IEEE Transactions on Industrial Electronics, Vol. 43, No. 2, 1996, pp. 309-320.

Lerb

Abstract — Permanent magnet synchronous machinesgenerate parasitic torque pulsations owing to distortion ofthe stator flux linkage distribution, variable magnetic re-luctance at the stator slots, and secondary phenomena. Theconsequences are speed oscillations which, although smallin magnitude, deteriorate the performance of the drive indemanding applications. The parasitic effects are analysedand modelled using the complex state-variable approach. Afast current control system is employed to produce high-frequency electromagnetic torque components for compen-sation. A self-commissioning scheme is described which iden-tifies the machine parameters, particularly the torque rip-ple functions which depend on the angular position of therotor. Variations of permanent magnet flux density withtemperature are compensated by on-line adaptation. Thealgorithms for adaptation and control are implemented in astandard microcontroller system without additional hard-ware. The effectiveness of the adaptive torque ripple com-pensation is demonstrated by experiments.

1. INTRODUCTION

AC machines with permanent magnet excitation are usedwith preference for applications in high-performance posi-tioning systems and machine tool spindle drives. They en-joy the unique advantage that the absence of separate exci-tation windings or magnetizing currents reduces the copperlosses considerably. The high efficiency of permanent magnetmachines permits a totally enclosed design with surfacecooling. The use of rare-earth permanent magnets enableshigh flux densities in the airgap, facilitating the construc-tion of motors of unsurpassed power density. A rapid dy-namic response is ensured by a high torque-to-inertia ratio,which is achieved using a slim rotor, or a disc-type rotorconstruction. These favorable properties make the perma-nent magnet motor an extremely fast, compact and robustmechanical actuator.

A persisting problem with permanent magnet machinesis the nonuniformity of the developed torque. The machinetorque changes periodically as the rotor advances during itsrotation. The resulting torque ripple is caused by deviationsfrom a sinusoidal flux density distribution around the air-gap, by deficiencies of feasible winding geometries, and bythe variable magnetic reluctance of the airgap due to thestator slots. The feeding power converter also contributesto the torque ripple owing to time harmonics in the currentwaveforms and to time-varying delays between the com-manded and the actual current. According to Jahns [1], “notechniques have been reported which can entirely eliminatethe presence of these fast torque transients under all operat-ing conditions”.

Identification and Compensation of Torque Ripple in High-Precision Permanent Magnet Motor Drives

Joachim Holtz, Fellow, IEEE, and Lothar SpringobUniversity of Wuppertal,

42097 Wuppertal – Germany

The effects of torque ripple are particularly undesirablein some demanding motion control and machine tool appli-cations. They lead to speed oscillations which cause deteri-oration in the performance. In addition, the torque ripplemay excite resonances in the mechanical portion of thedrive system, produce acoustic noise, and, in machine toolapplications, leave visible patterns in high-precision ma-chined surfaces.

This paper investigates the different sources of torqueripple in permanent magnet machines. Appropriate modelsare then defined for the ripple generating mechanisms. Aconcept for the compensation of torque ripple by a self-commissioning and adaptive control system is presented,and its effectiveness is demonstrated by experimental re-sults.

2. SOURCES OF TORQUE RIPPLE

Permanent magnet (PM) machines can be grouped intotwo categories, depending on their back-emf waveform,which is either trapezoidal or sinusoidal.

2.1 Trapezoidal emf machinesThe permanent magnets are mostly fixed to the rotor

surface, with groups of axially aligned parallel magnet stripsforming the individual machine poles. The airgap flux den-sity is constant under the poles, passing through zero be-tween two poles to assume the opposite direction under theadjacent pole. Ideally, the emf in the distributed statorwindings should be trapezoidal, and rectangular stator currentwaveforms would be required to produce torque which isconstant and independent of the respective angular rotorposition. In a real machine, the fringing fields at the rotorpole edges cause deviations from the ideal trapezoidal emfwaveform. Moreover, the currents are not strictly rectangu-lar. The commutation between the stator phase windingsrequires finite time intervals, during which the torque mag-nitude changes considerably [1, 2]. The dips of the torquemagnitude may reach up to 25% of rated torque [3]. Thetorque changes occur periodically in synchronism with therevolving rotor.

2.2 Sinusoidal emf machinesSinusoidal flux density distribution around the airgap is

difficult to achieve in a PM machine. What is important,however, is the resulting flux linkage with the stator wind-ing. Its spatial distribution can be adjusted by an appropri-ate stator winding geometry such that sinusoidal inducedvoltage waveforms are achieved with good accuracy. Un-like the trapezoidal PM motor, the sinusoidal version ope-rates as a revolving field machine and requires sinusoidal

- 2 -

voltage and current waveforms in the stator. The principaltorque component is proportional to the quadrature compo-nent of the armature current. Magnetic saliency of the rotorproduces a reluctance torque component which is low in aPM machine as the saliency is typically low. The reluctancetorque is proportional to the product of the quadrature cur-rent component iq and the direct-axis current component id.There are harmonic torque components in addition, whichresult from residual nonsinusoidal flux linkages and fromcurrent waveform distortions. In total, the torque ripple of asinusoidal PM machine is much less than that of a trapezoi-dal machine. Hence preference is given to sinusoidal PMmachines for drives with ultimate performance requirements.The following discussion concentrates on the sinusoidaltype of machine.

2.3 Stator slot harmonics and secondary effectsAnother cause for torque harmonics is the variable mag-

netic reluctance in the airgap, which changes periodicallywhen the stator teeth pass by the edges of the rotor magnets.Skewing the stator slots efficiently reduces stator slot rip-ple, although not completely. Residual torque pulsationsoccur at a frequency fsl = ω Nsl, which increases as themechanical speed ω increases. Nsl is the number of slots.Torque harmonics of lower frequency result from the interac-tion between unbalanced magnetization of the individualrotor poles with rotor eccentricity.

