Speed-sensorless vector control for permanent-magnetsynchronous motors based on instantaneousreactive power in the wide-speed region

Y.S. Kim, Y.K. Choi and J.H. Lee

Abstract: The paper proposes an implementation of the new sensorless speed-control scheme for apermanent-magnet synchronous motor (PMSM). In the proposed algorithm, the current observerestimates the line currents and the estimated speed can be yielded from the voltage equationbecause the information of speed is included in back EMF. However, the speed-estimation errorbetween the estimated and the actual speed arises because of the measurement errors of the motorparameters and the sensing errors of the line currents and input voltages. Because the parameterssuch as machine inertia or a viscosity-friction coefficient depend on the mechanical load which isnot constant in most applications, there are many restrictions in the actual implementation. Tominimise the speed-estimation error, the estimated speed error is compensated by the instantaneousreactive power. Moreover, to increase productivity, a flux-weakening control algorithm is added tothe proposed sensorless algorithm in the high-speed region. The proposed algorithm improvesrobustness by removing the mechanical equations in the observer and improves performance byadding flux weakening control in the high-speed region.

1 Introduction

The permanent-magnet synchronous motor (PMSM) isgaining much interest for many industrial applicationsbecause of its high torque-to-inertia ratio, superior powerdensity and high efficiency. To efficiently operate thePMSM and generate smooth torque, it is necessary toknow the absolute position of the rotor. This position canbe obtained by using a position encoder or resolvermounted to the motor shaft. However, a position sensorincreases the volume of the motor and, thus, reduces thereliability of the system, in some cases so much as to make itusable. These drawbacks may be eliminated if the rotorposition and speed are estimated by a sensorless algorithm.

In some studies, the rotor-flux quantities which areestimated by integration of the machine terminal voltagesare used to obtain the rotor-angle information [1].

In other studies, the rotor angle is detected directly fromthe terminal-current errors between the actual machinecurrents and the calculated currents on the machine model[2]. The state-estimation algorithm such as a state observeris adopted to estimate the rotor angle and speed [3–7].

The observer has been widely studied in the field of themotion control. The conventional observer is constructedby the voltage and mechanical equation. However, theparameters of the mechanical equation such as the inertia ofthe rotor or a viscosity-friction coefficient are difficult to

measure exactly, so the actual implementation of theobserver is limited.

To cope with the problems, this paper proposes a newsensorless control algorithm for a PMSM without theposition and speed sensor. In the proposed algorithm, theline currents are estimated by an observer and the estimatedspeed can be obtained from the voltage equation becausethe information about speed is included in the back EMF.However, the speed-estimation errors between the estimatedand the actual speeds arises due to measurement errors ofthe motor parameters and the sensing errors of the linecurrents and input voltages. To minimise the speed-estimation error, the estimated speed is compensated byan instantaneous reactive power. Moreover, to increase theproductivity, the speed range must be extended by adding aflux-weakening scheme to the proposed sensorless algorithmfor the high-speed region.

In this paper, the proposed algorithm improves robust-ness by removing the mechanical equation in the observerand improves performance by adding a flux-weakeningcontrol algorithm in the high-speed region. The sensorlessand flux-weakening control algorithms are implemented bythe DSP (TMS320C31) and adopted for an eight-polePMSM.

2 Sensorless control using instantaneous reactivepower

2.1 Estimation of stator currentThe voltage equation of a PMSM in the stationary referenceframe fixed to the stator is given by (1) and (2), magneticsaturation being neglected:

diSD

dt¼ � LS

RSiSD þ

KE

LSxr sin hr þ

1

LSVSD ð1Þ

The authors are with the Department of Electrical Engineering, Inha University,253, Yong Hyun-Dong, Nam-Gu, Inchon, 402-751, Korea

E-mail: yjhum@naver.com

r IEE, 2005

IEE Proceedings online no. 20055231

doi:10.1049/ip-epa:20055231

Paper first received 18th November 2004 and in final revised form 15th January2005. Originally published online: 5th July 2005

IEE Proc.-Electr. Power Appl., Vol. 152, No. 5, September 2005 1343

diSQ

dt¼ � LS

RSiSQ �

KE

LSxr cos hr þ

1

LSVSQ ð2Þ

where, VSD¼ stator D-axis voltage, VSQ¼ stator Q-axisvoltage, iSD¼ stator D-axis current, iSQ¼ stator Q-axiscurrent, xr ¼ rotor speed, hr ¼ rotor position, Rs¼ statorresistance, Ls¼ stator inductance and KE¼ back-EMFconstant.

