Direct Torque Control with Space Vector Modulation (DTC-SVM) of ...
Direct Torque Control with Space Vector Modulation (DTC-SVM) of ...
Direct Torque Control with Space Vector Modulation (DTC-SVM) of ...
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POLITECHNIKA<br />
WARSZAWSKA<br />
WARSAW UNIVERSITY OF TECHNOLOGY<br />
Faculty <strong>of</strong> Electrical Engineering<br />
ROZPRAWA DOKTORSKA<br />
Ph.D. Thesis<br />
Dariusz Świerczyński, M. Sc.<br />
<strong>Direct</strong> <strong>Torque</strong> <strong>Control</strong> <strong>with</strong><br />
<strong>Space</strong> <strong>Vector</strong> <strong>Modulation</strong> (<strong>DTC</strong>-<strong>SVM</strong>) <strong>of</strong> Inverter-Fed<br />
Permanent Magnet Synchronous Motor Drive<br />
WARSZAWA<br />
2005
WARSAW UNIVERSITY OF TECHNOLOGY<br />
Faculty <strong>of</strong> Electrical Engineering<br />
Institute <strong>of</strong> <strong>Control</strong> and Industrial Electronics<br />
Ph.D. Thesis<br />
M. Sc. Dariusz Świerczyński<br />
<strong>Direct</strong> <strong>Torque</strong> <strong>Control</strong> <strong>with</strong> <strong>Space</strong> <strong>Vector</strong><br />
<strong>Modulation</strong> (<strong>DTC</strong>-<strong>SVM</strong>) <strong>of</strong> Inverter-Fed<br />
Permanent Magnet Synchronous Motor Drive<br />
Thesis supervisor<br />
Pr<strong>of</strong>. Dr Sc. Marian P. Kaźmierkowski<br />
Warsaw, Poland - 2005
Contents<br />
Table <strong>of</strong> Contents<br />
Chapter 1 1<br />
INTRODUCTION<br />
Chapter 2 8<br />
MODELING AND CONTROL MODES OF PM SYNCHRONOUS DRIVES<br />
2.1 Mathematical model <strong>of</strong> PM synchronous motor 8<br />
2.1.1 Voltage and flux-current equations 9<br />
2.1.2 Instantaneous power and electromagnetic torque 17<br />
2.1.3 Mechanical motion equation 22<br />
2.2 Static characteristic under different control modes 25<br />
2.3 Summary 33<br />
Chapter 3 34<br />
VOLTAGE SOURCE PWM INVERTER FOR PMSM SUPPLY<br />
3.1 Introduction 34<br />
3.2 Voltage source inverter (VSI) 35<br />
3.3 <strong>Space</strong> vector based pulse width modulation (PWM) methods 46<br />
3.4 Summary 52<br />
Chapter 4 53<br />
CONTROL METHODS OF PM SYNCHRONOUS MOTOR<br />
4.1 Introduction 53<br />
4.2 Field oriented control (FOC) 54<br />
4.3 <strong>Direct</strong> torque control (<strong>DTC</strong>) 57<br />
4.4 Summary 64<br />
Chapter 5 65<br />
DIRECT TORQUE CONTROL WITH SPACE VECTOR MODULATION<br />
(<strong>DTC</strong>-<strong>SVM</strong>)<br />
5.1 Introduction 65<br />
5.2 Cascade structure <strong>of</strong> <strong>DTC</strong>–<strong>SVM</strong> scheme 66<br />
5.2.1 Digital flux control loop 68<br />
5.2.2 Digital torque control loop 82<br />
5.3 Parallel structure <strong>of</strong> <strong>DTC</strong>–<strong>SVM</strong> scheme 91<br />
5.3.1 Digital flux control loop 92<br />
5.3.2 Digital torque control loop 102<br />
5.4 Speed control loop for <strong>DTC</strong>–<strong>SVM</strong> structure control 113<br />
5.5 Summary 122<br />
Chapter 6 121<br />
DIRECT TORQUE CONTROL WITH SPACE VECTOR MODULATION (<strong>DTC</strong>-<br />
<strong>SVM</strong>) OF PMSM DRIVE WITHOUT MOTION SENSOR<br />
6.1 Introduction 121<br />
6.2 Initial rotor position estimation method 123<br />
6.3 Stator flux estimation methods 127<br />
6.3.1 Overview 127<br />
6.3.2 Current model based flux estimator 127<br />
6.3.3 Voltage model based flux estimator <strong>with</strong> ideal integrator 128
Contents<br />
6.3.4 Voltage model based flux estimator <strong>with</strong> low pas filter 129<br />
6.3.5 Improved voltage model based flux estimator 130<br />
6.4 Electromagnetic torque estimation 132<br />
6.5 Rotor speed estimation methods 132<br />
6.5.1 Overview 132<br />
6.5.2 Back electromotive force (BEMF) technique 133<br />
6.5.3 Stator flux based technique 133<br />
6.6 Summary 136<br />
Chapter 7 137<br />
DSP IMPLEMENTATION OF <strong>DTC</strong>-<strong>SVM</strong> CONTROL<br />
7.1 Description <strong>of</strong> the laboratory test-stand 137<br />
7.2 Steady state behaviour 140<br />
7.3 Dynamic behaviour 143<br />
7.3.1 Flux and torque control loop 143<br />
7.3.2 Speed control loop 151<br />
Chapter 8 161<br />
SUMMARY AND CLOSING CONCLUSIONS<br />
Appendices 163<br />
Picture <strong>of</strong> rotor and stator <strong>of</strong> PMSM machine<br />
Basic transformation<br />
Model <strong>of</strong> PM synchronous motor- SABER<br />
Parameters <strong>of</strong> PMSM machine<br />
Parameters <strong>of</strong> voltage source inverter<br />
PI speed controller<br />
PWM technique - overmodulation<br />
List <strong>of</strong> Symbols 170<br />
References 172
Introduction<br />
Chapter 1 INTRODUCTION<br />
Recently, an increased interest in application <strong>of</strong> permanent magnet synchronous motors<br />
(PMSM) in speed controlled drives has been observed. This is stimulated mainly by:<br />
• development <strong>of</strong> modern high switching frequency semiconductor power devices (as<br />
for example IGBT modules <strong>of</strong> 5-th generation),<br />
• new rare earth magnetic materials as samarium-cobalt (Sm-Co) or neodymium-ironboron<br />
(Nd-Fe-B),<br />
• specialized digital signal processor (DSP) for AC drive applications <strong>with</strong> integrated<br />
PWM function, A/D converters as well as processing <strong>of</strong> encoder signals (e.g<br />
ADMC401, TMS320FL24XX, TMS320FL28XX).<br />
Synchronous motors <strong>with</strong> an electrically excited rotor winding have a conventional threephase<br />
stator winding (called armature) and an electrically excited field winding on the rotor,<br />
which carries a DC current. The armature winding is similar to the stator <strong>of</strong> induction motor.<br />
The electrically excited field winding can be replaced by permanent magnet (PM) [1]. The use<br />
<strong>of</strong> permanent magnets has many advantages including the elimination <strong>of</strong> brushes, slip rings,<br />
and rotor copper losses in the field winding. It leads to higher efficiency. Additionally since<br />
the copper and iron losses are concentrated in the stator, cooling <strong>of</strong> machines through the<br />
stator is more effective. The lack <strong>of</strong> field winding and higher efficiency results in reduction <strong>of</strong><br />
the machine frame size and higher power/weight ratio.<br />
Figure. 1.1. General classification <strong>of</strong> AC synchronous motors.<br />
1
Introduction<br />
Generally, the permanent magnet AC machines can be classified into two types (Fig.1.1):<br />
trapezoidal type called “brushless DC machine” (BLDCM) and sinusoidal type called<br />
permanent magnet synchronous machine (PMSM). The BLDC machines operate <strong>with</strong><br />
trapezoidal back electromagnetic force (EMF) and require rectangular stator phase current.<br />
The PMSM’s generate sinusoidal EMF and operate <strong>with</strong> sinusoidal stator phase current.<br />
The PMSM can be further divided into two main groups in respect how the magnet bars have<br />
mounted in the rotor [6,7]. In the first group magnets are mounted in the rotor (Fig. 1.2 c-d)<br />
and this type is called interior permanent magnet synchronous motors (IPMSM). The second<br />
group is represented by surface permanent magnet synchronous motors (SPMSM). In the<br />
SPMSM magnet bars are mounted on the rotor surface (Fig. 1.2 a-b).<br />
q<br />
q<br />
SPMSM<br />
S N<br />
S N d<br />
a )<br />
b)<br />
S<br />
N<br />
S<br />
N<br />
d<br />
q<br />
q<br />
IPMSM<br />
S<br />
N<br />
c )<br />
d)<br />
S<br />
N<br />
d<br />
d<br />
Fig. 1.2. The cross section <strong>of</strong> the PMSM rotor shaft and the magnet bars placements:<br />
a),b),c) axial field direction, d) radial field direction.<br />
The magnets can be placed in many ways on the rotor (Fig. 1.2). In radial field fashion the<br />
magnet bars are along the radius <strong>of</strong> the machine and this arrangement provides the highest air<br />
gap flux density, but it has the drawback <strong>of</strong> lower structural integrity and mechanical<br />
robustness. Machines <strong>with</strong> this arrangement <strong>of</strong> magnets are not preferred for high-speed<br />
applications (higher than 3000 rpm). In axial field manner the magnets are placed parallel to<br />
the rotor shaft. This arrangement <strong>of</strong> magnets is much more robust mechanically as compared<br />
2
Introduction<br />
to surface-mounted machine. It makes possible to use IPMSM for higher-speed applications<br />
(contrary to SPMSM’s).<br />
Regardless <strong>of</strong> the fashion <strong>of</strong> mounting the PM, the basic principle <strong>of</strong> motor control is the same<br />
and the differences are only in particularities. An important consequence <strong>of</strong> the method <strong>of</strong><br />
mounting the rotor magnets is the difference in direct and quadrature axes inductance values.<br />
The direct axis reluctance is greater than the quadrature axis reluctance, because the effective<br />
air gap <strong>of</strong> the direct axis is multiple times that <strong>of</strong> the actual air gap seen by the quadrature<br />
axis. As consequence <strong>of</strong> such an unequal reluctance, the quadrature inductance is higher than<br />
direct inductance L q<br />
> L d<br />
. It produces reluctance torque in addition to the mutual torque.<br />
Reluctance torque is produced due to the magnet saliency in the quadrature and the direct axis<br />
magnetic paths. Mutual torque is produced due to the interaction <strong>of</strong> the magnet field and the<br />
stator current. In case where the magnets bars are mounted on the rotor surface the quadrature<br />
inductance is equal direct inductance L q<br />
= L d<br />
, because <strong>of</strong> the same flux paths in d and q axis.<br />
As result the reluctance torque disappears.<br />
Among the main advantage <strong>of</strong> PM machines are [12]:<br />
• high air gap flux density,<br />
• higher power/weight ratio,<br />
• large torque/inertia ratio,<br />
• small torque ripples,<br />
• high speed operation,<br />
• high torque capability (quick acceleration and deceleration),<br />
• high efficiency and high cosφ (low expense for the power supply),<br />
• compact design.<br />
Thanks to this advantages the PMSM’s are usually used in high performance servo drives, in<br />
special applications as computer peripheral equipment, robotics, ect. However, recently the<br />
PMSM are also used as adjustable–speed drives in variety <strong>of</strong> application such as fans, pumps,<br />
compressors, blowers. Another area is automotive application as an alternative drive in hybrid<br />
mode <strong>with</strong> classical engine. The power <strong>of</strong> <strong>of</strong>fered synchronous motors is in the range several<br />
kW to MW.<br />
3
Introduction<br />
The main requirements for high performance PWM inverter-fed PMSM drive can be<br />
formulated as follows:<br />
• operation <strong>with</strong> and <strong>with</strong>out mechanical motion sensor,<br />
• fast flux and torque response,<br />
• available maximum output torque in wide range <strong>of</strong> speed operation region,<br />
• constant switching frequency,<br />
• uni-polar voltage PWM,<br />
• low flux and torque ripples,<br />
• robustness to parameters variation,<br />
• four quadrant operation.<br />
To meet the above requirements, different control methods can be used [3,4,10].<br />
Variable<br />
Frequency<br />
<strong>Control</strong><br />
Scalar based<br />
controllers<br />
<strong>Vector</strong> based<br />
controllers<br />
V/Hz=const<br />
<strong>with</strong> stabilization<br />
loop<br />
Field<br />
Oriented<br />
(FOC)<br />
<strong>Direct</strong> <strong>Torque</strong><br />
<strong>Control</strong><br />
(<strong>DTC</strong>)<br />
PM (rotor)<br />
Flux Oriented<br />
(RFOC)<br />
Stator Flux<br />
Oriented<br />
(SFOC)<br />
<strong>Direct</strong> <strong>Torque</strong><br />
<strong>Control</strong> <strong>with</strong> <strong>Space</strong><br />
<strong>Vector</strong> <strong>Modulation</strong><br />
(<strong>DTC</strong>-<strong>SVM</strong>)<br />
Circular flux<br />
trajectory<br />
(Takahashi)<br />
Figure 1.3 Classification <strong>of</strong> PMSM control methods.<br />
The general classification <strong>of</strong> the variable frequency control for PMSM is presented in Fig. 1.3.<br />
The PMSM control methods can be divided into scalar and vector control. According to [3],<br />
in scalar control, which based on a relation valid for steady states, only the magnitude and<br />
frequency (angular speed) <strong>of</strong> voltage, currents, and flux linkage space vectors are controlled.<br />
Thus, the control system does not act on space vector position during transient. Therefore, this<br />
control is dedicated for application, where high dynamics is not demanded. Contrary, in<br />
4
Introduction<br />
vector control, which is based on relation valid for dynamics states, not just magnitude and<br />
frequency (angular speed), but also instantaneous position <strong>of</strong> voltage, current and flux space<br />
vectors are controlled. Thus, the control system adjust the position <strong>of</strong> the space vectors and<br />
guarantee their correct orientation for both steady states and transients.<br />
The scalar constant V/Hz control for PMSM <strong>with</strong>out damper winding (squire cage) is not<br />
simple as for induction motor. It requires additional stabilization control loop, which can be<br />
provide by feedback from: rotor velocity perturbation, active power or DC-link current<br />
perturbation [9].<br />
The most popular vector control method developed in 70s, known as field oriented control<br />
(FOC) [31] gives the permanent magnet synchronous motor high performance. In this method<br />
the motor equation are transformed in a coordinate system that rotates in synchronism <strong>with</strong><br />
permanent magnet flux. It allows separately and indirectly control flux and torque quantities<br />
by using current control loop <strong>with</strong> PI controllers like in well known DC machine control [3].<br />
In search <strong>of</strong> a simpler and more robust high performance control system in 80s new vector<br />
control called direct torque control (<strong>DTC</strong>) was developed [50]. It was innovative studies at<br />
this time and completely different approach which depart from the idea <strong>of</strong> coordinate<br />
transformation and the analogy <strong>with</strong> DC motor control. It allows direct control flux and torque<br />
quantities <strong>with</strong>out inner current control loops. Using bang-bang hysteresis controllers for flux<br />
and torque control loops made this control concept very fast and not complicated. However,<br />
the main disadvantage <strong>of</strong> <strong>DTC</strong> is fast sampling time required and variable switching<br />
frequency, because <strong>of</strong> hysteresis based control loops. In order to eliminate above<br />
disadvantages and kept basic control rules <strong>of</strong> classical <strong>DTC</strong>, at the beginning <strong>of</strong> 90’s a new<br />
developed control technique called direct torque control <strong>with</strong> space vector modulator (<strong>DTC</strong>-<br />
<strong>SVM</strong>) has been introduced [54,55]. However, from the formal consideration this method can<br />
also be viewed as stator flux oriented control (SFOC). This control employed instead <strong>of</strong><br />
hysteresis controller as for classical <strong>DTC</strong>, the PI controllers and space vector modulator<br />
(<strong>SVM</strong>). It allows to achieve fixed switching frequency, what considerably reduce switching<br />
losses as well as torque and current ripples. Also requirement <strong>of</strong> very fast sampling time is<br />
eliminated [113,115,117]. Therefore, this new method is subject <strong>of</strong> this thesis. In spite <strong>of</strong><br />
many control strategies there is no one which may be considered as standard solution.<br />
5
Introduction<br />
Therefore, the following thesis can be formulated:<br />
“In the view <strong>of</strong> commercial manufacturing process the most convenient control scheme<br />
for voltage source inverter-fed permanent magnet synchronous motor (PMSM) drives is<br />
direct torque control <strong>with</strong> space vector modulator (<strong>DTC</strong>-<strong>SVM</strong>)”.<br />
To prove the above thesis, the author used methodology based on an analyze and simulation<br />
as well as experimental verification on the laboratory setup <strong>with</strong> 3kW PMSM motor.<br />
Moreover, the presented control algorithm <strong>DTC</strong>-<strong>SVM</strong> has been introduced and used in serial<br />
commercial product <strong>of</strong> Polish manufacture TWERD, Toruń.<br />
In the author’s opinion the following results <strong>of</strong> the thesis are his original achievements:<br />
• development <strong>of</strong> a simulation algorithm in SABER package for the investigation<br />
<strong>of</strong> PWM inverter-fed PMSM control,<br />
• elaboration and experimental verification <strong>of</strong> digital flux and torque controller<br />
design based on the Z-transform approach for series (cascade) and parallel<br />
structure <strong>of</strong> <strong>DTC</strong>-<strong>SVM</strong> schemes,<br />
• implementation and verification <strong>of</strong> series (cascade) and parallel <strong>DTC</strong>-<strong>SVM</strong><br />
schemes on experimental laboratory setup <strong>with</strong> 3kW PM synchronous motor<br />
drive controlled by floating point DS1103 board.<br />
• bringing into production and testing <strong>of</strong> developed <strong>DTC</strong>-<strong>SVM</strong> algorithm in Polish<br />
industry.<br />
The thesis consists <strong>of</strong> eight chapters. Chapter 1 is an introduction. In Chapter 2 mathematical<br />
model <strong>of</strong> PM synchronous motor and his basic control modes are presented. Chapter 3 is<br />
devoted to voltage source inverter, his nonlinear characteristics and different PWM<br />
techniques. Chapter 4 gives brief review <strong>of</strong> PM synchronous motor control method such as<br />
FOC and classical <strong>DTC</strong>. In Chapter 5 two kind <strong>of</strong> <strong>DTC</strong>-<strong>SVM</strong> control schemes are presented.<br />
Also, the analysis and synthesis <strong>of</strong> digital flux, torque and speed controllers based on Z<br />
transform approach are given. Chapter 6 is devoted to initial rotor detection methods, stator<br />
flux vector and rotor speed estimation algorithms. In Chapter 7 experimental results are<br />
6
Introduction<br />
presented and studied. Chapter 8 includes the finally conclusions. Description <strong>of</strong> the SABER<br />
based control algorithm, basic coordinate transformations and parameters <strong>of</strong> used PM<br />
synchronous machine as well as inverter are given in Appendices.<br />
7
Modeling and control modes <strong>of</strong> PM synchronous motor drives<br />
Chapter 2<br />
MODELING AND CONTROL MODES OF PM SYNCHRONOUS DRIVES<br />
2.1 Mathematical model <strong>of</strong> PM synchronous motor<br />
Development <strong>of</strong> the machine model through the understanding <strong>of</strong> physics <strong>of</strong> the<br />
machine is the key requirement for any type <strong>of</strong> electrical machine control. Since in this<br />
project a Surface type Permanent Magnet Synchronous Motor (SPMSM) is used for the<br />
investigation [9,13,14,15,16]. The development <strong>of</strong> those models is under bellow<br />
assumptions as [3]:<br />
• three-phase motor is symmetrical,<br />
• only a fundamental harmonic <strong>of</strong> the magneto motive force (MMF) is taking in to<br />
account,<br />
• the spatially distributed stator and rotor winding are replaced by a concentrated<br />
coil,<br />
• an anisotropy effects, magnetic saturation, iron loses and eddy currents are not<br />
taking into considerations,<br />
• the coil resistances and reactances are taking to be constant,<br />
• in many cases, especially when is considered steady state, the currents and<br />
voltages are assumed to be sinusoidal,<br />
• thermal effect for permanent magnets is omitted.<br />
The synchronous motor model will be presented in space vector notation. <strong>Space</strong> vector<br />
form <strong>of</strong> the machine equations has many advantages such as compact notation, easy<br />
algebraic manipulation, and very simple graphical interpretation. Specially, this notation<br />
is very useful when analyzing the vector control based technique <strong>of</strong> the AC machines.<br />
The space vector representation <strong>of</strong> AC machine equations has been discussed in detail<br />
in number <strong>of</strong> text books ([3,4,12]).<br />
The instantaneous value <strong>of</strong> a three-phase system KA, KB,<br />
K<br />
C<br />
(such as currents, voltages<br />
and flux linkages) can be replaced by one resultant vector called the space vector,<br />
2<br />
K = ⎡ 1 ⋅ K + a⋅ K + a K<br />
3 ⎣<br />
2<br />
A B C<br />
⎤<br />
⎦<br />
(2.1)<br />
8
Modeling and control modes <strong>of</strong> PM synchronous motor drives<br />
2π<br />
2π<br />
4π<br />
j 1 3<br />
where:1,<br />
3 2<br />
− j j 1 3<br />
a= e =− + j , a = e<br />
3<br />
= e<br />
3<br />
=− − j - complex vectors, 2/3 –<br />
2 2<br />
2 2<br />
normalization factor (guarantee that for balanced sinusoidal waveforms the magnitude<br />
<strong>of</strong> the space vector is equal to the amplitude <strong>of</strong> that phase waveforms).<br />
The elements <strong>of</strong> this space vector satisfy the condition:<br />
KA + KB + KC<br />
= 0<br />
(2.2)<br />
and it means that we have three-phase system <strong>with</strong>out neutral wire.<br />
2.1.1 Voltage and current equations<br />
For idealized motor (Fig. 2.1), the following equations <strong>of</strong> the instantaneous stator phase<br />
voltages can be written [3]:<br />
B<br />
b<br />
Z sB<br />
I sB<br />
a<br />
U sB<br />
S<br />
N<br />
N<br />
S<br />
U sA<br />
γ m<br />
Z sA<br />
A<br />
I sA<br />
I sC<br />
Z sC<br />
U sC<br />
N<br />
S<br />
C<br />
c<br />
Figure 2.1. Layout and symbols for three-phase PMSM electric motor windings.<br />
dΨ<br />
dt<br />
sA<br />
sA<br />
=<br />
sA sA<br />
+ (2.3a)<br />
U I R<br />
U I R<br />
dΨ<br />
dt<br />
sB<br />
sB<br />
=<br />
sB sB<br />
+ (2.3b)<br />
U I R<br />
dΨ<br />
dt<br />
sC<br />
sC<br />
=<br />
sC sC<br />
+ (2.3c)<br />
9
Modeling and control modes <strong>of</strong> PM synchronous motor drives<br />
where<br />
U ,<br />
sA, U<br />
sB<br />
U<br />
sC<br />
are the instantaneous stator voltage values, I<br />
sA<br />
I<br />
sB<br />
, I<br />
sC<br />
, are<br />
instantaneous values <strong>of</strong> the current,<br />
R = R = R = R is the resistance <strong>of</strong> the stator<br />
s<br />
sA<br />
sB<br />
sC<br />
windings, and ΨsA,<br />
Ψ<br />
sB<br />
and Ψ<br />
sC<br />
are magnetic flux linkages stator windings A, B and<br />
C , respectively.<br />
Using the space vector theory to voltage equations we can written in vector form<br />
where:<br />
U<br />
d Ψ<br />
dt<br />
sABC<br />
sABC<br />
= Rs<br />
I<br />
sABC<br />
+ (2.4)<br />
2 2<br />
U<br />
sABC<br />
= (1 UsA + aUsB + a UsC<br />
) ,<br />
2 2<br />
2 2<br />
I<br />
sABC<br />
= (1 IsA + aIsB + a IsC<br />
) ,<br />
sABC (1<br />
sA<br />
a<br />
sB<br />
a<br />
sC )<br />
3<br />
3<br />
3<br />
stator voltage, current and flux space vectors, respectively.<br />
Ψ = Ψ + Ψ + Ψ are the<br />
The stator winding flux consist <strong>of</strong> rotor flux and stator flux linkages:<br />
where,<br />
Ψ<br />
sABC<br />
=Ψ<br />
ABC ( s) +Ψ<br />
ABC( r)<br />
(2.5)<br />
⎡ LsA MsAB MsAC⎤⎡IsA⎤<br />
⎢<br />
M L M<br />
⎥⎢<br />
I<br />
⎥<br />
Ψ<br />
ABC( s)<br />
= ⎢ sBA sB sBC ⎥⎢ sB ⎥<br />
⎢⎣ M<br />
sCA<br />
MsCB L ⎥⎢<br />
sC ⎦⎣I<br />
⎥<br />
sC ⎦<br />
(2.6)<br />
⎡<br />
⎤<br />
⎢ cosθ<br />
⎥<br />
r<br />
⎢<br />
⎥<br />
2π<br />
Ψ<br />
ABC ( r)<br />
=Ψ ⎢<br />
PM<br />
cos( θr<br />
− ) ⎥<br />
⎢ 3 ⎥<br />
⎢<br />
2π<br />
⎥<br />
cos( θr<br />
+ )<br />
⎢⎣<br />
3 ⎥⎦<br />
(2.7)<br />
and, θr<br />
is electrical rotor position. Mechanical rotor position is defined as:<br />
θ = p γ<br />
(2.8)<br />
r b m<br />
where: pb<br />
- number <strong>of</strong> pole pairs, γ<br />
m<br />
- mechanical position.<br />
In equation (2.6) LsA<br />
is the self-inductance <strong>of</strong> phase A winding, M<br />
sAB<br />
and M<br />
sAC<br />
are the<br />
mutual inductances between A and B phase, A and C phase, respectively. For self and<br />
mutual inductances <strong>of</strong> B and C phase the same notations used. In (2.7),<br />
Ψ<br />
PM<br />
is the<br />
10
Modeling and control modes <strong>of</strong> PM synchronous motor drives<br />
amplitude <strong>of</strong> the flux linkages established by the permanent magnet on the rotor. The<br />
inductances are described below.<br />
Due to the rotor saliency in IPMSM the air gap is not uniform and, therefore, the self<br />
and mutual inductances <strong>of</strong> stator windings are a function <strong>of</strong> the rotor position.<br />
The derivation <strong>of</strong> these rotor position dependent inductances is available in details in<br />
[5]. The results are summarized here as follows:<br />
The stator winding self-inductances are<br />
L = L + L − L cos 2θ<br />
(2.9a)<br />
sA ls A B r<br />
2π<br />
4π<br />
LsB = Lls + LA −LB cos2( θr − ) = Lls + LA −LB cos(2 θr<br />
− )<br />
3 3<br />
(2.9b)<br />
2π<br />
4π<br />
LsC = Lls + LA − LB cos 2( θr + ) = Lls + LA − LB cos(2 θr<br />
+ )<br />
3 3<br />
(2.9c)<br />
where, Lls<br />
is stator-winding leakage inductance and LA,<br />
LB<br />
are given by<br />
L<br />
A<br />
⎛ms<br />
⎞<br />
= ⎜<br />
2 ⎟<br />
⎝ ⎠<br />
2<br />
πµ rlε<br />
0 1<br />
(2.10a)<br />
L<br />
B<br />
1 ⎛ms<br />
⎞<br />
=<br />
2 ⎜<br />
2 ⎟<br />
⎝ ⎠<br />
2<br />
πµ rlε<br />
0 2<br />
(2.10b)<br />
where,<br />
m<br />
s<br />
is number <strong>of</strong> turns <strong>of</strong> each phase winding, r is radius, which is from center<br />
<strong>of</strong> machine to the inside circumference <strong>of</strong> the stator, and l is the axial length <strong>of</strong> the air<br />
gap <strong>of</strong> the machine, µ<br />
0<br />
is permeability <strong>of</strong> the air, ε<br />
1<br />
and ε<br />
2<br />
are defined as s:<br />
1 1 1<br />
ε<br />
1<br />
= ( + )<br />
(2.11a)<br />
2 g g<br />
min<br />
max<br />
1 1 1<br />
ε<br />
2<br />
= ( − )<br />
(2.11b)<br />
2 g g<br />
min<br />
max<br />
where,<br />
gmin<br />
is minimum air gap length and<br />
max<br />
g<br />
is maximum air gap length.<br />
The mutual inductances between stator phase are:<br />
1 π 1 2π<br />
MsAB = MsBA =− LA −LB cos 2( θr − ) =− LA −LB cos(2 θr<br />
− ) (2.12a)<br />
2 3 2 3<br />
11
Modeling and control modes <strong>of</strong> PM synchronous motor drives<br />
1 π 1 2π<br />
MsAC = MsCA =− LA − LB cos 2( θr + ) =− LA − LB cos(2 θr<br />
+ ) (2.12b)<br />
2 3 2 3<br />
1 1<br />
MsBC = MsCB =− LA − LB cos 2( θr + π) =− LA − LB cos(2θr<br />
+ 2 π)<br />
2 2<br />
1<br />
=− LA −LBcos 2θ<br />
r<br />
2<br />
(2.12c)<br />
Using the space vector theory, the flux linkage<br />
Ψ<br />
sABC<br />
space vector can be written as:<br />
3 3<br />
∗ j2θr<br />
jθr<br />
Ψ<br />
sABC<br />
= ( Lls + LA)<br />
IsABC − LB IsABC<br />
e +Ψ<br />
PM<br />
e<br />
(2.13)<br />
2 2<br />
where,<br />
2 2<br />
(1<br />
sA sB sC )<br />
IsABC<br />
= I + aI + a I ,<br />
3<br />
space vector and conjugate stator current space vector.<br />
2 2 (1<br />
sA sB sC )<br />
I ∗ sABC<br />
= I + a I + aI are the stator current<br />
3<br />
Taking into account that:<br />
Ld = Lls + Lmd<br />
(2.14a)<br />
Lq = Lls + Lmq<br />
(2.14b)<br />
3 3<br />
where, Lmd = ( LA + LB<br />
) , Lmq = ( LA − LB<br />
) are d and q magnetizing inductances and<br />
2<br />
2<br />
are defined as [5].<br />
Finally, equations (2.13) comes as:<br />
where, L<br />
d<br />
, L<br />
q<br />
are d and q inductances.<br />
Ld + Lq Lq −Ld ∗ j2θr<br />
jθ<br />
r<br />
Ψ<br />
sABC<br />
= ( ) I<br />
sABC<br />
− ( ) IsABC<br />
e +Ψ<br />
PM<br />
e<br />
(2.15)<br />
2 2<br />
<strong>Space</strong> vector form <strong>of</strong> machine equations (2.4, 2.15) becomes more compact, but the<br />
rotor position dependent parameters still exist in that form <strong>of</strong> expressions for the stator<br />
flux linkage space vector. Therefore, the space vector model is still not simple to use for<br />
the analysis. A simplification can be made if the space vector model is referred to a<br />
suitably selected rotating frame.<br />
12
Modeling and control modes <strong>of</strong> PM synchronous motor drives<br />
Figure 2.2 shows axes <strong>of</strong> reference for the three-stator phase ABC. , , It also shows a<br />
rotating set <strong>of</strong> x,<br />
y axes, where the angleθ K<br />
is position <strong>of</strong> x -axis in respect to the stator<br />
A phase axis. Variables along the AB , and C axes can be referred to the x − and<br />
y − axes by the expression:<br />
⎡K<br />
A ⎤<br />
⎡K<br />
x ⎤ 2 ⎡ cosθK cos( θK − 2 π / 3) cos( θK<br />
+ 2 π / 3) ⎤⎢<br />
K<br />
⎥<br />
⎢ B<br />
K<br />
⎥ =<br />
y 3<br />
⎢<br />
sinθK sin( θK 2 π / 3) sin( θK<br />
2 π / 3)<br />
⎥⎢ ⎥<br />
⎣ ⎦ ⎣− − − − + ⎦<br />
⎢⎣K<br />
⎥<br />
C ⎦<br />
(2.16)<br />
y<br />
K B<br />
K ABC<br />
K y<br />
K x<br />
Ω K<br />
x<br />
θ K<br />
K A<br />
K C<br />
Figure 2.2. Stator fixed three phase axes (A,B,C) and general rotating reference frame ( x,<br />
y ).<br />
Finally, the space vector in general rotating frame can be written as:<br />
j K<br />
K = K (cosΘ + jsin Θ ) = K e θ<br />
(2.17)<br />
ABCs K K<br />
K K<br />
In this case the voltage equation (2.4) using (2.17) can written as:<br />
jθK jθ d<br />
K jθK<br />
U<br />
sKe Rs IsKe (<br />
sKe<br />
)<br />
dt<br />
= + Ψ (2.18)<br />
Using chain rule, equation. (2.17) and divided by term<br />
j K<br />
e θ<br />
can be written as:<br />
d Ψ<br />
U<br />
sK<br />
R I j<br />
dt<br />
= + + Ω Ψ (2.19)<br />
sK<br />
s sK K sK<br />
where<br />
U<br />
sK<br />
rotating frame.<br />
, I sK , Ψ sK is the stator voltage, current and flux space vector in general<br />
Making similar arrangement like for the voltage equation the flux linkage vector in<br />
general reference frame can be expressed as:<br />
13
Modeling and control modes <strong>of</strong> PM synchronous motor drives<br />
Ld + Lq Lq −Ld j2( θr θK) j( θr θK)<br />
Ψ<br />
sK<br />
= ( ) IsK − ( ) I ∗ sK<br />
e − +Ψ<br />
PM<br />
e<br />
− (2.20)<br />
2 2<br />
Stator fixed system ( α,<br />
β )<br />
Taking the angular speed <strong>of</strong> the reference frame to be Ω<br />
K<br />
= 0 and θ<br />
K<br />
= 0 , the set <strong>of</strong><br />
synchronous machine vector equations (2.19) and (2.20) my be written as:<br />
U<br />
d Ψ<br />
sαβ<br />
sαβ<br />
= Rs<br />
Isαβ<br />
+ (2.21)<br />
dt<br />
Ld + Lq Lq −Ld ∗ j2θr<br />
jθ<br />
r<br />
Ψ<br />
sαβ = ( ) I<br />
sαβ − ( ) Isαβ<br />
e +Ψ<br />
PM<br />
e<br />
(2.22)<br />
2 2<br />
Substituting to above equations the following expressions for complex vectors<br />
U U jU<br />
= s sα<br />
+ , αβ sβ<br />
s sα<br />
sβ<br />
I αβ = I + jI , Ψ αβ =Ψ<br />
sα<br />
+ jΨ sβ<br />
and splitting into real and<br />
s<br />
imaginary parts one can obtain the scalar form <strong>of</strong> the machine equations in stationary<br />
α,<br />
β reference frame:<br />
U<br />
U<br />
dΨ<br />
dt<br />
sα<br />
sα<br />
= RsIsα<br />
+ (2.23a)<br />
dΨ<br />
sβ<br />
sβ<br />
= RsIsβ<br />
+ (2.23b)<br />
dt<br />
Ld + Lq Lq −Ld Lq −Ld<br />
( cos2 θ ) Is<br />
( )sin2θ Is<br />
cosθ<br />
2 2 2<br />
Ψ = − − +Ψ (2.24a)<br />
sα r α r β PM r<br />
Lq − Ld Ld + Lq Lq −Ld<br />
Ψ<br />
sβ =− ( )sin 2 θrIsα + [( ) + ( )cos2 θr] Isβ<br />
+Ψ<br />
PM<br />
sinθ<br />
(2.24b)<br />
r<br />
2 2 2<br />
Note, that in the flux-current equations (2.24a and b) still we can observe that value <strong>of</strong><br />
inductances depends on rotor position θ<br />
r<br />
.<br />
Stator flux fixed system ( x,<br />
y )<br />
In order to take advantage <strong>of</strong> the set <strong>of</strong> equations (2.19) and (2.20) in rotating coordinate<br />
system, one assumes that the coordinate system rotates <strong>with</strong> the stator flux linkage<br />
angular speed Ω<br />
K<br />
=Ω<br />
Ψ<br />
and θK<br />
= θ Ψ<br />
. As a<br />
sx s<br />
Ψ =Ψ , δ = −( θ − θ )<br />
Ψ<br />
r<br />
Ψ<br />
d Ψ<br />
U R I j<br />
dt<br />
sxy<br />
sxy<br />
=<br />
s sxy<br />
+ + ΩΨsΨ (2.25)<br />
sxy<br />
Ld + Lq Lq −Ld ∗ − j2δΨ<br />
− jδΨ<br />
Ψ<br />
sxy<br />
= ( ) Isxy − ( ) Isxy<br />
e +Ψ<br />
PM<br />
e<br />
(2.26)<br />
2 2<br />
14
Modeling and control modes <strong>of</strong> PM synchronous motor drives<br />
Substituting to above equations the following expressions for complex vectors<br />
U U jU<br />
sxy<br />
=<br />
sx<br />
+<br />
sy, Isxy<br />
Isx jIsy<br />
= + , Ψ =Ψ<br />
s<br />
and splitting into real and imaginary parts<br />
sxy<br />
one can obtain the scalar form <strong>of</strong> the machine equations in stationary x,<br />
y reference<br />
frame:<br />
U<br />
dΨ<br />
dt<br />
s<br />
sx<br />
= Rs Isx<br />
+ (2.27a)<br />
U = R I +ΩΨ<br />
Ψ (2.27b)<br />
sy s sy s<br />
s<br />
1 1<br />
Ψ<br />
s<br />
= [( Ld + Lq) −( Lq − Ld)cos 2 δΨ ] Isx<br />
+ ( Lq − Ld)sin 2δΨIsy<br />
+Ψ<br />
PM<br />
cosδΨ<br />
2 2<br />
(2.28a)<br />
1 1<br />
0 = ( Lq − Ld)sin2 δΨIsx<br />
+ [( Ld + Lq) + ( Lq −Ld)cos2 δΨ] Isy<br />
−Ψ<br />
PM<br />
sinδΨ<br />
2 2<br />
(2.28b)<br />
The current-flux equations can be expressed also in simplest form as:<br />
2 2<br />
s<br />
( Ld cos δΨ Lqsin δΨ) Isx<br />
( Lq Ld)sinδΨcosnδΨIsy<br />
PM<br />
cosδΨ<br />
Ψ = + + − +Ψ (2.29a)<br />
2 2<br />
q d<br />
δΨ δΨ sx d<br />
δΨ q<br />
δΨ sy PM<br />
δΨ<br />
0 ( L L )sin cos I ( L sin L cos ) I sin<br />
= − + + −Ψ (2.29b)<br />
Rotor flux fixed system ( dq) ,<br />
In order to take advantage <strong>of</strong> the set <strong>of</strong> equations (2.19) and (2.20) in rotating coordinate<br />
system, one assumes that the coordinate system rotates <strong>with</strong> the rotor flux angular speed<br />
Ω<br />
K<br />
= pbΩ m<br />
and θK = pbγm = θr<br />
d Ψ<br />
U<br />
sdq<br />
R I jp<br />
dt<br />
sdq<br />
s sdq<br />
b m<br />
= + + Ω Ψ (2.30)<br />
sdq<br />
Ld + Lq Lq −Ld<br />
∗<br />
Ψ<br />
sdq<br />
= ( ) Isdq − ( ) Isdq<br />
+Ψ<br />
PM<br />
(2.31)<br />
2 2<br />
Substituting the following expressions for complex vectors U = Usd + jUsq,<br />
I I jI<br />
sdq<br />
=<br />
sd<br />
+<br />
sq, sdq sd<br />
j<br />
sq<br />
Ψ =Ψ + Ψ to (2.30) and (2.31), and splitting for real and<br />
imaginary parts the scalar form <strong>of</strong> the machine equations in rotational fixed reference<br />
frame can be obtained:<br />
dΨ<br />
sd<br />
Usd = Rs Isd + − pbΩmΨ sq<br />
(2.32a)<br />
dt<br />
sdq<br />
15
Modeling and control modes <strong>of</strong> PM synchronous motor drives<br />
dΨsq<br />
Usq = Rs Isq + + pbΩmΨ sd<br />
(2.32b)<br />
dt<br />
where,<br />
Ψ<br />
sd<br />
= LI<br />
d sd<br />
+Ψ (2.33a)<br />
PM<br />
Ψ<br />
sq<br />
= LqI<br />
(2.33b)<br />
sq<br />
It should be noted that when transforming the flux linkage vector<br />
Ψ s<br />
to the dq ,<br />
reference frame the rotor position θ r<br />
dependent terms disappear it can be seen from<br />
equation (2.31). This is the main advantage <strong>of</strong> rotor-oriented representation.<br />
Substituting the relationship <strong>of</strong> (2.33a-b) into (2.32a-b), and also considering<br />
dΨ PM = 0 , the most common scalar form <strong>of</strong> the machine voltage equations in the rotor<br />
dt<br />
reference frame can be obtained as:<br />
dIsd<br />
Usd = RsIsd + Ld − pbΩ mLqIsq<br />
(2.34a)<br />
dt<br />
dIsq<br />
Usq = Rs Isq + Lq + pbΩmΨ PM<br />
+ pbΩ mLd Isd<br />
(2.34b)<br />
dt<br />
Based on the above voltage-current equations it is possible to draw the equivalent<br />
electrical circuit separately for d and q axes (Fig. 2.3).<br />
R s<br />
p Ω<br />
L I<br />
b m q sq<br />
R s<br />
p Ω<br />
L I<br />
b m d sd<br />
U<br />
sd<br />
I sd<br />
L d<br />
U<br />
sq<br />
I sq<br />
p Ω Ψ<br />
b m PM<br />
L q<br />
Figure 2.3. Equivalent circuit model <strong>of</strong> PMSM in the rotor reference frame. (a) Rotor d-axis<br />
equivalent circuit, (b) Rotor q-axis equivalent circuit.<br />
16
Modeling and control modes <strong>of</strong> PM synchronous motor drives<br />
2.1.2 Instantaneous power and electromagnetic torque<br />
The three-phase star-connection system <strong>with</strong>out neutral wire is shown in Fig. 2.4. This<br />
is classical configuration for AC motor windings connections.<br />
A<br />
I sA<br />
U sAC<br />
U sAB<br />
B<br />
U sA<br />
Z sC<br />
Z sA<br />
U sAB<br />
Z sB<br />
C<br />
U sBC<br />
I sC<br />
U sC U sB<br />
I sB<br />
Figure 2.4. Three-phase star connection system <strong>with</strong>out neutral wire.<br />
For this configuration the expression for instantaneous active power supplied to load<br />
can be expressed as:<br />
P= UsAIsA+ UsBIsB+ UsCIsC<br />
(2.35)<br />
Introducing space vector definition, after some arrangement and taking into account the<br />
relation: I + I + I = 0, the equation (2.35) can be written as:<br />
sA sB sC<br />
3 ∗<br />
P= Re[ U<br />
sABC<br />
IsABC<br />
]<br />
(2.36)<br />
2<br />
For dq , frame, the equation (2.35) for the active power can be expressed as:<br />
3<br />
P= ( UsdIsd + UsqIsq<br />
)<br />
(2.37)<br />
2<br />
Substituting voltage equation (2.4) into (2.36), and adopting Ω<br />
K<br />
= pbΩmone obtains<br />
3 ∗ d Ψ<br />
[Re(<br />
sABC ∗ ∗<br />
P= RsIsABC IsABC + IsABC − jpbΩmΨ sABCIsABC<br />
)] (2.38)<br />
2<br />
dt<br />
Note that<br />
sABC<br />
sABC<br />
2<br />
s<br />
I I ∗ = I and:<br />
3 2 d Ψ<br />
[ Re(<br />
sABC ∗<br />
∗<br />
P= Rs Is + IsABC ) + Re( −jpbΩmΨ sABC<br />
IsABC<br />
)] (2.39)<br />
2<br />
dt<br />
d ΨsABC<br />
Hence, neglecting the losses in stator resistance R<br />
s<br />
and assuming that = 0 , the<br />
dt<br />
electromagnetic power is expressed:<br />
17
Modeling and control modes <strong>of</strong> PM synchronous motor drives<br />
3<br />
∗<br />
Pe = pbΩmIm( Ψ<br />
sABC<br />
IsABC<br />
)<br />
(2.40)<br />
2<br />
In dq , frame the active power can be written:<br />
3<br />
Pe = pbΩm( ΨsdIsq −Ψ<br />
sqIsd)<br />
(2.41)<br />
2<br />
For the presented system (Fig. 2.4) the expression for instantaneous reactive power<br />