Flux linkage harmonics can reach typically 2 – 4% of therated torque, slot harmonics about 3%. The characteristicpattern of machine-produced harmonic torque may sub-stantially differ even when comparing identical machinesfrom the same production batch. In a given machine, thispattern varies with temperature, also irreversibly as in thecase of Nd-Fe-B magnets, or following a current overload.The predominant ripple frequency component is six timesthe stator frequency. Since speed fluctuations due to lowfrequency torque ripple are automatically compensated bythe drive control system, the critical torque harmonics rangetypically from 100 Hz – 2 kHz. Harmonics of higher fre-quency are sufficiently attenuated by the rotor inertia.

2.4 Time harmonicsTime harmonics are caused by current waveform distor-

tions in the feeding power converter. To minimize theireffect, the switching frequency must be high, and thepulsewidth modulation scheme must not generate subhar-monic currents [4]. The influence of the current dependentswitching delay time (dead-time effect) must be minimized[5].

3. REVIEW OF EARLIER WORK

The torque ripple content in sinusoidal PM machines canbe reduced using specific layouts of the three-phase statorwinding, [6]. Also an adequate choice of the relative widthof the magnet poles and a particular airgap profile underthe poles have an effect on the flux linkage distributionwhich leads to a reduction of undesired torque ripple [7].

Torque variations of very low frequency are normallyeliminated by the speed control system. Torque harmonicsof higher frequency can be compensated in principle bygenerating an inverse torque component through appropri-

ate modulation of the stator current. The compensatingcurrent can be obtained in different ways. Most of theexisting proposals are based on the evaluation of the back-emf waveform.

A method which relies on the manufacturing data of themotor is presented in [8]. The approach is based on theassumption that the airgap is uniform and that magneticmaterial is homogeneously magnetized. Manufacturing tol-erances and modelling errors will generally influence theflux density distributions to differ from those of a typicalmachine, or the actual machine, respectively. The ripplecompensation is then incomplete.

A different approach is based on the Fourier analysis ofthe torque equation, from which the required current com-pensation terms are synthesized, or on the Fourier analysisof the back-emf waveform [9, 10, 11, 12, 13]. The compu-tations are done off-line based on machine data measuredon a test bench. The synthesized compensation waveformis valid only for a particular machine.

A simpler method for off-line calculation of the compen-sation currents is based on the measured trajectory of theback-emf space vector in rotor coordinates [14].

None of these methods are suited for automated the ac-quisition of stator slot harmonics, a prerequisite for theimplementation in industrial drive systems.

Very fast current controllers are required for the practicalimplementation, as the torque harmonic frequencies in-crease proportional to the speed. Conventional PI controllersmay exhibit bandwidth limits and are not suited to coverthe full range of ripple frequencies. Feedforward techniqueshave been proposed in an effort to overcome the limitedbandwidth of PI current control systems [15]. The feedforwardsignals are read from memory tables which are addressedby the respective values of the state variables. The tablesare preprogrammed based on off-line calculations that makeuse of the differential equation of the motor, but do notinclude the harmonic phenomena. A large number of EPROMsare required to store the solutions of the pertinent equations.

The existing methods leave the following problems un-solved:

1. Torque harmonics caused by stator slots and rotor ec-centricity do not reflect in the induced voltage andhence are not compensated.

2. Owing to manufacturing tolerances, even identical ma-chines from the same production batch will differ intheir back-emf waveforms. This makes the harmoniccompensation necessarily incomplete.

3. The permenent magnet properties vary with tempera-ture. Their magnetization may also change due to arma-ture reaction in a current overload situation.

4. The compensation of high frequency torque ripple callsfor an extended bandwidth of the current control sys-tem.

4. DYNAMIC MODEL OF A PM MACHINE

The design of a permanent magnet motor drive for highperformance and minimum torque ripple is based on a ma-chine model representing the harmonic effects. The dynam-ic analysis is based on the space vector notation. In thestationary reference frame, the voltage equation of the sta-tor winding is

- 3 -

u is s s

s(S) (S)(S)

= +rd

dy

t , (1)

where us is the stator voltage, is is the stator current, rs isthe winding resistance, and ys is the stator flux linkage.The superscript (S) refers to the stationary reference frame.All quantities are normalized by their rated values. Notethat time is also normalized, τ = w sRt, [16], where w sR isthe nominal stator frequency.

The stator flux linkage is composed of a contributionfrom the stator current, and of the component ysf relatingto the permanent magnet field of the rotor,

y ys s s sf(S) (S) (S)( ) = ( ) + ( )d d dl i∗ , (2)

where the saliency of the rotor is expressed by the induct-ance tensor.

This tensor reflects the influence of the d-axis inductanceld and the q-axis inductance lq in the stator as a function ofthe angular rotor position d.

Since the magnet material in the wide airgap has smallpermeability and low conductivity, eddy currents at tran-sient operation are negligible. Hence a transient time con-stant does not really exist.

ls

d q d q d q

d q d q d q( ) =(S)d

d d

d d

l l l l l l

l l l l l l

++

− −

− +−

−2 2 2 2 2

2 2 2 2 2

cos sin

sin cos(3)

From (1) through (3) the voltage equation becomes

u i l is s s s s

sf(S) (S) (S) (S)

(S)

= + +( )

rd

dd

dt td

∗( ) y. (4)

The rotor-fixed reference frame (superscript (R)) is pre-ferred for further analysis as it decouples the variables andsimplifies the equations. The coordinate transformation leadsto

t t

yy

ss

s s s

ssf

sf

ss

(R)(R) (R)

(R)

(R)

(R)

j

j( )

∗ + = − ∗

− +

+

dd

rd

d r

ii i

u

t w

w wdd

1 1(5)

