Marcin Zelechowski

8/22/2019 Marcin Zelechowski

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Warsaw University of Technology

Faculty of Electrical EngineeringInstitute of Control and Industrial Electronics

Ph.D. Thesis

Marcin elechowski, M. Sc.

Space Vector Modulated Direct

Torque Controlled (DTC SVM)Inverter Fed Induction Motor Drive

Thesis supervisor

Prof. Dr Sc. Marian P. Kamierkowski

Warsaw Poland, 2005


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Acknowledgements

The work presented in the thesis was carried out during authors 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, Toru, Poland.First of all, I would like to express gratitude Prof. Marian P. Kamierkowski 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 Biaystok Technical University and

Prof. Wodzimierz 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 Pawe 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 wierczyki M.Sc.

Finally, I would like thank to my whole family, particularly my parents for their love

and patience.


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Contents

Pages

1. Introduction 12. Voltage Source Inverter Fed Induction Motor Drive 6

2.1. Introduction 62.2.Mathematical Model of Induction Motor 62.3.Voltage Source Inverter (VSI) 122.4.Pulse Width Modulation (PWM) 17

2.4.1. Introduction 172.4.2. Carrier Based PWM 182.4.3. Space Vector Modulation (SVM) 222.4.4. Relation Between Carrier Based and Space Vector Modulation 282.4.5. Overmodulation (OM) 312.4.6. Random Modulation Techniques 35

2.5.Summary 393. Vector Control Methods of Induction Motor 40

3.1. Introduction 403.2.Field Oriented Control (FOC) 403.3.Feedback Linearization Control (FLC) 453.4.Direct Flux and Torque Control (DTC) 49

3.4.1. Basics of Direct Flux and Torque Control 493.4.2. Classical Direct Torque Control (DTC) Circular Flux Path 533.4.3. Direct Self Control (DSC) Hexagon Flux Path 61

3.5.Summary 644. Direct Flux and Torque Control with Space Vector Modulation (DTC-SVM) 66

4.1. Introduction 664.2.Structures of DTC-SVM Review 66

4.2.1. DTC-SVM Scheme with Closed Loop Flux Control 664.2.2. DTC-SVM Scheme with Closed Loop Torque Control 684.2.3. DTC-SVM Scheme with Close Loop Torque and Flux Control

Operating in Polar Coordinates 69

4.2.4. DTC-SVM Scheme with Close Loop Torque and Flux Controlin Stator Flux Coordinates 70

4.2.5. Conclusions from Review of the DTC-SVM Structures 714.3.Analysis and Controller Design for DTC-SVM Method with

Close Loop Torque and Flux Control in Stator Flux Coordinates 71

4.3.1. Torque and Flux Controllers Design Symmetry Criterion Method 754.3.2. Torque and Flux Controllers Design Root Locus Method 784.3.3. Summary of Flux and Torque Controllers Design 88

4.4.Speed Controller Design 944.5.Summary 98


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Contents

5. Estimation in Induction Motor Drives 995.1. Introduction 995.2.Estimation of Inverter Output Voltage 1005.3.Stator Flux Vector Estimators 1045.4.Torque Estimation 1105.5.Rotor Speed Estimation 1105.6.Summary 112

6. Configuration of the Developed IM Drive Based on DTC-SVM 1136.1. Introduction 1136.2.Block Scheme of Implemented Control System 1136.3.Laboratory Setup Based on DS1103 1156.4.Drive Based on TMS320LF2406 118

7. Experimental Results 1227.1. Introduction 1227.2.Pulse Width Modulation 1227.3.Flux and Torque Controllers 1257.4.DTC-SVM Control System 129

8. Summary and Conclusions 138References 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


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1. IntroductionThe 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 dc-

link 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,


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1. Introduction

2

These features depend on the applied control strategy. The main goal of the chosen

control method is to provide the best possible parameters of drive. Additionally, a very

important requirement regarding control method is simplicity (simple algorithm, simple

tuning and operation with small controller dimension leads to low price of final

product).

A general classification of the variable frequency IM control methods is presented in

Fig. 1.1 [67]. These methods can be divided into two groups:scalarand vector.

Variable

Frequency Control

Scalar based

controllers

Vector based

controller

U/f=const.

Volt/Hertz

( )rs fi = Field Oriented

Feedback

Linearization

Scalar based

controllers

Direct Torque

Control

Rotor Flux

Oriented

Stator Flux

Oriented

Direct Torque

Space - Vector

Modulation

Passivity Based

Control

Circle flux

trajectory

(Takahashi)

Hexagon flux

trajectory

(Takahashi)

Direct(Blaschke)

Indirect(Hasse)

Closed Loop

Flux & Torque

Control

Open LoopNFO (Jonsson)o&&

Stator Current

Fig. 1.1. General classification of induction motor control methods

The scalar control methods are simple to implement. The most popular in industry is

constant Voltage/Frequency (V/Hz=const.) control. This is the simplest, which does not

provide a 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.


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1. Introduction

3

The first vector control method of induction motor was Field Oriented Control

(FOC) presented by K. Hasse (Indirect FOC) [45] and F. Blaschke (Direct FOC) [12] in

early of 70s. Those methods were investigated and discussed by many researchers and

have now become an industry standard. In this method the motor equations are

transformed into a coordinate system that rotates in synchronism with the rotor flux

vector. The FOC method guarantees flux and torque decoupling. However, the

induction motor equations are still nonlinear fully decoupled only for constant flux

operation.

