Warsaw University of Technology
Faculty of Electrical Engineering
Institute of Control and Industrial Electronics
Ph.D. Thesis
Marcin Żelechowski, M. Sc.
Space Vector Modulated – Direct
Torque Controlled (DTC – SVM)
Inverter – Fed Induction Motor Drive
Thesis supervisor
Prof. Dr Sc. Marian P. Kaźmierkowski
Warsaw – Poland, 2005
Acknowledgements
The work presented in the thesis was carried out during author’s Ph.D. studies at the
Institute of Control and Industrial Electronics in Warsaw University of Technology,
Faculty of Electrical Engineering. Some parts of the work were realized in cooperation
with foreign Universities:
• University of Nevada, Reno, USA (US National Science Foundation grant –
Prof. Andrzej Trzynadlowski),
• University of Aalborg, Denmark (Prof. Frede Blaabjerg),
and company:
• Power Electronics Manufacture – „TWERD”, Toruń, Poland.
First of all, I would like to express gratitude Prof. Marian P. Kaźmierkowski for the
continuous support and help during work of the thesis. His precious advice and
numerous discussions enhanced my knowledge and scientific inspiration.
I am grateful to Prof. Andrzej Sikorski from the Białystok Technical University and
Prof. Włodzimierz Koczara from the Warsaw University of Technology for their
interest in this work and holding the post of referee.
Specially, I am indebted to my friend Dr Paweł Grabowski for support and
assistance.
Furthermore, I thank my colleagues from the Intelligent Control Group in Power
Electronics for their support and friendly atmosphere. Specially, to Dr Dariusz Sobczuk,
Dr Mariusz Malinowski, Dr Mariusz Cichowlas, and Dariusz Świerczyńki M.Sc.
Finally, I would like thank to my whole family, particularly my parents for their love
and patience.
Contents
Pages
1. Introduction 1
2. Voltage Source Inverter Fed Induction Motor Drive 6
2.1. Introduction 6
2.2. Mathematical Model of Induction Motor 6
2.3. Voltage Source Inverter (VSI) 12
2.4. Pulse Width Modulation (PWM) 17
2.4.1. Introduction 17
2.4.2. Carrier Based PWM 18
2.4.3. Space Vector Modulation (SVM) 22
2.4.4. Relation Between Carrier Based and Space Vector Modulation 28
2.4.5. Overmodulation (OM) 31
2.4.6. Random Modulation Techniques 35
2.5. Summary 39
3. Vector Control Methods of Induction Motor 40
3.1. Introduction 40
3.2. Field Oriented Control (FOC) 40
3.3. Feedback Linearization Control (FLC) 45
3.4. Direct Flux and Torque Control (DTC) 49
3.4.1. Basics of Direct Flux and Torque Control 49
3.4.2. Classical Direct Torque Control (DTC) – Circular Flux Path 53
3.4.3. Direct Self Control (DSC) – Hexagon Flux Path 61
3.5. Summary 64
4. Direct Flux and Torque Control with Space Vector Modulation (DTCSVM) 66
4.1. Introduction 66
4.2. Structures of DTCSVM – Review 66
4.2.1. DTCSVM Scheme with Closed – Loop Flux Control 66
4.2.2. DTCSVM Scheme with Closed – Loop Torque Control 68
4.2.3. DTCSVM Scheme with Close – Loop Torque and Flux Control
Operating in Polar Coordinates 69
4.2.4. DTCSVM Scheme with Close – Loop Torque and Flux Control
in Stator Flux Coordinates 70
4.2.5. Conclusions from Review of the DTCSVM Structures 71
4.3. Analysis and Controller Design for DTCSVM Method with
Close – Loop Torque and Flux Control in Stator Flux Coordinates 71
4.3.1. Torque and Flux Controllers Design – Symmetry Criterion Method 75
4.3.2. Torque and Flux Controllers Design – Root Locus Method 78
4.3.3. Summary of Flux and Torque Controllers Design 88
4.4. Speed Controller Design 94
4.5. Summary 98
Contents
5. Estimation in Induction Motor Drives 99
5.1. Introduction 99
5.2. Estimation of Inverter Output Voltage 100
5.3. Stator Flux Vector Estimators 104
5.4. Torque Estimation 110
5.5. Rotor Speed Estimation 110
5.6. Summary 112
6. Configuration of the Developed IM Drive Based on DTCSVM 113
6.1. Introduction 113
6.2. Block Scheme of Implemented Control System 113
6.3. Laboratory Setup Based on DS1103 115
6.4. Drive Based on TMS320LF2406 118
7. Experimental Results 122
7.1. Introduction 122
7.2. Pulse Width Modulation 122
7.3. Flux and Torque Controllers 125
7.4. DTCSVM Control System 129
8. Summary and Conclusions 138
References 141
List of Symbols 151
Appendices 156
A.1. Derivation of Fourier Series Formula for Phase Voltage
A.2. SABER Simulation Model
A.3. Data and Parameters of Induction Motors
A.4. Equipment
A.5. dSPACE DS1103 PPC Board
A.6. Processor TMS320FL2406
1. Introduction
The Adjustable Speed Drives (ADS) are generally used in industry. In most drives
AC motors are applied. The standard in those drives are Induction Motors (IM) and
recently also Permanent Magnet Synchronous Motors (PMSM) are offered. Variable
speed drives are widely used in application such as pumps, fans, elevators, electrical
vehicles, heating, ventilation and airconditioning (HVAC), robotics, wind generation
systems, ship propulsion, etc. [16].
Previously, DC machines were preferred for variable speed drives. However, DC
motors have disadvantages of higher cost, higher rotor inertia and maintenance problem
with commutators and brushes. In addition they cannot operate in dirty and explosive
environments. The AC motors do not have the disadvantages of DC machines.
Therefore, in last three decades the DC motors are progressively replaced by AC drives.
The responsible for those result are development of modern semiconductor devices,
especially power Insulated Gate Bipolar Transistor (IGBT) and Digital Signal Processor
(DSP) technologies.
The most economical IM speed control methods are realized by using frequency
converters. Many different topologies of frequency converters are proposed and
investigated in a literature. However, a converter consisting of a diode rectifier, a dc
link and a Pulse Width Modulated (PWM) voltage inverter is the most applied used in
industry (see section 2.3).
The highperformance frequency controlled PWM inverter – fed IM drive should be
characterized by:
• fast flux and torque response,
• available maximum output torque in wide range of speed operation region,
• constant switching frequency,
• unipolar voltage PWM,
• low flux and torque ripple,
• robustness for parameter variation,
• fourquadrant operation,
1. Introduction
2
These features depend on the applied control strategy. The main goal of the chosen
control method is to provide the best possible parameters of drive. Additionally, a very
important requirement regarding control method is simplicity (simple algorithm, simple
tuning and operation with small controller dimension leads to low price of final
product).
A general classification of the variable frequency IM control methods is presented in
Fig. 1.1 [67]. These methods can be divided into two groups: scalar and vector.
Variable
Frequency Control
Scalar based
controllers
Vector based
controller
U/f=const.
Volt/Hertz
( )
rs
fi ω=
Field Oriented
Feedback
Linearization
Scalar based
controllers
Direct Torque
Control
Rotor Flux
Oriented
Stator Flux
Oriented
Direct Torque
Space  Vector
Modulation
Passivity Based
Control
Circle flux
trajectory
(Takahashi)
Hexagon flux
trajectory
(Takahashi)
Direct
(Blaschke)
Indirect
(Hasse)
Closed Loop
Flux & Torque
Control
Open Loop
NFO (Jonsson)
o
&&
Stator Current
Fig. 1.1. General classification of induction motor control methods
The scalar control methods are simple to implement. The most popular in industry is
constant Voltage/Frequency (V/Hz=const.) control. This is the simplest, which does not
provide a highperformance. The vector control group allows not only control of the
voltage amplitude and frequency, like in the scalar control methods, but also the
instantaneous position of the voltage, current and flux vectors. This improves
significantly the dynamic behavior of the drive.
However, induction motor has a nonlinear structure and a coupling exists in the
motor, between flux and the produced electromagnetic torque. Therefore, several
methods for decoupling torque and flux have been proposed. These algorithms are
based on different ideas and analysis.
1. Introduction
3
The first vector control method of induction motor was Field Oriented Control
(FOC) presented by K. Hasse (Indirect FOC) [45] and F. Blaschke (Direct FOC) [12] in
early of 70s. Those methods were investigated and discussed by many researchers and
have now become an industry standard. In this method the motor equations are
transformed into a coordinate system that rotates in synchronism with the rotor flux
vector. The FOC method guarantees flux and torque decoupling. However, the
induction motor equations are still nonlinear fully decoupled only for constant flux
operation.
An other method known as Feedback Linearization Control (FLC) introduces a new
nonlinear transformation of the IM state variables, so that in the new coordinates, the
speed and rotor flux amplitude are decoupled by feedback [81, 83].
A method based on the variation theory and energy shaping has been investigated
recently, and is called Passivity Based Control (PBC) [88]. In this case the induction
motor is described in terms of the EulerLagrange equations expressed in generalized
coordinates.
