On the Modeling of Brushless DC Motor with Surface-Mounted Magnets

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1 Osama Mohamed Arafa, 2Ahmed Ali Mansour Researcher at Electronics Research Institute - Egypt, oarafa@eri.sci.eg, mansour@eri.sci.eg 1 1,2 Abstract Back EMF of brushless DC motor BLDCM may considerably deviate from the ideal trapezoidal shape such that the flat top of the EMF waveform is shorter than 120 degrees. This deviation necessarily further enlarges the torque dips during sector to sector transition period. These dips take place even in ideal trapezoidal machines due to the commutation process itself where different decay and growth rates of outgoing and incoming phase currents diminish the electromagnetic developed torque. In this paper, an equation of the back EMF that is valid for BLDCM characterized with such deviation is presented. Also a torque equation capable of calculating the motor torque at all conditions including standstill and commutation periods in such machines is derived from first principles using the partial derivative of stator flux linkages with respect to rotor position. These two equations bring more accuracy to dynamic simulation model by addressing one of the root causes of torque dips. Simulation results based on the developed equations well demonstrate these features. Keywords: Brushless DC Motors, Non-Ideal Back EMF, Torque Dips 1. Introduction With recent developments in rare earth magnetic materials BLDCM and permanent magnet synchronous motors PMSM are offering high power densities and superior speed control characteristics [1-4]. Both motors have a permanent-magnet rotor, but in BLDCM the stator windings are concentrically wound such that the back electromotive force per phase EMF in the ideal case is trapezoidal. It therefore requires rectangular-shaped stator phase currents to produce constant torque. The trapezoidal back EMF implies that the flux linking the stator windings across a complete electrical cycle 2-pole pitch due to the rotor magnets and its rate of change are not sinusoidal. Although steady-state analysis, dynamic modeling, detailed simulation and experimental verification of BLDC motors has been done [5], the back EMF was very frequently assumed to follow the perfect trapezoidal shape with flat tops lasting for 120 electrical degrees in each half cycle as illustrated in Figure 1 and with two linear change intervals between the positive and the negative flat intervals in each half cycle each lasts for 0 electrical degrees. In [] a Fast Fourier Transform and its inverse FFT, IFFT is introduced as a generic technique for restoring the real shape of non-ideal back EMF of BLDCM no matter how it is deviated from ideality, however there is a lot of coefficients to be identified to make such technique usable and it needs intensive calculations to be applied in real time. Figure 1. Ideal Trapezoidal back EMF with 120 degree flat intervals Journal of Next Generation Information TechnologyJNIT Volume 5, Number 3, August

2 Also torque equation that is independent of rotor speed has not been covered in detail in the literature. In most literatures as in [5], [8], authors use the following equation to calculate the electromagnetic developed torque Tem eaia ebib ecic / r 1 Where: ei is the instantaneous value of phase i back EMF, ii is the instantaneous value of phase i current, suffix i is a, b and c. It is evident that the motor torque at standstill cant be calculated using this equation despite the fact that it has a specific value and direction at such condition. The value is depending on the torque coefficient, the magnitude and direction of the rotor currents and the actual rotor position. In direct torque control applications, it is required to on-line estimate the electromagnetic torque and this is done frequently using normalized back EMF functions saved in lookup tables [9]. Other researchers measure the back EMF offline of one phase at every angle at certain fixed speed. The values are saved in look-up table and the EMF of the three phases is obtained from the tables using on-line position measurements and corrected for speed difference [10]. It has been also reported that PI regulators with feed forward compensation of back-emf is widely applied in drive industry as a basic structure for current control [11]. Therefore closed form representation of EMF that is relatively accurate can be significantly important in many applications. The non-ideality addressed by this paper is limited to having a short flat interval. The purpose of this paper is to go through a systematic approach to model the back EMF as close as possible to the practical case shown in [10] and to derive an equation of the electromagnetic torque which can be used for ideal and non-ideal BLDCM to calculate the electromagnetic torque with reasonable accuracy under all normal operating conditions. The paper is organized as follows: in section 2, an expression of the electromotive force per phase is developed for a rotor with surface-mounted magnets rotating in the space confined by three-phase concentric-type windings. The flux linkage of stator phases due to rotation of rotor magnets are derived in section 3 based on the fact that they are time integrals of the electromotive forces. The electromagnetic developed torque is derived in section 4 by taking the partial derivative of the co-energy with respect to rotor mechanical position is presented. A dynamic simulation using the derived EMF and torque expression is presented in section 5. The conclusion is presented in section. 2. Back EMF representation The following assumptions are considered: 1 The motors stator is a star wound type and the neutral point is inaccessible meaning that the three phase currents sum to zero. 2 The motors three phases are concentrically- wound and symmetrical, including their resistances, self inductances and mutual inductances. 3 The rotor magnets are surface mounted uniform air gap and no magnetic saliency so that there is no change in inductance due to rotor position. The rotor angle θr as illustrated in Figure 2 is the angle between the magnetic axis of stator phase a and the quadratic axis of the rotor magnets. phase b + r q r phase a d phase c Figure 2. Rotor position angle definition 2

