Marcin Zelechowski

Warsaw University of Technology

Faculty of Electrical Engineering Institute of Control and Industrial Indust rial 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”, Toru ń, Poland.

First of all, I would like to express gratitude Prof. Marian P. Ka źmierkowski 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 Bia łystok 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 Pawe ł 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

is

Vector based controller

=

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, Toru ń. 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 R s +

d Ψ A

U B = I B R s +

d Ψ B

dt

dt

(2.1a)

(2.1b)

2.2. Mathematical Model of Induction Motor

U C = I C R s +

d Ψ C

(2.1c)

dt

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 3

[1 ⋅ k (t ) + a ⋅ k (t ) + a A

2

B

⋅ k C (t )]

(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 , 2

1, a, a – complex unit vectors, with a phase shift

2/3 – normalization factor.

Im

3 2

k a 2 k C (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 R s +

d Ψ s

U r = I r Rr +

d Ψ r

(2.3a)

dt

(2.3b)

dt

Ψs

= L s I s + Me jγ I r

(2.4a)

Ψr

= Lr I r + Me − jγ I s

(2.4b)

m

m

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 U s K R s I s K

U r K Rr I r K

d Ψ s K dt

Ω K .

j Ω K Ψ s K

d Ψ r K j Ω K pb Ωm dt

(2.5a)

Ψ r K

(2.5b)

Ψ s K

L s I s K

L M I r K

(2.6a)

Ψ r K

Lr I r K

L M I s K

(2.6b)

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

d Ωm dt

=

1

J

[ M e − M L − BΩm ]

(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


M e = − pb M e = pb

m s

(

LM Im I*s I r

2

m s

(

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:

d Ψ s K

U s K R s I s K

0 Rr I r K

dt

d Ψ r K

j

dt

j Ω K Ψ s K

(2.10a)

pb Ωm

Ω K

(2.10b)

Ψ r K

Ψ s K

L s I s K

L M I r K

(2.11a)

Ψ r K

Lr I r K

L M I s K

(2.11b)

d Ωm

1

dt

J

pb

m s 2

Im Ψ *s I s

M L

(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

Ω K

= 0 . The complex space

vector can be resolved into components α and β . U s K

U sα jU s β

I s K = I sα + j I s β , Ψ s K

= Ψ sα + jΨ s β ,

(2.13a) I r K I r α j I r β Ψ r K

Ψ r β

jΨ r β

(2.13b) (2.13c)

In α − β coordinate system the motor model equation can be written as: U sα = R s I sα +

d Ψ sα dt

(2.14a)

9

2. Voltage Source Inverter Fed Induction Motor Drive

U s β = R s I s β +

0 = Rr I r α +

0 = Rr I r β +

d Ψ s β

(2.14b)

dt

d Ψ r α dt d Ψ r β dt

+ pb ΩmΨ r β

(2.14c)

− pb ΩmΨ r α

(2.14d)

Ψ sα

= L s I sα + L M I r α

(2.15a)

Ψ s β

= L s I s β + L M I r β

(2.15b)

Ψ r α

= Lr I r α + L M I sα

(2.15c)

Ψ r β

= Lr I r β + L M I s β

(2.15d)

d Ωm dt

=

1  m s pb J  2

(Ψ

 

I − Ψ s β I sα ) − M L 

sα s β

(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: d Ψ s α

= U sα − R s I sα

(2.17a)

= U s β − R s I s β

(2.17b)

= − Rr I r α − pb ΩmΨ r β

(2.17c)

= − Rr I r β + pb ΩmΨ r α

(2.17d)

I sα =

L M 1 Ψ sα − Ψ r α σ L s σ L s Lr

(2.18a)

I s β =

L M 1 Ψ s β − Ψ r β σ Lr σ L s Lr

(2.18b)

dt d Ψ s β dt d Ψ r α dt d Ψ r β dt

10


I r α =

L M 1 Ψ r α − Ψ sα σ Lr σ L s Lr

(2.18c)

I r β =

L M 1 Ψ r β − Ψ s β σ Lr σ L s Lr

(2.18d)

d Ωm dt

=

1  m s pb J  2

(Ψ

 

I − Ψ s β I sα ) − M L 

sα s β

(2.19)

These equations can be represented in the block diagram as shown in Fig. 2.2. M L R s U sα

∫

Rr

∫

Ψ sα

I sα

1 σ L s

L M

L M

σ L s Lr

σ L s Lr

I r α

pb

m s M e 2

1 J

∫

Ωm

1 σ Lr

Ψ r α

pb

∫ Rr

U s β

∫

Ψ r β

I r β

1 σ Lr L M

L M

σ L s Lr

σ L s Lr

Ψ s β

1

I s β

σ L s

R s

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

SA+

2

T3 S B+

D1

T5 SC+

D3

D5

C 0

T2 SA-

U dc 2

T4 SB -

D2

T6 SC-

D4

D6

C

I A A

U AB

R A

L A E A

I B

I C B

R B U A

RC U B

L B E B

LC

U C

E C N

Fig. 2.3. Topology of the voltage source inverter

12

AC side

C

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). U A0

a)

1

1

2

3

4

5

6

U dc

2

0 1 2

π

2π

ω t

π

2π

ω t

π

2π

ω t

π

2π

ω t

π

2π

ω t

U dc

U B0

b)

1

U dc

2

0 1 2

U dc

U C0

c)

1

U dc

2

0 1 2

U dc

U AB

d)

U dc

2

U dc

3 1

U dc

3

0 1 3 2 3

U dc U dc U dc

U A

e)

2

U dc

3 1

U dc

3

0 1 3 2 3

U dc U dc

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]: U A =

2 π

∞

U dc

∞

1

∑ n sin(nω t ) = U ( ) ∑ sin(nω t ) m n

n =1

(2.20)

n =1

where: U dc - dc supply voltage,

13


U m n

2 nπ

U dc - 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. U1 (100)

a) U dc

U dc

A

B

C

U3 (010)

c)

B

C

A

B

C

A

B

C

A

B

C

U dc

A

B

C

U5 (001)

U6 (101)

f)

U dc

U dc

A

B

C

U0 (000)

g)

A

U4 (011)

d)

U dc

e)

U2 (110)

b)

U7 (111)

h)

U dc

U dc

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

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 dce 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 T D transistors cease to conduct. Two control signals SA+, SA- for transistors T1, T2 with dead-time T D are presented in Fig. 2.7. The duration of dead-time depends of the used transistor. Most of them it takes 13µ s.

SA+

t SAT D

T D

t

T s

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 U Ac, U Bc, U Cc Cc are compared with triangular carrier signal U t t, which is common to all three phases. In this way the logical signals S A, S B, S C are generated, which define the switching instants of the power transistors as is shown in Fig. 2.9.

18

2.4. Pulse Width Modulation (PWM) U dc U Ac

SA

U Bc

SB

U Cc

SC A

B

C

U t Carrier

N Fig. 2.8. Block scheme of carrier based sinusoidal PWM

U dc 2

U t

U Ac U Bc

0

U dc 2

U Cc

1

SA

0 1

SB

0 1

SC

U A

0

2 3U dc 1 3U dc0 0

−1 3U dc − 2 3U dc 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

U m

(2.23)

U m ( t )

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 maximum value of fundamental fundamental peak voltage voltage is

U dc

2

, 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. π U A 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]. U dc U Ac

U Ac*

SA

U Bc

U Bc*

SB

U Cc

U Cc*

SC A

B

C

U t

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 dc

2

. Therefore, ±

U dc 2

saturation limits

correspond to ±1. In Fig. 2.12 only phase “A” modulation waveform is shown as the modulation signals of phase “B” and “C” are identical waveforms with 120º phase shift. The modulated methods illustrated in Fig. 2.12 can be separated into two groups: continuous and discontinuous. In the continuous PWM (CPWM) methods, the modulation waveform are always within the triangular peak boundaries and in every carrier cycle triangle and modulation waveform intersections. Therefore, on and off switchings occur. In the discontinuous PWM (DPWM) methods a modulation waveform of a phase has a segment which is clamped to the positive or negative DC

