Torque Ripple Reduction in PM Motors using Current Harmonics Controller

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Torque Ripple Reduction in PM Motors using Current Harmonics Controller

Martin Sumega, Pavol Rafajdus, Stefan Kocan University of Zilina

Zilina, Slovak Republic [email protected]

Marek Stulrajter NXP Semiconductors Roznov pod Radhostem, Czechia

Abstract—This paper presents novel modification of Field Ori- ented Control (FOC) in order to reduce torque ripple in 3-phase Permanent Magnet (PM) motors caused by non-sinusoidal Back- EMF. A smooth torque operation is ensured by precise tracking of a modified q-axis current reference, which includes Back- EMF shape. Alternating component added into q-axis current reference has 6 times higher frequency than fundamental stator electrical frequency. Therefore precise tracking of such current reference is usually not feasible with conventional PI controller due to its limited bandwidth. This paper introduces Current Harmonics Controller (CHC) added into current control loop in FOC structure in order to increase performance of the current loop if alternating current reference or alternating disturbances occurs as a 6^thharmonic over one electrical revolution.

Index Terms—torque ripple, Back-EMF, PMSM, BLDC, torque ripple reduction

I. INTRODUCTION

Low-cost three-phase PM motors are widely used in many industry and automotive applications due to cost reduction trends. Disadvantage of simpler and thus cheaper construction of PM motor is in higher torque ripple.

Conventional FOC works with constant d-axis and q-axis currents and their references in order to achieve a smooth torque under condition of perfectly sinusoidal Back Electro- motive Force (Back-EMF). However, in practice Back-EMF shape varies from almost sinusoidal to almost trapezoidal, depending on the shape, arrangement and magnetization of the magnets [1]. Non-sinusoidal waveform of Back-EMF voltage leads to torque ripple if constant current reference is used and also to generation of harmonic components in currents. To ensure smooth torque under FOC operation, current reference must be properly modified and current control loops must be extended by advanced algorithms, because conventional PI current controllers are strongly limited by their bandwidth.

This significantly reduces ability of PI controllers for precise tracking of such alternating current reference [2], [3].

Most of the solutions for reduction of torque ripple caused by non-sinusoidal Back-EMF require information about Back- EMF shape of used PM motor. Based on on this waveform, dqcurrent references are modified to reduce torque ripple [4], [5], [6]. This basic approach is appropriate only at lower stator frequencies, because alternating component added into current

reference has few times higher frequency than fundamental stator electrical frequencyfe. Limited bandwidth of current PI controllers leads to insufficient reduction of torque ripple. Fre- quency of alternating component added into current reference to suppress impact of non-sinusoidal Back-EMF is 6 times higher than fundamental stator electrical frequencyfe. Precise tracking ofd-axis andq-axis current references is necessary in order to reduce torque ripple and therefore advanced control techniques must be used.

Several advanced algorithms to reduce torque ripple caused by non-sinusoidal Back-EMF have been introduced in the past.

High-bandwidth current controller for torque ripple reduction in PMSM was implemented based on dead-beat controller.

Control algorithm was extended by current predictor to ensure stability [2], [7]. Repetitive current controller is able to elim- inate periodic disturbances and thus accurately track periodic current references [3]. Multiple reference frame approach was used to transform alternating components ofdq currents into DC values, which were then controlled by conventional PI regulators [8], [9]. Resonant controller placed parallel to conventional PI controller significantly improves performance of current control loop [10]. Neural current or speed controllers are also able to achieve improvement in terms of torque ripple [11]. Modified field oriented control for smooth torque in drives with BLDC motor was successfully implemented with extended model of 3-phase PM machine and with modified Park transformation respecting Back-EMF shape [12], [13] .

This paper presents novel approach for precise control of current harmonics in dq frame. Current Harmonic Con- troller (CHC) extends conventional FOC structure in order to precisely tracks modified dq current references respecting shape of non-sinusoidal Back-EMF and ensures torque ripple minimization. Paper includes extended model of 3-phase PM motor and detailed description of proposed approach with successful experimental verification.

II. MODEL OF3-PHASEPMMOTOR INdqFRAME

Voltage equations of 3-phase PM motor indqframe:

ud=Rsid+dψd

dt −ωeψq (1) u_q=R_si_q+dψq

dt +ω_eψ_d (2)

(2)

where ud and uq are dq voltages, Rs is stator resistance, Ld andLq are dqinductances, id andiq are dq currents,ωe

is electric angular velocity, ψd and ψq are dq linkage fluxes.

