Direct Torque Control with Space Vector Modulation (DTC-SVM) of

POLITECHNIKA WARSZAWSKA

WARSAW UNIVERSITY OF TECHNOLOGY Faculty of Electrical Engineering

ROZPRAWA DOKTORSKA Ph.D. Thesis

Dariusz Świerczyński, M. Sc.

Direct Torque Control with Space Vector Modulation (DTC-SVM) of Inverter-Fed

Permanent Magnet Synchronous Motor Drive

WARSZAWA 2005

WARSAW UNIVERSITY OF TECHNOLOGY

Faculty of Electrical Engineering Institute of Control and Industrial Electronics

Ph.D. Thesis

M. Sc. Dariusz Świerczyński

Direct Torque Control with Space Vector Modulation (DTC-SVM) of Inverter-Fed

Permanent Magnet Synchronous Motor Drive

Thesis supervisor Prof. Dr Sc. Marian P. Kaźmierkowski

Warsaw, Poland - 2005

Contents

Table of Contents

Chapter 1 1 INTRODUCTION Chapter 2 8 MODELING AND CONTROL MODES OF PM SYNCHRONOUS DRIVES

2.1 Mathematical model of PM synchronous motor 8 2.1.1 Voltage and flux-current equations 9 2.1.2 Instantaneous power and electromagnetic torque 17 2.1.3 Mechanical motion equation 22

2.2 Static characteristic under different control modes 25 2.3 Summary 33

Chapter 3 34 VOLTAGE SOURCE PWM INVERTER FOR PMSM SUPPLY

3.1 Introduction 34 3.2 Voltage source inverter (VSI) 35 3.3 Space vector based pulse width modulation (PWM) methods 46 3.4 Summary 52

Chapter 4 53 CONTROL METHODS OF PM SYNCHRONOUS MOTOR

4.1 Introduction 53 4.2 Field oriented control (FOC) 54 4.3 Direct torque control (DTC) 57 4.4 Summary 64

Chapter 5 65 DIRECT TORQUE CONTROL WITH SPACE VECTOR MODULATION (DTC-SVM)

5.1 Introduction 65 5.2 Cascade structure of DTC–SVM scheme 66

5.2.1 Digital flux control loop 68 5.2.2 Digital torque control loop 82

5.3 Parallel structure of DTC–SVM scheme 91 5.3.1 Digital flux control loop 92 5.3.2 Digital torque control loop 102

5.4 Speed control loop for DTC–SVM structure control 113 5.5 Summary 122

Chapter 6 121

DIRECT TORQUE CONTROL WITH SPACE VECTOR MODULATION (DTC-SVM) OF PMSM DRIVE WITHOUT MOTION SENSOR

6.1 Introduction 121 6.2 Initial rotor position estimation method 123 6.3 Stator flux estimation methods 127

6.3.1 Overview 127 6.3.2 Current model based flux estimator 127 6.3.3 Voltage model based flux estimator with ideal integrator 128

Contents

6.3.4 Voltage model based flux estimator with low pas filter 129 6.3.5 Improved voltage model based flux estimator 130

6.4 Electromagnetic torque estimation 132 6.5 Rotor speed estimation methods 132

6.5.1 Overview 132 6.5.2 Back electromotive force (BEMF) technique 133 6.5.3 Stator flux based technique 133

6.6 Summary 136 Chapter 7 137 DSP IMPLEMENTATION OF DTC-SVM CONTROL

7.1 Description of the laboratory test-stand 137 7.2 Steady state behaviour 140 7.3 Dynamic behaviour 143

7.3.1 Flux and torque control loop 143 7.3.2 Speed control loop 151

Chapter 8 161 SUMMARY AND CLOSING CONCLUSIONS Appendices 163 Picture of rotor and stator of PMSM machine

Basic transformation

Model of PM synchronous motor- SABER

Parameters of PMSM machine

Parameters of voltage source inverter PI speed controller

PWM technique - overmodulation List of Symbols 170 References 172

Introduction

1

Chapter 1 INTRODUCTION

Recently, an increased interest in application of permanent magnet synchronous motors

(PMSM) in speed controlled drives has been observed. This is stimulated mainly by:

• development of modern high switching frequency semiconductor power devices (as

for example IGBT modules of 5-th generation),

• new rare earth magnetic materials as samarium-cobalt (Sm-Co) or neodymium-iron-

boron (Nd-Fe-B),

• specialized digital signal processor (DSP) for AC drive applications with integrated

PWM function, A/D converters as well as processing of encoder signals (e.g

ADMC401, TMS320FL24XX, TMS320FL28XX).

Synchronous motors with an electrically excited rotor winding have a conventional three-

phase stator winding (called armature) and an electrically excited field winding on the rotor,

which carries a DC current. The armature winding is similar to the stator of induction motor.

The electrically excited field winding can be replaced by permanent magnet (PM) [1]. The use

of permanent magnets has many advantages including the elimination of brushes, slip rings,

and rotor copper losses in the field winding. It leads to higher efficiency. Additionally since

the copper and iron losses are concentrated in the stator, cooling of machines through the

stator is more effective. The lack of field winding and higher efficiency results in reduction of

the machine frame size and higher power/weight ratio.

Figure. 1.1. General classification of AC synchronous motors.

Introduction

2

Generally, the permanent magnet AC machines can be classified into two types (Fig.1.1):

trapezoidal type called “brushless DC machine” (BLDCM) and sinusoidal type called

permanent magnet synchronous machine (PMSM). The BLDC machines operate with

trapezoidal back electromagnetic force (EMF) and require rectangular stator phase current.

The PMSM’s generate sinusoidal EMF and operate with sinusoidal stator phase current.

The PMSM can be further divided into two main groups in respect how the magnet bars have

mounted in the rotor [6,7]. In the first group magnets are mounted in the rotor (Fig. 1.2 c-d)

and this type is called interior permanent magnet synchronous motors (IPMSM). The second

group is represented by surface permanent magnet synchronous motors (SPMSM). In the

SPMSM magnet bars are mounted on the rotor surface (Fig. 1.2 a-b).

SPMSM

IPMSMd

q

d

q

NSNS d

q

NSNS

d

q

NSNS

)a )b

)c )d

Fig. 1.2. The cross section of the PMSM rotor shaft and the magnet bars placements:

a),b),c) axial field direction, d) radial field direction.

The magnets can be placed in many ways on the rotor (Fig. 1.2). In radial field fashion the

magnet bars are along the radius of the machine and this arrangement provides the highest air

gap flux density, but it has the drawback of lower structural integrity and mechanical

robustness. Machines with this arrangement of magnets are not preferred for high-speed

applications (higher than 3000 rpm). In axial field manner the magnets are placed parallel to

the rotor shaft. This arrangement of magnets is much more robust mechanically as compared

Introduction

3

to surface-mounted machine. It makes possible to use IPMSM for higher-speed applications

(contrary to SPMSM’s).

Regardless of the fashion of mounting the PM, the basic principle of motor control is the same

and the differences are only in particularities. An important consequence of the method of

mounting the rotor magnets is the difference in direct and quadrature axes inductance values.

The direct axis reluctance is greater than the quadrature axis reluctance, because the effective

air gap of the direct axis is multiple times that of the actual air gap seen by the quadrature

axis. As consequence of such an unequal reluctance, the quadrature inductance is higher than

direct inductance q dL L> . It produces reluctance torque in addition to the mutual torque.

Reluctance torque is produced due to the magnet saliency in the quadrature and the direct axis

magnetic paths. Mutual torque is produced due to the interaction of the magnet field and the

stator current. In case where the magnets bars are mounted on the rotor surface the quadrature

inductance is equal direct inductance q dL L= , because of the same flux paths in d and q axis.

As result the reluctance torque disappears.

Among the main advantage of PM machines are [12]:

• high air gap flux density,

• higher power/weight ratio,

• large torque/inertia ratio,

• small torque ripples,

• high speed operation,

• high torque capability (quick acceleration and deceleration),

• high efficiency and high cosφ (low expense for the power supply),

• compact design.

Thanks to this advantages the PMSM’s are usually used in high performance servo drives, in

special applications as computer peripheral equipment, robotics, ect. However, recently the

PMSM are also used as adjustable–speed drives in variety of application such as fans, pumps,

compressors, blowers. Another area is automotive application as an alternative drive in hybrid

mode with classical engine. The power of offered synchronous motors is in the range several

kW to MW.

Introduction

4

The main requirements for high performance PWM inverter-fed PMSM drive can be

formulated as follows:

• operation with and without mechanical motion sensor,

• 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 ripples,

• robustness to parameters variation,

• four quadrant operation.

To meet the above requirements, different control methods can be used [3,4,10].

VariableFrequency

Control

Vector basedcontrollers

Scalar basedcontrollers

V/Hz=constwith stabilization

loop

FieldOriented(FOC)

Direct TorqueControl(DTC)

PM (rotor) Flux Oriented

(RFOC)

Stator FluxOriented(SFOC)

Direct TorqueControl with Space Vector Modulation

(DTC-SVM)

Circular flux trajectory

(Takahashi)

Figure 1.3 Classification of PMSM control methods.

The general classification of the variable frequency control for PMSM is presented in Fig. 1.3.

The PMSM control methods can be divided into scalar and vector control. According to [3],

in scalar control, which based on a relation valid for steady states, only the magnitude and

frequency (angular speed) of voltage, currents, and flux linkage space vectors are controlled.

Thus, the control system does not act on space vector position during transient. Therefore, this

control is dedicated for application, where high dynamics is not demanded. Contrary, in

Introduction

5

vector control, which is based on relation valid for dynamics states, not just magnitude and

frequency (angular speed), but also instantaneous position of voltage, current and flux space

vectors are controlled. Thus, the control system adjust the position of the space vectors and

guarantee their correct orientation for both steady states and transients.

The scalar constant V/Hz control for PMSM without damper winding (squire cage) is not

simple as for induction motor. It requires additional stabilization control loop, which can be

provide by feedback from: rotor velocity perturbation, active power or DC-link current

perturbation [9].

The most popular vector control method developed in 70s, known as field oriented control

(FOC) [31] gives the permanent magnet synchronous motor high performance. In this method

the motor equation are transformed in a coordinate system that rotates in synchronism with

permanent magnet flux. It allows separately and indirectly control flux and torque quantities

by using current control loop with PI controllers like in well known DC machine control [3].

In search of a simpler and more robust high performance control system in 80s new vector

control called direct torque control (DTC) was developed [50]. It was innovative studies at

this time and completely different approach which depart from the idea of coordinate

transformation and the analogy with DC motor control. It allows direct control flux and torque

quantities without inner current control loops. Using bang-bang hysteresis controllers for flux

and torque control loops made this control concept very fast and not complicated. However,

the main disadvantage of DTC is fast sampling time required and variable switching

frequency, because of hysteresis based control loops. In order to eliminate above

disadvantages and kept basic control rules of classical DTC, at the beginning of 90’s a new

developed control technique called direct torque control with space vector modulator (DTC-

SVM) has been introduced [54,55]. However, from the formal consideration this method can

also be viewed as stator flux oriented control (SFOC). This control employed instead of

hysteresis controller as for classical DTC, the PI controllers and space vector modulator

(SVM). It allows to achieve fixed switching frequency, what considerably reduce switching

losses as well as torque and current ripples. Also requirement of very fast sampling time is

eliminated [113,115,117]. Therefore, this new method is subject of this thesis. In spite of

many control strategies there is no one which may be considered as standard solution.

Introduction

6

Therefore, the following thesis can be formulated:

“In the view of commercial manufacturing process the most convenient control scheme

for voltage source inverter-fed permanent magnet synchronous motor (PMSM) drives is

direct torque control with space vector modulator (DTC-SVM)”.

To prove the above thesis, the author used methodology based on an analyze and simulation

as well as experimental verification on the laboratory setup with 3kW PMSM motor.

Moreover, the presented control algorithm DTC-SVM has been introduced and used in serial

commercial product of Polish manufacture TWERD, Toruń.

In the author’s opinion the following results of the thesis are his original achievements:

• development of a simulation algorithm in SABER package for the investigation

of PWM inverter-fed PMSM control,

• elaboration and experimental verification of digital flux and torque controller

design based on the Z-transform approach for series (cascade) and parallel

structure of DTC-SVM schemes,

• implementation and verification of series (cascade) and parallel DTC-SVM

schemes on experimental laboratory setup with 3kW PM synchronous motor

drive controlled by floating point DS1103 board.

• bringing into production and testing of developed DTC-SVM algorithm in Polish

industry.

The thesis consists of eight chapters. Chapter 1 is an introduction. In Chapter 2 mathematical

model of PM synchronous motor and his basic control modes are presented. Chapter 3 is

devoted to voltage source inverter, his nonlinear characteristics and different PWM

techniques. Chapter 4 gives brief review of PM synchronous motor control method such as

FOC and classical DTC. In Chapter 5 two kind of DTC-SVM control schemes are presented.

Also, the analysis and synthesis of digital flux, torque and speed controllers based on Z

transform approach are given. Chapter 6 is devoted to initial rotor detection methods, stator

flux vector and rotor speed estimation algorithms. In Chapter 7 experimental results are

Introduction

7

presented and studied. Chapter 8 includes the finally conclusions. Description of the SABER

based control algorithm, basic coordinate transformations and parameters of used PM

synchronous machine as well as inverter are given in Appendices.

Modeling and control modes of PM synchronous motor drives

8

Chapter 2 MODELING AND CONTROL MODES OF PM SYNCHRONOUS DRIVES 2.1 Mathematical model of PM synchronous motor

Development of the machine model through the understanding of physics of the

machine is the key requirement for any type of electrical machine control. Since in this

project a Surface type Permanent Magnet Synchronous Motor (SPMSM) is used for the

investigation [9,13,14,15,16]. The development of those models is under bellow

assumptions as [3]:

• three-phase motor is symmetrical,

• only a fundamental harmonic of the magneto motive force (MMF) is taking in to

account,

• the spatially distributed stator and rotor winding are replaced by a concentrated

coil,

• an anisotropy effects, magnetic saturation, iron loses and eddy currents are not

taking into considerations,

• the coil resistances and reactances are taking to be constant,

• in many cases, especially when is considered steady state, the currents and

voltages are assumed to be sinusoidal,

• thermal effect for permanent magnets is omitted. The synchronous motor model will be presented in space vector notation. Space vector

form of the machine equations has many advantages such as compact notation, easy

algebraic manipulation, and very simple graphical interpretation. Specially, this notation

is very useful when analyzing the vector control based technique of the AC machines.

The space vector representation of AC machine equations has been discussed in detail

in number of text books ([3,4,12]).

The instantaneous value of a three-phase system , ,A B CK K K (such as currents, voltages

and flux linkages) can be replaced by one resultant vector called the space vector,

22 13 A B CK K a K a K⎡ ⎤= ⋅ + ⋅ +⎣ ⎦ (2.1)


9

where:1,23 1 3

2 2j

a e jπ

= = − + ,2 4

2 3 3 1 32 2

j ja e e j

π π−

= = = − − - complex vectors, 2/3 –

normalization factor (guarantee that for balanced sinusoidal waveforms the magnitude

of the space vector is equal to the amplitude of that phase waveforms).

The elements of this space vector satisfy the condition:

0A B CK K K+ + = (2.2)

and it means that we have three-phase system without neutral wire.

2.1.1 Voltage and current equations For idealized motor (Fig. 2.1), the following equations of the instantaneous stator phase

voltages can be written [3]:

Figure 2.1. Layout and symbols for three-phase PMSM electric motor windings.

sAsA sA sA

dU I RdtΨ

= + (2.3a)

sBsB sB sB

dU I RdtΨ

= + (2.3b)

sCsC sC sC

dΨU I Rdt

= + (2.3c)

NS

N

S

N

S

A

B

C

a

b

c

mγ

sBUsBI

sAU

sAI

sCI

sCU

sAZ

sBZ

sCZ


10

where sCsBsA UUU ,, are the instantaneous stator voltage values, sCsBsA III ,, are

instantaneous values of the current, sCsBsAs RRRR === is the resistance of the stator

windings, and ,sA sBΨ Ψ and sCΨ are magnetic flux linkages stator windings BA, and

C , respectively.

Using the space vector theory to voltage equations we can written in vector form

sABCsABCsABC s

dU R IdtΨ

= + (2.4)

where:

22 (1 )3 sA sB sCsABCU U aU a U= + + , 22 (1 )

3 sA sB sCsABCI I aI a I= + + , 22 (1 )3 sA sB sCsABC a aΨ = Ψ + Ψ + Ψ are the

stator voltage, current and flux space vectors, respectively.

The stator winding flux consist of rotor flux and stator flux linkages:

( ) ( )sABC ABC s ABC rΨ = Ψ +Ψ (2.5)

where,

( )

sA sAB sAC sA

ABC s sBA sB sBC sB

sCA sCB sC sC

L M M IM L M IM M L I

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥Ψ = ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

(2.6)

( )

cos2cos( )3

2cos( )3

r

ABC r PM r

r

θπθ

πθ

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥Ψ = Ψ −⎢ ⎥⎢ ⎥⎢ ⎥+⎢ ⎥⎣ ⎦

(2.7)

and, rθ is electrical rotor position. Mechanical rotor position is defined as:

r b mpθ γ= (2.8)

where: bp - number of pole pairs, mγ - mechanical position.

In equation (2.6) sAL is the self-inductance of phase A winding, sABM and sACM are the

mutual inductances between A and B phase, A and C phase, respectively. For self and

mutual inductances of B and C phase the same notations used. In (2.7), PMΨ is the


11

amplitude of the flux linkages established by the permanent magnet on the rotor. The

inductances are described below.

Due to the rotor saliency in IPMSM the air gap is not uniform and, therefore, the self

and mutual inductances of stator windings are a function of the rotor position.

The derivation of these rotor position dependent inductances is available in details in

[5]. The results are summarized here as follows:

The stator winding self-inductances are

cos 2sA ls A B rL L L L θ= + − (2.9a)

2 4cos2( ) cos(2 )3 3sB ls A B r ls A B rL L L L L L Lπ πθ θ= + − − = + − − (2.9b)

2 4cos 2( ) cos(2 )3 3sC ls A B r ls A B rL L L L L L Lπ πθ θ= + − + = + − + (2.9c)

where, lsL is stator-winding leakage inductance and ,A BL L are given by

2

0 12s

AmL rlπµ ε⎛ ⎞= ⎜ ⎟

⎝ ⎠ (2.10a)

2

0 212 2

sB

mL rlπµ ε⎛ ⎞= ⎜ ⎟⎝ ⎠

(2.10b)

where, sm is number of turns of each phase winding, r is radius, which is from center

of machine to the inside circumference of the stator, and l is the axial length of the air

gap of the machine, 0µ is permeability of the air, 1ε and 2ε are defined as s:

)11(21

maxmin1 gg

+=ε (2.11a)

)11(21

maxmin2 gg

−=ε (2.11b)

where, ming is minimum air gap length and maxg is maximum air gap length.

The mutual inductances between stator phase are:

1 1 2cos 2( ) cos(2 )2 3 2 3sAB sBA A B r A B rM M L L L Lπ πθ θ= = − − − = − − − (2.12a)


12

1 1 2cos 2( ) cos(2 )2 3 2 3sAC sCA A B r A B rM M L L L Lπ πθ θ= = − − + = − − + (2.12b)

1 1cos 2( ) cos(2 2 )2 2

1 cos 22

sBC sCB A B r A B r

A B r

M M L L L L

L L

θ π θ π

θ

= = − − + = − − +

= − − (2.12c)

Using the space vector theory, the flux linkage sABCΨ space vector can be written as:

23 3( )2 2

r rj jls A B PMsABC sABC sABCL L I L I e eθ θ∗Ψ = + − + Ψ (2.13)

where, 22 (1 )3 sA sB sCsABCI I aI a I= + + , 22 (1 )

3 sA sB sCsABCI I a I aI∗ = + + are the stator current

space vector and conjugate stator current space vector.

Taking into account that:

d ls mdL L L= + (2.14a)

q ls mqL L L= + (2.14b)

where, 3 ( )2md A BL L L= + , 3 ( )

2mq A BL L L= − are d and q magnetizing inductances and

are defined as [5].

Finally, equations (2.13) comes as:

2( ) ( )2 2

r rd q q d j jPMsABC sABC sABC

L L L LI I e eθ θ∗+ −

Ψ = − + Ψ (2.15)

where, dL , qL are d and q inductances.

Space vector form of machine equations (2.4, 2.15) becomes more compact, but the

rotor position dependent parameters still exist in that form of expressions for the stator

flux linkage space vector. Therefore, the space vector model is still not simple to use for

the analysis. A simplification can be made if the space vector model is referred to a

suitably selected rotating frame.


13

Figure 2.2 shows axes of reference for the three-stator phase , ,A B C . It also shows a

rotating set of ,x y axes, where the angle Kθ is position of x -axis in respect to the stator

A phase axis. Variables along the ,A B and C axes can be referred to the x − and

y − axes by the expression:

cos cos( 2 / 3) cos( 2 / 3)2sin sin( 2 / 3) sin( 2 / 3)3

Ax K K K

By K K K

C

KK

KK

K

θ θ π θ πθ θ π θ π

⎡ ⎤− +⎡ ⎤ ⎡ ⎤ ⎢ ⎥=⎢ ⎥ ⎢ ⎥ ⎢ ⎥− − − − +⎣ ⎦⎣ ⎦ ⎢ ⎥⎣ ⎦

(2.16)

AKCK

xKyK

Kθ

ABCK

x

y

BK

KΩ

Figure 2.2. Stator fixed three phase axes (A,B,C) and general rotating reference frame ( ,x y ). Finally, the space vector in general rotating frame can be written as:

(cos sin ) KjK KABCs K KK K j K e θ= Θ + Θ = (2.17)

In this case the voltage equation (2.4) using (2.17) can written as:

( )K K Kj j js sK sKsK

dU e R I e edt

θ θ θ= + Ψ (2.18)

Using chain rule, equation. (2.17) and divided by term Kje θ can be written as:

K

sKs sK sKsK

dU R I j

dtΨ

= + + Ω Ψ (2.19)

where sKU , sKI , sKΨ is the stator voltage, current and flux space vector in general

rotating frame.

Making similar arrangement like for the voltage equation the flux linkage vector in

general reference frame can be expressed as:


14

2( ) ( )( ) ( )2 2

r K r Kd q q d j jPMsK sK sK

L L L LI I e eθ θ θ θ∗ − −+ −

Ψ = − + Ψ (2.20)

Stator fixed system ( ,α β ) Taking the angular speed of the reference frame to be 0KΩ = and 0Kθ = , the set of

synchronous machine vector equations (2.19) and (2.20) my be written as:

ss ss

dU R I

dtαβ

αβαβ

Ψ= + (2.21)

2( ) ( )2 2

r rd q q d j jPMs s s

L L L LI I e eθ θ

αβ αβ αβ∗+ −

Ψ = − + Ψ (2.22)

Substituting to above equations the following expressions for complex vectors

s ssU U jUα βαβ = + , s ssI I jIα βαβ = + , s ss jα βαβΨ = Ψ + Ψ and splitting into real and

imaginary parts one can obtain the scalar form of the machine equations in stationary

,α β reference frame:

ss s s

dU R Idt

αα α

Ψ= + (2.23a)

ss s s

dU R I

dtβ

β β

Ψ= + (2.23b)

( cos 2 ) ( )sin 2 cos2 2 2

d q q d q ds r r PM rs s

L L L L L LI Iα α βθ θ θ

+ − −Ψ = − − + Ψ (2.24a)

( )sin 2 [( ) ( )cos2 ] sin2 2 2

q d d q q ds r r PM rs s

L L L L L LI Iβ α βθ θ θ

− + −Ψ = − + + + Ψ (2.24b)

Note, that in the flux-current equations (2.24a and b) still we can observe that value of

inductances depends on rotor position rθ .

Stator flux fixed system ( ,x y )

In order to take advantage of the set of equations (2.19) and (2.20) in rotating coordinate

system, one assumes that the coordinate system rotates with the stator flux linkage

angular speed K ΨΩ =Ω and Kθ θΨ= . As a sx sΨ = Ψ , ( )rδ θ θΨ Ψ= − −

sxys ssxy sxysxy

dU R I j

dt Ψ

Ψ= + + Ω Ψ (2.25)

2( ) ( )2 2

d q q d j jPMsxy sxy sxy

L L L LI I e eδ δΨ Ψ∗ − −+ −

Ψ = − + Ψ (2.26)


15

Substituting to above equations the following expressions for complex vectors

sx sysxyU U jU= + , sx sysxyI I jI= + , ssxyΨ = Ψ and splitting into real and imaginary parts

one can obtain the scalar form of the machine equations in stationary ,x y reference

frame:

ssx s sx

dU R IdtΨ

= + (2.27a)

ssy s sy sU R I Ψ= +Ω Ψ (2.27b)

1 1[( ) ( )cos 2 ] ( )sin 2 cos2 2s d q q d q d PMsx syL L L L I L L Iδ δ δΨ Ψ ΨΨ = + − − + − + Ψ (2.28a)

1 10 ( )sin 2 [( ) ( )cos2 ] sin2 2q d d q q d PMsx syL L I L L L L Iδ δ δΨ Ψ Ψ= − + + + − −Ψ (2.28b)

The current-flux equations can be expressed also in simplest form as:

2 2( cos sin ) ( )sin cosn coss d q q d PMsx syL L I L L Iδ δ δ δ δΨ Ψ Ψ Ψ ΨΨ = + + − + Ψ (2.29a)

2 20 ( )sin cos ( sin cos ) sinq d d q PMsx syL L I L L Iδ δ δ δ δΨ Ψ Ψ Ψ Ψ= − + + −Ψ (2.29b)

Rotor flux fixed system ( ,d q )

In order to take advantage of the set of equations (2.19) and (2.20) in rotating coordinate

system, one assumes that the coordinate system rotates with the rotor flux angular speed

K b mpΩ = Ω and K b m rpθ γ θ= =

sdqs b msdq sdqsdq

dU R I jp

dtΨ

= + + Ω Ψ (2.30)

( ) ( )2 2

d q q dPMsdq sdq sdq

L L L LI I ∗+ −

Ψ = − + Ψ (2.31)

Substituting the following expressions for complex vectors sd sqsdqU U jU= + ,

sd sqsdqI I jI= + , sd sqsdq jΨ = Ψ + Ψ to (2.30) and (2.31), and splitting for real and

imaginary parts the scalar form of the machine equations in rotational fixed reference

frame can be obtained:

sdsd s sd b m sq

dU R I pdtΨ

= + − Ω Ψ (2.32a)


16

sqsq s sq b m sd

dU R I p

dtΨ

= + + Ω Ψ (2.32b)

where,

sd d sd PML IΨ = +Ψ (2.33a)

sq q sqL IΨ = (2.33b)

It should be noted that when transforming the flux linkage vector sΨ to the ,d q

reference frame the rotor position rθ dependent terms disappear it can be seen from

equation (2.31). This is the main advantage of rotor-oriented representation.

Substituting the relationship of (2.33a-b) into (2.32a-b), and also considering

0PMddtΨ

= , the most common scalar form of the machine voltage equations in the rotor

reference frame can be obtained as:

sdsd s sd d b m q sq

dIU R I L p L Idt

= + − Ω (2.34a)

sqsq s sq q b m PM b m d sd

dIU R I L p p L I

dt= + + Ω Ψ + Ω (2.34b)

Based on the above voltage-current equations it is possible to draw the equivalent

electrical circuit separately for d and q axes (Fig. 2.3).

sR

sdI

b m q sqp L IΩ

dL

sR

sqI

qLsdU sqU

b m d sdp L IΩ

b m PMp ΨΩ

Figure 2.3. Equivalent circuit model of PMSM in the rotor reference frame. (a) Rotor d-axis equivalent circuit, (b) Rotor q-axis equivalent circuit.


17

2.1.2 Instantaneous power and electromagnetic torque The three-phase star-connection system without neutral wire is shown in Fig. 2.4. This

is classical configuration for AC motor windings connections.

A

B

C

sAZ

sBZsCZ

sAI

sBIsCI

sAU

sBUsCU

sABUsABU

sBCU

sACU

Figure 2.4. Three-phase star connection system without neutral wire.