supplied to the three-phase load system <strong>with</strong>out neutral wire can be calculated as:<br />
1<br />
Q= ( IsAUsBC + IsBUsCA + IsCUsAB<br />
)<br />
(2.42)<br />
3<br />
Introducing the space vector definition into equation (2.42), after some arrangement,<br />
and taking into account the relation: I + I + I = 0, one obtains:<br />
sA sB sC<br />
3 ∗<br />
Q= Im[ U<br />
sABC<br />
IsABC<br />
]<br />
(2.43)<br />
2<br />
In dq , frame the reactive power is expressed as:<br />
3<br />
Q= ( UsqIsd − UsdIsq<br />
)<br />
(2.44)<br />
2<br />
Substituting voltage equation (2.4) into (2.43), adopting Ω<br />
K<br />
= pbΩ m<br />
and made similar<br />
arrangements like for active power calculation, the final expression for reactive power<br />
is:<br />
3<br />
∗<br />
Q= pbΩmRe( Ψ<br />
sABC<br />
I<br />
sABC<br />
)<br />
(2.45)<br />
2<br />
In dq , frame the expression (2.45) for the reactive power becomes:<br />
3<br />
Q= pbΩm ( Ψ<br />
sd<br />
Isd +Ψ<br />
sqIsq<br />
)<br />
(2.46)<br />
2<br />
The important quantity <strong>of</strong> the drive is the power factor cosφ , which can be calculated<br />
as:<br />
Q<br />
cosφ = (2.47)<br />
S<br />
where S is module <strong>of</strong> apparent power vector S = P+ jQ:<br />
2 2<br />
S = P + Q<br />
(2.48)<br />
18
Modeling and control modes <strong>of</strong> PM synchronous motor drives<br />
The instantaneous electromagnetic torque developed by an electric motor can be defined<br />
as:<br />
P<br />
e<br />
M<br />
e<br />
= (2.49)<br />
Ω<br />
m<br />
where, P<br />
e<br />
is the electromagnetic power and<br />
Ωm<br />
is the mechanical angular rotor speed.<br />
Finally, taking into account equation (2.49) the expression for electromagnetic torque<br />
can be obtained as:<br />
and in dqframe: ,<br />
3<br />
∗<br />
Me = pbIm( Ψ<br />
sABC<br />
IsABC<br />
) , (2.50)<br />
2<br />
3<br />
M<br />
e<br />
= pb ( Ψsd Isq −Ψ<br />
sqIsd<br />
)<br />
(2.51)<br />
2<br />
Substituting Ψ , Ψ from (2.33a-b), the torque expression <strong>of</strong> equations (2.47)<br />
becomes:<br />
sd<br />
sq<br />
3<br />
M<br />
e<br />
= pb ( ΨPM Isq −( Lq − Ld ) Isd Isq<br />
)<br />
(2.52)<br />
2<br />
It can be seen from (2.52), that developed torque consist <strong>of</strong> two parts, one produced by<br />
the permanent magnet flux called synchronous torque ( M<br />
reluctance torque ( M<br />
er<br />
es<br />
) and the second called<br />
), which is produced by the difference <strong>of</strong> the inductance in rotor<br />
d- and q-axes. Expressions for those two torque components are:<br />
3<br />
M<br />
es<br />
= pbΨ PM<br />
Isq<br />
(2.53a)<br />
2<br />
3<br />
M<br />
er<br />
=− pb ( Lq − Ld ) Isd Isq<br />
(2.53b)<br />
2<br />
It should be mentioned that for SPMSM ( L d<br />
= L ) the reluctance torque does not exist<br />
q<br />
due to the same inductance paths in rotor d- and q-axes.<br />
The torque expression (2.52) can also be written in polar form using the current vector<br />
amplitude<br />
vector (Fig. 2.5.).<br />
I<br />
s<br />
and the torque angle δ I<br />
, i.e. angle between rotor d-axis and current<br />
19
Modeling and control modes <strong>of</strong> PM synchronous motor drives<br />
q−<br />
axis<br />
I sq<br />
I s<br />
Ω s<br />
δ I<br />
d<br />
− axis<br />
I sd<br />
Ψ PM<br />
Figure 2.5. Stator current vector in rotor reference frame.<br />
For two current components using trigonometrical rules we can write:<br />
I = I cosδ<br />
(2.54a)<br />
sd<br />
s<br />
I<br />
I = I sinδ<br />
(2.54b)<br />
sq<br />
s<br />
I<br />
Substituting I , I into equation (2.52), the torque expression can be obtain as:<br />
sd<br />
sq<br />
M 3 1<br />
2<br />
e = b[ PM sin I ( q d) sin2 I]<br />
2 p Ψ I s δ −<br />
2<br />
L − L I s δ<br />
(2.55)<br />
<br />
M<br />
es<br />
For given current amplitude the synchronous and reluctances torque varies according to<br />
the sine <strong>of</strong> torque angle δ<br />
I<br />
. The variation <strong>of</strong><br />
M<br />
er<br />
M<br />
es<br />
and M<br />
er<br />
and resultant torque M<br />
e<br />
<strong>with</strong><br />
torque angle are illustrated in Fig. 2.6. The IPMSM parameters used for this calculation<br />
are given in the Appendices.<br />
e [ ] M Nm<br />
M [ ] er<br />
Nm M<br />
es<br />
[ Nm ]<br />
[deg] δ I<br />
Figure 2.6. Variation <strong>of</strong> synchronous torque M es<br />
, reluctance torque M<br />
er<br />
and resultant<br />
torque M as a function <strong>of</strong> torque angle (for rated current amplitude).<br />
e<br />
20
Modeling and control modes <strong>of</strong> PM synchronous motor drives<br />
Referring to Fig. 2.7, stator flux components in rotor reference frame can be written as:<br />
where:<br />
Ψ = Ψ cos = LI +Ψ (2.56a)<br />
sd s<br />
δ Ψ d sd PM<br />
Ψ = Ψ sin = LI<br />
(2.56b)<br />
sq s<br />
δ Ψ q sq<br />
Ψ<br />
s<br />
is stator flux linkage amplitude, Ψ<br />
PM<br />
is rotor permanent magnet and δ Ψ<br />
is<br />
torque angle (angle between stator flux linkage vector and rotor permanent magnets flux<br />
vector).<br />
q−<br />
axis<br />
I sq<br />
Ψ sd<br />
Ψ PM<br />
I s<br />
Ω s<br />
Ψ s<br />
Ψ<br />
sq<br />
= LqI<br />
sq<br />
LI<br />
d<br />
sd<br />
δ Ψ<br />
I sd<br />
d<br />
− axis<br />
Ψ<br />
sd<br />
= LI<br />
d sd<br />
+ΨPM<br />
Figure 2.7. Rotor permanent magnet flux vector and stator flux linkage vector in rotor reference<br />
frame.<br />
From (2.56a) and (2.56b) the I<br />
sd<br />
and Isq<br />
can be obtained as:<br />
I<br />
I<br />
sd<br />
sd<br />
Ψs<br />
cosδ Ψ<br />
−ΨPM<br />
= (2.57a)<br />
L<br />
d<br />
Ψs<br />
sinδ = Ψ<br />
(2.57b)<br />
L<br />
q<br />
Substituting current components (2.57a), (2.57b) into equation (2.51), one can obtain<br />
another useful torque expressions:<br />
M<br />
e<br />
3 Ψ sin<br />
2 s<br />
ΨPM<br />
δΨ Ψs<br />
( Lq<br />
−Ld)sin2δ<br />
Ψ<br />
= pb[ −<br />
]<br />
2 Ld 2LdL<br />
<br />
q<br />
<br />
Mes<br />
M<br />
er<br />
(2.58)<br />
where:<br />
Ψ<br />
s<br />
stator flux linkage amplitude, and Ψ<br />
PM<br />
rotor flux, δ Ψ<br />
is torque angle, M<br />
es<br />
-<br />
synchronous torque,<br />
M<br />
er<br />
- reluctance torque.<br />
21
Modeling and control modes <strong>of</strong> PM synchronous motor drives<br />
For the PM synchronous motor the amplitude <strong>of</strong> stator flux<br />
Ψ<br />
s<br />
is established by<br />
permanent magnet. Operation <strong>with</strong> stator flux amplitude belong the nominal value <strong>of</strong><br />
rotor flux amplitude<br />
Ψ<br />
PM<br />
increases the amplitude <strong>of</strong> stator phase current. Please note<br />
that maximum amplitude <strong>of</strong> the stator current vector is calculated as:<br />
higher value may damage the PM (complete demagnetization).<br />
I<br />
s<br />
Ψ<br />
≤<br />
L<br />
PM<br />
d<br />
, and<br />
From the Fig. 2.8 it can be observed that rated torque is achieved for torque angle<br />
0< < 25 <br />
electrical degree.<br />
δ Ψ<br />
e[ ] M Nm<br />
M<br />
er<br />
[ Nm ]<br />
M<br />
es<br />
[ Nm ]<br />
δ Ψ<br />
[deg]<br />
Figure 2.8. Variation <strong>of</strong> synchronous torque M es<br />
, reluctance torque M<br />
er<br />
and resultant<br />
torque M as a function <strong>of</strong> torque angle (for constant stator flux equal value <strong>of</strong> PM).<br />
e<br />
2.1.3 Mechanical motion equation<br />
The equation <strong>of</strong> rotor motion dynamics describes the mechanical equilibrium <strong>of</strong> a drive<br />
system. Taking the moment <strong>of</strong> inertia to be constant ( J = const.<br />
) and neglecting friction<br />
and elastic torque we can write:<br />
where,<br />
M<br />
e<br />
= Ml + Md<br />
(2.59)<br />
Ml<br />
is the external torque on the motor shaft, and<br />
M<br />
dΩ<br />
dt<br />
where: J is total moment <strong>of</strong> inertia,<br />
M<br />
d<br />
the dynamic torque<br />
m<br />
d<br />
= J<br />
(2.60)<br />
Ω<br />
m<br />
angular speed <strong>of</strong> the rotor.<br />
22
Modeling and control modes <strong>of</strong> PM synchronous motor drives<br />
In general, for a drive system,<br />
J = J + J<br />
(2.61)<br />
m<br />
where: J<br />
m<br />
- motor inertia, J<br />
l<br />
load moments <strong>of</strong> inertia.<br />
l<br />
From equation (2.55) and (2.56) one can write:<br />
and, <strong>with</strong> (2.50),<br />
dΩ m 1 = ( M<br />
e − M<br />
l )<br />
(2.62)<br />
dt J<br />
dΩ m 1 3<br />
∗<br />
= ( pb<br />
Im( Ψ<br />
s<br />
Is<br />
) − Ml<br />
)<br />
(2.63)<br />
dt J 2<br />
Finally, the full mathematical model <strong>of</strong> PM synchronous machine which is used in<br />
simulation studies [Appendices] is described in dq , reference frame as:<br />
dΨ<br />
sd<br />
Usd = Rs Isd + − pbΩmΨ sq<br />
(2.64a)<br />
dt<br />
dΨ<br />
sq<br />
Usq = Rs Isq + + pbΩmΨ sd<br />
(2.64b)<br />
dt<br />
Ψ<br />
sd<br />
= LsdIsd +Ψ<br />
PM<br />
(2.65a)<br />
Ψ<br />
sq<br />
= LsqIsq<br />
(2.65b)<br />
dΩ m 1 = ( M<br />
e − M<br />
l )<br />
(2.66)<br />
dt J<br />
3 ∗ 3<br />
M<br />
e<br />
= pbIm( Ψ<br />
s<br />
Is) = pb( ΨsdIsq −Ψ<br />
sqIsd)<br />
(2.67)<br />
2 2<br />
Based on above equations we can create the block scheme <strong>of</strong> the PMSM machine (Fig.<br />
2.9), where the input signals are the voltage components in dq , reference frame<br />
U , U and the output signal is the mechanical speed <strong>of</strong> the rotor Ω<br />
m<br />
. As the external<br />
sd<br />
sq<br />
load torque<br />
M<br />
l<br />
is disturbance.<br />
23
Modeling and control modes <strong>of</strong> PM synchronous motor drives<br />
M l<br />
U sd<br />
+<br />
−<br />
pbΩmΨsd<br />
∫<br />
Ψ sd<br />
Ψ PM<br />
−<br />
1<br />
L d<br />
I sd<br />
Ψ<br />
sq<br />
I<br />
sd<br />
p b<br />
R s<br />
−<br />
3<br />
p<br />
2 b<br />
M e<br />
−<br />
1<br />
J<br />
∫<br />
Ω m<br />
R s<br />
U sq<br />
−<br />
−<br />
∫<br />
Ψ sq<br />
1<br />
L q<br />
I sq<br />
Ψ<br />
sd<br />
I<br />
sq<br />
pbΩmΨsq<br />
p b<br />
Figure 2.9. Block scheme <strong>of</strong> PM synchronous machine in rotating dq , frame.<br />
Based on equations (2.64-2.67) we can also draw the vector diagram <strong>of</strong> PM<br />
synchronous motor (Fig. 2.10). From this vector representation it can see the positions<br />
<strong>of</strong> the vectors (currents, voltages and fluxes). Especially, power angle φ (angle between<br />
voltage and current vectors) and torque angle defined in two manners: as an angle<br />
between current and rotor flux vectors -δ I<br />
, or as angle between stator flux and rotor<br />
flux vectors -δ Ψ<br />
.<br />
q−axis<br />
β<br />
U s<br />
RI<br />
s<br />
s<br />
I s<br />
Ψ sq<br />
I sq<br />
LI<br />
d sd<br />
ΩΨ<br />
s<br />
s<br />
φ<br />
Ψ s<br />
LqI<br />
sq<br />
I sd<br />
δ I<br />
δ Ψ<br />
θ r<br />
Ψ sd<br />
θ Ψs<br />
Ψ PM<br />
d −axis<br />
rotor<br />
α ( A)<br />
stator<br />
Figure 2.10. <strong>Vector</strong> diagram <strong>of</strong> PM synchronous motor in rotor reference frame dq. ,<br />
24
Modeling and control modes <strong>of</strong> PM synchronous motor drives<br />
2.2 Static characteristic under different control modes<br />
In this section, basic steady state properties <strong>of</strong> the PMSM under different control mode<br />
strategies will be study [6,9]. The key control strategies for the PMSM can be listed as<br />
follows:<br />
• Constant torque angle control (CTAC).<br />
• Maximum torque per ampere control (MTPAC)<br />
• Unity power factor control (UPFC)<br />
• Constant stator flux control (CSFC)<br />
Constant torque angle (CTA) control<br />
This control strategy for PMSM keeps the torque angle δ<br />
I<br />
(angle between stator current<br />
vector and rotor permanent magnet flux) at constant value 90 .<br />
q−<br />
axis<br />
Isq<br />
= I<br />
s<br />
δ = 90<br />
I<br />
<br />
d<br />
− axis<br />
Figure 2.11. Current vector and permanent magnet flux vector for constant torque angle<br />
operation (CTAC)<br />
Hence, this control can be achieved by controlling the d-axis current components to<br />
zero leaving the current vector on the rotor q-axis (see Fig. 2.11). Therefore, this<br />
strategy is also referred to as I<br />
sd<br />
= 0 control. The amplitude <strong>of</strong> rotor flux vector is<br />
constant and also the torque angle is constant. So, the torque depends only on the value<br />
<strong>of</strong> stator current amplitude. Therefore, this control strategy is not recommended for<br />
IPMSM <strong>with</strong> high saliency ratio. However, for SPMSM, this strategy is commonly<br />
used.<br />
Ψ PM<br />
The torque equation in this mode <strong>of</strong> operation becomes:<br />
3 3<br />
M<br />
e<br />
= pbΨ PMIsq = pbΨ PM<br />
Is<br />
(2.68)<br />
2 2<br />
25
Modeling and control modes <strong>of</strong> PM synchronous motor drives<br />
The steady state voltage components based on the equations (2.34a) and (2.34b) are:<br />
Usd =−pbΩ mLq Iqs =−pbΩ mLq Is<br />
(2.69a)<br />
U = R I + p Ω Ψ = R I + p Ω Ψ (2.69b)<br />
sq s qs b m PM s s b m PM<br />
The amplitude <strong>of</strong> stator voltage vector can be calculated as:<br />
s<br />
2 2<br />
sd sq<br />
U = U + U<br />
(2.70)<br />
The stator flux vector amplitude can be calculated from equations (2.65a-b) as:<br />
s<br />
2 2<br />
sd sq<br />
Ψ = Ψ +Ψ (2.71)<br />
The active and reactive power and also the power factor can be obtained from equations<br />
(2.41),(2.46), (2.47).<br />
Maximum torque per ampere (MTPA) control<br />
The main idea <strong>of</strong> this control is develop the torque using minimum value <strong>of</strong> stator<br />
current amplitude. In this case the I sd<br />
components is not equal zero, and may cancel the<br />
reluctance torque produced by high saliency ratio. Therefore, this control strategy is<br />
recommended for IPMSM.<br />
q−<br />
axis<br />
I s<br />
I sq<br />
δ >= 90<br />
I<br />
<br />
d<br />
− axis<br />
I sd<br />
Figure 2.12. Current vector I s and permanent magnet flux vector Ψ<br />
PM<br />
for maximum torque<br />
per ampere operation (MTPAC).<br />
In order to obtain the maximum torque per ampere we should solve the derivative <strong>of</strong><br />
torque equations (2.55) in respect to torque angle. Solving for torque angle α and taking<br />
into account that only negative sign should be considered for the solution, we can<br />
calculate torque angle as:<br />
Ψ PM<br />
−1 −1 1 1 2<br />
δ<br />
I<br />
= cos [ − + ( ) ]<br />
4( L −L ) I 2 4( L −L ) I<br />
d q s<br />
d q<br />
s<br />
(2.72)<br />
26
Modeling and control modes <strong>of</strong> PM synchronous motor drives<br />
From Fig. 2.8, it can be seen that<br />
M is maximum when torque angle is 90 < < 180<br />
e<br />
The relevant torque equation in this mode <strong>of</strong> operation becomes from (2.55).<br />
<br />
δ <br />
I<br />
.<br />
The steady state voltage equations can be written using the current vector amplitude<br />
I<br />
s<br />
and the torque angle δ<br />
I<br />
as:<br />
U = R I cosδ<br />
+ p Ω L I sinδ<br />
(2.73a)<br />
sd s s I b m q s I<br />
U = R I sinδ<br />
− p Ω L I cosδ<br />
+ p Ω Ψ (2.73b)<br />
sq s s I b m d s I b m PM<br />
The amplitude <strong>of</strong> stator voltage vector can be calculated from equation (2.70) and<br />
amplitude <strong>of</strong> stator flux vector from (2.71). The active and reactive power and also the<br />
power factor can be obtained from equations (2.41),(2.46), (2.47).<br />
Unity power factor (UPF) control<br />
Under this control strategy there is no phase different between the current vector and the<br />
voltage vector. Hence, power factor angle φ (see Fig. 2.13) becomes zero. Since only<br />
active power is supplied to the machine under unity power factor operation, the VA<br />
rating requirement <strong>of</strong> the inverter can be reduced.<br />
q − axis<br />
U s<br />
φ = 0<br />
I s<br />
δ I<br />
d − axis<br />
Figure 2.13. Current vector and permanent magnet flux vector under unity power factor<br />
operation (UPFC).<br />
Ψ PM<br />
In this case when φ = 0 we have the relationship:<br />
U<br />
U<br />
sq<br />
sd<br />
Isq<br />
= = tanδ<br />
I<br />
(2.74)<br />
I<br />
sq<br />
Substituting the voltage equations (2.69a-b) into (2.71) and made some simplifying, we<br />
can obtain:<br />
I L −L −Ψ + L I = (2.75)<br />
2<br />
s<br />
(<br />
d q)cos δI PM<br />
cosδI q s<br />
0<br />
27
Modeling and control modes <strong>of</strong> PM synchronous motor drives<br />
Solving for the torque angle δ<br />
I<br />
:<br />
2 2<br />
PM PM<br />
4 Is<br />
( Ld Lq)<br />
Lq<br />
Ψ − Ψ − −<br />
−1<br />
δ<br />
I<br />
= cos [ ]<br />
2( L − L ) I<br />
d q s<br />
(2.76)<br />
only positive sign should be take into consideration.<br />
After obtaining δ I<br />
the amplitude <strong>of</strong> stator voltage vector can be calculated from<br />
equation (2.70) and amplitude <strong>of</strong> stator flux vector from (2.71). The active and reactive<br />
power and also the power factor can be obtained from equations (2.41),(2.46), (2.47).<br />
Constant stator flux (CSF) control<br />
As it can be see from the torque expression (2.58) for a given stator flux amplitude<br />
the electromagnetic torque<br />
amplitude<br />
M<br />
e<br />
is a function <strong>of</strong> torque angle δ Ψ<br />
. The stator flux linkage<br />
Ψ<br />
s<br />
is kept constant <strong>of</strong> the permanent magnet flux amplitude Ψ<br />
PM<br />
.<br />
I s<br />
q − axis<br />
Ψs<br />
Ψ<br />
s<br />
δ I<br />
δ Ψ<br />
d − axis<br />
Figure 2.14. Flux vector and permanent magnet flux vector under constant stator flux operation<br />
(CSFC).<br />
The amplitude <strong>of</strong> the stator flux linkage vector is<br />
Ψ PM<br />
2 2 2 2<br />
Ψ<br />
s<br />
= Ψ<br />
sd<br />
+Ψ<br />
sq<br />
= ( LI<br />
q sq<br />
) + ( LI<br />
d sd<br />
+Ψ<br />
PM<br />
)<br />
(2.77)<br />
Equating<br />
Ψ =Ψ (2.78)<br />
s<br />
PM<br />
can be obtain the relationship for rotor frame currents as:<br />
2 2<br />
q sq d sd d PM sd<br />
( LI ) + ( LI ) + 2LΨ I = 0<br />
(2.79)<br />
This condition is true if I<br />
sd<br />
< 0 , because expression<br />
always positive values.<br />
2 2<br />
q sq<br />
LI<br />
d sd<br />
( LI ) + ( ) and L , Ψ are<br />
d<br />
PM<br />
28
Modeling and control modes <strong>of</strong> PM synchronous motor drives<br />
2 2 2 2 2 2<br />
d q s<br />
δI d PM s<br />
δI q s<br />
( L − L ) I cos + 2L Ψ I cos + L I = 0 (2.80)<br />
Solving for the torque angle δ<br />
I<br />
2 2 2 2 2 2<br />
d PM d PM s<br />
(<br />
d q<br />
)<br />
q<br />
−L Ψ ± L Ψ − I L −L L<br />
−1<br />
δ<br />
I<br />
= cos [ ]<br />
( L L ) I<br />
2 2<br />
d<br />
−<br />
q s<br />
(2.81)<br />
For givenδ I<br />
, the amplitude <strong>of</strong> stator voltage vector we can be calculated from equation<br />
(2.70) and amplitude <strong>of</strong> stator flux vector from (2.71). The active and reactive power<br />
and also the power factor can be obtained from equations (2.41),(2.46),(2.47),<br />
respectively for defined speed.<br />
Comparison study<br />
In order to compare the control strategies and to cancel dependence <strong>of</strong> machine power,<br />
per unit values defined as shown in Table 2.1 below have been introduced [3,9].<br />
The value <strong>of</strong> current vector:<br />
I<br />
sN<br />
= Is<br />
Is<br />
I<br />
= 2I<br />
(2.82)<br />
b<br />
srms( rated )<br />
The value <strong>of</strong> voltage vector: U<br />
s<br />
U<br />
s<br />
U<br />
sN<br />
= = (2.83)<br />
Ub ΩΨ<br />
b PM<br />
where: Ω b<br />
= 2 π f and b<br />
fb<br />
is rated frequency<br />
<strong>of</strong> the PM motor.<br />
The value <strong>of</strong> flux vector is: Ψ<br />
s<br />
Ψ<br />
s<br />
Ψ<br />
sN<br />
= =<br />
(2.84)<br />
Ψb<br />
ΨPM<br />
The value <strong>of</strong> torque is:<br />
Me<br />
Me<br />
M<br />
eN<br />
= =<br />
(2.85)<br />
M 3<br />
b pbΨ<br />
PMIb<br />
2<br />
The value <strong>of</strong> apparent power vector<br />
S S<br />
S = N<br />
S<br />
= 3<br />
(2.86)<br />
b UI<br />
b b<br />
2<br />
The value <strong>of</strong> active power<br />
P<br />
PN<br />
= (2.87)<br />
Sb<br />
The value <strong>of</strong> reactive power<br />
Q<br />
QN<br />
= (2.88)<br />
S<br />
b<br />
Table 2.1. Per unit values definition.<br />
In order to compare the steady state performance characteristic <strong>of</strong> the above discussed<br />
control strategies, for each <strong>of</strong> the control strategy some important quantities <strong>of</strong> the<br />
machine have been plotted as a function <strong>of</strong> the torque. The PMSM parameters, which<br />
are used for the calculations are given in Appendices.<br />
29
Modeling and control modes <strong>of</strong> PM synchronous motor drives<br />
The current requirement versus torque is illustrated in Fig. 2.15 for the different control<br />
strategies. It can be seen, that up to 1 pu torque, the requirement for current is lowest for<br />
CSF control. Highest than 1 pu torque the low current needs MTPA control requirement<br />
lowest current for a given torque.<br />
3<br />
I<br />
sN<br />
[ pu]<br />
2.5<br />
2<br />
1.5<br />
UPF<br />
CSF<br />
CTA<br />
MTPA<br />
1<br />
0.5<br />
0<br />
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2<br />
M [ ] eN<br />
pu<br />
Figure 2.15. Stator current amplitude under different control strategies versus electromagnetic<br />
torque.<br />
The voltage requirement versus torque for the different control strategies is illustrated in<br />
Fig. 2.16. It can be seen, that CSF requires the highest value <strong>of</strong> stator voltage.<br />
U<br />
sN<br />
[ pu]<br />
2.5<br />
2<br />
1.5<br />
1<br />
CSF<br />
CTA<br />
MTPA<br />
0.5<br />
UPF<br />
0<br />
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2<br />
M [ ] eN<br />
pu<br />
Figure 2.16. Stator voltage amplitude versus electromagnetic torque under different control<br />
strategies (at 1 pu rotor speed).<br />
3<br />
PN<br />
2<br />
[ pu]<br />
1<br />
UPF<br />
CSF<br />
CTA<br />
MTPA<br />
0<br />
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2<br />
M [ ] eN<br />
pu<br />
Figure 2.17. Active power versus electromagnetic torque under different control strategies.<br />
30
Modeling and control modes <strong>of</strong> PM synchronous motor drives<br />
The active power requirement as a function <strong>of</strong> torque is illustrated in Fig. 2.17 for the<br />
different control strategies. It can be seen, that all control strategies require<br />
approximately the same value <strong>of</strong> active power for a given torque. CSF control needs<br />
less active power in the region up to 1.3 pu torque.<br />
The reactive power requirement as a function <strong>of</strong> torque is illustrated in Fig. 2.18. It can<br />
be seen, that CTA control requires the highest value <strong>of</strong> active power for a given torque<br />
and the CSF control lowest.<br />
4<br />
QN<br />
[ pu]<br />
3<br />
2<br />
CTA<br />
MTPA<br />
1<br />
UPF<br />
CSF<br />
0<br />
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2<br />
M [ ] eN<br />
pu<br />
Figure 2.18. Reactive power versus electromagnetic torque under different control strategies.<br />
1.1<br />
1<br />
0.9<br />
UPF<br />
CSF<br />
cosφ<br />
0.8<br />
MTPA<br />
0.7<br />
0.6<br />
CTA<br />
0.5<br />
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2<br />
M [ ] eN<br />
pu<br />
Figure 2.19. Power factor as a function <strong>of</strong> electromagnetic torque under different control<br />
strategies.<br />
The power factor as a function <strong>of</strong> torque is illustrated in Fig. 2.19. It can be seen, that as<br />
it could be expected, UPF control requires constant power factor for a given torque.<br />
CSF control is very close to the unity power factor up to 1 pu torque.<br />
The above analysis can be summarized as shown in Table. 2.2.<br />
31
Modeling and control modes <strong>of</strong> PM synchronous motor drives<br />
Table. 2.2. Summary <strong>of</strong> voltage, power, power factor requirements under control modes.<br />
Requirement<br />
<strong>Control</strong><br />
method<br />
Voltage<br />
Current<br />
Power<br />
factor<br />
CTA middle low<br />
low<br />
MTP<br />
A<br />
low<br />
low<br />
middle<br />
UPF low high 1<br />
CSF high lowes up to 1.1<br />
pu torque<br />
Close to 1 up to<br />
1 pu torque<br />
From this comparison study it can be concluded that CSF control appears to be superior<br />
in terms <strong>of</strong> steady state performance characteristics compared to other methods under<br />
discussion.<br />
32
Modeling and control modes <strong>of</strong> PM synchronous motor drives<br />
2.3 Summary<br />
‣ There are different forms to express the PMSM equations, but the rotor<br />
reference frame equations are the most widely used. The simplification in rotor<br />
dq , reference frame equations results from the disappearance <strong>of</strong> position<br />
dependent inductances.<br />
‣ The electromagnetic torque <strong>of</strong> the IPMSM is not only produced by the<br />
permanent magnet flux, but also by the reluctance difference in rotor d- and q-<br />
axes.<br />
‣ Electromagnetic torque as cross vector product <strong>of</strong> the stator flux linkage and<br />
current space vectors or rotor and stator flux linkages is independent <strong>of</strong><br />
coordinate system selected. Therefore, can be expressed in stationary ( α,<br />
β ) or<br />
rotated ( dq) , coordinates.<br />
‣ For further control strategies consideration it is convenient to express the<br />
electromagnetic torque <strong>of</strong> PMSM machine by:<br />
• vector product <strong>of</strong> stator current and rotor flux vectors. The rotor flux<br />
vector in PMSM machine is constant, because <strong>of</strong> the PM. Therefore, to<br />
increase and decrease the torque, the current amplitude and the torque<br />
angle δ<br />
I<br />
should be changed (see Fig. 2.20a),<br />
• vector product <strong>of</strong> stator flux vector and rotor flux vector. Generally, the<br />
value <strong>of</strong> the stator flux amplitude is kept constant at value <strong>of</strong> rotor flux<br />
produced by permanent magnets. So, in this case to change the torque we<br />
should adjust the torque angle δ Ψ<br />
(see Fig. 2.20b).<br />
q<br />
β<br />
a )<br />
q β b)<br />
I s<br />
Ψ PM<br />
d<br />
Ψ s<br />
Ψ<br />
PM<br />
d<br />
δ I<br />
γ m<br />
α<br />
δ Ψ<br />
γ m<br />
α<br />
Fig. 2.20 <strong>Torque</strong> production: a) current control, b) flux control<br />
‣ Taking into account discussion regarding static characteristic under different<br />
control strategies it can be said that –depart from special requirements- the most<br />
suited for general application PMSM drives is constant stator flux (CSF)<br />
operation.<br />
33
Voltage source PWM inverter for PMSM supply<br />
Chapter 3 VOLTAGE SOURCE PWM INVERTER FOR PMSM SUPPLY<br />
3.1 Introduction<br />
The block scheme <strong>of</strong> an adjustable speed drive (ASD) commonly used in industrial<br />
applications to supply three-phase AC motor is presented in Fig. 3.1.<br />
Three-phase<br />
grid<br />
Rectifier<br />
DC link<br />
filter<br />
Inverter<br />
AC motor<br />
Choke<br />
Figure 3.1 Basic scheme <strong>of</strong> adjustable speed AC motor system.<br />
An ASD is supplied from three or single phase grid. It consists <strong>of</strong> a diode rectifier, DC<br />
link filter and an inverter. The rectifier converts supply AC voltage into DC voltage.<br />
The DC voltage is filtered by a capacitor in the DC link. The inverter converts the DC<br />
to an variable voltage, variable frequency AC for motor speed or (torque/current)<br />
control.<br />
The rectifier section <strong>of</strong> an ASD, called the front end, is responsible for generating<br />
current harmonics into the power supply system. Therefore, to reduce the total harmonic<br />
distortion (THD) <strong>of</strong> phase current it is necessary to add additional choke inductances.<br />
There are generally to way how insert choke inductances (see Fig. 3.2).<br />
a)<br />
b)<br />
L F<br />
D 1<br />
D3<br />
D5<br />
D 1<br />
D3<br />
D5<br />
L F<br />
L F<br />
C F<br />
U DC<br />
C F<br />
U DC<br />
L F<br />
D 2<br />
D4<br />
D6<br />
D 2<br />
D4<br />
D6<br />
L F<br />
Fig. 3.2 Three phase diode rectifier <strong>with</strong> smoothing choke: a) at the input b) at the DC link side.<br />
By adding a choke inductance at the input <strong>of</strong> rectifier gives the significant harmonic<br />
reduction. Some drive manufactures are starting to include this choke inductance in the<br />
DC link <strong>of</strong> the drive, providing the same harmonic current reduction benefit.<br />
Regarding to power electronics standards IEEE Std 519 it is recommended that<br />
production <strong>of</strong> harmonics should be less than 5%. It is a trend to replace the diode<br />
34
Voltage source PWM inverter for PMSM supply<br />
rectifier by fully controllable active rectifier [8] (see Fig. 3.3), which guaranties<br />
following futures:<br />
• power flow from AC/DC or DC/AC side (there is no need <strong>of</strong> break resistor ),<br />
• significant reduction <strong>of</strong> phase current THD,<br />
• unity power factor (phase voltage is in phase <strong>with</strong> current),<br />
• reduction <strong>of</strong> DC link capacitor,<br />
• controllable DC link voltage.<br />
Three-phase<br />
grid<br />
Active<br />
Rectifier<br />
DC link<br />
filter<br />
Inverter<br />
AC motor<br />
Choke<br />
Fig. 3.3 Modern AC/DC/AC converter topology <strong>of</strong> adjustable speed drives.<br />
In the next part <strong>of</strong> this thesis the author will be focus on voltage source inverter.<br />
3.2 Voltage source inverter (VSI)<br />
The made constant DC voltage by rectifier is delivered the to the input <strong>of</strong> inverter (Fig.<br />
3.4), which thanks to controlled transistor switches, converts this voltage to three-phase<br />
AC voltage signal <strong>with</strong> wide range variable voltage amplitude and frequency [3].<br />
Voltage Source Inverter<br />
C F<br />
T 1<br />
D 7<br />
T 3<br />
D 9<br />
T 5<br />
D 11<br />
U DC<br />
O<br />
C F<br />
T 2<br />
D 8<br />
T 4<br />
D 10<br />
T 6<br />
D 12<br />
A B C<br />
U<br />
sAN<br />
U<br />
sBN<br />
U<br />
sCN<br />
N<br />
Three-phase motor windings<br />
Figure 3.4 Basic scheme <strong>of</strong> voltage source inverter circuit.<br />
35
Voltage source PWM inverter for PMSM supply<br />
The one leg <strong>of</strong> inverter consists <strong>of</strong> two transistor switches. A simple transistor switch<br />
consist <strong>of</strong> feedback diode connected in anti-parallel <strong>with</strong> transistor. Feedback diode<br />
conducts current when the load current direction is opposite to the voltage direction.<br />
Assuming that the power devices are ideal: when they are conducting the voltage across<br />
them is zero and they present an open circuit in their blocking mode. Therefore, each<br />
inverter leg can be represented as an ideal switch. Its gives possibility to connect each<br />
<strong>of</strong> the three motor phase coils to a positive or negative voltage <strong>of</strong> the dc link ( U ).<br />
Thus the equivalent scheme for three-phase inverter and possible eight combinations <strong>of</strong><br />
the switches in the inverter are shown in Fig. 3.5.<br />
DC<br />
U<br />
7<br />
= 111<br />
U<br />
0<br />
= 000<br />
U DC<br />
1 1<br />
S<br />
A<br />
SB<br />
1<br />
S C<br />
U DC<br />
S<br />
A<br />
0 0<br />
SB<br />
0<br />
S C<br />
U<br />
1<br />
= 100<br />
A B C<br />
U<br />
2<br />
= 110<br />
A B C<br />
U<br />
3<br />
= 010<br />
U DC<br />
1<br />
SA<br />
0<br />
SB<br />
0<br />
S C<br />
U DC<br />
1 1<br />
S<br />
A<br />
SB<br />
0<br />
S C<br />
U DC<br />
0<br />
S<br />
A<br />
1<br />
SB<br />
0<br />
S C<br />
U<br />
4<br />
= 011<br />
A B C<br />
U<br />
5<br />
= 001<br />
A B C<br />
U<br />
6<br />
= 101<br />
A B C<br />
U DC<br />
0<br />
SA<br />
1<br />
SB<br />
1<br />
S C<br />
U DC<br />
0<br />
S<br />
A<br />
0<br />
SB<br />
1<br />
S C<br />
U DC<br />
1<br />
S<br />
A<br />
0<br />
SB<br />
1<br />
S C<br />
A B C<br />
A B C<br />
A B C<br />
Figure 3.5 Possible switches state in VSI.<br />
The six positions <strong>of</strong> switches ( U1 − U6) produce an output phase voltage equal ± 1/3 or<br />
± 2/3 <strong>of</strong> the DC voltage. The last two ( U0,<br />
U<br />
7) give zero output voltage. The output<br />
phase voltages produced by inverter are shown in Fig. 3.6a. and adequate line to line<br />
voltage calculated in bellow formula also are presented in Fig. 3.6b.<br />
UsAB = UsAN − UsBN<br />
(3.1a)<br />
UsBC = UsBN − UsCN<br />
(3.1b)<br />
UsCA = UsCN − UsAN<br />
(3.1c)<br />
36
Voltage source PWM inverter for PMSM supply<br />
a )<br />
b)<br />
U sAN<br />
U sAB<br />
2<br />
U<br />
3 DC<br />
U DC<br />
2<br />
− U<br />
3 DC<br />
2π<br />
Ωt<br />
−U DC<br />
2π<br />
Ωt<br />
2<br />
U<br />
3 DC<br />
U sBN<br />
U DC<br />
U sBC<br />
2<br />
− U<br />
3 DC<br />
2π<br />
Ωt<br />
−U DC<br />
2π<br />
Ωt<br />
2<br />
U<br />
3 DC<br />
U sCN<br />
U DC<br />
U sCA<br />
2<br />
− U<br />
3 DC<br />
U1<br />
U<br />
2<br />
U3<br />
U<br />
4<br />
U5<br />
U6<br />
2π<br />
Ωt<br />
−U DC<br />
a )<br />
b)<br />
U1<br />
U<br />
2<br />
U3<br />
U<br />
4<br />
U5<br />
U6<br />
Figure. 3.6 Three voltage waveforms generated by the inverter: a) phase voltages, b) line to line<br />
voltages.<br />
2π<br />
Ωt<br />
Form the Fourier analysis for phase voltage produced by inverter (Fig. 3.7) the<br />
maximum amplitude <strong>of</strong> fundamental phase voltage for a given DC link voltage is given<br />
by:<br />
U<br />
_ amp<br />
2<br />
= UDC<br />
(3.2)<br />
π<br />
U out<br />
2<br />
U<br />
3 DC<br />
1<br />
U<br />
3 DC<br />
1<br />
− U<br />
3 DC<br />
2<br />
− U<br />
3 DC<br />
2<br />
U DC<br />
π<br />
2π<br />
ωt<br />
Figure 3.7 Inverter phase voltage generated during six step operation (solid line), corresponding<br />
fundamental component <strong>of</strong> output voltage (dashed line) and harmonic spectrum <strong>of</strong> phase<br />
voltage.<br />
37
Voltage source PWM inverter for PMSM supply<br />
The three-phase output voltage <strong>of</strong> the inverter can be described by space vector<br />
definition as:<br />
U<br />
k<br />
⎧2<br />
⎪ U<br />
3<br />
⎪<br />
= ⎨<br />
⎪⎪⎪ 0<br />
⎩<br />
DC<br />
e<br />
π<br />
j( k−1) 3<br />
for k =1,2...,6.<br />
for k=0,7.<br />
, (3.3)<br />
where k denotes numbers vector.<br />
<strong>Vector</strong>s from 1-6 are called active vectors, whereas vectors 0,7 are called zero vectors<br />
or non active vectors. The voltage space vector<br />
U<br />
k<br />
in complex plane forms a regular<br />
hexagon and divides in into six equal sectors (one sector takes 60 electrical degree) Fig.<br />
3.8.<br />
Im<br />
U 4<br />
(011)<br />
U 3 (010)<br />
sector3<br />
sector4<br />
sector2<br />
sector5<br />
U (110) 2<br />
sector1<br />
U (000) U 1<br />
(100)<br />
0 U (111) 7 Re<br />
sector6<br />
2<br />
U<br />
3 DC<br />
U (001)<br />
U (101)<br />
5 6<br />
Figure 3.8 Representation <strong>of</strong> the inverter states in the complex space.<br />
In practice the real voltage source inverter has non-linear characteristic due to [19]:<br />
• the dead-time,<br />
• a voltage drop across the power switches,<br />
• pulsation <strong>of</strong> the DC link voltage.<br />
Dead time effect [17,20,27,30]<br />
Semiconductors power switches <strong>of</strong> voltage source inverter operate not ideally. They do<br />
not turn-on or turn-<strong>of</strong>f instantaneously. Therefore it is necessary to include a protection<br />
time to avoid a short circuit in the DC link, when two switching devices are in the same<br />
leg (see Fig. 3.9). This time T d<br />
is included in the control signals and it is called “dead<br />
38
Voltage source PWM inverter for PMSM supply<br />
time”. It guarantees safe operation <strong>of</strong> the inverter. The typical value is from 1µ s - 5µ s .<br />
When the lowest value is for small power IGBT and is growing in respect to increasing<br />
<strong>of</strong> IGBT power. More details about real IGBT module you can find in Appendices.<br />
The effect <strong>of</strong> dead time can be examined from one phase <strong>of</strong> PWM inverter. The basic<br />
configuration is shown in Fig. 3.9. Consist <strong>of</strong> upper and lower power devices T<br />
1<br />
and T<br />
2<br />
,<br />
and reverse recovery diodes D<br />
1<br />
and D<br />
2<br />
, connected between the positive and negative<br />
rails <strong>of</strong> power supply. The gating signals S<br />
A<br />
and S<br />
Ai<br />
come from control block. Output<br />
voltage terminal U<br />
0<br />
is connected to motor phase.<br />
S A<br />
T 1<br />
D 1<br />
U DC<br />
I sA<br />
S Ai<br />
T 2<br />
D 2<br />
LOAD<br />
U 0<br />
S<br />
Dead time<br />
Td<br />
Figure 3.9 Circuit diagram <strong>of</strong> one inverter leg.<br />
Fig. 3.10 shows the ideal control signals and real control signals <strong>with</strong> inserted dead time<br />
T<br />
d<br />
. As can be observed the time duration <strong>of</strong> real drive signal for upper transistor is<br />
shorted than ideal drive signal and for lower transistor is longer than ideal.<br />
Ideal drive signals<br />
Real drive signals<br />
S A<br />
S A<br />
T d<br />
S Ai<br />
S Ai<br />
T d<br />
Figure 3.10 Gate signals control <strong>of</strong> one inverter leg.<br />
39
Voltage source PWM inverter for PMSM supply<br />
As a consequence when the phase current I sA<br />
is positive, the output voltage is reduced,<br />
and when the current I<br />
sA<br />
is negative the output voltage is increased (see Fig. 3.11).<br />
U0<br />
I<br />
sA<br />
> 0<br />
ideal voltage<br />
real voltage<br />
U I < 0<br />
0<br />
sA<br />
U DC<br />
2<br />
U DC<br />
2<br />
−U DC<br />
2<br />
deacresing<br />
−U DC<br />
2<br />
increasing<br />
Figure 3.11 Dead time effect on the inverter output voltage: (fat line real voltage, doted line<br />
ideal voltage).<br />
Voltage drop across power devices<br />
In real voltage source inverter power switches do not conduct ideally. When they are<br />
conducting the voltage across them is not zero and equal the voltage drop on the<br />
conducted transistor V T<br />
. Also in blocking mode the power switches have voltage drop<br />
on the conducted diode V D<br />
. More details about real IGBT module you can find in<br />
Appendices.<br />