3 4 The flat top of each phase back EMF is lasting 2π/3-2δ, where 2δ is the deviation angle between the ideal trapezoidal BLDCM and the considered machine δ can be found by running the BLDCM as a generator at fixed speed and inspecting the back EMF waveforms using oscilloscope. Typical back EMF models as illustrated in Figure 1 have the following three characteristics: 1 The flat top value is determined by the product of the back EMF constant λ m and the rotor electrical speed ω r. 2 The transition between the positive and negative flat top regions is linear in terms of rotation angle. 3 The span of each of the positive and negative flat top is 2π/3 electrical degrees and the span of each linear transition region is π/3 electrical degrees. Investigating the waveforms of the back EMF of real non-ideal BLDCM reveals that ideal models introduce some approximations to simplify analysis especially concerning the second and the third characteristics. Transitions between flat intervals are not linear but exhibit some curvature in the vicinity of each interval boundary. This curvature is better approximated by a sinusoidal profile. Figure 3. Comparing linear and sinusoidal representation of the 0 degree transition interval In Figure 3 the instantaneous values of phase a back EMF at a given rotational speed are plotted according to linear change profile e linear and to sinusoidal change profile e sinusoidal between two consecutive flat regions. The percentage differences between the 2 profiles are also plotted against rotor position in electrical degrees. It is obvious that the linear representation of back EMF in the full transition range -30 to 30 electrical degrees doesn t introduce a big difference in back EMF value when compared to a sinusoidal representation over the same span at any given speed. It is indeed of no effect as phase windings undergoing such transitions are ideally unexcited with current during these transitions. The peak percentage difference [elinear esinusoidal/e] is about 1.81%. This difference is negative in the pre-zero crossing interval and positive in the post zero crossing interval. If sinusoidal change profile is considered, the back EMF of the three phases [ea e b e c] T could be simply modeled as per the following equation 2 which is arranged in seven piecewise-continuous intervals : K msin a d r / dt eabc m d r / dt m d r / dt 0 r 2-a 3

4 m d r / dt eabc K msin b d r / dt md r / dt K msin a d r / dt m d r / dt eabc m d r / dt m d r / dt m d r / dt eabc K msin c d r / dt m d r / dt eabc K msin b d r / dt m d r / dt K msin a d r / dt eabc m d r / dt m d r / dt 3 5 r 2-c 5 7 r 2-d 7 9 r 2-e 9 11 r 2-f r 2-g Where: θa = θr, θb = θr-2π/3, θc = θr+2π/3 and λ m is the back EMF constant. The constant K can be determined from any of the intersection points between the sinusoidal transition interval and the flat interval. If the span of the flat interval is complying with the ideal trapezoidal form i.e. 2π/3 the value of K can be found from: K msin r d r / dt m d r / dt 3 r / It follows that: K 2 In this ideal case, equation 2 can be used to determine back EMF at any given rotor position and speed, however, if the span of the flat interval is less than 2π/3 with an angle 2δ, the constant K will have a different value that can be found from the intersection point between the flat and sinusoidal interval r / in the same previous way which yields: K 1 / sin / 4 In this non-ideal case, equation 2 can no longer give the correct values of back EMF along the complete cycle because the seven piecewise-continuous definition ranges will be further segmented as illustrated in the thirteen intervals. Figure 4 shows the case δ = radians, i.e. the flat interval is 110 instead of 120 electrical degrees. Figure 4. Non-Ideal Trapezoidal Waveform conduction period per phase per half cycle=110 4