For this configuration the expression for instantaneous active power supplied to load

can be expressed as:

sA sA sB sB sC sCP U I U I U I= + + (2.35)

Introducing space vector definition, after some arrangement and taking into account the

relation: 0sA sB sCI I I+ + = , the equation (2.35) can be written as:

3 Re[ ]2 sABCsABCP U I ∗= (2.36)

For ,d q frame, the equation (2.35) for the active power can be expressed as:

3 ( )2 sd sd sq sqP U I U I= + (2.37)

Substituting voltage equation (2.4) into (2.36), and adopting K b mpΩ = Ω one obtains

3 [Re( )]2

sABCs b m sABCsABC sABC sABC sABC

dP R I I I jp I

dt∗ ∗ ∗Ψ

= + − Ω Ψ (2.38)

Note that 2ssABC sABCI I I∗ = and:

23 [ Re( ) Re( )]2

sABCs s b msABC sABC sABC

dP R I I jp I

dt∗ ∗Ψ

= + + − Ω Ψ (2.39)

Hence, neglecting the losses in stator resistance sR and assuming that 0sABCddtΨ

= , the

electromagnetic power is expressed:


18

3 Im( )2e b m sABC sABCP p I∗= Ω Ψ (2.40)

In ,d q frame the active power can be written:

3 ( )2e b m sd sq sq sdP p I I= Ω Ψ −Ψ (2.41)

For the presented system (Fig. 2.4) the expression for instantaneous reactive power

supplied to the three-phase load system without neutral wire can be calculated as:

1 ( )3 sA sBC sB sCA sC sABQ I U I U I U= + + (2.42)

Introducing the space vector definition into equation (2.42), after some arrangement,

and taking into account the relation: 0sA sB sCI I I+ + = , one obtains:

3 Im[ ]2 sABCsABCQ U I ∗= (2.43)

In ,d q frame the reactive power is expressed as:

3 ( )2 sq sd sd sqQ U I U I= − (2.44)

Substituting voltage equation (2.4) into (2.43), adopting K b mpΩ = Ω and made similar

arrangements like for active power calculation, the final expression for reactive power

is:

3 Re( )2 sABC sABCb mQ p I∗= Ω Ψ (2.45)

In ,d q frame the expression (2.45) for the reactive power becomes:

3 ( )2 b m sd sd sq sqQ p I I= Ω Ψ +Ψ (2.46)

The important quantity of the drive is the power factor cosφ , which can be calculated

as:

cos QS

φ = (2.47)

where S is module of apparent power vector S P jQ= + :

2 2S P Q= + (2.48)


19

The instantaneous electromagnetic torque developed by an electric motor can be defined

as:

ee

m

PM =Ω

(2.49)

where, eP is the electromagnetic power and mΩ is the mechanical angular rotor speed.

Finally, taking into account equation (2.49) the expression for electromagnetic torque

can be obtained as:

3 Im( )2e b sABC sABCM p I∗= Ψ , (2.50)

and in ,d q frame:

3 ( )2e b sd sq sq sdM p I I= Ψ −Ψ (2.51)

Substituting ,sd sqΨ Ψ from (2.33a-b), the torque expression of equations (2.47)

becomes:

3 ( ( ) )2e b PM sq q d sd sqM p I L L I I= Ψ − − (2.52)

It can be seen from (2.52), that developed torque consist of two parts, one produced by

the permanent magnet flux called synchronous torque ( esM ) and the second called

reluctance torque ( erM ), which is produced by the difference of the inductance in rotor

d- and q-axes. Expressions for those two torque components are:

32es b PM sqM p I= Ψ (2.53a)

3 ( )2er b q d sd sqM p L L I I= − − (2.53b)

It should be mentioned that for SPMSM ( d qL L= ) the reluctance torque does not exist

due to the same inductance paths in rotor d- and q-axes.

The torque expression (2.52) can also be written in polar form using the current vector

amplitude sI and the torque angle Iδ , i.e. angle between rotor d-axis and current

vector (Fig. 2.5.).


20

d axis−

q axis−

sI

sdI

sqI

PMΨ

sΩ

Iδ

Figure 2.5. Stator current vector in rotor reference frame.

For two current components using trigonometrical rules we can write:

cossd IsI I δ= (2.54a)

sinsq IsI I δ= (2.54b)

Substituting ,sd sqI I into equation (2.52), the torque expression can be obtain as:

23 1[ sin ( ) sin 2 ]2 2

es er

e b PM I q d Is sM p I L L I

M M

δ δ= Ψ − − (2.55)

For given current amplitude the synchronous and reluctances torque varies according to

the sine of torque angle Iδ . The variation of esM and erM and resultant torque eM with

torque angle are illustrated in Fig. 2.6. The IPMSM parameters used for this calculation

are given in the Appendices.

[]

erM

Nm

[]

esM

Nm

[]

eM

Nm

[deg]Iδ Figure 2.6. Variation of synchronous torque esM , reluctance torque erM and resultant torque eM as a function of torque angle (for rated current amplitude).


21

Referring to Fig. 2.7, stator flux components in rotor reference frame can be written as:

cossd d sd PMs L IδΨΨ = Ψ = +Ψ (2.56a)

sinsq q sqs L IδΨΨ = Ψ = (2.56b)

where: sΨ is stator flux linkage amplitude, PMΨ is rotor permanent magnet and δΨ is

torque angle (angle between stator flux linkage vector and rotor permanent magnets flux

vector).

d axis−

q axis−

sΨ

sdI

sqIsI

PMΨ

δΨ

sq q sqL IΨ =

d sdL I

sdΨ

sd d sd PML IΨ = +Ψ

sΩ

Figure 2.7. Rotor permanent magnet flux vector and stator flux linkage vector in rotor reference

frame.

From (2.56a) and (2.56b) the sdI and sqI can be obtained as:

cos PMssd

d

ILδΨΨ −Ψ

= (2.57a)

sinssd

qI

LδΨΨ

= (2.57b)

Substituting current components (2.57a), (2.57b) into equation (2.51), one can obtain

another useful torque expressions:

2 ( )sin 2sin3 [ ]2 2

q dsPMse b

d d q

es erM M

L LM p

L L Lδδ ΨΨ Ψ −Ψ Ψ

= − (2.58)

where: sΨ stator flux linkage amplitude, and PMΨ rotor flux, δΨ is torque angle, esM -

synchronous torque, erM - reluctance torque.


22

For the PM synchronous motor the amplitude of stator flux sΨ is established by

permanent magnet. Operation with stator flux amplitude belong the nominal value of

rotor flux amplitude PMΨ increases the amplitude of stator phase current. Please note

that maximum amplitude of the stator current vector is calculated as: PMs

d

ILΨ

≤ , and

higher value may damage the PM (complete demagnetization).

From the Fig. 2.8 it can be observed that rated torque is achieved for torque angle

0 25δΨ< < electrical degree.

[]

esM

Nm

[]

erM

Nm[

]e

MN

m

[deg]δΨ

Figure 2.8. Variation of synchronous torque esM , reluctance torque erM and resultant torque eM as a function of torque angle (for constant stator flux equal value of PM). 2.1.3 Mechanical motion equation

The equation of rotor motion dynamics describes the mechanical equilibrium of a drive

system. Taking the moment of inertia to be constant ( .constJ = ) and neglecting friction

and elastic torque we can write:

e l dM M M= + (2.59)

where, lM is the external torque on the motor shaft, and dM the dynamic torque

md

dM JdtΩ

= (2.60)

where: J is total moment of inertia, mΩ angular speed of the rotor.


23

In general, for a drive system,

m lJ J J= + (2.61)

where: mJ - motor inertia, lJ load moments of inertia.

From equation (2.55) and (2.56) one can write:

1 ( )me l

d M Mdt JΩ

= − (2.62)

and, with (2.50),

1 3( Im( ) )2

mb ls s

d p I Mdt J

∗Ω= Ψ − (2.63)

Finally, the full mathematical model of PM synchronous machine which is used in

simulation studies [Appendices] is described in ,d q reference frame as:

sdsd s sd b m sq

dU R I pdtΨ

= + − Ω Ψ (2.64a)

sqsq s sq b m sd

dU R I p

dtΨ

= + + Ω Ψ (2.64b)

sd sd sd PML IΨ = +Ψ (2.65a)

sq sq sqL IΨ = (2.65b)

1 ( )me l

d M Mdt JΩ

= − (2.66)

3 3Im( ) ( )2 2e b b sd sq sq sds sM p I p I I∗= Ψ = Ψ −Ψ (2.67)

Based on above equations we can create the block scheme of the PMSM machine (Fig.

2.9), where the input signals are the voltage components in ,d q reference frame

,sd sqU U and the output signal is the mechanical speed of the rotor mΩ . As the external

load torque lM is disturbance.


24

∫

∫

∫

sR

qL1

sR

dL1

−

+

−

−

sdU

sqU

PMΨ

sdI

sqI

sdΨ

sqΨ

sd sqIΨ

sq sdIΨ

eM

lM

32 bp

J1

b m sdp Ω Ψ

b m sqp Ω Ψ

mΩ

bp

bp

−

−

−

Figure 2.9. Block scheme of PM synchronous machine in rotating ,d q frame.

Based on equations (2.64-2.67) we can also draw the vector diagram of PM

synchronous motor (Fig. 2.10). From this vector representation it can see the positions

of the vectors (currents, voltages and fluxes). Especially, power angle φ (angle between

voltage and current vectors) and torque angle defined in two manners: as an angle

between current and rotor flux vectors - Iδ , or as angle between stator flux and rotor

flux vectors -δΨ .

axisd −

axisq−

PMΨ

q sqL IsΨ

sqI

Iδ

( )Aα

β

sdI

sI

d sdL I

sdΨ

sqΨ

δΨ

sU

ssΩΨ φ

ssR I

stator

rotorrθ

sθΨ

Figure 2.10. Vector diagram of PM synchronous motor in rotor reference frame ,d q .


25

2.2 Static characteristic under different control modes

In this section, basic steady state properties of the PMSM under different control mode

strategies will be study [6,9]. The key control strategies for the PMSM can be listed as

follows:

• Constant torque angle control (CTAC).

• Maximum torque per ampere control (MTPAC)

• Unity power factor control (UPFC)

• Constant stator flux control (CSFC)

Constant torque angle (CTA) control This control strategy for PMSM keeps the torque angle Iδ (angle between stator current

vector and rotor permanent magnet flux) at constant value 90 .

d axis−

q axis−

ssqI I=

PMΨ

90Iδ =

Figure 2.11. Current vector and permanent magnet flux vector for constant torque angle operation (CTAC) Hence, this control can be achieved by controlling the d-axis current components to

zero leaving the current vector on the rotor q-axis (see Fig. 2.11). Therefore, this

strategy is also referred to as 0sdI = control. The amplitude of rotor flux vector is

constant and also the torque angle is constant. So, the torque depends only on the value

of stator current amplitude. Therefore, this control strategy is not recommended for

IPMSM with high saliency ratio. However, for SPMSM, this strategy is commonly

used.

The torque equation in this mode of operation becomes:

3 32 2e b PM sq b PM sM p I p I= Ψ = Ψ (2.68)


26

The steady state voltage components based on the equations (2.34a) and (2.34b) are:

sd b m q qs b m q sU p L I p L I= − Ω = − Ω (2.69a)

sq s qs b m PM s b m PMsU R I p R I p= + Ω Ψ = + Ω Ψ (2.69b)

The amplitude of stator voltage vector can be calculated as:

2 2sd sqsU U U= + (2.70)

The stator flux vector amplitude can be calculated from equations (2.65a-b) as:

2 2sd sqsΨ = Ψ +Ψ (2.71)

The active and reactive power and also the power factor can be obtained from equations

(2.41),(2.46), (2.47).

Maximum torque per ampere (MTPA) control The main idea of this control is develop the torque using minimum value of stator

current amplitude. In this case the sdI components is not equal zero, and may cancel the

reluctance torque produced by high saliency ratio. Therefore, this control strategy is

recommended for IPMSM.

d axis−

q axis−

sqI

PMΨ

90Iδ >=

sI

sdI Figure 2.12. Current vector sI and permanent magnet flux vector PMΨ for maximum torque per ampere operation (MTPAC). In order to obtain the maximum torque per ampere we should solve the derivative of

torque equations (2.55) in respect to torque angle. Solving for torque angle α and taking

into account that only negative sign should be considered for the solution, we can

calculate torque angle as:

1 21 1 1cos [ ( ) ]4( ) 2 4( )I

d q d qs sL L I L L Iδ − −

= − +− −

(2.72)


27

From Fig. 2.8, it can be seen that eM is maximum when torque angle is 90 180Iδ< < .

The relevant torque equation in this mode of operation becomes from (2.55).

The steady state voltage equations can be written using the current vector amplitude sI

and the torque angle Iδ as:

cos sinsd s I b m q Is sU R I p L Iδ δ= + Ω (2.73a)

sin cossq s I b m d I b m PMs sU R I p L I pδ δ= − Ω + Ω Ψ (2.73b)

The amplitude of stator voltage vector can be calculated from equation (2.70) and

amplitude of stator flux vector from (2.71). The active and reactive power and also the

power factor can be obtained from equations (2.41),(2.46), (2.47).

Unity power factor (UPF) control Under this control strategy there is no phase different between the current vector and the

voltage vector. Hence, power factor angle φ (see Fig. 2.13) becomes zero. Since only

active power is supplied to the machine under unity power factor operation, the VA

rating requirement of the inverter can be reduced.

axisd −

axisq −

PMΨ

Iδ

sI

sU

0φ =

Figure 2.13. Current vector and permanent magnet flux vector under unity power factor operation (UPFC). In this case when 0φ = we have the relationship:

tansq sqI

sd sq

U IU I

δ= = (2.74)

Substituting the voltage equations (2.69a-b) into (2.71) and made some simplifying, we

can obtain:

2( )cos cos 0s sd q I PM I qI L L L Iδ δ− −Ψ + = (2.75)


28

Solving for the torque angle Iδ :

221

4 ( )cos [ ]

2( )PM PM d q qs

Id q s

I L L L

L L Iδ −

Ψ − Ψ − −=

− (2.76)

only positive sign should be take into consideration.

After obtaining Iδ the amplitude of stator voltage vector can be calculated from

equation (2.70) and amplitude of stator flux vector from (2.71). The active and reactive

power and also the power factor can be obtained from equations (2.41),(2.46), (2.47).

Constant stator flux (CSF) control As it can be see from the torque expression (2.58) for a given stator flux amplitude sΨ

the electromagnetic torque eM is a function of torque angle δΨ . The stator flux linkage

amplitude sΨ is kept constant of the permanent magnet flux amplitude PMΨ .

axisd −

axisq −

PMΨ

δΨ

sΨsI

Iδ

Figure 2.14. Flux vector and permanent magnet flux vector under constant stator flux operation (CSFC). The amplitude of the stator flux linkage vector is

2 2 2 2( ) ( )sd sq q sq d sd PMs L I L IΨ = Ψ +Ψ = + + Ψ (2.77)

Equating

PMsΨ = Ψ (2.78)

can be obtain the relationship for rotor frame currents as:

2 2( ) ( ) 2 0q sq d sd d PM sdL I L I L I+ + Ψ = (2.79)

This condition is true if 0sdI < , because expression 2 2( ) ( )q sq d sdL I L I+ and ,d PML Ψ are

always positive values.


29

2 22 2 2 2( ) cos 2 cos 0d q I d PM I qs s sL L I L I L Iδ δ− + Ψ + = (2.80)

Solving for the torque angle Iδ

22 2 2 2 21

2 2

( )cos [ ]

( )d PM d PM d q qs

Id q s

L L I L L L

L L Iδ −

− Ψ ± Ψ − −=

− (2.81)

For given Iδ , the amplitude of stator voltage vector we can be calculated from equation

(2.70) and amplitude of stator flux vector from (2.71). The active and reactive power

and also the power factor can be obtained from equations (2.41),(2.46),(2.47),

respectively for defined speed.

Comparison study In order to compare the control strategies and to cancel dependence of machine power,

per unit values defined as shown in Table 2.1 below have been introduced [3,9].

The value of current vector:

( )2s s

sNb srms rated

I II

I I= = (2.82)

The value of voltage vector: s ssN

b b PM

U UU

U= =

Ω Ψ (2.83)

where: 2b bfπΩ = and bf is rated frequency of the PM motor.

The value of flux vector is: s ssN

b PM

Ψ ΨΨ = =

Ψ Ψ (2.84)

The value of torque is: 32

e eeN

bb PM b

M MMM p I

= =Ψ

(2.85)

The value of apparent power vector 32

Nb

b b

S SSS U I

= = (2.86)

The value of active power N

b

PPS

= (2.87)

The value of reactive power N

b

QQS

= (2.88)

Table 2.1. Per unit values definition.

In order to compare the steady state performance characteristic of the above discussed

control strategies, for each of the control strategy some important quantities of the

machine have been plotted as a function of the torque. The PMSM parameters, which

are used for the calculations are given in Appendices.


30

The current requirement versus torque is illustrated in Fig. 2.15 for the different control

strategies. It can be seen, that up to 1 pu torque, the requirement for current is lowest for

CSF control. Highest than 1 pu torque the low current needs MTPA control requirement

lowest current for a given torque.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.5

1

1.5

2

2.5

3

[ ]eNM pu

[ ]sNI puCTA

MTPA

CSF

UPF

Figure 2.15. Stator current amplitude under different control strategies versus electromagnetic

torque. The voltage requirement versus torque for the different control strategies is illustrated in

Fig. 2.16. It can be seen, that CSF requires the highest value of stator voltage.

[ ]eNM pu0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0

0.5

1

1.5

2

2.5

[ ]sNU pu

UPF

CSFCTA

MTPA

Figure 2.16. Stator voltage amplitude versus electromagnetic torque under different control

strategies (at 1 pu rotor speed).

[ ]NP pu

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

1

2

3

[ ]eNM pu

CSF

UPF

CTAMTPA

Figure 2.17. Active power versus electromagnetic torque under different control strategies.


31

The active power requirement as a function of torque is illustrated in Fig. 2.17 for the

different control strategies. It can be seen, that all control strategies require

approximately the same value of active power for a given torque. CSF control needs

less active power in the region up to 1.3 pu torque.

The reactive power requirement as a function of torque is illustrated in Fig. 2.18. It can

be seen, that CTA control requires the highest value of active power for a given torque

and the CSF control lowest.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

1

2

3

4

[ ]eNM pu

UPF

CSF

CTAMTPA

[ ]NQ pu

Figure 2.18. Reactive power versus electromagnetic torque under different control strategies.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20.5

0.6

0.7

0.8

0.9

1

1.1

[ ]eNM pu

cosφ

UPF

CSF

MTPA

CTA

Figure 2.19. Power factor as a function of electromagnetic torque under different control

strategies.

The power factor as a function of torque is illustrated in Fig. 2.19. It can be seen, that as

it could be expected, UPF control requires constant power factor for a given torque.

CSF control is very close to the unity power factor up to 1 pu torque.

The above analysis can be summarized as shown in Table. 2.2.


32

Table. 2.2. Summary of voltage, power, power factor requirements under control modes.

Controlmethod

VoltageRequirement

CTA middle low

MTPA

low middle

UPF low high 1

CSF high lowes up to 1.1pu torque

Close to 1 up to1 pu torque

Current Powerfactor

low

low

From this comparison study it can be concluded that CSF control appears to be superior

in terms of steady state performance characteristics compared to other methods under

discussion.


33

2.3 Summary

There are different forms to express the PMSM equations, but the rotor

reference frame equations are the most widely used. The simplification in rotor

,d q reference frame equations results from the disappearance of position

dependent inductances.

The electromagnetic torque of the IPMSM is not only produced by the

permanent magnet flux, but also by the reluctance difference in rotor d- and q-

axes.

Electromagnetic torque as cross vector product of the stator flux linkage and

current space vectors or rotor and stator flux linkages is independent of

coordinate system selected. Therefore, can be expressed in stationary ( ,α β ) or

rotated ( ,d q ) coordinates.

For further control strategies consideration it is convenient to express the

electromagnetic torque of PMSM machine by:

• vector product of stator current and rotor flux vectors. The rotor flux

vector in PMSM machine is constant, because of the PM. Therefore, to

increase and decrease the torque, the current amplitude and the torque

angle Iδ should be changed (see Fig. 2.20a),

• vector product of stator flux vector and rotor flux vector. Generally, the

value of the stator flux amplitude is kept constant at value of rotor flux

produced by permanent magnets. So, in this case to change the torque we

should adjust the torque angle δΨ (see Fig. 2.20b).

Iδ

sIPMΨ

α

β

d

q

sΨ

δΨ

PMΨ

α

β

d

q

mγmγ

)a )b

Fig. 2.20 Torque production: a) current control, b) flux control

Taking into account discussion regarding static characteristic under different

control strategies it can be said that –depart from special requirements- the most

suited for general application PMSM drives is constant stator flux (CSF)

operation.

Voltage source PWM inverter for PMSM supply

34

Chapter 3 VOLTAGE SOURCE PWM INVERTER FOR PMSM SUPPLY 3.1 Introduction The block scheme of an adjustable speed drive (ASD) commonly used in industrial

applications to supply three-phase AC motor is presented in Fig. 3.1.

Three-phasegrid

Rectifier Inverter

AC motor

DC link filter

Choke

Figure 3.1 Basic scheme of adjustable speed AC motor system.

An ASD is supplied from three or single phase grid. It consists of a diode rectifier, DC

link filter and an inverter. The rectifier converts supply AC voltage into DC voltage.

The DC voltage is filtered by a capacitor in the DC link. The inverter converts the DC

to an variable voltage, variable frequency AC for motor speed or (torque/current)

control.

The rectifier section of an ASD, called the front end, is responsible for generating

current harmonics into the power supply system. Therefore, to reduce the total harmonic

distortion (THD) of phase current it is necessary to add additional choke inductances.

There are generally to way how insert choke inductances (see Fig. 3.2).

FC

4D 6D2D

3D 5D1D

DCUFLFL

FL

)a

FC

4D 6D2D

3D 5D1D

FL

DCU

FL

)b

Fig. 3.2 Three phase diode rectifier with smoothing choke: a) at the input b) at the DC link side.

By adding a choke inductance at the input of rectifier gives the significant harmonic

reduction. Some drive manufactures are starting to include this choke inductance in the

DC link of the drive, providing the same harmonic current reduction benefit.

Regarding to power electronics standards IEEE Std 519 it is recommended that

production of harmonics should be less than 5%. It is a trend to replace the diode


35

rectifier by fully controllable active rectifier [8] (see Fig. 3.3), which guaranties

following futures:

• power flow from AC/DC or DC/AC side (there is no need of break resistor ),

• significant reduction of phase current THD,

• unity power factor (phase voltage is in phase with current),

• reduction of DC link capacitor,

• controllable DC link voltage.

Three-phasegrid

ActiveRectifier Inverter

AC motor

DC link filter

Choke

Fig. 3.3 Modern AC/DC/AC converter topology of adjustable speed drives.

In the next part of this thesis the author will be focus on voltage source inverter.

3.2 Voltage source inverter (VSI)

The made constant DC voltage by rectifier is delivered the to the input of inverter (Fig.

3.4), which thanks to controlled transistor switches, converts this voltage to three-phase

AC voltage signal with wide range variable voltage amplitude and frequency [3].

1T7D

2T8D

3T9D

4T10D

5T11D

6T12D

FC

DCU

sANU sBNU sCNU

Three-phase motor windings

A B C

N

O

FC

Voltage Source Inverter

Figure 3.4 Basic scheme of voltage source inverter circuit.


36

The one leg of inverter consists of two transistor switches. A simple transistor switch

consist of feedback diode connected in anti-parallel with transistor. Feedback diode

conducts current when the load current direction is opposite to the voltage direction.

Assuming that the power devices are ideal: when they are conducting the voltage across

them is zero and they present an open circuit in their blocking mode. Therefore, each

inverter leg can be represented as an ideal switch. Its gives possibility to connect each

of the three motor phase coils to a positive or negative voltage of the dc link ( DCU ).

Thus the equivalent scheme for three-phase inverter and possible eight combinations of

the switches in the inverter are shown in Fig. 3.5.

DCU AS BS

7 111U =

1 1

A B C

CS1

DCU AS BS

0 000U =

0 0

A B C

CS

0

DCU AS BS

1 100U =

0

A B C

CS

0

1DCU AS BS

2 110U =

A B C

CS

0

1 1DCU AS BS

3 010U =

A B C

CS

0

1

0

DCU AS BS

4 011U =

A B C

CS1

0

1DCU AS BS

5 001U =

A B C

CS

0

1

0DCU AS BS

6 101U =

A B C

CS1

0

1

Figure 3.5 Possible switches state in VSI. The six positions of switches ( 1 6U U− ) produce an output phase voltage equal ± 1/3 or

± 2/3 of the DC voltage. The last two ( 0 7,U U ) give zero output voltage. The output

phase voltages produced by inverter are shown in Fig. 3.6a. and adequate line to line

voltage calculated in bellow formula also are presented in Fig. 3.6b.

sAB sAN sBNU U U= − (3.1a)

sBC sBN sCNU U U= − (3.1b)

sCA sCN sANU U U= − (3.1c)


37

sANU

23 DCU

23 DCU−

tΩ

23 DCU

23 DCU−

tΩ

sBNU

23 DCU

23 DCU−

tΩ

sCNU

2π

2π

2π

1U 2U 3U 4U 5U 6U

sABU

DCU

DCU−

tΩ

DCU

DCU−

tΩ

sBCU

DCU

DCU−

tΩ

sCAU

2π

2π

2π

1U 2U 3U 4U 5U 6U)a )b

)a )b

Figure. 3.6 Three voltage waveforms generated by the inverter: a) phase voltages, b) line to line voltages.

Form the Fourier analysis for phase voltage produced by inverter (Fig. 3.7) the

maximum amplitude of fundamental phase voltage for a given DC link voltage is given

by:

_2

amp DCU Uπ

= (3.2)

outU

23 DCU

tω2π13 DCU−

13 DCU

23 DCU−

2DCU

π

Figure 3.7 Inverter phase voltage generated during six step operation (solid line), corresponding fundamental component of output voltage (dashed line) and harmonic spectrum of phase voltage.


38

The three-phase output voltage of the inverter can be described by space vector

definition as:

where k denotes numbers vector.

Vectors from 1-6 are called active vectors, whereas vectors 0,7 are called zero vectors

or non active vectors. The voltage space vector kU in complex plane forms a regular

hexagon and divides in into six equal sectors (one sector takes 60 electrical degree) Fig.

3.8.

Figure 3.8 Representation of the inverter states in the complex space.

In practice the real voltage source inverter has non-linear characteristic due to [19]:

• the dead-time,

• a voltage drop across the power switches,

• pulsation of the DC link voltage.

Dead time effect [17,20,27,30]

Semiconductors power switches of voltage source inverter operate not ideally. They do

not turn-on or turn-off instantaneously. Therefore it is necessary to include a protection

time to avoid a short circuit in the DC link, when two switching devices are in the same

leg (see Fig. 3.9). This time dT is included in the control signals and it is called “dead

1(100)U0 (000)U

Im

Re7 (111)U

2 (110)U3 (010)U

5 (001)U

4 (011)U

6 (101)U

sec 1tor

sec 2tor

sec 3tor

sec 4tor

sec 5torsec 6tor

23 DCU

( 1)32

3

0

j k

DC

k

U e

U

π−⎧

⎪⎪⎪= ⎨⎪⎪⎪⎩

for k =1,2...,6.

for k=0,7., (3.3)


39

time”. It guarantees safe operation of the inverter. The typical value is from 1 sµ - 5 sµ .

When the lowest value is for small power IGBT and is growing in respect to increasing

of IGBT power. More details about real IGBT module you can find in Appendices.

The effect of dead time can be examined from one phase of PWM inverter. The basic

configuration is shown in Fig. 3.9. Consist of upper and lower power devices 1T and 2T ,

and reverse recovery diodes 1D and 2D , connected between the positive and negative

rails of power supply. The gating signals AS and AiS come from control block. Output

voltage terminal 0U is connected to motor phase.

Dead timeTd

S

AS

AiS

1T1D

2T2D 0U

sAIDCU

LOAD

Figure 3.9 Circuit diagram of one inverter leg.

Fig. 3.10 shows the ideal control signals and real control signals with inserted dead time

dT . As can be observed the time duration of real drive signal for upper transistor is

shorted than ideal drive signal and for lower transistor is longer than ideal.

AS

AiS

dTAS

AiS

Ideal drive signals Real drive signals

dT

Figure 3.10 Gate signals control of one inverter leg.


40

As a consequence when the phase current sAI is positive, the output voltage is reduced,

and when the current sAI is negative the output voltage is increased (see Fig. 3.11).

2DCU−

2DCU

0U 0sAI >ideal voltage real voltage

2DCU−

2DCU

0U 0sAI <

deacresing increasing

Figure 3.11 Dead time effect on the inverter output voltage: (fat line real voltage, doted line ideal voltage).

Voltage drop across power devices

In real voltage source inverter power switches do not conduct ideally. When they are

conducting the voltage across them is not zero and equal the voltage drop on the

conducted transistor TV . Also in blocking mode the power switches have voltage drop

on the conducted diode DV . More details about real IGBT module you can find in

Appendices.