Linkage fluxes indqframe and their derivations are described as follows:

ψd=Ldid+ ΨPM ⇒ dψd

dt =Ld

did

dt (3)

ψq=Lqiq ⇒ dψq

dt =Lq

diq

dt (4)

whereΨ_{P M} is a flux of PM. Linkage fluxes indqframe and their derivations (3) and (4) will be substituted into equations (1) and (2) as follows [1]:

u_d=R_si_d+L_ddi_d

dt −ω_eL_qi_q (5) u_q=R_si_q+L_qdi_q

dt +ω_e(L_di_d+ψ_PM) (6) Used 3-phase PM motor ACT 57BLF02 has a non- sinusoidal Back-EMF voltage as is shown in Fig. 1. Dominant 5^th and 7^th harmonic components in Back-EMF voltage in abc reference frame will lead to dominant 6^th harmonic component indqframe like it is shown in Fig. 2. This has to be considered in the model and therefore the rotational induced voltages from equations (5) and (6) will be adjusted to include this non-sinusoidal Back-EMF waveform [12].

ωe

−ψq

ψd

=ωe

Lq

Ld

−iq

id

+P(ϑe)



 ea

eb

e_c



=

=ω_e Lq

Ld

−iq

id

+ω_eK_e

fd(ϑe) fq(ϑe)

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wheree_a,e_b,e_crepresent phase Back-EMF voltages inabc frame, K_e represent voltage constant, f_d(ϑ_e)andf_q(ϑ_e)represent Back-EMF shape functions depending on the electrical positionϑ_eindqframe. Fig. 2 showsf_d(ϑ_e)andf_q(ϑ_e)Back- EMF shape functions with dominant 6^th harmonic components. P(ϑ_e)representsabctodqtransformation matrix.

P(ϑe) = 2 3

cos(ϑ_e) cos(ϑ_e−^2π₃ ) cos(ϑ_e+^2π₃)

−sin(ϑe) −sin(ϑe−^2π₃ ) −sin(ϑe+^2π₃)

(8) Voltage equations (5) and (6) respecting rotational induced voltages (7) will be modified as follows [14]:

u_d=R_si_d+L_ddi_d

dt −ω_eL_qi_q+ω_eK_ef_d(ϑ_e) (9) u_q=R_si_q+L_qdi_q

dt +ω_eL_di_d+ω_eK_ef_q(ϑ_e) (10) Mechanical equations of 3-phase PM motor are as follows:

dωe

dt = p

J(T_e−T_load) (11)

(a)

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 0

0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045

Harmonic component

e [V]

(b)

Fig. 1: a) Measured phase Back-EMF voltage b) FFT analysis of Back-EMF voltage

Fig. 2: Back-EMF shape functions depending on the electrical positionϑ_e indqframe

dϑ_e

dt =ω_e (12)

where J is inertia, T_e is electromagnetic torque, T_load is load torque andϑ_e is electrical angular position of the rotor.

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Electromagnetic torque for 3-phase PM motor with sinusoidal Back-EMF:

T_e=3

2p(ψ_di_q−ψ_qi_d) =3 2p

ψ_PMi_q+ (L_d−L_q)i_qi_d (13) Electromagnetic torque for 3-phase PM motor with non- sinusoidal Back-EMF respectingfd(ϑe)andfq(ϑe)Back-EMF shape functions:

Te= 3 2p

Kefd(ϑe)id+Kefq(ϑe)iq+ (Ld−Lq)iqid

(14) III. FOCRESPECTING NON-SINUSOIDALBACK-EMF Back-EMF shape functions fd(ϑe) and fq(ϑe) have to be suitably added into FOC structure to ensure production of the smooth electromagnetic torque Te. In a practice, impact of Back-EMF harmonics in d-axis only produces mainly 6^th harmonic component in d-axis current, but impact on the torque ripple is quite small and therefore f_d(ϑ_e) will not be added into d-axis current reference, which will remain set to i^∗_d = 0. Impact of Back-EMF harmonics in q-axis is much bigger, because it produces mainly 6^th harmonic component in q-axis current and also significant torque ripple, even in case that q-axis current harmonics are zero. Equation (14) shows, that fq(ϑe) function must be inversely added into q- axis current to counteract impact of non-sinusoidal Back-EMF in order to reduce torque ripple.

