Eddy Current Effects on Rotor Position Estimationfor Sensorless Control ofPM Synchronous Machine

Jiangang Hu and Longya Xu, Fellow IEEEDepartment of Electrical and Computer Engineering

The Ohio State UniversityColumbus, OH 43210, U.S.A.

hu. 1 58gosu.edu, xu. 12gosu.edu

Jingbo Liu, Member IEEEAdvanced TechnologyRockwell Automation

Milwaukee, WI 53204, U.S.A.jliu2gra.rockwell.com

Abstract- High-frequency current responses of a PMsynchronous machine with and without considering theeddy current effects are significantly different. Thispaper identifies and investigates the eddy current effectson rotor position estimation. Compensation method isdeveloped to improve high-frequency-injection-basedsensorless control of a PM synchronous machine at zeroand low speeds. The estimation and sensorless controlmethod, considering eddy current effects, has beenverified by experimental results.

Keywords- eddy current effects; rotor position estimation;sensorless control; PM synchronous machine

I. INTRODUCTIONRecently, considerable attention has been given to the

sensorless control of synchronous machine drives, especiallyto permanent magnet (PM) synchronous machines. As wellknown, most sensorless methods fail at low and zero speedsbecause the rotor position estimation fundamentally relies onthe back-EMF or speed dependent flux variables. Rotorestimation by high-frequency carrier voltage injection (UFI)to track rotor saliency has been successful in applicationsneeding zero speed starting (zero speed torque). Althoughsignificant research progress on the sensorless controlscheme of PM synchronous machines has been reported inthe literature, few publications so far have addressed theissues related to the effects of eddy current on the rotorposition estimation for sensorless control [1]-[6].

This paper investigates the eddy current effects andcontributes to the improved modeling and implementation ofrotor position estimation and sensorless control based onUFI. It is shown that the rotor estimation results of a PMsynchronous machine with and without considering the eddycurrent effects are significantly different. The improved UFIcontrol scheme is developed with compensation for the eddycurrent effects. Simulation results are presented anddiscussed. Experimental results show that the proposedapproach with compensation has significantly improvedsensorless control of PMSM in standstill and at very lowspeeds.

II. MODELING OF PM SYNCHRONOUS MACHINECONSIDERING EDDY CURRENTS

A PM synchronous machine is generally modeled as asynchronous machine with some rotor saliency. The eddycurrents can be modeled as the currents circulating through apair of short-circuited coils along the d- and q-axes [7], [8].Fig. 1 shows the basic structure of a PM synchronousmachine with shorted coils emulating the eddy currenteffects. The d-axis is aligned with the N-pole of the rotorand q-axis 90 degree apart from the d-axis.

The voltage and flux linkage equations in the stationary

d axis 7N0)

q axis

d axis shorted coil

q axis shorted coil

-aas

S

Cs $

Figure 1. PM synchrounous machine with shorted coils accounting foreddy currents

reference frame are given by

va3 = Rs iOCS + dtic3dtv~ =R i +

d

Vdqr Rr idqr + dt dqr

Aa8s = Ls (20) -iajs + Lsr(9)*idqr + Am (9)

(1)

(2)

(3)

(4)2dqr LTr (o). ia!s + Lr (o) idqr + 2m (0)

where

20341-4244-0365-0/06/$20.00 (c) 2006 IEEE

F 3 3L+ L+ L2L cos(20)

Ls (2&0 2 2s3 L2 sin(20)3L2 sin(20)

3 3L1 + L -L cos(20)is2Os2 1s

Lds sin0 Lrsin.0 , andLs,6() =

L Ldrcos60 Lq Cosaj

L Ldlr +Ldmr0

Lqlr +LLqmr

Without considering eddy current effects, the q- and d-axes currents in the related short-circuited coils in Eqs. (1)through (4) will be zero. Hence the voltage equationswithout considering eddy current effects are reduced to Eqs(5) and (6).

However, in the 14Ff estimation method, a set of highfrequency voltages, viewed as a voltage vector and notsynchronized with the rotor rotations, is always applied.Therefore, substantial eddy currents will be induced on thestator and rotor cores. It is conceivable that the eddy currents

v = R* i + d

2a~L (20W ~ sinL,t(20)-igA+mUOSk11a13 mLCos j

Lsincot)V alsS HFI = VHFI csni

-Za,s HFI =LS(2fr HFI )'iafs HFI +Lsr(r HFI) idqr HFI jvas HFI dt

ca/3s HFI= [Ls2Or HFI)] J Va/s HFI dt o(2)rHFI Lsr HFI) idqr HFI i=a/s HFI

_dqrHFI FI / dqr HFI

[LS 26r HFI)] [Aafls HFI -LS (26r HFI) ia/s HFI]

(5)

(6)

. F7>dqr_HFI

(7)

(8)

(9)

(10)