The voltage equation (5) is visualized by a complex first-order system [16] located in the shaded box of the signalflow graph Fig. 1. The feedback signal –jwts * is indicatesthat the stator winding rotates at the angular velocity –ωagainst the reference frame. The stator time constant isexpressed by the tensor

t s s s= 1

r l , (6)

where

l l C l Cs s

d

qs= = ( )(R) (S)

l

l

0

0= ∗ ∗d Τ (7)

is the tensor of the stator inductance in rotor coordinates. Itis obtained by rotating (3) with the help of the transforma-tion matrix

C( ) =δ

cos sin

sin cos

d dd d−

, (8)

which eliminates the dependency on the rotor position an-gle. The time constant of the stator winding as given in (6)exhibits a spatial orientation which models the saliency ofthe rotor. The resulting reluctance torque is independent ofthe rotor field; it is nonzero when both stator current com-ponents, id and iq, are nonzero. The back-emf componentwhich is induced at nonzero reluctance torque originatesfrom the complex factor ts in the feedback term –jwts * isshown in the machine model Fig. 1.

The magnet induced back-emf in (5) is

u u ui sf

sfi1 i hj

( )(R)

(R)

= + = +w wddy

ydd

. (9)

Note that ui refers only to the magnet induced back-emf,for which the expression back-emf will be used. Subse-quently, all equations will be referred to in rotor coordi-nates; the superscript (R) is then omitted for simplicity.

The first term in (9) is the fundamental back-emf compo-nent ui1; it is aligned with the q-axis, and its magnitude isproportional to the angular velocity w. The second termdescribes the variation of the flux linkage ysf with the rotorposition angle d. This variation is zero in rotor coordinateson condition that the spatial distribution of the permanentmagnet flux linkage ysf is sinusoidal. In such a case, the

Fig. 1: Complex signal flow graph of a PM machine; p: number of polepairs

flux linkage with the stator winding isindependent of the rotor position δ : thespace vector ysf

(R) is constant and hasonly a real component. Its derivativedysf/dτ = 0.

In a real PM machine, the flux densitydistribution around the airgap isapproximately rectangular. It is almostconstant under the magnet poles, andclose to zero in the gap between thepoles. Special winding geometries canbe used to establish a flux linkage withthe stator that is sinusoidally distributedin space. Residual deviations from thesinusoidal distribution can be describedby a flux linkage vector ysf(d) that,

LTeT

dw

F

1 p

1

2

j

idealized machine0

FT ( )d

( )d

w

Tsh( )d/p

t s

ls

tj s

r1s 1

us

ui ys

is

ysfysf

uih

u i1

tm

- 4 -

being expressed in rotor coordinates, varies with the rotorposition d. Such variations define the complex function ofthe flux linkage harmonics

F

yd

dd( ) =

ddsf( )

. (10)

A comparison with (9) shows that, at rated speed, thisfunction is formally identical with the space vector uih ofthe back-emf harmonics in the stator winding:

F d

dw( ) = ( )ui h . (11)

Its trajectory in rotor coordinates was computed from ameasurement in the no-load generator mode of the rotorinduced harmonic voltage and of speed. An 8-pole PMmachine was used, the data of which are given in the Ap-pendix. The trajectory in Fig. 2 is periodic with 360° electric,being detailed in Fig. 2(a) and Fig. 2(b) for 0 – 180° and180° – 360°, respectively. For comparison, the two-axescomponents of the back-emf vector (9) are oscillographedin Fig. 2(c).

It is important to note that, owing to the presence of fluxlinkage harmonics, field coordinates and rotor coordinatesdo not coincide in this type of synchronous machine.

The Fourier spectrum Fig. 2(d) of the complete trajectoryrepresented by Fig. 2(a) and Fig. 2(b) shows that spaceharmonics of 2nd and 6th order are predominant. Harmon-ics of higher order than 12 tend to be small, and theircontribution to torque ripple can be neglected. The exist-ence of even harmonics indicates the presence of a zerosequence system in the back-emf trajectory, while compo-nents having a negative sign relate to flux linkage harmon-ics of counter-clockwise rotation.

The electromagnetic torque generated by the fundamentaldistributions is expressed by

Te0 s s s s sf s= × = ∗ +( ) ×y yi l i i , (12)

where ysf is substituted from (2). In (12), the reluctancetorque is represented by (ls* is) × is.

Since the space vector approach cannot represent spatialdistributions other than sinusoidal, the space harmonics ofthe flux linkage distribution ysf(d) are expressed in thetime domain. Their effect is represented by the back-emfterm uih(t), which is then time-varying also in rotor coor-dinates. The torque contribution of the flux linkage har-monics is derived from the equivalence of electrical andmechanical power

TF d w t d( ) = ( ) = ( )1

u i iih s so oF . (13)

This equation holds regardless of whether the internaldistributions of voltages, currents and flux linkages aresinusoidal or not.

The mechanical system is described by the equations ofmotion

dd T T T Twt t t d d d= ( ) + ( ) + ( ) − ( )[ ]1

me0 s h LF (14)

0 π 2π

0

1

uid,uiq

d

uiq

uid

uik

0

0.01

0.02

0.03

1

–18

k–12 –6 181260

90°

30°

60° 0°

150°

180° 120°

djIm{ui}

1.08

0.92

1

Re{ui}0.1 –0.10

210°

240°

330°jIm{ui}

1.08

0.92

1

Re{ui}0.1 –0.10

d

a)

b)

c)

d)

300°360°180°

270°

ui1

ui1

Fig. 2: Effects of nonsinusoidal stator flux distribution; (a) and (b)complex function F(d) of the flux linkage harmonics, (c) two-axescomponents of the complex function F(d) in rotor coordinates, (d)harmonic spectrum of F(d) in rotor coordinates

- 5 -

dddt w= , (15)

where τm is the normalized mechanical time constant, Te0is the fundamental electromagnetic torque, TΦ is the harmonictorque caused by nonsinusoidal flux linkage distribution,Tsh is the torque due to stator slotting and rotor eccentricity(cogging torque), and TL is the load torque. In (14), thefundamental electromagnetic torque Te0 is a function of thetime dependent state variables in (12); the harmonic torquecomponents TΦ and Tsh depend on the rotor angle. Notethat the rotor angle d is expressed in terms of electricalradians. The same applies to the angular velocity w of therotor.

