FORCE RIPPLE COMPENSATION OF LINEAR SYNCHRONOUS MOTORS · Christof Röhrig: Force Ripple Compensation of Linear Synchronous Motors 3 The star connection is the most common configura-tion

FORCE RIPPLE COMPENSATION OF LINEAR

SYNCHRONOUS MOTORS

Christof Röhrig

Asian Journal of Control

Vol. 7, No.1, pp. 1-11, March 2005

Asian Journal of Control, Vol. 7, No.1, pp. 1-11, March 2005 1

Manuscript received A? 1?, 2004; accepted ??, 2004.

Christof Röhrig is with Department of Computer Science,

University of Applied Sciences Dortmund, D-44227 Dort-

mund, Germany.

Manuscript received April 9, 2004; accepted June 16, 2004.

The author is with Department of Computer Science, Uni-

versity of Applied Sciences Dortmund, D-44227 Dortmund,

Germany.

FORCE RIPPLE COMPENSATION OF LINEAR

SYNCHRONOUS MOTORS

Christof Röhrig

ABSTRACT

Linear synchronous motors are finding expanded use in high-perform-

ance applications where high speed and high accuracy is essential. The main

problem in improving the tracking performance of linear synchronous mo-

tors is the presence of force ripple caused by mismatched current waveforms

and unbalanced motor phases or amplifier gains. This paper presents a

method to optimize the waveform of the phase currents in order to generate

smooth force. The optimized current waveforms produces minimal copper

losses and maximize motor efficiency. The waveforms are implemented in a

waveform generator of the motion controller and approximated with Fourier

series. The optimization method consist of three stages. In every stage, dif-

ferent harmonics of the force ripple are reduced. A comparison of the track-

ing performance with optimized waveforms and with sinusoidal waveforms

shows the effectiveness of the proposed method.

KeyWords: Motor control, force ripple, linear synchronous motors, opti-

mization.

I. INTRODUCTION

Permanent magnet (PM) linear synchronous motors

(LSM) are beginning to find widespread industrial appli-

cations, particularly for tasks requiring a high precision in

positioning such as various semiconductor fabrication and

inspection processes [1]. PM LSMs have better perform-

ance and higher power density than their induction coun-

terparts [2]. PM LSMs have a significant application po-

tential for a wide range of industrial applications [3-5].

The main benefits of PM LSMs are the high force density

achievable and the high positioning precision and accu-

racy associated with the mechanical simplicity of such

systems. The electromagnetic force is applied directly to

the payload without any mechanical transmission such as

chains or screw couplings. This greatly reduces nonlin-

earities and disturbances caused by backlash and addi-

tional frictional forces [6]. Today’s state-of-the-art linear

motors can achieve velocities up to 10m/s and accelera-

tions of 25g [7].

The more predominant nonlinear effects underlying a

PM LSM system are friction and force ripple arising from

imperfections in the underlying components [8]. In order

to avoid force ripple, different methods have been devel-

oped. In [9], a force ripple model is developed and identi-

fication is carried out with a force sensor and a fric-

tionless air bearing support of the motor carriage. In [10],

several techniques of torque ripple minimization for ro-

tating motors are reviewed. The design of linear motors

can be adapted to reduce force ripple. The cost of reduc-

ing force ripple by means of motor design is a decrease of

the mean force [11]. In [12], a neuronal-network based

feedforward controller is proposed to reduce the effect of

force ripple. Position-triggered repetitive control is pre-

sented in [13]. Other approaches are based on disturbance

observers [14,15], iterative learning control [16] or adap-

tive control [17,8,18]. The main problem in adaptive con-

trol is the low signal-to-noise ratio at high motor speeds

[19]. In [20], a force ripple compensation method for PM

LSM systems with electronic commutated servo amplifi-

ers was presented. A model based method was chosen,

because force ripple is a highly reproducible and

time-invariant disturbance [21]. The parameters of the

force ripple are identified in an offline procedure. The

2 Asian Journal of Control, Vol. 7, No. 1, March 2005

proposed method applies a feedforward controller for

compensation of force ripple and friction. The proposed

compensation method reduces the force ripple produced

by the imperfections of the underlying components, par-

ticularly non-optimal commutation of phase currents in

the servo amplifier. The drawback of controller-based

compensation methods in conjunction with electronic

sensor-based commutation is the non optimal current ex-

citation of the motor. Imperfect commutation of the phase

currents produces additional heat in the phase windings

and therefore reduces the motor efficiency.

