Hybrid linear stepper motors by John B. Young

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

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Printed in Romania

CONTENTS 1. INTRODUCTION 1.1. STEPPER MOTORS 1.2. HYBRID LINEAR STEPPER MOTOR OPERATING PRINCIPLES 1.3. HYBRID LINEAR STEPPER MOTOR VARIANTS

1 1 4 7

2. THEORY AND PERFORMANCE 2.1. MATHEMATICAL MODEL 2.1.1. Field Submodel 2.1.2. Mechanical Submodel 2.2. NUMERICAL FIELD ANALYSIS APPROACH 2.3. AIR-GAP TEETH CONFIGURATION 2.4. EXPERIMENTAL RESULTS

10 11 14 19 21 26 29

3. CONTROL STRATEGIES 3.1. THEORETICAL APPROACH 3.2. COMPARISON OF DIFFERENT CONTROL METHODS 3.3. HYBRID LINEAR STEPPER MOTOR POSITIONING SYSTEM

32 33 42

4. DESIGN 4.1. DESIGN PROCEDURE 4.1.1. Prescription of the Basic Design Inputs 4.1.2. Motor Sizing 4.1.3. Optimization of the Magnetic Circuit 4.1.4. Final Analysis of the Motor 4.2. HYBRID LINEAR STEPPER MOTOR DESIGN EXAMPLES 4.3. COMPARISON OF THE THREE DESIGNED MOTORS

50 51 51 51 55 56 58 64

APPENDIX 1. EQUIVALENT VARIABLE AIR-GAP CALCULATION

REFERENCES LIST OF THE MAIN SYMBOLS INDEX TO THE READER

70 77 81 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-fieldmechanical 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

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

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 Figure 1.1 Switched reluctance motor 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 15ď&#x201A;°, 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 15ď&#x201A;°. 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

1.1. Stepper Motors

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. Closedloop 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

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.

Figure 1.2 Hybrid linear stepper motor

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

1.2. Hybrid Linear Stepper Motor Operating Principles

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 airgap magnetic reluctance.

Figure 1.3 Four positions of the mover

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 Figure 1.4 Tangential force variation under the are shown in Fig. 1.4. mover poles 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

1. INTRODUCTION

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 deenergized 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 Figure 1.5 The command currents teeth. sequence and the corresponding The sequences of the displacements 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 twophase 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.

1.3. Hybrid Linear Stepper Motor Variants

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.

Figure 1.6 The outer magnet type hybrid linear stepper motor

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.

Figure 1.7 The eight poles compact hybrid linear stepper motor

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.

1. INTRODUCTION

Figure 1.8 Tubular variant of the hybrid linear stepper motor

A tubular variant of the hybrid linear stepper motor is presented 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.

Figure 1.9 Surface hybrid linear stepper motor

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

1.3. Hybrid Linear Stepper Motor Variants

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.

Figure 2.1 Four pole hybrid linear stepper motor (1รท4 poles, A-B electromagnets with command coils, PM permanent magnet)

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

2.1. Mathematical Model

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: dA v A  R Ai A  (2.1) dt where v A is the input voltage, R A the coil resistance and i A the current. The flux linkage through the same coil is given by:



A  N A  N  CA   pm   CB



(2.2)

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: d v A  R Ai A  (2.3) N CA  dt A typical circuit-type equation is obtained by introducing the main and the leakage inductances ( L Am , respectively L A ):

2. THEORY AND PERFORMANCE

v A  R A i A  L A



di A d  L i dt dt Am A

where the leakage and main fluxes are given by: N CA  L A i A N CAm  L Am i A



(2.4) (2.5) (2.6)

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: L A  L (x ); x  f (t ) (2.7) m

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, considering 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 .

Figure 2.2 The standstill current decay test setup

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:

2.1. Mathematical Model

13 



R A  i A dt  A 0

0

(2.8)

The flux linkage through the coil A at the initial moment t  0 is: A0   pm  CA0 and when the current i A reaches zero, it becomes:

A   pm

(2.9) (2.10)

Assuming that the permanent magnet flux through the coil A is unchanged, then the following relation can be considered: 

CA0  R A  i A dt

(2.11)

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

2. THEORY AND PERFORMANCE

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.

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

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.

2.1. Mathematical Model

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.

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

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:

2. THEORY AND PERFORMANCE

where Pm pm   0 Fc

   0  Pm pm F pm

(2.12)

F pm  Rm pm   Fc

(2.13)

and Rm pm  1 Pm pm

is the permanent

magnet permeance and respectively reluctance, Rm Fe 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.

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

The two corresponding systems of equations are:

2.1. Mathematical Model

 1   2   5  0   3   4   5  0      0 7  1   4   8  0              0 6 7 10  2      (2.14) 1  Rm12  Rmg1  Rm13    2  Rm22  Rmg 2    7 Rm11  0        R  Rmg 2    3  Rm32  Rmg 3    5 Rm5  F pm  2  m22        3  Rm32  Rmg 3    4  Rm42  Rmg 4  Rm43    8 Rm41  0   9  0    Rmpm  F pm 10  11  0           0 2 5  1   3   4   5  0  1   7   9  0    0  4   8  11   2   6   7   9  0   (2.15) 1 Rm12  Rmg1  Rm13   2 Rm22  Rmg 2   7 Rm11  F1   2 Rm22  Rmg 2   3 Rm32  Rmg 3   5 Rm5   6 Rm pm  0    3 Rm  Rm   4 Rm42  Rmg 4  Rm43   8 Rm41  F2 32 g3    R   7 Rm11  F1  9 m 1    0  10   Rm 2  F2  8 Rm41  11

