Position Sensorless Control of Interior Permanent Magnet Synchronous MotorUsing Extended Electromotive Force

KOJI TANAKA and ICHIRO MIKIMeiji University, Japan

SUMMARY

Driving a permanent magnet synchronous motor(PMSM) requires the rotor position information to controlthe motor torque, and this is generally detected by mechani-cal position sensors such as an encoder or a resolver. How-ever, these sensors increase the machine size and the costof the drive, and reduce reliability of the system. Therefore,many papers about position sensorless drive method ofPMSM have been published. This paper presents a positionsensorless control of interior permanent magnet synchro-nous motor (IPMSM). A mathematical model of IPMSMusing the extended electromotive force (EMF) in the rotat-ing reference frame is utilized to estimate the rotor speedand position. This model has a simple structure integratingposition information into the extended EMF term. There-fore, the sensorless control based on the mathematicalmotor model can be implemented simply. The estimationmethod proposed is based on the principle that the error ofthe current is proportional to that of extended EMF. Thismethod was carried out using a 6-pole, 400-W, 1750 r/mintest motor system. It was found that sensorless speed con-trol was achieved from 80 r/min to 1800 r/min under 0 to100% loads. © 2007 Wiley Periodicals, Inc. Electr Eng Jpn,161(3): 41–48, 2007; Published online in Wiley Inter-Science (www.interscience.wiley.com). DOI 10.1002/eej.20406

Key words: permanent magnet synchronous mo-tor; sensorless control; position estimation; speed estima-tion.

1. Introduction

Permanent magnet synchronous motors (PMSM) arewidely used in industry at present because of advantagessuch as their being small and highly efficient, and beingeasy to control and maintain. PMSM are classified intosurface magnets and interior magnets based on the position

of the permanent magnet used in the rotor [1]. An interiorpermanent magnet synchronous motor (IPMSM), whichhas a structure in which the magnet is embedded in the rotor,has a saliency in the inductance, and can be expected toincrease efficiency and output through the use of a reluc-tance torque, and so has garnered a lot of attention.

Torque control in a PMSM requires control of thecurrent, depending on the rotor magnetic pole position. Asa result, ordinarily an encoder, resolver, or other positionsensor is used. However, given the need to improve envi-ronmental durability and reliability, and to reduce cost,there has been considerable research on position sensorcontrol [1–9]. Estimation methods for the magnetic poleposition and speed in an IPMSM often involve using theinduced voltage (speed electromotive force) or usingchanges in the inductance due to the saliency of the rotor[1]. The latter has frequently been proposed to estimate inan equivalent fashion the difference in the inductance afterintroducing a high-frequency signal [3, 4], and can takeestimates even during low-speed operation or when opera-tion has been stopped. However, this special signal must besuperimposed on the input signal, and this can cause vibra-tion, noise, and torque ripple in the machine [1]. On theother hand, methods to estimate the induced voltage basedon a mathematical model of the machine are commonplace[2, 3, 5–8], though difficulty in estimation and other prob-lems exist due to the induced voltage during low-speedoperation or when operation has been stopped dropping oreven reaching zero. Moreover, the machine model for anIPMSM is complicated compared to that of a surface per-manent magnet-type machine, and as a result approxima-tions that partially ignore saliency are often used. Problemssuch as a deterioration in control performance in machineswith a high saliency ratio have also been pointed out [6].

The estimation method proposed in this paper in-volves the use of the induced voltage in the motor, based ona motor model [5–8] which uses an extended inducedvoltage. In this model, the position information does notbecome complex, because the items in the extended in-

© 2007 Wiley Periodicals, Inc.

Electrical Engineering in Japan, Vol. 161, No. 3, 2007Translated from Denki Gakkai Ronbunshi, Vol. 125-D, No. 9, September 2005, pp. 833–838

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duced voltage are limited, and so can achieve sensorlesscontrol using a relatively simple setup. The sensorlesscontrol method using the extended induced voltage modelhas been proposed several times. Methods that use a modelon rotating coordinates have been attempted in addition tomethods to create a low-dimension observer using a modelon fixed orthogonal coordinates and estimate the extendedinduced voltage. Methods that use an observer and methodsthat directly calculate the extended induced voltage basedon mathematical formulas have also been proposed. All ofthese methods are in theory clearer compared to methodsthat use the conventional model, and they are simpler.