Equations (12), (14) and (15) are represented on theupper right-hand side of the signal flow diagram Fig. 1.The sources of torque ripple are visualized by the twononlinear functions below. The scalar function Tsh(d /p)represents the stator slot and eccentricity effects. Its argu-ment is d /p, where p is the number of pole pairs. Theargument takes into account that the rotor angle d of themachine model refers to an equivalent 2-pole machine,while the eccentricity effects in the real machine repeatevery full revolution of the rotor.

The complex function F(d) of the flux linkage harmo-nics models the effect of an existing nonsinusoidal fluxlinkage according to (11) and (13). It relies on the spacevector uih(τ) of the back-emf, which, in fact, symbolizes asinusoidal distribution in space [16, 17]. Although suchdistribution does not exist in this machine, the representation(13) of the harmonic torque is nevertheless correct, sincethe nonsinusoidal spatial distribution w F(d) from (11) hasthe same effect on the machine as the sinusoidal distributionuih.

5. THE CURRENT CONTROL SYSTEM

A very-high bandwidth current control system was de-signed for high-frequency torque ripple compensation. Itcomprises a dynamic feedforward structure for decoupledtorque control, an error prediction scheme, and a deadbeatcurrent controller [18]. The parameters of the current pre-dictor and the deadbeat controller must be accurately set.An automated self-commissioning scheme is provided forthis purpose. The deadbeat controller eliminates the currenterror within one subcycle of the pulsewidth modulator (e.g. 25 ms at 20 kHz switching frequency). The inherentsampling delay of digital signal processing does not limitthe effective bandwith since the compensation functionsare fully predictable. An underlying space vector modulatorensures low time harmonics and, having a reference voltageinput u*, favors self-commissioning.

The current control system forms part of the torque ripplecompensation scheme Fig. 9. Its performance is demon-strated in Fig. 3 in comparison with a PI current controllerplus feedforward decoupling of the machine dynamics.

6. SELF-COMMISSIONING SCHEME

The proposed torque harmonics suppression scheme andthe associated deadbeat current control system require theaccurate knowledge of the following machine properties:

• stator resistance rs,• stator inductances ld and lq,• the complex function F(d) of the flux linkage harmon-

ics,• the function Tsh(d) of the slot harmonic torque.

Usually these data are not, or only partially, included inthe data sheet of the machine. Particularly the harmonicflux linkage characteristic F(d) differs with every individ-ual machine, making the commissioning of such drive systema very complicated task. Therefore, a self-commissioningscheme is indispensable. It automatically determines thecorrect parameter settings, and the specific functions thatcharacterize the torque harmonics. As the permanent mag-net properties vary with temperature, and may also changefollowing a current overload situation, an on-line adaptationscheme for the flux linkage harmonics is required. At best,such identification schemes should work without hardwareadditions to an existing control system. This constraint isimportant for applications in an industrial environment.

A self-commissioning scheme as specified before is de-scribed next. It relies only on the name-plate data of themachine, which is entered by the operator through a userinterface. The following name-plate data are required: ratedcurrent, rated speed and the number of poles. In the caseconsidered here, the motor is a Mavilor Discodyn ac servo-motor type SE 808 which is already designed for minimumtorque ripple. Its cross-section is shown in Fig. 4.

a)

b) 10 2 5 ms3 4

0

1

i(kT)

t

iq

id

iq*

10 2 5 ms3 4

0

1

i(kT)

t

iq

id

iq*

Fig. 3: Comparison of current controllers, (a) PI-controller withfeedforward decoupling, (b) deadbeat controller; sampling frequen-cy 10 kHz

- 6 -

6.1 Measurement of the stator resistanceWhen the automated identification process is started, the

stator resistance is measured first. The pulsewidth modula-tor receives an input signal u* such that a voltage vector ofarbitrary, but constant phase angle is applied to the ma-chine. On condition that the variable inverter switchingdelay is accurately compensated [5], the average statorvoltage being applied to the machine equals the referencevoltage vector. During the test, the magnitude of the appliedvoltage vector is started at a minimal value and then in-creased slowly until the rated current flows in the statorwindings. The stator resistance is then taken as the ratiobetween the reference voltage and the measured current.Irregularities of the pulsewidth modulator and the inverterare compensated by repeating the measurement at differentphase angles of the voltage vector. The obtained results areaveraged. The measurement error with this method is lessthan 5%. Such accuracy is more than sufficient as the statorresistance is not a critical parameter in the proposed controlscheme. The error due to its thermal variations is muchhigher.

6.2 Identification of the stator inductanceThe measurement of the stator resistance terminates with

the machine carrying a dc stator current of about ratedmagnitude. The current is maintained stationary for an ade-quate duration of time so as to permit the rotor to align withthe stator field. The stator current then becomes the d-axiscurrent.

The saliency of the machine produces a reluctance torquecomponent that tends to deviate the rotor from that angularposition which is determined by the stator current spacevector, even at zero shaft torque. The reluctance torque wasfound negligible in our laboratory machine, and hence nofurther measures were taken. If necessary, however, thedeviation of the rotor position due to the reluctance torquecould be determined by tuning the average imaginary com-ponent of the measured back-emf vector (Fig. 7) to zero.This would require an additional test at nonzero speed.

