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
2.1. Introduction
2.2. Mathematical Model of Induction Motor
2.3. Voltage Source Inverter (VSI)
2.4. Pulse Width Modulation (PWM)
2.4.1. Introduction
2.4.2. Carrier Based PWM
2.4.3. Space Vector Modulation (SVM)
2.4.4. Relation Between Carrier Based and Space Vector Modulation
2.4.5. Overmodulation (OM)
2.4.6. Random Modulation Techniques
2.5. Summary
6
6
6
12
17
17
18
22
28
31
35
39
3. Vector Control Methods of Induction Motor
3.1. Introduction
3.2. Field Oriented Control (FOC)
3.3. Feedback Linearization Control (FLC)
3.4. Direct Flux and Torque Control (DTC)
3.4.1. Basics of Direct Flux and Torque Control
3.4.2. Classical Direct Torque Control (DTC) – Circular Flux Path
3.4.3. Direct Self Control (DSC) – Hexagon Flux Path
3.5. Summary
40
40
40
45
49
49
53
61
64
4. Direct Flux and Torque Control with Space Vector Modulation (DTC-SVM)
4.1. Introduction
4.2. Structures of DTC-SVM – Review
4.2.1. DTC-SVM Scheme with Closed – Loop Flux Control
4.2.2. DTC-SVM Scheme with Closed – Loop Torque Control
4.2.3. DTC-SVM Scheme with Close – Loop Torque and Flux Control
Operating in Polar Coordinates
4.2.4. DTC-SVM Scheme with Close – Loop Torque and Flux Control
in Stator Flux Coordinates
4.2.5. Conclusions from Review of the DTC-SVM Structures
4.3. Analysis and Controller Design for DTC-SVM Method with
Close – Loop Torque and Flux Control in Stator Flux Coordinates
4.3.1. Torque and Flux Controllers Design – Symmetry Criterion Method
4.3.2. Torque and Flux Controllers Design – Root Locus Method
4.3.3. Summary of Flux and Torque Controllers Design
4.4. Speed Controller Design
4.5. Summary
66
66
66
66
68
69
70
71
71
75
78
88
94
98
Contents
5. Estimation in Induction Motor Drives
5.1. Introduction
5.2. Estimation of Inverter Output Voltage
5.3. Stator Flux Vector Estimators
5.4. Torque Estimation
5.5. Rotor Speed Estimation
5.6. Summary
99
99
100
104
110
110
112
6. Configuration of the Developed IM Drive Based on DTC-SVM
6.1. Introduction
6.2. Block Scheme of Implemented Control System
6.3. Laboratory Setup Based on DS1103
6.4. Drive Based on TMS320LF2406
113
113
113
115
118
7. Experimental Results
7.1. Introduction
7.2. Pulse Width Modulation
7.3. Flux and Torque Controllers
7.4. DTC-SVM Control System
122
122
122
125
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 air-conditioning (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 dclink and a Pulse Width Modulated (PWM) voltage inverter is the most applied used in
industry (see section 2.3).
The high-performance 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,
•
uni-polar voltage PWM,
•
low flux and torque ripple,
•
robustness for parameter variation,
•
four-quadrant operation,
1. Introduction
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
U/f=const.
Volt/Hertz
Vector based
controller
i s = f (ωr )
Feedback
Linearization
Field Oriented
Stator Current
Rotor Flux
Oriented
Direct
(Blaschke)
Stator Flux
Oriented
Indirect
(Hasse)
Direct Torque
Control
Direct Torque
Space - Vector
Modulation
Open Loop
&o&
NFO (Jonsson)
Circle flux
trajectory
(Takahashi)
Passivity Based
Control
Hexagon flux
trajectory
(Takahashi)
Closed Loop
Flux & Torque
Control
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 high-performance. 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.
2
1. Introduction
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 Euler-Lagrange 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 on-off 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 (DTC-SVM) has been developed.
In this new method disadvantages of the classical DTC are eliminated. Basically, the
DTC-SVM strategies are the methods, which operates with constant switching
frequency. These methods are the main subject of this thesis. The DTC-SVM structures
3
1. Introduction
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 DTC-SVM technique has also simple structure and provide dynamic
behavior comparable with classical DTC. However, DTC-SVM 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 inverter-fed induction motor drives is direct
torque control with space vector modulation DTC-SVM”
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 DTC-SVM 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
DTC-SVM induction motor drives,
•
development of a SABER - based simulation algorithm for control and
investigation voltage source inverter-fed 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 DTC-SVM 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 DTC-SVM method.
Chapter 4 is devoted to analysis and synthesis of DTC-SVM control technique. The
flux, torque and speed controllers design are presented. In Chapter 5 the estimations
4
1. Introduction
algorithms are described and discussed. In Chapter 6 implemented DTC-SVM 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.
5
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 three-phase IM by a system of equations [66] the following
simplifying assumptions are made:
•
the three-phase 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, so-called 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:
U A = I A Rs +
dΨ A
dt
(2.1a)
U B = I B Rs +
dΨ B
dt
(2.1b)
2.2. Mathematical Model of Induction Motor
U C = I C Rs +
dΨ C
dt
(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 three-phase 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 flux-linkage. A space vector is
defined as:
k=
[
]
2
1 ⋅ k A (t ) + a ⋅ k B (t ) + a 2 ⋅ k C (t )
3
(2.2)
where: k A (t ), k B (t ), k C (t ) – arbitrary phase quantities in a system of natural
coordinates, satisfying the condition k A (t ) + k B (t ) + k C (t ) = 0 ,
1, a, a2 – complex unit vectors, with a phase shift
2/3 – normalization factor.
Im
3
k
2
B
a 2 kC (t )
a
k
ak B (t )
Re
1
k A (t )
A
a2
C
Fig. 2.1. Construction of space vector according to the definition (2.2)
7
2. Voltage Source Inverter Fed Induction Motor Drive
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:
U s = I s Rs +
dΨ s
dt
(2.3a)
U r = I r Rr +
dΨ r
dt
(2.3b)
Ψ s = Ls I s + Me jγ m I r
(2.4a)
Ψ r = Lr I r + Me − jγ m I s
(2.4b)
These are the voltage equations (2.3) and flux-current 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 ΩK .
U sK = Rs I sK +
dΨ sK
+ j ΩK Ψ sK
dt
(2.5a)
U rK = Rr I rK +
dΨ rK
+ j(ΩK − pb Ωm )Ψ rK
dt
(2.5b)
Ψ sK = Ls I sK + LM I rK
(2.6a)
Ψ rK = Lr I rK + LM I sK
(2.6b)
The equation of the dynamic rotor rotation can be expressed as:
dΩm 1
= [M e − M L − BΩm ]
dt
J
(2.7)
where: M e – electromagnetic torque,
M L – load torque,
B – viscous constant.
In further consideration the friction factor will be negated (B = 0 ) .
The electromagnetic torque M e can be expressed by the following formulas:
8
2.2. Mathematical Model of Induction Motor
M e = − pb
M e = pb
(
ms
LM Im I *s I r
2
(
ms
Im Ψ *s I s
2
)
(2.8)
)
(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:
U sK = Rs I sK +
0 = Rr I rK +
dΨ sK
+ j ΩK Ψ sK
dt
dΨ rK
+ j(ΩK − pb Ωm )Ψ rK
dt
(2.10a)
(2.10b)
Ψ sK = Ls I sK + LM I rK
(2.11a)
Ψ rK = Lr I rK + LM I sK
(2.11b)
(
)
dΩm 1  ms

Im Ψ *s I s − M L 
=  pb
dt
J
2

(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) α − β
in this case angular speed of the reference frame is zero ΩK = 0 . The complex space
vector can be resolved into components α and β .
U sK = U sα + jU sβ
I sK = I sα + j I sβ ,
(2.13a)
I rK = I rα + j I rβ
Ψ sK = Ψ sα + jΨ sβ , Ψ rK = Ψ rβ + jΨ rβ
(2.13b)
(2.13c)
In α − β coordinate system the motor model equation can be written as:
U sα = Rs I sα +
dΨ sα
dt
(2.14a)
9
2. Voltage Source Inverter Fed Induction Motor Drive
U sβ = Rs I sβ +
0 = Rr I rα +
0 = Rr I rβ +
dΨ sβ
dt
dΨ rα
+ pb ΩmΨ rβ
dt
dΨ rβ
dt
− pb ΩmΨ rα
(2.14b)
(2.14c)
(2.14d)
Ψ sα = Ls I sα + LM I rα
(2.15a)
Ψ sβ = Ls I sβ + LM I rβ
(2.15b)
Ψ rα = Lr I rα + LM I sα
(2.15c)
Ψ rβ = Lr I rβ + LM I sβ
(2.15d)
dΩm 1  ms
(Ψ sα I sβ − Ψ sβ I sα ) − M L 
=  pb
dt
J
2

(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 lay-out
Construction of a block diagram will depend on the chosen coordinate system and input
signals. For instance, if it is assumed in the stationary α − β coordinate system that the
input signal to the motor is the stator voltage, the equations (2.14-2.16) can be
transformed into:
dΨ sα
= U sα − Rs I sα
dt
dΨ sβ
= U sβ − Rs I sβ
(2.17b)
dΨ rα
= − Rr I rα − pb ΩmΨ rβ
dt
(2.17c)
dt
dΨ rβ
= − Rr I rβ + pb ΩmΨ rα
(2.17d)
I sα =
L
1
Ψ sα − M Ψ rα
σLs
σLs Lr
(2.18a)
I sβ =
1
L
Ψ sβ − M Ψ rβ
σLr
σLs Lr
(2.18b)
dt
10
(2.17a)
2.2. Mathematical Model of Induction Motor
I rα =
1
L
Ψ rα − M Ψ sα
σLr
σLs Lr
(2.18c)
I rβ =
1
L
Ψ rβ − M Ψ sβ
σLr
σLs Lr
(2.18d)
dΩm 1  ms
(Ψ sα I sβ − Ψ sβ I sα ) − M L 
=  pb
dt
J 2

(2.19)
These equations can be represented in the block diagram as shown in Fig. 2.2.
ML
Rs
U sα
∫
Ψ sα
LM
LM
σLs Lr
Rr
∫
I sα
1
σ Ls
I rα
σLs Lr
pb
ms M e
2
1
J
∫
Ωm
1
σ Lr
Ψ rα
pb
∫
Rr
Ψ rβ
I rβ
1
σ Lr
LM
LM
σLs Lr
U sβ
∫
Ψ sβ
σLs Lr
1
σ Ls
I sβ
Rs
Fig. 2.2. Block diagram of an induction motor in the stationary coordinate system
α −β
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
11
2. Voltage Source Inverter Fed Induction Motor Drive
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 three-phase 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 free-wheeling diodes. The
inverter is supplied by a voltage source composed of a diode rectifier with a C filter in
the dc-link. The capacitor C is typically large enough to obtain adequately low voltage
source impedance for the alternating current component in the dc-link.
DC side
PWM Converter
T1
U dc
2
S A+
C
T3
D1
SB +
D2
SB -
T5
D3
S C+
D4
S C-
D5
0
T2
U dc
2
S A-
C
T4
IA
UAB
A
RA
LA
EA
T6
IB
IC
B
RC
UB
LB
EB
LC
UC
EC
N
Fig. 2.3. Topology of the voltage source inverter
12
AC side
C
RB
UA
D6
IM
2.3. Voltage Source Inverter (VSI)
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 square-wave, or six-step, mode of operation.
The phase voltages are related to the dc-link center point 0 (see Fig. 2.3).
a)
UA0
1
2
3
4
5
6
1
U dc
2
0
π
2π
ωt
π
2π
ωt
π
2π
ωt
π
2π
ωt
π
2π
ωt
1
− U dc
2
b)
UB0
1
U dc
2
0
1
− U dc
2
c)
UC0
1
U dc
2
0
1
− U dc
2
d)
UAB
U dc
2
U dc
3
1
U dc
3
0
1
− U dc
3
2
− U dc
3
− U dc
e)
UA
2
U dc
3
1
U dc
3
0
1
− U dc
3
2
− U dc
3
Fig. 2.4. The output voltage waveforms in six-step mode
The phase voltage of an inverter fed motor (Fig. 2.4e) can be expressed by Fourier
series as [16, 66]:
UA =
∞
∞
1
U dc ∑ sin (nωt ) = U m (n ) ∑ sin (nωt )
π
n =1 n
n =1
2
(2.20)
where:
U dc - dc supply voltage,
13
2. Voltage Source Inverter Fed Induction Motor Drive
U m (n ) =
2
U dc - peak value of the n-th harmonic,
nπ
n = 1+6k, k = 0, ±1, ±2,…
Derivation of the formula (2.20) is presented in Appendix A.1.
U1 (100)
a)
Udc
Udc
A
B
C
U3 (010)
c)
B
C
A
B
C
A
B
C
A
B
C
Udc
A
B
C
U5 (001)
U6 (101)
f)
Udc
Udc
A
B
C
U0 (000)
g)
A
U4 (011)
d)
Udc
e)
U2 (110)
b)
U7 (111)
h)
Udc
Udc
A
B
C
Fig. 2.5. Switching states for the voltage source inverter
From the equation (2.20) the fundamental peak value is given as:
U m (1) =
14
2
π
U dc
(2.21)
2.3. Voltage Source Inverter (VSI)
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 inverter-leg 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 a-f) are active vectors and the last
two (Fig. 2.5 g-h) are zero vectors. The output voltage represented by space vectors is
defined as:
2
j ( v −1)π 3
 3 U dc e
Uv = 
0

v = 1...6
(2.22)
v = 0,7
The output voltage vectors are shown in Fig. 2.6.
Im
U3 (010)
U4 (011)
U2 (110)
U1 (100)
U0 (000)
Re
U7 (111)
U5 (001)
U6 (101)
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
15
2. Voltage Source Inverter Fed Induction Motor Drive
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 inverter-leg (Fig. 2.3) can be turned on at a time, to avoid a
short circuit in the dc-link. A delay time in the transistor switching signals must be
inserted. During this delay time, the dead-time TD transistors cease to conduct. Two
control signals SA+, SA- for transistors T1, T2 with dead-time TD are presented in Fig.
2.7. The duration of dead-time depends of the used transistor. Most of them it takes 13µs.
SA+
t
SATD
TD
t
Ts
Fig. 2.7. Dead-time 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 dead-time 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 dead-time effect [8]. It is especially important
16
2.4. Pulse Width Modulation (PWM)
for the low speed range when voltage is very low. The dead-time may also cause
instability in the induction motor [52].
Therefore, for correct operation of control algorithm proper compensation of deadtime is required. Many approaches are proposed to compensate of this effect [2, 3, 8, 29,
54, 64, 76].
The dead-time 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 three-phase ac power is
performed in the switched mode (Fig. 2.3). This mode consists in power semiconductors
switches are controlled in an on-off 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
17
2. Voltage Source Inverter Fed Induction Motor Drive
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 six-step 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 UAc, UBc, UCc are
compared with triangular carrier signal Ut, which is common to all three phases. In this
way the logical signals SA, SB, SC are generated, which define the switching instants of
the power transistors as is shown in Fig. 2.9.
18
2.4. Pulse Width Modulation (PWM)
Udc
UAc
SA
UBc
SB
UCc
SC
A
B
C
Ut
Carrier
N
Fig. 2.8. Block scheme of carrier based sinusoidal PWM
U dc 2
Ut
UAc UBc
0
− U dc 2
UCc
1
SA
0
1
SB
SC
UA
0
1
0
2 3Udc
1 3Udc0
0
−1 3Udc
− 2 3Udc
U dc
U AB
0
− U dc
0
0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02
Fig. 2.9. Basic waveforms of carrier based sinusoidal PWM
19
2. Voltage Source Inverter Fed Induction Motor Drive
The modulation index m is defined as:
m=
Um
U m(t )
(2.23)
where:
U m - peak value of the modulating wave,
U m (t ) - 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
U dc
, which is 78.55% of the peak voltage of the square wave (2.21).
2
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.
π UA
2 U dc
⋅ 100 [%]
100
90.7
SVPWM
or SPWM with ZSS
Six step
operation
OM
78.5
SPWM
m
1 1.155
0.785 0.907
3.24
M
1
Fig. 2.10. Output voltage of VSI versus modulation index for different PWM techniques
20
2.4. Pulse Width Modulation (PWM)
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 3-th 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].
Udc
UAc
UAc*
SA
UBc
UBc*
SB
UCc
UCc*
SC
A
B
C
Ut
Calculation
of ZSS
Carrier
N
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
U
U dc
. Therefore, ± dc saturation limits
2
2
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
21
2. Voltage Source Inverter Fed Induction Motor Drive
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].
a)
b)
SPWM
1
1
UA=UA0
0.8
0.6
0.4
0.4
0.2
0.2
0
UN0
SVPWM
0.8
UA0
0.6
0.2
0
0
-0.2
-0.2
UN0
-0.4
-0.6
-0.6
-0.6
-0.8
-0.8
-0.8
-1
-1
0.002 0.004 0.006 0.008
d)
0.01
Time
0.012 0.014 0.016 0.018
0.02
1
0.002 0.004 0.006 0.008
UA0
0.012 0.014 0.016 0.018
0.02
UA
UA
0.4
0
0
0
-0.2
-0.2
-0.2
-0.4
-0.4
-0.6
-0.8
-0.8
-1
0.2
0.002 0.004 0.006 0.008
0.01
Time
0.012 0.014 0.016 0.018
0.02
UA
-0.6
UN0
UN0
-0.8
-1
0
0.02
UA0
0.8
0.2
-0.6
0.012 0.014 0.016 0.018
0.6
0.4
UN0
0.01
Time
DPWM3
0.2
-0.4
0.002 0.004 0.006 0.008
1
0.6
0.4
0
f)
UA0
0.8
0.6
0.01
Time
DPWM2
1
0.8
UN0
-0.4
-1
0
e)
DPWM1
UA0
0.4
-0.4
0
UA
1
UA
0.8
0.6
-0.2
c)
THIPWM
-1
0
0.002 0.004 0.006 0.008
0.01
Time
0.012 0.014 0.016 0.018
0.02
0
0.002 0.004 0.006 0.008
0.01
Time
0.012 0.014 0.016 0.018
0.02
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 U1 - U6, and two zero vectors U0, U7 (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 Uc and eight voltage
vectors, which corresponds to the possible states of inverter. The six active vectors
22
2.4. Pulse Width Modulation (PWM)
divide a plane for the six sectors I - VI. In the each sector the reference voltage vector
Uc is obtained by switching on, for a proper time, two adjacent vectors. Presented in
Fig. 2.13 the reference vector Uc can be implemented by the switching vectors of U1, U2
and zero vectors U0, U7.
U3 (010)
II
U2 (110)
I
(t
2 /T
s )U
2
III
U4 (011)
Uc
U0 (000)
α
U7 (111)
(t1 /Ts )U1
IV
U1 (100)
VI
U5 (001)
V
U6 (101)
Fig. 2.13. Principle of the space vector modulation
The reference voltage vector Uc is sampled with the fixed clock frequency f s = 1 Ts ,
and next a sampled value U c (Ts ) is used for calculation of times t1, t2, t0 and t7. The
signal flow in space vector modulator is shown in Fig. 2.14.
Udc
fs
Uc
Uc(Ts)
SA
Sector
selection
t1 t2
SB
SC
t0 t7
A
B
C
Calculation
N
Fig. 2.14. Block scheme of the space vector modulator
23
2. Voltage Source Inverter Fed Induction Motor Drive
The times t1 and t2 are obtained from simple trigonometrical relationships and can be
expressed in the following equations:
t1 =
2 3
t2 =
π
2 3
π
MTs sin (π 3 − α )
(2.24a)
MTs sin (α )
(2.24b)
Where M is a modulation index, which for the space vector modulation is defined as:
M =
Uc
U 1( six − step )
=
Uc
2
U dc
(2.25)
π
where:
U c - vector magnitude, or phase peak value,
U 1( six − step )
- fundamental peak value (2U dc π ) of the square-phase voltage
wave.
The modulation index M varies from 0 to 1 at the square-wave output. The length of
the Uc vector, which is possible to realize in the whole range of α is equal to
3
U dc .
3
This is a radius of the circle inscribed of the hexagon in Fig. 2.13. At this condition the
modulation index is equal:
3
U dc
M = 3
= 0.907
2
U dc
(2.26)
π
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 t1 and t2 from equations (2.24) the residual sampling time is
reserved for zero vectors U0 and U7.
t 0,7 = Ts − (t1 + t 2 ) = t 0 + t 7
24
(2.27)
2.4. Pulse Width Modulation (PWM)
The equations for t1 and t2 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 t0 and t7 are equal:
t 0 = t 7 = (Ts − t1 − t 2 ) 2
(2.28)
For the first sector switching sequence can be written as follows:
U0 → U1 → U2 → U7 → U2 → U1 → U0
(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 U0, and in the
center of the period U7. In one sampling period all three phases are switched. To realize
the reference vector can also be used an other switching sequence, for example:
U0 → U1 → U2 → U1 → U0
(2.30)
U1 → U2 → U7 → U2 → U1
(2.31)
or
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 U0 (Fig.
2.15b) or U7 (Fig. 2.15c). This type of modulation is called discontinuous pulse width
modulation (DPWM).
a)
b)
c)
SA
0
1
1
1
1
1
1
0
SA
0
1
1
1
0
SA
1
1
1
1
1
1
1
1
SB
0
0
1
1
1
1
0
0
SB
0
0
1
0
0
SB
0
1
1
1
1
1
1
0
SC
0
0
0
1
1
0
0
0
SC
0
0
0
0
0
SC
0
0
1
1
1
1
0
0
t0/4
t1/2
t2/2
t0/4
t0/4
t1/2
t2/2
t0/4
t0/2
t1/2
t2
t1/2
t0/2
t1/2
t2/2
t2/2
t1/2
U2
U1
Ts
U0
U1
U2
U7
Ts
U7
U2
U1
U0
U0
U1
U2
t0
Ts
U1
U0
U1
U2
U7
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
25
2. Voltage Source Inverter Fed Induction Motor Drive
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.
a)
DPWM1
U3 (010)
t7= 0
t0 = 0
t0 = 0
1
U2 (110)
UA0
0.8
t7 = 0
0.6
UA
0.4
t7= 0
U4 (011)
t0= 0
0.2
U1 (100)
U0 (000)
0
U7 (111)
-0.2
t0= 0
t7= 0
UN0
-0.4
-0.6
t0 = 0
t7 = 0
t0 = 0
t7= 0
-0.8
U6 (101)
-1
U5 (001)
0
b)
0.002 0.004 0.006 0.008
0.01
Time
0.012 0.014 0.016 0.018
0.02
DPWM2
U3 (010)
1
U2 (110)
UA0
0.8
t7= 0
0.6
t0= 0
UA
0.4
t0 = 0
0.2
U0 (000)
U4 (011)
U1 (100)
0
U7 (111)
-0.2
-0.4
t7 = 0
t7= 0
-0.6
UN0
-0.8
t0= 0
U6 (101)
-1
U5 (001)
0
c)
0.002 0.004 0.006 0.008
0.01
Time
0.012 0.014 0.016 0.018
0.02
DPWM3
U3 (010)
t7 = 0
t0= 0
t7= 0
U2 (110)
t0 = 0
t0= 0
U4 (011)
1
UA0
0.8
0.6
0.4
t7= 0
0.2
U1 (100)
U0 (000)
UA
0
U7 (111)
-0.2
t7= 0
t0= 0
-0.4
-0.6
t0 = 0
t7= 0
t7 = 0
t0= 0
UN0
-0.8
U6 (101)
-1
U5 (001)
0
d)
0.002 0.004 0.006 0.008
0.01
Time
0.012 0.014 0.016 0.018
0.02
DPWM4
U3 (010)
1
U2 (110)
t7= 0
UA
0.8
t0= 0
UA0
0.6
0.4
t7 = 0
0.2
U4 (011)
U0 (000)
U1 (100)
U7 (111)
0
-0.2
t0 = 0
t0= 0
t7= 0
U5 (001)
-0.4
UN0
-0.6
-0.8
U6 (101)
-1
0
0.002 0.004 0.006 0.008
0.01
Time
0.012 0.014 0.016 0.018
Fig. 2.16. The discontinuous space vector modulation
26
0.02
2.4. Pulse Width Modulation (PWM)
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).
27
2. Voltage Source Inverter Fed Induction Motor Drive
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 t0 and t7.
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 UAc*, UBc*, UCc* are compared
with triangular carrier signal Ut, and in this way logical signals SA, SB, SC are generated.
In the space vector modulation duration time of active (t1, t2) and zero (t0, t7) vectors are
calculated, and from these times switching signals SA, SB, SC 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 t1 and t2 equations are identically for all space
vector modulation methods. But every method demand suitable equation for the zero
vectors t0 and t7.
The eight voltage vectors U0 - U7 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 U1 is presented.
28
2.4. Pulse Width Modulation (PWM)
SA
SB
SC
Carrir based PWM
b)
Carrir based PWM
a)
SA
SB
SC
UAc*
UBc*
UAc*
UCc*
UBc*
0
1
1
1
1
1
1
0
SB
0
0
1
1
1
1
0
0
SC
0
0
0
1
1
0
0
0
t0/4
t1/2
t2/2
t0/4
t0/4
t1/2
t2/2
t0/4
SA
0
0
1
1
1
1
0
0
SB
0
0
0
1
1
0
0
0
SC
0
0
0
0
0
0
0
0
t0/2
t1/2
t2
Ts
U0
U1
U2
U7
t1/2
t0/2
U1
U0
Space vector PWM
SA
Space vector PWM
UCc*
Ts
U7
U2
U1
U0
U0
U1
U2
Fig. 2.18. Comparison of carrier based PWM with space vector PWM a) SVPWM, b) DPWM1
A
U A0
U dc
2
UA
UN0
0
U dc
2
N
UB
U
UC
B0 =U
C0
B
C
Fig. 2.19. Equivalent circuit of VSI for the U1 vector
29
2. Voltage Source Inverter Fed Induction Motor Drive
Taking into consideration the electrical circuit in Fig. 2.19 the voltage distribution
can be obtained. The voltages can be written as:
2
1
1
U A = U dc ; U B = − U dc ; U C = − U dc
3
3
3
(2.32)
1
1
1
U A0 = U dc ; U B0 = − U dc ; U C0 = − U dc
2
2
2
(2.33)
1
U N0 = U A0 − U AN = − U dc
6
(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
U0
U1
U2
U3
U4
U5
U6
U7
U A0
1
− U dc
2
1
U dc
2
1
U dc
2
1
− U dc
2
1
− U dc
2
1
− U dc
2
1
U dc
2
1
U dc
2
U B0
1
− U dc
2
1
− U dc
2
1
U dc
2
1
U dc
2
1
U dc
2
1
− U dc
2
1
− U dc
2
1
U dc
2
U C0
1
− U dc
2
1
− U dc
2
1
− U dc
2
1
− U dc
2
1
U dc
2
1
U dc
2
1
U dc
2
1
U dc
2
UA
UB
0
0
2
U dc
3
1
U dc
3
1
− U dc
3
2
− U dc
3
1
− U dc
3
1
U dc
3
1
− U dc
3
1
U dc
3
2
U dc
3
1
U dc
3
1
− U dc
3
2
− U dc
3
0
0
UC
U N0
1
− U dc
0
2
1
1
− U dc − U dc
3
6
2
1
− U dc U dc
3
6
1
1
− U dc − U dc
3
6
1
1
U dc
U dc
3
6
2
1
U dc
U dc
3
6
1
1
U dc
U dc
3
6
1
U dc
0
2
The average value for sampling time of UNO voltage can be written as follows:
U N0 =
1 U dc 
1
1

