control of permanent magnet linear synchronous motor in motion control applications
DESCRIPTION
MechatronicsTRANSCRIPT
-
Lappeenranta University of Technology
Faculty of Technology
Department of Electrical Engineering
CONTROL OF PERMANENT MAGNET LINEAR
SYNCHRONOUS MOTOR IN MOTION CONTROL
APPLICATIONS
MASTERS THESIS
Examiners: Professor Juha Pyrhnen
D.Sc. Markku Niemel
Supervisors: D.Sc. Markku Niemel
M.Sc. Mari Haapala
Lappeenranta, May 20, 2009 Pavel Ponomarev
Karankokatu 4 C 2
53850 Lappeenranta
-
ABSTRACT Lappeenranta University of Technology
Faculty of Technology
Department of Electrical Engineering
Pavel Ponomarev
CONTROL OF PERMANENT MAGNET LINEAR SYNCHRONOUS MOTOR
IN MOTION CONTROL APPLICATIONS
Masters thesis
2009
64 pages, 50 figures, 7 tables and 2 appendices
Examiners: Professor Juha Pyrhnen, D.Sc. Markku Niemel
Keywords: Linear motor, Permanent magnet motor, Direct Thrust Force Control
(DTFC)
The aim of the thesis is to study the principles of the permanent magnet linear
synchronous motor (PMLSM) and to develop a simulator model of direct force
controlled PMLSM. The basic motor model is described by the traditional two-axis
equations. The end effects, cogging force and friction model are also included into the
final motor model. Direct thrust force control of PMLSM is described and modelled.
The full system model is proven by comparison with the data provided by the motor
manufacturer.
-
Acknowledgments The work was carried out at Lappeenranta University of Technology (LUT) during the
period from winter up to spring 2009. I would like to thank all people who made this
work possible.
I wish to express my deepest appreciation to my first supervisor professor Juha
Pyrhnen and to my second supervisor D.Sc. Markku Niemel for their guidance and
support.
I want to thank M.Sc. Mari Haapala for her help and guidance.
Special thanks to Julia Vauterin who has made my life and study in Lappeenranta
possible.
I also wish to express my appreciation to my sister Malukhina Elena and her husband
Malukhin Aleksey for their support.
I am grateful to my parents and my sweetheart Julia for their love and support.
Lappeenranta, May 2009
Pavel Ponomarev
-
TABLE OF CONTENTS List of Symbols and Abbreviations 5
1 Introduction
1.1 Comparison of linear actuators
1.2 Description of the motor system to be studied
1.3 Direct thrust force control method
1.4 Objectives and outline of the thesis
7
7
9
13
14
2 Modelling of a Permanent Magnet Linear Synchronous Motor
2.1 Space vector theory, coordinate systems and transformations
2.2 Basic motor model
2.3 Nonlinearities and mechanical model
2.3.1 Friction model
2.3.2 Cogging force
2.3.3 End-effect
16
16
21
24
27
27
29
3 Modelling of Direct Thrust Force Control of a PMLSM
3.1 VSI voltage vectors
3.2 Direct torque control
3.3 Estimations
31
31
33
40
4 Simulations
4.1 System overview
4.2 Simulation results
4.2.1 Conventional DTFC
4.2.2 DTFC with MTPA strategy
45
45
48
48
51
5 Results and Conclusions 53
References 55
Appendices 57
4
-
LIST OF SYMBOLS AND ABBREVIATIONS
Symbols
phase shift operator
Cfr Coulomb friction factor
Fdisturb force of disturbances
Ffriction friction force
Fthrust, FT thrust force
Fthrust.est thrust force estimation
id, iq direct-axis and quadrature-axis currents
ix, iy currents in stationary xy-reference frame
is stator current vector
isA, isB, isC stator current vector phase components
kend end-effect coefficient
L inductance
Ld, L q direct- and quadrature-axis inductances
Ls stator leakage inductance
mmov , mload mover and load masses
p number of pole pairs
Pelm electromagnetic power
R resistance
Sfr Stribeck friction factor
UDC DC-link voltage
us stator voltage vector
us_est stator voltage vector estimation
Vfr viscous friction factor
load angle
angle between ABC- and xy-reference frames
m air gap flux linkage vector
PM permanent magnet flux linkage
5
-
s stator flux linkage vector
sd, sq stator flux linkage vector dq-reference frame components
sx, sy stator flux linkage vector xy-reference frame components
el electrical frequency
flux linkage hysteresis comparator output signal
thrust force hysteresis comparator output signal
pole pitch
vlin linear velocity of the mover
Abbreviations
AC alternating current
DC direct current
DFLC direct flux linkage control
DTC direct torque control
DTFC direct thrust force control
FEM finite element method
ITC indirect torque control
LIM linear induction motor
LM linear motor
LSM linear synchronous motor
MTPA maximum thrust per ampere
PI proportional-integral
PM permanent magnet
PMLSM permanent magnet linear synchronous motor
VSI voltage source inverter
6
-
Chapter 1
Introduction
Most of linear actuators have traditionally been made by means of rotating machines
and special transmission devices such as ball rail, roller rail, linear shaft and ball screw
systems. In these devices a rotating motion produced by rotating machines is converted
into linear motion. Such a conversion decreases, and in many cases significantly, the
efficiency of the whole system. These mechanical rotary-to-linear transmission devices
also require a lot of maintenance which connected with wear outs of mechanical parts.
Linear motors are electromagnetic devices which can produce linear motion without
any intermediate gears, screws or crank shafts. The linear motion is obtained directly
by electromagnetic forces.
1.1 Comparison of linear actuators
A popular way to produce linear motion from a rotary motor is to use the belt and
pulley system, figure 1.1. This system has limited thrust force capability due to tensile
strength of the belt. It usually requires additional gear box to decrease rotational speed
to be suitable for the pulley. Mechanical windup, backlash of the gearbox and belt
stretching all these factors are contribute to inaccuracies in the system. Settling time
of the system is very poor due to extensibility of the belt. And system characteristics
become poorer with longer belts.
7
-
Figure 1.1. Belt and pulley system (Oriental Motor).
Another way to achieve linear motion from rotary motor is to use rack and pinion
system, figure 1.2. Such a system provides better thrust capability in comparison with
the belt and pulley system. Pinion gear and required gearbox cause inaccuracies in the
system, which will increase with wear of the system. Position accuracy and
repeatability of the system is influenced by the backlash of the gears.
Figure 1.2. Rack and pinion system (Oriental Motor).
Screw system is also popular way to translate rotary motion into linear, figure 1.3.
There are two screw systems: ball-screws and lead-screws. The very cheap lead-screw
system has very low mechanical efficiency of about 50 percents. This system is not
intended for high duty operational mode because of high wear.