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

U A=U A0

0.8

c)

THIPWM U A

0.8

SVPWM 0.8

U A0

0.6

0.6

0.4

0.4

0.2

0.2

0

0

0

-0.2

-0.2

U N0

-0.2

0.6

0.2

U N0

-0.4

-0.6

-0.6

-0.6

-0.8

-0.8

-0.8

-1

-1 0.002 0.004 0.006 0.008

0.01 0.012 0.014 0.016 0.018

0.02

0

0.006 0. 008

0.01

0.012 0. 014 0. 016 0.018

0.02

e)

1

f)

U A

0

0

0

-0.2

-0.2

-0.2

-0.4

-0.4

-0.6

-0.8

-0.8

-1

0.2

0.01 0.012 0.014 0.016 0.018 Time

0.02

0

U A

-0.6

U N0

U N0

-0.8

-1 0.002 0.004 0.006 0.008

0.02

0.4

0.2

U N0

0.016 0. 018

0.6

0.4

-0.6

0.012 0. 014

U A0

0.8

0.6

U A

0.01

DPWM3

0.2

0

0.008

1

U A0

0.8

0.6

-0.4

0.002 0. 004 0. 006

Time

DPWM2 1

U A0

0.4

0

Time

DPWM1 0.8

U N0

-0.4

-1 0.002 0. 004

Time

d)

U A0

0.4

-0.4

0

U A

1

-1 0.002 0. 004

0.006 0. 008

0.01

0.012 0. 014 0. 016 0.018

Time

0.02

0

0.002 0. 004

0.006 0. 008

0.01

0.012 0. 014 0. 016

0.018

0.02

Time

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. II

U3 (010)

U2 (110)

III

I U ) / T s ( t 2

2

Uc

U0 (000)

U4 (011)

U1 (100)

U7 (111)

(t 1 /T )U1 s

IV

VI

U5 (001)

U6 (101)

V

Fig. 2.13. Principle of the space vector modulation

The reference voltage vector Uc is sampled with the fixed clock frequency f s = 1 T s , and next a sampled value U c (T s ) is used for calculation of times t 1, t 2, t 0 and t 7 . The signal flow in space vector modulator is shown in Fig. 2.14. U dc f s SA Uc

Uc(T s)

Sector selection

t 1

t 2

t 0

t 7

SB SC A

B

C

Calculation

N Fig. 2.14. Block scheme of the space vector modulator

23


The times t 1 and t 2 are obtained from simple trigonometrical relationships and can be expressed in the following equations: t 1 =

2 3

t 2 =

2 3

π

π

MT s sin (π 3 − α )

MT s sin (α )

(2.24a)

(2.24b)

Where M is a modulation index, which for the space vector modulation is defined as: M =

U c U 1( six − step )

=

U c 2 U dc π

(2.25)

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 3

U dc .

This is a radius of the circle inscribed of the hexagon in Fig. 2.13. At this condition the modulation index is equal: 3 M = 3 2 π

U dc

= 0.907

(2.26)

U dc

This means that 90.7% of the fundamental at the square wave can be obtained. It extends the linear range of modulation in relation to 78.55% in the sinusoidal modulation techniques (Fig. 2.10). After calculation of t 1 and t 2 from equations (2.24) the residual sampling time is reserved for zero vectors U0 and U7. t 0, 7 = T s − (t 1 + t 2 ) = t 0 + t 7

24

(2.27)

2.4. Pulse Width Modulation (PWM) The equations for t 1 and t 2 are identically for all space vector modulation methods. The only difference between the other type of SVM is the placement of zero vectors at the sampling time. The basic SVM method is the modulation method with symmetrical spacing zero vectors (SVPWM). In this method times t 0 and t 7 are equal: t 0 = t 7 = (T s − t 1 − t 2 ) 2

(2.28)

For the first sector switching sequence can be written as follows: U0

U1

U2

U7

U2

U1

(2.29)

U0

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

U2

U7

U1

(2.30)

U0

or U2

(2.31)

U1

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

t 0/4

t 1/2

t 2/2

t 0/4

t 0/4

t 1/2

t 2/2

t 0/4

t 0/2

t 1/2

t 2

t 1/2

t 0/2

t 1/2

t 2/2

t 0

t 2/2

t 1/2

U0

U1

U2

U7

U7

U2

U1

U0

U0

U1

U2

U1

U0

U1

U2

U7

U2

U1

T s

T s

T s

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)

t 0= 0

t 7 = 0

t 0= 0

1

U2 (110)

U A0

0.8

t 7 = 0

0.6

U A

0.4

t 7 = 0

t 0= 0

U4 (011)

0.2

U1 (100)

U0 (000)

0

U7 (111)

-0.2

t 0= 0

t 7 = 0

U N0

-0.4 -0.6

t 0= 0

t 7 = 0 t 0= 0

t 7 = 0

-0.8

U6 (101)

-1

U5 (001)

0

0.002 0.004 0.006 0.008

0.01

0.012 0.014 0.016 0.018

0.02

Time

b)

DPWM2 U3 (010)

1

U2 (110)

U A0

0.8

t 7 = 0

0.6

t 0= 0

U A

0.4

t 0= 0

0.2

U0 (000)

U4 (011)

U1 (100)

0

U7 (111)

-0.2

t 7 = 0

-0.4

t 7 = 0

-0.6

U N0

-0.8

t 0= 0 U6 (101)

-1

U5 (001)

0

0.002 0.004 0.006 0.008

0.01

0.012 0.014 0.016 0.018

0.02

Time

c)

DPWM3 U3 (010)

t 7 = 0

t 0= 0

t 7 = 0

U2 (110)

1

U A0

0.8

t 0= 0

0.6 0.4

t 0= 0 U4 (011)

t 7 = 0

0.2

U1 (100)

U0 (000)

0

U7 (111)

U A

-0.2

t 0= 0

t 7 = 0

-0.4 -0.6

t 7 = 0

t 0= 0 t 7 = 0

t 0= 0

U N0

-0.8

U6 (101)

-1

U5 (001)

0

0.002 0.004 0.006 0.008

0.01

0.012 0.014 0.016 0.018

0.02

Time

d)

DPWM4 U3 (010)

1

U2 (110)

U A

0.8

t 0= 0

U A0

0.6

t 7 = 0

0.4

t 7 = 0

0.2

U4 (011)

U0 (000)

U1 (100)

U7 (111)

0 -0.2

t 0= 0

t 0= 0

-0.4

U N0

-0.6 -0.8

t 7 = 0 U6 (101) U5 (001)

-1 0

0.002 0.004 0.006 0.008

0.01

0.012 0.014 0.016 0.018

Time

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 t 0 and t 7. A comparison of carrier based method with SVM is shown in Fig 2.18. There is shown a carrier based modulation with triangular shape of ZSS with 1/4 peak value. This method corresponds to the space vector modulation (SVPWM) with symmetrical placement of zero vectors in sampling period. In Fig. 2.18b is presented discontinuous method DPWM1 for carrier based and for SVM techniques. In the carrier based methods three reference signals U Ac*, U Bc*, U Cc* are compared with triangular carrier signal U t, and in this way logical signals S A, SB, SC are generated. In the space vector modulation duration time of active (t 1, t 2) and zero (t 0, t 7) vectors are calculated, and from these times switching signals S A, 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 t 1 and t 2 equations are identically for all space vector modulation methods. But every method demand suitable equation for the zero vectors t 0 and t 7. The eight voltage vectors 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) a)

b)