5 K msin a d r / dt eabc m d r / dt m d r / dt K msin r d r / dt eabc m d r / dt K msin c d r / dt m d r / dt eabc m d r / dt K msin c d r / dt m d r / dt eabc K msin b d r / dt K msin c d r / dt m d r / dt eabc K msin b d r / dt m d r / dt K msin a d r / dt eabc K msin b d r / dt m d r / dt K msin a d r / dt m d r / dt eabc m d r / dt K msin a d r / dt eabc m d r / dt K msin c d r / dt m d r / dt m d r / dt eabc K msin c d r / dt m d r / dt eabc K msin b d r / dt K msin c d r / dt m d r / dt eabc K msin b d r / dt m d r / dt K msin a d r / dt eabc K msin b d r / dt m d r / dt 0 r 5-a r 5-b 3 r 5-c 3 3 r 5-d 3 5 r 5-e 5 5 r 5-f 5 7 r 5-g 7 7 r 5-h 7 9 r 5-i 9 9 r 5-j 9 11 r 5-k r 5-l 5

6 K msin a d r / dt eabc md r / dt d dt / m r 11 r 2 5-m The waveforms of Figure 4 are calculated using equation 5 which considers the general case, where δ can be zero ideal BLDCM or non-zero non-ideal BLDCM. The factor K in equation 5 is given by equation 4 and the thirteen intervals described by equation 5 are then reduced to the seven intervals of equation 3 once δ is substituted with zero every second interval is then shortened to zero. Figure 5 compares the transitional interval of phase "a" back EMF in ideal trapezoidal representation with the proposed sinusoidal representation given by equations 5 in the case of δ = radians i.e. the flat interval is 110 instead of 120 electrical degrees. It is interesting to observe that at any given speed, if the absolute value of flat interval EMF is E, the percentage error based on E between the two representations reaches a maximum absolute value of 12.83% at the two ends of the flat 120 interval due to only 10 shortening. It can be shown that a considerable torque calculation error in simulation can be obtained if simulation uses the EMF expressions based on the idealized linear approximation. This error increases as δ increases. Figure 5. The difference in EMF magnitude during the non-conducting interval between proposed sinusoidal change profile and linear change profile 3. Stator flux linkages due to rotor magnets For the part of flux linking the stator windings due to rotation of the rotor magnets λ m, it can be obtained via integrating the back EMF as given by equation 5 with respect to time. The integration should be performed on a piecewise continuity basis also and the unknown integration constants can be calculated using the piecewise continuity condition of the resulting waveforms. Figure shows the waveform of stator flux linkage due to rotor magnets along a complete electrical cycle upon carrying out such integration. In the waveform of each phase, two types of intervals can be observed. The thick line styled intervals represent linear functions of electrical rotor angle, therefore it produces EMF values that are proportional only to the electrical speed of the rotor giving flat EMF intervals at constant rotational speeds. The intervals plotted using thin line style represent sinusoidal functions of electrical rotor angle thus it produces non-linear EMF products of the rotor electrical speed and sinusoidal functions of the rotor electrical angle.

7 Figure. Stator Flux Linkages due to rotor magnets rotation Based on the following definitions: K c m 1 / sin / -a K m / 3 K c cos / -b -c After obtaining the integration constants, the stator flux linkages due to magnets rotation λm arranged as [λam λbm λ cm] T can be expressed as follows: K c cos a K m m r 0 r 7-a m r K K K c cos a K m m r K c cos c K m r 2 m m r K c cos c K m r 2 m K c cos b K K cos K c c m r m K c cos b K 5 m r r 7-b 3 r 7-c 3 3 r 7-d 3 5 r 7-e 7