VHFI LIS

w1XLls (Lls + 3Los )L+-LOS cos(o)jt)- 2 L2S COS(26r HFI -ct

3-LO5 )sin\ )it)-2- 'a/is HFI+ 2 Los sin(cot) - -L2s Slfl(2n0 HFI - co,tj

it VHFIa)6s HFI ViLIS (LlS + 3LoFIL) LIS

3+ L0Os2

) cos(wjt) - -L2 CoS(26r2 2

+ LO sin(a)it)- L2Ls CoS 2r2 )O 2

Demodulation

High Pass Filter (HPF)

Low Pass Filter(LPF)Compensator

I I I20LAEddy Currenxt 74 t .\ 2i l rHF IXjaWk_eqII ,

pOEf cs 4 eIc '~HFI

io ompensato o

Figure 2. Rotor position estimation with eddy current effects compensator based on high frequency injection method

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HHI-O)(1 1)

1, 0

'b 0il 10

Figure 3. Eddy current effects compensation

will be reflected into the stator windings and cannot beneglected. As an approximation, a PM synchronous machinewith short-circuited coils on the rotor can be utilized toaccount for eddy current effects. The improved rotor positionestimation and sensorless control scheme based on UFIconsidering the eddy current effects are shown in Fig. 2. Theinjected high frequency voltages are superimposed over thefundamental excitation and a signal demodulator is used toextract the rotor position information.

Note that the voltage and flux linkage equations for thePM synchronous machine with shorted coils are two ordershigher and more complicated than those for a conventionalPMSM. The additional components accounting for the eddycurrent are also rotor position dependent. Therefore, theeddy current effect compensation is needed to modify theconventional UFI-based method for rotor position estimation.

The injected high frequency voltages are expressed in Eq.

raqpha-hfi-ir

Figure 4. Simulation results of a -axis current, 20, HFI and 0, HFI for

the synchronous machine without shorted coils (V OV)

(7). In the high frequency voltage equations, the stator androtor resistance is negligible because for high frequencyinjection, c,L >> Rs . Eq. (8) shows the relationship betweenthe stator flux linkages, stator currents and rotor eddycurrents with high frequency voltage excitation. The statorcurrents can be derived using Eq. (9) and the componentsrepresenting the eddy current effects are iq. HFJ shown inEq. (10). With the eddy current effects properly modeledand compensated as shown in Fig. 3, we can obtain thedesired components (ia,s HFI ) described by Eq. (11) that areto be demodulated by the conventional UFI method. Then,the rotor position information can be extracted.

III. SIMULATION RESULTSFor comparison, a PM synchronous machine with and

without considering eddy current effects are simulated usingMATLAB/SIMULINK. To identify and investigate theeffects of eddy current, the rotor speed is maintained at aconstant speed of 50 rpm and stator windings are excitedwith a high frequency voltage of 500 Hz. Fig. 4 shows thesimulated phase current, the estimated rotor angle 20 HFJand . HFI for the PM synchronous machine without shortedcoils. Fig. 5 shows the similar results based on the dynamicequations ofPM synchronous machine considering the eddycurrent effects. As indicated by the high frequency currentenvelopes, the current responses in both Figs 4 and 5 containobvious information of the rotor positions. However, carefulexamination of the current responses also reveals the majordifferences in these two cases. In particular, the amplitude ofhigh frequency current in Fig. 4 is modulated by the rotorsaliency strongly and it is quite easy to identify the rotorposition for sensorless control. Nevertheless, in Fig. 5, boththe amplitude and phase angle of the phase current envelopare seriously interfered by the eddy current effects. As aresult, the rotor position estimation has a difference of 48degrees compared to that of the case in Fig. 4. Afteridentifying the eddy current effects, we can properly design acompensation scheme for the eddy current effects so as toextract the correct rotor positions, as indicated in Fig. 3.

hhh

Figure 5. Simulation results of a -axis current, 20, HFI and 0, HFI for

the synchronous machine with shorted coils (V OV)

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-T

4

2

0 0.2 0.4 0.6 0.8 1.2 1.4 1.6 1.8 2

th.t.-t

6- --

4

2

0

-T4 ) I-

2 i- - T-/ - - - - -

0 0.2 0.4 0.6 0.8 1.2 1.4 1.6 1.8 2

th.t.-t

6 --

4

2

0-

B: 2========.5 s1.00 V

C=3=====---==[C. 3=====.5 s1.00 V

:4========.5 s10.0mv-13.12mV

.5 s

1 .1 V2 .1 vE .1 V4 10 mv

BNLDC loDC loDC lo50Q

At 893.1 ms 't 1.1197 Hz

DC 4.29 V10 KS/s

STOPPED

Figure 6. Low speed (+1Hz) waveforms without load (top trace is HF, middel is 20 HFI and bottom trace is the phase A current (lOA/Div))

i:3=====1 s

1.00 V

D:i4======== = Y

1~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ s

1 s BNL

1..1 V DC ix52 .1 V DC 53.1 V DC .x DC 4.61 V

20 mV 50Q l STOPPEDFigure 7. Low speed (+1.25Hz) waveforms with load (top trace is HFI and bottom trace is the phase A current (20A/Div))