In the middle of 80s new strategies for the torque control of induction motor was
presented by I. Takahashi and T. Noguchi as Direct Torque Control (DTC) [97] and by
M. Depenbrock as Direct Self Control (DSC) [4, 31, 32]. Those methods thanks to the
other approach to control of IM have become alternatives for the classical vector control
– FOC. The authors of the new control strategies proposed to replace motor decoupling
and linearization via coordinate transformation, like in FOC, by hysteresis controllers,
which corresponds very well to onoff operation of the inverter semiconductor power
devices. These methods are referred to as classical DTC. Since 1985 they have been
continuously developed and improved by many researchers.
Simple structure and very good dynamic behavior are main features of DTC.
However, classical DTC has several disadvantages, from which most important is
variable switching frequency.
Recently, from the classical DTC methods a new control techniques called Direct
Torque Control – Space Vector Modulated (DTCSVM) has been developed.
In this new method disadvantages of the classical DTC are eliminated. Basically, the
DTCSVM strategies are the methods, which operates with constant switching
frequency. These methods are the main subject of this thesis. The DTCSVM structures
1. Introduction
4
are based on the same fundamentals and analysis of the drive as classical DTC.
However, from the formal considerations these methods can also be viewed as stator
field oriented control (SFOC), as shown in Fig. 1.1.
Presented DTCSVM technique has also simple structure and provide dynamic
behavior comparable with classical DTC. However, DTCSVM method is characterized
by much better parameters in steady state operation.
Therefore, the following thesis can be formulated: “The most convenient industrial
control scheme for voltage source inverterfed induction motor drives is direct
torque control with space vector modulation DTCSVM”
In order to prove the above thesis the author used an analytical and simulation based
approach, as well as experimental verification on the laboratory setup with 5 kVA and
18 kVA IGBT inverters with 3 kW and 15 kW induction motors, respectively.
Moreover, the control algorithm DTCSVM has been introduced used in a serial
commercial product of Polish manufacture TWERD, Toruń.
In the author’s opinion the following parts of the thesis are his original achievements:
• elaboration and experimental verification of flux and torque controller design for
DTCSVM induction motor drives,
• development of a SABER  based simulation algorithm for control and
investigation voltage source inverterfed induction motors,
• construction and practical verification of the experimental setups with 5 kVA and
18 kVA IGBT inverters,
• bringing into production and testing of developed DTCSVM algorithm in Polish
industry.
The thesis consist of eight chapters. Chapter 1 is an introduction. In Chapter 2
mathematical model of IM, voltage source inverter construction and pulse width
modulation techniques are presented. Chapter 3 describes basic vector control method
of IM and gives analysis of advantages and disadvantages for all methods. In this
chapter basic principles of direct torque control are also presented. Those basis are
common for classical DTC, which is presented in Chapter 3 and for DTCSVM method.
Chapter 4 is devoted to analysis and synthesis of DTCSVM control technique. The
flux, torque and speed controllers design are presented. In Chapter 5 the estimations
1. Introduction
5
algorithms are described and discussed. In Chapter 6 implemented DTCSVM control
algorithm and used hardware setup are presented. In Chapter 7 experimental results are
presented and studied. Chapter 8 includes a conclusion. Description of the simulation
program and parameters of the equipment used are given in Appendixes.
2. Voltage Source Inverter Fed Induction Motor Drive
2.1. Introduction
In this chapter the model of induction motor will be presented. This mathematical
description is based on space vector notation. In next part description of the voltage
source inverter is given. The inverter is controlled in Pulse Width Modulation fashion.
In last part of this chapter review of the modulation technique is presented.
2.2. Mathematical Model of Induction Motor
When describing a threephase IM by a system of equations [66] the following
simplifying assumptions are made:
• the threephase motor is symmetrical,
• only the fundamental harmonic is considered, while the higher harmonics of the
spatial field distribution and of the magnetomotive force (MMF) in the air gap
are disregarded,
• the spatially distributed stator and rotor windings are replaced by a specially
formed, socalled concentrated coil,
• the effects of anisotropy, magnetic saturation, iron losses and eddy currents are
neglected,
• the coil resistances and reactance are taken to be constant,
• in many cases, especially when considering steady state, the current and voltages
are taken to be sinusoidal.
Taking into consideration the above stated assumptions the following equations of
the instantaneous stator phase voltage values can be written:
dt
dΨ
RIU
A
sAA
+=
(2.1a)
dt
dΨ
RIU
B
sBB
+=
(2.1b)
2.2. Mathematical Model of Induction Motor
7
dt
dΨ
RIU
C
sCC
+=
(2.1c)
The space vector method is generally used to describe the model of the induction
motor. The advantages of this method are as follows:
•
reduction of the number of dynamic equations,
•
possibility of analysis at any supply voltage waveform,
•
the equations can be represented in various rectangular coordinate systems.
A threephase symmetric system represented in a neutral coordinate system by phase
quantities, such as: voltages, currents or flux linkages, can be replaced by one resulting
space vector of, respectively, voltage, current and fluxlinkage. A space vector is
defined as:
( ) ( ) ( )
[ ]
tktktk
CBA
⋅+⋅+⋅=
2
aa1k
3
2
(2.2)
where:
(
) ( ) ( )
tktktk
CBA
,, – arbitrary phase quantities in a system of natural
coordinates, satisfying the condition
(
)
(
) ( )
0
=
+
+
tktktk
CBA
,
1, a, a
2
– complex unit vectors, with a phase shift
2/3 – normalization factor.
I
m
)(
2
tka
C
)(tak
B
)(tk
A
Re
k
k
2
3
A
B
C
a
2
a
1
Fig. 2.1. Construction of space vector according to the definition (2.2)
2. Voltage Source Inverter Fed Induction Motor Drive
8
An example of the space vector construction is shown in Fig. 2.1.
Using the space vector method the IM model equation can be written as:
dt
d
R
s
s
ss
Ψ
IU
+=
(2.3a)
dt
d
R
r
r
rr
Ψ
IU
+=
(2.3b)
rss
IIΨ
m
j
s
MeL
γ
+=
(2.4a)
srr
IIΨ
m
j
r
MeL
γ−
+=
(2.4b)
These are the voltage equations (2.3) and fluxcurrent equations (2.4).
To obtain a complete set of electric motor equations it is necessary to, firstly,
transform the equations (2.3, 2.4) into a common rotating coordinate system and
secondly bring the rotor value into the stator side and thirdly. These equations are
written in the coordinate system
K
rotating with the angular speed
K
Ω
.
KK
K
KsK
Ω
dt
d
R
s
s
ss
Ψ
Ψ
IU j
++=
(2.5a)
( )
KmbK
K
KrK
ΩpΩ
dt
d
R
r
r
rr
Ψj
Ψ
IU
−++=
(2.5b)
KMKsK
LL
rss
IIΨ
+=
(2.6a)
KMKrK
LL
srr
IIΨ
+=
(2.6b)
The equation of the dynamic rotor rotation can be expressed as:
[ ]
mLe
m
BΩMM
J
dt
dΩ
−−=
1
(2.7)
where:
e
M
– electromagnetic torque,
L
M
– load torque,
B
– viscous constant.
In further consideration the friction factor will be negated
(
)
0
=
B
.
The electromagnetic torque
e
M
can be expressed by the following formulas:
2.2. Mathematical Model of Induction Motor
9
(
)
rs
II
*
Im
2
M
s
be
L
m
pM −=
(2.8)
( )
ss
IΨ
*
Im
2
s
be
m
pM =
(2.9)
Taking into consideration the fact that in the cage motor the rotor voltage equals zero
and the electromagnetic torque equation (2.9) a complete set of equations for the cage
induction motor can be written as:
KK
K
KsK
Ω
dt
d
R
s
s
ss
Ψ
Ψ
IU j
++=
(2.10a)
( )
KmbK
K
Kr
ΩpΩ
dt
d
R
r
r
r
Ψ
Ψ
I
−++=
j0 (2.10b)
KMKsK
LL
rss
IIΨ
+=
(2.11a)
KMKrK
LL
srr
IIΨ
+=
(2.11b)
(
)
−=
L
s
b
m
M
m
p
Jdt
dΩ
ss
IΨ
*
Im
2
1
(2.12)
Equations (2.10), (2.11) and (2.12) are the basis of further consideration.