 













2. THEORY AND PERFORMANCE

where the command coils MMFs are: F A  N Ai A FB  N B i B The resulting magnetic fluxes are given by:  j   j   j , j  1  11

(2.16) (2.17) (2.18)

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

2.1. Mathematical Model

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- Figure 2.7 Magnetic equivalent circuit of one gap variable equivalent permeance developed electromagnet 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: 2Zg  ge  (2.19) 2 Z   11  c cos     with the motor constant c : c 



 1   2Z  1 22Z    1



(2.20)

All the notations are given in Appendix 1, too. The air-gap permeance computed for a mover pole is: 0S p (2.21) Pmgj  ; j 1 4 ge j where S p is the pole area and 0  410 7 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

2. THEORY AND PERFORMANCE

former method is more reliable for this problem and it was adopted here. The tangential force under one mover pole j is given by:  W m j f t j    x

which leads to: ftj

N  

   

j 1 4

(2.22)

j  ct .

20S p

d   g (x )  dx  e j

j 1 4

(2.23)

The normal force developed by one mover pole MMF is:  W m j   f n j   j 1 4 (2.24)   g    ct . j

After some computations the following relation can be obtained: fnj

N  

20S p

j 1 4

(2.25)

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 homogeneous solid, the resulting tangential and normal forces being applied on its center ii) The resulting forces are obtained as an algebraic sum of the Figure 2.8 Simplified mechanical pole forces. model 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.

2.1. Mathematical Model The equation:

mechanical

submodel

d 2x



characterized

the



 ft  fn  G c f (2.26) dt 2 is the friction coefficient, G is the mover weight and m m

where c f

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 zcoordinate. This leads to a two-dimensional analysis. ii) Only the axially directed components of the magnetic vector potential and current density ( A z , 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).

2. THEORY AND PERFORMANCE

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: B x U (A z )      x B x  B rx dB x   D 0 (2.27) By       y  B y  B ry  dB y  J z A z dxdy  0  where  x  f (B x ) , y  f (B y ) and B rx , respectively B ry 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 ) , B r  0 and elsewhere in the considered domain D ,    0 and B r  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 energyrelated functional (Eq. 2.27) is minimized by a set of trial

2.2. Numerical Field Analysis Approach

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:

 Ad 



(2.28)



Here the tangential forces are computed by the surface integration of Maxwell's stress tensor: 1   f t    0 Bn  B   0 B 2n  d (2.29) 2   





where the closed surface  (having the unit outward vector normal n  ) surrounds the mover, passing through the centers of the air-gap mesh elements.

2. THEORY AND PERFORMANCE

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.

Figure 2.9 The discretized domain

As it can be seen, an adequate discretization was ensured, especially in and around the air-gaps.

Figure 2.10 Field distribution in the unexcited motor

2.2. Numerical Field Analysis Approach

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 halfaligned 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.

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

The above presented figures demonstrate that the results obtained via the FEM analysis perfectly agree with the theoretical anticipations.

2. THEORY AND PERFORMANCE

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.

Figure 2.13 The considered teeth configurations

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: g  01 . mm   2mm (2.30) The first variant (Fig. 2.13/a) has different tooth the platen and on the mover, in order to concentrate the flux into the head of the platen teeth. For the second version (Fig. 2.13/b) the tooth unequal to the slot width. As it was recommended in

width on magnetic width is [26], the

2.3. Air-Gap Teeth Configuration

optimum tooth width to tooth pitch ratio is about 0.42. So the width of the tooth and of the slot are: wt  0.85mm w s  115 . mm (2.31) 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: wt  1mm w s  1mm (2.32) 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:

 w  20 

f w  01 . mm

whence w is the slop of the wedge and f w

(2.33) 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.

Figure 2.14 The tangential forcedisplacement static characteristics

Figure 2.15 The normal forcedisplacement static characteristics

2. THEORY AND PERFORMANCE

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

2.3. Air-Gap Teeth Configuration

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 ( ď ´ )

2 mm

nr. of teeth per pole ( Z ) airgap ( g ) permanent magnet type residual flux density ( B r ) coercive force ( H c ) maximal tangential force ( f t max ) Table 2.1 The main data of the sample motor

4 0.1 mm AlNiCo200 0.95 T 50 kA/m 9N

2. THEORY AND PERFORMANCE

The mover was precisely placed in several positions at 0.05mm distance each of the other within a quarter tooth pitch. POSITON OF THE TANGENTIAL The command coil FORCE (N) MOVER (mm) was energized with the 0 0 maximum command 0.05 0.1 current determined for this 0.10 3.1 motor (150 mA). In each 0.15 4.7 position the tracking 0.20 5.4 0.25 5.9 tangential force was 0.30 6.6 measured using a force 0.35 7.9 transducer stamp fixed on 0.40 8.0 the mover. The results of 0.45 8.0 0.50 7.9 the measurements are Table 2.2 The experimental results 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 characteristics are shown in Fig. 2.18. The static characteristic obtained by simulation is plotted with continuous line. The asterisks (*) mark the points acquired experimentally. These Figure 2.18 Tangential force static characteristics obtained points are inter- experimentally and by numeric simulation polated by dash-

2.4. Experimental Results

ed line. The interpolation function is one of the logistic approximations and is given by: a  4bn y (2.34) 1  n  2 where: n e



x c d

The obtained coefficients are the followings: a  0.228 b  153 . c  28.373 d  0.669 e  43.864

(2.35)

(2.36)

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.