In this paper, a force ripple compensation method for

software commutated servo amplifiers is proposed. The

additional input signal of a software commutated ampli-

fier is used to optimize the operation of the motor. The

paper proposes a method for design of the phase current

commands for minimization of force ripple and maximi-

zation of motor efficiency. The waveforms of the phase

currents are optimized in order to get smooth force and

minimal copper losses. In order to optimize the current

waveforms, the force functions of the phases are identi-

fied. The identification is performed by measuring the

control signals in a closed position control loop. The

waveform generation is integrated in the software com-

mutation module of the motion controller. The optimized

current waveforms are approximated with Fourier series.

This paper extends the work the author has presented in

[22] and [23].

II. EXPERIMENTAL SETUP

2.1 Linear motor

The motors considered here are PM LSM with epoxy

cores. A PM LSM consists of a secondary and a moving

primary. There are two basic classifications of PM LSMs:

epoxy core (i.e. non-ferrous, slotless) and iron core. Ep-

oxy core motors have coils wound within epoxy support.

These motors have a closed magnetic path through the

gap since two magnetic plates “sandwich” the coil as-

sembly [24]. Epoxy core linear motors can reach very

high dynamics and accuracy. This type of motor have the

highest force-to-inertia ratio [1]. Epoxy core LSMs are

key components of semiconductor manufacturing [7].

Figure 1 shows an unmounted PM LSM with epoxy core.

The secondary induces a multipole magnetic field in the

air gap between the magnetic plates. The magnet assem-

bly consists of rare earth magnets, mounted in alternate

polarity on steel plates. The electromagnetic thrust force

is produced by the interaction between the permanent

magnetic field in the secondary and the magnetic field in

the primary driven by the phase currents of the servo am-

plifier. The linear motor under evaluation is a current-

controlled three-phase motor driving a carriage supported

by roller bearings. The motor drives a mass of total 1.5kg

and is vertical mounted (Fig. 2).

2.2 Servo amplifier

The servo amplifiers employed in the setup are PWM

(pulse width modulation) types with closed current

control loop. Software commutation of the three phases is

performed in the motion controller by means of a position

encoder. This commutation method requires two current

command signals from the controller. The initialization

routine for determining the phase relationship is part of

the motion controller. The third phase current depends on

the others, because of the star connection of the motor

windings.

C A Bi i i= − − (1)

Fig. 1. Anorad LE linear motor [24].

Fig. 2. Experimental setup.

Christof Röhrig: Force Ripple Compensation of Linear Synchronous Motors 3

The star connection is the most common configura-

tion in three-phase motors [25]. The neutral of the star is

not brought out. The three external lines of the star are

connected to a half bridge circuit. The collection of three

half bridges is called a three-phase bridge. Only six

power electronic devices are needed in a three-phase

bridge, this is minimal compared to any other number of

phases. A two-phase motor needs eight power electronic

devices.

Figure 3 shows the block diagram of a software

commutated servo amplifier. The maximum of the ampli-

fier input signals uA, uB (±10V) correlates to the peak

currents of the current loops. In the setup, the peak cur-

rents of the amplifier are 25A. The PWM works with a

switching frequency of 24kHz. The current loop band-

width is specified with 2.5kHz [26]. Dynamics of the

amplifier will not be considered as they play a minor role

in the dynamics of the whole system.

III. SYSTEM MODELING

The thrust force is produced by interaction between

the permanent magnetic field in the secondary and the

electromagnetic field of the phase windings. The thrust

force is proportional to the magnetic field and the phase

currents iA, iB, iC. The back-EMF (electromotive force)

induced in a phase winding (eA, eB, eC) is proportional to

the magnetic field and the speed of the motor. The total

thrust force Fthrust is the sum of the forces produced by

each phases:

( ) { }thrust p pp

x F e x i p A B C= ; ∈ , ,∑� (2)

The back-EMF waveforms p

e

x�

can also be interpreted

as the force functions of the phases (KMp(x)).

