Improved Rotor Position Estimation Of Salient-Pole PMSM Using High Frequency

Carrier Signal Injection O. Mansouri-Toudert1, H. Zeroug2, F. Auger3 and A. Chibah3

1 : Département Electrotechnique de la Faculté de Génie Electrique et Informatique, Mouloud Mammeri University, Tizi-Ouzou, Algeria (e-mail: toudert_ouiza@yahoo.fr). 2 : Industrial Electrical Systems Laboratory, Faculty of Electronics and Computing, Houari Boumediene University of Sciences and Technology, Bab-Ezzouar, Algiers, Algeria.. 3 : LUNAM University and with the « Institut de Recherches sur l’Energie Electrique » (IREENA), France (e-mail: Francois.Auger@univ-nantes.fr)

Abstract— This paper presents a new rotor position estimator

for salient-pole permanent magnet synchronous machines (PMSM) based on carrier signal injection. This method first uses a new way to extract the negative frequency component of the current, which contains the rotor position information. Compared to the classical approach, this method uses only one filter. It also performs an on line compensation of the phase shift of the filter used for this extraction. This method is used for sensorless field-oriented control, confirming then its effectiveness under various operating conditions.

Keywords— sensorless control, carrier signal injection, surface-mounted permanent magnet synchronous machines, demodulation.

I. NOMENCLATURE Ld, Lq : d-axis and q-axis stator inductance (H). Rs : stator resistance (Ω). vα, vβ, iα, iβ: voltages and currents in the Clark reference frame. va, vb, vc, ia, ib, ic: voltages and currents in the abc reference frame. vd, vq, id, iq: voltages and currents in the Park reference frame. s : derivative operator d/dt (Laplace variable). φm : rotor flux due to the permanent magnets (V.s)

rθ , r∧θ : real and estimated electrical angles between the stator

q-axis and the rotor q-axis (rad).

ωr and r∧ω : real and estimated angular frequency of the

fundamental electrical excitation (rad/s). ωc : carrier signal angular frequency (rad/s). fc : carrier signal frequency (Hz). ωm : mechanical rotor speed (rad/s).

*mω : desired mechanical rotor speed (rad/s).

Vc : injected carrier voltage (volt). p : number of pole pairs. J : total moment of inertia (kg m2). f : viscuous friction coefficient (Nm s) Tr : load torque (Nm). Te : electromagnetic torque (Nm).

II. INTRODUCTION ERMANENT magnet synchronous motors have recently come into wide use because of their high efficiency, large power density and control simplicity [1], [2]. The

control of these machines involves the knowledge of the exact rotor position, which is generally provided by a feedback sensor. This position sensor (or rotational transducer) not only increases cost, maintenance, and complexity, but also impairs robustness and reliability of the drive system [1], [3].

Therefore, many research studies are being conducted in order to eliminate the mechanical sensor. Conventional sensorless control schemes of the PMSM can be divided in two categories. Methods from the first category perform an estimation of the back-EMF [4], [5] using either voltage models, state observers or Kalman filters [6], [7]. Methods using a voltage model generally present a high sensitivity to parameter variations, which affects performance and may significantly degrade the rotor position estimation at low speed [7]. Methods using state observers or Kalman filters are less sensitive to parameter variations, but the common problem of all the mentioned methods is that for lower speeds, the back EMF has a very small amplitude, resulting in an inaccurate rotor position estimation [4], [7] and in an unstable speed control [8]. Methods of the second category try to solve this problem thanks to the injection of a high frequency voltage component used as a carrier signal [8], [9], [10]. These methods can be used when a motor saliency exists, even for surface-mounted permanent magnet synchronous motors [11]. Their advantage is that they are more devoted to accurately estimate the rotor position at low speeds, and even during standstill. In this category, however, several processings such as signal heterodyning and filtering [12] are necessary, in order to extract the position information contained in the negative frequency component of the current. Therefore, phase shifts are added to the rotor position, resulting in biased position estimation. The most commonly used method to extract the negative frequency component of the current involves three continuous-time filters (a band-pass filter, a high-pass filter and a low-pass filter) and two frequency shifts [13], [14], [15]. In this article, a simple method is introduced, which consists of only one continuous-time low-pass filter and one frequency

