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Sensorless Control of a PMSM Synchronous Motor at Low Speed R. S. Miranda, E. M. Fernandes, C. B. Jacobina, A. M. N. Lima, A. C. Oliveira, M. B. R. Correa Departamento de Engenharia El´etrica Universidade Federal de Campina Grande Caixa Postal 10.105 58109-970 Campina Grande - PB - Brazil E-mails:[rmiranda, eisenhawer, jacobina, amnlima, aco, mbrcorrea]@dee.ufcg.edu.br

Abstract - This paper presents a self-sensing technique for a surface-mounted permanent-magnet synchronous machine (SM-PMSM) based on high frequency voltage injection. The reactances of the high frequency model vary as a function of rotor position and a simple method based on measured currents and voltages can be used to identify those reactances. Using the motor reactances, the rotor position and velocity are determined. The proposed technique is relatively independent of the machine parameters and it can be applied to any type of synchronous machine. The proposed technique has been tested experimentally for a commercial SM-PMSM machine. The experimental results have demonstrated the feasibility of the proposed technique. I. I NTRODUCTION The control of permanent magnet synchronous machine (PMSM) is based on the information of rotor position and angular speed. Encoders and resolvers can be used to detect the rotor position and speed. However, such sensors present some drawbacks related to cost, size and the reduction of the reliability of the drive system. Therefore, efforts have been focused on investigating alternatives to eliminate such electromechanical sensors, which has become an interesting field of research. [1–15]. Essentially, there are two methods for position estimation: (i) back electromotive-force (EMF) methods, used for medium and high speeds [1–7] and (ii) spatial saliencies and signal injection methods, applied at low speeds and standstill [8–15]. The BEMF method is based on the motor equations to determine the induced electromotive-force, from current and voltage measurements. In the related literature we find several approaches, that include deterministic observers [4], [5], stochastic filtering [6], [16], and other techniques based on the flux linkage [17], [18]. Some techniques are quite simple to implement but rely on the machine parameter’s and exhibit poor performance at low speed since the amplitude of the back-EMF is proportional to the speed. To overcome such limitation at low speed, the detection of rotor position can be done by tracking the magnetic saliencies of the machine

1-4244-0136-4/06/$20.00 '2006 IEEE

by injecting high frequency signals. There are different characteristics that are usually used as sources of saliencies in PMSM. In interior permanent-magnet synchronous machine (IPM) the saliency can be attributed to the shape of rotor, while in the surface permanent-magnet synchronous machine there is some saliency due to saturation of the stator by the permanent magnet flux. The saliency of the machine is represented as different inductances with respect to a dq reference frame. The underlying principle behind almost all low and zero speed sensorless techniques is the use of additional excitation to extract the saliency information. A high frequency voltage or current is injected into the motor and the resulting signal is processed to determine the rotor position. The high frequency injected signal (carrier) can be rotating [9], [11], [15], or pulsating signal along a specific axis [13], [14], or a pulse-based excitation [7], [8]. The important feature is that the injected signal must be persistent and have sufficiently high frequency to provide the position estimation with high bandwidth. The self-sensing technique proposed in this paper is based on the signal injection method. However, the proposed technique does not require any demodulation scheme to extract the rotor position from the measured signals that is usual for the signal injection methods. In the proposed technique it is considered that all the information concerning the rotor position is fully embedded in the machine inductance. Thus, if by a simple procedure it is possible to estimate the machine inductances then the rotor position and shaft speed can be extracted from the estimated parameters. The rotor position can be obtained through the identification of the high frequency model machine parameters. Since this method employs only voltages and currents, it is fairly insensitive to parameter variations. Other advantage of this method is it can uses both information of rotor saliency and magnetic saturation, and therefore can be applied to different machines. The proposed estimation method was used to provided selfsensing operation of a SM-PMSM motor with or without load condition. Experimental results demonstrates the efficiency and proper operation of the proposed solution.

5069

b

d

III. PARAMETER AND P OSITION E STIMATION M ETHOD

q

The technique presented for estimating rotor position is based on-line parameter identification of high frequency model. The estimation of the position and velocity can be obtained from the identified parameters. The identification procedure can be described in terms of both the stator and the estimated rotor reference frames. In this work, the identification is performed in the estimated rotor reference frame.

d g qe

Fig. 1.

^ q r

qr

a

A. Parameter Estimation

Reference frames for a PMSM.

II. M ODEL OF P ERMANENT-M AGNET S YNCHRONOUS M ACHINE Usually, the SM-PMSM is considered as non-salient, but in practice, a small amount of magnetic saliency is detected that is attributed to rotor saliency or saturation. Therefore, the PMSM can be represented with different inductances in dq axes (Lq > Ld ). The voltage equations for permanent-magnet synchronous motor with a sinusoidal flux distribution can be written, in the stator reference frame, as vα pLαβ rs + pLα iα − sin θr = + ωr λm (1) vβ pLαβ rs + pLβ iβ cos θr where vα and vβ are α and β axis stator voltages in the stator reference frame, respectively; iα and iβ are α and β axis stator currents, respectively; rs is the stator resistance; ωr is the rotor angular velocity; λm is the permanent-magnet flux linkage; p is differential operator and the inductances Lα , Lβ , and Lαβ are given by Lα