M W P d e s a b r i r r a C

SA SB SC

M W P d e s a b r i r r a C

SA SB SC

U Ac* U Bc*

U Ac*

U Cc*

U Bc* U Cc*

SA

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

t 0/4

t 1/2

t 2/2

t 0/4

t 0/4

t 1/2

t 2/2

t 0/4

M W P r o t c e v e c a p S

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

t 0/2

t 1/2

t 2

T s U0

U1

U2

U7

t 1/2

t 0/2

U1

U0

M W P r o t c e v e c a p S

T s 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 A 0 U dc

U A

2

U N0

0

N

U B

U dc 2

U

U C

B 0 = U C 0

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: U A =

2

1 1 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

U B0

U C0

1

1

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

2 1

U dc −

2 1

U A

U B

U C

U N0

0

0

0

− U dc

U dc

1 U dc − U dc 2 2 3 3 1 1 1 1 U dc − U dc U dc U dc 2 3 3 2 1 1 1 2 U dc − U dc − U dc U dc 2 2 3 3 1 2 1 1 U dc U dc − U dc U dc 2 2 3 3 1 1 1 1 − U dc U dc − U dc − U dc 2 2 3 3 1 1 1 2 − U dc U dc U dc − U dc 2 2 3 3 1 1 U dc U dc 0 0 2 2

− U dc − U dc

2

1

1

2 1

− U dc − U dc 3 2

6 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 U NO voltage can be written as follows: U N0 =

1 U dc 

1 1   − t 0 − t 1 + t 2 + t 7  for the sectors I, III, V T s 2  3 3 

U N0 =

1 U dc  1 1   − t 0 − t 2 + t 1 + t 7  for the sectors II, IV, VI (2.36) T s 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 (t 0, t 7 ) 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 t 0 and t 7 t 0 =

T s  4  1 − M cos α  2  π 

for sectors I, III, V

t 0 =

T s  2  1 − M (cos α ) + 3 sin α  2  π 

for sectors II, IV, VI

t 7 = T s − t 0 − t 1 − t 2

THIPWM

Sinusoidal with 1/4 amplitude

t 0 =

1 T s  4    1 − M  cosα − cos 3α   for sectors I, III, V 2  π  4  

t 0 =

T s  2  1   1 − M  cosα + 3 sin α − cos 3α   2  π  2   for sectors II, IV, VI

t 7 = T s − t 0 − t 1 − t 2

SVPWM

Triangular with 1/4 amplitude

t 0 = t 7 = (T s − t 1 − t 2 ) 2

DPWM1

Discontinuous

t 0 = 0 t 7 = T s − t 1 − t 2

when

π

3

n ≤ α <

π

6

(2n + 1)

t 7 = 0 t 0 = T s − t 1 − t 2

when

π

6

π

(2n + 1) ≤ α < (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 3

U dc . 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 (at M = 1). U3 (010)

U2 (110)

U ) s / T ( t 2 U4 (011)

2

Uc

U0 (000) U7 (111)

Six-step mode M = 1 Overmodulation range 0.907 < M < 1 U1 (100)

(t 1 /T s )U1

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. U3 (010)

U2 (110)

Uc Uc* U4 (011)

U0 (000)

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 t 1 and t 2 are obtained from trigonometrical relationships and can be expressed in the following equations: t 1 = T s

3 cosα − sin α

(2.37a)

3 cosα + sin α

t 2 = T s − t 1

(2.37b)

t 0 = t 7 = 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) α h

Uc*

U4 (011)

U0 (000)

α

Uc

α ∗ α h

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 − α h π ∗ α =  π 6 − α h 6  π 3 

0 ≤ α ≤ α h for

α h < α < π 3 − α h

(2.38)

π 3 − α h ≤ α ≤ π 3

where: α h – 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 α h, the algorithm holds the vector Uc* at the vertex (U1). Next part is for α from α h 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 α h 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 T s and realized in the next switching time T sw are shown. In drive systems the controller mostly operates in synchronism with modulator and in RPWM 1 arises problems in the control system, when it works with variable sampling frequency. An additional control algorithm with variable sampling frequency is difficult tin a digital implementation. a)

U c(1)

K)

n −1)

n +1)

n)

U (c2)

U c(3)

Uc(

U c(

U c(

U (c

sampling cycles

1

2

3

...

n-1

n

...

switching cycles

1

2

3

...

n-1

n

...

U c(n )

U c( n

T s = T sw b)

U c(1)

−1)

+1)

U c( 2)

U c(3)

U c(K)

sampling cycles

1

2

3

...

n-1

n

...

switching cycles

1

2

3

...

n-1

n

...

U c( n

T s = T sw 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 α T s 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

T s

T s

T s

T s

SA SB SC

b)

α T s

α T s

T s

T s

SA SB SC T s

T s

T s

T s

T s

T s

T s

T s

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. T s U c(1)

sampling cycles

U c( 2)

U c(3)

Uc(K)

1

2

3

switching cycles

1 ∆t

2

U c( n

−1)

... 3

U (cn )

n-1

...

U c( n

+1)

...

n n-1

n

...

T sw

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 = rT s

(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 Ω K = Ω sr , which is defined as follows: Ω sr =

d γsr

(3.1)

dt

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 s K = U sd + jU sq I s K = I sd + j I sq , Ψ s K

= Ψ sd + jΨ sq ,

(3.2a) I r K = I rd + j I rq Ψ r K

= Ψ rd = Ψ r

(3.2b) (3.2c)

3.2. Field Oriented Control (FOC)

β

q

Is

I s β

Ω sr d

δ

I sq

Ψr

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 = R s I sd +

U sq = R s I sq +

0 = Rr I rd +

d Ψ sd dt d Ψ sq dt

− Ω sr Ψ sq

(3.3a)

+ Ω sr Ψ sd

(3.3b)

d Ψ r dt

(3.3c)

0 = Rr I rq + Ψ r (Ω sr − pb Ωm )

(3.3d)

Ψ sd = L s I sd + L M I rd

(3.4a)

Ψ sq = L s I sq + L M I rq

(3.4b)

Ψ r = Lr I rd + L M I sd

(3.4c)

0 = Lr I rq + L M I sq

(3.4d)

d Ωm dt

=

 1  m s L M − p Ψ I M r sq L  b J  2 Lr 

(3.5)

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

41

3. Vector Control Methods of Induction Motor d Ψ r dt

=

L M Rr R I sd − r Ψ r 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

m s L M

2 Lr

Ψ r I sq

(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

L M Rr

Ψ r

∫

Lr

M e

I sq

pb

m s

L M

2

Lr

M e

1 J

∫

Ω m

M L 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 s β must be transformed to the rotating system d − q . Similarly, the reference stator voltage vector components U sα c , U s β c , 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

I sd c

1

PI

L M

d − q

SA

U sα c

SVM M ec

2

I sq c

Lr 1

PI

pb m s L M Ψ rc

SB SC

U s β c

α − β

U sα γ sr

I sd

Flux Estimator

d − q

Voltage U s β Calculation

I sα 2

I sq

I s β

I s 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 = Ω sl + pb Ωm

(3.8)

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

1 Rr I sdc Lr

I sqc

(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

I sdc

1 L M

d − q

PI

SA

U sα c

SVM M ec

2

I sq c

Lr 1

SC

U s β c

PI

pb m s L M Ψ rc

SB

α − β γ sr

I sd

Rr 1

d − q

I sα 2

Lr I sdc I sq

∫

I s β

I s 3

α − β

Motor

Ω sr Ω sl

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 =

I sqc =

1 L M 2

Ψ r Lr 1

pb m s L M Ψ rc

(3.10)

M ec

(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 s β g β

(3.12)

where:

45


α Ψ r α

p b Ω mΨ r β

p b Ω mΨ r α αβ Ψ r α

f ( x )

α Ψ r β

µ ( Ψ r α I s β

0 ,0 ,

x

αβ Ψ r β Ψ r β I s α )

γ I sα γ I s β M L

(3.13)

J

T

1 σ L s

0 ,0 ,0 ,

g β

α L M I s β

β p b Ω mΨ r β

β p b Ω mΨ r α

gα

α L M I sα

,0 ,0

1 σ L s

(3.14)

T

,0

(3.15)

(3.16)

T

Ψ r α ,Ψ r β , I sα , I s β , Ωm

and Rr

α

(3.17)

