Hybrid Linear Stepper Motors

HYBRID LINEAR

STEPPER MOTORS

IOAN-ADRIAN, VIOREL, Ph.D. Professor

LORÁND, SZABÓ, Ph.D.

Associate Lecturer

Technical University of Cluj, Romania Electrical Machines Department

MEDIAMIRA Cluj-Napoca, Romania

MEDIAMIRA PUBLISHING COMPANY P.O. Box 117. Cluj-Napoca Romania Copyright © 1998 MEDIAMIRA ISBN 973-9358-12-8 All rights reserved. No part of this book may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without permission in writing from the Publishers. Printed in Romania

CONTENTS 1. INTRODUCTION 1

1.1. STEPPER MOTORS 1 1.2. HYBRID LINEAR STEPPER MOTOR

OPERATING PRINCIPLES 4 1.3. HYBRID LINEAR STEPPER MOTOR VARIANTS 7

2. THEORY AND PERFORMANCE 10

2.1. MATHEMATICAL MODEL 11 2.1.1. Field Submodel 14 2.1.2. Mechanical Submodel 19

2.2. NUMERICAL FIELD ANALYSIS APPROACH 21 2.3. AIR-GAP TEETH CONFIGURATION 26 2.4. EXPERIMENTAL RESULTS 29

3. CONTROL STRATEGIES 32 3.1. THEORETICAL APPROACH 33 3.2. COMPARISON OF DIFFERENT CONTROL METHODS 42 3.3. HYBRID LINEAR STEPPER MOTOR

POSITIONING SYSTEM 45 4. DESIGN 50

4.1. DESIGN PROCEDURE 51 4.1.1. Prescription of the Basic Design Inputs 51 4.1.2. Motor Sizing 51 4.1.3. Optimization of the Magnetic Circuit 55 4.1.4. Final Analysis of the Motor 56

4.2. HYBRID LINEAR STEPPER MOTOR DESIGN EXAMPLES 58 4.3. COMPARISON OF THE THREE DESIGNED MOTORS 64

APPENDIX 1. EQUIVALENT VARIABLE AIR-GAP CALCULATION 66

REFERENCES 70 LIST OF THE MAIN SYMBOLS 77 INDEX 81 TO THE READER 85

PREFACE

This book deals with a special group of electrical

machines called hybrid linear stepper motors, covering their

construction, principles, theory, control techniques and design

procedure.

In the first chapter a brief presentation of the

technological advancement of the stepping motors from the

rotational to the linear stepper motor is surveyed. The basic

types of linear stepper motors are presented. Some features of

the stepper motors from the view point of applications are

discussed and some hybrid linear stepper motor variants are

illustrated.

The second chapter deals about the theory and

performance of linear hybrid stepper motors. A mathematical

model is derived, consisting of three main submodels, whose

interaction is described in a block diagram of the circuit-field-

mechanical model. A numerical field analysis approach is

performed by FEM-analysis. Different airgap-teeth configurations

are investigated and compared to find out the variant with the

highest tangential force.

Control strategies for open-loop and closed-looped

systems are discussed in the third chapter. The authors show

that positioning capabilities and the dynamic performance for

linear hybrid stepper motors can be improved for a closed-loop

control, if an optimum control angle is determined. The

possibility of estimating the angular displacement by monitoring

the induced EMF in the control is presented, too. By computer

simulation the control strategies are compared.

The final chapter is devoted to the design of hybrid linear

stepper motors. The designing process consists of four steps:

PREFACE

ii

prescription of the basic design inputs, motor’s sizing,

optimisation of the magnetic circuit and thermal and

electromagnetic analysis of the motor. The accuracy is good and

easy to implement as computer program. Results for three

design variants are given and compared.

An appendix describing the equivalent variable airgap

calculation and a worthful collection of references conclude the

book.

The book may be recommended to all those who are

interested in the basic theory of hybrid linear stepper motors as

well as in modern techniques of control strategies and design

optimisation. In comparison to conventional literature new ideas

for a complete mathematical model, for closed-loop systems with

optimal control and for design optimisation by FEM-analysis are

given.

Univ.-Prof. Dr.-Ing. Dr. h. c. G. Henneberger

Institut für Elektrische Maschinen Rheinisch-Westfälische Technische

Hochschule Aachen

1. INTRODUCTION

The rapid development and application of high

technologies make new demands on precise linear incremental

positioning. In numerous branches as robotics, computer

peripherals, NC machine-tools, using ultraprecision techniques at

high speed, the linear positioning is realized by hybrid linear

stepper motors. This book deals with the construction, operating

principles, theory, control techniques and design procedure of the

hybrid linear stepper motor.

In this chapter an overview of the technological

advancement of the stepping motors, from rotational to the linear

stepper motors, will be surveyed. Next the basic type of the hybrid

linear stepper motor will be examined. Some features of the

stepper motors from the viewpoint of the applications will be

discussed, too. In the last section some hybrid linear stepper

motor variants will be presented.

1.1. STEPPER MOTORS

Rotational stepper motors were developed well before the

second world war. In fact the basic principle of a stepper motor is

the same as the principle of the synchronous machine. By

supplying the synchronous machine stator winding on a step by

step base from a DC source, a sequence of rotor positions will be

obtained. Therefore the development of the stepper motors was

mainly tied to the supply system improvements, the switched

reluctance motor being a valuable example.

1. INTRODUCTION

2

Fig. 1.1 illustrates the

cross-sectional structure of a

typical stepper motor, in fact a

switched reluctance motor.

The stator has eight salient

poles, while the rotor has six

poles. Four sets of windings

are disposed on the stator

poles. Each set (called a phase)

has two coils connected in

series, disposed on two

opposite poles. Consequently

this machine is a four-phase

motor. The command current

is supplied from a DC power

source. The rotor position

given in Fig 1.1 was obtained by supplying the first phase, coils 1

and 5. If the current flowing through the first phase is zero and

the next phase (coils 2 and 6) is supplied, then the rotor will rotate

to the right. In this case the step angle is 15, as one switching

operation is carried out. If phase two is de-energized and the third

phase (coils 3 and 7) is supplied, the rotor will travel another 15. The angular position of the rotor can thus be controlled in units of

the step angle by a switching process. If the switching is

accomplished in sequence, the rotor will rotate with a stepped

motion. The average speed can also be controlled by the switching

process.

As explained above, the stepper motor is an electrical

motor that converts a digital electric input into an incremental

mechanical motion. For a position or speed control an electric

drive system using stepper motors can be built up without

feedback loop. Such a system is compatible with the digital

Figure 1.1 Switched reluctance motor

1.1. Stepper Motors 3

equipment and benefits of the advantage that the positional error

is non-cumulative.

In addition to the above presented variable reluctance

stepper motors, several types of electromagnetic stepping motors

using permanent magnets were developed. The so called hybrid

stepper motor works upon the combined principles of the

permanent magnet motor and of the variable reluctance motor.

From the beginning of the 60's computer manufacturers

took note of the possible uses of stepper motors as actuators in

terminal devices, and they promoted the development of reliable,

high-performance motors. Such a motor is the hybrid linear

stepper motor, too, which is the main topic of this book. The

advancement of semiconductor technology, which seems to have

no end, enlarged the application domain and contributes to the

performance improvement of stepper motors.

Generally, stepper motors are operated by electronic

circuits, mostly from a DC power supply. Stepper motors, utilized

mainly in speed and position control systems, can operate with or

without feedback loops. Open-loop control is an economically

advantageous driving method, but it has some drawbacks. Closed-

loop control is a very effective driving mode, avoiding instability

and assuring quick acceleration.

The most important features of a stepper motor from the

viewpoint of application are:

i) Small step displacement.

ii) High positioning accuracy.

iii) High torque (or force) to inertia ratio.

The main types of stepper motors are the variable

reluctance motors and the hybrid permanent magnet variable

reluctance motors. The first type was described above, the second

type will be described next, taking as example a linear stepper

motor.

1. INTRODUCTION

4

1.2. HYBRID LINEAR STEPPER MOTOR OPERATING

PRINCIPLES

The hybrid linear stepper motor basic construction, shown

in Fig. 1.2, consists of a moveable armature (the mover)

suspended over a fixed part, the platen.

The platen is an equidistant toothed bar of any length

fabricated from high permeability cold-rolled steel. The mover

consists of two electromagnets having command coils and a

permanent magnet between them. The permanent magnet serves

as an excitation bias source and also separates the

electromagnets. Each electromagnet has two poles. All the poles

have the same number of teeth. The toothed structure in both

parts, mover and platen, has the same very fine tooth pitch. Each

of the two poles of an electromagnet is displaced with respect to

the platen slotting by half of tooth pitch, as it can be seen in

Fig. 1.2. The first pole of the right side electromagnet is displaced

by a quarter of tooth pitch with respect to the first pole of the left

side electromagnet. In absence of the command current, the flux

produced by the magnet flows through both poles of one

electromagnet. When a command coil is excited the flux is

Figure 1.2 Hybrid linear stepper motor

1.2. Hybrid Linear Stepper Motor Operating Principles 5

concentrated into one pole of the corresponding electromagnet.

The flux density in that pole becomes maximum, while the flux

density in the other pole is reduced to a negligible value. By

commuting this way the permanent magnet flux a tangential force

is developed, that tends to align the teeth of the pole where the

flux density is maximum with the platen teeth, minimizing the air-

gap magnetic reluctance.

For a displacement of one step to the right from the initial

position (position number one in Fig. 1.3) the right side command

coil must be excited in a way to concentrate the magnetic flux into

the pole number four. The mover will be driven to the right a

quarter tooth pitch (one step) and the teeth of pole number four

will be aligned with the platen teeth (position 2 in Fig. 1.3).

The variations

of the tangential forces

developed under the

four poles during the

above described step

from the initial position

are shown in Fig. 1.4.

The representation is

given for a simplified mode of the force variation, when the MMF

produced by the command coil is considered constant during the

motion. The tangential force developed under the fourth pole is the

greatest one at the beginning of the step and reaches zero at the

end of the step. The tangential force developed under pole number

Figure 1.3 Four positions of the mover

Figure 1.4 Tangential force variation under the mover poles

1. INTRODUCTION

6

one and two, which are at the beginning of the step aligned,

respectively unaligned, starts from zero value. These two forces,

one tracking and other backing, are quite equal in absolute value,

their sum being almost zero. As the magnetic flux through pole

number three is negligible, the developed tangential force is

insignificant, too. The total traction force is the sum of these four

tangential forces, and it can be considered equal with the force

developed by the fourth pole.

In order to continue

the displacement to the right,

the command coil of the right

electromagnet must be de-

energized and the other has to

be excited. The flux through

the pole number two will be

maximum and at the end of

the step the teeth of this pole

will be aligned to the platen

teeth.

The sequences of the

command coil currents for a

four-step displacement in two directions (to the right and

respectively to the left) are given in Fig. 1.5.

There is a possibility to supply in the same time both

command coils. In this case one must be energized with a sine

wave voltage and another with a cosine wave voltage. This two-

phase excitation mode, which will be adequately presented in

Chapter 3, provides high resolution.

The main disadvantage of this motor type is that the

maximal magnetic fluxes through the inner poles are a little larger

than that through the outer ones [38]. Therefore tangential force

unbalance can occur, causing step errors and undesirable

vibrations.

Figure 1.5 The command currents sequence and the corresponding displacements

1.3. Hybrid Linear Stepper Motor Variants 7

1.3. HYBRID LINEAR STEPPER MOTOR VARIANTS The basic construction of the hybrid linear stepper motor

presented above is very simple, but is not the single one existing.

An outer magnet type motor construction is presented in

Fig. 1.6. The motor has two permanent magnets placed on the top

of the two electromagnets [16, 38]. A back iron closes the magnetic

circuit of both mover parts. Each pole has its command coil. The

motor has good control facilities and the currents flow in the same

single direction in all the coils.

The motor shown in Fig. 1.7 contains 8 poles pieces and

two command coils in a very compact construction [65]. Between

the two half-parts of the motor is disposed an insulator for

magnetic separation purpose. The motor magnetic circuit is

characterized by low mass and small volume related to the holding

force.

Figure 1.6 The outer magnet type hybrid linear stepper motor

Figure 1.7 The eight poles compact hybrid linear stepper motor

1. INTRODUCTION

8

A tubular variant of the hybrid linear stepper motor is pre-

sented in Fig. 1.8. The motor has four poles, two coils and one ring

type permanent magnet [10, 38]. The outer cylindrical part is the

mover. The operating principle is the same as of the motor

presented in Fig. 1.2.

A motor that can travel in any direction on a stationary

base (a surface moving motor) can be realized by combining two

hybrid linear stepper motors in the way shown in Fig. 1.9 [12, 21].

One of the motors will produce force in the x-direction, and other

one in the y-direction.