 − t0 − t1 + t2 + t7  for the sectors I, III, V
Ts 2 
3
3

U N0 =
1 U dc 
1
1

 − t0 − t2 + t1 + t7  for the sectors II, IV, VI (2.36)
Ts 2 
3
3

(2.35)
and
30
2.4. Pulse Width Modulation (PWM)
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 (t0, t7) times equations for the most significant
modulation methods.
Table 2.2. Relation between carrier based and SVM methods
Modulation
method
Waveform of the
ZSS (Fig. 2.13)
SPWM
no signal
(U N0 = 0)
Calculation of t0 and t7
Ts  4

1 − M cos α 
2 π

T 
2

t0 = s 1 − M (cos α ) + 3 sin α 
2 π

t0 =
for sectors I, III, V
for sectors II, IV, VI
t7 = Ts − t0 − t1 − t2
THIPWM
Sinusoidal with
1/4 amplitude
Ts  4 
1

1 − M  cos α − cos 3α   for sectors I, III, V

2 π 
4

T  2 
1

t0 = s 1 − M  cos α + 3 sin α − cos 3α  
2
2 π 

t0 =
for sectors II, IV, VI
t7 = Ts − t0 − t1 − t2
SVPWM
Triangular with
1/4 amplitude
t0 = t7 = (Ts − t1 − t 2 ) 2
DPWM1
Discontinuous
t0 = 0
t7 = Ts − t1 − t 2
when
π
3
n ≤α <
π
6
(2n + 1)
t7 = 0
t0 = Ts − t1 − t 2
when
π
6
(2n + 1) ≤ α < π (n + 1)
3
n = 0, 1, 2, 3, 4, 5
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 six-step mode (see equation 2.21). The nonlinear
31
2. Voltage Source Inverter Fed Induction Motor Drive
range between this point and six-step 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 Uc possible to realization in whole range of α angle is equal
3
U dc . It is a radius of the circle inscribed of the hexagon. This value corresponds to
3
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 six-step mode (at M = 1).
U2 (110)
)U
2
U3 (010)
Overmodulation range
0.907 < M < 1
(t
2
/T
s
Uc
Six-step mode
M=1
U4 (011)
U0 (000)
α
U7 (111)
(t1 /Ts )U1
U1 (100)
Linear range
M ≤ 0.907
U5 (001)
U6 (101)
Fig. 2.20. Definition of the overmodulation range
32
2.4. Pulse Width Modulation (PWM)
If the value of the reference voltage beyond maximal value in the linear range vector
Uc can not be realized for whole range of α angle. However, average voltage value can
be obtained for modification of the reference voltage vector. Because of the modified
reference voltage vector overmodulation algorithms are not widely used in vector
control methods of drive. To modify the reference voltage vector different algorithm
may be applied. Overmodulation range can be considered as one region [33], or it can
be divided into two regions [16, 55, 75, 89].
In the algorithm where overmodulation region is considered as two regions two
modes depending on the reference voltage value were defined. In mode I the algorithm
modifies only the voltage vector amplitude, in mode II both the amplitude and angle of
the voltage vector.
Overmodulation mode I is shown in Fig. 2.21.
U2 (110)
U3 (010)
Uc
Uc*
U4 (011)
U0 (000)
α
θ
U1 (100)
U7 (111)
U5 (001)
U6 (101)
Fig. 2.21. Overmodulation mode I
In this mode voltage vector Uc 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.
33
2. Voltage Source Inverter Fed Induction Motor Drive
In the hexagon trajectory part only active vectors are used. The duration of these
vectors t1 and t2 are obtained from trigonometrical relationships and can be expressed in
the following equations:
t1 = Ts
3 cosα − sin α
3 cosα + sin α
(2.37a)
t2 = Ts − t1
(2.37b)
t0 = t7 = 0
(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.
U3 (010)
U2 (110)
αh
U4 (011)
U0 (000)
Uc*
∗
α α
Uc
αh
U1 (100)
U7 (111)
U5 (001)
U6 (101)
Fig. 2.22. Overmodulation mode II where both amplitude and angle is changed
34
2.4. Pulse Width Modulation (PWM)
The trajectory of Uc* is maintained on the hexagon which defines amplitude of the
reference voltage vector. The angle is calculated from the following equations:

0

 α −αh π
∗
α =
π 6 − α h 6

π 3

0 ≤ α ≤ αh
for
αh < α < π 3 −αh
(2.38)
π 3 −αh ≤ α ≤ π 3
where: αh – hold-angle.
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 αh, the algorithm holds the vector Uc* at the vertex (U1). Next part is for α from
αh to π 3 − α h . In this angle range the reference vector moves along the hexagon. In the
last range, from π 3 − α h to α h , the vector Uc* is held until the next vertex (U2).
The overmodulation mode II works up to the six-step mode for αh equal zero. The
six-step mode characterized by selection of the switching vector for one-sixth 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
35
2. Voltage Source Inverter Fed Induction Motor Drive
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 Uc, which are calculated
in one sampling time Ts and realized in the next switching time Tsw 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.
a)
U c(1)
U c( n−1)
U c( n+1)
U c( 2 )
U c( 3)
Uc(K)
U c(n )
sampling cycles
1
2
3
...
n-1
n
...
switching cycles
1
2
3
...
n-1
n
...
U c( n−1)
U c(n )
U c( n+1)
Ts = Tsw
b)
U c(1)
U c( 2)
U c( 3)
Uc(K)
sampling cycles
1
2
3
...
n-1
n
...
switching cycles
1
2
3
...
n-1
n
...
Ts = Tsw
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 lead-lag modulation (RLL). In this method pulse
position is either commencing at the beginning of the switching interval (leading-edge
36
2.4. Pulse Width Modulation (PWM)
modulation) or its tailing edge is aligned with the end of the interval (lagging-edge
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 αTs from the middle of the period.
The parameter α is varied randomly within a band limited by the maximum duty cycle.
The last presented method (Fig. 2.24c) is random distribution of the zero voltage
vector (RZD). Additionally distribution of the zero vectors can by different, until only
one zero vector in switching cycle in the discontinuous methods (Fig. 2.15b, c). This
fact is utilized in the random distribution of the zero voltage vector, where the
proportion between the time duration for the two zero vectors U0(000) and U7(111) is
randomized in the switching cycles.
a)
Lead
Lag
Lag
Lead
Ts
Ts
Ts
Ts
SA
SB
SC
b)
αTs
αTs
αTs
αTs
SA
SB
SC
Ts
Ts
Ts
Ts
Ts
Ts
Ts
Ts
c)
SA
SB
SC
Fig. 2.24. Different fixed switching random modulation schemes a) Random lead-leg modulation (RLL),
b) Random center displacement (RCD), c) Random zero vector distribution (RZD)
37
2. Voltage Source Inverter Fed Induction Motor Drive
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.
Ts
U c(1)
sampling cycles
U c( 2)
U c( 3)
U c(K)
1
2
3
switching cycles
1
∆t
2
U c( n−1)
...
3
U c(n )
n-1
...
U c( n+1)
...
n
n-1
n
...
Tsw
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:
∆t = rTs
(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
38
2.5. Summary
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 dead-time 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.
39
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 DTC-SVM 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.
PI-regulators.
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 d − q is
rotating with the angular speed equal to rotor flux vector angular speed ΩK = Ωsr ,
which is defined as follows:
Ωsr =
dγsr
dt
(3.1)
The rotating coordinate system d − q is shown in Fig. 3.1.
The voltage, current and flux complex space vector can be resolved into components
d and q.
U sK = U sd + jU sq
(3.2a)
I sK = I sd + j I sq ,
I rK = I rd + j I rq
(3.2b)
Ψ sK = Ψ sd + jΨ sq ,
Ψ rK = Ψ rd = Ψ r
(3.2c)
3.2. Field Oriented Control (FOC)
β
q
Is
I sβ
Ω sr
d
Ψr
δ
I sq
I sd
γ sr
I sα
α
Fig. 3.1. Vector diagram of induction motor in stationary α − β and rotating d − q coordinates
In d − q coordinate system the induction motor model equations (2.10-2.12) can be
written as follows:
U sd = Rs I sd +
U sq = Rs I sq +
dΨ sd
− ΩsrΨ sq
dt
dΨ sq
dt
+ ΩsrΨ sd
(3.3a)
(3.3b)
dΨ r
dt
(3.3c)
0 = Rr I rq + Ψ r (Ωsr − pb Ωm )
(3.3d)
Ψ sd = Ls I sd + LM I rd
(3.4a)
Ψ sq = Ls I sq + LM I rq
(3.4b)
Ψ r = Lr I rd + LM I sd
(3.4c)
0 = Lr I rq + LM I sq
(3.4d)
0 = Rr I rd +
dΩm 1
=
dt
J
 ms LM

Ψ r I sq − M L 
 pb
2 Lr


(3.5)
The equations 3.3c and 3.4c can be easy transformed to:
41
3. Vector Control Methods of Induction Motor
dΨ r LM Rr
R
=
I sd − r Ψ r
dt
Lr
Lr
(3.6)
The motor torque can by expressed by rotor flux magnitude Ψ r and stator current
component I sq as follows:
M e = pb
ms LM
Ψ r I sq
2 Lr
(3.7)
Equations (3.6) and (3.7) are used to construct a block diagram of the induction
motor in d − q coordinate system, which is presented in Fig. 3.2.
Rr
Lr
I sd
LM Rr
Lr
Ψr
∫
Me
I sq
pb
ms
2
LM
Lr
Me
1
J
∫
Ωm
ML
Fig. 3.2. Block diagram of induction motor in
d − q coordinate system
The main feature of the field oriented control (FOC) method is the coordinate
transformation. The current vector is measured in stationary coordinate α − β .
Therefore, current components I sα , I sβ must be transformed to the rotating system
d − q . Similarly, the reference stator voltage vector components U sαc , U sβc , must be
transformed from the system d − q to α − β . These transformations requires a rotor
flux angle γ sr . 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.
42
3.2. Field Oriented Control (FOC)
For DFOC an estimator or observer calculates the rotor flux angle γ sr . Inputs to the
estimator or observer are stator voltages and currents. An example of the DFOC system
is presented in Fig. 3.3.
U dc
Ψ rc
M ec
I sdc
1
LM
2 Lr 1
pb ms LM Ψ rc
I sqc
PI
d −q
SA
U s αc
SVM
PI
SC
U sβ c
α−β
γ sr
I sd
SB
Flux
Estimator
d −q
I sq
I sα
I sβ
U sα
Voltage
U sβ Calculation
2
Is
3
α−β
Motor
Fig. 3.3. Block diagram of the Direct Field Oriented Control (DFOC)
For the IFOC rotor flux angle γ sr is obtained from reference I sdc , I sqc currents. The
angular speed of the rotor flux vector speed can be calculated as follows:
Ωrs = Ωsl + pb Ωm
(3.8)
where Ωsl is a slip angular speed. It can be calculated from (3.3d) and (3.4d).
Ωsl =
1 Rr
I sqc
I sdc Lr
(3.9)
In Fig. 3.4 a block diagram of the IFOC is shown.
43
3. Vector Control Methods of Induction Motor
U dc
Ψ rc
M ec
I sdc
1
LM
SA
U s αc
SVM
I sqc
2 Lr 1
pb ms LM Ψ rc
d−q
PI
U sβc
PI
SB
SC
α−β
γ sr
I sd
Rr 1
Lr I sdc
d−q
I sα
I sq
∫
2
I sβ
Is
3
α−β
Motor
Ω sr
Ωsl
pb
Ωm
Fig. 3.4. Block diagram of the Indirect Field Oriented Control (IFOC)
In both presented examples reference currents in rotating coordinate system I sdc , I sqc
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:
I sdc =
1
Ψr
LM
(3.10)
I sqc =
2 Lr 1
M ec
pb ms LM Ψ rc
(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,
44
3.3. Feedback Linearization Control (FLC)
•
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 third-order 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
(2.10-2.12) can be written in the following form:
x& = f ( x ) + U sα gα + U sβ g β
(3.12)
where:
45
3. Vector Control Methods of Induction Motor
 − α Ψ rα − p b Ω mΨ rβ + α L M I sα 
 p Ω Ψ − αΨ + αL I 
rβ
M sβ 
 b m rα


f ( x ) =  αβ Ψ rα + β p b Ω mΨ rβ − γ I sα 
 − β p b Ω mΨ rα + αβ Ψ rβ − γ I sβ 

ML 
 µ ( Ψ rα I s β − Ψ r β I s α ) − J 


1
gα = 0 ,0 ,
,0 ,0
σLs


T

1 
,0
g β = 0 ,0 ,0 ,
σLs 

[
x = Ψ rα ,Ψ rβ , I sα , I sβ , Ωm
(3.13)
(3.14)
T
]T
(3.15)
(3.16)
and
α=
Rr
Lr
(3.17)
β=
LM
σLs Lr
(3.18)
γ=
2
Rs Lr + Rr LM
µ = pb
σLs Lr
2
2
ms LM
2J
(3.19)
(3.20)
Because Ωm ,Ψ rα ,Ψ rβ are not dependent on U sα ,U sβ it is possible to chose variable
dependent on x:
φ1 ( x ) = Ψ rα 2 + Ψ rβ 2 = Ψ r 2
(3.21)
φ2 ( x ) = Ωm
(3.22)
If it is assumed that φ1 ( x ) , φ2 ( x ) are output variables, the full definition of new
coordinates can be given by:
46
z1 = φ1 ( x )
(3.23a)
z2 = L f φ1 ( x )
(3.23b)
3.3. Feedback Linearization Control (FLC)
z3 = φ 2 ( x )
(3.23c)
z4 = L f φ2 ( x )
(3.23d)
Ψ
z5 = arctan rβ
 Ψ rα