= L0 + L1 cos(2θr )

(2)

= L0 − L1 cos(2θr ) (3) = L1 sin(2θr ) (4) Ld + Lq (5) L0 = 2 Ld − Lq (6) L1 = 2 This model can also be transformed to the rotor reference frame as vd rs + pLd −ωr Lq id 0 = + (7) vq ωr Ld rs + pLq iq ωr λm

If a high frequency voltage signal is injected in the machine (7), i.e., with frequency ωh , the voltage drop related to the resistive component and the back-electromotive force can be neglected. Thus, the high frequency model in steady-state can be expressed as Ld 0 idh vdh = jωh (9) vqh iqh 0 Lq where {vdh , vqh } and {idh , iqh } are the high frequency voltages and currents expressed in terms of dq components in the actual rotor reference frame, respectively. This expression (9) can be transformed to the estimated synchronous reference frame to obtain Lγ Lγδ iγh vγh = jωh (10) vδh Lγδ Lδ iδh where {vγh , vδh } and {iγh , iδh } are the high frequency voltages and currents expressed in terms of components of the estimated rotor reference frame, respectively. The terms Lγ , Lδ and Lγδ are the high frequency inductances in the estimated rotor reference frame. The machine reactances in this reference frame are given by yγ yδ

Lβ Lαβ

Figure 1 shows the different reference frames (αβ, γδ, and dq) that are used for the development of the estimation method proposed in this paper. The αβ axis corresponds to the stator reference frame, the dq axis corresponds to the rotor reference frame and the γδ axis is defined as the estimated rotor reference frame which is related with an angular error given by (8) θe = θr − θr where θr is the measured rotor position and θr is the estimated rotor position.

yγδ

= ωh L0 + ωh L1 cos(2θe ) = ωh L0 − ωh L1 cos(2θe )

(11) (12)

= ωh L1 sin(2θe )

(13)

Its worth noting that the machine reactance’s expressed in the γδ reference frame are functions of rotor position estimation error. Suppose that high frequency currents and voltages in the γδ reference frame at the k th sampling period are denoted k k , vδh }, respectively. Then currents and by {ikγh , ikδh } and {vγh th voltages at the next (k + 1) sampling period will be denoted k+1 k+1 k+1 by {ik+1 γh , iδh } and {vγh , vδh }, respectively. The actual and the next sampling variables can be related by k k iγh ikδh vγh yγ = (14) k+1 yγδ vγh ik+1 ik+1 γh δh k k iγh ikδh vδh yγδ = (15) k+1 k+1 k+1 yδ iγh iδh vδh It has been considered that the reactances are constant during two consecutive sampling intervals. This assumption is acceptable at low speeds and thus the reactances can be calculated

5070

Fig. 2.

by

Block diagram of the self-sensing PMSM drive system.

1 yγ = (16) yγδ k+1 ikγh iδh − ikδh ik+1 γh k k+1 1 iδh −ikδh vδh yγδ = k+1 k+1 (17) k yδ k+1 k+1 −i i v k k γh γh δh iγh iδh − iδh iγh ik+1 δh −ik+1 γh

−ikδh ikγh

k vγh k+1 vγh

The computational complexity of the proposed technique is relatively modest when compared to standard parameter estimation techniques. B. Position Estimation Method The estimated parameters are functions of the rotor position and thus we can estimate rotor position from these parameters. Particularly, yγδ is proportional to sin (θe ). Therefore, it is possible to estimate the actual rotor position by forcing it to be zero. Assume that the error signal ε is given by ε

= K sin(θe )

(18)

where K is a constant gain. If the position estimation error is sufficiently small, (18) can be approximated by K sin (θe ) ∼ = Kθe , which can be used to track the actual speed and rotor position via a controller that drives the estimation error towards zero. The block diagram shown in Fig. 3 illustrates how the rotor position can be estimated.

kp qe

Fig. 3.

ki

+ + 1 s

1 s

^ q

r

^r w

Block diagram of the rotor position and speed estimator.

The PI controller in Fig. 3 uses the error signal to track the shaft position. Here kp and ki are the gains that determine the dynamic behavior of the estimator. Therefore, the position and speed can determined by ω r

=

(kp +

ki )ε s

(19)

ω r (20) s Figure 2 shows the block diagram of the self-sensing PMSM drive system. The position and speed loop employ a P and PI control law respectively, while the current control is based on a synchronous PI controller with decoupling and back-EMF compensation. The vector control block defines the references currents for the dq axis. The reference q component is obtained from reference torque at the output of the speed controller. The reference d component is defined to be zero. The filtered currents of αβ axis are transformed to the estimated rotor reference by transformation matrix T . The injected high frequency signal is a rotating vector applied in stator reference frame. θr

=

IV. M EASURED REACTANCES To observe how the high frequency reactances change with position error, a simple procedure has been implemented. From the measurement of the high frequency currents and voltages in the estimated rotor reference frame, the reactances can be obtained directly from (16) and (17). The investigation has been carried out by fixing the position observer output. This constant value is used in the transformation for estimation of these quantities in the γδ axis. By driving the motor to a known velocity, the position error causes a cyclic variation in the reactances. The PMSM data used in tests are listed in the Table I. Figure 4 shows the procedure to obtain the high frequency reactances. In Fig. 4, T denotes the matrix transformation

5071

0.1

0.2

0.3

0.4

0.5

0.6 20 V

OHMS

OHMS

10 V

0.1

0.2

0.3

0.4

0.5

Block diagram for processing the high frequency reactances.