The great variety of the hybrid linear stepper motor

construction variants demonstrates that these motors have a lot of

advantages and can be used in a myriad of applications. As

example a xy plotter is given in Fig. 1.10. The plotter was

Figure 1.8 Tubular variant of the hybrid linear stepper motor

Figure 1.9 Surface hybrid linear stepper motor

1.3. Hybrid Linear Stepper Motor Variants 9

developed at the Technical University of Cluj-Napoca [48]. It has

two hybrid linear stepper motors, one for the x-direction, another

for the y-direction. An adequate test software was developed, too.

A test graph is shown in Fig. 1.11.

Figure 1.10 The xy plotter

Figure 1.11 The test graph obtained with the xy plotter

2. THEORY AND PERFORMANCE

The different types of linear stepper motors were presented

in the previous chapter. From the large variety of existing types

the simplest one, that shown in Fig. 2.1, will be considered as the

basic type. This simple constructed motor has all the features of

the class of motors represented by it. The theory, including the

mathematical model, will be developed for this basic type. It can

be easily applied with very few changes to any other motor types.

In the unexcited hybrid linear stepper motor the flux

generated by the permanent magnet flows into the core of one

electromagnet, passes through its poles and traverses the air-gap.

Then it flows through the platen, crosses the other gap, divides

evenly between the pole faces of the other electromagnet, and

closes its circuit at the opposite side of the permanent magnet.

The MMF produced in one of the command coils reinforces the

flux generated by the magnet in one pole face and diminishes it in

the other. The permanent magnet flux is effectively commuted

from one pole to the other of an electromagnet. It is obvious that

the permanent magnet has a double role, acting as a bias source

and separating the two electromagnets. It means that the hybrid

linear stepper motor is basically a variable reluctance

Figure 2.1 Four pole hybrid linear stepper motor (1÷4 poles, A-B electromagnets with command coils, PM permanent magnet)

2.1. Mathematical Model

11

synchronous motor. The traveling magnetic field is obtained by

switching the flux produced by the permanent magnet from one

pole to another using the command coil MMFs.

2.1. MATHEMATICAL MODEL The mathematical model of the hybrid linear stepper motor

seems trivial at the beginning, because the voltage equation of a

supplied command coil is very simple. This equation written for

command coil A is:

where v A is the input voltage, RA the coil resistance and i A the

current. The flux linkage through the same coil is given by:

N being the number of turns of the coil, CA the flux produced

by command current i A (having two components: CA the

leakage flux and CAm the main magnetizing flux), pm the flux

generated by the permanent magnet and CB the flux produced

by the current flowing through command coil B.

This circuit type model works only with the following

assumptions:

i) The flux generated by the permanent magnet is constant.

ii) The permanent magnet reluctance is so large that no

flux produced by a command coil from the other electromagnet

flows through the poles.

With these assumptions Eq. 2.1 becomes:

A typical circuit-type equation is obtained by introducing the main and the leakage inductances (L Am

, respectively L A ):

v R id

dtA A AA

(2.1)

A A CA pm CBN N (2.2)

v R iddt

NA A A CA (2.3)


12

where the leakage and main fluxes are given by:

The leakage inductance is considered constant (unaffected

by saturation and mover position). The main inductance is

affected by saturation and strongly depends of the mover position.

By neglecting the iron core saturation, the main inductance will

depend only on the mover position:

In the circuit-field-mechanical model that will be presented

further the iron core saturation and the permanent magnet

operating point changes will be fully taken into account.

It is quite difficult to obtain such

a relation as Eq. 2.7, and obviously it is

necessary to impose some simplified

assumptions. A possibility to determine the simplified A coil flux linkage CA

function of the mover position, consi-

dering the saturation effect, is by using

the standstill current decay test [50].

With the mover at standstill in a certain

position a DC current is applied to

coil A (Fig. 2.2). The coil is fed with a current i A .

The power transistor T is turned off. The current is continuing to

flow through diode D , until it reaches zero. After turning off the

transistor the following relation can be obtained by time

integration:

v R i Ldi

dtddt

L iA A A AA

A Am

(2.4)

N L iCA A A (2.5)

N L iCA A Am m (2.6)

L L x x f tAm ( ); ( ) (2.7)

Figure 2.2 The standstill current decay test setup


13

The flux linkage through the coil A at the initial moment

t 0 is:

and when the current i A reaches zero, it becomes:

Assuming that the permanent magnet flux through the

coil A is unchanged, then the following relation can be considered:

The test is performed for different mover positions and DC

coil currents. The variation of the flux linkage through coil A is

obtained function of the mover position and for each position

function of the current. Using these curves the flux linkage value

at a certain mover position and a given current value can be

determined. Through these curves the saturation of the iron core

is fully considered, but the permanent magnet flux is taken

constant.

Another way of obtaining the coil flux linkages is by solving

the field problem at different mover positions and coil current

values. If the main path flux linkages are computed, there is no

need to calculate the magnetizing inductance. It leads to the

circuit-field type model. The model covers accurately the effects of

the complex toothed configuration, the magnetic saturation of iron

core parts and the permanent magnet operating point change due

to air-gap variable reluctance and command MMF [57, 60].

The coupled circuit-field model can not be solved

analytically. The computational process consists of a

simultaneous iterative calculation of the circuit type equations

and of the field problem. In the particular case of the hybrid linear

R i dtA A A0 0

0

(2.8)

A pm CA0 0 (2.9)

A pm (2.10)

CA A AR i dt0

0

(2.11)


14

stepper motor a supplementary mechanical model has to be

solved simultaneously to determine the mover position at each

time moment. The block diagram of this model is shown in

Fig. 2.3, where the three main submodels with the connections

between them are presented.

The first submodel consists only of the circuit-type

equation (Eq. 2.3). The needed flux linkages must be computed in

another submodel.

2.1.1. Field Submodel

The field submodel is based on the equivalent magnetic

circuit of the motor. Obviously the field problem may be solved via

a numerical method, using finite elements or finite differences

models, as it will be presented later. In the case when the motor is

moving, it is necessary to solve the field problem for each

considered position. This is possible only by using the equivalent

magnetic circuit method, because of the short computational time.

The field submodel based on the equivalent magnetic circuit is

useful for both dynamic and steady-state motor regimes. In order

to compute the fluxes in a certain mover position the numerical

methods are recommended. They offer better accuracy, but at

longer computational time.

Figure 2.3 Block diagram of the circuit-field-mechanical model


15

In building up the magnetic equivalent circuit two

problems arise: the permanent magnet model to be adopted and

the calculation of the air-gap magnetic reluctance.

The permanent magnet is a source for its field and has a

large magnetic reluctance for the external fields. It means that a

magnetic circuit, like that given in Fig. 2.4/a, can be represented

by two magnetic equivalent circuits given in Fig. 2.4/b.

The permanent magnet, described by its second quadrant

characteristic (Fig. 2.4/c) can be represented by Norton's

equivalent circuit (Fig. 2.4/d) [63].

The relations that basically conduct to the equivalent

circuit are:

Figure 2.4 a) A simple configuration with iron core, permanent magnet and coil b) The equivalent magnetic circuits c) The second quadrant characteristic of the permanent magnet d) Norton's equivalent circuit of the permanent magnet


16

where P Fm cpm 0 and R Pm mpm pm

1 is the permanent

magnet permeance and respectively reluctance, RmFe is the iron

core reluctance and Fc is the permanent magnet coercive MMF.

In the first case the nonlinearities are taken fully into

account by considering the nonlinear iron core reluctance and

computing at each iteration the permanent magnet MMF, from its

second quadrant characteristic (Eq. 2.13). In fact in the second

case, when the permanent magnet is described by a unique

equivalent circuit, the computational process is quite the same,

because at any time moment the permanent magnet MMF must

be calculated (Eq. 2.13) by using the previously determined value

of the flux.

The two equivalent magnetic circuits obtained for the

hybrid linear stepper motor given in Fig. 2.1 are presented in

Fig. 2.5.

The two corresponding systems of equations are:

0 P Fm pmpm (2.12)

F R Fpm m cpm (2.13)

Figure 2.5 The magnetic equivalent circuit a) without command MMF b) with command MMF and permanent magnet reluctance


17

1 2 5

3 4 5

1 7

4 8

2 6 7 10

1 2 7

2 3 5

0

0

0

0

0

012 1 13 22 2 11

22 2 32 3 5

R R R R R R

R R R R R F

m m m m m m

m m m m m

g g

g g pm

m m m m m m

m pm

R R R R R R

R F

g g

pm

3 4 8

9

10

11

32 3 42 4 43 410

0

0

(2.14)

1 2 5

3 4 5

1 7 9

4 8 11

2 6 7 9

1 2 7 1

2 3

0

0

0

0

0

12 1 13 22 2 11

22 2 32 3

R R R R R R F

R R R R

m m m m m m

m m m m

g g

g g

5 6

3 4 8 2

9 7 1

10

8 11 2

5

32 3 42 4 43 41

1 11

41 2

0

0

R R

R R R R R R F

R R F

R R F

m m

m m m m m m

m m

m m

pm

g g

(2.15)


18

The resulting magnetic fluxes are given by:

In the magnetic equivalent circuit built up for the hybrid

linear stepper motor (Fig. 2.5) only one magnetic reluctance for

every pole was considered and in the platen only one magnetic

reluctance was taken for each flux path. These magnetic

reluctances are computed as a sum of many elementary

reluctances, but in the equivalent magnetic circuit only the result

of the computation is represented.

The nonlinear permeances of the iron core portions must

be computed by means of the corresponding field-dependent single valued permeability . The dependence of the permeability

of the flux or induction has to be given, or it has to be

computed at each iteration from the magnetizing characteristic of

the iron core material.

In order to obtain analytical results, which can be helpful

in elaborating the control strategy two assumptions must be

made:

i) The air-gap reluctances are much more larger than all

other reluctances, excepting that of the permanent magnet.

ii) The permanent magnet reluctance is so large that it will

separate the two electromagnets.

Therefore,

the equivalent

magnetic circuit

of the motor in

absence of the

control currents

is given in

where the command coils MMFs are: F N iA A A (2.16)

F N iB B B (2.17)

j j j j , 1 11 (2.18)

Figure 2.6 The equivalent magnetic circuit in absence of the command MMFs


19

Fig. 2.6. When the command coil B is supplied, the flux produced by its MMF, CB , can be computed by using the magnetic

equivalent circuit given in Fig. 2.7.

There are some different possibilities

to calculate the air-gap permeance when

both iron cores have teeth and slots in

cylindrical or linear machines [27]. The

method proposed here comes up as an

extension of the method to calculate the air-

gap variable equivalent permeance developed

first in the case of the induction machine

[61]. The computational process is fully described in Appendix 1.

For the variable equivalent air-gap the following expression was

obtained:

with the motor constant c :

All the notations are given in Appendix 1, too.

The air-gap permeance computed for a mover pole is:

where S p is the pole area and 074 10 H m is the free space

permeability.

2.1.2. Mechanical Submodel

The electromagnetic forces can be evaluated either from

the gradient of the magnetic co-energy with respect to a virtual

displacement or by Maxwell's stress tensor method [33]. The

Figure 2.7 Magnetic equivalent circuit of one electromagnet

gZg

Z ce

2

2 1 1 cos (2.19)

c

Z

Z

1 2 1

2 2 1 (2.20)

PS

gjm

p

egjj

0

1 4; (2.21)


20

former method is more reliable for this problem and it was

adopted here. The tangential force under one mover pole j is given by:

which leads to:

The normal force developed by one mover pole MMF is:

After some computations the following relation can be obtained:

The mechanical submodel considered here is a simplified

one. In order to obtain this model, given in Fig. 2.8, two

assumptions were made:

i) The motor is a homo-

geneous solid, the resulting tangen-

tial and normal forces being applied

on its center

ii) The resulting forces are

obtained as an algebraic sum of the

pole forces.

This simplified mechanical

model does not take into considerations the torques that exist.

These torques are produced by the normal forces that are not

applied in the center of the mover but in each pole axe.

fW

xjt

m

ctj

j

j

.

1 4 (2.22)

fN

Sddx

g x jtj

pej j

2

021 4

( ) (2.23)

fW

gjn

m

ctj

j

j

.

1 4 (2.24)

fN

Sjn

j

pj

2

021 4

(2.25)

Figure 2.8 Simplified mechanical model


21

The mechanical submodel is characterized by the

equation:

where c f is the friction coefficient, G is the mover weight and m

the mover mass. By solving the above force equation the velocity

and displacement are computed.

2.2. NUMERICAL FIELD ANALYSIS APPROACH

Previously a magnetic equivalent circuit approach was

considered to calculate the fluxes through the hybrid linear

stepper motor iron core and air-gap. The magnetic equivalent

circuit approach is less accurate then a numerical field analysis

approach, but it is requiring shorter computing time. To check

certain values of the motor characteristics in a stage of the design

procedure the numerical field analysis approach is recommended.