8 K c cos a K m K c cos b K 5 m r K c cos a K 7 m m r 5 m r K c cos a K 7 m m r K c cos c K 9 m r 7 m m r K c cos c K 9 m r m K c cos b K K cos K c c 9 m r m K c cos b K 11 m r K c cos a K m K c cos b K 11 m r K c cos a K 13 m m r 11 m r 5 5 r 7-f 5 7 r 7-g 7 7 r 7-h 7 9 r 7-i 9 9 r 7-j 9 11 r 7-k r 7-l 11 r 2 7-m Figure 7 illustrates the profiles of the back EMF of a set of virtual BLDC motors whose flat intervals are ranging from 120 to 0 and Figure 8 illustrates the corresponding flux linkages of stator windings due to rotor magnets. These waveforms are calculated using equations 5, and 7. According to the proposed model, these figures show that for the same magnet quality and back EMF constant λm the peak value of the rotor flux linkages as seen by the stator 8

9 windings is the highest when the flat EMF interval is the 120 and this peak is reduced as δ increases. Figure 7. Profile of Back EMF per phase with flat intervals shortened in steps of 5 degrees Figure 8. Profile of corresponding flux linkages of stator phases due to rotor magnets 4. Electromagnetic Developed Torque The stator total flux linkages equations expressed in matrix form is given by: λ abcs L 0 0 i a 0 L 0 ib λ m, L L M 0 0 L ic 8 Where: L is the self inductance per phase, M is the mutual inductance between 2 phases, iabcs are the stator currents and λ m is given by the thirteen piecewise continuous intervals of equation 7. Based on the assumptions made in section 2, the coupling field is conservative and the co-energy is given by: Wc i a λ a i b λ b i c λ c W pm L ia2 ib2 ic2 ia am ib bm ic cm W pm 9 Where: Wm is the permanent magnet energy. The electromagnetic torque is calculated by taking the partial derivative of Wc with respect to rotor mechanical angle θ m. Since stator currents are shaped by active switching, they are independent of rotor angle. Phase inductances are also independent of rotor angle due to absence of magnetic saliency. Magnet energy is independent of θ m as well. Hence, the electromagnetic torque is given by: 9

10 Tem Wc W i i i P / 2 c P / 2 a am b bm c cm m r r 10 Where: P is number of motor poles. Applying equation 10 on the thirteen intervals given by equation 7, we get the following Expression for Tem factor K is given by equation 4 Tem P / 2 m K sin a ia ib ic 0 r 11-a Tem P / 2 m K sin a ia ib K sin c ic Tem P / 2 m ia ib K sin Tem P / 2 m ia K sin b Tem P / 2 m ia K sin c ic ib K sin c ic b ib ic a ia K sin b ib ic Tem P / 2 m K sin a ia ib ic Tem P / 2 m K sin a ia ib K sin c ic Tem P / 2 m ia K sin b c ic ib K sin c ic Tem P / 2 m ia K sin b r 3 r ib i c Tem P / 2 m K sin a ia K sin b ib ic Tem P / 2 m K sin a ia ib ic 11-b 3 3 r 3 5 r 5 5 r 7 5 r 7 7 r 7 9 r 9 9 r 9 11 r r 11 r 2 Tem P / 2 m K sin Tem P / 2 m ia ib K sin 11-c 11-d 11-e 11-f 11-g 11-h 11-i 11-j 11-k 11-l 11-m 5. Accurate Simulation Results The BLDCM whose parameters are listed in Table 1 is simulated using the proposed EMF and torque expressions Equations 5 and 11 to show the effectiveness of such expressions in reflecting the torque dips due to deviation of actual EMF from the ideal model. The current is controlled using PI controllers in the three phase domain, currents are sampled at 20 KHz and modeled back EMF was added to the current PI controllers output feed forward compensation to enhance current control quality. Parameter Stator resistance per phase R Combined motor-load inertia coefficient J EMF shortening angle δ Rated speed ωr Table 1. BLDCM parameters Value Parameter Self minus mutual inductance per 0.5 Ω phase L kgm2 EMF constant λ m RPM Number of poles P Rated Torque Tem rated Value mh V/rad/s 2 0x10-3 N.m 10