In the identification process, the phase angle of the statorcurrent space vector is now set to arg(is) ≈ 30°. The reasonfor selecting this particular angular position is that thepulsewidth modulator works more accurately if its refer-ence vector is located half way between two active switch-ing state vectors. The error introduced by the minimum on-time condition of the inverter is then minimum, and us =u*.

After the alignment of the rotor to a defined position, themodulator is turned off, and the machine deenergizes.

A constant voltage vector, having the same phase angleas in the previous test, is now applied to the stator winding.Its predetermined magnitude is controlled by pulsewidthmodulation. To reduce the magnitude error of the pulsewidthmodulation process, the voltage amplitude is selected high.The stator current builds up exclusively in the d-axis, risingexponentially according to

i

ud s

s

d= − −

*expr

rl1 t , (16)

from which equation the d-axis inductance ld can be deter-mined. Fig. 5(a) shows the d-axis response in terms ofthose current components iα and iβ in stator coordinatesthat are acquired by the system. The stator current (16) issampled at those time instants at which its harmonic com-ponent equals zero. The sampling instants kTs are determinedby the pulsewidth modulator [4]; they are marked by therising edges of the clock signal in the lower trace of Fig.5(a). Ts is the sampling period.

Fig. 5(b) shows the q-axis response, from which lq isdetermined. The oscillogram was recorded in the same wayas in the previous test, but with the voltage reference vectoradvanced by 90°. To demonstrate the influence of satura-tion, the current slopes in Fig. 5 were arbitrarily increasedand the currents driven beyond their rated value. While theq-axis inductance appears fairly linear, saturation in the d-axis becomes visible at is > 2 isR, where isR is the ratedstator current.

The exponential characteristic of the response is barelyvisible in Fig. 5 since the stator time constant is muchlarger than the interval of measurement. This circumstancefavourably permits measuring the q-axis inductance by re-cording the step response of the q-axis current before theelectromagnetic torque produced by this current componentaccelerates the rotor appreciably. This also ensures that theback-emf remains near zero, and its influence on the cur-rent need not be considered.

The stator inductance values ld and lq are calculated fromn discrete samples is(S)(kT), k ∈ [1, n], of the stator current.A total number of n = 8 current samples proved to beFig. 4: Mechanical construction of the Mavilor Discodyn ac servo-

motor SE 808

- 7 -

sufficient for an accurate evaluation of the inductance. Theyare transformed to rotor coordinates, and a table containingthe values

ylk

sd, q

s

d, q= ( ) =1 1 8− −

∈[ ]ru

i kTr

kT k*

exp , ,

(17)

is established. The stator inductances ld and lq are thenobtained by exponential regression:

ld,q s

k k

= − ∈[ ]( )

=

=

= ==

−

−

∑∑

∑ ∑∑r k

n kT kT

n kT y kT y

k

n

k

n

k

n

k

n

k

n

2

1

2

1

1 11

1 8

ln ln

, , (18)

6.3 Identification of the mechanical time constantThe identified values of rs, ld and lq permit adjusting the

parameters of regular PI current controllers for the d- andq-axis currents. The controllers are then used to inject a q-axis current step of rated magnitude into the machine,producing unity torque, TeR = 1. The acceleration Dw /Dt of

the drive system is measured, and the mechanical timeconstant is obtained from

t Dt

Dwm e R= T . (19)

This value is used to set the parameters of the speed con-troller.

6.4 Identification of the flux linkage harmonicsThe effect of flux linkage harmonics is acquired in an

on-line process during normal operation of the drive system,thus permitting the adaptation of torque ripple compensa-tion to the prevailing conditions. The flux linkage harmon-ics are reflected in the distortion of the back-emf, (9) and(5),

u u i l

il ii s s s s

ss sj= − − ∗ − ∗r

ddt w , (20a)

where

w t

d= dd

. (20b)

Hence the flux linkage harmonics can be extracted fromthis waveform as a function of the rotor position angle d.The machine parameters rs and ls in (20) are known fromthe initial self-commissioning tests. The stator voltage us isa switched quantity and equals zero when the zero vector ison. Evaluating (20) during this time interval avoids themeasurement of us. However, a problem is associated withthe evaluation of the current derivative dis/dt. The numeri-cal differentiation is bound to be inaccurate in the presenceof noise.

To improve on this, all terms in (20) are averaged over aswitching subcycle of the pulsewidth modulator. The inte-gration of (20) leads to:

u u i l

il ii s s s s s

s s s s sT T T T T

dt dt r dtddt dt j dt∫ ∫ ∫ ∫ ∫= − − ∗ − ∗ w

(21a)

w d

T T

dtddt dt

s s

s∫ ∫= l (21b)

where Ts is the subcycle interval. Its short duration of 50 µsor less supports the following assumptions:

• The back-emf changes linearly.

• The average output voltage us equals the reference volt-age vector u*.

• The stator current derivative is constant. Hence the aver-age stator current equals the average of the two currentsamples at the beginning and the end of the subcycleinterval.

• The angular velocity ω of the rotor is constant.

Using these conditions we have from (21)

u u i l

i il ii k 1 s k s

k k 1

ss k

d d wk k r T j+

= − − ∗ − − ∗−

−−1

2 * (22a)

where i i ik k k 1= +( )−12

, (22b)

an w d d= − −k k

T1

s. (22c)

a)

b) 0.20 0.4 1 ms0.6 0.8

3

2

1

0

iα(t)

t

iβ(t)

Ts

iα , iβ

3

2

1

0

0.20 0.4 1 ms0.6 0.8t

iα(t)

iβ(t)

iα , iβ

Fig. 5: Identification of the stator inductances following a voltagestep input, (a) d-axis response, (b) q-axis response, lower trace:sampling signal

- 8 -

The relationship in (22) between the sampled and calcu-lated data is displayed in Fig. 6. Note that only current andspeed samples are taken, as in noߘal operation. All varia-bles are referred to in rotor coordinates.