The applied space vector method as a mathematical tool for the analysis of the
electric machines a complete set equations can be represented in various systems of
coordinates. One of them is the stationary coordinates system (fixed to the stator)
β
α
−
=
楮⁴桩猠捡獥湧畬慲灥敤i ⁴桥敦敲敮捥牡=攠楳⁺敲漠
0
=
K
Ω
. The complex space
vector can be resolved into components
α
湤=
β
⸠
βα
ssK
UU
j
+=
s
U
(2.13a)
βα ssK
II
j
+=
s
I
,
βα rrK
II j+=
r
I
(2.13b)
βα ssK
ΨΨ
j
+=
s
Ψ
,
ββ rrK
ΨΨ j+=
r
Ψ
(2.13c)
In
β
α
−
coordinate system the motor model equation can be written as:
dt
dΨ
IRU
s
sss
α
αα
+=
(2.14a)
2. Voltage Source Inverter Fed Induction Motor Drive
10
dt
dΨ
IRU
s
sss
β
ββ
+=
(2.14b)
β
α
α
rmb
r
rr
ΨΩp
dt
dΨ
IR
++=
0
(2.14c)
α
β
β
rmb
r
rr
ΨΩp
dt
dΨ
IR −+=
0
(2.14d)
ααα
rMsss
ILILΨ
+=
(2.15a)
βββ
rMsss
ILILΨ
+=
(2.15b)
ααα
sMrrr
ILILΨ
+=
(2.15c)
βββ sMrrr
ILILΨ
+=
(2.15d)
( )
−−=
Lssss
s
b
m
MIΨIΨ
m
p
Jdt
dΩ
αββα
2
1
(2.16)
The relations described above by the motor equations can be represented as a block
diagram. There is not just one block diagram of an induction motor. The layout
Construction of a block diagram will depend on the chosen coordinate system and input
signals. For instance, if it is assumed in the stationary
β
α
−
潯牤i湡瑥n獴敭⁴桡琠瑨==
楮灵琠獩杮慬⁴漠瑨攠i→瑯爠楳⁴桥瑡瑯爠 癯汴慧攬⁴桥煵慴楯湳
㈮ㄴⴲ⸱㘩慮攠
瑲慮獦潲→e搠楮瑯㨠
αα
α
sss
s
IRU
dt
dΨ
−=
(2.17a)
ββ
β
sss
s
IRU
dt
dΨ
−=
(2.17b)
βα
α
rmbrr
r
ΨΩpIR
dt
dΨ
−−=
(2.17c)
αβ
β
rmbrr
r
ΨΩpIR
dt
dΨ
+−=
(2.17d)
ααα
σσ
r
rs
M
s
s
s
Ψ
LL
L
Ψ
L
I
−=
1
(2.18a)
βββ
σσ
r
rs
M
s
r
s
Ψ
LL
L
Ψ
L
I
−=
1
(2.18b)
2.2. Mathematical Model of Induction Motor
11
ααα
σσ
s
rs
M
r
r
r
Ψ
LL
L
Ψ
L
I −=
1
(2.18c)
βββ
σσ
s
rs
M
r
r
r
Ψ
LL
L
Ψ
L
I −=
1
(2.18d)
( )
−−=
Lssss
s
b
m
MIΨIΨ
m
p
Jdt
dΩ
αββα
2
1
(2.19)
These equations can be represented in the block diagram as shown in Fig. 2.2.
βs
Ψ
m
Ω
b
p
αs
I
αr
I
αs
Ψ
αs
U
s
R
s
R
∫
∫
∫
s
Lσ
1
rs
M
LL
L
σ
rs
M
LL
L
σ
r
Lσ
1
r
R
αr
Ψ
∫
r
R
r
Lσ
1
βr
I
∫
s
R
s
Lσ
1
rs
M
LL
L
σ
rs
M
LL
L
σ
βr
Ψ
βs
U
2
s
b
m
p
e
M
∫
L
M
βs
I
J
1
Fig. 2.2. Block diagram of an induction motor in the stationary coordinate system
β
α
−
This representation of the induction motor is not good for use to design a control
structure, because the output signals flux, torque and speed depend on both inputs. From
the control point of view this system is complicated. That is the reason why there are a
2. Voltage Source Inverter Fed Induction Motor Drive
12
few methods proposed to decouple the flux and torque control. It is achieved, for
example, by the orientation of the coordinate system to the rotor or stator flux vectors.
Both control systems are described further in Chapter 3.
The equations (2.17), (2.18), (2.19) and the block diagram presented in the Fig. 2.2
can be used to build a simulation model of the induction motor. It was used in a
simulation model, which is presented in Appendix A.2.
2.3.
Voltage Source Inverter (VSI)
The threephase two level VSI consists of six active switches. The basic topology of
the inverter is shown in Fig. 2.3. The converter consists of the three legs with IGBT
transistors, or (in the case of high power) GTO thyristors and freewheeling diodes. The
inverter is supplied by a voltage source composed of a diode rectifier with a C filter in
the dclink. The capacitor C is typically large enough to obtain adequately low voltage
source impedance for the alternating current component in the dclink.
D
1
D
2
D
3
D
4
D
5
D
6
C
2
dc
U
2
dc
U
C
0
S
B
+
S
B

S
A
+
S
A

S
C
+
S
C

T
1
T
2
T
5
T
6
T
3
T
4
DC side
U
AB
A B C
N
I
A
I
B
I
C
U
A
R
A
L
A
E
A
U
B
R
B
L
B
E
B
U
C
R
C
L
C
E
C
AC side
IM
PWM Converter
Fig. 2.3. Topology of the voltage source inverter
2.3. Voltage Source Inverter (VSI)
13
The voltage source inverter (Fig. 2.3) makes it possible to connect each of the three
motor phase coils to a positive or negative voltage of the dc link. Fig. 2.4 explains the
fabrication of the output voltage waves in squarewave, or sixstep, mode of operation.
The phase voltages are related to the dclink center point
0
(see Fig. 2.3).
a)
0
U
B0
ωt
2π
dc
U
2
1
dc
U
2
1
−
π
0
U
A0
ωt
2π
1 2 3 4 5 6
dc
U
2
1
dc
U
2
1
−
π
0
U
C0
ωt
2π
dc
U
2
1
dc
U
2
1
−
π
dc
U
3
2
dc
U
3
2
−
0
U
AB
ωt
2π
dc
U
3
1
dc
U
3
1
−
dc
U
dc
U−
π
dc
U
3
2
dc
U
3
2
−
0
U
A
ωt
2π
dc
U
3
1
dc
U
3
1
−
π
b)
c)
d)
e)
Fig. 2.4. The output voltage waveforms in sixstep mode
The phase voltage of an inverter fed motor (Fig. 2.4e) can be expressed by Fourier
series as [16, 66]:
( )
( )
( )
∑∑
∞
=
∞
=
==
11
sinsin
12
n
nm
n
dcA
tnUtn
n
UU ωω
π
(2.20)
where:
dc
U  dc supply voltage,
2. Voltage Source Inverter Fed Induction Motor Drive
14
( )
dcnm
U
n
U
π
2
=
 peak value of the nth harmonic,
n = 1+6k, k = 0, ±1, ±2,…
Derivation of the formula (2.20) is presented in Appendix A.1.
a) b)
c) d)
e) f)
g) h)
U
1
(100)
A B C
U
dc
U
2
(110)
A B C
U
dc
U
3
(010)
A B C
U
dc
U
4
(011)
A B C
U
dc
U
5
(001)
A B C
U
dc
U
6
(101)
A B C
U
dc
U
0
(000)
A B C
U
dc
U
7
(111)
A B C
U
dc
Fig. 2.5. Switching states for the voltage source inverter
From the equation (2.20) the fundamental peak value is given as:
( )
dcm
UU
π
2
1
= (2.21)
2.3. Voltage Source Inverter (VSI)
15
This value will be used to define the modulation index
M
used in pulse width
modulation (PWM) methods (see section 2.4).
For the sake of the inverter structure, each inverterleg can be represented as an ideal
switch. The equivalent inverter states are shown in Fig. 2.5.
There are eight possible positions of the switches in the inverter. These states
correspond to voltage vectors. Six of them (Fig. 2.5 af) are active vectors and the last
two (Fig. 2.5 gh) are zero vectors. The output voltage represented by space vectors is
defined as:
=
=
=
−
7,00
6...1
3
2
3)1(
v
veU
vj
dc
v
π
U
(2.22)
The output voltage vectors are shown in Fig. 2.6.
U
1
(100)
U
2
(110)U
3
(010)
U
4
(011)
U
5
(001) U
6
(101)
U
7
(111)
U
0
(000)
Im
Re
Fig. 2.6. Output voltage represented as space vectors
Any output voltage can in average be generated, of course limited by the value of the
dc voltage. In order to realize many different pulse width modulation methods are
proposed [13, 27, 30, 38, 46, 47, 51, 52, 105] in history. However, the general idea is
2. Voltage Source Inverter Fed Induction Motor Drive
16
based on a sequential switching of active and zero vectors. The modulation methods are
widely described in the next section.
Only one switch in an inverterleg (Fig. 2.3) can be turned on at a time, to avoid a
short circuit in the dclink. A delay time in the transistor switching signals must be
inserted. During this delay time, the deadtime T
D
transistors cease to conduct. Two
control signals S
A
+, S
A
 for transistors T
1
, T
2
with deadtime T
D
are presented in Fig.
2.7. The duration of deadtime depends of the used transistor. Most of them it takes 1
3
µ
s.
t
t
T
s
T
D
T
D
S
A

S
A
+
Fig. 2.7. Deadtime effect in a PWM inverter
Although, this delay time guarantees safe operation of the inverter, it causes a serious
distortion in the output voltage. It results in a momentary loss of control, where the
output voltage deviates from the reference voltage. Since this is repeated for every
switching operation, it has significant influence on the control of the inverter. This is
known as the deadtime effect. This is important in applications like a sensorless direct
torque control of induction motor. These applications require feedback signals like:
stator flux, torque and mechanical speed. Typically the inverter output voltage is needed
to calculate it. Unfortunately, the output voltage is very difficult to measure and it
requires additional hardware. Because of that for calculation of feedback signals the
reference voltage is used. However, the relation between the output voltage and the
reference voltage is nonlinear due to the deadtime effect [8]. It is especially important
2.4. Pulse Width Modulation (PWM)
17
for the low speed range when voltage is very low. The deadtime may also cause
instability in the induction motor [52].