8
-
Figure 1.3. Lead screw system (Oriental Motor).
Linear motion electromagnetic devices allow high precision positioning. Performance
is limited only by the resolution of the linear encoder and the stability of the
mechanics. Since there is no backlash or mechanical windup, the linear motors have
great repeatability and precision. Since the load is directly connected to the mover, the
settling time of the system is very short. Performance characteristics remain unchanged
even in long travel configurations. The reliability of the linear drives is much higher
than the reliability of the linear actuator systems with rotating machines.
The comparison of the properties of the most popular linear actuator systems is shown
in the table 1.1.
Table 1.1. Properties comparison table.
Belt and pulley Rack and
pinion Lead screw Ball screw Linear
electrical motor
Accuracy -- - ++ + ++
Speed range + - -- - ++
Travel range - + -- -- ++
Thrust - + - + ++
Friction + + -- + ++
Maintenance + + - -- ++
Life time + - -- + ++
Price + ++ ++ + -
Efficiency + - -- - ++
9
-
1.2 Description of the motor system to be studied
Linear motors (LMs) fall into two main types: linear synchronous motors (LSM) and
linear induction motors (LIM). In linear induction motors the rotor magnetic field is
produced by the induced current, when in linear synchronous motors rotor field is
produced by the permanent magnets or by the independently generated current.
Linear motors could be applied in many areas. Industrial automation systems are the
main field of application of the linear motors. Linear induction motors have found
applications in the following areas: conveyor systems; liquid metal pumping; material
handling; low and medium speed trains. Linear synchronous motors are successfully
utilized in any application where high precision motion control is required: machining
tools; welding robots; laser scribing systems; industrial laser cutting systems; industrial
robots.
Linear motors are used for high-speed ground transportation as propulsion and
levitation systems. Also extensive use of LMs could be found in building and factory
transportation systems. LMs are applied as elevator hoisting machines, ropeless
elevators in ultra-high buildings, horizontal transportation systems at factories.
There are several types of LSMs which are classified according to the following items:
Short primary / short secondary Ironcore / ironless Moving primary / moving secondary Tubular / flat Permanent magnet excitation / electromagnetic excitation Single sided / double sided Transverse flux / longitudinal flux
Nowadays flat brushless permanent magnet (PM) linear synchronous motors are
dominating. The main reasons for the PM-machine popularity are higher force density,
efficiency and faster response due to existing magnetic field than in LIMs.
10
-
There are two design types of the flat PM linear synchronous motors: ironcore and
ironless. The ironless motor has no iron parts in the forcer, so the motor has no
attractive force or cogging which increases lifetime of the guideway bearings. Such a
motor is ideal for smooth velocity control but has low force output.
The ironcore flat linear PM motor has forcer with steel laminations. This configuration
allows increasing force output of the motor due to focusing the magnetic field created
by the windings. At the same time due to the strong attractive force between the
ironcore armature and the magnetic track this type of motors characterized by
increasing bearings wear. The strong cogging force causes difficulties in position
control.
In this thesis a flat single sided ironcore surface permanent magnet linear synchronous
motor (PMLSM) with short moving primary is under observation, figure 1.4. This type
of motors had been essentially developed for the factory automation.
Figure 1.4. The construction of a flat PMLSM (Tecnotion 2008).
A linear motor usually is a port of a bigger automation system. A complete linear
motor system (see figure 1.4) usually consists at least of following parts:
A mounting frame. A magnet track build up of magnet plates.
11
-
A set of linear guides or rails that support the slide. A positioning system consisting of a drive, a controller, a linear encoder for
position feedback to the controller.
A coil unit attached to the slide carried functional load. Safety end dumpers and switches to stop the movement in case of
malfunctions.
Cable chain to provide cabling to the coil unit.
The practical range of the travel of movers of the mentioned type of linear motors is,
generally, between 0.1m and 3m. For shorter lengths of travel, tubular configurations
are better suited. For longer distance travels the path with PMs becomes rather
expensive, and different machine types should be considered. Thrusts up to 10,000 N
at speeds up to 5 m/s are common for that type of PMLSM.
The first target of this M.Sc thesis is to study the principles of PMLSM. One
traditional way to describe the construction of PMLSM is to cut and unroll a PM rotary
synchronous motor to produce a flattened configuration. So we have a lying plate with
permanent magnets and a stator winding which slides in a straight line. The slider
which consists of a slotted armature and three-phase windings is termed moving
primary, or mover, or slide, while the fixed magnet plate could be named as secondary,
or exciter, or, somewhat confusingly, stator. The primary and the secondary are
separated by a flat air gap. The constant value of the air gap is maintained by guide rail
and bearings, Fig 1.5.
Figure 1.5. The construction of a PMLSM (Hirvonen 2006).
12
-
When the coils of the mover are excited by 3-phase alternating current an armature
magnetic field is generated. This field produces a travelling flux wave along the main
axis of the linear motor. This flux wave interacts with the flux produced by the
permanent magnets of the stationary fixed magnet plate. Interaction results in an
electromagnetic force which moves the primary in positive or negative direction in
dependence of the alignment of the permanent magnets along the magnet plate. In this
case such a force is called thrust, or traction force, or propulsion force. The linear
motion is obtained directly by electromagnetic force.
1.3 Direct thrust force control method
The second target of this M.Sc thesis is to study direct thrust force control technique.
The direct thrust force control (DTFC) is an implementation of direct torque control
(DTC) of rotating AC machines for linear AC motors. DTC is an advanced drive
technology used in variable frequency drives to control the torque (and consequently
the speed) in three-phase AC electric motors.
The idea of the DTC was introduced by Manfred Depenbrock in mid-1980s in
Germany, and at the same time in Japan by Isao Takahashi and Toshihiko Noguchi.
The idea was to influence on the flux linkage of the motor as directly as possible, and
thus on the torque produced by the machine. The principal operation of the DTC is
shown in the figure 1.6 as block diagram.
13
-
Figure 1.6. Direct Torque Control (ABB 1999).
DTC uses motor electric torque and stator flux linkage directly as control variables,
whether traditional field oriented current vector control uses currents as control
variables to indirectly influence on the electric torque of the machine. For the current
vector control the term ITC (Indirect Torque Control) is frequently used to contrast
with the direct torque control.
The DTC can be applied to any rotating field machine. Main advantages of the method
are minimum torque response time, absence of voltage modulator block, possibility of
sensorless control, control of the torque at low frequencies and relatively low annoying
noise. The disadvantage is presence of some difficulties in the flux linkage integration.
1.4 Objectives and outline of the thesis
The objectives of this work are as follows:
Study the principles of PMLSM. Study the applicability of DTC in direct thrust force control DTFC.