For the hybrid linear stepper motor both finite difference and finite

element methods are utilizable because the cross section has only

right lines.

The field model of the hybrid linear stepper motor, having

the x-y plane cross section shown in Fig. 2.1, is obtained using the

following assumptions:

i) The magnetic field quantities are independent of the z-

coordinate. This leads to a two-dimensional analysis.

ii) Only the axially directed components of the magnetic vector potential and current density ( Az , respectively J z ) exist.

iii) The iron parts are isotropic and the corresponding nonlinear B(H) characteristics are single-valued (i.e., hysteresis

effects are neglected).

md x

dtf f G ct n f

2

2 (2.26)


22

iv) The anisotropic permanent magnet reveals an elemental magnetic orthotropy for the easy (x ) and difficult (y )

magnetization axes. Accordingly, the permanent magnet behavior

in the easy x-axis being entirely explained in terms of demagnetization characteristic, B(H) , and in the difficult y-axis

being air-like.

v) The external contour of the motor is treated as a line of

zero vector potential, i.e. there is no field outside the motor

periphery.

vi) Eddy-current effects are neglected.

From Hamilton's principle applied to macroscopic

magnetostatics, the two-dimensional nonlinear variational field

model of the hybrid linear stepper motor involves the

minimization, with homogeneous boundary conditions, of the

following energy functional:

where x xf B ( ) , y yf B ( ) and Brx , respectively Bry define

the non-zero diagonal components of the reluctivity tensor and the

remanent magnetic flux density, respectively, corresponding to the

permanent-magnet easy x-axis and difficult y-axis of magnetization. In the iron core portions ( )B , Br 0 and

elsewhere in the considered domain D , 0 and Br 0 .

Even the finite difference method can be and was applied

with quite satisfactory results, here it is discussed only the finite

element approach, because there are a lot of specialized software

packages, that can be used in solving such a field problem.

By means of finite element method, FEM, the energy-

related functional (Eq. 2.27) is minimized by a set of trial

U A B B dB

B B dB J A dxdy

z x x r x

B

D

y y r y z z

B

x

x

y

y

( )

0

0

(2.27)

2.2. Numerical Field Analysis Approach

23

functions, approximating the magnetic field solution [39]. A usual

FEM package has three main parts: pre-processing, processing

and post-processing. Within the pre-processing sequence the next

steps will be covered:

i) The field domain D geometry is described and the

subdomains are precisely defined.

ii) The boundary and the symmetry conditions are

introduced.

iii) The field domain D is discretized into first-order

triangular finite elements. Usually the packages have automatic

mesh generators. iv) The material characteristics, B H( ) curves, are selected

from the package library or are defined for the subdomains.

The processing FEM sequence contains two phases: the

global system generation and its iterative solution. All these are

done automatically by the solver module.

In the post-processing part the obtained values of the

magnetic vector potential at each mesh node are used to compute

fluxes, magnetic energy, forces etc. All the packages have the

possibility to show the magnetic flux distribution given by the

magnetic potential constant lines.

In the post-processing FEM sequence the air-gap magnetic

flux of each pole can be computed using the following line integral:

Here the tangential forces are computed by the surface

integration of Maxwell's stress tensor:

where the closed surface (having the unit outward vector normal n ) surrounds the mover, passing through the centers of

the air-gap mesh elements.

Ad (2.28)

f Bn B B n dt

0 021

2

(2.29)


24

Next some results of the FEM analysis of a certain

sandwich magnet type hybrid linear stepper motor (having AlNiCo

magnet) will be presented. In Fig. 2.9 the automatic generated

mesh is shown.

As it can be seen, an adequate discretization was ensured,

especially in and around the air-gaps.

Figure 2.9 The discretized domain

Figure 2.10 Field distribution in the unexcited motor

2.2. Numerical Field Analysis Approach

25

The field distribution for the unexcited motor computed by

the FEM package is presented in Fig. 2.10. The magnetic flux

generated by the permanent magnet is almost concentrated in the

first pole having the teeth aligned with the platen teeth. As the

teeth of the poles of the right side electromagnet are both in a half-

aligned position, the flux distribution through these poles is near

the same.

For a better view of the field lines two zoomed figures of the

air-gap zones are illustrated next. Figs. 2.11 and 2.12 show the

constant potential vector lines of the air-gap portion of the aligned,

respectively half-aligned teeth.

The above presented figures demonstrate that the results

obtained via the FEM analysis perfectly agree with the theoretical

anticipations.

Figure 2.11 Field distribution in the air-gap area under an aligned pole

Figure 2.12 Field distribution in the air-gap area under a half-aligned pole


26

2.3. AIR-GAP TEETH CONFIGURATION

The tangential thrust force and the normal attraction force

are both dependents on the teeth configuration of the mover and

platen [4]. Therefore it is important to choose the best teeth

geometry (the tooth pitch, tooth shape and the tooth width to tooth

pitch ratio) in order to have a tangential force as great as possible

and the smallest possible normal attraction force.

Four teeth configurations are considered here (Fig. 2.13).

They cover adequately the most interesting cases. In all the cases

the air-gap length and the tooth pitch are the same:

The first variant (Fig. 2.13/a) has different tooth width on

the platen and on the mover, in order to concentrate the magnetic

flux into the head of the platen teeth.

For the second version (Fig. 2.13/b) the tooth width is

unequal to the slot width. As it was recommended in [26], the

Figure 2.13 The considered teeth configurations

g 01 2. mm mm (2.30)

2.3. Air-Gap Teeth Configuration

27

optimum tooth width to tooth pitch ratio is about 0.42. So the

width of the tooth and of the slot are:

The third teeth geometry (Fig. 2.13/c) is the "classical" one.

The rectangle teeth on each side of the motor have the same width

for the tooth and slot:

The last air-gap structure in study is that having wedge

headed teeth [47] and it is presented in Fig. 2.13/d. This structure

has two more design parameters in addition to the tooth and slot

width:

whence w is the slop of the wedge and wf is the flat width at the

wedge head.

In order to compare the different teeth configurations a

finite element method analysis was performed. As result the two

force-displacement static characteristics of the tangential and of

the normal force are presented.

w wt s 0 85 115. .mm mm (2.31)

w wt s 1 1mm mm (2.32)

w wf 20 01 . mm (2.33)

Figure 2.14 The tangential force-displacement static characteristics

Figure 2.15 The normal force-displacement static characteristics


28

As it can be seen in Fig. 2.14, the total tangential force has

its greatest peak value for the second and the fourth variant. The

static characteristics of the total normal forces, presented in

Fig. 2.15, show that the attraction force is reduced about to the

half for the tooth structure having wedge head teeth. The other

versions have almost the same normal force, but the second

variant is smaller.

The optimal tooth geometry has to be selected now from

only two variants, the second and the fourth.

Next the plots of the flux densities in the air-gap under the

poles will be studied. These figures were obtained from the above

mentioned FEM analysis, too. Two situations were considered:

when the mover teeth are aligned with the platen teeth (Fig. 2.16)

and when the teeth on both armatures are completely unaligned

(Fig. 2.17). The continuous line corresponds to the fourth variant

and the dashed line to the second one. As it can be seen from

Fig. 2.17 the wedge heads of the teeth of the fourth variant are

strongly saturated.

Figure 2.16 The flux density in the air-gap under an aligned pole

Figure 2.17 The flux density in the air-gap under an unaligned pole

2.3. Air-Gap Teeth Configuration

29

Finally it can be concluded that the variant having wedge

teeth has the greatest tangential force and far the less normal

force. On the other hand is more difficult to manufacture this

tooth construction than the other ones. Besides the heads of the

teeth are very saturated in the aligned position of the teeth.

So the use of the second variant is hardly recommended. It

has almost as great tangential force as the fourth version in study,

and its manufacture is more simple. In this case the shape of the

tangential force-displacement static characteristic is near optimal,

assuring high stiffness and therefore greater positional accuracy.

2.4. EXPERIMENTAL RESULTS

In order to validate the results obtained by means of

numeric simulation of the hybrid linear stepper motor the

tangential force static characteristic was determined

experimentally.

The sample motor was of sandwich magnet type. The

geometrical dimensions and the main data of the motor are given

in Table 2.1. ITEM VALUE

tooth width 1 mm

slot width 1 mm

tooth pitch ( ) 2 mm

nr. of teeth per pole ( Z ) 4 airgap (g ) 0.1 mm

permanent magnet type AlNiCo200 residual flux density ( Br ) 0.95 T coercive force ( H c ) 50 kA/m maximal tangential force ( f tmax

) 9 N

Table 2.1 The main data of the sample motor


30

The mover was precisely placed in several positions at

0.05mm distance each of the other within a quarter tooth pitch.

The command coil

was energized with the

maximum command

current determined for this

motor (150 mA). In each

position the tracking

tangential force was

measured using a force

transducer stamp fixed on

the mover. The results of

the measurements are

included in Table 2.2.

The same tangential

force static characteristic was determined using the analytical

model based on the equivalent magnetic circuit of the motor

described in detail in Section 2.1.1.

Both cha-

racteristics are

shown in

Fig. 2.18. The

static characte-

ristic obtained by

simulation is

plotted with con-

tinuous line. The

asterisks (*)

mark the points

acquired experi-

mentally. These

points are inter-

polated by dash-

POSITON OF THE MOVER (mm)

TANGENTIAL FORCE (N)

0 0 0.05 0.1 0.10 3.1 0.15 4.7 0.20 5.4 0.25 5.9 0.30 6.6 0.35 7.9 0.40 8.0 0.45 8.0 0.50 7.9

Table 2.2 The experimental results

Figure 2.18 Tangential force static characteristics obtained experimentally and by numeric simulation

2.4. Experimental Results

31

ed line. The interpolation function is one of the logistic approxi-

mations and is given by:

where:

The obtained coefficients are the followings:

As it can be seen the two characteristics are close enough.

This means that the mathematical model of the hybrid linear

stepper motor describes truly the behavior of the motor.

ya bn

n

4

1 2 (2.34)

n ex c

d

(2.35)

a b

c d

e

0228 153

28373 0 669

43 864

. .

. .

.

(2.36)

3. CONTROL STRATEGIES

Modern manufacturing technologies are characterized by

increasing quality requirements for the drive systems regarding

dynamic range, accuracy and reliability. The electrical machines

having high speed, accurate positioning capability, high servo

stiffness, smooth travel and fast settling times must be driven to

their fullest capability [42].

In the design of a high performance motion system the

most adequate control strategy is the key issue, as normally each

application presents specific requirements.

The hybrid linear stepper motor can be used either in an

open-loop or in a closed-loop control system.

In many cases the hybrid linear stepper motor is used in

open-loop control mode, the simplest way to command such a

motor. In this case the motor steps in response to a sequence of

command current pulses. The open-loop control mode has some

disadvantages, such as low efficiency, the tendency to mechanical

resonance or the peril of losing steps when the expected load is

exceeded.

The positioning capabilities and dynamic performance of

the motor can be improved by operating it under closed-loop

control. The control system has to offer, in certain limits, the

possibility to maintain a prescribed speed not depending of the

load. Thus the operating frequency is variable and depends only

on the motor capability to realize a certain displacement under

given conditions, as load and input source limits.

At the beginning of this chapter the theoretical bases of

different control strategies of the hybrid linear stepper motor are

presented. First, under certain simplifying assumptions, which do

3.1. Theoretical Approach

33

not affect basically the results, the total tangential force of the

motor will be expressed. Then, the optimum control angle will be

determined by imposing a maximum value for the average total

tangential force developed during a control sequence. The

possibility of estimating the angular displacement by monitoring

the EMF induced in the command coils will be presented, too. By

computer simulation the motor characteristics will be determined

for several different driving modes and all the control strategies in

study are compared. Finally an adjustable speed precise

positioning system using EMF sensing controlled hybrid linear

stepper motor will be suggested.

3.1. THEORETICAL APPROACH

In order to obtain the analytical expressions, which are

necessary in elaborating the control strategies, all the calculus

must lay basically on three assumptions:

i) The air-gap reluctances are much more greater than all

other reluctances, excepting that of the permanent magnet.

ii) The permanent magnet reluctance is so great that no

flux linkages produced by the command currents will pass from

one side of the permanent magnet to the other one.

iii) The iron core is not affected by saturation and the

permanent magnet operating point does not change. So the

superposition principle of the magnetic fluxes can be applied [54].

There is no difference between the basic motor variants in

study (i.e. the sandwich magnet type, Fig. 1.2, that with outer

magnets, Fig. 1.6, and the motor having four command coils,

Fig. 4.6), as far as the magnetic circuit is concerned, excepting of

the number of the command coils. So all the relations will be

expressed for the hybrid linear stepper motor having four

command coils, the most general construction. The other motor


34

versions can be considered as particular cases of the above

mentioned one.