Equation (22) is continuously evaluated during operationto be adapted to the actual magnetic properties of the ma-chine. The acquired data are used to compute the fluxlinkage harmonics from (9) and (11). The result is exemplifiedin Fig. 7(a). The remaining distortions are due to quantiza-tion effects, particularly of the pulsewidth modulator. Theyare eliminated by a nonrecursive filter (FIR-filter) of con-stant group velocity, defined by

F Fk k i ii

i N

N

( ) = −( ) ∈[ ]=−∑a (D) , ,1 8 , (23)

where F(d)(D) represents the distorted preliminary data.The values of the filter coefficients a i are calculated usingthe Hamming window function:

a aa

p

p

i i

ii N

N

N

f iki

iN i N

= = + ⋅

− +( )( ) ∈[ ]

−=−∑

2 1

2

0 54 0 461

0

sin

. . cos , ,

π c

max(24)

Here, kmax = 2048 is the number of samples taken per polepair, and fc is the cut-off frequency of the filter, which isset to the 13th harmonic at rated speed (2600 Hz). Thesmoothened flux linkage function F(d) is shown in Fig.7(b). The same function, but measured off-line on a ma-chine test bench is shown for comparison. The curves inFig. 7 were obtained on-line from zero initial values afterthree minutes of operation including noise reduction asdescribed by (24).

6.5 Identification of slot harmonicsThe slot harmonic torque function Tsh(d) is indirectly

acquired by operating the machine at very low speed. Thetorque harmonic frequency is then also very low and wellwithin the bandwidth of the speed control loop. Hence thetorque ripple is instantaneously compensated by the speedcontroller in its effort to maintain the speed constant. Theslot harmonic function Tsh(d) can be extracted from the

torque producing current iq(d), which is sampled and storedin a table. For simplicity, id = 0.

Readings are taken over more than one full revolution ofthe motor shaft with a view to remove statistical noise fromthe sampled data. Following the first revolution, the read-ings taken during the ensuing k revolutions are used tomodify the table according to

i

i g igq

q qkk 1 k

dd d

,, ,

( ) ⇐−( ) + ⋅ ( )

+1 . (25)

The weight factor g is selected in a compromise to obtain asmooth curve in minimum time. A value of g = 1.55 workedwell. The resulting waveform is shown in Fig. 8(a). Thiscurve represents the total machine torque. Its ac content isthe torque ripple, and the dc offset is the load torque in-cluding friction of the motor bearings. Hence

T i i dsh q qd d d d d

p

( ) = ( ) ⋅ ( ){ } − ( )∫Im F0

2

(26)

u_*

k–1

acquired

t

averagevoltages

u_

i,k–2

dk–2 dk–1 dk dk+1

2dk + dk–1

interruptinterruptinterrupt

data

u_

i,k–1 u_

i,k

u_*

k–2 u_*

k

ik–2

i_k–1 i

_k i

_k+1

ik+1ikik–1

Fig. 6: Timing diagram; the k-values number the sampling instants

Fig. 7: Components of the rotor induced back-emf; (a) measuredwaveform after transformation to rotor coordinates, superscript (D)

indicates distorted function; (b) rotor induced back-emf after non-recursive filtering, compared with the accurate waveform obtainedoff-line on a test bench

a)0 π 2πd

– 0.06

0

0.06

– 0.06

0.06

0

Re{F}(D)

Im{F}(D)

– 0.06

0

0.06

– 0.06

0.06

b)

0

0 π 2πd

Re{F}

Im{F}

identified on-linemeasured on test bench

- 9 -

is the normalized slot harmonic torque. The factor Im Im Im Im Im { F (d)}in (26) eliminates the flux linkage harmonics at id = 0,which is derived from (9), (11) and (12).

Fig. 8(b) shows the torque ripple resulting from slotharmonics as extracted from an 8-pole sinusoidal PM machineusing (26). There are 23 pulsations visible per rotor revolu-tion which relate to the slot effect. The additional low-frequency component results from different pole flux levelsof the four pole pairs, possibly combined with rotor eccen-tricity. The latter effect may also account for the varyingamplitude of the slot ripple function. The peak ripple am-plitude amounts to 2.5% of rated torque. Note that thismethod of acquiring the slot harmonics inherently includesany periodic torque ripple of the mechanical load in therecorded data, which will then be compensated as well.

Given the low speed of operation during this test, thesecond term in (26) represents basically Coulomb frictionas generated by the machine and the load. The Coulombfriction torque reverses sign at speed reversal, thus intro-ducing an instantaneous disturbance. During normal opera-tion, the disturbance is compensated using the identifiedfriction torque.

7. HARMONIC TORQUE COMPENSATION

The identified torque ripple functions are used for com-pensating the imperfections of the drive motor. Fig. 9 showsthe signal flow diagram of the compensation scheme. Theshaded block contains the current control system. The d-axis reference id* = 0, and hence the reluctance torque(ls* is) × is = 0, which term is contained in (12); The reluc-tance torque would be small anyway even if id = 0 sinceld ≈ lq. The q-axis reference iq* controls the torque andhence is derived from the speed controller. The additionalsignal iq comp = iq sh + iq F compensates the slot harmonics,and the flux linkage harmonics, respectively. The compen-sation signals are read from two tables, one being addressedby the mechanical rotor position angle, and the other by the

electrical rotor position angle.Fig. 10 shows the steady-state stator current

trajectory including the compensation currentfor torque harmonics. Four different patternsexist for an 8-pole machine, demonstratingthat the compensation function repeats everymechanical revolution.

The effect of slot harmonics compensationon the rotor speed is demonstrated in Fig.11(a). This oscillogram was recorded at no-load to eliminate the influence of flux link-age harmonics. The resolution of speed meas-urement is 4 .10–6.