From (1) and (2), the current observers can be expressedto as

d iSD

dt¼ � LS

RSiSD þ

KE

LSxr sin hr þ

1

LSVSD þ k1eSD ð3Þ

d iSQ

dt¼ � LS

RSiSQ �

KE

LSxr cos hr þ

1

LSVSQ þ k2eSQ ð4Þ

where, iSD¼ stator D-axis estimated current, iSQ¼ statorQ-axis estimated current and k1, k2¼ constant.

eSD ¼ iSD � iSD ð5Þ

eSQ ¼ iSQ � iSQ ð6ÞError dynamics are obtained by subtracting (1) and (2)from (3) and (4), respectively, and represented as (7) and (8):

deSD

dt¼ �RS

LSeSD þ

KE

LSðxr sin hr � xr sin hrÞ þ k1eSD ð7Þ

deSQ

dt¼ �RS

LSeSQ �

KE

LSðxr cos hr � xr cos hrÞ þ k2eSQ ð8Þ

From (7) and (8), when the back-emf estimation errors areminimised and k1 and k2 are determined to stabilise theerror equations, the current-estimation errors can beconverged to zero. The algorithm which minimises speed-estimation errors can be developed.

2.2 Speed estimation based oninstantaneous reactive powerIn the rotating reference frame fixed to the rotor, thevoltage equation of a PMSM can be expressed as

VSD ¼ RS iSd þ LSdiSd

dt� xrLS iSq ð9Þ

VSQ ¼ RS iSq þ LSdiSq

dtþ xrLS iSd þ KExr ð10Þ

From (10), (11) is derived:

xr ¼VSQ � RS iSq � LS

diSq

dtKE þ LS iSd

ð11Þ

If the speed of the rotor is estimated from (11), the speed-estimation error between the actual speed and the estimatedspeed arises because of errors in measuring the motorparameters and in sensing the line currents. Therefore, (11)has to be compensated to minimise the speed-estimationerror. Eqn. (12) compensates (11) and the value C isdetermined by an instantaneous reactive power:

xr ¼VSQ � RS iSq � LS

diSq

dtKE þ LS iSd

þ C ð12Þ

An instantaneous reactive power is defined by the cross-product of the line current iS and the back EMF ES :

qm ¼ iS � ES ð13Þ

where iS ¼ ½iSd ; iSq�ES ¼ ½ESd ;ESq�T

ESd ¼ 0

ESq ¼ KExr

An estimated instantaneous reactive power defined by theestimated current is

qm ¼ iS � ES ð14Þwhere, iS ¼ ½iSd ; iSq�.

In the rotating reference frame fixed to the rotor, if theestimated current has the position-estimation error Dh forthe actual current, (13) and (14) are represented as (15) and(16) from Fig. 1:

qm ¼ iSdKExr ð15Þ

qm ¼ iSdKExr ¼ ðiSd cosDhþ iSq sinDhÞKExr ð16ÞIf xr 6¼ 0 and Dh ffi 0 are satisfied, (16) can be approxi-mated as (17):

qm ¼ ðiSd þ iSqDhÞKExr ð17ÞFrom (15) and (17), the errors in instantaneous reactivepowers are derived as (18) and represented by the termwhich includes the information for the position-estimationerror.

Suppose that the sign of Dh is positive when theestimated current leads to the actual current. The value Chas to be determined by satisfying (19) and (20) and addingthe integral term to accomplish for stable compensation:

Dqm ¼ qm � qm ¼ iSqDhrKExr ð18Þ

Dhr40; KCP iSqDhrKExro0 ð19Þ

Dhro0; KCP iSqDhrKExr40 ð20Þ

C ¼ KCPDqm þ KCI

Z t

0

Dqmdt ð21Þ

where, KCP ;KCI : constant.If xr 6¼ 0 and Dh ffi 0 are satisfied, error dynamics can be

obtained by (22) and (23):

deSD

dt¼ �RS

LSeSD þ

KE

LSðxr � xrÞ sin hr þ k1eSD ð22Þ

deSQ

dt¼ �RS

LSeSQ �

KE

LSðxr � xrÞ cos hr þ k2eSQ ð23Þ

If the speed-estimation error is minimised by (21), (22) and(23) can be represented as (24) and (25), respectively.If k1 and k2 are determined to satisfy inequality (26), the

is is

d

d

∆�

∆�

�r

Fig. 1 Position-estimation error for actual and estimated currents

1344 IEE Proc.-Electr. Power Appl., Vol. 152, No. 5, September 2005

current-estimation errors of (24) and (25) can converge tozero.