(3.23e)
It should be mentioned that the goal of the control is to obtain constant flux
amplitude and to follow the reference angular speed.
The fifth variable cannot be fully linearized. Additionally, it is not controllable (the
fifth variable correspond to slip in the motor). Therefore, the last equation is not
considered. Then the dynamics of the system are given by:
2
U sα 
 &z&1   L f φ1 
=
 &z&   L2 φ  + DU 
 3   f 2 
 sβ 
(3.24)
L L φ
D =  gα f 1
 Lgα L f φ2
(3.25)
where
Lgβ L f φ1 

Lgβ L f φ2 
If φ1 ≠ 0 (the amplitude of flux is not zero) then det(D) ≠ 0 and it is possible to
define the linearization feedback as:
  2   v1  
U sα 
- 1  − L f φ1
=
D
 2  +   
U 
  − L f φ2  v2  
β
s


(3.26)
Then the resulting system is described by the equations:
z&1 = z2
(3.27a)
z&2 = v1
(3.27b)
z&3 = z 4
(3.27c)
z&4 = v2
(3.27d)
and the final block diagram of the induction motor with the new defined control
signals can be shown as in Fig. 3.5.
47
3. Vector Control Methods of Induction Motor
ν1
∫
z2
Ψr
∫
2
Ψr
Ωm
∫
ν2
∫
z4
Me
J
ML
Fig. 3.5. Block diagram of the induction motor with new v1 and v2 control signals
The control signals v1 , v2 are calculated by using linear feedback as follows:
v1 = k11 (z1 − z1ref ) − k12 z2
(3.28)
v2 = k21 (z3 − z3ref ) − k22 z4
(3.29)
where coefficients k11 , k12 , k 21 , k22 are chosen to receive reference close loop
system dynamics.
An example of a FLC system for PWM inverter-fed induction motor is presented in
Fig. 3.6.
The property of the FLC can be summarized as follows:
•
it guarantees exactly decoupling of the motor speed and rotor flux control in both
dynamic and steady state,
•
the method is implemented in a state variable control fashion and needs complex
signal processing,
•
full information about motor state variables and load torque is required,
•
there are no current controllers,
•
a PWM vector modulator is required, what further guarantee constant switching
frequency,
48
3.4. Direct Flux and Torque Control (DTC)
•
the stator currents are sinusoidal except of high frequency switching harmonics.
U dc
Ψ rc
2
ν1
Flux
Controller
Ωmc
ν2
Speed
Controller
Control
Signals
Transformation
U sβc
SA
U sαc
SB
Vector
Modulator S
C
Voltage
Calculation
I sα
z1
z2
z3
z4
z5
Feedback
Signals
Transformation
I sβ
Ψ̂ rα
Flux
Estimator
Û s
Is
Ψ̂ rβ
Motor
Ωm
Fig. 3.6. Block scheme of the feedback linearization control method
3.4.
Direct Flux and Torque Control (DTC)
3.4.1. Basics of Direct Flux and Torque Control
As it was mentioned in section 3.2 in the classical vector control strategy (FOC) the
torque is controlled by the stator current component I sq in accordance with equation
(3.7). This equation can also be written as:
M e = pb
ms LM
Ψ r I s sin δ
2 Lr
(3.30)
where:
δ - angle between rotor flux vector and stator current vector.
The formula (3.30) can be transformed into the equation:
M e = pb
ms
LM
Ψ sΨ r sin δ Ψ
2 Lr Ls − LM 2
(3.31)
where:
δ Ψ - angle between rotor and stator flux vectors.
49
3. Vector Control Methods of Induction Motor
It can be noticed that the torque depends on the stator and rotor flux magnitude as
well as the angle δ Ψ . The vector diagram of IM is presented in Fig. 3.7. The two angels
δ and δ Ψ are also shown in Fig. 3.7. The angle δ is important in FOC algorithms,
whereas δ Ψ in DTC techniques.
β
Is
Ψs
δ
δΨ
γ ss
γ sr
Ψr
α
Fig. 3.7. Vector diagram of induction motor
From the motor voltage equation (2.10a), for the omitted voltage drop on the stator
resistance, the stator flux can by expressed as:
dΨ s
= Us
dt
(3.32)
Taking into consideration the output voltage of the inverter in the above equation it
can be written as:
t
Ψ s = ∫ U v dt
(3.33)
0
where:
2
j ( v −1)π
 3 U dc e
Uv = 
0

3
v = 1...6
v = 0,7
(3.34)
Equation (3.33) describe eight voltage vectors which correspond to possible inverter
states. These vectors are shown in Fig. 3.8. There are six active vectors U1-U6 and two
zero vectors U0, U7.
50
3.4. Direct Flux and Torque Control (DTC)
Im
U2 (110)
U3 (010)
U4 (011)
U1 (100)
U0 (000)
Re
U7 (111)
U5 (001)
U6 (101)
Fig. 3.8. Inverter output voltage represented as space vectors
It can be seen from (3.33), that the stator flux directly depends on the inverter voltage
(3.34).
By using one of the active voltage vectors the stator flux vector moves to the
direction and sense of the voltage vector. It can be observed by simulation of six-step
mode (Fig. 3.9) and PWM operation (Fig. 3.10). In Fig. 3.9 is well shown how stator
flux changes direction for the cycle sequence of the active voltage vectors. Obviously,
the same effect is for the PWM operation (Fig. 3.10). However, in this case the control
algorithm choose correct voltage vectors, thanks to that waveform is close to be
sinusoidal. In this simulation a low sampling frequency is used (0.5kHz) for better
presenting the effect. A zoom part of the flux vector trajectory is shown in Fig. 3.11.
In induction motor the rotor flux is slowly moving but the stator flux can be changed
immediately. In direct torque control methods the angle between stator and rotor flux
δ Ψ can be used as a variable of torque control (3.31). Moreover stator flux can be
adjusted by stator voltage in simple way. Therefore, angle δ Ψ as well as torque can be
changed thanks to the appropriate selection of voltage vector.
There are the general bases of the direct flux and torque control methods. Those
consideration and above equations can be used in analysis of the classical DTC
algorithms as well as in new proposed methods. It is also bases of the DTC-SVM
methods, which are presented in Chapter 4.
51
3. Vector Control Methods of Induction Motor
a)
b)
Fig. 3.9. IM under six-step mode a) voltage and stator flux waveforms, b) stator flux trajectory
a)
b)
Fig. 3.10. IM under PWM operation a) voltage and stator flux waveforms, b) stator flux trajectory
52
3.4. Direct Flux and Torque Control (DTC)
voltage U4 applied
voltage U3 applied
voltage U4 applied
β
voltage U3 applied
voltage U4 applied
U2 (110)
U3 (010)
voltage U3 applied
voltage U2 applied
U4 (011)
voltage U3 applied
U1 (100)
U0 (000)
α
U7 (111)
U5 (001)
U6 (101)
Fig. 3.11. Forming of the stator flux trajectory by appropriate voltage vectors selection
3.4.2. Classical Direct Torque Control (DTC) – Circular Flux Path
The block diagram of classical DTC proposed by I. Takahashi and T. Nogouchi [97]
is presented in Fig. 3.12.
Flux
Controller
Ψ sc
Mec
Udc
dΨ
eΨ
dM
eM
Torque
Controller
Vector
Selection
Table
Ψ̂ s
SB
SC
γ ss (N)
Sector
Detection
M̂ e
SA
Ψ̂ sα
Voltage
Calculation
Ψ̂ sβ
Flux and
Torque
Estimator
Us
Is
Motor
Fig. 3.12. Block scheme of the direct torque control method
53
3. Vector Control Methods of Induction Motor
The stator flux amplitude Ψ sc and the electromagnetic torque M c are the reference
signals which are compared with the estimated Ψ̂ s and M̂ e values respectively. The
flux eΨ and torque eM errors are delivered to the hysteresis controllers. The digitized
output variables dΨ , d M and the stator flux position sector γ ss (N ) selects the
appropriate voltage vector from the switching table. Thus, the selection table generates
pulses SA, SB, SC to control the power switches in the inverter.
For the flux is defined two-level hysteresis controller, for the torque three-level, as it
is shown in Fig. 3.13.
a)
b)
dΨ
dM
eM
HΨ
eΨ
HM
Fig. 3.13. The hysteresis controllers a) two-level, b) three-level
The output signals dΨ , d M are defined as:
dΨ = 1 for eΨ > HΨ
(3.35a)
dΨ = 0 for eΨ < − HΨ
(3.35b)
d M = 1 for eM > H M
(3.36a)
d M = 0 for eM = 0
(3.36b)
d M = −1 for eM < − H M
(3.36c)
In the classical DTC method the plane is divided for the six sectors (Fig. 3.14),
which are defined as:
54
 π π
Sector 1: γ ss ∈  − ,+ 
 6 6
(3.37a)
 π π
Sector 2: γ ss ∈  + , 
 6 2
(3.37b)
 π 5π 
Sector 3: γ ss ∈  + ,+ 
6 
 2
(3.37c)
3.4. Direct Flux and Torque Control (DTC)
 5π 5π 
,− 
Sector 4: γ ss ∈  +
6 
 6
(3.37d)
 5π π 
,− 
Sector 5: γ ss ∈  −
2
 6
(3.37e)
 π π
Sector 6: γ ss ∈  − ,− 
 2 6
(3.37f)
β
Sector 3
U3 (010)
Sector 4
U4 (011)
Sector 2
U2 (110)
U1 (100)
U0 (000)
α
U7 (111)
Sector 1
U6 (101)
U5 (001)
Sector 5
Sector 6
Fig. 3.14. Sectors in the classical DTC method
For the stator flux vector laying in sector 1 (Fig. 3.15) in order to increase its
magnitude the voltage vectors U1, U2, U6 can be selected. Conversely, a decrease can be
obtained by selecting U3, U4, U5. By applying one of the zero vectors U0 or U7 the
integration in equation (3.33) is stopped. The stator flux vector is not changed.
For the torque control, angle between stator and rotor flux δ Ψ is used (equation
3.31). Therefore, to increase motor torque the voltage vectors U2, U3, U4 can be selected
and to decrease U1, U5, U6.
The above considerations allow construction of the selection table as presented in
Table 3.1.
55
3. Vector Control Methods of Induction Motor
β
U3
U2
U4
δΨ
Ψs
U1
U5
U6
α
Sector 1
Ψr
Fig. 3.15. Selection of the optimum voltage vectors for the stator flux vector in sector 1
Table 3.1. Optimum switching table
dΨ
1
0
dM
Sector 1
Sector 2
Sector 3
Sector 4
Sector 5
Sector 6
1
U2
U3
U4
U5
U6
U1
0
U7
U0
U7
U0
U7
U0
-1
U6
U1
U2
U3
U4
U5
1
U3
U4
U5
U6
U1
U2
0
U0
U7
U0
U7
U0
U7
-1
U5
U6
U1
U2
U3
U4
The signal waveforms for steady state operation of classical DTC method are shown
in Fig. 3.16.
The DTC was proposed as an analog control method. The implementation of the
hysteresis controller in the analog setup is easy and the control system works properly.
When the hysteresis controller is implemented in a digital signal processor (DSP), its
operation is quite different from that of the analog scheme [19]. The digital
implementation of the hysteresis controller is also called sampled hysteresis.
56
3.4. Direct Flux and Torque Control (DTC)
a)
b)
Fig. 3.16. Steady state operation for the classical DTC method ( f s = 40kHz )
a) signals in time domain, b) stator flux trajectory
In Fig. 3.17 are presented typical switching sequences of the torque hysteresis
controller for the analog (Fig. 3.17a) and for the digital (Fig. 3.17b) implementation.
57
3. Vector Control Methods of Induction Motor
a)
b)
S/H
Mc + Hm
Mc
Mc − Hm
t1
t2
t3
Ts
Ts
Ts
Fig. 3.17. Operating of the torque hysteresis controller a) analog, b) digital
In the analog implementation the torque ripple are kept exactly within the hysteresis
band and the switching instants are not equally spaced. The digital system operates at
fixed sampling time Ts and works like analog only for high sampling frequencies
fs =
1
.
Ts
For the lower sapling frequency the switching instants are not when the estimated
torque crosses the hysteresis band but on the sampling time. This situation is presented
in Fig. 3.17b. The simulation results illustrated control system behavior at lower
sampling frequency f s = 15kHz are given in Fig. 3.18. It can be seen that current and
torque ripples are bigger compare to this one operate with sampling frequency
f s = 40kHz (see Fig. 3.16).
The influence of the torque hysteresis band for the torque error and switching
frequency at different sampling frequencies is shown in Fig. 3.19 and Fig. 3.20. At low
sampling frequency fs = 20kHz (Fig. 3.19) the switching frequency and torque error are
not sensitive for hysteresis band. However, at the high sampling frequency fs = 80kHz
(Fig. 3.20) when the hysteresis band is increased the switching frequency decreases and
the torque error increases. Simulated results show that the hysteresis controllers need a
high sampling frequency to obtain a proper operation.
The torque and flux errors are calculated according to equations:
εψ =
s
58
Ψˆ s − Ψ sc
100%
Ψ sN
(3.38a)
3.4. Direct Flux and Torque Control (DTC)
εM =
Mˆ e − M ec
100%
M eN
(3.38b)
where: Ψ sN - nominal stator flux, M eN - nominal torque
Fig. 3.18. Steady state operation for the classical DTC method operating with lower
sampling frequency ( f s = 15kHz )
The average value of the flux and torque errors are calculated in a period of the
fundamental frequency.
59
3. Vector Control Methods of Induction Motor
a)
f sw [Hz]
25000
20000
15000
10000
4792
5000 5400
2750 2208 2367 2333
4567 4333 3508
0
0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 H m [Nm]
b)
ε Μ _avr [%]
14
11,06
12
10,68
11,97
9,43 9,93
10
11,00
10,17
8 9,65
6
4
2
0
0,0
0,5
1,0
1,5
2,0
2,5
3,0
3,5
12,03
4,0 H m [Nm]
Fig. 3.19. Simulated results for classical DTC a) switching frequency and b) torque error as a function of
the torque hysteresis band at sampling frequency fs = 20kHz
a)
f sw [Hz]
25000
20000
15000
19750
13317
10000
6142 5492 5450 5666
8233 7400
6666
5000
0
0,0
0,5
1,0
b) ε Μ _avr [%]
14
12
10
8
6
4
2,43
3,06
2 2,64
0
0,0
0,5
1,0
1,5
2,0
2,5
3,0
3,5
4,0 H m [Nm]
10,27
4,21
1,5
5,36
2,0
6,56
2,5
7,77
3,0
8,94
3,5
4,0 H m [Nm]
Fig. 3.20. Simulated results for classical DTC a) switching frequency and b) torque error as a function of
the torque hysteresis band at sampling frequency fs = 80kHz
60
3.4. Direct Flux and Torque Control (DTC)
The classical DTC method can be characterized as follows:
Advantages:
•
simple structure:
o no coordinate transformation,
o no separate voltage modulation block,
o no current control loops,
•
very good flux and torque dynamic performance,
Disadvantages:
•
variable switching frequency,
•
problems during starting and low speed operation,
•
high torque ripples,
•
flux and current distortion caused by stator flux vector sector position change
•
high sampling frequency is required for digital implementation.
3.4.3. Direct Self Control (DSC) – Hexagon Flux Path
The block diagram of the direct self control method proposed by M. Depenbrock [31,
32] is presented in Fig. 3.21. This method was mainly applied in high power
applications, which required fast torque dynamic and low switching frequency [96].
Based on the command stator flux Ψ sc and the actual phase components Ψ sA , Ψ sB ,
Ψ sC , the flux comparators generate digital variables d A , d B , d C , which corresponds to
active voltage vectors (U1 – U6). The hysteresis torque controller generates the signal
d m , which determines zero states. For the constant flux region, the control algorithm is
as follows:
S A = d C , S B = d A , S C = d B for d m = 1
(3.39a)
S A = 0 , S B = 0 , S C = 0 for d m = 0
(3.39b)
61
3. Vector Control Methods of Induction Motor
Flux
Comparators
ψ sc
Udc
dA
SB
dB
M ec
SC
SA
dm
dC
Torque
Controller
Voltage
Calculation
ψ̂ sC
ψ̂sB
ABC
ψ̂ sA
ψ̂sα
ψ̂sβ
α −β
M̂ e
Flux and U s
Torque
Estimator
Is
Motor
Fig. 3.21. Block diagram of Direct Self Control method
The signal waveforms for steady state operation of DSC method are shown in Fig.
3.22. It can be seen that the flux trajectory is identical with that for the six-step mode
(Fig. 3.9). This follows from the fact that the zero voltage vectors stop the flux vector,
but do not affect its trajectory. The dynamic performances of torque control for the DSC
are similar as for the classical DTC.
The property of the DSC can be summarized as follows:
•
hexagonal trajectory of the stator flux vector for PWM operation,
•
block type of PWM (not sinusoidal),
•
non-sinusoidal current waveforms,
•
switching selection table is not required,
•
low (minimum) inverter switching frequency (depended on hysteresis torque
band),
•
62
very good torque and flux control dynamics.
3.4. Direct Flux and Torque Control (DTC)
a)
b)
Fig. 3.22. Steady state operation for the DSC method
a) signals in time domain, b) stator flux trajectory
Several solutions have been proposed to improve the conventional DSC. For
instance, reduction of the current distortion has been achieved by introducing 12 stator
flux sectors [110] or by processing not only the stator flux value , but also the stator flux
63
3. Vector Control Methods of Induction Motor
angle [109]. Also solutions based on fuzzy logic and neural networks solutions were
proposed [85, 90].
3.5.
Summary
In this chapter review of significant vector control methods of IM has been
presented. The characteristic features for all control schemes were described.
The FLC structure guarantees exact decoupling of the motor speed and rotor flux
control in both dynamic and steady states. However, it is complicated and difficult to
implement in practice. This method requires complex computation and additionally it is
sensitive to changes of motor parameters. Because of these features this method was not
chosen for implementation.
Table 3.2 Comparison of control methods
FOC
DTC
¾ Modulator
9 Structure
Advantages
independent on
¾ Constant switching
rotor parameters,
frequency
universal for IM
¾ Unipolar inverter
and PMSM
output voltage
¾ Low switching
9 Simple
losses
implementation of
sensorless
¾ Low sampling
operation
frequency
¾ Current control
9 No coordinate
loops
transformation
9 No current control
loops
Disadvantages • Coordinate
• No modulator
transformation
• Bipolar inverter
output voltage
• A lot of control
loops
• Variable switching
frequency
• Control structure
depended on rotor • High switching
parameters
losses
• High sampling
frequency
DTC-SVM
9 Structure
independent on
rotor parameters,
universal for IM
and PMSM
9 Simple
implementation of
sensorless
operation
9 No coordinate
transformation
9 No current control
loops
¾ Modulator
¾ Constant switching
frequency
¾ Unipolar inverter
output voltage
¾ Low switching
losses
¾ Low sampling
frequency
Due to above mentioned facts the FOC and DTC methods were considered next.
Analysis of advantages and disadvantages of FOC and DTC methods resulted in a
search for method which will eliminate disadvantages and keep advantages of those
64
3.5. Summary
methods. Table 3.2 summarizes features of analyzed control methods. It can be seen a
combination of DTC and FOC leads to the direct torque control with space vector
modulation (DTC-SVM) method which is an effect of this search. In Table 3.2 also
characteristic performance of DTC-SVM was given.
The disadvantages of classical DTC are caused by hysteresis controllers and
switching table used in a structure. Therefore, new DTC-SVM method replaces
switching table by space vector modulator and linear PI controllers are used like in the
FOC scheme. However, the current control loops are eliminated. The DTC-SVM
methods are widely discussed in the Chapter 4 where a detailed description of those
features can be found.
65
4.
Direct Flux and Torque Control with Space Vector
Modulation (DTC-SVM)
4.1.
Introduction
Direct flux and torque control with space vector modulation (DTC-SVM) schemes
are proposed in order to improve the classical DTC. The DTC-SVM strategies operate
at a constant switching frequency. In the control structures, space vector modulation
(SVM) algorithm is used. The type of DTC-SVM strategy depends on the applied flux
and torque control algorithm. Basically, the controllers calculate the required stator
voltage vector and then it is realized by space vector modulation technique.
In the DTC-SVM methods several classes have evolved:
•
schemes with PI controllers [111],
•
schemes with predictive/dead-beat [74],
•
schemes based on fuzzy logic and/or neural networks [40],
•
variable-structure control (VSC) [72, 73, 112].
Different structures of DTC-SVM methods are presented in the next section. For
each of the control structures, different controller design methods are proposed.
The classical DTC algorithm is based on the instantaneous values and directly
calculated the digital control signals for the inverter. The control algorithm in DTCSVM methods are based on averaged values whereas the switching signals for the
inverter are calculated by space vector modulator. This is main difference between
classical DTC and DTC-SVM control methods.
4.2.
Structures of DTC-SVM – Review
4.2.1. DTC-SVM Scheme with Closed – Loop Flux Control
In the control structure of Fig. 4.1 the rotor flux is assumed as a reference [24]. The
reference stator flux components defined in the rotor flux coordinates Ψ sdc , Ψ sqc can be
calculated from the following equations:
4.2. Structures of DTC-SVM – Review