The principle of the estimation method is the use ofa property which states that the difference between thecurrent estimated based on the motor model using theextended induced voltage and the real current is propor-tional to the error in the estimated extended induced volt-age. Reference 2 recognizes this method as a way toestimate the induced voltage based on the error in thecurrents in the model, but extracting the salient positioninformation for use in the conventional model on the esti-mated rotating coordinates is problematic, and so approxi-mations are made by partially ignoring saliency byassuming the salient position error to be zero. Because theestimation method does not ignore saliency, there is no dropin control performance, even in machines with a highsaliency ratio, and an estimation precision equivalent to orgreater than that of conventional methods [2] can be ex-pected. Moreover, the estimation algorithm is clear andsimple, and the vector angle for the extended inducedvoltage shows the salient position information. As a result,the salient position information can be required obtained byestimating the extended induced voltage using the charac-teristics described above. This method is in theory simple,and its application to real systems is straightforward, mean-ing that it is a practical method.

When the authors performed experiments on theirestimation method using a 6-pole, 400-W, 1750 r/minIPMSM, they confirmed that normal operation occurredwithout incident over a range of 80 r/min to 1800 r/minunder a rated load of 0% to 100%. Moreover, good transientcharacteristics were obtained. Note that because this esti-mation method uses the induced voltage of the motor, stableestimation at low speed is problematic. The authors havepreviously reported in Ref. 9 on the estimation of salientposition when a motor has been stopped.

2. The Principle of Estimation

2.1 Motor coordinate system

Figure 1 shows the coordinate system for theIPMSM. The α–β coordinate axes represent the two-phase

orthogonal coordinate axes for the stator. The d axis followsthe direction of the magnetic pole of the rotor, and the anglefrom the α axis (electrical angle) is designated θ. Further-more, γ follows the direction of the estimated magneticpole for the controller. The angle from the α axis isdesignated θM, and the error with the d axis is ∆θ (= θ –θM).

2.2 The machine model using the extendedinduced voltage

The machine model on the rotating coordinates is ingeneral given by the equation

Here, vd and vq represent the d and q axis components ofthe rotor voltage; id and iq, the d and q axis components ofthe rotor; ω, the rotor angular velocity (electrical angle); R,the winding resistance; Ld and Lq, the d and q axis induc-tance; KE, the induced voltage constant; and p, the differ-ential operator (d/dt).

As can be seen in the first term on the right in Eq. (1),the IPMSM has saliency on the rotor, and the inductancecomponent for the d and q axes is asymmetric. Convertingthe above equation to a γ–δ estimated rotating coordinatesystem yields

Here, vγ and vδ represent the γ and δ axis components of thearmature voltage; iγ and iδ, the γ and δ axis components ofthe armature current, and ωM, the γ–δ axis rotational speed,

Fig. 1. Definition of coordinates.

(1)

(2)

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As can be seen in the equations above, the effect ofan axial error appears in the second and later terms, and theeffect of a speed error appears in the final term on the rightside. For a surface permanent magnet synchronous motor,Ld = Lq is assumed. As a result, the third and fourth termson the right side of the equation above are 0. An IPMSMhas saliency in the inductance. As a result, the model for theestimated coordinates is complex, as can be seen in thedescription above. In estimation methods that use the con-ventional model, extracting position information is exceed-ingly difficult. Consequently, a simplified model [2, 3] inwhich the third and later terms on the right side of Eq. (2)are ignored by assuming that the position error is suffi-ciently small is often used. However, this leads to thepossibility of a deterioration in control performance [6].

Thus, Eq. (1) is rewritten so that the saliency, that is,the difference in the d, q axis inductance, is concentrated inthe term for the induced voltage. The inductance compo-nent in the first term on the right side is made symmetric,and the difference in the d and q axis inductance is keptwithin the induced voltage item. This results in the follow-ing equation [5–8]:

Here, vdq = [vd vq]T (T refers to the transposed matrix), idq

= [id iq]T, edq = [ed eq]

T, and

Here, Eq. (4) is referred to as the extended induced voltage[5]. Converting Eq. (3) to a γ–δ estimated rotating coordi-nate system results in the following equation [6–8]:

Here, vγδ = [vγ vδ]T, iγδ = [iγ iδ]

T, and eγδ = [eγ eδ]T. Also,

Compared to the conventional estimated rotating coordi-nates model shown in Eq. (2), the above equation, which

uses the extended induced voltage, is very simple. More-over, in the parentheses in the first term on the right of Eq.(6), the magnitude of the d, q axis inductance is roughly thesame (the saliency ratio is thought not to reach 10 at mostordinarily, and a special machine with a saliency ratioaround 15 is not addressed here). As a result, during normaloperation if the time differential (p∆θ) for the position erroris ignored because it is assumed to be sufficiently smallcompared to the axial rotation speed, then the informationrelated to the rotor position converges into one term in theextended induced voltage in the second term on the right,and the equation becomes extremely clear. In addition, ascan be seen in Eq. (7), the vector angle for the extendedinduced voltage represents the magnetic pole position error.Consequently, the information related to the magnetic poleposition can be found by estimating the extended inducedvoltage. Note that because ∆θ = θ – θM, this is equivalentto the time differential (ω – ωM) for the position error.Therefore, the above approximation can be taken as mean-ing ω = ωM. This approximation has an effect when speedrapidly drops during low-speed operation. This will beanalyzed later.