An other method known asFeedback 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 asDirect Torque Control(DTC) [97] and by

M. Depenbrock asDirect 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-offoperation 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


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1. Introduction

4

are based on the same fundamentals and analysis of the drive as classical DTC.

However, from the formal considerations these methods can also be viewed as stator

field oriented control (SFOC), as shown in Fig. 1.1.

Presented 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, Toru.

In the authors opinion the following parts of the thesis are his original achievements:

elaboration and experimental verification of flux and torque controller design forDTC-SVM induction motor drives,

development of a SABER - based simulation algorithm for control andinvestigation voltage source inverter-fed induction motors,

construction and practical verification of the experimental setups with 5 kVA and18 kVA IGBT inverters,

bringing into production and testing of developed DTC-SVM algorithm in Polishindustry.

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


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1. Introduction

5

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.


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2. Voltage Source Inverter Fed Induction Motor Drive2.1. 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 MotorWhen 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:

dt

dRIU AsAA += (2.1a)

dt

dRIU BsBB += (2.1b)


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2.2. Mathematical Model of Induction Motor

7

dt

dRIU CsCC += (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 isdefined as:

( ) ( ) ( )[ ]tktktk CBA ++= 2aa1k3

2(2.2)

where: ( ) ( ) ( )tktktk CBA ,, arbitrary phase quantities in a system of natural

coordinates, satisfying the condition ( ) ( ) ( ) 0=++ tktktk CBA ,

1, a, a

2

complex unit vectors, with a phase shift

2/3 normalization factor.

Im

)(2 tka C

)(takB

)(tkA

Re

k

k2

3

A

C

a

2a

1

Fig. 2.1. Construction of space vector according to the definition (2.2)


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2. Voltage Source Inverter Fed Induction Motor Drive

8

An example of the space vector construction is shown in Fig. 2.1.

Using the space vector method the IM model equation can be written as:

dt

dRs

sss

IU += (2.3a)

dt

dRr

rrr

IU += (2.3b)

rss IImj

s MeL+= (2.4a)

srr IImj

r MeL+= (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 systemKrotating with the angular speed K .

KKK

KsK dt

dR s

sss

IU j (2.5a)

( ) KmbKKKrK pdt

dR rr

rr j

IU (2.5b)

KMKsK LL rss II + (2.6a)KMKrK LL srr II + (2.6b)

The equation of the dynamic rotor rotation can be expressed as:

[ ]mLem

B

MMJdt

d

=

1

(2.7)

where: eM electromagnetic torque,

LM load torque,

B viscous constant.

In further consideration the friction factor will be negated ( )0=B .

The electromagnetic torque eM can be expressed by the following formulas:


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9

( )rs II*Im2

Ms

be Lm

pM = (2.8)

( )ss I

*Im2

sbe

mpM = (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:

KKK

KsK dt

dR s

sss

IU j (2.10a)

( ) KmbKKKr pdt

dR r

rr

I j0 (2.10b)

KMKsK LL rss II + (2.11a)KMKrK LL srr II + (2.11b)( ) Lsbm MmpJdtd ss I*Im21 (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)

in this case angular speed of the reference frame is zero 0=K . The complex space

vector can be resolved into components and .

ssK UU jsU (2.13a)

ssK II j+=sI , rrK II jrI (2.13b)

ssK j+=s , rrK jr (2.13c)

In coordinate system the motor model equation can be written as:

dt

dIRU ssss

+=

(2.14a)


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10

dt

dIRU

s

sss

+= (2.14b)

rmbr

rr pdt

dIR ++=0 (2.14c)

rmb

r

rr pdt

dIR +=0 (2.14d)

rMsss ILIL += (2.15a)

rMsss ILIL += (2.15b)

sMrrr ILIL += (2.15c)

sMrrr ILIL += (2.15d)

( )

= Lssss

sb

m MIIm

pJdt

d

2

1(2.16)

The relations described above by the motor equations can be represented as a block

diagram. There is not just one block diagram of an induction motor. The 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 coordinate system that theinput signal to the motor is the stator voltage, the equations (2.14-2.16) can be

transformed into:

ssss IRU

dt

d= (2.17a)

sss

sIRU

dt

d= (2.17b)

rmbrrr

pIRdt

d= (2.17c)

rmbrr

rpIR

dt

d+= (2.17d)

r

rs

Ms

s

s LL

L

LI =

1(2.18a)

r

rs

Ms

r

s LL

LL

I = 1 (2.18b)


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11

s

rs

Mr

r

r LL

L

LI =

1(2.18c)

s

rs

Mr

r

r LL

L

LI =

1(2.18d)

( )

= Lssss

sb

m MIIm

pJdt

d

2

1(2.19)

These equations can be represented in the block diagram as shown in Fig. 2.2.

s

m

bp

sI

rI

ssU

sRsR

sL1

rs

M

LL

L

rs

M

LL

L

rL

1rR

r

rRrL

1rI

sR

sL

1

rs

M

LL

L

rs

M

LL

L

r

sU

2

sb

mp

eM

LM

sI

J

1

Fig. 2.2. Block diagram of an induction motor in the stationary coordinate system

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. Fromthe control point of view this system is complicated. That is the reason why there are a


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12

few methods proposed to decouple the flux and torque control. It is achieved, for

example, by the orientation of the coordinate system to the rotor or stator flux vectors.

Both control systems are described further in Chapter 3.

