Voltage Vector Selection Area of the Direct Torque Control for Permanent Magnet Synchronous Motor

Voltage Vector Selection Area of the Direct Torque Control for Permanent Magnet Synchronous Motor

Yaohua Li*, Jian Ma*, Qiang Yu*, and Jingyu Liu*

Abstract –The control of stator flux, torque angle, excitation torque, reluctance torque and total torque of the direct torque control (DTC) for a permanent magnet synchronous motor (PMSM) are studied in this paper. Simplified expressions to represent the changes of these variables due to the application of a voltage vector are given. Finally, a voltage vector selection area and the implementation of a voltage vector selection strategy are proposed.

Keywords: Voltage vector, Direct torque control, Permanent magnet synchronous motor

1. Introduction

The direct torque control (DTC) for permanent magnet synchronous motors (PMSM) possesses many advantages:

a simple control configuration, lesser parameters dependence and fast dynamic response^[1]-[4]. However, it also suffers from high stator current and torque ripple. The DTC selects the proper voltage vector to implement the control of stator flux and torque according to a switching table and the outputs of hysteresis comparators for stator flux and torque are shown in Fig. 1. Thus, in nature the DTC is a hysteresis control. A voltage vector selection strategy as a hysteresis control principle is critical to improving the control performance.

PMSM

SA

Stator flux comparator

+_-

+_-

Switching VSI tale

SB

SC

Torque comparator Stator flux sector

*

y)s

y⁾s

*

Te

Te

1-6

q

t f

Fig. 1. The DTC for a PMSM

Studies have shown that the switching table as a voltage vector selection strategy can’t always satisfy the control of torque and will cause torque ripple^[5]-[7]. In [8], the control of stator flux, torque angle and torque of the DTC for a

surface PMSM are analyzed. In [9]-[10], a simplified voltage vector selection strategy of the DTC for an interior PMSM was proposed based on the control of stator flux, torque angle and torque. However, as expressions to represent the change of stator flux, torque angle, excitation torque, reluctance torque and total torque of the PMSM due to the application of a voltage vector are very complicated, simple and clear control principles of these variables weren’t proposed. Thus finding a simple way to determine the voltage vector selection area and strategy of a PMSM DTC system is the main aim of this paper.

2. Control of Stator Flux and Torque Angle Neglecting voltage drop on stator resistance, after applying a voltage vector for Dt, stator flux is presented in Fig. 2. According to Fig. 2, the change of amplitude and angular position of the stator flux can be expressed in (1) and (2).

qs

D

y ¢rs

yrs a t Vr_s ×D

Fig. 2. The move of stator flux

s s

s s

s

s y V t y V t a y

y⁾ = ⁾ + ⁾ ×D + ⁾ × ⁾ ×D × - ⁾

D ² ( )² 2 cos (1)

a y

y q a

cos 2

) ( arcsin sin

2

2+ ×D + × ×D ×

× D

= ×

D V t V t

t V

s s s

s

s

s ) ) ) )

)

(2)

* School of Automobile, Chang’an University (nuaaliyaohua@126.

com)

Received 11 July 2011; Accepted 07 May 2012

Here we define q shown in (3). Substituting (3) into (1) and (2) and neglecting the move of rotor flux, we can use f and Dd to show the control of the stator flux and torque angle.

s

s t

q V y⁾ ) ×D

= (3)

1 cos 2

1+ ²+ -

= q q a

f (4)

a d a

cos 2 1

arcsin sin

2 q

q q

+

= +

D (5)

According to (4) and (5), when 0<q<0.01 and 0^°<a<360^°, f versus a and Dd versus a are shown in Fig. 3 and Fig. 4.

Fig. 3. f versus a

Fig. 4. Dd versus a

Here we define f^* and Dd^* shown in (6) and (7). The error rate of f and Dd are shown in (8) and (9).

a

* qcos

f = (6) a

d^*=arcsin ×sin

D q (7)

% 100

*

- ´

= f

f f

hf (8)

% 100

*

D ´ D -

= D

D d

d

h _d d (9)

According to (8) and (9), when 0<q<0.01 and 0^°<a<360^°, hf versus a and hDd versus a are shown in Fig.

5 and Fig. 6.

Fig. 5. hf versus a

Fig. 6. hDd versus a

The DTC is a hysteresis control, so there are only two control signals: to increase or to decrease. Thus, as long as the error rate is lower than 100%, f^* and Dd^* can be used to replace f and Dd.

Fig. 5 shows that only in the area near zero crossings of f^* (90^° and 180^°), the error rate is more than 100%, and this area is very small. As such, f^* can be used to represent the control of stator flux except for in the area near 90^° and 180^°.