deSD

dt¼ � RS

LS� k1

� �eSD ð24Þ

deSQ

dt¼ � RS

LS� k2

� �eSQ ð25Þ

k1oLS

RS

k2oLS

RS

ð26Þ

The block diagram for the proposed algorithm is shown inFig. 2. The algorithm transforms the actual and theestimated current from the stationary reference frame tothe rotating reference frame fixed to the rotor. Eqns. (13)and (14) yield real reactive power and estimated reactivepower, respectively, and then (21) determines the value Cfrom the difference in the reactive powers.

2.3 PMSM control in flux-weakening regionThe input voltage of a PMSM is increased with increasingrotor speed. A PMSM cannot be operated beyond the ratedspeed which corresponds to the rated input voltage of thePMSM without flux-weakening control [8].

In this paper, VSmax is identified as the rated voltageV rated of the PMSM and the maximum current iSmax isdetermined by the rated current of the motor. Also, themaximum voltage VSmax that the invertor can supply to thePMSM is limited by the DC-link voltage.

The voltage and current of the PMSM have to satisfy thelimit.

V2SD þ V2

SQ � V2Smax ð27Þ

i2Sd þ i2Sq � i2Smax ð28Þ

From (9) and (10), when the PMSM is operated at steadystate, the effect of the resistance term can be neglectedbecause the back EMF is much larger than the voltage ofresistance. Therefore, the voltage equation in the d�q frameis

VSD ¼ �xrLS iSq ð29Þ

VSQ ¼ xrLS iSd þ KExr ð30ÞFrom (27), (29) and (30), the condition for iSd and iSq can beexpressed as follows:

iSd þKE

LS

� �2

þi2Sq �VSmax

xrLS

� �2

ð31Þ

This inequality (31) expresses a circle with a radius VSmax=xrLS and its centre is located at ð�KE=LS ; 0Þ as shown inFig. 3.

Figure 3 shows that the voltage-limit circle decreases asthe speed is increased. The stator current satisfying bothconditions of (27) and (28) must be inside the current-limitcircle and the voltage-limit circle.

If maximum speed of constant-torque operation is x1,the stator current is limited by iSmax and the maximumtorque is obtained at point A.

The operating point of iSmax moves from A to B as speedincreases. To increase speed, a PMSM requires increasedinput voltage. If the rotor speed is higher than the ratedspeed, the input voltage has to be higher than the ratedvoltage. However, if d-axis current is injected in thedirection that d-axis flux is decreased in the rated voltage,a PMSM can be driven at a higher speed than the ratedspeed.

Figure 4 illustrates the block diagram of the flux-weakening control algorithm. In flux weakening region, ifthe input voltage exceeds the rated voltage, the controllercalculates the error of the voltage and injects the d-axiscurrent negatively. The q-axis current reference such that

the total current i�S ½¼ ðp

i�2Sd þ i�2SqÞ� does not exceed the

voltage limit

A

B

current limit

iSmax

d

q

KE

LS

, 0)(−

ωr = ω1

Fig. 3 Voltage- and current- limit diagrams for the maximumtorque in PMSM

+vSd

vSdvSq

vSq√

iSq max

0

limit PI

∗iSd ∗

−iS max

−

+

vS max

iSdiSmax −√ 2 2 22

Fig. 4 Block diagram of the flux-weakening control algorithm

current estimation

speed estimation

actual reactivepower

estimated reactivepower

compensation ofspeed-estimation error

co-ordinatetransformation

is

�r

vs

co-ordinatetransformation

C

�r

�r

− +

is

Fig. 2 Algorithm that uses an instantaneous reactive power forspeed and position estimation

IEE Proc.-Electr. Power Appl., Vol. 152, No. 5, September 2005 1345

current-limit value is calculated and used at the speedcontroller as a limit.