Therefore, for correct operation of control algorithm proper compensation of dead
time is required. Many approaches are proposed to compensate of this effect [2, 3, 8, 29,
54, 64, 76].
The deadtime compensation is directly connected with estimation of inverter output
voltage. Therefore, compensation algorithm, which is used in final control structure of
the inverter is presented in Chapter 5.
2.4.
Pulse Width Modulation (PWM)
2.4.1.
Introduction
In the voltage source inverter conversion of dc power to threephase ac power is
performed in the switched mode (Fig. 2.3). This mode consists in power semiconductors
switches are controlled in an onoff fashion. The actual power flow in each motor phase
is controlled by the duty cycle of the respective switches. To obtain a suitable duty
cycle for each switches technique pulse width modulation is used. Many different
modulation methods were proposed and development of it is still in progress [13, 27,
30, 38, 46, 47, 51, 52, 105].
The modulation method is an important part of the control structure. It should
provide features like:
•
wide range of linear operation,
•
low content of higher harmonics in voltage and current,
•
low frequency harmonics,
•
operation in overmodulation,
•
reduction of common mode voltage,
•
minimal number of switching to decrease switching losses in the power
components.
The development of modulation methods may improve converter parameters. In the
carrier based PWM methods the Zero Sequence Signals (ZSS) [46] are added to extend
2. Voltage Source Inverter Fed Induction Motor Drive
18
the linear operation range (see section 2.4.2). The carrier based modulation methods
with ZSS correspond to space vector modulation. It will be widely presented in section
2.4.4.
All PWM methods have specific features. However, there is not just one PWM
method which satisfies all requirements in the whole operating region. Therefore, in the
literature are proposed modulators, which contain from several modulation methods.
For example, adaptive space vector modulation [79], which provides the following
features:
•
full control range including overmodulation and sixstep mode, achieved by the
use of three different modulation algorithms,
•
reduction of switching losses thanks to an instantaneous tracking peak value of
the phase current.
The content of the higher harmonics voltage (current) and electromagnetic
interference generated in the inverter fed drive depends on the modulation technique.
Therefore, PWM methods are investigated from this point of view. To reduce these
disadvantages several methods have been proposed. One of these methods is random
modulation (RPWM). The classical carrier based method or space vector modulation
method are named deterministic (DEPWM), because these methods work with constant
switching frequency. In opposite to the deterministic methods, the random modulation
methods work with variable frequency, or with randomly changed switching sequence
(see section 2.4.6).
2.4.2.
Carrier Based PWM
The most widely used method of pulse width modulation are carrier based. This
method is also known as the sinusoidal (SPWM), triangulation, subharmonic, or
suboscillation method [16, 52]. Sinusoidal modulation is based on triangular carrier
signal as shown in Fig. 2.8. In this method three reference signals U
Ac
, U
Bc
, U
Cc
are
compared with triangular carrier signal U
t
, which is common to all three phases. In this
way the logical signals S
A
, S
B
, S
C
are generated, which define the switching instants of
the power transistors as is shown in Fig. 2.9.
2.4. Pulse Width Modulation (PWM)
19
U
dc
A
B C
N
Carrier
U
Ac
U
Bc
U
Cc
U
t
S
A
S
B
S
C
Fig. 2.8. Block scheme of carrier based sinusoidal PWM
U
t
U
Ac
U
Bc
U
Cc
0
1
0
1
0
1
0
S
B
S
C
0
dc
U32
dc
U31
dc
U32−
dc
U31−
0
dc
U
dc
U−
0
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02
2
dc
U−
S
A
A
U
AB
U
2
dc
U
Fig. 2.9. Basic waveforms of carrier based sinusoidal PWM
2. Voltage Source Inverter Fed Induction Motor Drive
20
The modulation index m is defined as:
)(
tm
m
U
U
m = (2.23)
where:
m
U  peak value of the modulating wave,
)(tm
U
 peak value of the carrier wave.
The modulation index m can be varied between 0 and 1 to give a linear relation
between the reference and output wave. At m=1, the maximum value of fundamental
peak voltage is
2
dc
U
, which is 78.55% of the peak voltage of the square wave (2.21).
The maximum value in the linear range can be increased to 90.7% of that of the
square wave by inserting the appropriate value of a triple harmonics to the modulating
wave. It is shown in Fig. 2.10, which presents the whole range characteristic of the
modulation methods [67]. This characteristic include also the overmodulation (OM)
region, which is widely described in section 2.4.5.
1
0.785 0.907 1
1.155 3.24
M
m
[ ]
%100
2
⋅
dc
A
U
U
π
78.5
90.7
100
SPWM
SVPWM
or SPWM with ZSS
OM
Six step
operation
Fig. 2.10. Output voltage of VSI versus modulation index for different PWM techniques
2.4. Pulse Width Modulation (PWM)
21
If the neutral point
N
on the AC side of the inverter is not connected with the DC
side midpoint
0
(Fig. 2.3), phase currents depend only on the voltage difference
between phases. Therefore, it is possible to insert an additional Zero Sequence Signal
(ZSS) of the 3th harmonic frequency, which does not produce phase voltage distortion
and without affecting load currents. A block scheme of the modulator based on the
additional ZSS is shown in Fig. 2.11 [46].
N
U
dc
A
B C
S
A
S
B
S
C
Carrier
U
t
Calculation
of ZSS
U
Ac
U
Bc
U
Cc
U
Ac
*
U
Bc
*
U
Cc
*
Fig. 2.11. Generalized PWM with additional Zero Sequence Signal (ZSS)
The type of the modulation method depends on the ZSS waveform. The most popular
PWM methods are shown in Fig. 2.12 where unity the triangular carrier waveform gain
is assumed and the signals are normalized to
2
dc
U
. Therefore,
2
dc
U
±
saturation limits
correspond to ±1. In Fig. 2.12 only phase “A” modulation waveform is shown as the
modulation signals of phase “B” and “C” are identical waveforms with 120º phase shift.
The modulated methods illustrated in Fig. 2.12 can be separated into two groups:
continuous and discontinuous. In the continuous PWM (CPWM) methods, the
modulation waveform are always within the triangular peak boundaries and in every
carrier cycle triangle and modulation waveform intersections. Therefore, on and off
switchings occur. In the discontinuous PWM (DPWM) methods a modulation
waveform of a phase has a segment which is clamped to the positive or negative DC
2. Voltage Source Inverter Fed Induction Motor Drive
22
bus. In this segments some power converter switches do not switch. Discontinuous
modulation methods give lower (average 33%) switching losses. The modulation
method with triangular shape of ZSS with 1/4 peak value corresponds to space vector
modulation (SVPWM) with symmetrical placement of the zero vectors in a sampling
period. It will be widely describe in section 2.4.4. In Fig. 2.12 is also shown sinusoidal
PWM (SPWM) and third harmonic PWM (THIPWM) with sinusoidal ZSS with 1/4
peak value corresponding to a minimum of output current harmonics [63].
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
0.018
0.02
1
0.8
0.6
0.4
0.2
0
0.2
0.4
0.6
0.8
1
Time
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
0.018
0.02
1
0.8
0.6
0.4
0.2
0
0.2
0.4
0.6
0.8
1
Time
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
0.018
0.02
1
0.8
0.6
0.4
0.2
0
0.2
0.4
0.6
0.8
1
Time
SPWM THIPWM SVPWM
U
A
U
N0
U
A0
U
A
=U
A0
U
N0
U
N0
U
A
U
A0
a) b) c)
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
0.018
0.02
1
0.8
0.6
0.4
0.2
0
0.2
0.4
0.6
0.8
1
Time
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
0.018
0.02
1
0.8
0.6
0.4
0.2
0
0.2
0.4
0.6
0.8
1
Time
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
0.018
0.02
1
0.8
0.6
0.4
0.2
0
0.2
0.4
0.6
0.8
1
Time
U
N0
U
N0
U
N0
U
A0
U
A0
U
A0
U
A
U
A
U
A
DPWM1 DPWM2 DPWM3
d) e) f)
Fig. 2.12. Waveforms for PWM with added Zero Sequence Signal a) SPWM, b)THIPWM, c) SVPWM,
d) DPWM1, e) DPWM2, f) DPWM3
2.4.3.
Space Vector Modulation (SVM)
The space vector modulation techniques differ from the carrier based in that way,
there are no separate modulators used for each of the three phases. Instead of them, the
reference voltages are given by space voltage vector and the output voltages of the
inverter are considered as space vectors (2.22). There are eight possible output voltage
vectors, six active vectors
U
1

U
6
, and two zero vectors
U
0
,
U
7
(Fig. 2.13). The
reference voltage vector is realized by the sequential switching of active and zero
vectors.
In the Fig. 2.13 there are shown reference voltage vector
U
c
and eight voltage
vectors, which corresponds to the possible states of inverter. The six active vectors
2.4. Pulse Width Modulation (PWM)
23
divide a plane for the six sectors I  VI. In the each sector the reference voltage vector
U
c
is obtained by switching on, for a proper time, two adjacent vectors. Presented in
Fig. 2.13 the reference vector
U
c
can be implemented by the switching vectors of
U
1
,
U
2
and zero vectors
U
0
,
U
7
.