13 800 Hz

0.6 30 V

0.1

0.2

0.3

0.4

0.5

0.6 40 V

OHMS

OHMS

16-17

Fig. 4.

8 7 6 5 4 0 8 7 6 5 4 0 8 7 6 5 4 0 8 7 6 5 4 0

0.1

0.2 t(s)

0.3

0.4

0.5

0.6

12

Fig. 6.

Reactance (Ohms)

11

High frequency yγ at various magnitudes of high frequency signal.

700 Hz

10 9

600 Hz 8 7 6

500 Hz

5 4 0

0.1

0.2

0.3 t(s)

0.4

0.5

0.6

Fig. 5. High frequency yγ at various frequencies of high frequency voltage.

related to the estimated rotor reference frame. On the other hand, Fig. 5 shows the high frequency reactance yγ at various frequencies of the injected signal. Note that the high frequency reactance yγ increase as the frequency of injected signal is increased. This occurs due to the dependence of magnitude reactance with the high frequency as defined in (11-13). In this results, the magnitude of high frequency injected signal is 10V . Figure 6 shows the high frequency reactance at various magnitudes of the injected signal. In this case, we can observe just a little change in the reactance. In this results, the frequency of the carrier signal is 500 Hz. The differences with respect to the nominal values are due to current measurement errors and to the inverter non-linear effects. V. E XPERIMENTAL R ESULTS A. System configuration The self-sensing control system was evaluated in the laboratory using a commercial PMSM drive system. The PMSM data are listed in the Table I. Figure 7 shows the configuration of the test setup. The test setup is composed by a microcomputer equipped with a specific data acquisition and control board and two SM-PMSM servodrives. The servodrives uses an inverter with 10kHz switching frequency with a 1µs dead time. The command board of one of the servodrives (CONVERTER 1) has been disabled and the control signals have been provided

Fig. 7.

Experimental test setup.

by microcomputer board. The command signals are generated by the microcomputer with a sampling time of 100µs. The data acquisition system employs Hall effect sensors and 12 bit A/D converters. The other servodrive (CONVERTER 2) as shown in Fig. 7 is used to simulate different load conditions for testing the proposed solution. B. Self-sensing position control The experimental results of the proposed self-sensing speed algorithm are shown in Figs. 8-11. In the figures, measured rotor position (θr ), estimated rotor position (θr ) and rotor position estimation error (θe ) are in radians. Figure 8 shows the performance of the proposed method when the reference speed is 50 rpm when no load is applied. The high frequency voltage applied has amplitude of 10 V and a frequency of 500 Hz. In this test the high frequency reference voltage has been used in the computations. The estimated electrical rotor position was calculated using the method proposed. The actual rotor position was measured using the resolver attached to motor

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TABLE I NAMEPLATE DATA FOR M ACHINE 1 Rated power (kW) Rated voltage (V) Phase resistance (Ω) Ld (mH) Lq (mH) Rated current (A) Maximum speed (rpm) Pole pairs Maximum Torque (Nm)

1.13 220 0.663 1.93 2.28 7.2 6000 4 2.5

shaft only for the sake comparisons. The position estimation error is nearly zero in steady state. Besides Fig. 9 illustrates the performance of the proposed self-sensing method when the reference speed varies from 30 rpm to 100 rpm at no load. The rotor position error increased after the transient but in steady state remains small, approximately less than 2 radians. The gains kp and ki was selected in a such a way to provide a minimal error position in steady state, there was no utilized any procedure to design them. Figures 10-11 present the estimation method behavior under load conditions. Figure 10 shows the response of the proposed method when the reference speed is 30 rpm in steady state under 80% rated load applied. As in the operations with no load, the high frequency voltage applied has amplitude of 10V and a frequency of 500 Hz, and the high frequency reference voltage has been used in the computations. The machine under test has 4 poles and rated rotor frequency of 400 Hz, thus, the reference speed of 30 rpm means that the rotor frequency is about 2 Hz, a very low frequency condition. The control algorithm does not lose the position estimation and the position estimation error is nearly zero in steady state. Figure 11 presents the performance of the proposed algorithm with a step change of the reference speed from 30 rpm to 100 rpm. The speed variation happens under 80% at load condition. One can observe that the position estimation error is very small in the transient period as well as in the steady state. Figures 8-11 demonstrate that the proposed rotor position estimation method is feasible at low-speed. VI. C ONCLUSIONS This paper has been presented how to combine rotor position based on signal injection with a model based parameter estimation technique. Note that the proposed method does not depend on motor parameters. Based on high frequency model, the high frequency reactances in the estimated rotor reference frame was obtained using measurement signals. It was shown that the reactances are function of the rotor position error even under different condition of injected signal. The proposed self-sensing control use this reactances to determine the position and speed. The performance of the self-sensing control was demonstrated by experiments with and without load. The presented results have demonstrated the feasibility of the proposed solution at low speed.