Lr L M

β

2

γ

(3.18)

σ L s Lr R s Lr

2

Rr L M

(3.19)

2

σ L s Lr

µ = pb

m s LM

(3.20)

2 J

Because Ωm ,Ψ r α ,Ψ r β are not dependent on U sα ,U s β it is possible to chose variable dependent on x: φ 1 ( x )

Ψ r α

φ 2 ( x )

Ωm

2

Ψ r β

2

2

Ψ r

(3.21) (3.22)

If it is assumed that φ 1 ( x ) , φ 2 ( x ) are output variables, the full definition of new coordinates can be given by:

46

z 1

φ 1 ( x )

(3.23a)

z 2

L f φ 1 ( x )

(3.23b)

3.3. Feedback Linearization Control (FLC) z 3

φ 2 ( x )

(3.23c)

z 4

L f φ 2 ( x )

(3.23d)

z 5

arctan

Ψ r β

(3.23e)

Ψ r α

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:

&&1 z

L2 f φ 1

&&3 z

L2 f φ 2

D

U sα

(3.24)

U s β

where L g α L f φ 1 L g β L f φ 1

D

If φ 1

(3.25)

L g α L f φ 2 L g β L f φ 2

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

0 and it is possible to

define the linearization feedback as: U sα

D

U s β

-1

L2 f φ 1

v1

L2 f φ 2

v2

(3.26)

Then the resulting system is described by the equations:

&1 z

z 2

(3.27a)

&2 z

v1

(3.27b)

&3 z

z 4

(3.27c)

&4 z

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


ν 1

∫

z 2

2

Ψ r

∫

Ψ r

Ω m

∫ ν 2

∫

z 4

M e

J

M L

Fig. 3.5. Block diagram of the induction motor with new v1 and v 2 control signals

The control signals v1 , v2 are calculated by using linear feedback as follows: v1

k 11 z 1 z 1ref

v2

k 21 z 3 z 3ref

k 12 z 2 k 22 z 4

(3.28) (3.29)

where coefficients k 11 , k 12 , k 21 , k 22 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

ν 1 Flux Controller

Ωmc

ν 2 Speed Controller

Control Signals Transformation

U s β c

SA

U sα c

SB Vector Modulator S C

Voltage Calculation I sα

z 1 z 2 z 3 z 4 z 5

I s β

Feedback Signals Transformation

ˆ Ψ r α

Flux Estimator

ˆ U s I s

ˆ Ψ r β 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

m s L M 2 Lr

Ψ r I s sin δ

(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

m s

L M

2 Lr L s − L M

2

Ψ sΨ r sin δ Ψ

(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 Ψ

δ

Ψr

γ ss γ sr

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: d Ψ s dt

= Us

(3.32)

Taking into consideration the output voltage of the inverter in the above equation it can be written as: t

Ψs

= ∫ U v dt

(3.33)

0

where:

2 j ( v −1)π 3 U e dc  Uv =  3 0 

v = 1...6 (3.34) v = 0,7

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


Im U3 (010)

U4 (011)

U2 (110)

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


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

U3 (010)

U2 (110)

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

M ec

U dc

d Ψ

eΨ

e M

d M

Vector Selection Table

SA SB SC

γ ss (N)

Torque Controller

ˆ M e ˆ Ψ s

Sector Detection ˆ ˆ Ψ Ψ s β sα Flux and Torque Estimator

Voltage Calculation U s I s

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 ˆ and M ˆ values respectively. The signals which are compared with the estimated Ψ s e flux eΨ and torque e M errors are delivered to the hysteresis controllers. The digitized output variables d Ψ , 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)

d Ψ

d M

e M H Ψ

eΨ

H M

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

The output signals d Ψ , d M are defined as: d Ψ = 1 for eΨ > H Ψ

(3.35a)

d Ψ = 0 for eΨ < − H Ψ

(3.35b)

d M = 1 for e M > H M

(3.36a)

d M = 0 for e M = 0

(3.36b)

d M = −1 for e M < − H M

(3.36c)

In the classical DTC method the plane is divided for the six sectors (Fig. 3.14), which are defined as:

 π π  ,+  6 6  

(3.37a)

 π π  ,  6 2  

(3.37b)

 π 5π  ,+   2 6 

(3.37c)

Sector 1: γ ss ∈  −

Sector 2: γ ss ∈  +


54


 5π 5π  ,−  6   6

(3.37d)

 5π π  ,−  6 2  

(3.37e)

 π π  ,−  2 6  

(3.37f)




β

Sector 3

U3 (010)

Sector 4

U4 (011)

Sector 2 U2 (110)

U1 (100)

U0 (000) U7 (111)

α

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


β U3

U2

U4

δ Ψ

Ψs

U1

U5

U6

Sector 1 α Ψr

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

Table 3.1. Optimum switching table

d Ψ

1

0

d M

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


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


a)

b)

S/H M c + H m M c M c − H m t 1

t 2

t 3

T s

T s

T s

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 T s and works like analog only for high sampling frequencies f s =

1 T s

.

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 f s = 20kHz (Fig. 3.19) the switching frequency and torque error are not sensitive for hysteresis band. However, at the high sampling frequency f s = 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: ε ψ s =

58

ˆ − Ψ Ψ s sc Ψ sN

100%

(3.38a)


ε M =

ˆ − M M e ec M eN

100%

(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 5000

4792 5400

4567 4333 3508

0 0,0 b)

0,5

1,0

1,5

2,0

2750 2208 2367 2333 2,5

3,0

3,5

4,0 H m [Nm]

ε Μ _avr [%]

14 11,06 12 10,68 9,93 11,97 9,43 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 f s = 20kHz

a)

f sw [Hz]

25000 20000 15000

19750 13317

10000 8233 7400 6666

5000

6142 5492 5450 5666

0 0,0 b)

0,5

1,0

1,5

2,0

2,5

3,0

3,5

4,0 H m [Nm]

ε Μ _avr [%]

14 12 10 8 6 4 2,43 3,06 2 2,64 0 0,0

0,5

1,0

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 f s = 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


Flux Comparators

ψ sc

U dc

d A

SB

d B

M ec

SC SA

d m d C

Torque Controller

Voltage Calculation ˆ sC ψ

ˆ sB ψ ˆ sA ψ

BC

ˆ sα ψ ˆ s β ψ

α − β

ˆ M e

Flux and U s Torque Estimator

I s

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),

• very good torque and flux control dynamics.

62


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

Advantages

FOC ¾ Modulator ¾ Constant switching frequency ¾ Unipolar inverter output voltage ¾ Low switching losses ¾ Low sampling frequency ¾ Current control loops

Disadvantages • Coordinate transformation • A lot of control loops • Control structure depended on rotor parameters

DTC 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 • No modulator • Bipolar inverter output voltage • Variable switching frequency • High switching 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 calculated from the following equations:

Ψ sdc , Ψ sqc

can be

4.2. Structures of DTC-SVM – Review L s 

L d Ψ  Ψ rc + r σ rc  L M  Rr dt 

Ψ sdc

=

Ψ sqc

=

2

Lr

pb m s L M

σ L s

(4.1a)

M ec

(4.1b)

Ψ rc

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

2

Ψ sc

 L M   L   2   (σ L s )2  r ec  =  s Ψ rc  +   L M   pb m s   L M Ψ rc 

The commanded value of stator flux

Ψ sdc

,

Ψ sqc

2

(4.2)

after transformation to stationary

ˆ . coordinate system α − β are compared with the estimated values Ψ ˆ sα , Ψ s β Ψ rc

M ec

Ψ sdc

Egs (4.1)

SA

d − q

Ψ sqc

Ψ sc

∆Ψ s

T s

α − β

SB

SVM

SC

R s

γ ˆ sr Rotor Flux Estimator

U sc

1

ˆ Ψ s

Stator Flux Estimator

Us

Voltage Calculation

Is

U dc I A

α − β

I B ABC

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 R s : U sc =

∆Ψ s

T s

+ R sI s

(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

Torque Controller M ec

PI

∆δ ψ

SA Eg. (4.4)

γ ˆ ss ˆ M e

Ψ sc

∆Ψ s

U sc

1 T s

ˆ Ψ s

SB

SVM

SC

R s Flux and Torque Estimator

Us

Is

Voltage Calculation α − β

U dc I A I B

ABC

Fig. 4.2. DTC-SVM scheme with closed-loop to rque 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

ss + ∆δ Ψ

= Ψ sc e j (γ ˆ

)

(4.4)

Next, reference stator flux vector is compared with the estimated value. The error of the flux

∆Ψ s

equation (4.3).