The equations (2.17), (2.18), (2.19) and the block diagram presented in the Fig. 2.2can 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 IGBTtransistors, 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.

D1

D2

D3

D4

D5

D6

C2

dcU

2dc

U

C

0

SB+

SB-

SA+

SA-

SC+

SC-

T1

T2

T5

T6

T3

T4

DC side

UABA B C

N

IA

IB

IC

UA

RA

LA

EA

UB

RB

LB

EB

UC

RC

LC

EC

AC side

IM

PWM Converter

Fig. 2.3. Topology of the voltage source inverter


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2.3. Voltage Source Inverter (VSI)

13

The voltage source inverter (Fig. 2.3) makes it possible to connect each of the three

motor phase coils to a positive or negative voltage of the dc link. Fig. 2.4 explains the

fabrication of the output voltage waves in 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)

0

UB0

t2

dcU2

1

dcU2

1

0

UA0

t2

1 2 3 4 5 6

dcU2

1

dcU2

1

0

UC0

t2

dcU

2

1

dcU2

1

dcU3

2

dcU3

2

0

UAB

t

2

dcU3

1

dcU31

dcU

dcU

dcU3

2

dcU3

2

0

UA

t2

dcU

3

1

dcU3

1

b)

c)

d)

e)

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]:

( ) ( ) ( )

=

=

==11

sinsin12

n

nm

n

dcA tnUtnn

UU

(2.20)

where:

dcU - dc supply voltage,


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14

( ) dcnm Un

U

2= - peak value of the n-th harmonic,n = 1+6k, k= 0, 1, 2,

Derivation of the formula (2.20) is presented in Appendix A.1.

a) b)

c) d)

e) f)

g) h)

U1(100)

A B C

Udc

U2

(110)

A B C

Udc

U3(010)

A B C

Udc

U4

(011)

A B C

Udc

U5(001)

A B C

Udc

U6

(101)

A B C

Udc

U0(000)

A B C

Udc

U7

(111)

A B C

Udc

Fig. 2.5. Switching states for the voltage source inverter

From the equation (2.20) the fundamental peak value is given as:

( ) dcm UU

21 = (2.21)


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2.3. Voltage Source Inverter (VSI)

15

This value will be used to define the modulation index 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:

=

==

7,00

6...13

2 3)1(

v

veU vjdcv

U (2.22)

The output voltage vectors are shown in Fig. 2.6.

U1

(100)

U2(110)U

3(010)

U4(011)

U5(001) U

6(101)

U7(111)

U0(000)

Im

Re

Fig. 2.6. Output voltage represented as space vectors

Any output voltage can in average be generated, of course limited by the value of the

dc voltage. In order to realize many different pulse width modulation methods are

proposed [13, 27, 30, 38, 46, 47, 51, 52, 105] in history. However, the general idea is


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16

based on a sequential switching of active and zero vectors. The modulation methods are

widely described in the next section.

Only one switch in an 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 beinserted. 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 1-

3s.

t

t

Ts

TD

TD

SA-

SA+

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


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2.4. Pulse Width Modulation (PWM)

17

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 dead-

time 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 thecarrier based PWM methods theZero Sequence Signals (ZSS) [46] are added to extend


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18

the linear operation range (see section 2.4.2). The carrier based modulation methods

with ZSS correspond to space vector modulation. It will be widely presented in section

2.4.4.

All PWM methods have specific features. However, there is not just one PWMmethod 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 PWMThe 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.


25/175


19

Udc

A B C

N

Carrier

UAc

UBc

UCc

Ut

SA

SB

SC

Fig. 2.8. Block scheme of carrier based sinusoidal PWM

Ut

UAc

UBc

UCc

0

1

0

1

0

1

0

SB

SC

0

dcU32

dcU31

dcU32dcU310

dcU

dcU

0

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02

2dcU

SA

AU

ABU

2dcU

Fig. 2.9. Basic waveforms of carrier based sinusoidal PWM


26/175


20

The modulation index m is defined as:

)(tm

m

U

Um = (2.23)

where:

mU - peak value of the modulating wave,

)(tmU - peak value of the carrier wave.

The modulation index m can be varied between 0 and 1 to give a linear relation

between the reference and output wave. At m=1, the maximum value of fundamental

peak voltage is 2

dcU

, which is 78.55% of the peak voltage of the square wave (2.21).

The maximum value in the linear range can be increased to 90.7% of that of the

square wave by inserting the appropriate value of a triple harmonics to the modulating

wave. It is shown in Fig. 2.10, which presents the whole range characteristic of the

modulation methods [67]. This characteristic include also the overmodulation (OM)

region, which is widely described in section 2.4.5.

1

0.785 0.907 1

1.155 3.24 M

m

[ ]%1002

dc

A

UU

78.5

90.7

100

SPWM

SVPWM

or SPWM with ZSS OM

Six step

operation

Fig. 2.10. Output voltage of VSI versus modulation index for different PWM techniques


27/175


21

If the neutral point N on the AC side of the inverter is not connected with the DC

side midpoint 0 (Fig. 2.3), phase currents depend only on the voltage difference

between phases. Therefore, it is possible to insert an additional Zero Sequence Signal(ZSS) of the 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].