14
-
Chapter 2 is dedicated to introducing a simulation model of a PMLSM. The modelling
of a PMLSM is based on the space vector theory which is introduced in Section 2.1.
The mathematical representation of a PMLSM is described in Section 2.2. Then a
simplified mechanical model and linear motor nonlinearities are described in Section
2.3.
At Chapter 3 a simulation model for direct thrust force control is described. The basic
theory of the method is introduced along with the simulation model.
Chapter 4 contains simulations. Section 4.1 contains description of some auxiliary
controllers required for the simulations. The data obtained from the developed
simulation model is compared with the data given by the manufacturer of the motor in
Section 4.2. Also the influence of id = 0 control is investigated.
The conclusions reached in the work and results are discussed in Chapter 5.
15
-
Chapter 2
Modelling of a Permanent Magnet Linear Synchronous Motor
2.1 Space vector theory, coordinate systems and transformations
In a three-phase machine, there is a phase shift of 120 electrical degrees between the
different phases. So we can introduce the phase-shift operator .
2j3 e= (2.1)
This operator could be applied to construct so called space vector of the stator current
from instantaneous values of currents of different phases.
0 1 2s sA sB sC2
( ) ( ) ( )3
i t i t i ti = + + (2.2)
This space vector illustrates the effect of the all three currents created together by the
windings. The factor 2/3 allows using space vector with parameters of real equivalent
circuit of the machine. Along with the stator current is the voltage space vector us, the
space vector of the stator current linkage s and the flux linkage space vector s could be constructed from the phase quantities.
0 1 2s sA sB s2( ) ( ) ( ) ( )3
u t u t u t u t= + + C (2.3)
0 1 2s sA sB s2( ) ( ) ( ) ( )3
t t t = + + C t (2.4)
0 1 2s sA sB s2( ) ( ) ( ) ( )3
t t t C t = + + (2.5)
Now we have a vector representation created from three phase quantities which could
be transformed into two-axis representation of the space vector. The stator windings
16
-
are fixed within the stator, so a two-axes stator-fixed orthogonal coordinate system
could be introduced, figure 2.1.
Figure 2.1. ABC and XY frames.
Next, we determine the mathematical transformations from the three-phase A, B and C
components into two-phase XY-axes components. The following equations are
performed with stator (mover in linear motors terminology) currents, but they also
valid for voltages and fluxes.
( ) ( )( ) ( )
sAsX
sBsY
sC
120 240
120 240
cos cos cossin sin sin
23
ii
ii
i
+ ++ +
=
o o
o o, (2.6)
where is an angle between A and X axes. If is set to zero we obtain next formula
sA
sXsB
sYsC
1 11
2 23 3
02 2
23
ii
ii
i
= . (2.7)
The Matlab/Simulink block for this transformation is depicted next in the figure 2.2.
17
-
Figure 2.2. ABC to XY transformation block.
Reverse transformation from two-phase quantities into three-phase quantities
( )( )
( )( )
sAsX
sBsY
sC
cos sin
cos 120 sin 120
cos 240 sin 240
ii
ii
i
+ ++ +
= o o
o o
. (2.8)
If is set to zero we obtain next formula
sA
sXsB
sYsC
01
1 3
2 21 32 2
ii
ii
i
=
. (2.9)
And correspondent Matlab/Simulink block is depicted in the figure 2.3.
18
-
Figure 2.3. XY to ABC transformation block.
The model of a linear synchronous motor with permanent magnets will be introduced
as a model in direct-quadrature (d-q) axes reference frame, figure 2.4. This frame
rotates relative to the stationary fixed XY frame with the angular synchronous speed
el. This el could be expressed from the linear velocity vlin using the equation
el lin v = . (2.10)
All variables are expressed on orthogonal direct and quadrature axis which rotates with
synchronous speed. d-axis is aligned along with the magnetic axis, and q-axis is
aligned 90 degrees ahead in the direction of rotation, which traditionally assumed to be
counter-clockwise.
19
-
Figure 2.4. dq reference frame.
Next equations are used to convert space vectors from fixed XY-frame into rotating
coordinate system of direct and quadrature axis:
d sx sy( ) cos sini t i i = + , (2.11) q sx sy( ) sin cosi t i i = + , (2.12)
where is an angle between the X- and d- axes.
Matlab/Simulink blocks for transformation are depicted in the figure 2.5.
Figure 2.5. XY-dq transformation block.
20
-
Figure 2.6. dq-XY transformation block.
2.2 Basic motor model
Control algorithms of AC motors frequently use the d-q axes model of AC machines.
To derive a d-q axes model of the motor let us introduce stator voltage equations:
dd d( )
du t Ridt q = + , (2.13)
qq q( )
du t Ri
dt d = + + , (2.14)
where ud and uq are the d-axis and q-axis components of the stator voltage space
vector, id and iq are the d-axis and q-axis components of the armature current vector, R
is an armature phase resistance. The armature winding d-axis and q-axis flux linkages
d and q in the previous equations are
d =Ldid + PM, (2.15) q =Lqiq, (2.16)
where Ld and Lq are the d-axis and q-axis armature inductances and PM is the flux linkage of the permanent magnet.
21
-
The instantaneous power input to a three-phase armature is
A A B B C C d d q q3 (2
P u i u i u i u i u i= + + = + ) , (2.17)
where uA, uB and uC are instantaneous phase voltages, iA, iB, and iC are instantaneous
phase currents, and ud and uq are d- and q-axis voltage components, id and iq are d- and
q-axis current components. The power balance equation is obtained from equations of
stator voltages (2.13) and (2.14).
q2 2dd d q q d d q q d q q d( )
ddu i u i Ri i Ri i i idt dt
+ = + + + + , (2.18)
The last term accounts for the electromagnetic power of a single phase, two pole
synchronous machine. For a three-phase machine
elm d q q d PM d q d q3 3( ) [ ( )2 2
P i i L L = = + ]i i , (2.19)
where Ld and Lq are the armature inductances. The electromagnetic thrust Fthurst of a
PMLSM with p pole pairs is the electromagnetic power Pelm in last equation divided by
the linear velocity vlin in equation (2.10) and multiplied by p.
thrust d q q d PM d q d q3 3 ( ) [ (2 2
) ]F p i i p L L i = = + i . (2.20)
The set of equations 2.13-2.16 and 2.20 comprises the permanent magnet linear
synchronous motor basic mathematical model. The Simulink block diagram of the
motor model is depicted in the figure 2.7.
22
-
Figure 2.7. Simulink motor model block.
Next, the parameters given by the manufacturer of the motor (see Appendix 2) are
presented in the table 2.1. PM flux linkage is calculated using motor force constant K
and equation 2.21.
thrustPM
s
2 2 3 3
F Kp i p
= = . (2.21)
Table 2.1. Parameters of the motor model.