Fig. 6 shows that Dd^* can be used to represent the control of torque angle.

According to (6), the voltage vector selection area needed to increase the amplitude of stator flux in terms of a is (-90^°, 90^°) and the voltage vector selection area needed to decrease the amplitude of stator flux in terms of a is (90^°, 270^°).

According to (7), the voltage vector selection area required to increase the torque angle in terms of a is (0^°, 180^°) and the voltage vector selection area needed to decrease the torque angle in terms of a is (180^°, 360^°).

3. Control of Torque

In the stator flux reference frame, torque of the PMSM in terms of the amplitude of stator flux and torque angle is

presented in (10). It consists of two terms: the first term is the excitation torque produced by the permanent magnet flux and the second term is the reluctance torque produced by the saliency of the motor.

f q

s d q d

s f

e L

L k L

L k T p

y d y

d y d

y ) ( ))

), cos sin 2 (sin

3 -

= -

= (10)

According to (1), (2) and (10), we can use M shown in (11) to represent the change of total torque of the PMSM due to the application of the voltage vector for Dt. It also consists of two terms: the first term is M1 to represent the change of excitation torque shown in (12) and the second term is M2 to represent the change of reluctance torque shown in (13).

2

1 M

M

M= + (11)

a d d a

a ) sin

cos 2 1 arcsin sin sin(

cos 2

1 2

2

1 -

+ + +

+ +

= q q

q q q

M (12)

] cos sin cos ) 2 1 arcsin sin cos(

cos ) 2 1 arcsin sin sin(

) cos 2 1 [(

2

2 2

2

d a d

d a

a d a

a + - + +

+ + +

+ + -

=

q q

q

q q q q

q k

M (13)

When q=0.01, 0^°<a<360^° and 0^°<d<120^°, M1 versus a is shown in Fig. 7.

Fig. 7. M1 versus a

Here we define M1* shown in (14). The error of M1 is shown in (15). According to (15), when d=70^°, q=0.01 and 0^°<a<360^°, hM1 versus a is shown in Fig. 8.

)

* sin(

1 = q× a +d

M (14)

% 100

1

* 1 1

1= - ´

M M M

hM (15)

Fig. 8. hM1 versus a @ d=70^°

Fig. 8 shows only at zero crossings of M1 (110 and 290^°), the error rate is more than 100%. So M1* can be used to represent the control of excitation torque except for 110^° and 270^°.

When q=0.01, 0^°<d<120^° and 0^°<a<360^°, hM1 versus a is shown in Fig. 9.

Fig. 9. hM1 versus a

Fig. 9 shows that only in the area near zero crossings of M1* (180^°-d and 360^°-d), the error rate is more than 100%

and this area is very small. Thus, M1* can be used to represent the control of the excitation torque in all areas except for the area near 180^°-d and 360^°-d.

According to (12), the voltage vector selection area to increase the excitation torque in terms of a is (-d, 180^°-d) and the voltage vector selection area to decrease the excitation torque in terms of a is (180^°-d, 360^°-d).

When q=0.01, k=1, 0^°<d<120^° and 0^°<a<360^°, M2

versus a is shown in Fig. 10.

Fig. 10. M2 versus a

Here we define M2* shown in (16); the error rate of M2

is shown in (17). According to (17), when d=30^°, k=1, q=0.01 and 0^°<a<360^°, hM2 versus a is shown in Fig. 11.

) 2

* sin(

2=-k×q× a + d

M (16)

% 100

2

* 2 2

2= - ´

M M M

hM (17)

Fig. 11. hM2 versus a @ d=30^°

Fig. 11 shows that only in the area near zero crossings of M2* (120^° and 300^°), the error rate is more than 100% and this area is very small, as shown in Fig. 12 and Fig. 13.

Fig. 12. hM2 versus a @ d=30^°

Fig. 13. hM2 versus a @ d=30^°

Thus, M2* can be used to represent the control of the reluctance torque except for the area near 120^° and 300^°.

When q=0.01, k=1, 0^°<d<120^° and 0^°<a<360^°, hM2 versus a is shown in Fig. 14.

Fig. 14. hM2 versus a

Fig. 14 shows that only in the area near zero crossings of M2* (180^°-2d and 360^°-2d), the error rate is more than 100%

and this area is very small. As such, M2* can be used to represent the control of the reluctance torque in all areas except for the area near 180^°-2d and 360^°-2d.

According to (16), the voltage vector selection area to decrease the reluctance torque in terms of a is (-2d, 180^°-2d) and the voltage vector selection area to increase the reluctance torque in terms of a is (180^°-2d, 360^°-2d).