The voltage drop across the power devices is dependent on the direction <strong>of</strong> the phase<br />
current. It has influence on the output voltage, especially at low speed operation <strong>of</strong><br />
motor and high load current [17,20,27,30]. Fig. 3.12 shows the voltage drop influence<br />
on the output voltage. Also shows that the output voltage is asymmetric (<strong>with</strong> <strong>of</strong>fset)<br />
and the voltage drop decreases the output voltage when the phase current is positive and<br />
increases the output voltage when the phase current is negative.<br />
U 0<br />
I > 0<br />
sA<br />
U 0<br />
I < 0<br />
sA<br />
U DC<br />
2<br />
V T<br />
U DC<br />
2<br />
V D<br />
V D<br />
V T<br />
−U DC<br />
2<br />
−U DC<br />
2<br />
Figure 3.12 Output voltage in voltage source inverter due to voltage drop across the power<br />
devices a) for I<br />
sA<br />
> 0 , b) I<br />
sA<br />
< 0 (fat line real voltage, doted line ideal voltage).<br />
The influence <strong>of</strong> dead time effect and voltage drop across power devices on the output<br />
voltage from inverter is illustrated in block diagram (Fig. 3.13). The ideal reference<br />
*<br />
voltage components in stationary reference frame ( U α<br />
,<br />
_ ideal<br />
*<br />
U β _ ideal<br />
) are equal real<br />
40
Voltage source PWM inverter for PMSM supply<br />
*<br />
( U α<br />
,<br />
_ real<br />
*<br />
U β _ real<br />
) and delivered to pulse width modulation (PWM) modulator block<br />
<strong>with</strong> real non-linear inverter. As a result the output voltages ( U α _ out<br />
, U β _ out<br />
) are distorted<br />
(green signals) and as consequence the phase currents ( I α _ out<br />
, I β _ out<br />
) in the load (red<br />
signals) are also distorted.<br />
*<br />
U *<br />
α _ideal<br />
U α _ real<br />
*<br />
*<br />
U β<br />
U<br />
_ideal<br />
β _ real<br />
PWM<br />
Modulator<br />
S<br />
A<br />
S B<br />
S C<br />
Inverter<br />
U α _out<br />
U β _out<br />
I α _ out<br />
I β _out<br />
Load<br />
Figure 3.13 Block diagram illustrated the dead time effect and voltage drop across power<br />
devices in three phase motor supplied from non-ideal voltage source inverter.<br />
Pulsation <strong>of</strong> the DC link voltage [19]<br />
In practice it should be take into account that the real input dc-link voltage required for<br />
supply VSI is not ideal. It has ripples and fluctuation, because <strong>of</strong> not ideal filtering and<br />
disadvantages <strong>of</strong> diode rectifier. Therefore, the quality <strong>of</strong> dc-link voltage has impact on<br />
the output voltage from inverter. If dc-link voltage will change we can observed the<br />
changing at the output <strong>of</strong> inverter. In order to overcome this problem:<br />
• in PWM modulator we can not assume a constant dc-link voltage and we should<br />
measured this voltage in order to calculate the modulation index (see subchapter<br />
3.3),<br />
41
Voltage source PWM inverter for PMSM supply<br />
• instate <strong>of</strong> diode rectifier will be use the active rectifier, which provided<br />
controllable DC-link voltage.<br />
• or used bigger capacitor in the DC-link side in order to increase possibility <strong>of</strong><br />
filtering.<br />
Let us summarize, adding influence <strong>of</strong> non-linear VSI causes by:<br />
• serious distortion in the inverter output voltage,<br />
• distorted machine currents,<br />
• torque pulsation,<br />
Additionally, also causes motor instability due to the interaction between motor and the<br />
PWM inverter, or the choice <strong>of</strong> the PWM strategy [25].<br />
Based on simulated and experimental observation one can say that the dead time effect<br />
is:<br />
• more visible in low speed operation <strong>of</strong> the motor,<br />
• may become significantly in drives where high switching frequency is required<br />
for good dynamics performances.<br />
In some applications such as sensor-less vector control, the inverter output voltages are<br />
needed to calculate the rotor or stator flux vectors. Unfortunately, it is very difficult to<br />
measure the output voltage and requires additional hardware. The most desirable<br />
method to obtain the output voltage feedback signal is to use the reference voltages<br />
instead. However, the relation between the output and reference voltage is nonlinear due<br />
to the dead-time effect and voltage drop across power devices. Thus, unless the properly<br />
dead-time and voltage drop compensation will be applied, the reference voltage can not<br />
be used instead <strong>of</strong> the inverter output voltage. Several compensation method were<br />
proposed to overcome this problem. One <strong>of</strong> them will be present bellow.<br />
Compensation based on modification <strong>of</strong> reference voltage waveform [17]<br />
The compensation process <strong>of</strong> dead time effect and voltage drop across power devices on<br />
the inverter output voltage from is illustrated in Fig. 3.14.<br />
42
Voltage source PWM inverter for PMSM supply<br />
Compensation<br />
block<br />
IsA<br />
IsB<br />
Compensation <strong>of</strong><br />
inverter<br />
I sC<br />
U DC<br />
T comp<br />
U α _comp<br />
U β _comp<br />
*<br />
U α _ideal<br />
*<br />
U β _ideal<br />
*<br />
U α _ real<br />
*<br />
U α _ real<br />
*<br />
U β _ real<br />
PWM<br />
Modulator<br />
S<br />
A<br />
S B<br />
S C<br />
Inverter<br />
*<br />
U β _ real<br />
U α _out<br />
U β _out<br />
I α _out<br />
I β _out<br />
Load<br />
Figure 3.14 Block diagram illustrating the dead time and voltage drop across power devices<br />
compensation method in three phase motor supplied from non-ideal voltage source inverter.<br />
In order to compensate the inverter non-linearity to the ideal reference voltage<br />
*<br />
components in stationary reference frame ( U α<br />
,<br />
(<br />
_ comp<br />
_ ideal<br />
*<br />
U β _ ideal<br />
) a compensation signal<br />
U α<br />
, U β _ comp<br />
) is added. As a consequence the real reference voltage components<br />
*<br />
( U α<br />
,<br />
_ real<br />
*<br />
U β _ real<br />
) are pre-distorted. Further those signals are delivered to PWM<br />
modulator <strong>with</strong> non-ideal inverter. As a result the output voltages are not distorted in<br />
Fig. 3.14 and thus phase currents in the load (red signals in Fig. 3.14) are almost<br />
sinusoidal.<br />
To calculate an average compensation voltages ( U α _ comp<br />
, U β _ comp<br />
), the parameters <strong>of</strong><br />
IGBT modules as:<br />
• dead time T<br />
d<br />
,<br />
• turn on T<br />
ON<br />
and turn <strong>of</strong>f T<br />
OFF<br />
<strong>of</strong> IGBT transistors,<br />
• and also on a voltage drop on diode V<br />
D<br />
and transistor<br />
should be know.<br />
V<br />
T<br />
,<br />
43
Voltage source PWM inverter for PMSM supply<br />
The total compensation time to compensate the non-linearity <strong>of</strong> inverter can be<br />
calculated as:<br />
U<br />
T = T + T − T + T<br />
(3.4)<br />
dp<br />
comp d ON OFF s<br />
U<br />
DC<br />
Where Udp = VD −ton ( VD − VT )/ Ts<br />
and t on<br />
is conducting time <strong>of</strong> IGBT devices in one<br />
sampling time.<br />
The compensation voltage vector can be obtained as:<br />
Tcomp<br />
Ucomp = 2 UDC s ign( Is ) = 2Uth s ign( Is<br />
)<br />
(3.5)<br />
Ts<br />
where<br />
and<br />
2<br />
2<br />
sign( Is) = ( sign( IsA) + asign( IsB) + a sign( IsC))<br />
,<br />
3<br />
⎧ sign( IsA) = 1 if IsA<br />
> 0<br />
sign( IsA)<br />
= ⎨<br />
⎩sign( IsA) = 0 if IsA<br />
< 0<br />
(3.6)<br />
The sign function for remain phase currents are calculated similarly.<br />
Solving equations (3.5) for real and imagine part in stationary frame, one obtains:<br />
1<br />
U U sign I sign I sign I<br />
3<br />
α _ comp<br />
= 2<br />
th<br />
(2 (<br />
sA) −0.5 (<br />
sB<br />
) − 0.5 (<br />
sC<br />
)) (3.7a)<br />
1<br />
U U sign I sign I<br />
3<br />
β _ comp<br />
= 2<br />
th<br />
( (<br />
sB<br />
) − (<br />
sC<br />
))<br />
(3.7b)<br />
The waveform <strong>of</strong> compensation voltages in stationary frame are shown in Fig. 3.15.<br />
44
Voltage source PWM inverter for PMSM supply<br />
4<br />
U<br />
3 th<br />
8<br />
U<br />
3 th<br />
2<br />
U<br />
3 th<br />
Figure 3.15 Voltage compensation components in stationary reference frame.<br />
From the top reference α and β components.<br />
In order to illustrate the effectiveness <strong>of</strong> the proposed compensation a simulation study<br />
has been performed. Fig. 3.16a shows the phase current in α,<br />
β frame and their<br />
hodograph <strong>with</strong>out compensation and Fig. 3.16b <strong>with</strong> proposed compensation method.<br />
a)<br />
a)<br />
b)<br />
b)<br />
Figure 3.16 Nonlinearity effect <strong>of</strong> voltage source inverter on phase current <strong>of</strong> AC machine:<br />
a) <strong>with</strong>out compensation, b) <strong>with</strong> compensation.<br />
45
Voltage source PWM inverter for PMSM supply<br />
3.3 <strong>Space</strong> vector based pulse width modulation (PWM) methods<br />
In voltage source inverter the transistors are controlled in a on-<strong>of</strong>f fashion. In order to<br />
obtain a suitable duty cycle for each switches the technique pulse <strong>with</strong> modulation is<br />
used. The modulation methods [18,21,22,23,24,26,28,29,30] have the influence on:<br />
wide range <strong>of</strong> linear operation, low content <strong>of</strong> higher harmonics in voltage and current,<br />
low frequency harmonics, minimal number <strong>of</strong> switching to decrease switching losses in<br />
the power components.<br />
The most important factor in PWM mode is modulation index defined as the ratio <strong>of</strong> the<br />
reference voltage amplitude value to the maximum voltage amplitude value during sixstep<br />
operation (see Fig 3.17) and is given by:<br />
M<br />
U<br />
ref<br />
= (3.8)<br />
2<br />
U<br />
DC<br />
π<br />
where the<br />
U<br />
DC<br />
is the DC link voltage (for three phase six diodes rectifier is 560 V ).<br />
The modulation index varies between 0-1 and can be divided into two regions: the<br />
linear ( 0< M ≤ 0.907) and the nonlinear ( 0.907 < M ≤ 1) as is shown in Fig. 3.17.<br />
Im<br />
U3(010)<br />
U 2(110)<br />
End <strong>of</strong> overmodulation<br />
region (six-step mode)<br />
2<br />
π<br />
Umax<br />
= U DC<br />
⇒ M = 1<br />
sector = 2<br />
sector = 3<br />
2<br />
U<br />
3 dc<br />
U ref<br />
End <strong>of</strong> linear region<br />
U<br />
max<br />
= U DC<br />
⇒ M = 0.907<br />
3<br />
U 4<br />
(011)<br />
sector = 4<br />
U 0<br />
(000)<br />
U 7<br />
(111)<br />
sector = 5<br />
θ ref<br />
T1<br />
U<br />
1<br />
Ts<br />
sector = 1<br />
sector = 6<br />
Re<br />
U (100) 1<br />
nonlinear<br />
(overmodulation) region<br />
linear region<br />
U 5 (001)<br />
U 6 (101)<br />
Figure 3.17 <strong>Space</strong> vector diagram <strong>of</strong> the available switching vectors.<br />
46
Voltage source PWM inverter for PMSM supply<br />
Linear range <strong>of</strong> operation ( 0> M
Voltage source PWM inverter for PMSM supply<br />
Next from<br />
U<br />
ref<br />
, α<br />
ref<br />
it is necessary to calculate the time interval for particular vectors.<br />
U 2<br />
U ref<br />
120°- α<br />
UU<br />
0, 7<br />
t 0<br />
t 2<br />
α ref<br />
t 1<br />
Using the low <strong>of</strong> sine it is possible to write:<br />
U 1<br />
120°<br />
Figure 3.18 One sector in voltage plane.<br />
U<br />
ref U1 U2<br />
= =<br />
sin120° sin(60 °−α<br />
) sinα<br />
ref<br />
ref<br />
(3.12)<br />
From this relations calculated value <strong>of</strong> vectors<br />
sin(60 °−αref<br />
) 2<br />
U1<br />
= Uref = Uref sin(60 °−αref<br />
)<br />
sin120°<br />
3<br />
(3.13a)<br />
sin<br />
ref 2<br />
U2<br />
= U α = U<br />
sin120°<br />
3<br />
sinα<br />
ref ref ref<br />
(3.13b)<br />
and respectively the normalized times are given:<br />
3 U<br />
1<br />
ref<br />
1<br />
= = sin(60 °− αref<br />
)<br />
(3.14a)<br />
2<br />
U<br />
U<br />
DC<br />
DC<br />
t<br />
3<br />
U<br />
3 U<br />
2<br />
ref<br />
2<br />
= = sinα<br />
ref<br />
(3.14b)<br />
2<br />
U<br />
U<br />
DC<br />
DC<br />
t<br />
3<br />
U<br />
Putting Eq. (3.13a-b) in to Eq. (3.14a-b) the normalized value can be presented as:<br />
2 3<br />
t1<br />
= Msin(60 °− α ref<br />
)<br />
(3.15a)<br />
π<br />
2 3<br />
t2<br />
= Msinα<br />
ref<br />
(3.15b)<br />
π<br />
or in other form:<br />
48
Voltage source PWM inverter for PMSM supply<br />
2 3<br />
t2<br />
= Msinα<br />
ref<br />
(3.16a)<br />
π<br />
3 1<br />
t1 = M cosαref<br />
− t2<br />
(3.16b)<br />
π<br />
2<br />
After t 1<br />
and t 2<br />
calculation, the remaining normalized time is reserved for zero vectors<br />
U<br />
0<br />
, U7<br />
<strong>with</strong> condition t 1<br />
+ t 2<br />
≤ 1.Therefore, the normalized total time for zero vectors<br />
becomes:<br />
t = t + t = 1 − ( t + t )<br />
(3.17)<br />
07 0 7 1 2<br />
The equations (3.15a-b) for time interval <strong>of</strong> active vectors and equation (3.17) for total<br />
time interval <strong>of</strong> zero vectors are identical for all variants <strong>of</strong> space vector modulation<br />
(<strong>SVM</strong>) techniques.<br />
The absence <strong>of</strong> neutral wire in star connected load provides a degree <strong>of</strong> freedom in<br />
selecting the partitioning (zero sequence signals -ZSS) time <strong>of</strong> the two zero vectors. It is<br />
equivalent to the freedom <strong>of</strong> injected signals in to phase signals. Therefore, it gives<br />
different equations <strong>of</strong> t 0<br />
and t 7<br />
for each PWM method, but normalized duration time <strong>of</strong><br />
must fulfill condition in Eq. 3.17. As a consequence is only in different placement <strong>of</strong><br />
zero vectors U<br />
0<br />
, U<br />
7<br />
. Therefore, we can introduce the portioning factor <strong>of</strong> zero vectors,<br />
which is defined as:<br />
t t<br />
k = =<br />
t + t t<br />
7 7<br />
0 7 07<br />
(3.18)<br />
Please note that, the zero sequence signals does not change the inverter output line-toline<br />
voltage.<br />
From knowledge <strong>of</strong> the neutral voltage U<br />
N 0<br />
(see Fig. 3.4) and information what kind <strong>of</strong><br />
zero sequence signal (ZSS) will be injected in each phase <strong>of</strong> motor it is possible to<br />
calculate normalized duration time <strong>of</strong> zero vectors t 0<br />
and t 7<br />
. In bellow Table 3.1 are<br />
summarized different three-phase modulation techniques and remarks. Also graphical<br />
interpretation are shown in Fig. 3.19.<br />
49
Voltage source PWM inverter for PMSM supply<br />
Zero Sequence Signal Time interval <strong>of</strong> zero vectors Remark<br />
SPWM<br />
ZSS =0<br />
for even sectors<br />
4<br />
t0<br />
= (1 − ( M cos θ )) / 2<br />
π<br />
for odd sectors<br />
4<br />
t0 = t07<br />
−(1 −( M cos θ )) / 2<br />
π<br />
and t7 = t07 − t0<br />
U<br />
DC<br />
/2 π<br />
M = = = 0.785<br />
2U<br />
DC 4<br />
π<br />
complicated calculation<br />
for time interval <strong>of</strong><br />
vectors<br />
THIPWM<br />
ZSS =sinusoidal signal<br />
<strong>with</strong> triple harmonic<br />
for even sectors<br />
4 1<br />
t0<br />
= (1 −( M cos θ) − cos 3 θ)) / 2<br />
π 6<br />
for odd sectors<br />
4 1<br />
t0 = t07<br />
−(1 −( M cos θ) − cos 3 θ)) / 2<br />
π 6<br />
and t7 = t07 − t0<br />
U<br />
DC<br />
/ 3 π<br />
M = = = 0.907<br />
2U<br />
DC 2 3<br />
π<br />
Increase the linearity <strong>of</strong><br />
inverter grater than 15%<br />
<strong>of</strong> SPWM<br />
SVPWM<br />
ZSS=triangle signal<br />
<strong>with</strong> triple harmonic<br />
0.906<br />
for all sectors<br />
M =<br />
t = t t t /2<br />
time interval<br />
7 07 0 07 t = t /2<br />
0 07 Simple calculation for<br />
k=0.5<br />
Table 3.1. Variants <strong>of</strong> three-phase space vector modulation techniques.<br />
a) b)<br />
c)<br />
the portioning<br />
factor <strong>of</strong> zero vectors<br />
Fig. 3.19 PWM techniques <strong>with</strong> various zero signal sequence shape: a) SPWM, b) THIPWM,<br />
c) SVPWM. The upper part <strong>of</strong> figure: phase voltage U<br />
AN<br />
(green), pole voltage U<br />
AO<br />
(blue),<br />
voltage between neutral points U<br />
NO<br />
(black). The lower part <strong>of</strong> figure shows the portioning<br />
factor <strong>of</strong> zero vectors for all PWM techniques.<br />
50
Voltage source PWM inverter for PMSM supply<br />
Except SPWM technique all PWM method guarantee that the ZSS extends the range <strong>of</strong><br />
modulation index from 0.78 to 0.906, i.e. 15% greater than that obtained <strong>with</strong> standard<br />
version <strong>of</strong> SPWM.<br />
The duty time cycles in sector 1 for each phase can be written:<br />
d = t + t + kt<br />
(3.19a)<br />
a<br />
b<br />
1 2 07<br />
d = t + kt<br />
(3.19b)<br />
dc<br />
2 07<br />
= kt<br />
(3.19c)<br />
07<br />
for all sectors the duty time calculations for each phase can be calculated:<br />
sec tor1 sec tor 2 sector 3 sec tor 4 sector5 sec tor 6<br />
⎡<br />
⎤<br />
⎡d 1 1 1 0 0 0 0 0 0 1 1 1<br />
a ⎤ k k k k k k t<br />
⎢<br />
⎥ ⎡ 1 ⎤<br />
⎢<br />
d<br />
⎥<br />
= ⎢0 1 k 1 1 k 1 1 k 1 0 k 0 0 k 0 0 k ⎥ ⎢<br />
t<br />
⎥<br />
b<br />
2<br />
⎢ ⎥ ⎢<br />
⎥ ⎢ ⎥<br />
⎢⎣d ⎥ 0 0 k 0 0 k 0 1 k 1 1 k 1 1 k 1 0 k ⎢t<br />
⎥<br />
c ⎦ ⎣ 07 ⎦<br />
⎢<br />
⎣<br />
T<br />
⎥<br />
⎦<br />
(3.20)<br />
Depending on the location <strong>of</strong> the space vector, the basic vectors must be chosen in order<br />
to get the minimum number <strong>of</strong> changes in the switches <strong>of</strong> the converter. The switching<br />
sequence for each sector and suitable pulse pattern for first sector are shown in Fig.<br />
3.20.<br />
Sector<br />
Three-phase <strong>Modulation</strong><br />
T S<br />
/2<br />
T S<br />
t<br />
0 t1<br />
2 7<br />
t<br />
t<br />
1 U0, U1, U2, U7, U2, U1,<br />
U<br />
0<br />
2 U0, U3, U2, U7, U2, U3,<br />
U<br />
0<br />
3 U0, U3, U4, U7, U4, U3,<br />
U<br />
0<br />
4 U0, U5, U4, U7, U4, U5,<br />
U<br />
0<br />
5 U0, U5, U6, U7, U6, U5,<br />
U<br />
0<br />
6 U0, U1, U6, U7, U6, U1,<br />
U<br />
0<br />
0 1 1 1 1 1 1 0<br />
0 0 1 1 1 1 0 0<br />
0 0 0 1 1 0 0 0<br />
U0<br />
U1<br />
U2<br />
U7<br />
U<br />
2<br />
U1<br />
U0<br />
d A<br />
d B<br />
d C<br />
Fig. 3.20 Switching sequence for three-phase PWM techniques (on the left ) and pulse pattern<br />
<strong>of</strong> three-phase vector modulator in sector 1 (on the right).<br />
51
Voltage source PWM inverter for PMSM supply<br />
3.4 Summary<br />
The general conclusions from this chapters can be summarized as follows:<br />
‣ to supply the voltage source inverter can be used diode rectifier or active<br />
rectifier <strong>with</strong> IGBT transistors,<br />
‣ supplied PMSM machine from VSI do not required mechanical comutator. It is<br />
thanks to electronic commutation. This overcome the problem <strong>with</strong> brushes and<br />
periodical service,<br />
‣ The voltage source inverter is non-linear power amplifier in respect to:<br />
o dead time effect,<br />
o voltage drop across the power devices,<br />
o DC link voltage pulsation.<br />
‣ Without appropriate dead-time and voltage drop compensation, the sensorless<br />
operation in low speed range is not possible.<br />
‣ The quality <strong>of</strong> DC link voltage has influence on proper operation <strong>of</strong> AC drive,<br />
‣ Sinusoidal modulation technique (SPWM) guarantees the inverter output voltage<br />
U<br />
DC<br />
amplitude from 0 until V. This correspond to changes modulation index<br />
2<br />
from 0-0.785.<br />
‣ <strong>Modulation</strong> techniques <strong>with</strong> zero sequence signals not equal zero guarantees the<br />
U<br />
DC<br />
inverter output voltage amplitude from 0 until V (it is 15% grater than<br />
3<br />
SPWM). This correspond to changes modulation index from 0-0.907.<br />
‣ In this work the PWM modulator <strong>with</strong> zero sequence signal <strong>of</strong> triple harmonics<br />
(SVPWM) will be used. Mainly because <strong>of</strong> simple calculation <strong>of</strong> zero vector<br />
duration and placement.<br />
52
<strong>Control</strong> methods <strong>of</strong> PM Synchronous motor<br />
Chapter 4 CONTROL METHODS OF PM SYNCHRONOUS MOTOR<br />
4.1 Introduction<br />
The basic block scheme <strong>of</strong> adjustable speed drive <strong>with</strong> control block for PMSM is<br />
presented in Fig. 4.2. It consists <strong>of</strong> two parts: power (fat line) and control part<br />
employed microprocessor (thin line). The first one previously have been explained in<br />
chapter 3. The second one will be described bellow.<br />
Figure 4.2. The basic block scheme <strong>of</strong> PMSM drive supplied voltage source inverter.<br />
The main task <strong>of</strong> control block is follow demand reference speed by motor and provide<br />
proper operation in static (insight <strong>of</strong> the limits) and dynamic states <strong>with</strong>out any<br />
instability. This is ensured through suitable generated gate signals for the IGBT<br />
transistor inside <strong>of</strong> the inverter. Therefore, to make good decision how to control power<br />
transistors in the inverter, the following feedback signals are measured and used:<br />
• DC link voltage,<br />
• motor phase currents,<br />
• speed or position <strong>of</strong> the rotor.<br />
This significantly improve dynamic behavior <strong>of</strong> the system (good performance <strong>of</strong> the<br />
torque and speed response, very fast dynamics response <strong>with</strong> fully controllable torque in<br />
wide speed range).<br />
The scalar control for PMSM <strong>with</strong>out damper winding (squire cage) is not simple as for<br />
induction motor [39,40]. It requires additional stabilization loop, which can be provide<br />
by feedback loop from: rotor velocity perturbation, active power or DC-link current<br />
perturbation [9].<br />
The vector control method will be described bellow.<br />
53
<strong>Control</strong> methods <strong>of</strong> PM Synchronous motor<br />
4.2 Field oriented control (FOC)<br />
During many years a DC motor has been mostly used. Because <strong>of</strong> simple control<br />
method, which based on fact that flux and torque can be controlled separately using<br />
current control loop <strong>with</strong> PI controllers. However the weak point <strong>of</strong> this drive was DC<br />
motor, which could not worked in aggressive or volatile environment and required<br />
cyclical maintenance. This disadvantages has been eliminated, when instead <strong>of</strong> DC<br />
machine a three phase PMSM motor were used.<br />
In searching new control method for induction machine in 1971 was developed vector<br />
control method known as field oriented control (FOC) [31,38,49]. This method allows<br />
control the flux and the torque in the AC machine in similar way as for DC motor. It<br />
was achieved by transform current vector in stationary reference frame ( α,<br />
β ) into new<br />
coordinate system ( dq) , <strong>with</strong> respect to rotor (magnet) flux vector. So the flux<br />
produced by permanent magnet is frozen to the direct axis <strong>of</strong> the rotor (see Fig. 4.4).<br />
q−axis<br />
β −axis<br />
I s<br />
I sq<br />
δ I<br />
γ I<br />
I sd<br />
γ m<br />
Ω s<br />
Ψ PM<br />
d−axis<br />
α −axis<br />
Figure 4.4 <strong>Vector</strong> diagram illustrated the principle <strong>of</strong> FOC.<br />
Further, stator current vector can be split into two current components: flux current I sd<br />
and torque producing current I sq<br />
. In analogy to separate commutator motor, the flux<br />
current components corresponds to excitation current and torque-producing current<br />
corresponds to the armature current. Therefore, the goal <strong>of</strong> the control system is to<br />
reference the<br />
I<br />
sd_<br />
ref<br />
,<br />
sq_<br />
ref<br />
I stator current components in respect to requirement <strong>of</strong><br />
references torque and flux. The flux and torque producing stator current references are<br />
obtained on the output <strong>of</strong> the reference current generation block (see Fig. 4.5).<br />
54
<strong>Control</strong> methods <strong>of</strong> PM Synchronous motor<br />
Reference Current<br />
Calculation<br />
Reference Current<br />
Calculation<br />
reference<br />
torque<br />
M e _ ref<br />
FG1<br />
FG2<br />
I sd _ ref<br />
reference<br />
d-axis current<br />
I sq _ ref<br />
reference<br />
q-axis current<br />
reference<br />
torque<br />
M e _ ref<br />
FG3<br />
FG3<br />
δ I<br />
I s<br />
γ I<br />
Figure 4.5. Reference current generator block for FOC technique<br />
a) in cartesian form, b) in polar form.<br />
γ m<br />
Function generation FG1 gives the relationship between the torque and the direct axis<br />
stator current component<br />
I<br />
sd_<br />
ref<br />
, and function generator FG2 gives the relationship<br />
between the torque and the quadrature axis stator current<br />
I<br />
sq_<br />
ref<br />
. His graphical<br />
illustration in Fig. 4.6 are presented.<br />
0.5<br />
0<br />
CTA<br />
I [ . ]<br />
sdN<br />
pu<br />
0.5<br />
1<br />
CSF<br />
MTPA<br />
1.5<br />
2<br />
UPF<br />
2.5<br />
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2<br />
M<br />
eN<br />
[ pu . ]<br />
2.5<br />
I [ . ]<br />
sqN<br />
pu<br />
2<br />
1.5<br />
1<br />
CSF<br />
CTA<br />
MTPA<br />
0.5<br />
UPF<br />
0<br />
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2<br />
M<br />
eN<br />
[ pu . ]<br />
Figure 4.6. Generated current components I<br />
sd<br />
and I sq<br />
dependent on required electromagnetic<br />
torque under difference control strategies. CTA-constant torque angle control, MTPA-maximum<br />
torque per ampere control, CSF-constant stator flux control, UPF – unity power factor control.<br />
55
<strong>Control</strong> methods <strong>of</strong> PM Synchronous motor<br />
For many years the CTA control ( I<br />
_<br />
= 0 ) method has been a popular technique for a<br />
sd<br />
ref<br />
long time because <strong>of</strong> simple control. This method was dedicated for surface permanent<br />
magnet synchronous motor (SPMSM), where the magnetic saliency does not exist<br />
[126]. So the maximum torque per ampere is obtained, when stator current vector is<br />
shifted in respect to rotor flux vector 90 degree. However, in IPMSM, maximum torque<br />
per ampere is obtained <strong>with</strong> torque angle more than 90 degree. This is because <strong>of</strong><br />
existence <strong>of</strong> reluctance torque component due to magnetic saliency (see subchapter<br />
2.2.2). Therefore, the I<br />
sd_<br />
ref<br />
should be negative value [44].<br />
The main question is how or in which manner produce the reference currents in d-q<br />
frame. Its leads to many realization <strong>of</strong> current control structure. Among them generally<br />
we can distinguish two structures <strong>of</strong> current control loop. One <strong>of</strong> them is hysteresis<br />
based control (Fig.4.7a) [3,52] and the second one is PI based current controllers<br />
(Fig.4.7b).<br />
Hysteresis based current control has following disadvantage such as [3]:<br />
• measurement <strong>of</strong> three phase currents are required,<br />
• three independent hysteresis current controllers are required,<br />
• variable switching frequency is achieved,<br />
• fast sampling time is required.<br />
All this listed above disadvantage can be eliminate, when the PI current control are<br />
used. This structure are mostly used in industrial application (Fig. 4.7b).<br />
a)<br />
U DC<br />
M e _ ref<br />
Fig. 4.5<br />
I d _ ref<br />
I q _ ref<br />
d,q/ABC<br />
I A _ ref<br />
I B _ ref<br />
I C _ ref<br />
−<br />
−<br />
S A<br />
S B<br />
S C<br />
Inverter<br />
current<br />
feedback<br />
−<br />
I A current<br />
I<br />
sensors<br />
B<br />
I C<br />
rotor<br />
position<br />
sensor<br />
PMSM<br />
56
<strong>Control</strong> methods <strong>of</strong> PM Synchronous motor<br />
b)<br />
U DC<br />
M e _ ref<br />
Fig. 4.5<br />
I sd _ ref<br />
I sq _ ref<br />
-<br />
e Isd<br />
-<br />
e Isq<br />
PI<br />
PI<br />
U sd<br />
U sq<br />
Reference<br />
Voltage<strong>Vector</strong><br />
Calculation<br />
U s _ ref<br />
ϕ Us _ ref<br />
<strong>Space</strong><br />
<strong>Vector</strong><br />
Modulator<br />
S A<br />
S B<br />
S C<br />
Inverter<br />
γ s<br />
I sd<br />
I sq<br />
dq/ABC<br />
current<br />
sensors<br />
I s<br />
γ m<br />
rotor<br />
position<br />
sensor<br />
PMSM<br />
Figure 4.7 <strong>Vector</strong> control structure for PM synchronous motor <strong>with</strong>: a) hysteresis current<br />
control, b) synchronous PI current control<br />
4.3 <strong>Direct</strong> torque control (<strong>DTC</strong>)<br />
The name direct torque control is deliver by the fact that, on the basis <strong>of</strong> the errors<br />
between the reference and the estimated values <strong>of</strong> torque and flux, it is possible to<br />
directly control the inverter states <strong>with</strong>out inner current control loop as for FOC<br />
[32,34,35,50] and [57-66].<br />
The basic idea <strong>of</strong> this control rely on stator voltage vector equation <strong>of</strong> AC motor.<br />
U<br />
dΨ<br />
s<br />
s<br />
= RsIs<br />
+ (4.1)<br />
dt<br />
Making the assumption that ohmic voltage drop on the stator resistor can be neglected<br />
the equation for stator flux vector takes the form:<br />
Ψ =∫ ( U ) dt<br />
(4.2)<br />
s<br />
s<br />
It can be said that the stator voltage vector has directly influence on control stator flux<br />
vector. Using a three phase voltage source inverter to supply the AC motor, there are six<br />
non-zero vectors and two zero voltage vectors. The active vectors change the amplitude<br />
and position <strong>of</strong> stator flux vector, while the zero vectors stop the stator flux vector as<br />
shown in Fig. 4.8.<br />
57
<strong>Control</strong> methods <strong>of</strong> PM Synchronous motor<br />
y<br />
moment <strong>with</strong> active<br />
forward vector<br />
stops <strong>with</strong> zero<br />
vector<br />
Ψ s<br />
moment <strong>with</strong> active<br />
backward vector<br />
γ s<br />
δ Ψ<br />
Ψ =Ψ<br />
r<br />
PM<br />
x<br />
rotates<br />
continuously<br />
α<br />
stator<br />
Figure. 4.7 Stator flux vector Ψ<br />
s<br />
movement relative to rotor flux vector Ψ<br />
r<br />
=ΨPM<br />
under the<br />
influence <strong>of</strong> active and zero inverter voltage vectors.<br />
Therefore, it is possible control the torque angle δ Ψ<br />
across control stator flux vector<br />
position in respect to rotor flux vector produced by permanent magnet Ψ<br />
r<br />
=Ψ<br />
PM<br />
Ψ<br />
s<br />
, what<br />
further allows to have impact on control the electromagnetic torque in accordance <strong>with</strong><br />
following formula:<br />
M<br />
e<br />
3 Ψ sin<br />
2 s<br />
ΨPM<br />
δΨ Ψs<br />
( Lq<br />
−Ld)sin2δ<br />
Ψ<br />
= pb[ − ]<br />
(4.3)<br />
2 L<br />
2L L<br />
d d q<br />
Generally the <strong>DTC</strong> technique operate at constant stator flux amplitude<br />
Ψ<br />
s<br />
, what<br />
correspond to CSF operation, because <strong>of</strong> simple reference stator flux amplitude equal<br />
nominal value <strong>of</strong> permanent magnet. For <strong>DTC</strong> technique can be also apply all control<br />
strategies discussed in Chapter 2.<br />
Using the block generator for reference stator flux amplitude and electromagnetic<br />
torque as is shown in Fig 4.8. it is possible to draw the relationship between required<br />
Reference<br />
Flux<br />
Calculation<br />
reference<br />
flux<br />
Ψ s _ ref<br />
M<br />
e_<br />
ref<br />
reference<br />
torque<br />
reference<br />
torque<br />
M e _ ref<br />
Figure 4.8. Reference flux and torque generator block for <strong>DTC</strong> technique.<br />
reference flux and torque. His graphical illustration in Fig. 4.9 is presented.<br />
58
<strong>Control</strong> methods <strong>of</strong> PM Synchronous motor<br />
Ψ<br />
sN<br />
[ pu]<br />
2<br />
1.5<br />
1<br />
CTA<br />
MTPA<br />
CSF<br />
0.5<br />
UPF<br />
0<br />
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2<br />
M<br />
eN<br />
[ pu]<br />
Figure 4.9. Generated stator flux amplitude Ψ s _ ref dependent on required electromagnetic<br />
torque under difference control strategies. CTA-constant torque angle control, MTPA-maximum<br />
torque per ampere control, CSF-constant stator flux control, UPF – unity power factor control.<br />
The basic structure <strong>of</strong> direct flux and torque control voltage-sourced PWM inverter-fed<br />
permanent magnet synchronous motor is shown in Fig.4.10.[3,36,37,41,45]<br />
Flux<br />
hysteresis<br />
U DC<br />
Ψ s _ ref<br />
M e _ ref<br />
−<br />
−<br />
H Ψ s<br />
H m<br />
d Ψ s<br />
d M e<br />
Switching<br />
Table<br />
<strong>Torque</strong><br />
sector γ<br />
histeresis s ( N )<br />
S A<br />
S B<br />
S C<br />
Inverter<br />
Ψ s<br />
M e<br />
Flux and<br />
<strong>Torque</strong><br />
Estimation<br />
I s<br />
γ m<br />
PMSM<br />
Figure. 4.10. Block diagram <strong>of</strong> switching table based direct torque control ST-<strong>DTC</strong>.<br />
The command stator flux amplitude<br />
Ψ<br />
s _ ref<br />
and electromagnetic torque<br />
e_<br />
ref<br />
M values<br />
are compared <strong>with</strong> the actual<br />
Ψ s<br />
and M e<br />
values, in hysteresis flux and torque<br />
controllers, respectively. The flux and the torque controllers are a two-level<br />
comparators.<br />
The digitized outputs signals <strong>of</strong> the flux controllers are defined as:<br />
d = 1 (increase flux) for Ψ > Ψ<br />
_<br />
+ (4.4a)<br />
Ψ s<br />
s s ref<br />
H Ψ<br />
d = 0 (decrease flux) for Ψ < Ψ<br />
_<br />
− (4.4b)<br />
Ψ s<br />
s s ref<br />
H Ψ<br />
59
<strong>Control</strong> methods <strong>of</strong> PM Synchronous motor<br />
and those <strong>of</strong> the torque controller as<br />
where<br />
d = 1(increase torque) for M > M<br />
_<br />
+ H<br />
(4.5a)<br />
M e<br />
e e ref m<br />
d = 0 (decrease torque) for M < M<br />
_<br />
− H<br />
(4.5b)<br />
M e<br />
e e ref m<br />
H<br />
m<br />
and H Ψ<br />
are hysteresis bands for torque and flux, respectively.<br />
The digitized variables d Ψ<br />
, d and the stator flux position sector γ M<br />
s( N )<br />
information<br />
s<br />
e<br />
create a digital word, which select appropriate voltage vector from the switching table.<br />
Next, from the selection table the proper voltage vectors are selected and the gate pulses<br />
SA, SB,<br />
S<br />
C<br />
to control the power switches in the inverter are generated.<br />
The circular stator flux vector trajectory can be divided into six symmetrical sectors<br />
(according to the non zero voltage vectors), which are defined as (see Fig. 4.11):<br />
sector 1 − 30°≤ γ s<br />
< 30°<br />
V 3 (010)<br />
V 2 (110)<br />
sector 2 30°< γ s<br />
≤ 90°<br />
sector 3 90°< γ s<br />
≤ 150°<br />
sector 4 150°< γ s<br />
≤ 210°<br />
sector 5 210°< γ s<br />
≤ 270°<br />
sector 6 270°< γ s<br />
<strong>Control</strong> methods <strong>of</strong> PM Synchronous motor<br />
β<br />
Ψ<br />
s<br />
↓<br />
M<br />
e<br />
↑<br />
Ψ<br />
s<br />
↑<br />
M<br />
e<br />
↑<br />
Ψ<br />
s<br />
↓<br />
Ψs<br />
↑<br />
Ψ s<br />
γ<br />
s γ<br />
s<br />
α<br />
Ψ<br />
s<br />
↓<br />
M<br />
e<br />
↓<br />
Ψ<br />
s<br />
↑<br />
M<br />
e<br />
↓<br />
Figure. 4.12 Voltage vector effects in sector 1 on stator flux and torque.<br />
The presented rule for first sector can be extended for other sectors, what further help<br />
construct the switching Tables 4.1 and 4.2 as below [56]<br />
Flux <strong>Torque</strong> sector 1 sector 2 sector 3 sector 4 sector 5 sector 6<br />
d ψs =1<br />
d me. =1 V<br />
2<br />
V<br />
3<br />
V<br />
4<br />
V<br />
5<br />
V<br />
6<br />
V<br />
1<br />
d me =0 V<br />
6<br />
V<br />
1<br />
V<br />
2<br />
V<br />
3<br />
V<br />
4<br />
V<br />
5<br />
d ψs =0<br />
d me. =1 V<br />
3<br />
V<br />
4<br />
V<br />
5<br />
V<br />
6<br />
V<br />
1<br />
V<br />
2<br />
d me =0 V<br />
5<br />
V<br />
6<br />
V<br />
1<br />
V<br />
2<br />
V<br />
3<br />
V<br />
4<br />
Table 4.1. Switching table for <strong>DTC</strong> <strong>with</strong> active vectors.<br />
Flux <strong>Torque</strong> sector 1 sector 2 sector 3 sector 4 sector 5 sector 6<br />
d ψs =1<br />
d me. =1 V<br />
2<br />
V<br />
3<br />
V<br />
4<br />
V<br />
5<br />
V<br />
6<br />
V<br />
1<br />
d me =0 V<br />
7<br />
V<br />
0<br />
V<br />
7<br />
V<br />
0<br />
V<br />
7<br />
V<br />
0<br />
d ψs =0<br />
d me. =1 V<br />
3<br />
V<br />
4<br />
V<br />
5<br />
V<br />
6<br />
V<br />
1<br />
V<br />
2<br />
d me =0 V<br />
0<br />
V<br />
7<br />
V<br />
0<br />
V<br />
7<br />
V<br />
0<br />
V<br />
7<br />
Table 4.2. Switching table for <strong>DTC</strong> <strong>with</strong> zero and active vectors.<br />
Tables 4.1 and 4.2 represent the eight and six voltage-vectors switching tables.<br />
61
<strong>Control</strong> methods <strong>of</strong> PM Synchronous motor<br />
The <strong>DTC</strong> has lesser parameter dependence and fast torque when compare <strong>with</strong> the<br />
torque control via PWM current control.<br />
Among the well-know advantages <strong>of</strong> the <strong>DTC</strong> scheme are the following:<br />
• Simple control,<br />
• Excellent torque dynamics,<br />
• Absence <strong>of</strong> coordinate transformations,<br />
• Absence <strong>of</strong> separate voltage modulation block,<br />
• Absence <strong>of</strong> voltage decoupling circuits,<br />
• There are no current control loops, hence, the current is not regulated directly,<br />
• Stator flux vector and torque estimation is required.<br />
Among the well-know disadvantages <strong>of</strong> the <strong>DTC</strong> scheme are the following:<br />
• variable switching frequency (difficulties <strong>of</strong> LC input EMI filter design),<br />
• high sampling time is required (fast microprocessor and A/D converter ),<br />
• inverter switching frequency depending on: flux and torque hysteresis bands,<br />
machine parameters, sampling frequency,<br />
• violence <strong>of</strong> polarity consistency rules (huge voltage stress for IGBT transistor),<br />
