A flux estimation method for a permanent-magnet synchronous motor

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Journal of Magnetism and Magnetic Materials 282 (2004) 355–359

A ﬂux estimation method for a permanent-magnet

synchronous motor

Shyh-Jier Wang

a,b,

*, Chich-Hsing Fang

a

, Shir-Kuan Lin

a a

Department of Electrical and Control Engineering, National Chiao Tung University, 1001 Da Hsueh Road, Hsinchu 300, Taiwan b

Magnetics Department OES/ITRI, Bldg. 78, 195-8 Sec. 4, Chung Hsing Road, Chutung, Hsinchu 310, Taiwan Available online 4 May 2004

Abstract

This paper deals with the ﬂux estimation of a permanent-magnet synchronous motor (PMSM). Contrary to the conventional no-load test, the proposed method needs no extra servomotor. It is simply to drive the PMSM in a single-phase mode with the currents large enough to make the PMSM rotate in the same direction. Under such a condition, the ﬂux of the PMSM can be easily estimated.

PACS: 84.50.+d

Keywords: PMSM; No-load test; Emf; Flux estimation

1. Introduction

The permanent-magnet synchronous motor (PMSM) has been used broadly in industry, because of its easy controllability and fast response. An accurate estimation of the ﬂux is useful for the design and the servo control of a PMSM. Several researches have dealt with the estimation of the ﬂux of a PMSM[1–3]. It is of no doubt that the no-load test method[1]has become the most popular one. In this method, an auxiliary motor is required to drive the PMSM at constant

speed. The windings of PMSM are at open-circuit so that the ﬂux can be estimated by the emf of the PMSM.

This paper tries to propose an alternative estimation method, which does not require the aid of an auxiliary servomotor. The proposed method is simply to drive the PMSM by exciting a single phase, say, let ia¼ ib; and ic¼ 0; where ia;

ib; and ic are the currents of phases a, b, c,

respectively. The ﬂux is then easily estimated using the measurement of three terminal voltages va; vb;

and vc:

A spindle motor in a 50 XCD-ROM driver is taken as an example to compare the estima-tion results between the proposed method and the conventional no-load test. It will be shown that the experiment results of both methods are very close.

*Corresponding author. Department of Electrical and Con-trol Engineering, National Chiao Tung University, 1001 Da Hsueh Road, Hsinchu 300, Taiwan. Tel.: +886-3-5913049; fax: +886-3-5918095.

E-mail address:[email protected] (S.-J. Wang).

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2. Proposed method

Consider a PMSM with three-phase and Y-connected windings, whose models can be de-scribed as [4] vas vbs vcs 2 6 4 3 7 5 ¼ rs 0 0 0 rs 0 0 0 rs 2 6 4 3 7 5 ia ib ic 2 6 4 3 7 5 þ Ls M M M Ls M M M Ls 2 6 4 3 7 5 dia dt dib dt dic dt 2 6 6 6 6 6 6 4 3 7 7 7 7 7 7 5 þ orlr cos yr cos yr 2p 3 cos yrþ 2p 3 2 6 6 6 6 6 4 3 7 7 7 7 7 5 ; ð1Þ

where vas; vbs; and vcs; are the terminal voltages

with respect to the neutral point s (seeFig. 1(a)), ia;

ib; and ic are the currents of phases a, b, c,

respectively, rsis the winding resistance per phase,

Ls is the self-inductance per phase, M is the

mutual inductance per phase, or is the electric

angular speed of the rotor, lris the maximum ﬂux

induced by the rotor magnet, and yr is the

rotational electrical angle of the rotor. Note that yr¼ 0 is the position where the intersection line of

the N–S and the S–N magnet is in alignment with the centerline of a tooth of phase a (seeFig. 1(b)). Furthermore, the output torque Te of the motor

can be obtained as Te¼ P 2lr ia ib 2 ic 2 cos yrþ ffiffiffi 3 p 2 ðib icÞsin yr ! ¼2J P dor dt þ 2Bm P orþ TL; ð2Þ where J is the inertia, P is the number of poles of the magnet, Bmis the damping ratio, and TLis the

