a power control scheme of a medium frequency induction...
TRANSCRIPT
A Power Control Scheme of a Medium Frequency
Induction Furnace
Submitted by:
Muhammad Nawaz
2009-PhD-Elect-05
Supervised by:
Dr. Muhammad Asghar Saqib
Department of Electrical Engineering
University of Engineering and Technology
Lahore
2017
i
A Power Control Scheme of a Medium Frequency
Induction Furnace
Submitted to the faculty of the Electrical Engineering Department of the
University of Engineering and Technology Lahore
In partial fulfillment of the requirements for the Degree of
Doctor of Philosophy
in
Electrical Engineering
Submitted by
Muhammad Nawaz 2009-PhD-Elect-05
Thesis approved on 11-08-2017
Dr. Muhammad Asghar Saqib
(Internal Examiner)
Dr. Nisar Ahmed
(External Examiner)
Dr. Intesar Ahmed
(External Examiner)
(Dr. K. M Hasan)
For Chairman
Department of Electrical Engineering
(Dr. Suhail Aftab Qureshi)
Dean
Faculty of Electrical Engineering
Department of Electrical Engineering
University of Engineering and Technology, Lahore
August-2017
ii
a. From within the Country
i) Prof. Dr. Nisar Ahmed,
Dean, Faculty of Electrical Engineering,
GIK Institute of Engineering Sciences and Technology,
Topi 23640, District Swabi, Khyber Pakhtunkhwa.
E-mail:[email protected]
ii) Prof. Dr. Intesar Ahmed,
Department of Electrical Engineering,
Lahore College for Women University, Lahore.
E-mail:[email protected]
b. From Abroad
i) Prof. Dr.-Ing. Bernhard Arndt,
Faculty of Electrical Engineering,
University of Applied Sciences FHWS,
Würzburg-Schweinfurt, Germany.
E-mail: [email protected]
ii) Dr. Kamran Iqbal,
Professor, University of Arkansas at Little Rock,
Systems Engineering Department.
EIT 518, 2801 South University Avenue,
Little Rock, AR 72204-1099, USA.
E-mail:[email protected]
iii) Dr. Mahmood Nagrial,
Associate Professor, Engineering and Construction Management,
Western Sydney University, Locked Bag 1797,
Penrith NSW 2751, Australia.
E-mail:[email protected]
iii
Declaration
I hereby declare that the work contained in this thesis is my own, except where explicitly stated
otherwise. Moreover, this work has not been submitted to obtain another degree or professional
qualification.
Signed: ________________
Date: _________________
iv
Acknowledgments
I bow my head in gratitude to ALMIGHTY ALLAH whose blessings and exaltations flourished
my childhood wish, who enabled me in the accomplishment of this task and Who is the source of
all knowledge and wisdom. Countless thanks to ALLAH (S.W.T), The Beneficent, The Merciful
for it is under His grace that we live, learn, prosper and flourish.
Blessings and salutation on PROPHET MUHAMMAD (S.A.W) who is forever a torch of
knowledge and tower of guidance to humanity.
It is a matter of utmost pleasure for me to extend my gratitude and give due credit to my supervisor
Dr. Muhammad Asghar Saqib whose support has always been there in time of need. Without his
valuable suggestions and immense encouragement it would not have been possible for me to attain
this goal. In spite of his extremely busy schedule, he always spared his precious time to discuss
the research results, review of papers and directed me to improve the research writing skills.
Almighty Allah may bless him with good health.
I would also like to pay my appreciation to Dr. Syed Abdul Rehman Kashif, Assistant Professor,
Electrical Engineering Department, for his technical help in form of system modelling and
guidance to enhance the quality of research.
My acknowledgement also goes to one of highly genius Engineer Mr. Mehr Hussain at CPC for
his enormous contribution in form of practical and theoretical knowledge during this research
work. My appreciations are also due to all my seniors especially Mr. Muhammad Mubashar Amin
Manager (Electrical Development) at CPC for their remarkable co-operation during this project.
Most importantly, all my prayers are for my late parents, no doubt they nourished me with love
and care. Almighty Allah bless them. I am also deeply indebted to my entire family who missed
me a lot and always encouraged me to achieve this goal. Thank you very much.
Finally, I am also obliged to Higher Education Commission (HEC), Government of Pakistan for
providing me the financial support to carry out this research activity.
v
This work is dedicated to my honourable parents who are not with me today
vi
Table of Contents
Declaration .................................................................................................................................. iii
Acknowledgments .................................................................................................................... iv
Table of Contents ........................................................................................................................... vi
List of Figures ............................................................................................................................ ix
List of Tables ............................................................................................................................ xii
List of Acronyms ......................................................................................................................... xiii
Abstract ............................................................................................................................... xviii
Chapter 1 Introduction .............................................................................................................. 1
1.1 Overview .......................................................................................................................... 1
1.2 Historical Background...................................................................................................... 1
1.3 Induction Heating Present-Day Structure ........................................................................ 2
1.4 Control Techniques .......................................................................................................... 3
1.5 Thesis Outline .................................................................................................................. 5
Chapter 2 Literature Review .................................................................................................... 6
2.1 Overview .......................................................................................................................... 6
2.2 Induction Heating Control Strategies ............................................................................... 6
2.3 Problem Statement ......................................................................................................... 12
2.3.1 Space-vector PWM based PI control ...................................................................... 12
2.3.2 Model predictive control ......................................................................................... 12
Chapter 3 Medium-Frequency Resonant Inverter ............................................................... 14
3.1 Overview ........................................................................................................................ 14
3.2 Resonance Frequency ..................................................................................................... 14
3.3 Inverter Topologies ........................................................................................................ 15
3.3.1 Quarter-bridge inverter ........................................................................................... 15
3.3.2 Half-bridge inverter ................................................................................................ 15
3.3.3 Full-bridge inverter ................................................................................................. 16
3.4 Resonant Tank Circuit .................................................................................................... 17
3.5 Parallel-Resonant Inverter: Modeling and Analysis ...................................................... 19
3.5.1 Quality factor .......................................................................................................... 21
3.5.2 Power analysis ........................................................................................................ 23
3.6 Soft Switching ................................................................................................................ 24
Chapter 4 Current-Source Converter in Induction Heating: Theory and Modeling ........ 26
4.1 Overview ........................................................................................................................ 26
4.2 Structure of Induction Furnace ....................................................................................... 26
vii
4.3 Front-End Rectifier ........................................................................................................ 27
4.3.1 Uncontrolled rectifier .............................................................................................. 27
4.3.2 Controlled rectifier .................................................................................................. 28
4.4 Constant-Source Topologies (CST) ............................................................................... 28
4.4.1 Voltage-source converter (VSC) ............................................................................. 29
4.4.2 Current-source converter (CSC) ............................................................................. 29
4.5 Structure of a Current-Source Converter Feeding an Induction Heating Load.............. 31
4.6 Mathematical Description .............................................................................................. 33
4.6.1 CSC model in abc- reference frame ........................................................................ 34
4.6.2 CSC model in αβ- reference frame ......................................................................... 35
4.6.3 CSC model in dq- reference frame ......................................................................... 36
4.6.4 Model decoupling and linearization ........................................................................ 40
4.7 Power Analysis ............................................................................................................... 41
Chapter 5 Field Oriented Control .......................................................................................... 43
5.1 Overview ........................................................................................................................ 43
5.2 Line-Commutated Rectifier............................................................................................ 43
5.3 Forced-Commutated Rectifier ........................................................................................ 44
5.3.1 Pulse width modulation (PWM) based control ....................................................... 46
5.3.2 Sinusoidal pulse width modulation (SPWM) based control ................................... 46
5.3.3 Space-vector based PWM control ........................................................................... 47
5.4 Space Vector PWM Based PI Control ........................................................................... 52
5.5 Load Model .................................................................................................................... 53
Chapter 6 Model Predictive Control (MPC) ......................................................................... 55
6.1 Overview ........................................................................................................................ 55
6.2 Predictive and Non-Predictive Controllers .................................................................... 55
6.3 Mathematical Model ...................................................................................................... 57
6.3.1 Linear continuous time state-space model .............................................................. 58
6.3.2 Discrete-time state-space model ............................................................................. 59
6.3.3 System mathematical model ................................................................................... 59
6.4 Linear Quadratic Regulator (LQR) ................................................................................ 61
6.5 Model Predictive Control ............................................................................................... 62
6.6 Generalized Predictive Control (GPC) ........................................................................... 65
6.6.1 Model prediction ..................................................................................................... 66
6.6.2 Cost function ........................................................................................................... 68
6.6.3 System constraints .................................................................................................. 71
6.6.4 Constraints benefits ................................................................................................. 72
6.6.5 Significant features ................................................................................................. 73
Chapter 7 Control Algorithms: Results and Discussions ..................................................... 74
7.1 Overview ........................................................................................................................ 74
viii
7.2 Dynamic Behaviour of a Current-Fed Inverter .............................................................. 74
7.3 SVPWM-PI Based DC-link Current Control ................................................................. 83
7.3.1 Fixed load................................................................................................................ 83
7.3.2 Dynamic load .......................................................................................................... 87
7.4 GPC based DC-Link Current Control ............................................................................ 91
7.5 Variation in GPC’s Parameters ...................................................................................... 96
Chapter 8 Conclusions ........................................................................................................... 100
8.1 Conclusions .................................................................................................................. 100
8.2 Future Works ................................................................................................................ 101
References ................................................................................................................................ 102
Appendix-A ................................................................................................................................ 121
A.1 abc-Reference Frames into dq-Coordinates ................................................................. 121
A.2 dq-Coordinates into abc-Reference Frame ................................................................... 123
Appendix-B ................................................................................................................................ 125
B.1 Parallel Resonant Load Model ..................................................................................... 125
B.2 CSC with Equivalent Load Model ............................................................................... 126
Appendix-C ................................................................................................................................ 128
C.1 Switching Devices ........................................................................................................ 128
C.1.1 Thyristor ................................................................................................................ 128
C.1.2 Transistors ............................................................................................................. 129
Research Papers ........................................................................................................................ 131
ix
List of Figures
Figure 1.1: Basic block diagram of an induction heating system. .................................................. 2
Figure 1.2: Circuit configurations: a) series resonant load, b) parallel resonant load. ................... 3
Figure 1.3: Control techniques in induction heating. ...................................................................... 4
Figure 3.1: Quarter bridge inverter. .............................................................................................. 15
Figure 3.2: Half bridge inverter. ................................................................................................... 16
Figure 3.3: Current-fed full bridge inverter. ................................................................................. 17
Figure 3.4: Parallel-resonant load a) Circuit diagram, b) Output waveforms. ............................. 17
Figure 3.5: Series-resonant load a) Circuit diagram, b) Output waveforms. ................................ 18
Figure 3.6: Basic scheme of a current-fed resonant inverter. ....................................................... 19
Figure 3.7: Equivalent model of a CFI. ........................................................................................ 20
Figure 3.8: Phase angle control scheme. ....................................................................................... 25
Figure 4.1: A basic diagram of induction furnace structure. ........................................................ 27
Figure 4.2: Uncontrolled rectifier. ................................................................................................ 28
Figure 4.3: Thyristor based controlled rectifier. ........................................................................... 28
Figure 4.4: a) Voltage-source converter, b) Current source converter. ........................................ 30
Figure 4.5: A current source converter feeding an induction-heating load. ................................. 31
Figure 4.6: Equivalent circuit of coil and work piece ................................................................... 32
Figure 4.7: Equivalent model of CSC with induction heating load. ............................................. 34
Figure 4.8: Clark-transformation in graphical form. .................................................................... 36
Figure 4.9: Park-transformation. ................................................................................................... 37
Figure 5.1: Phase angle control of a controlled rectifier feeding an induction heating load ........ 44
Figure 5.2: PWM based voltage-source converter. ....................................................................... 45
Figure 5.3: PWM based current-source converter. ....................................................................... 46
Figure 5.4: The switching states diagram of a CSC...................................................................... 49
Figure 5.5: Switched vectors and sectors for a CSC. .................................................................... 50
Figure 5.6: Reference vector Iref in sector i. .................................................................................. 51
Figure 5.7: SVPWM based control design for a current-source converter (CSC)........................ 52
Figure 5.8: PI based DC-link current with SVPWM. ................................................................... 53
Figure 5.9: CSC with a constant heating load............................................................................... 54
Figure 5.10: Dynamic heating load. .............................................................................................. 54
Figure 6.1: Block diagram of a) non-predictive controller b) predictive controller ..................... 57
Figure 6.2: Fundamental structure of a liner quadratic regulator (LQR). ..................................... 62
Figure 6.3: Model predictive control a) Functioning pattern, b) Control structure ...................... 64
Figure 6.4: General block diagram of generalized predictive control. ......................................... 65
Figure 6.5: Generalized predictive controller without constraints. ............................................... 71
Figure 7.1: Inverter input current. ................................................................................................. 75
Figure 7.2: Output voltages a) Transient state, b) Steady state. ................................................... 76
Figure 7.3: Effective load voltages a) Transient state, b) Steady state ......................................... 77