Symbol Value Parameter
R 1.6 phase resistance Ld 13 mH d-axis inductance
Lq 13 mH q-axis inductance
p 1 number of pole pairs
0.012 m PM pole pitch
K 93 N/A Motor force constant
PM 0.237 Wb PM flux linkage
The model corresponds to the linear synchronous motor with surface mounted PMs.
The relative permeability of the permanent magnets is almost unity (1.04), which
23
-
means that permanent magnets are like air in the magnetic circuit and parameters of
the magnetic circuit are equal in d- and q-axes. It causes in equal and quite low d- and
q- axes inductances.
2.3 Nonlinearities and mechanical model
In motion control applications the nonlinearities of a linear motor could bring
significant tracking error or increase settling time. To avoid such effects this
nonlinearities should be carefully modelled and taken into account while designing the
control system. Next we introduce the mechanical model of the motor. The dynamic
behavior of the linear motor system could be expressed by following equation:
thrust tot load friction disturb( ) ( ) ( ) ( )dvF t m F t F v F xdt
= + + + , (2.22)
where mtot is a sum of the mover mass mmov and the mass of the load mload; v is a linear
mover speed; Ffriction is a force which takes into consideration viscous, Stribeck and
Coulomb effects; Fload is an additional force produced by the load; Fdisturb is a force
which accounts the effects of cogging and flux non-uniformity at the ends of the
mover.
Let us next consider all four elements of the right part of the equation 2.22. The first
term is indispensable part of any physical dynamic system associated with inertia. The
second term is application related. The last two terms are discussed in the following
three subsections.
24
-
2.3.1 Friction model
In linear motors of observed type friction is very important due to significant attraction
force between permanent magnets and iron parts of the mover. This attraction force
should be considered for the mechanical design of PMLSM, in particular related to
noise, vibration and linear bearing design. Next the friction model will be discussed.
The friction force is composed of Coulomb, viscous and Stribeck effects and could be
described by the figure 2.8.
Figure 2.8. Viscous, Coulomb and Stribeck effects of friction.
The Stribeck friction is the negatively sloped characteristics taking place at low
velocities. The Coulomb friction results in a constant force at any velocity. The viscous
friction opposes motion with the force proportional to the velocity. The friction force
as a function of mover velocity:
friction fr fr fr( ) sign( ) sign( )k vF v C v V v S e= + + v . (2.23)
The parameters of the friction model used in the simulation model are represented in
the table 2.2. The behavior of the friction model with these parameters is similar to
25
-
friction model described by Hirvonen (2006). However, exact values should be
separately determined in every particular case, and should be proved by tests.
Table 2.2. Parameters of the friction model.
Symbol Value Parameter
Cfr 30 N Coulomb coefficient Vfr 3 Ns/m viscous coefficient Sfr 10 N Stribeck coefficient k 10 s/m Stribeck speed factor
The Matlab/Simulink block for the friction model is shown in the figure 2.9. And
friction model ramp response is shown in the figure 2.10.
Figure 2.9. Simulink block of the friction model.
Figure 2.10. Friction model ramp response.
26
-
2.3.2 Cogging force
The last term of the equation 2.22 composed by two parts: cogging force and a force
caused by end effects.
disturb cogging end_effect( ) ( )F x F x F= + . (2.24)
Cogging force is caused by interaction between the iron slots of the mover and the
permanent magnets of the track. Cogging force is presented even when there is no
motor current. Due to the slotted nature of the primary core, the cogging force is
periodic and has two components related to the primary core length and PM pole
pitching. The first component of cogging force could be reduced by modifying the core
shape and optimizing the length of the mover. And the second component of the
cogging force can be reduced substantially by skewing either the primary teeth or the
permanent magnets (Jung et al. 1999). But these measures can reduce the maximum
thrust force and the efficiency of the motor and increase the complexity of the motor
structure. To achieve high positioning performance the cogging force should be
carefully modelled and compensated.
It should be mentioned that cogging force is very insignificant when configuration
with aircore windings is used. But this configuration produces lower force output.
The mathematical representation of the cogging force as a function of the mover
position (Hirvonen 2006):
[ ]cogging s 1 r1 r2 2( ) sin( 2) sin( 2)F x K x A A x= + , (2.25)
where Ks is a scaling factor, 1 and 2 are wavenumbers which are related to the and
the primary core length, Ar1 and Ar2 are amplitudes of two harmonics.
27
-
The Simulink block diagram of the cogging force model according the mathematical
model 2.25 is represented in figure 2.11
Figure 2.11. Cogging force Simulink block.
Parameters of the model are represented in the table 2.3.
Table 2.3. Cogging force Simulink block parameters.
Symbol Value Parameter
Ks 0.7 scaling factor
1 41.7 m-1 1st wavenumber
2 4.1 m-1 2nd wavenumber
Ar1 35 N 1st harmonic
Ar2 15 N 2nd harmonic
Figure 2.12. shows cogging force simulation results.
28
-
Figure 2.12. Simulated cogging force .
2.3.3 End-effect
End-effect is a special phenomena because of the limited length of the mover.
Generally it is difficult to describe the end-effect with exact mathematical model
(Jiefan et al. 2004). The powerful finite element method (FEM) should be applied to
analyze the magnetic field to determine the role of the end-effect in any particular case.
In practice many researchers use multiplying coefficient kend or function which takes
into account the thrust reduction due to the end-effect (Gieras et al. 2000, 138). It is
evident that the impact of the end-effect on the thrust is reduced with increasing
number of poles of the mover. The value of the coefficient can be determined
experimentally.
end_effect T( )F t F= . (2.26)
In this work the end-effect is described by the coefficient kend. The value of the
coefficient is suggested to be 0.01.
end_effect end TF k F= . (2.27)
29
-
The Simulink block diagram of the dynamic model including friction, cogging force
and end-effect is depicted on the figure 2.13.
Figure 2.13. Dynamic model Simulink block.
Table 2.4. Parameters of the dynamic model used in the simulations.
Symbol Value Parameter
mmov 4.8 kg mover mass
mload 10.8 kg load mass
30
-
Chapter 3
Modelling of Direct Thrust Force Control of a PMLSM
Direct thrust force control (DTFC) method is a DTC applied to the linear motors. The
only difference between the DTC and the DTFC is that in the rotating field machines
we deal with torque and angular velocity, whether in the linear machines these
replaced by thrust force and linear velocity. Next the DTC will be observed in details.
But first of all the concept of VSI voltage vectors should be introduced.
3.1 VSI voltage vectors
At present, the voltage source inverter (VSI) drives are most widely used. The VSIs
are applied to the control of all kinds of rotating field machines. There are mainly two
types of VSI drives available: two-level and three level devices. The typical structure
of the two-level VSI drive is shown in the figure 3.1 (Mohan et al. 2003).