• current and torque distortion caused by sector changes,<br />
• start and low speed operation problems,<br />
• high sampling frequency needed for digital implementation <strong>of</strong> hysteresis<br />
comparators,<br />
• high noisy level,<br />
• high current and torque ripple.<br />
Many modifications <strong>of</strong> the basic switching table based direct torque control (ST-<strong>DTC</strong>)<br />
scheme at improving starting, overload condition, very low speed operation, torque<br />
ripple reduction, variable switching frequency functioning, and noise level attenuation<br />
have been proposed during last decade.<br />
In the last five years, many researcher have been carried out to try solve the above<br />
mentioned problems <strong>of</strong> ST-<strong>DTC</strong> scheme. Therefore, the following solutions have been<br />
developed in order to eliminated mentioned before problems:<br />
62
<strong>Control</strong> methods <strong>of</strong> PM Synchronous motor<br />
• Use <strong>of</strong> improved switching table,<br />
• Use <strong>of</strong> comparators <strong>with</strong> and <strong>with</strong>out hysteresis, at two or three levels,<br />
• Use <strong>of</strong> multi-level inverter,<br />
• Introduction <strong>of</strong> fuzzy or neuro-fuzzy techniques,<br />
• Use <strong>of</strong> sophisticated flux estimators to improve the low speed behavior,<br />
• Implementation <strong>of</strong> <strong>DTC</strong> schemes <strong>with</strong> constant switching frequency operation<br />
In multi-level inverter there will be more voltage vectors available to control the flux<br />
and torque. Therefore, a smoother torque can be expected. However, more power<br />
switches are needed to achieved a lower ripple, which will increase the system cost and<br />
complexity.<br />
All this contributions allow the <strong>DTC</strong> performance to be improved, but at the same time<br />
they lead to more complex schemes. As expected, conventional <strong>DTC</strong> is growing in<br />
field-oriented control area and the so-called improved <strong>DTC</strong> <strong>with</strong> space vector<br />
modulation (<strong>SVM</strong>). Let us call it <strong>DTC</strong>-<strong>SVM</strong>. This control concept will be deeply<br />
discussed in the next Chapter.<br />
63
<strong>Control</strong> methods <strong>of</strong> PM Synchronous motor<br />
4.4 Summary<br />
‣ In FOC drive flux linkage and electromagnetic torque are controlled indirectly<br />
and independently by PI controllers <strong>with</strong> space vector modulator (<strong>SVM</strong>). In this<br />
control concept current control loop is required,<br />
‣ In <strong>DTC</strong> drive, flux linkage and electromagnetic torque are controlled directly<br />
and independently by hysteresis controllers and selection <strong>of</strong> optimum inverter<br />
switching modes. In this control concept flux and torque control loop is required,<br />
‣ In <strong>DTC</strong> all switch changes <strong>of</strong> the inverter are based on the electromagnetic state<br />
<strong>of</strong> the motor.<br />
‣ The <strong>DTC</strong> technique is different from the traditional methods <strong>of</strong> controlling<br />
torque, where the current controllers in the rotor reference frame are used. It is<br />
completely different control concept (approach) from FOC. The new control<br />
technology was characterized by simplicity, good performance and robustness,<br />
because <strong>of</strong> bang-bang hysteresis control. Using <strong>DTC</strong> it is possible to obtain a<br />
good dynamic control <strong>of</strong> the torque <strong>with</strong>out current controllers and any<br />
mechanical transducers on the machine shaft. Moreover, in this control structure<br />
the PWM modulator is not required. Its is occupied by variable switching<br />
frequency.<br />
‣ The flux weakening control becomes easier because stator flux linkage can be<br />
controlled directly in the <strong>DTC</strong> system <strong>of</strong> PMSM.<br />
64
<strong>Direct</strong> <strong>Torque</strong> <strong>Control</strong> <strong>with</strong> <strong>Space</strong> <strong>Vector</strong> <strong>Modulation</strong><br />
Chapter 5<br />
DIRECT TORQUE CONTROL WITH SPACE VECTOR MODULATION<br />
(<strong>DTC</strong>-<strong>SVM</strong>)<br />
5.1 Introduction<br />
The <strong>DTC</strong>-<strong>SVM</strong> greatly improves torque and flux performance by:<br />
• Achieved fixed switching frequency,<br />
• Reduced torque and flux ripples.<br />
The main idea <strong>of</strong> <strong>DTC</strong>-<strong>SVM</strong> is based on analyze <strong>of</strong> the torque equation<br />
M<br />
Assuming that the Ld = Lq = Ls<br />
e<br />
3 Ψ sin<br />
2 s<br />
ΨPM<br />
δΨ Ψs<br />
( Lq<br />
−Ld)sin2δ<br />
Ψ<br />
= pb[ − ]<br />
(5.1)<br />
2 L<br />
2L L<br />
d d q<br />
M<br />
e<br />
3 Ψs<br />
ΨPM<br />
sinδ = pb[ Ψ<br />
]<br />
(5.2)<br />
2 L<br />
From equation (5.2) we can see that for constant stator flux amplitude<br />
by permanent magnet<br />
Ψ<br />
PM<br />
s<br />
Ψ<br />
s<br />
and flux produced<br />
, the electromagnetic torque can be changed by control <strong>of</strong> the<br />
torque angle δ Ψ<br />
. This is the angle between the stator and rotor flux linkage, when the stator<br />
resistance is neglected. The torque angle, in turn, can be changed by changing position <strong>of</strong> the<br />
stator flux vector θ Ψ s<br />
in respect to PM vector using the actual voltage vector supplied by<br />
PWM inverter.<br />
In the steady state, δ Ψ<br />
is constant and corresponds to a load torque, whereas stator and rotor<br />
flux rotate at synchronous speed. In transient operation, δ Ψ<br />
varies and the stator and rotor<br />
flux rotate at different speeds (Fig. 5.1).<br />
65
<strong>Direct</strong> <strong>Torque</strong> <strong>Control</strong> <strong>with</strong> <strong>Space</strong> <strong>Vector</strong> <strong>Modulation</strong><br />
q − axis<br />
β<br />
δ Ψ<br />
Ψ s _ ref<br />
∆δ Ψ<br />
θ Ψ<br />
s<br />
Ψ s<br />
Ψ PM<br />
d − axis<br />
θ r<br />
Figure 5.1. <strong>Space</strong> vector diagram illustrating torque control conditions.<br />
α<br />
5.2 Cascade structure <strong>of</strong> <strong>DTC</strong>–<strong>SVM</strong> scheme<br />
The structure <strong>of</strong> proposed control scheme is shown in the Fig. 5.2. [11,33,42,48,51,53,54]<br />
U DC<br />
Ψ s _ ref<br />
U<br />
s α _ ref<br />
S A<br />
S B<br />
M e _ ref<br />
e M<br />
∆δ Ψ<br />
U<br />
s β _ ref<br />
S C<br />
θ Ψs Ψs<br />
I s<br />
M e<br />
I s<br />
γ m<br />
Figure 5.2. Cascade structure <strong>of</strong> <strong>DTC</strong>-<strong>SVM</strong> scheme.<br />
The error between reference and measured torque can be expressed as:<br />
Ψ Ψ sin( δ +∆δ ) Ψ Ψ sinδ<br />
eM = M − M = p<br />
− (5.3)<br />
3 s_<br />
ref PM Ψ Ψ<br />
_<br />
[<br />
s PM Ψ<br />
e ref e b<br />
]<br />
2<br />
Ls<br />
Ls<br />
From equation (5.3) we can see that the relation between torque error and increment <strong>of</strong> load<br />
angel<br />
∆δ Ψ<br />
is nonlinear. Therefore, we used PI controller which generates the load angel<br />
66
<strong>Direct</strong> <strong>Torque</strong> <strong>Control</strong> <strong>with</strong> <strong>Space</strong> <strong>Vector</strong> <strong>Modulation</strong><br />
increment required to minimize the instantaneous error between reference<br />
M and actual<br />
e_<br />
ref<br />
M<br />
e<br />
torque.<br />
In control scheme <strong>of</strong> Fig. 5.2 the torque error signal e M<br />
is delivered to the PI controller,<br />
which determines the increment <strong>of</strong> torque angle<br />
∆ δ Ψ<br />
. Based on this signal and reference<br />
amplitude <strong>of</strong> stator flux<br />
Ψ , the reference voltage vector in stator coordinates α,<br />
β is<br />
s _ ref<br />
calculated. The calculation block <strong>of</strong> reference voltage vector also uses information about the<br />
actual stator flux vector (amplitude<br />
Ψ<br />
s<br />
and position θ Ψ s<br />
) as well as measured current vector<br />
I<br />
s<br />
. The reference stator voltage vector is delivered to space vector modulator (<strong>SVM</strong>), which<br />
generates the switching signals<br />
S , S , S for power transistors <strong>of</strong> inverter.<br />
A<br />
B<br />
C<br />
The calculation block <strong>of</strong> reference voltage vector is shown in Fig. 5.3.<br />
Ψ s _ ref<br />
RsI s α<br />
Ψ<br />
s α _ ref<br />
∆Ψ sα<br />
U α<br />
_<br />
s<br />
_ ref<br />
∆δ Ψ<br />
−<br />
Ψ<br />
s β _ ref<br />
∆Ψ sβ<br />
−<br />
U<br />
s β ref<br />
θ Ψs<br />
Ψ sα<br />
Ψ sβ<br />
Rs<br />
Is<br />
β<br />
Figure 5.3. Calculation block <strong>of</strong> reference voltage vector.<br />
Based on<br />
∆ signal, reference <strong>of</strong> stator flux amplitude Ψ<br />
s _ ref<br />
and measured stator flux<br />
δ Ψ<br />
vector position<br />
θ Ψ s<br />
(Fig. 5.3.), the reference flux components<br />
Ψ Ψ in stator<br />
s α _ ref<br />
,<br />
sβ<br />
_ ref<br />
coordinate system are calculated as:<br />
Ψ = Ψ cos( θ +∆δ<br />
)<br />
sα<br />
_ ref s_<br />
ref Ψs<br />
Ψ = Ψ sin( θ +∆δ<br />
)<br />
sβ<br />
_ ref s _ ref Ψs<br />
Ψ<br />
Ψ<br />
(5.4)<br />
Pleas note that for constant flux operation region the reference value <strong>of</strong> stator flux amplitude<br />
Ψ is equal flux amplitude <strong>of</strong> permanent magnet Ψ<br />
PM<br />
.<br />
s _ ref<br />
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<strong>Direct</strong> <strong>Torque</strong> <strong>Control</strong> <strong>with</strong> <strong>Space</strong> <strong>Vector</strong> <strong>Modulation</strong><br />
The references <strong>of</strong> stator flux components (see Fig. 5.3) are compared <strong>with</strong> estimated value:<br />
Ψ<br />
Ψ<br />
= Ψ<br />
cos θ ,<br />
sα<br />
s Ψs<br />
= Ψ<br />
sin θ ,<br />
sβ<br />
s Ψs<br />
The command voltage can be calculated from flux errors in<br />
(5.5)<br />
α, β coordinate system as<br />
follows:<br />
U<br />
U<br />
∆Ψ<br />
= + R I<br />
sα<br />
sα<br />
_ ref s sα<br />
Ts<br />
∆Ψ<br />
= + R I<br />
sβ<br />
sβ<br />
_ ref s sβ<br />
Ts<br />
(5.6)<br />
Where: T<br />
s<br />
is sampling time,<br />
∆Ψ = Ψ − Ψ , ∆Ψ =Ψ<br />
_<br />
−Ψ .<br />
sα sα_<br />
ref sα<br />
sβ sβ ref sβ<br />
The presented bellow design methodology for flux and torque control loops based on the<br />
approach presented in literature [11,43].<br />
5.2.1 Digital flux control loop<br />
The flux control loop is based on the voltage equations <strong>of</strong> PMSM machine in stator<br />
coordinates.<br />
U<br />
U<br />
dΨ<br />
dt<br />
sα<br />
sα<br />
= RsIsα<br />
+ (5.7a)<br />
dΨ<br />
sβ<br />
sβ<br />
= RsIsβ<br />
+ (5.7b)<br />
dt<br />
Using Laplace transformation the above equations can be written as:<br />
U = sΨ + R I<br />
(5.8a)<br />
sα sα s sα<br />
U = sΨ + R I<br />
(5.8b)<br />
sβ sβ s sβ<br />
It corresponds to flux model <strong>of</strong> PMSM machine in α,<br />
β system presented in Fig. 5.4.<br />
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<strong>Direct</strong> <strong>Torque</strong> <strong>Control</strong> <strong>with</strong> <strong>Space</strong> <strong>Vector</strong> <strong>Modulation</strong><br />
R I α<br />
U<br />
s α<br />
U<br />
s β<br />
s<br />
−<br />
s<br />
1<br />
s<br />
1<br />
s<br />
−<br />
R I β<br />
s<br />
s<br />
Ψ sα<br />
Ψ sβ<br />
Figure 5.4. Flux model <strong>of</strong> PMSM in stator coordinates.<br />
In order to control the flux components in α,<br />
β frame the bellow control structure can be<br />
applied.<br />
Ψ<br />
s α _ ref<br />
Ψ<br />
s β _ ref<br />
Flux control<br />
part<br />
−<br />
∆Ψ sα<br />
−<br />
∆Ψ sβ<br />
P block<br />
P=<br />
1/ Ts<br />
P block<br />
P=<br />
1/ Ts<br />
∆Ψ<br />
s α<br />
T s<br />
∆Ψ<br />
s β<br />
T s<br />
RI<br />
s s α<br />
U<br />
s α _ ref<br />
U<br />
s β _ ref<br />
RI<br />
s s β<br />
RI<br />
s s α<br />
−<br />
1<br />
s<br />
1<br />
s<br />
−<br />
RI β<br />
s<br />
s<br />
Flux PMSM<br />
model<br />
Ψ<br />
sα<br />
Ψ sβ<br />
Ψ sα<br />
Ψ sβ<br />
Figure 5.5. Flux control loop <strong>with</strong> two P controller in α,<br />
β reference frame.<br />
Pleas note that regarding to Fig. 5.1 the following rules are keeping:<br />
Ψ<br />
sα _ ref<br />
= Ψ<br />
s _ ref<br />
cos( θΨs<br />
+∆ δ )<br />
(5.9a)<br />
Ψ<br />
sβ _ ref<br />
= Ψ<br />
s _ ref<br />
sin( θΨs<br />
+∆ δ )<br />
(5.9b)<br />
and<br />
sα<br />
s_<br />
refcosθ Ψ s<br />
Ψ = Ψ (5.10a)<br />
sβ<br />
s_<br />
refsinθ Ψ s<br />
Ψ = Ψ (5.10b)<br />
In order to find the formula for tuning the P controllers in the flux loop, the following<br />
assumptions should be made:<br />
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<strong>Direct</strong> <strong>Torque</strong> <strong>Control</strong> <strong>with</strong> <strong>Space</strong> <strong>Vector</strong> <strong>Modulation</strong><br />
• increment <strong>of</strong> torque angle ∆δ Ψ<br />
coming from torque control loop (see Fig.5.2.) is equal<br />
zero. It means that the torque is not produced,<br />
• stator flux vector position θ Ψ s<br />
and rotor flux vector position θ r<br />
are equal zero. It<br />
corresponds to situation, where the those two flux vectors lie along the α axis.<br />
In this special case the reference stator flux amplitude Ψ<br />
_<br />
=Ψ<br />
α _<br />
can be controlled<br />
s ref s ref<br />
trough the reference stator voltage component U<br />
α _<br />
= U<br />
_<br />
, when the voltage drop on the<br />
s ref s ref<br />
stator resistances in α,<br />
β axes are neglected (see Fig. 5.5). Therefore, the simplified flux<br />
control loop can be shown in Fig. 5.6.<br />
Ψ =Ψ α<br />
s _ ref s _ ref<br />
s sα<br />
Ψ =Ψ<br />
−<br />
∆Ψ sα<br />
controller<br />
P<br />
U<br />
sα _ ref<br />
= Us _ ref<br />
U β<br />
=<br />
s _ ref<br />
0<br />
PMSM<br />
Ψ sα<br />
Ψ sβ<br />
Continuous s-domain<br />
Figure 5.6. Simplified flux control loop in α,<br />
β coordinates.<br />
Simplified flux control loop in s domain is shown in Fig. 5.7, where CΨ ( s)<br />
is a transfer<br />
function <strong>of</strong> the P controller given by:<br />
CΨ() s = KpΨ<br />
(5.11)<br />
The transfer function between stator flux amplitude Ψ<br />
s<br />
=Ψ sα and stator voltage amplitude<br />
U<br />
s<br />
can be expressed as:<br />
G<br />
Ψ<br />
Ψ<br />
s 1<br />
() s = = (5.12)<br />
U s<br />
s<br />
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<strong>Direct</strong> <strong>Torque</strong> <strong>Control</strong> <strong>with</strong> <strong>Space</strong> <strong>Vector</strong> <strong>Modulation</strong><br />
<strong>Control</strong> Plant<br />
P controller<br />
Ψ s_ref<br />
C<br />
Ψ<br />
() s<br />
U s<br />
G<br />
Ψ<br />
() s<br />
Ψ s<br />
Figure 5.7. Block diagram <strong>of</strong> flux controller in s domain.<br />
Hence the transfer function <strong>of</strong> the closed stator flux amplitude control loop is obtained as:<br />
G<br />
Ψ _ closed<br />
Ψ<br />
s<br />
() s<br />
_ ref CΨ() s GΨ()<br />
s<br />
() s = =<br />
Ψ () s 1 + C () s G () s<br />
s<br />
Ψ<br />
Ψ<br />
(5.12)<br />
Substituting transfer function for CΨ ( s)<br />
and GΨ ( s)<br />
one becomes:<br />
⎛1⎞ ⎛1⎞<br />
KpΨ⎜ ⎟ KpΨ⎜ ⎟<br />
s s K<br />
p<br />
GΨ _ closed<br />
() s =<br />
⎝ ⎠<br />
=<br />
⎝ ⎠<br />
=<br />
⎛1<br />
⎞ s+ KpΨ<br />
s+<br />
K<br />
1+ K<br />
pΨ<br />
⎜ ⎟<br />
⎝s<br />
⎠ s<br />
Ψ<br />
pΨ<br />
(5.13)<br />
Discrete design<br />
The transfer function for P controller in discrete system is expressed as:<br />
CΨ( z) = KpΨ<br />
(5.14)<br />
Ψ<br />
s_ ref( z)<br />
C<br />
Ψ<br />
( z)<br />
U<br />
= sα U<br />
s<br />
Dz ( )<br />
z −1<br />
GΨ<br />
( z)<br />
}<br />
ZOH<br />
1<br />
s<br />
Ψ<br />
s<br />
( z)<br />
Figure 5.8. Block diagram <strong>of</strong> flux controller in discrete domain.<br />
Where, CΨ ( z)<br />
is discrete transfer function for P controller, Dz ( )<br />
delay for voltage generation from PWM .<br />
1<br />
z −<br />
is for one sampling time<br />
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<strong>Direct</strong> <strong>Torque</strong> <strong>Control</strong> <strong>with</strong> <strong>Space</strong> <strong>Vector</strong> <strong>Modulation</strong><br />
The GΨ ( z)<br />
is discrete transfer function for voltage-flux relationship <strong>with</strong> zero hold order<br />
(ZOH) can be calculated as:<br />
−1<br />
GΨ<br />
() s z−1 1<br />
GΨ<br />
( z) = (1 − z ) Z[ ] = Z[ ]<br />
(5.15)<br />
2<br />
s z s<br />
Using table <strong>of</strong> Z transformation [2]. Finally, it gives<br />
G<br />
Ψ<br />
( z −1)<br />
zTs<br />
AΨ<br />
d<br />
2<br />
z ( z −1) ( z−1)<br />
( z)<br />
= =<br />
(5.16)<br />
Where A = Ψ d<br />
T and s<br />
T<br />
s<br />
is sampling time <strong>of</strong> the discrete system.<br />
Hence, the closed loop transfer function between Ψ<br />
s<br />
( z)<br />
and Ψ ( z)<br />
is obtained as:<br />
_ ref<br />
s<br />
G<br />
Ψ _ closed<br />
K<br />
pΨ<br />
Ψ<br />
s<br />
( z) _ ref CΨ( z) GΨ( z) D( z)<br />
( z)<br />
= =<br />
Ψ ( z) 1 + C ( z) G ( z) D( z)<br />
A<br />
Ψd<br />
( −1)<br />
z z<br />
K A<br />
= =<br />
K A z z K A<br />
1+<br />
z z<br />
pΨ<br />
Ψd<br />
2<br />
pΨ Ψd − +<br />
pΨ Ψd<br />
( −1)<br />
s<br />
Ψ<br />
Ψ<br />
(5.16)<br />
The flux step response depended on poles placement <strong>of</strong> closed flux control loop. The pole<br />
placement can be selected by setting the<br />
K<br />
p Ψ<br />
.<br />
Assuming, that CΨd = KpΨAΨd<br />
the GΨ _ closed<br />
( z)<br />
expressed by equations (5.16) will take the<br />
following form:<br />
G<br />
CΨd<br />
( z) =<br />
z − z+<br />
C<br />
Ψ _ closed<br />
2<br />
Ψd<br />
(5.17)<br />
The nomogram <strong>of</strong> Fig. 5.9 shows the relationship between overshoot M<br />
p[%]<br />
, rise time t r<br />
and<br />
settling time t s<br />
in respect to<br />
C Ψ d<br />
.<br />
Please not that t r<br />
is time calculate from 10% to 90% <strong>of</strong> output signals and t s<br />
is the time it<br />
takes the system transient to decay +-1%.<br />
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<strong>Direct</strong> <strong>Torque</strong> <strong>Control</strong> <strong>with</strong> <strong>Space</strong> <strong>Vector</strong> <strong>Modulation</strong><br />
Figure 5.9. The relationship between overshoot, rise time and settling time versus to<br />
flux amplitude control loop.<br />
C Ψ d<br />
for stator<br />
Now from few values <strong>of</strong> C Ψ d<br />
=[0.4046, 0.2688, 0.1720] we can choose C Ψ d<br />
=0.2688, which<br />
guaranties overshoot about 0% and settling time about 10 times <strong>of</strong> sampling time.<br />
1<br />
2<br />
3<br />
1<br />
Figure 5.10. Step flux response for different to<br />
C Ψ d<br />
=0.2688, black line (3)<br />
d<br />
C Ψ d<br />
: red line(1) C Ψ d<br />
=0.4046, blue line (2)<br />
C Ψ<br />
=0.1720.<br />
It corresponds to the transfer function <strong>of</strong> closed stator flux control loop as:<br />
G<br />
C<br />
0.2688<br />
( z)<br />
= =<br />
z − z+ C z −z+ 0.2688<br />
Ψd<br />
Ψ _ closed<br />
2 2<br />
Ψd<br />
(5.18)<br />
Using digitalized motor parameters A = Ψ d<br />
T and chosen s<br />
C Ψ d<br />
value we can calculate the<br />
parameters <strong>of</strong> P digital flux controller as:<br />
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<strong>Direct</strong> <strong>Torque</strong> <strong>Control</strong> <strong>with</strong> <strong>Space</strong> <strong>Vector</strong> <strong>Modulation</strong><br />
K<br />
pΨ<br />
C<br />
A<br />
Ψd<br />
Ψd<br />
= = (5.19)<br />
Ψd<br />
C<br />
T<br />
s<br />
For example let us assume that sampling time in digital flux control loop is equal T = 200µ<br />
s.<br />
The gain <strong>of</strong> P controller is:<br />
s<br />
K<br />
pΨ<br />
0.2688 0.2688<br />
= = = 1344<br />
(5.20)<br />
A 0.0002<br />
Ψd<br />
In digital control when the sampling time changes the parameters <strong>of</strong> digitalized plant control<br />
A Ψ d<br />
will also change. Therefore, to keep closed loop transfer function as close as possible to<br />
G<br />
C<br />
0.2688<br />
( z)<br />
= =<br />
, the gain <strong>of</strong> P flux controller should also be<br />
z − z+ C z −z+ 0.2688<br />
Ψd<br />
Ψ _ closed<br />
2 2<br />
Ψd<br />
changed (see Table 5.1.).<br />
Keeping constant transfer function GΨ _ closed<br />
( z)<br />
the flux step response for different sampling<br />
times<br />
T<br />
s<br />
= 50µ s , 100µ s , 200µ s , 400µ s , which correspond to switching frequency f s<br />
=<br />
20kHz, 10kHz, 5kHz, 2.5kHz are presented in Fig. 5.11.<br />
Figure 5.11. Flux tracking performance for different sampling times T<br />
s<br />
= 50µ s (blue line -1), 100µ<br />
s<br />
(green line -2), 200µ s (red line -3), 400µ s (light blue line -4).<br />
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<strong>Direct</strong> <strong>Torque</strong> <strong>Control</strong> <strong>with</strong> <strong>Space</strong> <strong>Vector</strong> <strong>Modulation</strong><br />
From Fig. 5.11 it can observed that overshoot is around 0% and the settling time took 10<br />
times <strong>of</strong> microprocessor sampling time. So, it is possible control the flux amplitude in 10<br />
samples<br />
The settings <strong>of</strong> P flux controller for different sampling time T<br />
s<br />
= 50 µ s,100 µ s,<br />
200 µ s,400µ<br />
s<br />
are summarized in Table 5.1.<br />
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<strong>Direct</strong> <strong>Torque</strong> <strong>Control</strong> <strong>with</strong> <strong>Space</strong> <strong>Vector</strong> <strong>Modulation</strong><br />
76
<strong>Direct</strong> <strong>Torque</strong> <strong>Control</strong> <strong>with</strong> <strong>Space</strong> <strong>Vector</strong> <strong>Modulation</strong><br />
The behavior <strong>of</strong> the flux control loop was tested using SABER simulation package. The<br />
model created in SABER takes into account the whole control system, which include<br />
real models <strong>of</strong> inverter and permanent magnet synchronous motor.<br />
The flux step response is shown in Fig. 5.12., when parameters <strong>of</strong> P flux controller<br />
designed for sampling time T s<br />
= 200µ s were used for control plant for different<br />
sampling times T s<br />
= 50µ s , 100µ s , 200µ s , 400µ s , which correspond to switching<br />
frequency f<br />
s<br />
= 20kHz, 10kHz, 5kHz, 2.5kHz.<br />
Figure 5.12. Flux tracking performance for different sampling time T<br />
s<br />
= 50µ s , 100µ s ,<br />
200µ s , 400µ s ( f<br />
s<br />
= 20kHz, 10kHz, 5kHz,2.5kHz) using gain <strong>of</strong> P controller designed<br />
for T<br />
s<br />
= 200µ s ( f<br />
s<br />
= 5kHz).<br />
After modification <strong>of</strong> P flux controller gain according to Table 5.1 it is possible to<br />
achieve better results as shown in Fig. 5.13, what confirms proper flux tracking<br />
performance in steady and dynamics state.<br />
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<strong>Direct</strong> <strong>Torque</strong> <strong>Control</strong> <strong>with</strong> <strong>Space</strong> <strong>Vector</strong> <strong>Modulation</strong><br />
Figure 5.13. Flux tracking performance for different sampling time T s<br />
= 50µ s , 100µ s ,<br />
200µ s , 400µ s ( f<br />
s<br />
= 20kHz, 10kHz, 5kHz,2.5kHz) using designed gain <strong>of</strong> P controller<br />
calculated individually (see table 5.1.)<br />
The simulation results in SABER package for Ts=200us is presented in Fig. 5.14.<br />
Figure 5.14. Simulated (SABER) flux tracking performance for step change from 70% - 100%<br />
<strong>of</strong> nominal flux.<br />
As we can observed from Fig. 5.14 that overshoot is around M<br />
p<br />
= 0% and settling time<br />
took about 10 sampling time <strong>of</strong> microprocessor, what proved the design procedure <strong>of</strong> P<br />
digital flux controller gain.<br />
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<strong>Direct</strong> <strong>Torque</strong> <strong>Control</strong> <strong>with</strong> <strong>Space</strong> <strong>Vector</strong> <strong>Modulation</strong><br />
5.2.2 Digital torque control loop<br />
The considered torque control loop is shown in Fig. 5.15.<br />
Ψ s _ ref<br />
M e _ ref<br />
−<br />
<strong>Torque</strong><br />
controller<br />
PI<br />
∆δ Ψ<br />
Reference<br />
flux generator<br />
In stator frame<br />
Ψ<br />
s α _ ref<br />
Ψ<br />
s β _ ref<br />
−<br />
∆Ψ sα<br />
−<br />
∆Ψ sβ<br />
Flux<br />
controllers<br />
In stator frame<br />
U<br />
s α _ ref<br />
U<br />
s β _ ref<br />
M e<br />
θ Ψs<br />
Ψ sα<br />
Ψ sβ<br />
Figure 5.15. <strong>Torque</strong> control loop <strong>with</strong> PI controller.<br />
Based on the equation (5.9a-b and 5.10a-b) the stator flux errors in α,<br />
β coordinates can<br />
be calculated as:<br />
∆Ψ<br />
α<br />
= Ψ cos( θ + ∆δ ) − Ψ cos θ = Ψ [cos( θ + ∆δ ) − cos θ ]<br />
s s_ ref Ψs Ψ s_ ref Ψs s_<br />
ref Ψs Ψ Ψs<br />
(5.21a)<br />
∆Ψ<br />
β<br />
= Ψ sin( θ +∆δ ) − Ψ sin θ = Ψ [sin( θ +∆δ ) − sin θ ]<br />
s s_ ref Ψs Ψ s_ ref Ψs s_<br />
ref Ψs Ψ Ψs<br />
(5.21b)<br />
Assuming that for small changes <strong>of</strong> ∆δ Ψ<br />
the cos ∆δ Ψ<br />
≅ 1and sin ∆δ Ψ<br />
≅∆ δ<br />
Ψ<br />
, the equations<br />
(5.21a) and (5.21b) are given by:<br />
∆Ψ<br />
α _<br />
= − Ψ<br />
_<br />
∆ δ sinθ<br />
(5.22a)<br />
s ref s ref Ψ Ψs<br />
∆Ψ<br />
β _<br />
= Ψ<br />
_<br />
∆ δ cosθ<br />
(5.22b)<br />
s ref s ref Ψ Ψs<br />
In order to design the PI torque controller the following assumption are made:<br />
• stator flux vector position θ Ψ s<br />
and rotor flux vector position θ<br />
r<br />
are equal zero. It<br />
correspond to situation, when those two flux vectors lie along theα axis,<br />
• the reference stator flux amplitude is equal value <strong>of</strong> permanent magnet flux<br />
Ψ<br />
s _ ref<br />
=Ψ<br />
PM<br />
,<br />
• stator resistance is neglected.<br />
Therefore, the error stator fluxes in α,<br />
β coordinates are calculated as:<br />
∆Ψ = , (5.23a)<br />
sα<br />
_ ref<br />
0<br />
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<strong>Direct</strong> <strong>Torque</strong> <strong>Control</strong> <strong>with</strong> <strong>Space</strong> <strong>Vector</strong> <strong>Modulation</strong><br />
∆Ψ = Ψ ∆ . (5.23b)<br />
sβ<br />
_ ref PM<br />
δ Ψ<br />
Stator voltage components using equations (2.23a-b) can be expressed as:<br />
U α<br />
= (5.24a)<br />
s _ ref<br />
0<br />
U<br />
= ∆Ψ K<br />
(5.24b)<br />
sβ _ ref sβ _ ref pΨ<br />
And further because <strong>of</strong> U<br />
β _<br />
= U<br />
_<br />
s ref s ref<br />
U<br />
=Ψ ∆ δ K<br />
(5.25)<br />
s PM pΨ<br />
So, the transfer function between stator voltage amplitude<br />
angle ∆ δ Ψ<br />
can be written as:<br />
U<br />
s<br />
G<br />
δ<br />
() s = = K<br />
ΨΨ<br />
∆δ<br />
M p PM<br />
Ψ<br />
U<br />
s<br />
and increment <strong>of</strong> torque<br />
(5.26)<br />
Where<br />
K<br />
p Ψ<br />
is the gain <strong>of</strong> stator flux P controller.<br />
For example for sampling time T = 200µ<br />
s, calculated<br />
s<br />
0.2688<br />
K = pΨ 1344<br />
T<br />
= (see<br />
s<br />
Table5.1.) and nominal value <strong>of</strong> Ψ<br />
PM<br />
= 0.264Wb<br />
the calculated GMδ () s = 354,82V / rad .<br />
The obtained transfer function between electromagnetic torque<br />
amplitude<br />
U<br />
s<br />
is (see equation 5.72):<br />
M<br />
e<br />
and stator voltage<br />
G<br />
M<br />
M () s A s<br />
() s = =<br />
U s s + B s+ C<br />
(5.27)<br />
e<br />
M<br />
2<br />
s<br />
()<br />
M M<br />
Where<br />
A<br />
M<br />
2<br />
3pbΨ<br />
PM<br />
RsΨ<br />
3 Ψ<br />
PM<br />
s<br />
ΨPM pb<br />
= and BM<br />
= CM<br />
=<br />
2L<br />
Ψ L<br />
2JL<br />
s<br />
Using the motor parameters (see Appendices), one obtains:<br />
A = 198 and B = 115.3 C = 9065<br />
M<br />
M<br />
M<br />
s<br />
s<br />
s<br />
Continuous s-domain<br />
The torque control loop is shown in Fig. 5.16, where C ( s ) is a transfer function <strong>of</strong> the<br />
PI controller given by [105]:<br />
M<br />
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<strong>Direct</strong> <strong>Torque</strong> <strong>Control</strong> <strong>with</strong> <strong>Space</strong> <strong>Vector</strong> <strong>Modulation</strong><br />
C<br />
M<br />
⎛ K ⎞<br />
iM<br />
KpM<br />
s+<br />
⎜<br />
K ⎟<br />
pM<br />
() s =<br />
⎝ ⎠<br />
(5.28)<br />
s<br />
where<br />
K<br />
iM<br />
=<br />
K<br />
T<br />
pM<br />
iM<br />
M e _ ref<br />
() ∆δ U = sβ Us<br />
CM s Ψ<br />
G ()<br />
GM<br />
() s<br />
Mδ<br />
s<br />
M e<br />
Figure 5.16. Block diagram <strong>of</strong> torque control loop represented in s-domain.<br />
Hence, the transfer function between M<br />
_<br />
() s and M ( s ) is obtained as:<br />
e<br />
ref<br />
e<br />
G<br />
M _ closed<br />
() s<br />
M () s C () s G () s G () s<br />
e_<br />
ref M M Mδ<br />
= = (5.29)<br />
M<br />
e() s 1 + CM() z GM() s GMδ<br />
() s<br />
Substituting transfer function for CM<br />
( s)<br />
and GM<br />
( s ) equation (5.29) becomes:<br />
KiM<br />
K Ψ p PMAMKpM( s +<br />
Ψ<br />
)<br />
K<br />
pM<br />
GM<br />
_ closed()<br />
s = =<br />
2<br />
s + ( B + K A ) s+ C + K A<br />
M pM M M iM M<br />
(5.30)<br />
Discrete design<br />
The transfer function (equations 5.28) for PI controller in discrete system using<br />
z −1<br />
backward difference method for discretization process ( s = ) [2] is expressed as:<br />
Tz<br />
s<br />
K<br />
pM<br />
( z − )<br />
KpM<br />
+ KiM<br />
CM( z) = ( KpM + KiM)<br />
( z −1)<br />
(5.31)<br />
where:<br />
K<br />
iM<br />
K<br />
pM<br />
= Ts<br />
,<br />
s<br />
TiM<br />
T - sampling time, C ( z ) is the discrete transfer function <strong>of</strong><br />
M<br />
torque PI controller, Dz ( )<br />
1<br />
z −<br />
is one sampling time delay for voltage generation from<br />
PWM, and<br />
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<strong>Direct</strong> <strong>Torque</strong> <strong>Control</strong> <strong>with</strong> <strong>Space</strong> <strong>Vector</strong> <strong>Modulation</strong><br />
D( z)<br />
} G ( ) M<br />
z<br />
M<br />
e_ ref( z)<br />
C ( ) M<br />
z<br />
∆δ Ψ<br />
G<br />
Mδ<br />
( z)<br />
U s<br />
z −1<br />
AM<br />
s<br />
ZOH<br />
2<br />
s + B s+<br />
C<br />
M<br />
M<br />
M ( ) e<br />
z<br />
Figure 5.17. Block diagram <strong>of</strong> torque control loop in discrete domain.<br />
G ( z)<br />
= K Ψ (5.32)<br />
Mδ<br />
pΨ<br />
PM<br />
is discrete transfer function for relation between stator voltage amplitude<br />
increment <strong>of</strong> torque angle ∆ δ Ψ<br />
(see Fig. 5.17)<br />
U s<br />
and<br />
The discrete transfer function G ( z ) for voltage-torque relationship <strong>with</strong> zero order<br />
hold (ZOH) can be calculated as:<br />
M<br />
G () s z −1<br />
A<br />
G z = − z Z = Z<br />
−1<br />
M<br />
M<br />
M<br />
( ) (1 ) [ ] [ ]<br />
2<br />
s z s + BMs+<br />
CM<br />
(5.33)<br />
Finally, the discrete transfer function <strong>of</strong> controlled plant G ( z ) can be written as:<br />
M<br />
⎛ A ⎞<br />
Md<br />
GM<br />
( z) = ( z−1) ⎜ 2<br />
⎟<br />
⎝ z − BMd<br />
z+<br />
CMd<br />
⎠<br />
(5.34)<br />
BM<br />
2<br />
A<br />
− T<br />
M<br />
s<br />
B<br />
2<br />
M<br />
Where: AM _ d<br />
= e sin( T )<br />
2<br />
s<br />
CM<br />
− ,<br />
B<br />
4<br />
M<br />
CM<br />
−<br />
4<br />
BM<br />
2<br />
− Ts<br />
B<br />
2<br />
M<br />
BM _ d<br />
= 2e cos( Ts CM<br />
− ), CM<br />
_<br />
4<br />
d<br />
BM<br />
Ts<br />
= e − , and<br />
s<br />
T is sampling time.<br />
Hence, the transfer function <strong>of</strong> closed torque control loop is obtained in the following<br />
form:<br />
G<br />
M _ closed<br />
( z)<br />
M ( z) C ( z) G ( z) D( z) G ( z)<br />
M ( z) 1 + C ( zG ) ( zDzG ) ( ) ( z)<br />
e_<br />
ref M M Mδ<br />
= = =<br />
e M M Mδ<br />
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<strong>Direct</strong> <strong>Torque</strong> <strong>Control</strong> <strong>with</strong> <strong>Space</strong> <strong>Vector</strong> <strong>Modulation</strong><br />
K<br />
K Ψ A ( K + K )( z− )( z−1)<br />
pM<br />
pΨ<br />
PM M _ d pM iM<br />
KpM<br />
+ KiM<br />
3 2<br />
M _ d M _ d pM iM M _ d M _ d pM<br />
= =<br />
[ z − B z + [ A ( K + K ) + C ] z− A K ]( z−1)<br />
K<br />
K Ψ A ( K + K )( z−<br />
)<br />
pM<br />
pΨ<br />
PM M _ d pM iM<br />
KpM<br />
+ KiM<br />
3 2<br />
−<br />
M _ d<br />
+ [<br />
M _ d( pM<br />
+<br />
iM) +<br />
M _ d]<br />
−<br />
M _ d pM<br />
z B z A K K C z A K<br />
(5.35)<br />
Selecting<br />
K Ψ<br />
, K Ψ<br />
will influence poles placement <strong>of</strong> closed torque control loop and as<br />
p<br />
i<br />
a consequence also torque step responses can be selected.<br />
The transfer function <strong>of</strong> closed torque control loop is more complicated than flux<br />
control loop (see design <strong>of</strong> P-flux controller – section 5.2.1). One possibility is use to<br />
the SISO tools from Matlab package to tune parameters <strong>of</strong> PI torque controller [106].<br />
a) b)<br />
Figure 5.18. a) <strong>Torque</strong> step response for sampling time T = 200µ<br />
s, b) <strong>with</strong> denoted rise time,<br />
overshoot and settling time.<br />
As can be observed in (Fig. 5.18) torque response is characterized by overshoot about<br />
40%, rise time 4 samples and settling time 17 samples.<br />
s<br />
To eliminate high overshoot it is recommended to insert at the input prefilter (see<br />
Fig.5.19 ) <strong>with</strong> transfer function:<br />
z−<br />
b z−<br />
0.6878<br />
PM<br />
( z)<br />
= K = K<br />
K<br />
pM z − 0.855<br />
( z − )<br />
K + K<br />
1<br />
where K =<br />
=0.466 is gain <strong>of</strong> the prefilter.<br />
z − 0.6878<br />
lim<br />
z→1<br />
z − 0.855<br />
pM<br />
iM<br />
(5.36)<br />
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<strong>Direct</strong> <strong>Torque</strong> <strong>Control</strong> <strong>with</strong> <strong>Space</strong> <strong>Vector</strong> <strong>Modulation</strong><br />
Dz ( )<br />
G ( ) M<br />
z<br />
M<br />
e_ ref( z)<br />
P ( ) M<br />
z<br />
C ( ) M<br />
z<br />
∆δ Ψ<br />
G<br />
Mδ<br />
( z)<br />
U s<br />
z −1<br />
AM<br />
s<br />
ZOH<br />
2<br />
s + B s+<br />
C<br />
M<br />
M<br />
M ( ) e<br />
z<br />
Figure 5.19. Block diagram <strong>of</strong> torque control loop <strong>with</strong> prefilter (discrete domain).<br />
Finally, the step response <strong>of</strong> closed torque control loop <strong>with</strong> prefilter at the input is<br />
presented bellow:<br />
a) b)<br />
Figure 5.20. a) <strong>Torque</strong> step response for sampling time T = 200µ<br />
s, b) <strong>with</strong> denoted rise time,<br />
overshoot and settling time.<br />
s<br />
As it can be observed, the response is now characterized by overshoot about 2%, rise<br />
time 5 samples and the settling time 15 samples.<br />
In digital control system when the sampling time is changed the parameters <strong>of</strong><br />
digitalized control plant AMd, BMd,<br />
C<br />
Md<br />
will also change. Therefore, the parameters <strong>of</strong> PI<br />
torque controller will change also (see Table 2).<br />
Simulation results for digital torque control loop in SABER package for 5KHz <strong>with</strong> and<br />
<strong>with</strong>out prefilter are shown in Fig. 5.21. Also, the torque step response for different<br />
level <strong>of</strong> reference torque are presented in Fig. 5.22.<br />
The settings for PI torque controller for different sampling time T s<br />
= 50µ s , 100µ s ,<br />
200µ s , 400µ s are summarized in Table 5.2.<br />
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<strong>Direct</strong> <strong>Torque</strong> <strong>Control</strong> <strong>with</strong> <strong>Space</strong> <strong>Vector</strong> <strong>Modulation</strong><br />
a)<br />
b)<br />
Figure 5.21. <strong>Torque</strong> step response: a) <strong>with</strong>out prefilter, b) <strong>with</strong> prefilter.<br />
Figure 5.22. <strong>Torque</strong> step response <strong>with</strong> prefilter (from 0 to 25%, 50%, 75% and 100% <strong>of</strong><br />
nominal torque).<br />
85
<strong>Direct</strong> <strong>Torque</strong> <strong>Control</strong> <strong>with</strong> <strong>Space</strong> <strong>Vector</strong> <strong>Modulation</strong><br />
86
<strong>Direct</strong> <strong>Torque</strong> <strong>Control</strong> <strong>with</strong> <strong>Space</strong> <strong>Vector</strong> <strong>Modulation</strong><br />
The torque step response is shown in Fig. 5.23, when parameters <strong>of</strong> PI torque controller<br />
designed for sampling time T s<br />
= 200µ s were used for control plant for different sampling<br />
times<br />
T<br />
s<br />
= 50µ s , 100µ s , 200µ s , 400µ s , which correspond to switching frequency f s<br />
=<br />
20kHz, 10kHz, 5kHz, 2.5kHz.<br />
Figure 5.23. <strong>Torque</strong> tracking performance for different sampling time T s<br />
= 50µ s , 100µ s , 200µ s ,<br />
400µ s ( f<br />
s<br />
= 20kHz, 10kHz, 5kHz,2.5kHz) using parameters <strong>of</strong> PI controller designed for<br />
T<br />
s<br />
= 200µ s ( f<br />
s<br />
= 5kHz). (Please note that for 2.5kHz the system was unstable).<br />
After modification <strong>of</strong> PI torque controller parameters according to Table 2 it is possible to<br />
achieve better results as shown in Fig. 5.23, what confirms and proper torque tracking<br />
performance in steady and dynamics state.<br />
Figure 5.24. <strong>Torque</strong> tracking performance for different sampling time T<br />
s<br />
= 50µ s , 100µ s , 200µ s ,<br />
400µ s ( f<br />
s<br />
= 20kHz, 10kHz, 5kHz,2.5kHz) using designed parameters <strong>of</strong> PI controller calculated<br />
individually (see table 5.2)<br />
87
<strong>Direct</strong> <strong>Torque</strong> <strong>Control</strong> <strong>with</strong> <strong>Space</strong> <strong>Vector</strong> <strong>Modulation</strong><br />
5.3 Parallel structure <strong>of</strong> <strong>DTC</strong>–<strong>SVM</strong> scheme<br />
Block scheme <strong>of</strong> the control structure is shown in Fig. 5.25. Two PI controllers are used for<br />
regulation torque and flux magnitude loops [11,55].<br />
U DC<br />
Ψ s _ ref<br />
e Ψ<br />
ss<br />
U sx _ ref<br />
s<br />
_ ref<br />
S A<br />
S B<br />
M e _ ref<br />
e M<br />
U sy _ ref<br />
U α<br />
U<br />
s β _ ref<br />
S C<br />
θ Ψs<br />
Ψ<br />