P

978-1-4673-1388-9/13/$31.00 ©2013 IEEE 761

shift. Moreover, the phase shift of the low-pass filter is compensated, in order to nullify the position estimation bias. This method has shown its effectiveness in providing accurate rotor position estimations under various operating conditions, and particularly in the low speed range. The aim of the approach presented here may seem close to a previous work published several years ago [16]. But in that paper, unbiased position estimation of an interior PMSM was derived from a state estimator of the saliency-based back-EMF. In our study, the position estimation is derived from a high frequency current injection, and the position estimation bias is cancelled thanks to a phase compensation scheme. This paper is organized as follows: In section II, a simplified model of a PMSM is developed, and a high-frequency model is presented in section III. In section IV, two methods for the extraction of the negative frequency component of the complex current signal are presented: the most often used method and a new one. In section V, the rotor position estimator is presented and a new scheme for the compensation of the phase shift of the filters is given. Simulation results are shown in section VI.

III. PMSM MODEL The d-q model for a PM synchronous machine is very adequate to present the analytical basis of the proposed estimation method. Under the assumptions that the machine contains only one single sinusoidally distributed saliency and that only the inductances are considered in the high frequency model, the stator voltage model in the rotor reference frame is given by

⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡ −+⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡

qs

ds

r

r

qs

ds

s

s

qs

ds

ss

ii

rr

vv

φφ

ωω

00

(1)

And the magnetic flux is given by

⎥⎦

⎤⎢⎣

⎡+⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡00

0 m

qs

ds

qs

ds

qs

ds

ii

LL φ

φφ

(2)

When transformed into a stationary reference frame αβ, the above equations can be rewritten as [17]

⎥⎦

⎤⎢⎣

⎡+⎥

⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡

s

s

s

ss

s

s

ii

rvv

β

α

β

α

β

α

φφ

dtd

(3)

with ⎥⎦

⎤⎢⎣

⎡+⎥

⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡)sin()cos(

)(r

rm

s

sr

s

s

ii

Lθθ

φθφφ

β

ααβ

β

α (4)

and ⎥⎦

⎤⎢⎣

⎡Δ+Δ−

Δ−Δ−=

)2cos()2sin()2sin()2cos(

)(rr

rrr LLL

LLLL

θθθθ

θαβ

where ( ) 2/qd LLL += and ( ) 2/dq LLL −=Δ are

respectively the average inductance and the zero-to-peak

differential inductance. Of course, the machine is not powered

to produce heat, but to generate an electromagnetic torque that

can be modelized as

( )[ ] qsdsqdme IILLpT −+= φ (5)

This torque allows us to act on the rotor speed as shown by the mechanical equation

rerr TTpfpJ −=+ ωω

dtd

(6)

Therefore, this electromagnetic torque produced by the stator currents may be used either to compensate the load torque and the friction losses or to produce a speed variation.

IV. PRINCIPLE OF THE CARRIER SIGNAL INJECTION BASED ESTIMATION METHODS

If Ld differs from Lq, it is possible to estimate the rotor position by superimposing to the low frequency excitation of the PMSM motor terminals a balanced set of high frequency low magnitude sinusoidal voltages [9], [18] :

( )

( )⎟⎠⎞

⎜⎝⎛ +

=⎥⎦

⎤⎢⎣

⎡−=⎥

⎦

⎤⎢⎣

⎡= 2

_

__ V

cosVsinV πω

β

ααβ ω

ω tj

ccc

cc

c

cc

c

ett

vv

v (7)

with ωc=2πfc. The frequency fc (around 1 kHz) and the magnitude Vc (around 10 V) of this carrier signal are chosen to ensure an acceptable increase of the total harmonic distorsion (THD) of the phase current [10], [19] and an insignificant torque ripple resulting from Eq. (5). At a high frequency, the PMSM model equations (3)-(4) can be simplified to [3]:

( ) ( )