IV. EXPERIMENTAL RESULTSExperimental verification of rotor position estimation

including eddy current effects has been accomplished basedon a six-pole 5 UP PM synchronous machine. Theverification has been focused on for the PM synchronousmachine at zero and low speeds. In the experiment, the rotor

of the machine is commanded rotating at very low speedsand subsequently reverses its rotating direction. Fig. 6 showsthe estimated rotor position waveforms of S HFJ and 2 HFI

with the phase current in a very low speed range (±20rpm) inno load condition. Fig. 7 shows the estimated rotor positionof HFI and phase current with load (±25rpm). Fig. 8

shows the experimental results when the PM synchronous

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-T- l-- T--

Mln

machine is running at a constant low speed (45rpm) withregulated currents of its fundamental frequency. The resultsshow that the rotor position obtained through the proposedscheme works stably in the very low speed region.

D:4=========.2 s10. 0 Av-4.37mV

.2 s1.00 V

.2 s1.00 V

-781 mV

.5 s BhL1 .1 V DC x12 .1 v DC 103 .1 V DC x

E 20 mV 50Q

In the experimental testing, the rotor position iscompensated with the proposed eddy current effects as

shown in Fig. 9. In the figure, ,r encoder is the real rotor

position measured by the encoder. The rotor position

At 221.9 ms 1At 4.507 Hz

-F DC 3.94 V10 KS/s

[ STOPPED

Figure 8. Low speed (2.25Hz) with fundamental current waveforms (top trace is 20 HFI!' middel is HFI and bottom is the phase A current (lOA/Div))

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

-0.1

-*htheta r HFI

error(theta_r_HFI-theta_r_encoder)error(thetar_H F Icps-theta_r_encoder)

theta_r encoder

-theta_r_HFl_cps

Figure 9. Rotor position compensation

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I-

/ I I 1~ /1 /H} /

I

recorded by the encoder is considered a reference forcomparison. r HFI is the estimated rotor position usingUFI with eddy current compensation while HFI withoutconsidering eddy current effects. As shown clearly, withoutcompensation, the rotor position estimation error is as largeas about 50 degrees, indicating the necessity of consideringeddy current effects.

V. CONCLUSIONS

High-frequency current responses of a PM synchronousmachine with and without considering the eddy currenteffects are considerably different. This paper identifies andinvestigates the eddy current effects on rotor positionestimation and contributes to an improved high-frequency-injection-based sensorless control of a PM synchronousmachine at zero and low speeds. Both computer simulationand experimental results have proven that the proposed highfrequency injection method with compensation works verywell. It should also be pointed that for a conventionalsalient-pole synchronous machine, the proposed rotorposition estimation and sensorless control approach in thispaper is valid. This is because the currents in the damperwindings of a conventional salient-pole synchronou machinefunction identically as those eddy currents during the UFImode.

REFERENCES

[1] Takashi Aihara, Akio Toba, Takao Yanase, Akihide Mashimo andKenji Endo, "Sensorless torque control of salient-pole synchronousmotor at zero-speed operation", IEEE Trans. on Power Electronics,Vol. 14, No. 1, pp. 202-208, Jan. 1999.

[2] Holtz, J., Hangwen Pan, "Acquisition of rotor anisotropy signals insensorless position control systems," IEEE Trans. on IndustryApplications, Vol. 40, No. 5, pp. 1379-1387, Sept./Oct. 2004.

[3] S. Morimoto, K Kawamoto, M. Sanada, Y. Takeda, "Sensorlessconreol strategy for salient-pole PMSM based on extended EMF inrotating reference frame," IEEE Trans. On Industry Applications,Vol. 38, No. 4, pp. 1054-1061, Jul./Aug. 2002.

[4] Linke, M., Kennel, R., Holtz, J., "Sensorless speed and positioncontrol of synchronous machines using alternating carrier injection,"Electric Machines and Drives Conference, 2003. IEMDC'03. IEEEInternational, Vol. 2, pp:1211- 1217, June 2003.

[5] Matthew J. Corey and Robert D. Lorenz, "Rotor position and velocityestimation for a salient-pole permanent magnet synchronous machineat standstill and high speeds", IEEE Trans. on Industry Applications,Vol. 34, No. 4, pp. 784-789, Jul./Aug. 1998.

[6] Linke, M., Kennel, R., Holtz, J., "Sensorless position control ofpermanent magnet synchronous machines without limitation at zerospeed," IECON 02, Industrial Electronics Society, IEEE 2002 28thAnnual Conference of the, Vol. 1, pp:674 - 679, Nov. 2002.

[7] L. Xu, X. Xu, T. Lipo and D. Novotny, "Vector control of asynchronous reluctance motor including saturation and iron losses,"IEEE Trans. On Industry Applications, Vol. 27, No. 54, pp. 977-985,Sept./Oct. 1991.

[8] I. Boldea and S. A. Nassar, "Unified treatment of core losses andsaturation in orthorgonal-axis model of electric machines," Proc. Inst.Elec. Eng., vol. 134, pt. B, no. 2. pp. 355-363, Nov. 1987.

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