Requiredq-axis currenti^∗_q consists of DC parti^∗_{qP I}, which is output from speed PI controller and from AC parti^∗_{qBEM F}, which containsfq(ϑe)Back-EMF shape function.

i^∗_q =i^∗_{qP I}+i^∗_{qBEM F} (15) i^∗_{qBEM F} = (1−f_q(ϑ_e))i^∗_{qP I} (16) Alternating part i^∗_{qBEM F} in q-axis current reference i^∗_q will have 6 times higher frequency than fundamental stator electrical frequency fe, therefore precise tracking of such q- axis current reference is usually not feasible with conventional PI controllers. In addition, evendq current harmonics(mainly 6^th) caused by non-sinusoidal Back-EMF voltage is usually not feasible to suppress with conventional PI controllers. Fig.

3 shows current loop bode plot, where its bandwidth is tuned for 500 Hz. In case thatf_e= 50Hz, theni^∗_{qBEM F} added into q-axis current reference for compensation of non-sinusoidal Back-EMF will have frequency 300 Hz. Fig. 3 shows significant phase delay at frequency 300 Hz. Phase delay between alternating q-axis current reference and alternating part of q- axis current leads to insufficient reduction of the torque ripple caused by non-sinusoidal Back-EMF. Limited bandwidth is reason, why conventional PI controller is usually not suitable for precise control of such harmonic signals and successful minimization of the torque ripple. Therefore conventional FOC structure have to extended.

Fig. 4 shows advanced algorithm, which has to be added into FOC structure in order to ensure precise control of dq

current harmonics and reduce 6^th harmonic in torque ripple in whole speed range. In Fig. 4 inverter switching signals are indicated bySa,Sb,Scand mechanical angular velocity isωr.

−30

−20

−10 0 10

Magnitude (dB)

10¹ 10² 10³ 10⁴

−90

−45 0

System: CL Frequency (Hz): 300 Phase (deg): −19.3

Phase (deg)

Frequency (Hz)

Fig. 3: Bode plot of closed current loop with conventional PI controller

* 0

id

dq αβ

u*

u*

*

ud

*

uq

PM motor

αβ abc dq

αβ

ia

ib

ic

i

i_

*

r

r ^e

id

iq

*

iq PI

e

e Speed

observer

e

* q BEMF

i

q- current harmonics controller

* 6

uq iq

e

Equation (16)

PIq

PId

PI

d- current harmonics controller id

e

* 6

ud

UDC

SVM 3-phase power inverter Sc

Sb

Sa

Fig. 4: Proposed modification of FOC

IV. CURRENT HARMONICS CONTROLLER

Precise tracking of current references i^∗_q and i^∗_d without phase shift or amplitude attenuation have to be ensured for successful compensation of the non-sinusoidal Back-EMF voltage in order to reduce torque ripple under FOC operation. However, this is usually not possible with conventional current PI controllers, because they must control signals, that have a frequency 6 times higher, than the stator fundamental frequencyfe, which is mentioned in the section above.

Conventional FOC algorithm with current PI controllers is extended with novel Current Harmonics Controller (CHC) algorithm for control of 6^thharmonic component indqcurrents.

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CHC is located parallel to bothd andqPI current controllers.

Fig. 4 shows conventional FOC extended by CHC algorithm (blue blocks) placed to both current loops. CHC calculates 6^th harmonic voltages, which must be added intod-axis and q-axis voltage. Proper amplitude and phase of 6^th harmonic component added into d andqvoltages will lead to reducing the error between reference and feedback indqcurrents. Then extended current controllers are able to precisely track 6^th harmonic component in their reference.

Input to CHC algorithm inq-axis is error between reference and feedback q-axis current and rotor electrical position ϑ_e. Phase φ and amplitude x_q6 of compensation voltage u^∗_q6 in q-axis are adjusted during operation, based on input variables to achieve precise control of 6^th harmonic in q-axis current.