Figure 7.4: Output voltages and load current with phase angles a) In phase, b) Leading. ........... 78
x
Figure 7.5: Coil current a) Transient state, b) Steady state. .......................................................... 79
Figure 7.6: Capacitor current response a) Transient state, b) Steady state. .................................. 80
Figure 7.7: Output voltage and load current with different quality factors a) Qf = 7, b) Qf = 8,
c) Qf = 9, d) Qf = 10.48. ................................................................................................................ 82
Figure 7.8: Rectifier output a) DC current, b) DC voltage. .......................................................... 84
Figure 7.9: (a) Three-phase line currents, b) Three-phase line voltages. ..................................... 84
Figure 7.10: DC current and reference. ........................................................................................ 85
Figure 7.11: DC voltage. ............................................................................................................... 85
Figure 7.12: Current-fed resonant inverter a) Output voltage and current, b) Active component of
output voltage, c) Reactive component of output voltage, d) Resonant current flowing through the
coil................................................................................................................................................. 87
Figure 7.13: Load dynamic model. ............................................................................................... 88
Figure 7.14: a) DC current, b) DC Voltage with change in reference at t = 0.4 s and t = 0.7 s. .. 89
Figure 7.15: Three-phase line current when reference changes at t = 0.4 s. ................................. 89
Figure 7.16: a) DC current and b) DC voltage with a constant load. ........................................... 90
Figure 7.17: DC a) Current and b) Voltage where load changes at t= 0.3 s and t= 0.5 s. ............ 90
Figure 7.18: a) Tracking of DC current reference b) Tracking of reactive component of supply
current, c) Response of direct-axis current Icd at input, d) Response of quadrature-axis current Icq
at input. ......................................................................................................................................... 92
Figure 7.19: Error signals at peak load in a) DC current, b) Reactive component of supply
current. .......................................................................................................................................... 92
Figure 7.20: Tracking of references in defined constraints at output a) DC current, b) Reactive
component of supply current. ....................................................................................................... 93
Figure 7.21: Control response under input constraints a) Input current, b) DC current. .............. 94
Figure 7.22: Output and Input variables without constraints a) DC current, b) Reactive component
of supply current, c) Direct-axis component of control signal Icd, d) Quadrature-axis component
of control signal Icq. ..................................................................................................................... 95
Figure 7.23: a) Magnitude of control signal ................................................................................. 95
Figure 7.24: System response with a large sampling time a) DC current, b) Reactive component of
supply current, c) Direct-axis component of control signal Icd, d) Quadrature-axis component of
control signal Icq. ........................................................................................................................... 97
Figure 7.25: Impact of weight factor a) DC current, b) Reactive component of supply current, c)
Direct-axis component of control signal Icd, d) Quadrature-axis component of control
signal Icq. ....................................................................................................................................... 98
Figure 7.26: Variation in power’s level a) DC current, b) Reactive component of supply current,
c) Direct-axis component of control signal Icd, d) Quadrature-axis component of control
signal Icq. ....................................................................................................................................... 99
Figure A.1.1: Simulation of abc-reference frame into αβ- transformation. ................................ 121
Figure A.1.2: Simulation of αβ -reference frame into dq- transformation. ................................ 121
Figure A.1.3: Simulation of abc -reference frame into dq- transformation. ............................... 122
xi
Figure A.2.1: Simulation of dq-reference frame into αβ - transformation. ................................ 123
Figure A.2.2: Simulation of αβ -reference frame into abc- transformation. ............................... 123
Figure A.2.3: Simulation of dq -reference frame into abc- transformation. ............................... 124
Figure B.1.1: Simulation model of the parallel resonant load. ................................................... 125
Figure B.2.1: Simulation model of the non-linear model of CSC with equivalent load............. 126
Figure B.2.2: Simulation model of the linearized model of CSC with equivalent load. ............ 127
Figure C.1.1: a) Silicon controlled rectifier, b) Metal oxide semiconductor field effect transistor,
c) Insulated gate bipolar transistor. ............................................................................................. 129
xii
List of Tables
Table 3.1: Characteristics of bridge resonant inverter. ................................................................. 18
Table 5.1: Switching states. .......................................................................................................... 48
Table 7.1: Parallel resonant load circuit parameters. .................................................................... 75
Table 7.2: Summary of obtained results at inverter output. .......................................................... 82
Table 7.3: System parameters. ...................................................................................................... 83
Table 7.4: PI controller tuning parameters. .................................................................................. 88
Table C.1: Frequency and power ranges of switching devices in induction heating inverters. .. 130
xiii
List of Acronyms
IH Induction heating
AC Alternating current
DC Direct current
PWM Pulse width modulation
VFPC Variable frequency power control
PDM Pulse density modulation
PSM Phase shift modulation
FM Frequency modulation
SFG Swept-frequency generator
RL Resonant load
MOSFET Metal oxide semiconductor field effect transistor
IGBT Insulated gate bipolar transistor
SCR Silicon controlled rectifier
SPWM Sinusoidal pulse width modulation
SVPWM Space vector pulse width modulation
DSP Digital signal processor
PAC Phase angle control
MV Medium voltage
FPGA Field programmable gate array
PLL Phase-locked loop
PI Proportional integral
PID Proportional integral derivative
MPC Model predictive control
GPC Generalized predictive control
xiv
CFI Current-fed inverter
L Inductance of induction coil
C Capacitance of capacitor bank
R Equivalent load resistance
fQ Quality factor
LX Inductive reactance
cX Capacitive reactance
oV Inverter’s output voltage
rV Voltage across resistance
LV Voltage across inductance
invI Inverter’s output current
Li Heating coil’s current
ci Capacitor’s current
DCI DC-link current
DCV DC-link voltage
DCL DC reactor
rf Resonant frequency
Angular frequency
oP Effective power at inverter’s output
Phase angle between inverter output current and voltage
mA Signal’s amplitude
xv
qf Quiescent frequency
sh Input sensitivity parameter
(t)o Output signal
CST Constant source topologies
VSC Voltage-source converter
CSC Current-source converter
D Diode
SW General switch
T Thyristor
Qi IGBT, 1,2,3...i
wpR Work-piece’s resistance
wpX Work-piece’s reactance
wpA Work-piece’s cross sectional area
r Relative permeability
gA Gap area between coil and work-piece
rk Coil correction factor
cL Coil length
0 Permeability of free space
Resistance per unit area
icN Number of induction coil turns
icd Diameter of induction coil
xvi
Depth of penetration
,p q Functions for a solid piece
, ,a b cv v v Instantaneous three phase voltages
, ,a b ci i i Instantaneous three phase currents
,V V αβ- components of voltage
,i i αβ- components of current
,sd sqV V dq- components of source voltage
,sd sqI I dq- components of source current
,cd cqV V dq- components of capacitor voltage
,cd cqI I dq- components of capacitor current at input
ACG AC gain
m Modulating vector
cI Magnitude of control signal
ACP Power at ac-side
DCP Power at DC-side
, ,r y bi i i Instantaneous three phase currents
pK Proportional gain
iK Integral gain
A System matrix
B Input matrix coefficient
xvii
oC Output matrix coefficient
x k State vector
v k Input disturbance
u k Input vector
y k Output vector
Δu k Incremental control signal
Ricatti-equation solution
LQR LQR gain matrix
LQRJ LQR cost function
Q State weight matrix
Control weight matrix
k Sampling instant
pH Length of the prediction horizon
cH Length of the control horizon
|k h k Value predicted for the time k h
r k Desired reference value
xviii
Abstract
Induction heating (IH) works on the principle of electromagnetic induction and, in it, the energy is
transferred from a work coil to a work piece. Unlike a transformer in which energy is transferred
from the primary to the secondary winding at 50 or 60 Hz, the transfer of energy in induction
heating typically takes place at higher frequency. The induction heating is being used extensively
in domestic, industrial and medical applications. In industry, this heating process is accomplished
through induction furnaces which are used for different heating jobs e.g. melting, annealing,
forging and tempering etc. The structure of the IH system can be built up with the help of a current-
source converter, DC reactor, inverter and parallel arrangement of capacitor and inductor to form
the resonant circuit. The work piece is inserted into the induction coil for heating or melting and
power will flow through this work piece. The system load can be seen from the converter side,
therefore, power is controlled by current-source converter using a suitable control method.
In this thesis, power control of a current-source converter feeding an induction furnace is focused
by two different control approaches; the space-vector based PI technique and generalized predictive
control (GPC) algorithm. The PI control scheme with space-vector PWM pattern regulates the
power of a high Qf- resonant load by controlling the DC link current according to the defined target.
The PI controller adjusts the DC voltage by SVPWM in such a way that the error signal is reduced
to a minimum value and a constant current is maintained uninterruptedly to the load. The key
objective of this strategy is to avoid the fluctuation of the DC link current as load varies in heating
process.
The generalized predictive control (GPC) predicts the states and future control sequence of the
system; then achieves on-line optimization with a reduced error by manipulated variables. The
state-space model of the current-source converter with resonant load is developed and generalized
predictive control (GPC) for the power regulation of induction furnace is presented. The obtained
results show the effectiveness of the GPC to control the power of the heating load and its regulation
within the defined constraints. The PI control method has tracked the reference effectively but it
cannot handle the constraints. Generalized predictive control has a potential of reference tracking
and constraints handling that is considered a major benefit over conventional field-oriented
controllers.
1
Chapter 1
Introduction
1.1 Overview
Induction heating, by electromagnetic-induction phenomenon, is used for different heating
purposes. It has several advantages over other nonelectric heat-treating process such as short
response, higher efficiency, cleanliness and well controlled heating [1]-[3].This chapter briefs the
historical background of the induction heating and its expansion to the present day structure.
Various control techniques employed with different topologies in induction heating are also
discussed.
1.2 Historical Background
In early 1900’s, a medium-frequency induction furnace was developed by Dr. Northrup through
spark-gap generators [4]. In 1927, first medium-frequency furnace was installed by the Electric
Furnace Company at Sheffield [5]. Midvale Steel (1927) and Ohio Crank Shaft Company (mid-
1930’s) had introduced this technology for steel hardening [1], [4]. Since then, induction-heating
technology spread in various heat-treatment areas. The major developments arose in this heat
pattern are: spark-gap generators, motor-alternator sets, and solid-state semiconductor technology.
After early development in spark gap generators, the motor-generator sets were mainly involved
in induction heating [6]-[7]. The motor-generator sets were replaced by solid-state technology in
late 1960’s and it was considered a major breakthrough in induction heat technology [8]-[11]. The
working principle of a motor-generator set was different than the today solid-state switching
devices. In a motor-generator set, three-phase electrical power was supplied to an AC motor. The
output of the motor was coupled with a generator which produced a fixed high frequency
(multiples of the supply frequency) output. This higher-frequency power was then applied to heat
the material. Power was kept controlled through field supply variation of the motor and the
generator output was used as feedback for regulation. The main disadvantages of a motor-
generator set were fixed frequency, higher operating cost, lower efficiency, heavy weight and
bigger in size [5], [12].
2
The semiconductor devices were used to build both front end rectifiers and high-frequency
inverters. The power control was achieved either through a controlled rectifier at input or by
inverter switching using a suitable method. In induction heating, silicon controlled rectifier (SCR)
initially was introduced for high-power and low-frequency applications while metal oxide
semiconductor field effect transistor (MOSFET) was used for low-power, fast-switching
applications. Insulated gate bipolar transistor (IGBT) is the current trend to cope both high-power
and high-frequency applications [13]-[17].
Several induction furnaces have been developed with different operating frequencies; up for loads
in MW range [18]. The appropriate selection of frequency depends on the type of the material to
be heated. Induction heating is being used extensively in both domestic and industrial applications,
e.g. cooking, melting, heating, hardening, forging, brazing, soldering and tempering [3], [19]-[23].
1.3 Induction Heating Present-Day Structure
The revolution of semiconductor technology boosted the inducting-heat treatment tremendously
in different aspects. Typically, an induction heating system (IHS) can be divided into four main
sections: rectification, DC filters, inverter and resonant tank. The purpose of the rectification is to
convert AC supply into DC either through a controlled or an uncontrolled manner. The rectified
DC power is then filtered out through DC filters followed by an inverter [24]-[27]. A basic block
diagram of an induction heating system is shown in Figure 1.1.
Figure 1.1: Basic block diagram of an induction heating system.
3
Two types of power supplies, voltage-fed and current-fed sources, are being used at the input of
inverter [2], [28]. The selection of a particular source for an application depends upon its cost (both
initial and operating), effectiveness, reliability and the impact of a furnace supply on a utility’s
power quality. The power flowing into the inverter is then fed to the load which has two main
arrangements, i.e. series-resonant load and parallel-resonant load as shown in Figure 1.2.
(a)
(b)
Figure 1.2: Circuit configurations: a) series resonant load, b) parallel resonant load [2].
1.4 Control Techniques
In early days of semiconductor technology, two main power control scheme were used: swept-
frequency generator and resonant-load generator [5]. The former type, also called variable-
frequency power control, was used in series-resonant loads while the later type was employed in
parallel-resonant load configurations. After then new modulation and control algorithms were
proposed. These control techniques were mainly involved for power control through various means
either directly or indirectly [29]-[33]. Major classical control techniques are given below.
1. By adjusting input of an inverter through DC link control using [13], [34]:
i. Controlled rectifier
ii. DC-DC switch mode regulator
4
2. By inverter duty cycle control methods [4], [35]-[41].
i. Variable frequency power control (VFPC)
ii. Pulse width modulation (PWM)
iii. Pulse density modulation (PDM)
iv. Phase shift modulation (PSM)
The controlled rectifier controls the output power either by firing-angle method or modulation
based switching pattern. In inverter duty cycle control, usually uncontrolled rectifier is used at the
front end and then a constant voltage is maintained through DC filter capacitor at input of the
inverter. Therefore, it is easy to modulate the voltage signal and switching is performed
accordingly. A block diagram of typical control methods with respective topologies is given in
Figure 1.3.
Figure 1.3: Control techniques in induction heating [4].
5
1.5 Thesis Outline
The introduction of induction heating with historical background is discussed in this chapter.
Literature review and problem statement is given in chapter 2. In chapter 3, the topologies of
induction furnace resonant inverters are presented. Two major topologies and their features are
compared. An analytical model of a current-fed inverter, with a parallel resonant load, is
extensively explained. Based on the selection of a current parallel resonant load configuration
investigated in chapter 3, a current-source converter feeding induction heating load is described in
chapter 4. A mathematical model of the CSC is realized. The issues associated with decoupling
and nonlinearity of the model are resolved and finally a linear state-space model is formed.
A DC link control design, using space-vector PWM based PI control, is presented in chapter 5.
The SVPWM pattern generates the switching pulses for the converter. An advanced control
technique for power control in induction heating, based on model predictive control is presented
in chapter 6. The discrete-time state-space model is used to design the control law and constraints
based MPC control is realized. The results are presented in chapter 7. At first, the output
parameters’ responses are shown based on the analytical discussion of a parallel-resonant load fed
by a constant current at the input of the inverter, discussed in chapter 3. Then the SVPWM based
PI control to maintain a constant DC current both for fixed and dynamic loads is presented. Finally,
the results of the model predictive control are shown. Target tracking, output and manipulated
variables’ graphical view with defined constraints and without constraints are illustrated.
Chapter 8 concludes the thesis work, emphasizes the significance of the author’s contribution and
suggests the future works.
6
Chapter 2
Literature Review
2.1 Overview
This chapter discusses a detailed literature review of induction heating. The technology
transfer from a motor-generator set to solid-state converter with control schemes is presented.
The problem statement and contribution of the author are also summarized at the end of
chapter.
2.2 Induction Heating Control Strategies
Since the invention of induction-heating technology, the researchers have been exploring it in
terms of response, efficiency, and cost etc. The first main breakthrough happened in transfer
of technology from motor-alternator sets to solid-state semiconductor devices in 1960’s. Then
expansions in the control of power electronics increased steadily and in recent years, thanks
to commercially available digital signal processors (DSP), various control strategies have been
reported. The digital processors have provided many advantages such as flexibility,
effectiveness and implementation of computational algorithms; hence, several control
algorithm have been proposed in literature to achieve good response and high performance.
In all the stated techniques, the front-end rectifier and inverter were considered the integral
parts of the induction heating systems.