I
II
III
IV
V
VI
U
7
(111)
U
0
(000)
U
1
(100)
U
2
(110)U
3
(010)
U
4
(011)
U
5
(001) U
6
(101)
α
U
c
(t
1
/T
s
)U
1
(
t
2
/
T
s
)
U
2
Fig. 2.13. Principle of the space vector modulation
The reference voltage vector
U
c
is sampled with the fixed clock frequency
ss
Tf 1
=
Ⱐ
慮搠湥硴慭p汥搠癡汵攠
(
)
s
T
c
U
is used for calculation of times t
1
, t
2
, t
0
and t
7
. The
signal flow in space vector modulator is shown in Fig. 2.14.
U
dc
Sector
selection
S
A
S
B
S
C
Calculation
t
1
t
2
t
0
t
7
f
s
U
c
A
B C
N
U
c
(T
s
)
Fig. 2.14. Block scheme of the space vector modulator
2. Voltage Source Inverter Fed Induction Motor Drive
24
The times t
1
and t
2
are obtained from simple trigonometrical relationships and can be
expressed in the following equations:
( )
απ
π
−= 3sin
32
1
s
MTt
(2.24a)
( )
α
π
sin
32
2
s
MTt = (2.24b)
Where M is a modulation index, which for the space vector modulation is defined as:
dc
c
stepsix
c
U
U
U
U
M
π
2
)(1
==
−
(2.25)
where:
c
U  vector magnitude, or phase peak value,
)(1
stepsix
U
−
 fundamental peak value
(
)
π
dc
U2 of the squarephase voltage
wave.
The modulation index M varies from 0 to 1 at the squarewave output. The length of
the
U
c
vector, which is possible to realize in the whole range of
α
猠敱畡氠瑯l
dc
U
3
3
.
This is a radius of the circle inscribed of the hexagon in Fig. 2.13. At this condition the
modulation index is equal:
907.0
2
3
3
==
dc
dc
U
U
M
π
(2.26)
This means that 90.7% of the fundamental at the square wave can be obtained. It
extends the linear range of modulation in relation to 78.55% in the sinusoidal
modulation techniques (Fig. 2.10).
After calculation of t
1
and t
2
from equations (2.24) the residual sampling time is
reserved for zero vectors
U
0
and
U
7
.
70217,0
)( ttttTt
s
+=+−=
(2.27)
2.4. Pulse Width Modulation (PWM)
25
The equations for t
1
and t
2
are identically for all space vector modulation methods.
The only difference between the other type of SVM is the placement of zero vectors at
the sampling time.
The basic SVM method is the modulation method with symmetrical spacing zero
vectors (SVPWM). In this method times t
0
and t
7
are equal:
( )
2
2170
ttTtt
s
−
−=
=
(2.28)
For the first sector switching sequence can be written as follows:
U
0
→
U
1
→
U
2
→
U
7
→
U
2
→
U
1
→
U
0
(2.29)
This vector switching sequence in the SVPWM method is shown in Fig. 2.15a. In
this case zero vectors are placed in the beginning and in the end of period
U
0
, and in the
center of the period
U
7
. In one sampling period all three phases are switched. To realize
the reference vector can also be used an other switching sequence, for example:
U
0
→
U
1
→
U
2
→
U
1
→
U
0
(2.30)
or
U
1
→
U
2
→
U
7
→
U
2
→
U
1
(2.31)
These sequences are shown in Fig. 2.15b and 2.15c respectively. In these cases only
two phases switch in one sampling time, and only one zero vector is used
U
0
(Fig.
2.15b) or
U
7
(Fig. 2.15c). This type of modulation is called discontinuous pulse width
modulation (DPWM).
1
0 1
0 0 1
1
01
001
1 1 1
1 1
1
111
11
1
T
s
U
7
t
0
t
2
/2
S
A
S
B
S
C
U
1
U
2
U
2
U
1
t
1
/2
t
1
/2 t
2
/2
0
0 0
0
00
0 1 1
0 1
0
01
0
T
s
U
0
U
1
U
2
U
1
U
0
t
2
t
1
/2 t
0
/2
t
0
/2 t
1
/2
S
A
S
B
S
C
0
0 0
0 0 0
0
00
000
1 1 1
1 1
1
111
11
1
T
s
U
0
U
1
U
2
U
7
U
7
U
2
U
1
U
0
t
0
/4 t
1
/2 t
2
/2 t
0
/4
t
0
/4 t
1
/2 t
2
/2 t
0
/4
S
A
S
B
S
C
a) b) c)
Fig. 2.15. Space vectors in the sampling period a) SVPWM, b), c) DPWM
The idea of discontinuous modulation is based on the assumption that one phase is
clamped by 60
°
to lower or upper of the dc bus voltage. It gives only one zero state per
sampling period (Fig. 2.15b, c). The discontinuous modulation provides 33% reduction
2. Voltage Source Inverter Fed Induction Motor Drive
26
of the effective switching frequency and switching losses. The discontinuous space
vector modulation techniques, like all the space vector methods, correspond to the
carrier based modulation method. It will be widely described in the next section.
DPWM4
U
7
(111)
U
0
(000)
U
1
(100)
U
2
(110)
U
3
(010)
U
4
(011)
U
5
(001)
U
6
(101)
t
7
= 0
t
7
= 0
t
7
= 0
t
0
= 0
t
0
= 0
t
0
= 0
DPWM1
U
7
(111)
U
0
(000)
U
1
(100)
U
2
(110)
U
3
(010)
U
4
(011)
U
5
(001)
U
6
(101)
t
7
= 0
t
7
= 0
t
7
= 0
t
0
= 0
t
0
= 0
t
0
= 0
t
0
= 0
t
7
= 0
t
0
= 0
t
7
= 0
t
7
= 0
t
0
= 0
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
0.018
0.02
1
0.8
0.6
0.4
0.2
0
0.2
0.4
0.6
0.8
1
Time
U
N0
U
A0
U
A
DPWM2
U
7
(111)
U
0
(000)
U
1
(100)
U
2
(110)
U
3
(010)
U
4
(011)
U
5
(001)
U
6
(101)
t
7
= 0
t
7
= 0t
7
= 0
t
0
= 0 t
0
= 0
t
0
= 0
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
0.018
0.02
1
0.8
0.6
0.4
0.2
0
0.2
0.4
0.6
0.8
1
Time
U
N0
U
A0
U
A
DPWM3
U
7
(111)
U
0
(000)
U
1
(100)
U
2
(110)
U
3
(010)
U
4
(011)
U
5
(001)
U
6
(101)
t
7
= 0
t
7
= 0
t
7
= 0
t
0
= 0
t
0
= 0
t
0
= 0
t
0
= 0
t
7
= 0
t
0
= 0
t
7
= 0
t
7
= 0
t
0
= 0
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
0.018
0.02
1
0.8
0.6
0.4
0.2
0
0.2
0.4
0.6
0.8
1
Time
U
N0
U
A0
U
A
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
0.018
0.02
1
0.8
0.6
0.4
0.2
0
0.2
0.4
0.6
0.8
1
Time
U
A
U
A0
U
N0
a)
b)
c)
d)
Fig. 2.16. The discontinuous space vector modulation
2.4. Pulse Width Modulation (PWM)
27
In the Fig. 2.16 there are shown several different kinds of space vector discontinues
modulation. It can be seen that the type of method depends on the moved do not switch
sectors. These sectors are adequately moved on 0
°
, 30
°
, 60
°
, 90
°
and denoted as
DPWM1, DPWM2, DPWM3 and DPWM4. Fig. 2.16 also shows voltage waveforms for
each methods. For the carrier based methods with ZSS these waveforms are identical
(Fig. 2.12).
From the type of modulation it depends also harmonic content, what is presented in
Fig. 2.17 for the SVPWM and DPWM1 methods.
Fig. 2.17. The output line to line voltage harmonics content a) SVPWM, b) DPWM 1
In Fig. 2.17 harmonics of output line to line voltage are shown. The voltage
frequency domain representation is composed of the series discrete harmonics
components. These are clustered about multiplies of the switching frequency. In this
case the switching frequency was 5 kHz. Spectrum for every modulation methods is
different. In Fig. 2.17 the differences between SVPWM and DPWM1 modulation
method can be seen. However, characteristic feature for all methods, which work with
constant switching frequency is clustered higher harmonics round multiplies of the
switching frequency. These type of modulation methods are named deterministic PWM
(DEPWM). The modulation method influence also for current distortion, torque ripple
and acoustic noise emitted from the motor. Modulation techniques are still being
improved for reduction of these disadvantages. One of the proposed methods is a
random PWM (RPWM) (see section 2.4.6).
2. Voltage Source Inverter Fed Induction Motor Drive
28
2.4.4.
Relation Between Carrier Based and Space Vector Modulation
All the carrier based methods have equivalent to the space vector modulation
methods. The type of carrier based method depends on the added ZSS, as shown in
section 2.4.2, and type of the space vector modulation depending on the time of zero
vectors t
0
and t
7
.
A comparison of carrier based method with SVM is shown in Fig 2.18. There is
shown a carrier based modulation with triangular shape of ZSS with 1/4 peak value.
