and reluctance torque ripples. For example, when the space harmonics ( and ) are beyond control or the ripples of the mutual and reluctance torque can be reduced by controlling the waveform of the current . The flux linkage of the IPM motor under study can be obtained by an integral of the flux density due to the permanent magnet in air gap as follows: (20) where is the number of coils per phase per pole pairs and a half of the slot pitch. The flux density in the middle of air(16) where is harmonic coefficients of the self inductance and those of mutual inductance. Variations of inductance with respect to the rotor position are caused by the magnetic saturation of the rotor core. Therefore, in IPM motors the inductance matrix should be obtained by the FEM or measurements. The currents supplied to the IPM motors are often not a pure harmonic function of time and, hence, can be represented by Fourier series as follows: (17) where the leading angle is an angle between the fundamental component of the flux fields by the magnets and that by the currents. The electrical angle is also given by the source frequency multiplied by the time, i.e., is equal to .3486IEEE TRANSACTIONS ON MAGNETICS, VOL. 40, NO. 6, NOVEMBER 2004Fig. 5. Harmonic coefficients of flux linkage .Fig. 6. Self inductance Lby measurements.Fig. 7. (a) Waveform and (b) harmonic coefficients of current at 500 r/min.gap due to the permanent magnets can be obtained by (9) and can be rewritten as (21) Therefore, the harmonic coefficients of the flux linkages around the phase , , and can be derived by substituting (21) into (20) as follows: (22) Fig. 5 shows harmonic coefficients of the flux linkage and Fig. 6 measured self inductance . The inductance is close to a sinusoidal wave and, hence, higher harmonics of the inductance except the fundamental can be neglected. The IPM motor under study is for washing machines and runs at 500 r/min in the slow washing mode and at 10 000 r/min in the fast dehydration mode. Fig. 7 shows the current at 500 r/min under the load of 9.6 kg cm and Fig. 8 harmonic with components of the mutual, reluctance, and total torque, where it can be seen that the reluctance torque which does not exist in the SPM motors, resulted in increase of the static torque by about component by 19.7% and, surprisingly, decrease of the about 47%. Here, the stands for the rotation frequency, which is twice the source frequency for a 4-pole IPM motor. Yet, the component, which does not show up in the SPM motors, showed up undesirably. Fig. 9 shows waveform in time domain and harmonic coefficients of currents for the motor running at under no-load. The output torque 10 000 r/min with and the ripples are shown in Fig. 10, where it can be seen that not only and component but also component hasFig. 8. Harmonic components of output torque at 500 r/min when lead angle is 30 .0shown up, and the reluctance torque has contributed to decrease of the static torque as well as the dynamic torque. C. Fluctuation of Attractive Forces Between the Rotor and Stator Excitation sources explained in Section II-A and B are variations with time of the output torques, which were classified into cogging torques independent of the electric current and ripples of the mutual and reluctance torque due to the currents. In this subsection, another type of excitation source is discussed, which is related to the spatial distribution of the radial attractive forces between the stator and the rotor. The radial attractive force or so called the Maxwell stress on the inner surface of the stator can be written as [4]: (23)KO AND KIM: CHARACTERIZATION OF NOISE AND VIBRATION SOURCES IN IPM BLDC MOTORS3487Fig. 11.Distributions of radial flux density on inner surface of stator.Fig. 9. Waveform and harmonic coefficients of current at 10 000 r/min (a) waveform (b) harmonic coefficients.Fig. 12. Radial attractive force at given stator’s slot with respect to rotor positions.Fig. 10. Harmonic components of output torque at 10 000 r/min when lead angle is 15 .Since permeability of the iron in the rotor and stator is extremely large compared with that of the air, the stress due to the flux , which is inversely proportional to the density in the iron, permeability of the iron, can be neglected. Therefore, the radial attractive force on the end surfaces of the stator’s teeth can be written as (24) The equivalent air gap of the SPM motor given by is rather large compared with the pure air gap since the relative recoil permeability of the magnets is approximately 1. Therefore, the magnetic flux in the air gap by the currents in the stator can be neglected. The air gap