Figure 3.1. Two-level VSI drive.
The voltage source inverter considered together with 3-phase winding can generate
voltage vectors. The voltage vectors are defined by combinations of switch positions
(SA, SB, SC). For two-level VSI there are 23 possible combinations of power switches
which can produce 6 active voltage vectors and 2 zero voltage vectors. Voltage vectors
of a two-level VSI in a stationary XY-reference frame are presented in the figure 3.2.
31
-
Voltage polarities at the three-phase terminals of the inverter are marked near the
vectors. The voltage polarities are representing the power switch statuses of all three
phases.
Figure 3.2. Two-level VSI voltage vectors.
The stator voltage space vector us can take one of the following eight instantaneous
values in dependence of polarities of A-, B- and C-phases which are shown in round
brackets:
us(---) = u0 = 0;
us(+--) = u1 = UDC 0;
us(++-) = u2 = UDC ( 0 + 1);
us(-+-) = u3 = UDC 1;
us(-++) = u4 = UDC ( 1 + 2); (3.1)
us(--+) = u5 = UDC 2;
us(+-+) = u6 = UDC ( 2 + 0);
us(+++) = u7 = 0.
The Simulink block diagram of the inverter is shown in the figure 3.3.
32
-
Figure 3.3. VSI Simulink block diagram.
3.2 Direct Torque Control
The Direct Torque Control is a continuation of the vector control of the AC motors.
The foundations of the DTC were firstly described by I. Takahashi and T. Noguchi in
middle 80s. And in middle 90s the first industrial DTC utilized drives were introduced
by ABB.
The main task of the DTC is to supply quick electromagnetic torque response of the
motor. Unlike the vector control, where the torque is altered by the influence on the
stator current vector, in the DTC the control variable is the stator flux linkage vector
s, figure 3.4 (because of that the better name for the DTC is Direct Flux Linkage
Control, DFLC). The change of the flux linkage is achieved by the optimal switchings
of the power switches of the VSI which feed the motor.
33
-
Figure 3.4. DTC block diagram (Luukko 2000).
Let us next consider theoretical foundations of the DTC. For this purpose next
equation should be introduced (Niemel 1999, 13).
elm s ms
3 1 sin( )2
T pL
= , (3.2)
where Ls is the stator leakage inductance, m is the air gap flux linkage vector, and
is an angle between the stator and the air gap flux linkages.
This equation justifies the direct torque control. It shows that the motor torque is
proportional to the sinus of the angle . Hence, if the magnitudes of the stator and air
gap flux linkages are constant, the motor torque could be controlled by the variation of
the angle .
34
-
s s s s( ) ( )t u R i dt s0 = + . (3.3)
The flux linkage estimation equation 3.3 is the key element of the DTC. It gives a
connection between the stator voltage vector us and the flux linkage vector s. This
equation gives an opportunity to control the stator flux linkage vector s, its position,
and magnitude, by controlling the stator voltage vector us. And consequently it allows
to control the angle .
It should be noticed that the winding voltage drop (term Rsis in eq. 3.3) is usually
relatively small and could be neglected and next equation could be written.
s s( ) ( )t u dt s0 = + . (3.4)
So the trajectory of the flux linkage moves in the direction of the applied voltage
vector of the VSI with the speed proportional to the applied voltage. If one of the zero
voltage vectors u0 or u7 is applied the locus of the flux linkage vector almost stands
still because of the small value of the resistive winding voltage drop Rsis. This allows
to freely control the rotating velocity of the flux linkage vector by changing the ratio
between zero voltage vectors and active voltage vectors.
Figure 3.5 shows the plane where the basic active VSI voltage vectors u1 - u6 are
located in the stationary XY-axes reference frame. In the same stationary frame the
flux linkage vector s is shown.
35
-
Figure 3.5. Six sectors of the flux circle.
The flux plane is divided into six sectors of equal sizes of 60 electrical degrees in such
a way that the sectors are bisected by the basic VSI voltage vectors (Mohan 2001).
Let us first consider the torque control. In PM synchronous machine the torque is
proportional to the sinus of the angle between the PM flux linkage PM (d-axis in the
d-q reference frame) and the stator flux linkage s. Torque could be effectively
controlled by varying this angle. There are three possible situations.
First, if the actual torque of the machine is too small, then the stator flux linkage vector
must advance to increase the angle . It means that if we want to increase the torque we
should choose voltage vector which is located ahead in the direction of rotation. For
example, if the flux linkage vector is in the first sector, as it shown in the figure 3.5,
then the voltage vectors u2 and u3 are suitable.
36
-
Second, if the actual torque is in tolerance borders, then the flux vector must keep its
position, angle should not be changed. And we should select a zero voltage vector u0
or u7.
Third, if the actual torque is too great, then the flux vector must go backward, angle
should be decreased. It means that if we want to decrease the torque than the behind
located voltage vector must be selected. For example, if the flux linkage vector is in
the first sector, then the voltage vectors u6 and u5 are suitable. If the rotating speed of
the motor is significant, then the zero voltage vectors could also be used to decrease
torque.
This logic can be implemented by the double hysteresis of the error signal as it shown
in the figure 3.6. The output signal of the torque hysteresis comparator can possess
three different values: 1, -1 and 0. Value = 1 corresponds to the instant when the
torque increase is required; at = -1 the torque must be decreased; = 0 means that the
torque is lying in the tolerance borders.
Figure 3.6. Torque double hysteresis.
37
-
In the same time the magnitude of the flux vector must be kept in necessary borders. In
each sector two voltage vectors can be used in both rotation directions to decrease or
increase the flux linkage.
The regulation of the flux linkage could be implemented with hysteresis of the flux
linkage error signal as it shown in the figure 3.7.
Figure 3.7. Flux linkage hysteresis.
The output signal of the flux hysteresis comparator can possess two values 1 and 0.
If the current magnitude of the stator flux linkage is significantly less than the
reference value, then flux linkage magnitude should be increased which corresponds to
= 1. But if the magnitude of the flux linkage is significantly bigger than the reference
signal, then flux linkage magnitude should be decreased which corresponds to = 0.
Signals from torque and flux linkage hysteresis controllers and as well as the
position information (the sector number) about flux linkage vector are indexing
elements of so-called optimal switching table, the core component of the DTC. This
table provides an optimal selection of voltage vectors. Selected voltage vector next
applies to the power switches and provides desired change of the flux vector to achieve
optimal performance. The control of the power switches is adjusted only when the
torque or absolute value of the flux linkage differs too much from the reference value.