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

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

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.

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

The tangential force can be determined by using the airgap flux under each pole. This flux is a sum of two fluxes, one produced 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]: F pm Pme 0j  (3.1) 1  c cos  j , j = 1  4 4 where F pm 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:

3.1. Theoretical Approach

  which means:

2



(3.2)

1   2     3     2 4     2

(3.3)

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:

   FCBM sint   B 

FC1  FCA  FCAM sin t   A

FC 2  0

FC 4  FCB

FC 3  0

(3.4)

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]: FCA Pme  CA  1  c cos 1 1  c cos 1 4 (3.5) FCB Pme  CB  1  c cos  3 1  c cos  3 4 As it can be seen in Fig. 3.2 the resulting flux through the poles is a sum of two fluxes:













Figure 3.2 Magnetic flux pattern

3. CONTROL STRATEGIES

1   01   CA  F pm Pme 1  c cos   1  k FA sin t  A 1  c cos    4  2   02   CA  F pm Pme

































1  c cos   1  k FA sin t  A 1  c cos   4  3   03   CB  F pm Pme 1  c sin   1  k FB sin t  B 1  c sin    4  4   04   CB  



F pm Pme

where kFA

(3.6)

1  c sin   1  k FB sin t  B 1  c sin   4 and kFB are the command MMF factors: kF A 

FCAM

FCB M

kF B 

F pm

(3.7)

F pm

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:









f t1  k f t sin  1  k FA sin t   A 1  c cos  









f t2  k f t sin  1  k FA sin t   A 1  c cos  









f t3  k f t cos  1  k FB sin t   B 1  c sin  









f t4  k f t cos  1  k FB sin t   B 1  c sin  

(3.8)

where k f t is the tangential force coefficient: 2

 F pm  2  Pm c k f t   e   4 

(3.9)

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.

3.1. Theoretical Approach

The resulting total tangential force of the motor is given by the sum of tangential forces under each pole:







f t  2k f t 2k FA sin  sin t   A 





 2k FB cos a sin t   B 





(3.10)







  c sin 2 k F2 sin 2 t   A  k F2 sin 2 t   B  A B 

Based upon this relation the total unitary tangential force is: f t* 

ft  k FA sin  sin t   A  4k f t





 k FB cos a sin t   B  



(3.11)

c sin 2 k F2 sin 2 t   A  k F2 sin 2 t   B A B 2









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 sin t 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: t =   A = 0 (3.12) In this case the unitary tangential force will be:





f t*  k FA sin 2   k FB cos a sin    B  



c sin 2 k F2 sin 2   k F2 sin 2    B A B 2





(3.13)

It was demonstrated [41] that the greatest average unitary tangential force is obtained if the two command MMFs are in quadrature. It means that: (3.14) kFA  kFB  kF B   2 Under these conditions the expression of the unitary tangential force is:

3. CONTROL STRATEGIES

c   f t*  k F 1  k F sin 4    4

(3.15)

In the above mentioned article [41] the effect of the MMF factor k F 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: FC1  C1FC1M FC 2  C 2 FC 2M (3.16) FC 3  C 3 FC 3M FC 4  C 4 FC 4M The coefficient Cj , j  1  4 , is taken one if coil j is supplied and it is nil

Figure 3.3 Magnetic flux pattern

if it is not energized. By feeding the command coils 2 and 4 ( C 2  C 4  1 and C1  C 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: 1   01   C 2  F pm Pme

















1  c cos   1  C 2k F2 1  c cos   4  2   02   C 2  F pm Pme 1  c cos   1  C 2k F2 1  c cos    4  3   03   C 4  

F pm Pme

1  c sin   1  C 4 k F4 1  c sin   4  4   04   C 4  F pm Pme 1  c sin   1  C 4 k F4 1  c sin    4 

(3.17)

3.1. Theoretical Approach

The unitary tangential force in this case is: f t*  C 2k F2 sin   C 4 k F4 cos a  

(3.18)

c sin 2  C 2k F2  C 4 k F2  2 4 2

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: f t*m

which gives:





0   4

 f t* d

(3.19)

0   4





2 2 C 2k F2 sin  0  C 4 k F4 cos a 0    c   sin 2 0  C 2k F2  C 4 k F2   2 4  2 Its derivative function of  0 is: f t*m 

df t*m d 0



2

 





2 C 2k F2 cos  0  C 4 k F4 sin a 0 

  c cos 2 0  C 2k F2  C 4 k F2   2 4 

(3.20)

(3.21)

A particular case can be considered when only one coil is supplied: C2 = 0 C4 = 1 (3.22) Under this condition the optimal commutation position results:

3. CONTROL STRATEGIES





 2   1  1  2ck F4   0  arcsin  2 op 4ck F4      

 

(3.23)

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: e j  N

d j

, j 14 (3.24) dt The expressions of the EMF induced in the command coils

are:



 ddt 

 e1  e 2  ke c sin  1  2C 2k F2 c cos   dk F2    C 2 1  c 2 cos 2  dt   e 3  e 4  ke c cos  1  2C 4 k F4 c sin   dk F4    C 4 1  c 2 sin 2  dt 









 ddt 

(3.25)

3.1. Theoretical Approach

where

d 2 dx 2   v  dt  dt v being the mover speed and ke the EMF coefficient: ke  N

(3.26)

Pme

Fpm (3.27) 4 For the particular case considered above ( C 2  0, C 4  1 ) e 2 becomes: e 2  ke c sin 

d dt

(3.28)

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.

