hybrid linear stepper motors
DESCRIPTION
The book may be recommended to all those who are interested in the basic theory of hybrid linear stepper motorsas 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.TRANSCRIPT
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)
2. THEORY AND PERFORMANCE
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
2.1. Mathematical Model
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)
2. THEORY AND PERFORMANCE
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
2.1. Mathematical 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
2. THEORY AND PERFORMANCE
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
2.1. Mathematical Model
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)
2. THEORY AND PERFORMANCE
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
2.1. Mathematical Model
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)
2. THEORY AND PERFORMANCE
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
2.1. Mathematical 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)
2. THEORY AND PERFORMANCE
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)
2. THEORY AND PERFORMANCE
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
2. THEORY AND PERFORMANCE
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
2. THEORY AND PERFORMANCE
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
2. THEORY AND PERFORMANCE
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
3. CONTROL STRATEGIES
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)
3.1. Theoretical Approach
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
3. CONTROL STRATEGIES
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)
3.1. Theoretical Approach
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)
3. CONTROL STRATEGIES
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)
3.1. Theoretical Approach
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)
3. CONTROL STRATEGIES
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)
3.1. Theoretical Approach
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)
3. CONTROL STRATEGIES
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)
3. CONTROL STRATEGIES
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
3. CONTROL STRATEGIES
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
3.3. Hybrid Linear Stepper Motor 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.
3. CONTROL STRATEGIES
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
3.3. Hybrid Linear Stepper Motor Positioning System
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:
4.1. Design Procedure
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)
4.1. Design Procedure
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)
4.1. Design Procedure
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)
4.2. Hybrid Linear Stepper Motor Design Examples
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
4.2. Hybrid Linear Stepper Motor Design Examples
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)
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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]
LIST OF THE MAIN SYMBOLS
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]
LIST OF THE MAIN SYMBOLS
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]
LIST OF THE MAIN SYMBOLS
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