When the hysteresis limit is reached the next voltage vector is selected from the
38
-
optimal switching table to bring the flux linkage vector into the right direction. Figure
3.8 depicts an optimal switching table introduced by I. Takahashi and T. Noguchi
(1986).
Figure 3.8. Optimal switching table.
The selection of the voltage vector is made so as to restrict the errors of the flux and
torque within the hysteresis bands and to obtain the fastest torque response and highest
efficiency at every instant.
Typically the flux linkage reference signal for flux hysteresis controller is kept
constant for the speed range below the rated speed. Suitable value for the flux linkage
reference signal is flux linkage of the permanent magnets. If the motor intended to run
above the rated speed the field weakening technique could be applied. The reference
signal for the torque hysteresis controller is obtained from the speed PI type controller.
The actual values of the torque and flux linkage are estimated in the estimator. The
Simulink block diagram of the DTC controller is depicted in the figure 3.9.
Figure 3.9. DTC controller model.
39
-
3.3 Estimations
The actual values of the thrust force and the stator flux linkage for the hysteresis
comparators are obtained by estimations. The estimation of the stator flux linkage
could be performed by measurement of the stator voltage and current using expression
3.3.
In the real drive there are no phase voltage measurement devices. Instead of direct
measurement of the stator voltage the estimation of the stator voltage us_est is used.
Estimated voltage is constructed according to the information about the actual position
of the power switches (SA, SB, SC) and measured value of the DC-link voltage UDC by
using formula 3.5.
0 1s_est DC A B C2 (3
u u S S S= + + 2 ) . (3.5) The Simulink block for the voltage estimation is depicted in the figure 3.10.
Figure 3.10. Voltage estimation block.
There is a valuable drawback in using formula 3.3 in flux estimation. Voltage
integration is very sensitive for inaccuracies in parameters. Unaccounted voltage drop
in power switches, stator current measurement errors, DC-link voltage measurement
errors, and errors in the stator resistance estimation all this factors cause errors in the
flux estimation. So estimated flux linkage could much deviate from the real value (see
figure 3.14). Stator current measurement errors, DC-link voltage measurement errors,
40
-
stator resistance estimation error, inaccuracies in determination of PM flux linkage and
d- and q-axes inductances - all these inaccuracies are included in the Simulink model
(see Appendix 1).
It should be mentioned that to increase accuracy of the flux estimation the temperature
related resistance variation should be taken into account and resistance estimation
model which uses information from a temperature sensor should be created. In this
work, the temperature related stator resistance variations are neglected.
One possible way to eliminate this error is to use current model for the flux linkage
correction. Current model gives accurate estimation of the stator flux linkage, but
needs accurate inductance model. Block diagram which illustrates calculation of the
correction terms using current model is depicted in the figure 3.11.
Figure 3.11. Flux linkage correction terms calculation using current model.
Correction is occurred in the appointed time instances because current model
calculation is a time consuming task. The Simulink block diagram of the corrector is
depicted in the figure 3.12.
41
-
Figure 3.12. Corrector.
These correction terms are then applied to the voltage model in the estimator, figure
3.13.
Figure 3.13. Estimator.
Figure 3.14 shows real and estimated flux linkage trajectories in XY coordinate system
when estimated phase resistance is slightly bigger than real value and flux linkage
correction is switched off. When the flux linkage correction is switched on the
trajectories are almost identical.
42
-
Figure 3.14. Real (left) and estimated (right) flux linkage trajectories without correction.
The X-axes and Y-axes components of the flux linkage x and y are the output of the
voltage model. These values are used to calculate thrust force estimation, flux linkage
magnitude estimation, and the sector of the flux linkage vector position.
Thrust force is calculated as a cross product of estimated flux linkage vector and
measured motor current vector in accordance with next formula:
thrust.est x y y x3 (2
)F p i i = . (3.6)
The Simulink block diagram of the force estimation is depicted in the figure 3.15.
Figure 3.15. Force estimation.
Calculation block of flux linkage estimate modulus is depicted in the figure 3.16.
43
-
Figure 3.16. Calculation of flux linkage estimate modulus.
The sector estimation block is shown in the figure 3.17.
Figure 3.17. Sector estimation.
44
-
Chapter 4
Simulations
4.1 System overview
The whole drive Simulink model is depicted in the figure 4.1. Inverter is assumed to be
supplied from constant DC voltage.
Figure 4.1. Drive model. The model of the whole system is represented in the figure 4.2.
Figure 4.2. System model. A DTFC controller inverter unit requires several auxiliary controllers. The thrust force
reference signal is either the thrust force reference from the speed controller or an
external thrust force reference. Thrust force reference must be limited in order not to
exceed the maximum allowed peak thrust of the motor and consequently the permitted
45
-
currents for the inverter. The basic algorithm of the speed controller is a PI control
algorithm. The speed PI controller Simulink model with anti-windup which produces
thrust force reference signal for the DTFC is represented in the figure 4.3.
Figure 4.3. Speed PI controller.
The values of the coefficients are represented in the table 4.1. The speed controller had
been tuned experimentally by simulations.
Table 4.1. Speed controller coefficients.
Symbol Value Parameter speed_PID_P 6000 Proportional gain speed_PID_I 1000 Integration gain speed_saturation 760 N Saturation border Speed_antiwindup_gain 0.19 Antiwindup gain
The speed reference signal is either the speed reference from the position controller or
an external speed reference. The basic algorithm of the position controller is a PI
control algorithm. The position PI controller Simulink model with anti-windup which
produces speed reference signal for the speed controller is represented in the figure 4.4.
Figure 4.4. Position PI controller.
46
-
The values of the coefficients are represented in the table 4.2. The position controller,
as well as speed controller, had been tuned experimentally by simulations.
Table 4.2. Position PI controller coefficients.
Symbol Value Parameter position_PID_P 23 Proportional gain sposition_PID_I 3 Integration gain position_saturation 4.5 m/s Saturation border Position_antiwindup_gain 0.17 Antiwindup gain
The flux linkage reference signal control block produces reference signal for the flux
linkage hysteresis controller for the DTFC. Suitable value for the reference signal is
the flux linkage magnitude of the permanent magnets. In order to increase speed range
of the motor above the rated speed the field weakening technique could be
implemented in this control block. Also some optimization strategies, like maximum
thrust per ampere (MTPA) or id = 0 control, could be implemented to increase
efficiency of the drive in the speed range below the rated speed.
In the surface mounted permanent magnet linear synchronous motor the direct-axis and
quadrature-axis inductances are approximately equal, Ld Lq. In the steady state the
thrust equation 2.20 can be simplified in the form
thrust.est PM q3 2
F p i . (4.1)
This equation shows that the direct-axis current id does not have any effect on the
thrust. And for the given thrust force the minimum stator current and in most cases
maximum efficiency are reached when id = 0. This creates the basis for the id = 0
control, which in linear synchronous drive with surface mounted PMs is equal to the
MTPA strategy (Abroshan et al. 2008).