in theIPM motors, however, is just because the magnets are embedded into the rotor. As a consequence, it is essential to take the magnetic field by the currents into consideration to analyze the effects of the attractive forces on noise and vibration in the IPM motor. Fig. 11 shows distributions of the radial flux density on the inner surface of the stator when the magnitude of currents is 2.5 A. The maximum flux density by both magnets and currents is three times larger than that by the magnets alone. Fig. 12 shows variations of the radial attractive force on a given stator teeth with movement of the rotor with respect to the stator. Fig. 13 shows harmonic components of the corresponding attractive forces, where it can be seen that integer multiples of component show up and the radial attractive force at by both magnets and currents is about 14 times larger than that by the magnets alone. Therefore, it can be claimed in general that the radial attractive forces in the IPM motors are far larger than those in the SPM motors regarding the noise and vibration and that the motor structure will be excited at harmonics of the frequency of rotor rotation multiplied by the number of poles or twice the number of pair of poles. In summary of this section, it is claimed that the electric current in the stator in the IPM BLDC motors is far more strongly responsible for noise and vibration than in the SPM motors and that the frequency characteristics of the electromagnetic excitation sources in the IPM BLDC motors can be described as follows.3488IEEE TRANSACTIONS ON MAGNETICS, VOL. 40, NO. 6, NOVEMBER 2004Fig. 13.Harmonic components of radial attractive force.1) Cogging torque: the lowest common multiple of numbers of slots and poles times the rotating frequency and its higher harmonics. and its 2) Ripples of mutual and reluctance torque: higher harmonics. 3) Fluctuations of radial attractive force: number of poles times and its higher harmonics. III. NOISE AND VIBRATION OF MOTOR UNDER OPERATION In this section, the noise and vibration measured for an IPM motor running are presented and discussed for the purpose of supporting the claims in Section II. For the IPM motor, which had noticeable noise problems at 10 000 r/min, measurements were made with power on and immediately after disconnection of the power in order to investigate contribution of the electromagnetic excitation sources. The spectrum of an acceleration signal measured from a point on the outer surface of the stator is shown in Fig. 14. Since the axial length of the stator is short relative to the diameter, transverse modes were not observed in the frequency range shown in Fig. 14 but the acceleration signal was taken at one position on the center plane where the vibrations were largest. After the disconnection of the electric power, the rotating frequency decreased slightly from 188 Hz (11 200 r/min) to 168 Hz (10 000 r/min) and, as can be seen in the figure, most of the peaks with power on disappeared after power off, which are believed to be related to the electromagnetic excitations. Although the peak at was reduced in its magnitude by power off, it did not disappear completely because this peak was contributed by fluctuations of the radial attractive force due to the permanent magnets. A power spectrum of sound pressure level was measured at 10 000 r/min with power on and is shown in Fig. 15, where the first peak at 168 Hz, which was observed also in the acceleration shown in Fig. 14, is believed to be due to the rotor unbalance. Comparing the peak frequencies of the sound pressure level spectrum in Fig. 15 with those in Figs. 4, 10, and 13 allows the source of each peak to be understood. That is, the peaks at , , , and are due to variation of the radial attractive forces with rotation of rotors with four poles, the peak at due to the ripples of torque, and the peaks at and due to both fluctuation of the attractive force and ripples of the mutual and reluctance torque. The peaks at and seem to have been magnified by resonance because natural modes happened to exist at these frequencies, 1.34 and 2.67 kHz, respectively, which were found atFig. 