L dΨ rc 

Ψ rc + r σ
Rr
dt 

Ψ sdc =
Ls
LM
Ψ sqc =
M
2 Lr
σLs ec
pb ms LM
Ψ rc
(4.1a)
(4.1b)
Formulas (4.1) can be derived from the equations (3.3), (3.4) and (3.7). The
equations (3.3), (3.4) and (3.7) describe the motor model in the rotor flux coordinate
system d − q .
The amplitude of the reference stator flux, using equations (4.1) can by expressed as:
2
Ψ sc
 L
  2
=  s Ψ rc  + 
 LM
  pb m s
2

 L M ec 
 (σLs )2  r


 LM Ψ rc 
2
(4.2)
The commanded value of stator flux Ψ sdc , Ψ sqc after transformation to stationary
coordinate system α − β are compared with the estimated values Ψ̂ sα , Ψ̂ sβ .
Ψ rc
M ec
Ψ sdc
Egs (4.1)
Ψ sqc
SA
d −q
Ψ sc
∆Ψ s
α −β
SB
SVM
SC
Rs
γˆsr
Rotor
Flux
Estimator
U sc
1
Ts
Ψ̂ s
Stator
Flux
Estimator
Us
Is
Voltage
Calculation
α −β
ABC
U dc
IA
IB
Fig. 4.1. DTC-SVM scheme with closed flux control
The reference voltage vector depends on the increment stator flux ∆Ψ s and voltage
drop on the stator winding resistance Rs :
U sc =
∆Ψ s
+ Rs I s
Ts
(4.3)
In this DTC-SVM structure the rotor flux magnitude is regulated. Thanks of them
increase the torque overload capability is possible [19, 24]. However, the drawback of
67
4. Direct Flux and Torque Control with Space Vector Modulation (DTC-SVM)
this algorithm is that it requires all the motor parameters and moreover it is very
sensitive to their variation.
4.2.2. DTC-SVM Scheme with Closed – Loop Torque Control
The method with close-loop torque control was originally proposed for the
permanent magnet synchronous motor (PMSM) [35, 36, 37]. However, the DTC basics
for both IM and PMSM are identical and therefore the method can also be used for the
IM [126]. The block scheme of the control structure DTC-SVM with close-loop torque
control is presented in Fig. 4.2.
Ψ sc
M ec
Torque
Controller
∆δψ
PI
SA
Eg. (4.4)
γˆss
M̂ e
Ψ sc
∆Ψ s
Ψ̂ s
U sc
1
Ts
SB
SVM
SC
Rs
Flux and
Torque
Estimator
Us
Is
Voltage
Calculation
α −β
ABC
U dc
IA
IB
Fig. 4.2. DTC-SVM scheme with closed-loop torque control
For the torque regulation a PI controller is applied. Output of this PI controller is an
increment of torque angle ∆δ Ψ (Fig. 4.3). In this way the torque is controlled by
changing the angle between stator and rotor fluxes according to the basics of DTC (see
section 3.4.2).
The reference stator flux vector is calculated as follows:
Ψ sc = Ψ sc e j (γˆss + ∆δΨ )
(4.4)
Next, reference stator flux vector is compared with the estimated value. The error of
the flux ∆Ψ s is used, for calculation of the reference voltage vector, according to the
equation (4.3).
68
4.2. Structures of DTC-SVM – Review
β
Ψ sc
Ψ̂ s
∆δΨ
δˆΨ
γˆss
Ψ̂ r
γˆsr
α
Fig. 4.3. Vector diagram
The presented method has simple structure and only one PI torque controller. It
makes the tuning procedure easier. The flux is adjusted in open-loop fashion.
4.2.3. DTC-SVM Scheme with Close – Loop Torque and Flux Control
Operating in Polar Coordinates
When both torque and flux magnitudes are controlled in a closed-loop way, the
strategies provide further improvement. The method operating in polar coordinates is
shown in Fig. 4.4 [49].
Flux
Controller
Ψ sc
M ec
kΨ
P
PI
∆γ sd
Torque
Controller
SA
∆γ s
∆γ ss
∆Ψ s
Eg. (4.7)
U sc
1
Ts
γˆss
SB
SVM
SC
Rs
Ψ̂ s
M̂ e
Flux and
Torque
Estimator
Us
Is
Voltage
Calculation
α −β
ABC
U dc
IA
IB
Fig. 4.4. DTC-SVM scheme operated in stator flux polar coordinates
The error of the stator flux vector ∆Ψ s is calculated from the outputs kΨ and ∆γ s
of the flux and torque controllers as follows:
69
4. Direct Flux and Torque Control with Space Vector Modulation (DTC-SVM)
∆Ψ s (k ) = Ψ s (k ) − Ψ s (k − 1)
(
)
= [1 + kΨ (k )]⋅ e j∆γ s (k ) − 1 ⋅ Ψ s (k − 1)
(4.5)
With the approximation
e j∆γ s (k ) ≅ 1 + j∆γ s (k )
(4.6)
The equation (4.5) can be written in the form
∆Ψ s (k ) = [kΨ (k ) + j∆γ s (k )]⋅ Ψ s (k − 1)
(4.7)
The commanded stator voltage vector is calculated according to equation (4.3). To
improve the dynamic performance of the torque control, the angle increment ∆γ s is
composed of two parts: the dynamic part ∆γ sd delivered by the torque controller and
the stationary part ∆γ ss generated by a feedforward loop.
4.2.4. DTC-SVM Scheme with Close – Loop Torque and Flux Control
in Stator Flux Coordinates
A block diagram of the method with close-loop torque and flux control in stator flux
coordinate system [111] is presented in Fig. 4.5. The output of the PI flux and torque
controllers can be interpreted as the reference stator voltage components U sxc , U syc in
the stator flux oriented coordinates ( x − y ).
Flux
Controller
Ψ sc
PI
U sxc
x− y
SA
U sc
M ec
PI
U syc
Torque
Controller
M̂ e
SC
α−β
γˆss
Ψ̂ s
SB
SVM
Flux and
Torque
Estimator
Us
Is
Voltage
Calculation
α −β
ABC
Fig. 4.5. DTC-SVM scheme operated in stator flux cartesian coordinates
70
U dc
IA
IB
4.3. Analysis and Controller Design for DTC-SVM Method
These dc voltage commands are then transformed into stationary frame ( α − β ), the
commanded values U sαc , U sβc are delivered to SVM.
4.2.5. Conclusions from Review of the DTC-SVM Structures
In the three first presented structures (Fig. 4.1, Fig. 4.2 and Fig. 4.4) the calculation
of reference voltage vector is based on demanded ∆Ψ s according to equation (4.3).
This differentiation algorithm is very sensitive to disturbances. In case of errors in the
feedback signals the differentiation algorithm may not be stable. This is very serious
drawback of these methods.
The methods presented in Fig. 4.1 and Fig. 4.2 do not have close-loop flux control.
In these methods stator flux magnitude is only adjusted.
The last presented method (Fig. 4.5) eliminates problems with differentiation
algorithm. Moreover, this method controls torque and flux in close-loop fashion.
Therefore, this scheme will be selected for experimental realization. In the next subsection controller design for flux and torque closed loops will be discussed.
4.3.
Analysis and Controller Design for DTC-SVM Method with Close – Loop
Torque and Flux Control in Stator Flux Coordinates
The compete set of motor model equations can be written in stator flux coordinate
system x − y . This system of coordinates x − y rotates with the stator flux angular
speed ΩK = Ωss . This angular speed is defined as follows:
Ωss =
dγ ss
dt
(4.8)
where: γ ss is a stator flux vector angle.
The complex space vector can be resolved into components x and y .
U sK = U sx + jU sy
(4.9a)
I sK = I sx + j I sy , I rK = I rx + j I ry
(4.9b)
71
4. Direct Flux and Torque Control with Space Vector Modulation (DTC-SVM)
Ψ sK = Ψ sx = Ψ s , Ψ rK = Ψ rx + jΨ ry
(4.9c)
The motor model equations (2.10-2.12) in x − y coordinate system can be written as:
U sx = Rs I sx +
dΨ s
dt
U sy = Rs I sy + ΩssΨ s
0 = Rr I rx +
0 = Rr I ry +
dΨ rx
+ Ψ ry ( pb Ωm − Ωss )
dt
dΨ ry
dt
+ Ψ rx (Ωss − pb Ωm )
(4.10a)
(4.10b)
(4.11a)
(4.11b)
Ψ s = Ls I sx + LM I rx
(4.12a)
0 = Ls I sy + LM I ry
(4.12b)
Ψ rx = Lr I rx + LM I sx
(4.12c)
Ψ ry = Lr I ry + LM I sy
(4.12d)
dΩm 1  ms

Ψ s I sy − M L 
=  pb
dt
J
2

(4.13)
The electromagnetic torque can be expressed by the following formula:
M e = pb
ms
Ψ s I sy
2
(4.14)
Based on the equations (4.10-4.14) the block diagram of induction motor can be
constructed (Fig. 4.6).
The block scheme presented in Fig. 4.6 is a full model of an induction motor. As can
be seen, this model is quite complicated and therefore difficult to analyze. However,
taking into consideration the stator voltage equations (4.10) and torque equation (4.14),
the motor can be described as follows:
dΨ s
= U sx − Rs I sx
dt
Me =
72
m
1
pb s Ψ s (U sy − Ω ssΨ s )
Rs
2
(4.15)
(4.16)
4.3. Analysis and Controller Design for DTC-SVM Method
ML
Rs
U sx
∫
I sx
1
2
Ls Lr − Lm
Lr
Ψs
LM
Ls
U sy
Ωss
÷
Rs
Rr
∫
Ψs
pb
ms M e
2
1
J
∫
Ωm
LM
I sy
I rx
LM
2
Ls Lr − Lm
1
σ Lr
Ψ rx
pb
∫
Rr
Ψ ry
I ry
1
σ Lr
Fig. 4.6. Complete block diagram of an induction motor in the stator flux oriented coordinates x − y
The block diagram of induction motor based on equations (4.15) and (4.16) is shown
in Fig. 4.7.
Rs I sx
U sx
∫
Ψs
Ωss
U sy
pb
ms 1
2 Rs
Me
Fig. 4.7. Simplified (rotor equation omitted) induction motor block diagram in the stator flux oriented
coordinates x − y
73
4. Direct Flux and Torque Control with Space Vector Modulation (DTC-SVM)
Different control structures based on the above induction motor model are proposed
in literature [73, 111, 112]. One of them is a method with two PI controllers [111],
which is presented in Fig. 4.5.
Considering a simple model of IM (Fig. 4.7), Fig. 4.8 shows the flux and torque
control loops for the method shown in Fig. 4.5. In Fig. 4.8 the dashed line represents the
IM model.
Rs I sx
Ψ sc
PI
U sx
∫
Ψs
Ωss
M ec
PI
U sy
pb
ms 1
2 Rs
Me
Fig. 4.8. Control loops with two PI controllers and simplified IM model of Fig. 4.7
In the next parts two approaches to a controller design will be presented and
compared. Both of them are based o the assumption that control loop can be considered
as quasi-continuous (fast sampling). The first method is based on simple symmetric
criterion [66], the second one uses root locus technique [34, 86].
PI Controllers
The transfer function of PI controllers is given as follows:
G R (s ) =

1 + sTi
U (s )
1 
 = K p
= K p 1 +
E (s )
sTi
 sTi 
where: K p - controller gain, Ti - controller integrating time.
The PI controller scheme is presented in Fig. 4.9.
74
(4.17)
4.3. Analysis and Controller Design for DTC-SVM Method
E (s )
1
Kp
U (s )
1
Ti s
Fig. 4.9. Block diagram of PI controller
Presented above model of the controller was used in DTC-SVM control method with
two PI controllers.
4.3.1. Torque and Flux Controllers Design – Symmetry Criterion Method
Flux Controller Design
The block diagram of the flux control loop is shown in Fig. 4.10. This control loops
is based on the model presented in Fig. 4.8. The voltage drop on the stator resistance is
neglected. In the stator flux control loop the inverter delay is taken into consideration.
Ψ sc
PI
U sx
1
1 + sT1
1
s
Ψs
Fig. 4.10. Stator flux magnitude control loops
For the flux controller parameter design the symmetry criterion can by applied [66].
In accordance with the symmetry criterion the plant transfer function can be written as:
G (s ) =
K c e − sτ 0
sT2 (1 + sT1 )
(4.18)
where: K c = 1 is the inverter gain, τ 0 is dead time of the inverter ( τ 0 = 0 ideal
converter), T2 = 1 , and T1 = Ts is a sum of small time constants, which includes
statistical delay of the PWM generation and signal processing delay. The optimal
controller parameters can be calculated as:
75
4. Direct Flux and Torque Control with Space Vector Modulation (DTC-SVM)
K pΨ =
1
T2
=
2 K c (T1 + τ 0 ) 2Ts
(4.19)
TiΨ = 4(T1 + τ 0 ) = 4Ts
(4.20)
In Table 4.1 are shown flux controller parameters calculated according to equations
(4.19) and (4.20). The considered range of the sampling frequency was form 2.5kHz to
10kHz. In Table 4.1 are also shown parameters of the step flux response obtained in
simulation, tnΨ - time when the actual flux is first time equal reference value and pΨ overshoot. The results of simulation are presented in Fig. 4.11.
Table 4.1. Flux controller parameters calculated according to symmetric optimum criterion
fs
10.0 kHz
5.0 kHz
2.5 kHz
KpΨ
5000
2500
1250
TiΨ
0.00040
0.00080
0.00160
tnΨ
0.00150 s
0.00180 s
0.00200 s
pΨ
1.60 %
2.37 %
9.33 %
a)
b)
c)
Fig. 4.11. Simulated flux response for controller parameters calculated according to symmetric optimum
criterion at different sampling frequency a) f s = 10kHz , b) f s = 5kHz , c) f s = 2.5kHz
76
4.3. Analysis and Controller Design for DTC-SVM Method
Presented in Fig. 4.11 simulation results confirm proper operation of the flux
controller for the different sampling frequency. The symmetric optimum criterion can
be apply to tune flux controller in analyzed DTC-SVM structure.
Torque Controller Design
The block diagram of the torque control loop is shown in Fig. 4.12. The same like for
flux this control loops is based on the model presented in Fig. 4.8. However, coupling
between torque and flux is omitted. Because of that very simple model is obtained and
for this model any criterion cannot be applied.
M ec
PI
U sy
1
1 + sTs
pb
Me
ms 1
Ψs
2 Rs
Fig. 4.12. Block diagram of the torque control loops
In this case the simple (practical) way to design torque controller can be used.
Starting from the initial values e.g. K pM = 1 ,
TiM = 4Ts the proportional gain K pM is
increasing cyclically as it is shown in Fig. 4.13. From these oscillograms the best value
of K pM for the fast torque response without oscillation and small overshoot can be
selected. In Fig. 4.13 the chosen simulation results for 5kHz and 10kHz sampling
frequencies are shown. For the sampling frequency 5kHz the best value of proportional
gain is K pM = 17 and for 10kHz K pM = 24 .
The finally obtained in this way parameters of the torque controller are shown in
Table 4.2. There are also shown parameters of the step torque response obtained in
simulation, tnM - time when the actual torque achieves first time reference value and
pM - overshoot.
Table 4.2. Torque controller parameters
fs
10.0 kHz
5.0 kHz
K pM
24
17
T iM
0.0004
0.0008
t nM
0.0007 s
0.0008 s
pΜ
8.39 %
18.53 %
77
4. Direct Flux and Torque Control with Space Vector Modulation (DTC-SVM)
b)
a)
K pM = 4
K pM = 4
K pM = 10
K pM = 10
K pM = 17
K pM = 24
Fig. 4.13. Torque response for selected controller gain K pM values, at different sampling frequency
a) f s = 5kHz (TiM = 800µs ) , b) f s = 10kHz (TiM = 400µs )
4.3.2. Torque and Flux Controllers Design – Root Locus Method
A root-locus analysis is used for tuning the flux and torque controllers. This
technique shows how the changes in the system’s open-loop characteristics influences
the closed-loop dynamic characteristics. This method allows to plot the locus of the
closed-loop roots in s-plane as an open-loop parameters varies, thus producing a root
locus.
The damping factor, overshoot and settling time [106] limit the allowable area of
existence of the close-loop roots. The border of each of these parameters can be
represented in s-plane as a straight line.
The allowable area of existence for the close-loop roots limited by dumping and
settling time is shown in Fig. 4.14.
78
4.3. Analysis and Controller Design for DTC-SVM Method
damping
settling
time
Im
α
α
Re
damping
Fig. 4.14. Allowable area of existence for the close-loop roots in s-plane
To plot and analyze the locus of the root in s-plane SISO Design Tool Control
System Toolbox v 5.0 the MathWorks, Inc. was used [84].
The SISO Design Tool is a Graphical User Interface (GUI) that allows to analyze
and tune the Single Input Single Output (SISO) feedback control systems. Using the
SISO Design Tool, it is possible to graphically tune the gains and dynamics of the
compensator (C) and prefilter (F), using a mix of root locus and loop shaping
techniques. The example window of the SISO Design Tool is shown in Fig. 4.15. In the
upper right area of the window, the currently tested control structure is displayed. More
on the left the values of the compensator parameters are visible, and below them the
resulting root-locus of the system is shown. In the root locus diagram, two lines
corresponding to the inserted values of settling time and the overshoot are also visible.
79
4. Direct Flux and Torque Control with Space Vector Modulation (DTC-SVM)
Fig. 4.15. SISO Design Tool
Configuration of the system structure is possible by importing transfer functions of
each block from the workspace. This is shown in Fig. 4.16.
Fig. 4.16. Import system data
80
4.3. Analysis and Controller Design for DTC-SVM Method
The plant (G) is a transfer function of the motor torque or flux and the compensator
(C) is a transfer function of the PI controller.
In the cases of flux and torque control, the open-loop consists of a PI controller and
plant transfer function, according to scheme (Fig. 4.8). The plant transfer function for
the flux and the torque are calculated separately based on the motor model equation in
the stator flux reference frame (4.10 - 4.12).
Flux Controller Design
Based on the motor model equations (4.10 - 4.12), the following equation can be
obtained:
2

d
d

d 
(
)
+
=
+
+
+
R
L
σ
L
L
U
R
R
R
L
R
L
σ
L
L
 r s
 sx  s r
 Ψ s
s r
r s
s r
s r
dt 
dt

 dt  

+ Rs I syσLs Lr (Ωss − pb Ωm )
(4.21)
2
L
where: σ = 1 − M
Ls Lr
Under the assumption that the last term in the equation (4.21) is very small:
Rs I syσLs Lr (Ω ss − pb Ωm ) ≈ 0
(4.22)
the equation (4.21) becomes:
2

d
d

d 
 Rr Ls + σLs Lr U sx =  Rs Rr + (Rr Ls + Rs Lr ) + σLs Lr   Ψ s
dt 
dt

 dt  

(4.23)
Based on the equations (4.23) the open-loop flux transfer function can be obtained as
follows:
GΨ (s ) =
where: AΨ =
Ψs
A +s
= 2 Ψ
U sx s + BΨ s + CΨ
(4.24)
RR
R L + R s Lr
Rr
; BΨ = r s
; CΨ = s r
σ Lr
σL s Lr
σL s L r
The flux control loop is shown in Fig. 4.17, where GRΨ (s ) is a transfer function of
the PI controller given by equation (4.17).
81
4. Direct Flux and Torque Control with Space Vector Modulation (DTC-SVM)
Ψ sc
U sx
GRΨ (s )
Ψs
GΨ (s )
Fig. 4.17. Flux control loop
The input data to the SISO Design Tool are obtained based on equations (4.17) and
(4.24). The parameter values are calculated for a 3 kW motor. The motor data are given
in appendix A.3. Required control parameters are set as follows: settling time < 0.003,
overshoot < 4.33%. For these parameters a root loci of the close-loop is obtained, see
Fig. 4.18.
Root Locus Editor (C)
0.93
0.78
0.87
0.46
0.64
0.24
1500
0.97
1000
0.992
Imag Axis
500
4e+003
0
3e+003
2e+003
1e+003
-500
0.992
-1000
0.97
-1500
0.93
-4500
-4000
0.87
-3500
-3000
-2000
0.46
0.64
0.78
-2500
-1500
-1000
0.24
-500
0
Real Axis
Fig. 4.18. Root loci of the close-loop stator flux control system
From the position of the poles, the parameters of the PI flux controller are obtained:
K pΨ = 2531 , TiΨ = 0.00074 .
The behaviour of the flux control loop with parameters like above was tested using
SABER simulation package. The model created in SABER takes into account the full
control system, including the models of inverter and induction motor (see appendix
A.2). The flux step response is presented in Fig. 4.19. The simulation result confirms a
good dynamics of the flux and proper operation in the steady state.
82
4.3. Analysis and Controller Design for DTC-SVM Method
Fig. 4.19. Simulated (SABER) flux response for controller parameters designed
according to root locus method
Torque Controller Design
Based on motor model equations (4.10 - 4.12), the following equation can be
obtained:
d