In the early days of solid-state technology swept-frequency generator (SFG) was developed,
having power up to a few hundred kilo watts, and used for small melting jobs. In this control
method, the input AC power was rectified through an uncontrolled rectifier followed by a DC
capacitor in order to maintain DC voltage at the input of inverter. A local oscillator was used
to control the switching frequency of the inverter. The power regulation had been achieved
through a function of frequency and a constant power was delivered to the load through
frequency regulation generally referred as variable frequency power control. In the same
years, the resonant load (RL) method was also developed where the load circuit was used for
7
thyristor commutation. Power was controlled through a variable DC source by switching the
semiconductor power devices in a rectifier bridge. In swept-frequency method, power control
was limited by the quality factor of the load and it was Qf-dependent power control. This
method was complex and inefficient compared to that of the load resonant generator. In
resonant load method, range of power control was independent of the quality factor; this
scheme became much popular for high-power applications. However, initially the RL method
had faced the start-up problem and an additional auxiliary circuit was essential to start the
oscillation [5], [12]. Hu et al., in reference [42], had proposed that the start-up of a constant-
current resonant inverter could be achieved during transient period by utilizing the ramp-up
delay of a practical DC supply. The major benefits of this ramp up delay start-up were its
reliability and on-line start-up without any additional cost.
Various control strategies were being applied for output power control of inverters by phase-
locked loop (PLL) and PI controllers with different modulation schemes i.e. pulse width
modulation (PWM), phase shift control (PSC), frequency modulation (FM) and pulse density
modulation (PDM) etc. [30]-[31], [33], [35]- [37], [40]-[41], [43] –[50]. The following
paragraph discusses these methods briefly.
References [31], [35], [36], [44]-[45] have described the use of a diode-bridge rectifier to
rectify the AC supply voltage into DC and then a DC capacitor was connected at the input of
an inverter to maintain a constant voltage source. Power regulation was achieved by duty cycle
adjustment of the inverter, commonly known as PWM control. A phase shift control method
was applied to a high-frequency voltage-fed inverter in induction heating [30], [37], [46]–
[48]. References [32], [40], [43] explore asymmetrical-frequency modulation and pulse-
frequency modulation based power control strategy for a voltage-fed inverter in induction
heating. A power-control strategy using pulse-density modulation for high-frequency
induction heating was described in references [33], [41], [49]–[50]. In this pulse-density
modulation technique, power regulation was achieved by adjusting the density of voltage
pulses. Cheng et al. in reference [51] have proposed a fuzzy controller for temperature control
in induction heating. This temperature detection based control method was employed to
generate the PWM signals for the inverter. A few hybrid control methods were also reported
extracting the good features of different strategies and then utilized in a hybrid-control
8
method. Shen et al. have proposed a pulse-density and phase-shifting modulation based hybrid
power control for series-resonant inverters [52]. Similarly, references [53] has discussed the
phase-shift and asymmetric PWM based adapted control method to regulate the load resonant
current for a low-power application. These methods are helpful to some extent but they
enhance the complexity of design and implementation.
The proposed modulation schemes have certain drawbacks and can be summarized as follows:
Frequency-modulation control has electromagnetic noises and high-switching losses [33]. In
pulse-width modulation, hard switching is a major issue and can deteriorate the switches
efficiency and may reduce their life period. In phase-shifting modulation, the main problem
is the distortion of current waveforms which cannot be utilized for a constant current load
where uniform heating is desired. Pulse-density modulation has fluctuation of current and
discontinuous power regulation problem [52]. A common drawback in these modulation based
methods is that they are suitable for the heating applications which are mostly configured as
series-resonant load with constant input voltage.
DC link switch mode regulator was also proposed to control the power in induction heating
(IH). A full-controlled rectifier was connected at the system input to rectify the AC line power
into DC. Then a DC link regulator was used to interface the front-end rectifier to the resonant
inverter. This regulator controlled the power of inverter and also maintained a proper
switching sequence of the bridge [34], [54]. Chen et al., have proposed a buck regulator to
control the power by regulating the output voltage and PLL was used to control the frequency
of the series-resonant inverter [55]. Likewise, a boost regulator followed by a controlled
rectifier was designed for a 250 kW induction furnace where IGBT switches were used instead
of SCRs. The purpose of DC-link regulator was to achieve high-level voltage at the input of
the inverter. The low level voltage was conventionally attained by controlled rectifier through
phase-angle control (PAC) [18]. The disadvantage of the presented schemes was extra
regulating circuit desired to regulate the output power, hence complexity of the system
definitely increased [52].
Line commutated thyristors have been considered the fundamental topology of the controlled
rectifier since the emergence of semiconductor technology. PAC method was practiced widely
in controlled rectifier for the power control in induction heating. In this method, field oriented
9
control was utilized through analog circuits by measuring and controlling different
parameters. A power control method for a parallel compensated load was presented. Load
voltage and load current were fed back in a pulse-suppressing circuit where their product was
compared with a reference power. Pulse generator was used to synchronize the inverter
switching with output frequency [9].
References [13], [56] have discussed a power-control method by generating the pulses for
controlled rectifier. The power was sensed from output voltage and current. The control
decision was then taken accordingly.
A microprocessor based DC power control by generating the pulses for controlled rectifier
was discussed in reference [57].
In reference [58], a power control for a series load resonant inverter was proposed by two
methods: indirect power control using output voltage across the heating coil and direct power
control (i.e. product of the output voltage and current). A phase-locked loop was also
employed for the frequency tracking of resonant load and operating the inverter at zero current
crossing in order to maintain a soft switching.
A temperature based power control was explored in reference [59]. In this method, phase-
locked loop had been used to lock the switching frequency of the inverter according to the
resonant frequency. Power control had been achieved by sensing the temperature of the work
piece and based on this temperature; firing angle of the converter was adjusted.
Tan et al. [60] have discussed a DC link current control technique for a coreless induction
furnace. The DC current reference was compared with the actual current flowing at the output
of the rectifier and error signal was compensated by a PI controller. The firing pulses were
generated to control the power. In this scheme, the controlled rectifier delivered a DC current
in such a way that output power was adjusted as desired reference. Digital phase-shift trigger
circuit has also been presented in reference [61] for three-phase controlled rectifier. This
method worked like a phase-angle control scheme to control the power of the converter by
adjusting the DC voltage.
10
Conventionally, phase-angle control by the controlled rectifier was used for a parallel-
resonant load. However, Zhu et al. had proposed a power-control method for a series-resonant
load by the controlled rectifier and phase-locked loop was utilized to track the resonant
frequency of the resonant load in order to operate the inverter according to the tracked
frequency [62]. The major issues in this PAC technique were line distortion and power factor.
Phase-angle control method was then replaced by PWM based control approach [63]-[66].
The advances in semiconductor technology and digital controllers, control algorithms with
different modulation patterns, spread out in voltage-source and current-source converters to
provide a desired power for various types of load i.e. MV drives, STATCOM and wind
turbines etc. [65], [67]-[72]. The space-vector based PWM for current-source converter (CSC)
has been used for DC motor drives and induction heating systems [73]-[76]. An adaptive back-
stepping approach for a three-phase voltage-source converter, feeding induction heating, for
a series resonant load was proposed in reference [66]. The space-vector pulse-width-
modulation (SVPWM) technique has been used and the controller maintained the DC voltage
according to the desired reference. These PWM-based converters have overcome the line
distortion and power factor problem of conventional phase-control technique.
Zone-control heating has also been reported for a series-resonant inverter [77]. A multiphase
induction heating system was developed, instead of a single phase, at the output. Three
resonant current inverters were connected in series and a single unit CSC was used to feed
them in order to reduce the number of elements. The PI-control strategy with FPGA
implementation has been used to control the power flowing through a current-source converter
[78]. A three-phase inverter configuration followed by an uncontrolled rectifier has been
discussed for a series-resonant load. Output-power control was achieved through PI controller
by varying phase angle between inverter output voltage and current [79]. A power control
method for multi coils connected with H-bridge inverters was discussed in reference [80].
This arrangement has been applied for a zone-control heating to get a uniform temperature
profile. The reported control technique controls the power by real and imaginary components
of inverter voltage and coil current. Sarnago et al. have also described a direct AC-AC
converter for a domestic heating system in reference [81].
11
The control strategies i.e. phase-locked loop (PLL) and PI controllers with different
modulation schemes -phase angle and PWM based PI control, and DC-link switch mode
regulators-have been reviewed for the power control of induction-heating systems. All these
control schemes do not have the prediction and optimization strategies to operate the system
in an efficient manner. They take action when an error appears in a system against some set
point value [82]. Conversely, the predictive controllers predict the future behaviour of a
system and perform on-line optimization by manipulating the present input of the system to
achieve a minimum cost function [83]-[84]; therefore, the performance of MPC has been
found better than PI control [85]-[89].
The other major development in the modern-control theory, besides PID controllers, was the
optimal control representation by Kalman in early 1960s. Then linear quadratic optimal
controller (LQOC) was developed but later on some shortcomings were observed in it [90].
The input, output and state constraints were not addressed in LQOC, and nonlinearities of the
process and model disturbances could not be handled properly [91]. Zerouali et al. have also
proposed a linear quadratic regulator (LQR) for a series-resonant load in induction heating
[92]. The optimization problem was solved by a fixed window with infinite prediction
horizon. However, ill-conditioning problems were observed due to infinite prediction horizon.
The other drawback was system performance around the given initial state due to a fixed
window [90]. Model predictive control (MPC) has solved these problems in a moving time-
horizon window. The main target of this moving-time horizon window is to achieve the real-
time optimization with defined constraints on system variables [93]-[99]. More recently,
various predictive control algorithms have been applied successfully in different electrical and
power electronics applications such as multi-level converters, uninterruptible power supplies
(UPS), machine drives and power plants etc. [88], [100]-[109].
The MPC has replaced conventional field oriented PID controllers thanks to fast response,
slight overshoot, and relatively easy tuning [86]- [87], [110]. Compared to other areas, limited
research has been done for application of MPC especially in induction heating [91], [111]-
[112]. In various predictive algorithms, generalized predictive control (GPC) is considered an
efficient and flexible predictive control approach [96], [113], and is extensively being used in
different control purposes [88], [114]-[120].
12
In this work, power control of a current-source converter feeding an induction furnace is
focused. Two different techniques are presented. The main objective of the work is to get a
good control response of the system’s variables.
2.3 Problem Statement
The phase angle control scheme employed in a line commutated rectifier feeding induction
heating load has certain poor quality problems. Similarly, inverter duty cycle control methods
i.e. pulse density modulation, phase shift modulation and frequency-modulation etc. were
applied to control the power of the heating load. These modulation techniques are suitable for
the low power applications and are applicable in a series fed resonant inverter. A detail
discussion is addressed in section 2.2. For a current fed resonant inverter, an efficient control
technique is still desired to regulate the power of the heating load in an effective manner. Two
approaches: space-vector PWM based PI control and model predictive control are presented
in this work. Both techniques operate the current source converter by maintaining a constant
DC link current to the heating load and also regulate the power of the system effectively. A
short description of the presented methods is given in the following sub-sections.
2.3.1 Space-vector PWM based PI control
The PI control scheme with space-vector PWM pattern regulates the power of a high Qf-
resonant load by controlling the DC current according to the defined target. The PI controller
adjusts the manipulated variable by SVPWM in such a way that the error signal is reduced to
a minimum value and a constant current is maintained uninterruptedly for the load. The key
objective of this method is to avoid the fluctuation in DC current as load varies in heating
process, hence the anti-disturbance ability is achieved through this simple and active
approach. SVPWM based control is a better technique than the phase-angle control technique:
it enhances the power factor and reduces the line distortion.
2.3.2 Model predictive control
The MPC predicts the states of a system and then achieves on-line optimization with a reduced
error by manipulated variables. This work has presented the application of generalized
predictive control (GPC) for power tracking of a resonant-heating load. Initially, the
13
mathematical model of the system is developed. In this model, coupling and non-linearity of
the state variables exist which then were resolved and a linear state-space model of the system
was formed. The model predictive control law is applied to control the power of the system.
This model based predictive controller outperforms the field oriented controllers in terms of
prediction and constraints handling.
In this work, both transient and steady-state analysis of a parallel-resonant inverter in
induction heating are presented also. This analysis is not a part of presented control
techniques. The purpose of this analysis is to emphasize the importance of a constant current
feeding at the input of the resonant inverter configured with a parallel load. The resonant
frequency is formed by resonant load pattern and is tracked to maintain a constant switching
angle of the inverter. The response of the output parameters for a current-fed resonant inverter
is analyzed keeping a constant phase difference to operate the inverter with a desired angle.
14
Chapter 3
Medium-Frequency Resonant Inverter
3.1 Overview
The main configurations of a resonant-inverter are quarter-bridge, half-bridge and full-bridge
topologies connected with a resonant tank circuit. A resonant circuit has two major
arrangements, i.e. parallel or series. Each arrangement has its own features, benefits and
specific purposes which are discussed in the following sections. The mathematical expressions
presented in this chapter are based on resonant circuit theory extracted from classical books
and papers to give a view to the reader but not in the proposed control design of the presented
work.
3.2 Resonance Frequency
The electrical elements, i.e. resistors, capacitors and inductors with different combinations,
when used in a circuit, will produce the maximum output at a certain frequency: this frequency
is called ‘resonance frequency’ of that circuit. The maximum output response may be current
or voltage depending upon the circuit configuration being analyzed.
An AC circuit includes capacitive and inductive reactances which vary with the frequency:
this variation results in the change of the combined or overall reactance of the circuit which,
in turn, changes the natural frequency of the circuit. The capacitive reactance has the inverse
relation with the frequency while the inductive reactance is directly proportional. At zero
frequency, the capacitive reactance approaches infinity but at very high frequency a capacitor
is considered to possess a zero reactance. On the other hand, the inductive reactance
approaches zero at zero frequency and has a high value at high frequency. Hence, the
resonance frequency occurs at a point where capacitive and inductive reactances cancel each
other (i.e. XL = XC) and circuit impedance will be equal to the resistance of the circuit.
This resonant frequency is developed with respect to the load arrangement connected at the
output, i.e. the parallel-resonant load or the series-resonant load. At resonance, the circuit
15
either parallel or series loaded forces the voltage and current to pass through zero or if
thyristors are used they exhibit self-commutation
3.3 Inverter Topologies
An inverter produces AC power at the output and has three common configurations, i.e.
quarter-bridge, half-bridge and full-bridge. These configurations are used either with a series
or parallel resonant load. A short description of each configuration is given below.