( ) { }pthrust M p

p

F K x i p A B C= ; ∈ , ,∑ (3)

There are two types of position dependent distur-

bances: cogging force and force ripple. Cogging is a

magnetic disturbance force, that is caused by attraction

between the PMs and the iron part of the primary. The

force depends on the relative position of the primary with

respect to the magnets, and it is independent of the motor

current. Cogging is negligible in motors with iron-less

primaries [24].

Force ripple is an electromagnetic effect and causes a

periodic variation of the force constant. There are two

physical phenomena which lead to force ripple: Reluc-

tance force and harmonics in the electromagnetic force.

Reluctance force occurs only in motors with interior

mounted PMs. In this type of motor, the reluctance of the

motor is a function of position. The self inductance of the

phase windings varies with position of the primary with

respect to the secondary. When current flows, this causes

a position dependent force. If PMs are surface mounted,

reluctance is constant and reluctance force is negligible

[11].

In ironless and slotless motors, the only source of

force ripple are harmonics in the electromagnetic force.

Only if the back-EMF waveforms are sinusoidal and bal-

anced, symmetric sinusoidal commutation of the phase

currents produces smooth force. Force ripple occurs if the

motor current is different from zero, and its absolute

value depends on the required thrust force and the relative

position of the primary with respect to the secondary.

There are several sources of force ripple:

• Motor

− Harmonics of back-EMFs

− Amplitude imbalance of back-EMFs

− Phase imbalance of back-EMFs

• Amplifier

− Offset currents

− Imbalance of current gains

Fig. 3. Block diagram of a software commutated servo amplifier.

uA

Current Measurement

uB

CurrentController

PowerAmplifierPWM Linear MotorCurrent

Controller

Phase CurrentCommands

iA

iB

iC


Offset currents lead to force ripple with the same pe-

riod as the commutation period. This force ripple is inde-

pendent of the desired thrust force. Amplitude or phase

imbalance of the motor and imbalance of amplifier gains

lead to force ripple with half commutation period which

scale in direct proportion to the desired thrust force. A kth

order harmonic of a back-EMF produces (k − 1)th

and (k

+ 1)th order harmonic force ripple, if sinusoidal currents

are applied.

Force ripple with the same period as the commuta-

tion period is independent of the desired current, all

higher order harmonics scale in direct proportion to the

desired current, because of the linearity of the force

Eq. (2). Figure 4 shows the block diagram of a servo sys-

tem with PM LSM. The friction force is modeled with a

kinetic friction model. In the kinetic friction model, the

friction force is a function of velocity only.

IV. COMPENSATION OF FORCE RIPPLE

The method for force ripple compensation consist of

three stages. In every stage, different harmonics of the

ripple spectrum are reduced. In order to optimize the

waveforms of the currents, identification of the force

functions KMp(x) is essential. The main idea of the pro-

posed method is to identify the force functions in a closed

position control loop by measuring the control signal u of

the position controller at a constant load force Floak as a

function of the position x. In the experimental setup, the

constant load force is produced by the force of gravitation.

In order to avoid inaccuracy by stiction, the measurement

is achieved with moving carriage. The position of the

carriage is obtained from an incremental linear optical

encoder with a measurement resolution of 0.1µm.

4.1 Experimental analysis of force ripple

In order to analyze the force ripple of the motor, a

sinusoidal reference current is applied:

2( ) sin( ( ))

3A Au u u x o,ϑ = ϑ +

2 2π( ) sin ( )

3 3B Bu u u x o

⎛ ⎞,ϑ = ϑ + +⎜ ⎟⎝ ⎠

0with ( ) ( )p

x x x

πϑ = −τ

(4)

where uA, uB are the current commands of the two phases,

u is the output of the position controller (force command),

x is the position of the carriage, τp is the pole pitch and x0

is the zero position with maximum force. In the first stage,

the DC components of the command signals (oA, oB) are

chosen equal zero. Figure 5 shows the control signals u,

uA, uB versus the position x. The ripple on the force com-

mand signal u is caused by force ripple. The controller

compensates the force ripple by changing the force com-

mand over the position. If the speed of the motor is high,

the force ripple increases the tracking error.