I s<br />
M e<br />
γ<br />
m<br />
Figure 5.25. Parallel structure <strong>of</strong> <strong>DTC</strong>-<strong>SVM</strong> scheme.<br />
In this control scheme the reference stator flux magnitude<br />
Ψ and reference<br />
s _ ref<br />
electromagnetic torque<br />
M are compared <strong>with</strong> estimated values, respectively. The flux and<br />
e_<br />
ref<br />
torque errors eΨ , e are delivered to PI controllers, which generate command value the stator<br />
s<br />
M<br />
voltage components in stator flux coordinates<br />
U , U<br />
_<br />
sx_<br />
ref<br />
sy<br />
ref<br />
. This voltage signals are<br />
transformed to stationary coordinates using the stator flux position angle θ Ψ s<br />
. The reference<br />
stator voltage vector ( U<br />
s α _ ref<br />
, U<br />
s β _<br />
ref<br />
) is delivered to space vector modulator (<strong>SVM</strong>), which<br />
generates the switching signals<br />
S , S , S to control power transistors <strong>of</strong> the inverter.<br />
A<br />
B<br />
C<br />
The presented control strategy is based on simplified stator voltage equations described in<br />
stator flux oriented x-y coordinates (equations 2.27a-b):<br />
U<br />
d<br />
Ψ<br />
s<br />
sx<br />
= Rs Isx<br />
+ (5.37)<br />
dt<br />
U = R I +Ω Ψ = R I + E = k M + E<br />
(5.38)<br />
sy s sy Ψs s s sy sy s e sy<br />
88
<strong>Direct</strong> <strong>Torque</strong> <strong>Control</strong> <strong>with</strong> <strong>Space</strong> <strong>Vector</strong> <strong>Modulation</strong><br />
where:<br />
k<br />
s<br />
=<br />
2<br />
3<br />
R<br />
p<br />
= Ψ and Esy = ΩΨs Ψ<br />
s<br />
.<br />
s<br />
b<br />
Ψ , 2<br />
M<br />
e<br />
pb s<br />
Isy<br />
s<br />
3<br />
The above equations show that the<br />
U<br />
sx<br />
component has influence only on the change <strong>of</strong> stator<br />
flux magnitude<br />
Ψ<br />
s<br />
, and the component<br />
Ω<br />
U<br />
sy<br />
– if the term<br />
Ψs s<br />
Ψ is decoupled – can be<br />
used for torque adjustment. Therefore, the flux and torque quantities can be controlled as<br />
shown in Fig. 5.26.<br />
Note, that this <strong>DTC</strong>-<strong>SVM</strong> scheme formally corresponds to the stator flux oriented voltage<br />
source inverter-fed drive induction motor. The block diagram <strong>of</strong> the <strong>DTC</strong>-<strong>SVM</strong> scheme <strong>with</strong><br />
two PI controllers is shown in Fig. 5.26 The dashed line represents the PMSM part [124,125].<br />
Ψ s _ ref<br />
Ψ s<br />
_<br />
R I<br />
s<br />
sx<br />
U sx<br />
R I<br />
s<br />
sx<br />
∫<br />
Ψ s<br />
R I<br />
s<br />
sy<br />
Ω Ψs<br />
⊗<br />
Ω<br />
Ψ<br />
Ψs s<br />
M e _ ref<br />
_<br />
E sy<br />
U sy<br />
1 Isy<br />
R s<br />
3<br />
2<br />
p<br />
b<br />
Ψ<br />
s _ ref<br />
M e<br />
M e<br />
5.3.1 Digital flux control loop<br />
Figure 5.26. Block diagram <strong>of</strong> the scheme presented in Fig. 5.27<br />
Putting the stator x-axis current expression from equation 2.28a under the assumption Ld<br />
= L<br />
q<br />
into equation (5.37) one can obtaines<br />
U<br />
I<br />
sx<br />
Ψs<br />
−ΨPM<br />
cos<br />
= (5.39)<br />
L<br />
s<br />
δ Ψ<br />
Ψs −ΨPM cosδΨ<br />
d Ψs R d Ψ<br />
s s RsΨ<br />
PM<br />
= R ( ) + = Ψ + − cosδΨ<br />
(5.40)<br />
L dt L dt L<br />
sx s s<br />
s s s<br />
Using Laplace transformation to equation 5.40 can be written as:<br />
U<br />
sx<br />
Rs RsΨ<br />
PM<br />
=Ψ<br />
s<br />
( s+ ) − cosδ Ψ<br />
. (5.41)<br />
L L<br />
s<br />
s<br />
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<strong>Direct</strong> <strong>Torque</strong> <strong>Control</strong> <strong>with</strong> <strong>Space</strong> <strong>Vector</strong> <strong>Modulation</strong><br />
Assuming small changes <strong>of</strong> δ Ψ<br />
, the cosδ Ψ<br />
≅ 1, and equations (5.41) reduces to:<br />
Rs<br />
Usx<br />
=Ψ<br />
s<br />
( s+ ) + WΨ<br />
(5.42)<br />
L<br />
s<br />
RsΨ<br />
PM<br />
Where WΨ<br />
=<br />
L<br />
s<br />
The transfer function between stator flux amplitude<br />
Ψ<br />
s<br />
and x-axis <strong>of</strong> stator voltage is:<br />
G<br />
Ψ<br />
Ψ<br />
s Ls<br />
1 1<br />
() s = = = =<br />
U R<br />
sx<br />
+ WΨ<br />
sLs + Rs<br />
s<br />
s +<br />
s+<br />
A<br />
L<br />
s<br />
Ψ<br />
(5.43)<br />
Where W<br />
Ψ<br />
Rs<br />
Rs<br />
≅ ΨPM<br />
cosδΨ<br />
≅ Ψ<br />
L<br />
L<br />
s<br />
Rs<br />
For motor parameters (see Appendices ): A = Ψ<br />
115.333<br />
L<br />
= .<br />
s<br />
PM<br />
Continuous s-domain<br />
The flux control loop is shown in Fig. 5.27, where CΨ ( s)<br />
is a transfer function <strong>of</strong> the PI<br />
controller given by [105]:<br />
C<br />
Ψ<br />
s<br />
⎛ K ⎞<br />
iΨ<br />
KpΨ<br />
s+<br />
⎜<br />
K ⎟<br />
pΨ<br />
() s =<br />
⎝ ⎠<br />
(5.45)<br />
s<br />
where<br />
K<br />
iΨ<br />
=<br />
K<br />
T<br />
pΨ<br />
iΨ<br />
Ψ s_ref<br />
C<br />
Ψ<br />
() s<br />
U sx<br />
W Ψ<br />
()<br />
Rs<br />
Ψ<br />
L<br />
s<br />
PM<br />
cos<br />
δ Ψ<br />
+<br />
G<br />
Ψ<br />
s<br />
Ψ s<br />
Figure 5.27. Block diagram <strong>of</strong> flux control loop in s-domain.<br />
Hence, the transfer function <strong>of</strong> the closed stator flux amplitude control loop is obtained as:<br />
90
<strong>Direct</strong> <strong>Torque</strong> <strong>Control</strong> <strong>with</strong> <strong>Space</strong> <strong>Vector</strong> <strong>Modulation</strong><br />
G<br />
Ψ _ closed<br />
Ψ<br />
s<br />
() s<br />
_ ref CΨ() s GΨ()<br />
s<br />
() s = =<br />
Ψ () s 1 + C () z G () s<br />
s<br />
Ψ<br />
Ψ<br />
(5.46)<br />
Substituting transfer function for CΨ ( s)<br />
and GΨ ( s)<br />
one becomes<br />
G<br />
_ closed<br />
() s<br />
⎛ K ⎞<br />
iΨ<br />
KpΨ<br />
s+<br />
⎜ K ⎟<br />
⎝ pΨ<br />
⎠ ⎛ 1 ⎞<br />
⎛ K ⎞<br />
iΨ<br />
⎜ ⎟ KpΨ<br />
s+<br />
s s A<br />
⎜ K ⎟<br />
⎝ + ⎠ pΨ<br />
=<br />
⎝ ⎠<br />
⎛ K ⎞<br />
s + ( A + Kp<br />
) s+<br />
K<br />
iΨ<br />
KpΨ<br />
s+<br />
⎜ K ⎟<br />
pΨ<br />
⎛ 1 ⎞<br />
1+ ⎝ ⎠<br />
⎜ ⎟<br />
s ⎝s+<br />
AΨ<br />
⎠<br />
Ψ<br />
Ψ<br />
=<br />
2<br />
Ψ Ψ iΨ<br />
(5.47)<br />
Discrete design<br />
z −1<br />
Using backward difference method for discretization process ( s = ) [2] the transfer<br />
Tz<br />
function <strong>of</strong> equation (5.45) for flux PI controller in discrete system is expressed as:<br />
s<br />
K<br />
pΨ<br />
( KpΨ<br />
+ KiΨ)( z−<br />
)<br />
Tz<br />
K<br />
s<br />
pΨ<br />
+ KiΨ<br />
CΨ( z) = KpΨ(1 + ) =<br />
T ( z−1) ( z−1)<br />
iΨ<br />
(5.48)<br />
Where:<br />
K<br />
s<br />
K<br />
pΨ<br />
iΨ<br />
= Ts; s<br />
Ti<br />
Ψ<br />
Ψ<br />
_<br />
( z)<br />
ref<br />
T - sampling time.<br />
C<br />
Ψ<br />
( z)<br />
W( z)<br />
U sx<br />
Dz ( )<br />
z −1<br />
GΨ<br />
( z)<br />
}<br />
ZOH<br />
1<br />
s+<br />
A Ψ<br />
Ψ<br />
s<br />
( z)<br />
Figure 5.28. Block diagram <strong>of</strong> flux control loop in discrete domain.<br />
Where: CΨ ( z)<br />
discrete transfer function <strong>of</strong> PI controller, Dz ( )<br />
1<br />
z − - one sampling time delay<br />
for voltage generation from PWM, and W( z)<br />
- disturbance voltage due to cross coupling<br />
between x-y axis (see Fig. 5.28).<br />
The GΨ ( z)<br />
is discrete transfer function <strong>of</strong> voltage-flux relationship <strong>with</strong> zero order hold<br />
(ZOH) block can be calculated as:<br />
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<strong>Direct</strong> <strong>Torque</strong> <strong>Control</strong> <strong>with</strong> <strong>Space</strong> <strong>Vector</strong> <strong>Modulation</strong><br />
−1 GΨ() s z −1<br />
AΨ<br />
GΨ<br />
( z) = (1 − z ) Z[ ] = Z[ ] =<br />
s z A s( s+<br />
A )<br />
( z − )<br />
1 1 AΨ<br />
Z[ ]<br />
z A s( s+<br />
A )<br />
Ψ<br />
Ψ<br />
Ψ<br />
Ψ<br />
(5.49)<br />
Using table <strong>of</strong> Z transformation [2] one can calculate:<br />
G<br />
Ψ<br />
( z − )<br />
− AΨTs<br />
1 1 z(1 − e ) AΨ<br />
d<br />
( z)<br />
=<br />
− A T<br />
z A<br />
Ψ s<br />
=<br />
z 1( z−e<br />
) z − B<br />
Ψ<br />
( − )<br />
Ψd<br />
(5.50)<br />
Where:<br />
A<br />
Ψd<br />
− AΨT (1 − e s<br />
)<br />
− A Ts<br />
= , BΨ d<br />
e Ψ<br />
A<br />
= and T<br />
s<br />
is sampling time.<br />
Ψ<br />
Hence, the transfer function <strong>of</strong> closed stator flux control loop can be expressed in the<br />
following form:<br />
G<br />
Ψ _ closed<br />
Ψ<br />
s<br />
( z) _ ref CΨ( z) GΨ( z) D( z)<br />
( z)<br />
= =<br />
Ψ ( z) 1 + C ( z) G ( z) D( z)<br />
s<br />
K<br />
pΨ<br />
( KpΨ + KiΨ) AΨd( z−<br />
)<br />
KpΨ<br />
+ KiΨ<br />
=<br />
K<br />
pΨ<br />
zz ( −1)( z− BΨd) + ( KpΨ + KiΨ) AΨd( z−<br />
)<br />
K + K<br />
Ψ<br />
Ψ<br />
pΨ<br />
iΨ<br />
(5.51)<br />
Now selecting<br />
K Ψ<br />
, K Ψ<br />
is possible to obtain poles placement, which define the dynamic <strong>of</strong><br />
p<br />
i<br />
closed torque control loop.<br />
Assuming, that<br />
B<br />
Ψd<br />
=<br />
K<br />
K<br />
pΨ<br />
pΨ<br />
+ K<br />
iΨ<br />
KpΨ − BΨdKpΨ KpΨ<br />
⇒ KiΨ<br />
= = (1 − BΨd)<br />
B B<br />
Ψd<br />
Ψd<br />
and the transfer function <strong>of</strong> closed stator flux control loop will take the following form:<br />
G<br />
( KpΨ + KiΨ)<br />
AΨd<br />
( z)<br />
=<br />
z − z+ ( K + K ) A<br />
Ψ _ closed<br />
2<br />
pΨ iΨ Ψd<br />
(5.52)<br />
Putting into above equation<br />
K<br />
K<br />
pΨ<br />
pΨ<br />
+ KiΨ<br />
= one obtains:<br />
BΨd<br />
G<br />
K<br />
A<br />
CΨd<br />
K z − z+<br />
C<br />
pΨ<br />
Ψd<br />
BΨd<br />
Ψ _ closed<br />
( z)<br />
= =<br />
2<br />
2<br />
pΨ<br />
z − z+<br />
AΨ<br />
d<br />
BΨd<br />
Ψd<br />
(5.53)<br />
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<strong>Direct</strong> <strong>Torque</strong> <strong>Control</strong> <strong>with</strong> <strong>Space</strong> <strong>Vector</strong> <strong>Modulation</strong><br />
K<br />
pΨ<br />
where CΨd<br />
= AΨd<br />
BΨd<br />
The bellow diagrams shows the relationship between overshoot<br />
time t s<br />
as function<br />
C Ψ d<br />
value.<br />
M<br />
p<br />
, rise time t r<br />
and settling<br />
Please note that t r<br />
is time calculated from 10% to 90% <strong>of</strong> output signals and t s<br />
is the time in<br />
witch the system transient decay to +-1%.<br />
Figure 5.29. The relationship between overshoot, rise time and settling time versus<br />
amplitude control loop.<br />
From a few values <strong>of</strong><br />
C Ψ d<br />
=[0.4046, 0.2688, 0.1720] we can selected C Ψ d<br />
guaranties overshoot 0% and settling time about 10 samples.<br />
C Ψ d<br />
for stator flux<br />
=0.2688, which<br />
1<br />
2<br />
3<br />
Figure 5.30. Flux step response for different values <strong>of</strong><br />
C Ψ d<br />
=0.2688, black line (3) C Ψ d<br />
=0.1720.<br />
C Ψ d<br />
: red line (1) C Ψ d<br />
=0.4046, blue line (2)<br />
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<strong>Direct</strong> <strong>Torque</strong> <strong>Control</strong> <strong>with</strong> <strong>Space</strong> <strong>Vector</strong> <strong>Modulation</strong><br />
It corresponds to the transfer function <strong>of</strong> closed stator flux control loop:<br />
G<br />
C<br />
0.2688<br />
( z)<br />
= =<br />
z − z+ C z −z+ 0.2688<br />
Ψd<br />
Ψ _ closed<br />
2 2<br />
Ψd<br />
(5.54)<br />
Using digitalized motor parameters A<br />
can calculate the parameters <strong>of</strong> digital PI flux controller as:<br />
Ψd<br />
− AΨT (1 − e s<br />
)<br />
− A Ts<br />
= , BΨ d<br />
e Ψ<br />
A<br />
= and chosen C Ψ d<br />
value, we<br />
Ψ<br />
K<br />
pΨ<br />
C B<br />
A<br />
Ψd<br />
Ψd<br />
= (5.55a)<br />
Ψd<br />
T<br />
iΨ<br />
K<br />
T<br />
pΨ<br />
s<br />
= (5.55b)<br />
K<br />
iΨ<br />
K<br />
iΨ<br />
K<br />
pΨ<br />
= (1 − BΨd)<br />
(5.55c)<br />
B<br />
Ψd<br />
For example <strong>with</strong> sampling time T = 200µ<br />
s, parameters <strong>of</strong> PI controller are:<br />
s<br />
K<br />
pΨ<br />
0.2688BΨd<br />
0.2688*0.9772<br />
= = = 1328.64<br />
(5.56a)<br />
A 0.0001977<br />
Ψd<br />
T<br />
iΨ<br />
KpΨTs<br />
1328.64*200µ<br />
s<br />
= = = 8572µ<br />
s<br />
(5. 56b)<br />
K 30.999<br />
iΨ<br />
K<br />
iΨ<br />
K<br />
pΨ<br />
1328.64<br />
= (1 − BΨd) = *(1-0.9772) = 30.999 (5. 56c)<br />
B<br />
0.9772<br />
Ψd<br />
For different sampling time the closed transfer function GΨ _ closed<br />
( z)<br />
<strong>of</strong> digital flux control<br />
loop should be kept to:<br />
G<br />
C<br />
0.2688<br />
( z)<br />
= =<br />
z − z+ C z −z+ 0.2688<br />
Ψd<br />
Ψ _ closed<br />
2 2<br />
Ψd<br />
(5.57)<br />
In order to find the original function <strong>of</strong> Z transfer function GΨ _ closed<br />
( z)<br />
using the Z properties<br />
as (sum transformations) [2]:<br />
n<br />
z<br />
z<br />
∑ f kT Z F z Z G z<br />
k = 0<br />
z−1 z−1<br />
−1<br />
z a<br />
= Z [ ]<br />
2<br />
z−1( z − z+<br />
a)<br />
−1 −1<br />
(<br />
s) = [ ( )] = [<br />
Ψ _ closed( )]<br />
(5.58)<br />
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<strong>Direct</strong> <strong>Torque</strong> <strong>Control</strong> <strong>with</strong> <strong>Space</strong> <strong>Vector</strong> <strong>Modulation</strong><br />
As an example the calculated response for 4 samples are given:<br />
n<br />
∑<br />
k = 0<br />
f kT az az a a az<br />
−2 −3 −4<br />
(<br />
s<br />
) = + 2 − (1 + ) + 2 + .......<br />
which gives:<br />
f(0) = f( k) = 0<br />
f(1 T ) = f( k+ 1) = 0<br />
s<br />
f(2 T ) = f( k+ 2) = a=<br />
0.2688<br />
s<br />
f(3 T ) = f( k+ 3) = a( b+ 1) = 2a=<br />
0.5376<br />
s<br />
f T f k a b c a b a a a<br />
2<br />
(4<br />
s<br />
) = ( + 4) =− ( + ) + ( + 1) =− (1 + ) + 2 = 0.7342<br />
Keeping constant<br />
C Ψ d<br />
in equation (5.57) the flux step response for different sampling time<br />
T<br />
s<br />
= 50 µ s,<br />
100 µ s,<br />
200 µ s,<br />
400µ s , which correspond to switching frequency f s<br />
= 20kHz,<br />
10kHz, 5kHz, 2.5kHz are presented.<br />
1<br />
2<br />
3<br />
4<br />
Figure 5.31.Flux step response for different sampling time T<br />
s<br />
= 50 µ s,<br />
100 µ s,<br />
200 µ s,<br />
400µ<br />
s<br />
(switching frequency f<br />
s<br />
20kHz (1 -blue line), 10kHz (2 -green line), 5kHz (3 -red line),2.5kHz<br />
(4 -light blue line).<br />
We may observe from Fig. 5.31 that overshoot is 0% and the settling time is 10 samples.<br />
Selected parameters <strong>of</strong> PI flux controller for sampling time T s<br />
= 50 µ s,<br />
100 µ s,<br />
200 µ s,<br />
400µ s are summarized in Table 5.3<br />
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<strong>Direct</strong> <strong>Torque</strong> <strong>Control</strong> <strong>with</strong> <strong>Space</strong> <strong>Vector</strong> <strong>Modulation</strong><br />
96
<strong>Direct</strong> <strong>Torque</strong> <strong>Control</strong> <strong>with</strong> <strong>Space</strong> <strong>Vector</strong> <strong>Modulation</strong><br />
The behavior <strong>of</strong> the flux control loop was tested using SABER simulation package. The<br />
model created in SABER takes into account the whole control system, which include real<br />
models <strong>of</strong> inverter and permanent magnet synchronous motor.<br />
The flux step response is shown in Fig. 5.32., when parameters <strong>of</strong> PI flux controller designed<br />
for sampling time T s<br />
= 200µ s were used for control plant for different sampling times<br />
T<br />
s<br />
= 50µ s , 100µ s , 200µ s , 400µ s , which correspond to switching frequency f<br />
s<br />
= 20kHz,<br />
10kHz, 5kHz, 2.5kHz.<br />
Figure 5.32. Flux tracking performance for different sampling time T<br />
s<br />
= 50µ s , 100µ s , 200µ s ,<br />
400µ s ( f<br />
s<br />
= 20kHz, 10kHz, 5kHz,2.5kHz) using parameters <strong>of</strong> PI controller designed for<br />
T<br />
s<br />
= 200µ s ( f<br />
s<br />
= 5kHz).<br />
After modification <strong>of</strong> PI flux controller parameters according to Table 5.31 it is possible to<br />
achieve better results as shown in Fig. 5.33, what confirms proper flux tracking performance<br />
in steady and dynamics state.<br />
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<strong>Direct</strong> <strong>Torque</strong> <strong>Control</strong> <strong>with</strong> <strong>Space</strong> <strong>Vector</strong> <strong>Modulation</strong><br />
Figure 5.33. Flux tracking performance for different sampling time T<br />
s<br />
= 50µ s , 100µ s , 200µ s ,<br />
400µ s ( f<br />
s<br />
= 20kHz, 10kHz, 5kHz,2.5kHz) using parameters <strong>of</strong> PI controller calculated individually<br />
(see table 5.3.)<br />
The simulation results in SABER package for Ts=200us is presented in Fig. 5.34.<br />
Figure 5.34. Simulated (SABER) flux tracking performance for step change from 70% - 100% <strong>of</strong><br />
nominal flux.<br />
As we can observed from Fig. 5.34 that overshoot is around M<br />
p<br />
= 0% and settling time took<br />
about 10 sampling time <strong>of</strong> microprocessor, what proved the design procedure <strong>of</strong> PI digital<br />
flux controller parameters.<br />
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<strong>Direct</strong> <strong>Torque</strong> <strong>Control</strong> <strong>with</strong> <strong>Space</strong> <strong>Vector</strong> <strong>Modulation</strong><br />
5.3.2 Digital torque control loop<br />
The PMSM equations (2.27a,b-2.28a,b) in stator flux coordinates under the assumption<br />
L<br />
d<br />
= L can be written as:<br />
q<br />
U = R I +ΩΨ<br />
Ψ (5.59)<br />
sy s sy s s<br />
0 LI sin<br />
= −Ψ (5.60)<br />
s sy PM<br />
δ Ψ<br />
3<br />
M<br />
e<br />
= pb Ψ I<br />
s sy<br />
(5.61)<br />
2<br />
dΩ m 1 = ( M<br />
e − M<br />
l )<br />
(5.62)<br />
dt J<br />
The load angle can be expressed (Fig. 5.1):<br />
δ = θ − p γ , (5.63)<br />
Ψ<br />
Ψs b m<br />
Where: δ Ψ<br />
is torque angle, θ Ψ s<br />
is stator flux vector position, and γ<br />
m<br />
is rotor position in stator<br />
α,<br />
β coordinates,<br />
p<br />
b<br />
is number <strong>of</strong> pole pars.<br />
After differentiation equation (5.63) can be written as:<br />
dδ<br />
dθ<br />
dγ<br />
dt dt dt<br />
Ψ Ψs<br />
m<br />
= − pb<br />
(5.64)<br />
δ Ψ<br />
d<br />
dt<br />
δ Ψ<br />
d<br />
=ΩΨ<br />
−p<br />
s bΩm<br />
⇒Ω Ψ s<br />
= + pbΩ m<br />
(5.65)<br />
dt<br />
Putting equations (5.64) and (5.65) into voltage equation (5.59) one obtains:<br />
dδ Usy = RsIsy +Ω ( )<br />
s s<br />
RsI Ψ<br />
Ψ<br />
Ψ =<br />
sy<br />
+ Ψ<br />
s<br />
+ pbΩ m<br />
(5.66)<br />
dt<br />
From equation 0= LI −Ψ sin <strong>with</strong> assumption that for small angle δ = sinδ<br />
, the<br />
s sy PM<br />
torque angle can be expressed as:<br />
δ Ψ<br />
Ψ<br />
Ψ<br />
L I<br />
s sy<br />
δ Ψ<br />
= (5.67)<br />
Ψ<br />
PM<br />
So, the voltage equation (5.59) becomes:<br />
L dI<br />
s sy<br />
Usy = Rs Isy +ΩΨ<br />
Ψ ( )<br />
s s<br />
= Rs Isy + Ψ<br />
s<br />
+ pbΩm<br />
Ψ dt<br />
PM<br />
(5.68)<br />
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<strong>Direct</strong> <strong>Torque</strong> <strong>Control</strong> <strong>with</strong> <strong>Space</strong> <strong>Vector</strong> <strong>Modulation</strong><br />
After differentiating <strong>of</strong> the above equation one obtains:<br />
2<br />
sy sy Ls<br />
sy<br />
= Rs + Ψ<br />
s<br />
+ pb<br />
Ψ<br />
PM<br />
dU dI d I dΩm<br />
( )<br />
dt dt dt dt<br />
Take into account that from equation (5.61) the y-axis current is equal<br />
I<br />
sy<br />
(5.69)<br />
2M<br />
e<br />
=<br />
3p<br />
Ψ ,<br />
b<br />
s<br />
dΩ m 1 = ( M<br />
e − M<br />
l ) and under assumption that the motor is no loaded equation (5.69) takes<br />
dt J<br />
form:<br />
dU dM L d M p<br />
( )<br />
dt p dt p dt J<br />
2<br />
sy 2 e s 2<br />
e b<br />
= Rs + Ψ<br />
s<br />
+ M<br />
e<br />
3<br />
b<br />
Ψs ΨPM 3<br />
b<br />
Ψs<br />
(5.70)<br />
Using Laplace transformation and after some arrangements the equation (5.70) can be written:<br />
2Ls<br />
2 2R<br />
Ψs<br />
p<br />
s<br />
b<br />
sU<br />
sy<br />
= M<br />
e<br />
( s + s + )<br />
3p Ψ 3p Ψ J<br />
b PM b s<br />
(5.71)<br />
Hence, the transfer function between electromagnetic torque<br />
be obtained as:<br />
G<br />
M<br />
e<br />
M<br />
2<br />
sy()<br />
M M<br />
M<br />
e<br />
and y-axis voltage<br />
U<br />
sy<br />
can<br />
M () s A s<br />
() s = =<br />
U s s + B s+ C<br />
(5.72)<br />
Where:<br />
A<br />
M<br />
2<br />
3pbΨ<br />
PM<br />
RsΨ<br />
3 Ψ<br />
PM<br />
s<br />
ΨPM pb<br />
= and BM<br />
= CM<br />
=<br />
2L<br />
Ψ L<br />
2JL<br />
s<br />
s<br />
s<br />
s<br />
Using the motor parameters (see Appendices) we may calculates:<br />
A = 198 , B = 115.3 and C = 9065<br />
M<br />
M<br />
M<br />
Continuous s-domain<br />
The torque control loop <strong>of</strong> the block scheme <strong>DTC</strong>-<strong>SVM</strong> from Fig. 5.25 is shown in Fig. 5.35,<br />
where CM<br />
( s ) is a transfer function <strong>of</strong> the PI controller given by equation 5.28:<br />
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<strong>Direct</strong> <strong>Torque</strong> <strong>Control</strong> <strong>with</strong> <strong>Space</strong> <strong>Vector</strong> <strong>Modulation</strong><br />
M e _ ref<br />
U sy<br />
C () M<br />
s GM<br />
() s<br />
M e<br />
Figure 5.35. Block diagram <strong>of</strong> torque control loop in s-domain.<br />
The transfer function <strong>of</strong> torque control loop is obtained as:<br />
G<br />
M _ closed<br />
() s<br />
M () s C () s G () s<br />
e_<br />
ref M M<br />
= = (5.73)<br />
M<br />
e() s 1 + CM() z GM()<br />
s<br />
Substituting in equation (5.73) transfer function for CM<br />
( s)<br />
-Eq.5.28 and GM<br />
( s)<br />
- Eq.5.72 we<br />
may calculate:<br />
KiM<br />
AMKpM( s+<br />
)<br />
K<br />
pM<br />
GM<br />
_ closed()<br />
s = =<br />
2<br />
s + ( B + K A ) s+ C + K A<br />
M pM M M iM M<br />
(5.74)<br />
Discrete design<br />
z −1<br />
Using backward difference method for discretization process ( s = ) the transfer function<br />
Tz<br />
for discrete PI controller is expressed as:<br />
s<br />
K<br />
pM<br />
( z − )<br />
Tz<br />
KpM<br />
+ K<br />
s<br />
iM<br />
CM( z) = KpM(1 + ) = ( KpM + KiM)<br />
T ( z−1) ( z−1)<br />
iM<br />
(5.75)<br />
Where:<br />
K<br />
iM<br />
K<br />
pM<br />
= Ts<br />
- integration gain;<br />
s<br />
TiM<br />
T - sampling time<br />
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<strong>Direct</strong> <strong>Torque</strong> <strong>Control</strong> <strong>with</strong> <strong>Space</strong> <strong>Vector</strong> <strong>Modulation</strong><br />
PI controller<br />
Time delay<br />
G ( ) M<br />
z<br />
Dz ( )} Plant<br />
M<br />
e_ ref( z)<br />
C ( ) M<br />
z<br />
U sy<br />
z −1<br />
AM<br />
s<br />
ZOH<br />
2<br />
s + B s+<br />
C<br />
M<br />
M<br />
M ( ) e<br />
z<br />
Figure 5.36. Block diagram <strong>of</strong> torque control loop in discrete domain.<br />
Where: CM<br />
( z)<br />
- discrete transfer function for PI controller, Dz ( )<br />
delay for voltage generation from PWM block (see Fig. 5.36).<br />
1<br />
z − - one sampling time<br />
The discrete transfer function G ( z ) for voltage-torque relationship <strong>with</strong> zero order hold<br />
(ZOH) can be calculated as:<br />
M<br />
G () s z −1<br />
A<br />
G z z Z Z<br />
−1<br />
M<br />
M<br />
M<br />
( ) = (1 − ) [ ] = [ ]<br />
2<br />
s z s + BMs+<br />
CM<br />
⎡<br />
⎤<br />
⎡ ⎤ ⎢ ⎥<br />
⎢ ⎥<br />
z−1 AM<br />
z−1<br />
⎢ A<br />
⎥<br />
M<br />
= Z ⎢<br />
⎥ = Z<br />
2<br />
⎢<br />
2<br />
2<br />
⎥ =<br />
z ⎢⎛ B 2<br />
M ⎞ B ⎥ z<br />
M ⎢ B ⎛<br />
M 2 B ⎞ ⎥<br />
⎢⎜s+ ⎟ + CM<br />
− ⎥<br />
M<br />
2 4 ⎢( s+ ) + CM<br />
−<br />
⎥<br />
⎣⎝ ⎠ ⎦<br />
⎢ 2 ⎜ 4 ⎟<br />
⎣ ⎝ ⎠ ⎥⎦<br />
.<br />
⎡<br />
⎤<br />
2<br />
⎢<br />
B ⎥<br />
M<br />
CM<br />
−<br />
z −1 A ⎢<br />
M<br />
Z<br />
4<br />
⎥<br />
= ⎢<br />
2<br />
2<br />
⎥<br />
z B 2<br />
M<br />
⎢<br />
C B ⎛<br />
M 2 B ⎞ ⎥<br />
M<br />
M<br />
−<br />
4<br />
⎢( s+ ) + CM<br />
−<br />
⎥<br />
⎢ 2 ⎜ 4 ⎟<br />
⎣ ⎝ ⎠ ⎥⎦<br />
(5.76)<br />
Assuming that<br />
have:<br />
B<br />
a = M<br />
and<br />
2<br />
2<br />
BM<br />
b= CM<br />
− , and using table <strong>of</strong> Z transformation [2] we<br />
4<br />
−aTs<br />
⎡ b ⎤ ze sin( bTs<br />
)<br />
Z ⎢ 2 2 2 aTs<br />
( s a) b<br />
⎥ =<br />
−<br />
⎣ + + ⎦ z − 2 e (cos( bTs<br />
)) z+<br />
e<br />
−2aTs<br />
(5.77)<br />
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<strong>Direct</strong> <strong>Torque</strong> <strong>Control</strong> <strong>with</strong> <strong>Space</strong> <strong>Vector</strong> <strong>Modulation</strong><br />
Finally, the discrete transfer function <strong>of</strong> controlled plant G ( z ) can be written as:<br />
M<br />
Where:<br />
C<br />
M _ d<br />
⎛ A ⎞<br />
Md<br />
GM<br />
( z) = ( z−1) ⎜ 2<br />
⎟<br />
⎝ z − BMd<br />
z+<br />
CMd<br />
⎠<br />
2<br />
A<br />
BM<br />
A = e sin( T C − ) ,<br />
4<br />
BM<br />
− T<br />
M<br />
s<br />
2<br />
M _ d<br />
2<br />
s M<br />
BM<br />
CM<br />
−<br />
4<br />
BM<br />
Ts<br />
= e − and T<br />
s<br />
is sampling time.<br />
(5.78)<br />
BM<br />
2<br />
− Ts<br />
B<br />
2<br />
M<br />
BM _ d<br />
= 2e cos( Ts CM<br />
− )<br />
4<br />
Hence, the transfer function <strong>of</strong> closed torque control loop is obtained as:<br />
G<br />
M _ closed<br />
( z)<br />
M ( z) C ( z) G ( z) D( z)<br />
M ( z) 1 + C ( z) G ( z) D( z)<br />
e_<br />
ref M M<br />
= = =<br />
e M M<br />
K<br />
A ( K + K )( z−<br />
)<br />
pM<br />
M _ d pM iM<br />
KpM<br />
+ KiM<br />
3 2<br />
−<br />
M _ d<br />
+ [<br />
M _ d( pM<br />
+<br />
iM) +<br />
M _ d]<br />
−<br />
M _ d pM<br />
=<br />
z B z A K K C z A K<br />
(5.79)<br />
Selecting<br />
K Ψ<br />
, K Ψ<br />
will influence poles placement <strong>of</strong> closed torque control loop and as a<br />
p<br />
i<br />
consequence also torque step responses can be selected.<br />
The transfer function <strong>of</strong> closed torque control loop is more complicated than flux control loop<br />
(see design <strong>of</strong> PI-flux controller – section 5.3.1). One possibility is use to the SISO tools from<br />
Matlab package to tune parameters <strong>of</strong> PI torque controller [106].<br />
a) b)<br />
Figure 5.37. a) <strong>Torque</strong> step response for sampling time T = 200µ<br />
s, b) <strong>with</strong> denoted rise time,<br />
overshoot and settling time.<br />
As can be observed the response is characterized by overshoot about 40%, rise time 4 samples<br />
and settling time 17 samples.<br />
s<br />
103
<strong>Direct</strong> <strong>Torque</strong> <strong>Control</strong> <strong>with</strong> <strong>Space</strong> <strong>Vector</strong> <strong>Modulation</strong><br />
The torque control loop should be as fast as possible even <strong>with</strong> some overshoot. This improve<br />
response to disturbance (for example from flux control loop –see Fig. 5.38).<br />
M<br />
e_ ref( z)<br />
C ( ) M<br />
z<br />
U sy<br />
Dz ( )<br />
z −1<br />
G ( ) M<br />
z<br />
}<br />
AM<br />
s<br />
ZOH<br />
2<br />
s + B s+<br />
C<br />
M<br />
M<br />
M ( ) e<br />
z<br />
Figure 5.38. Block diagram <strong>of</strong> torque controller in discrete domain <strong>with</strong> disturbance.<br />
a )<br />
b)<br />
Figure 5.39. Disturbance rejection in torque control loop: a) short voltage impulse, b) voltage<br />
step.<br />
To improve reference tracking performance (<strong>with</strong>out any overshoot) it is recommended to<br />
insert a input prefilter (see Fig. 5.40 ) described by transfer function:<br />
z−b z−b z−0.6663<br />
PM<br />
( z)<br />
= K = K = K<br />
(5.80)<br />
K<br />
pM ( z−a) z−0.855<br />
( z − )<br />
K + K<br />
pM<br />
iM<br />
Where:<br />
K<br />
1<br />
=<br />
=0.43413 is gain <strong>of</strong> the prefilter.<br />
z − 0.6663<br />
lim<br />
z→1<br />
z − 0.855<br />
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<strong>Direct</strong> <strong>Torque</strong> <strong>Control</strong> <strong>with</strong> <strong>Space</strong> <strong>Vector</strong> <strong>Modulation</strong><br />
M<br />
e_ ref( z)<br />
P ( ) M<br />
z<br />
C ( ) M<br />
z<br />
U sy<br />
Dz ( )<br />
z −1<br />
G ( ) M<br />
z<br />
}<br />
AM<br />
s<br />
ZOH<br />
2<br />
s + B s+<br />
C<br />
M<br />
M<br />
M ( ) e<br />
z<br />
Figure 5.40. Block diagram <strong>of</strong> torque control loop <strong>with</strong> prefilter in discrete domain.<br />
Finally, the reference tracking performance <strong>of</strong> closed torque control loop <strong>with</strong> prefilter is<br />
presented in Fig. 5.41.<br />
Figure 5.41. a) Reference tracking performance <strong>of</strong> the torque control loop for sampling time<br />
Ts<br />
= 200µ<br />
s, b) <strong>with</strong> denoted rise time, overshoot and settling time.<br />
In digital control when the sampling time is changing the parameters <strong>of</strong> digitalized plant<br />
control AMd, BMd,<br />
C<br />
Md<br />
will be also changes. It is normally that the parameters <strong>of</strong> PI torque<br />
control will be changes also (see Table 5.4.).<br />
Simulation for PI flux calculated parameters <strong>with</strong> and <strong>with</strong>out prefilter are shown in Fig.<br />
5.42a-b. Also torque step response for different level <strong>of</strong> reference torque are presented in Fig.<br />
5.43.<br />
105
<strong>Direct</strong> <strong>Torque</strong> <strong>Control</strong> <strong>with</strong> <strong>Space</strong> <strong>Vector</strong> <strong>Modulation</strong><br />
a)<br />
b)<br />
Figure 5.42. <strong>Torque</strong> step response: a) <strong>with</strong>out prefilter, b) <strong>with</strong> prefilter.<br />
Figure 5.43. <strong>Torque</strong> step response <strong>with</strong> prefilter (from 0 to 25%, 50%, 75% and 100% <strong>of</strong> nominal<br />
torque).<br />
106
<strong>Direct</strong> <strong>Torque</strong> <strong>Control</strong> <strong>with</strong> <strong>Space</strong> <strong>Vector</strong> <strong>Modulation</strong><br />
107
<strong>Direct</strong> <strong>Torque</strong> <strong>Control</strong> <strong>with</strong> <strong>Space</strong> <strong>Vector</strong> <strong>Modulation</strong><br />
The behavior <strong>of</strong> the torque control loop was tested using SABER simulation package.<br />
The model created in SABER takes into account the whole control system, which<br />
include real models <strong>of</strong> inverter and permanent magnet synchronous motor.<br />
The torque step response is shown in Fig. 5.44, when parameters <strong>of</strong> PI torque controller<br />
designed for sampling time T s<br />
= 200µ s were used for control plant for different<br />
sampling times T s<br />
= 50µ s , 100µ s , 200µ s , 400µ s , which correspond to switching<br />
frequency f<br />
s<br />
= 20kHz, 10kHz, 5kHz, 2.5kHz.<br />
Figure 5.44. <strong>Torque</strong> tracking performance for different sampling time T s<br />
= 50µ s , 100µ s ,<br />
200µ s , 400µ s ( f<br />
s<br />
= 20kHz, 10kHz, 5kHz,2.5kHz) using parameters <strong>of</strong> PI controller<br />
designed for T<br />
s<br />
= 200µ s ( f<br />
s<br />
= 5kHz). Pleas note that for 2.5kHz the system was unstable.<br />
After modification <strong>of</strong> PI flux controller parameters according to Table 5.4 it is possible<br />
to achieve better results as shown in Fig. 5.45, what confirms proper torque tracking<br />
performance in steady and dynamics state.<br />
108
<strong>Direct</strong> <strong>Torque</strong> <strong>Control</strong> <strong>with</strong> <strong>Space</strong> <strong>Vector</strong> <strong>Modulation</strong><br />
Figure 5.45. <strong>Torque</strong> tracking performance for different sampling time T s<br />
= 50µ s , 100µ s ,<br />
200µ s , 400µ s ( f<br />
s<br />
= 20kHz, 10kHz, 5kHz,2.5kHz) using parameters <strong>of</strong> PI controller<br />
calculated individually (see Table 5.4.)<br />
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<strong>Direct</strong> <strong>Torque</strong> <strong>Control</strong> <strong>with</strong> <strong>Space</strong> <strong>Vector</strong> <strong>Modulation</strong><br />
5.4 Speed control loop for <strong>DTC</strong>-<strong>SVM</strong> structure<br />
The structure <strong>of</strong> speed control loop for cascade (a) and parallel (b) <strong>DTC</strong>-<strong>SVM</strong> scheme<br />
is shown in the Fig. 5.46.<br />
a)<br />
U DC<br />
Ψ s _ ref<br />
U<br />
s α _ ref<br />
S A<br />
S B<br />
Ω m _ ref<br />
M e _ ref<br />
e M<br />
∆δ Ψ<br />
U<br />
s β _ ref<br />
S C<br />
θ Ψs Ψs<br />
I s<br />
M e<br />
I s<br />
d<br />
dt<br />
γ m<br />
U DC<br />
b)<br />
Ψ s _ ref<br />
e Ψ<br />
ss<br />
U sx _ ref<br />
s<br />
_ ref<br />
S A<br />
S B<br />
Ω m _ ref<br />
M e _ ref<br />
e M<br />
U sy _ ref<br />
U α<br />
U<br />
s β _ ref<br />
S C<br />
θ Ψs<br />
Ψ<br />
I s<br />
M e<br />
d<br />
dt<br />
Figure 5.46. Speed control loop for: a) cascade structure <strong>of</strong> <strong>DTC</strong>-<strong>SVM</strong> scheme, b) parallel<br />
structure <strong>of</strong> <strong>DTC</strong>-<strong>SVM</strong> scheme.<br />
γ<br />
m<br />
The mechanical speed equation (2.66) for the PMSM is:<br />
dΩ<br />
M M J dt<br />
m<br />
e<br />
−<br />
L<br />
= (5.81)<br />
Taking Laplace transformation to equation (5.81) one obtains:<br />
M () s − M () s = JsΩ () s<br />
(5.82)<br />
e L m<br />
The transfer function between mechanical rotor speed<br />
M<br />
e<br />
can be expressed as:<br />
Ω<br />
m<br />
and electromagnetic torque<br />
110
<strong>Direct</strong> <strong>Torque</strong> <strong>Control</strong> <strong>with</strong> <strong>Space</strong> <strong>Vector</strong> <strong>Modulation</strong><br />
Ωm<br />
1 1<br />
GΩ<br />
() s = = = (5.83)<br />
M Js Js<br />
e<br />
Continuous s-domain<br />
The block diagram <strong>of</strong> speed control loop is shown in Fig. 5.47, where CΩ ( s)<br />
is a<br />
transfer function <strong>of</strong> the PI speed controller given by:<br />
KiΩ<br />
KpΩ( s+<br />
)<br />
1<br />
K<br />
pΩ<br />
CΩ() s = KpΩ(1 + ) = (5.84)<br />
T s s<br />
iΩ<br />
and DΩ ( s)<br />
is approximated transfer function <strong>of</strong> closed torque control loop for cascade<br />
or parallel <strong>DTC</strong>-<strong>SVM</strong> structure.<br />
M L<br />
Ω m _ ref<br />
M e ref<br />
_<br />
CΩ () s<br />
D () s<br />
G () s<br />
Ω<br />
M e<br />
Ω<br />
Ω m<br />
Figure 5.47. Block diagram <strong>of</strong> speed control loop in s-domain.<br />
Discrete design<br />
The transfer function for PI controller in discrete system using backward difference<br />
method for discretization process is expressed as:<br />
K<br />
pΩ<br />
( z − )<br />
KpΩ<br />
+ KiΩ<br />
CΩ( z) = ( KpΩ + KiΩ)<br />
(5.85)<br />
( z −1)<br />
K<br />
pΩ<br />
Where: KiΩ<br />
= Ts- integration and K<br />
p Ω<br />
proportional gain <strong>of</strong> speed controller, T s<br />
-<br />
T<br />
sampling time.<br />
iΩ<br />
111
<strong>Direct</strong> <strong>Torque</strong> <strong>Control</strong> <strong>with</strong> <strong>Space</strong> <strong>Vector</strong> <strong>Modulation</strong><br />
Ω m _ ref<br />
( z)<br />
C<br />
Ω<br />
( z)<br />
Me_ ref()<br />
z<br />
D ( z)<br />
Ω<br />
M ( ) e<br />
z<br />
M ( L<br />
z )<br />
GΩ<br />
( z)<br />
}<br />
ZOH<br />
1<br />
Js<br />
Ω ( z m<br />
)<br />
Figure 5.48. Block diagram <strong>of</strong> speed control loop in discrete domain.<br />
The closed torque control loop transfer function (see for cascade <strong>DTC</strong>-<strong>SVM</strong> -Table 5.2<br />
or parallel <strong>DTC</strong>-<strong>SVM</strong> - Table 5.4) is:<br />
0.466*0.37267 a<br />
DΩ ( z) = GM<br />
_ closed( z)<br />
≅ = (5.86)<br />
2 2<br />
z - 1.289z + 0.4633) z +bz + c<br />
The GΩ ( z)<br />
is discrete transfer function for torque-speed relationship <strong>with</strong> zero order<br />
hold (ZOH). The G ( z)<br />
can be calculated as:<br />
Ω<br />
G () s<br />
1<br />
G z z Z z Z<br />
s<br />
( Js)<br />
s<br />
−1 Ω<br />
−1<br />
Ω( ) = (1 − ) [ ] = (1 − ) [ ]<br />
−1<br />
1 1 1 Ts<br />
= (1 − z ) Z [ ] = J s<br />
2 J ( z −1)<br />
(5.87)<br />
Finally, it can be expressed as:<br />
AΩ<br />
d<br />
GΩ ( z)<br />
=<br />
( z −1)<br />
(5.88)<br />
Ts<br />
Where: A = Ω d<br />
J<br />
, and T<br />
s<br />
is sampling time.<br />
Using the sampling time T = 200µ<br />
s and motor parameters J = 0.0173 the GΩ ( z)<br />
can<br />
be calculated as:<br />
s<br />
G ( z)<br />
Ω<br />
=<br />
0.01156<br />
( z −1)<br />
Hence, the transfer function <strong>of</strong> closed speed control loop can be written:<br />
G<br />
Ω _ closed<br />
Ωm( z) CΩ( z) DΩ( z) GΩ( z)<br />
( z)<br />
= =<br />
Ω ( z) 1 + C ( z) D ( z) G ( z)<br />
m_<br />
ref<br />
Ω Ω Ω<br />
(5.89)<br />
(5.90)<br />
And finally<br />
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<strong>Direct</strong> <strong>Torque</strong> <strong>Control</strong> <strong>with</strong> <strong>Space</strong> <strong>Vector</strong> <strong>Modulation</strong><br />
G<br />
Ω _ closed<br />
( z)<br />
=<br />
K<br />
pΩ<br />
( K + K )( z−<br />
) aA<br />
K + K<br />
pΩ iΩ Ωd<br />
pΩ<br />
iΩ<br />
2<br />
pΩ<br />
( z−1)( z − bz+ c)( z− 1) + ( KpΩ + KiΩ)( z−<br />
) aAΩ<br />
d<br />
KpΩ<br />
+ KiΩ<br />
K<br />
(5.91)<br />
In many practical cases the digital filter is used in speed measurement loop (see Fig.<br />
5.49).<br />
Ω m _ ref<br />
( z)<br />
PI controller GΩ<br />
( z)<br />
C ( z)<br />
Ω<br />
Me_ ref()<br />
z<br />
D ( z)<br />
Ω<br />
M ( ) e<br />
z<br />
M ( ) L<br />
z<br />
ZOH<br />
<strong>Control</strong> Plant<br />
}<br />
1<br />
Js<br />
Ω ( z m<br />
)<br />
<strong>Torque</strong> control loop<br />
F ( z)<br />
Ω<br />
Digital Filter<br />
Figure 5.49. Block scheme <strong>of</strong> speed control <strong>with</strong> digital filter in speed measurement loop<br />
(discrete domain).<br />
The transfer function FΩ ( s)<br />
<strong>of</strong> first order low pass filter in s domain is expressed as:<br />
FΩ () s =<br />
1<br />
s<br />
+ 1<br />
ω<br />
i<br />
(5.92)<br />
2 ω<br />
Where ωc<br />
= 2π<br />
fc<br />
and f<br />
c<br />
is cut <strong>of</strong>f frequency and tan(<br />
cTs<br />
ω<br />
i<br />
= ) [2]. In practice f<br />
c<br />
T 2<br />
is selected in the range 20-250Hz<br />
2( z −1)<br />
Using the Tutsins’s approximation method s = for discretization process, the<br />
Ts<br />
( z+<br />
1)<br />
discrete transfer function <strong>of</strong> first order low pass filter can be expressed as:<br />
Tsωi<br />
2 −Tsωi<br />
Where: a1<br />
= , b1<br />
=<br />
2 + T ω 2 + T ω<br />
s<br />