( ) (8) dtd

dtd

dtd

I

_

_

_

_

_

_2

_

_

⎥⎦

⎤⎢⎣

⎡≈

⎥⎦

⎤⎢⎣

⎡+⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+=⎥

⎦

⎤⎢⎣

⎡

c

cr

c

cr

c

crs

c

c

ii

L

ii

LiiL

rvv

β

ααβ

β

ααβ

β

ααβ

β

α

θ

θθ

where I2 is the two-dimensional identity matrix. This equation shows that a current ___ ccc ijii βααβ += is induced by this

carrier voltage injection. The stator current can hence be decomposed into three components [10]: the first one is a positive frequency component that rotates in the same direction as the injected voltage, the second one is a negative frequency component that rotates in the opposite direction as the injected voltage, and the third one is the low frequency excitation component, resulting from the low frequency controlling voltage and from the stator flux and from the rotor flux:

s

tj

cn

tj

cp ieieiircc

_2

2 2

αβ

πθωπω

αβ ++≈⎟⎠⎞

⎜⎝⎛ ++−⎟

⎠⎞

⎜⎝⎛ −

, (9)

where the amplitude of the positive and negative frequency components are respectively given by

c

ccp LL

Liω)(

V. 22 Δ−= ,

)2)((V. 22

rc

ccn LL

Liωω −Δ−

Δ= (10)

From (10), one can see that only the negative frequency

component 2 2 ⎟

⎠⎞

⎜⎝⎛ ++−

πθω rctj

cn ei contains the information about the rotor position θr. As a consequence, so as to perform rotor

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position estimation, it is necessary to use some appropriate signal processings to extract this component [2], [10].

V. STATOR CURRENT SIGNAL PROCESSING Several methods using analog filters and frequency shifts

may be used to isolate the rotor position information θr contained in the second term of (9) and to get rid of the other undesirable components. In this paper, only two methods will be presented and compared. An emphasis is made on the second one, which is original, more simple and more effective.

Fig. 1. Most frequently used method for rotor position estimation

A. Previously proposed method One of the most widely used demodulation method [3],

[14], [15], [18], [19] first consists (see Fig. 1) of an analog band-pass filter (BPF) centered on the carrier frequency fc. This filter aims at preserving the two low amplitude high frequency components of the measured stator current and at canceling the low frequency excitation current. The resulting complex valued current is multiplied by tj ce ω − (this operation is sometimes called a heterodyning process), to perform a frequency translation of cω − . As a result, the negative frequency

component is moved around cω2 − , and the positive frequency component becomes a DC component, which can then be removed by a high-pass filter. The output of this high-pass filter is multiplied by tj ce ω2 + , so as to translate the negative frequency component around zero and to provide a signal proportional to rje θ2 . This signal is cleaned by a low-pass filter, and a phase extraction is finally used to provide the rotor position estimation [13].

Fig. 2. Simple method proposed for rotor position estimation

B. Proposed processing The review of the previous method raises questions about the real need for as many steps to achieve the final result. Therefore, we tried to find a simpler method. This new method (see Fig. 2) first consists of a multiplication of the measured stator current αβi by tj ce ω + , to perform a

frequency translation of cω + . As a result, the negative frequency component is moved around zero, and the remaining two components are located at frequencies higher than the carrier frequency:

( ) tjs

tj

cp

j

cntj c

crc eieieiei ω

αβ

πωπθω

αβ _2

22

2 ++≈

⎟⎠⎞

⎜⎝⎛ −⎟

⎠⎞

⎜⎝⎛ +

These last two components can be removed with a low-pass filter, providing again a signal demi _αβ (see Fig. 2)

proportional to rje θ2 .

In both cases, Bessel filters should be used, because of their maximally-flat group delay (or maximally linear phase response), leading to a position estimation error proportional to the rotor speed. Their coefficients are computed as shown in [20], [22]. After numerous trials, we set the cutoff frequency of all the low-pass filters to 40 Hz. The band-pass filter is centered at fc and its bandwidth has been set to 400 Hz. The cutoff frequency of the highpass filter has been set to 10 Hz. In the previously proposed demodulation method, second-order filters can be used successfully. By cons, a fourth-order low-pass-filter is used in the second method, to sufficiently attenuate the low frequency excitation component. This filter is realized with two second-order low pass-filters in series.

Fig. 3 shows the frequency response of the two demodulators. This figure shows that they have approximately the same bandwidth (±40 Hz around -1000 Hz). The classical demodulation process provides a perfect rejection around DC and around +1000 Hz, but the proposed method provides a higher attenuation anywhere else. Fig. 3 also shows that the slope of the phase around -1000 Hz is higher for the proposed demodulator than for the classical method, justifying therefore the need for a phase compensation scheme.

-1500 -1000 -500 0 500 1000 1500-140

-120

-100

-80

-60

-40

-20

0

mod

ulus

(dB

)

-1500 -1000 -500 0 500 1000 1500-1000

-800

-600

-400

-200

0

200

400

frequency (Hz)

phas

e (d

eg)

Fig. 3. Frequency responses of the two demodulators: proposed method

(blue dashed lines) and classical demodulation process (red solid lines).