When q=0.01, k=1, 0^°<a<360^° and 0^°<d<120^°, M versus a is shown in Fig. 15.

Fig. 15. M versus a

According to (14) and (16), we define M^* shown in (18).

] sin ) cos(

) cos 1 )(

[sin(

) 2 sin(

) sin(

* 2

* 1

*

d d a d

d a

d a d

a

+

× -

× - +

=

+

×

× - +

×

= +

=

k k

q

q k q

M M M

(18)

Here we define l shown in (19), and (18) can be rewritten into (20).

2

2 ( sin )

) cos 1

( arcsin sin

d d

l d

× +

× -

= ×

k k

k (19)

) sin(

) sin ( ) cos 1

( ^{2 2}

* =q× -k× d + k× d a+d-l

M (20)

The error rate of M is shown in (21). According to (19) and (20), when k=1 and d=40^°, (d-l)=-30^°. Thus, when

d=40, k=1, q=0.01 and 0<a<360, hM versus a is shown in Fig. 16.

% 100

*

- ´

= M

M M

hM (21)

Fig. 16. hM versus a @ d=40^°

Fig. 16 shows that only in the area near zero crossings of M^* (30^° and 210^°), the error rate is more than 100% and this area is very small, as shown in Fig. 17 and Fig. 18.

Fig. 17. hM versus a @ d=40^°

Fig. 18. hM versus a @ d=40^°

Thus, M^* can be used to represent the control of torque except for the area near 30^° and 210^°.

When q=0.01, k=1, 0^°<d<120^° and 0^°<a<360^°, hM versus a is shown in Fig. 19.

Fig. 19. hM versus a

Fig. 19 shows that only in the area near zero crossings of M^* (l-d and 180^°+l-d), the error rate is more than 100%

and this area is very small. Therefore, M^* can be used to represent the control of torque in all areas except for the area near l-d and 180^°+l-d.

According to (17), the voltage vector selection area to increase torque in terms of a is (l-d, 180^°+l-d) and the voltage vector selection area to decrease torque in terms of a is (180^°+l-d, 360^°+l-d).

4. Voltage Vector Selection Area

According to the control of the stator flux, torque angle and torque, the voltage vector area of the PMSM DTC system is shown as follows, where a1 to a4 are shown in (22):

The selection area of the voltage vector V11 to increase the stator flux, torque angle and torque is (a1, 90^°).

The selection area of the voltage vector V01 to decrease the stator flux and increase the torque angle and torque is (90^°, a2).

The selection area of the voltage vector V00 to decrease the stator flux, torque angle and torque is (a3, 270^°).

The selection area of the voltage vector V10 to increase the stator flux and decrease the torque angle and torque is (270^°, a4).

ïï î ïï í ì

+ -

=

+ -

=

+ -

=

-

=

°

°

°

°

°

°

°

) 360 ,

360 min(

) 180 ,

180 max(

) 180 ,

180 min(

) , 0 max(

4 3 2 1

d l a

d l a

d l a

d l

a (22)

5. Voltage Vector Selection Strategy Any voltage vector in the selection area can be used to control the stator flux and torque, so there are unlimited voltage vector selection strategies. As well, space vector

modulation (SVM) must be used to generate the applying voltage vector.

The implementation of the voltage vector selection strategy of the PMSM DTC system is based on the following steps and inputs:

1) k: it is determined by the amplitude of the stator flux and parameters of the PMSM.

2) torque angle: it can be calculated from (10) or obtained from a lookup table representing (10).

3) (d-l): it can be calculated from (19).

4) a1 to a4: they can be calculated from (22) using (d-l).

5) voltage vector selection area: it can be obtained using a1 to a4, and outputs of hysteresis comparators for the stator flux and torque.

6) angular position of applying the voltage vector: it can be obtained using the angular position of the stator flux in the stationary reference frame, voltage vector selection area and strategy.

7) applying voltage vector: SVM is used to generate it.

Thus the diagram of the DTC for the PMSM using the voltage vector selection strategy is shown in Fig. 20.

PMSM

SA

Stator flux comparator

+_-

+_- Voltage SVM VSI

vector selection

strategy

S_B SC

Torque Comparator

Lookup table

Lookup table

*

y)s

y⁾s

*

Te

Te

y⁾s

Te

d k

) (d -l

t f

Vrs

Ð

qs

Fig. 20. The DTC for the PMSM using voltage vector Selection stratage

6. Conclusions

Simplified expressions to represent the change of stator flux, torque angle, excitation torque, reluctance torque and total torque of a PMSM DTC system due to the application of a voltage vector are proposed in this paper. Based on these expressions, the voltage vector selection area of these variables are given as follows:

The voltage vector selection area needed to increase the amplitude of stator flux in terms of a is (-90^°, 90^°) and the voltage vector selection area needed to decrease the amplitude of the stator flux in terms of a is (90^°, 270^°).