The compensation of flux linkage harmon-ics is demonstrated in Fig. 11(b) with ratedtorque applied to the motor shaft. The slotharmonic torque compensation was enabledthroughout this measurement, which separatesthe slot effect. A water cooled, torque con-trolled iron-powder brake was used for thistest to ensure that the ripple torque added bythe load is minimum. A dc machine provedto be inadequate for this purpose. Some ofthe residual speed ripple which is seen in Fig.

-4%

0

4%

iq

dmech0 π 2π

dmech0 π 2π

-4%

0

4%

Tsh

b)

a)

Fig. 8: Identification of slot harmonic torque, (a) q-axis currentversus rotor position angle, (b) extracted slot harmonics

Ts

ud

1 p

currentpredictor

= 0

deadbeatcontroller

current controller

ejd

e-jd

3~M

=~PWM

us

w

*w

d

*id

*iq

is(R)is1

*u

*iq comp

*iq sh

F*iq

Fig. 9: Signal flow diagram of the speed and torque control system(d in degrees electrical)

- 10 -

11 with the compensation activated could be contributed bythe load.

To demonstrate the performance of the current controlsystem, the electromagnetic torque of the machine wasestimated from terminal quantities. The estimated torquecomprises flux linkage harmonics, but excludes slot har-monics. PI-controllers fail to suppress the torque harmonicsdue to their limited bandwidth and sampling delay effects.These lead to instabilities which amplify the torque ripple,when given the task to compensate the flux linkage har-monics. The performance of the PI-controller is shown inFig. 12(a). The deadbeat controller performs much betterowing to its higher bandwidth. This is demonstrated in Fig.12(b).

The estimated electromagnetic torque without ripple com-pensation is shown in Fig. 12(c) for comparison. This oscil-logram was recorded while forcing pure sinusoidal statorcurrents by deadbeat current control. Operating at forcedsinusoidal stator voltages basically reproduces the back-emf waveform Fig. 7 as a harmonic torque waveform.

Note that the motor used for the experiments is an inher-ently smooth machine, designed for minimum torque ripple.

8. PRACTICAL IMPLEMENTATION

A standard 80C166 microcontroller system was used forthe implementation in hardware. A 64-kB EPROM is pro-vided to store the object code. The on-chip CAPCOM unitand A/D converters were used for PWM pattern generation

iα

jiβ

–1 0 1

0

1

–1

iα

jiβ

–1 0 1

0

1

–1

iα

jiβ

–1 0 1

0

1

–1

iα

jiβ

–1 0 1

0

1

–1

a)

1

2

3

4

t

t

t

t

b)

c)

d)

Fig 10: Recorded stator current trajectories at steady-state, compen-sation of flux linkage harmonics. Each trajectory relates to a quarterrevolution of the 8-pole machine

1000 200 500 ms300 400t

84

90

96

87

93

rpm

w

1000 200 500 ms300 400t

75

w

compensation on

71

83

79

67

rpm

compensation on

Fig. 11: Effect harmonic compensation; (a) compensation of slotharmonics, (b) compensation of flux linkage harmonics.

- 11 -

and feedback signal conversion. Pulsewidth modulation wasprogrammed in software. The sampling rate for currentcontrol was set to 10 kHz, a hardware limit for the responsetime of the current control loop. A higher bandwidth can beobtained using a DSP system.

9. SUMMARY

Residual torque pulsations in PM synchronous machinesof the sinusoidal flux linkage type impair their perform-ance in critical applications. Torque pulsations are generatedby deviations from the sinusoidal flux linkage distribution,and by stator slotting effects, unbalanced magnetization,and secondary phenomena. The critical ripple frequenciesrange up to 2 kHz.

The design of a fast ripple compensation scheme is based

0.92

1.0

0 π 2πd

1.08

c)

b) 0.92

1.0

1.08

0.92

1.0

a)

1.08

T̂Φ

T̂Φ

T̂Φ

Fig. 12: Estimated machine torque at rated speed (slot ripple notincluded) with compensation of flux linkage harmonics, (a) PI cur-rent controller and feedforward decoupling scheme, (b) deadbeatcurrent controller, (c) without compensation

on a model of the parasitic effects, for which the complexstate-variable approach is used. Its restricted validity, whichextends only to sinusoidal distributions in space, is over-come by expressing space harmonic effects in the timedomain. An automated self-commissioning scheme extractsthe pertinent fundamental and harmonic parameters fromthe drive system. This is indispensable, since even ma-chines from the same production batch may exhibit majordifferences. Variations of the back-emf due to varying magnetproperties are continuously adapted during normal opera-tion. High-bandwidth ripple compensation is enabled usinga deadbeat current controller and current predictor. Theunderlying space vector modulator facilitates data acquisi-tion without additional hardware and ensures low time har-monics. The inherent sampling delay of digital signal process-ing does not limit the effective bandwidth, since the har-monic compensation functions are fully predictable.

10. NOMENCLATURE

us stator voltage vectorysf stator flux linkage produced byis stator current vector

permanent magnetsrs winding resistancels stator inductance tensorys stator flux linkage vectorld, lq d-axis and q-axis inductancet normalized timed angular rotor position

wsR nominal stator frequencyts stator time constantui back-emf vectorTe0 electromagnetic torque excludingui1 fundamental of back-emf vector

harmonicsF normalized flux linkage harmonicsTeR rated torque

tm mechanical time constantTF flux linkage harmonic torquep number of pole pairsTsh slot harmonic torqueuih vector of back-emf harmonicsTL load torqueai filter coefficients of FIR filteru* reference voltage vectorfc cut-off frequencyTs sampling time

11. APPENDIX

The experiments described in this paper where carriedout using the following PM machine:

rated power 1700 Wnumber of poles 8rated speed 3000 rpmmax. speed 6000 rpmrated current 6.8 Amax. current 64 A

- 12 -

rated torque @100˚ C 5.6 Nmmax. torque 52 Nmrated back-emf 148 Vstator resistance 0.9 Ohmd-axis inductance 3.1 mHq-axis inductance 3.4 mHnumber of rotor slots 23

12. REFERENCES

1. T. M. Jahns, “Motion Control with Permanent-MagnetAC Machines”, Proceedings of the IEEE, Vol. 82, No.8, Aug. 1994, pp. 1241-1252.