Frequency domain analyses of the force command

indicates, that the fundamental corresponds to the com-

mutation period 2τp. In order to estimate the parameters

of the ripple, a least square estimation of the model

parameters (5) was applied. A least square estimation is

chosen, because noise overlays the force command signal.

1 2( ) θ θf x x, = +θ

2 1 2 2

1

θ sin π θ cos πτ τ

N

k kk p p

x xk k+ +

=

⎛ ⎞⎛ ⎞ ⎛ ⎞⎜ ⎟+ +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟

⎝ ⎠ ⎝ ⎠⎝ ⎠∑ (5)

where θk are the estimated parameters. The sum of load

force and friction is estimated in θ1. The modeled spring

force (θ2) is necessary, because the control signal rises

with rising positions (this is caused by wire chains).

Figure 6 shows the amplitudes of the sinusoids

2 2

2 1 2 2θ θk k+ ++ versus the order of the harmonics k.

Fig. 4. Nonlinear model of a PM LSM servo system.

1

m

Fthrust

Ffriction

Fload

xx x

Position

Phase CurrentCommands

ServoAmplifier

F=f(i ,i ,i ,x)A B Cu

B

uA

iA

iB

iC


Fig. 5. Phase currents at load force Fload = 20N.

Fig. 6. Spectrum of the force command signal.

The fundamental period (k = 1) corresponds to 2τp (30mm). The amplitude of this sinusoid is independent of

the force command. The higher order harmonics (k > 1)

scale in direct proportion to the force command, because

of the linearity of the force Eq. (3).

4.2 Compensation of command independent force

ripple

The curve of the phase current command uA, in Fig. 5,

shows, that this command independent ripple produces a

DC component of the current command. This DC com-

ponent compensates offsets in the analog circuits of the

servo amplifier. In the first stage, the DC components of

the phase current commands are calculated and applied in

(4) as

π πcos α sin α

6 3 63A

a a

o

⎛ ⎞ ⎛ ⎞= − − −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

2 πsin α

3 6B

a

o

⎛ ⎞= −⎜ ⎟⎝ ⎠

(6)

where

2 2 4

3 4

3

θand α arctanθ θ

θa = + =

In a servo system with iron-less motor, the DC com-

ponents compensate the offsets of the amplifier. If an iron

motor is employed, the DC components generate a sinu-

soidal force which compensates the fundamental of the

cogging force.

4.3 Identification of force functions

In the next stage, the force functions of the phases

are identified. In order to optimize the current waveforms,

it is essential to identify the amplitudes and phases of the

force functions properly. The motor efficiency depends

directly on a properly identified commutation zero posi-

tion x0, which depends on the phase lag of the force func-

tions. A square-wave commutation is applied, in order to

identify the force functions independently:

π0 : 0

6

π 5π:

6 63

5π 7π( ) 0 :

6 6

7π 11π:

6 63

11π0 : 2π

6

A

u

u u

u

⎧ ≤ ϑ <⎪⎪⎪ ≤ ϑ <⎪⎪⎪,ϑ = ≤ ϑ <⎨⎪⎪

− ≤ ϑ <⎪⎪⎪

≤ ϑ <⎪⎩

(7)

The second phase command signal is 2

3

π apart:

2π( )

3B Au u u u

⎛ ⎞,ϑ = ,ϑ +⎜ ⎟⎝ ⎠

(8)

In Fig. 7, a comparison of sinusoidal commutation

with square-wave commutation is given. Figure 8 shows

the phase currents iA, iB, iC versus the position x, when

square-wave commutation is applied. The minimum val-

ues of the phase currents correlate to the maximum values

of the force functions. In square-wave commutation, al-

ways two phases share the same absolute value of current,

−25 −20 −15 −10 −5 0 5 10 15−1.5

−1

−0.5

0

0.5

1

1.5

2

x [mm]

u [V

]

u u

A u

B

0 1 2 3 4 5 6 7 8 9 100

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

k

Am

plitu

de


Fig. 7. Sinusoidal and square-wave commutation.

which depends on the force command u and the current

gains of the amplifier KSp.