The voltage vector selection area needed to increase the torque angle in terms of a is (0^°, 180^°) and the voltage vector selection area needed to decrease the amplitude of the stator flux in terms of a is (180^°, 360^°).

The voltage vector selection area needed to increase the excitation torque in terms of a is (-d, 180^°-d) and the voltage vector selection area needed to decrease the excitation torque in terms of a is (180^°-d, 360^°-d).

The voltage vector selection area needed to decrease the reluctance torque in terms of a is (-2d, 180^°-2d) and voltage vector selection area needed to increase the reluctance torque in terms of a is (180^°-2d, 360^°-2d).

The voltage vector selection area needed to increase torque in terms of a is (l-d, 180^°+l-d) and the voltage vector selection area needed to decrease torque in terms of a is (180^°+l-d, 360^°+l-d).

The voltage vector selection area of the PMSM DTC system and the implementation of the voltage vector selection strategy of the PMSM DTC system are studied.

Acknowledgements

This work was supported by the China Postdoctoral Science Foundation (20110491636) and the Special Fund for Basic Scientific Research of Central Colleagues, Chang’an University (CHD2011JC136).

References

[1] M. F. Rahman, L. Zhong, W. Y. Hu, K. W. Lim, and M. A.

Rahman, “A direct torque controller for permanent magnet synchronous motor drives,” IEEE Transactions on Energy Conversion, vol. 14, pp. 637-642, Sept. 1997.

[2] M. F. Rahman, L. Zhong, and K. W. Lim, “A direct torque controlled interior magnet synchronous motor drive incorporating field weakening,” IEEE Transactions on Industry Applications, vol. 34, pp. 1246-1253, Nov./Dec.

1998.

[3] L. Zhong, M. F. Rahman, W. Y. Hu, and K. W. Lim,

“Analysis of direct torque control in permanent magnet synchronous motor drives,” IEEE Transactions on Power Electronics, vol. 12, pp. 528-536, May. 1997.

[4] M. F. Rahman, M. E. Haque, L. X. Tang, and L. Zhong,

“Problems associated with the direct torque control of an interior permanent-magnet synchronous motor drive and their remedies,” IEEE Transactions on Industrial Electronics, vol.

51, pp. 799-809, Aug. 2004.

[5] Y. H. Li, “Direct Torque Control of Permanent Magnet Synchronous Machine,” Shaker Verlag Gmbh, 2010.

[6] Y. H. Li, W. G. Liu, “Simulation Study on the Effect of Voltage Vector on Torque in Direct Torque Control System of Permanent Magnet Synchronous Motor,” in Second IEEE Conference on Industrial Electronics and Applications, pp.

1521-1525, 2007.

[7] Y. H. Li, D. Gerling, W. G. Liu, “A Novel Switching Table to Suppress Unreasonable Torque Ripple for the PMSM DTC Drives,” in 11th International Conference on Electrical Machines and Systems, pp. 972-977, 2008.

[8] Y. H. Li, D. Gerling, W. G. Liu, “The Control of Stator Flux and Torque in the Surface Permanent Magnet Synchronous

Conference on Industrial Electronics and Applications, pp.

2004-2009, 2009.

[9] Y. H. Li, D. Gerling, W. G. Liu, “A Simplified Voltage Vector Selection Strategy for the Permanent Magnet Synchronous Motor Direct Torque Control Drive withLow Torque Ripple and Fixed Switching Frequency,” in 13rd International Conference on Electrical Machines and Systems, pp. 674-679, 2010.

[10] Y. H. Li, D. Gerling, W. G. Liu, “The Comparison of Control Strategies for the interior PMSM drive used in the Electric Vehicle,” in 25th World Battery, Hybrid and Fuel Cell Electric Vehicle Symposium & Exhibition, Report number:

4C-7, 2010.

Yaohua Li received his Dr.-Ing degree in electrical engineering from Univeristy of Federal Defense, Munich, Germany. His research interests are electrical vehicles, high-performance motor control and advanced digital control with real-time implementation.

Jian Ma received his Ph. D. degree in vehicle engineering from Chang’an University, China.

Qiang Yu received his Ph. D. degree in vehicle engineering from Chang’an University, China.

Jingyu Liu received her Ph. D. degree in vehicle engineering from Chang’an University, China.