3. CONTROL STRATEGIES

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 geoITEM VALUE tooth width ( wt ) metrical dimen1.16 mm slot width ( w s ) sions and the 0.84 mm tooth pitch ( ď ´ ) 2 mm main data of the nr. of teeth per pole ( Z ) 5 hybrid linear stepair-gap ( g ) 0.1 mm per motor, having permanent magnet type VACOMAX-145 four command residual flux density ( B r ) 0.9 T coils, are given in coercive force ( H c ) 650 kA/m Table 3.1. number of coil turns ( N ) 200 In order to motor's constant ( c ) 0.244 emphasize the Table 3.1 Parameters and leading dimensions of the considered sample motor 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 ( k F ď&#x201A;ť 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

3.2. Comparison of Different Control Methods

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].

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

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

3. CONTROL STRATEGIES

 op . In the first

after moving a whole step), respectively with  0

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. CHARACTERISTICS Maximal tangential force [N] Minimal tangential force [N] Maximum velocity [m/s] Medium tangential force [N] Medium velocity [m/s] Final displacement [mm]

CONTROL METHOD OPENCLOSED-LOOP  0 LOOP    0  35.46 -34.55 0.33 3.31 0.05 4.11

35.46 -0.75 0.48 19.92 0.33 4.63

35.46 21.78 0.87 31.15 0.59 12.02

Table 3.2 Comparison between the characteristics obtained by computer simulation for the sample motor (different control methods)

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).

3.3. Hybrid Linear Stepper Motor Positioning System

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

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 Figure 3.7 The control unit of the positioning unenergized command system 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

3.3. Hybrid Linear Stepper Motor Positioning System

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 ( V CC ) 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 ď&#x20AC;¨ď Ą 0 ď&#x20AC;Š 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 SEQUENCE NUMBER I II III IV

ENERGIZED MONITORED COIL COIL 4 1 2 4 3 2 1 3

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

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 moverâ&#x20AC;&#x2122;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.

3.3. Hybrid Linear Stepper Motor Positioning System

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

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.

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 wellestablished 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

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 t max ), ii) the resolution of the positioning (the step length), function of the selected control strategy ( x i ), iii) the length ( l r ) 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: ď ´ ď&#x20AC;˝ 4x i (4.1)

4. DESIGN 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. Rareearth 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 moverâ&#x20AC;&#x2122;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 ( B s ) is obtained similarly. The permanent magnet working point on the straight demagnetization characteristic throughout the second quadrant ( B pm , H 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

S pm min  k p l pm  k x

f t max

(4.2)

B p B pm

B p Br

(4.3) H c B r  B pm where B r and H c are the remanent flux density and the coercive





force of the selected magnet. The two designing constants ( k p and k x ) 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 ( w pm ) is taken equal to the prescribed width of the running track.

Figure 4.1 Diagram for selecting the design constant k p

Figure 4.2 Diagram for selecting the design constant kx

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 onehalf tooth pitch. The following expression must be considered: l e  4k  1



k N

     wt   k 2 4 2

The mover's pole length is given by:

(4.4)

4. DESIGN

lp 

where S p

(4.5)

w pm

is the pole area, computed using the following

expression: Sp 

B pm S pm

(4.6)

The number of the pole teeth can be calculated by: lp Z 



(4.7)

The most appropriate integer number will be chosen. Having the number of teeth selected, the final value of the pole width can be computed: l p  Zwt   Z  1w s (4.8) where wt is the tooth width and w s 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: 2Zg  Fc  Ni   pm  S p 0 2Z    1

(4.9)

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: k N

  l y  k    w t  2  k 2

(4.10)

4.1. Design Procedure

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: N 300 ď&#x201A;Ł ď&#x201A;Ł 500 (4.11) i 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.

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: pdt  Gc h d   S dt (4.12) where

p [W] is the total loss in the body,

G [Kg] and

c h [Ws/KgC] are the mass, respectively the specific heat capacity

of the body and  [W/m2C] 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:

4.1. Design Procedure

 p w1 dt  cw S cw c dt  G w c w dw1    t S wat  l S wal w1 dt   wc S wc w1 dt   p dt   S  dt  G c d cw cw c w w w2   w2   t S wat  l S wal w2 dt   wc S wc w2 dt  (4.13)   p c dt   wc S wc w  w dt   sc S sc s dt  G c c Fe dc  1 2    2cw S cw c dt  t S cat  l S cal c dt  cs S cs c dt   p s dt  cs S cs c dt    G s c Fe ds   sa S sa s dt   sc S sc s dt  where p w1 and p w2` are the losses in the coils, p c and p s 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.