{ }3

pp S

u

i K p A B| | = ; ∈ , (9)

Since only two phase currents are independent, the force

equation can be described as:

( ) ( )thrust A A B BF K u K u= ϑ + ϑ (10)

where KA(ϑ) and KB(ϑ) are the force functions of the cur-

rent commands uA and uB respectively.

π π( ) :

6 6

π π( ) :

3 6 2

( ) 5π 7π( ) :

6 6

7π 9π( ) :

6 6

B

A

thrust

B

A

K

KF

uK

K

⎧ ϑ − ≤ ϑ <⎪⎪⎪ ϑ ≤ ϑ <⎪⎪= ⎨ϑ ⎪− ϑ ≤ ϑ <⎪⎪⎪− ϑ ≤ ϑ <⎪⎩

(11)

Figure 9 shows the identified force functions. Sinu-

soids are applied to approximate the force functions, be-

cause the back-EMFs of the motor are nearly sinusoidal.

4.4 Current waveform optimization

Aim of the current waveform optimization is to ob-

tain reference waveforms of the phase currents, which

generate smooth force. Inspection of the force Eq. (10)

reveals that there are an infinite number of waveforms

that generates smooth force. Therefore a secondary con-

dition has to be applied. Since one problem of epoxy-core

PM LSM is overheating, the logical choice is to minimize

Fig. 8. Square-wave commutation.

the winding losses. The problem is formulated as con-

strained optimization. The constraint of the optimization

is a position independent force

( ) ( ) ( )thrust A A B B FF K u K u K u f= ϑ + ϑ = ≠ ϑ (12)

where KF is a freely eligible constant and u is the force

command. The winding losses can be written as

2( ) ( ) { }cu p p

p

P x R i x p A B C= ; ∈ , ,∑ (13)

where Rp is the resistance of phase winding p. In assump-

tion of symmetric winding resistances RA = RB = RC and

servo amplifier gains KSA = KSB

, the functional to be

minimized can be written as

2 2( )A B A B A Bf u u u u u u, = + + (14)

If the resistances are unsymmetric and/or the gains

are unequal and known, (14) has to be adapted to meet the

requirements. In case of small unsymmetric resistances

and gains, the minimization of (14) minimizes the wind-

ing losses approximately. After substituting (12) in (14)

the optimized waveforms can be obtained by minimizing

(14)

( )0A B

A

A

f u uu

u

∂ ,= =∂

( )0B B

B

B

f u uu

u

∂ ,= =∂

(15)

as

1

2

2 2

( ) ( )( )

( ) ( ) ( ) ( )

A B

A F A

A B A B

K Ku u K u o

K K K K

ϑ − ϑ,ϑ = +

ϑ + ϑ − ϑ ϑ

0 30 60 90 120 150 180 210 240 270 300 330 360

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

ϑ [mm]

uA(1

, ϑ)

square−wave commutationsinusoidal commutation

−25 −20 −15 −10 −5 0 5 10 15−3

−2

−1

0

1

2

3

x [mm]

i A, i

B, i

C [A

]

iA

iB

iC


1

2

2 2

( ) ( )( )

( ) ( ) ( ) ( )

B A

B F B

A B A B

K Ku u K u o

K K K K

ϑ − ϑ,ϑ = +

ϑ + ϑ − ϑ ϑ

(16)

where oA, oB are the DC components obtained form (6).

In Fig. 10, the optimized phase current waveforms

are shown. Figure 11 compares the ripple of the force

commands by use of different commutation functions.

The dashed line shows the force command when a sinu-

soidal commutation plus offset compensation is employed

(Eq. (4)). The solid line shows the force command when

optimized waveforms are applied. The ripple of the force

command is significantly reduced when the optimized

waveforms are applied. The optimized waveforms maxi-

mize the motor efficiency by minimizing the ohmic

winding losses.