The voltage drop across the power devices is dependent on the direction of the phase

current. It has influence on the output voltage, especially at low speed operation of

motor and high load current [17,20,27,30]. Fig. 3.12 shows the voltage drop influence

on the output voltage. Also shows that the output voltage is asymmetric (with offset)

and the voltage drop decreases the output voltage when the phase current is positive and

increases the output voltage when the phase current is negative.

2DCU−

2DCU

0U0sAI >

2DCU−

2DCU

0U0sAI <

TV

DVTV

DV

Figure 3.12 Output voltage in voltage source inverter due to voltage drop across the power devices a) for 0sAI > , b) 0sAI < (fat line real voltage, doted line ideal voltage).

The influence of dead time effect and voltage drop across power devices on the output

voltage from inverter is illustrated in block diagram (Fig. 3.13). The ideal reference

voltage components in stationary reference frame ( *_ idealU α , *

_ idealU β ) are equal real


41

( *_ realU α , *

_ realU β ) and delivered to pulse width modulation (PWM) modulator block

with real non-linear inverter. As a result the output voltages ( _ outUα , _ outUβ ) are distorted

(green signals) and as consequence the phase currents ( _ outIα , _ outIβ ) in the load (red

signals) are also distorted.

*_ idealU α

*_ idealU β

*_ realU α

*_ realU β

AS

BS

CS

PWMModulator Inverter

Load

_ outIα

_outUβ

_outIβ

_outUα

Figure 3.13 Block diagram illustrated the dead time effect and voltage drop across power devices in three phase motor supplied from non-ideal voltage source inverter.

Pulsation of the DC link voltage [19]

In practice it should be take into account that the real input dc-link voltage required for

supply VSI is not ideal. It has ripples and fluctuation, because of not ideal filtering and

disadvantages of diode rectifier. Therefore, the quality of dc-link voltage has impact on

the output voltage from inverter. If dc-link voltage will change we can observed the

changing at the output of inverter. In order to overcome this problem:

• in PWM modulator we can not assume a constant dc-link voltage and we should

measured this voltage in order to calculate the modulation index (see subchapter

3.3),


42

• instate of diode rectifier will be use the active rectifier, which provided

controllable DC-link voltage.

• or used bigger capacitor in the DC-link side in order to increase possibility of

filtering.

Let us summarize, adding influence of non-linear VSI causes by:

• serious distortion in the inverter output voltage,

• distorted machine currents,

• torque pulsation,

Additionally, also causes motor instability due to the interaction between motor and the

PWM inverter, or the choice of the PWM strategy [25].

Based on simulated and experimental observation one can say that the dead time effect

is:

• more visible in low speed operation of the motor,

• may become significantly in drives where high switching frequency is required

for good dynamics performances.

In some applications such as sensor-less vector control, the inverter output voltages are

needed to calculate the rotor or stator flux vectors. Unfortunately, it is very difficult to

measure the output voltage and requires additional hardware. The most desirable

method to obtain the output voltage feedback signal is to use the reference voltages

instead. However, the relation between the output and reference voltage is nonlinear due

to the dead-time effect and voltage drop across power devices. Thus, unless the properly

dead-time and voltage drop compensation will be applied, the reference voltage can not

be used instead of the inverter output voltage. Several compensation method were

proposed to overcome this problem. One of them will be present bellow.

Compensation based on modification of reference voltage waveform [17]

The compensation process of dead time effect and voltage drop across power devices on

the inverter output voltage from is illustrated in Fig. 3.14.


43

*_ idealU α

*_ idealU β

sAI sBI

Compensation of inverter

*_ realU α

*_ realU β

AS

BS

CS

PWMModulator Inverter

Load

sCI

DCU

compT

_compUα _compUβ

_outIα

_outUβ

_outIβ

_outUα

Compensationblock

*_ realU α

*_ realU β

Figure 3.14 Block diagram illustrating the dead time and voltage drop across power devices compensation method in three phase motor supplied from non-ideal voltage source inverter.

In order to compensate the inverter non-linearity to the ideal reference voltage

components in stationary reference frame ( *_ idealU α , *

_ idealU β ) a compensation signal

( _ compUα , _ compUβ ) is added. As a consequence the real reference voltage components

( *_ realU α , *

_ realU β ) are pre-distorted. Further those signals are delivered to PWM

modulator with non-ideal inverter. As a result the output voltages are not distorted in

Fig. 3.14 and thus phase currents in the load (red signals in Fig. 3.14) are almost

sinusoidal.

To calculate an average compensation voltages ( _ compUα , _ compUβ ), the parameters of

IGBT modules as:

• dead time dT ,

• turn on ONT and turn off OFFT of IGBT transistors,

• and also on a voltage drop on diode DV and transistor TV ,

should be know.


44

The total compensation time to compensate the non-linearity of inverter can be

calculated as:

dpcomp d ON OFF s

DC

UT T T T T

U= + − + (3.4)

Where ( ) /dp D on D T sU V t V V T= − − and ont is conducting time of IGBT devices in one

sampling time.

The compensation voltage vector can be obtained as:

2 s ( ) 2 s ( )compcomp DC s th s

TU U ign I U ign I

Ts= = (3.5)

where 22( ) ( ( ) ( ) ( ))3s sA sB sCsign I sign I asign I a sign I= + + ,

and

sA

sA

( ) 1 if I 0( )

( ) 0 if I 0sA

sAsA

sign Isign I

sign I= >⎧

= ⎨ = <⎩ (3.6)

The sign function for remain phase currents are calculated similarly.

Solving equations (3.5) for real and imagine part in stationary frame, one obtains:

_12 (2 ( ) 0.5 ( ) 0.5 ( ))3comp th sA sB sCU U sign I sign I sign Iα = − − (3.7a)

_12 ( ( ) ( ))3comp th sB sCU U sign I sign Iβ = − (3.7b)

The waveform of compensation voltages in stationary frame are shown in Fig. 3.15.


45

43 thU

83 thU

23 thU

Figure 3.15 Voltage compensation components in stationary reference frame.

From the top reference α and β components.

In order to illustrate the effectiveness of the proposed compensation a simulation study

has been performed. Fig. 3.16a shows the phase current in ,α β frame and their

hodograph without compensation and Fig. 3.16b with proposed compensation method. )a

)b

Figure 3.16 Nonlinearity effect of voltage source inverter on phase current of AC machine:

a) without compensation, b) with compensation.

a)

b)


46

3.3 Space vector based pulse width modulation (PWM) methods In voltage source inverter the transistors are controlled in a on-off fashion. In order to

obtain a suitable duty cycle for each switches the technique pulse with modulation is

used. The modulation methods [18,21,22,23,24,26,28,29,30] have the influence on:

wide range of linear operation, low content of higher harmonics in voltage and current,

low frequency harmonics, minimal number of switching to decrease switching losses in

the power components.

The most important factor in PWM mode is modulation index defined as the ratio of the

reference voltage amplitude value to the maximum voltage amplitude value during six-

step operation (see Fig 3.17) and is given by:

2ref

DC

UM

Uπ

= (3.8)

where the DCU is the DC link voltage (for three phase six diodes rectifier is 560 V ).

The modulation index varies between 0-1 and can be divided into two regions: the

linear ( 0 0.907M< ≤ ) and the nonlinear ( 0.907 1M< ≤ ) as is shown in Fig. 3.17.

Re

Immax

2DCU U

π=

max 3DCUU =

refU

refθ

End of overmodulation region (six-step mode)

1M⇒ =

0.907M⇒ =

nonlinear (overmodulation) region

23 dcU

linear region

End of linear region

3 (010)U 2 (110)U

1(100)U

6 (101)U5 (001)U

4 (011)U0 (000)U

7 (111)U1

1s

T UT

sec 1tor =

sec 2tor =

sec 3tor =

sec 4tor =

sec 5tor =

sec 6tor =

Figure 3.17 Space vector diagram of the available switching vectors.


47

Linear range of operation ( 0 0.907M> <= )

In linear region, shown in Fig. 3.17, the rotating reference voltage vector refU remains

within the hexagon. It means that the maximum amplitude reference voltage is equal

3DCU and adequately modulation index is equal:

/ 3 0.90682 2 3DC

DC

UM Uπ

π

= = = (3.9)

The active vectors ( 1 2 3 4 5 6, , , , ,U U U U U U ) are used to change the position of voltage

vector. The zero vectors ( 0 7,U U ) are using to increase or decrease the amplitude of

voltage vector in one sampling time.

The desired voltage refU is approximated by a time average of selected voltage vectors.

These selected vectors are non-zero vectors which are adjacent to the refU .

Reference voltage vector control section is sampled at the fixed clock frequency of 2 sf

(where sf is the switching frequency). After that, the reference stator voltage vector

magnitude and position are calculated.

The principle for the implemented space vector modulation is described below. First,

based on the position of reference voltage vector is calculated the sector with help of

accordance below equations.

sec int( ) 1/3

reftorθπ

= + (3.10)

Where refθ is position of reference voltage vector refU in respect to real axis in complex

plane.

Next based on the sector information angle refα in respect to adjacent vector is

calculated (see Fig 3.18).

(sec 1)3ref ref tor πα θ= − − (3.11)


48

Next from refU , refα it is necessary to calculate the time interval for particular vectors.

120°

refU

1t

2t

0t

1U

2U

0, 7U U refα

α120°-

Figure 3.18 One sector in voltage plane.

Using the low of sine it is possible to write:

1 2

sin120 sin(60 ) sinref

ref ref

U U Uα α

= =° ° −

(3.12)

From this relations calculated value of vectors

1

sin(60 ) 2 sin(60 )sin120 3

refref ref refU U U

αα

° −= = ° −

° (3.13a)

2

sin 2 sinsin120 3

refref ref refU U U

αα= =

° (3.13b)

and respectively the normalized times are given:

11

3sin(60 )2

3

refref

DCDC

UUt

UUα= = ° − (3.14a)

22

3sin2

3

refref

DCDC

UUt

UUα= = (3.14b)

Putting Eq. (3.13a-b) in to Eq. (3.14a-b) the normalized value can be presented as:

12 3 sin(60 )reft M απ

= ° − (3.15a)

22 3 sin reft M απ

= (3.15b)

or in other form:


49

22 3 sin reft M απ

= (3.16a)

1 23 1cos

2reft M tαπ

= − (3.16b)

After 1t and 2t calculation, the remaining normalized time is reserved for zero vectors

0U , 7U with condition 1 2 1t t+ ≤ .Therefore, the normalized total time for zero vectors

becomes:

07 0 7 1 21 ( )t t t t t= + = − + (3.17)

The equations (3.15a-b) for time interval of active vectors and equation (3.17) for total

time interval of zero vectors are identical for all variants of space vector modulation

(SVM) techniques.

The absence of neutral wire in star connected load provides a degree of freedom in

selecting the partitioning (zero sequence signals -ZSS) time of the two zero vectors. It is

equivalent to the freedom of injected signals in to phase signals. Therefore, it gives

different equations of 0t and 7t for each PWM method, but normalized duration time of

must fulfill condition in Eq. 3.17. As a consequence is only in different placement of

zero vectors 0U , 7U . Therefore, we can introduce the portioning factor of zero vectors,

which is defined as:

7 7

0 7 07

t tkt t t

= =+

(3.18)

Please note that, the zero sequence signals does not change the inverter output line-to-

line voltage.

From knowledge of the neutral voltage 0NU (see Fig. 3.4) and information what kind of

zero sequence signal (ZSS) will be injected in each phase of motor it is possible to

calculate normalized duration time of zero vectors 0t and 7t . In bellow Table 3.1 are

summarized different three-phase modulation techniques and remarks. Also graphical

interpretation are shown in Fig. 3.19.


50

Zero Sequence Signal Time interval of zero vectors Remark

SPWM

ZSS =0

for even sectors

04(1 ( cos )) / 2t M θπ

= −

for odd sectors

0 074(1 ( cos )) / 2t t M= − − θπ

and 7 07 0t t t= −

/ 2 0.7852 4

DC

DC

UM U= = =π

π

complicated calculation

for time interval of

vectors

THIPWM

ZSS =sinusoidal signal

with triple harmonic

for even sectors

04 1(1 ( cos ) cos3 )) / 2

6t M θ θ

π= − −

for odd sectors

0 074 1(1 ( cos ) cos3 )) / 2

6t t M= − − −θ θ

π

and 7 07 0t t t= −

/ 3 0.9072 2 3DC

DC

UM Uπ

π

= = =

Increase the linearity of

inverter grater than 15%

of SPWM

SVPWM

ZSS=triangle signal

with triple harmonic

for all sectors

0 07 / 2t t=

7 07 0 07 / 2t t t t= − =

0.906M =

Simple calculation for

time interval

k=0.5 the portioning

factor of zero vectors

Table 3.1. Variants of three-phase space vector modulation techniques.

b)a) c)

Fig. 3.19 PWM techniques with various zero signal sequence shape: a) SPWM, b) THIPWM, c) SVPWM. The upper part of figure: phase voltage ANU (green), pole voltage AOU (blue), voltage between neutral points NOU (black). The lower part of figure shows the portioning factor of zero vectors for all PWM techniques.


51

Except SPWM technique all PWM method guarantee that the ZSS extends the range of

modulation index from 0.78 to 0.906, i.e. 15% greater than that obtained with standard

version of SPWM.

The duty time cycles in sector 1 for each phase can be written:

1 2 07ad t t kt= + + (3.19a)

2 07bd t kt= + (3.19b)

07cd kt= (3.19c)

for all sectors the duty time calculations for each phase can be calculated:

sec 1 sec 2 sec 3 sec 4 sec 5 sec 6

1

2

07

1 1 1 0 0 0 0 0 0 1 1 1

0 1 1 1 1 1 1 0 0 0 0 0

0 0 0 0 0 1 1 1 1 1 1 0

Ttor tor tor tor tor tor

a

b

c

d k k k k k k t

d k k k k k k t

d k k k k k k t

=

⎡ ⎤⎡ ⎤ ⎡ ⎤⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎢ ⎥⎣ ⎦

(3.20)

Depending on the location of the space vector, the basic vectors must be chosen in order

to get the minimum number of changes in the switches of the converter. The switching

sequence for each sector and suitable pulse pattern for first sector are shown in Fig.

3.20.

Sector Three-phase Modulation

1 0 1 2 7 2 1 0, , , , , ,U U U U U U U

2 0 3 2 7 2 3 0, , , , , ,U U U U U U U

3 0 3 4 7 4 3 0, , , , , ,U U U U U U U

4 0 5 4 7 4 5 0, , , , , ,U U U U U U U

5 0 5 6 7 6 5 0, , , , , ,U U U U U U U

6 0 1 6 7 6 1 0, , , , , ,U U U U U U U

0t 7t1t 2t

ST

0 1 1 1 1 1 1 0

0 0 1 1 1 1 0 0

0 0 0 1 1 0 0 00 0 0 1 1 0 0 0

Ad

Bd

Cd

/ 2ST

0U 1U 2U 7U 2U 1U 0U

Fig. 3.20 Switching sequence for three-phase PWM techniques (on the left ) and pulse pattern of three-phase vector modulator in sector 1 (on the right).


52

3.4 Summary

The general conclusions from this chapters can be summarized as follows:

to supply the voltage source inverter can be used diode rectifier or active

rectifier with IGBT transistors,

supplied PMSM machine from VSI do not required mechanical comutator. It is

thanks to electronic commutation. This overcome the problem with brushes and

periodical service,

The voltage source inverter is non-linear power amplifier in respect to:

o dead time effect,

o voltage drop across the power devices,

o DC link voltage pulsation.

Without appropriate dead-time and voltage drop compensation, the sensorless

operation in low speed range is not possible.

The quality of DC link voltage has influence on proper operation of AC drive,

Sinusoidal modulation technique (SPWM) guarantees the inverter output voltage

amplitude from 0 until 2DCU V. This correspond to changes modulation index

from 0-0.785.

Modulation techniques with zero sequence signals not equal zero guarantees the

inverter output voltage amplitude from 0 until 3

DCU V (it is 15% grater than

SPWM). This correspond to changes modulation index from 0-0.907.

In this work the PWM modulator with zero sequence signal of triple harmonics

(SVPWM) will be used. Mainly because of simple calculation of zero vector

duration and placement.

Control methods of PM Synchronous motor

53

Chapter 4 CONTROL METHODS OF PM SYNCHRONOUS MOTOR 4.1 Introduction The basic block scheme of adjustable speed drive with control block for PMSM is

presented in Fig. 4.2. It consists of two parts: power (fat line) and control part

employed microprocessor (thin line). The first one previously have been explained in

chapter 3. The second one will be described bellow.

Figure 4.2. The basic block scheme of PMSM drive supplied voltage source inverter.

The main task of control block is follow demand reference speed by motor and provide

proper operation in static (insight of the limits) and dynamic states without any

instability. This is ensured through suitable generated gate signals for the IGBT

transistor inside of the inverter. Therefore, to make good decision how to control power

transistors in the inverter, the following feedback signals are measured and used:

• DC link voltage,

• motor phase currents,

• speed or position of the rotor.

This significantly improve dynamic behavior of the system (good performance of the

torque and speed response, very fast dynamics response with fully controllable torque in

wide speed range).

The scalar control for PMSM without damper winding (squire cage) is not simple as for

induction motor [39,40]. It requires additional stabilization loop, which can be provide

by feedback loop from: rotor velocity perturbation, active power or DC-link current

perturbation [9].

The vector control method will be described bellow.


54

4.2 Field oriented control (FOC) During many years a DC motor has been mostly used. Because of simple control

method, which based on fact that flux and torque can be controlled separately using

current control loop with PI controllers. However the weak point of this drive was DC

motor, which could not worked in aggressive or volatile environment and required

cyclical maintenance. This disadvantages has been eliminated, when instead of DC

machine a three phase PMSM motor were used.

In searching new control method for induction machine in 1971 was developed vector

control method known as field oriented control (FOC) [31,38,49]. This method allows

control the flux and the torque in the AC machine in similar way as for DC motor. It

was achieved by transform current vector in stationary reference frame ( ,α β ) into new

coordinate system ( ,d q ) with respect to rotor (magnet) flux vector. So the flux

produced by permanent magnet is frozen to the direct axis of the rotor (see Fig. 4.4).

sqI

axisα−

axisβ −

sdI

sI

PMΨ

sΩ

Iδ

d axis−

q axis−

mγ

Iγ

Figure 4.4 Vector diagram illustrated the principle of FOC.

Further, stator current vector can be split into two current components: flux current sdI

and torque producing current sqI . In analogy to separate commutator motor, the flux

current components corresponds to excitation current and torque-producing current

corresponds to the armature current. Therefore, the goal of the control system is to

reference the _sd refI , _sq refI stator current components in respect to requirement of

references torque and flux. The flux and torque producing stator current references are

obtained on the output of the reference current generation block (see Fig. 4.5).


55

reference torque

reference d-axis current

reference q-axis current

_sd refI

_sq refI_e refM

Reference CurrentCalculation

FG1

FG2

reference torque

sI

Iδ_e refM

FG3

FG3

Reference CurrentCalculation

mγ

Iγ

Figure 4.5. Reference current generator block for FOC technique

a) in cartesian form, b) in polar form.

Function generation FG1 gives the relationship between the torque and the direct axis

stator current component _sd refI , and function generator FG2 gives the relationship

between the torque and the quadrature axis stator current _sq refI . His graphical

illustration in Fig. 4.6 are presented.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 22.5

2

1.5

1

0.5

0

0.5

[ . ]eNM p u

[ . ]sdNI p u

UPF

CSF

MTPA

CTA

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.5

1

1.5

2

2.5

[ . ]eNM p u

[ . ]sqNI p u

UPF

CSF

CTA

MTPA

Figure 4.6. Generated current components sdI and sqI dependent on required electromagnetic torque under difference control strategies. CTA-constant torque angle control, MTPA-maximum torque per ampere control, CSF-constant stator flux control, UPF – unity power factor control.


56

For many years the CTA control ( _ 0sd refI = ) method has been a popular technique for a

long time because of simple control. This method was dedicated for surface permanent

magnet synchronous motor (SPMSM), where the magnetic saliency does not exist

[126]. So the maximum torque per ampere is obtained, when stator current vector is

shifted in respect to rotor flux vector 90 degree. However, in IPMSM, maximum torque

per ampere is obtained with torque angle more than 90 degree. This is because of

existence of reluctance torque component due to magnetic saliency (see subchapter

2.2.2). Therefore, the _sd refI should be negative value [44].

The main question is how or in which manner produce the reference currents in d-q

frame. Its leads to many realization of current control structure. Among them generally

we can distinguish two structures of current control loop. One of them is hysteresis

based control (Fig.4.7a) [3,52] and the second one is PI based current controllers

(Fig.4.7b).

Hysteresis based current control has following disadvantage such as [3]:

• measurement of three phase currents are required,

• three independent hysteresis current controllers are required,

• variable switching frequency is achieved,

• fast sampling time is required.

All this listed above disadvantage can be eliminate, when the PI current control are

used. This structure are mostly used in industrial application (Fig. 4.7b).

current feedback

current sensors

rotorpositionsensor

_A refI

_B refI

_C refI

AI

BI

CI

d,q/ABC

_d refI

_q refI

−

−

−

PMSM

Inverter

DCU)a

AS

BS

CSFig. 4.5_e refM


57

-

Space Vector

Modulator

dq/ABC

_sq refI

_sd refI

Isqe

AS

BS

CS

sdU

sqIsI

sdIe-

sdI

PI

sqU

mγ

PI

PMSM

Inverter

DCU

current sensors

rotorpositionsensor

)b

sγ

_s refU

_Us refϕReference

VoltageVectorCalculation

Fig. 4.5_e refM

Figure 4.7 Vector control structure for PM synchronous motor with: a) hysteresis current

control, b) synchronous PI current control 4.3 Direct torque control (DTC)

The name direct torque control is deliver by the fact that, on the basis of the errors

between the reference and the estimated values of torque and flux, it is possible to

directly control the inverter states without inner current control loop as for FOC

[32,34,35,50] and [57-66].

The basic idea of this control rely on stator voltage vector equation of AC motor.

ss s s

dU R I

dtΨ

= + (4.1)

Making the assumption that ohmic voltage drop on the stator resistor can be neglected

the equation for stator flux vector takes the form:

( )s sU dtΨ = ∫ (4.2)

It can be said that the stator voltage vector has directly influence on control stator flux

vector. Using a three phase voltage source inverter to supply the AC motor, there are six

non-zero vectors and two zero voltage vectors. The active vectors change the amplitude

and position of stator flux vector, while the zero vectors stop the stator flux vector as

shown in Fig. 4.8.


58

stops with zerovector

moment with activeforward vector

moment with activebackward vector

rotatescontinuously

Ψδ

sΨ

x

α stator

y

r PMΨ = Ψ

sγ

Figure. 4.7 Stator flux vector sΨ movement relative to rotor flux vector r PMΨ =Ψ under the

influence of active and zero inverter voltage vectors.

Therefore, it is possible control the torque angle δΨ across control stator flux vector sΨ

position in respect to rotor flux vector produced by permanent magnet r PMΨ =Ψ , what

further allows to have impact on control the electromagnetic torque in accordance with

following formula:

2 ( )sin 2sin3 [ ]2 2

q dsPMse b

d d q

L LM p


= − (4.3)

Generally the DTC technique operate at constant stator flux amplitude sΨ , what

correspond to CSF operation, because of simple reference stator flux amplitude equal

nominal value of permanent magnet. For DTC technique can be also apply all control

strategies discussed in Chapter 2.

Using the block generator for reference stator flux amplitude and electromagnetic

torque as is shown in Fig 4.8. it is possible to draw the relationship between required

referencetorque

referenceflux

ReferenceFlux

Calculation

referencetorque_e refM _e refM

_s refΨ

Figure 4.8. Reference flux and torque generator block for DTC technique.

reference flux and torque. His graphical illustration in Fig. 4.9 is presented.


59

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.5

1

1.5

2

[ ]sNΨ pu

[ ]eNM pu

UPF

MTPA

CTA

CSF

Figure 4.9. Generated stator flux amplitude _s refΨ dependent on required electromagnetic

torque under difference control strategies. CTA-constant torque angle control, MTPA-maximum torque per ampere control, CSF-constant stator flux control, UPF – unity power factor control.

The basic structure of direct flux and torque control voltage-sourced PWM inverter-fed

permanent magnet synchronous motor is shown in Fig.4.10.[3,36,37,41,45]

SwitchingTable

−

−

Flux andTorque

Estimation

mγ

sΨ

eM

sector

sdΨ

eMd

Fluxhysteresis

Torquehisteresis ( )s Nγ

Inverter

DCU

PMSM

sI

_s refΨ

_e refM

ASBSCS

sHΨ

mH

Figure. 4.10. Block diagram of switching table based direct torque control ST-DTC.

The command stator flux amplitude _s refΨ and electromagnetic torque _e refM values

are compared with the actual sΨ and eM values, in hysteresis flux and torque

controllers, respectively. The flux and the torque controllers are a two-level

comparators.

The digitized outputs signals of the flux controllers are defined as:

1=Ψsd (increase flux) for _s s ref HΨΨ > Ψ + (4.4a)

0=Ψsd (decrease flux) for _s s ref HΨΨ < Ψ − (4.4b)


60

and those of the torque controller as

1eMd = (increase torque) for _e e ref mM M H> + (4.5a)

0eMd = (decrease torque) for _e e ref mM M H< − (4.5b)

where mH and HΨ are hysteresis bands for torque and flux, respectively.

The digitized variables ,s eMd dΨ and the stator flux position sector ( )s Nγ information

create a digital word, which select appropriate voltage vector from the switching table.

Next, from the selection table the proper voltage vectors are selected and the gate pulses

, ,A B CS S S to control the power switches in the inverter are generated.

The circular stator flux vector trajectory can be divided into six symmetrical sectors

(according to the non zero voltage vectors), which are defined as (see Fig. 4.11):

sector 1 30 30sγ− ° ≤ < °

sector 2 30 90sγ° < ≤ °

sector 3 90 150sγ° < ≤ °

sector 4 150 210sγ° < ≤ °

sector 5 210 270sγ° < ≤ °

sector 6 270 30sγ° < < − °

Sector 1

Sector 2Sector 3

Sector 4

Sector 5 Sector 6

V1 (100)

V3 (010)

V5 (001)

V2 (110)

V4 (011)

V6 (101)

Vo (000)V7 (111)

Figure. 4.11 Sector definition for DTC

In each region, two adjacent voltage vectors, which give the minimum switching

frequency, may be selected to increase or decrease the amplitude of stator flux and

electromagnetic torque.

The selection table is created accordance with following formula explained for first

sector as is illustrated in Fig. 4.12 .

When stator flux vector is located in first sector sΨ and rotates clockwise in order to

increase the torque ( 1eMd = ) the vectors V2,V3 can be used. In order to reduce torque

the opposite two vectors are used.


61

sΨ

sΨ ↓

eM ↑

sΨ ↑

eM ↑

sΨ ↑

eM ↓sΨ ↓

eM ↓

sΨ ↓sΨ ↑

α

β

sγsγ

Figure. 4.12 Voltage vector effects in sector 1 on stator flux and torque.

The presented rule for first sector can be extended for other sectors, what further help

construct the switching Tables 4.1 and 4.2 as below [56]

Flux Torque sector 1 sector 2 sector 3 sector 4 sector 5

sector 6

dme.=1 2V 3V 4V 5V 6V 1V dψs=1 dme=0 6V 1V 2V 3V 4V 5V


Table 4.1. Switching table for DTC with active vectors.

Flux Torque sector 1 sector 2 sector 3 sector 4 sector 5

sector 6



Table 4.2. Switching table for DTC with zero and active vectors.

Tables 4.1 and 4.2 represent the eight and six voltage-vectors switching tables.


62

The DTC has lesser parameter dependence and fast torque when compare with the

torque control via PWM current control.

Among the well-know advantages of the DTC scheme are the following:

• Simple control,

• Excellent torque dynamics,

• Absence of coordinate transformations,

• Absence of separate voltage modulation block,

• Absence of voltage decoupling circuits,

• There are no current control loops, hence, the current is not regulated directly,

• Stator flux vector and torque estimation is required.

Among the well-know disadvantages of the DTC scheme are the following:

• variable switching frequency (difficulties of LC input EMI filter design),

• high sampling time is required (fast microprocessor and A/D converter ),

• inverter switching frequency depending on: flux and torque hysteresis bands,

machine parameters, sampling frequency,

• violence of polarity consistency rules (huge voltage stress for IGBT transistor),

• current and torque distortion caused by sector changes,

• start and low speed operation problems,

• high sampling frequency needed for digital implementation of hysteresis

comparators,

• high noisy level,

• high current and torque ripple.