4. DESIGN

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: f t max  50N x i  0.5mm (4.14) l r  200mm wr  85mm 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]: B r  0.9 T H c  650 kA m (4.15) The working point of the permanent magnet is given by: B pm  0.9B r  0.81T B r  B pm (4.16) H pm  H c  65 kA m Br The two designing constants k x and k p were determined from Figs. 4.1 and 4.2:

4.2. Hybrid Linear Stepper Motor Design Examples k x  175

k p  8.6  10 6

59 (4.17)

With these constants the minimal active surface and thickness of the permanent magnet were found out using Eqs. 4.2, respectively 4.3: 50 S pm min  8.6  10  6  6.25cm 2 0.85  0.81 (4.18) 0.85  0.9 l pm  175  2.28mm 3 650  10 0.9  0.81 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: l pm  2mm h pm  9mm w pm  83mm (4.19) S pm  h pm w pm  747mm 2 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: (4.20) wr  w pm The pole area is obtained from Eq. 4.6: 0.81  747 Sp   711.84mm 2 (4.21) 0.85 With this the pole length can be computed using Eq. 4.5: 711.84 lp   8.57mm (4.22) 83 The number of the pole teeth was calculated using Eq. 4.7: 8.57 Z   4.28 (4.23) 2 Five teeth per pole were selected. Having the teeth number the precise pole length must be recomputed utilizing Eq. 4.8: (4.24) l p  5  0.84  4  116 .  8.84mm Next the command coil sizing was performed. In order to evaluate the command coil MMF the equivalent air-gap must be

4. DESIGN

determined. Using Eqs. A1.3 through A1.5 the following results were obtained:   6.465   0.415 u  11685 . (4.25)   0.917 kc  1477 . The equivalent air-gap was calculated using Eq. A1.8: 2 g   1477  01 . .  0.218mm

(4.26)

Eq. 4.9 yields to the necessary command coil MMF: Fc  12612 .  130Aturns (4.27) 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.

4.2. Hybrid Linear Stepper Motor Design Examples

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

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 considered during the iterative process of the dynamic simulation, the heating curves represented in Fig. 4.4 were Figure 4.4 The heating curves obtained by the obtained. thermal checking of the sandwich magnet type As it can be seen hybrid linear stepper motor 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.

4. DESIGN

The presented computer aided design methodology not only shortened 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. CHARACTERISTICS

VALUES

Tooth width ( wt )

0.84 mm

Slot width ( w s )

1.16 mm

Number of teeth on one pole ( Z )

Pole length ( l p )

8.84 mm

Pole area ( S p )

7.11 cm2

Permanent magnet width ( w pm )

83 mm

Permanent magnet length ( l pm )

2 mm

Table 4.1 The most important parameters of the designed motors

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.

4.2. Hybrid Linear Stepper Motor Design Examples

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

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.

4. DESIGN

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

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. 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

[cm3]

82.96

131.38

124.25

1.49

19.29

19.23

41.21

848

1257

1373

Core volume

Magnet volume

[cm3]

Winding volume Mover mass [g]

[cm3]

Table 4.2 Comparison of the main characteristics of the designed motor variants

4.3. Comparison of the Three Designed Motors

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: Pm (x )  Pm1 (x )Pm2 (x )g

(A1.1)

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]:





1 1    1 Z 1  cos x kc g 1   Z 1 cos x   2k g 1    1 c   x   Z , Z 1   Z 1  , Z Pm2 (x )   1  1    1 Z 1  cos x   kc g  x  Z 1  , Z 1   Pm1 (x ) 





             

(A1.2)

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.

Equivalent Variable Air-Gap Calculation

The equivalent variable air-gap permeance variation is given in Fig. A1.1.

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

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]:

=



 g     2

 kc sin  kc =

where:

  - g

(A1.3) (A1.4)

APPENDIX 1.

 2  ws    ws  3.2  w s      ln 1    arctan   2g  2g    2g    1  u  2  21 u2



u



ws  ws    1  2g  2g 

(A1.5)

 being the tooth pitch and w s 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: Pm* e 

( Z 1) Z   ( Z 1) 1   (A1.6) P ( x ) dx  P ( x ) dx  P ( x ) dx  m  m  2Z   m 1 1 ( ) ( ) Z Z Z         

and after some computations it results: Pm* e 

1 2Zg 

   2 1   2Z  1 cos   2Z    1  

(A1.7)

where the equivalent air-gap g  is: g   kc2g

(A1.8)

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.

Equivalent Variable Air-Gap Calculation

Figure A1.2 The average specific permeance variation

The equivalent variable air-gap is: ge 

2Zg  2Z    11  c cos  

(A1.9)

with the motor constant c given by: c 





 1   2Z  1 22Z    1

(A1.10)

Next the equivalent magnetic permeance of electromagnet A (see Fig. 2.1) can be expressed using Eq. 2.21 as: Pm A  Pmg1  Pmg 2  

 0S p 2Zg 



2Z    1 2  c  cos 1  cos 2 



(A1.11)

resulting in:

Pme  Pm A  PmB 

 0S p Zg 

2Z    1

(A1.12)

As it can be seen the equivalent magnetic permeance of one electromagnet does not depend on the mover's position.

REFERENCES 1. ACARNLEY P.P. et al.: Detection of Rotor Position in Stepping and Switched Motors by Monitoring of Current Waveforms, Transactions on Industrial Electronics, vol. IE-32. (1985), no. 3., pp. 215-222. 2. ACARNLEY P.P.: Stepping Motors: A Guide to Modern Theory and Practice, IEE Control Engineering Series 19., Peter Peregrinus, 1984. 3. ASTROM K.J - WITTENMARK B: Computer - Controlled Systems - Theory and Design, Prentice Hall, 1990. 4. BERTOLINI T.: The Calculation of Stepper-Motors Torque in Regard to the Geometry of the Stator and Rotor-Teeth, Proceedings of the International Conference on Electrical Machines , Pisa, 1988, pp.161-164. 5. BINNS K.J. et al.: Hybrid Permanent - Magnet Synchronous Motors, Proceedings IEE, vol. 125, No. 3. (march 1978), pp. 203-208. 6. BINNS K.J. et al.: Implicit Rotor Position Sensing for Permanent Magnet Self - Commutating Machine Drives, Proceedings of International Conference on Electrical Machines, Cambridge, 1990, pp. 1231-1236. 7. BINNS K.J. - LAWRENSON P.J.: Analysis and Computation of Electric and Magnetic Field Problems, Pergamon Press, 1973. 8. BIRĂ&#x201C; K. et al.: Dynamic Modelling of Inverter-Fed Permanent Magnet Synchronous Motors, Proceedings of the International Conference on the Evolution and Modern Aspects of Synchronous Machines, ZĂźrich, 1991, pp. 229-233. 9. COLBY R.S. - NOVOTNY D.W.: An Efficiency Optimizing Permanent Magnet Synchronous Motor Drive, IEEE