loading torque. Suppose that the PMSM is operated in a single-phase mode, i.e., phase c is open, such that ia¼ ib; and ic¼ 0: Such a kind

of operation can be achieved by manipulating four transistors on legs, Leg2 and Leg3, in Fig. 1(a),

while turning those on Leg1 always open. Under

this situation, Eq. (1) is simpliﬁed in the form of vas ¼ va vs¼ rsi þ Ls di dtþ M di dtþ orlrcos yr; vbs ¼ vb vs¼ rsi Ls di dt M di dt þ orlrcos yr 2p 3 ; vcs ¼ vc vs¼ orlrcos yrþ 2p 3 ; ð3Þ

where va; vb; and vcare the terminal voltages, and

vs is the neutral voltage. Similarly, Eq. (2) is also

simplified as Te ¼ ffiffiffi 3 p P 2 lriðtÞcos yrþ p 6 ¼2J P dor dt þ 2Bm P orþ TL: ð4Þ According to Eq. (4), we can make the motor rotate continuously in one direction by simply

Fig. 1. (a) A three-phase PMSM with Y-connected windings and its driver. (b) The relative position between the stator and the rotor at yr¼ 0:

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assigning the current iðtÞ in phase with cos(yr+p/

6), and large enough to ensure Te> TL: Since the

values of vas; vbs; and vcsare not available and only

va; vb; and vccan be measured, we introduce a new

variable voðtÞ deﬁned as voðtÞ ðvaþ vb 2vcÞ 3 ¼ orlrcos yrþ 2p 3 : ð5Þ It follows from Eq. (5) and or¼ dyr=dt that

voðtÞdt ¼ lrcos yrþ

2p 3

dyr: ð6Þ

Integrating both sides of Eq. (6) leads to CðtÞ Z t 0 voðtÞdt ¼ Z yrð Þt yrð Þ0 lrcos t þ 2p 3 dt ¼ lrsin yrðtÞ þ 2p 3 lrsin yrð0Þ þ 2p 3 ¼ lrsin yrðtÞ þ 2p 3 þ C0 ð7Þ

where C0 is a constant. Consequently, the ﬂux

estimation method is that the PMSM is operated in a single-phase mode with the current in phase with cos(yr+p/6) and large enough to overcome

the load TL; and the measured histograms of va; vb;

and vc are used to calculate voðtÞ in Eq. (5) and

then CðtÞ ¼R₀tvoðtÞdt: The ﬂux lrof the PMSM is

then the amplitude of the sinusoidal part of CðtÞ according to Eq. (7).

3. Implementation

A three-phase, 12-pole, and nine-slot DC brushless motor used as a spindle motor of a 50 XCD-ROM is taken as an example. The motor with lr¼ 7:9 104Wb-turn has the

sur-face-mounted NdFeB magnet rotor. There are three Hall elements Ha; Hb; and Hc mounted on

the stator (seeFig. 1(b)), each of which generates a pair of differential signals, e.g., Hþ

a and Ha; as the

feedback signals for the motor driver. Now, we want to establish a driving system that drives the PMSM only in a single-phase mode as described in Section 2. First, the winding labeled c in

Fig. 1(a) should be ﬂoated from the transistors.

When transistors Tr4 and Tr5 in Fig. 1(a)are on and Tr3 and Tr6 are off, the currents of the PMSM are ia¼ ib; and ic¼ 0 as shown inFig.

2(c). On the other hand, the currents are reversed when Tr3 and Tr6 are on and Tr4 and Tr5 are off as shown in Fig. 2 (b). Both cases of the single-phase mode can be implemented with a BA6849 chipof the Rohm Company.