Fig. 8. Performance of the proposed algorithm with 50 rpm at no load: (a) measured and estimated position, (b) rotor position estimation error.

Fig. 9. Performance of the proposed algorithm with a step in the reference speed at no load: (a) measured and estimated rotor position, (b) rotor position estimation error

ACKNOWLEDGEMENTS The authors would like to thank the support provided by the Conselho Nacional de Desenvolvimento Cient´ıfico e Tecnol´ogico (CNPq), by the Coordenac¸a˜ o de Aperfeic¸oamento de Pessoal de N´ıvel Superior (CAPES) and by WEG ´ ˜ . INDUSTRIAS S.A. - AUTOMAC¸AO R EFERENCES [1] R. Wu and G. R. Slemon, “A permanent magnet motor drive without a shaft sensor,” Conf. Rec. IEEE-IAS Annu. Meeting, pp. 553–558, 1990. [2] N. Matsui and M. Shigyo, “Brushless dc motor without position and speed sensors,” IEEE Trans. Ind. Applicat., vol. 28, pp. 120–127, Jan./Feb. 1992. [3] N. Matsui, T. TAkeshita, and K. Yasuda, “A new sensorless drive of brushless dc motor,” Proc. IECON’92, pp. 430–435, 1992. [4] H. Kim, M. C. Harke, and R. Lorenz, “Sensorless control of interior permanent-magnet machine drive with zero-phase lag position estimation,” IEEE Trans. Ind. Applicat., vol. 39, pp. 784–789, Nov./Dec. 2003.

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Fig. 10. Performance of the proposed algorithm with 30 rpm under 80% rated load condition: (a) measured and estimated speed, (b) rotor position estimation error).

[12] P. L. Jansen, M. Corley, and R. Lorenz, “Flux, position, and velocity estimation in ac machines at zero speed via tracking of high frequency saliencies,” Proc. EPE’95, pp. 154–160, 1995. [13] M. Linke, R. Kennel, and J. Holtz, “Sensorless speed an position control of permanent magnet synchronous machines,” Proc. IECON’02, vol. 34, pp. 784–789, Nov. 2002. [14] J.-H. Jang, J.-I. Ha, M. Ohto, K. Ide, and S.-K. Sul, “Analysis of permanent-magnet machine for sensorless control based on highfrequency signal injection,” IEEE Trans. Ind. Applicat., vol. 39, pp. 1595–2004, May/June 2003. [15] H. Kim and R. D. Lorenz, “Carrier signal injection based sensorless control methods for ipm synchronous machine drives,” Conf. Rec. IEEEIAS Annu. Meeting, vol. 2, pp. 977–984, 2003. [16] S. Bolognani, R. Oboe, and M. Zigliotto, “Sensorless full-digital pmsm drive with emf estimation of speed and rotor position.,” IEEE Trans. Ind. Electr., vol. 46, pp. 184–191, 1999. [17] N. Ertugrul and P. P. acarnely, “A new algorithm for sensorless operation of permanent magnet motors,” IEEE Trans. Ind. Applicat., vol. 30, pp. 126–133, Jan./Feb. 1994. [18] H. Rasmussen, V. Vadstrup, and H. Borsting, “Sensorless field oriented control of pm motor including zero speed.,” Proc. IEMDC’03, 2003.

Fig. 11. Performance of the proposed algorithm with a step of the reference speed (from 30 to 100 rpm) from under 80% rated load condition: (a) measured and estimated position, (b) rotor position estimation error.

[5] S. Ichikawa, C. Zhiqian, M. Tomita, S. Doki, and S. Okuma, “Sensorless control of an interior permanent magnet synchronous motor on the rotating coordinate using an extended electromotive force,” Proc. IECON’01, vol. 3, pp. 1667–1672, Dec. 2001. [6] S. Marimoto, K. Kawamoto, M. Sanada, and Y. Takeda, “Sensorless control strategy for salient-pole pmsm based on extended emf in rotating reference frame,” IEEE Trans. Ind. Applicat., vol. 38, pp. 1054–1061, July/Aug. 2002. [7] M. Schroedl, “Sensorless control of permanent synchronous motors,” Electric Machines and Power Systems, vol. 22, pp. 173–185, 1994. [8] T. Aihara, A. Toba, T. Yanase, A. Mashimo, and K. Endo, “Sensorless torque control of salient-pole synchronous motor at zero speed operation,” Proc. APEC’97, vol. 2, pp. 715–720, Feb. 1997. [9] A. Consoli, G. Scarcella, and A. Testa, “Industy application of zerospeed sensorless control techniques for pm synchronous motors,” IEEE Trans. Ind. Applicat., vol. 37, pp. 806–812, March/April 2001. [10] H. Akagi and S. Ogasawara, “Implementation and position control performance of a ipm motor drive system based on magnetic saliency,” IEEE Trans. Ind. Applicat., pp. 806–812, July/Aug. 1998. [11] M. J. Corley and R. D. Lorenz, “Rotor position and velocity estimation for a salient-pole permanent magnet synchronous machine at standstill and high speed,” IEEE Trans. Ind. Applicat., vol. 34, pp. 784–789, July/Aug. 1998.