REFERENCES

Transactions on Industry Applications, vol. 24, no. 3. (may-june 1988), pp. 462-469. 10. CORDA J.: Linear Variable - Reluctance Actuator in Cylindrical Form, Proceedings of the PCIM Conference, Nürnberg, 1994, Vol. Intelligent Motion, pp. 133-140. 11. EBIHARA D.: Design of a PM Type Linear Stepping Motor for Automatic Conveyer System, Proceedings of the International Conference on Electrical Machines Design and Applications, London, 1985, pp. 265-269. 12. EBIHARA D. - WATADA M.: The Study on Basic Structure of Surface Actuators, Digests of the INTERMAG '89 Conference, pp. HB-9. 13. EMERALD P.: Power Multi-Chip Modules Reawaken HighSpeed, Efficient PWM Operation of Unipolar Stepper Motors, PCIM, may 1995, pp. 25-33. 14. EMERALD P. - SASAKI M.: Multi-Chip Modules for Stepper Motors Integrate PWM Control with Power FETs for Superb Performance, Proceedings of the PCIM Conference, Nürnberg, 1995, Vol. Intelligent Motion, pp. 471-480. 15. FINCH J.W. et. al.: Normal Force in Linear Stepping Motors, Proceedings of the International Conference on Small and Special Electrical Machines, London, 1981, pp. 78-81. 16. FU Z.X. - NASAR S.A.: Analysis of a Hybrid Linear Stepper Motor, Proceedings of the Annual Symposium on Incremental Motion Control Systems & Devices, Urbana-Champaign, 1992, pp. 234-240. 17. GODDIJN B.H.A.: New hybrid stepper motor design, Electronic Components and Applications, vol. 3. (1980), no. 1. 18. GRANTHAM W.J. - VINCENT T.L.: Modern Control Systems Analysis and Design, John Wiley & Sons, 1993. 19. HANITSCH R.: Electromagnetic Machines with Nd-Fe-B Magnets, Journal of Magnetism and Magnetic Materials, vol.80. (1980), pp. 119-130.

REFERENCES

20. HAYT W.H. Jr.: Engineering Electromagnetics, McGraw-Hill, 1974. 21. HOFFMAN B.D.: The Use of 2-D Linear Motors in Surface Mount Technology, Proceedings of the Surface Mount Conference, Boston, 1990, pp. 183-189. 22. HONG Y. et al.: The Latest Study on CAD of Stepper Motors, Proceedings of the International Conference on Electrical Machines, Manchester, 1992, pp. 642-646. 23. JENKINS et al.: An Improved Design Procedure for Hybrid Stepper Motors, IEEE Transactions on Magnetics, Vol. 26., No.5. (september 1990), pp. 2535-2538. 24. JONES D.I. - FINCH J.W.: Computer Aided Synthesis of Optimal Position and Velocity Profiles for a Stepping Motor, Proceedings of the International Conference on Electrical Machines, Cambridge, 1990, pp. 836-841. 25. JUFER M. et al.: On the Switched Reluctance Motor AirGap Permeance Calculation, Proceedings of the ELECTROMOTION International Symposium, Cluj, Romania, 1995, pp. 141-146. 26. KENJO T.: Stepping Motors and their Microprocessor Controls, Monographs in Electrical and Electronic Engineering, Vol. 16., Oxford Science Publications, 1985. 27. KUO B.C. - HUH U.Y.: Permeance Models and their Applications to Step Motor Design, Proceedings of the Annual Symposium on Incremental Motion Control Systems & Devices, Urbana-Champaign, 1986, pp. 351-369. 28. LEU M.C. et. al.: Characteristics and Optimal Design of Variable Airgap Linear Force Motors, IEE Proceedings, vol. 135., Pt. B, No. 6. (november 1988), pp. 341-345. 29. LOFTHUS R.M. et al: Processing Back EMF Signals of Hybrid Step Motors, Control Engineering Practice, vol. 3., no. 1. (1995), pp. 1-10.