Many modifications of the basic switching table based direct torque control (ST-DTC)

scheme at improving starting, overload condition, very low speed operation, torque

ripple reduction, variable switching frequency functioning, and noise level attenuation

have been proposed during last decade.

In the last five years, many researcher have been carried out to try solve the above

mentioned problems of ST-DTC scheme. Therefore, the following solutions have been

developed in order to eliminated mentioned before problems:


63

• Use of improved switching table,

• Use of comparators with and without hysteresis, at two or three levels,

• Use of multi-level inverter,

• Introduction of fuzzy or neuro-fuzzy techniques,

• Use of sophisticated flux estimators to improve the low speed behavior,

• Implementation of DTC schemes with constant switching frequency operation

In multi-level inverter there will be more voltage vectors available to control the flux

and torque. Therefore, a smoother torque can be expected. However, more power

switches are needed to achieved a lower ripple, which will increase the system cost and

complexity.

All this contributions allow the DTC performance to be improved, but at the same time

they lead to more complex schemes. As expected, conventional DTC is growing in

field-oriented control area and the so-called improved DTC with space vector

modulation (SVM). Let us call it DTC-SVM. This control concept will be deeply

discussed in the next Chapter.


64

4.4 Summary

In FOC drive flux linkage and electromagnetic torque are controlled indirectly

and independently by PI controllers with space vector modulator (SVM). In this

control concept current control loop is required,

In DTC drive, flux linkage and electromagnetic torque are controlled directly

and independently by hysteresis controllers and selection of optimum inverter

switching modes. In this control concept flux and torque control loop is required,

In DTC all switch changes of the inverter are based on the electromagnetic state

of the motor.

The DTC technique is different from the traditional methods of controlling

torque, where the current controllers in the rotor reference frame are used. It is

completely different control concept (approach) from FOC. The new control

technology was characterized by simplicity, good performance and robustness,

because of bang-bang hysteresis control. Using DTC it is possible to obtain a

good dynamic control of the torque without current controllers and any

mechanical transducers on the machine shaft. Moreover, in this control structure

the PWM modulator is not required. Its is occupied by variable switching

frequency.

The flux weakening control becomes easier because stator flux linkage can be

controlled directly in the DTC system of PMSM.

Direct Torque Control with Space Vector Modulation

65

Chapter 5 DIRECT TORQUE CONTROL WITH SPACE VECTOR MODULATION (DTC-SVM)

5.1 Introduction

The DTC-SVM greatly improves torque and flux performance by:

• Achieved fixed switching frequency,

• Reduced torque and flux ripples.

The main idea of DTC-SVM is based on analyze of the torque equation

2 ( )sin 2sin3 [ ]2 2

q dsPMse b

d d q

L LM p


= − (5.1)

Assuming that the d q sL L L= =

sin3 [ ]2

PMse b

sM p

LδΨΨ Ψ

= (5.2)

From equation (5.2) we can see that for constant stator flux amplitude sΨ and flux produced

by permanent magnet PMΨ , the electromagnetic torque can be changed by control of the

torque angle δΨ . This is the angle between the stator and rotor flux linkage, when the stator

resistance is neglected. The torque angle, in turn, can be changed by changing position of the

stator flux vector sθΨ in respect to PM vector using the actual voltage vector supplied by

PWM inverter.

In the steady state, δΨ is constant and corresponds to a load torque, whereas stator and rotor

flux rotate at synchronous speed. In transient operation, δΨ varies and the stator and rotor

flux rotate at different speeds (Fig. 5.1).


66

axisd −

axisq −

sΨ

α

β

rθ

δΨ sθΨ

_s refΨ

δΨ∆

PMΨ

Figure 5.1. Space vector diagram illustrating torque control conditions.

5.2 Cascade structure of DTC–SVM scheme

The structure of proposed control scheme is shown in the Fig. 5.2. [11,33,42,48,51,53,54]

_e refM

_s refΨ

Me

AS

BS

CS

eM

sθΨsΨ

_s refU α

_s refU βδΨ∆

DCU

mγ

sI

sI

Figure 5.2. Cascade structure of DTC-SVM scheme.

The error between reference and measured torque can be expressed as:

__

sin( ) sin3 [ ]2

s ref PM s PMe ref e b

s s

eM M M pL L

δ δ δΨ Ψ ΨΨ Ψ +∆ Ψ Ψ= − = − (5.3)

From equation (5.3) we can see that the relation between torque error and increment of load

angel δΨ∆ is nonlinear. Therefore, we used PI controller which generates the load angel


67

increment required to minimize the instantaneous error between reference _e refM and actual

eM torque.

In control scheme of Fig. 5.2 the torque error signal Me is delivered to the PI controller,

which determines the increment of torque angle δΨ∆ . Based on this signal and reference

amplitude of stator flux _s refΨ , the reference voltage vector in stator coordinates ,α β is

calculated. The calculation block of reference voltage vector also uses information about the

actual stator flux vector (amplitude sΨ and position sθΨ ) as well as measured current vector

sI . The reference stator voltage vector is delivered to space vector modulator (SVM), which

generates the switching signals CBA SSS ,, for power transistors of inverter.

The calculation block of reference voltage vector is shown in Fig. 5.3.

δΨ∆

_s refΨ

sθ Ψ

_s refαΨ

_s refβΨ

−

sαΨ

sβΨ

sα∆Ψ

−sβ∆Ψ

_s refU α

_s refU β

s sR I α

s sR I β

Figure 5.3. Calculation block of reference voltage vector.

Based on δΨ∆ signal, reference of stator flux amplitude _s refΨ and measured stator flux

vector position sθΨ (Fig. 5.3.), the reference flux components refsrefs __ , βα ΨΨ in stator

coordinate system are calculated as:

_ _

_ _

cos( )

sin( )

s ref s ref s

s ref s ref s

α

β

θ δ

θ δ

Ψ Ψ

Ψ Ψ

Ψ = Ψ + ∆

Ψ = Ψ +∆ (5.4)

Pleas note that for constant flux operation region the reference value of stator flux amplitude

_s refΨ is equal flux amplitude of permanent magnet PMΨ .


68

The references of stator flux components (see Fig. 5.3) are compared with estimated value:

cos ,

sin ,s s s

s s s

α

β

θ

θΨ

Ψ

Ψ = Ψ

Ψ = Ψ (5.5)

The command voltage can be calculated from flux errors in βα , coordinate system as

follows:

_

_

ss ref s s

s

ss ref s s

s

U R IT

U R IT

αα α

ββ β

∆Ψ= +

∆Ψ= +

(5.6)

Where: sT is sampling time, _s s ref sα α α∆Ψ = Ψ −Ψ , _s s ref sβ β β∆Ψ = Ψ −Ψ .

The presented bellow design methodology for flux and torque control loops based on the

approach presented in literature [11,43].

5.2.1 Digital flux control loop The flux control loop is based on the voltage equations of PMSM machine in stator

coordinates.

ss s s

dU R Idt

αα α

Ψ= + (5.7a)

ss s s

dU R I

dtβ

β β

Ψ= + (5.7b)

Using Laplace transformation the above equations can be written as:

s s s sU s R Iα α α= Ψ + (5.8a)

s s s sU s R Iβ β β= Ψ + (5.8b)

It corresponds to flux model of PMSM machine in ,α β system presented in Fig. 5.4.


69

sU α

sU β

s sR I α

−

−

s sR I β

1s

1s

sαΨ

sβΨ

Figure 5.4. Flux model of PMSM in stator coordinates.

In order to control the flux components in ,α β frame the bellow control structure can be applied.

_s refαΨ

_s refβΨ

−

−

sα∆Ψ

sβ∆Ψ

s sR I α

s sR I β

_s refU α

_s refU β

sαΨ

sβΨ

sαΨ

sβΨ

s

sTα∆Ψ

s sR I α

−

−

s sR I β

1s

1s

Flux PMSM model

Flux control part

s

sTβ∆Ψ

1/ sP T=

1/ sP T=

P block

P block

Figure 5.5. Flux control loop with two P controller in ,α β reference frame.

Pleas note that regarding to Fig. 5.1 the following rules are keeping:

_ _ cos( )s ref s ref sα θ δΨΨ = Ψ + ∆ (5.9a)

_ _ sin( )s ref s ref sβ θ δΨΨ = Ψ + ∆ (5.9b)

and

_ coss s ref sα θΨΨ = Ψ (5.10a)

_ sins s ref sβ θΨΨ = Ψ (5.10b)

In order to find the formula for tuning the P controllers in the flux loop, the following

assumptions should be made:


70

• increment of torque angle δΨ∆ coming from torque control loop (see Fig.5.2.) is equal

zero. It means that the torque is not produced,

• stator flux vector position sθΨ and rotor flux vector position rθ are equal zero. It

corresponds to situation, where the those two flux vectors lie along the α axis.

In this special case the reference stator flux amplitude _ _s ref s refαΨ = Ψ can be controlled

trough the reference stator voltage component _ _s ref s refU Uα = , when the voltage drop on the

stator resistances in ,α β axes are neglected (see Fig. 5.5). Therefore, the simplified flux

control loop can be shown in Fig. 5.6.

ssα

Ψ=Ψ

−

sα∆Ψ

_ 0s refU β =

P

controller

PMSM

sαΨ

sβΨ

_ _s ref s refαΨ = Ψ_ _s ref s refU Uα =

Figure 5.6. Simplified flux control loop in ,α β coordinates.

Continuous s-domain

Simplified flux control loop in s domain is shown in Fig. 5.7, where ( )C sΨ is a transfer

function of the P controller given by:

( ) pC s KΨ Ψ= (5.11)

The transfer function between stator flux amplitude s sαΨ = Ψ and stator voltage amplitude

sU can be expressed as:

1( ) s

s

G sU sΨ

Ψ= = (5.12)


71

sΨsUs_refΨ( )C sΨ

P controller

( )G sΨ

Control Plant

Figure 5.7. Block diagram of flux controller in s domain.

Hence the transfer function of the closed stator flux amplitude control loop is obtained as:

__

( ) ( ) ( )( )( ) 1 ( ) ( )

s refclosed

s

s C s G sG ss C s G s

Ψ ΨΨ

Ψ Ψ

Ψ= =

Ψ + (5.12)

Substituting transfer function for ( )C sΨ and ( )G sΨ one becomes:

_ ( )closedG sΨ =

1 1

11

p pp

p pp

K K Ks ss K s KK

s s

Ψ ΨΨ

Ψ ΨΨ

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠= =

+ +⎛ ⎞+ ⎜ ⎟⎝ ⎠

(5.13)

Discrete design

The transfer function for P controller in discrete system is expressed as:

( ) pC z KΨ Ψ= (5.14)

( )sΨ zs sU Uα =_ ( )s refΨ z

1z−( )C zΨ

( )G zΨ( )D z

ZOH

1s

Figure 5.8. Block diagram of flux controller in discrete domain.

Where, ( )C zΨ is discrete transfer function for P controller, ( )D z 1z− is for one sampling time

delay for voltage generation from PWM .


72

The ( )G zΨ is discrete transfer function for voltage-flux relationship with zero hold order

(ZOH) can be calculated as:

12

( ) 1 1( ) (1 ) [ ] [ ]G s zG z z Z Zs z s

− ΨΨ

−= − = (5.15)

Using table of Z transformation [2]. Finally, it gives

( )( ) ( )2

1( )

11s dz zT AG z

z zzΨ

Ψ

−= =

−− (5.16)

Where d sA TΨ = and sT is sampling time of the discrete system.

Hence, the closed loop transfer function between _

( )s refzΨ and ( )s zΨ is obtained as:

( )

( )

__

2

( ) ( ) ( ) ( )( )( ) 1 ( ) ( ) ( )

1

11

s refclosed

s

p d

p d

p d p d

z C z G z D zG zz C z G z D z

K AK Az z

K A z z K Az z

Ψ ΨΨ

Ψ Ψ

Ψ Ψ

Ψ Ψ

Ψ Ψ Ψ Ψ

Ψ= =

Ψ +

−= =

− ++−

(5.16)

The flux step response depended on poles placement of closed flux control loop. The pole

placement can be selected by setting the pK Ψ .

Assuming, that d p dC K AΨ Ψ Ψ= the _ ( )closedG zΨ expressed by equations (5.16) will take the

following form:

_ 2( ) dclosed

d

CG zz z C

ΨΨ

Ψ

=− +

(5.17)

The nomogram of Fig. 5.9 shows the relationship between overshoot [%]pM , rise time rt and

settling time st in respect to dCΨ .

Please not that rt is time calculate from 10% to 90% of output signals and st is the time it

takes the system transient to decay +-1%.


73

Figure 5.9. The relationship between overshoot, rise time and settling time versus to dCΨ for stator flux amplitude control loop. Now from few values of dCΨ =[0.4046, 0.2688, 0.1720] we can choose dCΨ =0.2688, which guaranties overshoot about 0% and settling time about 10 times of sampling time.

1

1

2

3

Figure 5.10. Step flux response for different to dCΨ : red line(1) dCΨ =0.4046, blue line (2)

dCΨ =0.2688, black line (3) dCΨ =0.1720. It corresponds to the transfer function of closed stator flux control loop as:

_ 2 2

0.2688( )+ 0.2688

dclosed

d

CG zz z C z z

ΨΨ

Ψ

= =− + −

(5.18)

Using digitalized motor parameters d sA TΨ = and chosen dCΨ value we can calculate the parameters of P digital flux controller as:


74

d dp

d s

C CKA TΨ Ψ

ΨΨ

= = (5.19)

For example let us assume that sampling time in digital flux control loop is equal 200sT sµ= .

The gain of P controller is:

0.2688 0.2688 13440.0002p

d

KAΨΨ

= = = (5.20)

In digital control when the sampling time changes the parameters of digitalized plant control

dAΨ will also change. Therefore, to keep closed loop transfer function as close as possible to

_ 2 2

0.2688( )+ 0.2688

dclosed

d

CG zz z C z z

ΨΨ

Ψ

= =− + −

, the gain of P flux controller should also be

changed (see Table 5.1.).

Keeping constant transfer function _ ( )closedG zΨ the flux step response for different sampling

times sT = 50 sµ , 100 sµ , 200 sµ , 400 sµ , which correspond to switching frequency sf =

20kHz, 10kHz, 5kHz, 2.5kHz are presented in Fig. 5.11.

Figure 5.11. Flux tracking performance for different sampling times sT = 50 sµ (blue line -1), 100 sµ (green line -2), 200 sµ (red line -3), 400 sµ (light blue line -4).


75

From Fig. 5.11 it can observed that overshoot is around 0% and the settling time took 10

times of microprocessor sampling time. So, it is possible control the flux amplitude in 10

samples

The settings of P flux controller for different sampling time sT = 50 ,100 ,s sµ µ 200 ,400s sµ µ

are summarized in Table 5.1.


76


77

The behavior of the flux control loop was tested using SABER simulation package. The

model created in SABER takes into account the whole control system, which include

real models of inverter and permanent magnet synchronous motor.

The flux step response is shown in Fig. 5.12., when parameters of P flux controller

designed for sampling time sT = 200 sµ were used for control plant for different

sampling times sT = 50 sµ , 100 sµ , 200 sµ , 400 sµ , which correspond to switching

frequency sf = 20kHz, 10kHz, 5kHz, 2.5kHz.

Figure 5.12. Flux tracking performance for different sampling time sT = 50 sµ , 100 sµ , 200 sµ , 400 sµ ( sf = 20kHz, 10kHz, 5kHz,2.5kHz) using gain of P controller designed for sT = 200 sµ ( sf = 5kHz).

After modification of P flux controller gain according to Table 5.1 it is possible to

achieve better results as shown in Fig. 5.13, what confirms proper flux tracking

performance in steady and dynamics state.


78

Figure 5.13. Flux tracking performance for different sampling time sT = 50 sµ , 100 sµ , 200 sµ , 400 sµ ( sf = 20kHz, 10kHz, 5kHz,2.5kHz) using designed gain of P controller calculated individually (see table 5.1.)

The simulation results in SABER package for Ts=200us is presented in Fig. 5.14.

Figure 5.14. Simulated (SABER) flux tracking performance for step change from 70% - 100% of nominal flux.

As we can observed from Fig. 5.14 that overshoot is around 0%pM = and settling time

took about 10 sampling time of microprocessor, what proved the design procedure of P

digital flux controller gain.


79

5.2.2 Digital torque control loop The considered torque control loop is shown in Fig. 5.15.

δΨ∆ Referenceflux generatorIn stator frame

_sre

fΨ

sθ Ψ

_s refαΨ

_s refβΨ−

sαΨ

sβΨ

sα∆Ψ

−

sβ∆ΨFlux

controllers In stator frame

_s refU α

_s refU βPI

Torque controller

_e refM

eM

−

Figure 5.15. Torque control loop with PI controller.

Based on the equation (5.9a-b and 5.10a-b) the stator flux errors in ,α β coordinates can

be calculated as:

_ _ _cos( ) cos [cos( ) cos ]s s ref s s ref s s ref s sα θ δ θ θ δ θΨ Ψ Ψ Ψ Ψ Ψ∆Ψ = Ψ + ∆ − Ψ = Ψ + ∆ −

(5.21a)

_ _ _sin( ) sin [sin( ) sin ]s s ref s s ref s s ref s sβ θ δ θ θ δ θΨ Ψ Ψ Ψ Ψ Ψ∆Ψ = Ψ + ∆ − Ψ = Ψ + ∆ −

(5.21b)

Assuming that for small changes of δΨ∆ the cos 1δΨ∆ ≅ and sin δ δΨ Ψ∆ ≅ ∆ , the equations

(5.21a) and (5.21b) are given by:

_ _ sins ref s ref sα δ θΨ Ψ∆Ψ = − Ψ ∆ (5.22a)

_ _ coss ref s ref sβ δ θΨ Ψ∆Ψ = Ψ ∆ (5.22b)

In order to design the PI torque controller the following assumption are made:

• stator flux vector position sθΨ and rotor flux vector position rθ are equal zero. It

correspond to situation, when those two flux vectors lie along theα axis,

• the reference stator flux amplitude is equal value of permanent magnet flux

_s ref PMΨ = Ψ ,

• stator resistance is neglected.

Therefore, the error stator fluxes in ,α β coordinates are calculated as:

_ 0s refα∆Ψ = , (5.23a)


80

_s ref PMβ δΨ∆Ψ = Ψ ∆ . (5.23b)

Stator voltage components using equations (2.23a-b) can be expressed as:

_ 0s refU α = (5.24a)

_ _s ref s ref pU Kβ β Ψ= ∆Ψ (5.24b)

And further because of _ _s ref s refU Uβ =

s PM pU Kδ Ψ= Ψ ∆ (5.25)

So, the transfer function between stator voltage amplitude sU and increment of torque

angle δΨ∆ can be written as:

( ) sM p PM

UG s Kδ δ Ψ

Ψ

= = Ψ∆

(5.26)

Where pK Ψ is the gain of stator flux P controller.

For example for sampling time 200sT sµ= , calculated 0.2688 1344ps

KTΨ = = (see

Table5.1.) and nominal value of 0.264PM WbΨ = the calculated ( ) 354,82MG sδ = /V rad .

The obtained transfer function between electromagnetic torque eM and stator voltage

amplitude sU is (see equation 5.72):

2

( )( )( )

e MM

s M M

M s A sG sU s s B s C

= =+ + (5.27)

Where 32b PM

Ms

pALΨ

= and s PMM

s s

RBL

Ψ=

Ψ

232

s PM bM

s

pC

JLΨ Ψ

=

Using the motor parameters (see Appendices), one obtains:

198MA = and 115.3MB = 9065MC =

Continuous s-domain

The torque control loop is shown in Fig. 5.16, where ( )MC s is a transfer function of the

PI controller given by [105]:


81

( )

iMpM

pMM

KK sK

C ss

⎛ ⎞+⎜ ⎟⎜ ⎟

⎝ ⎠= (5.28)

where pMiM

iM

KK

T=

eMs sU Uβ =_e refM ( )MC s ( )MG sδΨ∆( )MG sδ

Figure 5.16. Block diagram of torque control loop represented in s-domain.

Hence, the transfer function between _ ( )e refM s and ( )eM s is obtained as:

__

( ) ( ) ( ) ( )( )( ) 1 ( ) ( ) ( )

e ref M M MM closed

e M M M

M s C s G s G sG sM s C z G s G s

δ

δ

= =+

(5.29)

Substituting transfer function for ( )MC s and ( )MG s equation (5.29) becomes:

_ ( )M closedG s = 2

( )

( )

iMp PM M pM

pM

M pM M M iM M

KK A K sK

s B K A s C K A

ΨΨ +=

+ + + + (5.30)

Discrete design

The transfer function (equations 5.28) for PI controller in discrete system using

backward difference method for discretization process ( 1

s

zsT z−

= ) [2] is expressed as:

( )( ) ( )

( 1)

pM

pM iMM pM iM

Kz

K KC z K K

z

−+

= +−

(5.31)

where: pMiM s

iM

KK T

T= , sT - sampling time, ( )MC z is the discrete transfer function of

torque PI controller, ( )D z 1z− is one sampling time delay for voltage generation from

PWM, and


82

( )eM zsU_ ( )e refM z1z−( )MC z

( )MG z( )D z

ZOH2

M

M M

A ss B s C+ +

δΨ∆

( )MG zδ

Figure 5.17. Block diagram of torque control loop in discrete domain.

( )M p PMG z Kδ Ψ= Ψ (5.32)

is discrete transfer function for relation between stator voltage amplitude sU and

increment of torque angle δΨ∆ (see Fig. 5.17)

The discrete transfer function ( )MG z for voltage-torque relationship with zero order

hold (ZOH) can be calculated as:

12

( ) 1( ) (1 ) [ ] [ ]M MM

M M

G s AzG z z Z Zs z s B s C

− −= − =

+ + (5.33)

Finally, the discrete transfer function of controlled plant ( )MG z can be written as:

( ) 2( ) 1 MdM

Md Md

AG z zz B z C

⎛ ⎞= − ⎜ ⎟− +⎝ ⎠

(5.34)

Where:2

2_ 2

sin( )4

4

Ms

B TM M

M d s M

MM

A BA e T CBC

−= −

−

,

22

_ 2 cos( )4

Ms

B TM

M d s MBB e T C

−= − , _

M sB TM dC e−= , and sT is sampling time.

Hence, the transfer function of closed torque control loop is obtained in the following

form:

__

( ) ( ) ( ) ( ) ( )( )( ) 1 ( ) ( ) ( ) ( )

e ref M M MM closed

e M M M

M z C z G z D z G zG zM z C z G z D z G z

δ

δ

= = =+


83

_

3 2_ _ _ _

_

3 2_ _ _ _

( )( )( 1)

[ [ ( ) ] ]( 1)

( )( )

[ ( ) ]

pMp PM M d pM iM

pM iM

M d M d pM iM M d M d pM

pMp PM M d pM iM

pM iM


KK A K K z z

K Kz B z A K K C z A K z

KK A K K z

K Kz B z A K K C z A K

Ψ

Ψ

Ψ + − −+

= =− + + + − −

Ψ + −+

− + + + −

(5.35)

Selecting pK Ψ , iK Ψ will influence poles placement of closed torque control loop and as

a consequence also torque step responses can be selected.

The transfer function of closed torque control loop is more complicated than flux

control loop (see design of P-flux controller – section 5.2.1). One possibility is use to

the SISO tools from Matlab package to tune parameters of PI torque controller [106].

Figure 5.18. a) Torque step response for sampling time 200sT sµ= , b) with denoted rise time,

overshoot and settling time.

As can be observed in (Fig. 5.18) torque response is characterized by overshoot about

40%, rise time 4 samples and settling time 17 samples.

To eliminate high overshoot it is recommended to insert at the input prefilter (see

Fig.5.19 ) with transfer function:

0.6878( )0.855( )

MpM

pM iM

z b zP z K KK zzK K

− −= =

−−+

(5.36)

where

1

1 =0.4660.6878lim0.855z

K zz→

=−−

is gain of the prefilter.

a) b)


84

( )eM zsU1z−( )MC z

( )MG z( )D z

ZOH 2M

M M

A ss B s C+ +

δΨ∆( )MG zδ

_ ( )e refM z( )MP z

Figure 5.19. Block diagram of torque control loop with prefilter (discrete domain).

Finally, the step response of closed torque control loop with prefilter at the input is

presented bellow:



As it can be observed, the response is now characterized by overshoot about 2%, rise

time 5 samples and the settling time 15 samples.

In digital control system when the sampling time is changed the parameters of

digitalized control plant , ,Md Md MdA B C will also change. Therefore, the parameters of PI

torque controller will change also (see Table 2).

Simulation results for digital torque control loop in SABER package for 5KHz with and

without prefilter are shown in Fig. 5.21. Also, the torque step response for different

level of reference torque are presented in Fig. 5.22.

The settings for PI torque controller for different sampling time sT = 50 sµ , 100 sµ ,

200 sµ , 400 sµ are summarized in Table 5.2.

a) b)


85

Figure 5.21. Torque step response: a) without prefilter, b) with prefilter.

Figure 5.22. Torque step response with prefilter (from 0 to 25%, 50%, 75% and 100% of nominal torque).

a)

b)


86


87

The torque step response is shown in Fig. 5.23, when parameters of PI torque controller

designed for sampling time sT = 200 sµ were used for control plant for different sampling

times sT = 50 sµ , 100 sµ , 200 sµ , 400 sµ , which correspond to switching frequency sf =

20kHz, 10kHz, 5kHz, 2.5kHz.

Figure 5.23. Torque tracking performance for different sampling time sT = 50 sµ , 100 sµ , 200 sµ ,

400 sµ ( sf = 20kHz, 10kHz, 5kHz,2.5kHz) using parameters of PI controller designed for

sT = 200 sµ ( sf = 5kHz). (Please note that for 2.5kHz the system was unstable).

After modification of PI torque controller parameters according to Table 2 it is possible to

achieve better results as shown in Fig. 5.23, what confirms and proper torque tracking


Figure 5.24. Torque tracking performance for different sampling time sT = 50 sµ , 100 sµ , 200 sµ ,

400 sµ ( sf = 20kHz, 10kHz, 5kHz,2.5kHz) using designed parameters of PI controller calculated

individually (see table 5.2)


88

5.3 Parallel structure of DTC–SVM scheme

Block scheme of the control structure is shown in Fig. 5.25. Two PI controllers are used for

regulation torque and flux magnitude loops [11,55].

_e refM

_s refΨ

Me

AS

BS

CS

_sx refU

eMsI

seΨ

sΨ

_sy refU

DCU

_s refU α

_s refU β

mγ

sθΨ

Figure 5.25. Parallel structure of DTC-SVM scheme.

In this control scheme the reference stator flux magnitude _s refΨ and reference

electromagnetic torque _e refM are compared with estimated values, respectively. The flux and

torque errors ,s Me eΨ are delivered to PI controllers, which generate command value the stator

voltage components in stator flux coordinates _sx refU , _sy refU . This voltage signals are

transformed to stationary coordinates using the stator flux position angle sθΨ . The reference

stator voltage vector ( _s refU α , _s refU β ) is delivered to space vector modulator (SVM), which

generates the switching signals CBA SSS ,, to control power transistors of the inverter.

The presented control strategy is based on simplified stator voltage equations described in

stator flux oriented x-y coordinates (equations 2.27a-b):

ssx s sx

dU R I

dtΨ

= + (5.37)

sy s sy s s s sy sy s e syU R I R I E k M EΨ= +Ω Ψ = + = + (5.38)


89

where: 23

ss

b s

Rkp

=Ψ

, 23e b s syM p I= Ψ and sy s sE Ψ= Ω Ψ .

The above equations show that the sxU component has influence only on the change of stator

flux magnitude sΨ , and the component syU – if the term s sΨΩ Ψ is decoupled – can be

used for torque adjustment. Therefore, the flux and torque quantities can be controlled as

shown in Fig. 5.26.

Note, that this DTC-SVM scheme formally corresponds to the stator flux oriented voltage

source inverter-fed drive induction motor. The block diagram of the DTC-SVM scheme with

two PI controllers is shown in Fig. 5.26 The dashed line represents the PMSM part [124,125].

refsbp _23

Ψ

∫

eM

eM

_e refM

s sxR I

⊗

sxU

syU

sR1 syI

_

_

s sxR I

s syR I

syEs sΨΩ Ψ

sΨΩ

_s refΨsΨ

sΨ

Figure 5.26. Block diagram of the scheme presented in Fig. 5.27

5.3.1 Digital flux control loop Putting the stator x-axis current expression from equation 2.28a under the assumption d qL L=

coss PMsx

s

IL

δΨΨ −Ψ= (5.39)

into equation (5.37) one can obtaines

cos( ) coss PM s ss s PM

sx s ss s s

d dR RU RL dt L dt L

δδΨΨ

Ψ −Ψ Ψ Ψ Ψ= + = Ψ + − (5.40)

Using Laplace transformation to equation 5.40 can be written as:

( ) coss s PMsx s

s s

R RU sL L

δΨΨ

= Ψ + − . (5.41)


90

Assuming small changes of δΨ , the cos 1δΨ ≅ , and equations (5.41) reduces to:

( )ssx s

s

RU s WL Ψ= Ψ + + (5.42)

Where s PM

s

RWLΨΨ

=

The transfer function between stator flux amplitude sΨ and x-axis of stator voltage is:

1 1( ) s s

ssx s s

s

LG s RU W sL R s AsL

ΨΨ Ψ

Ψ= = = =

+ + ++ (5.43)

Where coss sPM PM

s s

R RWL L

δΨ Ψ≅ Ψ ≅ Ψ

For motor parameters (see Appendices ): 115.333s

s

RALΨ = = .