2.3 Principle of estimation

If the approximation described above in Eq. (6) isused, is rewritten using the Euler approximation as anequation for a discrete system with sampling period T, andthen solved for the current at the sample point n, thefollowing equation results:

Here, ωM(n − 1) is the rotational speed of the γ–δ axis at thesample point (n – 1).

On the other hand, the estimated current calculatedusing the controller can be represented with the followingequation using the estimated extended induced voltageeMgδ(n − 1) at the sample point (n – 1):

Therefore, the error between the real current in Eq. (8) andthe estimated current in Eq. (9) is given by

(3)

(6)

(7)

(8)

(9)

(10)

(4)

(5)

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Note that

According to Eq. (10), the estimated current error and theestimated extended induced voltage error are proportional.Thus, the estimated extended induced voltage can be com-pensated as shown below using the difference between thesampled real current and the estimated current calculatedfrom Eq. (9):

Based on the relationship in Eq. (7), the estimated positionerror can be found from the equation below using theestimated extended induced voltage obtained from theequation above:

2.4 Estimation of speed and position [5–8]

Speed and position can be estimated as shown in Fig.2. First, the speed is adjusted using the PI compensatorshown in the equation below so that the estimated positionerror obtained from Eq. (13) is zero.

Here, Kp is the proportional gain, and Ki is the integral gain.This is used for speed control after it is passed through alow-pass filter. Next, the position is estimated by integrat-ing the speed as follows:

Here, the equivalent block diagram shown in Fig. 3 can bedrawn for speed and position estimation. Therefore, thetransmission function is given by

Based on the above equation, the transmission charac-teristics show a second-order response. As a result, the gainfor the PI compensator can be set by using this as a metric.Here, the gain is determined by using the attenuation coef-ficient ζn and the normal mode of vibration ωn shownbelow:

The attenuation coefficient ζn affects the form of the re-sponse. When it is below 1, oscillation results, and when itis greater than 1, convergence results. The normal mode ofvibration ωn affects the speed of the response.

3. Experimental Results

3.1 Experimental setup

The authors performed experiments using a 400-W,1750 r/min, 2.18 Nm IPMSM. Figure 4 shows the setup ofthe system, and Table 1 lists the various parameters of thecontroller. The “*” in the figure indicates a command value.The current is converted to voltage by a current sensor, andthen quantized by a 12-bit A/D converter. The voltage usedin the position error estimation is the command values vγ

∗

and vδ∗ resulting from the current controller.

3.2 Position and speed estimation

Figure 5 shows the estimated speed and position at1750 r/min, and Fig. 6 shows a comparison of the estimatedvalue and the measured values for the mechanical speed andd axis position at 100 r/min during a rated load. Based onFig. 5, the estimated speed and the estimated position at1750 r/min operation resulted in values close to the meas-ured values. Moreover, as can be seen in Fig. 6, there is alarge ripple in the estimated speed at 100 r/min. This isthought to be a deterioration of the estimation precision dueto the induced voltage becoming smaller at lower speeds.The authors confirmed normal operation from 80 r/min to1800 r/min at a rated 100% load. However, the maximum

(11)

(12)

(13)

(14)

(15)

Fig. 2. Estimation of speed and position.

Fig. 3. Equivalent block diagram of speed and positionestimation.

(16)

(17)

(18)

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estimated position error was about 20 degrees, and theestimation precision worsened as the speed dropped.

Figure 7 shows the transient characteristics for whena speed command in a step form was given for 1500 r/min→ 1800 r/min → 1500 r/min. The estimated speed errorwas 20 r/min, and the estimated position error was about 10degrees, representing good responses. Figure 8 shows theestimated values and measured values for the speed whenaccelerating to 1000 r/min from a normal operating state of100 r/min, and also shows the estimated position error. InSection 2.2, the authors performed an approximation inwhich the time differential for the position error in Eq. (6)was ignored as sufficiently small compared to the axialrotating speed. The effect of this approximation appearsduring low-speed operation and rapid acceleration or decel-eration. In Fig. 8, the speed error was 40 r/min at a maxi-mum, and the estimated position error was 15 degrees atmost. The effects of the approximation are clearly visiblecompared to medium or high speeds as shown in Fig. 7.However, no significant effects on operation were seen.

Figure 9 shows the response when a load torque wasapplied in increments of 0% → 100% → 0% at rated speed.The speed error when the load was varied was 10 r/min at

Fig. 4. Block diagram of the proposed sensorless control system.