(Rs Lr + Rr Ls ) + σLs Lr dt  I sy = LrU sy − LrΨ s pb Ωm + I sxσLs Lr (Ω ss − pb Ωm )
(4.25)
where: σ = 1 −
2
LM
Ls Lr
Under the assumption that the last term in equation (4.25) is very small:
I sxσLs Lr (Ωss − pb Ωm ) ≈ 0
(4.26)
the equation (4.25) becomes:
d

(
)
R
L
+
R
L
+
σ
L
L
I sy = LrU sy − LrΨ s pb Ωm
s
r
r
s
s
r

dt 
(4.27)
The additional assumption is that the motor is not loaded M L = 0 .
Under those assumptions the rotor speed can be expressed:
83
4. Direct Flux and Torque Control with Space Vector Modulation (DTC-SVM)
dΩm 1
m
= p b s Ψ s I sy
dt
J
2
(4.28)
From equation (4.14) current I sy can be expressed as follows:
I sy = M e
2
pb m sΨ s
(4.29)
If both sides of equation (4.27) are differentiated, this equation becomes:
2

dU sy
dΩm
d
d 
− LrΨ s p b
(Rs Lr + Rr Ls ) + σLs Lr    I sy = Lr
dt
dt
dt
 dt  

(4.30)
Based on the equations (4.30), (4.28) and (4.29) the open-loop torque transfer
function can be obtained as follows:
GM (s ) =
where: AM =
Me
AM s
= 2
U sy s + BM s + CM
(4.31)
2
2
R L + Rr Ls
pb msΨ s
p mΨ
; BM = s r
; CM = b s s
2σLs J
2σLs
σLs Lr
The torque control loop is shown in Fig. 4.20, where GRM (s ) is a transfer function of
the PI controller given by equation (4.17).
M ec
GRM (s )
U sy
GM (s )
Me
Fig. 4.20. Torque control loop
The input data to the SISO Design Tool are obtained in the same way like for the
flux. The transfer functions are calculated for the 3 kW motor from the equation (4.17)
and (4.31). The required control parameters are set as follows: settling time < 0.0015,
overshoot < 2%. For these parameters a root loci of the close-loop is obtained, see Fig.
4.21. From the position of the poles (Fig. 4.21), the parameters of the PI torque
controller are obtained: K pM = 33.21 , TiM = 0.00045 .
84
4.3. Analysis and Controller Design for DTC-SVM Method
Root Locus Editor (C)
0.93
2500
2000
0.78
0.87
0.48
0.66
0.24
0.97
1500
1000
0.992
Imag Axis
500
7e+003
0
5e+003
6e+003
4e+003
3e+003
2e+003
1e+003
-500
-1000
0.992
-1500
-2000
0.97
-2500
0.93
0.78
0.87
-7000
-6000
-5000
-4000
Real Axis
0.24
0.48
0.66
-3000
-2000
-1000
0
Fig. 4.21. Root loci of the close-loop torque control system
The transfer function of the close loop torque control shown in Fig. 4.20 is given as:
AM K pM
M
GMc (s ) = e =
M ec
TiM
(TiM s + 1)
s + (AM K pM + BM )s + CM +
2
(4.32)
AM K pM
TiM
The SISO Design Tool enables to observe the step response of the investigated
control system. In the Fig. 4.22 is shown the step response of the torque control system
from Fig. 4.20 described by equation (4.32), with the PI controller parameters setting as:
K pM = 33.21 , TiM = 0.00045 .
Step Response
From: r
1.4
1.2
0.8
To: y
Amplitude
1
0.6
0.4
0.2
0
0
0.5
1
1.5
2
Time (sec)
2.5
3
3.5
4
-3
x 10
Fig. 4.22. Simulated (Matlab) step response of the system from Fig. 4.20 described by transfer
function given by equation (4.32)
85
4. Direct Flux and Torque Control with Space Vector Modulation (DTC-SVM)
It should be note that moment of inertia J can change during drive operation (for
example in still industry systems). However, the value of coefficient CM , in equation
(A
(4.32) normally is several order lower in comparison with
M
K pM TiM ) . Therefore,
it’s influence on torque close loop dynamic can be neglected.
Because of the forcing element in transfer function (4.32) the step response presented
in Fig. 4.22 characterized much higher overshoot then the assumed 2%.
To compensate the forcing element in the numerator (4.32) a prefilter is inserted into
the reference channel of the torque controller. The transfer function of the prefilter is
given as:
GFM (s ) =
1
TF s + 1
(4.33)
The time constant of the prefilter is equal time constant of the torque controller
TF = TiM .
The full control loop of torque with prefilter is shown in Fig. 4.23. The step response
of this control loop is presented in Fig. 4.24.
M ec
GFM (s )
U sy
GRM (s )
GM (s )
Fig. 4.23. Torque control loop with prefilter
Step Response
From: r
1.4
1.2
0.8
To: y
Amplitude
1
0.6
0.4
0.2
0
0
0.5
1
1.5
2
2.5
Time (sec)
3
3.5
4
4.5
x 10
-3
Fig. 4.24. Simulated (Matlab) step response of the system from Fig. 4.23
86
Me
4.3. Analysis and Controller Design for DTC-SVM Method
Figure 4.24 shows that the torque control loop with a prefilter incorporated into the
reference channel reduces considerably the overshoot.
The behaviour of the torque control loop with the same settings of the parameters
was also tested in SABER simulation model. The torque step response is presented in
Fig. 4.25. The result of simulation confirms a good dynamics of the torque and proper
operation in the steady state.
Fig. 4.25. Simulated (SABER) torque response
Torque Controller Design for High Power Motor
The same method of tuning the controllers was used for a 90 kW motor. The
parameters of this motor can be found in appendix A.3. The required control parameters
are set as follows: for the flux settling time < 0.003, overshoot < 4.33% and for the
torque settling time < 0.0015, overshoot < 2%. The parameters of the controllers are
obtained as follows: flux controller K pΨ = 2592 , TiΨ = 0.00076 and torque controller
K pM = 1.8492 , TiM = 0.00046 .
The simulation model of drive with a 90 kW motor was also build in the SABER
package.
The flux step response is presented in Fig. 4.26. The control loop of the flux is
identical for both motors (Fig. 4.8) and does not depend on the motor parameters.
Therefore, the parameters of the flux controller and the result of simulation (Fig. 4.26)
is very similar to the result for the 3 kW motor (Fig. 4.19).
87
4. Direct Flux and Torque Control with Space Vector Modulation (DTC-SVM)
The torque response for the 90 kW motor is presented in Fig. 4.27. The results of the
simulations (Fig. 4.26, 4.27), similarly like in the case of the small power ratting motor,
confirm a good dynamics of the torque and a proper operation in the steady state.
Fig. 4.26. Simulated (SABER) flux response for 90 kW motor
Fig. 4.27. Simulated (SABER) torque response for 90 kW motor
4.3.3. Summary of Flux and Torque Controllers Design
In the Fig. 4.28 a full control structure of the DTC-SVM scheme is shown. This
scheme is completed on the prefilter, compared to the basic scheme form Fig. 4.5.
88
4.3. Analysis and Controller Design for DTC-SVM Method
The presented above controller tuning algorithm is based on the open-loop transfer
function for the flux (equation 4.24) and for the torque (equation 4.31). These transfer
functions are obtained under the assumptions (4.22) and (4.26) respectively. Because of
the assumed simplifications, the results of full model simulations are slightly differ form
the initially expected values.
Flux
Controller
Ψ sc
PI
U sxc
x− y
SA
U sc
M ec
F
PI
Prefilter
Torque
Controller
U syc
M̂ e
SC
α −β
γˆss
Ψ̂ s
SB
SVM
Us
Flux and
Torque
Estimator
Is
Voltage
Calculation
α −β
ABC
U dc
IA
IB
Fig. 4.28. Full scheme of the DTC-SVM control method
Additional assumption for the torque controller analysis is that the stator flux
magnitude is constant. Therefore, decoupling between flux and torque control loops is
important. In Fig. 4.29 the torque step response (Fig. 4.29a) and magnitude stator flux
step response (Fig. 4.29b) are shown. From Fig. 4.29 can be seen that both controllers
are very fast and decoupling between flux and torque is correct.
The full control structure (Fig. 4.28) is different from the basic scheme, which can be
seen in Fig. 4.8. In the torque reference channel a prefilter is incorporated. The basic
structure assumed four controllers parameters: K pΨ , TiΨ , K pM and TiM . The addition
of the prefilter does not introduce any additional parameters, because the time constant
of the prefilter is equal to the torque controller integrating time TiM (see equation 4.33).
Thus the control methods needs only four parameters.
Additionally, if a very fast torque response is not required, the prefilter time constant
can be increased independently from the torque controller parameters in order to
improve the stability of the system.
89
4. Direct Flux and Torque Control with Space Vector Modulation (DTC-SVM)
a)
b)
Fig. 4.29. Dynamic tests a) torque step change, b) flux step change. From the top: reference and estimated
torque, reference and estimated stator flux
In section 4.3 two methods of flux and torque controller design for DTC-SVM are
presented. The comparison of the result obtained in two methods is summarized in
Table 4.3. The summary is done for the 3kW motor and sampling frequency
f s = 10kHz . The first method uses simplified IM model and is based on symmetric
optimum criterion. However, this approach gives good results only for flux control loop.
The second approach uses dynamic model of IM including rotor parameters and is
based on root locus method. The results obtained in simulation are good for both flux
and torque controllers. However, it is much more complicated than first method.
The dynamic of the flux control loop is very similar in both cases. Therefore, to tune
flux controller symmetry criterion should be used because it is simpler.
90
Ts
pb , ms ,Ψ s , Rs
Torque
Flux
Torque
pM = 8.39%
K pM = 24.00 t nΨ = 0.0015s t nM = 0.0007 s
Torque
Dynamic parameters
TiΨ = 0.00040 TiM = 0.00080 pΨ = 1.6%
K pΨ = 5000
Flux
Controller parameters
L p mΨ
R
AM = r b s s
Root Locus AΨ = r
σLr
2σLs Lr
Method
R L + Rs Lr
R L + Rr Ls K pΨ = 2531 K pM = 33.21 tnΨ = 0.0019s t nM = 0.0009s
BΨ = r s
BM = s r
σLs Lr
σLs Lr
TiΨ = 0.00074 TiM = 0.00045 pΨ = 1.49%
pM = 1.04%
2
2
RR
p mΨ L
CΨ = s r
CM = b s s r
σLs Lr
2σLs Lr J
Symmetry Ts
Criterion T = 1
Method
pure integrator
Flux
Model parameters
Table 4.3. Summary of controller design
4.3. Analysis and Controller Design for DTC-SVM Method
91
4. Direct Flux and Torque Control with Space Vector Modulation (DTC-SVM)
All simulation results for root locus method presented in section 4.3.2 were done at
sampling frequency
f s = 10kHz . However, presented controller design method
provides to obtain controller parameters for different sampling frequency. This aspect
will be presented for the torque controller. When the sampling frequency is changed the
input parameters: settling time and overshoot must be modified. For lower sampling
frequency the dynamic of control loop is decreasing [34]. Thus, for the continuous
analysis, which is used in root locus method, the settling time should be increased and
overshoot reduced.
Table 4.4 shows torque controller parameters calculated for three sampling frequency
values: f s = 10kHz , f s = 5kHz and f s = 2.5kHz .
Table 4.4. Torque controller parameters for different sampling frequency
fs
settling time overshoot
10.0 kHz
0.0015
2%
0.0030
1%
5.0 kHz
0.0060
1%
2.5 kHz
KpΜ
TiΜ
33.21 0.00045
15.88 0.00098
7.12 0.00180
Simulated results obtained for parameters presented in Table 4.4 are shown in Fig.
4.30. The result of simulation confirms a good behavior of the system for all three
sampling frequencies.
The root locus method gives proper results for different motor type. It confirms
results obtained for the 90 kW motor.
The very important features of the DTC-SVM in comparison with classical DTC are
performance in steady state. In the Fig. 4.31 the steady state operation of the DTC-SVM
control system is shown. It can be seen that the line current is sinusoidal and voltage has
an unipolar waveform. Presented in Fig. 4.31 can be compared with simulation results
for classical DTC from Fig. 3.16, where controller just select voltage vectors to reduce
instantaneous flux and torque errors, and does not implement the true PWM. Therefore,
inverter output voltage is not unipolar. This increase switching losses of the
semiconductor power devices.
92
4.3. Analysis and Controller Design for DTC-SVM Method
a)
b)
c)
Fig. 4.30. 3 kW motor torque response for controller parameters calculated according to root locus
method at different sampling frequency a) f s = 10kHz , b) f s = 5kHz , c) f s = 2.5kHz
93
4. Direct Flux and Torque Control with Space Vector Modulation (DTC-SVM)
Fig. 4.31. Steady state operation. From the top: line to line voltage, line current
The features of the DTC-SVM method can be summarized as follows:
•
good dynamic control of flux and torque,
•
constant switching frequency,
•
unipolar voltage thanks to use of PWM block (SVM),
•
low flux and torque ripple,
•
sinusoidal stator currents.
4.4.
Speed Controller Design
If the stator flux is assumed constant, Ψ s = const. , that based on the equations (4.13)
and (4.14) dynamic of IM can be described as:
dΩm 1
= [M e − M L ]
dt
J
(4.34)
A block diagram of the speed control loop is shown in Fig. 4.32, where GRS (s ) is a
transfer function of PI controller (see equation 4.17) and GM (s ) is a transfer function of
'
full torque control loop. In the speed controller design process the filter for the
measured value should be taken into consideration. T f is a time constant of the filter.
The low pass filter is necessary in hardware setup.
94
4.4. Speed Controller Design
ML
Ωmc
GRS (s )
M ec
Me
GM (s )
'
1
J
1
s
Ωm
1
Tf s + 1
Fig. 4.32. Block diagram of the speed control loop
The transfer function of the full torque control loop (Fig. 4.23) can be calculated as:
G M (s ) =
'
Me
= GFM (s ) ⋅ GMc (s )
M ec
(4.35)
where: GMc (s ) - torque control loop transfer function given by equation (4.32),
GFM (s ) - prefilter transfer function given by equation (4.33).
The transfer function GM (s ) can by expressed as:
'
G M (s ) =
'
AM K pM
'
where: AM =
'
AM
' 2
'
BM s + CM s + 1
CM TiM + AM K pM
(4.36)
'
; BM =
TiM (AM K pM + BM )
TiM
'
; CM =
CM TiM + AM K pM
C M TiM + AM K pM
The torque control loop can be approximate by first order integrating part, because
of:
'
BM ≈ 0
(4.37)
The simplified transfer function can be written as:
GM (s ) =
'
'
AM
'
CM s + 1
(4.38)
For the torque controller parameters K pM = 15.87 , TiM = 0.00087 obtained in section
4.3.3 at the sampling frequency f s = 5kHz the transfer function parameters have values:
95
4. Direct Flux and Torque Control with Space Vector Modulation (DTC-SVM)
'
'
'
AM = 0.9944 , BM = 3.563e − 007 , C M = 0.0009329 . Those parameters confirm that
assumption (4.37) is correct.
The step response of the full and simplified transfer function are shown in Fig. 4.33.
25
full transfer
function
20
simplified
transfer function
15
10
5
0
-5
0
0.005
0.01
0.015
Time
0.02
0.025
0.03
Fig. 4.33. Torque response for full and simplified transfer function
For the speed controller parameter design the symmetry criterion can by applied [66].
In accordance with the symmetry criterion the plant transfer function can be written as:
G (s ) =
K c e − sτ 0
sT2 (1 + sT1 )
(4.39)
where: K c = AM ' is gain of the plan, τ 0 is dead time of the inverter ( τ 0 = 0 ideal
converter), T2 = J , and T1 = C + T f is a sum of small time constants. The optimal
controller parameters can be calculated as:
K ps =
T2
J
=
2 K c (T1 + τ 0 ) 2(C + T f
)
Tis = 4(T1 + τ 0 ) = 4(C + T f )
(4.40)
(4.41)
For the filter frequency f f = 25Hz where:
Tf =
96
1
2πf f
(4.42)
4.4. Speed Controller Design
the speed controller parameters are obtained as follows: K ps = 1.33 ; Tis = 0.0292 .
Fig. 4.34, 4.35 and 4.36 show simulation and experimental results for the system
operated with speed controller parameters obtained above. The speed reversals are
presented in Fig. 4.34 and 4.35 for high and small reference speed differences
respectively. The step change of the load torque at constant speed is presented in Fig.
4.36. All presented in Fig. 4.34, 4.35 and 4.36 results confirm proper operation of the
speed control loop.
a)
b)
Fig. 4.34. Speed reversal Ωm = ±100rad / s a) simulated (SABER), b) experimental 1) reference speed
(75 (rad/s)/div), 2) actual speed (75 (rad/s)/div), 3) reference torque (20 Nm/div)
a)
b)
Fig. 4.35. Speed reversal - small signal Ωm = ±5rad / s a) simulated (SABER), b) experimental 1)
reference speed (7.5 (rad/s)/div), 2) actual speed (7.5 (rad/s)/div), 3) reference torque (20 Nm/div)
97
4. Direct Flux and Torque Control with Space Vector Modulation (DTC-SVM)
a)
b)
Fig. 4.36. Load torque step change at Ωm = 100rad / s a) simulated (SABER), b) experimental
1) reference speed (30 (rad/s)/div), 2) actual speed (30 (rad/s)/div), 3) estimated torque (20 Nm/div)
4.5.
Summary
This chapter gives review of DTC-SVM control methods. To analysis and
implementation was chosen DTC-SVM method with close-loop torque and flux control
in stator flux coordinates. Full mathematical analysis of IM drive working with this
control method is presented. Two different flux and torque controllers design algorithm
are analyzed and discussed. Furthermore, speed controller tuning methods is shown.
The flux and torque controller design methods for sampling frequency changes and
different motor power are discussed. The analysis presented in this chapter give
complex knowledge about control structure and controller design methods. Obtained
parameters provide good dynamic and steady state operation of a drive. It is confirmed
by simulation and experimental results presented in this chapter and in Chapter 7.
98
5.
Estimation in Induction Motor Drives
5.1.
Introduction
The vector control methods of induction motor require feedback signals. This is an
information about flux, torque and mechanical speed in drives operated without
mechanical sensor (sensorless operation mode).
There are many different method to obtain these state variables of induction motor.
Basic methods can be divided into three main group [87]:
•
physical methods – based on nonlinear construction of IM [60, 77, 113],
•
mathematical models – used mathematical description of IM and control theory,
•
neural network methods – based on the artificial intelligence techniques [9, 91,
95].
The general classification of the state variables calculation methods is presented in
Fig. 5.1 [87].
Induction motor state variables
calculation methods
Physical
methods
Estimators of
state variables
Mathematical
models
Observer of
state variables
Neural network
methods
Kalman Filter
Fig. 5.1. Classification of induction state variables calculation methods
The mathematical models is based on the space vector equations, which describe
induction motors. Fig. 5.1 shows division of these methods into three groups:
•
estimators of state variables,
•
observer of state variables,
5. Estimation in Induction Motor Drives
•
Kalman filter.
The DTC-SVM method is based on the information about stator flux vector (see
section 4.3). Therefore, it is the most important variable of the motor. Measurement of
flux in motor is difficult and demands special sensor. This solution is very expensive
and complicated. Because of that a method of calculation motor flux was developed.
In vector control methods this part of algorithm is especially important. Estimation
algorithm uses as input signals values, which are simple to measure. There are current
and voltage signals. Obviously new methods aim at reducing number of sensors for
more reliable operation and lower price of a drive.
The motor flux is the main component to calculate torque and speed. Therefore,
accuracy of the estimation flux is very important. Flux estimation is a significant task in
implementing of high-performance motor drives.
The advanced state variables calculation algorithm is characterized by:
•
accuracy in steady and dynamic states,
•
robustness for motor parameters variation,
•
minimal number of sensor,
•
operation in whole speed range,
•
low calculation demanded.
All estimation algorithms based on the motor parameters. These parameters change
in time work of the drive. For instance, with change the temperature. Therefore,
estimation algorithm have to be less sensitive to the parameters variations.
All presented flux estimation algorithms are shown as stator flux estimators, because
of these algorithms work with DTC-SVM structure. In some algorithm rotor flux
estimation is required, but in this case it is convert on stator flux.
5.2.
Estimation of Inverter Output Voltage
Input signals for the estimators are measurements of stator currents and voltages
which are recreated from the switching signals. Switch signals for the each inverter
phase are obtained by control algorithm. The reference voltage vector is realized by
100
5.2. Estimation of Inverter Output Voltage
modulator (see section 2.4). However, duty times are modified by dead-time, which is
requisite for correct inverter operation (see section 2.3). Because of this modification
delivered to the motor voltage is different from reference. To eliminate dead-time effect
there is a special part for compensation of dead-time in control algorithms. Obtained by
vector modulator duty cycles, represented by switching signals SA, SB, SC are modified
to SA', SB', SC' (Fig. 5.2). This modification depends on the phase current direction and is
realized for each phase. Many different dead-time compensation methods are presented
in literature [2, 3, 8, 29, 64, 76]. Thanks to this modification after change signals by
dead-time, a correct voltage vector obtained by controller is delivered to the motor.
Because of that signals SA, SB, SC are used to recreate voltage values. The voltage is
calculated form the equations:
U sα =
2
U dc (D A − 0 . 5 (D B + D C
3
U sβ =
))
(5.1a)
3
U dc (D B − D C )
3
(5.1b)
where DA, DB, DC are duty cycles corresponding to the switching signals SA, SB, SC
and U dc is the voltage of inverter dc-link.
U dc
SA
U sβc
U sαc
Dead
Time
&
Voltage
Drop
Compensation
Vector
Modulator
SB
SC
U sα
U sβ
Voltage
Calculation
SA+
SA-
SA'
S B'
S C'
Dead
Time
S B+
S BS C+
S C-
U dc
Is
Is
Motor
Fig. 5.2. Input signals for the estimators
101
5. Estimation in Induction Motor Drives
In Fig. 5.2 voltage calculation block diagram is shown. Simultaneously with deadtime compensation a voltage drop compensation algorithm is realized. It is especially
important for low speed operation range, when voltage is very low.
The main assumption in voltage calculation method is that identical voltage vector,
which is calculated by a controller is delivered to the motor. It means, proper
information about voltage depends on correct implementation dead-time and voltage
drop compensation algorithms.
Dead – Time Compensation
In order to prevent shortcircuiting an inverter leg, there should be a dead-time (TD)
between the turn-off one switch (IGBT) and the turn-on of the next one (from the same
leg). TD should be larger than the maximum storage time of the switching device. The
effect of the dead-time is a voltage distortion delivered to the motor. The voltage
distortion ∆U is depending on current sign, as can be seen in Fig. 5.3.
a)
b)
T1
U dc
2
S A+
C
IA > 0
A
0
T1
U dc
2
D1
SA+
U dc
2
C
IA < 0
A
0
T2
S A-
D1
C
T2
D2
SA-
U dc
2
C
IA > 0
D2
IA < 0
SA
SA
t
SA+
TD
t
SA+
TD
t
SAUA0
TD
SAt
1
U dc
2
0
1
− U dc
2
t
TD
UA0
t
0
1
− U dc
2
Fig. 5.3. Dead-time effect for different current sing a) I A > 0 , b) I A < 0
102
t
1
U dc
2
t
5.2. Estimation of Inverter Output Voltage
So the real voltage vector across the motor can be expressed as:
U mot = U sc − ∆U
(5.2)
The voltage distortion ∆U can be written as:
∆U = TD f sU dc sign(I s )
(5.3)
where: f s - sampling frequency,
sign(
) - signum function.
The dead-time compensation can be implemented by adjusting the phase duty cycles
as following:
Dk = Dk + TD f s sign(I k )
'
(5.4)
where: k = A, B, C .
This means that the on-time of the upper bridge arm switch is shortened by TD and
for positive current it is increased by the same amount for negative current.
Because of the current has ripple around zero-crossing the algorithm should be
modified. One of the possible solutions is method with current level. In this method the
current level (I level ) is defined, which describes zone around the zero current as:
− I level > I k > I level
(5.5)
If the condition (5.6) is performed the duty cycles are modified as follows:
'
Dk = Dk +
Ik
I level
TD f s sign(I k )
(5.6)
In the other cases the duty cycles are modified according to the equation (5.4).
The value of the current level (I level ) depends on the motor power and can be
deducted experimentally. For 3kW drive the optimal value of current level was
I level = 0.1 A .
The simulated results for the dead-time compensation algorithms are presented in
Fig. 5.4. In this test drive operates with scalar control (U/f=const.) algorithm at
fundamental frequency f = 2 Hz .
103
5. Estimation in Induction Motor Drives
a)
b)
Fig. 5.4. Simulated U/f=const. control method at frequency f = 2 Hz a) without dead-time compensation,
b) with dead-time compensation
From Fig. 5.4a it can be seen that without dead-time compensation the output
currents are considerably distorted and has reduced value. Fig. 5.4b shown simulated
result with dead-time compensation algorithm. Thanks of the compensation proper
voltage is delivered to the motor. Therefore, currents have correct value and currents
waveforms are sinusoidal.
Presented dead-time compensation algorithm was implemented in final control
system.
5.3.
Stator Flux Vector Estimators
The flux vector estimator algorithms can be divided into two groups in terms of the
input signal. The currents and voltages are the input signals to the voltage models (VM),
while the currents and speed or position information are input signals to the current
models (CM). Obviously, for sensorless control structures general voltage models with
many different modifications and improvements are used.
The stator flux can be directly obtained from the motor model equation (2.10a) as
follows:
Ψ̂ s = ∫ (U s − Rs I s )dt
104
(5.7)
5.3. Stator Flux Vector Estimators
This is a classical voltage model of stator flux vector estimation, which obtain flux
by integrating the motor back electromagnetic force (EMF). The block diagram of this
estimator is shown in the Fig. 5.5.
Is
Rs
Us
Ψ̂ s
∫
Fig. 5.5. Voltage model based estimator with pure integrators
This method is sensitive for only one motor parameter, stator resistance. However,
the implementation of pure integrator is difficult because of dc drift and initial value
problems. Moreover, when estimator based on pure integrator in control structure are
additional disadvantages. Using a pure integrator to estimate the stator flux it is not
possible to magnetize the machine if a zero torque command is applied [25]. Moreover,
the dynamic performance is lower and torque oscillations are bigger than in another
stator flux estimation method. Because of that many different stator flux estimation
algorithms based on the voltage model were proposed, which does not approach to the
pure integrator [15, 53, 54, 57, 58].
Voltage Model with Low – Pass Filter (VM-LPF)
The simplest method, which eliminates problems with initial conditions and dc drift,
which appear in pure integrator, is a method with low-pass filter. In this case the
equation (5.7) can be transformed as follows:
ˆ
dΨ
s
ˆ
ˆ −R I − 1 Ψ
= U
s s
s
s
dt
TF
(
)
(5.8)
The block diagram of the method with low-pass filter is presented in Fig. 5.6.
Is
Us
Rs
1
TF
1
s
Ψ̂ s
Fig. 5.6. Flux estimator based on voltage model with low-pass filter
105
5. Estimation in Induction Motor Drives
The estimator stabilization time depends on the low-pass filter time constant TF.
Obviously, the low-pass filter produces some errors in phase angle and a magnitude of
stator flux, especially when the motor frequency is lower than the cutoff frequency of
the filter. Therefore, flux estimator with low-pass filter can be used successfully only in
a limited speed range.
Voltage Model with Compensated Low – Pass Filter (VM-CLPF)
One way to overcome the errors introduced by low-pass filter is compensated
algorithm [48]. The block diagram of flux estimator based on a voltage model with
compensated low-pass filter is presented in Fig. 5.7.
Us
1 − j λ sign ( Ωˆ ss )
1
s + Ωˆ ss λ
Ψ̂ s
γ̂
Ω̂ ss
Ψ̂ s
ss
γ̂
ss
s
Fig. 5.7. Flux estimator based on voltage model with compensated low-pass filter
In presented method the compensation is carried out before low-pass filtering. The
stator flux is given by equation:
ˆ
1 − jλsign(Ωˆ ss )
Ψ
s
=
Es
s + λ Ωˆ ss
(5.9)
where: λ is a positive constant.
The complex-valued gain, instead of calculating the phase error and the gain error, is
used to compensation. Moreover, due to shifting the poles of pure integration from the
origin to − λ Ω̂ ss , the drift problems are avoided. The λ factor can be selected in range
from 0.1 to 0.5. For lower λ the transient performance is better, but a higher value of λ
allows bigger system inexactness.
106
5.3. Stator Flux Vector Estimators
Voltage Model with Reference Flux (VM-RF)
The block diagram of the estimator based on voltage model with reference flux is
presented in Fig. 5.8 [25].
Us
Lr
LM
Is
1 + sτ
Rs
Lsσ
Ψ̂ r
Lsσ
s
Ψ rc
Is
τ
e
1
1 + sτ
jγˆ sr
γˆ
Ψ̂ s
LM
Lr
sr
Fig. 5.8. Flux estimator based on voltage model with rotor flux assumed as reference
This estimator calculates rotor and stator flux vector on the basis of stator voltages
and currents, and simultaneously the difference between reference and estimated rotor
flux magnitude is utilizing to correction estimated values.
In this estimator first a rotor flux vector is calculated based on the equation:
ˆ
dΨ
r
ˆ − Ψ e jγˆsr )
= E r + K (Ψ
rc
r
dt
(5.10)
where K is the gain factor and E r is the rotor back EMF defined as:
Er =
dI
Lr
( U s − R s I s − σL s s )
Lm
dt
Then assuming K = −
ˆ =
Ψ
r
τ
1 + sτ
1
τ
(5.11)
the equation (5.10) can be rewritten yielding:
Er +
1
Ψ rc e jγˆ sr
1 + sτ
(5.12)
where:
s=
d
dt
(5.13)
107
5. Estimation in Induction Motor Drives
From the equation describing the IM in α − β coordinate system (2.15) formulas for
calculation stator flux vector Ψ s are obtained.
ˆ = Lm Ψ
ˆ + σL I
Ψ
s
r
s s
Lr
(5.14)
This estimator works correctly for a wide speed range, ensures good dynamic
performance, eliminates influence of non correct initial values of the flux level.
Moreover, in this algorithm rotor flux is calculated, which is necessary for rotor speed
calculation (see section 5.5). It is important advantage of this estimator.
The flux estimator based on voltage model with reference flux was selected for the
implementation DTC-SVM control structure in sensorless operation mode (see section
6.2). Presented algorithm is compromise between precision of rotor and stator flux
estimation and computing demand.
Current Model in Rotor Coordinated (CM-RC)
The measured currents and mechanical speed are the input signals for the flux
estimator based on the current model in rotor coordinate.
Coordinate system d ′ − q ′ rotates with the angular speed of the motor shaft Ωm ,
which can be defined as follows:
Ωm =
dγm
dt
(5.15)
Taking into consideration number of pole pairs pb angular speed of the coordinate
system d ′ − q ′ is equal ΩK = pb Ωm .
The voltage, currents and fluxes complex space vector can be resolved into
components d ′ and q ′ .
U sK = U sd ′ + jU sq′
I sK = I sd ′ + j I sq′ ,
(5.16a)
I rK = I rd ′ + j I rq′
Ψ sK = Ψ sd ′ + jΨ sq′ , Ψ rK = Ψ rd ′ + jΨ rq′
108
(5.16b)
(5.16c)
5.3. Stator Flux Vector Estimators
The complete set of equations for IM (2.10-2.12) can be transformed to the d ′ − q ′
coordinate system. In this coordinate system the motor model equation can be written as
follows:
U sd ′ = Rs I sd ′ +
U sq′ = Rs I sq′ +
dΨ sq′
dt
+ pb ΩmΨ sd ′
dΨ rd ′
dt
0 = Rr I rd ′ +
0 = Rr I rq′ +
dΨ sd ′
− pb ΩmΨ sq′
dt
(5.17a)
(5.17b)
(5.17c)
dΨ rq′
(5.17d)
dt
Ψ sd ′ = Ls I sd ′ + LM I rd ′
(5.18a)
Ψ sq′ = Ls I sq′ + LM I rq′
(5.18b)
Ψ rd ′ = Lr I rd ′ + LM I sd ′
(5.18c)
Ψ rq′ = Lr I rq′ + LM I sq′
(5.18d)
dΩm 1  ms
(Ψ sd ′ I sq′ − Ψ sq′ I sd ′ ) − M L 
=  pb
dt
J
2