Continuous s-domain

The flux control loop is shown in Fig. 5.27, where ( )C sΨ is a transfer function of the PI

controller given by [105]:

( )

ip

p

KK sK

C ss

ΨΨ

ΨΨ

⎛ ⎞+⎜ ⎟⎜ ⎟

⎝ ⎠= (5.45)

where pi

i

KK

TΨ

ΨΨ

=

sΨsxUs_refΨ

( )C sΨ

WΨ

( )G sΨ

cossPM

s

RL

δΨΨ

+

Figure 5.27. Block diagram of flux control loop in s-domain.

Hence, the transfer function of the closed stator flux amplitude control loop is obtained as:


91

__

( ) ( ) ( )( )( ) 1 ( ) ( )

s refclosed

s

s C s G sG ss C z G s

Ψ ΨΨ

Ψ Ψ

Ψ= =

Ψ + (5.46)

Substituting transfer function for ( )C sΨ and ( )G sΨ one becomes

_ ( )closedG sΨ = 2

1

( )

11

ip

p ip

p

p iip

p

KK sK KK s

Ks s As A K s KKK s

Ks s A

ΨΨ

Ψ ΨΨ

ΨΨ

Ψ Ψ ΨΨΨ

Ψ

Ψ

⎛ ⎞+⎜ ⎟⎜ ⎟ ⎛ ⎞⎛ ⎞⎝ ⎠ +⎜ ⎟⎜ ⎟ ⎜ ⎟+⎝ ⎠ ⎝ ⎠=

+ + +⎛ ⎞+⎜ ⎟⎜ ⎟ ⎛ ⎞⎝ ⎠+ ⎜ ⎟+⎝ ⎠

(5.47)

Discrete design

Using backward difference method for discretization process ( 1

s

zsT z−

= ) [2] the transfer

function of equation (5.45) for flux PI controller in discrete system is expressed as:

( )( )( ) (1 )

( 1) ( 1)

pp i

p isp

i

KK K z

K KT zC z KT z z

ΨΨ Ψ

Ψ ΨΨ Ψ

Ψ

+ −+

= + =− −

(5.48)

Where: pi s

i

KK T

TΨ

ΨΨ

= ; sT - sampling time.

( )sΨ zsxU_ ( )s refΨ z1z−( )C zΨ

( )W z

( )G zΨ( )D z

ZOH

1

s AΨ+

Figure 5.28. Block diagram of flux control loop in discrete domain.

Where: ( )C zΨ discrete transfer function of PI controller, ( )D z 1z− - one sampling time delay

for voltage generation from PWM, and ( )W z - disturbance voltage due to cross coupling

between x-y axis (see Fig. 5.28).

The ( )G zΨ is discrete transfer function of voltage-flux relationship with zero order hold

(ZOH) block can be calculated as:


92

( )

1 ( ) 1( ) (1 ) [ ] [ ]( )

1 1 [ ]( )

G s AzG z z Z Zs z A s s A

z AZz A s s A

− Ψ ΨΨ

Ψ Ψ

Ψ

Ψ Ψ

−= − = =

+

−+

(5.49)

Using table of Z transformation [2] one can calculate:

( )( )

1 1 (1 )( )1 ( )

s

A Ts

A Td

d

z Az eG zz A z Bz z e

Ψ

− Ψ

−Ψ

ΨΨ Ψ

− −= =

−− − (5.50)

Where: (1 )sA T

deAA

Ψ−

ΨΨ

−= , sA T

dB e Ψ−Ψ = and sT is sampling time.

Hence, the transfer function of closed stator flux control loop can be expressed in the

following form:

__

( ) ( ) ( ) ( )( )( ) 1 ( ) ( ) ( )

( ) ( )

( 1)( ) ( ) ( )

s refclosed

s

pp i d

p i

pd p i d

p i

z C z G z D zG zz C z G z D z

KK K A z

K KK

z z z B K K A zK K

Ψ ΨΨ

Ψ Ψ

ΨΨ Ψ Ψ

Ψ Ψ

ΨΨ Ψ Ψ Ψ

Ψ Ψ

Ψ= =

Ψ +

+ −+

=− − + + −

+

(5.51)

Now selecting pK Ψ , iK Ψ is possible to obtain poles placement, which define the dynamic of

closed torque control loop.

Assuming, that pd

p i

KB

K KΨ

ΨΨ Ψ

=+

⇒ (1 )p d p pi d

d d

K B K KK B

B BΨ Ψ Ψ Ψ

Ψ ΨΨ Ψ

−= = −

and the transfer function of closed stator flux control loop will take the following form:

_ 2

( )( )

( )p i d

closedp i d

K K AG z

z z K K AΨ Ψ Ψ

ΨΨ Ψ Ψ

+=

− + + (5.52)

Putting into above equation pp i

d

KK K

BΨ

Ψ ΨΨ

+ = one obtains:

_ 22

( )

pd

d dclosed

p dd

d

KA

B CG z K z z Cz z AB

ΨΨ

Ψ ΨΨ

Ψ ΨΨ

Ψ

= =− +− +

(5.53)


93

where pd d

d

KC A

BΨ

Ψ ΨΨ

=

The bellow diagrams shows the relationship between overshoot pM , rise time rt and settling

time st as function dCΨ value.

Please note that rt is time calculated from 10% to 90% of output signals and st is the time in

witch the system transient decay to +-1%.

Figure 5.29. The relationship between overshoot, rise time and settling time versus dCΨ for stator flux amplitude control loop. From a few values of dCΨ =[0.4046, 0.2688, 0.1720] we can selected dCΨ =0.2688, which

guaranties overshoot 0% and settling time about 10 samples.

Figure 5.30. Flux step response for different values of dCΨ : red line (1) dCΨ =0.4046, blue line (2)

dCΨ =0.2688, black line (3) dCΨ =0.1720.

1

2 3


94

It corresponds to the transfer function of closed stator flux control loop:

_ 2 2

0.2688( )+ 0.2688

dclosed

d

CG zz z C z z

ΨΨ

Ψ

= =− + −

(5.54)

Using digitalized motor parameters (1 )sA T

deAA

Ψ−

ΨΨ

−= , sA T

dB e Ψ−Ψ = and chosen dCΨ value, we

can calculate the parameters of digital PI flux controller as:

d dp

d

C BKAΨ Ψ

ΨΨ

= (5.55a)

p si

i

K TT

KΨ

ΨΨ

= (5.55b)

(1 )pi d

d

KK B

BΨ

Ψ ΨΨ

= − (5.55c)

For example with sampling time 200sT sµ= , parameters of PI controller are:

0.2688 0.2688*0.9772 1328.640.0001977

dp

d

BKA

ΨΨ

Ψ

= = = (5.56a)

1328.64*200 857230.999

p si

i

K T sT sK

µ µΨΨ

Ψ

= = = (5. 56b)

1328.64(1 ) *(1-0.9772) 30.9990.9772

pi d

d

KK B

BΨ

Ψ ΨΨ

= − = = (5. 56c)

For different sampling time the closed transfer function _ ( )closedG zΨ of digital flux control

loop should be kept to:

_ 2 2

0.2688( )+ 0.2688

dclosed

d

CG zz z C z z

ΨΨ

Ψ

= =− + −

(5.57)

In order to find the original function of Z transfer function _ ( )closedG zΨ using the Z properties

as (sum transformations) [2]:

1 1_

0

12

( ) [ ( )] [ ( )]1 1

[ ]1 ( )

n

s closedk

z zf kT Z F z Z G zz z

z aZz z z a

− −Ψ

=

−

= =− −

=− − +

∑ (5.58)


95

As an example the calculated response for 4 samples are given:

2 3 4

0

( ) 2 (1 ) 2 .......n

sk

f kT az az a a az− − −

=

= + − + + +∑

which gives:

(0) ( ) 0f f k= =

(1 ) ( 1) 0sf T f k= + =

(2 ) ( 2) 0.2688sf T f k a= + = =

(3 ) ( 3) ( 1) 2 0.5376sf T f k a b a= + = + = =

2(4 ) ( 4) ( ) ( 1) (1 ) 2 0.7342sf T f k a b c a b a a a= + = − + + + = − + + =

Keeping constant dCΨ in equation (5.57) the flux step response for different sampling time

sT = 50 ,sµ 100 ,sµ 200 ,sµ 400 sµ , which correspond to switching frequency sf = 20kHz,

10kHz, 5kHz, 2.5kHz are presented.

Figure 5.31.Flux step response for different sampling time sT = 50 ,sµ 100 ,sµ 200 ,sµ 400 sµ

(switching frequency sf 20kHz (1 -blue line), 10kHz (2 -green line), 5kHz (3 -red line),2.5kHz (4 -light blue line).

We may observe from Fig. 5.31 that overshoot is 0% and the settling time is 10 samples.

Selected parameters of PI flux controller for sampling time sT = 50 ,sµ 100 ,sµ 200 ,sµ

400 sµ are summarized in Table 5.3

1

2

3

4


96


97

The behavior of the flux control loop was tested using SABER simulation package. The

model created in SABER takes into account the whole control system, which include real

models of inverter and permanent magnet synchronous motor.

The flux step response is shown in Fig. 5.32., when parameters of PI flux controller designed

for sampling time sT = 200 sµ were used for control plant for different sampling times

sT = 50 sµ , 100 sµ , 200 sµ , 400 sµ , which correspond to switching frequency sf = 20kHz,

10kHz, 5kHz, 2.5kHz.

Figure 5.32. Flux tracking performance for different sampling time sT = 50 sµ , 100 sµ , 200 sµ ,

400 sµ ( sf = 20kHz, 10kHz, 5kHz,2.5kHz) using parameters of PI controller designed for

sT = 200 sµ ( sf = 5kHz).

After modification of PI flux controller parameters according to Table 5.31 it is possible to

achieve better results as shown in Fig. 5.33, what confirms proper flux tracking performance

in steady and dynamics state.


98

Figure 5.33. Flux tracking performance for different sampling time sT = 50 sµ , 100 sµ , 200 sµ ,

400 sµ ( sf = 20kHz, 10kHz, 5kHz,2.5kHz) using parameters of PI controller calculated individually

(see table 5.3.)

The simulation results in SABER package for Ts=200us is presented in Fig. 5.34.

Figure 5.34. Simulated (SABER) flux tracking performance for step change from 70% - 100% of

nominal flux.

As we can observed from Fig. 5.34 that overshoot is around 0%pM = and settling time took

about 10 sampling time of microprocessor, what proved the design procedure of PI digital

flux controller parameters.


99

5.3.2 Digital torque control loop The PMSM equations (2.27a,b-2.28a,b) in stator flux coordinates under the assumption

d qL L= can be written as:

ssy s sy sU R I Ψ= +Ω Ψ (5.59)

0 sins sy PML I δΨ= −Ψ (5.60)

32e b sysM p I= Ψ (5.61)

1 ( )me l

d M Mdt JΩ

= − (5.62)

The load angle can be expressed (Fig. 5.1):

s b mpδ θ γΨ Ψ= − , (5.63)

Where: δΨ is torque angle, sθΨ is stator flux vector position, and mγ is rotor position in stator

,α β coordinates, bp is number of pole pars.

After differentiation equation (5.63) can be written as:

s mb

d dd pdt dt dt

θ γδ ΨΨ = − (5.64)

s b md pdtδΨ

Ψ= Ω − Ω ⇒s b m

d pdtδΨ

ΨΩ = + Ω (5.65)

Putting equations (5.64) and (5.65) into voltage equation (5.59) one obtains:

( )ssy s sy s s sy s b m

dU R I R I pdtδΨ

Ψ= +Ω Ψ = + Ψ + Ω (5.66)

From equation 0 sins sy PML I δΨ= −Ψ with assumption that for small angle sinδ δΨ Ψ= , the

torque angle can be expressed as:

s sy

PM

L IδΨ =

Ψ (5.67)

So, the voltage equation (5.59) becomes:

( )s

syssy s sy s s sy s b m

PM

dILU R I R I pdtΨ= +Ω Ψ = + Ψ + Ω

Ψ (5.68)


100

After differentiating of the above equation one obtains:

2

( )sy sy sys ms s b

PM

dU dI d IL dR pdt dt dt dt

Ω= + Ψ +

Ψ (5.69)

Take into account that from equation (5.61) the y-axis current is equal 23

esy

b s

MIp

=Ψ

,

1 ( )me l

d M Mdt JΩ

= − and under assumption that the motor is no loaded equation (5.69) takes

form:

22 2( )3 3

sy e s e bs s e

b s PM b s

dU dM L d M pR M

dt p dt p dt J= + Ψ +

Ψ Ψ Ψ (5.70)

Using Laplace transformation and after some arrangements the equation (5.70) can be written:

22 2( )3 3

s bs ssy e

b PM b s

pL RsU M s sp p J

Ψ= + +

Ψ Ψ (5.71)

Hence, the transfer function between electromagnetic torque eM and y-axis voltage syU can

be obtained as:

2

( )( )( )

e MM

sy M M

M s A sG sU s s B s C

= =+ + (5.72)

Where: 32b PM

Ms

pALΨ

= and s PMM

s s

RBL

Ψ=

Ψ

232

s PM bM

s

pC

JLΨ Ψ

=

Using the motor parameters (see Appendices) we may calculates:

198MA = , 115.3MB = and 9065MC =

Continuous s-domain

The torque control loop of the block scheme DTC-SVM from Fig. 5.25 is shown in Fig. 5.35,

where ( )MC s is a transfer function of the PI controller given by equation 5.28:


101

eMsyU_e refM

( )MC s ( )MG s

Figure 5.35. Block diagram of torque control loop in s-domain.

The transfer function of torque control loop is obtained as:

__

( ) ( ) ( )( )( ) 1 ( ) ( )

e ref M MM closed

e M M

M s C s G sG sM s C z G s

= =+

(5.73)

Substituting in equation (5.73) transfer function for ( )MC s -Eq.5.28 and ( )MG s - Eq.5.72 we

may calculate:

_ ( )M closedG s = 2

( )

( )

iMM pM

pM

M pM M M iM M

KA K sK

s B K A s C K A

+=

+ + + + (5.74)

Discrete design

Using backward difference method for discretization process ( 1

s

zsT z−

= ) the transfer function

for discrete PI controller is expressed as:

( )( ) (1 ) ( )

( 1) ( 1)

pM

pM iMsM pM pM iM

iM

Kz

K KT zC z K K KT z z

−+

= + = +− −

(5.75)

Where: pMiM s

iM

KK T

T= - integration gain; sT - sampling time


102

( )eM zsyU_ ( )e refM z

1z−( )MC z

PI controllerTime delay ( )MG z

( )D z

ZOH2

M

M M

A ss B s C+ +

Plant

Figure 5.36. Block diagram of torque control loop in discrete domain.

Where: ( )MC z - discrete transfer function for PI controller, ( )D z 1z− - one sampling time

delay for voltage generation from PWM block (see Fig. 5.36).

The discrete transfer function ( )MG z for voltage-torque relationship with zero order hold

(ZOH) can be calculated as:

1

2

( ) 1( ) (1 ) [ ] [ ]M MM

M M

G s AzG z z Z Zs z s B s C

− −= − =

+ +

2 22 22

2

22 22

1 1

( )2 4 2 4

1 4

( )4 2 4

M M

M M M MM M

MM

M

M M MM M

A Az zZ Zz zB B B Bs C s C

BCAz Zz B B BC s C

⎡ ⎤⎡ ⎤ ⎢ ⎥⎢ ⎥ ⎢ ⎥− −⎢ ⎥= = =⎢ ⎥⎢ ⎥⎛ ⎞ ⎛ ⎞⎢ ⎥+ + −⎢ ⎥⎜ ⎟ + + −⎜ ⎟⎢ ⎥⎝ ⎠ ⎜ ⎟⎣ ⎦ ⎢ ⎥⎝ ⎠⎣ ⎦

⎡ ⎤⎢ ⎥

−⎢ ⎥−= ⎢ ⎥

⎛ ⎞⎢ ⎥− + + −⎜ ⎟⎢ ⎥⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

.

(5.76)

Assuming that 2MBa = and

2

4M

MBb C= − , and using table of Z transformation [2] we

have:

22 2 2

sin( )( ) 2 (cos( ))

s

s s

aTs

aT aTs

ze bTbZs a b z e bT z e

−

− −

⎡ ⎤=⎢ ⎥+ + − +⎣ ⎦

(5.77)


103

Finally, the discrete transfer function of controlled plant ( )MG z can be written as:

( ) 2( ) 1 MdM

Md Md

AG z zz B z C

⎛ ⎞= − ⎜ ⎟− +⎝ ⎠

(5.78)

Where: 2

2_ 2

sin( )4

4

Ms

B TM M

M d s M

MM

A BA e T CBC

−= −

−

, 2

2_ 2 cos( )

4

Ms

B TM

M d s MBB e T C

−= −

_M sB T

M dC e−= and sT is sampling time. Hence, the transfer function of closed torque control loop is obtained as:

__

( ) ( ) ( ) ( )( )( ) 1 ( ) ( ) ( )

e ref M MM closed

e M M

M z C z G z D zG zM z C z G z D z

= = =+

_

3 2_ _ _ _

( )( )

[ ( ) ]

pMM d pM iM

pM iM


KA K K z

K Kz B z A K K C z A K

+ −+

=− + + + −

(5.79)

Selecting pK Ψ , iK Ψ will influence poles placement of closed torque control loop and as a

consequence also torque step responses can be selected.

The transfer function of closed torque control loop is more complicated than flux control loop

(see design of PI-flux controller – section 5.3.1). One possibility is use to the SISO tools from

Matlab package to tune parameters of PI torque controller [106].



As can be observed the response is characterized by overshoot about 40%, rise time 4 samples

and settling time 17 samples.

a) b)


104

The torque control loop should be as fast as possible even with some overshoot. This improve

response to disturbance (for example from flux control loop –see Fig. 5.38).


1z−( )MC z

( )MG z( )D z

ZOH 2M

M M

A ss B s C+ +

Figure 5.38. Block diagram of torque controller in discrete domain with disturbance.

)a )bFigure 5.39. Disturbance rejection in torque control loop: a) short voltage impulse, b) voltage step. To improve reference tracking performance (without any overshoot) it is recommended to

insert a input prefilter (see Fig. 5.40 ) described by transfer function:

0.6663( )( ) 0.855( )

MpM

pM iM

z b z b zP z K K KK z a zzK K

− − −= = =

− −−+

(5.80)

Where:

1

1 =0.434130.6663lim0.855z

K zz→

=−−

is gain of the prefilter.


105


1z−( )MC z

( )MG z( )D z

ZOH2

M

M M

A ss B s C+ +

( )MP z

Figure 5.40. Block diagram of torque control loop with prefilter in discrete domain.

Finally, the reference tracking performance of closed torque control loop with prefilter is

presented in Fig. 5.41.

Figure 5.41. a) Reference tracking performance of the torque control loop for sampling time

200sT sµ= , b) with denoted rise time, overshoot and settling time.

In digital control when the sampling time is changing the parameters of digitalized plant

control , ,Md Md MdA B C will be also changes. It is normally that the parameters of PI torque

control will be changes also (see Table 5.4.).

Simulation for PI flux calculated parameters with and without prefilter are shown in Fig.

5.42a-b. Also torque step response for different level of reference torque are presented in Fig.

5.43.


106

Figure 5.42. Torque step response: a) without prefilter, b) with prefilter.

Figure 5.43. Torque step response with prefilter (from 0 to 25%, 50%, 75% and 100% of nominal

torque).

a)

b)


107


108

The behavior of the torque control loop was tested using SABER simulation package.

The model created in SABER takes into account the whole control system, which

include real models of inverter and permanent magnet synchronous motor.

The torque step response is shown in Fig. 5.44, when parameters of PI torque controller

designed for sampling time sT = 200 sµ were used for control plant for different

sampling times sT = 50 sµ , 100 sµ , 200 sµ , 400 sµ , which correspond to switching

frequency sf = 20kHz, 10kHz, 5kHz, 2.5kHz.

Figure 5.44. Torque tracking performance for different sampling time sT = 50 sµ , 100 sµ ,

200 sµ , 400 sµ ( sf = 20kHz, 10kHz, 5kHz,2.5kHz) using parameters of PI controller

designed for sT = 200 sµ ( sf = 5kHz). Pleas note that for 2.5kHz the system was unstable.

After modification of PI flux controller parameters according to Table 5.4 it is possible

to achieve better results as shown in Fig. 5.45, what confirms proper torque tracking



109

Figure 5.45. Torque tracking performance for different sampling time sT = 50 sµ , 100 sµ ,

200 sµ , 400 sµ ( sf = 20kHz, 10kHz, 5kHz,2.5kHz) using parameters of PI controller

calculated individually (see Table 5.4.)


110

5.4 Speed control loop for DTC-SVM structure

The structure of speed control loop for cascade (a) and parallel (b) DTC-SVM scheme

is shown in the Fig. 5.46.

_e refM

_s refΨ

Me

AS

BS

CS

eM

sθΨsΨ

_s refU α

_s refU βδΨ∆

DCU

mγ

sI

sI

_m refΩ

ddt

)a

_e refM

_s refΨ

Me

AS

BS

CS

_sx refU

eMsI

seΨ

sΨ

_sy refU

DCU

_s refU α

_s refU β

mγ

sθΨ

_m refΩ

ddt

)b

Figure 5.46. Speed control loop for: a) cascade structure of DTC-SVM scheme, b) parallel structure of DTC-SVM scheme.

The mechanical speed equation (2.66) for the PMSM is:

me L

dM M JdtΩ

− = (5.81)

Taking Laplace transformation to equation (5.81) one obtains:

( ) ( ) ( )e L mM s M s Js s− = Ω (5.82)

The transfer function between mechanical rotor speed mΩ and electromagnetic torque

eM can be expressed as:


111

1 1( ) m

e

G sM Js JsΩ

Ω= = = (5.83)

Continuous s-domain

The block diagram of speed control loop is shown in Fig. 5.47, where ( )C sΩ is a

transfer function of the PI speed controller given by:

( )1( ) (1 )

ip

pp

i

KK sK

C s KT s s

ΩΩ

ΩΩ Ω

Ω

+= + = (5.84)

and ( )D sΩ is approximated transfer function of closed torque control loop for cascade

or parallel DTC-SVM structure.

_m refΩ ( )C sΩ ( )G sΩ

mΩ( )D sΩ

LM

eM_e refM

Figure 5.47. Block diagram of speed control loop in s-domain.

Discrete design

The transfer function for PI controller in discrete system using backward difference

method for discretization process is expressed as:

( )( ) ( )

( 1)

p

p ip i

Kz

K KC z K K

z

Ω

Ω ΩΩ Ω Ω

−+

= +−

(5.85)

Where: pi s

i

KK T

TΩ

ΩΩ

= - integration and pK Ω proportional gain of speed controller, sT -

sampling time.


112

( )m zΩ1Js

_ ( )m ref zΩ( )C zΩ

( )G zΩ

( )D zΩ ZOH

( )LM z

( )eM z

_ ( )e refM z

Figure 5.48. Block diagram of speed control loop in discrete domain.

The closed torque control loop transfer function (see for cascade DTC-SVM -Table 5.2

or parallel DTC-SVM - Table 5.4) is:

_ 2 2

0.466*0.37267 a( ) ( )z - 1.289z + 0.4633) z +bz + cM closedD z G zΩ = ≅ = (5.86)

The ( )G zΩ is discrete transfer function for torque-speed relationship with zero order

hold (ZOH). The ( )G zΩ can be calculated as:

1 1

12

( ) 1( ) (1 ) [ ] (1 ) [ ]( )

1 1 1(1 ) [ ]( 1)

s

G sG z z Z z Zs Js s

Tz ZJ s J z

− −ΩΩ

−

= − = −

= − =−

(5.87)

Finally, it can be expressed as:

( )( 1)

dAG zz

ΩΩ =

− (5.88)

Where: sd

TAJΩ = , and sT is sampling time.

Using the sampling time 200sT sµ= and motor parameters 0.0173J = the ( )G zΩ can

be calculated as:

0.01156( )( 1)

G zzΩ =−

(5.89)

Hence, the transfer function of closed speed control loop can be written:

__

( ) ( ) ( ) ( )( )( ) 1 ( ) ( ) ( )

mclosed

m ref

z C z D z G zG zz C z D z G z

Ω Ω ΩΩ

Ω Ω Ω

Ω= =Ω +

(5.90)

And finally


113

_

2

( )

( )( )

( 1)( )( 1) ( )( )

closed

pp i d

p i

pp i d

p i

G z

KK K z aA

K KK

z z bz c z K K z aAK K

Ω

ΩΩ Ω Ω

Ω Ω

ΩΩ Ω Ω

Ω Ω

=

+ −+

− − + − + + −+

(5.91)

In many practical cases the digital filter is used in speed measurement loop (see Fig.

5.49).

( )m zΩ1Js

_ ( )m ref zΩ( )C zΩ

PI controller ( )G zΩ

( )D zΩ

Control Plant

ZOH

Torque control loop

( )LM z

( )eM z

_ ( )e refM z

( )F zΩ

Digital Filter

Figure 5.49. Block scheme of speed control with digital filter in speed measurement loop

(discrete domain).

The transfer function ( )F sΩ of first order low pass filter in s domain is expressed as:

1( )1

i

F s sω

Ω =+

(5.92)

Where 2c cfω π= and cf is cut off frequency and 2 tan( )2c s

is

TT

ωω = [2]. In practice cf

is selected in the range 20-250Hz

Using the Tutsins’s approximation method 2( 1)( 1)s

zsT z

−=

+ for discretization process, the

discrete transfer function of first order low pass filter can be expressed as:

1

1

( 1)2 ( 1)( ) 2

2

s i

s i

s i

s i

T zT a zF z T z bz

T

ωω

ωω

Ω

++ +

= =− −−+

(5.93)

Where: 1 2s i

s i

TaTωω

=+

, 122

s i

s i

TbTωω

−=

+


114

The discrete transfer function of digital filter for 200sT sµ= and 25cf Hz= can be

calculated as:

0.015466 (z+1)( )(z-0.9691)

F zΩ = (5.94)

Hence, the transfer function speed control loop with digital filter:

__

( ) ( ) ( ) ( )( )( ) 1 ( ) ( ) ( ) ( )

mclosed

m ref

z C z D z G zG zz C z D z G z F c

Ω Ω ΩΩ

Ω Ω Ω

Ω= =Ω +

(5.95)

And finally

_

1

21 1

( )

( )( )( )

( )( 1)( 1)( ) ( )( ) ( 1)

closed

pp i d

p i

pp i d

p i

G z

KK K z b z aA

K KK

z bz c z z z b K K z aA a zK K

Ω

ΩΩ Ω Ω

Ω Ω

ΩΩ Ω Ω

Ω Ω

=

+ − −+

=− + − − − + + − +

+

(5.96)

Selecting pK Ω , iK Ω will influence poles placement of closed speed control loop and as

a consequence also speed step responses can be selected.

In order to select the best value of PI speed controller it is recommended to use the

SISO tools from Matlab package to tune the parameter of PI speed controller.

The speed response with digital filter simulated in SIMULINK is shown in Fig. 5.50

and simulated in SABER in Fig. 5.51 is presented.


115

Figure 5.50. Simulated (SIMULINK) speed response with digital filter

Figure 5.51. Simulated (SABER) speed response with digital filter in feedback. From the top

reference torque, measured speed.

However, the speed respond is characterized by large overshoot. Therefore, the prefilter

will be applied in order to reduce overshoot (see Fig. 5.52). The discrete transfer

function of prefilter ( )P z can be expressed as”

_ 0.005( )_ 0.995

KK sP zz bb s z

= =− −

(5.97)

where 1

_ lim( 0.995)=0.005z

KK s z→

= − is gain of the prefilter.