Fig. 9. Identified force functions.

Fig. 10. Optimized waveforms.

Fig. 11. Force command signals with sinusoidal versus opti-

mized waveforms.

4.5 Unbalanced sinusoidal force functions

In special case of unbalanced sinusoidal force func-

tions

( ) sin( α )A k kK aϑ = ϑ + , ( ) sin( β )B k kK bϑ = ϑ + (17)

sinusoidal commutation functions can produce smooth

force:

( ) sin( α )A u u Au u u a o,ϑ = ϑ + +

( ) sin( β )B u u Bu u ub o,ϑ = ϑ + + (18)

In consideration of (12), we obtain:

k u k ua a b b=

α α β β πk u k u+ = + ± (19)

which leads to

2

3

k

u

k

ba

a= ,

2

3

k

u

k

ab

b= (20)

and

3πα β

2u k= + ,

πβ α

2u k= + (21)

Figure 12 compares the unbalanced sinusoidal com-

mutation functions with the optimal commutation func-

tions. In both cases, the functions produces smooth force.

In Fig.13, the force commands for both cases are com-

pared. The optimized commutation functions lead to a

slightly lower ripple.

−25 −20 −15 −10 −5 0 5 10 15−15

−10

−5

0

5

10

15

x [mm]

kA, k

B [N

/ V

]

kA

kB

−25 −20 −15 −10 −5 0 5 10 15

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

x [mm]

uA, u

B

uA

uB

−25 −20 −15 −10 −5 0 5 10 151.2

1.25

1.3

1.35

1.4

1.45

1.5

1.55

1.6

x [mm]

u [V

]

sinusoidal waveformsoptimized waveforms


4.6 Fine tuning of the waveforms

After optimization of the waveforms, the motor effi-

ciency is maximized, but the force command still consists

of some higher order ripple. The higher order ripple is

caused by unmodeled harmonics of the force functions. In

order to reduce some of the higher order ripple, a fine

tuning of the waveforms is performed. The main idea of

the fine tuning algorithm is to measure the phase current

control signals at constant load force. The fine tuning is

performed with previously optimized waveforms, in order

to maximize motor efficiency. The still remaining higher

order ripple of the force command modulates the opti-

mized waveforms. The fine tuning algorithm approxi-

mates the shapes of the measured phase command signals

Fig. 12. Commutation functions in case of sinusoidal force

functions.

Fig. 13. Command signals with unbalanced sinusoidal com-

mutation functions and optimal commutation func-

tions.

with Fourier series. Figure 14 compares the measured

phase command signals with the approximation. In Fig.

15, a comparison of commutation functions of the second

stage with the third stage is given.

3 3

2( ) (sin( ) sin(3 α )

3Au u u a,ϑ = ϑ + ϑ +

5 5sin(5 α )) Aa o+ ϑ + +

2 2π( ) sin

3 3Bu u u

⎛ ⎛ ⎞,ϑ = ϑ +⎜ ⎜ ⎟⎝ ⎠⎝

3 3 5 5sin(3 β ) sin(5 β ) Bb b o⎞+ ϑ + + ϑ+ +⎟⎠

(22)

Figure 16 compares the command signal spectrum of

the optimization process. The upper left graph shows the

spectrum before any optimization is applied. The upper

right graph shows the spectrum after the offset compensa-

tion is applied. The fundamental is significantly reduced.

Fig. 14. Fine tuned waveforms.

Fig. 15. Force command signals with optimized versus fine

tuned waveforms.

−25 −20 −15 −10 −5 0 5 10 15

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

x [mm]

uA, u

B

uA optimized

uB optimized

uA sinusoidal

uB sinusoidal

−25 −20 −15 −10 −5 0 5 10 151.2

1.25

1.3

1.35

1.4

1.45

1.5

1.55

1.6

x [mm]

u [V

]

unbalanced sinusoidaloptimized

−25 −20 −15 −10 −5 0 5 10 15

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

x [mm]

uA, u

B [V

]

measurement approximation

−25 −20 −15 −10 −5 0 5 10 151.2

1.25

1.3

1.35

1.4

1.45

1.5

1.55

1.6

x [mm]

u [V

]

optimizationfine tuning


Fig. 16. Spectrum of the command signal.