68

is used, for calculation of the reference voltage vector, according to the

4.2. Structures of DTC-SVM – Review β

Ψ sc

ˆ Ψ s

∆δ Ψ

δ ˆΨ

ˆ Ψ r

γ ˆ ss γ ˆ sr

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

k Ψ

P

SA ∆Ψs

Eg. (4.7) M ec

PI

∆γ sd

Torque Controller

T s

∆γ s ∆γ ss

U sc

1

γ ˆ ss

SB

SVM

SC

R s ˆ Ψ s ˆ M e

Flux and Torque Estimator

Us

Is


ABC

U dc I A I B

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 k Ψ 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 ) = Ψs (k ) − Ψs (k − 1) = [1 + k (k )]⋅ e j

∆γ s

Ψ

( 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 ) = [k (k ) + j ∆γ s (k )]⋅ Ψ s (k − 1)

(4.7)

Ψ

The commanded stator voltage vector is calculated according to equation (4.3). To improve the dynamic performance of the torque control, the angle increment composed of two parts: the dynamic part the stationary part

∆γ ss

∆γ sd

∆γ s

is

delivered by the torque controller and

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

ˆ M e

SC

α − β γ ˆ ss

ˆ Ψ s

SB

SVM


Us

Is

Voltage Calculation

α − β

U dc I A I B

ABC

Fig. 4.5. DTC-SVM scheme operated in stator flux cartesian coordinates

70

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 sα c , U s β c 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 speed

Ω K

− y rotates with the stator flux angular

= Ω ss . This angular speed is defined as follows: Ω ss

=

d γ 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 s K = U sx + jU sy

(4.9a)

I s K = I sx + j I sy , I r K = I rx + j I ry

(4.9b)

71

4. Direct Flux and Torque Control with Space Vector Modulation (DTC-SVM) Ψ s K

= Ψ sx = Ψ s ,

Ψ r K

= Ψ rx + jΨ ry

The motor model equations (2.10-2.12) in U sx = R s I sx +

(4.9c)

− y coordinate system can be written as:

d Ψ s

(4.10a)

dt

U sy = R s I sy + Ω ssΨ s

0 = Rr I rx +

0 = Rr I ry + Ψ s

d Ψ rx dt d Ψ ry dt

(4.10b)

+ Ψ ry ( pb Ωm − Ω ss )

(4.11a)

+ Ψ rx (Ω ss − pb Ωm )

(4.11b)

= L s I sx + L M I rx

(4.12a)

0 = L s I sy + L M I ry

(4.12b)

Ψ rx

= Lr I rx + L M I sx

(4.12c)

Ψ ry

= Lr I ry + L M I sy

(4.12d)

d Ωm dt

=

1  m s  Ψ s I sy − M L pb   J  2 

(4.13)

The electromagnetic torque can be expressed by the following formula: M e = pb

m s 2

Ψ s I sy

(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: d Ψ s dt

= U sx − R s I sx

M e =

72

1 R s

p b

m s 2

Ψ s

(4.15)

(U

sy

− Ω ssΨ s )

(4.16)

4.3. Analysis and Controller Design for DTC-SVM Method

M L R s U sx

∫

I sx

1 L s Lr − Lm

2

Lr

Ψ s

Ψ s

L M pb

L s

U sy

Ω ss

÷ R s

m s M e 2

1 J

∫

Ω m

LM

I sy L M

Rr

∫

I rx

L s Lr − Lm

1

2

σ L r Ψ rx

pb

∫ Rr

Ψ ry

I ry

1 σ L r

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. R s I sx U sx

∫

Ψ s

Ω ss

U sy

pb

m s 1

M e

2 R s

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.

R s I sx Ψ sc

PI

U sx

∫

Ψ s

Ω ss

M ec

PI

U sy

pb

m s 1

M e

2 R s

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 + sT i 1   = K p = K p 1 + E ( s ) sT i  sT i  U ( s )

where: K p - controller gain, T i - controller integrating time. The PI controller scheme is presented in Fig. 4.9.

74

(4.17)


1 E ( s )

U ( s )

K p 1 T s i

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

1 + sT 1

s

Ψ s

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

(4.18)

sT 2 (1 + sT 1 )

where: K c = 1 is the inverter gain,

τ 0

is dead time of the inverter ( τ 0 = 0 ideal

converter), T 2 = 1 , and T 1 = T s 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 =

T 2 2 K c (T 1 + τ 0 )

=

1

(4.19)

2T s

T iΨ = 4(T 1 + τ 0 ) = 4T s

(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, t nΨ - 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

f s

K p Ψ

10.0 kHz 5.0 kHz 2.5 kHz

5000 2500 1250

T i Ψ 0.00040 0.00080 0.00160

t n Ψ 0.00150 s 0.00180 s 0.00200 s

p Ψ 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

U sy

PI

1 1 + sT s

pb

m s 1 2 R s

M e

Ψ s

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 ,

T iM = 4T s 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, t nM - time when the actual torque achieves first time reference value and p M - overshoot. Table 4.2. Torque controller parameters

f s 10.0 kHz 5.0 kHz

K pM 24 17

T iM 0.0004 0.0008

t nM 0.0007 s 0.0008 s

p Μ 8.39 % 18.53 %

77


a)

b)

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 (T iM = 800µ s ) , b) f s = 10kHz (T iM = 400µ s )

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 settling time damping

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


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    Rr L s + σ L s Lr U sx =  R s Rr + ( Rr L s + R s Lr ) + σ L s Lr   Ψ s dt  dt   dt   

+ R s I syσ L s Lr (Ω ss − pb Ωm )

(4.21)

2

where: σ = 1 −

L M

L s Lr

Under the assumption that the last term in the equation (4.21) is very small: R s I syσ L s Lr (Ω ss − p b Ωm ) ≈ 0

(4.22)

the equation (4.21) becomes: 2  d  d   d    Rr L s + σ L s Lr U sx =  R s Rr + ( Rr L s + R s Lr ) + σ L s 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 =

Ψ s

U sx

Rr σ Lr

=

AΨ + s

(4.24)

s + BΨ s + C Ψ 2

; BΨ =

Rr L s + R s Lr σ L s Lr

; C Ψ =

R s Rr σ L s Lr

The flux control loop is shown in Fig. 4.17, where G RΨ ( s ) is a transfer function of the PI controller given by equation (4.17).

81


Ψ sc

U sx

G RΨ ( s )

Ψ s

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

0.87

0.78

0.64

0.46

0.24 0. 24

1500

0.97

1000

0.992 500

s i x A

g a m I

4e+003

0

3e+003

2e+003

1e+003

-500 0.992

-1000

0.97

-1500 0.93 -4500

- 4000

0.87 -3500

-3000

0.78 -2500

-2000

0. 64 -1500

0.46 -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 , T iΨ = 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


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   R L R L σ L L I sy = Lr U sy − Lr Ψ s p b Ωm + I sxσ L s Lr (Ω ss − p b Ωm ) ( ) + + s r r s s r  dt 

(4.25) 2

where: σ = 1 −

L M

L s Lr

Under the assumption that the last term in equation (4.25) is very small: I sxσ L s Lr (Ω ss − pb Ωm ) ≈ 0

(4.26)

the equation (4.25) becomes: d   ( ) + + R L R L σ L L I sy = Lr U sy − Lr Ψ 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) d Ωm dt

=

m p b s Ψ s I sy J 2 1

(4.28)

From equation (4.14) current I sy can be expressed as follows: I sy = M e

2

(4.29)

p b m sΨ s

If both sides of equation (4.27) are differentiated, this equation becomes: 2  dU sy d Ω m d  d   ( ) + + = − Lr Ψ s pb R L R L σ L L I L   sy  s r r s s r  r 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: G M ( s ) =

where: A M =

M e U sy

pb m sΨ s 2σ L s

=

A M s

(4.31)

s 2 + B M s + C M

; B M =

R s Lr + Rr L s σ L s Lr

2

; C M =

pb m sΨ s

2

2σ L s J

The torque control loop is shown in Fig. 4.20, where G RM ( s ) is a transfer function of the PI controller given by equation (4.17).