(5.19)
From the equations (5.17-5.17) formulas for the estimated rotor flux can be obtained
[66].
dΨˆ rd ′ 1
=
LM I sd ′ − Ψˆ rd ′
dt
Tr
(
)
(5.20a)
dΨˆ rq′
(
)
(5.20b)
dt
where: Tr =
=
1
LM I sq′ − Ψˆ rq′
Tr
Lr
Rr
The current vector is measured in stationary coordinate α − β . Therefore, current
components I sα , I sβ must be transformed to the system d ′ − q ′ . Similarly, the
estimated rotor flux vector Ψ r , must be transformed from the system d ′ − q ′ to α − β .
109
5. Estimation in Induction Motor Drives
Stator flux vector Ψ s is calculated from the equation (5.14).
Block diagram of the whole stator flux estimator is shown in Fig. 5.9.
I sα
I sα
α −β
I sβ
I sd ′
I sq′
d ′ − q′
LM
1
Tr
LM
1
Tr
∫
∫
Lsσ
Ψˆ rd ′ d ′ − q′
Ψ̂ rα
LM
Lr
Ψ̂ sα
Ψˆ rq′
Ψ̂ rβ
LM
Lr
Ψ̂ sβ
α −β
Isβ
γm
Lsσ
Fig. 5.9. Block diagram of the current model flux estimator in rotor coordinates
This flux estimator model ensures good accuracy over the entire frequency range. It
has a very good behavior in steady and dynamic state. Also it has resistant to wrong
initial conditions. Its disadvantage is sensitive on change motor parameters.
This estimator was selected for the implementation DTC-SVM control structure in
sensor operation mode (see section 6.2).
5.4.
Torque Estimation
The induction motor output torque is calculated based on the equation (2.9), which
for stationary coordinate system α − β can be written as follows:
M e = pb
(
)
(
ms
ˆ * I = p ms Ψˆ I − Ψˆ I
Im Ψ
s s
b
sα sβ
sβ sα
2
2
)
(5.21)
It can be seen that the calculated torque is depended on the current measurement
accuracy and stator flux estimation method.
5.5.
Rotor Speed Estimation
If a flux estimator works properly and rotor flux is accurately calculated mechanical
speed can be obtained from simple motor model equation [87]. If in control structure the
110
5.5. Rotor Speed Estimation
stator flux estimator is applied rotor flux can be calculated based on the equations
(5.14).
In the IM mechanical speed is defined as difference between synchronous speed and
sleep frequency:
Ωm =
1
(Ωsr − Ωsl )
pb
(5.22)
where: Ω sr - rotor synchronous speed,
Ω sl - slip frequency,
pb - number of pole pairs.
The rotor synchronous speed is equal angular speed of the rotor flux vector and can
be calculated as:
Ωsr =
dγ sr
dt
(5.23)
The slip frequency of induction motor is defined as follows [66]:
Ωsl = Ωsr − pb Ωm
(5.24)
Based on the equations (3.3d) and (3.4d) in rotor flux coordinate system the slip
frequency can be expressed:
Ωsl = Rr
LM 1
I sq
Lr Ψ r
(5.25)
Taking into consideration the torque equations (3.7) and (5.25) the estimated sleep
frequency can be calculated as follows:
Ωsl =
(
Rr ˆ
Ψ sα I sβ − Ψˆ sβ I sα
2
ˆ
Ψr
)
(5.26)
Finally mechanical motor speed is calculated from the equation (5.22).
111
5. Estimation in Induction Motor Drives
5.6.
Summary
In this chapter estimation algorithms of flux, torque and rotor speed are presented.
The estimators provide feedback signals for DTC-SVM control scheme. Algorithms
selected to the implementation in final structure are described and discussed.
The speed estimator is based on the estimated stator and rotor fluxes. The mechanical
speed can be calculated in a simple way if motor flux is properly estimated. Therefore,
flux estimation algorithm is the most important part of sensorless control scheme.
Selected flux estimator for the sensorless mode is based on the voltage model. Thus
algorithm is sensitive on accuracy of inverter output voltage calculation. The voltages
are reconstructed from switching signals. In this method dead-time compensation
algorithm is significant. The dead-time effect and compensation algorithm was
presented.
The presented estimation methods are implemented in final DTC-SVM control
structure. The experimental results, presented in Chapter 7 confirm proper operation of
selected estimation methods.
112
6.
Configuration of the Developed IM Drive Based on
DTC-SVM
6.1.
Introduction
In this chapter a whole implemented control system will be presented. In the first
part, the configuration of the system and operation modes are described. In the next
parts, two hardware setups, which were used to verify DTC-SVM control structure are
presented. To development work was used laboratory setup based on dSPACE company
control board DS1103 PPC. This board has powerful microprocessor and special inputoutput interface. The laboratory setup and control board DS1103 will be widely
described in section 6.3. The control algorithm was also implemented in a setup based
on a microcontroller TMS320LF2406 from Texas Instruments company. The
TMS320LF2406 is a 16-bits, fixed point microcontroller devoted for drive application
(see section 6.4).
6.2.
Block Scheme of Implemented Control System
The IM drive based on DTC-SVM control structure can operate in three modes:
•
scalar control,
•
sensor vector control,
•
sensorless vector control.
The inverter operate in a mode which is required by application. The system
configuration depends on the switches position, see Fig. 6.1. The most advanced is the
sensorless vector control mode.
In the scalar control mode algorithm obtains command voltage vector based on the
reference frequency. The command voltage vector is realized by space vector modulator
(SVM).
The reference speed in the command signal in the vector control modes. Depending
on mode the reference speed is compared with measured (sensor vector control mode)
or estimated (sensorless vector control mode) speed signal.
6. Configuration of the Developed IM Drive Based on
Reference
Frequency
Reference
Speed
Speed
Controller
DTC-SVM
Switch 1
Scalar
Control
SVM
References
Value
Torque
and Flux
Controller
Inverter
Measurements
Signals
Estimations
Value
Torque
and Flux
Estimator
Switch 2
Estimation
Speed
Measurment
Speed
Speed
Estimator
Speed
Sensor
Motor
Fig. 6.1. Block scheme of implemented control algorithm
Based on the speed error speed controller calculates reference torque value. The
commanded flux is obtained from the reference speed and selected characteristic, which
depends on the application. The reference values of torque and flux are compared with
estimated values. Based on the errors flux and torque controllers calculate command
voltage vector. The command voltage vector is realized by the same space vector
modulator (SVM) algorithm, which is used in scalar control mode. Therefore, depended
on application requirements change between scalar and vector mode is simple.
The measured current and reconstructed voltage are input signals for the estimation
algorithms (see Chapter 5).
An inverter control structure presented in Fig. 6.1 was implemented for IM.
However, this structure can be also used for Permanent Magnet Synchronous Motor
(PMSM) [129].
All presented in Fig. 6.1 blocks are described in previous chapter of the thesis. The
torque, flux and speed controllers are discussed in Chapter 4. The estimation algorithms
are shown in Chapter 5 and different modulation techniques are presented in Chapter 2.
The experimental results for all three operating modes are presented in Chapter 7.
114
6.3. Laboratory Setup Based on DS1103
6.3.
Laboratory Setup Based on DS1103
The basic structure of the laboratory setup is depicted in Fig. 6.1. The motor setup
consist of induction motor and DC motor, which is used for the loading. The induction
motor is fed by the frequency inverter controlled directly by the DS1103 board. The
dSPACE DS1103 PPC is plugged in the host PC. The DC motor is supplied by a torque
controlled rectifier. The encoder is used for the measure mechanical speed. The DSP
Interface – a set of eurocards mounted in a 19” rack with the main purpose to provide
galvanic isolation to all signals connected to the DS1103 PPC controller.
3
Rectifier
grid
measured
DC line voltage
2
Inverter
SA
SB
SC
3
Rectifier
measured
phase
current
DSP
Interface
encoder
Measurement
AC motor
DC motor
DS1103 dSPACE
Master : PowerPC 604e
Slave: DSP TMS320F240
PC
Fig. 6.2. Structure of the laboratory setup
Fig. 6.3. Laboratory setup
115
6. Configuration of the Developed IM Drive Based on
DTC-SVM
In Fig. 6.3 view of the laboratory setup is shown. All parts of the laboratory setup
can be seen in this picture.
dSPACE DS1103 PPC Board
The dSPACE DS1103 PPC is a mixed RISC/DSP digital controller providing a very
powerful processor for floating point calculations as well as comprehensive I/O
capabilities. Here are the most relevant features of the controller:
•
Motorola PowerPC 604e running at 333 MHz,
•
Slave DSP TI's TMS320F240 Subsystem,
•
16 channels (4 x 4ch) ADC, 16 bit , 4 µs, ±10 V,
•
4 channels ADC, 12 bit , 800 ns, ± 10V,
•
8 channels (2 x 4ch) DAC, 14 bit , ±10 V,6 µs,
•
Incremental Encoder Interface -7 channels
•
32 digital I/O lines, programmable in 8-bit groups,
•
Software development tools (Matlab/Simulink, RTI, RTW, TDE, Control Desk)
The DS1103 PPC card is pluged in one of the ISA slot of the motherboard of a host
computer of the type PIII/900MHz, 512 MBRAM, 40GB HDD, Windows 2000. All the
connections are made through six flat cables (50 wires each) available at the backside of
the desktop computer.
The DS1103 PPC is a very flexible and powerful system featuring both high
computational capability and comprenhensive I/O periphery. The board can be
programmed in C language. Additionally, it features a software SIMULINK interface
that allows all applications to be developed in the Matlab/Simulink user friendly
environment. All compiling and downloading processes are carried out automatically in
the background. An experimenting software called Control Desk, allow real-time
management of the running process by providing a virtual control panel with
instruments and scopes.
The detailed parameters of the dSPACE DS1103 PPC board are given in Appendix
A5.
116
6.3. Laboratory Setup Based on DS1103
Experimenting Software – Control Desk
Control Desk experiment software provides all the functions for controlling,
monitoring, and automation of real-time experiments and makes the development of
controllers more effective. A Control Desk experiment layout for controlling an
induction motor with DTC-SVM control methods is shown in Fig. 6.5.
Fig. 6.4. Control Desk experiment layout
Control Desk package consists of the following modules:
•
The Experiment Management - assures a consistent data management controlling
all the data relevant for an experiment. The experiment can be loaded as a
complete set of data with a single operation. The content of the experiment can
be defined by the user.
•
The Hardware Management - allows you to configure the dSPACE hardware and
to handle real-time applications with a graphical user interface.
•
The Instrumentation Kits - offer a variety of virtual instruments to build and
configure virtual instrument panels according to your special needs.
117
6. Configuration of the Developed IM Drive Based on
DTC-SVM
Using data acquisition instruments you can capture data from the model running on
the real-time hardware. Changing parameter values is performed by operating input
instruments. The integrated Parameter Editor allows you to read the current parameter
values from the hardware and to change a parameter set in one step.
6.4.
Drive Based on TMS320LF2406
DTC-SVM control algorithm was implemented in the drive based on microcontroller
TMS320LF2406. Setup consists of 18 kVA IGBT inverter and 15 kW induction motor.
The view of inverter is shown in Fig. 6.5. In this picture main control board of the
inverter with microprocessor module can be seen.
Fig. 6.5. 18 kVA inverter controlled by TMS320FL2406 processor
118
6.4. Drive Based on TMS320LF2406
The motor set (Fig. 6.6), which was used in tests consists of 15 kW induction motor
and 22 kW DC motor. The induction motor data are given in appendix A.3. The DC
motor works as a load and it is supply from the controlled rectifier.
Fig. 6.6. Motor set. From the left 22 kW DC motor and 15 kW IM motor.
Fig. 6.7. TMS320LF2406 microprocessor board
119
6. Configuration of the Developed IM Drive Based on
DTC-SVM
The microprocessor board shown in the Fig. 6.7 was used to control the inverter. The
sizes of the processor module are 53x56mm. This board contains microcontroller
TMS320LF2406 and required equipment. The communication with main inverter board
by three connectors (2x20pins and 1x26pins) is provided.
The TMS320Lx240xA series of devices are members of the TMS320 family of
digital signal processors (DSPs) designed to meet a wide range of digital motor control
(DMC) and other embedded control applications [99, 100]. This series is based on the
C2xLP 16-bit, fixed-point, low-power DSP CPU, and is complemented with a wide
range of on-chip peripherals and on-chip ROM or flash program memory, plus on-chip
dual-access RAM (DARAM).
The TMS320 family consists of fixed-point, floating-point, multiprocessor digital
signal processors (DSPs), and fixed-point DSP controllers. TMS320 DSPs have an
architecture designed specifically for real-time signal processing. The 240xA series of
DSP controllers combine this real-time processing capability with controller peripherals
to create an ideal solution for control system applications. There are short characteristics
of the TMS320 family:
•
flexible instruction set,
•
operational flexibility,
•
high-speed performance
•
Innovative parallel architecture,
•
cost effectiveness.
Devices within a generation of a TMS320 platform have the same CPU structure but
different on-chip memory and peripheral configurations. Spin-off devices use new
combinations of on-chip memory and peripherals to satisfy a wide range of needs in the
worldwide electronics market. By integrating memory and peripherals onto a single
chip, TMS320 devices reduce system costs and save circuit board space.
The detailed parameters of the TMS320FL2406 microprocessor are given in
Appendix A6.
The important feature of the TMS320FL246 microprocessor is the bootloader.
Thanks to that it is possible to program the device using Serial Communications
120
6.4. Drive Based on TMS320LF2406
Interface (SCI) or Serial Peripheral Interface (SPI). Therefore, program can be loaded
from the PC via standard serial port (RS232).
This way of programming was used during the implementation of DTC-SVM control
algorithm. Thus it was possible to work with the processor without using the expensive
tools like JTAG.
121
7.
Experimental Results
7.1.
Introduction
In this chapter selected experimental results obtained in the system described in
Chapter 6 are shown. All tests was done for 3 kW induction motor, which parameters
are given in Appendix A3.
7.2.
Pulse Width Modulation
In Fig. 7.1 – 7.5 different modulation method are presented. All test was measured at
frequency f = 40 Hz .
In Fig. 7.1 space vector modulation method with symmetrical zero vectors placement
– SVPWM is shown (see section 2.4.3).
Fig. 7.1. Space vector modulation (SVPWM) at frequency f = 40 Hz 1) switching signal SA,
2) pole voltage UA0 (150 V/div), 3) phase voltage UA (150 V/div), 4) output current IA (5 A/div)
In Fig. 7.2 discontinuous pulse width modulation – DPWM2 is shown (see section
2.4.3). It can be observe differences in pole voltage waveforms and switching signal in
Fig. 7.1 and 7.2. DPWM2 modulation method has 60º no switch sectors. However,
phase voltage and output current have sinusoidal waveforms.
7.2. Pulse Width Modulation
Fig. 7.2. Discontinuous modulation (DPWM2) at frequency f = 40 Hz 1) switching signal SA,
2) pole voltage UA0 (150 V/div), 3) phase voltage UA (150 V/div), 4) output current IA (5 A/div)
In Fig. 7.3 and 7.4 overmodulation (OM) algorithm is shown (see section 2.4.5).
Fig. 7.3. Overmodulation mode I at frequency f = 40 Hz 1) switching signal SA, 2) pole voltage
UA0 (150 V/div), 3) phase voltage UA (150 V/div), 4) output current IA (5 A/div)
123
7. Experimental Results
Fig. 7.4. Overmodulation mode II at frequency f = 40 Hz 1) switching signal SA, 2) pole voltage
UA0 (150 V/div), 3) phase voltage UA (150 V/div), 4) output current IA (5 A/div)
The results for six-step mode are presented in Fig. 7.5.
Fig. 7.5. Six-step mode at frequency f = 40 Hz 1) switching signal SA, 2) pole voltage UA0 (150 V/div),
3) phase voltage UA (150 V/div), 4) output current IA (10 A/div)
Results presented in Fig. 7.3 – 7.5 ware obtained at decreased dc-link voltage.
Therefore, overmodulation and six-step operation modes can be shown with frequency
124
7.3. Flux and Torque Controllers
f = 40 Hz like the other results. Thanks to it, current and voltage waveforms can be
better compared.
Experimental results presented in Fig. 7.1 – 7.5 confirm proper operation all type
modulation algorithms.
7.3.
Flux and Torque Controllers
Dynamic tests for the flux and torque controller were done for different sampling
frequencies values and the same condition like for simulation presented in section 4.3
(motor speed Ωm = 0 ). The flux controller parameters were calculated according to
symmetric optimum criterion (see section 4.3.1) and torque controller parameters were
calculated according to root locus method (see section 4.3.2).
In Fig. 7.6 – 7.8 are presented stator flux step response at sampling frequency
f s = 10 kHz , f s = 5 kHz , f s = 2.5 kHz respectively. Those results can be compared
with simulation results presented in Fig. 4.11.
Fig. 7.6. Stator flux response at sampling frequency f s = 10 kHz 1) reference flux (0.15 Wb/div),
2) estimated flux (0.15 Wb/div)
125
7. Experimental Results
Fig. 7.7. Stator flux response at sampling frequency f s = 5 kHz 1) reference flux (0.15 Wb/div),
2) estimated flux (0.15 Wb/div)
Fig. 7.8. Stator flux response at sampling frequency f s = 2.5 kHz 1) reference flux (0.15 Wb/div),
2) estimated flux (0.15 Wb/div)
Presented in Fig. 7.6 – 7.8 experimental results confirm proper operation of the flux
control loop at different sampling frequency.
126
7.3. Flux and Torque Controllers
The experimental results of torque controller dynamic test are shown in Fig. 7.9 –
7.11. Presented results were obtain at sampling frequency f s = 10 kHz (Fig. 7.9),
f s = 5 kHz (Fig. 7.10), f s = 2.5 kHz (Fig. 7.11).
Fig. 7.9. Torque response at sampling frequency f s = 10 kHz 1) reference torque (4.5 Nm/div),
3) estimated torque (4.5 Nm/div)
Fig. 7.10. Torque response at sampling frequency f s = 5 kHz 1) reference torque (4.5 Nm/div),
3) estimated torque (4.5 Nm/div)
127
7. Experimental Results
Fig. 7.11. Torque response at sampling frequency f s = 2.5 kHz 1) reference torque (4.5 Nm/div),
3) estimated torque (4.5 Nm/div)
The result from Fig. 7.9 – 7.11 can be compared with simulation results presented in