This method corresponds to the space vector modulation (SVPWM) with symmetrical
placement of zero vectors in sampling period. In Fig. 2.18b is presented discontinuous
method DPWM1 for carrier based and for SVM techniques.
In the carrier based methods three reference signals U
Ac
*
, U
Bc
*
, U
Cc
*
are compared
with triangular carrier signal U
t
, and in this way logical signals S
A
, S
B
, S
C
are generated.
In the space vector modulation duration time of active (t
1
, t
2
) and zero (t
0
, t
7
) vectors are
calculated, and from these times switching signals S
A
, S
B
, S
C
are obtained. The gate
pulses generated by both methods are identical.
The carrier based PWM methods are simple to implement in hardware. Through the
compare reference signals with triangular carrier signal it receives gate pulses.
However, a PWM inverter is generally controlled by a microprocessor/controller
nowadays. Thanks to the representation of command voltages as space vector, a
microprocessor using suitable equations can calculate duration time and realize
switching sequence easily.
It is possible to implement all carrier based modulation methods using the space
vector technique. The active vector times t
1
and t
2
equations are identically for all space
vector modulation methods. But every method demand suitable equation for the zero
vectors t
0
and t
7
.
The eight voltage vectors
U
0

U
7
correspond to the possible states of the inverter
(Fig. 2.13). Each of these states can be composed by a different equivalent electrical
circuit. In Fig 2.19 the circuit for the vector
U
1
is presented.
2.4. Pulse Width Modulation (PWM)
29
0
0 0
0 0 0
0
00
000
0 1 1
0 1
0
011
01
0
T
s
U
0
U
1
U
2
U
1
U
0
t
2
t
1
/2 t
0
/2
t
0
/2 t
1
/2
S
A
S
B
S
C
U
Ac
*
U
Bc
*
U
Cc
*
S
A
S
B
S
C
b)
Carrir based PWMSpace vector PWM
U
Ac
*
0
0 0
0 0 0
0
00
000
1 1 1
1 1
1
111
11
1
T
s
U
0
U
1
U
2
U
7
U
7
U
2
U
1
U
0
t
0
/4 t
1
/2 t
2
/2 t
0
/4
t
0
/4 t
1
/2 t
2
/2 t
0
/4
U
Bc
*
U
Cc
*
S
A
S
B
S
C
S
A
S
B
S
C
a)
Carrir based PWMSpace vector PWM
Fig. 2.18. Comparison of carrier based PWM with space vector PWM a) SVPWM, b) DPWM1
U
A
U
B
U
C
U
N0
A
B C
N
0
2
dc
U
2
dc
U
U
B
0
=
U
C
0
U
A
0
Fig. 2.19. Equivalent circuit of VSI for the U
1
vector
2. Voltage Source Inverter Fed Induction Motor Drive
30
Taking into consideration the electrical circuit in Fig. 2.19 the voltage distribution
can be obtained. The voltages can be written as:
dcA
UU
3
2
=
;
dcB
UU
3
1
−=
;
dcC
UU
3
1
−=
(2.32)
dcA0
UU
2
1
=
;
dcB0
UU
2
1
−=
;
dcC0
UU
2
1
−=
(2.33)
dcANA0N0
UUUU
6
1
−=−=
(2.34)
This analysis may be repeated for all vectors provided to obtain voltages presented in
Table 2.1.
Table 2.1. The voltages for the eight converter output vectors
A0
U
B0
U
A
U
B
U
C
U
C0
U
N0
U
0
U
1
U
2
U
3
U
4
U
5
U
6
U
7
U
dc
U
2
1
dc
U
2
1
−
dc
U
3
2
dc
U
3
1
−
dc
U
2
1
−
dc
U
3
1
−
dc
U
6
1
−
dc
U
2
1
dc
U
3
2
−
dc
U
3
1
dc
U
2
1
−
dc
U
6
1
dc
U
2
1
dc
U
2
1
−
dc
U
3
2
dc
U
3
1
−
dc
U
2
1
−
dc
U
3
1
−
dc
U
6
1
−
dc
U
2
1
dc
U
3
1
dc
U
2
1
dc
U
2
1
−
dc
U
3
2
−
dc
U
3
1
dc
U
2
1
dc
U
3
1
dc
U
6
1
dc
U
2
1
−
dc
U
2
1
−
dc
U
3
2
dc
U
3
1
−
dc
U
2
1
dc
U
3
1
−
dc
U
6
1
dc
U
2
1
dc
U
2
1
−
dc
U
3
1
dc
U
3
1
dc
U
2
1
dc
U
3
2
−
dc
U
6
1
dc
U
2
1
−
dc
U
2
1
−
0 0
dc
U
2
1
−
0
dc
U
2
1
−
dc
U
2
1
dc
U
2
1
0 0
dc
U
2
1
0
dc
U
2
1
The average value for sampling time of U
NO
voltage can be written as follows:
++−−=
7210
dc
s
N0
tttt
U
T
U
3
1
3
1
2
1
for the sectors I, III, V (2.35)
and
++−−=
7120
dc
s
N0
tttt
U
T
U
3
1
3
1
2
1
for the sectors II, IV, VI (2.36)
2.4. Pulse Width Modulation (PWM)
31
From the above equations and taking into consideration equations (2.24) and (2.27)
the zero vectors times for different kinds of modulation can be calculated.
Relations between carrier based and SVM methods are presented in Table 2.2. This
table presents also the zero vector (t
0
, t
7
) times equations for the most significant
modulation methods.
Table 2.2. Relation between carrier based and SVM methods
−= α
π
cos
4
1
2
M
T
t
s
0
( )
+−= αα
π
sin3cos
2
1
2
M
T
t
s
0
Calculation of t
0
and t
7
for sectors I, III, V
for sectors II, IV, VI
210s7
tttTt
−
−
−
=
Waveform of the
ZSS (Fig. 2.13)
( )
0=
N0
U
Modulation
method
SPWM
Sinusoidal with
1/4 amplitude
no signal
THIPWM
−−= αα
π
3cos
4
1
cos
4
1
2
M
T
t
s
0
−+−= ααα
π
3cos
2
1
sin3cos
2
1
2
M
T
t
s
0
for sectors II, IV, VI
210s7
tttTt
−
−
−
=
for sectors I, III, V
Triangular with
1/4 amplitude
Discontinuous
SVPWM
DPWM1
(
)
2
21s70
ttTtt
−
−
=
=
0
=
0
t
21s7
ttTt
−
−
=
0
=
7
t
21s0
ttTt
−
−
=
when
( )
12
63
+<≤ nn
π
α
π
when
( )
( )
1
3
12
6
+<≤+ nn
π
α
π
5,4,3,2,1,0
=
n
Waveforms of the ZSS presented in Table 2.2 are shown in Fig. 2.12.
2.4.5.
Overmodulation (OM)
At the end of the linear range (Fig. 2.10) the inverter output voltage is 90.7% of the
maximum output peak voltage in sixstep mode (see equation 2.21). The nonlinear
2. Voltage Source Inverter Fed Induction Motor Drive
32
range between this point and sixstep mode is called overmodulation. This part of the
modulation techniques is not so important in vector controlled drives methods for the
sake of big distortion current and torque. For example, the overmodulation can be
applied in drives working in open loop control mode to increase the value of inverter
output voltage.
The overmodulation has been widely discussed in the literature [16, 33, 55, 75, 89].
Some of methods are proposed as extensions of the carrier based modulation and others
as extensions of space vector modulation. In the carrier based methods overmodulation
algorithm is realized by increasing reference voltage beyond the amplitude of the
triangular carrier signal. In this case some switching cycles are omitted and each phase
is clamped to one of the dc busses.
The overmodulation region for space vector modulation is shown in Fig. 2.20. The
maximum length of vector
U
c
possible to realization in whole range of
α
慮杬攠楳煵慬a
dc
U
3
3
. It is a radius of the circle inscribed of the hexagon. This value corresponds to
the modulation index equal to 0.907 (see equation 2.26). To realize higher values a
voltage overmodulation algorithm has to be applied. At the end of the overmodulation
region is a sixstep mode (at M = 1).
U
7
(111)
U
0
(000)
U
1
(100)
U
2
(110)U
3
(010)
U
4
(011)
U
5
(001) U
6
(101)
α
U
c
(t
1
/T
s
)U
1
(
t
2
/
T
s
)
U
2
Overmodulation range
0.907 < M < 1
Sixstep mode
M = 1
Linear range
M ≤ 0.907
Fig. 2.20. Definition of the overmodulation range
2.4. Pulse Width Modulation (PWM)
33
If the value of the reference voltage beyond maximal value in the linear range vector
U
c
can not be realized for whole range of
α
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m潤楦i敳湬礠瑨攠癯汴慧e⁶散瑯±=灬楴畤eⰠ 楮潤攠䥉潴栠瑨攠慭灬楴畤攠慮搠慮pl攠潦e
瑨攠癯汴慧攠癥捴潲⸠
佶敲O潤畬→瑩潮潤攠䤠楳桯睮渠䙩朮′⸲ㄮ→
U
0
(000)
U
7
(111)
α
U
c
U
c
*
U
1
(100)
U
2
(110)U
3
(010)
U
4
(011)
U
5
(001) U
6
(101)
θ
=
䙩朮g㈮㈱⸠2ve±m潤畬慴i潮潤→⁉=
=
䥮⁴桩猠I潤攠癯汴慧攠癥捴潲→
U
c
crosses the hexagon boundary at two points in each
sector. There is a loss of fundamental voltage in the region where reference vector
exceeds the hexagon boundary. To compensate for this loss, the reference vector
amplitude is increased in the region where the reference vector is in hexagon boundary.