Compensation voltage u^∗_q6 is described as follows:

u^∗_q6=x_q6sin(6ϑ_e+φ) =

x_q6[sin(6ϑ_e)cos(φ) +sin(φ)cos(6ϑ_e)] (17) Injected compensation q-axis voltage can be divided into sine component and cosine component:

u^∗_q6 =x_q6sin(6ϑ_e+φ) =a_q6sinsin(6ϑ_e) +a_q6coscos(6ϑ_e) (18) where a_q6sin and a_q6cos represent amplitude coefficients of sine component and cosine component. Changes ina_q6sinand a_q6cos will lead to changes in phase and amplitude of injected u^∗_q6 voltage. These coefficients converge to DC values in steady state and their proper values ensure minimum error between reference and feedback of 6^th harmonic component in q current. Value of aq6sin and aq6cos are determined by integration with constant gain kCHC:

aq6sin=eq_sin

kCHC

s a_q6cos=e_q_cosk_CHC

s

(19)

Input for integrators are determined as follows:

eq_sin=e_i_qsin(6ϑ_e)

e_q_cos =e_i_qcos(6ϑ_e) (20) where eiq is error between reference and feedback q-axis current. Backward-Euler methods≈^z−1_T

sz is used to transform integrators in equations (19) into discrete domain. Then amplitude coefficients ink^thstep will be implemented in micro- controller as follows:

aq6sin(k) =aq6sin(k−1) +eq_sin(k)kCHCTs

a_q6cos(k) =a_q6cos(k−1) +e_q_cos(k)k_CHCT_s (21) wherekdescribes actual step andTs= 0.0001srepresents a sampling time. Integrators in equations (21) will behave same as a low-pass filter with the time constant equal to

1

kCHC. Integrator gain k_CHC = 10was chosen in this article.

This ensures convergence to 63.3% of steady state values for amplitude coefficients in 0.1 second.

Fig. 5 shows CHC algorithm applied intoqaxis to calculate compensation q voltage harmonics. The same approach with CHC is applied ford-axis.

Fig. 5: Current harmonic controller applied for qaxis

0 0.005 0.01 0.015 0.02 0.025 0.03

−2 0 2

time [s]

Iabc [A]

CHC OFF

I_a I_b I_c

0 0.005 0.01 0.015 0.02 0.025 0.03

−2 0 2

time [s]

Iabc [A]

CHC ON

I_a I_b I_c

(a)

0 0.005 0.01 0.015 0.02 0.025 0.03

−0.5 0 0.5

time [s]

I d [A]

CHC OFF CHC ON

0 0.005 0.01 0.015 0.02 0.025 0.03 2.2

2.4 2.6

time [s]

Iq [A]

CHC OFF CHC ON

(b)

Fig. 6: a) abcb)dqcurrents when CHC is applied without modifiedq-axis current reference

If CHC is used as shown in Fig. 4 withoutfq(ϑe)function added into i^∗_q current reference (i^∗_q = i^∗_{qP I}), then CHC is able to suppress only dq current harmonics caused by dead- time and non-sinusoidal Back-EMF. Ripple in electromagnetic torqueT_ecan be completely reduce only by includingf_q(ϑ_e)

(5)

function into i^∗_q. Fig. 6 shows dq and abc currents strongly deformed due to non-sinusoidal Back-EMF and dead-time. 6^th harmonic in dq currents is almost completely minimized, if CHC is turned on (CHC ON). Then dominant 5^th and 7^th harmonics no longer appear in abc currents. Fig. 7 shows transient state in sine and cosine components of amplitude coefficients and continuous minimization of 6^th harmonic components in d-axis and q-axis currents after activating the CHC algorithm.

0 0.2 0.4 0.6 0.8 1

−0.2

−0.1 0 0.1 0.2

time [s]

a [−]

a_d6sin a_d6cos a_q6sin a_q6cos

0 0.2 0.4 0.6 0.8 1

−0.5 0 0.5

time [s]

Idq [A]

I_d fbck.

I_d ref.

I_q fbck.

I_q ref.

CHC ON

Fig. 7: Transient state of amplitude coefficients andd-axis andq-axis currents after turning on CHC

V. TORQUE RIPPLE REDUCTION USINGCHC In the first place, FOC with the conventional current PI controllers has been evaluated in terms of capability to precisely track i^∗_q and i^∗_d current references. In the other words, its capability to suppress alternating input error. As expected, limited bandwidth of PI current controllers causes, that it doesn’t track dq current references properly. Fig. 8 shows significant steady-state error in dq currents as a consequence of insufficient current loop bandwidth. The feedback q-axis current shows the impact of the imprecisely tracked reference i^∗_{qBEM F} and the non-sinusoidal Back-EMF voltage, which acts as a disturbance input as well as for d-axis current.