Fig. 4.30. Experimental results presented in Fig. 7.9 – 7.11 confirm proper operation of
the torque control loop at different sampling frequency.
The decoupling between flux and torque control loops is presented in Fig. 7.12. The
torque step response (Fig. 7.12a) and magnitude stator flux step response (Fig. 7.12b)
are shown.
a)
128
7.4. DTC-SVM Control System
b)
Fig. 7.12. Dynamic tests a) torque step change, b) flux step change
1) reference torque (9 Nm/div), 2) estimated torque (9 Nm/div),
3) reference flux (0.3 Wb/div), 4) estimated flux (0.3 Wb/div)
The results from Fig. 7.12 can be compared with simulation results presented in Fig.
4.29. From Fig. 7.12 can be seen that decoupling between flux and torque is correct.
7.4.
DTC-SVM Control System
In this section the experimental result for three possible drive operation modes,
which are described in Chapter 6 are shown. Therefore, comparison of a system
behavior in different modes is possible.
In Fig. 7.13 – 7.16 results for scalar control mode are presented. Fig. 7.13 gives
result for system startup to frequency f = 40 Hz (motor speed Ωm = 125rad / s ).
129
7. Experimental Results
Fig. 7.13. Scalar control mode - Startup from 0 to f = 40 Hz 1) reference frequency (25 Hz/div),
2) actual speed (30 (rad/s)/div, 4) phase current (10 A/div)
The load torque step change at frequency f = 25Hz is shown in Fig. 7.14.
Fig. 7.14. Scalar control mode - Load torque step change from 0 to M L = M N at frequency f = 25Hz
1) reference frequency (25 Hz/div), 2) actual speed (30 (rad/s)/div), 3) torque (20 Nm/div),
4) phase current (10 A/div)
In Fig. 7.15 and 7.16 result of speed reverses are shown ( f = ±25Hz ). The reverse
time is 0.5s (Fig. 7.15) and 5s (Fig. 7.16).
130
7.4. DTC-SVM Control System
Fig. 7.15. Scalar control mode - Speed reversal f = ±25 Hz (reverse time 0.5s) 1) reference frequency
(25 Hz/div), 2) actual speed (30 (rad/s)/div), 4) phase current (10 A/div)
Fig. 7.16. Scalar control mode - Speed reversal f = ±25 Hz (reverse time 5s) 1) reference frequency
(25 Hz/div), 2) actual speed (30 (rad/s)/div), 4) phase current (10 A/div)
In Fig. 7.17 – 7.20 results for sensor vector control mode are presented. Fig. 7.17
gives result for system startup to speed Ωm = 120 rad / s .
131
7. Experimental Results
Fig. 7.17. Vector control mode with speed sensor - Startup from 0 to Ωm = 120 rad / s 1) reference speed
(30 (rad/s)/div), 2) actual speed (30 (rad/s)/div, 4) phase current (10 A/div)
The load torque step change at speed Ωm = 75 rad / s is shown in Fig. 7.18.
Fig. 7.18. Vector control mode with speed sensor - Load torque step change from 0 to M L = M N at
speed Ωm = 75 rad / s 1) reference speed (30 (rad/s)/div), 2) actual speed (30 (rad/s)/div),
3) torque (20 Nm/div), 4) phase current (10 A/div)
In Fig. 7.19 and 7.20 result of speed reverses are shown ( Ωm = ±75rad / s ). The
reverse time is 0.5s (Fig. 7.19) and 5s (Fig. 7.20).
132
7.4. DTC-SVM Control System
Fig. 7.19. Vector control mode with speed sensor - Speed reversal Ωm = ±75rad / s (reverse time 0.5s)
1) reference speed (30 (rad/s)/div), 2) actual speed (30 (rad/s)/div), 4) phase current (10 A/div)
Fig. 7.20. Vector control mode with speed sensor - Speed reversal Ωm = ±75rad / s (reverse time 5s)
1) reference speed (30 (rad/s)/div), 2) actual speed (30 (rad/s)/div), 4) phase current (10 A/div)
In sensorless vector control mode the accuracy of the speed estimation algorithm
is important. Therefore, static and dynamic error of estimated speed were
investigated. The error of estimated speed can be written as:
133
7. Experimental Results
ε Ωm =
Ωm − Ωˆ m
100%
Ωm
(7.1)
where:
Ωm - actual speed, Ω̂m - estimated speed.
In Fig. 7.21 speed estimation error as the function of mechanical speed in steady
state is presented.
ε Ωm [%]
50
45
40
error_omega [%]
35
30
25
20
15
10
5
0
0
5
10
15
20
25
omega_m [rad/s]
30
35
40
45
50
Ωm [rad/s]
Fig. 7.21. Estimated speed error as the function of mechanical speed in steady state.
The results of speed estimator dynamic test are presented in Fig. 22. In this test speed
controller operates with the sensor and speed estimator work in open loop fashion.
134
7.4. DTC-SVM Control System
Fig. 7.22. Dynamic test of the speed estimation - Speed reversal Ωm = ±50rad / s 1) reference speed
(30 (rad/s)/div), 2) actual speed (30 (rad/s)/div), 3) estimated speed (30 (rad/s)/div),
4) error of estimated speed (25 %/div)
In Fig. 7.23 – 7.26 results for sensorless vector control mode are presented. Fig. 7.23
gives result for system startup to speed Ωm = 120 rad / s .
Fig. 7.23. Sensorless vector control mode - Startup from 0 to Ωm = 120 rad / s 1) reference speed
(30 (rad/s)/div), 2) actual speed (30 (rad/s)/div, 4) phase current (10 A/div)
The load torque step change at speed Ωm = 75 rad / s is shown in Fig. 7.24.
135
7. Experimental Results
Fig. 7.24. Sensorless vector control mode - Load torque step change from 0 to M L = M N at speed
Ωm = 75 rad / s 1) reference speed (30 (rad/s)/div), 2) actual speed (30 (rad/s)/div),
3) torque (20 Nm/div), 4) phase current (10 A/div)
In Fig. 7.25 and 7.26 result of speed reverses are shown ( Ωm = ±75rad / s ). The
reverse time is 0.5s (Fig. 7.25) and 5s (Fig. 7.26).
Fig. 7.25. Sensorless vector control mode - Speed reverse Ωm = ±75rad / s (reverse time 0.5s)
1) reference speed (30 (rad/s)/div), 2) actual speed (30 (rad/s)/div), 4) phase current (10 A/div)
136
7.4. DTC-SVM Control System
Fig. 7.26. Sensorless vector control mode - Speed reverse Ωm = ±75rad / s (reverse time 5s)
1) reference speed (30 (rad/s)/div), 2) actual speed (30 (rad/s)/div), 4) phase current (10 A/div)
137
8.
Summary and Conclusions
In this thesis the most convenient industrial control scheme for voltage source
inverter-fed induction motor drives was searched for, based on the existing control
methods. This method should provide: operation in wide power range, guarantee good
and repeatable parameters of drive. It is required by a serial production of a drive. To
achieve a low costs the control system should be implemented in simple
microprocessor. The analysis of existing methods were done in order to chose the
industrial oriented universal scheme.
The most important control techniques of IM were presented in Chapter 3: Field
Oriented Control (FOC), Feedback Linearization Control (FLC) and Direct Torque
Control (DTC). The FLC structure guarantees exact decoupling of the motor speed and
rotor flux control in both dynamic and steady states. However, it is complicated and
difficult to implement in practice. This method requires complex computation and
additionally it is sensitive to changes of motor parameters. Because of these features
this method was not chosen for implementation. In next step FOC and DTC methods
were analyzed. Characteristics of those methods were done on the basis of the literature,
simulation and experimental investigation. The conclusions of those consideration were
shown in section 3.5.
Analysis of advantages and disadvantages of FOC and DTC methods resulted in a
search for method which will eliminate disadvantages and keep advantages of those
methods. The direct torque control with space vector modulation (DTC-SVM) is an
effect of this search. The main features of this method can be summarized as:
•
Space vector modulator,
•
Constant switching frequency,
•
Unipolar voltage thanks to use of PWM block (SVM),
•
Sinusoidal waveform of stator currents,
•
Algorithm operates with torque and flux value – implementation in
manufacturing process is easier,
•
Good dynamic control of flux and torque. The step responses are slower than in
classical DTC, because PI controllers are slower than hysteresis controllers,
8. Summary and Conclusions
which are used in classical DTC. However, obtained dynamic (response time for
the torque 1.5-2ms) is sufficient for general purpose drives.
•
High sampling frequency is not required. The DTC-SVM algorithm works
properly at sampling frequency f s = 5kHz whereas DTC requires sampling
frequency at least 25 − 40kHz .
•
Low flux and torque ripple than in classical DTC. The torque ripples in DTC-SVM
at sampling frequency f s = 5kHz are ten times lower than presented in section
3.4.2 torque ripples for classical DTC at sampling frequency f s = 40kHz .
The DTC-SVM scheme is based only on the analysis of stator equations like classical
DTC, therefore control algorithm is not sensitive to rotor parameters changes. This
method can be applied also for surface mounted permanent magnet (PM) synchronous
motors [129]. The PM synchronous motors of this type are more frequently used in
standard speed drives as interior PM. Hence, DTC-SVM method allows universal drive
building for both types of AC motors.
The very important part of DTC-SVM scheme is a space vector modulator. The
different modulation techniques can be applied in the system. Therefore, a drive has
additional advantages. The most important is full range of voltage control and reduction
of switching losses. For instance, reduction of switching losses can be obtained by
implementation of discontinuous PWM methods. These modulation techniques were
described and characterized in section 2.4. The experimental results for the
implemented modulation methods were shown in Chapter 7.
The short review of DTC-SVM methods proposed in literature were given in section
4.2. For further consideration the DTC-SVM method with close-loop torque and flux
control in stator flux Cartesian coordinates have been chosen. In author opinion this
method is best suited for commercial manufactured drives. For chosen scheme two
controller design procedures were proposed. Those analysis were presented in Chapter 4.
Also correction of controllers parameters for sampling frequency changes was discussed.
In adjustable speed drive superior speed controller is used. The analysis of speed
control loop and controller tuning were presented in section 4.4. Correctness of used
method was confirmed by simulation and experimental results.
139
8. Summary and Conclusions
The quality of regulation process depends on an accuracy of feedback signals. In the
vector control of induction motor those signals are provided by flux and torque
estimators and, in sensorless operation mode, by a speed estimator. The precision of
estimated signals depends on:
•
exact knowledge of motor parameters,
•
good dead-time and voltage drop compensation algorithms,
•
well realized measurements,
•
implementation of on-line adaptation of motor parameters.
Those features are common for all vector control methods. Therefore, if feedback
signals are estimated accurately, the control scheme should be as simple as possible.
The DTC-SVM has a simple structure and it can be analyzed and implemented in a
simple way. It is very important feature of DTC-SVM.
Estimation problems in a drive with induction motor were discussed in Chapter 5.
Following estimation algorithms, selected for implementation, were presented: voltage
estimator with dead-time compensation algorithm, stator flux estimator, torque
estimator and mechanical speed estimator.
All parts of control scheme were verified in simulation and experiment. The whole
scheme consists of: flux and torque controllers, speed controller, estimation of flux,
torque and speed and compensation algorithms. Those complete structure was presented
in Chapter 6. Proposed solution was implemented in 3 kW experimental and 15 kW
industrial drives. The laboratory setups were also presented in Chapter 6.
Presented in Chapter 7 experimental results confirm proper operation of developed
control system.
Thus, thesis shows the process to select and develop the most convenient control
scheme for voltage source inverter-fed induction motor drives. Whole problems of
direct flux and torque control with space vector modulation (DTC-SVM) were analyzed
and investigated in simulation and experiment.
Finally, it should be stressed that the developed system was brought into serial
production. Presented algorithm has been used in new family of inverter drives
produced by Polish company Power Electronic Manufacture – „TWERD”, Toruń.
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References
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Warsaw University of Technology, Warsaw, May 2001, pp.370-372.
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[123] M. Żelechowski, M. Malinowski, P. Kaczyński, W. Kołomyjski, M. Twerd, J. Załęski, "DSP
Based Sensorless Direct Torque Control – Space Vector Modulated (DTC-SVM) for Inverter Fed
Induction Motor Drives", Problems of Automated Electrodrives Theory and Practice, Crimea,
Ukraine, Sep. 2003, pp.90-92.
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and Torque Control of Space Vector Modulated AC/DC/AC Converter - Fed Induction Motor",
Control in Power Electronics & Electrical Drives, SENE 2003, Łódź, Nov. 2003, pp.179-185.
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Przegląd Elektrotechniczny, No. 1/2004, pp.6-10. (in Polish)
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Control – Space Vector Modulated (DTC-SVM) for Inverter Fed Induction Motor Drives", IV
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Estonia, June 2004, pp.77-79.
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Nordick Network for Multi Disciplinary Optimised Electric Drives, Tallinn, Estonia, June 2004,
pp.101-107.
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control both induction and PM synchronous motor”, In Proc. of the EPE- PEMC, Riga, Latvia,
Sep. 2004.
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M.P. Kaźmierkowski, M. Żelechowski, "Unified Scheme of Direct Power and
Torque Control for Space Vector Modulated AC/DC/AC Converter- Fed Induction Motor", In
Proc. of the EPE- PEMC, Riga, Latvia, Sep. 2004.
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Induction and PM Synchronous Motor", XVI International Conference on Electrical Machines
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Industrial Application", IX Sympozjum - Energoelektronika w Nauce i Dydaktyce ENID’2004,
Poznań, Sep. 2004, pp. 115-122.
150
List of Symbols
1
3
a = e j2 π 3 = − + j
2
2
B - viscous constant
f - frequency
f s - sampling frequency
f sw - switching frequency
I - current, absolute value
I A , I B , I C - instantaneous values of stator phase currents
I r - rotor current space vector
I s - stator current space vector
I sα , I sβ - stator voltage vector components in stationary α − β coordinate
system
I rα , I rβ - rotor voltage vector components in stationary α − β coordinate system
k - space vector, generally
K p - controller gain
K pM - torque controller gain
K pΨ - flux controller gain
L - inductance, absolute value
LM - main, magnetizing inductance
Ls - stator winding self-inductance
Lr - rotor winding self-inductance
M - mutual inductance, absolute value
List of symbols
M - torque, absolute value
M e - electromagnetic torque
M L - load torque
M , m - modulation index
ms - number of phase windings
pb - number of pole pairs
SA, SB, SC - switching states for the voltage source inverter
R - resistance, absolute value
Rr - rotor phase windings resistance
Rs - stator phase windings resistance
Ti - controller integrating time
TiM - torque controller integrating time
TiΨ - flux controller integrating time
TD - dead time of inverter
Tr =
Lr
- rotor time constant
Rr
Ts - sampling time
Tsw - switching time
U - voltage, absolute value
U A , U B , U C - instantaneous values of the stator phase voltages
U s - stator voltage space vector
U r - rotor voltage space vector
U ν - inverter output voltage space vectors, ν = 0,...,7
U c - reference voltage vector
152
List of symbols
U sα ,U sβ - stator voltage vector components in stationary α − β coordinate
system
U sαc ,U sβc - reference stator voltage vector components in stationary α − β
coordinate system
U sdc ,U sqc - reference stator voltage vector components in rotating d − q
coordinate system
U dc - inverter dc link voltage
U m (n ) - peak value of the n-th harmonic, n = 1, 2, 3,…
U Ac , U Bc , U Cc - reference stator phase voltages
U t - triangular carrier signal
U AB , U BC , U CA - line to line voltages
Ψ - flux linkage, absolute value
Ψ A , Ψ B , Ψ C - flux linkages of the stator phase windings
Ψ s - space vector of the stator flux linkage
Ψ r - space vector of the rotor flux linkage
Ψ s - stator flux amplitude
Ψ r - rotor flux amplitude
Ψ sα ,Ψ sβ - stator flux vector components in stationary α − β coordinate system
Ψ rβ ,Ψ rβ - rotor flux vector components in stationary α − β coordinate system
γ m - motor shaft position angle
γ sr - rotor flux vector angle
γ ss - stator flux vector angle
Ω - angular speed, absolute value
153
List of symbols
ΩK - angular speed of the coordinate system
Ωm - angular speed of the motor shaft Ωm =
dγ m
dt
Ωsr - angular speed of the rotor flux vector Ωsr =
dγ sr
dt
Ωss - angular speed of the stator flux vector Ωss =
dγ ss
dt
Ωsl - slip frequency
2
L
σ = 1 − M - total leakage factor
Ls Lr
Superscript
^ - estimated value
Subscripts
..c - reference value
Rectangular coordinate systems
α − β - stator oriented, stationary coordinate system
d ' − q ' - rotor oriented, rotated coordinate system
x − y - stator flux oriented, rotated coordinate system
d − q - rotor flux oriented, rotated coordinate system
Abbreviations
IM – Induction Motor
MMF – Magnetomotive Force
PWM – Pulse Width Modulation
154
List of symbols
ZSS – Zero Sequence Signals
SPWM – Sinusoidal (triangulation) Pulse Width Modulation
SVPWM – Space Vector Pulse Width Modulation
THIPWM – Third Harmonic Pulse Width Modulation
DPWM – Discontinues Pulse Width Modulation
SVM – Space Vector Modulation
OM – Overmodulation
RPWM – Random Pulse Width Modulation
RLL – Random Lead-Lag Modulation
RCD – Random Center Pulse Displacement
RZD – Random Distribution of the Zero Voltage Vector
155
Appendices
A.1.
Derivation of Fourier Series Formula for Phase Voltage
If function f is a periodic, piecewise continuous and an odd, then its trigonometric
Fourier series is given by [56]:
∞
f (ωt ) = ∑ bn sin (nωt )
(A.1.1)
n =1
where, for n = 1, 2, 3, …
2
bn =
π
π
∫ f (ωt )sin (nωt )d (ωt )
(A.1.2)
0
Function which describes phase inverter voltage is shown in the Fig. A.1.1
UA
2
U dc
3
1
U dc
3
0
π
3
1
− U dc
3
2
− U dc
3
2π
3
π
4π
3
5π
3
2π
ωt
Fig. A.1.1. Phase voltage of the inverter
Taking into consideration this function coefficient bn can be written as follows:
bn =
2
π
π
∫ U (t )sin (nωt )d (ωt )
A
0
2π
π
3
23 1
2
=  ∫ U dc sin (nωt )d (ωt ) + ∫ U dc sin (nωt )d (ωt ) +
π 0 3
π 3
3