The initial position of the mover is that one given in

Fig. 2.1 (the teeth of the first pole are aligned with the platen

teeth). The displacement is considered to be performed to the

right, thus increasing the x-coordinate value. The simplified

equivalent magnetic circuit of the hybrid linear stepper motor with

four command coils is given in Fig. 3.1.

The tangential force can be determined by using the air-

gap flux under each pole. This flux is a sum of two fluxes, one pro-

duced by the permanent magnet and another produced by the

command coils.

The magnetic fluxes through the poles that originate from

the permanent magnet when the command coils are not energized

can be expressed as [54, 55]:

where Fpm is the permanent magnet MMF, c is the motor

constant given by Eq. 2.20 and Pme is the equivalent magnetic

permeance of an electromagnet (Eq. A1.12) [56]. The angular

displacement of the mover is given by:

Figure 3.1 Simplified magnetic circuit of the motor with four command coils

0 41 1 4

je

F Pc j =

pm mj cos , (3.1)


35

which means:

Next the two most usual control possibilities of the hybrid

linear stepper motor will be considered. In the first case the

supplying command currents are sinusoidal. In the second case

the command coils are supplied by a square wave pulse sequence.

If the supplying command currents are sinusoidal only one

coil is fed on each electromagnet (coil number one, respectively

number four). The frequencies of the sinusoidal wave command

currents are the same and only their phases differ:

The flux produced by the command amperturns passes

only through the respective electromagnet, the air-gaps and the

platen (see Fig. 2.7). The following expressions can be obtained for

the left side, A, respectively for the right side, B, electromagnet

[56]:

As it can be

seen in Fig. 3.2 the re-

sulting flux through the

poles is a sum of two

fluxes:

2

x (3.2)

1 2

3 42 2

(3.3)

F F F t F

F F F t F

C CA CA A C

C CB CB B C

M

M

1 2

4 3

0

0

sin

sin

(3.4)

CACA m

CBCB m

F Pc c

F Pc c

e

e

41 1

41 1

1 1

3 3

cos cos

cos cos

(3.5)

Figure 3.2 Magnetic flux pattern


36

where kFA and kFB

are the command MMF factors:

The tangential forces under the poles can be expressed by

substituting the relations for the magnetic fluxes given by Eq. 3.6

in Eq. 2.23:

where k ft is the tangential force coefficient:

As it was demonstrated, [55, 56], the equivalent magnetic permeance Pme

is independent of the mover’s position, therefore

the tangential force coefficient is independent of the displacement,

too.

1 01

2 02

3 03

4 04

41 1 1

41 1 1

41 1 1

41 1 1

CA

pm mF A

CA

pm mF A

CB

pm mF B

CB

pm mF B

F Pc k t c

F Pc k t c

F Pc k t c

F Pc k t c

e

A

eA

eB

eB

cos sin cos

cos sin cos

sin sin sin

sin sin sin

(3.6)

kF

Fk

F

FFCA

pmF

CB

pmA

MB

M (3.7)

f k k t c

f k k t c

f k k t c

f k k t c

t f F A

t f F A

t f F B

t f F B

t A

t A

t B

t B

1

2

3

4

1 1

1 1

1 1

1 1

2

2

2

2

sin sin cos

sin sin cos

cos sin sin

cos sin sin

(3.8)

kF

P cfpm

mt e

4

22

(3.9)


37

The resulting total tangential force of the motor is given by

the sum of tangential forces under each pole:

Based upon this relation the total unitary tangential force

is:

As it can be seen the total tangential force can be

computed only if besides the command current frequency, phase and amplitude, the time variation of the displacement is

known. In the most favorable situation sin and sint has the

same variation function of time. In this case the motor is moving

synchronized with the command current.

So the following obligatory conditions will be imposed:

In this case the unitary tangential force will be:

It was demonstrated [41] that the greatest average unitary

tangential force is obtained if the two command MMFs are in

quadrature. It means that:

Under these conditions the expression of the unitary

tangential force is:

f k k t

k a t

c k t k t

t f F A

F B

F A F B

t A

B

A B

2 2

2

2 2 2 2 2

sin sin

cos sin

sin sin sin

(3.10)

ff

kk t

k a t

ck t k t

tt

fF A

F B

F A F B

tA

B

A B

* sin sin

cos sin

sin sin sin

4

22 2 2 2 2

(3.11)

t = = 0A (3.12)

f k k a

ck k

t F F B

F F B

A B

A B

* sin cos sin

sin sin sin

2

2 2 2 22

2

(3.13)

k k kF F F BA B 2 (3.14)


38

In the above mentioned article [41] the effect of the MMF factor Fk on the unitary tangential force was studied, too. It was

pointed out that if the MMF factor is raised, the unitary tangential

force is increased too, but also the force ripples are much greater.

It was also demonstrated that the peak value of the unitary tangential force is obtained at = /8- .

The second command possibility is that of supplying the

command coils with square wave current pulses. In this case the

four command MMFs are:

The coefficient Cj j, 1 4 , is taken one if

coil j is supplied and it is nil

if it is not energized. By feeding the command coils 2 and 4 (C C2 4 1 and C C1 3 0 )

the flux pattern presented in Fig. 3.3 will be obtained.

In this situation the magnetic fluxes through the four poles

can be expressed as:

f kc

kt F F* sin

14

4 (3.15)

F C F F C F

F C F F C FC C C C

C C C C

M M

M M

1 1 1 2 2 2

3 3 3 4 4 4

(3.16)

Figure 3.3 Magnetic flux pattern

1 01 2

2

2 02 2

2

3 03 4

4

4 04 4

4

41 1 1

41 1 1

41 1 1

41 1 1

2

2

4

4

C

pm mF

C

pm mF

C

pm mF

C

pm mF

F Pc C k c

F Pc C k c

F Pc C k c

F Pc C k c

e

e

e

e

cos cos

cos cos

sin sin

sin sin

(3.17)


39

The unitary tangential force in this case is:

If the command coil currents have an ideally square shape

the tangential force depends mainly on the mover position. The

control system has to assure the change of excitation through the command coils at a specific displacement (0 ) in order to keep the

tangential force at a certain value. The maximum value of the

average tangential force is obtained if the command current pulses

are commuted at the optimal value of the angular displacement

[55].

The unitary average tangential force is:

which gives:

Its derivative function of 0 is:

A particular case can be considered when only one coil is

supplied:

Under this condition the optimal commutation position

results:

f C k C k a

cC k C k

t F F

F F

* sin cos

sin

2 4

22

42

2 4

2 422

(3.18)

f f dt tm* *

2

0

0

4

4

(3.19)

f C k C k a

cC k C k

t F F

F F

m* sin cos

sin

22

22

2 0 4 0

0 22

42

2 4

2 4

(3.20)

df

dC k C k a

c C k C k

tF F

F F

m*

cos sin

cos

02 0 4 0

0 22

42

22

2

2 4

2 4

(3.21)

C = C =2 40 1 (3.22)


40

As it can be seen the optimal commutation position for a

given motor depends only on the MMF factor. It is very important

the detection of the mover position for establishing the command

current commutation moment. The mover displacement can be

obtained by using a conventional transducer. Another possibility

is presented in [67]: a special built transducer allows detecting the

displacement together with an adaptive control system based on

the model reference adaptive control (MRAC) method. The adaptive

algorithm can regulate the system in real time to adapt the

changes of the parameters and to make the system follow the

desired response. These two methods require extra components,

which means supplementary costs.

Monitoring the induced EMF in the command coils are an

efficient mode of detecting the mover’s position.

The general formula of the EMF induced in the N turn coil j is:

The expressions of the EMF induced in the command coils

are:

0

2

21 1 2

44

4op

F

F

ck

ck

arcsin (3.23)

e Nd

dtjj

j

, 1 4 (3.24)

e e k c C k cddt

C cdk

dt

e e k c C k cddt

C cdk

dt

e F

F

e F

F

1 2 2

22 2

3 4 4

42 2

1 2

1

1 2

1

2

2

4

4

sin cos

cos

cos sin

sin

(3.25)


41

where

v being the mover speed and ek the EMF coefficient:

For the particular case considered above (C C2 40 1 , )

e2 becomes:

The theoretical results obtained can provide a reliable

control method based on monitoring the back EMF generated in

an unenergized coil [59]. The speed can be determined by

integrating the acceleration signal obtained from a piezoelectrical

accelerometer placed on the mover. The moment of the command

current commutation can be obtained by dividing the measured

EMF in an unsupplied coil by the velocity signal. The current

controller has to assure a certain command current in order to

obtain the imposed velocity.

This theoretical development was the fundament of

designing a model reference controller, described in detail in [58].

It is based on the Nerandra model reference adaptive control

method. The reference model gives the desired response of the

adjustable system.

ddt

dxdt

v

2 2

(3.26)

k NP

Fem

pme

4 (3.27)

e k cddte2 sin

(3.28)


42

3.2. COMPARISON OF DIFFERENT CONTROL

METHODS

In order to sustain the theoretical results presented

previously several motor characteristics (forces, acceleration,

speed, displacement, command currents, back EMF) are

determined by computer simulation using the coupled circuit-field

model presented in Chapter 2.

The geo-

metrical dimen-

sions and the

main data of the

hybrid linear step-

per motor, having

four command

coils, are given in

Table 3.1.

In order to

emphasize the

differences that

exist between open-loop and different closed-loop driving modes

the total tangential force, the velocity and the displacement of the

motor plotted against time are given. The conditions are identical: the load is the same, there is no current control (kF 1 ) and the

simulated run time is 25ms.

The results obtained for the open-loop drive mode are

presented in Fig. 3.4. The input frequency is constant and equal to

50Hz. It is easy to see, that at the beginning, the total tangential

force has great values. The velocity and the displacement are

increasing quite uniformly. But at a certain moment, because of

the increased speed, the control angle gets out of range and the

tangential force becomes negative. So the synchronism is lost and

ITEM VALUE tooth width (wt ) 1.16 mm slot width (ws ) 0.84 mm tooth pitch ( ) 2 mm nr. of teeth per pole ( Z ) 5 air-gap (g ) 0.1 mm permanent magnet type VACOMAX-145 residual flux density ( Br ) 0.9 T coercive force ( H c ) 650 kA/m number of coil turns (N ) 200 motor's constant (c ) 0.244 Table 3.1 Parameters and leading dimensions of the considered sample motor

3.2. Comparison of Different Control Methods

43

the speed decreases to zero. Of course, there is a possibility to find

another frequency for which the motor characteristics are

improved. It is not a valuable solution because it will work just for

a certain load and command current value. Therefore it can be

pointed out that the open-loop drive mode does not satisfy the

expectations of a high precision system [55].

Fig. 3.5 and Fig. 3.6 contain the same characteristics of

the motor controlled in closed-loop mode. The motor is operated

with control angle zero (the command currents are commuted

Figure 3.4 Results of simulation for the open-loop drive mode

Figure 3.5 Results of simulation for the closed-loop drive mode (commutation at 0 )

Figure 3.6 Results of simulation for the closed-loop drive mode (commutation at

0 op)


44

after moving a whole step), respectively with 0 op. In the first

case the total tangential force has no negative values, and the

commuting moment takes place at the zero value of the force. The

force ripples are great. Because of a small medium value of the

tangential force, the velocity increases slowly.

In the second case the force ripples are much more

smaller, the medium value of the tangential force is greater and

the characteristics of the motor are good. The speed increases fast

due to the enhanced acceleration characteristics of the motor.

Some values obtained by computer simulation are given in

the Table 3.2.

The above presented motor characteristics obtained via

computer simulation stand by to sustain the theoretical results

and to confirm that the control strategy has to be that presented

in this chapter (commutation at the optimum value of the control

angle and velocity control via the command current).

CONTROL METHODCHARACTERISTICS OPEN- CLOSED-LOOP

LOOP 0 0 opMaximal tangential force [N] 35.46 35.46 35.46

Minimal tangential force [N] -34.55 -0.75 21.78

Maximum velocity [m/s] 0.33 0.48 0.87

Medium tangential force [N] 3.31 19.92 31.15

Medium velocity [m/s] 0.05 0.33 0.59

Final displacement [mm] 4.11 4.63 12.02 Table 3.2 Comparison between the characteristics obtained by computer simulation for the sample motor (different control methods)

3.3. Hybrid Linear Stepper Motor Positioning System

45

3.3. HYBRID LINEAR STEPPER MOTOR

POSITIONING SYSTEM

It is an increasing brisk to automate the factories using

precise variable speed linear positioning systems. For these

purposes the hybrid linear stepper motor is a good choice because

of its high positioning accuracy at significant speeds and its

capability of developing great linear thrust. It is suitable for

precise acceleration, deceleration, and stopping at arbitrary

points. There are no complications involved in using a rotary

motor with rotary to linear gearing, as wear, losses and backlash,

besides the associated extra costs.