N

Udc

A B C

SA

SB

SC

Carrier

UtCalculation

of ZSS

UAc

UBc

UCc

UAc

*

UBc*

UCc

*

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 to2

dcU . Therefore,2

dcU saturation limits

correspond to 1. In Fig. 2.12 only phase A modulation waveform is shown as the

modulation signals of phase B and C are identical waveforms with 120 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


28/175


22

bus. In this segments some power converter switches do not switch. Discontinuous

modulation methods give lower (average 33%) switching losses. The modulation

method with triangular shape of ZSS with 1/4 peak value corresponds to space vector

modulation (SVPWM) with symmetrical placement of the zero vectors in a sampling

period. It will be widely describe in section 2.4.4. In Fig. 2.12 is also shown sinusoidal

PWM (SPWM) and third harmonic PWM (THIPWM) with sinusoidal ZSS with 1/4

peak value corresponding to a minimum of output current harmonics [63].

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Time

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Time

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Time

SPWM THIPWM SVPWM

UA

UN0

UA0

UA

=UA0

UN0

UN0

UA

UA0

a) b) c)

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Time

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Time

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Time

UN0UN0

UN0

UA0

UA0U

A0

UA

UA

UA

DPWM1 DPWM2 DPWM3d) e) f)

Fig. 2.12. Waveforms for PWM with added Zero Sequence Signal a) SPWM, b)THIPWM, c) SVPWM,

d) DPWM1, e) DPWM2, f) DPWM3

2.4.3. Space Vector Modulation (SVM)The space vector modulation techniques differ from the carrier based in that way,

there are no separate modulators used for each of the three phases. Instead of them, the

reference voltages are given by space voltage vector and the output voltages of the

inverter are considered as space vectors (2.22). There are eight possible output voltage

vectors, six active vectors 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


29/175


23

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 vectorUc can be implemented by the switching vectors ofU1, U2

and zero vectors U0, U7.

I

II

III

IV

V

VI

U7(111)

U0(000) U

1(100)

U2

(110)U3(010)

U4(011)

U5(001) U

6(101)

U

c

(t1

/Ts)U

1

(t2/T s

)U 2

Fig. 2.13. Principle of the space vector modulation

The reference voltage vectorUc is sampled with the fixed clock frequency ss Tf 1= ,

and next a sampled value ( )sTcU 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

Sector

selection

SA

SB

SC

Calculation

t1

t2

t0

t7

fs

Uc

A B C

N

Uc(Ts)

Fig. 2.14. Block scheme of the space vector modulator


30/175


24

The times t1 and t2 are obtained from simple trigonometrical relationships and can be

expressed in the following equations:

( )

= 3sin32

1 sMTt (2.24a)

( )

sin32

2 sMTt = (2.24b)

WhereMis a modulation index, which for the space vector modulation is defined as:

dc

c

stepsix

c

U

U

U

UM

2)(1

==

(2.25)

where:

cU - vector magnitude, or phase peak value,

)(1 stepsixU - fundamental peak value ( )dcU2 of the square-phase voltagewave.

The modulation indexMvaries 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 is equal to dcU3

3.

This is a radius of the circle inscribed of the hexagon in Fig. 2.13. At this condition the

modulation index is equal:

907.0

2

3

3

==

dc

dc

U

U

M

(2.26)

This means that 90.7% of the fundamental at the square wave can be obtained. It

extends the linear range of modulation in relation to 78.55% in the sinusoidal

modulation techniques (Fig. 2.10).

After calculation of t1 and t2 from equations (2.24) the residual sampling time is

reserved for zero vectors U0 and U7.

70217,0 )( ttttTt s +=+= (2.27)


31/175


25

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 t7are equal:

( ) 22170 ttTtt s == (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)or

U1 U2 U7 U2 U1 (2.31)These sequences are shown in Fig. 2.15b and 2.15c respectively. In these cases only

two phases switch in one sampling time, and only one zero vector is used U0 (Fig.

2.15b) orU7 (Fig. 2.15c). This type of modulation is called discontinuous pulse width

modulation (DPWM).

1

0 1

0 0 1

1

01

001

1 1 1

1 1

1

111

11

1

Ts

U7

t0

t2/2

SA

SB

SC

U1

U2

U2

U1

t1/2t

1/2 t

2/2

0

0 0

0

00

0 1 1

0 1

0

01

0

Ts

U0

U1

U2

U1

U0

t2

t1/2 t

0/2t

0/2 t

1/2

SA

SB

SC

0

0 0

0 0 0

0

00

000

1 1 1

1 1

1

111

11

1

Ts

U0

U1

U2

U7

U7

U2

U1

U0

t0/4 t

1/2 t

2/2 t

0/4t

0/4 t

1/2 t

2/2 t

0/4

SA

SB

SC

a) b) c)

Fig. 2.15. Space vectors in the sampling period a) SVPWM, b), c) DPWM

The idea of discontinuous modulation is based on the assumption that one phase is

clamped by 60 to lower or upper of the dc bus voltage. It gives only one zero state per

sampling period (Fig. 2.15b, c). The discontinuous modulation provides 33% reduction


32/175


26

of the effective switching frequency and switching losses. The discontinuous space

vector modulation techniques, like all the space vector methods, correspond to the

carrier based modulation method. It will be widely described in the next section.