VI. UNBIASED SPEED AND POSITION ESTIMATION

A. Speed and position estimator The demodulation of the extracted negative frequency

component provides the signals Icn cos(2θr) and Icn sin(2θr). The rotor position could be estimated from these signals with

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an inverse tangent function, but a more noise insensitive estimation is obtained by a non-linear second-order angle tracking observer [9], [10] suited to this case, as shown in Fig. 4. This observer provides an estimation of both the position and the speed of the rotor. The setting of this observer can be done as proposed in [21]. From the maximal electromagnetic torque of the machine Temax, the highest angular acceleration can be computed as

JTemax=α (11)

The first coefficient Kb (see Fig. 4) can then be chosen as

r_max~θ

α=bK , (12)

where r_max~θ is the maximum permissible position estimation

error resulting from such an angular acceleration. The second coefficient Ka can be computed as

ba KmK 2= , (13)

where m is a damping ratio chosen by the user. This parameter can be chosen so as to obtain a desired peak overshoot when the actual rotor position abruptly changes. For example, choosing an overshoot of 5% leads to m=1.945 [21].

Fig. 4. Block diagram of the nonlinear second-order angle tracking observer

B. Compensation of the position estimation error An extensive use of this estimator revealed that a biased position estimation is obtained when the angle tracking observer (Fig. 4) follows the negative frequency component extractor (Fig. 1 or 2). This bias comes from the phase shift ϕ(ω) of the filters used during the demodulation process. Since the expressions of these phase shifts are known and since the angle tracking observer also estimates the rotor speed, one may attempt to remove this estimation bias by subtracting ( )rc ˆ2ωωϕ +− to ( )rc 2ωωϕ +− . This compensation scheme is more precisely described by the block diagram shown in Fig. 5. In this figure, ϕBP(ω), ϕHP(ω), ϕLP2(ω) and ϕLP4(ω) are respectively the arguments of the band-pass filter, high-pass filter, second-order low-pass filter and fourth-order low-pass filter. The angle ϕ is the sum of the phase shifts of the filters used for the negative frequency component extraction. This compensation scheme can hence be used for any of the demodulation methods presented in section IV : when the “classical” method is used, K1=K2=K3=1

and K4=0, whereas when the proposed method is used, K1=K2=K3=0 and K4=1.

Fig. 5. The proposed estimation error compensation scheme.

Fig. 6. Block diagram of the sensorless vector control of a salient PMSM using high frequency signal injection.

VII. SIMULATION RESULTS To evaluate the performances of the described methods, a set of experiments have been performed on a MATLAB/ SIMULINK simulation. The numerical values of the parameters of the simulated machine come from the identification of an industrial 4.4 kW SM-PMSM (Yaskawa SGMGH-44DCA6F) and are given in Table I. The motor is fed by a power converter which receives three control signals from a classical vector control ensured by three PI controllers. The speed and position estimations are performed by a negative frequency component extractor and demodulator, an angle tracking observer and a phase shifts compensator.

Fig. 7 and Fig. 8 show the estimation error of the rotor position obtained with the “classical” method when the reference of the speed controller is constant and respectively equal to 10 r/min and 50 r/min. These results show that without compensation, the estimation error increases with speed. This may lead to an unstable speed control, since the estimated position is used in the abc/dq and dq/αβ transformations of the vector control, as shown in Fig. 6. When using a compensation of the phase shifts, the estimation error becomes lower than one degree in both cases. Fig. 8, 9 and 10 show the results obtained under the same conditions with the proposed method for the extraction and demodulation of the negative frequency component. These results show that without compensation, the estimation bias is higher than with the “classical” method. But once the phase shifts are compensated, the estimation error still remains below one

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degree. This means that with the proposed method, a highly stable speed control is obtained at the price of a demodulator with a lower complexity.

Fig. 11 shows the true speed obtained when the reference of the speed control is quickly varying. This figure shows that the PI controllers of the vector control have been set for the actual speed to closely follow the desired speed. Fig. 12 and 13 respectively show the speed estimation error and the position estimation error obtained with both the “classical” and the proposed demodulator, both with phase shifts compensation. These figures show that when the phase shifts are compensated, the “classical” demodulator provides a slightly more accurate estimation of speed and position, but at the price of a more complex process. It should be underlined that the classical demodulator is generally used without phase shifts compensation.

