Non-Sinusoidal Electromotive Force Compensation of the PMLSM with Multiple-Frequency Resonant Controller

Ghislain REMY*, Jia ZENG*, Pierre-Jean BARRE*†, Philippe DEGOBERT*, and Jean-Paul HAUTIER*

* Ecole Nationale Supérieure d’Arts et Métiers, Lille, France, Laboratoire d’Electrotechnique et d’Electronique de Puissance de Lille1 (L2EP), Tel: (33)3.20.62.22.46 – Fax: (33)3.20.62.27.59

E-mail: †barre@lille.ensam.fr

Abstract This paper presents how to compensate for non-sinusoidal electromotive force (EMF) of the permanent magnet linear synchronous motor (PMLSM) with multiple-frequency resonant controller. After the modeling of non-sinusoidal EMF in a PMLSM, the multiple-frequency resonant controller is proposed to control the PMLSM in order to compensate for the negative influence of such EMF. The effectiveness of the suggested method is verified on a laboratory system equipped with a Rexroth LSP120C linear motor and a dSPACE DS1005 real-time controller board.

Keywords: PMLSM, Non-Sinusoidal Electromotive Force, Resonant Controller, Thrust Control

1. Introduction Nowadays, with its attractive characteristics, the

Permanent Magnet Linear Synchronous Motor (PMLSM) has been used in the high speed and high precision drive systems (1). Generally, the field-oriented control with PI controllers in synchronous d-q reference frame is used to control such systems. Nevertheless, this method is not capable of compensating for the negative influence of non sinusoidal EMF, which will bring on undesired ripples in the thrust.

For a PMLSM with non-sinusoidal back-EMF, the excitation currents should comprise appropriate harmonics in order to suppress the thrust ripple components (2)(3). Direct Torque Control (DTC) and the Multiple Reference Frame (MRF) theory have been applied to resolve this problem (4)(5). However, each method has its troublesome drawbacks: large torque ripple at low speeds, variable switching frequency (6) for DTC and complex control structure and large computation effort requirement for the MRF (7). 1

In this paper, we will propose to use the multiple-frequency resonant controller to compensate for the influence of the non-sinusoidal EMF. By associating an arbitrary number of resonant elements, this type of controller can eliminate the steady-state control errors of

1 G. Remy, J. Zeng, Ph. Degobert P. J. Barre, work in the research team of “Commande et Entraînement de Machines-Outils à DYNamique Elevée” (CEMODYNE), which is within the Laboratoire d’Electrotechnique et d’Electronique de Puissance de Lille (L2EP), ENSAM, 8 Bd Louis XIV, 59046 Lille Cedex, France, Tel: +33 3 20 62 22 46, Fax: +33 3 20 62 27 59 (e-mail: barre@lille.ensam.fr). This team is constituted of teachers - researchers of the L2EP, the LAGIS and the LML.

the fundamental component as well as the harmonics components at the same resonant frequencies (8).

After the modeling of the PMLSM with non-sinusoidal back EMF, optimal current waveforms are suggested to reduce the thrust ripple caused by the harmonics of the back-EMF. By using the multi-frequency resonant controllers, the load currents in the PMLSM can follow the reference currents perfectly, and thereby, significantly reduce the thrust ripple of the PMLSM.

2. Influence of Non-sinusoidal EMF on Thrust In this section, we present the model of a PMLSM with a

non-sinusoidal back EMF. To adapt the proposed control structure, this model is established in the Concordia Reference Frame (Fig. 1).

When the back-EMF wavefoms are half-wave

symmetrical, they have no even harmonics and can be expressed by:

[ ]( )( )( )( )( )

20 11

sin 2 1ˆ( ) sin 2 1 2 / 3

sin 2 1 4 / 3

p

f pn

p

n

n N xe t v n N x

n N xφ π

πλ

∞

=−

⎡ ⎤⎡ − ⎤⎣ ⎦⎢ ⎥= − ⋅ ⋅ ⎡ − − ⎤⎣ ⎦⎢ ⎥⎢ ⎥⎡ − − ⎤⎣ ⎦⎣ ⎦

∑ ........ (1)

Wherein ˆfφ denotes the maximum value of magnetic

excitation flux per phase in a natural a-b-c reference frame.

Fig. 1. Principle schematic of PMLSM

The coefficients 2 1nλ − represent the relative magnitudes of the (2n-1)’th back-EMF harmonic. Np = π / τ is the electrical position constant of the PMLSM (9) (τ: step between two consecutive magnetic poles of the secondary) (10). Finally, ν denotes the linear speed of the mobile part.

Then, the voltage equations of the PMLSM in the Concordia reference frame can be expressed as:

[ ] [ ] [ ] [ ] [ ] [ ]

( )( )

2 1

2 1 2 11

sin 2 13 ˆ2 cos 2 1

n

p f n nn

d L d IV R I Ldt dt

K nN v

K n

αβ αβαβ αβ αβ

α

β

θφ λ

θ

−∞

− −=

⎛ ⎞= + ⋅ + ⋅⎜ ⎟⎝ ⎠

⎡ ⎤⋅ ⎡ − ⎤⎣ ⎦+ ⋅ ⋅ ⋅ ⎢ ⎥⋅ ⎡ − ⎤⎢ ⎥⎣ ⎦⎣ ⎦∑

. (2)

R represents the stator resistances; Vαβ and Iαβ denote the voltage and current vectors, respectively.