A modified reference voltage trajectory proceeds partly on the hexagon and partly on
the circle. This trajectory is shown in Fig. 2.21.
2. Voltage Source Inverter Fed Induction Motor Drive
34
In the hexagon trajectory part only active vectors are used. The duration of these
vectors t
1
and t
2
are obtained from trigonometrical relationships and can be expressed in
the following equations:
αα
αα
sincos3
sincos3
+
−
=
s1
Tt (2.37a)
1s2
tTt
−=
(2.37b)
0
==
70
tt (2.37c)
The output voltage waveform is given approximately by linear segments for the
hexagon trajectory and sinusoidal segments for the circular trajectory. Boundary of the
segments is determined by a crossover angle
θ
which depends on the reference voltage
value. As known from Fig. 2.21 the upper limit in mode I is when
θ
= 0
°
. Then the
voltage trajectory is fully on the hexagon. The fundamental peak value generated in this
way voltage is 95% of the peak voltage of the square wave [75]. It gives modulation
index M = 0.952.
For the modulation index higher then 0.952 the overmodulation mode II is applied.
The overmodulation mode II is shown in Fig. 2.22. In this mode not only the reference
vector amplitude is modified but also an angle. The reference angle from
α
to
α
*
is
changed.
U
0
(000)
U
7
(111)
α
U
c
U
c
*
U
1
(100)
U
2
(110)U
3
(010)
U
4
(011)
U
5
(001) U
6
(101)
h
α
h
α
∗
α
Fig. 2.22. Overmodulation mode II where both amplitude and angle is changed
2.4. Pulse Width Modulation (PWM)
35
The trajectory of
U
c
*
is maintained on the hexagon which defines amplitude of the
reference voltage vector. The angle is calculated from the following equations:
≤≤−
−<<
−
−
≤≤
=
∗
333
3for
66
00
πααππ
απαα
π
απ
αα
αα
α
h
hh
h
h
h
(2.38)
where: α
h
– holdangle.
This angle uniquely controls the fundamental voltage. It is a nonlinear function of the
modulation index [16, 55].
In Fig. 2.22 is shown the reference vector trajectory generated for the first sector.
This trajectory is obtained in three steps. First part, if angle α is less than the respective
value of α
h
, the algorithm holds the vector
U
c
*
at the vertex (
U
1
). Next part is for α from
α
h
to
h
α
π
−3. In this angle range the reference vector moves along the hexagon. In the
last range, from
h
α
π
−3 to
h
α
, the vector
U
c
*
is held until the next vertex (
U
2
).
The overmodulation mode II works up to the sixstep mode for α
h
equal zero. The
sixstep mode characterized by selection of the switching vector for onesixth of the
fundamental period. In this case the maximum possible inverter output voltage is
generated.
2.4.6.
Random Modulation Techniques
The pulse width modulation technique is important for drive performance in respect
to voltage and current harmonics, torque ripple, acoustic noise emitted from an
induction motor and also electromagnetic interference (EMI). Different approaches
were used in PWM techniques for reduction of these disadvantages. One of the
proposed methods is random pulse width modulation (RPWM) [5, 7, 11, 14, 61, 68,
104].
Previously presented modulation methods were named deterministic pulse width
modulation (DEPWM), because of constant sampling and switching frequency and all
2. Voltage Source Inverter Fed Induction Motor Drive
36
cycles the switching sequence is deterministic. In RPWM methods the switching
frequency or the switching sequence change randomly.
One of the proposed random modulation techniques is a method with randomly
varied lengths of coincident switching and sampling time of the modulator. This method
was named RPWM 1. The sampling and switching cycles in DEPWM with RPWM 1 is
comparable shown in Fig. 2.23. The reference voltage vectors
U
c
, which are calculated
in one sampling time
T
s
and realized in the next switching time
T
sw
are shown. In drive
systems the controller mostly operates in synchronism with modulator and in RPWM 1
arises problems in the control system, when it works with variable sampling frequency.
An additional control algorithm with variable sampling frequency is difficult tin a
digital implementation.
1 2 3
...
n1 n
...
sampling cycles
switching cycles
1 2 3
...
n1 n
...
)1(
c
U
)2(
c
U
)3(
c
U
)(K
c
U
)1( −n
c
U
)1( +n
c
U
)(n
c
U
sampling cycles
switching cycles
...
1 2
...
n1 n
...
1 2
...
n1 n
3
3
)1(
c
U
)2(
c
U
)3(
c
U
)(K
c
U
)1( −n
c
U
)1( +n
c
U
)(n
c
U
a)
b)
sws
TT
=
獷s
θθ =
Fig. 2.23. Sampling and switching cycles a) DEPWM, b) RPWM 1
For elimination of these disadvantages random modulation techniques were
proposed, which operate with a fixed switching and sampling frequency. These methods
randomly change switching sequence in the interval. Three of these methods are shown
in Fig. 2.24 [6].
First of them (Fig. 2.24a) is random leadlag modulation (RLL). In this method pulse
position is either commencing at the beginning of the switching interval (leadingedge
2.4. Pulse Width Modulation (PWM)
37
modulation) or its tailing edge is aligned with the end of the interval (laggingedge
modulation). A random number generator controls the choice between leading and
legging edge modulation.
In Fig. 2.24b is shown a random center pulse displacement (RCD) method. In this
technique pulses are generated identically as in the SVPWM method (Fig. 2.15), but
common pulse center is displaced by the amount
s
T
α
from the middle of the period.
The parameter
α
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U
0
(000) and
U
7
(111) is
randomized in the switching cycles.
S
A
S
B
S
C
T
s
T
s
T
s
T
s
s
T
α
s
θ
α
s
θ
α
s
θ
α
S
A
S
B
S
C
T
s
T
s
T
s
T
s
Lead Lag Lag Lead
S
A
S
B
S
C
T
s
T
s
T
s
T
s
a)
b)
c)
Fig. 2.24. Different fixed switching random modulation schemes a) Random leadleg modulation (RLL),
b) Random center displacement (RCD), c) Random zero vector distribution (RZD)
2. Voltage Source Inverter Fed Induction Motor Drive
38
The main disadvantage of the RPWM 1 method (Fig. 2.23b) is variable switching
frequency. For elimination of this disadvantage RPWM 2 [119] was proposed, which
operates with fixed sampling frequency and variable switching frequency. The principle
of this method is shown in Fig. 2.25.
1 2 3
...
n1 n
...
...
1 2 3
...
n1 n
sampling cycles
switching cycles
)1(
c
U
)2(
c
U
)3(
c
U
)(K
c
U
)1( −n
c
U
)1( +n
c
U
)(n
c
U
sw
T
s
T
t∆
Fig. 2.25. Sampling and switching cycles in RPWM 2 technique
In this method the start of each switching cycles is delayed with respect to that of the
coincident sampling cycle by a random varied time interval
t
∆
. It is given as:
s
rTt
=∆ (2.39)
where
r
denotes a random number between 0 and 1. Time interval
t
∆
is limited for
the sake of minimum switching time of inverter.
Fig. 2.26. The output line to line voltage harmonics content a) RPWM 1, b) RPWM 2
Corresponding spectra for the RPWM 1 and RPWM 2 techniques are shown in Fig.
2.26a and 2.26b respectively. It can be seen that the harmonic clusters typical for the
determination modulation (compared to Fig. 2.17) are practically eliminated by the
2.5. Summary
39
random modulation techniques. Simulation result presented in both figures (Fig. 2.17
and Fig. 2.26) was done at the same conditions: sampling frequency 5 kHz, output
frequency 50 Hz.
2.5.
Summary
In this chapter mathematical description of IM based on complex space vectors was
presented. The complete equations set is the basis of further consideration of control
and estimation methods.
The structure of two levels voltage source inverter was presented. The main features
and voltage forming methods were described. For the sake of deadtime and voltage
drop on the semiconductor devices the inverter has nonlinear characteristic. Therefore,
in control scheme compensation algorithms are needed.
The inverter is controlled by pulse width modulation (PWM) technique. The
modulation methods are divided into two groups: triangular carrier based and space
vector modulation. Between those two groups there are simple relations. All the carrier
based methods have equivalent to the space vector modulation methods. The type of
carrier based method depends on the added ZSS and type of the space vector
modulation depends on the placement of zero vectors in the sampling period. Presented
modulation methods will be used in the final drive.
This chapter contains compete review of the modulation techniques, including some
random modulation methods. Those methods have very interesting advantages and can
be implemented in special application of IM drives. Currently they have not been
implemented in a presented serially produced drive. However, it will be offered as an
option in a near future. Some experimental results for the implemented modulation
methods are shown in Chapter 7.
3. Vector Control Methods of Induction Motor
3.1. Introduction
In this chapter review of the most significant IM vector control method is presented.