2. M. Lajoie-Mazene, R. Carlson, J. C. Fagundes, “Analysisof Torque Ripple due to Phase Commutation in BrushlessDC Machines”, IEEE Ind. Appl. Soc. Ann. Meet., Seattle,1990, pp. 287-292.

3. J. Cros et al, “A Novel Current Control Strategy inTrapezoidal EMF Actuators to Minimize Torque Ripplesdue to Phase Commutation”, EPE European Conf. onPower Electronics and Applications (EPE), BrightonUK, 1993, Vol. 4, pp. 266-271.

4. J. Holtz, “Pulsewidth Modulation – A Survey”. IEEETransactions on Industrial Electronics, Oct. 1992, Vol.39, No. 5, pp. 410-420.

5. Y. Murai, T. Watanabe, H. Iwasaki: Waveform DistortionCorrection Circuit for PWM Inverters with SwitchingLag-Times. IEEE Transactions on Industry Applications,Vol. 23, No.5, 1987, pp. 881-886.

6. Kempkes, “Improvement and Assessment of ParasiticSpeed Variations in High-Precision Permanent MagnetSynchronous Motors” (in German), Ph.-D. Thesis,Aachen, 1991.

7. Marinescu, “Influence of Magnet Pole Size and AirgapDimensions on Torque Ripple in Permanent MagnetSynchronous Machines” (in German),etz-Archiv, Vol.10, No. 3, 1988, pp. 83-88.

8. H. Grotstollen, “Suppression of Torque Harmonics inSynchronous Motors by Injected Current Harmonics”,Archiv für Elektrotechnik, Vol. 67, 1984, pp. 17-27.

9. H. Le Huy, R. Perret, R. Feuillet: “Minimization ofTorque Ripple in Brushless DC Motor Drives”, IEEEInd. Appl. Soc. Ann. Meet., Toronto, 1985, pp. 790-797.

10. E. Favre, L. Cardoletti, M. Jufer: “Permanent-MagnetSynchronous Motors: A Comprehensive Approach toCogging Torque Suppression”, IEEE Trans. on Ind.Appl., Vol. 29, No. 6, Nov/Dec 1993, pp. 1141-1149.

11. Le-Huy, “Torque Characteristics of Brushless DC Motorswith Imposed Current Waveforms”, IEEE Ind. Appl.Soc. Ann. Meeting, Denver, 1986, pp. 176-181.

12. Amrhein, “Motor – Electronic Control – Precision ofRotation: Contributions for the Design of PermanentMagnet Excited Drive Systems with Minimum ParasiticSpeed Variations (in German)”, Verlag der Fachvereinean den Schweizerischen Hochschulen und Techniken,Zürich, 1989.

13. J. Y. Hung, Z. Ding, “Design of Currents to ReduceTorque Ripple in Brushless Permanent Magnet Motors”,IEE Proceedings, Vol. 140, No. 4, July 1993, pp. 260-266.

14. B.-J. Brunsbach, G. Henneberger, Th. Klepsch, “Com-pensation of Torque Ripple”, Electrical Machines andDrives Conf., Oxford, 1993, pp. 588-593.

15. H. Le-Huy, K. Slimani, P. Viarouge: “A PredictiveCurrent Controller for Synchronous Servo Drives”.European Conference on Power Electronics and

Applications (EPE), Florenz, 1991, pp. 2.114-2.119.16. J. Holtz, “The Representation of AC Machine Dynam-

ics by Complex Signal Flow Graphs”. IEEE Transac-tions on Industrial Electronics, Vol. 42, No. 3, 1995,pp. 263-271.

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Figure Captions

Fig. 1: Complex signal flow graph of a PM machine; p:number of pole pairs

Fig. 2: Effects of nonsinusoidal stator flux distribution; (a)and (b) complex function F (d) of the flux linkage harmo-nics, (c) two-axes components of the complex functionF (d) in rotor coordinates, (d) harmonic spectrum of F (d)in rotor coordinates

Fig. 3: Performance comparison of current controllers, (a)PI-controller with feedforward decoupling, (b) deadbeatcontroller; sampling frequency 10 kHz

Fig. 4: Mechanical construction of the Mavilor Discodynac servomotor SE 808

Fig. 5: Identification of the stator inductances following avoltage step input, (a) d-axis response, (b) q-axis response,lower trace: sampling signal

Fig. 6: Timing diagram; the k-values number the samplinginstants

Fig. 7: Components of the rotor induced back-emf; (a)measured waveform after transformation to rotor coordinates,superscript (D) indicates distorted function; (b) rotor inducedback-emf after nonrecursive filtering, compared with theaccurate waveform obtained off-line on a test bench

Fig. 8: Identification of slot harmonic torque, (a) q-axiscurrent versus rotor position angle, (b) extracted slotharmonics

Fig. 9: Signal flow diagram of the speed and torque controlsystem (d in degrees electrical)

Fig 10: Recorded stator current trajectories at steady-state,compensation of flux linkage harmonics. Each trajectoryrelates to a quarter revolution of the 8-pole machine

Fig. 11: Effect harmonic compensation; (a) compensationof slot harmonics, (b) compensation of flux linkage harmonics.

Fig. 12: Estimated machine torque at rated speed (slot ripplenot included) with compensation of flux linkage harmonics,(a) PI current controller and feedforward decoupling scheme,(b) deadbeat current controller, (c) without compensation