The coefficients 2 1nKα− and 2 1nKβ

− are triply periodic functions. Table 1 gives their values based on the independent periods n:

Table 1. Value of two periodic functions

independent periods n 2 1nKα− 2 1nKβ

−

1 -1 1 2 0 0 3 -1 -1 4 -1 1

The inductance matrix is given by:

[ ] 00

s s

s s

L ML

L Mαβ

−⎡ ⎤= ⎢ ⎥−⎣ ⎦

................................................ (2)

The electromagnetic thrust generated by the PMLSM can be derived by using co-energy techniques (11), which are created by the mutual coupling between the winding currents and the permanent magnetic field. It may be expressed as:

( ){ } ( ){ }( ){ } ( ){ }

6 1 6 10 1

6 1 6 10 1

3 ˆ2

sin 6 1 sin 6 1.

cos 6 1 cos 6 1

e p f

n nn n

n nn n

T N i i

n n

n n

α βφ

λ θ λ θ

λ θ λ θ

∞ ∞

+ −= =

∞ ∞

+ −= =

= ⋅ ⋅ ⋅⎡ ⎤⎣ ⎦

⎡ ⎤− ⋅ ⎡ + ⎤ − ⋅ ⎡ − ⎤⎢ ⎥⎣ ⎦ ⎣ ⎦⎢ ⎥⎢ ⎥⋅ ⎡ + ⎤ − ⋅ ⎡ − ⎤⎣ ⎦ ⎣ ⎦⎢ ⎥⎣ ⎦

∑ ∑

∑ ∑

(3)

The studied PMLSM is a Rexroth LSP120C linear motor.

The coefficients of EMF harmonics are experimentally identified (12) and are listed in Table 2.

Table 2. Coefficients of Electromotive Force harmonics

Coefficients names 1λ 5λ 7λ 11λ

Coefficients values 1 -0.0267 0.000423 0.000459

If we neglect the effects of EMF harmonics, the

waveforms of currents iα and iβ should be sinusoidal. By substituting these coefficients into (3), we can notice that thrust ripples will be introduced by non-sinusoidal EMF.

The thrust ripple caused by the 5th EMF harmonic could reach 2.67%, which is the dominant ripple source. However, those caused by the 7th and 11th harmonics are so slight (0.1% in total) that their influence can be neglected.

3. Thrust Control Scheme In this section, we establish the PMLSM thrust control

scheme in the Concordia reference frame (Fig. 2).

Wherein eT ∗ represents the thrust command. The

reference excitation currents ( iα∗ and iβ∗ ) are generated according to the mover position θ and to the torque command eT ∗ , which are given by (4) and implemented in the block “Excitation Currents Generator”.

( ) ( )[ ]( )

( ) ( )[ ]( )

*5*

25

*5*

25

sin sin 53 ˆ 12 .cos cos 53 ˆ 12

e

p f

e

p f

Ti

N

Ti

N

α

β

θ λ θ

φ λ

θ λ θ

φ λ

⎧ ⋅ − + ⋅=⎪

⋅ ⋅ ⋅ −⎪⎪⎨

⋅ + ⋅⎪ =⎪⋅ ⋅ ⋅ −⎪⎩

................................ (4)

The three-phase load currents are measured, tranformed and regulated by two multiple-frequency resonant controllers in the Concordia reference frame. The mover position is detected and fed into the block of “Excitation Currents Generator” to generate instantaneous thrust command. The angular speed ω is estimated and fed into the multiple-frequency resonant controllers so that they can adapt to the reference currents with time-varying frequency.

The general transfer function of a multiple-frequency

resonant controller is given by:

( )( )

2

0

2 2

1

kj

jj

k

ii

b sC s

s ω

=

=

=+

∑

∏.................................................. (5)

Wherein k denotes the number of associated resonant elements and iω correspond to the concerned resonant frequencies. The design of these controllers relies on the parameters of the respective regulated systems and the configurations of the respective commands.

3.1 Subsystem parameters In considering the self and mutual inductances as time constants, the PMLSM model can be treated as two decoupled subsystems in αβ axes. Therefore, the two controllers should be separately designed according to their respective resistance and inductances.

*abcV

ω

θ

ai

bi

*Vα

*Vβ

*iα

*iβ

iα

iβ

( )2 2

ii

ik

b ss

α

ω∈Α

+∑

∏

abc

αβ

αβ

abc

eT ∗

(speed)vpN

θ

( )2 2

B

ii

ik

b ss

β

ω∈

+∑

∏

Fig. 2. PMLSM thrust control scheme using multiple-frequency resonant controllers

3.2 Resonant frequencies The numbers and frequencies of associated resonant elements depend on the references and/or the characteristics of each decoupled subsystem.

In this way, the tracking of the reference currents and the rejection of disturbances from non-sinusoidal back-EMF can be simultaneously realised.