REFERENCES

30. MARLIN T.E.: Process Control - Designing Processes and Control Systems for Dynamic Performance, McGraw-Hill International, 1995. 31. McLEAN G.W. et al.: Design of a Permanent Magnet Linear Synchronous Motor Drive for a Position/Velocity Actuator, Proceedings of the International Conference on Electrical Machines, Cambridge, 1990, pp. 1136-1141. 32. McLEAN G.W.: Review of Recent Progress in Linear Motors, IEE Proceedings, Vol. 135., Pt. B, No. 6. (november 1988), pp. 385-417. 33. MULLER W.: Comparison of Different Methods of Force Calculation, IEEE Transactions on Magnetics, Vol. 26., No.2. (march 1990), pp. 1058-1061. 34. NASAR S.A. - BOLDEA I.: Linear Motion Electric Machines, John Wiley & Sons, 1976. 35. NORDQUIST J.I. - SMIT P.M.: A Motion-Control System for (Linear) Stepper Motors, Proceedings of the Annual Symposium on Incremental Motion Control Systems & Devices, UrbanaChampaign, 1985, pp. 215-231. 36. MOHAN N. et al: Power Electronics - Converters, Applications and Design, John Wiley & Sons, 1995. 37. PARKER R.J.: Advances in Permanent Magnetism, John Wiley & Sons, 1990. 38. SANADA M. et al.: Cylindrical Linear Pulse Motor with Laminated Ring Teeth, Proceedings of the International Conference on Electrical Machines, Cambridge, 1990, pp. 693-698. 39. SILVESTER P.P. - FERRARI R.L.: Finite Elements for Electrical Engineers, Cambridge University Press, 1983. 40. SZABĂ&#x201C; L. et al.: Computer Aided Design of a Linear Positioning System, Proceedings of the Power Electronics, Motion Control Conference (PEMC), Budapest, 1996, vol II., pp. 263-267.

REFERENCES

41. SZABÓ L.: Computer Aided Design of Variable Reluctance Permanent Magnet Synchronous Machines, Ph.D. Thesis, Technical University of Cluj-Napoca, Romania, 1995 (in Romanian). 42. SZABÓ L. et al.: Computer Simulation of a Closed-Loop Linear Positioning System, Proceedings of the PCIM Conference, Nürnberg, 1993, Vol. Intelligent Motion, pp. 142-151. 43. SZABÓ L. et al.: Computer Simulation of a Constant Velocity Contouring System Using x-y Surface Motor, Proceedings of the PCIM Conference, Nürnberg, 1995, Vol. Intelligent Motion, pp. 375-384. 44. SZABÓ L. et al.: E.M.F. Sensing Controlled Variable Speed Drive System of a Linear Stepper Motor, Proceedings of the Power Electronics, Motion Control Conference (PEMC), Warsaw, 1994, pp. 366-371. 45. SZABÓ L. et al.: Variable Speed Conveyer System Using E.M.F. Sensing Controlled Linear Stepper Motor, Proceedings of the PCIM Conference, Nürnberg, 1994, Vol. Intelligent Motion, pp. 183-190. 46. SUN L.I.: A New Type of Closed Loop Operation Mode of PM Hybrid Step Motor, Proceedings of the International Conference on the Evolution and Modern Aspects of Synchronous Machines, Zürich, 1991, pp. 533-537. 47. TAKEDA Y. et al.: Optimum Tooth Design for Linear Pulse Motor, Conference Record of the 1989 IEEE Industry Applications Society Annual Meeting, pp. 272-277. 48. TRIFA V. - KOVÁCS Z.: Numerical Simulation of Linear Stepping Motor for X-Y Plotter, Proceedings of the Third National Conference on Electrical Drives, Brasov, Romania, 1980, vol. I., pp. A17-A22. 49. VAC Vacuumschmelze: Rare - Earth Permanent Magnet Materials: VACODYM and VACOMAX, (Catalogue), 1992.

REFERENCES

50. VAS P.: Parameter Estimation, Condition and Diagnosis of Electrical Machines, Oxford University Press, 1993. 51. VEIGNAT N. et al.: Stepper Motors Optimisation Use, Proceedings of the International Conference on Electrical Machines, Paris, 1994, pp. 13-16. 52. VIOREL I.A. et al.: Combined Field-Circuit Model for Induction Machine Performance Analysis, Proceedings of the International Conference on Electrical Machines, München, 1986, pp. 476-479. 53. VIOREL I.A. et al.: Dynamic Modelling of a Closed-Loop Drive System of a Sawyer Type Linear Motor, Proceedings of PCIM Conference, Nürnberg, 1992, Vol. Intelligent Motion, pp. 251-257. 54. VIOREL I.A. et al.: On the Linear Stepper Motor Control Basics, Proceedings of the PCIM Conference, Nürnberg, 1995, Vol. Intelligent Motion, pp. 481-494. 55. VIOREL I.A. - SZABÓ L.: Permanent - magnet variable - reluctance linear motor control, Electromotion, vol. 1., nr. 1. (1994), pp. 31-38. 56. VIOREL I.A. et al.: Quadrature Field-Oriented Control of a Linear Stepper Motor, Proceedings of the PCIM Conference, Nürnberg, 1993, Vol. Intelligent Motion, pp. 64-73. 57. VIOREL I.A. et al.: Sawyer Type Linear Motor Modelling, Proceedings of the International Conference on Electrical Machines, Manchester, 1992, pp. 697-701. 58. VIOREL I.A. et al.: Sensorless Adaptive Control of a Linear Stepper Motor, Proceedings of the PCIM Conference, Nürnberg, 1997, Vol. Intelligent Motion, pp. 429-438. 59. VIOREL I.A. - SZABÓ L.: The Control Strategy of a Drive System with Variable Reluctance Permanent - Magnet Linear Motor, Proceedings of the 5th International Conference on Optimization of Electric and Electronic Equipments, OPTIM, Brasov, Romania, 1996, pp. 1593-1600.