In the lower left graph, the spectrum of the force com-

mand of the optimized waveforms is shown. This stage

reduces the 2nd

order harmonics. The lower right graph

shows the spectrum of the force command for fine tuned

waveforms. This last stage reduces the 2nd

and 4th

order

harmonics. After the fine tuning of the waveforms, no

dominant harmonic exists in the force command.

4.7 Controller design

Figure 17 shows the block diagram of the servo con-

trol system. In order to achieve a better tracking per-

formance, a feedforward controller is applied. Feedback

control without feedforward control always introduces a

phase lag in the command response. Feedforward control

sends an additional output, besides the feedback output,

to drive the servo amplifier input to desired thrust force.

The feedforward control compensates the effect of the

carriage mass and the friction force. The friction force is

modeled by a kinetic friction model and identified with

experiments at different velocities. The mass of the car-

riage is identified with a dynamic least square algorithm.

The stability of the system is determined by the feedback

loop. The compensation of the force ripple is performed

in the waveform generator with Fourier series approxima-

tion. Figure 18 compares the tracking error of a move-

ment without ripple compensation with the movement

with ripple compensation. In this measurement, the car-

riage moves from position − 25mm to position 15mm and

back to position − 25mm with vmax

= 200mm/s. If the rip-

ple compensation is applied, the tracking error is reduced

significantly.

Fig. 17. Controller design.

1 2 3 4 5 6 7 8 90

2

4

6

8

10

12Without Optimization

Am

plitu

de [%

]

1 2 3 4 5 6 7 8 90

2

4

6

8

10

12Step 1

Am

plitu

de [%

]

1 2 3 4 5 6 7 8 90

2

4

6

8

10

12Step 2

Am

plitu

de [%

]

1 2 3 4 5 6 7 8 90

2

4

6

8

10

12Step 3

Am

plitu

de [%

]

CurrentsPhase CurrentCommands

x

PositionPositionFeedbackControl

WaveformGenerator

ReferencePositionx

ref ServoAmplifier

LinearMotor

FeedforwardControl

ufb

uff

e

uB

uA

iA

iB

iC


Fig. 18. Tracking error.

V. CONCLUSION

In this paper, a method for optimization of the cur-

rent waveforms for PM LSMs is presented. The opti-

mized waveforms generate smooth force and produce

minimal copper losses which maximizes motor effi-

ciency. The optimized current shapes are valid for any

velocity and any desired thrust force. Since the wave-

forms can be easily adapted to any shape of the back-

EMF, motor design can be focused on increasing the

mean force of the motor and reducing the cost of pro-

duction. Experiments show, that the tracking perform-

ance is significantly improved, if the optimized wave-

forms are applied.

REFERENCES

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Clarendon Press, Oxford, UK (1996).

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0 0.1 0.2 0.3 0.4 0.5−8

−6

−4

−2

0

2

4

6

8

t [s]

e [µ

m]

sinusoidal commutationforce ripple compensation


18. Huang, S.N., K.K. Tan, and T.H. Lee, “Adaptive

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(1998).

Christof Röhrig received his Di-

ploma degree from the University

of Bochum, Germany, in 1993, and

his Doctor degree from the Univer-

sity of Hagen, Germany, in 2003,

both in electrical engineering. Be-

tween 1993 and 1997 he was Man-

ager Automated Systems Engineer-

ing at Reinoldus Transport- und Robotertechnik GmbH

Dortmund, Germany. From 1997 until 2003 he was with

the Control Systems Engineering Group at University of

Hagen. His research was focused on motor control,

eLearning and remote laboratories. Since 2003, he is

Professor of Computer Science at the University of Ap-

plied Sciences Dortmund, Germany. His current re-

search interests include motor control, robotics and

telematics.

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