11 Figure 8. Profiles of electromagnetic developed torque upper trace, back EMF per phase middle trace and three phase current lower trace all taken at fixed torque command & fixed speed The three current controller outputs are then converted using Clark s transform to drive a space vector modulation switcher center-aligned pulses at 20 KHz with DC voltage set to Volts. The speed controller was set to a fixed torque command to avoid possible reaction of the speed controller with the torque dips which may mitigate them. Figure 8 shows how the electromagnetic developed torque undergoes considerable dips exceeding 14% of mean commanded torque which is set equal to 1.5 of the rated torque mainly due to the non-ideal profile of the motor back EMF in the vicinity of every commutation instant. A minor contributor to torque dips in this case is the unequal rates of decay and growth of commutating currents.. Conclusion Accurate modeling of Brushless DC motor should consider the fact that flat intervals of EMF may be shorter than 120 degrees due to mechanical limitations. This shortening affects the torque developed by such machines and results in considerable dips in the vicinity of commutation instants. The magnitude and durations of torque dips are functions of such shortening angle. In current control schemes which are utilizing feed forward compensation using estimated back EMF, accurate modeling of back EMF in terms of rotor speed and position certainly support better control quality. During commutation periods, all three phase currents coexist and contribute to torque development. Hence; accurate torque equation is necessary as well for realistic simulation and for precise on-line estimation of torque whenever necessary. This paper covers such topics and introduces expressions for EMF and torque that take into consideration the non-ideal case of BLDCM and yet valid for ideal ones. 7. References [1] [2] [3] [4] J. F. Gieras, Permanent Magnet Motor Technology - Design and Applications CRC Press, R. Krishnan, Permanent Magnet Synchronous and Brushless DC Motor Drives CRC Press, R. Krishnan, Electric Motor Drives, Modeling, Analysis and Control Prentice Hall, Pragasen Pillay and Ramo Krishnan, "Modeling, Simulation and Analysis of Permanent-Magnet Motor Drives, Part II: The Brushless DC Motor Drive", IEEE Transactions on Power Electronics, Vol. 25, No. 2, PP , March/April [5] H. K. Saitha Ransara and Udaya Madawala,"A Technique for Torque Ripple Compensation of a Low Cost BLDC Motor Drive", Proceedings of IEEE International Conference on Industrial Technology ICIT2013, South Africa, PP , 25-28Feb [] Y. S. Jeon, H. S. Mok, G. H. Choe, D. K. Kim and J.S. Ryu, "A New Simulation Model of BLDC Motor with Real Back EMF Waveform", Proceedings of the 7th workshop on Computers in Power Electronics, COMPEL 2000, Blacksburg, VA, PP , July

12 [7] Salaheddin A. Zabalawi and Adel Nasiri,"State Space Modeling and Simulation of Sensorless Control of Brushless DC Motors Using Instantaneous Rotor Position Tracking", Proceeding of Vehicle Power and Propulsion Conference, VPPC 2007, IEEE, Beijing, China PP Sept [8] Byoung-Hee Kang, Choel-Ju Kim, Hyung-Su Mok and Gyu-Ha Choe, "Analysis of Torque Ripple in BLDC Motor with Commutation Time", Proceedings of International Symposium on Industrial Electronics, ISIE 2001, South Korea, Vol.2, PP , 12-1 June [9] Mourad Masmoudi, Bassem El Badsi and Ahmed Masmoudi, "DTC of B4-Inverter Fed BLDC Motor Drives with Reduced Torque Ripple During Sector to Sector Commutations", IEEE Transactions on Power Electronics, Issue 99, Publication year 2013, Early Access Article. [10] Jiancheng Fang, Haitoa Li and Bangcheng Han, "Torque Ripple Reduction in BLDC Torque Motor with Non-Ideal Back EMF", IEEE Transactions on Power Electronics, Vol. 27, Issue:11, PP , Nov [11] Jong-Woo Choi and Seung-Ki Sul, A novel 3-phase ac Current Control Technique with Fast Response and High Accuracy, Proceedings of Applied Power Electronics Twelfth Annual Conference and Exposition, Atlanta, GA, USA,Vol.1, PP , APEC 97, Feb