DPWM4

U7(111)

U0(000) U

1(100)

U2(110)

U3(010)

U4(011)

U5(001)

U6(101)

t7= 0

t7= 0

t7= 0

t0= 0

t0= 0

t0= 0

DPWM1

U7(111)

U0(000) U1 (100)

U2(110)

U3(010)

U4(011)

U5(001)

U6(101)

t7= 0

t7= 0

t7= 0

t0= 0

t0= 0

t0= 0

t0= 0

t7= 0

t0= 0

t7= 0

t7= 0

t0= 0

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Time

UN0

UA0

UA

DPWM2

U7(111)

U0(000) U

1(100)

U2(110)

U3(010)

U4(011)

U5(001)

U6(101)

t7= 0

t7= 0t

7= 0

t0= 0 t

0= 0

t0= 0

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Time

UN0

UA0

UA

DPWM3

U7(111)

U0(000) U1 (100)

U2(110)

U3(010)

U4(011)

U5(001)

U6(101)

t7= 0

t7= 0

t7= 0

t0= 0

t0= 0

t0= 0

t0= 0

t7= 0

t0= 0

t7= 0

t7= 0

t0= 0

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Time

UN0

UA0

UA

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Time

UA

UA0

UN0

a)

b)

c)

d)

Fig. 2.16. The discontinuous space vector modulation


33/175


27

In the Fig. 2.16 there are shown several different kinds of space vector discontinues

modulation. It can be seen that the type of method depends on the moved do not switch

sectors. These sectors are adequately moved on 0, 30, 60, 90 and denoted as

DPWM1, DPWM2, DPWM3 and DPWM4. Fig. 2.16 also shows voltage waveforms for

each methods. For the carrier based methods with ZSS these waveforms are identical

(Fig. 2.12).

From the type of modulation it depends also harmonic content, what is presented in

Fig. 2.17 for the SVPWM and DPWM1 methods.

Fig. 2.17. The output line to line voltage harmonics content a) SVPWM, b) DPWM 1

In Fig. 2.17 harmonics of output line to line voltage are shown. The voltage

frequency domain representation is composed of the series discrete harmonics

components. These are clustered about multiplies of the switching frequency. In this

case the switching frequency was 5 kHz. Spectrum for every modulation methods is

different. In Fig. 2.17 the differences between SVPWM and DPWM1 modulation

method can be seen. However, characteristic feature for all methods, which work with

constant switching frequency is clustered higher harmonics round multiplies of the

switching frequency. These type of modulation methods are named deterministic PWM

(DEPWM). The modulation method influence also for current distortion, torque ripple

and acoustic noise emitted from the motor. Modulation techniques are still being

improved for reduction of these disadvantages. One of the proposed methods is a

random PWM (RPWM) (see section 2.4.6).


34/175


28

2.4.4. Relation Between Carrier Based and Space Vector ModulationAll 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 vectorU1is presented.


35/175


29

0

0 0

0 0 0

0

00

000

0 1 1

0 1

0

011

01

0

Ts

U0

U1

U2

U1

U0

t2

t1/2 t

0/2t

0/2 t

1/2

SA

SB

SC

UAc

*

UBc

*

UCc

*

SA

SB

SC

b)

CarrirbasedPWM

SpacevectorPWM

UAc

*

0

0 0

0 0 0

0

00

000

1 1 1

1 1

1

111

11

1

Ts

U0

U1

U2

U7

U7

U2

U1

U0

t0/4 t

1/2 t

2/2 t

0/4t

0/4 t

1/2 t

2/2 t

0/4

UBc

*

UCc

*

SA

SB

SC

SA

SB

SC

a)

CarrirbasedPWM

SpacevectorPWM

Fig. 2.18. Comparison of carrier based PWM with space vector PWM a) SVPWM, b) DPWM1

UA

UB

UC

UN0

A

B C

N0

2dcU

2dc

U

UB0=U

C0

UA0

Fig. 2.19. Equivalent circuit of VSI for the U1 vector


36/175


30

Taking into consideration the electrical circuit in Fig. 2.19 the voltage distribution

can be obtained. The voltages can be written as:

dcA UU

3

2= ; dcB UU

3

1= ; dcC UU

3

1= (2.32)

dcA0 UU2

1= ; dcB0 UU

2

1= ; dcC0 UU

2

1= (2.33)

dcANA0N0 UUUU6

1== (2.34)

This analysis may be repeated for all vectors provided to obtain voltages presented in

Table 2.1.

Table 2.1. The voltages for the eight converter output vectors

A0U B0U AU BU CUC0U N0U

0U

1U

2U

3U

4U

5U

6U

7U

dcU2

1dcU

2

1

dcU3

2dcU

3

1dcU

2

1

dcU3

1 dcU

6

1

dcU2

1dcU

3

2dcU

3

1dcU

2

1 dcU

6

1

dcU21dcU

21 dcU3

2dcU31dcU21 dcU3

1 dcU61

dcU2

1dcU

3

1

dcU2

1dcU

2

1 dcU

3

2

dcU3

1dcU

2

1dcU

3

1dcU

6

1

dcU2

1 dcU

2

1 dcU

3

2dcU

3

1dcU

2

1dcU

3

1 dcU

6

1

dcU2

1dcU

2

1 dcU

3

1dcU

3

1dcU

2

1dcU

3

2 dcU

6

1

dcU2

1 dcU

2

1 0 0 dcU

2

10dcU

2

1

dcU2

1dcU

2

10 0 dcU

2

10dcU

2

1

The average value for sampling time ofUNO voltage can be written as follows:

++= 7210

dc

s

N0 ttttU

TU

3

1

3

1

2

1for the sectors I, III, V (2.35)

and

++= 7120

dc

s

N0 ttttU

TU 3

1

3

1

2

1for the sectors II, IV, VI (2.36)


37/175


31

From the above equations and taking into consideration equations (2.24) and (2.27)

the zero vectors times for different kinds of modulation can be calculated.