116

( )m zΩ1Js

_ ( )m ref zΩ ( )C zΩ

( )G zΩ

( )D zΩ ZOH

( )LM z

( )eM z

_ ( )e refM z

( )F zΩ

( )P zΩ

Figure 5.52. Speed response with digital filter in feedback ( )F zΩ and prefilter ( )P zΩ at the

input.

The speed response with and without prefilter are shown in Fig. 5.53.

With prefilter

Without prefilter

Figure 5.53. Speed response: blue signal without prefilter and green signal with prefilter at the

input.

Design parameters of PI speed controller for sampling time sT = 50 ,sµ 100 ,sµ 200 ,sµ

400 sµ are summarized in Table 5.5.


117


118

Simulated results for speed tracking performance for different reference speed level in

Fig. 5.54 are shown.

Figure 5.54. Simulated speed tracking performance with prefilter at the input for 10%, 20%,

50%, 100% of nominal speed. From the top actual speed, reference torque.

Investigation for influence of load torque in Fig. 5.55 is presented.

Figure 5.55. Simulated disturbance rejection performance of speed control loop for step load

change 50% of nominal torque. From the top electromagnetic torque, measured speed

Simulation results for speed control loop in Saber package for sampling time

200sT sµ= and PI speed parameters controller: 1.1940pK Ω = , 0.0398iTΩ = (see Table

5.5) in Fig. 5.56 is shown.


119

Figure 5.56. Simulated speed tracking performance to step of speed from 0 to 1000rpm.

The presented simulation results confirm well proper operation and design methodology

for digital speed control loop.


120

5.5 Summary

This Chapter presents design of discrete control loops for two DTC-SVM schemes:

series (cascade) and parallel structures of flux and torque controllers, Fig. 5.2 and

Fig. 5.25, respectively. The cascade structure operates with P-flux controller and PI-

torque controller whereas in parallel structure two PI controllers are used. In the first

step of design calculation of discrete Z- transfer function from continuous s- domain

transfer function using zero order hold (ZOH) method of discretization has been

performed. The continuous PI controller transfer function has been discretized using

backward difference approach. Secondly, a SISO tool from Matlab package for

digital controller parameter calculation has been applied. The results of design were

verified by Simulink (using simplified discrete transfer function) and Saber (using

full motor and inverter model) simulation. Also, the influence of sampling time

selection on controller parameters have been discussed. Finally, also the speed

control loop was synthesized using similar methodology.

Sensorless Speed Direct Torque Control with Space Vector Modulation (DTC-SVM)

121

Chapter 6 DIRECT TORQUE CONTROL WITH SPACE VECTOR MODULATION (DTC-SVM) OF PMSM DRIVE WITHOUT MOTION SENSOR 6.1 Introduction

Many motion control applications, such as material handling, packaging and hydraulic

or pneumatic cylinder replacement, require the use of a position transducer for speed or

position feedback, such as an encoder or resolver. In addition, permanent magnet

synchronous motors require position feedback to perform commutation. Some of

systems utilize velocity transducer as well. These sensor add cost, weight, and reduce

the reliability of the system. Also, a special mechanical arrangement needs to be made

for mounting the position sensors. An extra signal wires are required from the sensor to

the controller. Additionally, some type of position sensors are temperature sensitive and

their accuracy degrades, when the system temperature exceed the limits. Therefore, the

research in the area of sensorless speed control of PMSM is beneficial because of the

elimination of the feedback wiring, reduced cost, and improved reliability.

Sensorless speed DTC-SVM control block scheme is presented in Fig. 6.1.

Figure 6.1. Block scheme of DTC-SVM for PMSM drive without motion sensor.

As we can see the operation of speed controlled PMSM drive without mechanical

motion sensor is based only on measurement of following signals, which are available

in every PWM inverter-fed drive system as:

• DC link voltage,

• motor phase currents,


122

Based on this signals, the state variable of the drive can be indirectly calculated or

estimated what further allow to achieve the estimated (actual) rotor speed of PMSM.

Two motion sensorless control schemes of DTC-SVM for PMSM drive are presented in

Fig. 6.2

_e refM

_s refΨ

eMe

eMsI

_s refU α

_s refU β

sθΨ

AS

BS

CS

DCU

sU

δ∆

sΨ sI

_m refΩ

_m estΩ

)a

_e refMeMe

eM sI

seΨ

sΨ

_s refU α

_s refU β

sθΨ

_sx refU

_sy refU

AS

BS

CS

DCU

sU

_s refΨ

_m refΩ

_m estΩ

)b

Figure 6.2. The DTC-SVM block schemes of PMSM without motion sensor: a) cascade

structure and b) parallel structure.

In motion sensorless PMSM drives, as shown in Fig.6.2, the position or speed

transducer (see Fig. 5.52) is replaced by a speed estimation block, which generates the

speed feedback signal _m estΩ into the control systems and stator flux model.


123

The problem associated with speed sensorless operation of PMSM drive sourced from

VSI is listed below:

• initial rotor flux detection at start up of PMSM controlled drive,

• stator flux estimation without measured speed or position signal from sensor,

• rotor speed estimation based on state variable of the PMSM, especially in low

speed operation region.

Therefore, in this Section the initial rotor position detection method of permanent

magnets, as well as stator flux and rotor speed estimation techniques will be discussed.

6.2 Initial rotor detection method

In a PMSM drive the detection of initial flux position is an important task. The initial

position of the rotor must be detected correctly in order to initialize the flux estimation

procedure. In case of wrong detection the control algorithm has incorrect information

and the rotor shaft can be rotated through few second in positive or negative direction.

This situation is not acceptable in any drive system. Therefore, for the stable starting of

PMSM drive without the temporary reversal rotation, the initial rotor position

estimation is proposed.

The simplest method to achieved the initial rotor flux position is based on the following

rule. For short time the stator winding is supplied by the DC voltage. It impress the DC

current, which generates the magnetic field. The permanent magnet of PMSM sets

accordance with this field line. This position of PM flux is used to set initial values for

the stator flux estimation algorithm.

This method is very simple and not complicate. However, has disadvantage that during

this process the rotor can be moved in unknown direction depending on:

• position of PM before initial detection procedure,

• direction of DC voltage supply into the motor phase.

In order to make the initial rotor flux position correct without any movement (at

standstill) the following algorithms can be used in the literature [78,90,93,95,99,101]


124

To estimate the initial rotor position before starting DTC-SVM control, two kind of the

rectangular pulsewise voltages are applied from the inverter to the motor:

• one is the short pulsewise voltage,

• and another is the longer one.

Short pulsewise voltage test

This test based on general principle that the three-phase winding inductances of PMSM

are a function of the mechanical rotor position. Therefore, from the line current

responses in stator oriented coordinates ,α β under the short pulse wise voltage (see

Fig. 6.3) the position of PM can be estimated.

Figure 6.3. Voltage pulse wise during short time voltage test.

During the short time (100 sµ ) the vector 1V =(100) and opposite 4V =(011) is

generated by voltage source inverter. The achieved current responses in ,α β system for

two type of PMSM during this test in respect to mechanical rotor position are presented

in Fig. 6.4.

Figure 6.4. Current components in stator oriented coordinates ,α β under supplied voltage

vector to the motor for very short time : a) for d qL L= and b) for q dL L> .

a) b)


125

The stator current trajectory can be divided in to eight sectors (see Fig. 6.5a).

Sector 5

++++

−+

−+

−−

−− +−

+− 8π

8π

−

83π

85π

87π

89π

811π

813π

815π

o5.22

o5.202

o5.67

o5.247

o5.112

o5.292

o5.157

o5.337

Sector 1

Sector 2

Sector 3

Sector 4

Sector 6

Sector 7

Sector 8

sI

Figure 6.5. Stator current trajectory.

Based on the measured response of phase currents in ,α β coordinates, the

( )ssign I + and ( )ssign I − is calculated from following formulas:

( )s s s ssign I I I Iα β+ = + − (6.1)

( )s s s ssign I I I Iα β− = − − (6.2)

The possible combinations of ( )ssign I + and ( )ssign I − are shown in Fig. 6.6:

( )ssign I + + + − −

( )ssign I − + − − +

Figure 6.6. Possible combination of ( )ssign I + and ( )ssign I − under short pulse supply.

Let us assuming, for example, the case where the ( )ssign I + and ( )ssign I − have positive

sign. The position mγ exist in the domain of ~8 8π π

− or 7 9~8 8π π and two estimated

position can be obtained.

The mathematical analysis of ,s sI Iα β waveforms leads to following equations:

cos 2s s s mI I Iα γ= + ∆ (6.3)


126

sin 2s s mI Iβ γ= ∆ (6.4)

where sI is DC component in sI α and sI∆ amplitude of fluctuated component.

The current components in ,α β can be modeled as an average value of sI plus some

offset value sI∆ , as a function of the mechanical position mγ . Fig. 6.4 shows the

current components given in Eq. (6.1-2), which are function of the phase angle 2 mγ .

Solving those Eqs. (6.1 and 6.2) in respect to mechanical rotor position, two domains of

mechanical rotor position can be obtained as:

1 2( )s

ms s

II I

=−β

α

γ (6.5)

2 2( )s

ms s

II I

= +−β

α

γ π (6.6)

The estimated rotor positions for other combination of ( )ssign I + and ( )ssign I − are

summarized in Table 6.7.

158 8π π−

7 98 8π π−

38 8π π−

9 118 8π π−

3 58 8π π−

11 138 8π π−

5 78 8π π−

13 158 8π π−

βα sss III +− βα sss III −−

)(2 ss

s

III−α

β

πα

β +− )(2 ss

s

III

+ +

+ −

− −

+−

42π

β

α ++

−s

ssI

II

2)(2π

α

β +− ss

s

III

23

)(2π

α

β +− ss

s

III

45

2π

β

α ++

−s

ssI

II

43

2π

β

α ++

−s

ssI

II

47

2π

β

α ++

−s

ssI

II

mγ

Table 6.7. Mechanical rotor position calculations.

Long pulsewise voltage test

This test help us to choose the proper estimated value of mechanical rotor position from

two values calculated during the short pulse wise voltage test.


127

This test is based on the saturation effect of magnetic circuit. If the long pulse will send

in direction of north pole of PM, the current response will be slower, and, if current

response will be faster that the previous, it means that it is south pole.

6.3 Stator flux estimation methods

Flux estimation is an important task in implementation of high-performance DTC-SVM

motor drives. Vector control method of PMSM drive needs knowledge about actual

value of the stator flux magnitude and position as well electromagnetic torque. Also, the

flux estimation is needed to calculate the actual rotor speed for sensorless operation.

6.3.1 Overview

Many different technique has been developed for PMSM flux estimation [107].

Generally, they may be divided into two groups: open loop estimators and closed loop

estimators/observers. Most of these method are based on so called “current model” or

“voltage model” [110,113]. In fact closed loop estimators/observers are based on the

current or voltage model with an error correction loop, which drives error between two

flux models to zero in steady state. However, an observer has its own dynamics, is

sensitive to parameter changes, and has to be carefully designed for individual drives.

Therefore, for commercially manufactured drives is to complicated and impractical.

This is the reason why in this Chapter only open loop flux estimators will be

considered.

6.3.2 Current model based flux estimator The block scheme of the current based flux model is presented in Fig. 6.7. It requires:

• knowledge of PMSM machine inductance ,d qL L ,

• speed or position signal,

• PMSM phase currents.

This kind of flux estimator served in experimental test as a master (standard) to run the

DTC-SVM scheme with speed sensor.


128

mγ

αβ

dq

sI α

sI β

sdI

sqI

PMΨ

sdΨ

sqΨdL

qL

αsΨ

βsΨαβ

dq

αβ

ABCsAI

sBI

sΨ

sθΨ

Figure 6.7. Current model based stator flux estimator.

6.3.3 Voltage model based flux estimator with ideal integrator

The stator flux linkage can be obtained by using terminal voltages and currents. It is the

integral of terminal voltages minus the resistance voltage drop:

( )ss s s

d U R Idt

αα α

Ψ= − (6.7)

( )ss s s

dU R I

dtβ

β β

Ψ= − (6.8)

However, at low speed (frequencies) some problems arise, when this technique is

applied, since the stator voltage becomes very small and the resistive voltage drops

become dominant, requiring very accurate knowledge of the stator resistance sR and

very accurate integration. The stator resistance can vary due to temperature changes.

This effect can also be taken into consideration by using the thermal model of the

machine. Drifts and offsets can greatly influence the precision of integration. The

overall accuracy of the estimated flux linkage vector will also depend on the accuracy

of the monitored voltages and currents.

The most know classical voltage model obtains the flux components in stator

coordinates ( ,α β ) by integrating the motor back electromotive force ,s sE Eα β (see Fig.

6.8). The method is sensitive for only one motor parameter, stator resistance sR .

However, the application of pure integrator is difficult because of dc drift and initial

value problems.


129

sU α

sU βαβ

ABCsAU

sBU

sI α

sI βαβ

ABCsAI

sBI

sR

sRsE α

sE β ∫

∫ αsΨ

βsΨ

sΨ

sθΨ

Figure 6.8. Voltage model based estimator with ideal integrator.

There are proposed many improvements of the classical voltage model. Some of them

are presented bellow.

6.3.4 Voltage model based flux estimator with low pas filter

A common way to improve the stator flux voltage based model is to use a first-order

low-pass filter (LP) instead of the pure integrator. The equations (6.7 and 6.8) are

transferred to the form:

( )ss s s c s

d U R I Fdt

αα α α

Ψ= − + Ψ (6.9)

( )ss s s c s

dU R I F

dtβ

β β β

Ψ= − + Ψ (6.10)

The block diagram of the estimator is presented in Fig. 6.9. Discrete time

implementation of the integrator becomes:

( ) ( ) ( )s s s s s sz z z U R I Tα α α αΨ = Ψ + − (6.11)

( ) ( ) ( )s s s s s sz z z U R I Tβ β β βΨ = Ψ + − (6.12)

A LP filter does not give high accuracy at frequencies lower than cutoff frequency

2c cFω π= . There will be errors both in the magnitude and in the phase angle. As

results, the proposed voltage estimator with LP filter can be used successfully only in a

limited speed range above cutoff frequency


130

sU α

sU βαβ

ABCsAU

sBU

sI α

sI βαβ

ABCsAI

sBI

sR

-

-

sRsE α

sE β ∫

∫ αsΨ

βsΨ

Stator flux estimator (improved voltage model)

-cF

-

cF

sΨ

sθΨ

CartesianTo

Polar

Figure 6.9. Voltage model based estimator with low-pass filter.

Discrete time implementation of the LP filter becomes:

( ) ( ) ( ) ( )s s s s s s c sz z z U R I T F zα α α α αΨ = Ψ + − + Ψ (6.13)

( ) ( ) ( ) ( )s s s s s s c sz z z U R I T F zβ β β β αΨ = Ψ + − − Ψ (6.14)

6.3.5 Improved voltage model based flux estimator

Many other methods were developed in order to eliminate dc-offset and initial values

problems [107]. In general, the output Y of these new integrators (Fig. 6.10) is

expressed as:

lim1c

c c

Y X Ys sωω ω

= ++ +

(6.15)

Where X is the input and Y is output of the integrator respectively. The limY is a

compensation signal used as a feedback and cω is cutoff frequency.

1

cs ω+X Y

c

csωω+

lim

lim−

limY

Com

pens

atio

n si

gnal

Figure 6.10. Improved integration method with saturation block.


131

The first part of the equation represents a LP filter. The second part realizes a feedback,

which is used to compensate the error in the output. The block diagram of new

integration algorithm with saturation block is shown in see Fig. 6.11.

sU α

sU βαβ

ABCsAU

sBU

sI α

sI βαβ

ABCsAI

sBI

sR

sR

sE α

sE β

αsΨ

βsΨ

sΨ

sθΨ1

cs ω+

c

csωω+

lim

lim−

limY

1

cs ω+

c

csωω+

lim

lim−

limY

_ compαΨ

_s compβΨ

Figure 6.11. Full block diagram of voltage model based estimator with saturation block on the

,α β components.

The main task of saturation block is to stop the integration when the output signal sαΨ

or sβΨ exceed the reference value of stator flux amplitude. Please note that if the

compensation signal is set to zero, the improved integrator represents a first-order LP-

filter. If the compensation signal _s compαΨ or _s compβΨ is not zero the improved

integrator operates as a pure integrator.

Discrete time implementation of the improved integrator becomes:

_ lim( ) ( ) ( ) ( ( ) ( ))cs s s s s s s s

s

z z z U R I T z zTα α α α α αω

Ψ = Ψ + − + Ψ −Ψ (6.16)

_ lim( ) ( ) ( ) ( ( ) ( ))cs s s s s s s s

s

z z z U R I T z zTβ β β β β βω

Ψ = Ψ + − + Ψ −Ψ (6.17)

The output of saturation block can be described as:

sαβ_ lim

sαβ

( ) f (Ψ (z))<lim( )

lim f (Ψ (z))>=lims

s

z iz

iαβ

αβ

Ψ⎧Ψ = ⎨

⎩ (6.18)


132

Where lim is the limited value. Please note that lim should be set at reference stator

flux amplitude _lim s ref= Ψ equal PMΨ .

From Eq. 6.18 it can be observed that when one of the stator flux linkage components

sαΨ or sβΨ exceeds the limit, it causes in distortion of the output waveform.

6.4 Electromagnetic torque estimation

The PMSM motor output torque is calculated based on the equations (2.51), (2.52),

(2.55), (2.58) presented in Chapter 2, which for stator oriented coordinate system can be

written as follows:

3 ( )2e b s s s sM p I Iα β β α= Ψ −Ψ (6.19)

It can be seen that calculated torque is dependent on the current measurement accuracy

and stator flux estimation method. In practice current measurement is performed with

high accuracy (≤ 1% with 150kHz frequency bandwidth) using, for example, LEM

sensors.

6.5 Rotor speed estimation methods

6.5.1 Overview

High performance operation of motion sensorless PMSM drives depends mainly on

accurate knowledge of rotor PM flux magnitude, position and speed. The rotor position

estimation methods can be classified into two major groups:

• motor model based,

• rotor saliency based techniques.

The rotor saliency based approach is suitable only for the Interior PMSM (see Fig. 1.2 c

and d). Motor model based approach detect the back EMF vector, which includes

information about position and speed, using either open loop models/estimators

[81,85,86] or closed loop estimators/observers [70, 73,74,96,97,100]. Also adaptive

observers [72,92,98,83] and Extended Kalman Filters (EKF) [67,73] have been

proposed for motor position and speed estimation. These methods, however, are

computationally intensive and require careful design and proper initialization.


133

Therefore, for commercially manufactured drives are impractical and further a simple

open loop based techniques will be considered.

6.5.2 Back electromotive force (BEMF) technique

This technique uses the back electromotive force to estimate the rotor speed [70]. The

velocity signal could be integrated to generate a position estimate. However, this signal

is sensitive to parameter variations and tends to drift and have offset problem. Another

problem with using BEMF to estimate position is that at zero speed the BEMF goes to

zero and at low speed the signal to noise ratio can not be ignored.

6.5.3 Stator flux based technique

Generally, the calculation of rotor speed is based on the simple relationship:

r s sθ θ δΨ Ψ= − , (6.20)

where rθ is electrical position, sθΨ is stator flux position and sδΨ is torque angle.

After differentiation equation (6.20) and taking into account that r b mpθ γ= the

mechanical speed of PMSM rotor can be expressed as:

/sm b

d d pdt dtθ δΨ Ψ⎛ ⎞Ω = −⎜ ⎟

⎝ ⎠ , (6.21)

where ss

ddtθΨ

ΨΩ = is angular speed of stator flux vector and δΨ is torque angle.

As we can observe form equation (6.21) in order to calculate the mechanical rotor speed

it is necessary to calculate separately two components. One of them is angular speed of

stator flux vector sΨΩ and the second one is change of the load angle ddtδΨ (see Fig.

6.12).


134

sU α

sU β

sI α

sI βsαΨ

sβΨ

ss

ddtθΨ

ΨΩ =mΩ

ddtδΨ

1

bp

−

Figure 6.12. Block diagram of stator flux vector based rotor speed estimator.

The synchronous speed sΨΩ is calculated based on the stator flux estimator:

( )ss

s

arctg β

α

θΨ

Ψ=

Ψ (6.22)

and the calculation of sΨΩ can be done as:

( ) ( 1)s k s kss

s

ddt T

θ θθ Ψ Ψ −ΨΨ

−Ω = = (6.23)

The estimation of the synchronous speed sΨΩ based on the derivative of the position of

stator flux space vector can be modified taking in to account equation (6.22), which

finally gives [12]:

2 2

s ss s

ss s

d ddt dt

β αα β

α βΨ

Ψ ΨΨ −Ψ

Ω =Ψ +Ψ

(6.24)

Digital implementation of equation (6.24) can be written as:

( 1) ( ) ( 1) ( )( ) 2

s k s k s k s ks k

s sTα β β α− −

Ψ

Ψ Ψ −Ψ ΨΩ =

Ψ (6.25)

Also from equation (2.27b) in stator flux coordinate system the synchronous speed sΨΩ

can be obtained as:

sy s sy sys

s s

U R I EΨ

−Ω = =

Ψ Ψ (6.26)


135

Calculation of ddtδΨ

Based on the flux-current (Eq. 2.29b) and torque (Eq. 2.58) equations in stator flux

coordinates under consideration that for surface PMSM machine d qL L= and making

assumption that for small changes of torque angleδΨ , the sinδ δΨ Ψ= , the equations can

be written as:

0 sins sy PML I δΨ= −Ψ (6.27)

32e b PM syM p I= Ψ (6.28)

the torque angle δΨ can be calculated as:

23

s sy e s

PM b s PM

L I M Lp

δΨ = =Ψ Ψ Ψ

(6.29)

Further, the ddtδΨ is calculated as:

( ) ( 1)k k

s

ddt T

δ δδ Ψ Ψ −Ψ−

= (6.30)

Figure 6.13. Simulated oscillograms of rotor speed estimation according to block scheme from

Fig. 6.12 (in Saber). From the top: synchronous speed sΨΩ , the ddtδΨ signal, the measured and

estimated rotor speed, the measured and estimated electromagnetic torque.


136

The speed estimation problem is still open, especially at low and zero speed operations.

The accuracy of presented method depends on accuracy of applied stator flux estimation

and differentiation algorithm. It allows, however, for robust start, closed loop operation

above 10% of nominal speed, and braking the drive to zero speed.

6.5 Summary

The main problems associated with PMSM sensorless speed operation are presented

in this Chapter. For robust start of PMSM without the temporary rotor reversal a

special initialization algorithm has been used. This algorithm performs two test:

short and the longer voltage generated by the PWM inverter. The used speed

estimation algorithm is based on stator flux vector and torque angle estimation and

does not operate accurately around zero speed region. However, it allows robust

start and closed speed operation in the speed range above 10% of nominal speed.

The effectiveness of DTC-SVM with and without motion sensor has been proved by

simulation and experimental results (see Chapter 7)

DSP implementation of DTC-SVM control

137

Chapter 7 DSP IMPLEMENTATION OF DTC-SVM CONTROL 7.1 Description of the laboratory test-stand

The basic structure of laboratory setup is presented in Fig. 7.1 and the photo of

laboratory setup is shown in Fig. 7.2. The motor setup consists of 3kW permanent

magnet synchronous motor and DC motor, which is used as a load. The PMSM machine

is supplied by PWM inverter, which is controlled by digital signal processor (DSP)

based on DS1103 board. The voltage inverter is supplied from three-phase diodes

rectifier. The DSP interface is used in order to separate the high power from the low

power circuit (computer part). Please note that the DS1103 is inserted inside the PC

computer.

Figure 7.1. Block scheme of laboratory setup.

Figure 7.2. Laboratory setup. 1-voltage inverter, 2-control for DC motor, 3- PMSM machine, 4

– DSP interface.


138

The detailed power circuit of the laboratory setup is shown in Fig. 7.3.

three-phasesupply network

1T1D

2T2D

3T3D

4T4D

5T5D

6T6D

FC PMSMABC

4D2D

3D 5D1D

Rectifier Inverter

AS AS BS BS CS CS

Reference speedDTC-SVM

speed or position sensor

microprocessor

6D

AU

BU

CU

current sensors

1103DS

Figure 7.3. Power circuit of the laboratory setup.

In presented system the actual two currents and DC link voltage are measured by LEM

sensors and processed by A/D converter. The rotor position and speed are obtained with

an encoder of 2500 pulse per revolution. All internal data of DSP can be sent through a

D/A converter and displayed in the scope. All data of the PMSM and inverter are given

in the Appendices.

The control algorithm for PMSM machine was written in C language and was

implemented inside the processor. Also, a simple dead-time compensation method and

voltage drop on the semiconductor elements are implemented.

The phase voltage of the motor are reconstructed inside the processor using the

measured DC-link voltage and duty cycles of PWM for each phases. Motor and PI

controller model are given in Appendices.

Various tests have been carried out in order to investigate the drive performance and to

characterize the steady-state and transient behavior.


139

Control Desk experiment software run on the PC computer provides all function for

controlling, monitoring and automation of real-time experiments and makes the

development of controllers more effective. A Control Desk experiment layout for

control the PMSM machine using DTC-SVM control method is presented in Fig. 7.4.

Figure 7.4. Performed Control Desk experimental layout for control of PMSM drive.


140

7.2 Steady state behaviour

The experimental steady state no load operation at 25Hz stator frequency for

conventional ST-DTC (Fig. 4.10) and DTC-SVM (Fig. 5.46b) control is presented in

Fig. 7.5. The sampling time has been set at 200sT sµ= for DTC-SVM and 25sT sµ= for

hysteresis based ST-DTC method, respectively.

Figure 7.5. No load experimental steady state oscillograms at stator frequency 25Hz.

(a) ST-DTC for 25sT sµ= (b) DTC-SVM for 200sT sµ= .

From the top: line to line voltage, phase current, amplitude of stator flux, motor torque.

As we can observed from Fig. 7.5a the motor phase current characterized by high

current ripples. This is mainly because the inductances of the PMSM is smaller than an

equivalent power induction motor IM. In order to reduce the current ripples the

a)

b)


141

sampling time of microcontroller should be decrease. However there is hardware

limitation. The loaded program to microprocessor can not run faster.

Using the space vector modulation (SVM) based DTC much better results can be

obtained. Note, that in spite of lower switching frequency DTC-SVM guarantees less

current and torque ripple. This is mainly because contrary to hysteresis operation with

SVM operation, the inverter output voltage is unipolar (compare output voltage

waveform in Fig. 7.5a with Fig. 7.5b). This also reduces semiconductor device voltage

stress and instantaneous current reversal in DC link.

The presented experimental results (Fig. 7.6-7.9) are measured in the system with

measured speed taken to the feedback. These investigations have been performed to

show the behaviour of the DTC-SVM system without influence of the speed estimation.

In Fig. 7.6 and Fig. 7.7 steady state operation for different values of the mechanical

speed and load torque are shown.

Figure 7.6. Experimental steady state operation of PMSM controlled via DTC-SVM with the

encoder speed signal taken to the feedback ( 300m rpmΩ = , 0lM = ).

From the top: 1) amplitude of stator flux (0.25Wb/div), 2) electromagnetic torque (10Nm/div),

3) line to line voltage (1000V/div), 4) stator phase current (10A/div).


142


encoder speed signal taken to the feedback ( 300m rpmΩ = , 10lM Nm= -50% of nominal

torque). From the top: 1) amplitude of stator flux (0.25Wb/div), 2) electromagnetic torque

(10Nm/div), 3) line to line voltage (1000V/div), 4) stator phase current (10A/div).


encoder speed signal taken to the feedback ( 1500m rpmΩ = , 0lM = ). From the top: 1)

amplitude of stator flux (0.25Wb/div), 2) electromagnetic torque (10Nm/div), 3) line to line

voltage (1000V/div), 4) stator phase current (10A/div).


143


encoder speed signal taken to the feedback ( 1500m rpmΩ = , 10lM Nm= -50% of nominal

torque). From the top: 1) amplitude of stator flux (0.25Wb/div), 2) electromagnetic torque

(10Nm/div), 3) line to line voltage (1000V/div), 4) stator phase current (10A/div).

7.3 Dynamic behaviour

The experimental results of flux and torque control loop obtained in dynamic states for

PMSM machine controlled via two different DTC-SVM schemes are presented.

7.3.1 Flux and torque control loop Cascade DTC–SVM control scheme (Fig. 5.46a) In order to show behaviour of the system the dynamic testes for the flux and torque

controllers has been carried out for sampling time, 200sT sµ= . It corresponds to

switching frequency 5sf kHz= . Please note that the flux digital controller parameters

were selected according to Table 5.1 and the torque digital controller parameters were

selected from Table 5.2. (see Chapter 5.2).

In Fig. 7.10 stator flux tracking performance is presented. This result is comparable

with simulation results presented in Fig. 5.15.