Variable speed and high precision positioning are the two

basic and fundamentally conflicting requirements for the motion

controller that has to coordinate the variable speed linear

positioning system. In open-loop drive mode the hybrid linear

stepper motor command current pulses frequency is given by an

external source. However, if the load is varying and the frequency

is not getting in accord with the load modifications, the step

capability of the motor can be exceeded. Dynamic instabilities and

loss of synchronism between the motor position and the excitation

sequence are resulting, and the mover vibrations are amplified.

The total positioning capabilities and dynamic

performances of the motor can be improved by operating it under

closed-loop control via monitoring the induced EMF in

unenergized command coils. This is more expensive than the

open-loop control system because of the required feedback loops,

but enables significant motor efficiency, eliminates mechanical

resonances, allows stable operation at high speed. This control

method also offers the possibility to maintain, in certain limits, a

prescribed motor speed not depending of the load. In this case the

operating frequency will depend only on the capability of the motor


46

to realize a step under given conditions as load and input source

limits.

The positioning system has to ensure in step mode a

controlled motion over a preset distance. In position target mode

the motor has to be moved to an adequately specified location. In

true speed mode the motor has to be driven at a constant speed

irrespective of changing loads. The positioning system has to be

operated in position maintenance mode, too. In this case the

motor position is held to within a closed tolerance under load

fluctuations.

The control unit

of the precise linear

positioning system,

presented in Figure 3.7,

is a combination of an

intelligent controller, of

four circuits for the

captation of the induced

EMF through the

unenergized command

coils and of two dual motion control integrated circuits for the

efficient PWM current control [40].

The proposed intelligent motion controller, the "brain" of

the entire control system, operates based on the control method

proposed in Section 3.1. It coordinates the movement of the motor

in function of the unique external input signal, the prescribed

speed (v * ), and generates the four imposed command coil current

signals ( i * ) in dependence of the detected EMF (e ) and

acceleration (a ). The back EMF generated in the unenergized coil

of the motor is monitored to determine the current commutation

moment. The acceleration signal obtained from a piezoelectrical

accelerometer disposed on the mover can be integrated in order to

Figure 3.7 The control unit of the positioning system


47

compute the actual speed of the motor. By integrating the speed

signal the motor displacement is obtained [43, 44].

The measured EMF divided by velocity determines the

mover displacement as it is indicated by equation (3.28). This is

compared with a prescribed reference. When these two values are

equals, the command current is commuted to another coil. This

way step integrity is guaranteed under all load conditions, because

the start of each step is delayed until the previous step has been

satisfactorily completed. The controller compares the prescribed

speed with the actual motor speed. The information thus collected

provides the imposed currents for the current controllers.

The command coils are fed by two specialized control

integrated circuits (of SLA7024M type, produced by Allegro

MicroSystems Inc. U.S.A.), which enable efficient PWM motor

control [13]. They require beside a few external resistors and

capacitors only a current sensing resistor ( R ), a single fixed reference input (VCC ) and a logical input ( IN ) [14]. The amplitude

of the current pulses is determined by the reference input and

their duration by the logical input.

For high efficiency the commutation of the command

currents must be made in a way as to keep the average value of

the tangential force at its maximal value. Therefore the current

must be commuted before the mover is reaching an intermediate equilibrium position, at the optimal commutation angle 0 op

indicated by equation 3.23. This way the tangential force ripples

are as small as possible [45, 59].

At each time a single command coil is supplied. For each

supplied winding corresponds an unenergized coil at which the

induced EMF is monitored. The correspondence between the

supplied and unenergized coils at each sequence is given in

Table 3.3.


48

The basic characteristics of the adjustable speed linear

positioning system are obtained by dynamic simulation, an

accurate tool for designers because they can try out many

different control algorithms without prototyping hardware [53].

The sample motor is that described in Section 3.2.

In Figure 3.8 some results of the dynamic simulation (the

velocity, the command current in coil one, the tangential force and

the mover’s displacement versus time) are presented in the case of

an adjustable speed linear positioning system with hybrid linear

stepper motor. The system is controlled via the EMF detection

based method.

The simulated task for the positioning system was the

following: the motor moves 5mm to the right with no-load at a

speed of 0.8m/s. It stays stopped 10ms. Following the motor is

moved 3mm with a 0.5kg load at a lower (0.5m/s) speed.

Trapezoidal velocity profiles were adopted [24].

As it can be seen from Fig. 3.8 the command current and

the resulting thrust has great values during the two accelerations.

When the motor is moving at slew speed, the tangential force is

near constant. Negative tangential forces decelerate the motor.

During the motion the displacement is quite linear.

SEQUENCE NUMBER

ENERGIZED COIL

MONITORED COIL

I 4 1 II 2 4 III 3 2 IV 1 3

Table 3.3 Correspondence between the energized coil and the monitored one at each sequence


49

These results show that the imposed task was successfully

fulfilled by the adjustable speed linear positioning system and the

selected control strategy is well suited for such applications.

Figure 3.8 Simulation results of the proposed positioning task (coil number one current, resulting tangential force, motor speed and displacement)

4. DESIGN

Electrical machine design has more than a hundred years

background. The hybrid linear stepper motor design is quite a

different thing than the classical electric motor design. The

complex toothed configuration, the magnetic saturation of the iron

cores and the permanent magnet operating point change due to

air-gap variable reluctance and command MMF arise a lot of

problems to the designer. Therefore establishing an accurate

designing methodology and developing computer programs based

on this is a step toward the direction of cutting down drastically

the number of experiments.

In this chapter a design algorithm of the hybrid linear

stepper motor will be presented. The design method follows a well-

established procedure having four main parts. These are in fact

the stages that one has to go through in the designing process:

i) Establishing the required basic design inputs. Anyway

the requirements for the motor impose particular specifications on

the design inputs.

ii) Calculating the motor main dimensions.

iii) Optimization of a part of the motor dimensions to

increase the final performances and to reduce the costs.

iv) Thermal and electromagnetic analysis of the designed

motor.

The proposed design procedure is based on several

relationships obtained from a simplified analytical motor model

and on some experience resulted values for the important motor

dimension ratios. The last part of the design methodology is built

up around the previously presented coupled circuit-field motor

model. This way the accuracy is good and the required

computation time is short. The design algorithm and the motor

4.1. Design Procedure

51

analyzing procedures can be easily implemented in flexible and

easy-to-use computer programs. The programs allow the designer

to consider iron-core saturation and permanent magnet working

point variation.

4.1. DESIGN PROCEDURE

4.1.1. Prescription of the Basic Design Inputs

In the first phase of the hybrid linear stepper motor design

procedure the required design inputs must be prescribed

depending on the needs of the machine in which the motor will be

used. The basic design inputs are the following:

i) the maximal tangential (traction) force developed by the motor ( f tmax

),

ii) the resolution of the positioning (the step length), function of the selected control strategy (xi ),

iii) the length (lr ) and the width (wr ) of the running track.

These four parameters represent the starting point of the

whole design procedure.

4.1.2. Motor Sizing

In the second stage of the hybrid linear stepper motor

design each of the utilized ferromagnetic materials must be chosen

and all of the motor dimensions must be established.

At the beginning the sizes of the toothed air-gap structure

must be computed. The tooth pitch is given by the imposed

positioning resolution: 4xi (4.1)

4. DESIGN

52

The air-gap length (g ) must be as small as possible.

Normally it is in the range of 0.05...0.1 mm, being limited only by

the mechanical constrains and the cost of manufacturing.

The selection of the best tooth geometry is very important.

As it was previously presented in Section 2.3, the best choice is

that of rectangular teeth having the same width on both

armatures. The optimal tooth width to tooth pitch ratio is 0.42.

The permanent magnet selection, the most expensive and

sensitive assembly of the motor, is extremely important. Rare-

earth magnets are needed to meet the high thrust per unit volume

necessities [37]. It is very important to make a careful choice

between SmCo5 and NdFeB magnets, taking into account the

imposed temperature rise in the mover and the motor cost to

performance ratio.

The mover armature must be made of 0.35mm thick

silicon steel laminated sheets, having high saturation level and low

specific losses. The platen has to be fabricated of soft iron. The flux density in the mover’s poles (B p ) is limited only

by the saturation of the teeth. Excessive saturation absorbs too

much of the excitation MMF or gives rise to extreme heating due to

core losses. As the tooth width is approximately half of the tooth

pitch, the maximum pole flux density can not be much above the

half of the saturation flux density of the steel lamination. The maximum flux density in the platen ( Bs ) is obtained similarly.

The permanent magnet working point on the straight

demagnetization characteristic throughout the second quadrant ( B Hpm pm, ) ensures the desired flux density levels in the mover

and platen cores. The permanent magnet dimensions can be

determined by computing its minimal active surface and

thickness, in order to operate at the imposed working point:


53

where Br and H c are the remanent flux density and the coercive

force of the selected magnet. The two designing constants (kp and kx ) have to be

determined conditionally on the selected air-gap length and tooth

width to tooth pitch ratio from the two diagrams shown in Fig. 4.1

and Fig. 4.2. The initially width of the permanent magnet (wpm ) is

taken equal to the prescribed width of the running track.

The distance between the two electromagnets (le) must be

long enough to avoid the magnetic coupling through the leakage

flux. Beside this the two electromagnets must be displaced by one-

half tooth pitch. The following expression must be considered:

The mover's pole length is given by:

S kf

B Bpm pt

p pmminmax (4.2)

l kB B

H B Bpm x

p r

c r pm

(4.3)

Figure 4.1 Diagram for selecting the design constant kp

Figure 4.2 Diagram for selecting the design constant kx

l k we tk

k N

4 14 2 2

(4.4)

4. DESIGN

54

where S p is the pole area, computed using the following

expression:

The number of the pole teeth can be calculated by:

The most appropriate integer number will be chosen.

Having the number of teeth selected, the final value of the

pole width can be computed:

where wt is the tooth width and ws the slot width.

The mover core has a constant cross-section equal to the

computed pole area, avoiding local iron-core saturations.

The command coil design is made function of its MMF. It

must ensure the necessary command magnetic flux throughout

the poles. This is half of the magnetic flux generated by permanent magnet ( pm ), as shown in Chapter 3.

The command coil MMF can be expressed by:

The command coil sizing procedure follows the well-known

step-by-step outline of designing the winding of a naturally cooled

transformer.

The length of the coil (practically the length of the yoke)

must ensure a displacement equal to a quarter tooth pitch

between the two poles:

lS

wpp

pm (4.5)

SB S

Bppm pm

p (4.6)

Zl p

(4.7)

l Zw Z wp t s 1 (4.8)

F NiZg

S Zc pmp

2

2 10 (4.9)

l k wy tk

k N

2 2

(4.10)


55

The resulting height of the command coil defines the

height of the poles. The ratio of the two terms that form the

command MMF must be in the following range:

With this the motor sizing can be considered finished.

4.1.3. Optimization of the Magnetic Circuit

The main factors that need to be improved to make the

hybrid linear stepper motors more attractive are the cost and

efficiency. To achieve this, an optimization of the motor magnetic

circuit has to be done. The selection of the best tooth geometry

was the first step toward this purpose.

The best results in cost improvements can be made by the

permanent magnet volume optimization. As the computed sizes of

the permanent magnet must be rounded to the sizes included in

the catalogues, myriads of possibilities do exist to select the three

magnet sizes to achieve the same magnetic load. Several

combinations of these three sizes must be considered to obtain the

best solution, for which the magnet has its minimal volume and of

course the less cost.

Another way to decrease the mover's core volume is to

determine the optimum width to length ratio of the command coil (kcoil ). This factor influences the yoke length, the height of the

poles and the sizes of the command coil. Selecting several values

for this ratio the volume of the magnetic circuit and those of the

coils must be examined. For its optimal value both the magnetic

circuit and the winding volume are minimal.

300 500 Ni

(4.11)

4. DESIGN

56

4.1.4. Final Analysis of the Motor

The last step of the design procedure is the

electromagnetic and thermal examination of the designed motor.

Using the previously presented motor mathematical model

the maximal tangential force and the highest flux densities in

different motor portions can be calculated for the greatest

expected command current.

Finally, the motor thermal analysis is performed in order to

determine the temperature distribution over the whole motor

cross-section. It is very important to check the permanent magnet

and the coil insulation temperature.

In general form the thermal equilibrium at a given time of

an ideal homogeneous body is described by:

where p [W] is the total loss in the body, G [Kg] and

ch [Ws/KgC] are the mass, respectively the specific heat capacity

of the body and [W/m2C] is the heat transfer coefficient.

Solving the differential equation the temperature raises in the

body (the difference between the internal and external

temperatures) can be obtained. As it appears in the above

equation the temperature in each point of the motor depends not

only on the losses in that point, but also on the heat generated in

the surrounding area, as well as on the heat flow path throughout

the motor.