According to the classification presented in Chapter 1. The theoretical basis and short
characteristic for all methods are given. The direct torque control (DTC) method creates
a base for further analyze of DTCSVM algorithms. Therefore, DTC is more detailed
discussed (see section 3.4).
3.2. Field Oriented Control (FOC)
The principle of the field oriented control (FOC) is based on an analogy to the
separately excited dc motor. In this motor flux and torque can be controlled
independently. The control algorithm can be implemented using simple regulators, e.g.
PIregulators.
In induction motor independent control of flux and torque is possible in the case of
coordinate system is connected with rotor flux vector. A coordinate system
qd −
is
rotating with the angular speed equal to rotor flux vector angular speed
srK
ΩΩ =,
which is defined as follows:
dt
dγ
Ω
sr
sr
=
(3.1)
The rotating coordinate system
qd
−
is shown in Fig. 3.1.
The voltage, current and flux complex space vector can be resolved into components
d
and
q
.
sqsdK
UU j
+
=
s
U (3.2a)
sqsdK
II j+=
s
I
,
rqrdK
II j+=
r
I
(3.2b)
sqsdK
ΨΨ j
+
=
s
Ψ,
rrdK
ΨΨ ==
r
Ψ (3.2c)
3.2. Field Oriented Control (FOC)
41
α
β
r
Ψ
d
q
s
I
βs
I
αs
I
sd
I
sq
I
sr
γ
δ
獲
Ω
Fig. 3.1. Vector diagram of induction motor in stationary
β
α
−
=搠rotati湧n
qd −
coordinates
In
qd
− coordinate system the induction motor model equations (2.102.12) can be
written as follows:
sqsr
sd
sdssd
ΨΩ
dt
dΨ
IRU −+=
(3.3a)
sdsr
sq
sqssq
ΨΩ
dt
dΨ
IRU ++=
(3.3b)
dt
dΨ
IR
r
rdr
+=
0 (3.3c)
( )
mbsrrrqr
ΩpΩΨIR
−
+
=
0 (3.3d)
rdMsdssd
ILILΨ +=
(3.4a)
rqMsqssq
ILILΨ +=
(3.4b)
sdMrdrr
ILILΨ +=
(3.4c)
sqMrqr
ILIL +
=
0 (3.4d)
−=
Lsqr
r
M
s
b
m
MIΨ
L
L
m
p
Jdt
dΩ
2
1
(3.5)
The equations 3.3c and 3.4c can be easy transformed to:
3. Vector Control Methods of Induction Motor
42
r
r
r
sd
r
rMr
Ψ
L
R
I
L
RL
dt
dΨ
−=
(3.6)
The motor torque can by expressed by rotor flux magnitude
r
Ψ
and stator current
component
sq
I
as follows:
sqr
r
Ms
be
IΨ
L
Lm
pM
2
=
(3.7)
Equations (3.6) and (3.7) are used to construct a block diagram of the induction
motor in qd − coordinate system, which is presented in Fig. 3.2.
r
Ψ
m
Ω
∫
2
s
b
m
p
e
M
∫
L
M
γ
1
r
rM
L
RL
r
r
L
R
r
M
L
L
sd
I
sq
I
e
M
Fig. 3.2. Block diagram of induction motor in
qd
−
coordinate system
The main feature of the field oriented control (FOC) method is the coordinate
transformation. The current vector is measured in stationary coordinate
β
α
−.
Therefore, current components
α
s
I,
β
s
I must be transformed to the rotating system
qd −
. Similarly, the reference stator voltage vector components
cs
U
α
,
cs
U
β
, must be
transformed from the system
qd −
to
β
α
−
⸠周敳攠瑲慮獦潲oa瑩潮猠牥煵楲敳潴潲t
晬畸湧汥f
sr
γ
. Depending on calculations of this angle two different kind of field
oriented control methods maybe considered. Those are Direct Field Oriented Control
(DFOC) and Indirect Field Oriented Control (IFOC) methods.
3.2. Field Oriented Control (FOC)
43
For DFOC an estimator or observer calculates the rotor flux angle
sr
γ
. Inputs to the
estimator or observer are stator voltages and currents. An example of the DFOC system
is presented in Fig. 3.3.
PI
SVM
S
A
S
B
S
C
sqc
I
Flux
Estimator
U
dc
αs
U
βs
U
s
I
cs
U
α
cs
U
β
αs
I
βs
I
sr
γ
牣
Ψ
ec
M
sd
I
sq
I
sdc
I
PI
β
α
−
qd
−
β
α
−
qd
−
3
2
Motor
rcM
r
sb
ΨL
L
mp
12
M
L
1
Voltage
Calculation
Fig. 3.3. Block diagram of the Direct Field Oriented Control (DFOC)
For the IFOC rotor flux angle
sr
γ
is obtained from reference
sdc
I,
sqc
I currents. The
angular speed of the rotor flux vector speed can be calculated as follows:
mbslrs
ΩpΩΩ += (3.8)
where
sl
Ω is a slip angular speed. It can be calculated from (3.3d) and (3.4d).
sqc
r
r
sdc
sl
I
L
R
I
Ω
1
= (3.9)
In Fig. 3.4 a block diagram of the IFOC is shown.
3. Vector Control Methods of Induction Motor
44
PI
SVM
S
A
S
B
S
C
sqc
I
U
dc
s
I
cs
U
α
cs
U
β
αs
I
βs
I
sr
γ
牣
Ψ
ec
M
sd
I
sq
I
sdc
I
PI
β
α
−
qd
−
β
α
−
qd
−
Motor
rcM
r
sb
ΨL
L
mp
12
M
L
1
3
2
m
Ω
sr
Ω
sl
Ω
sdcr
r
IL
R 1
∫
b
p
Fig. 3.4. Block diagram of the Indirect Field Oriented Control (IFOC)
In both presented examples reference currents in rotating coordinate system
sdc
I,
sqc
I
are calculated from the reference flux and torque values. Taking into consideration the
equations describing IM in field oriented coordinate system (3.6) and (3.7) at steady
state the formulas for the reference currents can be written as follows:
r
M
sdc
Ψ
L
I
1
=
(3.10)
ec
rcM
r
sb
sqc
M
ΨL
L
mp
I
12
= (3.11)
The property of the FOC methods can be summarized as follows:
•
the method is based on the analogy to control of a DC motor,
•
FOC method does not guarantee an exact decoupling of the torque and flux
control in dynamic and steady state operation,
•
relationship between regulated value and control variables is linear only for
constant rotor flux amplitude,
3.3. Feedback Linearization Control (FLC)
45
•
full information about motor state variable and load torque is required (the
method is very sensitive to rotor time constant),
•
current controllers are required,
•
coordinate transformations are required,
•
a PWM algorithm is required (it guarantees constant switching frequency),
•
in the DFOC rotor flux estimator is required,
•
in the IFOC mechanical speed is required,
•
the stator currents are sinusoidal except of high frequency switching harmonics.
3.3.
Feedback Linearization Control (FLC)
The transformation of the induction motor equations in the field coordinates has a
good physical basis because it corresponds to the decoupled torque production in a
separately excited DC motor. However, from the theoretical point of view, other types
of coordinates can be selected to achieve decoupling and linearization of the induction
motor equations.
In [28] it is shown that a nonlinear dynamic model of IM can be considered as
equivalent to two thirdorder decoupled linear systems. In [70] a controller based on a
multiscalar motor model has been proposed. The new state variables have been chosen.
In result the motor speed is fully decoupled from the rotor flux. In [82] the authors
proposed a nonlinear transformation of the motor states variables, so that in the new
coordinates, the speed and rotor flux amplitude are decoupled by feedback. Others
proposed also modified methods based on Feedback Linearization Control like in [93,
94].
In the example given new quantities for control of rotor flux magnitude and
mechanical speed were chosen [93]. For this purpose the induction motor equations
(2.102.12) can be written in the following form:
ββαα
gg)x(x
ss
UUf
+
+
=
&
(3.12)
where:
3. Vector Control Methods of Induction Motor
46
−−
−+−
−+
+−
+−−
=
J
M
IΨIΨ
IΨΨΩp
IΨΩpΨ
ILΨΨΩp
ILΨΩpΨ
f
L
srsr
srrmb
srmbr
sMrrmb
sMrmbr
)(
)(
αββα
ββα
αβα
ββα
αβα
µ
γαββ
γβαβ
αα
αα
x
(3.13)
T
g
= 00
1
00,,,,
s
Lσ
α
(3.14)
T
g
= 0
1
000,,,,
s
Lσ
β
(3.15)
[ ]
T
x
mssrr
ΩIIΨΨ,,,,
βαβα
=
(3.16)
and
r
r
L
R
=
α
(3.17)
rs
M
LL
L
σ
β
= (3.18)
2
22
rs
Mrrs
LL
LRLR
σ
γ
+
=
(3.19)
J
Lm
p
Ms
b
2
=
µ
(3.20)
Because
βα rrm
ΨΨΩ,, are not dependent on
βα ss
UU, it is possible to chose variable
dependent on x:
222
1
)x(
rrr
ΨΨΨ =+=
βα
φ
(3.21)
m
Ω=)x(
2
φ
†⠳⸲㈩(
䥦琠楳獳=me搠瑨慴d
)•(
1
φ
Ⱐ
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