3.3.1 Quarter-bridge inverter
A simple circuit of a quarter-bridge inverter, utilizing a controlled switch and a diode, is shown
in Figure 3.1. A large inductor is connected at the input, so it is categorized as current-fed
inverter, and the load coil is connected in series with a capacitor at the output which is also in
contrast compared to the output circuit of a conventional current-fed inverter. This topology
has been used typically at 10 kHz–30 kHz frequency range for heat-treating jobs. For the
resultant sinusoidal load current, half of this wave flows through the controlled switch and
other half through the diode. Unregulated DC voltages are applied at the input and the output
power is regulated by the firing angle of the inverter. INDUCTOHEAT UNISCAN and
UNIPOWER induction-hardening machines are examples of this type of inverter [121].
Figure 3.1: Quarter bridge inverter [121].
3.3.2 Half-bridge inverter
It consists of two switches and two capacitors which are connected in such a way that for each
T0/2 time one switch and one capacitor provide the path for the flow of current while in
16
remaining period current flows in the second pair of switch and capacitor. The physical
appearance of this arrangement shows it as a half bridge and hence its name. This
configuration is depicted in Figure 3.2 and it is required mostly for low-voltage or low-power
applications [122]. In the first half cycle, switch SW1 with capacitor C1 provides the power
to the load, while in second half cycle capacitor C2 with switch SW2 supplies the power to
the load circuit. Switches have been shown with open states; however, they can be replaced
with any switching device such as SCR or transistor.
Figure 3.2: Half bridge inverter.
3.3.3 Full-bridge inverter
For a high output power a full-bridge configuration, given in Figure 3.3, is frequently used. It
consists of four switches which make two diagonal pairs: each pair is operated successively.
The arrangement of these four switches also presents it as an H-bridge with two switches
connected in the upper side and the remaining two in the lower side, where as the load is
connected between the center points of the legs (left and right-side legs of the letter H). The
sequence of switching in a full bridge topology is such that the switches connected on the
same leg should never be ON simultaneously otherwise short circuiting is happened in the
system. A time delay is ensured to operate the diagonal pairs at same time. The time delay
depends on the type of semiconductor switch which is being utilized. A symmetrical current
is produced at the output and the direction of current flow in each diagonal pair is opposite to
that of the other as depicted in Figure 3.3.
17
Figure 3.3: Current-fed full bridge inverter.
3.4 Resonant Tank Circuit
The resonant load circuit consists of a capacitor, furnace coil and the work piece. The work
piece is to be heated inside a crucible and the heating coil surrounds this crucible with a small
distance in order to maintain the insulation. A converter circuit is used to deliver power to the
tank circuit about its resonance frequency. It gives unity power factor when being operated at
resonance frequency; however, it never runs at this frequency due to the variation of work
piece parameters and system limitations. Typically two load arrangements are being used in
induction furnaces, i.e. parallel-resonant load shown in Figure 3.4 and series-resonant load
shown in Figure 3.5 [4]-[5].
Figure 3.4: Parallel-resonant load a) Circuit diagram, b) Output waveforms.
18
Figure 3.5: Series-resonant load a) Circuit diagram, b) Output waveforms [4].
In the parallel-load configuration as shown in Figure 3.4, capacitor ‘C’ and inductor ‘L’ have
been connected in parallel. This load needs a constant current; hence a current-fed inverter is
used to operate it. Series load is the combination of capacitor and inductor in series. A voltage-
fed inverter is used to supply this load as it needs constant voltage. Other features of current-
and voltage-fed inverters are given in Table 3.1 [2], [123]-[127].
Table 3.1: Characteristics of bridge resonant inverter.
Current-fed inverter Voltage-fed inverter
Parallel resonant load arrangement
Inverter is operated in capacitive
mode
Current and voltage have square and
sinusoidal waveforms respectively
DC choke or reactor at input of the
inverter
Square current and sinusoidal
voltage waveforms at output of the
inverter
Suitable for high Qf-loads
Series resonant load arrangement
Inverter is operated in inductive
mode
Current and voltage have sinusoidal
and square waveforms respectively
DC capacitor at input of the inverter
is desired
Square voltage and sinusoidal
current waveforms at output of the
inverter
Suitable for low Qf-loads
19
Series or parallel circuit based resonant inverter has its own benefits and features. Current-fed
inverter, in induction heating, is a more flexible topology in the adaption of heating coil
compared to a voltage-fed inverter [15]. Moreover, a parallel-resonant inverter offers good
short-circuit protection handling, paralleling capability and suitability for high QF-resonant
load compared to a voltage-fed inverter [128]; hence parallel resonant inverter topology is
focused upon in this research work. However, the starting of a parallel resonant load inverter
was a major issue in the early furnaces which now has been resolved by different techniques
[42], [129].
3.5 Parallel-Resonant Inverter: Modeling and Analysis
The main elements of the circuit are varying input DC supply, a large DC current limiting
reactor, switching bridge and a parallel resonant tank. This parallel resonant load structure
consists of a capacitor bank and a heating coil. The current at the input side essentially remains
constant due to a large current-limiting reactor; hence, the circuit is also called current-fed
resonant inverter. The resonance is developed due to the parallel combination of inductor and
capacitor so the other name of the converter is parallel-resonant inverter. Both names are
frequently used in induction heating. In this bridge, diagonal pairs are switched with a phase
difference of 180°. One pair is Q1Q3 and other is Q2Q4 as shown in Figure 3.6.
Figure 3.6: Basic scheme of a current-fed resonant inverter.
20
‘Vs’ is the rectified source voltage, ‘LDC’ is DC reactor, ‘IDC’ is DC link current, ‘Iinv’ is the
current flowing out from inverter bridge, ‘Vo’ is output voltage, ‘ic’ and ‘il’ are the capacitor
and inductor currents. The system model can be represented by a simplified circuit as shown
in Figure 3.7 where the heating coil ‘L’ is connected in parallel with the capacitor bank ‘C’.
‘R’ shows the equivalent resistance of coil and work piece.
Figure 3.7: Equivalent model of a CFI.
Mathematical expressions for this configuration are as under:
DCs DC o
dIV L v
dt (3.1)
DC o s
Dc DC
dI v V
dt L L (3.2)
lo l
div L Ri
dt (3.3)
l o ldi v Ri
dt L L (3.4)
c DC li I i (3.5)
where ‘ic’ and ‘il’ are the rms values of capacitor and inductor currents.
1 1oDC l
dvI i
dt C C (3.6)
21
0 0
10 0 1
10 0
01 1
0
DCDCDC DC
l l s
LLI I
d Ri i V
dt L Lv v
C C
(3.7)
The equations (3.1)-(3.6) can be used to analyse the behaviour of the model. For induction
heating, capacitor and inductor forms a resonant frequency which can be expressed as:
1
2rf
LC (3.8)
The output current of the inverter flowing into the resonant load is:
0 2
2 2
dc
dc
inv
I ft
I fI
t
(3.9)
3.5.1 Quality factor
Conventionally in the induction heating or melting, the load circuit has a small resistance
which is characterized by its quality factor. The quality factor of a resonant circuit illustrates
the maximum energy stored to the energy dissipated in a circuit during one oscillation period.
It is a dimensionless quantity and typically lies in the range 5-20 for induction melting [127].
Mathematically, it can be represented as:
2
2
1
22
1 1 2
f
r
LI
I R
Q
f
(3.10)
‘L’ is the energy storing element, ‘I’ is the current flowing into the circuit, ‘R’ is dissipating
resistance and ‘fr’ is the resonant frequency.
22
fQ can be simplified as:
2 rfr LL
fQR R
(3.11)
The quality factor is considered a key parameter in induction heating and is very helpful to
examine the electrical quantities i.e. voltage, current and power etc.
The active ‘Vr’ and reactive ‘VL’ component of the output voltage can be related using Qf-
factor as:
,
,21
o peak
r peak
f
VV
Q
(3.12)
, ,L peak f r peakV Q V (3.13)
The inductor and capacitor currents at the output can be related with the inverter current
through Qf- factor as follows:
L f invi Q I (3.14)
C f invi Q I (3.15)
Similarly, the ‘fQ ’ can be defined in terms of active ‘kW’ and reactive ‘kVAR’ power as:
f WkVAR Q k (3.16)
kVAR can also be found easily by the design data of the furnace and is given by:
2
1000
rCkV
VAR
(3.17)
where ‘V’ is the voltage across the capacitor bank.
23
3.5.2 Power analysis
As the purpose of the proposed control strategy is to supply a constant current to the load,
hence the power analysis of the parallel resonant load is presented here to examine the
effectiveness of the proposed control scheme if a constant current is provided to the inverter.
The inverter input side power (i.e., DC power) can be measured easily and in industry, it is
considered approximately equal to the output power by ignoring electrical and thermal losses.
The reason of this approximation is easy measuring and control of the DC parameters. This
DC power can be represented with this simple relation:
DC DC DCP V I (3.18)
Similarly, the effective power at the inverter output can also be defined as:
2
,
2
r peak
o
VP
R
(3.19)
oP is the output power and ,r peakV is the peak value of voltage across equivalent resistance of
the induction coil and work piece.
2
,
22 (1 )
o peak
o
f
VP
Q R
(3.20)
‘Vo,peak’ is the peak value of output voltage.
In the same way, the current flowing through the coil or load can also be used to find the
output power as:
2
,
2
L p k
o
eaiP R (3.21)
The inverter instantaneous current, voltage and power can be represented in the following
relations [130]-[131]:
1
sin(k t)inv inv
k
i i
(3.22)
24
1
sin(k t )o o
k
v v
(3.23)
1 1 1
sin(k t ) sin(k t) coso o inv
k k k
P v i
(3.24)
This equation shows that the output power is a function of the current flowing through the
inverter invi , output voltage vo, and their phase difference 𝜑.
3.6 Soft Switching
Soft switching is an important feature of a resonant inverter control strategy to operate the
switches at zero crossing of voltage and current. In case, the switching of current or voltage
away from zero crossing is done, current spikes are produced which will result in thermal and
insulation stresses on the power switches. These stresses may cause sudden failures of the
switches or the reduction of their life. The inverter current and output voltage are sensed with
high frequency transducers and then their phase angle is detected. In a current-fed inverter,
the inverter is required to operate with capacitive mode, hence the presented scheme operates
the inverter accordingly. The phase angle control architecture is shown in Figure 3.8 where
phase difference ‘φ’ for output voltage and current is detected by an XOR. This phase
difference is passed through a low pass filter and then compared with a reference phase
difference ‘φ*’. The phase error signal *( ) is fed to a PI controller and its transfer function
can be written as:
*1(1 )( )s K
sT
(3.25)
‘ K ’ is the gain of the controller.
The phase angle reference * is set to such a value that optimum power may be delivered to
the load considering the switches turn off delay. The PI controller is then followed by a
voltage-controlled oscillator (VCO). The VCO produces frequency signal which is desired to
drive the inverter and is stated as:
25
0
(t) cos 2 (t) 2 (t)dt
t
o q sA f h v
(3.26)
where A is the output signal amplitude, qf quiescent frequency, (t)v is input signal,
represents phase quantity, sh is input sensitivity parameter, and (t)o is the output signal.
Figure 3.8: Phase angle control scheme.
In this scheme, there is also a flexibility to operate the inverter either leading, lagging or in
phase. Positive reference value operates the inverter in leading mode while negative reference
value operates it in lagging mode. This lagging mode can be used in a series-resonant inverter
switching by just changing the reference value from positive to negative.
26
Chapter 4
Current-Source Converter in Induction
Heating: Theory and Modeling
4.1 Overview
The structure of the heating system is comprised of a current-source converter (CSC), DC reactor,
inverter-bridge and combination of capacitors and inductor. The work piece is inserted into the
coil for heating and electrical power will flow through this work piece. The system load can be
seen from the converter side, therefore, power is controlled by CSC using a suitable control
method. A sketch of the inverter side has been discussed in previous chapter; now in this chapter
the current-source converter is focused upon. The dq-model of the current-source converter, with
equivalent load circuit, is developed.
4.2 Structure of Induction Furnace
Various parameters such as production rate (in kg/hr), geometry of the induction-heating system,
desired temperature and the system efficiency determine the required power to heat a material. It
may vary from 10-20 kW for a little job and up to several megawatts for large applications [4]. A
general block diagram, comprising the main components of a medium-frequency induction
furnace, is depicted in Figure 4.1. The AC source, input filter, front end rectifier, DC filters,
inverter-bridge and resonant tank are shown in this diagram. It is essential to mention that this
diagram is presenting a general block diagram which can be configured as a current fed or voltage
fed converter. For a defined configuration, we have to select the type of filters, rectifier and
resonant load.
27
Figure 4.1: A basic diagram of induction furnace structure.
Each block has its own features which vary according to the topology for compatibility and proper
interfacing to meet the end-user’s requirements. Topologies with their types and usage are given
in the following sections.
4.3 Front-End Rectifier
A three-phase AC supply is fed to the rectifier [132]. The basic purpose of a rectifier is to convert
the AC supply into a DC one. If it is compared with a motor-generator set, then someone may say
that the motor has been replaced by rectifier and generator part by inverter. Both controlled and
uncontrolled rectifiers are still being employed in induction-heating applications.
4.3.1 Uncontrolled rectifier
In this configuration, diodes (D1-D6) are used for rectification as shown in Figure 4.2. When
diodes conduct there is no way to control their output DC voltage as fixed voltages are produced,
and this type of rectification is called ‘uncontrolled rectification’ [17]. In induction heating, this
type of rectifier is still in use and series-fed inverters are employed it mostly as front end rectifier.
Control objectives are achieved by various modulation based duty cycle operational control. In
diode rectification, input fuses are used to protect devices from short circuiting currents.
28
Figure 4.2: Uncontrolled rectifier.
4.3.2 Controlled rectifier
Unlike a diode bridge rectifier, a controlled rectifier uses semiconductor switching devices such
as thyristors, MOSFETs and IGBTs. A thyristor-based controlled rectifier is shown in Figure 4.3.
In a phase-angle control technique, SCRs (T1-T6) are fired for an interval with a pre-defined
sequence. Some modern rectification-control techniques such as sinusoidal-pulse width
modulation (SPWM), space-vector pulse modulation (SVPWM) and predictive control are
replacing the conventional-control techniques. A PWM based technique improves the line
distortion and input power factor. The PWM based controlled rectifier can be further divided into
two main topologies based on the kinds of sources illustrated in next section.
Figure 4.3: Thyristor based controlled rectifier.
4.4 Constant-Source Topologies (CST)
Constant-voltage and constant-current topologies are commonly used in the induction-heating
furnaces [4], [133]. However, the control techniques and load configurations are changed as the
power feeding topology is changed.