To determine flux linkage reference signal next equation could be used.
47
-
( )2 2s d d PM q q( )L i L i + + . (4.2)
As the direct-axis current id is equal to zero this equation transforms to
2s PM q q( )L i + 2 , (4.3)
where quadrature-axis current iq is obtained from the equation 4.1.
The Simulink model of the id = 0 controller which produces flux linkage reference
signal for the DTFC is represented in the figure 4.5.
Figure 4.5. id = 0 flux linkage controller.
4.2 Simulation results
In this section the simulation results are represented.
4.2.1 Conventional DTFC
Let us first compare simulation model behaviour with the data provided by the
manufacturer of the linear motor (see Appendix 2). Selected movement profile almost
the same as in the data provided by the manufacturer.
Figure 4.6 shows reference value and position response for the motor.
48
-
Figure 4.6. Position chart.
Figure 4.7 shows reference value generated by the PI position controller and speed
response for the motor.
Figure 4.7. Speed diagram. Figure 4.8 shows acceleration of the mover. The acceleration diagram resembles the
trust force produced by the mover with the account of end effects, cogging force and
friction, figure 4.9.
49
-
Figure 4.8. Acceleration diagram.
Figure 4.9. Thrust force diagram.
Figure 4.10 and figure 4.11 show currents during acceleration of the mover, currents at
steady speed of 4.5 m/s, and deceleration currents. If we compare figure 4.11 with the
current diagram provided by the manufacturer (see Appendix 2) we could see that the
simulated current is slightly smaller. It could be the result of underestimation of end
effects in simulation, or inexact cogging force and friction models (see Chapter 2.3).
Also it could be because the manufacturer had used different from the DTC method to
perform measurements.
50
-
Figure 4.10. d- and q-axes currents.
Figure 4.11. Phase currents of the motor. 4.2.2 DTFC with MTPA control strategy
During MTPA control, the reference signal of the flux linkage is adjusted by the way
to produce required thrust force by minimum current. This strategy allows to increase
efficiency of the motor and to minimize losses in copper.
Figure 4.12 shows flux linkage reference signal adjustments during id = 0 control. The
permanent magnets flux linkage is also shown in the figure.
51
-
Figure 4.12. Flux linkage reference signal, MTPA control.
Figures 4.13 and 4.14 show currents during acceleration of the mover, currents at
steady speed of 4.5 m/s, and deceleration currents, when maximum thrust per ampere
control strategy is applied. Comparison of figures 4.10 and 4.13 gives us clear vision
of advantages provided by id=0 control technique. The same thrust force is achieved by
smaller current.
Figure 4.13. d- and q-axes currents, MTPA control.
Figure 4.14. Phase currents of the motor, MTPA control.
52
-
Chapter 5
Results and Conclusions
The objectives of this thesis were focused on the development of a simulation model of
the direct thrust force controlled permanent magnet linear synchronous motor. A flat
single sided ironcore surface permanent magnet linear synchronous type motor had
been chosen for modelling. All the equations required for the basic motor model were
introduced. The basic permanent magnet linear synchronous motor model represents
conventional two-axes rotary type permanent magnet synchronous motor model. The
end-effect, the specific for linear motors phenomenon, was accounted in enlarged
motor model as well as friction and cogging force.
Direct thrust force control of PMLSM was described. It represents the traditional direct
torque control (DTC) applied to linear motors. All equations which justify DTC and
consequently DTFC were introduced. The foundations of DTC were carefully
described. The DTFC was modelled. Flux linkage correction was modelled using
current model.
The auxiliary modules for DTFC controller which are required for the speed and
position control were introduced as well. These controllers represent simple PI
regulators with anti-windup. The parameters for these controllers were experimentally
tuned.
The movement profile provided by the manufacturer was almost resembled by the
simulation. It shows that PI speed and position controllers have sufficient performance
and their tuning was performed properly.
The simulation results were compared with the data provided by the manufacturer.
Generally, results and data are equal. It proves the full system model. The simulated
currents were slightly smaller than currents of provided motor data. It could be the
result of underestimation of end effects in simulation, or inexact cogging force and
53
-
friction models. Also it could be because the manufacturer had used different from the
DTC method to perform measurements.
Also the implementation of maximum thrust per ampere control strategy was
modelled. For the considered motor type the MTPA strategy and id=0 control are the
same. The results clearly show that MTPA strategy allows to decrease current required
for the given thrust in comparison with conventional DTFC.
There are few issues related for future development. Most significant one is
transformation of continues simulation model into discrete-time system where
calculation time and time required for analog-to-digital and back transformations could
be taken into account. Field weakening must be modelled if speed above the rated
speed of the motor should be considered. Another future work is to apply FEM to
determine precise influence of end-effects.
For modelling precision position control more accurate friction model and cogging
force model should be used. Also these models should be included into control
algorithm to reduce the influence of friction and cogging force. The position and speed
PI controllers should be tuned analytically or even replaced by more intelligent ones to
achieve better performance.
Further work should also include tests with real linear motor to practically prove
simulator model.
Another potential direction of work is considering determination of initial angle
between XY-reference frame and dq-reference frame.
54
-
REFERENCES
ABB. 1999. Technical Guide No.1: Direct Torque Control. [Online document].
[Accessed 8 March 2009]. Available at http://www.abb-drives.com/
StdDrives/RestrictedPages/Marketing/Documentation/files/PRoducts/DTCTechGuide1
.pdf
Abroshan, M. & Malekian, K & Milimonfared, J. & Varmiab, B.A. 2008. An Optimal
Direct Thrust Force Control for Interior Permanent Magnet Linear Synchronous
Motors Incorporating Field Weakening. IEEE, SPEEDAM, pages 130-135.
Boldea, Ion & Nasar, S.A. 2001. Linear Motion Electromagnetic Devices. USA.
Sheridan Books, Ann Arbor, MI. 269 p. ISBN 90-5699-702-5.
Gieras, Jacek F. & Piech, Zbigniew J. 2000. Linear Synchronous motors:
Transportation and Automation Systems. USA. CRC press LLC. 327 p. ISBN 0-8493-
1859-9.
Hirvonen, Markus. 2006. On the Analysis and Control of a Linear Synchronous
Servomotor with a Flexible Load. Dissertation. Lappeenranta University of
Technology, Lappeenranta, Finland. 120 p. ISBN 952-214-320-0.