In a simplified form the hybrid linear stepper motor can be

considered as an assembly of four basic bodies: the two command

coils, the mover, respectively the platen core. The temperature rise

in the four parts of the motor can be computed by solving the

differential equation system that describes the heat equilibrium in

the motor:

pdt Gc d S dth (4.12)


57

where pw1 and pw2`

are the losses in the coils, pc and ps the

iron-core losses in the mover, respectively in the stator. The other

terms in the left part of the equations are the heat quantities

received from the contiguous bodies. The first terms in the right

part of the equation are the expressions of the heat quantities

accumulated by the bodies, followed by the terms corresponding

to the heat quantities transferred to the neighboring bodies,

respectively to the external environment.

The computer program for the thermal analysis can be

integrated in a global program based on the coupled circuit-field

mathematical model of the motor, presented in Chapter 2. Using

this model the dynamic simulation of the motor is possible. By

solving the above mentioned system at each time step considered

during the iterative process of the dynamic simulation the heating

curves can be obtained. The major temperature limit is that of the

permanent magnet (its maximum temperature without the risk of

damaging its magnetic properties) [37]. The greatest admitted coil

temperature is in relation with its electric insulation.

If the designed hybrid linear stepper motor fulfills the

required performances and the imposed magnetic and thermal

limits, the motor can be considered as designed suitably.

p dt S dt G c d

S S dt S dt

p dt S dt G c d

S S dt S dt

p dt S dt S dt G c d

S dt S S dt

w cw cw c w w w

t wa l wa w wc wc w

w cw cw c w w w

t wa l wa w wc wc w

c wc wc w w sc sc s c Fe c

cw cw c t ca l ca c

t l

t l

t l

1 1

1 1

2 2

2 2

1 2

2

cs cs c

s cs cs c

s Fe s sa sa s sc sc s

S dt

p dt S dt

G c d S dt S dt

(4.13)

4. DESIGN

58

4.2. HYBRID LINEAR STEPPER MOTOR DESIGN

EXAMPLES

The above mentioned design procedure has been used to

design three types of hybrid linear stepper motors: a so-called

sandwiched magnet type, an outer magnet type and one having

four command coils. They differ in the placements of the

permanent magnet and in the number of command coils. These

motor versions are characterized by different efficiencies, costs,

dynamic performances and command possibilities. Each designed

motor satisfies the same required basic input:

The design of the so-called sandwich magnet type hybrid

linear stepper motor will be presented in detail.

The tooth pitch forced upon the imposed step length is of

2mm. The air-gap length was selected of 0.1mm.

As it was presented in Section 2.3 the best choice is to use

the rectangular teeth having the same width on both armatures.

The optimal tooth width to tooth pitch ratio must be of 0.42. This

conducts to a tooth width of 0.84mm and a slot width of 1.16mm.

The flux density in the four mover poles is imposed to be of

0.85T and that in the platen of 0.6T.

The permanent magnet is of VACOMAX-145 type [49]:

The working point of the permanent magnet is given by:

The two designing constants kx and kp were determined

from Figs. 4.1 and 4.2:

f x

l wt i

r r

max.

50 05

200 85

N mm

mm mm (4.14)

B Hr c 09 650. T kA m (4.15)

B B

HB B

BH

pm r

pmr pm

rc

09 081

65

. . T

kA m (4.16)

4.2. Hybrid Linear Stepper Motor Design Examples

59

With these constants the minimal active surface and

thickness of the permanent magnet were found out using Eqs. 4.2,

respectively 4.3:

Next the permanent magnet volume optimization was

performed as it was described in Section 4.1.3. The minimal

magnet volume was found of 1.494cm3. The optimized sizes of the

permanent magnet are:

The width of the magnet will determine the width of the

whole motor. As it can be seen, the width of the motor is in

accordance with the width of the running track:

The pole area is obtained from Eq. 4.6:

With this the pole length can be computed using Eq. 4.5:

The number of the pole teeth was calculated using Eq. 4.7:

Five teeth per pole were selected. Having the teeth number the

precise pole length must be recomputed utilizing Eq. 4.8:

Next the command coil sizing was performed. In order to

evaluate the command coil MMF the equivalent air-gap must be

k kx p 175 8 6 10 6. (4.17)

S

l

pm

pm

min.

. ..

. .

. ..

8 6 1050

0 85 0 81625

1750 85 0 9

650 10 0 9 0 81228

6

3

cm

mm

2

(4.18)

l h w

S h wpm pm pm

pm pm pm

2 9 83

747

mm mm mm

mm2 (4.19)

w wr pm (4.20)

S p

0 81 747

0 8571184

..

. mm2 (4.21)

l p 71184

83857

.. mm (4.22)

Z 857

2428

.. (4.23)

l p 5 0 84 4 116 8 84. . . mm (4.24)

4. DESIGN

60

determined. Using Eqs. A1.3 through A1.5 the following results

were obtained:

The equivalent air-gap was calculated using Eq. A1.8:

Eq. 4.9 yields to the necessary command coil MMF:

This MMF can be assured by energizing a 200 turns coil

with a 0.65A command current.

In the following stage the optimization of the command coil

sizes was performed. These sizes determine the length of the yokes

and the height of the poles. Studying several values for the width to length ratio of the command coil (kcoil ) its optimal value was

found out as to be 3. For this value the mover volume was

minimal (82.96cm3). Taking into account Eq. 4.10, too, the coil

length was computed to be of 18.2mm.

The coils are made of winding conductor having 0.56mm

diameter. 30 turns can be placed on each of the 7 layers. The

resistance of the command coils was estimated to be 2.74.

The height of the command coil and the width of the yoke

(taken equal to the pole length) determined the pole height to be of

18mm.

With this all the motor dimensions were computed. The

cross-sectional view of the designed sandwich magnet type hybrid

linear stepper motor is given in Fig. 4.3. with all of the dimensions

in millimeters.

6465 0415 11685

0917 1477

. . .

. .

u

kc (4.25)

g 1477 01 02182. . . mm (4.26)

Fc 12612 130. Aturns (4.27)


61

Finally the electromagnetic and thermal checking of the

designed motor was performed.

Using a separate computer program, the maximal

tangential force and the highest flux densities in different motor

portions were calculated. All these parameters were found as in

accordance with the imposed data.

By solving the

differential equation

system, which describes

the heat equilibrium of

the motor (Eq. 4.13), at

each time step consi-

dered during the itera-

tive process of the dyna-

mic simulation, the

heating curves repre-

sented in Fig. 4.4 were

obtained.

As it can be seen

the major temperature

limits were not reached. Consequently, of the very low losses in

the motor, it can be cooled sufficiently by natural air convection.

Figure 4.3 The main dimensions of the designed sandwich magnet type motor

Figure 4.4 The heating curves obtained by the thermal checking of the sandwich magnet type hybrid linear stepper motor

4. DESIGN

62

The presented computer aided design methodology not only shor-

tened the design process, but also gave more economical, efficient

and higher quality alternatives. It was used with insignificant

changes for the design of the other two types of hybrid linear

stepper motors.

The design procedure of the other two types of hybrid

linear stepper motors is almost the same as that presented above.

Some of the most important parameters that can be considered

common for all the motors to be designed are presented in

Table 4.1.

In the case of the outer magnet type hybrid linear stepper

motor (shown in Fig. 1.6) the permanent magnet (of the same size

as in the previous case) is detached in two pieces. The command

coils are also divided in two and they are wound round the four

poles. The cross-section of the back iron, which closes the

magnetic circuit, is taken equal to the area of the platen. The

outline diagram of the designed outer magnet type motor is given

in Fig. 4.5.

CHARACTERISTICS VALUES

Tooth width (wt ) 0.84 mm

Slot width (ws ) 1.16 mm

Number of teeth on one pole ( Z ) 5

Pole length (l p ) 8.84 mm

Pole area (S p ) 7.11 cm2

Permanent magnet width (wpm ) 83 mm

Permanent magnet length (l pm ) 2 mm

Table 4.1 The most important parameters of the designed motors


63

Designing a hybrid linear stepper motor having four

command coils differs in a small extent of the first motor variant

sizing. The difference consists in the number and in the placement

of the command coils. This motor version has four coils, placed on

each pole, every one ensuring the command MMF.

This motor construction can avoid the main disadvantage

of the sandwich magnet type motor, that maximal magnetic fluxes

through the inner poles are a little larger than that through the

outer ones [38]. Increasing the number of turns of the coils of the

outer poles, the flux throughout them will be equalized with that

of the inner poles. The cross-sectional view of the designed motor

is shown in Fig. 4.6.

Figure 4.5 The outline with the main dimensions of the designed outer magnet type motor

4. DESIGN

64

4.3. COMPARISON OF THE THREE DESIGNED

MOTORS

As the initial design input was the same for all the three

designed hybrid linear stepper motor variants, they can be easily

compared.

The most important comparison characteristics are

included in Table 4.2. The first variant is that with the sandwich

magnet type, the second one is that having outer magnets and the

third one is the sandwich type with four command coils.

Figure 4.6 The cross-sectional view with the main sizes of the sandwich type motor with four command coils

CHARACTERISTICS Variant 1 Variant 2 Variant 3

Mover length [mm] 78.42 72.34 92.34

Mover height [mm] 23.8 38.00 23.00

Mover width [mm] 83.00 83.00 83.00

Core volume [cm3] 82.96 131.38 124.25

Magnet volume [cm3] 1.49 1.49 1.49

Winding volume [cm3] 19.29 19.23 41.21

Mover mass [g] 848 1257 1373 Table 4.2 Comparison of the main characteristics of the designed motor variants

4.3. Comparison of the Three Designed Motors

65

As it can be observed, the mass of the first version is the

smallest one, so its dynamic characteristics are the best ones. Its

main detriment, as it was shown previously, consists in the

difference of the maximal magnetic flux through the outer and the

inner poles. This drawback was solved at the second motor variant

by placing the permanent magnet symmetrically above the poles

and for the last version by increasing the number of turns of the

command coils disposed on the outer poles.

The motor having four command coils can be commanded

by unipolar (only positive) current pulses, a fact that simplifies

very much the control circuits. On the other hand its winding

volume is twice as much as that of the other types.

Finally, it can be concluded that neither one of the motors

taken in study is superior from all points of view. Therefore the

choice of the most suitable motor must be made taking into

account the needs of the electromechanical system in which they

are operating (accuracy, dynamic characteristics, the character of

the load and the type of positioning), as well as the possibilities of

the available control systems.

The design algorithm presented in this chapter can lay on

the basis of a CAD software. The programs can have the ability to

create the motor design almost totally, to optimize some motor

components (to attain the best achievements possible with a good

performance to cost ratio) and to display any of the computed

motor dimensions and each of the estimated motor performances.

The design procedure, as it was formulated in this chapter,

is valid with a few modifications for other permanent magnet

excited synchronous motors, too.

APPENDIX 1. EQUIVALENT VARIABLE AIR-GAP CALCULATION The equivalent variable air-gap permeance calculation is

based on two assumptions [61]:

i) The iron-core magnetic permeability is much more

greater than the air-gap one.

ii) The harmonics with harmonic order greater than the

pole number of teeth are neglected.

If the air-gap MMF is equal with a unit, then the air-gap

equivalent variable permeance is:

where g is the air-gap length, Pm1 and Pm2

are the air-gap

equivalent variable magnetic permeances calculated considering

slots and teeth only on the platen, respectively only on the mover

[41]:

where is the variable displacement between the axes considered on the mover and platen, kc is the Carter's factor, Z is the

number of the mover’s pole teeth and is the variable permeance

coefficient.

P x P x P x gm m m( ) ( ) ( )1 2

(A1.1)

P xk g

x

P x

k gx

x Z Z Z Z

k gx

x Z Z

mc

Z

m

c

Z

c

Z

1

2

11 1

12

1 1

1 11

1 1

1 1

1

1

1

( ) cos

( )

cos

, ,

cos

,

(A1.2)

Equivalent Variable Air-Gap Calculation

67

The equivalent variable air-gap permeance variation is

given in Fig. A1.1.

Two situations were studied: odd number of mover teeth

(Fig. A1./a and c), respectively even number (Fig A1./b and d).

First only the mover teeth were considered (Fig. A1./a and b). In

the second case only the platen teeth were taken into account

(Fig. A1./c and d). In both cases the axes of the mover and of the

platen are co-linear, which means that the variable displacement is zero.

The equivalent variable air-gap permeance coefficient ( ) and the Carter's factor (kc ) can be calculated with the following

relations [25]:

where:

Figure A1.1 The air-gap equivalent variable permeance variation

=

5k

g2c sin

(A1.3)

ck =- g

(A1.4)

APPENDIX 1. 68

being the tooth pitch and ws the slot width. In the above given

relations the tooth pitch and tooth width of the mover and platen

were considered the same [52].

The average specific permeance under one pole is given by:

and after some computations it results:

where the equivalent air-gap g is:

The average specific permeance variation under the motor

poles is given in Fig. A1.2. As it was expected, the permeance

maximum value occurs when the pole teeth are aligned with the

platen teeth.