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Abstract - This paper presents a self-sensing technique for a surface-mounted permanent-magnet synchronous machine (SM-PMSM) based on high frequency voltage injection. The reactances of the high frequency model vary as a function of rotor position and a simple method based on measured currents and voltages can be used to identify those reactances. Using the motor reactances, the rotor position and velocity are determined. The proposed technique is relatively independent of the machine parameters and it can be applied to any type of synchronous machine. The proposed technique has been tested experimentally for a commercial SM-PMSM machine. The experimental results have demonstrated the feasibility of the proposed technique. I. I NTRODUCTION The control of permanent magnet synchronous machine (PMSM) is based on the information of rotor position and angular speed. Encoders and resolvers can be used to detect the rotor position and speed. However, such sensors present some drawbacks related to cost, size and the reduction of the reliability of the drive system. Therefore, efforts have been focused on investigating alternatives to eliminate such electromechanical sensors, which has become an interesting field of research. [1–15]. Essentially, there are two methods for position estimation: (i) back electromotive-force (EMF) methods, used for medium and high speeds [1–7] and (ii) spatial saliencies and signal injection methods, applied at low speeds and standstill [8–15]. The BEMF method is based on the motor equations to determine the induced electromotive-force, from current and voltage measurements. In the related literature we find several approaches, that include deterministic observers [4], [5], stochastic filtering [6], [16], and other techniques based on the flux linkage [17], [18]. Some techniques are quite simple to implement but rely on the machine parameter’s and exhibit poor performance at low speed since the amplitude of the back-EMF is proportional to the speed. To overcome such limitation at low speed, the detection of rotor position can be done by tracking the magnetic saliencies of the machine

1-4244-0136-4/06/$20.00 '2006 IEEE

by injecting high frequency signals. There are different characteristics that are usually used as sources of saliencies in PMSM. In interior permanent-magnet synchronous machine (IPM) the saliency can be attributed to the shape of rotor, while in the surface permanent-magnet synchronous machine there is some saliency due to saturation of the stator by the permanent magnet flux. The saliency of the machine is represented as different inductances with respect to a dq reference frame. The underlying principle behind almost all low and zero speed sensorless techniques is the use of additional excitation to extract the saliency information. A high frequency voltage or current is injected into the motor and the resulting signal is processed to determine the rotor position. The high frequency injected signal (carrier) can be rotating [9], [11], [15], or pulsating signal along a specific axis [13], [14], or a pulse-based excitation [7], [8]. The important feature is that the injected signal must be persistent and have sufficiently high frequency to provide the position estimation with high bandwidth. The self-sensing technique proposed in this paper is based on the signal injection method. However, the proposed technique does not require any demodulation scheme to extract the rotor position from the measured signals that is usual for the signal injection methods. In the proposed technique it is considered that all the information concerning the rotor position is fully embedded in the machine inductance. Thus, if by a simple procedure it is possible to estimate the machine inductances then the rotor position and shaft speed can be extracted from the estimated parameters. The rotor position can be obtained through the identification of the high frequency model machine parameters. Since this method employs only voltages and currents, it is fairly insensitive to parameter variations. Other advantage of this method is it can uses both information of rotor saliency and magnetic saturation, and therefore can be applied to different machines. The proposed estimation method was used to provided selfsensing operation of a SM-PMSM motor with or without load condition. Experimental results demonstrates the efficiency and proper operation of the proposed solution.

5069

b

d

III. PARAMETER AND P OSITION E STIMATION M ETHOD

q

The technique presented for estimating rotor position is based on-line parameter identification of high frequency model. The estimation of the position and velocity can be obtained from the identified parameters. The identification procedure can be described in terms of both the stator and the estimated rotor reference frames. In this work, the identification is performed in the estimated rotor reference frame.

d g qe

Fig. 1.

^ q r

qr

a

A. Parameter Estimation

Reference frames for a PMSM.

II. M ODEL OF P ERMANENT-M AGNET S YNCHRONOUS M ACHINE Usually, the SM-PMSM is considered as non-salient, but in practice, a small amount of magnetic saliency is detected that is attributed to rotor saliency or saturation. Therefore, the PMSM can be represented with different inductances in dq axes (Lq > Ld ). The voltage equations for permanent-magnet synchronous motor with a sinusoidal flux distribution can be written, in the stator reference frame, as vα pLαβ rs + pLα iα − sin θr = + ωr λm (1) vβ pLαβ rs + pLβ iβ cos θr where vα and vβ are α and β axis stator voltages in the stator reference frame, respectively; iα and iβ are α and β axis stator currents, respectively; rs is the stator resistance; ωr is the rotor angular velocity; λm is the permanent-magnet flux linkage; p is differential operator and the inductances Lα , Lβ , and Lαβ are given by Lα

= L0 + L1 cos(2θr )

(2)