29
4.4.1 Voltage-source converter (VSC)
A voltage-source power supply is typically distinguished by the use of a DC capacitor at the output
of the converter in order to maintain a constant voltage as the converter name suggests. It is
depicted in Figure 4.4 (a) where ‘Li’ the line reactor, Q1-Q6 are IGBT switches and ‘CDC’ is the
DC filter capacitor. In this converter, the constant voltage is fed to the inverter with a series
connected resonant load at the output [2], [66], [134]. The power regulation can be attained by
various means as discussed in chapter 1.
4.4.2 Current-source converter (CSC)
A current-source converter is a dual of a voltage-source converter. A resonant load is supplied by
a current-source converter [4], [34]. This configuration differs from that of a VSC by a large reactor
at the output instead of a capacitor as shown in Figure 4.4(b) where input filter is the combination
of line reactor ‘Li’ and capacitor ‘Ci’ . The output power is regulated by the DC link current. This
current can be maintained constant by varying DC voltage which is accomplished through
switching of the converter. Compared to a current-source converter, voltage-source converter is
very famous due to its heavy usage in machine drives but the CSC has good features in induction
heating.
(a)
30
(b)
Figure 4.4: a) Voltage-source converter, b) Current source converter [132].
The main differences in the voltage-source and current-source converters can be enlisted as [67],
[69]:
Conventionally, only inductive-filter is used at the input of a VSC and DC capacitor at
output; in a CSC, second order LC-filter is employed at the input and DC- choke is installed
at the output.
A CSC needs semiconductor switches having bidirectional voltage blocking capability
while a VSC does not have such a limitation.
In a CSC, a diode is connected in series with power semiconductor switches to cope the
problem of bidirectional voltage blocking as described earlier. In a VSC topology, diodes
are connected in anti-parallel with power switches.
In a VSC, three switches are kept ON at any time while in a CSC only two switches are
turned ON.
In a VSC eight switching states are formed by SVPWM, while in a CSC nine switching
states are formed.
Both topologies’ names symbolize their use; a current-source converter delivers a constant
current while a voltage-source converter provides a fix DC link voltage.
31
A voltage-source converter boosts the voltage at output, its other name is boost converter
and a current-source converter reduces the voltage at output hence is called buck converter.
The disadvantage of a CSC is its DC link reactor losses compared to the DC-link capacitor
losses in a VSC.
The advantages of a CSC are its simple converter configuration and inherent short-circuit
protection capability.
4.5 Structure of a Current-Source Converter Feeding an Induction Heating
Load
The structure of a parallel resonant circuit in induction-heating system is comprised on a current-
source converter followed by a DC choke and inverter as shown in Figure 4.5 [127].
Figure 4.5: A current source converter feeding an induction-heating load.
The work-piece is heated by inserting it into the induction coil. The constant current, provided by
the inverter, energizes the coil and causes eddy currents into the work-piece [5]. The work piece
and coil parameters, i.e. geometry, conductivity, and permeability etc. tend to change during the
heating process: they result the change in equivalent system load, hence power fluctuates. An
equivalent model of the coil and work piece can be represented as shown in Figure 4.6 [5], [20].
32
Figure 4.6: Equivalent circuit of coil and work piece [20].
Mathematical expressions for the equivalent-circuit parameters are given by:
Work piece (WP) parameters:
wp r wpR pA (Ω) (4.1)
wp r wpX qA (Ω) (4.2)
Induction coil (IC) parameters:
2
r icic
k dR
(Ω) (4.3)
2
r icic
k dX
(Ω) (4.4)
Air-gap reactance between work piece and induction coil is also described as:
g gX A (Ω) (4.5)
2
02 icr
Nf
l
(Ω/m2) (4.6)
where ‘i’ is the current flowing into the coil; wpR , wpX and wpA are the resistance, reactance and
cross sectional area of the work-piece, respectively; r is the relative permeability, gA is the gap
area, rk is coil correction factor ( range is 1-1.5), is the resistance per unit area , icd is the
diameter of induction coil, p and q are function for a solid piece, is the depth of penetration, 0
33
is the permeability of free space, rf is resonant frequency, l is the gap length and icN is the
number of induction coil turns. These equations give resistive effect on the system and are useful
to describe the equivalent load circuit.
At resonant condition, coil capacitorX X the equivalent load can be modeled assuming only
resistive behaviour of the load.
4.6 Mathematical Description
It is essential for the control-system design to develop a mathematical model for the system, but
the converter consists of IGBT modules. It is very difficult to include all nonlinearities of the
modules; hence following assumptions were made in analysis and design:
Switching losses ignored.
Off/dead time delay of the transistors is zero.
Control signals delay also ignored.
A stable balanced power supply voltage at input is assumed, i.e. having amplitude of the
three phase voltages and each phase voltage displaced by 120° from another phase voltage.
The equivalent circuit of a current-source converter (CSC) employed for a parallel-resonant load,
in induction heating applications, is illustrated in Figure 4.7 [20], [135]-[136]. The model
parameters are input supply voltage sV , supply current si , capacitor voltage cV , bridge input
current ii , output current DCI , input resistance iR , inductance iL , capacitance iC , DC reactor
DCL and a constant current feeding the heating load at the output.
34
Figure 4.7: Equivalent model of CSC with induction heating load.
4.6.1 CSC model in abc- reference frame
In a balanced three-phase system, instantaneous phase voltages ( , ,a b cv v v ) can be represented as
functions of line-to-line voltages ( llv ):
sin3
ll peak
a
vv t
(4.7)
2sin
33
ll peak
b
vv t
(4.8)
2sin
33
ll peak
c
vv t
(4.9)
Similarly three-phase currents flowing through each phase by assuming 0a b ci i i can be
illustrated as:
sina peaki I t (4.10)
2sin
3b peaki I t
(4.11)
2sin
3c peaki I t
(4.12)
35
4.6.2 CSC model in αβ- reference frame
Clarke’s transformation is applied on this three-phase system to generate αβ-plane which is a two-
dimensional system and can be used to develop space-vector approach [137]. Clark-transformation
in its graphical form is drawn in Figure 4.8.
v v jv (4.13)
where,
2 2 1 11 cos cos 1
3 3 2 2 2
32 2 3 30 sin sin 0
3 3 2 2
a a
b b
c c
v vv
K v vv
v v
(4.14)
The expressions in terms of stationary reference frame are as:
22
3a b cv v v v (4.15)
Similarly,
22
3a b ci i i i (4.16)
where je and 23
36
Figure 4.8: Clark-transformation in graphical form.
4.6.3 CSC model in dq- reference frame
In Clark transformation, abc frame quantities are transformed into a single complex quantity with
the same angular frequency and is known as stationary reference frame. The Park transformation
is used to place some other base vectors on the complex plane of Clark transformation as shown
in Figure 4.9. The new basis vectors are called direct axis and quadratic axis. They rotate around
the αβ- plane, hence, this frame is named as rotating frame [137]-[138].
sin cosdV v v (4.17)
cos sinqV v v (4.18)
or,
sin cos
cos sin
d
q
V v
V v
(4.19)
and αβ -frame is simply obtained as:
sin cos
cos sin
d
q
Vv
Vv
(4.20)
37
The dq- model can be obtained from abc- frame directly as:
2 1 1
sin cos 3 3 3.
1 1cos sin0
3 3
a
d
b
q
c
vV
vV
v
(4.21)
Three-phase quantities can be found from dq-reference variables and the expression is given by:
sin cos
2 2sin cos
3 3
2 2sin cos
3 3
a
d
b
q
c
vV
vV
v
(4.22)
Similarly, three-phase supply currents can be transformed into dq- current form as:
2 2cos cos cos
3 32
3 2 2sin sin sin
3 3
a
sd
b
sq
c
iI
iI
i
(4.23)
Figure 4.9: Park-transformation.
Mathematical model of the current source converter can be described in dq-transformation. One
main advantage of dq-transformation is to present three-phase AC-quantities as DC-quantities.
38
The second advantage is independently control of the direct and quadratic axis components of a
variable.
From Figure 4.7, the model variables can be expressed in the following mathematical descriptions
[139]-[141]:
ss s i i c
div i R L v
dt (4.24)
1 1s is c s
i i i
di Ri v v
dt L L L (4.25)
The voltages and currents are in ordinary coordinates and can be transformed into dq-reference
frame. The source current si is decomposed into its dq-components sdI and sqI . The objective of
the presented work is the regulation of the active power at the output of the converter through DC
link current DCI and control of the reactive power flow by sqI to ensure the unity power factor at
the input. This unity power factor is achieved when source current quadrature axis component sqI
approach to zero i.e. source current aligns with source voltage, hence the regulation of reactive
power flow is achieved by dq-model.
1 1sd isd sq cd sd
i i i
I Rd I V V
dt L L L (4.26)
1 1sq isd sq cq sq
i i i
I Rd I V V
dt L L L (4.27)
where cdV and cqV are the dq-components of the capacitor voltage; sdV and sqV are the source-
voltage components and is the angular frequency of the input supply.
As illustrated in Figure 4.7, the source current si supplies the current to the capacitor and converter.
It can be expressed as:
s c ii i i (4.28)
39
cs i i
dvi C i
dt (4.29)
The converter input current ii can be related with the converter output current i.e. DCI as follows
[139]:
i AC DCabc abci G m I (4.30)
where GAC is the AC gain (i.e. GAC = 1) of a PWM technique and is also assumed here 1, m is the
modulating vector; putting the value of converter input current ii into equation (4.29):
1cs DC
i i
dv mi I
dt C C (4.31)
Again decomposing the above equation (4.31) into dq-form, we then get:
1cd dsd cq DC
i i
V md I V I
dt C C (4.32)
1cq q
sq cd DC
i i
V md I V I
dt C C (4.33)
The converter output side relation is given by:
DC
DC DC DC DC
dIV L I R
dt (4.34)
1DC DCDC DC
DC DC
dI RV I
dt L L (4.35)
Like currents relation given in equation (4.30), the DC voltage at converter output can also be
written in terms of the input side voltage i.e. cv and then is transformed into dq-form as follows
[139]-[141]:
3 3
2 2
DC dcd cd q cq DC
DC DC dc
I Rd m V m V I
dt L L L (4.36)
40
The equations (4.26), (4.27), (4.32), (4.33) and (4.36) can be expressed in matrix form as:
1
11
1
0 0
0
1 1
1 10
3 30
2 2
0 0
0d0 0
dt0 0
0 0 0
0 0
0
i
i i
iisd sd
i i
sq sq
idcd cd
i i
cq cq
qDC DCi i
DCd q
DC DC DC
R
L L
RLI I
L LI I
LmV VC C
V V
mI IC C
Rm m
L L L
sd
sq
V
V
(4.37)
4.6.4 Model decoupling and linearization
In the mathematical model given in equation (4.37), coupling and non-linearity of the state
variables exist that can be resolved in the following way. Capacitor current ci has a relation with
DC current DCI and is given by c DCi mI where m is the modulating vector.
This relation can be extended into dq- form and is written as:
&
cd d DC
cq q DC
I m I
I m I
(4.38)
Similarly, DCI is also considered a source of nonlinearity in equation (4.38) of the model and is
needed to be linearized. Ignoring losses in resistance and converter switches, the power balance
equation can be used to settle down this nonlinearity as follows:
DC ACP P (4.39)
41
3
2DC DC sd sdV I V I (4.40)
3
2
DCDC DC DC DC sd sd
dIL I R I V I
dt
(4.41)
22 2 3DCDC DC DC DC sd sd
dIL I I R V I
dt (4.42)
22 3 2DcDC DC sd sd DC DC
dIL I V I R I
dt
(4.43)
2
23 2DC sd DC
sd DC
DC DC
d I V RI I
dt L L (4.44)
Assuming direct-axis voltage is coincident with the system voltage sd sV v and quadrature-axis
voltage is zero i.e. 0sqV , hence the model is:
2 2
0 0
0
0 0 0
0d0
1
01
0
11
0
11
0 0
30 0 0 2
0dt
00
0 0
i
i i
isd sd
i i
sq sq
cd
icd cd
ci
cq cq
iDC DCi
DCsd
DC DC
R
L L
RI I
L LI I
ICV V
ICV V
CI IC
RV
L L
0
0
0 0
0 0
0 0
1
0
i
sd
q sq
L
V
V
(4.45)
4.7 Power Analysis
The power delivered to the load can be viewed at different sections of the system. Conventionally,
a three phase AC power is supplied to the front end rectifier which converts it into DC. This DC
power is further transformed into high frequency AC power. The inverter is used for this purpose
and in induction heating switching is performed according to the resonant load circuit frequency,
42
commonly known as resonant frequency of the circuit. In an induction furnace, a few kilo watt to
Mega-watt power is transformed from line supply AC to DC and then to high frequency AC.
The instantaneous power at the source side is obtained by summing up the instantaneous powers
of all phases. It can be expressed as:
r r y y b bP v i v i v i (4.46)
For a balanced three-phase system, instantaneous power is constant. In dq-transformation, the
active and reactive powers can be written by taking the direct and quadratic axis of the component
as:
*3 3( )
2 2e dq dq d d q qP R V I V I V I (4.47)
3
2sd sd sq sqP V I V I (4.48)
*3 3( )
2 2m dq dq q d d qQ I V I V I V I (4.49)
3
2sq sd sd sqQ V I V I (4.50)
The active and reactive components are actually part of the apparent power. How much a power
is leading or lagging, it depends upon its angle.
To achieve a unity power factor, the q-component of supply current is set to zero i.e. 0sqI and
assuming that the source voltage sV is aligned with d-axis, hence sqV = 0. We then get:
3
2sd sdP V I (4.51)
0Q (4.52)
43
Chapter 5
Field Oriented Control
5.1 Overview
There are different topologies for feeding power to the load with various arrangement. Each
configuration has its own components and features. Major classical control techniques used in
resonant inverters are based on the load arrangement at the output of an inverter. Series resonant
load is fed by a voltage-fed inverter while parallel load is supplied by a current-fed inverter. The
control techniques remained involved for power control through various means; either directly or
indirectly. This research work focuses on the parallel-resonant load circuit feeding by a current-
source converter. In this chapter, phase angle control technique is discussed at first which is still
being used in industry for high power heating applications and then PWM based control is
presented.
5.2 Line-Commutated Rectifier
Line-commutated rectifiers generally use thyristor switches. A positive short duration gate-to-
cathode signal is provided to turn a thyristor ON. However, no external signal is provided for
switching OFF a thyristor (T), hence is called line-commutated rectifier. Phase-angle control
(PAC) technique is used to trigger a thyristor at a desired angle. A typical PAC strategy is shown
in Figure 5.1. In this control method firing of the angle can be from 0 to 180° theoretically, but
practically it is opted maximum up to 160°. Thyristors are turned on and off once in a cycle. The
power flow is controlled by the firing angle of the rectifier through PI controller [142].