=
2π
π

2 1
π
U dc  − cos(nωt ) 03 − 2 cos(nωt ) π3 − cos(nωt ) 2π
3π n
3
3

=

2 1
 2 
 π
U dc 1 − cos(nπ ) + cos n  − cos n π  
3π n
 3 
 3



1
(
)
(
)
U
sin
n
ω
t
d
ω
t

dc
∫
2π 3

3

π




(A.1.3)
Appendices
for even n:
 π
 2 
1 − cos(nπ ) + cos n  − cos n π 
 3
 3 
π
 π

= 1 − 1 + cos n  − cos nπ − n  = 0
3
3



(A.1.4)
and for uneven n:
π
 π
 2 
 π

1 − cos(nπ ) + cos n  − cos n π  = 1 + 1 + cos n  − cos π + (n − 1)π − n 
3
 3
 3 
 3


 π 
(A.1.5)
= 21 + cos n  
 3 

From above formulas the Fourier series for UA is given by:
UA =
=
∞
4
1
 π 
U dc ∑ 1 + cos n   sin (nωt )
3π
 3 
n =1 n 
∞
1
U dc ∑ sin (nωt )
π
n =1 n
2
(A.1.6)
where:
n=1+6k, k=0, ±1, ±2,…
157
Appendices
A.2.
SABER Simulation Model
The control structures of IM were implemented in SABER v.2.4 Synopsys Inc.
package. SABER provides analysis behavior of the complete analog and mixed-signal
systems including electrical subsystem. SABER model scheme is presented in Fig.
A.2.1.
Fig. A.2.1. SABER model
The SABER package include the electrical and mechanical elements library. The
scheme of inverter (Fig. A.2.2) is based on the transistors and diodes models from
library.
The user of SABER package can create own model using mathematical equation. In
this way is build model of induction motor. The equations (2.14-2.16) described
induction motor in α − β
coordinates system are written in properly form in
“motor.sin” SABER file. The content of this file is shown in Fig. A.2.3
158
Appendices
Fig. A.2.2. Model of inverter
The control algorithm of induction motor has been written in MAST SABER
programming language. The code in MAST language is connected to “Control Block”,
which is shown in Fig. A.2.1. The MAST programming language is very similar to C
language. Therefore, implementation in laboratory setup of simulated structure is easier.
159
Appendices
#motor.sin
template motor t1 t2 t3 t0 = rs,rr,ls,lr,lm,ml,,j
electrical t1, t2, t3, t0
{
<consts.sin
values {
vt1=v(t1)-v(t0)
vt2=v(t2)-v(t0)
vt3=v(t3)-v(t0)
va=(1/3)*(2*vt1-vt2-vt3)
vb=(vt2-vt3)/sqrt(3)
fsa = ls*isa + lm*ira
fsb = ls*isb + lm*irb
fra = lr*ira + lm*isa
frb = lr*irb + lm*isb
}
equations {
isb: vb - rs*isb = d_by_dt(fsb)
isa: va - rs*isa = d_by_dt(fsa)
irb: - rr*irb + p*omega_m*fra = d_by_dt(frb)
ira: - rr*ira - p*omega_m*frb = d_by_dt(fra)
omega_m: (1/j ) * ( te - ml )= d_by_dt(omega_m)
i(t1->t0)+=it1
it1: it1=isa
i(t2->t0)+=it2
it2: it2=0.5*(-isa + sqrt(3)*isb)
i(t3->t0)+=it3
it3: it3=0.5*(-isa - sqrt(3)*isb)
}
}
Fig. A.2.3. SABER file „motor.sin”
160
Appendices
A.3.
Data and Parameters of Induction Motors
Table A.3.1. Data of 3 kW induction motor
Power
PN = 3 kW
Voltage
UN = 380 V
Current
IN = 6.9 A
Frequency
fN = 50 Hz
Base speed
Ω N = 1415 rpm
Number of pole pairs
pb = 2
Moment of inertia
J = 0.007 kgm2
Nominal torque
MN = 20 Nm
Nominal stator flux
Ψ sN = 0.98 Wb
Table A.3.2. Parameters of 3 kW induction motor
Stator winding resistance
Rs = 1.85 Ω
Rotor winding resistance
Rr = 1.84 Ω
Stator inductance
Ls = 170 mH
Rotor inductance
Lr = 170 mH
Mutual inductance
LM = 160 mH
Table A.3.3. Data of 15 kW induction motor
Power
PN = 15 kW
Voltage
UN = 380 V
Current
IN = 28.9 A
Frequency
fN = 50 Hz
Base speed
Ω N = 1460 rpm
Number of pole pairs
pb = 2
Moment of inertia
J = 0.875 kgm2
Nominal torque
MN = 98 Nm
Nominal stator flux
Ψ sN = 0.98 Wb
161
Appendices
Table A.3.4. Parameters of 15 kW induction motor
Stator winding resistance
Rs = 0.28 Ω
Rotor winding resistance
Rr = 0.26 Ω
Stator inductance
Ls = 63.5 mH
Rotor inductance
Lr = 63.5 mH
Mutual inductance
LM = 58.1 mH
Table A.3.5. Data of 90 kW induction motor
Power
PN = 90 kW
Voltage
UN = 380 V
Current
IN = 158 A
Frequency
fN = 50 Hz
Base speed
Ω N = 1483 rpm
Number of pole pairs
pb = 2
Moment of inertia
J = 1.50 kgm2
Nominal torque
MN = 580 Nm
Nominal stator flux
Ψ sN = 0.98 Wb
Table A.3.6. Parameters of 90 kW induction motor
162
Stator winding resistance
Rs = 0.020 Ω
Rotor winding resistance
Rr = 0.016 Ω
Stator inductance
Ls = 16.36 mH
Rotor inductance
Lr = 16.74 mH
Mutual inductance
LM = 16 mH
Appendices
A.4.
Equipment
Table A.4.1. List of equipment
Instrument
Type
Digital oscilloscope
Tektronix TDS3034 300MHz
Analyzer
NORMA D6000 Lem
Voltage differential probe
Tektronix P5200
Current probe
Tektronix TCP A300
Simulation program
SABER 2002.4 Synopsys, Inc.
Simulation program
Matlab 6.1 MathWorks, Inc.
163
Appendices
A.5.
dSPACE DS1103 PPC Board
Physically, DS1103 is built as a PC card that can be mounted into an ISA slot of a
regular PC. The I/O capability is rather impressive providing 300 signals. In order to
simplify the interface, 60 signals out of 300 are selected for further processing and then
connected to the SCU for signal conditioning. The selection is carried out in the
DEMUX card, which was fitted in a shielded box for EMC consideration.
The DS1103 is a single board system based on the Motorola PowerPC 604e/333MHz
processor (PPC), which forms the main processing unit.
I/O Units
A set of on-board peripherals frequently used in digital control systems has been
added to the PPC. They include: analog-digital and digital-analog converters, digital I/O
ports (Bit I/O), and a serial interface. The PPC can also control up to six incremental
encoders, which allow the development of advanced controllers for robots.
DSP Subsystem
The DSP subsystem, based on the Texas Instruments TMS320F240 DSP fixed-point
processor, is especially designed for the control of electric drives. Among other I/O
capabilities, the DSP provides 3-phase PWM generation making the subsystem useful
for drive applications.
CAN Subsystem
A further subsystem, based on Siemens 80C164 micro-controller (MC), is used for
connection to a CAN bus.
Master PPC Slave DSP Slave MC
The PPC has access to both the DSP and the CAN subsystems. Spoken in terms of
inter-processor communication, the PPC is the master, whereas the DSP and the CAN
MC are slaves.
Fig. A.5.14 gives an overview of the functional units of the DS1103 PPC.
164
Appendices
Fig. A.5.1. Block diagram of the dSPACE DS1103 board
The DS1103 PPC Controller Board provides the following features summarized in
alphabetical order:
A/D Conversion
•
4 parallel A/D-converters, multiplexed to 4 channels each, 16-bit resolution, 4 µs
sampling time, ± 10V input voltage range,
•
4 parallel A/D-converters with 1 channel each, 12-bit resolution, 800 ns sampling
time ± 10V input voltage range,
•
Slave DSP ADC Unit providing.
•
2 parallel A/D converters, multiplexed to 8 channels each, 10-bit resolution, 6 µs
sampling time ± 10V input voltage range,
Digital I/O
165
Appendices
•
32-bit input/output, configuration byte-wise,
•
Slave DSP Bit I/O-Unit providing,
•
19-bit input/output, configuration bit-wise,
CAN Support
•
Slave MC fulfilling CAN Specifications 2.0 A and 2.0 B, and ISO/DIS 11898.
D/A Conversion
•
2 D/A converters with 4 channels each, 14-bit resolution ±10 V voltage range
Incremental Encoder Interface
•
1 analog channel with 22/38-bit counter range,
•
1 digital channel with 16/24/32-bit counter range,
•
5 digital channels with 24-bit counter range.
Interrupt Control - Interrupt Handling.
Serial I/O
•
standard UART interface, alternatively RS-232 or RS-422 mode.
Timer Services
•
32-bit downcounter with interrupt function (Timer A),
•
32-bit upcounter with pre-scaler and interrupt function,
•
32-bit downcounter with interrupt function (PPC built-in Decrementer),
•
32/64-bit timebase register (PPC built-in Timebase Counter).
Timing I/O
•
4 PWM outputs accessible for standard Slave DSP PWM Generation,
•
3 x 2 PWM outputs accessible for Slave DSP PWM3 Generation and Slave DSP
PWM-SV Generation,
•
4 parallel channels accessible for Slave DSP Frequency Generation,
•
4 parallel channels accessible for Slave DSP Frequency Measurement (F2D) and
Slave DSP PWM Analysis (PWM2D).
166
Appendices
A.6.
Processor TMS320FL2406
Fig. A.6.1 gives overview of the TMS320FL2406 structure.
DARAM (B0)
256 Words
C2xx
DSP
Core
10 bit ADC
PLL Clock
DARAM (B1)
256 Words
SCI
SPI
DARAM (B2)
32 Words
CAN
Watchdog
SARAM (2K Words)
Digital I/O
Flash
(32K Words)
JTAG Port
Event Manager A
- Capture Inputs
- Com pare/PWM Outputs
- GP Tim ers/ PWM
Event Manager B
- Capture Inputs
- Com pare/PWM Outputs
- GP Tim ers/ PWM
Fig. A.6.1. TMS320F2406 device overview
The features of the TMS320FL2406 processor [101] can be summarized as:
•
•
•
High-Performance Static CMOS Technology:
•
25-ns Instruction Cycle Time (40 MHz),
•
40-MIPS Performance,
•
Low-Power 3.3-V Design.
Based on TMS320C2xx DSP CPU Core:
•
Code-Compatible With F243/F241/C242,
•
Instruction Set and Module Compatible With F240/C240.
On-Chip Memory:
•
32K Words x 16 Bits of Flash EEPROM (4 Sectors),
•
Programmable "Code-Security" Feature for the On-Chip Flash,
•
2.5K Words x 16 Bits of Data/Program RAM,
167
Appendices
•
•
544 Words of Dual-Access RAM,
•
2K Words of Single-Access RAM.
Boot ROM:
•
•
SCI/SPI Bootloader,
Two Event-Manager (EV) Modules (EVA and EVB), Each Includes:
•
Two 16-Bit General-Purpose Timers,
•
Eight 16-Bit Pulse-Width Modulation (PWM) Channels Which Enable:
•
Three-Phase Inverter Control,
•
Center- or Edge-Alignment of PWM Channels,
•
Emergency PWM Channel Shutdown With External PDPINTx\
Pin,
•
Programmable Deadband (Deadtime) Prevents Shoot-Through Faults,
•
Three Capture Units for Time-Stamping of External Events,
•
Input Qualifier for Select Pins,
•
On-Chip Position Encoder Interface Circuitry,
•
Synchronized A-to-D Conversion.
•
Watchdog (WD) Timer Module,
•
10-Bit Analog-to-Digital Converter (ADC):
•
16 Multiplexed Input Channels,
•
375 ns or 500 ns MIN Conversion Time,
•
Selectable Twin 8-State Sequencers Triggered by Two Event Managers,
•
Controller Area Network (CAN) 2.0B Module,
•
Serial Communications Interface (SCI),
•
16-Bit Serial Peripheral Interface (SPI),
•
Phase-Locked-Loop (PLL)-Based Clock Generation,
168
Appendices
•
40 Individually Programmable, Multiplexed General-Purpose Input/Output
(GPIO) Pins,
•
Five External Interrupts (Power Drive Protection, Reset, Two Maskable
Interrupts),
•
•
Power Management:
•
Three Power-Down Modes,
•
Ability to Power Down Each Peripheral Independently,
Real-Time JTAG-Compliant Scan-Based Emulation, IEEE Standard 1149.1
(JTAG),
•
Development Tools Include:
•
Texas Instruments (TI) ANSI C Compiler, Assembler/Linker, and Code
Composer Studio (CCS) Debugger,
•
Evaluation Modules,
•
Scan-Based Self-Emulation (XDS510™),
•
Broad Third-Party Digital Motor Control Support,
Package 100-Pin LQFP PZ.
169