= L0 − L1 cos(2θr ) (3) = L1 sin(2θr ) (4) Ld + Lq (5) L0 = 2 Ld − Lq (6) L1 = 2 This model can also be transformed to the rotor reference frame as vd rs + pLd −ωr Lq id 0 = + (7) vq ωr Ld rs + pLq iq ωr λm

If a high frequency voltage signal is injected in the machine (7), i.e., with frequency ωh , the voltage drop related to the resistive component and the back-electromotive force can be neglected. Thus, the high frequency model in steady-state can be expressed as Ld 0 idh vdh = jωh (9) vqh iqh 0 Lq where {vdh , vqh } and {idh , iqh } are the high frequency voltages and currents expressed in terms of dq components in the actual rotor reference frame, respectively. This expression (9) can be transformed to the estimated synchronous reference frame to obtain Lγ Lγδ iγh vγh = jωh (10) vδh Lγδ Lδ iδh where {vγh , vδh } and {iγh , iδh } are the high frequency voltages and currents expressed in terms of components of the estimated rotor reference frame, respectively. The terms Lγ , Lδ and Lγδ are the high frequency inductances in the estimated rotor reference frame. The machine reactances in this reference frame are given by yγ yδ

Lβ Lαβ

Figure 1 shows the different reference frames (αβ, γδ, and dq) that are used for the development of the estimation method proposed in this paper. The αβ axis corresponds to the stator reference frame, the dq axis corresponds to the rotor reference frame and the γδ axis is defined as the estimated rotor reference frame which is related with an angular error given by (8) θe = θr − θr where θr is the measured rotor position and θr is the estimated rotor position.

yγδ

= ωh L0 + ωh L1 cos(2θe ) = ωh L0 − ωh L1 cos(2θe )

(11) (12)

= ωh L1 sin(2θe )

(13)

Its worth noting that the machine reactance’s expressed in the γδ reference frame are functions of rotor position estimation error. Suppose that high frequency currents and voltages in the γδ reference frame at the k th sampling period are denoted k k , vδh }, respectively. Then currents and by {ikγh , ikδh } and {vγh th voltages at the next (k + 1) sampling period will be denoted k+1 k+1 k+1 by {ik+1 γh , iδh } and {vγh , vδh }, respectively. The actual and the next sampling variables can be related by k k iγh ikδh vγh yγ = (14) k+1 yγδ vγh ik+1 ik+1 γh δh k k iγh ikδh vδh yγδ = (15) k+1 k+1 k+1 yδ iγh iδh vδh It has been considered that the reactances are constant during two consecutive sampling intervals. This assumption is acceptable at low speeds and thus the reactances can be calculated

5070

Fig. 2.

by

Block diagram of the self-sensing PMSM drive system.

1 yγ = (16) yγδ k+1 ikγh iδh − ikδh ik+1 γh k k+1 1 iδh −ikδh vδh yγδ = k+1 k+1 (17) k yδ k+1 k+1 −i i v k k γh γh δh iγh iδh − iδh iγh ik+1 δh −ik+1 γh

−ikδh ikγh

k vγh k+1 vγh

The computational complexity of the proposed technique is relatively modest when compared to standard parameter estimation techniques. B. Position Estimation Method The estimated parameters are functions of the rotor position and thus we can estimate rotor position from these parameters. Particularly, yγδ is proportional to sin (θe ). Therefore, it is possible to estimate the actual rotor position by forcing it to be zero. Assume that the error signal ε is given by ε

= K sin(θe )

(18)

where K is a constant gain. If the position estimation error is sufficiently small, (18) can be approximated by K sin (θe ) ∼ = Kθe , which can be used to track the actual speed and rotor position via a controller that drives the estimation error towards zero. The block diagram shown in Fig. 3 illustrates how the rotor position can be estimated.

kp qe

Fig. 3.

ki

+ + 1 s

1 s

^ q

r

^r w

Block diagram of the rotor position and speed estimator.

The PI controller in Fig. 3 uses the error signal to track the shaft position. Here kp and ki are the gains that determine the dynamic behavior of the estimator. Therefore, the position and speed can determined by ω r

=

(kp +

ki )ε s

(19)

ω r (20) s Figure 2 shows the block diagram of the self-sensing PMSM drive system. The position and speed loop employ a P and PI control law respectively, while the current control is based on a synchronous PI controller with decoupling and back-EMF compensation. The vector control block defines the references currents for the dq axis. The reference q component is obtained from reference torque at the output of the speed controller. The reference d component is defined to be zero. The filtered currents of αβ axis are transformed to the estimated rotor reference by transformation matrix T . The injected high frequency signal is a rotating vector applied in stator reference frame. θr

=

IV. M EASURED REACTANCES To observe how the high frequency reactances change with position error, a simple procedure has been implemented. From the measurement of the high frequency currents and voltages in the estimated rotor reference frame, the reactances can be obtained directly from (16) and (17). The investigation has been carried out by fixing the position observer output. This constant value is used in the transformation for estimation of these quantities in the γδ axis. By driving the motor to a known velocity, the position error causes a cyclic variation in the reactances. The PMSM data used in tests are listed in the Table I. Figure 4 shows the procedure to obtain the high frequency reactances. In Fig. 4, T denotes the matrix transformation

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0.1

0.2

0.3

0.4

0.5

0.6 20 V

OHMS

OHMS

10 V

0.1

0.2

0.3

0.4

0.5

Block diagram for processing the high frequency reactances.