DC errors in dqcurrents are always perfectly reduced by the conventional PI controllers.

Under the same load condition, CHC has been placed parallel to dq currents PI controllers, as shown in scheme of Fig. 4. Fig. 9 shows significantly increased performance of the resulting current controllers, showing almost perfect match betweendqfeedback currents and their references even at f_e= 80Hz, while alternating input dqcurrent errors have frequency6f_e= 480Hz.

Electromagnetic torque Te has been estimated based on equation (14) in order to verify the proposed approach with CHC. Fig. 10 shows the estimated electromagnetic torque T_e, with a average value 0.18 Nm (45% of nominal torque).

Conventional PI current controllers achieved peak ripple about 5%, whereas torque ripple is approximately 5-times lower with the proposed CHC approach.

0 0.002 0.004 0.006 0.008 0.01 0.012

−0.5 0 0.5 1

time [s]

I d [A]

I_d feedback I_d reference

0 0.002 0.004 0.006 0.008 0.01 0.012

2.8 3 3.2 3.4

time [s]

Iq [A]

I_q feedback I_q reference

0 0.002 0.004 0.006 0.008 0.01 0.012

−2 0 2 4

time [s]

Iabc [A]

I_a I_b I_c

Fig. 8: Tracking ofdqcurrent references with conventional FOC and resulting abccurrents

VI. EXPERIMENTAL SETUP

Fig. 11 shows experimental setup used for all experi- ments shown above. Setup consists of control board with micro-controller NXP MPC5643L running FOC application [15], three-phase PM motor ACT 57BLF02 together with encoder (512 pulses per revolution), NXP three-phase low voltage power stage and power source (12V). Software NXP FreeMASTER for real-time data (position, speed, currents, ...) visualization was used.

TABLE I: Parameters of used BLDC motor ACT 57BLF02

Rated power PN[W] 125

Rated current IN[A] 7.8 Nominal voltage U_N[V] 24 Rated speed n_N[rpm] 3000 Rated torque TN[N m] 0.4 Number of poles 2p[−] 8 Stator resistance Rs[Ω] 0.242277 D-axis inductance Ld[H] 0.000140521 Q-axis inductance Lq[H] 0.000197 Motor inertia J[kgm⁻2] 0.000017 Back-EMF constant Ke[_rad^{V s}] 0.01

VII. CONCLUSION

This paper presents novel approach to reduce torque ripple in 3-phase PM motors under FOC operation caused by non- sinusoidal Back-EMF. Current harmonics controller was introduced together with modification of q-axis current reference in order to counteract impact of non-sinusoidal Back-EMF.

CHC was placed parallel to current PI controllers in FOC structure and significantly increased performance of current

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0 0.002 0.004 0.006 0.008 0.01 0.012

−0.5 0 0.5 1

time [s]

Id [A]

Id feedback I

d reference

0 0.002 0.004 0.006 0.008 0.01 0.012

2.8 3 3.2 3.4

time [s]

Iq [A]

I_q feedback I_q reference

0 0.002 0.004 0.006 0.008 0.01 0.012

−2 0 2 4

time [s]

I abc [A]

I_a I_b I_c

Fig. 9: Tracking ofdqcurrent references with FOC extended by CHC and resulting abccurrents

0 0.002 0.004 0.006 0.008 0.01 0.012 0

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

time [s]

Te [Nm]

CHC OFF CHC ON

Fig. 10: Estimated electromagnetic torqueT_ein conventional FOC vs. FOC with CHC

loops. Therefore this extended current controller was able to precisely track periodic alternating signal in q-axis current reference and also suppress all disturbances, which acts as 6^th harmonic component over one electrical revolution. Successful experimental validation with design of proposed method is included in the paper.

ACKNOWLEDGMENT

This work was supported by Slovak Scientific Grant Agency VEGA No. 1/0615/19, and by projects ITMS:26220120046, cofounded from EU sources and European Regional Develop- ment Fund.

Fig. 11: Used experimental setup

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