144

Figure 7.10. Experimental flux tracking performance of PM synchronous motor for zero speed at sampling time 200sT sµ= . Reference flux from70% to 100% of nominal value . From the top: 1-reference stator flux amplitude(0.05Wb/div), 2- stator flux amplitude (0.05Wb/div), 3- electromagnetic torque (2Nm/div), 4- motor phase current (10A/div).

In Fig. 7.11 torque tracking performance is presented. The achieved result is

comparable with simulation results presented in Fig. 5.21b.

Figure 7.11. Experimental torque tracking performance of PM synchronous motor for zero speed at sampling time 200sT sµ= . Reference torque from 0 to nominal value. From the top:1- reference torque (10Nm/div), 2 - estimated torque (10Nm/div), 3 - stator flux amplitude (0.1Wb/div), 4- motor phase current (10A/div)


145

Figure 7.12. Experimental torque tracking performance of PM synchronous motor for zero speed (zoom) at sampling time 200sT sµ= . Reference torque from 0 to nominal value. From the top:1- reference torque (10Nm/div), 2 - estimated torque (10Nm/div), 3 - stator flux amplitude (0.1Wb/div), 4- motor phase current (10A/div).

All experimental results presented in Fig. 7.10-7.12 confirm very well proper and stable

operation of flux and torque control loops for cascade DTC-SVM structure.


146

Influence of sampling time for torque control loop in cascade DTC-SVM

The influence of sampling time on experimental torque tracking performance is

illustrated in Fig. 7.13-7.15. The dynamic test has been carried out for the same

condition ( 0m rpmΩ = ) as for simulation shown in Fig. 5.24. The controller parameters

has been set according to Table 5.2. In all oscilograms we may see proper operation of

the torque control loop for different sampling time used in practice.

Figure 7.13. Experimental torque tracking performance of PM synchronous motor for zero speed at sampling time 100sT sµ= ( 10sf kHz= ). Reference torque from 0 to nominal value. From the top:1- reference torque (4Nm/div), 2 - estimated torque (4Nm/div)

Figure 7.14. Experimental torque tracking performance of PM synchronous motor for zero speed at sampling time 200sT sµ= ( 5sf kHz= ). Reference torque from 0 to nominal value. From the top:1- reference torque (4Nm/div), 2 - estimated torque (4Nm/div)


147

Figure 7.15. Experimental torque tracking performance of PM synchronous motor for zero

speed at sampling time 400sT sµ= ( 2.5sf kHz= ). Reference torque from 0 to nominal value.

From the top:1- reference torque (4Nm/div), 2 - estimated torque (4Nm/div).

Parallel structure of DTC–SVM scheme (Fig. 5.46b) Dynamic testes for the flux and torque controllers were carried out for sampling time

200sT sµ= , which corresponds to switching frequency 5sf kHz= . Please not that the

digital flux controller parameters were selected according to Table 5.3 and the digital

torque controller parameters were selected from Table 5.4. (see Chapter 5.3).

In Fig. 7.16 stator flux tracking performance is presented. This result is comparable

with simulation results presented in Fig. 5.33 for flux and Fig. 5.45 for torque loop,

respectively.


148

Figure 7.16. Experimental flux tracking performance of PM synchronous motor for zero speed at sampling time 200sT sµ= . Reference flux from70% to 100% of nominal value . From the top: 1-reference stator flux amplitude(0.05Wb/div), 2- stator flux amplitude (0.05Wb/div), 3- electromagnetic torque (2Nm/div), 4- motor phase current (10A/div).

It can be observed that achieved result is comparable with simulation results presented

in Fig. 5.34.

Figure 7.17. Experimental torque tracking performance of PM synchronous motor for zero speed at sampling time 200sT sµ= . Reference torque from 0 to nominal value. From the top:1- reference torque (10Nm/div), 2 - estimated torque (10Nm/div), 3 - stator flux amplitude (0.1Wb/div), 4- motor phase current (10A/div)


149

Figure 7.18. Experimental torque tracking performance of PM synchronous motor for zero speed (zoom) at sampling time 200sT sµ= . Reference torque from 0 to nominal value. From the top:1- reference torque (10Nm/div), 2 - estimated torque (10Nm/div), 3 - stator flux amplitude (0.1Wb/div), 4- motor phase current (10A/div).

The achieved result is comparable with simulation results presented in Fig. 5.42b.

Experimental results presented in Fig. 7.16-7.18 confirm very well the effectiveness of

controller design and proper operation of flux and torque control loops for DTC-SVM

structure.


150

Influence of sampling time for torque control loop in parallel DTC-SVM

The influence of sampling time on experimental torque tracking performance is

illustrated in Fig. 7.19-7.21. The dynamic test has been carried out for the same

condition ( 0m rpmΩ = ) as for simulation shown in Fig. 5.45. The controller parameters

has been set according to Table 5.4. In all oscilograms we may see proper operation of

the torque control loop for different sampling time used in practice.

Figure 7.19. Experimental torque tracking performance of PM synchronous motor for zero speed at sampling time 100sT sµ= . Reference torque from 0 to nominal value. From the top:1- reference torque (4Nm/div), 2 - estimated torque (4Nm/div)

Figure 7.20. Experimental torque tracking performance of PM synchronous motor for zero speed at sampling time 200sT sµ= . Reference torque from 0 to nominal value. From the top:1- reference torque (4Nm/div), 2 - estimated torque (4Nm/div).


151

Figure 7.21. Experimental torque tracking performance of PM synchronous motor for zero speed at sampling time 400sT sµ= . Reference torque from 0 to nominal value. From the top:1- reference torque (4Nm/div), 2 - estimated torque (4Nm/div).

In this Chapter the results of experimental verification 3kW PMSM drive with two

DTC-SVM schemes has been presented. As shown the drive performance confirms

applied design methodology. The performance of both cascade and parallel DTC-SVM

control structure are similar. However, parallel structure has been selected for industrial

manufacturing because of :

• less noisy control algorithm (differentiation required in cascade structure –

equation (5.6) is eliminated),

• stator flux control in closed loop,

• the same structure can be used for IM and PMSM control (universal control for

AC motors).

7.3.2 Speed control loop 0peration with speed sensor

Dynamic testes for the speed control loop were measured for sampling time 200sT sµ= .

Please note that the digital speed controller parameters were selected according to Table

5.5 (see Chapter 5.4). In Fig. 7.22-7.26 rotor speed tracking performance are presented.


152

Figure 7.22. Experimental start up and breaking to zero speed of PMSM motor controlled via

DTC-SVM with the encoder speed signal taken to the feedback ( 0 300m rpm rpmΩ = → ). From

the top: 1- reference speed (180rpm/div), 2- measured speed (180rpm/div), 3 - electromagnetic

torque (20Nm/div), 4- motor phase current (20A/div).

Figure 7.23. Experimental speed reversal for PMSM motor controlled via DTC-SVM with the

encoder speed signal taken to the feedback ( 300 300m rpm rpmΩ = − → ). From the top: 1-

reference speed (360rpm/div), 2- measured speed (360rpm/div), 3 - electromagnetic torque

(20Nm/div), 4- motor phase current (20A/div).


153


DTC-SVM with the encoder speed signal taken to the feedback ( 0 1500m rpm rpmΩ = → ).From

the top: 1- reference speed (900rpm/div), 2- measured speed (900rpm/div), 3 - electromagnetic







154



stator flux component sαΨ (0.25Wb/div), 2- measured speed (360rpm/div), 3 - electromagnetic


Figure 7.27. Experimental response to load torque step change of PMSM motor controlled via

DTC-SVM with the encoder speed signal taken to the feedback at 0m rpmΩ = .

( 0 10lM Nm Nm= → ) From the top: 1- reference speed (180rpm/div), 2- measured speed

(180rpm/div), 3 - electromagnetic torque (5Nm/div), 4- motor phase current (10A/div).


155



( 0 10lM Nm Nm= → ).From the top: 1- reference speed (180rpm/div), 2- measured speed




( 0 10lM Nm Nm= → ). From the top: 1- reference speed (900rpm/div), 2- measured speed



156

Experimental results presented in Figures 7.22-7.29 well confirm the effectiveness of

developed controller synthesis methodology and proper operation of speed control loop

for parallel DTC-SVM structure.

Sensorless speed operation

Dynamic tests for the speed control loop without motion sensor were measured for

sampling time 200sT sµ= . Please note that the digital speed controller parameters were

selected exactly like for operation with speed sensor according to Table 5.5 (see

Chapter 5.4).

The results of speed estimator dynamic test are presented in Fig. 7.30. In this test speed

controller operates with the encoder signal in feedback and speed estimator works in

open loop fashion.

Figure 7.30. Experimental dynamic test of the speed estimation. Speed reversal 300m rpmΩ = ±

From the top: 1- reference speed (180rpm/div), 2- measured speed (180rpm/div), 3 - estimated

speed (180rpm/div), 4- error of estimated speed (5%/div).

The typical dynamic performance tests of sensorless DTC-SVM drive has been

illustrated in Fig. 7.31-7.36. Start up and breaking to zero speed for different speed level

are shown in Fig. 7.31 and 7.33. Also, the speed reversal for low (Fig. 7.32) and

nominal (Fig. 7.34) speed are presented. The half load torque step change tests are

shown in Fig. 7.36 and 7.37.


157


DTC-SVM with the estimated speed signal taken to the feedback ( 0 300m rpm rpmΩ = → ).

From the top: 1- reference speed (180rpm/div), 2- measured speed (180rpm/div), 3 -

electromagnetic torque (20Nm/div), 4- motor phase current (20A/div).


estimated speed signal taken to the feedback ( 300 300m rpm rpmΩ = − → ). From the top: 1-




158

Figure 7.33. Experimental speed step response for PMSM motor controlled via DTC-SVM with

the estimated speed signal taken to the feedback ( 0 1500m rpm rpmΩ = → ).From the top: 1-








159



stator flux component sαΨ (0.25Wb/div), 2- measured speed (360rpm/div), 3 - electromagnetic



DTC-SVM with the estimated speed signal taken to the feedback at 300m rpmΩ = .




160


DTC-SVM with the estimated speed signal taken to the feedback at 1500m rpmΩ = .



Experimental results presented in Figures 7.31-7.37 confirm very well the effectiveness

of speed estimation algorithm of speed control loop for parallel DTC-SVM structure.

Summary and closing conclusion

161

Chapter 8 SUMMARY AND CLOSING CONCLUSIONS This thesis studied basic problems related to selection, investigation and implementation

of the PWM inverter-fed permanent magnet synchronous motor (PMSM) drives suitable

for serial manufacturing. The selected method should provide: robust start and operation

in wide speed control range including zero speed, with and without mechanical motion

sensor; guarantee good and repeatable parameters of PMSM drive for wide power range

(1-100kW). The control and protection algorithm should be implemented in simple and

cheap microprocessor.

To solve so formulated general task several related problems had to be solved. At first,

the space vector based mathematical description and static characteristic of PMSM

under different control modes were studied (Chapter 2). Secondly, the three phase

voltage source inverter model including nonlinearities (dead time, semiconductor

voltage drop, DC link pulsation) and pulse with modulation PWM techniques were

presented (Chapter 3). Next, based on the study of most important high performance

control methods as field oriented control (FOC), and direct torque control (DTC), the

method called direct torque control with space vector modulator (DTC-SVM) has been

selected for further consideration (Chapter 4). This methods combines main advantage

of FOC (space vector modulator and fixed switching frequency) and DTC (simple

structure, rotor parameter independent), as well as eliminates disadvantages like:

coordinate transformation, the need of internal current control loops, high sampling

time, high torque and current ripple at steady state operation, etc.

Consequently the most important contribution of this thesis is included in the Chapter 5,

where the two basic variants of DTC-SVM schemes: series (cascade) and parallel

structures of flux and torque controllers are presented (Fig. 5.2 and Fig. 5.25). Also, a

systematic methodology of digital controller design for these both DTC-SVM variants

have been given. This methodology has been verified by Simulink (using simplified

discrete transfer function) and Saber (using full motor and inverter model) simulation

studies. The influence of sampling time selection on controller design has also been

discussed.

The main problems associated with PMSM sensorless speed operation are presented in

the Chapter 6. It should be noted that PMSM differ from IM drives mainly in:

• PMSM parameters strongly depend on construction ,

Summary and closing conclusion

162

• position of PM flux has to be known prior to start up to achieve smooth

operation.

Therefore, for robust starting of PMSM without the temporary rotor reversal a simple

initialization algorithm has been used. This algorithm performs two tests: short and the

longer voltage pulses generated by the PWM inverter. The used speed estimation

algorithm is based on stator flux vector and torque angle estimation and does not

operate accurately in zero speed region. However, it allows robust start and closed

speed operation in the speed range above 10% of nominal speed. For application where

high performance operation around zero speed are required, the DTC-SVM drive with

motion sensor (encoder) is recommended. The effectiveness of DTC-SVM scheme with

and without motion sensor has been proved by simulation and experimental results

(Chapter 7).

Simulation study and experimental results have shown that from two variants of DTC-

SVM schemes the parallel structure is more flexible in torque and flux controller

design. Also, because of lack of the differentiation in the main control path (compare

Fig. 5.2 and Fig. 5.25), it is less sensitive to noise which inherently associates signal

processing in power electronic converters. Therefore, the parallel structure has been

selected for industrial manufacturing and implemented on digital signal processor

(DSP).

Thanks to direct torque and flux control structure the described control is suitable to

almost all – industrial applications including electrical vehicles (for example hybrid

cars).

Finally, it should be stressed that the developed system was brought into serial

production. Presented algorithm DTC-SVM has been used in new generation of inverter

drives produced by Polish company Power Electronic Manufacture – “TWERD”,

Toruń.

Appendices

163

APPENDICES A1 Rotor and stator of PMSM machine

A1.1. View of rotor (on the left side) and stator armature (on the right side) of PMSM. A2 Basic transformation

AK

CK

Kβ

Kα

xKyK

β

Kθ

Kθ

K

x

y

BK

α

Fig. A2.1. Space vector representation in stationary ,α β coordinates and synchronous rotating ,x y coordinates.

, , ,A B C x y⇒ 2 [ cos cos(2 / 3 ) cos(2 / 3 )]3x A K B K C KK K K Kθ π θ π θ= + − + +

2 [ cos( / 2 ) cos(2 / 3 ( / 2 )) cos(2 / 3 / 2 )]3y A K B K C KK K K Kπ θ π π θ π π θ= + + − + + + +

Appendices

164

2 [ cos cos( 2 / 3) cos( 2 / 3)]3x A K B K C KK U K Kθ θ π θ π= + − + +

2 [ sin sin( 2 / 3) sin( 2 / 3)]3y A K B K C KK K K Kθ θ π θ π= − − − − +

cos cos( 2 / 3) cos( 2 / 3)sin sin( 2 / 3) sin( 2 / 3)

Ax K K K

By K K K

C

KK

KK

K

θ θ π θ πθ θ π θ π

⎡ ⎤− +⎡ ⎤ ⎡ ⎤ ⎢ ⎥=⎢ ⎥ ⎢ ⎥ ⎢ ⎥− − − − +⎣ ⎦⎣ ⎦ ⎢ ⎥⎣ ⎦

, , ,A B C α β⇒ 0Kθ =

2 1 1( )3 2 2A B CK K K Kα = − −

2 3 3 1( ) ( )3 2 2 3B C B CK K K K Kβ = − = −

1 112 2 23 3 30

2 2

A

B

C

KK

KK

K

α

β

⎡ ⎤ ⎡ ⎤− −⎢ ⎥⎡ ⎤ ⎢ ⎥⎢ ⎥=⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦ ⎢ ⎥− ⎣ ⎦⎢ ⎥⎣ ⎦

, , ,x y A B C⇒

cos cos( / 2 ) cos sinA x K y K x K y KK K K K Kθ π θ θ θ= + + = − cos(2 / 3 ) cos(2 / 3 ( / 2 )) cos( 2 / 3) sin( 2 / 3)B x K y K x K y KK K K K Kπ θ π π θ θ π θ π= − + − + = − − −

cos(2 / 3 ) cos(2 / 3 ( / 2 )) cos( 2 / 3) sin( 2 / 3)C x K y K x K y KK K K K Kπ θ π π θ θ π θ π= + + + + = + − +

cos sincos( 2 / 3) sin( 2 / 3)cos( 2 / 3) sin( 2 / 3)

A K Kx

B K Ky

C K K

KK

KK

K

θ θθ π θ πθ π θ π

−⎡ ⎤ ⎡ ⎤⎡ ⎤⎢ ⎥ ⎢ ⎥= − − − ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎣ ⎦⎢ ⎥ ⎢ ⎥+ − +⎣ ⎦ ⎣ ⎦

, , ,A B Cα β ⇒ 0Kθ =

AK Kα=

1 32 2BK K Kα β= − +

1 32 2CK K Kα β= − −

Appendices

165

1 0

1 32 21 32 2

A

B

C

KK

KK

K

α

β

⎡ ⎤⎢ ⎥

⎡ ⎤ ⎢ ⎥⎡ ⎤⎢ ⎥ ⎢ ⎥= − ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎣ ⎦⎢ ⎥ ⎢ ⎥⎣ ⎦

⎢ ⎥− −⎢ ⎥⎣ ⎦

, ,x y α β⇒

cos cos( / 2 ) cos sinx K y K x K y KK K K K Kα θ π θ θ θ= + + = − cos( / 2 ) cos sin cosx K y K x K y KK K K K Kβ π θ θ θ θ= − + = +

cos sinsin cos

xK K

yK K

K KK K

α

β

θ θθ θ

−⎡ ⎤ ⎡ ⎤⎡ ⎤=⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦⎣ ⎦ ⎣ ⎦

, ,x yα β ⇒

cos cos( / 2 ) cos sinx K K K KK K K K Kα β α βθ π θ θ θ= + − = + cos( / 2 ) cos sin cosy K K K KK K K K Kα β α βπ θ θ θ θ= + + = − +

cos sinsin cos

x K K

y K K

K KK K

α

β

θ θθ θ

⎡ ⎤ ⎡ ⎤⎡ ⎤=⎢ ⎥ ⎢ ⎥⎢ ⎥−⎣ ⎦⎣ ⎦ ⎣ ⎦

A3 Model of PM synchronous motor # This template models the permanent magnet synchronous motor(pmsm) # t1,t2 and t3 are motor input terminals # rotor speed (in rad/s) is the output # rs-Stator windings' resistence per phase(in Ohms) # ld-d_axis inductance(in H) # lq-q_axis inductance(in H) # pm-Rotor magnet flux(Wb) # j-Moment of inertia(in kgm2) # d- Damping constant (Nm/rad/s) # tl-motor load (Nm) # p-Number of pole pairs # power-The total input power (W) # Assumptions:No core losses,no saturation,thermal effects #(rs,ld,lq and pm values are constants) element template pmsm_dtc t1 t2 t3 t0 speed out_me out_psi out_thetam out_ualf out_ubet out_psia out_psib out_ialf out_ibet out_theta out_tl=rs,ld,lq,pm,d,tl,j,p,init_theta_m,init_omega_m,omega_m_const electrical t1,t2,t3,t0 # motor input terminals and stator neutral point output nu speed, out_me,out_psi,out_thetam,out_ualf,out_ubet,out_psia,out_psib,out_ialf,out_ibet,out_theta,out_tl number rs=0.692,ld=6m,lq=6m,pm=0.26379,d=0.002044,tl=0.0,j=0.003,p=3.0, init_theta_m=0.0,init_omega_m=0.0,omega_m_const=0.0

Appendices

166

<consts.sin val v vq,vd,va,vb val v vt1,vt2,vt3,v0 # Stator phase voltages val tq_Nm te # Electro magnetic torque val f fd,fq # d and q axis fluxes val i ialf,ibet val f psia,psib, psi val p power val ang_rad theta var w_radps omega_m var ang_rad theta_m var i iq,id var i it1,it2,it3 # stator phase currents number y parameters y=2*math_pi/3 values vt1=v(t1)-v(t0) vt2=v(t2)-v(t0) vt3=v(t3)-v(t0) va=2.0*(vt1-0.5*(vt2+vt3))/3.0 vb=(vt2-vt3)/sqrt(3) #ialf=2.0*(it1-0.5*(it2+it3))/3.0 #ibet=(it2-it3)/sqrt(3) ialf=id*cos(p*theta_m)-iq*sin(p*theta_m) ibet=id*sin(p*theta_m)+iq*cos(p*theta_m) vd=2*(vt1*cos(p*theta_m)+vt2*cos(p*theta_m-y)+vt3*cos(p*theta_m+y))/3 #d_axis voltage vq=2*(-vt1*sin(p*theta_m)-vt2*sin(p*theta_m-y)-vt3*sin(p*theta_m+y))/3 #q_axis voltage fd=ld*id+pm #d_axis flux fq=lq*iq #q_axis flux psi=sqrt(fd*fd+fq*fq) psia=fd*cos(p*theta_m)-fq*sin(p*theta_m) psib=fd*sin(p*theta_m)+fq*cos(p*theta_m) te=1.5*p*(pm*iq+(ld-lq)*id*iq) #electromagnetic torque power=3.0*(vd*id+vq*iq)/2.0 theta=p*theta_m control_section initial_condition(theta_m,init_theta_m) initial_condition(omega_m,init_omega_m*math_pi/30.0) equations id: vd=rs*id + d_by_dt(fd)-p*omega_m*fq iq: vq=rs*iq + d_by_dt(fq)+p*omega_m*fd

Appendices

167

i(t1->t0)+=it1 it1: it1=id*cos(p*theta_m)-iq*sin(p*theta_m) i(t2->t0)+=it2 it2: it2=id*cos(p*theta_m-y)-iq*sin(p*theta_m-y) i(t3->t0)+=it3 it3: it3=id*cos(p*theta_m+y)-iq*sin(p*theta_m+y) omega_m: (te-tl-d*omega_m)/j=d_by_dt(omega_m) #omega_m: omega_m=omega_m_const*math_pi/30.0 theta_m: omega_m=d_by_dt(theta_m) speed: speed=omega_m out_me: out_me=te out_psi: out_psi=psi out_thetam:out_thetam=theta_m out_ualf: out_ualf=va out_ubet: out_ubet=vb out_psia: out_psia=psia out_psib: out_psib=psib out_ialf: out_ialf=ialf out_ibet: out_ibet=ibet out_theta: out_theta=theta out_tl: out_tl=tl A4 Motor parameters Surface type motor Power P 3kW Number of pole pairs p 3 Phase current I(rms) 6.9A Phase voltage U(rms) 70V Magnetic flux-linkage PMΨ 0.264 Wb Rotor speed mΩ 3000rpm Nominal torque Me 20Nm Moment of the inertia J 0.0174kgm2 Stator winding resistance Rs 0.692Ω Stator d-axis inductance Ld 6mH Stator d-axis inductance Lq 6mH Interior type motor Power P 2,2kW Number of pole pairs p 3 Phase current I(rms) 4.1A Rated voltage U(rms) 380V Magnetic flux-linkage PMΨ 0.4832 Wb Rotor speed mΩ 1750rpm Nominal torque Me 12Nm Moment of the inertia J 0.010074kgm2 Stator winding resistance Rs 3.3Ω Stator d-axis inductance Ld 41.59mH Stator d-axis inductance Lq 57.06mH

Appendices

168

A5 Voltage Source Inverter parameters Detailed date of IGBT transistors (module TOSHIBA M675Q2YS50):

1200CEU V= , 75CI A= 2.8 3.6CEsatU V= − , forward diode voltage 2.4 3.5V−

Turn on time 0.2ONt sµ= , Turn off time 0.6OFFt sµ= Delay of IGBT drivers 0.5ONdt sµ= 1OFFdT sµ=

0.7ON ON ONdT t t sµ= + = total turn on time of IGBT 1.6OFF OFF OFFdT t t sµ= + = total turn off time of IGBT

Dead time 2.5dT sµ=

A6 PI speed controller

The commonly used in industrial application speed controller is a Proportional-Integral

PI controller thanks to possibility to reduce the speed error between the reference ( refX )

and actual rotor speed ( mX ) to zero (see Fig. A6.1). The output signal of controller is a

reference torque, which has upper and lower limitation for this value equal the nominal

torque or more than 130% of nominal torque. The output of the speed controller acts as

a current reference command for the current controllers. This current command is

limited to a nominal current of the motor.

The speed controller demands produce proper electromagnetic torque.

_X ref

_X m

−Kp

i

KpT

LY

∫

Reference signal

Feed

back

sign

al

Controller output

error signal

a)

_X ref

_X m

−Kp

i

KpT

LYlim_max

lim_min

−

∫1

iT

NLY

EY

Reference signal

Feed

back

sign

al

Controller output

error signal

b)

A.6.1. General structure of Proportional- Integral controller without antiwindup (a) and with antwindup (b).

Appendices

169

A7 PWM technique – six step mode

Six-stepped-voltage waveforms are rich in harmonics. These time harmonics produce

respective stator current harmonics, which in turn interact with fundamental air gap

flux, generating harmonics torque pulsations. The torque pulsations are undesirable:

they generate audible noise, speed pulsations, and losses. In case of supplied motor by

using only active vectors (six step mode) we can observed non sinusoidal current, which

generates torque ripples with frequency of six time fundamental frequency of supplied phase

voltages.

Fig. A7.1. Experimental operation in six step mode. From the left side stator voltages in α , β coordinates, From right side voltage trajectory.

Fig. A7.2. Experimental operation in six step mode. From the left side stator currents in α , β coordinates, From right side stator current trajectory.

Appendices

170

Fig. A7.3. Experimental operation in six step mode. From the top: α stator voltage, electromagnetic torque in machine, phase current.

List of symbol

171

List of symbols Symbol (general) X - instantaneous value

NX - normalized value X - vector X ∗ - conjugate vector X - amplitude of vector

Re( X ) – real part of X Im( X ) – imaginary part of X Symbol (special)

,α β - stator fixed system ,d q - rotor reference system ,x y - general reference system sL - stator inductance sZ - stator impedance

sM - mutual inductance sI - phase current value sU - phase voltage value sΨ - phase flux value

P -active power Q - reactive power S - apparent power

eP - electro-magnetic power mΩ - mechanical rotor speed sΩ - synchronous speed

cosφ - power factor ,Iδ δΨ - torque angle

φ - power angle sR - stator resistance dL , dL - direct and quadrature inductances rθ - electrical rotor position mγ - mechanical rotor position bp - number of pole pairs PMΨ - rotor flux of permanent magnets eM - electromagnetic torque esM - synchronous torque erM - reluctance torque lM - load torque (external load torque) dM - dynamic torque

mJ - motor moment of inertia lJ - load moment of inertia

J - moment of inertia of total system (sum of mJ and lJ )

List of symbol

172

Subscripts A , B , C - denote arbitrary phase quantities in a system of natural coordinate , ,A B C . d , q - arbitrary direct and quadrature components in a system of rotor coordinate ,d q . α , β - arbitrary alpha and beta components in a system of stator coordinate ,α β . x , y - denote arbitrary components in a system of general coordinate ,x y . .. r - denotes value of rotor .. s - denotes value of stator .. _max - maximum value .. _min – minimum value .. _ ref - reference value .. _ est - estimated value .. _ amp -amplitude value .. _ rms - root mean square value .. _ LL - line to line value * - reference value ^ - estimated value Abbreviations RSM – reluctance synchronous motor BLDCM – blushless DC motor PMSM – permanent magnet synchronous motor IPMSM - interior permanent magnet synchronous motor SPMSM - surface permanent magnet synchronous motor EMF – electro-magnetic force VSI - voltage source inverter SVM – space vector modulator PWM – pulse width modulation PWM-VSI – voltage source inverter with PWM DTC - direct torque control DTC-SVM - direct torque control with space vector modulator RFOC - rotor field oriented control SFOC - stator field oriented control CTAC - constant torque angle control MTPAC - maximum torque per ampere control UPFC - unity power factor control CSFC - constant stator flux control

References

173

Books and PhD-Thesis

1. I. Boldea, S.A. Nasar, “Electrical drives”, CRC Press, 1999. 2. G. F. Franklin, J. D. Powell, “Feedback Control of Dynamic Systems”, Addison-

Wesley Publishing Company, 1994. 3. M. P. Kaźmierkowski, H. Tunia, “Automatic Control of Converter-Fed drives”,

Elsevier, 1994. 4. M. P. Kaźmierkowski, R. Krishnan, F. Blaabjerg, “Control in Power Electronics”,

Academic Press, 2002. 5. P.C. Krause, O. Wasynczuk, S.D. Sudhoff, “Analysis of Electric Machinery”, IEEE

Press, 1995. 6. R. Krishnan, “Electric Motor Drives – Modeling, Analysis, and Control”, Prentice Hall,

2001. 7. T. Kaczmarek, K. Zawirski, “Układy napędowe z silnikiem synchronicznym”,

Politechnika Poznańska, 2000. 8. M. Malinowski, "Sensorless Control Strategies for Three-Phase PWM Rectifiers", PhD

Thesis, Warsaw University of Technology, 2001. 9. Ch. Perera, “Sensorless Control of Permanent Magnet Synchronous Motor Drives”,

PhD thesis, Institute of Energy Technology, 2002. 10. K. Rajashekara, A. Kawamura, K. Matsuse, “Sensorless Control of AC Motor Drives”,

IEEE Press,1996, USA. 11. M. Zelechowski, “Space Vector Modulated – Direct Torque Controlled (DTC-SVM)

Inverter-Fed Induction Motor Drive”, Phd Thesis , Warsaw University of Technology, 2005.