Jiefan, Cui & Chengyuan, Wang & Junyou, Yang & Lifeng, Liu. 2004. Analysis of
Direct Thrust Force Control for Permanent Magnet Linear Synchronous Motor. IEEE,
Proceeding of the 5th World Congress on Intelligent Control and Automation, June 15-
19, Hangzhou, P.R. China, pages 4418-4421.
Jung, In-Soung & Yoon, Sang-Baeck & Shim, Jang-Ho & Hyun, Dong-Seok. 1999.
Analysis of Forces in a Short Primary Type and a Short Secondary Type Permanent
Magnet Linear Synchronous Motor. IEEE Transactions on Energy Conversion, Vol.
14, No. 4, December, pages 1265-1270.
55
-
Luukko, Julius. 2000. Direct Torque Control of Permanent Magnet Synchronous
Machines Analysis and Implementation. Dissertation. Lappeenranta University of
Technology, Lappeenranta, Finland. 172 p. ISBN 951-764-438-8.
Mohan, Ned. 2001. Advanced Electrical Drives: Analysis, Control and Modeling using
Simulink. USA. MNPERE. 184 p. ISBN 0-9715292-0-5.
Mohan, Ned & Undeland, Tore M. & Robbins, William P. 2003. Power Electronics:
Converters, Applications, and Design. Third edition. USA. John Wiley & Sons, Inc.
802p. ISBN 0-471-22693-9.
Niemel, Markku. 1999. Position Sensorless Electrically Excited Synchronous Motor
Drive for Industrial Use Based on Direct Flux Linkage and Torque Control.
Dissertation. Lappeenranta University of Technology, Lappeenranta, Finland. 144 p.
ISBN 951-764-314-4.
Oriental Motor. [Online Document]. [Accessed 17 April 2009]. Available at
http://www.orientalmotor.com.my/support/request.php
Pyrhnen, Juha. 2008. Electrical Drives 2008-2009. Lecture Notes. Lappeenranta
University of Technology. Department of Electrical Engineering.
Tecnotion. 2008. A primer of Tecnotion linear motors. Version 2.1. Issue date:
September 2008. Document nr. 4022.363.4187.2. [Online document]. [Accessed 10
March 2009]. Available at http://www.tecnotion.com/ Products.php?Keuze=Techdocs
56
-
Appendix 1 _ Appendices Appendix 1 Simulation model m-file listing open('DTC_PMLSM.mdl'); F_load = 0 %N, load force t_load = 0 %s, start load time %========================================================== % sampling parameters %========================================================== sampling_base = 25e-6 % switching_sample = 25e-6 % mechanics_sample = 4*sampling_base %========================================================== % motor parameters %========================================================== % PSI_PM=tau/pi*2/3*F/I, where F/I - motor force constant PSI_PM = 0.237 %Wb, flux linkage of the PM ( 0.4741, 0.35, 0.2368) Ld = 0.013 % H, d-axis inductance (m.b. *1.5) Lq = 0.013 % H, q-axis inductance (m.b. *1.5) p = 1 %number of pole-pairs tau = 0.012 % m, pole pitch Rs = 1.6 %Ohm, stator winding resistance %========================================================== % mechanical parameters %========================================================== M_coil_unit = 4.8 % kg, mover mass M_application = 10.8 % kg, load mass M_tot = M_coil_unit + M_application % Friction model: %C_fr*sign(speed) + V_fr*speed + S_fr*exp(-k_Str*abs(speed))*sign(speed) C_fr = 30 % N, Coulomb coefficient S_fr = 10 % N, Stribeck coefficient V_fr = 3 % N/(m/s), viscouse k_Str= 10 % s/m % End-effect: K_end = 0.01 % Cogging force model: % Ks*sin(wn1*x*2pi)[Ar1 + Ar2*sin(wn2*x*2pi)] Ks = -0.7 %scaling factor wn1 = 1/tau %1/m, 1st wavenumber 1/tau wn2 = 1/0.244 %1/m, 2nd wavenumber 1/l Ar1 = 25 %N, 1st harmonic Ar2 = 15 %N, 2nd harmonic Attraction_force = 3400 % N %========================================================== % inverter parameters %========================================================== U_DC = 560 %volts
-
_ Appendix 1 continued %========================================================== % control parameters %========================================================== V_max = 4.5 % m/s base speed Psi_hyst = 0.02*PSI_PM force_hyst = 60%0.02*T_peak theta_initial = 0 PSI_REF = PSI_PM %========================================================== % Id=0 and FW control and machine limitations %========================================================= I_constr = 8.2 %A %max continuous current V_base = 6 % m/s base speed PSI_base = 0.3482 %Wb %(U_DC-Rs*I_constr)/(p*pi*V_base/tau) F_base = 763% % 3/2*p*pi/tau*PSI_PM*I_constr F_peak = 1600 mtpa_sw= 1 % id=0 control on/off (1/0) %========================================================== % estimator model params %========================================================== PSI_PM_est = PSI_PM*0.98 %Wb, flux linkage of the PM Ld_est = Ld*1.01 % H, d-axis inductance (m.b. *1.5) Lq_est = Lq*1.01 % H, q-axis inductance (m.b. *1.5) Rs_est = Rs*1.1 %Ohm, stator winding resistance correction_sw = 1% Correction on/off (1/0) U_DC_meas = U_DC*0.99 K_corr = 1/switching_sample % flux correction coefficient corrector_sample = 0.001 %s, correction interval(default 0.001 s.) %========================================================== % position controller %========================================================== position_PID_P = 23 position_PID_I = 3 position_saturation = V_max %m/s, speed border pos_acc=0.00005 %m Pos_antiwindup_gain = 0.17 speed_ref = 3 %========================================================== % speed controller %========================================================== speed_PID_P = 6000 speed_PID_I = 1000 speed_saturation = F_base% %N, Thrust (current) limitation speed_acc=0.0003 %m/s Speed_antiwindup_gain = 0.19%0.02 %========================================================== clear switching_table %(force,psi,sector) %[-Psi 0 ; Psi+ 1] %goes back0 zerovector1 goes forward2 switching_table( : , : ,1)=[5 7 3 ; 6 0 2]; % sector 0 switching_table( : , : ,2)=[6 0 4 ; 1 7 3]; % sector 1 switching_table( : , : ,3)=[1 7 5 ; 2 0 4]; % sector 2 switching_table( : , : ,4)=[2 0 6 ; 3 7 5]; % sector 3 switching_table( : , : ,5)=[3 7 1 ; 4 0 6]; % sector 4 switching_table( : , : ,6)=[4 0 2 ; 5 7 1]; % sector 5
-
Appendix 2 _ Appendix 2 Motor data
-
Appendix 2 _
-
_ Appendix 2 continued
-
_ Appendix 2 continued
-
_ Appendix 2 continued
-
_ Appendix 2 continued
1.pdf2.pdf3.pdf