13 800 Hz

0.6 30 V

0.1

0.2

0.3

0.4

0.5

0.6 40 V

OHMS

OHMS

16-17

Fig. 4.

8 7 6 5 4 0 8 7 6 5 4 0 8 7 6 5 4 0 8 7 6 5 4 0

0.1

0.2 t(s)

0.3

0.4

0.5

0.6

12

Fig. 6.

Reactance (Ohms)

11

High frequency yγ at various magnitudes of high frequency signal.

700 Hz

10 9

600 Hz 8 7 6

500 Hz

5 4 0

0.1

0.2

0.3 t(s)

0.4

0.5

0.6

Fig. 5. High frequency yγ at various frequencies of high frequency voltage.

related to the estimated rotor reference frame. On the other hand, Fig. 5 shows the high frequency reactance yγ at various frequencies of the injected signal. Note that the high frequency reactance yγ increase as the frequency of injected signal is increased. This occurs due to the dependence of magnitude reactance with the high frequency as defined in (11-13). In this results, the magnitude of high frequency injected signal is 10V . Figure 6 shows the high frequency reactance at various magnitudes of the injected signal. In this case, we can observe just a little change in the reactance. In this results, the frequency of the carrier signal is 500 Hz. The differences with respect to the nominal values are due to current measurement errors and to the inverter non-linear effects. V. E XPERIMENTAL R ESULTS A. System configuration The self-sensing control system was evaluated in the laboratory using a commercial PMSM drive system. The PMSM data are listed in the Table I. Figure 7 shows the configuration of the test setup. The test setup is composed by a microcomputer equipped with a specific data acquisition and control board and two SM-PMSM servodrives. The servodrives uses an inverter with 10kHz switching frequency with a 1µs dead time. The command board of one of the servodrives (CONVERTER 1) has been disabled and the control signals have been provided

Fig. 7.

Experimental test setup.

by microcomputer board. The command signals are generated by the microcomputer with a sampling time of 100µs. The data acquisition system employs Hall effect sensors and 12 bit A/D converters. The other servodrive (CONVERTER 2) as shown in Fig. 7 is used to simulate different load conditions for testing the proposed solution. B. Self-sensing position control The experimental results of the proposed self-sensing speed algorithm are shown in Figs. 8-11. In the figures, measured rotor position (θr ), estimated rotor position (θr ) and rotor position estimation error (θe ) are in radians. Figure 8 shows the performance of the proposed method when the reference speed is 50 rpm when no load is applied. The high frequency voltage applied has amplitude of 10 V and a frequency of 500 Hz. In this test the high frequency reference voltage has been used in the computations. The estimated electrical rotor position was calculated using the method proposed. The actual rotor position was measured using the resolver attached to motor

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TABLE I NAMEPLATE DATA FOR M ACHINE 1 Rated power (kW) Rated voltage (V) Phase resistance (Ω) Ld (mH) Lq (mH) Rated current (A) Maximum speed (rpm) Pole pairs Maximum Torque (Nm)

1.13 220 0.663 1.93 2.28 7.2 6000 4 2.5

shaft only for the sake comparisons. The position estimation error is nearly zero in steady state. Besides Fig. 9 illustrates the performance of the proposed self-sensing method when the reference speed varies from 30 rpm to 100 rpm at no load. The rotor position error increased after the transient but in steady state remains small, approximately less than 2 radians. The gains kp and ki was selected in a such a way to provide a minimal error position in steady state, there was no utilized any procedure to design them. Figures 10-11 present the estimation method behavior under load conditions. Figure 10 shows the response of the proposed method when the reference speed is 30 rpm in steady state under 80% rated load applied. As in the operations with no load, the high frequency voltage applied has amplitude of 10V and a frequency of 500 Hz, and the high frequency reference voltage has been used in the computations. The machine under test has 4 poles and rated rotor frequency of 400 Hz, thus, the reference speed of 30 rpm means that the rotor frequency is about 2 Hz, a very low frequency condition. The control algorithm does not lose the position estimation and the position estimation error is nearly zero in steady state. Figure 11 presents the performance of the proposed algorithm with a step change of the reference speed from 30 rpm to 100 rpm. The speed variation happens under 80% at load condition. One can observe that the position estimation error is very small in the transient period as well as in the steady state. Figures 8-11 demonstrate that the proposed rotor position estimation method is feasible at low-speed. VI. C ONCLUSIONS This paper has been presented how to combine rotor position based on signal injection with a model based parameter estimation technique. Note that the proposed method does not depend on motor parameters. Based on high frequency model, the high frequency reactances in the estimated rotor reference frame was obtained using measurement signals. It was shown that the reactances are function of the rotor position error even under different condition of injected signal. The proposed self-sensing control use this reactances to determine the position and speed. The performance of the self-sensing control was demonstrated by experiments with and without load. The presented results have demonstrated the feasibility of the proposed solution at low speed.