i<br />
Tsωi<br />
( z + 1)<br />
2 + T ω a ( z+<br />
1)<br />
FΩ<br />
( z)<br />
= =<br />
z<br />
2 T ω<br />
s<br />
s<br />
s i<br />
1<br />
(5.93)<br />
2 −Tsωi<br />
−<br />
z−<br />
b1<br />
+<br />
i<br />
s<br />
i<br />
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<strong>Direct</strong> <strong>Torque</strong> <strong>Control</strong> <strong>with</strong> <strong>Space</strong> <strong>Vector</strong> <strong>Modulation</strong><br />
The discrete transfer function <strong>of</strong> digital filter for T = 200µ<br />
s and f = 25Hz<br />
can be<br />
calculated as:<br />
0.015466 (z+1)<br />
FΩ ( z)<br />
= (5.94)<br />
(z-0.9691)<br />
Hence, the transfer function speed control loop <strong>with</strong> digital filter:<br />
s<br />
c<br />
G<br />
Ω _ closed<br />
Ωm( z) CΩ( z) DΩ( z) GΩ( z)<br />
( z)<br />
= =<br />
Ω ( z) 1 + C ( z) D ( z) G ( z) F( c)<br />
m_<br />
ref<br />
Ω Ω Ω<br />
(5.95)<br />
And finally<br />
G<br />
=<br />
Ω _ closed<br />
( z)<br />
=<br />
K<br />
( K + K )( z−b)( z−<br />
) aA<br />
pΩ<br />
pΩ iΩ 1<br />
Ωd<br />
KpΩ<br />
+ KiΩ<br />
K<br />
z bz c z z z b K K z aA a z<br />
2<br />
pΩ<br />
( − + )( −1)( −1)( −<br />
1) + (<br />
pΩ +<br />
iΩ)( − )<br />
Ωd<br />
1( + 1)<br />
KpΩ<br />
+ KiΩ<br />
(5.96)<br />
Selecting<br />
K Ω<br />
, K Ω<br />
will influence poles placement <strong>of</strong> closed speed control loop and as<br />
p<br />
i<br />
a consequence also speed step responses can be selected.<br />
In order to select the best value <strong>of</strong> PI speed controller it is recommended to use the<br />
SISO tools from Matlab package to tune the parameter <strong>of</strong> PI speed controller.<br />
The speed response <strong>with</strong> digital filter simulated in SIMULINK is shown in Fig. 5.50<br />
and simulated in SABER in Fig. 5.51 is presented.<br />
114
<strong>Direct</strong> <strong>Torque</strong> <strong>Control</strong> <strong>with</strong> <strong>Space</strong> <strong>Vector</strong> <strong>Modulation</strong><br />
Figure 5.50. Simulated (SIMULINK) speed response <strong>with</strong> digital filter<br />
Figure 5.51. Simulated (SABER) speed response <strong>with</strong> digital filter in feedback. From the top<br />
reference torque, measured speed.<br />
However, the speed respond is characterized by large overshoot. Therefore, the prefilter<br />
will be applied in order to reduce overshoot (see Fig. 5.52). The discrete transfer<br />
function <strong>of</strong> prefilter Pz ( ) can be expressed as”<br />
KK _ s 0.005<br />
Pz ( ) = =<br />
z−bb_ s z−0.995<br />
(5.97)<br />
where<br />
KK _ s = lim( z − 0.995)=0.005 is gain <strong>of</strong> the prefilter.<br />
z→1<br />
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<strong>Direct</strong> <strong>Torque</strong> <strong>Control</strong> <strong>with</strong> <strong>Space</strong> <strong>Vector</strong> <strong>Modulation</strong><br />
Ω m _ ref<br />
( z)<br />
P ( z)<br />
Ω<br />
C ( z)<br />
Ω<br />
M<br />
_<br />
() z<br />
e ref<br />
D ( z)<br />
Ω<br />
M ( ) e<br />
z<br />
M ( L<br />
z )<br />
GΩ<br />
( z)<br />
}<br />
ZOH<br />
1<br />
Js<br />
Ω ( z m<br />
)<br />
F ( z)<br />
Ω<br />
Figure 5.52. Speed response <strong>with</strong> digital filter in feedback FΩ ( z)<br />
and prefilter PΩ ( z)<br />
at the<br />
input.<br />
The speed response <strong>with</strong> and <strong>with</strong>out prefilter are shown in Fig. 5.53.<br />
Without prefilter<br />
With prefilter<br />
Figure 5.53. Speed response: blue signal <strong>with</strong>out prefilter and green signal <strong>with</strong> prefilter at the<br />
input.<br />
Design parameters <strong>of</strong> PI speed controller for sampling time T<br />
s<br />
= 50 µ s,<br />
100 µ s,<br />
200 µ s,<br />
400µ s are summarized in Table 5.5.<br />
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<strong>Direct</strong> <strong>Torque</strong> <strong>Control</strong> <strong>with</strong> <strong>Space</strong> <strong>Vector</strong> <strong>Modulation</strong><br />
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<strong>Direct</strong> <strong>Torque</strong> <strong>Control</strong> <strong>with</strong> <strong>Space</strong> <strong>Vector</strong> <strong>Modulation</strong><br />
Simulated results for speed tracking performance for different reference speed level in<br />
Fig. 5.54 are shown.<br />
Figure 5.54. Simulated speed tracking performance <strong>with</strong> prefilter at the input for 10%, 20%,<br />
50%, 100% <strong>of</strong> nominal speed. From the top actual speed, reference torque.<br />
Investigation for influence <strong>of</strong> load torque in Fig. 5.55 is presented.<br />
Figure 5.55. Simulated disturbance rejection performance <strong>of</strong> speed control loop for step load<br />
change 50% <strong>of</strong> nominal torque. From the top electromagnetic torque, measured speed<br />
Simulation results for speed control loop in Saber package for sampling time<br />
Ts<br />
= 200µ<br />
sand PI speed parameters controller: K Ω<br />
= 1.1940 , T Ω<br />
= 0.0398 (see Table<br />
5.5) in Fig. 5.56 is shown.<br />
p<br />
i<br />
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<strong>Direct</strong> <strong>Torque</strong> <strong>Control</strong> <strong>with</strong> <strong>Space</strong> <strong>Vector</strong> <strong>Modulation</strong><br />
Figure 5.56. Simulated speed tracking performance to step <strong>of</strong> speed from 0 to 1000rpm.<br />
The presented simulation results confirm well proper operation and design methodology<br />
for digital speed control loop.<br />
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<strong>Direct</strong> <strong>Torque</strong> <strong>Control</strong> <strong>with</strong> <strong>Space</strong> <strong>Vector</strong> <strong>Modulation</strong><br />
5.5 Summary<br />
This Chapter presents design <strong>of</strong> discrete control loops for two <strong>DTC</strong>-<strong>SVM</strong> schemes:<br />
series (cascade) and parallel structures <strong>of</strong> flux and torque controllers, Fig. 5.2 and<br />
Fig. 5.25, respectively. The cascade structure operates <strong>with</strong> P-flux controller and PItorque<br />
controller whereas in parallel structure two PI controllers are used. In the first<br />
step <strong>of</strong> design calculation <strong>of</strong> discrete Z- transfer function from continuous s- domain<br />
transfer function using zero order hold (ZOH) method <strong>of</strong> discretization has been<br />
performed. The continuous PI controller transfer function has been discretized using<br />
backward difference approach. Secondly, a SISO tool from Matlab package for<br />
digital controller parameter calculation has been applied. The results <strong>of</strong> design were<br />
verified by Simulink (using simplified discrete transfer function) and Saber (using<br />
full motor and inverter model) simulation. Also, the influence <strong>of</strong> sampling time<br />
selection on controller parameters have been discussed. Finally, also the speed<br />
control loop was synthesized using similar methodology.<br />
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Sensorless Speed <strong>Direct</strong> <strong>Torque</strong> <strong>Control</strong> <strong>with</strong> <strong>Space</strong> <strong>Vector</strong> <strong>Modulation</strong> (<strong>DTC</strong>-<strong>SVM</strong>)<br />
Chapter 6<br />
DIRECT TORQUE CONTROL WITH SPACE VECTOR MODULATION (<strong>DTC</strong>-<br />
<strong>SVM</strong>) OF PMSM DRIVE WITHOUT MOTION SENSOR<br />
6.1 Introduction<br />
Many motion control applications, such as material handling, packaging and hydraulic<br />
or pneumatic cylinder replacement, require the use <strong>of</strong> a position transducer for speed or<br />
position feedback, such as an encoder or resolver. In addition, permanent magnet<br />
synchronous motors require position feedback to perform commutation. Some <strong>of</strong><br />
systems utilize velocity transducer as well. These sensor add cost, weight, and reduce<br />
the reliability <strong>of</strong> the system. Also, a special mechanical arrangement needs to be made<br />
for mounting the position sensors. An extra signal wires are required from the sensor to<br />
the controller. Additionally, some type <strong>of</strong> position sensors are temperature sensitive and<br />
their accuracy degrades, when the system temperature exceed the limits. Therefore, the<br />
research in the area <strong>of</strong> sensorless speed control <strong>of</strong> PMSM is beneficial because <strong>of</strong> the<br />
elimination <strong>of</strong> the feedback wiring, reduced cost, and improved reliability.<br />
Sensorless speed <strong>DTC</strong>-<strong>SVM</strong> control block scheme is presented in Fig. 6.1.<br />
Figure 6.1. Block scheme <strong>of</strong> <strong>DTC</strong>-<strong>SVM</strong> for PMSM drive <strong>with</strong>out motion sensor.<br />
As we can see the operation <strong>of</strong> speed controlled PMSM drive <strong>with</strong>out mechanical<br />
motion sensor is based only on measurement <strong>of</strong> following signals, which are available<br />
in every PWM inverter-fed drive system as:<br />
• DC link voltage,<br />
• motor phase currents,<br />
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Sensorless Speed <strong>Direct</strong> <strong>Torque</strong> <strong>Control</strong> <strong>with</strong> <strong>Space</strong> <strong>Vector</strong> <strong>Modulation</strong> (<strong>DTC</strong>-<strong>SVM</strong>)<br />
Based on this signals, the state variable <strong>of</strong> the drive can be indirectly calculated or<br />
estimated what further allow to achieve the estimated (actual) rotor speed <strong>of</strong> PMSM.<br />
Two motion sensorless control schemes <strong>of</strong> <strong>DTC</strong>-<strong>SVM</strong> for PMSM drive are presented in<br />
Fig. 6.2<br />
a)<br />
U DC<br />
Ω m _ ref<br />
Ψ s _ ref<br />
M e _ ref<br />
e Me<br />
∆δ<br />
U<br />
s α _ ref<br />
U<br />
s β _ ref<br />
S<br />
A<br />
S B<br />
S C<br />
θ Ψ s<br />
Ψ<br />
s<br />
I<br />
s<br />
U s<br />
M e<br />
I s<br />
Ω m _ est<br />
b)<br />
U DC<br />
Ω m _ ref<br />
Ψ s _ ref<br />
M e _ ref<br />
e Ψ s<br />
e M e<br />
U sx _ ref<br />
U sy _ ref<br />
U<br />
s α _ ref<br />
U<br />
s β _ ref<br />
S A<br />
S B<br />
S C<br />
Ψ<br />
s<br />
M<br />
e<br />
θ Ψs<br />
U s<br />
I<br />
s<br />
Ω m _ est<br />
Figure 6.2. The <strong>DTC</strong>-<strong>SVM</strong> block schemes <strong>of</strong> PMSM <strong>with</strong>out motion sensor: a) cascade<br />
structure and b) parallel structure.<br />
In motion sensorless PMSM drives, as shown in Fig.6.2, the position or speed<br />
transducer (see Fig. 5.52) is replaced by a speed estimation block, which generates the<br />
speed feedback signal Ω into the control systems and stator flux model.<br />
m_<br />
est<br />
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Sensorless Speed <strong>Direct</strong> <strong>Torque</strong> <strong>Control</strong> <strong>with</strong> <strong>Space</strong> <strong>Vector</strong> <strong>Modulation</strong> (<strong>DTC</strong>-<strong>SVM</strong>)<br />
The problem associated <strong>with</strong> speed sensorless operation <strong>of</strong> PMSM drive sourced from<br />
VSI is listed below:<br />
• initial rotor flux detection at start up <strong>of</strong> PMSM controlled drive,<br />
• stator flux estimation <strong>with</strong>out measured speed or position signal from sensor,<br />
• rotor speed estimation based on state variable <strong>of</strong> the PMSM, especially in low<br />
speed operation region.<br />
Therefore, in this Section the initial rotor position detection method <strong>of</strong> permanent<br />
magnets, as well as stator flux and rotor speed estimation techniques will be discussed.<br />
6.2 Initial rotor detection method<br />
In a PMSM drive the detection <strong>of</strong> initial flux position is an important task. The initial<br />
position <strong>of</strong> the rotor must be detected correctly in order to initialize the flux estimation<br />
procedure. In case <strong>of</strong> wrong detection the control algorithm has incorrect information<br />
and the rotor shaft can be rotated through few second in positive or negative direction.<br />
This situation is not acceptable in any drive system. Therefore, for the stable starting <strong>of</strong><br />
PMSM drive <strong>with</strong>out the temporary reversal rotation, the initial rotor position<br />
estimation is proposed.<br />
The simplest method to achieved the initial rotor flux position is based on the following<br />
rule. For short time the stator winding is supplied by the DC voltage. It impress the DC<br />
current, which generates the magnetic field. The permanent magnet <strong>of</strong> PMSM sets<br />
accordance <strong>with</strong> this field line. This position <strong>of</strong> PM flux is used to set initial values for<br />
the stator flux estimation algorithm.<br />
This method is very simple and not complicate. However, has disadvantage that during<br />
this process the rotor can be moved in unknown direction depending on:<br />
• position <strong>of</strong> PM before initial detection procedure,<br />
• direction <strong>of</strong> DC voltage supply into the motor phase.<br />
In order to make the initial rotor flux position correct <strong>with</strong>out any movement (at<br />
standstill) the following algorithms can be used in the literature [78,90,93,95,99,101]<br />
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Sensorless Speed <strong>Direct</strong> <strong>Torque</strong> <strong>Control</strong> <strong>with</strong> <strong>Space</strong> <strong>Vector</strong> <strong>Modulation</strong> (<strong>DTC</strong>-<strong>SVM</strong>)<br />
To estimate the initial rotor position before starting <strong>DTC</strong>-<strong>SVM</strong> control, two kind <strong>of</strong> the<br />
rectangular pulsewise voltages are applied from the inverter to the motor:<br />
• one is the short pulsewise voltage,<br />
• and another is the longer one.<br />
Short pulsewise voltage test<br />
This test based on general principle that the three-phase winding inductances <strong>of</strong> PMSM<br />
are a function <strong>of</strong> the mechanical rotor position. Therefore, from the line current<br />
responses in stator oriented coordinates α,<br />
β under the short pulse wise voltage (see<br />
Fig. 6.3) the position <strong>of</strong> PM can be estimated.<br />
Figure 6.3. Voltage pulse wise during short time voltage test.<br />
During the short time (100µ s ) the vector V 1<br />
=(100) and opposite V 4<br />
=(011) is<br />
generated by voltage source inverter. The achieved current responses in α,<br />
β system for<br />
two type <strong>of</strong> PMSM during this test in respect to mechanical rotor position are presented<br />
in Fig. 6.4.<br />
a) b)<br />
Figure 6.4. Current components in stator oriented coordinates α,<br />
β under supplied voltage<br />
vector to the motor for very short time : a) for Ld = Lq<br />
and b) for Lq > Ld.<br />
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Sensorless Speed <strong>Direct</strong> <strong>Torque</strong> <strong>Control</strong> <strong>with</strong> <strong>Space</strong> <strong>Vector</strong> <strong>Modulation</strong> (<strong>DTC</strong>-<strong>SVM</strong>)<br />
The stator current trajectory can be divided in to eight sectors (see Fig. 6.5a).<br />
7π<br />
8<br />
o<br />
157 .5<br />
Sector 4<br />
− +<br />
5π<br />
8<br />
o<br />
112 .5<br />
Sector 3<br />
− −<br />
3π<br />
8<br />
+ −<br />
o<br />
67 .5<br />
Sector 2<br />
I s<br />
o<br />
22.5<br />
π<br />
8<br />
+ +<br />
Sector 5<br />
o<br />
202 .5<br />
9π<br />
8<br />
Sector 6<br />
+ −<br />
o<br />
247 .5<br />
11π<br />
8<br />
− −<br />
Sector 7<br />
− +<br />
o<br />
292 .5<br />
13π<br />
8<br />
+ +<br />
Sector 8<br />
Sector 1<br />
15π<br />
8 π<br />
−<br />
8<br />
o<br />
337 .5<br />
Figure 6.5. Stator current trajectory.<br />
Based on the measured response <strong>of</strong> phase currents in α,<br />
β coordinates, the<br />
( ) sign I s<br />
+ and sign( I s<br />
)<br />
− is calculated from following formulas:<br />
sign( I )<br />
+ = I + I − I<br />
(6.1)<br />
s sα<br />
sβ<br />
s<br />
sign( I )<br />
− = I −I − I<br />
(6.2)<br />
s sα<br />
sβ<br />
s<br />
The possible combinations <strong>of</strong> sign( I s<br />
)<br />
+ and sign( I s<br />
)<br />
− are shown in Fig. 6.6:<br />
Figure 6.6. Possible combination <strong>of</strong> sign( I s<br />
)<br />
+<br />
sign( I s<br />
) + + − −<br />
−<br />
sign( I s<br />
) + − − +<br />
+ and sign( I s<br />
)<br />
− under short pulse supply.<br />
Let us assuming, for example, the case where the sign( I s<br />
)<br />
+ and sign( I s<br />
)<br />
− have positive<br />
π π<br />
sign. The position γ m<br />
exist in the domain <strong>of</strong> − ~ or 7 π 9 π<br />
~ and two estimated<br />
8 8 8 8<br />
position can be obtained.<br />
The mathematical analysis <strong>of</strong> I , I<br />
sα<br />
sβ waveforms leads to following equations:<br />
I = I +∆ I cos 2γ<br />
(6.3)<br />
sα<br />
s s m<br />
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Sensorless Speed <strong>Direct</strong> <strong>Torque</strong> <strong>Control</strong> <strong>with</strong> <strong>Space</strong> <strong>Vector</strong> <strong>Modulation</strong> (<strong>DTC</strong>-<strong>SVM</strong>)<br />
I<br />
=∆ I sin 2γ<br />
(6.4)<br />
sβ<br />
s m<br />
where I<br />
s<br />
is DC component in Is<br />
α<br />
and<br />
∆ Is<br />
amplitude <strong>of</strong> fluctuated component.<br />
The current components in α,<br />
β can be modeled as an average value <strong>of</strong> I<br />
s<br />
plus some<br />
<strong>of</strong>fset value<br />
∆<br />
Is<br />
, as a function <strong>of</strong> the mechanical position γ m<br />
. Fig. 6.4 shows the<br />
current components given in Eq. (6.1-2), which are function <strong>of</strong> the phase angle 2γ m<br />
.<br />
Solving those Eqs. (6.1 and 6.2) in respect to mechanical rotor position, two domains <strong>of</strong><br />
mechanical rotor position can be obtained as:<br />
Isβ<br />
γ<br />
m1<br />
=<br />
(6.5)<br />
2( I − I )<br />
sα<br />
s<br />
= Isβ<br />
γ<br />
m2<br />
2( I − I )<br />
+ π (6.6)<br />
sα<br />
s<br />
The estimated rotor positions for other combination <strong>of</strong> sign( I s<br />
)<br />
summarized in Table 6.7.<br />
+ and sign( I s<br />
)<br />
− are<br />
15π<br />
π<br />
−<br />
8 8<br />
7π<br />
9π<br />
−<br />
8 8<br />
π 3π<br />
−<br />
8 8<br />
9π<br />
11π<br />
−<br />
8 8<br />
3π<br />
5π<br />
−<br />
8 8<br />
11π<br />
13π<br />
−<br />
8 8<br />
5π<br />
7π<br />
−<br />
8 8<br />
13π<br />
15π<br />
−<br />
8 8<br />
I sα − I s + I sβ<br />
+ +<br />
+ −<br />
−<br />
−<br />
−<br />
I sα<br />
− I s − I sβ<br />
γ m<br />
+<br />
I sβ<br />
2(<br />
I sα<br />
− I s )<br />
I sβ<br />
+ π<br />
2(<br />
I sα<br />
− I s )<br />
I sα<br />
+ I s π<br />
− +<br />
2I<br />
sβ<br />
4<br />
I sα<br />
+ I s 5π<br />
− +<br />
2I<br />
sβ<br />
4<br />
I sβ<br />
π<br />
+<br />
2( I s α − I s ) 2<br />
I sβ<br />
3π<br />
+<br />
2( I s α − I s ) 2<br />
I sα<br />
+ I s 3π<br />
− +<br />
2I<br />
sβ<br />
4<br />
I sα<br />
+ I s 7π<br />
− +<br />
2I<br />
sβ<br />
4<br />
Table 6.7. Mechanical rotor position calculations.<br />
Long pulsewise voltage test<br />
This test help us to choose the proper estimated value <strong>of</strong> mechanical rotor position from<br />
two values calculated during the short pulse wise voltage test.<br />
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Sensorless Speed <strong>Direct</strong> <strong>Torque</strong> <strong>Control</strong> <strong>with</strong> <strong>Space</strong> <strong>Vector</strong> <strong>Modulation</strong> (<strong>DTC</strong>-<strong>SVM</strong>)<br />
This test is based on the saturation effect <strong>of</strong> magnetic circuit. If the long pulse will send<br />
in direction <strong>of</strong> north pole <strong>of</strong> PM, the current response will be slower, and, if current<br />
response will be faster that the previous, it means that it is south pole.<br />
6.3 Stator flux estimation methods<br />
Flux estimation is an important task in implementation <strong>of</strong> high-performance <strong>DTC</strong>-<strong>SVM</strong><br />
motor drives. <strong>Vector</strong> control method <strong>of</strong> PMSM drive needs knowledge about actual<br />
value <strong>of</strong> the stator flux magnitude and position as well electromagnetic torque. Also, the<br />
flux estimation is needed to calculate the actual rotor speed for sensorless operation.<br />
6.3.1 Overview<br />
Many different technique has been developed for PMSM flux estimation [107].<br />
Generally, they may be divided into two groups: open loop estimators and closed loop<br />
estimators/observers. Most <strong>of</strong> these method are based on so called “current model” or<br />
“voltage model” [110,113]. In fact closed loop estimators/observers are based on the<br />
current or voltage model <strong>with</strong> an error correction loop, which drives error between two<br />
flux models to zero in steady state. However, an observer has its own dynamics, is<br />
sensitive to parameter changes, and has to be carefully designed for individual drives.<br />
Therefore, for commercially manufactured drives is to complicated and impractical.<br />
This is the reason why in this Chapter only open loop flux estimators will be<br />
considered.<br />
6.3.2 Current model based flux estimator<br />
The block scheme <strong>of</strong> the current based flux model is presented in Fig. 6.7. It requires:<br />
• knowledge <strong>of</strong> PMSM machine inductance L , L ,<br />
• speed or position signal,<br />
• PMSM phase currents.<br />
d<br />
q<br />
This kind <strong>of</strong> flux estimator served in experimental test as a master (standard) to run the<br />
<strong>DTC</strong>-<strong>SVM</strong> scheme <strong>with</strong> speed sensor.<br />
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Sensorless Speed <strong>Direct</strong> <strong>Torque</strong> <strong>Control</strong> <strong>with</strong> <strong>Space</strong> <strong>Vector</strong> <strong>Modulation</strong> (<strong>DTC</strong>-<strong>SVM</strong>)<br />
Ψ PM<br />
I sA<br />
ABC<br />
Is<br />
α<br />
αβ<br />
Isd<br />
L d<br />
Ψ sd<br />
dq<br />
Ψ sα<br />
Ψ s<br />
I sB<br />
αβ<br />
Is<br />
β<br />
dq<br />
I sq<br />
L q<br />
Ψ sq<br />
αβ<br />
Ψ sβ<br />
θ Ψs<br />
γ m<br />
Figure 6.7. Current model based stator flux estimator.<br />
6.3.3 Voltage model based flux estimator <strong>with</strong> ideal integrator<br />
The stator flux linkage can be obtained by using terminal voltages and currents. It is the<br />
integral <strong>of</strong> terminal voltages minus the resistance voltage drop:<br />
dΨ sα<br />
= ( Us α − RsIs<br />
α<br />
)<br />
(6.7)<br />
dt<br />
dΨ sβ<br />
= ( Usβ<br />
− RsIsβ<br />
)<br />
(6.8)<br />
dt<br />
However, at low speed (frequencies) some problems arise, when this technique is<br />
applied, since the stator voltage becomes very small and the resistive voltage drops<br />
become dominant, requiring very accurate knowledge <strong>of</strong> the stator resistance R s<br />
and<br />
very accurate integration. The stator resistance can vary due to temperature changes.<br />
This effect can also be taken into consideration by using the thermal model <strong>of</strong> the<br />
machine. Drifts and <strong>of</strong>fsets can greatly influence the precision <strong>of</strong> integration. The<br />
overall accuracy <strong>of</strong> the estimated flux linkage vector will also depend on the accuracy<br />
<strong>of</strong> the monitored voltages and currents.<br />
The most know classical voltage model obtains the flux components in stator<br />
coordinates ( α,<br />
β ) by integrating the motor back electromotive force E , E<br />
sα<br />
sβ (see Fig.<br />
6.8). The method is sensitive for only one motor parameter, stator resistance R s<br />
.<br />
However, the application <strong>of</strong> pure integrator is difficult because <strong>of</strong> dc drift and initial<br />
value problems.<br />
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Sensorless Speed <strong>Direct</strong> <strong>Torque</strong> <strong>Control</strong> <strong>with</strong> <strong>Space</strong> <strong>Vector</strong> <strong>Modulation</strong> (<strong>DTC</strong>-<strong>SVM</strong>)<br />
I sA<br />
ABC<br />
Is<br />
α<br />
I sB<br />
αβ<br />
Is<br />
β<br />
R s<br />
U sA<br />
U sB<br />
ABC<br />
αβ<br />
U<br />
s α<br />
U<br />
s β<br />
R s<br />
Es<br />
α<br />
Es<br />
β<br />
∫<br />
∫<br />
Ψ sα<br />
Ψ sβ<br />
Ψ s<br />
θ Ψs<br />
Figure 6.8. Voltage model based estimator <strong>with</strong> ideal integrator.<br />
There are proposed many improvements <strong>of</strong> the classical voltage model. Some <strong>of</strong> them<br />
are presented bellow.<br />
6.3.4 Voltage model based flux estimator <strong>with</strong> low pas filter<br />
A common way to improve the stator flux voltage based model is to use a first-order<br />
low-pass filter (LP) instead <strong>of</strong> the pure integrator. The equations (6.7 and 6.8) are<br />
transferred to the form:<br />
dΨ sα<br />
= ( Us α − RsIs α)<br />
+ Fc Ψ<br />
sα<br />
(6.9)<br />
dt<br />
dΨ sβ<br />
= ( Usβ − RsIsβ)<br />
+ Fc Ψ<br />
sβ<br />
(6.10)<br />
dt<br />
The block diagram <strong>of</strong> the estimator is presented in Fig. 6.9. Discrete time<br />
implementation <strong>of</strong> the integrator becomes:<br />
zΨ ( z) =Ψ ( z) + ( U − R I ) T<br />
(6.11)<br />
sα sα sα s sα<br />
s<br />
zΨ ( z) =Ψ ( z) + ( U − R I ) T<br />
(6.12)<br />
sβ sβ sβ s sβ<br />
s<br />
A LP filter does not give high accuracy at frequencies lower than cut<strong>of</strong>f frequency<br />
ω = 2π F . There will be errors both in the magnitude and in the phase angle. As<br />
c<br />
c<br />
results, the proposed voltage estimator <strong>with</strong> LP filter can be used successfully only in a<br />
limited speed range above cut<strong>of</strong>f frequency<br />
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Sensorless Speed <strong>Direct</strong> <strong>Torque</strong> <strong>Control</strong> <strong>with</strong> <strong>Space</strong> <strong>Vector</strong> <strong>Modulation</strong> (<strong>DTC</strong>-<strong>SVM</strong>)<br />
I sA<br />
ABC<br />
Is<br />
α<br />
I sB<br />
αβ<br />
Is<br />
β<br />
R s<br />
F c<br />
U sA<br />
U sB<br />
ABC<br />
αβ<br />
U<br />
s α<br />
U<br />
s β<br />
-<br />
-<br />
R s<br />
Es<br />
α<br />
Es<br />
β<br />
-<br />
-<br />
∫<br />
∫<br />
Ψ sα<br />
Ψ sβ<br />
Cartesian<br />
To<br />
Polar<br />
Ψ s<br />
θ Ψs<br />
Stator flux estimator (improved voltage model)<br />
F c<br />
Figure 6.9. Voltage model based estimator <strong>with</strong> low-pass filter.<br />
Discrete time implementation <strong>of</strong> the LP filter becomes:<br />
zΨ ( z) =Ψ ( z) + ( U − R I ) T + FΨ ( z)<br />
(6.13)<br />
sα sα sα s sα s c sα<br />
zΨ ( z) =Ψ ( z) + ( U −R I ) T −FΨ ( z)<br />
(6.14)<br />
sβ sβ sβ s sβ s c sα<br />
6.3.5 Improved voltage model based flux estimator<br />
Many other methods were developed in order to eliminate dc-<strong>of</strong>fset and initial values<br />
problems [107]. In general, the output Y <strong>of</strong> these new integrators (Fig. 6.10) is<br />
expressed as:<br />
ωc<br />
1<br />
Y = X + Y<br />
s+ ω s+<br />
ω<br />
c<br />
c<br />
lim<br />
(6.15)<br />
Where X is the input and Y is output <strong>of</strong> the integrator respectively. The Y lim<br />
is a<br />
compensation signal used as a feedback and ω<br />
c<br />
is cut<strong>of</strong>f frequency.<br />
X<br />
1<br />
s + ω c<br />
Y<br />
Compensation<br />
signal<br />
ωc<br />
s +ω<br />
c<br />
Y lim<br />
− lim<br />
lim<br />
Figure 6.10. Improved integration method <strong>with</strong> saturation block.<br />
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Sensorless Speed <strong>Direct</strong> <strong>Torque</strong> <strong>Control</strong> <strong>with</strong> <strong>Space</strong> <strong>Vector</strong> <strong>Modulation</strong> (<strong>DTC</strong>-<strong>SVM</strong>)<br />
The first part <strong>of</strong> the equation represents a LP filter. The second part realizes a feedback,<br />
which is used to compensate the error in the output. The block diagram <strong>of</strong> new<br />
integration algorithm <strong>with</strong> saturation block is shown in see Fig. 6.11.<br />
I sA<br />
ABC<br />
Is<br />
α<br />
I sB<br />
αβ<br />
Is<br />
β<br />
R s<br />
ωc<br />
s +ω<br />
c<br />
Y lim<br />
− lim<br />
lim<br />
U sA<br />
U sB<br />
ABC<br />
αβ<br />
U<br />
s α<br />
U<br />
s β<br />
R s<br />
Es<br />
α<br />
Es<br />
β<br />
1<br />
s + ω c<br />
1<br />
s + ω c<br />
Ψ α _ comp<br />
Ψ s α<br />
Ψ sβ<br />
Ψ s<br />
θ Ψs<br />
Ψ<br />
s β _ comp<br />
ωc<br />
s +ω<br />
c<br />
Y lim<br />
− lim<br />
lim<br />
Figure 6.11. Full block diagram <strong>of</strong> voltage model based estimator <strong>with</strong> saturation block on the<br />
α,<br />
β components.<br />
The main task <strong>of</strong> saturation block is to stop the integration when the output signal<br />
Ψ<br />
sα<br />
or<br />
Ψ<br />
sβ<br />
exceed the reference value <strong>of</strong> stator flux amplitude. Please note that if the<br />
compensation signal is set to zero, the improved integrator represents a first-order LPfilter.<br />
If the compensation signal<br />
integrator operates as a pure integrator.<br />
Ψ or Ψ<br />
β _<br />
is not zero the improved<br />
sα<br />
_ comp<br />
s<br />
comp<br />
Discrete time implementation <strong>of</strong> the improved integrator becomes:<br />
ω<br />
zΨ ( z) =Ψ ( z) + ( U − R I ) T + ( Ψ ( z) −Ψ ( z))<br />
(6.16)<br />
c<br />
sα sα sα s sα s sα _lim<br />
sα<br />
Ts<br />
ω<br />
zΨ ( z) =Ψ ( z) + ( U − R I ) T + ( Ψ ( z) −Ψ ( z))<br />
(6.17)<br />
c<br />
sβ sβ sβ s sβ s sβ _lim<br />
sβ<br />
Ts<br />
The output <strong>of</strong> saturation block can be described as:<br />
⎧Ψ<br />
sαβ<br />
( z) if (Ψ<br />
sαβ(z))=lim<br />
sαβ<br />
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Sensorless Speed <strong>Direct</strong> <strong>Torque</strong> <strong>Control</strong> <strong>with</strong> <strong>Space</strong> <strong>Vector</strong> <strong>Modulation</strong> (<strong>DTC</strong>-<strong>SVM</strong>)<br />
Where lim is the limited value. Please note that lim should be set at reference stator<br />
flux amplitude<br />
lim =Ψ<br />
s _ ref<br />
equal Ψ<br />
PM<br />
.<br />
From Eq. 6.18 it can be observed that when one <strong>of</strong> the stator flux linkage components<br />
Ψ<br />
sα<br />
or Ψ<br />
sβ<br />
exceeds the limit, it causes in distortion <strong>of</strong> the output waveform.<br />
6.4 Electromagnetic torque estimation<br />
The PMSM motor output torque is calculated based on the equations (2.51), (2.52),<br />
(2.55), (2.58) presented in Chapter 2, which for stator oriented coordinate system can be<br />
written as follows:<br />
3<br />
M<br />
e<br />
= pb( Ψs αIsβ −Ψ<br />
sβIsα<br />
)<br />
(6.19)<br />
2<br />
It can be seen that calculated torque is dependent on the current measurement accuracy<br />
and stator flux estimation method. In practice current measurement is performed <strong>with</strong><br />
high accuracy ( ≤ 1% <strong>with</strong> 150kHz frequency bandwidth) using, for example, LEM<br />
sensors.<br />
6.5 Rotor speed estimation methods<br />
6.5.1 Overview<br />
High performance operation <strong>of</strong> motion sensorless PMSM drives depends mainly on<br />
accurate knowledge <strong>of</strong> rotor PM flux magnitude, position and speed. The rotor position<br />
estimation methods can be classified into two major groups:<br />
• motor model based,<br />
• rotor saliency based techniques.<br />
The rotor saliency based approach is suitable only for the Interior PMSM (see Fig. 1.2 c<br />
and d). Motor model based approach detect the back EMF vector, which includes<br />
information about position and speed, using either open loop models/estimators<br />
[81,85,86] or closed loop estimators/observers [70, 73,74,96,97,100]. Also adaptive<br />
observers [72,92,98,83] and Extended Kalman Filters (EKF) [67,73] have been<br />
proposed for motor position and speed estimation. These methods, however, are<br />
computationally intensive and require careful design and proper initialization.<br />
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Sensorless Speed <strong>Direct</strong> <strong>Torque</strong> <strong>Control</strong> <strong>with</strong> <strong>Space</strong> <strong>Vector</strong> <strong>Modulation</strong> (<strong>DTC</strong>-<strong>SVM</strong>)<br />
Therefore, for commercially manufactured drives are impractical and further a simple<br />
open loop based techniques will be considered.<br />
6.5.2 Back electromotive force (BEMF) technique<br />
This technique uses the back electromotive force to estimate the rotor speed [70]. The<br />
velocity signal could be integrated to generate a position estimate. However, this signal<br />
is sensitive to parameter variations and tends to drift and have <strong>of</strong>fset problem. Another<br />
problem <strong>with</strong> using BEMF to estimate position is that at zero speed the BEMF goes to<br />
zero and at low speed the signal to noise ratio can not be ignored.<br />
6.5.3 Stator flux based technique<br />
Generally, the calculation <strong>of</strong> rotor speed is based on the simple relationship:<br />
θ = θ − δ , (6.20)<br />
r Ψs Ψs<br />
where θ<br />
r<br />
is electrical position, θ Ψ s<br />
is stator flux position and δ Ψ s<br />
is torque angle.<br />
After differentiation equation (6.20) and taking into account that θ r<br />
= p b<br />
γ m<br />
the<br />
mechanical speed <strong>of</strong> PMSM rotor can be expressed as:<br />
⎛dθ<br />
⎜<br />
⎝ dt<br />
Ψs<br />
Ω<br />
m<br />
= −<br />
dδΨ<br />
⎞<br />
⎟/<br />
pb<br />
, (6.21)<br />
dt ⎠<br />
dθ s<br />
where Ω Ψ<br />
Ψs<br />
= is angular speed <strong>of</strong> stator flux vector and δ Ψ<br />
is torque angle.<br />
dt<br />
As we can observe form equation (6.21) in order to calculate the mechanical rotor speed<br />
it is necessary to calculate separately two components. One <strong>of</strong> them is angular speed <strong>of</strong><br />
stator flux vector Ω<br />
Ψs<br />
and the second one is change <strong>of</strong> the load angle d δ Ψ (see Fig.<br />
dt<br />
6.12).<br />
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Sensorless Speed <strong>Direct</strong> <strong>Torque</strong> <strong>Control</strong> <strong>with</strong> <strong>Space</strong> <strong>Vector</strong> <strong>Modulation</strong> (<strong>DTC</strong>-<strong>SVM</strong>)<br />
Is<br />
α<br />
Is<br />
β<br />
U<br />
s α<br />
Ψ<br />
sα<br />
Ψ sβ<br />
Ω =<br />
Ψs<br />
d<br />
dt<br />
θ Ψ s<br />
dδ Ψ<br />
dt<br />
−<br />
1<br />
p b<br />
Ω m<br />
U<br />
s β<br />
Figure 6.12. Block diagram <strong>of</strong> stator flux vector based rotor speed estimator.<br />
The synchronous speed<br />
Ω<br />
Ψs<br />
is calculated based on the stator flux estimator:<br />
θ Ψ s<br />
Ψ<br />
sβ<br />
= arctg( )<br />
Ψ<br />
sα<br />
(6.22)<br />
and the calculation <strong>of</strong><br />
ΩΨs<br />
can be done as:<br />
The estimation <strong>of</strong> the synchronous speed<br />
dθ θ<br />
s Ψs( k) −θ<br />
Ψ<br />
Ψs( k−1)<br />
Ω<br />
Ψs<br />
= = (6.23)<br />
dt T<br />
s<br />
Ω<br />
Ψs<br />
based on the derivative <strong>of</strong> the position <strong>of</strong><br />
stator flux space vector can be modified taking in to account equation (6.22), which<br />
finally gives [12]:<br />
Ψ<br />
Ω =<br />
Ψs<br />
sα<br />
dΨ<br />
sβ<br />
−Ψ<br />
dt<br />
Ψ +Ψ<br />
sβ<br />
2 2<br />
sα<br />
sβ<br />
dΨ<br />
dt<br />
sα<br />
(6.24)<br />
Digital implementation <strong>of</strong> equation (6.24) can be written as:<br />
Ψ Ψ −Ψ Ψ<br />
Ω =<br />
Ψ T<br />
sα( k−1) sβ( k) sβ( k−1) sα( k)<br />
Ψs( k) 2<br />
s<br />
s<br />
(6.25)<br />
Also from equation (2.27b) in stator flux coordinate system the synchronous speed<br />
can be obtained as:<br />
Ω<br />
Ψs<br />
U − R I E<br />
Ω<br />
Ψs<br />
= =<br />
Ψ Ψ<br />
sy s sy sy<br />
s<br />
s<br />
(6.26)<br />
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Sensorless Speed <strong>Direct</strong> <strong>Torque</strong> <strong>Control</strong> <strong>with</strong> <strong>Space</strong> <strong>Vector</strong> <strong>Modulation</strong> (<strong>DTC</strong>-<strong>SVM</strong>)<br />
Calculation <strong>of</strong> d δ Ψ<br />
dt<br />
Based on the flux-current (Eq. 2.29b) and torque (Eq. 2.58) equations in stator flux<br />
coordinates under consideration that for surface PMSM machine L d<br />
= L q<br />
and making<br />
assumption that for small changes <strong>of</strong> torque angleδ Ψ<br />
, the sinδ<br />
be written as:<br />
Ψ<br />
= δ , the equations can<br />
Ψ<br />
0= LI −Ψ sin<br />
(6.27)<br />
s sy PM<br />
δ Ψ<br />
3<br />
M<br />
e<br />
= pbΨ PMIsy<br />
(6.28)<br />
2<br />
the torque angle δ Ψ<br />
can be calculated as:<br />
LI<br />
s sy 2ML<br />
e s<br />
δ Ψ<br />
= =<br />
Ψ 3p<br />
Ψ Ψ<br />
PM b s PM<br />
(6.29)<br />
Further, the d dt<br />
δ Ψ is calculated as:<br />
( ) ( 1)<br />
δΨ<br />
Ψ k Ψ k−<br />
d<br />
dt<br />
δ −δ<br />
= (6.30)<br />
T<br />
s<br />
Figure 6.13. Simulated oscillograms <strong>of</strong> rotor speed estimation according to block scheme from<br />
Fig. 6.12 (in Saber). From the top: synchronous speed Ω<br />
Ψs<br />
, the d signal, the measured and<br />
dt<br />
estimated rotor speed, the measured and estimated electromagnetic torque.<br />
δ Ψ<br />
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Sensorless Speed <strong>Direct</strong> <strong>Torque</strong> <strong>Control</strong> <strong>with</strong> <strong>Space</strong> <strong>Vector</strong> <strong>Modulation</strong> (<strong>DTC</strong>-<strong>SVM</strong>)<br />
The speed estimation problem is still open, especially at low and zero speed operations.<br />
The accuracy <strong>of</strong> presented method depends on accuracy <strong>of</strong> applied stator flux estimation<br />
and differentiation algorithm. It allows, however, for robust start, closed loop operation<br />
above 10% <strong>of</strong> nominal speed, and braking the drive to zero speed.<br />
6.5 Summary<br />
The main problems associated <strong>with</strong> PMSM sensorless speed operation are presented<br />