Relations between carrier based and SVM methods are presented in Table 2.2. This

table presents also the zero vector (t0, t7) times equations for the most significant

modulation methods.

Table 2.2. Relation between carrier based and SVM methods

=

cos4

12

MT

t s0

( )

+= sin3cos2

12 M

T

ts

0

Calculation oft0

and t7

for sectors I, III, V

for sectors II, IV, VI

210s7 tttTt =

Waveform of the

ZSS (Fig. 2.13)

( )0=N0U

Modulation

method

SPWM

Sinusoidal with

1/4 amplitude

no signal

THIPWM

=

3cos

4

1cos

41

2M

Tt s0

+=

3cos

2

1sin3cos

21

2M

Tt s0

for sectors II, IV, VI

210s7 tttTt =

for sectors I, III, V

Triangular with

1/4 amplitude

Discontinuous

SVPWM

DPWM1

( ) 221s70 ttTtt ==

0=0t

21s7 ttTt =

0=7t

21s0 ttTt =

when ( )1263

+


38/175


32

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 vectorUc possible to realization in whole range of angle is equal

dcU3

3. It is a radius of the circle inscribed of the hexagon. This value corresponds to

the modulation index equal to 0.907 (see equation 2.26). To realize higher values a

voltage overmodulation algorithm has to be applied. At the end of the overmodulation

region is a six-step mode (atM= 1).

U7(111)

U0(000) U

1(100)

U2(110)U

3(010)

U4(011)

U5(001) U

6(101)

U

c

(t1 /Ts)U1

(t2/T

s)U

2 Overmodulation range

0.907


39/175


33

If the value of the reference voltage beyond maximal value in the linear range vector

Uc can not be realized for whole range of 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.

U0(000)

U7(111)

Uc

Uc*

U1 (100)

U2(110)U

3(010)

U4(011)

U5(001) U

6(101)

Fig. 2.21. Overmodulation mode I

In this mode voltage vectorUc 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.


40/175


34

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:

sincos3

sincos3

+= s1 Tt (2.37a)

1s2 tTt = (2.37b)

0== 70 tt (2.37c)

The output voltage waveform is given approximately by linear segments for the

hexagon trajectory and sinusoidal segments for the circular trajectory. Boundary of the

segments is determined by a crossover angle which depends on the reference voltage

value. As known from Fig. 2.21 the upper limit in mode I is when = 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 to * is

changed.

U0(000)

U7(111)

Uc

Uc*

U1(100)

U2(110)U

3(010)

U4(011)

U5(001) U

6(101)

h

h

Fig. 2.22. Overmodulation mode II where both amplitude and angle is changed


41/175


35

The trajectory ofUc* is maintained on the hexagon which defines amplitude of the

reference voltage vector. The angle is calculated from the following equations:


42/175


36

cycles the switching sequence is deterministic. In RPWM methods the switching

frequency or the switching sequence change randomly.

One of the proposed random modulation techniques is a method with randomly

varied lengths of coincident switching and sampling time of the modulator. This method

was named RPWM 1. The sampling and switching cycles in DEPWM with RPWM 1 is

comparable shown in Fig. 2.23. The reference voltage vectors 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.

1 2 3 ... n-1 n ...sampling cycles

switching cycles 1 2 3 ... n-1 n ...

)1(

cU)2(

cU)3(

cU)(K

cU)1( n

cU)1( +n

cU)(n

cU

sampling cycles

switching cycles ...1 2 ... n-1 n

...1 2 ... n-1 n

3

3

)1(

cU)2(

cU)3(

cU)(K

cU)1( n

cU)1( +n

cU)(n

cU

a)

b)

sws TT =

sws TT =

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


43/175


37

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 sT from the middle of the period.

The parameter 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.

SA

SB

SC

Ts

Ts

Ts

Ts

sT sT sT sT

SA

SB

SC

Ts

Ts

Ts

Ts

Lead Lag Lag Lead

SA

SB

SC

Ts

Ts

Ts

Ts

a)

b)

c)

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)


44/175


38

The main disadvantage of the RPWM 1 method (Fig. 2.23b) is variable switching

frequency. For elimination of this disadvantage RPWM 2 [119] was proposed, which

operates with fixed sampling frequency and variable switching frequency. The principle

of this method is shown in Fig. 2.25.

1 2 3 ... n-1 n ...

...1 2 3 ... n-1 n

sampling cycles

switching cycles

)1(

cU)2(

cU)3(

cU)(K

cU)1( n

cU)1( +n

cU)(n

cU

swT

sT

t

Fig. 2.25. Sampling and switching cycles in RPWM 2 technique

In this method the start of each switching cycles is delayed with respect to that of the

coincident sampling cycle by a random varied time interval t . It is given as:

srTt= (2.39)

where rdenotes 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


45/175

2.5. Summary

39

random modulation techniques. Simulation result presented in both figures (Fig. 2.17

and Fig. 2.26) was done at the same conditions: sampling frequency 5 kHz, output

frequency 50 Hz.

2.5. SummaryIn 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 voltagedrop 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.


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3. Vector Control Methods of Induction Motor3.1. 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 qd is

rotating with the angular speed equal to rotor flux vector angular speed srK = ,

which is defined as follows:

dt

d srsr = (3.1)

The rotating coordinate system qd is shown in Fig. 3.1.