M ec

G RM ( s )

U sy

G M ( s )

M e

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 , T iM = 0.00045 .

84

4.3. Analysis and Controller Design for DTC-SVM Method Root Locus Editor (C) 0.93

2500

2000

0.87

0.78

0.66

0.48

0.24

0.97

1500

1000

0.992

500

s i x A

g a m I

7e+003

0

6e+003

5e+003

4e+003

3e+003

2e+003

1e+003

-500

-1000

0.992

-1500

-2000

0.97

-2500

0.93

0.87

-7000

-6000

0.78

-5000

-4000 Real Axis

0.66 -3000

0.48 -2000

0.24 -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: A M K pM G Mc ( s ) =

M e M ec

T iM

=

(T iM s + 1)

s 2 + ( A M K pM + B M ) s + C M +

(4.32)

A M K pM T iM

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 , T iM = 0.00045 . Step Response From: r 1.4

1.2

1

0.8

e d y u t i : l o p T m A 0.6

0.4

0.2

0

0

0.5

1

1.5

2 Time (sec )

2.5

3

3.5

4 x 10

-3

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 C M , in equation (4.32) normally is several order lower in comparison with

A M K pM T iM . 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: G FM ( s ) =

1

(4.33)

T F s + 1

The time constant of the prefilter is equal time constant of the torque controller T F = T iM . 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

G FM ( s )

U sy

G RM ( s )

G M ( s )

Fig. 4.23. Torque control loop with prefilter

Step Response From: r 1.4

1.2

1

0.8

e d y u t i l : o p T m A 0.6

0.4

0.2

0

0

0.5

1

1.5

2

2.5

Time (sec)

3

3.5

4

4.5 -3

x 10

Fig. 4.24. Simulated (Matlab) step response of the system from Fig. 4.23

86

M e

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 , T iΨ = 0.00076 and torque controller K pM = 1.8492 , T iM = 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

ˆ M e

SC

α − β γ ˆ ss

ˆ Ψ s

SB

SVM


Us

Is


U dc I A I B

ABC 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 , T iΨ , K pM and T iM . 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 T iM (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


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


s

e u s r q e r t o e T m a r a p c i m x a u n l y F D

7 0 0 0 . 0 =

M

t n s 5 1 0 0 . 0

s

% 9 3 . 8 =

M p

% 6 . 1

9 0 0 0 . 0 =

M

t n s

9 1 0 0 . 0

% 4 0 . 1 =

M p

% 9 4 . 1

=

=

=

=

Ψ

p

Ψ

p

t n 0 0 . 4 2

Ψ

0 8 0 0 0 . 0

t n 1 2 . 3 3

Ψ

5 4 0 0 0 . 0

n s r e e u g q t i s e r = o = e = = T d m M M p p r a r M M i i e a l K T K T l p o r r 0 4 e t 7 l 1 0 4 n l 0 0 3 0 o o 0 r x 0 0 c t 5 u 5 0 2 0 f n l . . 0 o o F = 0 = y = = Ψ Ψ r C p p a Ψ Ψ i i K T K T m m r s u L L s r r 2 s J S r r R . Ψ L s L L 3 s e s + s Ψ . s s m u R L L L 4 b r , m q σ s σ L σ 2 p r e s b 2 l r o r 2 s Ψ p b t L R , s T a e e = = = T m m , M M M a s b r p A B C T a p r l L r s e r o d R t L o a + s r r r s L M x g L R u σ s s e r r L l t r F R L n R L σ i σ R 1 e = = = r = u Ψ Ψ Ψ s C T T p A B s u y r c t n o d e i o d L o o m r t h e h t t t o e m i o y r e S C M R M

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

f s

settling time overshoot

10.0 kHz 5.0 kHz 2.5 kHz

0.0015 0.0030 0.0060

2% 1% 1%

K p Μ 33.21 15.88 7.12

T i Μ 0.00045 0.00098 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


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


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: d Ωm dt

=

1 J

[ M e − M L ]

(4.34)

A block diagram of the speed control loop is shown in Fig. 4.32, where G RS ( s ) is a transfer function of PI controller (see equation 4.17) and G M ( 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

M L Ωmc

M ec

G RS ( s )

M e

G M ( s ) '

1

1

J

s

Ωm

1 T f 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 ) = '

M e M ec

= G FM ( s ) ⋅ G Mc ( s )

(4.35)

where: G Mc ( s ) - torque control loop transfer function given by equation (4.32), G FM ( s ) - prefilter transfer function given by equation (4.33). The transfer function G M ( s ) can by expressed as: '

'

' M

G

( s ) =

where: A M =

A M '

'

A M K pM

'

(4.36)

B M s 2 + C M s + 1

C M T iM + A M K pM

; B M = '

T iM C M T iM + A M K pM

; C M = '

T iM A M K pM + B M C M T iM + A M K pM

The torque control loop can be approximate by first order integrating part, because of: B M ≈ 0 '

(4.37)

The simplified transfer function can be written as: '

G ( s ) = ' M

A M

C M s + 1 '

(4.38)

For the torque controller parameters K pM = 15.87 , T iM = 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) A M = 0.9944 , B M = 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

0.02

0.025

0.03

Time

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 ce

− sτ 0

(4.39)

sT 2 (1 + sT 1 )

where: K c = A M ' is gain of the plan,

τ 0

is dead time of the inverter ( τ 0 = 0 ideal

converter), T 2 = J , and T 1 = C + T f is a sum of small time constants. The optimal controller parameters can be calculated as: K ps =

T 2 2 K c (T 1 + τ 0 )

=

J 2(C + T f )

T is = 4(T 1 + τ 0 ) = 4 C + T f

(4.40)

(4.41)

For the filter frequency f f = 25 Hz where: T f =

96

1 2π f f

(4.42)

4.4. Speed Controller Design the speed controller parameters are obtained as follows: K ps = 1.33 ; T is = 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 S A, S B, S C 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 3

U s β =

U dc ( D A − 0 .5 ( D B + D C )) 3

3

(5.1a)

U dc ( D B − D C )

(5.1b)

where D A, D B, DC are duty cycles corresponding to the switching signals SA, SB, S C and U dc is the voltage of inverter dc-link. U dc

U s β c

SA

U sα c

SB Vector Modulator S C

Dead Time & Voltage Drop Compensation

SA+ SA-

SA' S B' S C'

Dead Time

SB+ SBSC+ SC-

U sα U s β

Voltage Calculation

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 ( T D) between the turn-off one switch (IGBT) and the turn-on of the next one (from the same leg). T D 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

SA+

D1

C

I A > 0

A

0

T1 U dc 2

SA+

C

U dc

T2 D2

SA-

U dc

C

2

I A < 0

A

0

T2 S A-

D1

D2

C

2

I A > 0

I A < 0

SA

SA

t SA+

T D

t SA+

T D

t SA-

U A0 1

T D

SA-

t

0 1 2

U dc

0 1 2

U dc

Fig. 5.3. Dead-time effect for different current sing a) I A > 0 , b) I A < 0

102

t

U dc

2

t

T D

U A0 1

U dc

2

t

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

The voltage distortion ∆U

∆U

(5.2) can be written as:

= T D 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 + T D 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 T D 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 + '

I k I level

T D 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: ˆ Ψ s

104

= ∫ (U s − R s I s )dt

(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

R s

Us

ˆ Ψ 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 d Ψ s dt

ˆ − R I ) − = (U s s s

1 ˆ Ψs T F

(5.8)

The block diagram of the method with low-pass filter is presented in Fig. 5.6. Is

Us

1 R s

T F 1

ˆ Ψ s

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 T F . 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. ˆ ) 1 − j λ si gn ( Ω ss

Us

1 ˆ λ s + Ω ss

ˆ Ψ

γ ˆ

ˆ Ω ss

ˆ Ψ

s

s

ss

γ ˆ

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 Ψ s

1 − jλ sign (Ωˆ ss ) = 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 Ωˆ 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 Is

Lr

τ

L M

1 + sτ

R s

Is

L sσ ˆ Ψ r

L r

L sσ

s

Ψ rc

e

ˆ Ψ s

L M

1

jγ ˆ sr

1 + sτ

γ ˆ 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 d Ψ r

ˆ − Ψ e jγ ˆ ) = E r + K (Ψ r rc

(5.10)

sr

dt

where K is the gain factor and E r is the rotor back EMF defined as: Er =

Lr Lm

( U s − R s I s − σ L s

Then assuming K = −

ˆ Ψ r

=

τ 1 + sτ

1 τ

d I s dt

)

(5.11)

the equation (5.10) can be rewritten yielding:

Er +

1 1 + sτ

Ψ rc e

jγ ˆ sr

(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 Ψ s

=

Lm ˆ Ψ r + σ L s I 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 =

d γm

(5.15)

dt

Taking into consideration number of pole pairs pb angular speed of the coordinate system d ′ − q′ is equal Ω K = pb Ωm . The voltage, currents and fluxes complex space vector can be resolved into components d ′ and q′ . U s K = U sd ′ + jU sq′ I s K = I sd ′ + j I sq′ , Ψ s K

108

= Ψ sd ′ + jΨ sq′ ,

(5.16a) I r K = I r d ′ + j I r q ′ Ψ r K

= Ψ r d ′ + jΨ r q′

(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 ′ = R s I sd ′ +

U sq′ = R s I sq′ +

0 = Rr I r d ′ +

0 = Rr I r q′ +

d Ψ sd ′ dt d Ψ sq′ dt

− pb ΩmΨ sq′

(5.17a)

+ pb ΩmΨ sd ′

(5.17b)

d Ψ r d ′

(5.17c)

dt d Ψ r q′

(5.17d)

dt

Ψ sd ′ = L s I sd ′ + L M I r d ′

(5.18a)

Ψ sq′ = L s I sq ′ + L M I r q′

(5.18b)

Ψ r d ′ = Lr I r d ′ + L M I sd ′

(5.18c)

Ψ r q′ = Lr I r q′ + L M I sq′

(5.18d)

d Ωm dt

=

1  m s (Ψ sd ′ I sq′ − Ψ sq′ I sd ′ ) − M L  pb  J  2 

(5.19)

From the equations (5.17-5.17) formulas for the estimated rotor flux can be obtained [66]. ˆ d Ψ r d ′ dt ˆ d Ψ r q′ dt where: T r =

=

1

( L I T

− Ψ ˆ r d ′ )

(5.20a)

1

− Ψ ˆ r q′ )

(5.20b)

M sd ′

r

=

( L I T

M sq′

r

L r R r

The current vector is measured in stationary coordinate α − β . Therefore, current components I sα , I s β 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 s β

I sd ′

I sq′ d ′ − q′

L M

1 T r

L M

1 T r

∫ ∫

ˆ ′ ′ Ψ r d ′ d − q

ˆ Ψ r α

L sσ ˆ Ψ sα

L M L r

ˆ Ψ r q′

ˆ Ψ r β α − β

ˆ Ψ s β

L M L r

I s β γ m

L sσ

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

m s

ˆ * I = p m s Ψ ˆ I − Ψ ˆ I Im Ψ s s b sα s β s β 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 (Ω sr − Ω sl ) pb

(5.22)

where: Ω sr - rotor synchronous speed, Ω sl - slip frequency,

p b - number of pole pairs.

The rotor synchronous speed is equal angular speed of the rotor flux vector and can be calculated as: Ω sr =

d γ sr

(5.23)

dt

The slip frequency of induction motor is defined as follows [66]: Ω sl = Ω sr − pb Ωm

(5.24)

Based on the equations (3.3d) and (3.4d) in rotor flux coordinate system the slip frequency can be expressed: Ω sl = Rr

L M 1 Lr Ψ r

I sq

(5.25)

Taking into consideration the torque equations (3.7) and (5.25) the estimated sleep frequency can be calculated as follows: Ω sl =

Rr ˆ ˆ I Ψ sα I s β − Ψ s β sα 2 ˆ Ψ

(

)

(5.26)

r

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

DTC-SVM

Switch 1

Scalar Control SVM

Reference Speed

Speed Controller

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

3

Inverter

SA

SB

SC

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


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 TMS320F240 Subsystem,

•

16 channels (4 x 4ch) ADC, 16 bit , 4 µs, ±10 V,

•

4 channels ADC, 12 bit , 800 ns,

•

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)

±

10V,

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 Software – Control Desk 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


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


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 U A0 (150 V/div), 3) phase voltage U A (150 V/div), 4) output current I A (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 U A0 (150 V/div), 3) phase voltage U A (150 V/div), 4) output current I A (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 U A0 (150 V/div), 3) phase voltage U A (150 V/div), 4) output current I A (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 U A0 (150 V/div), 3) phase voltage U A (150 V/div), 4) output current I A (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 U A0 (150 V/div), 3) phase voltage U A (150 V/div), 4) output current I A (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



=

5 kHz 1) reference flux (0.15 Wb/div),



=

2.5 kHz 1) reference 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 f s

=

5 kHz (Fig. 7.10), f s

=

= 10

kHz (Fig. 7.9),

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)


=

5 kHz 1) reference torque (4.5 Nm/div),


127



=

2.5 kHz 1) reference 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 different modes is 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

Ω m = 125rad / s ).

129


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 = 25 Hz 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 = 25 Hz

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 = ±25 Hz ). The reverse time is 0.5s (Fig. 7.15) and 5s (Fig. 7.16).

130


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

Ω m = 120

rad / s .

131


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 speed

Ωm =

=

M N at

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 reverse time is 0.5s (Fig. 7.19) and 5s (Fig. 7.20).

132

= ±75rad / s

). The


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)


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


εΩ

m

=

ˆ Ωm − Ω m Ωm

100%

(7.1)

where: Ωm

- actual speed, Ωˆ m - estimated speed.

In Fig. 7.21 speed estimation error as the function of mechanical speed in steady state is presented. εΩ

m

[%]

50

45

40

35

] 30 % [ a g e m 25 o r_ o r r e

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


Fig. 7.22. Dynamic test of the speed estimation - Speed reversal

Ωm = ±50 rad / 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

Ω m = 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


Fig. 7.24. Sensorless vector control mode - Load torque step change from 0 to M L Ωm =

=

M N at speed

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)


136


Fig. 7.26. Sensorless vector control mode - Speed reverse

Ωm = ±75rad / s

(reverse time 5s)


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 frequency at least 25

•

−

=

5kHz whereas DTC requires sampling

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”, Toru ń.

140

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K.L. Shi, T.F. Chan, Y.K, Wong, S.L. Ho, "Direct self control of induction motor based on neural network", IEEE Transactions on Industry Applications, Vol. 37, Issue: 5, Sept.-Oct. 2001, pp.1290-1298.