In the DC link current control technique, the DC current reference is compared with the actual DC
current flowing towards the heating load. The comparator produces a difference signal which is
compensated by a PI controller. The firing pulses are basically generated to control the DC power
that is considered approximately equal to the load power. In this control scheme, the controlled
rectifier is triggered in such a way that DC current may track its desired reference; hence, the
output power is controlled indirectly by switching of the rectifier [60].
44
Figure 5.1: Phase angle control of a controlled rectifier feeding an induction heating load [142].
5.3 Forced-Commutated Rectifier
A forced-commutated rectifier differs from a line-commutated rectifier with the following main
reason: semiconductor switches have a provision of gate turn-off capability and can be turned on
and off when desired; hence, fully controlled rectifier was established through forced
commutation. The major benefit of forced commutation over line commutation is the
implementation of different modulation strategies. In a PWM converter, switches can be turned
on/off hundreds times in one time period whereas in a line commutated rectifier it is not possible
to do so and thyristor is turned on once in a cycle.
PWM-based converter has several advantages and following actions can be performed [69], [142]:
Reduction of harmonics both in voltage and current.
Active and reactive power are controlled through PWM forced commutation. Leading
power factor mode is also possible.
45
Voltage-source or current-source converters can be established easily.
Higher controlling capability.
Reversal of power is possible.
In forced commutation, a three-phase converter further can be configured into two types: voltage-
source converter (VSC) and current-source converter (CSC) as depicted in Figure 5.2 and 5.3.
Figure 5.2: PWM based voltage-source converter.
46
Figure 5.3: PWM based current-source converter.
In a current source converter, power can be reversed by the reversal of the DC voltage. In a voltage-
source converter, power is reversed by the DC link current reversal. The DC link current is
controlled through a feedback control loop. This method actually varies the voltage at the rectifier
output. A brief discussion of different PWM methods is given in the next section.
5.3.1 Pulse width modulation (PWM) based control
It is already discussed that in a pulse width modulation control, the power switches are turned on
and off several times in a cycle compared to a phase-angle control. A triangular wave is compared
with a DC signal and then switching pulses for converter are generated. In a simple PWM, the
pulses have the same width and are equally distributed. The output quantity i.e. voltage or current
is controlled by varying the width of the pulses.
5.3.2 Sinusoidal pulse width modulation (SPWM) based control
In a sinusoidal PWM modulation the pulses, having different widths and unequally distributed, are
compared to a simple PWM pattern. The sinusoidal PWM eliminates the lower-order harmonics
and is considered a better modulation pattern than a simple PWM control. In SPWM, a triangular
wave is compared with a sinusoidal waveform and then signals are generated to drive the converter
47
accordingly. Sinusoidal PWM also offers a higher power factor than that of a simple PWM
technique.
5.3.3 Space-vector based PWM control
The space vector-PWM is an advanced digital modulation scheme which works with vector-time
averaging approach and generates desired pulses for converter’s switches [63], [74], [132], [143].
The SVPWM modulation has some dominant features over other modulation techniques such as
optimized state selection, lower harmonics and easy implementation through a microprocessor
[72]; hence, this modulation strategy is selected and is illustrated below in detail.
The space-vector pulse width modulation is frequently used both for voltage-source and current
source topologies. For a current source converter, there is a slight modification of the conventional
SVPWM used in a voltage-source configuration. Like a voltage-source converter, there are six
active vectors in a current source converter, but three zero vectors are formed to freewheel the DC
current through the bridge. In a voltage-source topology, at any instant three phases are utilized by
the space-vector PWM, whereas in a current-source topology only two phases are used [70], [144].
Switching states:
In the current source operation , one switch in the upper legs (Q1, Q3, and Q5) and one switch in the
lower legs (Q2, Q4, and Q6) of the converter as shown in Figure 5.3 must be switched on at any
instant of time to ensure source connection to the load and obey:
3 5 2 4 61 1 and 1 Q Q Q QQ Q
These constraints give nine possible switching states for a current source configuration as shown
in Table 5.1.
48
Table 5.1: Switching states.
State Q1 Q3 Q5 Q2 Q4 Q6 ir iy ib
1 1 0 0 0 0 1 IDC 0 - IDC
2 0 1 0 0 0 1 0 IDC - IDC
3 0 1 0 1 0 0 - IDC IDC 0
4 0 0 1 1 0 0 - IDC 0 IDC
5 0 0 1 0 1 0 0 - IDC IDC
6 1 0 0 0 1 0 IDC - IDC 0
7 1 0 0 1 0 0 0 0 0
8 0 1 0 0 1 0 0 0 0
9 0 0 1 0 0 1 0 0 0
Based on this possible on/off combination of power devices, the switching states diagram of a CSC
is drawn in Figure 5.4.
One could notice that, at any instant, no two switches of upper or lower leg of the converter can
ever be ‘ON’ to avoid the short-circuiting of input phases.
The line currents can be written in a balanced three-phase system as:
cos tr pi I (5.1)
2cos t
3y pi I
(5.2)
2cos t
3b pi I
(5.3)
49
Current space vector is then defined as:
svI I jI (5.4)
where,
2 1 1(i )
3 2 2r y bI i i (5.5)
2 3 3( )
3 2 2y bI i i (5.6)
The converter states with respective line currents given in Table 5.1 can be used in equations (5.4)-
(5.6) to express the αβ- components which produce possible current vectors as follows:
61
2
3
j
DCI I e
(5.7)
Figure 5.4: The switching states diagram of a CSC.
50
36
2
2
3
j
DCI I e
(5.8)
The remaining current vectors 3 4 5, ,I I I and 6I can be found in the same way. The zero current
vectors are 7 8 9 0I I I .
These current vectors can be described in a general expression as follows:
(2 1)6
21,2,3,4,5,6
3
0 7,8,9
j i
DC
i
I e iI
i
(5.9)
Six active vectors generate hexagon in αβ-reference frame. This hexagon can be divided into six
sectors, as shown in Figure 5.5.
Figure 5.5: Switched vectors and sectors for a CSC.
51
Switching time calculations:
The space vector approach is based on the time calculations of the vectors in each sector. In Figure
5.6, assuming Iref is lying in a sector ‘i’, the adjacent active vectors Ii, Ii+1, and zero vector Iz with
their respective turn-on times Ti, Ti+1 and T0, are used to evaluate the reference current vector.
1 1 0. . . .ref s i i i i zI T I T I T I T (5.10)
1 0s i iT T T T (5.11)
A general relation for the desired vector turn-on times can be established by using equations (5.9)-
(5.11) as follows [12]:
_1
0 1
sin(2 ) cos(2 )6 6
sin(2 ) cos(2 )6 6
DC
refi s
refi
s i i
i iIT T
IT Ii i
T T T T
(5.12)
where, Iα_ref and Iβ_ref are the horizontal and vertical components of the reference current vector
Iref.
Figure 5.6: Reference vector Iref in sector i.
52
5.4 Space Vector PWM Based PI Control
A three-phase current-source converter with a feedback loop is depicted in Figure 5.7. A PI
controller can give a better response and removes the control error when desired target is constant
(i.e. DC value) in steady state.
Figure 5.7: SVPWM based control design for a current-source converter (CSC).
Hence, in this represented model, a PI controller is an effective solution with satisfactory
outcomes. It can be expressed as [145]:
0
1e(t) ( )d
t
t
i
u k eT
(5.13)
where (t) r(t) y(t)e , y(t) is the output DC current IDC and r(t) is the reference signal of DC
current.
The control strategy is shown in Figure 5.8 where DC current signal is compared with the reference
signal and their difference gives an error signal. The error signal is then fed to the PI controller.
The PI controller with SVPWM generates the gate drive pulses for the converter and produces
such a value of manipulated variable that the output current may track the reference value
53
accurately. As in a current-source converter, a DC current is desired to feed the load circuit, so the
control scheme supplies the DC current to the load. Power control has a significant role to achieve
an efficient and valuable product.
Figure 5.8: PI based DC-link current with SVPWM.
The power control of the converter is realized through space-vector switching of the converter with
a PI control technique; hence a controlled power is delivered to the heating load.
5.5 Load Model
The induction coil consists of inductive reactance and resistance. Both the inductive reactance and
the resistance of the coil have non-linear behaviour with respect to several parameters such as
frequency, properties of the charge material, and geometry of the coil as well as the work piece.
All these parameters ultimately change the resistance and inductive reactance of the material.
Magnetic permeability and electrical resistivity change non-linearly with the change in
temperature; hence, during heating cycle these both vary [146]. It is the desire of an end user that
the furnace should operate effectively in case of change in the size of material, production mixture
and properties of the material. At resonant condition coil capacitorX X , the equivalent load is modeled
assuming only resistive behaviour of the load as shown in Figure 5.9. However, the load changes
during heat treatment process due to variation in the parameters of the work piece and the coil
[42], [66]. The effect of this load variation is modeled by instantly connecting three different
parallel loads at different times and is shown in Fig. 5.10. The impact of this dynamic load
behaviour has also been observed through this control strategy.
54
Figure 5.9: CSC with a constant heating load.
Figure 5.10: Dynamic heating load.
55
Chapter 6
Model Predictive Control (MPC)
6.1 Overview
Model predictive control (MPC) is an advanced concept in modern control that optimizes the
process variables not only at present time but also considers these for future course of time [85],
[103], [147]. The predictive control law works on the mathematical model of a system, hence, a
plant’s model is essential to develop the MPC.This chapter first addresses the types of modelling
techniques to develop the system model then extends the discussion towards the linear quadratic
regulator and control strategy of the predictive control.
6.2 Predictive and Non-Predictive Controllers
The MPC is an increasingly growing advanced control technique which has been applied
successfully in thousands of applications. The reason of the MPC growing popularity is its
potential for the operational-constraints accommodation and future prediction of variables. On the
other hand, the control strategy of a non-predictive controller such as PID does not have such
characteristics. A short review of the development of control techniques is given in the following
paragraphs.
PID control was first formulated in 1922 by Minorsky and its practical control tuning was
presented in 1942 by Ziegler and Nichols [148]. In this type, only current process variables are
observed while in a model predictive control strategy both current as well as future process
variables are considered for the control action. Figure 6.1 illustrates the difference between the
non-predictive and predictive control methodologies [84]. It is shown that the model predictive
control works on current and future manipulated variables, observable disturbances and reference
signals etc. while a non-predictive control scheme works without any optimizer or predictive
block. The PID controllers have been employed in industry for a long period due to their simplicity,
technology maturity and easy implementation [149]; however, they do not perform effectively
because of their inheritance structure and approach. They were initially replaced by robust
56
controllers [145] and are now being replaced by model based predictive controllers [85]-[89]. The
model predictive control algorithm offers several advantages over PID control such as easy tuning,
minor overshoot, effective constraints handling, production’s cost reduction, good robust
behaviour in case of disturbances or parameters changes and nonlinear process control etc. It gives
better performance over a PID control when the system contains non-periodic features, dead time
and high frequency trajectories [85]-[86], [88], [150].
(a)
57
(b)
Figure 6.1: Block diagram of a) non-predictive controller b) predictive controller [84].
The linear quadratic regulator (LQR) is a classical control technique which optimizes a system
control efficiently to some extent but still needs improvement due to the following reasons: it
solves an optimization problem by a fixed window, and uses a long prediction horizon which may
create ill-conditioning problem. The other drawback in LQR is constraints handling [90]-[91].
The model based predictive controller was then introduced in control systems. It has some
advantages such as on-line problem optimization, computation of process input or manipulated
variables in each control interval, model dynamics information in a convenient mathematical form,
and anticipation as well as prevention of future constraints violation etc. Based on these significant
advantages the model predictive controller has expanded in a wide range of applications in
numerous areas, e.g. aerospace, food processing, chemical, metallurgy, automotive and furnaces
etc. [91].
6.3 Mathematical Model
A mathematical model of a system is basically the demonstration of the actual system in
mathematical language. How accurately a system performs? It depends on its mathematical model
58
which estimates the behaviour of the system. Generally, there are three ways to develop a system
model to design the predictive control [151]:
Unit step or LTI model – used in earlier modelling of the MPC and was applicable for a
stable system.
Transfer function model – used for both stable and unstable systems; however, this method
was not effective for a multi-variable system.
State-space model – very attractive technique; this design method replaced previous two
modelling approaches.
In this work, state-space formulations are employed to develop the MPC.
6.3.1 Linear continuous time state-space model
For a linear time-invariant (LTI) multivariable system, the continuous time state-space equations
are:
d
dx tAx t Bu t B v t
dt (6.1)
oy t C x t (6.2)
where x (t) is the state vector, u (t) is the input vector, v t is the input disturbance, and y (t) is the
output vector. Equations (6.1) and (6.2) are known as state equation and output equation of a
system respectively.
The general solutions of these equations are:
0 0
0
t tA t A tAt
dx t e x e Bu d e B dv
(6.3)
where Ate is the state transition matrix. The solution is formed by three terms: initial condition of
the model, control signal and constant term.
59
0 0
0
t tA t A tAt
o o o dy t C e x C e Bu e vd C B d
(6.4)
6.3.2 Discrete-time state-space model
The major benefit of a discrete-time representation is its easy computation on a digital computer.
Like a linear time-invariant system, the discrete-time state-space model for a multivariable system
can be written as:
1 1 111 12 1 11 12 1
2 2 221 22 2 21 22 2
1 2 1 2
11 12 1
21 22 2
1
1
1
1
s n
s n
s s ss s s sns s n
d
d
s
x k x k u ka a a b b b
x k x k u ka a a b b b
a a a b b bx k x k u k
b b b
b b b
b
1
2
2s sd d
v k
v k
b b v k
(6.5)
1 111 12 1
2 221 22 2
1 2
s
s
o o oso s
y k x kc c c
y k x kc c c
c c cy k x k
(6.6)
Here subscripts s, n and o are used for the state, input and output variables. Coefficient matrices
A, B and Co have s s , s n and o s dimensions respectively.
6.3.3 System mathematical model
The model developed in chapter 4 can be shaped into the state space model taking the state, output
and input variables from the matrix. Recalling the system model from Equation (4.45) which is
given by:
60
2 2
0 0
0
0 0 0
0d0
1
01
0
11
0
11
0 0
30 0 0 2
0dt
00
0 0
i
i i
isd sd
i i
sq sq
cd
icd cd
ci
cq cq
iDC DCi
DCsd
DC DC
R
L L
RI I
L LI I
ICV V
ICV V
CI IC
RV
L L
0
0
0 0
0 0
0 0
1
0
i
sd
q sq
L
V
V
(6.7)
where state variables 2T
sd sq cd cq DCI I V V I x , cdI and cqI are the control or input variables
T
cd cqu I I , sdv and sqv are system input disturbances
T
sd sqv V V ,
2
DCI and sqI are chosen
as controller outputs i.e. 2T
sq DCy I I .