VIII. CONCLUSION This paper has presented a new position estimator of the rotor position of a salient permanent magnet synchronous machine (PMSM) using carrier signal injection. The commonly used method generally contains three analog filters. Since the proposed one uses only one analog filter, it is simple and effective, and leads to a lower implementation processing time. It was shown that when the compensation scheme takes into account the operating speed, filter phase shift is greatly reduced, thus leading to an accurate rotor position estimation. Simulation results presented in this paper demonstrate the effectiveness of the proposed method. Further researches include the validation of this method on an experimental testbed, and a comparison with the approach presented in [23]. As mentioned in [5], carrier signal injection based methods provide an estimation of twice the electrical angle. As a result, an ambiguity of 180 degrees affects the determination of the rotor position, and this ambiguity requires to be properly managed.

REFERENCES [1] T. Ané and L. Loron, "Sensorless control of a PMSM using an improved

implementation of Matsui’s method," Proc. IEEE MELECON 2008. [2] B. H. Kenny and P. E. Kascak, "Sensorless control of permanent magnet

machine for Nasa flywheel technology development," 37th Intersociety IEEE Energy Conv. Engineering Conf. (IECEC), 2002.

[3] A. S. Budden, R. Wrobel, D. Holliday and P.H. Mellor, "Sensorless control of permanent magnet machine drives for aerospace applications," Proc. IEEE PEDS, pp. 372-377, 2005.

[4] Z. Feng, W. Xuhui, X. Shan and G. Jinwen, " Sensorless control of interior permanent magnet machine at standstill and low speed," in Proc. EPE- IEEE PEMC (Portoroz, Slovenia), pp. 1289-1294, 2006.

[5] F. Parasiliti, R. Petrela and M. Tursini, " Sensorless speed control of salient rotor PM synchronous motor based on high frequency signal injection and Kalman filter," IEEE ISIE, vol. 2, pp. 623 – 628, 2002.

[6] N. Mutasim, N. Y. Ee and R. Eyad, " Sensorless speed and position control of permanent magnet synchronous motor using hybrid high frequency signal injection method," in Proc. third int. conf. on modeling, simulation and applied optimization (ICMSAO), 2009.

[7] J. H. Jang, S. K, J. I. Ha and M. Sawamura, "Sensorless drive of surface-mounted permanent-magnet motor by high-frequency signal injection based on magnet saliency," IEEE Trans. Indust. Applic., vol. 39, N° 4, pp. 1031-557, Jul/Aug. 2003.

[8] M. Mamo, K. Ide, M. Sawamura and J. O. Oyama, "Encoderless position estimation for symmetric cage induction machines under loaded conditions," IEEE Trans. on Indust. App., vol. 37, N° 6, pp. 1793-1800, Nov/Dec. 2001.

[9] P. L. Jansen and R. D. Lorentz, "Transducerless field orientation concepts employing saturation-induced in induction machine," IEEE Trans. on Indust. App., vol. 32, pp. 1380-1393, Nov/Dec. 1996.

[10] X. Yunxiang, Z. Ping and N. Haiqing, " A study on high frequency signal injection method of aiming at detecting the rotor position of the salient-pole brushless DC motor,” Proc. ICEM 2005, vol. 3, pp. 2369 – 2372, 2005.

[11] K. Ide, M. Oto, M. Inazumi and Y. Yamamoto, "Sensorless drive technology for permanent magnet synchronous motors," Yaskawa technical review, Vol. 69, No 2, pp. 1-6, sept. 2005.

[12] J. Mazumdar, R. G. Harley, and G. K. Venayagamoorthy, "Synchronous reference frame based active filter current reference generation using neural Networks," Proc. IEEE IECON 2006, pp. 4404-4409, Nov. 2006.

[13] J. Hu, "Sensorless control of AC machines for integrated started generator application," Ph.D. dissertation, Dept. Aerosp. Eng., Univ. Ohio state, 2007.

[14] L. Baghli, I. Al-Rouh, and A. Rezzoug, "Signal analysis and identification for induction motor sensorless control," Elsevier Control Engineering Practice, vol. 14, pp. 1313-1324, nov 2006.