In our case, the two controllers have identical structures with two resonant frequencies – the fundamental and the 5th harmonic. Fig. 3 illustrates the bode diagrams of the open-loop transfer function of the control system using a two frequency resonant controller. We can note that infinite gains are produced at the concerned frequencies ( , 5p pω ω ), which ensures that the steady-state error at these frequencies can be completely eliminated.

The bode diagram of the closed-loop transfer function

(Fig. 4) presents a characteristic of unity gains (0dB) and zero phases at the selected frequencies, further confirming that the control system can perform zero steady-state error at these frequencies.

4. Experimental Results The proposed approach is experimentally verified on a

laboratory test system equipped with a Rexroth LSP120C linear motor (Fig. 5). Table 3 lists the specifications of the test system. The control scheme depicted in Fig. 2 is implemented in a dSPACE DS1005 real-time digital control card to drive the PMLSM through an IGBT inverter. We have used a Heindenhain exposed linear encoder with a grating period of 20µm, which is a high precision incremental encoder, to detect the mover position.

Fig. 6 and Fig. 7 present the experimental references, measurments and estimations of currents and thrust.

Table 3. Specifications and Parameters of Test System

Specifications Parameter Value

Inductance 16.2[ ]sL mH=

Resistance 1.1[ ]sR = ΩMax value of magnet excitation flux per phase

ˆ 0.65[ ]f Wbφ =

Pole pitch [ ]37.5 mmτ =

Electrical position constant 183.8pN mm−⎡ ⎤= ⎣ ⎦

PMLSM

Mass of mobile part 220[ ]M kg=

Switching Frequency of IGBT 10 kHz[ ]

Fig. 6 we have used a AC current control system using

one-frequency resonant controllers. Fig. 7 we have used an AC current control system using two-frequency resonant controllers, Fig. 7. In both cases, the load currents are very close to their referencies. The maximal delay of the load currents stays under 0.5ms, even if brutal changes occur in current references. When the non-sinsuoidal emf is not compensated (Fig. 5), we notice a ripple of 5% on thrust estimation. After the compensation using two-frequency resonant controllers, the ripple is reduced to 1% of the estimated thrust. Nevertheless, the load current in the second case is noisier (by over 300%), because we need to inject the measure of the 5th harmonic of the currents.

5. Conclusion In this paper, we have first analysed the influence of

non-sinusoidal back-EMF on the thrust generated by the PMLSM. Based on the proposed optimal excitation current waveforms, the thrust control scheme is established in the Concordia reference frame. The multiple-frequency resonant controllers are used to regulate the load currents, and morever, they compensate for the negative influence of back-EMF harmonics. Thanks to this type of controller, the load currents can perfectly track their respective references, and the thrust ripple caused by the back-EMF harmonics can be significantly reduced. The experimental results obtained from a laboratory test system verify the effectiveness of the suggested approach. The next paper will be on the implementation of resonant controllers inside an industrial drive with a classical PI cascaded structure.

Fig. 5. Linear Motor Rexroth LSP120C

102103 104101

Frequency (rad/sec)

-225

-180

-45

0

100

200

105

102 103 104101

Frequency (rad/sec)

105

pω

pω 5 pω

5 pω

300

-135

-90

0

400

-100

Fig. 4. Bode diagram of closed-loop transfer function

Phas

e (d

eg)

Mag

nitu

de (d

B)

pω

pω

5 pω

5 pω

Fig. 3. Bode diagram of open-loop transfer function

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(5) Jawad Faiz, B. Rezaei-Alam, “Control of a linear Permanent Magnet Synchronous Motor using Multiple Reference Frame Theory”, Maglev’2002, September 2002, Lausanne, Switzerland, Pages: 97-101.

(6) Jun-Koo Kang, Seung-Ki Sul, “New direct torque control of induction motor for minimum torque ripple and constant switching frequency”, IEEE Transactions on Industry Applications, Volume: 35, Issue: 5, Sept.-Oct. 1999, Pages: 1076-1082.

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(10) J. Zeng, P.J. Barre, P. Degobert, “Modeling and Thrust Control of PMLSM using Principle of Local Energy,” ICEMS2003, Beijing, P.R. China, 2003.

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(12) G. Remy, A. Tounzi, P.-J. Barre, F. Francis, J.-P. Hautier, "Finite-Element Analysis of Non-Sinusoidal Electromotive Force in a Permanent Magnet Linear Synchronous Motor", The Fifth International Symposium on Linear Drives for Industry Applications LDIA2005, kobe-Awaji, japan, 25-28 Sept. 2005.

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)i β

ref/ i

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eas

(A)

i αre

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αm

eas

(A)

Time (s)

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f/ T

e es

t(N

)i β

ref/ i

βm

eas

(A)

i αre

f/ i

αm

eas

(A)

Time (s)

Fig. 7. AC current control system using two-frequency resonant controllers

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ref/ i

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eas

(A)

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eas

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Time (s)

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1450

1500

1550

T e re

f/ T

e es

t(N

)i β

ref/ i

βm

eas

(A)

i αre

f/ i

αm

eas

(A)

Time (s)

Fig. 6. AC current control system using one-frequency resonant controllers