Fig. 8. Performance of the proposed algorithm with 50 rpm at no load: (a) measured and estimated position, (b) rotor position estimation error.

Fig. 9. Performance of the proposed algorithm with a step in the reference speed at no load: (a) measured and estimated rotor position, (b) rotor position estimation error

ACKNOWLEDGEMENTS The authors would like to thank the support provided by the Conselho Nacional de Desenvolvimento Cient´ıfico e Tecnol´ogico (CNPq), by the Coordenac¸a˜ o de Aperfeic¸oamento de Pessoal de N´ıvel Superior (CAPES) and by WEG ´ ˜ . INDUSTRIAS S.A. - AUTOMAC¸AO R EFERENCES [1] R. Wu and G. R. Slemon, “A permanent magnet motor drive without a shaft sensor,” Conf. Rec. IEEE-IAS Annu. Meeting, pp. 553–558, 1990. [2] N. Matsui and M. Shigyo, “Brushless dc motor without position and speed sensors,” IEEE Trans. Ind. Applicat., vol. 28, pp. 120–127, Jan./Feb. 1992. [3] N. Matsui, T. TAkeshita, and K. Yasuda, “A new sensorless drive of brushless dc motor,” Proc. IECON’92, pp. 430–435, 1992. [4] H. Kim, M. C. Harke, and R. Lorenz, “Sensorless control of interior permanent-magnet machine drive with zero-phase lag position estimation,” IEEE Trans. Ind. Applicat., vol. 39, pp. 784–789, Nov./Dec. 2003.

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Fig. 10. Performance of the proposed algorithm with 30 rpm under 80% rated load condition: (a) measured and estimated speed, (b) rotor position estimation error).

[12] P. L. Jansen, M. Corley, and R. Lorenz, “Flux, position, and velocity estimation in ac machines at zero speed via tracking of high frequency saliencies,” Proc. EPE’95, pp. 154–160, 1995. [13] M. Linke, R. Kennel, and J. Holtz, “Sensorless speed an position control of permanent magnet synchronous machines,” Proc. IECON’02, vol. 34, pp. 784–789, Nov. 2002. [14] J.-H. Jang, J.-I. Ha, M. Ohto, K. Ide, and S.-K. Sul, “Analysis of permanent-magnet machine for sensorless control based on highfrequency signal injection,” IEEE Trans. Ind. Applicat., vol. 39, pp. 1595–2004, May/June 2003. [15] H. Kim and R. D. Lorenz, “Carrier signal injection based sensorless control methods for ipm synchronous machine drives,” Conf. Rec. IEEEIAS Annu. Meeting, vol. 2, pp. 977–984, 2003. [16] S. Bolognani, R. Oboe, and M. Zigliotto, “Sensorless full-digital pmsm drive with emf estimation of speed and rotor position.,” IEEE Trans. Ind. Electr., vol. 46, pp. 184–191, 1999. [17] N. Ertugrul and P. P. acarnely, “A new algorithm for sensorless operation of permanent magnet motors,” IEEE Trans. Ind. Applicat., vol. 30, pp. 126–133, Jan./Feb. 1994. [18] H. Rasmussen, V. Vadstrup, and H. Borsting, “Sensorless field oriented control of pm motor including zero speed.,” Proc. IEMDC’03, 2003.

Fig. 11. Performance of the proposed algorithm with a step of the reference speed (from 30 to 100 rpm) from under 80% rated load condition: (a) measured and estimated position, (b) rotor position estimation error.

[5] S. Ichikawa, C. Zhiqian, M. Tomita, S. Doki, and S. Okuma, “Sensorless control of an interior permanent magnet synchronous motor on the rotating coordinate using an extended electromotive force,” Proc. IECON’01, vol. 3, pp. 1667–1672, Dec. 2001. [6] S. Marimoto, K. Kawamoto, M. Sanada, and Y. Takeda, “Sensorless control strategy for salient-pole pmsm based on extended emf in rotating reference frame,” IEEE Trans. Ind. Applicat., vol. 38, pp. 1054–1061, July/Aug. 2002. [7] M. Schroedl, “Sensorless control of permanent synchronous motors,” Electric Machines and Power Systems, vol. 22, pp. 173–185, 1994. [8] T. Aihara, A. Toba, T. Yanase, A. Mashimo, and K. Endo, “Sensorless torque control of salient-pole synchronous motor at zero speed operation,” Proc. APEC’97, vol. 2, pp. 715–720, Feb. 1997. [9] A. Consoli, G. Scarcella, and A. Testa, “Industy application of zerospeed sensorless control techniques for pm synchronous motors,” IEEE Trans. Ind. Applicat., vol. 37, pp. 806–812, March/April 2001. [10] H. Akagi and S. Ogasawara, “Implementation and position control performance of a ipm motor drive system based on magnetic saliency,” IEEE Trans. Ind. Applicat., pp. 806–812, July/Aug. 1998. [11] M. J. Corley and R. D. Lorenz, “Rotor position and velocity estimation for a salient-pole permanent magnet synchronous machine at standstill and high speed,” IEEE Trans. Ind. Applicat., vol. 34, pp. 784–789, July/Aug. 1998.

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