[15] N. Teske, G. M. Asher, M. Sumner, and K. J. Bradley, "Encoderless position estimation for symmetric cage induction machines under loaded conditions," IEEE Trans. on Indust. App., Vol. 37, No 6, pp. 1793-1800, Nov/Dec. 2001.

[16] H. Kim, M. C. Harke and R. D. Lorentz, "Sensorless control of interior-permanent magnet machine drives with zero-phase lag position estimation," IEEE Trans. on Indust. Applic., vol. 39, No 6, pp. 1726-1733, Nov/Dec. 2003.

[17] D. Saltiveri, A. Arias, G. Asher, M. Sumner, "Sensorless control of surface-mounted permanent magnet synchronous motors using matrix converters," Electrical power quality and utilization journal, vol. X, No 1, pp. 59-67, 2006.

[18] C. Ortega, A. Arias, C. Caruana, C. Staines, J. Balcells, J. Cilia, "Sensorless direct torque control of a surface mounted PMSM using high frequency injection, "IEEE ISIE 2006, July 9-12, 2006, Montreal, Quebec, Canada., pp. 2332-2337, 2006.

[19] H. Zeroug, L. Hocine, "Sensorless control of surface mounted permanent magnet synchronous motor by signal injection associated with space vector modulation technique, " Proc. Electric Machines and Drives Conf., pp. 1573-1580, 2009.

[20] J. Karki, "Active low-pass filter design," Texas Instruments, Application report. SLOA049B, sept. 2002.

[21] F. Auger, O. Mansouri-Toudert, A. Chibah, "Design of advanced resolver-to digital converters," Proc. Electrimacs 2011, Cergy-Pontoise (France), 6-8 june 2011.

[22] A.B. Williams, F.J. Taylor, “Electronic filter design handbook, 4th edition,” Mc Graw Hill, 2006.

[23] T. Ohnuma, S. Doki, S. Okuma, “Extended EMF observer for wide speed range sensorless control of salient-pole synchronous motor drives,” proc. IEEE ICEM, sept. 2010.

TABLE I. MOTOR MODEL PARAMETERS Parameter Value Rated flux

Rated output power Rated torque Rated speed

Rated voltage Rated current Load torque

rs Ld Lq P

0.32 Wb 4.4 kW

28.4 Nm 1500 r/min

400 V 16.5 A 0 Nm

0.25 Ω 4.8 mH 4.1 mH

4

765

0 0.05 0.1 0.15-10

-5

0

5

time (s)

Posi

tion

erro

r with

the

clas

sica

l met

hod

(deg

)

Fig. 7. Position estimation error using the classical method without phase compensation (blue dashed lines) and with phase compensation (red solid line). The speed reference is constantly equal to 10 r/min.

0 0.05 0.1 0.15-10

-8

-6

-4

-2

0

2

4

6

8

10

time (s)

Posi

tion

erro

r with

cla

ssic

al m

etho

d (d

eg)

Fig. 8. Position estimation error using the classical method without phase compensation (blue dashed lines) and with phase compensation (red solid line). The speed reference is constantly equal to 50 r/min.

0 0.05 0.1 0.15-10

-5

0

5

time (s)

Posi

tion

erro

r with

sim

ple

met

hod

(deg

)

Fig. 9. Position estimation error using the proposed method without phase compensation (blue dashed lines) and with phase compensation (red solid line). The speed reference is constantly equal to 10 r/min.

0 0.05 0.1 0.15-15

-10

-5

0

5

10

15

time (s)

Posi

tion

erro

r with

sim

ple

met

hod

(deg

)

Fig. 10. Position estimation error using the proposed method without phase compensation (blue dashed line) and with phase compensation (red solid line). The speed reference is constantly equal to 50 r/min.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6-20

-15

-10

-5

0

5

10

15

time (s)

desi

red

and

mea

sure

d sp

eed(

r/min

)

Fig. 11. Desired (red solid line) and true (blue dotted line) rotor speed.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6-4

-3

-2

-1

0

1

2

3

4

time (s)

spee

d es

timat

ion

erro

r (r

/min

)

Fig. 12. Speed estimation error obtained with the classical method (blue dotted lines) and proposed method (red solid line).

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0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6-5

-4

-3

-2

-1

0

1

2

3

4

5

time (s)

Posi

tion

estim

atio

n er

ror (

deg)

Fig. 13. Position estimation error obtained with the classical method (blue dotted lines) and simple method (red solid line), both with phase compensatio

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