Then the state space model in a typical matrix form can be written as:
dx t Ax t Bu t B v t (6.8)
oy t xC t Du t (6.9)
where 5 1x represents the state vector, 2 1u the input vector and 2 1y is the output
vector.
61
1
011
0
101
0
11
0 0
30 0
0 0
0
00 0 0
0 00 0 ,
0 0
0 0 00
0 00 0
0 2
i
i
d
i
i
ii i
i
i
i
i
DCsd
DC DC
A B and B
R
L L
R
LL L
CC
CC
RV
L L
There is not direct feeding hence second term of the output equation (6.9), commonly known as
direct transmission part, is ignored.
6.4 Linear Quadratic Regulator (LQR)
Before starting the model predictive control, a short review of liner quadratic regulator (LQR) is
presented which is considered a well-known optimal control regulator that can be used to minimize
the objective function [152]-[153]. The LQR control law can be found through the state-space
model in discrete form assuming one single time delay and is illustrated in Figure 6.2. It can be
described as:
LQRu k x k (6.10)
where x k is the state variable matrix and LQR presents the state feedback gain matrix that is
obtained through Ricatti- equation solution [100] and is given by:
1B( )LQR
T TB B A
(6.11)
1( B )( )T TTQ B B AA B
(6.12)
where Q and are the weight matrices that penalize the state dynamics and actuation effort. The
solution of equation (6.12) is achieved through recursive calculations.
62
Figure 6.2: Fundamental structure of a liner quadratic regulator (LQR).
The cost function of a linear quadratic regulator assesses the distance of system states from their
desired target and then reduces the error through manipulated variables adjustment. One main
difference between MPC and LQR is their cost function implementation over a prediction horizon.
In LQR, this prediction horizon is infinite which may cause the ill-conditioning issue and is given
by in the following expression [152]:
1
.x | x | | |1 1T
LQR
h
J k h k Q k h k u k h k u k h k
(6.13)
The second drawback of LQR is constraints handling problem. If constraints become active in a
multivariable system, then state feedback controller deviates from its desired behaviour and the
plant may move towards the destabilization.
6.5 Model Predictive Control
Model predictive control is a forward-looking type optimal control technique in which a set of
manipulated variables are computed in such a manner that they operate the process in optimum
steady state without violating constraints [82], [95]-[97], [154].
The MPC development can be classified into four generations [155]. First generation started from
1970’s which used step and impulse response linear models, quadratic cost function and
constraints treatment on ad-hoc. Second generation evolved using state-space models for a linear
system: quadratic cost function and constraint problems were solved by quadratic programming.
63
In early 1990’s third generation was established by insertion of soft and hard constraints. In late
1990’s the fourth generation was developed for nonlinear systems, and the system stability was
also guaranteed through this development. Initially, MPC was employed in process industry where
it was considered suitable for the slow process control. Different aspects of the predictive
controller were changed from slow process applications to medium- and then fast-dynamic
applications. During the last decade, the MPC with fast algorithms was introduced and then it
started spreading to various applications.
The reason of the MPC growing popularity over conventional PID and PI controllers is its
capability for operational-constraints accommodation and good performance for a designed system
[86]-[89]. It works primarily on receding horizon strategy as shown in Figure 6.3(a). The control
error is minimized not only at the current time point but also at many steps ahead from the current
time. A sequence of control inputs is predicted and the first element of this sequence is applied as
input while other elements are discarded and time is moved ahead one step. Similarly, in the next
time k+1, an updated optimal sequence of control is predicted and again its first element is
implemented and others are rejected. It is repeated till the end of the job. This phenomenon is
known as ‘receding horizon control’ [83], [103]-[106], [156].The benefit of the predictive strategy
is to operate the scheme towards the desired reference trajectory in order to make changes in future
manipulated variables and controlled signals.
The main elements of the MPC are the model of a system, optimizer and constraints as shown in
Figure 6.3(b). A system’s model is developed by applying physical laws or using system
identification techniques. The prediction based model depends upon the prediction attained from
system variables. In optimizer, manipulation of control signals are adjusted in such a way that the
cost function should be reduced as low as possible [84]-[85], [88]-[89], [115]-[118], [157].
64
(a)
(b)
Figure 6.3: Model predictive control a) Functioning pattern [85], b) Control structure [87].
65
6.6 Generalized Predictive Control (GPC)
Generalized predictive control is an efficient control technique that works on the principle of
receding horizon control [114]-[115] and its general block diagram is shown in Figure 6.4. It
predicts the future response of a system and accordingly reduces the error function.
Figure 6.4: General block diagram of generalized predictive control [115].
The state-space model in discrete form is recalled again:
1 dx k Ax k Bu k B v k
oy k C x k Du k
In order to ensemble the design with an embedded integrator, the discrete model is changed into
the augmented model which is basically an incremental model and can be illustrated by difference
equations. The augmented discrete-time state-space model is then used to develop the generalized
predictive control.
The discrete equation, in terms of difference equations, is given by:
1 dx k A x k B u k B v k (6.14)
66
where
1 1 ,x k x k x k
1 ,u k u k u k
1v k v k v k
As augmented model uses embedded integrator so the input disturbance term v k is assumed a
constant term and omitted for simplification of the analysis. The output variables can be formulated
in terms of the state variables and control signals as follows:
1 1oy k C x k (6.15)
1 oy k C A x k B u k (6.16)
Finally, the augmented state-space model in terms of new state variables is formed as:
1 0
1
T
s
o oo o o
x k x k BAu k
y k y k C BC A I
(6.17)
0s n n
o
x ky k I
y k
(6.18)
where 0s is a zero row vector matrix with o s size.
The resulted state-space model is attained and control method now takes control update Δu k as
input rather than u k [85]-[88], [116], [151].
6.6.1 Model prediction
The developed state-space model is used to predict the look-ahead response of the model through
future control parameters. The state variables are determined by successive calculations of the state
equations using elements of future control sequence u in a defined horizon CH :
67
1x k k Ax k B u k
2 1 1x k k Ax k k B u k
….. 2 1A x k AB u k B u k
1 2
1 1p p p p cH H H H H
p cx k H k A x k A B u k A B u k A B u k H
(6.19)
From the predicted state variables, the output variables are determined as:
1 o oy k k C Ax k C B u k
22 1o o oy k k C A x k C AB u k C B u k
1 2
1 1p p p p cH H H H H
p o o o o cy k H k C A x k C A B u k C A B u k C A B u k H
(6.20)
Then output and control sequence vectors are:
1 2T
pY y k k y k k y k H k
1 1T
cU u k u k u k H
The output equation based on the control sequence U can be written in a matrix form as under
[85], [151], [158]:
Y x k U (6.21)
68
where,
2
3
o
o
o
Hp
o
C A
C A
C A
C A
, 2
1 2 3
0 0 0
0 .. 0
.. 0
. P C
o
o o
o o o
H HHp Hp Hp
o o o o
C B
C AB C B
C A B C AB C B
C A C A B C A B C A B
1 1T
T T T
cU u k u k u k H
(6.22)
CH and PH represent control and prediction horizon respectively; prediction horizon PH is
always kept greater than or equal to control horizon CH [159]. The incremental control signal
u is considered only up to CH samples and is assumed zero for the remaining samples.
These tuning parameters are adjusted in such a mode that the manipulated and controlled variables
will give good performance results. Simulation outcomes, in reference [84], demonstrate that
overshoot may occur for a high-order system if prediction horizon is too short. On the other hand,
slow control response will result if prediction horizon is too long. A long range value of prediction
horizon gives fast control with a minor overshoot.
6.6.2 Cost function
The quadratic cost function in various forms is actually used in predictive control design [158].
For a desired set point tracking, an optimized cost function in MPC includes a penalty on the
predicted error and manipulated variables.
MPC computes the control-signal sequence for the future trajectory in a defined prediction horizon
pH and control moves CH for manipulated variables pcH H . It implements only the first
element of this sequence [85].
69
The cost function is given by:
1
1
0
| | | |
| |
HpT
MPC
h
HcT
h
J y k h k r k h k Q y k h k r k h k
u k h k u k h k
(6.23)
where |k h k symbolizes the value predicted for time k h , 1| oy k h k is the predicted
value of output, 1| or k h k is the desired reference value and 1| ou k h k is the
future input control update i.e. 1| 2 | |CU u k k u k k u k H k &
| 0u k h k ch H
Using this prediction technique, output and manipulated variables of the system can be predicted.
The core objective of the first term in this cost function is to minimize the difference between the
predicted output and the reference while the second term gives its reflection to the dimension of
Δu in order to decrease the cost function. The advantage of this predictive method is that it is easy
to develop an optimized cost function. On the other hand, its practical implementation is a little bit
hard for ordinary controller due to mathematical iterations; however, due to the development of
advanced controllers such as DSP and FPGA controllers, its execution is not an issue [82]. The
induction furnace is not a fast response process where line-commutated rectifiers are still being
used in industry for large-power applications, a predictive control is a step ahead control technique
to enhance the system performance.
The cost function J can also be defined in terms of equation (6.13):
1
1
0
| | | |
| |
HpT
MPC
h
HcT
h
J y k h k r k h k Q y k h k r k h k
u k h k u k h k
(6.24)
Weight matrices are block diagonals matrices and are given by:
1 2 pHQ diag Q Q Q (6.25)
70
1 2 1cHdiag (6.26)
In the cost function (6.24), the future output error depends on the first term while control action
on the second term [159]. The weight matrix is defined by a user and it shows the effect of input
and output of a system on the cost function. The predictive system always tries to bring the
predicted output equal to the set point signal so that the error function may approach zero.
An optimal u , which will minimize the cost function ‘ J ’ at a reduced value, can be found simply
using and .
1
T TQu x kQ
(6.27)
Interconnecting the control signal sequence with receding horizon principle and is given by:
1
0 0 T TQ Qu I x k
(6.28)
Considering these initial terms equal to the MPC gain and then the expression will be:
mpcu x k (6.29)
This MPC unconstrained control expression looks like a state feedback control law given in
equation (6.10) except the difference of the prediction horizon.
The main purpose of the term 1
TQ
given in equation (6.28) is to reduce the error function
to a least value. The actual input signal applied to the plant at any sampling instant k is the
addition of previous input sample (k-1) with increment u k as:
(k) (k 1) (k)u u u (6.30)
Here ‘n’ is the number of inputs. The control move is shifted and this optimization procedure is
repeated for the next sampling instant and so on. A block diagram of generalized predictive
controller without constraints is shown in Figure 6.5.
71
Figure 6.5: Generalized predictive controller without constraints.
6.6.3 System constraints
System constraints are significant features of the modern control architecture and express a
difference between field-oriented control and MPC. Each control application has certain
limitations; these have to be satisfied. In MPC, it is possible to outline the boundaries for control
signal, incremental control and output. A quadratic cost function with linear constraints are
addressed in reference [160]-[162]. There are three kinds of constraints; hard, soft and set point
approximation [91]. Hard constraints demonstrate the physical boundaries of a process such as
actuator extreme points and must be avoided from any violation of these limits. Compared to hard
constraints, soft constraints violation may happen but minimize them at the expense of objective
functions penalty, product quality or product cost etc. Set point approximation is used to handle
each soft constraint and quadratic penalty is applied on both sides of the constraint.
The cost function with system constraints is described as:
1
1
0
| | | |
| |
HpT
MPC
h
HcT
h
J y k h k r k h k Q y k h k r k h k
u k h k u k h k
(6.31)
Subject to
min max 0, 1cu u k u h Hh
min max 0, 1cu u h k u h H
72
min max 1, py hk y Hy h
In this model, constraints are only imposed on the control signals and output variables.These
constraints are defined as follows:
The system output constraints:
0.5 0.5 (A)sqI 0 1050 (A)DCI
The system input constraints:
0.1 0.1 (A)cqI 0 1 (A)cdI
6.6.4 Constraints benefits
The major benefit of model predictive control over other control laws is its constraints handling.
These constraints benefits are outlined below.
In this work, output constraints are used to counter the following two incorrect conditions in order
to prevent the system from damage and control loss:
1) The operator may wrongly operate the system or the output quantity such as DC current
may cross the upper limit.
2) If this current-fed inverter is used for melting of different materials, then these materials
may attract more power than the rated power. In such a situation, output constraint prevents
it to meet the full needed power of the material and runs it under the maximum defined
constraints, so the power remains within the defined limits.
Similarly for input and incremental constraints major benefits are:
1) If inverter is operated with incorrect firing signal sequence then inverter might fail and
power supply is short circuited; the input constraints do not allow high current flow than
the defined constrained current.
2) Similarly, constraints protects the rectifier from short circuiting due to incorrect firing
signal sequence.
73
3) Filter failure may also create short circuiting; input and incremental constraints prevent the
system from collapsing.
6.6.5 Significant features
Main attractive features of GPC over other conventional field oriented controllers are as under:
MPC is an advanced concept in modern control that optimizes the process variables not
only at present time but also considers these for future course of time. In a conventional
field oriented PI controller, only current process variables are observed.
MPC has a potential for the operational-constraints accommodation compared to a PI
controller.
MPC controls multi-input multi-output (MIMO) system, while PI controller cannot do so.
Other advantages of MPC over PI control are its easy tuning, good reference tracking, more
robust behaviour and its suitability for non-linear process etc.
74
Chapter 7
Control Algorithms: Results and Discussions
7.1 Overview
The main objectives in an induction heating, are power control and inverter switching. As
discussed in earlier chapters, a constant current is desired to feed the parallel resonant load circuit.
The aim of control algorithms presented in this work is to maintain a constant current to the heating
load and power regulation by DC-link current. The importance of the constant current can be
viewed by simply energizing the resonant-load circuit through an unvarying current source. Hence,
at first, the dynamic behaviour of the parallel resonant circuit is analysed in section 7.1 by
supplying a DC current at the input of the inverter. Then control techniques (i) DC link current
control by SVPWM-PI scheme and (ii) DC link current control by generalized predictive control
(GPC) are illustrated in the rest of this chapter. The results of the control algorithms with their
features have been discussed in detail. The advantage of smooth DC current through these control
techniques is actually provision of a uniform heating to the work piece and is desired for a good
quality product.
The presented control algorithms can be validated due to the availability of an extensive choice of
design tools. For example, one excellent opportunity to design and simulate a system with bit and
cycle accuracy is MATLAB-Simulink which offers a good picture of system response [163].