Fig. 5. Estimated speed and position (100% load, 1750 r/min).

Table 1. Parameters of controller

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a maximum, and the position error was 9 degrees at most,representing good response characteristics.

Note that when the authors attempted an experimentusing a method [2] that employed the conventional model,precision roughly equivalent to their proposed method wasobtained. This seems to mean that no major differenceappeared, and that the saliency ratio for the test motor wasat most about 1.5.

Moreover, the authors’ estimation method is thoughtto have fluctuations that influence the estimations as a resultof calculations performed using the motor parametersshown in, for instance, Eq. (9). When the authors performedexperiments by varying the parameters for the windingresistance and inductance of the controller by ±25%, no

Fig. 6. Estimated speed and position (100% load, 100 r/min).

Fig. 7. Responses for step change of speed reference.

Fig. 8. Responses for step change of speed reference atlow speed. Fig. 9. Responses for step change of load torque.

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significant difference was seen in the test motor. This resultrepresents a topic for the future.

4. Conclusion

The authors proposed a position sensorless controlmethod for an interior permanent magnet synchronousmotor. They used a simple mathematical model with ex-tended induced voltage and created a sensorless controlsystem with a relatively straightforward structure. Thisestimation method takes advantage of the fact that thedifference between the real current and the model currentis proportional to the estimated extended induced voltageerror. The estimation algorithm is also simple and clear evenwhile taking saliency into consideration. The method canalso be seen as simple and practical for use in real systems.Based on experiments, the authors confirmed operationfrom 80 r/min to 1800 r/min at a rated load of 100%. Also,transient characteristics were good. Future topics includethe effects that varying the parameters of the motor have onthe estimations.

REFERENCES

1. Rotating Motor Technology Committee, IndustrialApplications Division, Institute of Electrical Engi-neers of Japan. Electric motor and control systems forreluctance torque. IEEJ Tech Rep, No. 719, 1999.

2. Takeshita T, Ichikawa M, Lee J, Matsui N. Back EMFestimation-based sensorless salient-pole brushless

DC motor drives. Trans IEE Japan 1997;117-D:98–104. (in Japanese)

3. Takeshita T, Usui A, Matsui N. Sensorless salient-pole PM synchronous motor drives in all speedranges. Trans IEE Japan 2000;120-D:240–247. (inJapanese)

4. Ogasawara S, Kurokawa H, Akagi H. A position-sen-sorless IPM motor drive system using detection ofcircuit variations. IEEJ Trans Ind Appl 2003;123:667–674. (in Japanese)

5. Chen Z, Tomita M, Doki S, Okuma S. An extendedelectromotive force model for sensorless control ofinterior permanent-magnet synchronous motors.IEEE Trans Ind Electron 2003;50:288–295.

6. Morimoto S, Kawamoto K, Sanada M, Takeda Y.Sensorless control strategy for salient-pole PMSMbased on extended EMF in rotating reference frame.IEEE Trans Ind Appl 2002;38:1054–1061.

7. Sakamoto K, Iwaro Z, Endo T. Position sensorlesscontrol of an IPM motor using direct estimation ofthe axial error. IEEJ Semiconductor Power Conver-sion and Industrial Power Applications Report, SPC-00-67, IEA-00-42, p 73–77, 2000.

8. Tanaka K, Miki I. Position sensorless control ofinterior permanent magnet synchronous motor. Pa-pers of Technical Meeting on Rotating Machinery,IEE Japan, RM-03-30, p 29–32, 2003. (in Japanese)

9. Tanaka K, Moriyama R, Miki I. Initial rotor positionestimation of interior permanent magnet synchro-nous motor using optimal voltage vector. IEEJ TransInd Appl 2004;124:101–107. (in Japanese)

AUTHORS

Koji Tanaka (member) graduated from the Department of Electrical Engineering at Meiji University in 1997, completedthe first half of his doctoral studies in 1999, and joined Mitsubishi Heavy Industries, Ltd. He left Mitsubishi in 2000. He beganthe second half of his doctoral studies at Meiji University in 2001, and completed them in 2005. He is primarily pursuing researchrelated to control technology for AC motors. He holds a D.Eng. degree.

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AUTHORS (continued)

Ichiro Miki (member) completed his coursework for his doctorate at Meiji University in 1978 and became a lecturer withthe Faculty of Engineering. After serving as an instructor and an assistant professor, he became a professor in the Faculty ofScience. From 1987 to 1988, he was a visiting researcher at Kentucky University and the University of California at Davis. Heholds a D.Eng. degree, and is a member of IEEE, the Society of Instrument and Control Engineers, the Japan Society for FuzzyTheory and Intelligent Informatics, and the Institute of Electrical Installation Engineers of Japan.

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