14.Distributions of radial flux density on inner surface of stator.Fig. 15.Noise of IPM motor at 10 000 r/min.the stage of modal testing and operational deflection shape analysis for investigation of possible coincidence between excitation frequencies and modal properties of the structure. The natural frequencies and mode shapes were obtained from the measurements along the centerline on the surface of the stator and are shown in Fig. 16. The first mode at 865 Hz looks like a rigid body motion of the stator relative to the rotor and the modes at 1.34 and 2.64 kHz the first and the second elastic mode, respectively. Fig. 17 shows operational deflection shapes of the stator at major peak frequencies in Fig. 15. The deflection shape at 168 Hz, the rotating frequency of the rotor seems to be a rigid body motion where the stator itself whirls. The deflection shapes at both 1.34 and 2.67 kHz coincided with mode shapes at the corresponding natural frequencies, as could be expected.KO AND KIM: CHARACTERIZATION OF NOISE AND VIBRATION SOURCES IN IPM BLDC MOTORS3489Fig. 16.Mode shapes of IPM motor under study.attractive forces due to the magnetic flux by the permanent magnets in the rotor and electric currents in the stator was computed by the FEM to include nonlinear effects, where significance of the magnetic flux due to the electric current that is often neglected in the SPM motors was pointed out. In an illustrative investigation into an IPM motor, peak frequencies in the spectrum of the sound pressure level could be linked with such excitation sources and modal characteristics of the motor structure as well. REFERENCES[1] T. J. E. Miller, Brushless Permanent-Magnet and Reluctance Motor Drives. New York: Oxford Univ. Press, 1989. [2] Z. Q. Zhu and D. Howe, “Analytical prediction of the cogging torque in radial-field permanent magnet brushless motors,” IEEE Trans. Magn., vol. 28, pp. 1371–1374, Mar. 1992. [3] A. B. Proca, A. Keyhani, and A. EL-Antably, “Analytical model for permanent magnet motors with surface mounted magnets,” in Proc. IEMD ’99, pp. 767–769. [4] K. T. Kim, K. S. Kim, S. M. Hwang, T. J. Kim, and Y. H. Jung, “Comparison of magnetic forces for IPM and SPM motor with rotor eccentricity,” IEEE Trans. Magn., vol. 37, pp. 3448–3451, Sept. 2001. [5] D. Howe and Z. Q. Zhu, “The influence of finite element discretization on the prediction of cogging torque in permanent magnet excited motors,” IEEE Trans. Magn., vol. 28, pp. 1080–1083, Mar. 1992. [6] Z. Q. Zhu and D. Howe, “Instantaneous magnetic field distribution in brushless permanent magnet dc motor, part III: Effect of stator slotting,” IEEE Trans. Magn., vol. 29, pp. 143–151, Jan. 1993. [7] T. S. Low, K. J. Tseng, T. H. Lee, K. W. Lim, and K. S. Lock, “Strategy for the instantaneous torque control of permanent-magnet brushless DC drives,” Proc. Inst. Elect. Eng. , vol. 137, pp. 355–363, Nov. 1990. [8] P. C. Kraus, Analysis of Electric Machines. New York: McGraw-Hill, 1987.Fig. 17.Operational deflection shapes at 10 000 r/min.IV. CONCLUSION Analysis of electromagnetic excitation sources in the IPM motors is far more difficult than in the SPM motors because magnetic saturations in the rotor core are more likely to occur in the former. In this paper, such sources were classified into three types and efficient methods were presented to characterize each source, and then contribution of the sources to the noise and vibration was investigated for an IPM motor. An efficient technique was presented for computation of the cogging torque, where magnetic saturations in the rotor can be taken into account by employing the FEM just once for the slotless stator and effects of the stator slot are reflected by the concept of relative permeance. A formula was derived for representation of ripples of the torque based on the output torque formula. It was shown that the space and time harmonics are responsible for the torque ripples at three times the source frequency and their integer multiples. Then distribution of radialHong-Seok Ko received the B.S. degree in mechanical engineering from Korea University, Seoul, Korea, in 1991, and the M.S. and Ph.D. degrees in mechanical engineering from Korea Advanced Institute of Science and Technology (KAIST), Daejon, in 1993 and 2003, respectively. From 1993 to 1998, he was with LG Innotec Corporation. He is currently with Samsung Electronics Company Ltd., Suwon, Korea. His academic interests involve the noise and vibration induced by the electromagnetic excitation sources.Kwang-Joon Kim received the B.S. and M.S. degrees in mechanical engineering from Seoul National University, Seoul, Korea, in 1976 and 1978, respectively, and the Ph.D. degree from the University of Wisconsin, Madison, in 1982. He is a Professor in the Department of Mechanical Engineering at the Korea Advanced Institute of Science and Technology (KAIST), Daejon. His current interests include the application of viscoelastic materials for vibration control, vibration isolation based on power transmission approach, modal testing and operational deflection shape analysis, and noise and vibration of electric motors.。