322 2

12

1

21

21

2

2

2

2

2

.arctan ln

w

g

w

g

w

g

u

u

uw

g

w

g

s s s

s s

(A1.5)

PZ

P x dx P x dx P x dxm mZ

Z

mZ

Z

mZ

Z

e*

( )

( )

( )

( )

( ) ( ) ( )

12

1

1

1

1

(A1.6)

PZg

Z Zme* cos

1

2 21 2 1 2 1

(A1.7)

g k gc2 (A1.8)

Equivalent Variable Air-Gap Calculation

69

The equivalent variable air-gap is:

with the motor constant c given by:

Next the equivalent magnetic permeance of electromagnet

A (see Fig. 2.1) can be expressed using Eq. 2.21 as:

resulting in:

As it can be seen the equivalent magnetic permeance of

one electromagnet does not depend on the mover's position.

Figure A1.2 The average specific permeance variation

gZg

Z ce

2

2 1 1 cos (A1.9)

c

Z

Z

1 2 1

2 2 1 (A1.10)

P P P

S

ZgZ c

m m m

p

A g g

1 2

01 22

2 1 2

cos cos (A1.11)

P P PS

ZgZm m m

p

e A B

02 1 (A1.12)

REFERENCES

1. ACARNLEY P.P. et al.: Detection of Rotor Position in

Stepping and Switched Motors by Monitoring of Current

Waveforms, Transactions on Industrial Electronics, vol. IE-32.

(1985), no. 3., pp. 215-222.

2. ACARNLEY P.P.: Stepping Motors: A Guide to Modern Theory

and Practice, IEE Control Engineering Series 19., Peter

Peregrinus, 1984.

3. ASTROM K.J - WITTENMARK B: Computer - Controlled Sys-

tems - Theory and Design, Prentice Hall, 1990.

4. BERTOLINI T.: The Calculation of Stepper-Motors Torque in

Regard to the Geometry of the Stator and Rotor-Teeth,

Proceedings of the International Conference on Electrical

Machines , Pisa, 1988, pp.161-164.

5. BINNS K.J. et al.: Hybrid Permanent - Magnet Synchronous

Motors, Proceedings IEE, vol. 125, No. 3. (march 1978),

pp. 203-208.

6. BINNS K.J. et al.: Implicit Rotor Position Sensing for

Permanent Magnet Self - Commutating Machine Drives,

Proceedings of International Conference on Electrical Machines,

Cambridge, 1990, pp. 1231-1236.

7. BINNS K.J. - LAWRENSON P.J.: Analysis and Computation of

Electric and Magnetic Field Problems, Pergamon Press, 1973.

8. BIRÓ K. et al.: Dynamic Modelling of Inverter-Fed

Permanent Magnet Synchronous Motors, Proceedings of the

International Conference on the Evolution and Modern Aspects of

Synchronous Machines, Zürich, 1991, pp. 229-233.

9. COLBY R.S. - NOVOTNY D.W.: An Efficiency Optimizing

Permanent Magnet Synchronous Motor Drive, IEEE

REFERENCES

71

Transactions on Industry Applications, vol. 24, no. 3. (may-june

1988), pp. 462-469.

10. CORDA J.: Linear Variable - Reluctance Actuator in Cylin-

drical Form, Proceedings of the PCIM Conference, Nürnberg,

1994, Vol. Intelligent Motion, pp. 133-140.

11. EBIHARA D.: Design of a PM Type Linear Stepping Motor

for Automatic Conveyer System, Proceedings of the

International Conference on Electrical Machines Design and

Applications, London, 1985, pp. 265-269.

12. EBIHARA D. - WATADA M.: The Study on Basic Structure of

Surface Actuators, Digests of the INTERMAG '89 Conference,

pp. HB-9.

13. EMERALD P.: Power Multi-Chip Modules Reawaken High-

Speed, Efficient PWM Operation of Unipolar Stepper Motors,

PCIM, may 1995, pp. 25-33.

14. EMERALD P. - SASAKI M.: Multi-Chip Modules for Stepper

Motors Integrate PWM Control with Power FETs for Superb

Performance, Proceedings of the PCIM Conference, Nürnberg,


15. FINCH J.W. et. al.: Normal Force in Linear Stepping Motors,

Proceedings of the International Conference on Small and Special

Electrical Machines, London, 1981, pp. 78-81.

16. FU Z.X. - NASAR S.A.: Analysis of a Hybrid Linear Stepper

Motor, Proceedings of the Annual Symposium on Incremental

Motion Control Systems & Devices, Urbana-Champaign, 1992,

pp. 234-240.

17. GODDIJN B.H.A.: New hybrid stepper motor design,

Electronic Components and Applications, vol. 3. (1980), no. 1.

18. GRANTHAM W.J. - VINCENT T.L.: Modern Control Systems

Analysis and Design, John Wiley & Sons, 1993.

19. HANITSCH R.: Electromagnetic Machines with Nd-Fe-B

Magnets, Journal of Magnetism and Magnetic Materials, vol.80.

(1980), pp. 119-130.

REFERENCES

72

20. HAYT W.H. Jr.: Engineering Electromagnetics, McGraw-Hill,

1974.

21. HOFFMAN B.D.: The Use of 2-D Linear Motors in Surface

Mount Technology, Proceedings of the Surface

Mount Conference, Boston, 1990, pp. 183-189.

22. HONG Y. et al.: The Latest Study on CAD of Stepper

Motors, Proceedings of the International Conference on Electrical

Machines, Manchester, 1992, pp. 642-646.

23. JENKINS et al.: An Improved Design Procedure for Hybrid

Stepper Motors, IEEE Transactions on Magnetics, Vol. 26., No.5.

(september 1990), pp. 2535-2538.

24. JONES D.I. - FINCH J.W.: Computer Aided Synthesis of

Optimal Position and Velocity Profiles for a Stepping Motor,


Machines, Cambridge, 1990, pp. 836-841.

25. JUFER M. et al.: On the Switched Reluctance Motor Air-

Gap Permeance Calculation, Proceedings of the

ELECTROMOTION International Symposium, Cluj, Romania,

1995, pp. 141-146.

26. KENJO T.: Stepping Motors and their Microprocessor

Controls, Monographs in Electrical and Electronic Engineering,

Vol. 16., Oxford Science Publications, 1985.

27. KUO B.C. - HUH U.Y.: Permeance Models and their Appli-

cations to Step Motor Design, Proceedings of the Annual

Symposium on Incremental Motion Control Systems & Devices,

Urbana-Champaign, 1986, pp. 351-369.

28. LEU M.C. et. al.: Characteristics and Optimal Design of

Variable Airgap Linear Force Motors, IEE Proceedings, vol. 135.,

Pt. B, No. 6. (november 1988), pp. 341-345.

29. LOFTHUS R.M. et al: Processing Back EMF Signals of

Hybrid Step Motors, Control Engineering Practice, vol. 3., no. 1.

(1995), pp. 1-10.

REFERENCES

73

30. MARLIN T.E.: Process Control - Designing Processes and

Control Systems for Dynamic Performance, McGraw-Hill

International, 1995.

31. McLEAN G.W. et al.: Design of a Permanent Magnet Linear

Synchronous Motor Drive for a Position/Velocity Actuator,


Machines, Cambridge, 1990, pp. 1136-1141.

32. McLEAN G.W.: Review of Recent Progress in Linear Motors,

IEE Proceedings, Vol. 135., Pt. B, No. 6. (november 1988),

pp. 385-417.

33. MULLER W.: Comparison of Different Methods of Force

Calculation, IEEE Transactions on Magnetics, Vol. 26., No.2.

(march 1990), pp. 1058-1061.

34. NASAR S.A. - BOLDEA I.: Linear Motion Electric Machines,

John Wiley & Sons, 1976.

35. NORDQUIST J.I. - SMIT P.M.: A Motion-Control System for

(Linear) Stepper Motors, Proceedings of the Annual Symposium

on Incremental Motion Control Systems & Devices, Urbana-

Champaign, 1985, pp. 215-231.

36. MOHAN N. et al: Power Electronics - Converters, Appli-

cations and Design, John Wiley & Sons, 1995.

37. PARKER R.J.: Advances in Permanent Magnetism, John

Wiley & Sons, 1990.

38. SANADA M. et al.: Cylindrical Linear Pulse Motor with

Laminated Ring Teeth, Proceedings of the International

Conference on Electrical Machines, Cambridge, 1990,

pp. 693-698.

39. SILVESTER P.P. - FERRARI R.L.: Finite Elements for Electri-

cal Engineers, Cambridge University Press, 1983.

40. SZABÓ L. et al.: Computer Aided Design of a Linear Posi-

tioning System, Proceedings of the Power Electronics, Motion

Control Conference (PEMC), Budapest, 1996, vol II., pp. 263-267.

REFERENCES

74

41. SZABÓ L.: Computer Aided Design of Variable Reluctance

Permanent Magnet Synchronous Machines, Ph.D. Thesis,

Technical University of Cluj-Napoca, Romania, 1995 (in

Romanian).

42. SZABÓ L. et al.: Computer Simulation of a Closed-Loop

Linear Positioning System, Proceedings of the PCIM Conference,

Nürnberg, 1993, Vol. Intelligent Motion, pp. 142-151.

43. SZABÓ L. et al.: Computer Simulation of a Constant

Velocity Contouring System Using x-y Surface Motor,

Proceedings of the PCIM Conference, Nürnberg, 1995,

Vol. Intelligent Motion, pp. 375-384.

44. SZABÓ L. et al.: E.M.F. Sensing Controlled Variable Speed

Drive System of a Linear Stepper Motor, Proceedings of the

Power Electronics, Motion Control Conference (PEMC), Warsaw,

1994, pp. 366-371.

45. SZABÓ L. et al.: Variable Speed Conveyer System Using

E.M.F. Sensing Controlled Linear Stepper Motor, Proceedings

of the PCIM Conference, Nürnberg, 1994, Vol. Intelligent Motion,

pp. 183-190.

46. SUN L.I.: A New Type of Closed Loop Operation Mode of PM

Hybrid Step Motor, Proceedings of the International Conference

on the Evolution and Modern Aspects of Synchronous Machines,

Zürich, 1991, pp. 533-537.

47. TAKEDA Y. et al.: Optimum Tooth Design for Linear Pulse

Motor, Conference Record of the 1989 IEEE Industry Applications

Society Annual Meeting, pp. 272-277.

48. TRIFA V. - KOVÁCS Z.: Numerical Simulation of Linear

Stepping Motor for X-Y Plotter, Proceedings of the Third

National Conference on Electrical Drives, Brasov, Romania, 1980,

vol. I., pp. A17-A22.

49. VAC Vacuumschmelze: Rare - Earth Permanent Magnet

Materials: VACODYM and VACOMAX, (Catalogue), 1992.

REFERENCES

75

50. VAS P.: Parameter Estimation, Condition and Diagnosis of

Electrical Machines, Oxford University Press, 1993.

51. VEIGNAT N. et al.: Stepper Motors Optimisation Use, Pro-

ceedings of the International Conference on Electrical Machines,

Paris, 1994, pp. 13-16.

52. VIOREL I.A. et al.: Combined Field-Circuit Model for Induc-

tion Machine Performance Analysis, Proceedings of the

International Conference on Electrical Machines, München, 1986,

pp. 476-479.

53. VIOREL I.A. et al.: Dynamic Modelling of a Closed-Loop

Drive System of a Sawyer Type Linear Motor, Proceedings of

PCIM Conference, Nürnberg, 1992, Vol. Intelligent Motion,

pp. 251-257.

54. VIOREL I.A. et al.: On the Linear Stepper Motor Control

Basics, Proceedings of the PCIM Conference, Nürnberg, 1995,

Vol. Intelligent Motion, pp. 481-494.

55. VIOREL I.A. - SZABÓ L.: Permanent - magnet variable - re-

luctance linear motor control, Electromotion, vol. 1., nr. 1.

(1994), pp. 31-38.

56. VIOREL I.A. et al.: Quadrature Field-Oriented Control of a

Linear Stepper Motor, Proceedings of the PCIM Conference,

Nürnberg, 1993, Vol. Intelligent Motion, pp. 64-73.

57. VIOREL I.A. et al.: Sawyer Type Linear Motor Modelling,


Machines, Manchester, 1992, pp. 697-701.

58. VIOREL I.A. et al.: Sensorless Adaptive Control of a Linear

Stepper Motor, Proceedings of the PCIM Conference, Nürnberg,


59. VIOREL I.A. - SZABÓ L.: The Control Strategy of a Drive

System with Variable Reluctance Permanent - Magnet Linear

Motor, Proceedings of the 5th International Conference on

Optimization of Electric and Electronic Equipments, OPTIM,

Brasov, Romania, 1996, pp. 1593-1600.