in this Chapter. For robust start <strong>of</strong> PMSM <strong>with</strong>out the temporary rotor reversal a<br />
special initialization algorithm has been used. This algorithm performs two test:<br />
short and the longer voltage generated by the PWM inverter. The used speed<br />
estimation algorithm is based on stator flux vector and torque angle estimation and<br />
does not operate accurately around zero speed region. However, it allows robust<br />
start and closed speed operation in the speed range above 10% <strong>of</strong> nominal speed.<br />
The effectiveness <strong>of</strong> <strong>DTC</strong>-<strong>SVM</strong> <strong>with</strong> and <strong>with</strong>out motion sensor has been proved by<br />
simulation and experimental results (see Chapter 7)<br />
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DSP implementation <strong>of</strong> <strong>DTC</strong>-<strong>SVM</strong> control<br />
Chapter 7 DSP IMPLEMENTATION OF <strong>DTC</strong>-<strong>SVM</strong> CONTROL<br />
7.1 Description <strong>of</strong> the laboratory test-stand<br />
The basic structure <strong>of</strong> laboratory setup is presented in Fig. 7.1 and the photo <strong>of</strong><br />
laboratory setup is shown in Fig. 7.2. The motor setup consists <strong>of</strong> 3kW permanent<br />
magnet synchronous motor and DC motor, which is used as a load. The PMSM machine<br />
is supplied by PWM inverter, which is controlled by digital signal processor (DSP)<br />
based on DS1103 board. The voltage inverter is supplied from three-phase diodes<br />
rectifier. The DSP interface is used in order to separate the high power from the low<br />
power circuit (computer part). Please note that the DS1103 is inserted inside the PC<br />
computer.<br />
Figure 7.1. Block scheme <strong>of</strong> laboratory setup.<br />
Figure 7.2. Laboratory setup. 1-voltage inverter, 2-control for DC motor, 3- PMSM machine, 4<br />
– DSP interface.<br />
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DSP implementation <strong>of</strong> <strong>DTC</strong>-<strong>SVM</strong> control<br />
The detailed power circuit <strong>of</strong> the laboratory setup is shown in Fig. 7.3.<br />
Rectifier<br />
Inverter<br />
three-phase<br />
supply<br />
network<br />
U A<br />
U B<br />
U C<br />
D 1<br />
D3<br />
D5<br />
CF<br />
T 1<br />
D 1<br />
T 3<br />
D 3<br />
T 5<br />
D 5<br />
current<br />
sensors<br />
A<br />
B<br />
C<br />
PMSM<br />
D 2<br />
D 4<br />
D 6<br />
T 2<br />
D 2<br />
T 4<br />
D 4<br />
T 6<br />
D 6<br />
speed or position<br />
sensor<br />
SA<br />
SA<br />
SB<br />
SB<br />
SC<br />
SC<br />
Reference speed<br />
<strong>DTC</strong>-<strong>SVM</strong><br />
DS1103<br />
microprocessor<br />
Figure 7.3. Power circuit <strong>of</strong> the laboratory setup.<br />
In presented system the actual two currents and DC link voltage are measured by LEM<br />
sensors and processed by A/D converter. The rotor position and speed are obtained <strong>with</strong><br />
an encoder <strong>of</strong> 2500 pulse per revolution. All internal data <strong>of</strong> DSP can be sent through a<br />
D/A converter and displayed in the scope. All data <strong>of</strong> the PMSM and inverter are given<br />
in the Appendices.<br />
The control algorithm for PMSM machine was written in C language and was<br />
implemented inside the processor. Also, a simple dead-time compensation method and<br />
voltage drop on the semiconductor elements are implemented.<br />
The phase voltage <strong>of</strong> the motor are reconstructed inside the processor using the<br />
measured DC-link voltage and duty cycles <strong>of</strong> PWM for each phases. Motor and PI<br />
controller model are given in Appendices.<br />
Various tests have been carried out in order to investigate the drive performance and to<br />
characterize the steady-state and transient behavior.<br />
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DSP implementation <strong>of</strong> <strong>DTC</strong>-<strong>SVM</strong> control<br />
<strong>Control</strong> Desk experiment s<strong>of</strong>tware run on the PC computer provides all function for<br />
controlling, monitoring and automation <strong>of</strong> real-time experiments and makes the<br />
development <strong>of</strong> controllers more effective. A <strong>Control</strong> Desk experiment layout for<br />
control the PMSM machine using <strong>DTC</strong>-<strong>SVM</strong> control method is presented in Fig. 7.4.<br />
Figure 7.4. Performed <strong>Control</strong> Desk experimental layout for control <strong>of</strong> PMSM drive.<br />
139
DSP implementation <strong>of</strong> <strong>DTC</strong>-<strong>SVM</strong> control<br />
7.2 Steady state behaviour<br />
The experimental steady state no load operation at 25Hz stator frequency for<br />
conventional ST-<strong>DTC</strong> (Fig. 4.10) and <strong>DTC</strong>-<strong>SVM</strong> (Fig. 5.46b) control is presented in<br />
Fig. 7.5. The sampling time has been set at T = 200µ<br />
s for <strong>DTC</strong>-<strong>SVM</strong> and T = 25µ<br />
s for<br />
hysteresis based ST-<strong>DTC</strong> method, respectively.<br />
a)<br />
s<br />
s<br />
b)<br />
Figure 7.5. No load experimental steady state oscillograms at stator frequency 25Hz.<br />
(a) ST-<strong>DTC</strong> for T = 25µ<br />
s (b) <strong>DTC</strong>-<strong>SVM</strong> for T = 200µ<br />
s.<br />
s<br />
From the top: line to line voltage, phase current, amplitude <strong>of</strong> stator flux, motor torque.<br />
s<br />
As we can observed from Fig. 7.5a the motor phase current characterized by high<br />
current ripples. This is mainly because the inductances <strong>of</strong> the PMSM is smaller than an<br />
equivalent power induction motor IM. In order to reduce the current ripples the<br />
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DSP implementation <strong>of</strong> <strong>DTC</strong>-<strong>SVM</strong> control<br />
sampling time <strong>of</strong> microcontroller should be decrease. However there is hardware<br />
limitation. The loaded program to microprocessor can not run faster.<br />
Using the space vector modulation (<strong>SVM</strong>) based <strong>DTC</strong> much better results can be<br />
obtained. Note, that in spite <strong>of</strong> lower switching frequency <strong>DTC</strong>-<strong>SVM</strong> guarantees less<br />
current and torque ripple. This is mainly because contrary to hysteresis operation <strong>with</strong><br />
<strong>SVM</strong> operation, the inverter output voltage is unipolar (compare output voltage<br />
waveform in Fig. 7.5a <strong>with</strong> Fig. 7.5b). This also reduces semiconductor device voltage<br />
stress and instantaneous current reversal in DC link.<br />
The presented experimental results (Fig. 7.6-7.9) are measured in the system <strong>with</strong><br />
measured speed taken to the feedback. These investigations have been performed to<br />
show the behaviour <strong>of</strong> the <strong>DTC</strong>-<strong>SVM</strong> system <strong>with</strong>out influence <strong>of</strong> the speed estimation.<br />
In Fig. 7.6 and Fig. 7.7 steady state operation for different values <strong>of</strong> the mechanical<br />
speed and load torque are shown.<br />
Figure 7.6. Experimental steady state operation <strong>of</strong> PMSM controlled via <strong>DTC</strong>-<strong>SVM</strong> <strong>with</strong> the<br />
encoder speed signal taken to the feedback ( Ω = 300rpm<br />
, M = 0 ).<br />
m<br />
From the top: 1) amplitude <strong>of</strong> stator flux (0.25Wb/div), 2) electromagnetic torque (10Nm/div),<br />
3) line to line voltage (1000V/div), 4) stator phase current (10A/div).<br />
l<br />
141
DSP implementation <strong>of</strong> <strong>DTC</strong>-<strong>SVM</strong> control<br />
Figure 7.7. Experimental steady state operation <strong>of</strong> PMSM controlled via <strong>DTC</strong>-<strong>SVM</strong> <strong>with</strong> the<br />
encoder speed signal taken to the feedback ( Ω = 300rpm<br />
, M = 10Nm<br />
-50% <strong>of</strong> nominal<br />
torque). From the top: 1) amplitude <strong>of</strong> stator flux (0.25Wb/div), 2) electromagnetic torque<br />
(10Nm/div), 3) line to line voltage (1000V/div), 4) stator phase current (10A/div).<br />
m<br />
l<br />
Figure 7.8. Experimental steady state operation <strong>of</strong> PMSM controlled via <strong>DTC</strong>-<strong>SVM</strong> <strong>with</strong> the<br />
encoder speed signal taken to the feedback ( Ω = 1500rpm<br />
, M = 0 ). From the top: 1)<br />
amplitude <strong>of</strong> stator flux (0.25Wb/div), 2) electromagnetic torque (10Nm/div), 3) line to line<br />
voltage (1000V/div), 4) stator phase current (10A/div).<br />
m<br />
l<br />
142
DSP implementation <strong>of</strong> <strong>DTC</strong>-<strong>SVM</strong> control<br />
Figure 7.9. Experimental steady state operation <strong>of</strong> PMSM controlled via <strong>DTC</strong>-<strong>SVM</strong> <strong>with</strong> the<br />
encoder speed signal taken to the feedback ( Ω = 1500rpm<br />
, M = 10Nm<br />
-50% <strong>of</strong> nominal<br />
torque). From the top: 1) amplitude <strong>of</strong> stator flux (0.25Wb/div), 2) electromagnetic torque<br />
(10Nm/div), 3) line to line voltage (1000V/div), 4) stator phase current (10A/div).<br />
m<br />
l<br />
7.3 Dynamic behaviour<br />
The experimental results <strong>of</strong> flux and torque control loop obtained in dynamic states for<br />
PMSM machine controlled via two different <strong>DTC</strong>-<strong>SVM</strong> schemes are presented.<br />
7.3.1 Flux and torque control loop<br />
Cascade <strong>DTC</strong>–<strong>SVM</strong> control scheme (Fig. 5.46a)<br />
In order to show behaviour <strong>of</strong> the system the dynamic testes for the flux and torque<br />
controllers has been carried out for sampling time, T = 200µ<br />
s. It corresponds to<br />
switching frequency f = 5kHz<br />
. Please note that the flux digital controller parameters<br />
s<br />
were selected according to Table 5.1 and the torque digital controller parameters were<br />
selected from Table 5.2. (see Chapter 5.2).<br />
In Fig. 7.10 stator flux tracking performance is presented. This result is comparable<br />
<strong>with</strong> simulation results presented in Fig. 5.15.<br />
s<br />
143
DSP implementation <strong>of</strong> <strong>DTC</strong>-<strong>SVM</strong> control<br />
Figure 7.10. Experimental flux tracking performance <strong>of</strong> PM synchronous motor for zero speed<br />
at sampling time Ts<br />
= 200µ<br />
s. Reference flux from70% to 100% <strong>of</strong> nominal value . From the<br />
top: 1-reference stator flux amplitude(0.05Wb/div), 2- stator flux amplitude (0.05Wb/div), 3-<br />
electromagnetic torque (2Nm/div), 4- motor phase current (10A/div).<br />
In Fig. 7.11 torque tracking performance is presented. The achieved result is<br />
comparable <strong>with</strong> simulation results presented in Fig. 5.21b.<br />
Figure 7.11. Experimental torque tracking performance <strong>of</strong> PM synchronous motor for zero<br />
speed at sampling time Ts<br />
= 200µ<br />
s. Reference torque from 0 to nominal value. From the top:1-<br />
reference torque (10Nm/div), 2 - estimated torque (10Nm/div), 3 - stator flux amplitude<br />
(0.1Wb/div), 4- motor phase current (10A/div)<br />
144
DSP implementation <strong>of</strong> <strong>DTC</strong>-<strong>SVM</strong> control<br />
Figure 7.12. Experimental torque tracking performance <strong>of</strong> PM synchronous motor for zero<br />
speed (zoom) at sampling time Ts<br />
= 200µ<br />
s. Reference torque from 0 to nominal value. From<br />
the top:1- reference torque (10Nm/div), 2 - estimated torque (10Nm/div), 3 - stator flux<br />
amplitude (0.1Wb/div), 4- motor phase current (10A/div).<br />
All experimental results presented in Fig. 7.10-7.12 confirm very well proper and stable<br />
operation <strong>of</strong> flux and torque control loops for cascade <strong>DTC</strong>-<strong>SVM</strong> structure.<br />
145
DSP implementation <strong>of</strong> <strong>DTC</strong>-<strong>SVM</strong> control<br />
Influence <strong>of</strong> sampling time for torque control loop in cascade <strong>DTC</strong>-<strong>SVM</strong><br />
The influence <strong>of</strong> sampling time on experimental torque tracking performance is<br />
illustrated in Fig. 7.13-7.15. The dynamic test has been carried out for the same<br />
condition ( Ω = 0rpm<br />
) as for simulation shown in Fig. 5.24. The controller parameters<br />
m<br />
has been set according to Table 5.2. In all oscilograms we may see proper operation <strong>of</strong><br />
the torque control loop for different sampling time used in practice.<br />
Figure 7.13. Experimental torque tracking performance <strong>of</strong> PM synchronous motor for zero<br />
speed at sampling time Ts<br />
= 100µ<br />
s( fs<br />
= 10kHz<br />
). Reference torque from 0 to nominal value.<br />
From the top:1- reference torque (4Nm/div), 2 - estimated torque (4Nm/div)<br />
Figure 7.14. Experimental torque tracking performance <strong>of</strong> PM synchronous motor for zero<br />
speed at sampling time Ts<br />
= 200µ<br />
s( fs<br />
= 5kHz<br />
). Reference torque from 0 to nominal value.<br />
From the top:1- reference torque (4Nm/div), 2 - estimated torque (4Nm/div)<br />
146
DSP implementation <strong>of</strong> <strong>DTC</strong>-<strong>SVM</strong> control<br />
Figure 7.15. Experimental torque tracking performance <strong>of</strong> PM synchronous motor for zero<br />
speed at sampling time T = 400µ<br />
s( f = 2.5kHz<br />
). Reference torque from 0 to nominal value.<br />
s<br />
From the top:1- reference torque (4Nm/div), 2 - estimated torque (4Nm/div).<br />
s<br />
Parallel structure <strong>of</strong> <strong>DTC</strong>–<strong>SVM</strong> scheme (Fig. 5.46b)<br />
Dynamic testes for the flux and torque controllers were carried out for sampling time<br />
Ts<br />
= 200µ<br />
s, which corresponds to switching frequency f = 5kHz<br />
. Please not that the<br />
digital flux controller parameters were selected according to Table 5.3 and the digital<br />
torque controller parameters were selected from Table 5.4. (see Chapter 5.3).<br />
In Fig. 7.16 stator flux tracking performance is presented. This result is comparable<br />
<strong>with</strong> simulation results presented in Fig. 5.33 for flux and Fig. 5.45 for torque loop,<br />
respectively.<br />
s<br />
147
DSP implementation <strong>of</strong> <strong>DTC</strong>-<strong>SVM</strong> control<br />
Figure 7.16. Experimental flux tracking performance <strong>of</strong> PM synchronous motor for zero speed<br />
at sampling time Ts<br />
= 200µ<br />
s. Reference flux from70% to 100% <strong>of</strong> nominal value . From the<br />
top: 1-reference stator flux amplitude(0.05Wb/div), 2- stator flux amplitude (0.05Wb/div), 3-<br />
electromagnetic torque (2Nm/div), 4- motor phase current (10A/div).<br />
It can be observed that achieved result is comparable <strong>with</strong> simulation results presented<br />
in Fig. 5.34.<br />
Figure 7.17. Experimental torque tracking performance <strong>of</strong> PM synchronous motor for zero<br />
speed at sampling time Ts<br />
= 200µ<br />
s. Reference torque from 0 to nominal value. From the top:1-<br />
reference torque (10Nm/div), 2 - estimated torque (10Nm/div), 3 - stator flux amplitude<br />
(0.1Wb/div), 4- motor phase current (10A/div)<br />
148
DSP implementation <strong>of</strong> <strong>DTC</strong>-<strong>SVM</strong> control<br />
Figure 7.18. Experimental torque tracking performance <strong>of</strong> PM synchronous motor for zero<br />
speed (zoom) at sampling time Ts<br />
= 200µ<br />
s. Reference torque from 0 to nominal value. From<br />
the top:1- reference torque (10Nm/div), 2 - estimated torque (10Nm/div), 3 - stator flux<br />
amplitude (0.1Wb/div), 4- motor phase current (10A/div).<br />
The achieved result is comparable <strong>with</strong> simulation results presented in Fig. 5.42b.<br />
Experimental results presented in Fig. 7.16-7.18 confirm very well the effectiveness <strong>of</strong><br />
controller design and proper operation <strong>of</strong> flux and torque control loops for <strong>DTC</strong>-<strong>SVM</strong><br />
structure.<br />
149
DSP implementation <strong>of</strong> <strong>DTC</strong>-<strong>SVM</strong> control<br />
Influence <strong>of</strong> sampling time for torque control loop in parallel <strong>DTC</strong>-<strong>SVM</strong><br />
The influence <strong>of</strong> sampling time on experimental torque tracking performance is<br />
illustrated in Fig. 7.19-7.21. The dynamic test has been carried out for the same<br />
condition ( Ω = 0rpm<br />
) as for simulation shown in Fig. 5.45. The controller parameters<br />
m<br />
has been set according to Table 5.4. In all oscilograms we may see proper operation <strong>of</strong><br />
the torque control loop for different sampling time used in practice.<br />
Figure 7.19. Experimental torque tracking performance <strong>of</strong> PM synchronous motor for zero<br />
speed at sampling time Ts<br />
= 100µ<br />
s. Reference torque from 0 to nominal value. From the top:1-<br />
reference torque (4Nm/div), 2 - estimated torque (4Nm/div)<br />
Figure 7.20. Experimental torque tracking performance <strong>of</strong> PM synchronous motor for zero<br />
speed at sampling time Ts<br />
= 200µ<br />
s. Reference torque from 0 to nominal value. From the top:1-<br />
reference torque (4Nm/div), 2 - estimated torque (4Nm/div).<br />
150
DSP implementation <strong>of</strong> <strong>DTC</strong>-<strong>SVM</strong> control<br />
Figure 7.21. Experimental torque tracking performance <strong>of</strong> PM synchronous motor for<br />
zero speed at sampling time Ts<br />
= 400µ<br />
s. Reference torque from 0 to nominal value.<br />
From the top:1- reference torque (4Nm/div), 2 - estimated torque (4Nm/div).<br />
In this Chapter the results <strong>of</strong> experimental verification 3kW PMSM drive <strong>with</strong> two<br />
<strong>DTC</strong>-<strong>SVM</strong> schemes has been presented. As shown the drive performance confirms<br />
applied design methodology. The performance <strong>of</strong> both cascade and parallel <strong>DTC</strong>-<strong>SVM</strong><br />
control structure are similar. However, parallel structure has been selected for industrial<br />
manufacturing because <strong>of</strong> :<br />
• less noisy control algorithm (differentiation required in cascade structure –<br />
equation (5.6) is eliminated),<br />
• stator flux control in closed loop,<br />
• the same structure can be used for IM and PMSM control (universal control for<br />
AC motors).<br />
7.3.2 Speed control loop<br />
0peration <strong>with</strong> speed sensor<br />
Dynamic testes for the speed control loop were measured for sampling timeT<br />
= 200µ<br />
s.<br />
Please note that the digital speed controller parameters were selected according to Table<br />
5.5 (see Chapter 5.4). In Fig. 7.22-7.26 rotor speed tracking performance are presented.<br />
s<br />
151
DSP implementation <strong>of</strong> <strong>DTC</strong>-<strong>SVM</strong> control<br />
Figure 7.22. Experimental start up and breaking to zero speed <strong>of</strong> PMSM motor controlled via<br />
<strong>DTC</strong>-<strong>SVM</strong> <strong>with</strong> the encoder speed signal taken to the feedback ( Ω<br />
m<br />
= 0rpm → 300rpm<br />
). From<br />
the top: 1- reference speed (180rpm/div), 2- measured speed (180rpm/div), 3 - electromagnetic<br />
torque (20Nm/div), 4- motor phase current (20A/div).<br />
Figure 7.23. Experimental speed reversal for PMSM motor controlled via <strong>DTC</strong>-<strong>SVM</strong> <strong>with</strong> the<br />
encoder speed signal taken to the feedback ( Ω<br />
m<br />
=−300rpm → 300rpm<br />
). From the top: 1-<br />
reference speed (360rpm/div), 2- measured speed (360rpm/div), 3 - electromagnetic torque<br />
(20Nm/div), 4- motor phase current (20A/div).<br />
152
DSP implementation <strong>of</strong> <strong>DTC</strong>-<strong>SVM</strong> control<br />
Figure 7.24. Experimental start up and breaking to zero speed <strong>of</strong> PMSM motor controlled via<br />
<strong>DTC</strong>-<strong>SVM</strong> <strong>with</strong> the encoder speed signal taken to the feedback ( Ω<br />
m<br />
= 0rpm → 1500rpm<br />
).From<br />
the top: 1- reference speed (900rpm/div), 2- measured speed (900rpm/div), 3 - electromagnetic<br />
torque (20Nm/div), 4- motor phase current (20A/div).<br />
Figure 7.25. Experimental speed reversal for PMSM motor controlled via <strong>DTC</strong>-<strong>SVM</strong> <strong>with</strong> the<br />
encoder speed signal taken to the feedback ( Ω<br />
m<br />
=−1500rpm → 1500rpm<br />
). From the top: 1-<br />
reference speed (1800rpm/div), 2- measured speed (1800rpm/div), 3 - electromagnetic torque<br />
(20Nm/div), 4- motor phase current (20A/div).<br />
153
DSP implementation <strong>of</strong> <strong>DTC</strong>-<strong>SVM</strong> control<br />
Figure 7.26. Experimental speed reversal for PMSM motor controlled via <strong>DTC</strong>-<strong>SVM</strong> <strong>with</strong> the<br />
encoder speed signal taken to the feedback ( Ω<br />
m<br />
=−1200rpm → 1200rpm<br />
). From the top: 1-<br />
stator flux component Ψ<br />
sα<br />
(0.25Wb/div), 2- measured speed (360rpm/div), 3 - electromagnetic<br />
torque (20Nm/div), 4- motor phase current (20A/div).<br />
Figure 7.27. Experimental response to load torque step change <strong>of</strong> PMSM motor controlled via<br />
<strong>DTC</strong>-<strong>SVM</strong> <strong>with</strong> the encoder speed signal taken to the feedback at Ω = 0rpm<br />
.<br />
( M = 0Nm → 10Nm<br />
) From the top: 1- reference speed (180rpm/div), 2- measured speed<br />
l<br />
(180rpm/div), 3 - electromagnetic torque (5Nm/div), 4- motor phase current (10A/div).<br />
m<br />
154
DSP implementation <strong>of</strong> <strong>DTC</strong>-<strong>SVM</strong> control<br />
Figure 7.28. Experimental response to load torque step change <strong>of</strong> PMSM motor controlled via<br />
<strong>DTC</strong>-<strong>SVM</strong> <strong>with</strong> the encoder speed signal taken to the feedback at Ω = 300rpm<br />
.<br />
( M = 0Nm → 10Nm<br />
).From the top: 1- reference speed (180rpm/div), 2- measured speed<br />
l<br />
(180rpm/div), 3 - electromagnetic torque (5Nm/div), 4- motor phase current (10A/div).<br />
m<br />
Figure 7.29. Experimental response to load torque step change <strong>of</strong> PMSM motor controlled via<br />
<strong>DTC</strong>-<strong>SVM</strong> <strong>with</strong> the encoder speed signal taken to the feedback at Ω = 1500rpm<br />
.<br />
( M = 0Nm → 10Nm<br />
). From the top: 1- reference speed (900rpm/div), 2- measured speed<br />
l<br />
(900rpm/div), 3 - electromagnetic torque (5Nm/div), 4- motor phase current (10A/div).<br />
m<br />
155
DSP implementation <strong>of</strong> <strong>DTC</strong>-<strong>SVM</strong> control<br />
Experimental results presented in Figures 7.22-7.29 well confirm the effectiveness <strong>of</strong><br />
developed controller synthesis methodology and proper operation <strong>of</strong> speed control loop<br />
for parallel <strong>DTC</strong>-<strong>SVM</strong> structure.<br />
Sensorless speed operation<br />
Dynamic tests for the speed control loop <strong>with</strong>out motion sensor were measured for<br />
sampling time T = 200µ<br />
s. Please note that the digital speed controller parameters were<br />
s<br />
selected exactly like for operation <strong>with</strong> speed sensor according to Table 5.5 (see<br />
Chapter 5.4).<br />
The results <strong>of</strong> speed estimator dynamic test are presented in Fig. 7.30. In this test speed<br />
controller operates <strong>with</strong> the encoder signal in feedback and speed estimator works in<br />
open loop fashion.<br />
Figure 7.30. Experimental dynamic test <strong>of</strong> the speed estimation. Speed reversal Ω =± 300rpm<br />
From the top: 1- reference speed (180rpm/div), 2- measured speed (180rpm/div), 3 - estimated<br />
speed (180rpm/div), 4- error <strong>of</strong> estimated speed (5%/div).<br />
m<br />
The typical dynamic performance tests <strong>of</strong> sensorless <strong>DTC</strong>-<strong>SVM</strong> drive has been<br />
illustrated in Fig. 7.31-7.36. Start up and breaking to zero speed for different speed level<br />
are shown in Fig. 7.31 and 7.33. Also, the speed reversal for low (Fig. 7.32) and<br />
nominal (Fig. 7.34) speed are presented. The half load torque step change tests are<br />
shown in Fig. 7.36 and 7.37.<br />
156
DSP implementation <strong>of</strong> <strong>DTC</strong>-<strong>SVM</strong> control<br />
Figure 7.31. Experimental start up and breaking to zero speed <strong>of</strong> PMSM motor controlled via<br />
<strong>DTC</strong>-<strong>SVM</strong> <strong>with</strong> the estimated speed signal taken to the feedback ( Ω<br />
m<br />
= 0rpm → 300rpm<br />
).<br />
From the top: 1- reference speed (180rpm/div), 2- measured speed (180rpm/div), 3 -<br />
electromagnetic torque (20Nm/div), 4- motor phase current (20A/div).<br />
Figure 7.32. Experimental speed reversal for PMSM motor controlled via <strong>DTC</strong>-<strong>SVM</strong> <strong>with</strong> the<br />
estimated speed signal taken to the feedback ( Ω<br />
m<br />
=−300rpm → 300rpm<br />
). From the top: 1-<br />
reference speed (360rpm/div), 2- measured speed (360rpm/div), 3 - electromagnetic torque<br />
(20Nm/div), 4- motor phase current (20A/div).<br />
157
DSP implementation <strong>of</strong> <strong>DTC</strong>-<strong>SVM</strong> control<br />
Figure 7.33. Experimental speed step response for PMSM motor controlled via <strong>DTC</strong>-<strong>SVM</strong> <strong>with</strong><br />
the estimated speed signal taken to the feedback ( Ω<br />
m<br />
= 0rpm → 1500rpm<br />
).From the top: 1-<br />
reference speed (900rpm/div), 2- measured speed (900rpm/div), 3 - electromagnetic torque<br />
(20Nm/div), 4- motor phase current (20A/div).<br />
Figure 7.34. Experimental speed reversal for PMSM motor controlled via <strong>DTC</strong>-<strong>SVM</strong> <strong>with</strong> the<br />
estimated speed signal taken to the feedback ( Ω<br />
m<br />
=−1500rpm → 1500rpm<br />
). From the top: 1-<br />
reference speed (1800rpm/div), 2- measured speed (1800rpm/div), 3 - electromagnetic torque<br />
(20Nm/div), 4- motor phase current (20A/div).<br />
158
DSP implementation <strong>of</strong> <strong>DTC</strong>-<strong>SVM</strong> control<br />
Figure 7.35. Experimental speed reversal for PMSM motor controlled via <strong>DTC</strong>-<strong>SVM</strong> <strong>with</strong> the<br />
estimated speed signal taken to the feedback ( Ω<br />
m<br />
=−1200rpm → 1200rpm<br />
). From the top: 1-<br />
stator flux component Ψ<br />
sα<br />
(0.25Wb/div), 2- measured speed (360rpm/div), 3 - electromagnetic<br />
torque (20Nm/div), 4- motor phase current (20A/div).<br />
Figure 7.36. Experimental response to load torque step change <strong>of</strong> PMSM motor controlled via<br />
<strong>DTC</strong>-<strong>SVM</strong> <strong>with</strong> the estimated speed signal taken to the feedback at Ω = 300rpm<br />
.<br />
( M = 0Nm → 10Nm<br />
). From the top: 1- reference speed (180rpm/div), 2- measured speed<br />
l<br />
(180rpm/div), 3 - electromagnetic torque (10Nm/div), 4- motor phase current (10A/div).<br />
m<br />
159
DSP implementation <strong>of</strong> <strong>DTC</strong>-<strong>SVM</strong> control<br />
Figure 7.37. Experimental response to load torque step change <strong>of</strong> PMSM motor controlled via<br />
<strong>DTC</strong>-<strong>SVM</strong> <strong>with</strong> the estimated speed signal taken to the feedback at Ω = 1500rpm<br />
.<br />
( M = 0Nm → 10Nm<br />
). From the top: 1- reference speed (900rpm/div), 2- measured speed<br />
l<br />
(900rpm/div), 3 - electromagnetic torque (10Nm/div), 4- motor phase current (10A/div).<br />
m<br />
Experimental results presented in Figures 7.31-7.37 confirm very well the effectiveness<br />
<strong>of</strong> speed estimation algorithm <strong>of</strong> speed control loop for parallel <strong>DTC</strong>-<strong>SVM</strong> structure.<br />
160
Summary and closing conclusion<br />
Chapter 8<br />
SUMMARY AND CLOSING CONCLUSIONS<br />
This thesis studied basic problems related to selection, investigation and implementation<br />
<strong>of</strong> the PWM inverter-fed permanent magnet synchronous motor (PMSM) drives suitable<br />
for serial manufacturing. The selected method should provide: robust start and operation<br />
in wide speed control range including zero speed, <strong>with</strong> and <strong>with</strong>out mechanical motion<br />
sensor; guarantee good and repeatable parameters <strong>of</strong> PMSM drive for wide power range<br />
(1-100kW). The control and protection algorithm should be implemented in simple and<br />
cheap microprocessor.<br />
To solve so formulated general task several related problems had to be solved. At first,<br />
the space vector based mathematical description and static characteristic <strong>of</strong> PMSM<br />
under different control modes were studied (Chapter 2). Secondly, the three phase<br />
voltage source inverter model including nonlinearities (dead time, semiconductor<br />
voltage drop, DC link pulsation) and pulse <strong>with</strong> modulation PWM techniques were<br />
presented (Chapter 3). Next, based on the study <strong>of</strong> most important high performance<br />
control methods as field oriented control (FOC), and direct torque control (<strong>DTC</strong>), the<br />
method called direct torque control <strong>with</strong> space vector modulator (<strong>DTC</strong>-<strong>SVM</strong>) has been<br />
selected for further consideration (Chapter 4). This methods combines main advantage<br />
<strong>of</strong> FOC (space vector modulator and fixed switching frequency) and <strong>DTC</strong> (simple<br />
structure, rotor parameter independent), as well as eliminates disadvantages like:<br />
coordinate transformation, the need <strong>of</strong> internal current control loops, high sampling<br />
time, high torque and current ripple at steady state operation, etc.<br />
Consequently the most important contribution <strong>of</strong> this thesis is included in the Chapter 5,<br />
where the two basic variants <strong>of</strong> <strong>DTC</strong>-<strong>SVM</strong> schemes: series (cascade) and parallel<br />
structures <strong>of</strong> flux and torque controllers are presented (Fig. 5.2 and Fig. 5.25). Also, a<br />
systematic methodology <strong>of</strong> digital controller design for these both <strong>DTC</strong>-<strong>SVM</strong> variants<br />
have been given. This methodology has been verified by Simulink (using simplified<br />
discrete transfer function) and Saber (using full motor and inverter model) simulation<br />
studies. The influence <strong>of</strong> sampling time selection on controller design has also been<br />
discussed.<br />
The main problems associated <strong>with</strong> PMSM sensorless speed operation are presented in<br />
the Chapter 6. It should be noted that PMSM differ from IM drives mainly in:<br />
• PMSM parameters strongly depend on construction ,<br />
161
Summary and closing conclusion<br />
• position <strong>of</strong> PM flux has to be known prior to start up to achieve smooth<br />
operation.<br />
Therefore, for robust starting <strong>of</strong> PMSM <strong>with</strong>out the temporary rotor reversal a simple<br />
initialization algorithm has been used. This algorithm performs two tests: short and the<br />
longer voltage pulses generated by the PWM inverter. The used speed estimation<br />
algorithm is based on stator flux vector and torque angle estimation and does not<br />
operate accurately in zero speed region. However, it allows robust start and closed<br />
speed operation in the speed range above 10% <strong>of</strong> nominal speed. For application where<br />
high performance operation around zero speed are required, the <strong>DTC</strong>-<strong>SVM</strong> drive <strong>with</strong><br />
motion sensor (encoder) is recommended. The effectiveness <strong>of</strong> <strong>DTC</strong>-<strong>SVM</strong> scheme <strong>with</strong><br />
and <strong>with</strong>out motion sensor has been proved by simulation and experimental results<br />
(Chapter 7).<br />
Simulation study and experimental results have shown that from two variants <strong>of</strong> <strong>DTC</strong>-<br />
<strong>SVM</strong> schemes the parallel structure is more flexible in torque and flux controller<br />
design. Also, because <strong>of</strong> lack <strong>of</strong> the differentiation in the main control path (compare<br />
Fig. 5.2 and Fig. 5.25), it is less sensitive to noise which inherently associates signal<br />
processing in power electronic converters. Therefore, the parallel structure has been<br />
selected for industrial manufacturing and implemented on digital signal processor<br />
(DSP).<br />
Thanks to direct torque and flux control structure the described control is suitable to<br />
almost all – industrial applications including electrical vehicles (for example hybrid<br />
cars).<br />
Finally, it should be stressed that the developed system was brought into serial<br />
production. Presented algorithm <strong>DTC</strong>-<strong>SVM</strong> has been used in new generation <strong>of</strong> inverter<br />
drives produced by Polish company Power Electronic Manufacture – “TWERD”,<br />
Toruń.<br />
162
Appendices<br />
APPENDICES<br />
A1 Rotor and stator <strong>of</strong> PMSM machine<br />
A1.1. View <strong>of</strong> rotor (on the left side) and stator armature (on the right side) <strong>of</strong> PMSM.<br />
A2 Basic transformation<br />
y<br />
β<br />
K B<br />
K<br />
θ K<br />
K β<br />
K y<br />
K<br />
x<br />
K α<br />
θ K<br />
α<br />
x<br />
K A<br />
K C<br />
Fig. A2.1. <strong>Space</strong> vector representation in stationary α,<br />
β coordinates and synchronous<br />
rotating x,<br />
y coordinates.<br />
A, BC , ⇒ xy ,<br />
2<br />
Kx = [ KA cos θ<br />
K<br />
+ KB cos(2 π / 3 − θK ) + KC cos(2 π / 3 + θK<br />
)]<br />
3<br />
2<br />
Ky = [ KA cos( π /2 + θK ) + KB cos(2 π /3 − ( π /2 + θK )) + KC cos(2 π /3 + π /2 + θK<br />
)]<br />
3<br />
163
Appendices<br />
2<br />
Kx = [ UA cos θK + KB cos( θK − 2 π / 3) + KC cos( θK<br />
+ 2 π / 3)]<br />
3<br />
2<br />
Ky = [ −KA sin θK −KB sin( θK − 2 π / 3) − KC sin( θK<br />
+ 2 π / 3)]<br />
3<br />
⎡K<br />
A ⎤<br />
⎡K<br />
x ⎤ ⎡ cosθK cos( θK − 2 π / 3) cos( θK<br />
+ 2 π / 3) ⎤⎢<br />
K<br />
⎥<br />
⎢ B<br />
K<br />
⎥ = ⎢<br />
y sinθK sin( θK 2 π / 3) sin( θK<br />
2 π / 3)<br />
⎥⎢ ⎥<br />
⎣ ⎦ ⎣− − − − + ⎦<br />
⎢⎣<br />
K ⎥<br />
C ⎦<br />
ABC , , ⇒ α,<br />
β θ = 0<br />
2 1 1<br />
K = ( KA B C)<br />
3 − α<br />
K K<br />
2 − 2<br />
2 3 3 1<br />
K = ( KB C) (<br />
B C)<br />
3 2 − β<br />
K K K<br />
2 = 3<br />
−<br />
⎡ 1 1 ⎤<br />
1 − − ⎡ K<br />
A ⎤<br />
⎡Kα<br />
⎤ 2 ⎢ 2 2 ⎥<br />
⎢<br />
K<br />
⎥<br />
⎢ B<br />
K<br />
⎥ = ⎢<br />
⎥<br />
β 3 3 3 ⎢ ⎥<br />
⎣ ⎦ ⎢<br />
⎥<br />
⎢0<br />
− ⎢⎣K<br />
⎥<br />
C ⎦<br />
⎣ 2 2<br />
⎥<br />
⎦<br />
x, y⇒<br />
A, B,<br />
C<br />
K<br />
KA = Kxcosθ K<br />
+ Kycos( π / 2 + θK) = KxcosθK − KysinθK<br />
KB = Kxcos(2 π / 3 − θK) + Kycos(2 π / 3 − ( π / 2 + θK)) = Kxcos( θK −2 π / 3) −Kysin( θK<br />
−2 π / 3)<br />
K = K cos(2 π / 3 + θ ) + K cos(2 π / 3 + ( π / 2 + θ )) = K cos( θ + 2 π / 3) − K sin( θ + 2 π / 3)<br />
C x K y K x K y K<br />
⎡K<br />
A⎤ ⎡ cosθK −sinθK<br />
⎤<br />
⎢ K<br />
x<br />
K<br />
⎥ ⎢<br />
B<br />
cos( θK 2 π / 3) sin( θK<br />
2 π / 3)<br />
⎥ ⎡ ⎤<br />
⎢ ⎥<br />
=<br />
⎢<br />
− − −<br />
⎥⎢ K<br />
⎥<br />
y<br />
⎢K<br />
⎥ ⎢<br />
C<br />
cos( θK + 2 π / 3) − sin( θK<br />
+ 2 π / 3) ⎥ ⎣ ⎦<br />
⎣ ⎦ ⎣ ⎦<br />
α, β ⇒ ABC , , θ = 0<br />
K<br />
KA<br />
=<br />
K α<br />
1 3<br />
KB<br />
=− Kα<br />
+ K<br />
2 2<br />
1 3<br />
KC<br />
=− Kα<br />
− K<br />
2 2<br />
β<br />
β<br />
164
Appendices<br />
⎡ ⎤<br />
⎢ 1 0 ⎥<br />
⎡K<br />
A ⎤ ⎢ ⎥<br />
⎢ 1 3 K<br />
K<br />
⎥ ⎢ ⎥ ⎡ α ⎤<br />
⎢ B ⎥<br />
=<br />
⎢<br />
−<br />
2 2 ⎥⎢ K<br />
⎥<br />
β<br />
⎢K<br />
⎣ ⎦<br />
⎣ ⎥<br />
C ⎦ ⎢ ⎥<br />
⎢ 1 3⎥<br />
⎢<br />
− −<br />
⎣ 2 2 ⎥⎦<br />
xy , ⇒ α,<br />
β<br />
Kα = Kxcosθ K<br />
+ Kycos( π / 2 + θK) = KxcosθK − KysinθK<br />
Kβ = Kxcos( π / 2 − θK) + KycosθK = KxsinθK + KycosθK<br />
⎡Kα<br />
⎤ ⎡cosθK<br />
−sinθ<br />
K<br />
K⎤⎡ x ⎤<br />
⎢<br />
K<br />
⎥ = ⎢<br />
β sinθK<br />
cosθ<br />
⎥⎢ K<br />
⎥<br />
⎣ ⎦ ⎣<br />
K ⎦⎣ y ⎦<br />
α, β ⇒ x,<br />
y<br />
Kx = Kα cosθ K<br />
+ Kβ cos( π / 2 − θK) = Kα cosθK + Kβ<br />
sinθK<br />
Ky = Kα cos( π / 2 + θK) + Kβ cosθK =− Kα sinθK + Kβ<br />
cosθK<br />
⎡Kx ⎤ ⎡ cosθK sinθ<br />
K<br />
K ⎤⎡ α ⎤<br />
⎢<br />
K<br />
⎥ = ⎢<br />
y sinθK cosθ<br />
⎥⎢ K<br />
⎥<br />
⎣ ⎦ ⎣−<br />
K⎦⎣ β ⎦<br />
A3 Model <strong>of</strong> PM synchronous motor<br />
# This template models the permanent magnet synchronous motor(pmsm)<br />
# t1,t2 and t3 are motor input terminals<br />
# rotor speed (in rad/s) is the output<br />
# rs-Stator windings' resistence per phase(in Ohms)<br />
# ld-d_axis inductance(in H)<br />
# lq-q_axis inductance(in H)<br />
# pm-Rotor magnet flux(Wb)<br />
# j-Moment <strong>of</strong> inertia(in kgm2)<br />
# d- Damping constant (Nm/rad/s)<br />
# tl-motor load (Nm)<br />
# p-Number <strong>of</strong> pole pairs<br />
# power-The total input power (W)<br />
# Assumptions:No core losses,no saturation,thermal effects<br />
#(rs,ld,lq and pm values are constants)<br />
element template pmsm_dtc t1 t2 t3 t0 speed out_me out_psi out_thetam out_ualf out_ubet out_psia<br />
out_psib out_ialf out_ibet out_theta<br />
out_tl=rs,ld,lq,pm,d,tl,j,p,init_theta_m,init_omega_m,omega_m_const<br />
electrical t1,t2,t3,t0 # motor input terminals and stator neutral point<br />
output nu speed,<br />
out_me,out_psi,out_thetam,out_ualf,out_ubet,out_psia,out_psib,out_ialf,out_ibet,out_theta,out_tl<br />
number rs=0.692,ld=6m,lq=6m,pm=0.26379,d=0.002044,tl=0.0,j=0.003,p=3.0,<br />
init_theta_m=0.0,init_omega_m=0.0,omega_m_const=0.0<br />
{<br />
165
Appendices<br />
Appendices<br />
i(t1->t0)+=it1<br />
it1: it1=id*cos(p*theta_m)-iq*sin(p*theta_m)<br />
i(t2->t0)+=it2<br />
it2: it2=id*cos(p*theta_m-y)-iq*sin(p*theta_m-y)<br />
i(t3->t0)+=it3<br />
it3: it3=id*cos(p*theta_m+y)-iq*sin(p*theta_m+y)<br />
omega_m: (te-tl-d*omega_m)/j=d_by_dt(omega_m)<br />
#omega_m: omega_m=omega_m_const*math_pi/30.0<br />
theta_m: omega_m=d_by_dt(theta_m)<br />
speed: speed=omega_m<br />
out_me: out_me=te<br />
out_psi: out_psi=psi<br />