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149

References [122] M. Żelechowski, D. Świerczyński, M. P. Kazmierkowski, J. Załę ski, "Universal inverter drives controlled by new generation microprocessors", Elektroinfo Nr 6 (17) 2003, pp.26-28. (in Polish) [123] M. Żelechowski, M. Malinowski, P. Kaczyński, W. Kołomyjski, M. Twerd, J. Załę ski, "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. [124] M. Jasiński, D. Świerczyński, M. P. Kaźmierkowski, M. Żelechowski, "Sensorless Direct Power 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. [125] M. Żelechowski, P. Kaczyński, "Automatic measurement of induction motor parameters", Przeglą d Elektrotechniczny, No. 1/2004, pp.6-10. (in Polish) [126] D. Świerczyński, M. Żelechowski, "Universal structure of direct torque control for AC motor drives", Przeglą d Elektrotechniczny, No. 5/2004, pp.489-492. [127] M. Żelechowski, W. Kolomyjski, M. Twerd, "Industrial Application of Sensorless Direct Torque Control – Space Vector Modulated (DTC-SVM) for Inverter Fed Induction Motor Drives", IV Summer Seminar on Nordick Network for Multi Disciplinary Optimised Electric Drives, Tallinn, Estonia, June 2004, pp.77-79. [128] M. Cichowlas, M. Żelechowski, "PWM Rectifier with active filtering", IV Summer Seminar on Nordick Network for Multi Disciplinary Optimised Electric Drives, Tallinn, Estonia, June 2004, pp.101-107. [129] M.P. Kaźmierkowski, M. Żelechowski, D. Świerczynski, "DTC-SVM an efficient method for control both induction and PM synchronous motor”, In Proc. of the EPE- PEMC, Riga, Latvia, Sep. 2004. [130] M. Jasiński,

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. [131] M.P. Kaźmierkowski, M. Żelechowski, D. Świerczynski, "Simple DTC-SVM Control Scheme for Induction and PM Synchronous Motor", XVI International Conference on Electrical Machines ICEM’2004, Krakow, Poland, Sep. 2004. [132] M. Jasiński, M. P. Kaźmierkowski, M. Żelechowski, "Direct Power and Torque Control Scheme for Space Vector Modulated AC/DC/AC Converter- Fed Induction Motor", XVI International Conference on Electrical Machines ICEM’2004, Krakow, Poland, Sep. 2004. [133] M. Malinowski, W. Kołomyjski, M. Żelechowski, P. Wójcik, "New Space Vector Modulator in Industrial Application", IX Sympozjum - Energoelektronika w Nauce i Dydaktyce ENID’2004, Poznań, Sep. 2004, pp. 115-122.

150

List of Symbols a = e j 2

π

3

= −

1 2

+

j

3 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 s β - stator voltage vector components in stationary α − β coordinate

system I r α , I r β - 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 L M - main, magnetizing inductance L s - stator winding self-inductance Lr - rotor winding self-inductance - mutual inductance, absolute value

List of symbols - torque, absolute value M e - electromagnetic torque M L - load torque M , m - modulation index m s - 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 R s - stator phase windings resistance T i - controller integrating time T iM - torque controller integrating time T iΨ - flux controller integrating time T D - dead time of inverter T r =

Lr R r

- rotor time constant

T s - sampling time T sw - 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, U c - reference voltage vector

152

ν

=

0 ,...,7

List of symbols

U sα ,U s β - stator voltage vector components in stationary α − β coordinate system U sα c ,U s β c - 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

Ψs

- space vector of the stator flux linkage

Ψr

- space vector of the rotor flux linkage

Ψ s

- stator flux amplitude

Ψ r

- rotor flux amplitude

Ψ sα ,Ψ s β

- stator flux vector components in stationary α − β coordinate system

Ψ r β ,Ψ r β

- 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 Ω K

- angular speed of the coordinate system

Ωm

- angular speed of the motor shaft

Ω sr

- angular speed of the rotor flux vector Ω sr =

Ω ss

- angular speed of the stator flux vector Ω ss

Ω sl

- slip frequency

Ωm

=

d γ m dt

=

2

σ = 1 −

L M

L s Lr

- total leakage factor

Superscript

^ - estimated value

Subscripts

..c - reference value

Rectangular coordinate systems

α − β - stator oriented, stationary coordinate system d ' − q ' - rotor oriented, rotated coordinate system −

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

d γ sr dt d γ ss dt

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 (ω t ) =

∑ b sin(nω t )

(A.1.1)

n

n =1

where, for n = 1, 2, 3, … 2

bn =

π

∫ f (ω t )sin (nω t )d (ω t )

(A.1.2)

π 0

Function which describes phase inverter voltage is shown in the Fig. A.1.1 U A 2

U dc

3 1

U dc

3

0 1 3 2 3

π 3

2π 3

π

4π 3

5π 3

2π

ω t

U dc U dc

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 (nω t )d (ω t ) A

0

2π  π 3  π 3   2 1 2 1 =  ∫ U dc sin (nω t )d (ω t ) + ∫ U dc sin (nω t )d (ω t ) + ∫ U dc sin (nω t )d (ω t ) π  0 3 π 3 2π 3   3 3 

2π π  π  = U dc  − cos(nω t ) 03 − 2 cos(nω t ) π 3 − cos(nω t ) 2π  3π n 3  3 

2 1

=

2 1 3π n



 π   2    − cos n π    3   3  

U dc 1 − cos(nπ ) + cos n



(A.1.3)

Appendices for even n:

 π   2   − cos n π   3   3 

1 − cos(nπ ) + cos n

π   π   = 1 − 1 + cos n  − cos nπ − n  = 0 3   3  

(A.1.4)

and for uneven n: π   π   2   π    − 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    

1 − cos(nπ ) + cos n

From above formulas the Fourier series for U A is given by: U A =

=

4 3π 2 π

1   π   1 + cos n   sin (nω t )  3   n =1 n  ∞

U dc

∑

∞

U dc

1

∑ n sin(nω t )

(A.1.6)

n =1

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 { 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

P N = 3 kW

Voltage

U N = 380 V

Current

I N = 6.9 A

Frequency

f = 50 Hz N

Base speed

Ω N = 1415 rpm

Number of pole pairs

pb = 2

Moment of inertia

J = 0.007 kgm 2

Nominal torque

M N = 20 Nm

Nominal stator flux

Ψ sN = 0.98 Wb

Table A.3.2. Parameters of 3 kW induction motor

Stator winding resistance

R s = 1.85 Ω

Rotor winding resistance

Rr = 1.84 Ω

Stator inductance

L s = 170 mH

Rotor inductance

Lr = 170 mH

Mutual inductance

L M = 160 mH

Table A.3.3. Data of 15 kW induction motor

Power

P N = 15 kW

Voltage

U N = 380 V

Current

I N = 28.9 A

Frequency

f = 50 Hz N

Base speed

Ω N = 1460 rpm


pb = 2

Moment of inertia

J = 0.875 kgm 2

Nominal torque

M N = 98 Nm

Nominal stator flux

Ψ sN = 0.98 Wb

161

Appendices Table A.3.4. Parameters of 15 kW induction motor


R s = 0.28 Ω


Rr = 0.26 Ω

Stator inductance

L s = 63.5 mH

Rotor inductance

Lr = 63.5 mH

Mutual inductance

L M = 58.1 mH

Table A.3.5. Data of 90 kW induction motor

Power

P N = 90 kW

Voltage

U N = 380 V

Current

I N = 158 A

Frequency

f = 50 Hz N

Base speed

Ω N = 1483 rpm


pb = 2

Moment of inertia

J = 1.50 kgm 2

Nominal torque

M N = 580 Nm

Nominal stator flux

Ψ sN = 0.98 Wb

Table A.3.6. Parameters of 90 kW induction motor

162


R s = 0.020

Ω


Rr = 0.016

Ω

Stator inductance

L s = 16.36 mH

Rotor inductance

Lr = 16.74 mH

Mutual inductance

L M = 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

Eve nt Manager A - Capture Inputs - Compare/PWM Outputs - GP Timers/ PWM

Event Manager B - Capture Inputs - Compare/PWM Outputs - GP Time rs / 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

Marcin Zelechowski

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