The logic table of the BA6849 is shown in

Table 1. It is apparent that the states 1 and 4 meet the requirements of the circuits ofFig. 2(b) and (c), respectively. In these two states, the inputs H_a; H_bþ; H_b; and Hcshould be retained at the level of

M ¼ 2:5 V, while those of H_aþ and H_cþ are in opposite levels, i.e., H_cþ ¼ L; when Hþ

a ¼ H: The

implementation circuit is then connected to the input pins H_a; H_bþ; Hb, and Hc to a power

source of the voltage level 2.5 V, and to pass the signal of H_aþalso to pin H_cþvia a NOT device (see

Fig. 2(a)). The output signals of the Hall sensor Ha

are operated by a comparator to generate the signal for the input pin Hþ

a of the BA6849. If the

positive end of the sensor Ha has a higher voltage

level than the negative end, the output of the comparator is 2.6 V. If the positive end has a lower voltage level, then the output of the comparator is 2.4 V. The rotor ﬂux linkage measured by the Hall sensor Ha is so arranged to be proportional

Fig. 2. (a) The modiﬁed driven circuits of the Hall elements. (b) The PMSM is operated by state 1. (c) The PMSM is operated by state 4.

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to sin(yr+2p/3)=cos(yr+p/6). Thus, the input

signal of H_aþ is in phase of cos(yr+p/6), so that

the current i is in phase of cos(yr+p/6).

An experiment is conducted by the implemented system for the PMSM described above. The current i generated by the BA6849 is measured and shown in Fig. 3(a), while the input signal of pin H_aþ is shown in Fig. 3(b). It is apparent that the current i is positive when the state of H_aþis L; and negative when that of H_aþ is H. The voltages of va; vb; and vc are also measured to

compute vo by Eq. (5) and thus C by Eq. (7),

whose results are shown in Fig. 4(a) and (b), respectively.

The constant part of C in Fig. 4(b) is about C0¼ 7:8 104Wb-turn, while the amplitude

of the sinusoidal part of C is 7:7 104Wb-turn. Thus, lr¼ 7:7 104Wb-turn. The conventional

method with no-load test uses a servomotor to drive a PMSM at a constant speed. Since the PMSM is free to run, the three phase currents of the PMSM are all zero, i.e., ia¼ ib¼ ic¼ 0:

Under such a situation, the emf va and vb of

phases a and b are measured to compute vab¼

vas vbs¼ va vb¼

ffiffiffi 3 p

lrorcosðyrþp=6Þ; which

follows from Eq. (1) for ia¼ ib¼ ic¼ 0: Because

the speed of the PMSM is known, lr is then

ð1=ðpffiffiffi3orÞ times the amplitude of the sinusoidal

vab: Fig. 4(c) shows the experiment results of the

Fig. 3. (a) The i2t plot of a PMSM in the single-phase rotation. (b) The plot of the input signal of pin Hþ

a and time t:

Fig. 4. (a) The plot of vo2t:(b) The plot of C2t: (c) The line

emf vabof the PMSM driven by an auxiliary motor.

Table 1

The logic table of BA6849

N Tr1 Tr2 Tr3 Tr4 Tr5 Tr6

Output table

1 On Off On Off Off On

2 On Off Off On Off On

3 On Off Off On On Off

4 Off On Off On On Off

5 Off On On Off On Off

6 Off On On Off Off On

N Hþ a Ha Hbþ Hb Hcþ Hcþ Input table 1 H M M M L M 2 M M H M L M 3 L M H M M M 4 L M M M H M 5 M M L M H M 6 H M L M M M H ¼ 2:6 V, M ¼ 2:5 V, L ¼ 2:4 V.

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conventional method, which allows us to calculate out lrE7:62 104Wb-turn. This veriﬁes that the

proposed method is reliable.

4. Conclusion

This paper proposes a ﬂux estimation method for a PMSM. The PMSM is operated in a single-phase mode and the currents are controlled so that the PMSM rotates in the same direction. Under such a condition, calculating out CðtÞ in Eq. (7) allows us to obtain the ﬂux lras the amplitude of

the sinusoidal part of CðtÞ: An implemented system with a BA6849 chipis established to verify the proposed method.

Acknowledgements

The paper was in part supported by the National Science Council, Taiwan under grant no. NSC 92-2213-E-009-057.

References

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[3] T. Hisashi, K. Takashi, T. Hirokazu, in: Proceedings of the Power Conversion Conference, Vol. 2, 1997, p. 673. [4] D.W. Novotny, T.A. LIPO, Vector Control and Dynamics