3 Experimental results

Figure 5 shows that the overall system consists of a voltagesource inverter, a 1.8kW 8 pole PMSM and a dynam-ometer. The proposed algorithm was implemented by usingthe DSP(TMS320C31) and the overall calculations wereperformed by DSP with the sampling time of 160ms. Athree-phase voltage-source invertor was constructed by theIGBT, and the gate-firing logic to implement the space–vector modulation was developed using the motioncoprocessor (ADMC200). The actual speed and positioninformation were obtained from the encoder only formonitoring purposes. Also, to align the initial position ofthe rotor flux at the stator D-axis where the rotor positionwas zero, the voltage pulse was injected into phase a. Theparameters of the PMSM are listed in Table 1.

Figure 6 shows the actual and estimated speeds and thespeed-estimation error of the proposed algorithm with 60%load. The speed-estimation-error waveform is enlarged tentimes compared with the others.

At the start, the about 100 rev/min maximum speed-estimation errors appeared, but the estimated speed quicklyconverged to the actual speed and, after about 0.4 s, theestimation error of the speed showed that the estimatedspeed matched well the actual speed.

The estimation performance of the proposed algorithmwhen the motor was started with no load and the load waschanged from 0% to 100% at 500 [rev/min] is illustrated inFig. 7.

As soon as the load torque was applied by thedynamometer, the speed decreased. The speed and currentcontrollers provided the torque to converge the speed of therotor to the reference speed. After about 3 s, the speedreached steady state. Therefore, the proposed sensorlesssystem was robust for a disturbance.

Figure 8a shows the actual and estimated speed andspeed-estimation error when the speed was changed from1500 [rev/min] to –1500 [rev/min] with 60% load, andFig. 8b shows the actual and estimated position under thesame conditions. The speed-estimation-error waveform is

enlarged ten times compared with the others. As shown inFig. 8, the speed suddenly varied, but the position wasestimated well by the proposed algorithm.

Figure 9 represents the actual speed and estimated speed,and the actual and estimated position on the proposedalgorithm at speed 50 [rev/min]. To examine the dynamicbehaviour of drives, the speed reference was reversed from50 [rev/min] to –50 [rev/min].

flux-weakening controliSd

vSd

iSa

iSb

SVPWM

iSd

iSqiSQ

vSQ

vSQ

iSD

vSDvSqPI

PI

PI

PI

limit

D − Q

D − Q

D − Q

a − b − c

d − q

d − q

vSD

iSd

ωr

−ωr LS iq

ωr

iSq max iq

vSd

vSq

ωr LS iq−−

−

++

+

+

+

+

+

+

KE ωr INV

PMSM

dynamometer

proposed algorithm

Fig. 5 Block diagram of the proposed algorithm

a

time

b

c

Fig. 6 Actual and estimated speed and speed-estimation errorwhen the PMSM was derived at 1000 [rev/min] with 60% load

Table 1: Motor parameters

Rated power 1.8 kW

Pole number 8 poles

Stator resistance 0.22 O

Stator inductance 0.88 mH

Torque rating 5.88 Nm

Rotor inertia 0.00186 [kgm2]

Back-EMF constant 0.498 V/rads

1346 IEE Proc.-Electr. Power Appl., Vol. 152, No. 5, September 2005

a

b

time

Fig. 7 Line current and estimated speed when the load waschanged from 0 to 100% and returned to 0 at 500 [rev/min]a Line currentb Estimated speed (500 rev/min per division)Time: 2 s per division

atime

(i)

(i)

(ii)

(ii)

(iii)

timeb

Fig. 8 Speed and position when the speed was reversed from 1500[rev/min] to �1500 [rev/min] with 60% loada Actual speed, estimated speed and speed-estimation error(i) Actual speed (1000 rev/min per division)(ii) Estimated speed (1000 rev/min per division)(iii) Speed-estimation error (100 rev/min per division)Time: 1 s per divisionb Actual and estimated position(i) Actual position (500 rev/min per division)(ii) Estimated position (500 rev/min per division)

atime

btime

(i)

(i)

(ii)

(ii)

Fig. 9 Speed and position when the speed was reversed from 50[rev/min] to �50 [rev/min] with no loada Actual and estimated speed(i) Actual speed (50 rev/min per division)(ii) Estimated speed (50 rev/min per division)Time: 200 s per divisionb Actual and estimated position(i) Actual position(ii) Estimated position

a

b

time

Fig. 10 Actual and estimated speed when the load was changedfrom 0 to 34% and returned to 0 at 50 [rev/min]a Actual speed (50 rev/min per division)b Estimated speed (50 rev/min per division)Time: 2 s per division

IEE Proc.-Electr. Power Appl., Vol. 152, No. 5, September 2005 1347

(i)