out_thetam:out_thetam=theta_m<br />
out_ualf: out_ualf=va<br />
out_ubet: out_ubet=vb<br />
out_psia: out_psia=psia<br />
out_psib: out_psib=psib<br />
out_ialf: out_ialf=ialf<br />
out_ibet: out_ibet=ibet<br />
out_theta: out_theta=theta<br />
out_tl: out_tl=tl<br />
}<br />
}<br />
A4 Motor parameters<br />
Surface type motor<br />
Power P 3kW<br />
Number <strong>of</strong> pole pairs p 3<br />
Phase current I(rms) 6.9A<br />
Phase voltage U(rms) 70V<br />
Magnetic flux-linkage Ψ 0.264 Wb<br />
Rotor speed<br />
Ω<br />
PM<br />
m<br />
3000rpm<br />
Nominal torque Me 20Nm<br />
Moment <strong>of</strong> the inertia J 0.0174kgm 2<br />
Stator winding resistance Rs 0.692 Ω<br />
Stator d-axis inductance Ld 6mH<br />
Stator d-axis inductance Lq 6mH<br />
Interior type motor<br />
Power P 2,2kW<br />
Number <strong>of</strong> pole pairs p 3<br />
Phase current I(rms) 4.1A<br />
Rated voltage U(rms) 380V<br />
Magnetic flux-linkage Ψ 0.4832 Wb<br />
Rotor speed<br />
Ω<br />
PM<br />
m<br />
1750rpm<br />
Nominal torque Me 12Nm<br />
Moment <strong>of</strong> the inertia J 0.010074kgm 2<br />
Stator winding resistance Rs 3.3 Ω<br />
Stator d-axis inductance Ld 41.59mH<br />
Stator d-axis inductance Lq 57.06mH<br />
167
Appendices<br />
A5 Voltage Source Inverter parameters<br />
Detailed date <strong>of</strong> IGBT transistors (module TOSHIBA M675Q2YS50):<br />
U = 1200V<br />
, I = 75A<br />
CE<br />
C<br />
UCEsat<br />
= 2.8 − 3.6V<br />
, forward diode voltage 2.4 − 3.5V<br />
Turn on time tON<br />
= 0.2µ<br />
s, Turn <strong>of</strong>f time tOFF<br />
= 0.6µ<br />
s<br />
Delay <strong>of</strong> IGBT drivers tONd<br />
= 0.5µ<br />
s TOFFd<br />
= 1µ<br />
s<br />
TON = tON + tONd<br />
= 0.7µ<br />
s total turn on time <strong>of</strong> IGBT<br />
TOFF = tOFF + tOFFd<br />
= 1.6µ<br />
s total turn <strong>of</strong>f time <strong>of</strong> IGBT<br />
Dead time T = 2.5µ<br />
s<br />
A6 PI speed controller<br />
d<br />
The commonly used in industrial application speed controller is a Proportional-Integral<br />
PI controller thanks to possibility to reduce the speed error between the reference (<br />
and actual rotor speed (<br />
X<br />
ref<br />
)<br />
X<br />
m<br />
) to zero (see Fig. A6.1). The output signal <strong>of</strong> controller is a<br />
reference torque, which has upper and lower limitation for this value equal the nominal<br />
torque or more than 130% <strong>of</strong> nominal torque. The output <strong>of</strong> the speed controller acts as<br />
a current reference command for the current controllers. This current command is<br />
limited to a nominal current <strong>of</strong> the motor.<br />
The speed controller demands produce proper electromagnetic torque.<br />
X_<br />
ref<br />
Reference signal<br />
−<br />
error signal<br />
Kp<br />
Y L<br />
<strong>Control</strong>ler output<br />
a)<br />
Feedback signal<br />
X_<br />
m<br />
Kp<br />
T<br />
i<br />
∫<br />
X _ ref<br />
Reference signal<br />
b)<br />
−<br />
Feedback signal<br />
error signal<br />
X _ m<br />
Kp<br />
Kp<br />
T<br />
i<br />
∫<br />
1<br />
Y NL<br />
lim_max<br />
lim_ min<br />
−<br />
Y E<br />
Y L<br />
<strong>Control</strong>ler output<br />
T i<br />
A.6.1. General structure <strong>of</strong> Proportional- Integral controller <strong>with</strong>out antiwindup (a) and <strong>with</strong><br />
antwindup (b).<br />
168
Appendices<br />
A7 PWM technique – six step mode<br />
Six-stepped-voltage waveforms are rich in harmonics. These time harmonics produce<br />
respective stator current harmonics, which in turn interact <strong>with</strong> fundamental air gap<br />
flux, generating harmonics torque pulsations. The torque pulsations are undesirable:<br />
they generate audible noise, speed pulsations, and losses. In case <strong>of</strong> supplied motor by<br />
using only active vectors (six step mode) we can observed non sinusoidal current, which<br />
generates torque ripples <strong>with</strong> frequency <strong>of</strong> six time fundamental frequency <strong>of</strong> supplied phase<br />
voltages.<br />
Fig. A7.1. Experimental operation in six step mode. From the left side stator voltages in α , β<br />
coordinates, From right side voltage trajectory.<br />
Fig. A7.2. Experimental operation in six step mode. From the left side stator currents in α , β<br />
coordinates, From right side stator current trajectory.<br />
169
Appendices<br />
Fig. A7.3. Experimental operation in six step mode. From the top: α stator voltage,<br />
electromagnetic torque in machine, phase current.<br />
170
List <strong>of</strong> symbol<br />
List <strong>of</strong> symbols<br />
Symbol (general)<br />
X - instantaneous value<br />
X<br />
N<br />
- normalized value<br />
X - vector<br />
X ∗ - conjugate vector<br />
X - amplitude <strong>of</strong> vector<br />
Re( X ) – real part <strong>of</strong> X<br />
Im( X ) – imaginary part <strong>of</strong> X<br />
Symbol (special)<br />
α,<br />
β - stator fixed system<br />
dq , - rotor reference system<br />
x,<br />
y - general reference system<br />
Ls<br />
- stator inductance<br />
Zs<br />
- stator impedance<br />
M<br />
s<br />
- mutual inductance<br />
Is<br />
- phase current value<br />
U<br />
s<br />
- phase voltage value<br />
Ψ<br />
s<br />
- phase flux value<br />
P -active power<br />
Q - reactive power<br />
S - apparent power<br />
Pe<br />
- electro-magnetic power<br />
Ωm<br />
- mechanical rotor speed<br />
Ωs<br />
- synchronous speed<br />
cosφ - power factor<br />
δ<br />
I<br />
, δ Ψ<br />
- torque angle<br />
φ - power angle<br />
R<br />
s<br />
- stator resistance<br />
L<br />
d<br />
, L<br />
d<br />
- direct and quadrature inductances<br />
θr<br />
- electrical rotor position<br />
γ<br />
m<br />
- mechanical rotor position<br />
pb<br />
- number <strong>of</strong> pole pairs<br />
Ψ<br />
PM<br />
- rotor flux <strong>of</strong> permanent magnets<br />
M<br />
e<br />
- electromagnetic torque<br />
M<br />
es<br />
- synchronous torque<br />
M<br />
er<br />
- reluctance torque<br />
M<br />
l<br />
- load torque (external load torque)<br />
M<br />
d<br />
- dynamic torque<br />
Jm<br />
- motor moment <strong>of</strong> inertia<br />
Jl<br />
- load moment <strong>of</strong> inertia<br />
J - moment <strong>of</strong> inertia <strong>of</strong> total system (sum <strong>of</strong> J<br />
m<br />
and J<br />
l<br />
)<br />
171
List <strong>of</strong> symbol<br />
Subscripts<br />
A , B , C - denote arbitrary phase quantities in a system <strong>of</strong> natural coordinate ABC. , ,<br />
d , q - arbitrary direct and quadrature components in a system <strong>of</strong> rotor coordinate dq. ,<br />
α , β - arbitrary alpha and beta components in a system <strong>of</strong> stator coordinate α,<br />
β .<br />
x , y - denote arbitrary components in a system <strong>of</strong> general coordinate x,<br />
y .<br />
.. r - denotes value <strong>of</strong> rotor<br />
.. s - denotes value <strong>of</strong> stator<br />
.. _max - maximum value<br />
.. _min – minimum value<br />
.. _ ref - reference value<br />
.. _ est - estimated value<br />
.. _ amp -amplitude value<br />
.. _ rms - root mean square value<br />
.. _ LL - line to line value<br />
* - reference value<br />
^ - estimated value<br />
Abbreviations<br />
RSM – reluctance synchronous motor<br />
BLDCM – blushless DC motor<br />
PMSM – permanent magnet synchronous motor<br />
IPMSM - interior permanent magnet synchronous motor<br />
SPMSM - surface permanent magnet synchronous motor<br />
EMF – electro-magnetic force<br />
VSI - voltage source inverter<br />
<strong>SVM</strong> – space vector modulator<br />
PWM – pulse width modulation<br />
PWM-VSI – voltage source inverter <strong>with</strong> PWM<br />
<strong>DTC</strong> - direct torque control<br />
<strong>DTC</strong>-<strong>SVM</strong> - direct torque control <strong>with</strong> space vector modulator<br />
RFOC - rotor field oriented control<br />
SFOC - stator field oriented control<br />
CTAC - constant torque angle control<br />
MTPAC - maximum torque per ampere control<br />
UPFC - unity power factor control<br />
CSFC - constant stator flux control<br />
172
References<br />
Books and PhD-Thesis<br />
1. I. Boldea, S.A. Nasar, “Electrical drives”, CRC Press, 1999.<br />
2. G. F. Franklin, J. D. Powell, “Feedback <strong>Control</strong> <strong>of</strong> Dynamic Systems”, Addison-<br />
Wesley Publishing Company, 1994.<br />
3. M. P. Kaźmierkowski, H. Tunia, “Automatic <strong>Control</strong> <strong>of</strong> Converter-Fed drives”,<br />
Elsevier, 1994.<br />
4. M. P. Kaźmierkowski, R. Krishnan, F. Blaabjerg, “<strong>Control</strong> in Power Electronics”,<br />
Academic Press, 2002.<br />
5. P.C. Krause, O. Wasynczuk, S.D. Sudh<strong>of</strong>f, “Analysis <strong>of</strong> Electric Machinery”, IEEE<br />
Press, 1995.<br />
6. R. Krishnan, “Electric Motor Drives – Modeling, Analysis, and <strong>Control</strong>”, Prentice Hall,<br />
2001.<br />
7. T. Kaczmarek, K. Zawirski, “Układy napędowe z silnikiem synchronicznym”,<br />
Politechnika Poznańska, 2000.<br />
8. M. Malinowski, "Sensorless <strong>Control</strong> Strategies for Three-Phase PWM Rectifiers", PhD<br />
Thesis, Warsaw University <strong>of</strong> Technology, 2001.<br />
9. Ch. Perera, “Sensorless <strong>Control</strong> <strong>of</strong> Permanent Magnet Synchronous Motor Drives”,<br />
PhD thesis, Institute <strong>of</strong> Energy Technology, 2002.<br />
10. K. Rajashekara, A. Kawamura, K. Matsuse, “Sensorless <strong>Control</strong> <strong>of</strong> AC Motor Drives”,<br />
IEEE Press,1996, USA.<br />
11. M. Zelechowski, “<strong>Space</strong> <strong>Vector</strong> Modulated – <strong>Direct</strong> <strong>Torque</strong> <strong>Control</strong>led (<strong>DTC</strong>-<strong>SVM</strong>)<br />
Inverter-Fed Induction Motor Drive”, Phd Thesis , Warsaw University <strong>of</strong> Technology,<br />
2005.<br />
12. P. Vas, “Sensorless <strong>Vector</strong> and <strong>Direct</strong> <strong>Torque</strong> <strong>Control</strong>”, Oxford University Press, 1998.<br />
Model <strong>of</strong> PMSM<br />
13. C. Bowen; Z. Jihua; R. Zhang; “Modeling and simulation <strong>of</strong> permanent magnet<br />
synchronous motor drives”, Electrical Machines and Systems, ICEMS 2001, Vol. 2, 18-<br />
20 Aug. 2001, pp. 905 – 908.<br />
14. P. Pillay, R. Krishnan, “Modeling <strong>of</strong> permanent magnet motor drives”, Transactions on<br />
Industrial Electronics, Vol. 35, Nov. 1988, pp. 537 – 541.<br />
15. N. Urasaki; T. Senjyu; K Uezato, “An Accurate Modeling for Permanent Magnet<br />
Synchronous Motor Drives”, Applied Power Electronics Conference and Exposition,<br />
APEC 2000, Vol. 1, 6-10 Feb. 2000, pp:387 – 392.<br />
16. A.H. Wijenayake, P.B. Schmidt, “Modeling and analysis <strong>of</strong> permanent magnet<br />
synchronous motor by taking saturation and core loss into account”, International<br />
Conference on Power Electronics and Drive Systems, Vol. 2, 26-29 May 1997, pp. 530<br />
– 534.<br />
Inverter and PWM <strong>Modulation</strong><br />
17. L. Ben-Brahim, "The analysis and compensation <strong>of</strong> dead-time effects in three phase<br />
PWM inverters", Industrial Electronics Society, 1998. IECON '98. Proceedings <strong>of</strong> the<br />
24th Annual Conference <strong>of</strong> the IEEE, Vol. 2d, 31 Aug.-4 Sept. 1998, pp.792-797.<br />
18. V. Blasko, "Analysis <strong>of</strong> a hybrid PWM based on modified space-vector and trianglecomparison<br />
methods", IEEE Transactions on Industry Applications, Vol. 33, Issue: 3,<br />
May-June 1997, pp.756-764.<br />
19. F.Blaabjerg, J.K. Pedersen, P. Thoegersen, “Improved modulation techniques for<br />
PWM-VSI drives”, IEEE Transactions on Industrial Electronics, Feb. 1997, Vol. 44, pp.<br />
87 – 95.<br />
20. J.W. Choi; S.K. Sul, "Inverter output voltage synthesis using novel dead time<br />
compensation", IEEE Transactions on Power Electronics, Vol. 11, Issue: 2, March<br />
1996, pp.221-227.<br />
173
References<br />
21. D.W. Chung, J. Kim, S.K. Sul, "Unified voltage modulation technique for real-time<br />
three-phase power conversion", IEEE Transactions on Industry Applications, Vol. 34,<br />
Issue: 2, March-April 1998, pp.374-380.<br />
22. A. Diaz, E.G. Strangas, "A novel wide range pulse width overmodulation method [for<br />
voltage source inverters]", Applied Power Electronics Conference and Exposition,<br />
APEC 2000.<br />
23. A.M. Hava, R.J. Kerkman, T.A. Lipo, "A high performance generalized discontinuous<br />
PWM algorithm", Applied Power Electronics Conference and Exposition, APEC '97<br />
Conference Proceedings 1997, Twelfth Annual, Vol. 2, 23-27 Feb. 1997, pp.886-894.<br />
24. D.G. Holmes, "The significance <strong>of</strong> zero space vector placement for carrier-based PWM<br />
schemes", IEEE Transactions on Industry Applications, Vol. 32, Issue: 5, Sept.-Oct.<br />
1996, pp.1122-1129.<br />
25. J. Holtz, "Pulsewidth modulation for electronic power conversion", Proceedings <strong>of</strong> the<br />
IEEE, Vol. 82, Issue: 8, Aug. 1994, pp.1194-1214.<br />
26. J. Holtz, W. Lotzkat, A.M. Khambadkone, "On continuous control <strong>of</strong> PWM inverters in<br />
the overmodulation range including the six-step mode", IEEE Transactions on Power<br />
Electronics, Vol. 8, Issue: 4, Oct. 1993, pp.546-553.<br />
27. J.L. Lin, "A new approach <strong>of</strong> dead-time compensation for PWM voltage inverters",<br />
IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications,<br />
Vol. 49, Issue: 4, April 2002, pp.476-483.<br />
28. D.C. Lee, G.M. Lee, "A novel overmodulation technique for space-vector PWM<br />
inverters", IEEE Transactions on Power Electronics, Vol. 13, Issue: 6, Nov. 1998,<br />
pp.1144-1151.<br />
29. M. Malinowski "Adaptive modulator for three-phase PWM rectifier/inverter", in proc.<br />
EPE-PEMC Conf., Kosice, 2000, pp.1.35-1.41.<br />
30. N. Urasaki, T. Senjyu, K. Uezato, T. Funabashi, “On-line dead-time compensation<br />
method for permanent magnet synchronous motor drive”, IEEE International<br />
Conference on Industrial Technology, CIT '02, 11-14 Dec. 2002, Vol. 1, pp. 268 – 273.<br />
FOC, <strong>DTC</strong> and <strong>DTC</strong>-<strong>SVM</strong><br />
31. F. Blaschke, "The principle <strong>of</strong> fields-orientation as applied to the Transvector closedloop<br />
control system for rotating-field machines", in Siemens Reviev 34, 1972, pp.217-<br />
220.<br />
32. G.S. Buja, M.P. Kazmierkowski, "<strong>Direct</strong> <strong>Torque</strong> <strong>Control</strong> <strong>of</strong> PWM Inverter-Fed AC<br />
Motors-A Survey", IEEE Transactions on Industrial Electronics, Vol. 51, Aug. 2004,<br />
pp.744-757.<br />
33. D. Casadei, G. Serra, A. Tani, L. Zarri, F. Pr<strong>of</strong>umo, “Performance analysis <strong>of</strong> a speedsensorless<br />
induction motor drive based on a constant-switching-frequency <strong>DTC</strong><br />
scheme”; IEEE Transactions on Industry Applications, March-April 2003, Volume 39,<br />
pp. 476 – 484.<br />
34. D. Casadei, F. Pr<strong>of</strong>umo, G. Serra, A. Tani, “FOC and <strong>DTC</strong>: two viable schemes for<br />
induction motors torque control”; IEEE Transactions on Power Electronics, Sept. 2002,<br />
Vol. 17, pp. 779 – 787.<br />
35. T. Chun; W.Y. Hu, “Research on the direct torque control in electromagnetic<br />
synchronous motor drive”, Power Electronics and Motion <strong>Control</strong> Conference, PEMC<br />
2000, 15-18 Aug. 2000. Vol. 3, pp. 1262 – 1265.<br />
36. S.K. Chung, H.S. Kim, Ch. G. Kim, M.-J. Youn, “A New Instantaneous <strong>Torque</strong> <strong>Control</strong><br />
<strong>of</strong> PM Synchronous Motor for High-Performance <strong>Direct</strong>-Drive Applications” IEEE<br />
Transaction on Power Electronics, May 1998, Vol. 12, No. 3.<br />
37. Ch. French, P. Acarnley, “<strong>Direct</strong> <strong>Torque</strong> <strong>Control</strong> <strong>of</strong> Permanent Magnet Drivers”, IEEE<br />
Transaction on Industrial Application, Vol. 32, No. 5, 1996, pp. 1080-1088.<br />
38. L.-H. Hoang, “Comparison <strong>of</strong> field-oriented control and direct torque control for<br />
induction motor drives”, Industry Applications Conference, 1999. Thirty-Fourth IAS<br />
Annual Meeting, 3-7 Oct. 1999, Vol. 2, pp.1245 – 1252.<br />
174
References<br />
39. J.-I. Itoh, N. Nomura, H. Ohsawa, “A comparison between V/f control and positionsensorless<br />
vector control for the permanent magnet synchronous motor”, Proceedings <strong>of</strong><br />
the Power Conversion Conference, PCC Osaka 2-5 April 2002, Vol. 3, pp. 1310 – 1315.<br />
40. T.M. Jahns, “Motion control <strong>with</strong> permanent-magnet AC machines”, Proceedings <strong>of</strong> the<br />
IEEE, Aug. 1994, Vol. 82, pp. 1241 – 1252.<br />
41. C. Lascu, I. Boldea, F. Blaabjerg, “Variable-structure direct torque control - a class <strong>of</strong><br />
fast and robust controllers for induction machine drives”, IEEE Transactions on<br />
Industrial Electronics, Aug. 2004,Vol. 51, pp. 785 – 792.<br />
42. T. Lixin; Z. Limin; M.F. Rahman, Y. Hu, “A novel direct torque control for interior<br />
permanent-magnet synchronous machine drive <strong>with</strong> low ripple in torque and flux-a<br />
speed-sensorless approach”, IEEE Transactions on Industry Applications, Nov.-Dec.<br />
2003,Vol. 39, pp. 1748 – 1756.<br />
43. J. Lee; Ch.-G. Kim; M.-J. Youn, “A dead-beat type digital controller for the direct<br />
torque control <strong>of</strong> an induction motor”, IEEE Transactions on Power Electronics, Sept.<br />
2002, Vol. 17, pp. 739 – 746.<br />
44. H. Rasmussen, P. Vadstrup, H. Børsting, "Adaptive sensorless field oriented control <strong>of</strong><br />
PM motors including zero speed", IEEE International Symposium on Industrial<br />
Electronics, Ajaccio, France, May 2004.<br />
45. M. F. Rahman, L. Zhong, K. W. Lim, “A direct torque controlled interior permanent<br />
magnet synchronous motor drive incorporating field weakening”, IEEE-IAS’97, 1997 ,<br />
p. 67 - 74.<br />
46. M.F. Rahman, L. Zhong, K.W. Lim, “An Investigation <strong>of</strong> <strong>Direct</strong> and Indirect <strong>Torque</strong><br />
Conttrollers for PM Synchronous Motor Drivers”, Conference PEDS’97, May,<br />
Singapure, pp. 519-523.<br />
47. M. F. Rahman, L. Zhong, “Comparison <strong>of</strong> torque response <strong>of</strong> the interior permanent<br />
magnet motor under PWM current and direct torque control”, IEEE, 1999 , p. 1464 -<br />
1470.<br />
48. J. Rodriguez, J. Pontt, C. Silva, S. Kouro, H. Miranda, “A novel direct torque control<br />
scheme for induction machines <strong>with</strong> space vector modulation”; Power Electronics<br />
Specialists Conference, PESC’04, 20-25 June 2004, Vol. 2, pp. 1392 – 1397.<br />
49. D. Telford, M.W. Dunnigan, B.W. Williams, “A comparison <strong>of</strong> vector control and<br />
direct torque control <strong>of</strong> an induction machine”, 2000 IEEE 31st Annual Power<br />
Electronics Specialists Conference, 2000. PESC 00, 18-23 June 2000, Vol. 1, pp. 421 –<br />
426.<br />
50. L. Takahashi, T. Noguchi, “A new quick response and high efficiency strategy <strong>of</strong><br />
induction motor”, IAS, 1985 , p. 495-502.<br />
51. L. Tang; L. Zhong; F. Rahman, “Modeling and experimental approach <strong>of</strong> a novel direct<br />
torque control scheme for interior permanent magnet synchronous machine drive”,<br />
IECON 02, 5-8 Nov. 2002, Vol. 1, pp. 235 – 240.<br />
52. M.N. Uddin, T.S. Radwan, G.H. George, M.A. Rahman, “Performance <strong>of</strong> current<br />
controllers for VSI-fed IPMSM drive”, IEEE Transactions on Industry Applications,<br />
Volume 36, Issue 6, Nov.-Dec. 2000 Page(s):1531 – 1538.<br />
53. L. Xu, M. Fu “A Novel Sensorless <strong>Control</strong> Technique for Permanent Magnet<br />
Synchronous Motors (PMSM) using Digital Signal Processor (DSP)”, NEACON’97,<br />
Dayton, Ohio, July 14-17, 1997.<br />
54. L. Xu, M. Fu “A Sensorless <strong>Direct</strong> <strong>Torque</strong> <strong>Control</strong> Technique for Permanent Magnet<br />
Synchronous Motors”, IEEE Industrial Applications Conferences,1999, Vol. 1.<br />
55. Y. Xue, X. Xu, T.G. Habetler, D.M. Divan, “A low cost stator flux oriented voltage<br />
source variable speed drive”, Industry Applications Society Annual Meeting, 7-12 Oct.<br />
1990, Vol.1, pp. 410 – 415.<br />
56. H. Yuwen,T. Cun, G. Yikang,Y. Zhiqing, L.X. Tang, M.F. Rahman “In-depth Reserch<br />
on <strong>Direct</strong> <strong>Torque</strong> <strong>Control</strong> <strong>of</strong> Permanent Magnet Synchronous Motor”, IEEE Conference<br />
IECON 2002 ,November, Sevilla, Spain.<br />
175
References<br />
57. L. Zhong, M.F. Rahman, W.Y. Hu, Lim, K.W., Rahman, M.A., “A direct torque<br />
controller for permanent magnet synchronous motor drives”, IEEE Transactions on<br />
Energy Conversion, Sept. 1999, Vol. 14, pp. 637 – 642.<br />
58. L. Zhong, M.F. Rahman, W.Y.Hu, K.W. Lim, “Analysis <strong>of</strong> <strong>Direct</strong> <strong>Torque</strong> <strong>Control</strong> in<br />
Permanent Magnet Synchronous Motor Drives”, IEEE Transaction on Power<br />
Electronics, Vol. 12, No. 13, May, 1997.<br />
59. L. Zhong, M.F. Rahman, „Voltage Switching Tables for <strong>Control</strong>led Interior Permanent<br />
Magnet Motor“, IEEE Conference IECON’99, Vol. 3, pp. 1445-1451.<br />
60. L.Zhong, M. F. Rahman, “A direct torque controller for permanent magnet synchronous<br />
motor drives <strong>with</strong>out a speed sensor”, IEEE Trans. On Energy Conversion, Vol.14,<br />
No.3 ,September , 1999.<br />
61. M.R Zolghardi, D.Diallo, D.Roye, “<strong>Direct</strong> <strong>Torque</strong> <strong>Control</strong> System for Synchronous<br />
Machine”, Europen Conference on Power Electronics and Application,<br />
EPE’97,Trondheim, Norway,Vol. 3, pp. 694-699.<br />
62. M.R. Zolghardi, E. M. Olasagasti, D. Roye, “Steady State <strong>Torque</strong> Correction <strong>of</strong> <strong>Direct</strong><br />
<strong>Torque</strong> <strong>Control</strong>led PM Synchronous Machine”, IEEE International Conference on<br />
Electric Machines Drivers, 1997.<br />
63. M.R Zolghardi, J.Guiraud, J.Davoine, D. Roye, “A DSP <strong>Direct</strong> <strong>Torque</strong> <strong>Control</strong>ler for<br />
Permanent Magnet Synchronous Motor Drivers”, IEEE Power Electronics Conference,<br />
PESC’98, Vol.2, pp. 2055-2061.<br />
64. M.R Zolghardi, D. Roye, “Sensorless <strong>Direct</strong> <strong>Torque</strong> <strong>Control</strong> <strong>of</strong> Synchronous Motor<br />
Drive” Proceedings <strong>of</strong> International Conference on Electrical Machines ICEM’ 98, Vol.<br />
2, 1998, Istanbul, Turkey.<br />
65. M.R Zolghardi, D. Diallo, D. Roye, “Start up Strategies for <strong>Direct</strong> <strong>Torque</strong> <strong>Control</strong>led<br />
Synchronous Machine ”, Europen Conference on Power Electronics and Application,<br />
EPE’97,Trondheim, Norway,Vol. 3, pp. 689-693.<br />
66. M.R Zolghardi, C. Pelissu, D. Roye, “Start up <strong>of</strong> a global <strong>Direct</strong> <strong>Torque</strong> <strong>Control</strong><br />
System”, IEEE Power Electronic Conference PESC’96, Vol. 1, pp. 370-374.<br />
Sensorless <strong>Control</strong> and Initial Position Estimation<br />
67. S. Bolognani, R. Oboe, M. Zigliotto, “Sensorless full-digital PMSM drive <strong>with</strong> EKF<br />
estimation <strong>of</strong> speed and rotor position”, IEEE Transactions on Industrial Electronics,<br />
Vol. 46, Issue 1, Feb. 1999, pp. 184 – 191.<br />
68. S. Brock, J. Deskur, K. Zawirski, , “Robust speed and position control <strong>of</strong> PMSM”,<br />
Proceedings <strong>of</strong> the IEEE International Symposium on Industrial Electronics, 1999, ISIE<br />
'99. 12-16 July 1999, Vol. 2, pp. 667 – 672.<br />
69. M.J. Corley, R.D. Lorenz, “Rotor position and velocity estimation for a salient-pole<br />
permanent magnet synchronous machine at standstill and high speeds”, IEEE<br />
Transactions on Industry Applications, July-Aug. 1998, Vol. 34, pp. 784 – 789.<br />
70. Z. Chen, M. Tomita, S. Ichikawa, S. Doki, S. Okuma, “Sensorless control <strong>of</strong> interior<br />
permanent magnet synchronous motor by estimation <strong>of</strong> an extended electromotive<br />
force”; Industry Applications Conference, 2000, Vol. 3, 8-12 Oct. 2000 pp. 1814 –<br />
1819.<br />
71. Z. Chen, M. Tomita, S. Doki, S. Okuma, “An extended electromotive force model for<br />
sensorless control <strong>of</strong> interior permanent-magnet synchronous motors”, IEEE<br />
Transactions on Industrial Electronics, Vol. 50, Issue 2, April 2003, pp. 288 – 295.<br />
72. Z. Chen, M. Tomita, S. Doki, S. Okuma, “New adaptive sliding observers for<br />
position- and velocity-sensorless controls <strong>of</strong> brushless DC motors”, IEEE Transactions<br />
on Industrial Electronics, Vol. 47, Issue 3, June 2000, pp. 582 – 591.<br />
73. R. Dhaouadi, N. Mohan, L. Norum, “Design and implementation <strong>of</strong> an extended<br />
Kalman filter for the state estimation <strong>of</strong> a permanent magnet synchronous motor”, IEEE<br />
Transactions on Power Electronics, Vol. 6, Issue 3, July 1991, pp. 491 – 497.<br />
176
References<br />
74. N. Ertugrul, P. Acarnley, “A new algorithm for sensorless operation <strong>of</strong> permanent<br />
magnet motors”; IEEE Transactions on Industry Applications, Jan.-Feb. 1994, Vol. 30,<br />
pp. 126 – 133.<br />
75. C. French, P. Acarnley, I. Al-Bahadly, “Sensorless position control <strong>of</strong> permanent<br />
magnet drives”; Industry Applications Conference, IAS '95, 8-12 Oct. 1995, Vol. 1, pp.<br />
61 – 68.<br />
76. M. Fu, L. Xu, “A novel sensorless control technique for permanent magnet synchronous<br />
motor (PMSM) using digital signal processor (DSP)”, Proceedings <strong>of</strong> the IEEE 1997<br />
National Aerospace and Electronics Conference, 14-17 July 1997, NAECON 1997,Vol.<br />
1, pp. 403 – 408.<br />
77. J. Hu, B. Wu, “New integration algorithms for estimating motor flux over a wide speed<br />
range”, IEEE Transactions on Power Electronics, , Sept. 1998, Vol. 13, pp. 969 – 977.<br />
78. Y. Jeong, R.D. Lorenz, T.M. Jahns, S. Sul, “Initial rotor position estimation <strong>of</strong> an<br />
interior permanent magnet synchronous machine using carrier-frequency injection<br />
methods”, IEEE International Electric Machines and Drives Conference, IEMDC'03, 1-<br />
4 June 2003, Vol. 2,pp. 1218 – 1223.<br />
79. L.A. Jones, J.H. Lang, “A state observer for the permanent-magnet synchronous<br />
motor”, IEEE Transactions on Industrial Electronics, Vol. 36, Issue 3, Aug. 1989 pp.<br />
374 – 382.<br />
80. J.P. Johnson, M. Ehsani, Y. Guzelgunler, “Review <strong>of</strong> sensorless methods for<br />
brushless DC”, Industry Applications Conference, 1999. Thirty-Fourth IAS Annual<br />
Meeting. Conference Record <strong>of</strong> the 1999 IEEE, 3-7 Oct. 1999, Vol. 1, pp. 143 – 150.<br />
81. J. S. Kim; S. K. Sul, “New stand-still position detection strategy for PMSM drive<br />
<strong>with</strong>out rotational transducers”, Applied Power Electronics Conference and Exposition,<br />
APEC '94, 13-17 Feb. 1994, Vol.1, pp. 363 – 369.<br />
82. J.-S. Kim, S.-K. Sul, “New approach for high-performance PMSM drives <strong>with</strong>out<br />
rotational position sensors”, IEEE Transactions on Power Electronics, Sept. 1997, Vol.<br />
12, pp. 904 – 911.<br />
83. T.-S. Low, T.-H. Lee, K.-T. Chang, “A nonlinear speed observer for permanentmagnet<br />
synchronous motors”, IEEE Transactions on Industrial Electronics, Vol. 40,<br />
Issue 3, June 1993, pp. 307 – 316.<br />
84. M. Linke, R. Kennel, J. Holtz, “Sensorless position control <strong>of</strong> permanent magnet<br />
synchronous machines <strong>with</strong>out limitation at zero speed”, IECON 02, 5-8 Nov. 2002,<br />
Vol. 1, pp. 674 – 679.<br />
85. N. Matsui, “Sensorless PM brushless DC motor drives”, Transactions on Industrial<br />
Electronics, IEEE , Vol. 43, April 1996, pp. 300 – 308.<br />
86. N. Matsui, M. Shigyo, “Brushless DC motor control <strong>with</strong>out position and speed<br />
sensors”; IEEE Transactions on Industry Applications, Jan.-Feb. 1992, Vol. 28 pp. 120<br />
– 127.<br />
87. N. Matsui, “Sensorless PM brushless DC motor drives”, IEEE Transactions on<br />
Industrial Electronics, April 1996, Vol. 43, pp. 300 – 308.<br />
88. S. Nakashima, Y. Inagaki, I. Miki, “Sensorless initial rotor position estimation <strong>of</strong><br />
surface permanent-magnet synchronous motor”, IEEE Transactions on Industry<br />
Applications, Nov.-Dec. 2000, Vol. 36, pp. 1598 – 1603.<br />
89. T. Noguchi, K. Yamada, S. Kondo, I. Takahashi, “Rotor position estimation method <strong>of</strong><br />
sensorless PM motor at rest <strong>with</strong> no sensitivity to armature resistance”, Industrial<br />
Electronics, <strong>Control</strong>, and Instrumentation, Proceedings <strong>of</strong> the 1996 IEEE IECON 1996,<br />
5-10 Aug. 1996,Vol. 2, pp. 1171 – 1176.<br />
90. S. Ostlund, M. Brokemper, “Initial rotor position detections for an integrated PM<br />
synchronous motor drive”, Industry Applications Conference, IAS '95, 8-12 Oct. 1995,<br />
Vol. 1, pp. 741 – 747.<br />
91. G. Pesse, T. Paga, “A permanent magnet synchronous motor flux control scheme<br />
<strong>with</strong>out position sensor”, Conference proceedings on the EPE’97, Trondheim 1997,<br />
Vol. 4, pp. 553 – 556.<br />
177
References<br />
92. R.B. Sepe, J.H. Lang, “Real-time observer-based (adaptive) control <strong>of</strong> a permanentmagnet<br />
synchronous motor <strong>with</strong>out mechanical sensors”, IEEE Transactions on<br />
Industry Applications, Vol. 28, Issue 6, Nov.-Dec. 1992, pp. 1345 – 1352.<br />
93. P.B. Schmidt, M.L. Gasperi, G. Ray, A.H. Wijenayake, “Initial rotor angle detection<br />
<strong>of</strong> a nonsalient pole permanent magnet synchronous machine”, Industry Applications<br />
Conference, IAS '97., 5-9 Oct. 1997, Vol. 1, pp. 459 – 463.<br />
94. J.X. Shen, Z.Q. Zhu, D. Howe, “Improved speed estimation in sensorless PM<br />
brushless AC drives”, Transactions on Industry Applications, July-Aug. 2002, Vol. 38,<br />
pp. 1072 – 1080.<br />
95. Yen-Shin Lai; Fu-San Shyu; Shian Shau Tseng, “New initial position detection<br />
technique for three-phase brushless DC motor <strong>with</strong>out position and current sensors”,<br />
Transactions on Industry Applications, , March-April 2003, Vol. 39, pp. 485 – 491.<br />
96. J. Solsona, M.I. Valla, C. Muravchik, “A nonlinear reduced order observer for<br />
permanent magnet synchronous motors”, IEEE Transactions on Industrial Electronics,<br />
Aug. 1996, Vol. 43, pp. 492 - 497<br />
97. J.A. Solsona, M.I. Valla, “Disturbance and nonlinear Luenberger observers for<br />
estimating mechanical variables in permanent magnet synchronous motors under<br />
mechanical parameters uncertainties”, IEEE Transactions on Industrial Electronics, Vol.<br />
50, Issue 4, Aug. 2003 pp. 717 – 725.<br />
98. J. Solsona, M.I. Valla, C. Muravchik, “Nonlinear control <strong>of</strong> a permanent magnet<br />
synchronous motor <strong>with</strong> disturbance torque estimation”, International Conference on<br />
Industrial Electronics, <strong>Control</strong> and Instrumentation, 1997. IECON 97, Vol. 1, 9-14 Nov.<br />
1997, pp. 120 – 125.<br />
99. T. Takeshita, N. Matsui, “Sensorless control and initial position estimation <strong>of</strong> salientpole<br />
brushless DC motor”, Advanced Motion <strong>Control</strong>, 1996. AMC '96, 18-21 March<br />
1996, Vol. 1, pp. 18 – 23.<br />
100. M. Tomita, T. Senjyu, S. Doki, S. Okuma, “New sensorless control for brushless DC<br />
motors using disturbance observers and adaptive velocity estimations”, IEEE<br />
Transactions on Industrial Electronics, Vol. 45, Issue 2, April 1998, pp. 274 – 282.<br />
101. M. Tursini, R. Petrella, F. Parasiliti, “Initial rotor position estimation method for PM<br />
motors”; IEEE Transactions on Industry Applications, Nov.-Dec. 2003, Vol. 39, pp.<br />
1630 – 1640.<br />
102. R. Wu, G.R. Slemon, “A permanent magnet motor drive <strong>with</strong>out a shaft sensor”,<br />
Industry Applications Society Annual Meeting, 1990., Conference Record <strong>of</strong> the 1990<br />
IEEE, 7-12 Oct. 1990, Vol.1, pp. 553 – 558.<br />
103. D. Yousfi, M. Azizi, A. Saad, “Sensorless position and speed detection for permanent<br />
magnet synchronous motor”; Power Electronics and Motion <strong>Control</strong> Conference,<br />
PIEMC 2000, 15-18 Aug. 2000,Vol. 3, pp. 1224 – 1229.<br />
104. D. Yousfi, M. Azizi, A. Saad, “Robust position and speed estimation algorithm for<br />
permanent magnet synchronous drives”, Industry Applications Conference, 8-12 Oct.<br />
2000, Vol.3, pp. 1541 – 1546.<br />
Others<br />
105. H.-B. Shin, “New antiwindup PI controller for variable-speed motor drives”<br />
IEEE Transactions on Industrial Electronics, June 1998, Vol. 45, pp. 445 – 450.<br />
106. MathWorks, Inc, "Matlab® The Language <strong>of</strong> Technical Computing", Release 12,<br />
2000.<br />
Papers written during work on this thesis<br />
107. D. L. Sobczuk, D. Świerczyński "DSP implementation <strong>of</strong> vector controlled inverter<br />
fed induction motor <strong>with</strong> neural network speed estimator" SENE '99 tom. 2 str. 601-<br />
606.<br />
178
References<br />
108. D. Świerczyński, Marian P. Kaźmierkowski, Paweł Z. Grabowski "Nowe algorytmy<br />
estymacji strumienia stojana w silnikach indukcyjnych klatkowych zasilanych z<br />
falowników PWM" SENE '99 tom. 2 str. 649-654.<br />
109. M. Angielczyk, M. Malinowski, D. Świerczyński, „Stanowisko laboratoryjne oparte<br />
o kartę DSP do sterowania prostownikiem PWM oraz silnikiem PMSM zasilanym z<br />
falownika PWM”, Kamień Śląski 2000, Międzynarodowe XII Sympozjum<br />
Mikromaszyny i Serwonapędy, str. 480-487.<br />
110. D. Świerczyński, M. P. Kaźmierkowski.:„Bezpośrednie sterowanie momentu silnika<br />
synchronicznego o magnesach trwałych zasilanym z falownika PWM”, SENE 2001<br />
,tom II, str. 629<br />
111. D. Świerczyński, M. P. Kaźmierkowski."<strong>Direct</strong> torque control <strong>of</strong> permanent magnet<br />
synchronous motor (PMSM) using space vector modulation (<strong>DTC</strong>-<strong>SVM</strong>)", SME'2002,<br />
pp. 245<br />
112. D. Świerczyński, M. P. Kaźmierkowski, Frede Blaabjerg "DSP Based <strong>Direct</strong> <strong>Torque</strong><br />
<strong>Control</strong> <strong>of</strong> Permanent Magnet Synchronous Motor (PMSM) Using <strong>Space</strong> <strong>Vector</strong><br />
<strong>Modulation</strong> (<strong>DTC</strong>-<strong>SVM</strong>)", IEEE-ISIE 2002, L’Aquila, Italy, pp. 723-727.<br />
113. D. Świerczyński, M. Żelechowski "Studia symulacyjne nad universalną strukturą<br />
bezpośredniego sterowania momentem i strumieniem dla silników synchronicznych o<br />
magnesach trwałych oraz silników asynchronicznych ", Modelowania i Symulacja MiS<br />
2002, Zakopane.<br />
114. D. Świerczyński, M. P. Kaźmierkowski, Frede Blaabjerg "<strong>Direct</strong> <strong>Torque</strong> <strong>Control</strong><br />
<strong>with</strong> <strong>Space</strong> <strong>Vector</strong> <strong>Modulation</strong> (<strong>DTC</strong>-<strong>SVM</strong>) for Permanent Magnet Synchronous<br />
Motor (PMSM)", EPE-PEMC 2002, Dubrovnik & Cavtat, Croatia.<br />
115. D. Świerczyński, M. Żelechowski "Universalną strukturą bezpośredniego sterowania<br />
momentem i strumieniem dla silników synchronicznych o magnesach trwałych oraz<br />
silników asynchronicznych ", Mikromaszyny i Serwonapędy MiS 2002, Krasiczyn.<br />
116. D. Świerczyński, M. P. Kaźmierkowski " <strong>Direct</strong> <strong>Torque</strong> <strong>Control</strong> <strong>of</strong> Permanent<br />
Magnet Synchronous Motor (PMSM) Using <strong>Space</strong> <strong>Vector</strong> <strong>Modulation</strong> (<strong>DTC</strong>-<strong>SVM</strong>) –<br />
Simulation and Experimental Results", IECON 2002, Sevilla, Spain.<br />
117. D.Świerczyński, M. Żelechowski „Bezpośrednie sterowanie momentem i<br />
strumieniem dla silników synchronicznych i asynchronicznych”, II Krajowa<br />
Konferencja MiS-2 Modelowanie i Symulacja, Kościelisko, 24-28 czerwca 2002, pp.<br />
187-194.<br />
118. D. Świerczyński, M. Żelechowski ”Universal structure <strong>of</strong> direct torque control for<br />
AC motor drives”, III Summer Seminar on Nordick Network for Multi Disciplinary<br />
Optimised Electric Drives, June 21-23, 2003, Zegrze, Poland<br />
119. M. Jasiński D. Świerczyński, M. P. Kazmierkowski, M. Malinowski ”<strong>Direct</strong> <strong>Control</strong><br />
<strong>of</strong> an AC-DC-AC converter ”, III Summer Seminar on Nordick Network for Multi<br />
Disciplinary Optimised Electric Drives, June 21-23, 2003, Zegrze, Poland<br />
120. M. Żelechowski, D. Świerczyński, M. P. Kazmierkowski, J. Załęski „Uniwersalne<br />
napędy falownikowe ze sterowaniem mikroporcesorowym nowej generacji”,<br />
Elektro.info Nr 6 (17) 2003, str. 26-28<br />
121. D. Świerczyński, M. Żelechowski, „Universal Structure <strong>of</strong> <strong>Direct</strong> <strong>Torque</strong> <strong>Control</strong> for<br />
AC Motor Drives”, V Power Electronics Seminar, 7-8 July, 2003, Berlin, Niemcy<br />
122. M. Jasinski, D. Swierczynski, M. P. Kazmierkowski "Novel Sensorless <strong>Direct</strong> Power<br />
and <strong>Torque</strong> <strong>Control</strong> <strong>of</strong> <strong>Space</strong> <strong>Vector</strong> Modulated AC/DC/AC Converter", In Proc. <strong>of</strong> the<br />
ISIE 2004 Conf., 4-7 May, Ajaccio, France, pp.1147- 1152.<br />
123. D. Świerczyński, M. Żelechowski ”Universal structure <strong>of</strong> direct torque control for<br />
AC motor drives”, Przegląd Elektrotechniczny, May, 2004, Poland, pp. 489-492.<br />
124. M. P. Kaźmierkowski , M. Żelechowski, D. Świerczyński „Simple <strong>DTC</strong>-<strong>SVM</strong><br />
<strong>Control</strong> for Induction and PM Synchronous Motor”, In Proc. <strong>of</strong> the ICEM 2004 Conf.,<br />
5-8 September, Krakow, Poland, on CD.<br />
125. M. P. Kaźmierkowski , M. Żelechowski, D. Świerczyński „<strong>DTC</strong>-<strong>SVM</strong> an Efficient<br />
Method for <strong>Control</strong> Both Induction and PM Synchronous Motor”, In Proc. <strong>of</strong> the EPE-<br />
PEMC 2004 Conf., 2-4 September, Riga, Latvia, on CD.<br />
179
References<br />
126. B. W. Mehdawi, D. Świerczyński, M. P. Kaźmierkowski „Design and Investigation<br />
<strong>of</strong> <strong>Vector</strong> <strong>Control</strong> Scheme PWM Inverter-fed Nonsalient Permanent Magnet<br />
Synchronous Motor”, In Proc. <strong>of</strong> the PPEE 2005 Conf., 2-5 April, Poland, Wisla, on<br />
CD.<br />
180