REFERENCES

60. VIOREL I.A. et al.: Transformer Transient Behavior Simulation by a Coupled Circuit-Field Model, Proceedings of the International Conference on Electrical Machines, Paris, 1994, pp. 654-659. 61. VIOREL I.A. - RÃDULESCU M.M.: On the Induction Motor Variable Equivalent Air-Gap Permeance Coefficients Computation, EEA Electrotehnica vol. 32., nr. 3., 1984, pp. 108-111 (in Romanian). 62. VIOREL I.A. et al.: Magnetic Field Numerical Analysis of a Variable Reluctance Hybrid Linear Motor, Proceedings of the Romanian Conference on Electrical Engineering, 1984, Craiova, Romania, vol. 6. (1982), pp. 173-182 (in Romanian). 63. WAGNER J.A. - CORNWELL W.J.: A Study of Normal Force in a Toothed Linear Reluctance Motor, Electric Machines and Electromechanics, vol. 7, pp. 143-153. 64. WAVRE N. - VAUCHER J-M.: Direct Drive with Servo Linear Motors and Torque Motors, Proceedings of the European Power Electric Chapter Symposium (Electric Drive Design and Application), Lausanne, 1994, pp. 151-155. 65. WENDORFF E. - KALLENBACH E.: Direct Drives for Positioning Tasks Based on Hybrid Stepper Motors, Proceedings of the PCIM Conference, Nürnberg, 1993, Vol. Intelligent Motion, pp. 38-52. 66. XIONG G. - NASAR S.A.: Analysis of Field and Forces in a Permanent Magnet Linear Synchronous Machine Based on the Concept of Magnetic Charge, IEEE Transactions on Magnetics, vol. 25., no. 3., may 1989, pp. 2713-2719. 67. XU S. et al.: Adaptive Control of Linear Stepping Motor, Proceedings of the International Conference on Electrical Machines, Paris, 1994, pp. 178-181.

LIST OF THE MAIN SYMBOLS

acceleration [m/s2] z-axis component of the magnetic vector potential [Wb/m] flux density in the mover poles [T]

B pm

permanent magnet flux density [T]

Br Bs

permanent magnet remanent flux density [T] flux density in the platen [T]

c cf

motor constant [-] friction coefficient [-]

ch Cj

specific heat capacity [Ws/Kg°C] square wave MMF factor (j=1÷4) [-]

induced EMF in coil j (j=1÷4) [V]

a Az

FA , F B , Fc command coil MMF [A turns] FCA , FCB sinusoidal command MMF [A turns] FCAM , FCB M peak value of the sinusoidal command MMF

FC j

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

F pm

permanent magnet MMF [A turns]

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

fn j ft j

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

f t*

unitary total tangential force [-]

f t*m

unitary average tangential force [-]

f t max

maximal tangential force [N]

fw g

flat width of the wedged head teeth [m] air-gap length [m]

mass [Kg]

LIST OF THE MAIN SYMBOLS

78 g ge j Hc H pm

equivalent air-gap [m] equivalent variable air-gap under pole j (j=1÷4) [m] permanent magnet coercive force [A/m] permanent magnet magnetic field intensity [A/m]

i* imposed command coil current [A] i A , i B current in command coil A, respectively B [A] j pole number [-]

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] kF A , kFB , kF sinusoidal command MMF factor [-] Jz

kF j

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

k ft

tangential force coefficient [N]

designing constant of the permanent magnet thickness

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

[Wb/kg] L Am , L Bm main inductance of the command coils [H] L A , L B leakage inductance of the command coils [H] le distance between the two electromagnets [m] lp pole length [m] l pm

permanent magnet thickness [m]

running track length [m] yoke length [m]

m mover’s mass [Kg] N , N A , N B command coil turns [-] n unit outward vector normal [-] p pc Pm

total losses in the body [W] iron losses in the mover [W] air-gap equivalent variable permeance [Wb/A]

LIST OF THE MAIN SYMBOLS

Pme , Pm A , PmB equivalent magnetic permeance of an

electromagnet [m] Pm* e

average specific permeance of the air-gap under one pole

Pm pm

[1/m] permanent magnet permeance [Wb/A]

Pmg

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

platen core iron losses [W] pw1 , pw2 coil losses [W] ps

RA , RB command coil resistance [] Rm Fe iron core reluctance [A/Wb] Rm pm permanent magnet reluctance [A/Wb] Sp

pole area [m2]

S pm min permanent magnet minimal active surface [m2] u v

equivalent air-gap permeance computation factor [-] mover speed [m/s]

v* imposed speed [m/s] v A , v B command coils input voltage[V] w pm permanent magnet width [m]

running track width [m] slot width [m] tooth width [m]

Z xi

number of the pole teeth step length [m]

  j

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

wr ws

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

CAm , CB m 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

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

j

[Wb] 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 [-]

  0  x , y

equivalent variable air-gap permeance coefficient[-] magnetic permeability [H/m] free space permeability [H/m] axial components of the magnetic reluctivity [m/H]

 w

over-temperature [ºC] 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

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

INDEX 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 Hamiltonâ&#x20AC;&#x2122;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 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 Maxwellâ&#x20AC;&#x2122;s stress tensor 19, 23 measured EMF 41, 47 measurement 30 mechanical model 13, 20 model reference adaptive control (MRAC) 40 monitoring 47

83 moveable armature (mover) 4 moverâ&#x20AC;&#x2122;s displacement 47-48 normal attractive force 26 Nortonâ&#x20AC;&#x2122;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

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

INDEX 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: ioan.adrian.viorel@mae.utcluj.ro lorand.szabo@mae.utcluj.ro

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 fĂźr Elektrische Maschinen Rheinisch-WestfĂ¤lische Technische Hochschule Aachen