(ii)

(i)

(ii)

timea

time

b

Fig. 11 Speed and position when the speed was reversed from 20[rev/min] to –20 [rev/min] with no loada Actual and estimated speed(i) Actual speed (50 rev/min per division)(ii) Estimated speed (50 rev/min per division)Time: 200 s per divisionb Actual and estimated position(i) Actual position(ii) Estimated position

a

time

b

Fig. 12 Actual and estimated speed when the load was changedfrom 0 to 25.5% and returned to 0 at 20 [rev/min]a Actual speed (5 rev/min per division)b Estimated speed (5 rev/min per division)Time: 2 s per4 division

a

time

b

time

(i)

(i)

(ii)

(ii)

Fig. 13 Estimated speed, d-axis current and positions when thespeed was reversed from 3450 [rev/min] to –3450 [rev/min] with noloada Estimated speed and d-axis current(i) Estimated speed (2000 rev/min per division)(ii) d-axis currentTime: 4 s per divisionb Actual and estimated position(i) Actual position(ii) Estimated position

spee

d-es

timat

ion

erro

r, %

speed-estimation error for the reference speed

speed, rev/min

050 15001200900600300

1.0

2.0

3.0

4.0

Fig. 14 Percentage estimation errors of the rotor speed in thesteady state

1348 IEE Proc.-Electr. Power Appl., Vol. 152, No. 5, September 2005

Figure 10 shows the speed-estimation performance whichwas obtained when the load was changed from 0% to 34%and returned to 0% at 50 [rev/min].

Figures 11 and 12 show the performances of the speedand position estimation at 20 [rev/min]. The incrementalencoder measures the actual speed and has an output of4096 pulses per revolution per minute. The counter devicecounts four times for each output pulse. During onesampling time at 20 [rev/min] pulses increased by an averageof 0.67. The per-sampling time of 0–1 pulse is increased.Therefore, in Fig. 11, the actual speed has poor resolution.

As shown in Figs. 9 and 11, the control performance bythe proposed algorithm in the low-speed region was lowerthan that in the high-speed region. This result is due to poorobservability of the rotor angle at low speed because of thelack of back EMF under operating condition.

In Figs. 10 and 12, the rotor speed is shown with a stepload at the low speed. The proposed algorithm gave goodestimations at low speed.

Figure 13a shows the estimated speed when the speedwas changed from 3450 [rev/min] to –3450 [rev/min] andFig. 13b shows the actual and estimated positions for thesame condition. When the input voltage exceeded the ratedvoltage, the controller calculated the errors of the voltageand injected the d-axis currents negatively. In the high-speedregion, Fig. 13b shows that the proposed algorithmestimated the rotor position well.

Figure 14 shows the absolute values of the percentageerrors in the rotor-speed estimation obtained experimen-tally. The percentage errors for the estimation of each rotorspeed were excellent except at the low speed.

4 Conclusions

In this paper, a new sensorless algorithm using aninstantaneous reactive power is proposed. In the experi-mental results, although the speed varied suddenly, theproposed algorithm estimated the position very well. Also,

the robustness of the proposed algorithm for load-torquevariations was demonstrated and a flux-weakening controlalgorithm to extend the speed range was applied in the high-speed region.

Because the parameters of the mechanical equations suchas machine inertia or a viscosity friction coefficient aredifficult to measure exactly, actual implementation of thealgorithm was limited. The proposed algorithm improvedrobustness because the mechanical equations in the observerwere removed, and improved performance because aflux-weakening control algorithm was included for thehigh-speed region. The experimental results verified theeffectiveness of the proposed algorithm.

5 Acknowledgments

This work was supported by Inha University ResearchGrant.

6 References

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4 Sepe, R.B., and Lang, J.H.: ‘Real-time observer-based (Adaptive)control of a permanent-magnet synchromous motor without mechan-ical sensor’, IEEE Trans., 1992, IA-28, (6), pp. 1345–1352

5 Furuhshi, T., Sangwongwanich, S., and Okuma, S.: ‘A position-and-velocity sensorless control for brushless DC motors using an adaptivesliding mode observer’, IEEE Trans., 1992, IE-39, (2), pp. 89–95

6 Dhaouadi, R., Mohan, N., and Norum, L.: ‘Design and Implementa-tion of an extended Kalman filter for state estimation of a permanentmagnet synchronous motor’, IEEE Trans., 1991, PE-6, (3), pp. 491–497

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