12. P. Vas, “Sensorless Vector and Direct Torque Control”, Oxford University Press, 1998.

Model of PMSM

13. C. Bowen; Z. Jihua; R. Zhang; “Modeling and simulation of permanent magnet synchronous motor drives”, Electrical Machines and Systems, ICEMS 2001, Vol. 2, 18-20 Aug. 2001, pp. 905 – 908.

14. P. Pillay, R. Krishnan, “Modeling of permanent magnet motor drives”, Transactions on Industrial Electronics, Vol. 35, Nov. 1988, pp. 537 – 541.

15. N. Urasaki; T. Senjyu; K Uezato, “An Accurate Modeling for Permanent Magnet Synchronous Motor Drives”, Applied Power Electronics Conference and Exposition, APEC 2000, Vol. 1, 6-10 Feb. 2000, pp:387 – 392.

16. A.H. Wijenayake, P.B. Schmidt, “Modeling and analysis of permanent magnet synchronous motor by taking saturation and core loss into account”, International Conference on Power Electronics and Drive Systems, Vol. 2, 26-29 May 1997, pp. 530 – 534.

Inverter and PWM Modulation

17. L. Ben-Brahim, "The analysis and compensation of dead-time effects in three phase

PWM inverters", Industrial Electronics Society, 1998. IECON '98. Proceedings of the 24th Annual Conference of the IEEE, Vol. 2d, 31 Aug.-4 Sept. 1998, pp.792-797.

18. V. Blasko, "Analysis of a hybrid PWM based on modified space-vector and triangle-comparison methods", IEEE Transactions on Industry Applications, Vol. 33, Issue: 3, May-June 1997, pp.756-764.

19. F.Blaabjerg, J.K. Pedersen, P. Thoegersen, “Improved modulation techniques for PWM-VSI drives”, IEEE Transactions on Industrial Electronics, Feb. 1997, Vol. 44, pp. 87 – 95.

20. J.W. Choi; S.K. Sul, "Inverter output voltage synthesis using novel dead time compensation", IEEE Transactions on Power Electronics, Vol. 11, Issue: 2, March 1996, pp.221-227.

References

174

21. D.W. Chung, J. Kim, S.K. Sul, "Unified voltage modulation technique for real-time three-phase power conversion", IEEE Transactions on Industry Applications, Vol. 34, Issue: 2, March-April 1998, pp.374-380.

22. A. Diaz, E.G. Strangas, "A novel wide range pulse width overmodulation method [for voltage source inverters]", Applied Power Electronics Conference and Exposition, APEC 2000.

23. A.M. Hava, R.J. Kerkman, T.A. Lipo, "A high performance generalized discontinuous PWM algorithm", Applied Power Electronics Conference and Exposition, APEC '97 Conference Proceedings 1997, Twelfth Annual, Vol. 2, 23-27 Feb. 1997, pp.886-894.

24. D.G. Holmes, "The significance of zero space vector placement for carrier-based PWM schemes", IEEE Transactions on Industry Applications, Vol. 32, Issue: 5, Sept.-Oct. 1996, pp.1122-1129.

25. J. Holtz, "Pulsewidth modulation for electronic power conversion", Proceedings of the IEEE, Vol. 82, Issue: 8, Aug. 1994, pp.1194-1214.

26. J. Holtz, W. Lotzkat, A.M. Khambadkone, "On continuous control of PWM inverters in the overmodulation range including the six-step mode", IEEE Transactions on Power Electronics, Vol. 8, Issue: 4, Oct. 1993, pp.546-553.

27. J.L. Lin, "A new approach of dead-time compensation for PWM voltage inverters", IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, Vol. 49, Issue: 4, April 2002, pp.476-483.

28. D.C. Lee, G.M. Lee, "A novel overmodulation technique for space-vector PWM inverters", IEEE Transactions on Power Electronics, Vol. 13, Issue: 6, Nov. 1998, pp.1144-1151.

29. M. Malinowski "Adaptive modulator for three-phase PWM rectifier/inverter", in proc. EPE-PEMC Conf., Kosice, 2000, pp.1.35-1.41.

30. N. Urasaki, T. Senjyu, K. Uezato, T. Funabashi, “On-line dead-time compensation method for permanent magnet synchronous motor drive”, IEEE International Conference on Industrial Technology, CIT '02, 11-14 Dec. 2002, Vol. 1, pp. 268 – 273.

FOC, DTC and DTC-SVM

31. F. Blaschke, "The principle of fields-orientation as applied to the Transvector closed-

loop control system for rotating-field machines", in Siemens Reviev 34, 1972, pp.217-220.

32. G.S. Buja, M.P. Kazmierkowski, "Direct Torque Control of PWM Inverter-Fed AC Motors-A Survey", IEEE Transactions on Industrial Electronics, Vol. 51, Aug. 2004, pp.744-757.

33. D. Casadei, G. Serra, A. Tani, L. Zarri, F. Profumo, “Performance analysis of a speed-sensorless induction motor drive based on a constant-switching-frequency DTC scheme”; IEEE Transactions on Industry Applications, March-April 2003, Volume 39, pp. 476 – 484.

34. D. Casadei, F. Profumo, G. Serra, A. Tani, “FOC and DTC: two viable schemes for induction motors torque control”; IEEE Transactions on Power Electronics, Sept. 2002, Vol. 17, pp. 779 – 787.

35. T. Chun; W.Y. Hu, “Research on the direct torque control in electromagnetic synchronous motor drive”, Power Electronics and Motion Control Conference, PEMC 2000, 15-18 Aug. 2000. Vol. 3, pp. 1262 – 1265.

36. S.K. Chung, H.S. Kim, Ch. G. Kim, M.-J. Youn, “A New Instantaneous Torque Control of PM Synchronous Motor for High-Performance Direct-Drive Applications” IEEE Transaction on Power Electronics, May 1998, Vol. 12, No. 3.

37. Ch. French, P. Acarnley, “Direct Torque Control of Permanent Magnet Drivers”, IEEE Transaction on Industrial Application, Vol. 32, No. 5, 1996, pp. 1080-1088.

38. L.-H. Hoang, “Comparison of field-oriented control and direct torque control for induction motor drives”, Industry Applications Conference, 1999. Thirty-Fourth IAS Annual Meeting, 3-7 Oct. 1999, Vol. 2, pp.1245 – 1252.

References

175

39. J.-I. Itoh, N. Nomura, H. Ohsawa, “A comparison between V/f control and position-sensorless vector control for the permanent magnet synchronous motor”, Proceedings of the Power Conversion Conference, PCC Osaka 2-5 April 2002, Vol. 3, pp. 1310 – 1315.

40. T.M. Jahns, “Motion control with permanent-magnet AC machines”, Proceedings of the IEEE, Aug. 1994, Vol. 82, pp. 1241 – 1252.

41. C. Lascu, I. Boldea, F. Blaabjerg, “Variable-structure direct torque control - a class of fast and robust controllers for induction machine drives”, IEEE Transactions on Industrial Electronics, Aug. 2004,Vol. 51, pp. 785 – 792.

42. T. Lixin; Z. Limin; M.F. Rahman, Y. Hu, “A novel direct torque control for interior permanent-magnet synchronous machine drive with low ripple in torque and flux-a speed-sensorless approach”, IEEE Transactions on Industry Applications, Nov.-Dec. 2003,Vol. 39, pp. 1748 – 1756.

43. J. Lee; Ch.-G. Kim; M.-J. Youn, “A dead-beat type digital controller for the direct torque control of an induction motor”, IEEE Transactions on Power Electronics, Sept. 2002, Vol. 17, pp. 739 – 746.

44. H. Rasmussen, P. Vadstrup, H. Børsting, "Adaptive sensorless field oriented control of PM motors including zero speed", IEEE International Symposium on Industrial Electronics, Ajaccio, France, May 2004.

45. M. F. Rahman, L. Zhong, K. W. Lim, “A direct torque controlled interior permanent magnet synchronous motor drive incorporating field weakening”, IEEE-IAS’97, 1997 , p. 67 - 74.

46. M.F. Rahman, L. Zhong, K.W. Lim, “An Investigation of Direct and Indirect Torque Conttrollers for PM Synchronous Motor Drivers”, Conference PEDS’97, May, Singapure, pp. 519-523.

47. M. F. Rahman, L. Zhong, “Comparison of torque response of the interior permanent magnet motor under PWM current and direct torque control”, IEEE, 1999 , p. 1464 - 1470.

48. J. Rodriguez, J. Pontt, C. Silva, S. Kouro, H. Miranda, “A novel direct torque control scheme for induction machines with space vector modulation”; Power Electronics Specialists Conference, PESC’04, 20-25 June 2004, Vol. 2, pp. 1392 – 1397.

49. D. Telford, M.W. Dunnigan, B.W. Williams, “A comparison of vector control and direct torque control of an induction machine”, 2000 IEEE 31st Annual Power Electronics Specialists Conference, 2000. PESC 00, 18-23 June 2000, Vol. 1, pp. 421 – 426.

50. L. Takahashi, T. Noguchi, “A new quick response and high efficiency strategy of induction motor”, IAS, 1985 , p. 495-502.

51. L. Tang; L. Zhong; F. Rahman, “Modeling and experimental approach of a novel direct torque control scheme for interior permanent magnet synchronous machine drive”, IECON 02, 5-8 Nov. 2002, Vol. 1, pp. 235 – 240.

52. M.N. Uddin, T.S. Radwan, G.H. George, M.A. Rahman, “Performance of current controllers for VSI-fed IPMSM drive”, IEEE Transactions on Industry Applications, Volume 36, Issue 6, Nov.-Dec. 2000 Page(s):1531 – 1538.

53. L. Xu, M. Fu “A Novel Sensorless Control Technique for Permanent Magnet Synchronous Motors (PMSM) using Digital Signal Processor (DSP)”, NEACON’97, Dayton, Ohio, July 14-17, 1997.

54. L. Xu, M. Fu “A Sensorless Direct Torque Control Technique for Permanent Magnet Synchronous Motors”, IEEE Industrial Applications Conferences,1999, Vol. 1.

55. Y. Xue, X. Xu, T.G. Habetler, D.M. Divan, “A low cost stator flux oriented voltage source variable speed drive”, Industry Applications Society Annual Meeting, 7-12 Oct. 1990, Vol.1, pp. 410 – 415.

56. H. Yuwen,T. Cun, G. Yikang,Y. Zhiqing, L.X. Tang, M.F. Rahman “In-depth Reserch on Direct Torque Control of Permanent Magnet Synchronous Motor”, IEEE Conference IECON 2002 ,November, Sevilla, Spain.

References

176

57. L. Zhong, M.F. Rahman, W.Y. Hu, Lim, K.W., Rahman, M.A., “A direct torque controller for permanent magnet synchronous motor drives”, IEEE Transactions on Energy Conversion, Sept. 1999, Vol. 14, pp. 637 – 642.

58. L. Zhong, M.F. Rahman, W.Y.Hu, K.W. Lim, “Analysis of Direct Torque Control in Permanent Magnet Synchronous Motor Drives”, IEEE Transaction on Power Electronics, Vol. 12, No. 13, May, 1997.

59. L. Zhong, M.F. Rahman, „Voltage Switching Tables for Controlled Interior Permanent Magnet Motor“, IEEE Conference IECON’99, Vol. 3, pp. 1445-1451.

60. L.Zhong, M. F. Rahman, “A direct torque controller for permanent magnet synchronous motor drives without a speed sensor”, IEEE Trans. On Energy Conversion, Vol.14, No.3 ,September , 1999.

61. M.R Zolghardi, D.Diallo, D.Roye, “Direct Torque Control System for Synchronous Machine”, Europen Conference on Power Electronics and Application, EPE’97,Trondheim, Norway,Vol. 3, pp. 694-699.

62. M.R. Zolghardi, E. M. Olasagasti, D. Roye, “Steady State Torque Correction of Direct Torque Controlled PM Synchronous Machine”, IEEE International Conference on Electric Machines Drivers, 1997.

63. M.R Zolghardi, J.Guiraud, J.Davoine, D. Roye, “A DSP Direct Torque Controller for Permanent Magnet Synchronous Motor Drivers”, IEEE Power Electronics Conference, PESC’98, Vol.2, pp. 2055-2061.

64. M.R Zolghardi, D. Roye, “Sensorless Direct Torque Control of Synchronous Motor Drive” Proceedings of International Conference on Electrical Machines ICEM’ 98, Vol. 2, 1998, Istanbul, Turkey.

65. M.R Zolghardi, D. Diallo, D. Roye, “Start up Strategies for Direct Torque Controlled Synchronous Machine ”, Europen Conference on Power Electronics and Application, EPE’97,Trondheim, Norway,Vol. 3, pp. 689-693.

66. M.R Zolghardi, C. Pelissu, D. Roye, “Start up of a global Direct Torque Control System”, IEEE Power Electronic Conference PESC’96, Vol. 1, pp. 370-374.

Sensorless Control and Initial Position Estimation

67. S. Bolognani, R. Oboe, M. Zigliotto, “Sensorless full-digital PMSM drive with EKF

estimation of speed and rotor position”, IEEE Transactions on Industrial Electronics, Vol. 46, Issue 1, Feb. 1999, pp. 184 – 191.

68. S. Brock, J. Deskur, K. Zawirski, , “Robust speed and position control of PMSM”, Proceedings of the IEEE International Symposium on Industrial Electronics, 1999, ISIE '99. 12-16 July 1999, Vol. 2, pp. 667 – 672.

69. M.J. Corley, R.D. Lorenz, “Rotor position and velocity estimation for a salient-pole permanent magnet synchronous machine at standstill and high speeds”, IEEE Transactions on Industry Applications, July-Aug. 1998, Vol. 34, pp. 784 – 789.

70. Z. Chen, M. Tomita, S. Ichikawa, S. Doki, S. Okuma, “Sensorless control of interior permanent magnet synchronous motor by estimation of an extended electromotive force”; Industry Applications Conference, 2000, Vol. 3, 8-12 Oct. 2000 pp. 1814 – 1819.

71. Z. Chen, M. Tomita, S. Doki, S. Okuma, “An extended electromotive force model for sensorless control of interior permanent-magnet synchronous motors”, IEEE Transactions on Industrial Electronics, Vol. 50, Issue 2, April 2003, pp. 288 – 295.

72. Z. Chen, M. Tomita, S. Doki, S. Okuma, “New adaptive sliding observers for position- and velocity-sensorless controls of brushless DC motors”, IEEE Transactions on Industrial Electronics, Vol. 47, Issue 3, June 2000, pp. 582 – 591.

73. R. Dhaouadi, N. Mohan, L. Norum, “Design and implementation of an extended Kalman filter for the state estimation of a permanent magnet synchronous motor”, IEEE Transactions on Power Electronics, Vol. 6, Issue 3, July 1991, pp. 491 – 497.

References

177

74. N. Ertugrul, P. Acarnley, “A new algorithm for sensorless operation of permanent magnet motors”; IEEE Transactions on Industry Applications, Jan.-Feb. 1994, Vol. 30, pp. 126 – 133.

75. C. French, P. Acarnley, I. Al-Bahadly, “Sensorless position control of permanent magnet drives”; Industry Applications Conference, IAS '95, 8-12 Oct. 1995, Vol. 1, pp. 61 – 68.

76. M. Fu, L. Xu, “A novel sensorless control technique for permanent magnet synchronous motor (PMSM) using digital signal processor (DSP)”, Proceedings of the IEEE 1997 National Aerospace and Electronics Conference, 14-17 July 1997, NAECON 1997,Vol. 1, pp. 403 – 408.

77. J. Hu, B. Wu, “New integration algorithms for estimating motor flux over a wide speed range”, IEEE Transactions on Power Electronics, , Sept. 1998, Vol. 13, pp. 969 – 977.

78. Y. Jeong, R.D. Lorenz, T.M. Jahns, S. Sul, “Initial rotor position estimation of an interior permanent magnet synchronous machine using carrier-frequency injection methods”, IEEE International Electric Machines and Drives Conference, IEMDC'03, 1-4 June 2003, Vol. 2,pp. 1218 – 1223.

79. L.A. Jones, J.H. Lang, “A state observer for the permanent-magnet synchronous motor”, IEEE Transactions on Industrial Electronics, Vol. 36, Issue 3, Aug. 1989 pp. 374 – 382.

80. J.P. Johnson, M. Ehsani, Y. Guzelgunler, “Review of sensorless methods for brushless DC”, Industry Applications Conference, 1999. Thirty-Fourth IAS Annual Meeting. Conference Record of the 1999 IEEE, 3-7 Oct. 1999, Vol. 1, pp. 143 – 150.

81. J. S. Kim; S. K. Sul, “New stand-still position detection strategy for PMSM drive without rotational transducers”, Applied Power Electronics Conference and Exposition, APEC '94, 13-17 Feb. 1994, Vol.1, pp. 363 – 369.

82. J.-S. Kim, S.-K. Sul, “New approach for high-performance PMSM drives without rotational position sensors”, IEEE Transactions on Power Electronics, Sept. 1997, Vol. 12, pp. 904 – 911.

83. T.-S. Low, T.-H. Lee, K.-T. Chang, “A nonlinear speed observer for permanent-magnet synchronous motors”, IEEE Transactions on Industrial Electronics, Vol. 40, Issue 3, June 1993, pp. 307 – 316.

84. M. Linke, R. Kennel, J. Holtz, “Sensorless position control of permanent magnet synchronous machines without limitation at zero speed”, IECON 02, 5-8 Nov. 2002, Vol. 1, pp. 674 – 679.

85. N. Matsui, “Sensorless PM brushless DC motor drives”, Transactions on Industrial Electronics, IEEE , Vol. 43, April 1996, pp. 300 – 308.

86. N. Matsui, M. Shigyo, “Brushless DC motor control without position and speed sensors”; IEEE Transactions on Industry Applications, Jan.-Feb. 1992, Vol. 28 pp. 120 – 127.

87. N. Matsui, “Sensorless PM brushless DC motor drives”, IEEE Transactions on Industrial Electronics, April 1996, Vol. 43, pp. 300 – 308.

88. S. Nakashima, Y. Inagaki, I. Miki, “Sensorless initial rotor position estimation of surface permanent-magnet synchronous motor”, IEEE Transactions on Industry Applications, Nov.-Dec. 2000, Vol. 36, pp. 1598 – 1603.

89. T. Noguchi, K. Yamada, S. Kondo, I. Takahashi, “Rotor position estimation method of sensorless PM motor at rest with no sensitivity to armature resistance”, Industrial Electronics, Control, and Instrumentation, Proceedings of the 1996 IEEE IECON 1996, 5-10 Aug. 1996,Vol. 2, pp. 1171 – 1176.

90. S. Ostlund, M. Brokemper, “Initial rotor position detections for an integrated PM synchronous motor drive”, Industry Applications Conference, IAS '95, 8-12 Oct. 1995, Vol. 1, pp. 741 – 747.

91. G. Pesse, T. Paga, “A permanent magnet synchronous motor flux control scheme without position sensor”, Conference proceedings on the EPE’97, Trondheim 1997, Vol. 4, pp. 553 – 556.

References

178

92. R.B. Sepe, J.H. Lang, “Real-time observer-based (adaptive) control of a permanent-magnet synchronous motor without mechanical sensors”, IEEE Transactions on Industry Applications, Vol. 28, Issue 6, Nov.-Dec. 1992, pp. 1345 – 1352.

93. P.B. Schmidt, M.L. Gasperi, G. Ray, A.H. Wijenayake, “Initial rotor angle detection of a nonsalient pole permanent magnet synchronous machine”, Industry Applications Conference, IAS '97., 5-9 Oct. 1997, Vol. 1, pp. 459 – 463.

94. J.X. Shen, Z.Q. Zhu, D. Howe, “Improved speed estimation in sensorless PM brushless AC drives”, Transactions on Industry Applications, July-Aug. 2002, Vol. 38, pp. 1072 – 1080.

95. Yen-Shin Lai; Fu-San Shyu; Shian Shau Tseng, “New initial position detection technique for three-phase brushless DC motor without position and current sensors”, Transactions on Industry Applications, , March-April 2003, Vol. 39, pp. 485 – 491.

96. J. Solsona, M.I. Valla, C. Muravchik, “A nonlinear reduced order observer for permanent magnet synchronous motors”, IEEE Transactions on Industrial Electronics, Aug. 1996, Vol. 43, pp. 492 - 497

97. J.A. Solsona, M.I. Valla, “Disturbance and nonlinear Luenberger observers for estimating mechanical variables in permanent magnet synchronous motors under mechanical parameters uncertainties”, IEEE Transactions on Industrial Electronics, Vol. 50, Issue 4, Aug. 2003 pp. 717 – 725.

98. J. Solsona, M.I. Valla, C. Muravchik, “Nonlinear control of a permanent magnet synchronous motor with disturbance torque estimation”, International Conference on Industrial Electronics, Control and Instrumentation, 1997. IECON 97, Vol. 1, 9-14 Nov. 1997, pp. 120 – 125.

99. T. Takeshita, N. Matsui, “Sensorless control and initial position estimation of salient-pole brushless DC motor”, Advanced Motion Control, 1996. AMC '96, 18-21 March 1996, Vol. 1, pp. 18 – 23.

100. M. Tomita, T. Senjyu, S. Doki, S. Okuma, “New sensorless control for brushless DC motors using disturbance observers and adaptive velocity estimations”, IEEE Transactions on Industrial Electronics, Vol. 45, Issue 2, April 1998, pp. 274 – 282.

101. M. Tursini, R. Petrella, F. Parasiliti, “Initial rotor position estimation method for PM motors”; IEEE Transactions on Industry Applications, Nov.-Dec. 2003, Vol. 39, pp. 1630 – 1640.

102. R. Wu, G.R. Slemon, “A permanent magnet motor drive without a shaft sensor”, Industry Applications Society Annual Meeting, 1990., Conference Record of the 1990 IEEE, 7-12 Oct. 1990, Vol.1, pp. 553 – 558.

103. D. Yousfi, M. Azizi, A. Saad, “Sensorless position and speed detection for permanent magnet synchronous motor”; Power Electronics and Motion Control Conference, PIEMC 2000, 15-18 Aug. 2000,Vol. 3, pp. 1224 – 1229.

104. D. Yousfi, M. Azizi, A. Saad, “Robust position and speed estimation algorithm for permanent magnet synchronous drives”, Industry Applications Conference, 8-12 Oct. 2000, Vol.3, pp. 1541 – 1546.

Others

105. H.-B. Shin, “New antiwindup PI controller for variable-speed motor drives”

IEEE Transactions on Industrial Electronics, June 1998, Vol. 45, pp. 445 – 450. 106. MathWorks, Inc, "Matlab® The Language of Technical Computing", Release 12,

2000.

Papers written during work on this thesis

107. D. L. Sobczuk, D. Świerczyński "DSP implementation of vector controlled inverter fed induction motor with neural network speed estimator" SENE '99 tom. 2 str. 601-606.

References

179

108. D. Świerczyński, Marian P. Kaźmierkowski, Paweł Z. Grabowski "Nowe algorytmy estymacji strumienia stojana w silnikach indukcyjnych klatkowych zasilanych z falowników PWM" SENE '99 tom. 2 str. 649-654.

109. M. Angielczyk, M. Malinowski, D. Świerczyński, „Stanowisko laboratoryjne oparte o kartę DSP do sterowania prostownikiem PWM oraz silnikiem PMSM zasilanym z falownika PWM”, Kamień Śląski 2000, Międzynarodowe XII Sympozjum Mikromaszyny i Serwonapędy, str. 480-487.

110. D. Świerczyński, M. P. Kaźmierkowski.:„Bezpośrednie sterowanie momentu silnika synchronicznego o magnesach trwałych zasilanym z falownika PWM”, SENE 2001 ,tom II, str. 629

111. D. Świerczyński, M. P. Kaźmierkowski."Direct torque control of permanent magnet synchronous motor (PMSM) using space vector modulation (DTC-SVM)", SME'2002, pp. 245

112. D. Świerczyński, M. P. Kaźmierkowski, Frede Blaabjerg "DSP Based Direct Torque Control of Permanent Magnet Synchronous Motor (PMSM) Using Space Vector Modulation (DTC-SVM)", IEEE-ISIE 2002, L’Aquila, Italy, pp. 723-727.

113. D. Świerczyński, M. Żelechowski "Studia symulacyjne nad universalną strukturą bezpośredniego sterowania momentem i strumieniem dla silników synchronicznych o magnesach trwałych oraz silników asynchronicznych ", Modelowania i Symulacja MiS 2002, Zakopane.

114. D. Świerczyński, M. P. Kaźmierkowski, Frede Blaabjerg "Direct Torque Control with Space Vector Modulation (DTC-SVM) for Permanent Magnet Synchronous Motor (PMSM)", EPE-PEMC 2002, Dubrovnik & Cavtat, Croatia.

115. D. Świerczyński, M. Żelechowski "Universalną strukturą bezpośredniego sterowania momentem i strumieniem dla silników synchronicznych o magnesach trwałych oraz silników asynchronicznych ", Mikromaszyny i Serwonapędy MiS 2002, Krasiczyn.

116. D. Świerczyński, M. P. Kaźmierkowski " Direct Torque Control of Permanent Magnet Synchronous Motor (PMSM) Using Space Vector Modulation (DTC-SVM) – Simulation and Experimental Results", IECON 2002, Sevilla, Spain.

117. D.Świerczyński, M. Żelechowski „Bezpośrednie sterowanie momentem i strumieniem dla silników synchronicznych i asynchronicznych”, II Krajowa Konferencja MiS-2 Modelowanie i Symulacja, Kościelisko, 24-28 czerwca 2002, pp. 187-194.

118. D. Świerczyński, M. Żelechowski ”Universal structure of direct torque control for AC motor drives”, III Summer Seminar on Nordick Network for Multi Disciplinary Optimised Electric Drives, June 21-23, 2003, Zegrze, Poland

119. M. Jasiński D. Świerczyński, M. P. Kazmierkowski, M. Malinowski ”Direct Control of an AC-DC-AC converter ”, III Summer Seminar on Nordick Network for Multi Disciplinary Optimised Electric Drives, June 21-23, 2003, Zegrze, Poland

120. M. Żelechowski, D. Świerczyński, M. P. Kazmierkowski, J. Załęski „Uniwersalne napędy falownikowe ze sterowaniem mikroporcesorowym nowej generacji”, Elektro.info Nr 6 (17) 2003, str. 26-28

121. D. Świerczyński, M. Żelechowski, „Universal Structure of Direct Torque Control for AC Motor Drives”, V Power Electronics Seminar, 7-8 July, 2003, Berlin, Niemcy

122. M. Jasinski, D. Swierczynski, M. P. Kazmierkowski "Novel Sensorless Direct Power and Torque Control of Space Vector Modulated AC/DC/AC Converter", In Proc. of the ISIE 2004 Conf., 4-7 May, Ajaccio, France, pp.1147- 1152.

123. D. Świerczyński, M. Żelechowski ”Universal structure of direct torque control for AC motor drives”, Przegląd Elektrotechniczny, May, 2004, Poland, pp. 489-492.

124. M. P. Kaźmierkowski , M. Żelechowski, D. Świerczyński „Simple DTC-SVM Control for Induction and PM Synchronous Motor”, In Proc. of the ICEM 2004 Conf., 5-8 September, Krakow, Poland, on CD.

125. M. P. Kaźmierkowski , M. Żelechowski, D. Świerczyński „DTC-SVM an Efficient Method for Control Both Induction and PM Synchronous Motor”, In Proc. of the EPE-PEMC 2004 Conf., 2-4 September, Riga, Latvia, on CD.

References

180

126. B. W. Mehdawi, D. Świerczyński, M. P. Kaźmierkowski „Design and Investigation of Vector Control Scheme PWM Inverter-fed Nonsalient Permanent Magnet Synchronous Motor”, In Proc. of the PPEE 2005 Conf., 2-5 April, Poland, Wisla, on CD.

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Direct Torque Control with Space Vector Modulation (DTC-SVM) of