The voltage, current and flux complex space vector can be resolved into components

dand q.

sqsdK UU j+=sU (3.2a)

sqsdK II j+=sI , rqrdK II j+=rI (3.2b)

sqsdK j+=s , rrdK ==r (3.2c)


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3.2. Field Oriented Control (FOC)

41

r

d

q sI

sI

sI

sdIsqI

sr

sr

Fig. 3.1. Vector diagram of induction motor in stationary and rotating qd coordinates

In qd coordinate system the induction motor model equations (2.10-2.12) can be

written as follows:

sqsrsd

sdssd dt

dIRU += (3.3a)

sdsr

sq

sqssq dt

dIRU ++= (3.3b)

dt

dIR rrdr +=0 (3.3c)

( )mbsrrrqr pIR +=0 (3.3d)

rdMsdssd

ILIL += (3.4a)

rqMsqssq ILIL += (3.4b)

sdMrdrr ILIL += (3.4c)

sqMrqr ILIL +=0 (3.4d)

= Lsqr

r

Msb

m MIL

Lmp

Jdt

d

2

1(3.5)

The equations 3.3c and 3.4c can be easy transformed to:


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3. Vector Control Methods of Induction Motor

42

r

r

rsd

r

rMr L

RI

L

RL

dt

d= (3.6)

The motor torque can by expressed by rotor flux magnitude r and stator current

component sqI as follows:

sqr

r

Msbe I

L

LmpM

2= (3.7)

Equations (3.6) and (3.7) are used to construct a block diagram of the induction

motor in qd coordinate system, which is presented in Fig. 3.2.

r

m

2

s

b

m

p

eM

LM

J

1

r

rM

L

RL

r

r

L

R

r

M

L

L

sdI

sqI

eM

Fig. 3.2. Block diagram of induction motor in qd coordinate system

The main feature of the field oriented control (FOC) method is the coordinate

transformation. The current vector is measured in stationary coordinate .

Therefore, current components sI , sI must be transformed to the rotating system

qd . Similarly, the reference stator voltage vector components csU , csU , must be

transformed from the system qd to . These transformations requires a rotor

flux angle sr . Depending on calculations of this angle two different kind of field

oriented control methods maybe considered. Those are Direct Field Oriented Control

(DFOC) andIndirect Field Oriented Control(IFOC) methods.


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3.2. Field Oriented Control (FOC)

43

For DFOC an estimator or observer calculates the rotor flux angle sr . Inputs to the

estimator or observer are stator voltages and currents. An example of the DFOC system

is presented in Fig. 3.3.

PI

SVM

SA

SB

SCsq c

I

FluxEstimator

Udc

sU

sU

sI

csU

csU

sI

sI

sr

rc

ecM

sdI

sqI

sd cI

PI

qd

qd

3

2

Motor

rcM

r

sb L

L

mp

12

ML

1

VoltageCalculation

Fig. 3.3. Block diagram of the Direct Field Oriented Control (DFOC)

For the IFOC rotor flux angle sr is obtained from reference sdcI , sqcI currents. The

angular speed of the rotor flux vector speed can be calculated as follows:

mbslrs p += (3.8)

where sl is a slip angular speed. It can be calculated from (3.3d) and (3.4d).

sqc

r

r

sdc

sl IL

R

I

1= (3.9)

In Fig. 3.4 a block diagram of the IFOC is shown.


50/175


44

PI

SVM

SA

SB

SCsq c

I

Udc

sI

csU

csU

sI

sI

sr

rc

ecM

sdI

sqI

sdcI

PI

qd

qd

Motor

rcM

r

sb L

L

mp

12

ML

1

3

2

msr

sl

sdcr

r

IL

R 1

bp

Fig. 3.4. Block diagram of the Indirect Field Oriented Control (IFOC)

In both presented examples reference currents in rotating coordinate system sdcI , sqcI

are calculated from the reference flux and torque values. Taking into consideration the

equations describing IM in field oriented coordinate system (3.6) and (3.7) at steady

state the formulas for the reference currents can be written as follows:

r

M

sdc L

I1

= (3.10)

ec

rcM

r

sb

sqc ML

L

mpI

12= (3.11)

The property of the FOC methods can be summarized as follows:

the method is based on the analogy to control of a DC motor, FOC method does not guarantee an exact decoupling of the torque and flux

control in dynamic and steady state operation,

relationship between regulated value and control variables is linear only forconstant rotor flux amplitude,


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3.3. Feedback Linearization Control (FLC)

45

full information about motor state variable and load torque is required (themethod 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:

gg)x(x ss UUf +& (3.12)where:


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46

++

=

J

MII

Ip

Ip

ILp

ILp

f

Lsrsr

srrmb

srmbr

sMrrmb

sMrmbr

)(

)(

x (3.13)

T

g

= 00100 ,,,,sL

(3.14)

T

g

= 01000 ,,,,sL

(3.15)

[ ]Tx mssrr II ,,,, (3.16)

and

r

r

L

R= (3.17)

rs

M

LL

L

= (3.18)

2

22

rs

Mrrs

LL

LRLR

+= (3.19)

J

Lmp Msb

2= (3.20)

Because rrm ,, are not dependent on ss UU , it is possible to chose variable

dependent on x:

222

1 )x( rrr = (3.21)m)x(2 (3.22)

If it is assumed that )x(1 , )x(2 are output variables, the full definition of new

coordinates can be given by:

)x(11 z (3.23a)

)x(12 fLz = (3.23b)


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3.3. Feedback Linearization Control (FLC)

47

)x(23 z (3.23c)

)x(24 fLz = (3.23d)

=

r

r

z arctan5 (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:

+

=

s

s

f

f

U

U

L

L

z

zD

2

2

1

2

3

1

&&

&&(3.24)

where

=

22

11

fgfg

fgfg

LLLL

LLLLD (3.25)

If 01

(the amplitude of flux is not zero) then 0D)

det( and it is possible todefine the linearization feedback as:

+

=

2

1

2

2

1

2

v

v

L

L

U

U

f

f

s

s

1-D (3.26)

Then the resulting system is described by the equations:

21 zz =& (3.27a)12 vz =& (3.27b)43 zz =& (3.27c)24 vz =& (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.