REFERENCES

76

60. VIOREL I.A. et al.: Transformer Transient Behavior Simula-

tion by a Coupled Circuit-Field Model, Proceedings of the

International Conference on Electrical Machines, Paris, 1994,

pp. 654-659.

61. VIOREL I.A. - RÃDULESCU M.M.: On the Induction Motor

Variable Equivalent Air-Gap Permeance Coefficients

Computation, EEA Electrotehnica vol. 32., nr. 3., 1984,

pp. 108-111 (in Romanian).

62. VIOREL I.A. et al.: Magnetic Field Numerical Analysis of a

Variable Reluctance Hybrid Linear Motor, Proceedings of the

Romanian Conference on Electrical Engineering, 1984, Craiova,

Romania, vol. 6. (1982), pp. 173-182 (in Romanian).

63. WAGNER J.A. - CORNWELL W.J.: A Study of Normal Force

in a Toothed Linear Reluctance Motor, Electric Machines and

Electromechanics, vol. 7, pp. 143-153.

64. WAVRE N. - VAUCHER J-M.: Direct Drive with Servo Linear

Motors and Torque Motors, Proceedings of the European Power

Electric Chapter Symposium (Electric Drive Design and

Application), Lausanne, 1994, pp. 151-155.

65. WENDORFF E. - KALLENBACH E.: Direct Drives for Positio-

ning Tasks Based on Hybrid Stepper Motors, Proceedings of the

PCIM Conference, Nürnberg, 1993, Vol. Intelligent Motion,

pp. 38-52.

66. XIONG G. - NASAR S.A.: Analysis of Field and Forces in a

Permanent Magnet Linear Synchronous Machine Based on the

Concept of Magnetic Charge, IEEE Transactions on Magnetics,

vol. 25., no. 3., may 1989, pp. 2713-2719.

67. XU S. et al.: Adaptive Control of Linear Stepping Motor,


Machines, Paris, 1994, pp. 178-181.

LIST OF THE MAIN SYMBOLS

a acceleration [m/s2] Az z-axis component of the magnetic vector potential [Wb/m]

Bp flux density in the mover poles [T]

Bpm permanent magnet flux density [T]

Br permanent magnet remanent flux density [T]

Bs flux density in the platen [T]

c motor constant [-] c f friction coefficient [-]

ch specific heat capacity [Ws/Kg°C]

C j square wave MMF factor (j=1÷4) [-]

e j induced EMF in coil j (j=1÷4) [V]

FA , FB , Fc command coil MMF [A turns]

FCA , FCB sinusoidal command MMF [A turns]

FCAM, FCBM

peak value of the sinusoidal command MMF

[A turns] FC j

square wave command MMF (j=1÷4) [A turns]

Fpm permanent magnet MMF [A turns]

fn total normal force [N]

fn j normal force under pole j (j=1÷4) [N]

ft total tangential force [N]

f t j tangential force under pole j (j=1÷4) [N]

ft* unitary total tangential force [-]

ftm* unitary average tangential force [-]

ftmax maximal tangential force [N]

fw flat width of the wedged head teeth [m]

g air-gap length [m]

G mass [Kg]


78

g equivalent air-gap [m]

ge j equivalent variable air-gap under pole j (j=1÷4) [m]

Hc permanent magnet coercive force [A/m]

H pm permanent magnet magnetic field intensity [A/m]

i * imposed command coil current [A] iA , iB current in command coil A, respectively B [A]

j pole number [-]

J z z-axis component of the current density [A/m2]

kc Carter’s factor [-]

kcoil width to length ratio of the command coil [-]

ke EMF coefficient [V]

kFA, kFB

, kF sinusoidal command MMF factor [-]

kFj square wave command MMF factor (j=1÷4) [-]

kft tangential force coefficient [N]

kx designing constant of the permanent magnet thickness

[m2/H] kp designing constant of the permanent magnet active surface

[Wb/kg] LAm

, LBmmain inductance of the command coils [H]

LA , LB leakage inductance of the command coils [H]

le distance between the two electromagnets [m]

lp pole length [m]

lpm permanent magnet thickness [m]

lr running track length [m]

ly yoke length [m]

m mover’s mass [Kg] N , N A , N B command coil turns [-]

n unit outward vector normal [-]

p total losses in the body [W]

pc iron losses in the mover [W]

Pm air-gap equivalent variable permeance [Wb/A]


79

Pme, PmA

, PmBequivalent magnetic permeance of an

electromagnet [m]

Pme* average specific permeance of the air-gap under one pole

[1/m] Pmpm

permanent magnet permeance [Wb/A]

Pmg j air-gap permeance under pole j (j=1÷4) [Wb/A]

ps platen core iron losses [W]

pw1, pw2

coil losses [W]

RA , RB command coil resistance []

RmFe iron core reluctance [A/Wb]

Rmpm permanent magnet reluctance [A/Wb]

Sp pole area [m2]

Spmminpermanent magnet minimal active surface [m2]

u equivalent air-gap permeance computation factor [-]

v mover speed [m/s]

v * imposed speed [m/s] vA , vB command coils input voltage[V]

wpm permanent magnet width [m]

wr running track width [m]

ws slot width [m]

wt tooth width [m]

Z number of the pole teeth xi step length [m]

mover’s angular displacement [rad]

heat transfer coefficient [W/m2°C] j relative angular displacement of the pole axis (j=1÷4) [rad]

0 commutation angular position [rad]

( )0 op commutation optimal angular position [rad]

equivalent air-gap permeance computation factor [-]

CA , CB magnetic flux produced by the command coils [Wb]


80

CAm , CBm main magnetizing flux produced by the command

coils [Wb] CA , CB leakage flux produced by the command coils [Wb]

i magnetic flux generated only by the permanent magnet

(i=1÷11) [Wb] i magnetic flux generated only by the command coils (i=1÷11)

[Wb] j magnetic flux through pole j (j=1÷4) [Wb]

pm magnetic flux generated by the permanent magnet [Wb]

0 j magnetic flux through poles j without command currents

(j=1÷4) [Wb] equivalent air-gap permeance computation factor [-]

equivalent variable air-gap permeance coefficient[-] magnetic permeability [H/m]

0 free space permeability [H/m]

x , y axial components of the magnetic reluctivity [m/H]

over-temperature [ºC] w slop of the teeth wedge[°]

closed surface tooth pitch [m] A , B total flux linkage through the command coils [Wb]

INDEX

acceleration signal 42, 46

accurate positioning 32

adjustable speed linear

positioning system 48

air-gap

equivalent variable 69

length 52, 66

magnetic reluctance 5

MMF 66

permeance 19

variable equivalent

permeance 19, 66

analytical results 18

angular position 2

anisotropic permanent

magnet 21

average

specific permeance 68

speed 2

total tangential force 32, 47

unitary tangential force 37

back EMF 41-42, 46

backing 6

block diagram 14

boundary conditions 23

CAD software 65

circuit-field model 13

circuit-field-mechanical

model 12, 14

circuit-type model

(equation) 11

closed-loop control 3, 32

coil

command 4, 11, 30

de-energized 6

energized 6, 30

command coil 4, 11, 30

command current 11, 30, 48

control

algorithm 48

closed-loop 3, 32

integrated circuit 47

model reference adaptive

(MRAC) 40

open-loop 3, 32

PWM 47

strategy 18

core losses 52

cosine wave voltage 6

de-energized coil 6

demagnetization

characteristic 52

design

algorithm 50, 65

constant 53

method 50

procedure 51

INDEX

82

digital electric input 2

displacement of the motor 42

dynamic

regime 14

simulation 48

eddy current 22

electromagnetic checking 61

electromagnets 4

equivalent magnetic

permeance 69

electro-motive force (EMF)

back 41-42, 46

coefficient 41

detection based method 48

induced 33, 40, 45, 47

measured 41, 47

sensing 33

energized coil 6, 30

energy functional 22

equivalent magnetic

circuit 14-15

permeance of an

electromagnet 69

equivalent variable air-gap 69

experimental results 29

feedback loop 2

finite differences

methods 21-22

models 14

finite elements

methods 21

models 14

flux linkage 13

force

normal 26

ripples 44

tangential 5

total tangential 42, 48

transducer 30

gradient of the magnetic

co-energy 19

Hamilton’s principle 22

hardware simulation 48

heat

capacity 56

transfer coefficient 56

heating curves 61

homogenous boundary

conditions 22

hybrid stepper motor 3

hybrid linear stepper motor

with four command

coils 33-34, 58, 63-65

with outer

magnet 7, 58, 62-64

with sandwich

magnet 16, 29, 58, 60-64

hysteresis effects 21

incremental mechanical

motion 2

induced EMF 47

INDEX

83

inductance

leakage 11

main 11-12

induction machine 19

interpolation function 31

leakage inductance 11

logistic approximation 31

loss of synchronism 45

magnetic

energy 23

reluctance 18

magneto-motive force (MMF)

54

air-gap 66

command 50, 55, 63

excitation 52

factor 36, 38, 40

of the command

coil 10-11, 54, 59-60

permanent magnet 34

main

path flux 13

inductance 11-12

mathematical model 31

Maxwell’s stress tensor 19, 23

measured EMF 41, 47

measurement 30

mechanical model 13, 20

model reference adaptive

control (MRAC) 40

monitoring 47

moveable armature (mover) 4

mover’s displacement 47-48

normal attractive force 26

Norton’s equivalent circuit 15

number of pole teeth 54

numeric simulation 30

numerical method 14

open-loop control 3, 32

optimization 65

of the magnetic circuit 55

optimum control angle 32

outer magnet type

motor 7, 58, 62- 64

permanent magnet 7, 10-11,

5-16

anisotropic 21

flux 5, 10, 13, 34

MMF 16, 34

operating point 12-13,33,

50-52

reluctance 11, 16, 18, 33

ring type 8

working point 52

permeance

average specific 68

variable equivalent of

the air-gap 19, 66

piezoelectrical accelerometer

41, 46

platen 4

INDEX

84

position

accuracy 3

error 3

maintenance mode 46

target mode 46

positioning system 48

PWM motor control 47

reluctance stepper motor 3

reluctivity tensor 22

ring type permanent magnet 8

rotor 2

running track 51

salient poles 2

sample motor 29, 48

sandwich magnet type

motor 16, 29, 58, 60-64

saturation 12, 52

simulation task 48

sine wave voltage 6

slots 19

specific heat capacity 56

standstill current decay test 12

static characteristics 30

stator 2

steady-state regime 14

step

angle 2

integrity 47

length 51

stepper motor 1

superposition principle 33

surface hybrid linear

stepper motor 8

switched reluctance motor 1

symmetry conditions 23

synchronism 42

synchronous machine 1

tangential force 5

average total 32, 47

average unitary 37

tangential trust force 26

test

graph 9

software 9

thermal

analysis 57

equilibrium 56

checking 61

tooth 19

pitch 4

toothed structure 4

total tangential force 42, 48

tracking 6

traction force 6

tubular hybrid linear

stepper motor 8

unitary tangential force 37

unenergized coil 41

variational field model 22

velocity 42, 48

voltage equation 11

TO THE READER

The purpose of this book is to give its readers an

understanding and familiarity to the behaviour, uses, control

strategy and design of the hybrid linear stepper motor.

The author’s researches were focused on the hybrid

linear stepper motor for quite a long time. During this time some

pertinent results were obtained and spread off in various

scientific papers. These theoretical and practical results

constitute the base of the present work.

The authors are grateful to their colleagues for the

assistance received in various ways when clarifying some

research aspects or preparing the text.

The authors are particularly indebted to Prof. Radu

Munteanu, Technical University of Cluj, Romania, for his

generous encouragement and valuable help. Also we owe a great

deal to the support and friendly counsel of Prof. Ion Boldea,

Technical University of Timisoara, Romania.

The authors are deeply indebted to Prof. Gerhard

Henneberger, RWTH Aachen, Germany, who gave generously of

his time to read through the manuscript, made many useful

suggestions and accepted to introduce to the reader our work.

All errors, ambiguities and imperfections are our

responsibility and we want to know about them. Therefore all

corrections, clarifications, additions or suggestions are cheerfully

welcomed.

Ioan-Adrian Viorel, Szabó Loránd

Technical University of Cluj P.O. Box 358. 3400 Cluj, Romania e-mail: [email protected] [email protected]

The book may be recommended to all those who are

interested in the basic theory of hybrid linear stepper motors

as well as in modern techniques of control strategies and

design optimisation. In comparison to conventional literature

new ideas for a complete mathematical model, for closed-loop

systems with optimal control and for design optimisation by

FEM-analysis are given.

Univ.-Prof. Dr.-Ing. Dr. h. c. G. Henneberger

Institut für Elektrische Maschinen Rheinisch-Westfälische Technische

Hochschule Aachen

Documents

Hybrid Linear Stepper Motors