54/175


48

m

LM

eM2

4z

J

r1

2z2

r

Fig. 3.5. Block diagram of the induction motor with new 1v and 2v control signals

The control signals 1v , 2v are calculated by using linear feedback as follows:

( ) 21211111 zkzzkv ref (3.28)( ) 42233212 zkzzkv ref (3.29)

where coefficients 11k , 12k , 21k , 22k 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 bothdynamic and steady state,

the method is implemented in a state variable control fashion and needs complexsignal 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 switchingfrequency,


55/175

3.4. Direct Flux and Torque Control (DTC)

49

the stator currents are sinusoidal except of high frequency switching harmonics.

Speed

Controller

FluxController Vector

Modulator

Control

Signals

Transfor-

mation

Feedback

Signals

Transfor-mation

1

2

sI

sI

r

r

Flux

Estimator

m

2

rc

5z

SA

SB

SC

Voltage

Calculation

mc

1z

2z

3z

4z

csU

Motor

sU

sI

csU

dcU

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 sqI in accordance with equation

(3.7). This equation can also be written as:

sin2

sr

r

Msbe I

L

LmpM = (3.30)

where:

- angle between rotor flux vector and stator current vector.

The formula (3.30) can be transformed into the equation:

rs

Msr

Msbe

LLL

LmpM sin

22

= (3.31)

where:

- angle between rotor and stator flux vectors.


56/175


50

It can be noticed that the torque depends on the stator and rotor flux magnitude as

well as the angle . The vector diagram of IM is presented in Fig. 3.7. The two angels

and are also shown in Fig. 3.7. The angle is important in FOC algorithms,

whereas in DTC techniques.

sI

s

r

ss

sr

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:

s

s U

=dt

d(3.32)

Taking into consideration the output voltage of the inverter in the above equation it

can be written as:

=t

vdt0

Us (3.33)

where:

=

==

7,00

6...13

2 3)1(

v

veU vjdcv

U (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.


57/175


51

U1

(100)

U2

(110)U3(010)

U4(011)

U5(001) U

6(101)

Im

ReU7(111)

U0(000)

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

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 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.


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52

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


59/175


53

U1(100)

U2(110)U

3(010)

U4(011)

U5(001) U6(101)

U7(111)

U0(000)

voltage U3

applied

voltage U2applied

voltage U3

applied

voltage U3applied

voltage U4

applied

voltage U4

applied

voltage U3

applied

voltage U4applied

Fig. 3.11. Forming of the stator flux trajectory by appropriate voltage vectors selection

3.4.2. Classical Direct Torque Control (DTC) Circular Flux PathThe block diagram of classical DTC proposed by I. Takahashi and T. Nogouchi [97]

is presented in Fig. 3.12.

s

SA

SB

SC

Udc

Motor

Torque

Controller

Flux

Controller

sc d

Md

Voltage

Calculation

Vector

Selection

Table

(N)ss

Sector

Detection

Flux and

Torque

Estimator

Mec

Us

Is

eM s

s

Me

e

Fig. 3.12. Block scheme of the direct torque control method


60/175


54

The stator flux amplitude sc and the electromagnetic torque cM are the reference

signals which are compared with the estimated s and eM

values respectively. The

flux e and torque Me errors are delivered to the hysteresis controllers. The digitized

output variables d , Md and the stator flux position sector ( )Nss 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)

e

d

H

b)

M

d

Me

MH

Fig. 3.13. The hysteresis controllers a) two-level, b) three-level

The output signals d , Md are defined as:

1=d for He > (3.35a)

0=d for He < (3.35b)

1=Md for MM He > (3.36a)

0=Md for 0=Me (3.36b)

1=Md for MM He < (3.36c)

In the classical DTC method the plane is divided for the six sectors (Fig. 3.14),

which are defined as:

Sector 1:

+

6,

6

ss (3.37a)

Sector 2:

+

2,

6

ss (3.37b)

Sector 3:

++

65,

2ss (3.37c)


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55

Sector 4:

+

6

5,

6

5 ss (3.37d)

Sector 5:

2

,

6

5 ss (3.37e)

Sector 6:

6,

2

ss (3.37f)

U7(111)

U0(000) U1 (100)

U2(110)

U3(010)

U4(011)

U5(001)

U6(101)

Sector 1

Sector 2Sector 3

Sector 4

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 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.


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56

s

r

Sector 1

U2

U3

U1

U4

U5

U6

Fig. 3.15. Selection of the optimum voltage vectors for the stator flux vector in sector 1

Table 3.1. Optimum switching table

d Md Sector 1 Sector 2 Sector 3 Sector 4 Sector 5 Sector

Marcin Zelechowski

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