7.2 Dynamic Behaviour of a Current-Fed Inverter
Analytical illustrations of the model given in chapter 3 can be investigated in Simulink to observe
the transient and steady-state behaviour. It is essential to mention here that analysis of the current
fed inverter in this section is shown only to emphasize the behaviour of the load by a constant
current if fed at the input of the inverter and is not a part of contribution in this thesis. A few
kilowatt load circuit is connected at the output of the inverter. The parallel resonant circuit is
developed by connecting capacitor and inductor in parallel. The inverter input is replaced with an
equivalent current source which supplies a constant current to the resonant inverter configured
75
with parallel load. The time domain results have been presented. The values of the circuit elements
are given in Table 7.1. Based on these values the input current, inverter output voltage, effective
voltage, phase angle modes of operation, coil and capacitor currents are shown in Figures 7.1-7.6.
A constant input current is supplied to the inverter as shown in Figure 7.1. The maximum voltage
at inverter output reached a value of 417 V depicted in Figure 7.2. The quality factor provides a
relation between the output and effective voltage across the load, as discussed in the parallel load
circuit analysis, hence the effective voltage value ,eff peakV = 39.6 V is achieved. This value can
be verified from Figure 7.3. Figure 7.4 shows the switching of the inverter at in phase and leading
mode.
Table 7.1: Parallel resonant load circuit parameters.
Parameter Value Parameter Value
Input current 20 A Equivalent load resistance, R 0.15 Ω
Frequency, rf 10 kHz Capacitance, C 10.1 μF
Quality factor, fQ 10.48 Equivalent inductance, Leq 25 μH
Figure 7.1: Inverter input current.
76
Figure 7.2: Output voltages a) Transient state, b) Steady state.
77
Figure 7.3: Effective load voltages a) Transient state, b) Steady state
78
Figure 7.4: Output voltages and load current with phase angles a) In phase, b) Leading.
As Figure 7.3 presents , 39.6eff peakV V so corresponding power is 5.2 kW. This power can be
verified by coil current i.e. ,L peakI = 264 A in Figure 7.5. In Figures 7.5 and 7.6, one can also see
that the coil and capacitor currents have opposite peaks at the same time once the inverter is
triggered.
Figure 7.7 presents the variation of the quality factor from 7 to 10.48 with respective change in
resistance values of 0.224, 0.196, 0.174, and 0.15 keeping other parameters same. The change in
load resistance from high to a low value gives respective low to high quality factors as described
79
in system modeling. Similarly, change in quality factor provides a direct relation with power, hence
analytical analysis are validated in Figure 7.7 where power is varied from 3.5 kW to 5.2 kW.
Figure 7.5: Coil current a) Transient state, b) Steady state.
80
Figure 7.6: Capacitor current response a) Transient state, b) Steady state.
81
82
Figure 7.7: Output voltage and load current with different quality factors a)Qf = 7, b) Qf = 8,
c) Qf = 9, d) Qf = 10.48.
It can be concluded from these results that a smooth and sharp response of the model without any
spike or overshoot has been attained in transient state, which indicates that the simulation outcomes
offer a good agreement with analytic results stated in chapter 3. A summary of obtained results at
inverter output are given in Table 7.2 in response of 20 A DC current at the input.
Table 7.2: Summary of obtained results at inverter output.
Parameters Mathematical expressions Theoretical results Simulation results
Quality factor 1f
LQ
R C
10.48 10.48
Effective load
voltage
,
,21
o peak
r peak
f
VV
Q
39.6 V 39.0V
Output voltage 2 2
0 L RV V V 415 V 417 V
Output Power 2
2
LoP R
i
5.2 kW 5.2 kW
83
7.3 SVPWM-PI Based DC-link Current Control
This section presents a power control scheme of a current-source converter (CSC) which delivers
a constant current to the load for induction-melting applications. The proposed control scheme
with SVPWM pattern regulates the power of a high Qf - resonant load by controlling the DC current
according to the defined target. The PI controller adjusts the manipulated variable by SVPWM in
such a way that the error signal is reduced to a minimum value and a constant current is maintained
uninterruptedly for the load. In order to validate this constant current requirement to the load, the
output power analysis of the resonant inverter is also carried out in this work. The system’s
parameters are given in Table 7.3.
Table 7.3: System parameters.
Parameter Value Parameter Value
Input voltage (3-phase) 380 V DC smoothing reactor 25 mH
Frequency 50 Hz DC load resistance 0.8 Ω
Input filter inductance 6.05µH PI controller gain term 1.3
Input resistance 20 mΩ PI controller integral term 56
Input capacitance 50 µF Coil inductance 452 µH
Equivalent load resistance 0.02 Ω Capacitance bank 8.95 µH
Switching frequency 10 kHz Modulation index 0.93
Resonant frequency 2.5 kHz Sample time 2e-6 s
7.3.1 Fixed load
Figure 7.8 shows the DC voltage DCV and current DCI at the rectifier output. The DC current
reference is set at 320 A; the actual DC current reaches its steady state value within a short time
and the respective rectified DC voltage also attains its steady state value i.e. 256 V; hence, power
84
Figure 7.8: Rectifier output a) DC current, b) DC voltage.
is obtained simply by the multiplication of the voltage and current. Three-phase line current and
voltages are also shown in Figure 7.9 assuming a constant reference.
Figure 7.9: (a) Three-phase line currents, b) Three-phase line voltages.
Figure 7.10 shows the reference current and the actual current: this illustrates the effectiveness of
the control scheme for the target’s tracking.
85
Figure 7.10: DC current and reference.
Similarly, it can be seen clearly in Figure 7.11, with the change in reference value of the DC
current, respective DC voltage is also altered; and ultimately power is moved to a new value
according to the relation DC DC DCP V I . At the beginning i.e. t = 0 s, reference current was set at
320 A; then at 0.5 s, it was reduced to 50 % i.e. 160 A. One could notice from the figure that a new
power value is achieved by varying the DC current reference.
Figure 7.11: DC voltage.
Figure 7.12 illustrates the output quantities of the inverter which are helpful for the validation of
parameters analysis given in the following equations.
86
,
,21
o peak
r peak
f
VV
Q
(7.1)
, ,L peak f r peakV Q V (7.2)
The output power is found as follows:
2
,
2
r peak
o
VP
R
(7.3)
2
,
22 (1 )
o peak
o
f
VP
Q R
(7.4)
Similarly, the current flowing through the coil is used to find the output power as:
2
,
2
L peak
oP Ri
(7.5)
Here, peak values are considered for accurate analysis; however, in a practical system effective
values of these quantities are preferred. In Figure 7.12 (a) the square waveform represents the
current flowing through the inverter and sinusoidal waveform expresses the output voltage which
is 405 V. In Figure 7.12 (b) & (c) voltages across active and reactive components are seen as
57 V and 401 V respectively. Putting respective quantities in equations (7.1) and (7.2) same results
are found as in Figures. Similarly, looking at Figure 7.12 (d), inductor current has been observed
as 2855 A. The output power i.e. 81 kW is determined by using the power relations presented in
equations (7.3)-(7.5).
87
Figure 7.12: Current-fed resonant inverter a) Output voltage and current, b) Active component of output
voltage, c) Reactive component of output voltage, d) Resonant current flowing through the coil.
The above analysis presents the response of the system variables by assuming a fixed load
connected at the output. However, load parameters vary during the heat process, a dynamic load
is modelled in the next section.
7.3.2 Dynamic load
The previous section analysis has focused on the power control study for a constant load while the
following results show that CSC supplies a constant current to the induction heating even in
changing load conditions. The output load is instantly varied through a dynamic load model as
88
shown in Figure 7.13. The control scheme regulates the power at desired target by DC current
reference adjustment and also maintains an uninterrupted constant current supply to the dynamic
load. The main benefit of this analysis is to observe the anti-disturbance ability.
Figure 7.13: Load dynamic model.
The system parameters are same as given in Table 7.3 except the PI tuning parameters and the
equivalent load resistance i.e. 0.8 Ω with 25 % variation. The tuning parameters of PI controller
are presented in Table 7.4.
Table 7.4: PI controller tuning parameters.
Parameter Value
Kp 1.27
Ki 62
Initially, the control response of the system is investigated with the change in step reference. The
change in target value at t = 0.4 s and at t = 0.7 s is shown in Figure 7.14: both the DC current and
DC voltage change accordingly.
89
Figure 7.14: a) DC current, b) DC Voltage with change in reference at t = 0.4 s and t = 0.7 s.
As change in reference value either decreases or increases the DC current, so the respective change
in the input line current at the same time setting can be observed as depicted in Figure 7.15
according to the expectation.
Figure 7.15: Three-phase line current when reference changes at t = 0.4 s.
A 0.8 Ω load resistance is connected at the output of the system. The DC current and voltage
waveforms are shown in Figure 7.16 where smooth responses without any overshoot are observed
from transient to the steady state value.
90
Figure 7.16: a) DC current and b) DC voltage with a constant load.
To observe the dynamic load model impact on the system’ control, the output load has been
decreased up to 25 % of the nominal value through a step change. The system response is
presented in Figure 7.17, where load has been changed at t = 0.3 s and t =0.7 s. The control scheme
is supplying constant current continuously to the load as in Figure 7.17 (a), whereas DC voltage is
decreased and increased with the respective decrease and increase in resistance value shown in
Figure 7.17 (b), so power is preserved by DC voltage. This makes sense; hence, one can see clearly
that in spite of load variation, the presented control is remained stable.
Figure 7.17: DC a) Current and b) Voltage where load changes at t= 0.3 s and t= 0.5 s.
91
7.4 GPC based DC-Link Current Control
In this power control application, the first objective is to follow the operator set-point DC current
in order to meet the heat process requirement and the second is to maintain the system response
within the defined constraints. For this, the reactive component of the supply current always
approach towards a minimum value ideally equal to zero and the DC current approaches its desired
reference i.e. a positive value. The results are discussed in three aspects: reference tracking,
response of the system with defined constraints and model results without constraints. The system
parameters are: input voltage 380 V, frequency 50 Hz, input filter inductance 6.05 µH, input
resistance 20 mΩ, input capacitance 50 µF, DC smoothing reactor 25 mH, sample time 0.1 s and
equivalent load resistance 0.8 Ω. The GPC’s parameters are: Hp =10, Hc = 4, Q = 200 for reactive
component of input current, Q = 20 for DC current, = 0.1 for both input variables. Initially, the
current sqI is set at 0 while DCI is set at 500 and 1000 A in different time instants. The tracking of
DC current and reactive component of supply current with respective response of control signals
are shown in Figure 7.18. One can see clearly that fast and stable responses have been achieved.
The desired heating power at output is controlled through DC current flowing into the heating load.
92
Figure 7.18: a) Tracking of DC current reference b) Tracking of reactive component of supply current,
c) Response of direct-axis current Icd at input, d) Response of quadrature-axis current Icq at input.
The reference value of DCI is set at 1000 A i.e. peak load of the system and observe the error
signals in steady state. It has been seen in Figure 7.19 that steady state error exist below 0.4 %,
which is a minute error. The error in reactive component of the supply current is also very low. It
shows that GPC offers an efficient tracking of the target.
Figure 7.19: Error signals at peak load in a) DC current, b) Reactive component of supply current.
93
The constraints for the quadrature current and DC current at output are defined as:
0.5 0.5sqA I A 0 1050DCI A
The DC current reference is then set at a 1200 A which is higher than the defined constraint. It can
be seen in Figure 7.20 that current tracks the desired trajectory effectively up to its defined
constraints i.e. 1050 A but did not violate the constraint.
Figure 7.20: Tracking of references in defined constraints at output a) DC current, b) Reactive component
of supply current.
The constraints for the control signal currents are defined as:
0.1 0.1cqA I A 0 1cdI A
To see the effectiveness of the input constraints, the output constraints are disabled temporarily
and the DC current reference is set at 1300 A. It can be observed from Figure 7.21 that the output
variable DCI was limited by input constraint cdI . It has tracked the desired reference up to its
defined constraint and did not violate the constraint.
94
Figure 7.21: Control response under input constraints a) Input current, b) DC current.
The predictive control significant advantage over other field-oriented control is its constraints
handling as discussed earlier. If this feature is disabled, then it only tracks the desired reference
like ordinary controllers. The results of the GPC without any constraints is also shown in Figure
7.22 where it tracks DC current 1200 A i.e. higher than the defined constraint of DCI in previous
case. The magnitude of control signal at peak load is shown in Figure 7.23.
95
Figure 7.22: Output and Input variables without constraints a) DC current, b) Reactive component of
supply current, c) Direct-axis component of control signal Icd, d) Quadrature-axis component of control
signal Icq.
Figure 7.23: a) Magnitude of control signal
The discussed results represent that both PID and MPC have reduced the error between desired
trajectory and system actual output. Model predictive controller outperforms compared to a
conventional field-oriented controllers in response and tuning. Moreover, MPC’s additional
potential over PID is constraints handling, so it is considered a better control solution.
96
7.5 Variation in GPC’s Parameters
The results discussed in section 7.4 are based on the optimized parameters obtained from analysis
of the model response. In this section, we express the response of the model with variation in
parameters’ values. At first, sampling time is changed from 0.1 sec to 1 sec. The results are shown
in Figure 7.24 where slow response is received in all results. However, sampling time depends
upon the type of application and switching limitations of hard devices.
The weight factors are tuning parameters of GPC to adjust the variables under the defined targets.
If a system gives a poor response for its output and manipulated variables, then the weight
parameters are changed into high values. A low value of Q is selected for the reactive component
of supply current while other parameters are kept same as in previous section. It is seen that sqI
has gone out from the defined constraint as depicted in Figure 7.25. Hence, adjustment of weight
elements is essential to get the desired response.
97
Figure 7.24: System response with a large sampling time a) DC current, b) Reactive component of supply
current, c) Direct-axis component of control signal Icd, d) Quadrature-axis component of control signal Icq.
98
Figure 7.25: Impact of weight factor a) DC current, b) Reactive component of supply current, c) Direct-
axis component of control signal Icd, d) Quadrature-axis component of control signal Icq.
The presented induction heating process takes a certain time to complete the process. Initially, it
is started at a low value power and is increased gradually with the pre-defined intervals.
Simultaneously, the temperature of the work piece continuously increases until the desired
temperature is achieved and the process is stayed there for a short time. Then it is again decreased
to a low power value. This trend is shown in Figure 7.26 with a varying reference of DC current
at different time instants in order to get the different power levels.
99
Figure 7.26: Variation in power’s level a) DC current, b) Reactive component of supply current, c) Direct-
axis component of control signal Icd, d) Quadrature-axis component of control signal Icq.
100
Chapter 8
Conclusions
8.1 Conclusions
The main objective of this research has been to develop a power control strategy of a medium
frequency induction furnace. The work done in this thesis along with control patterns are
summarized in following paragraphs:
At first, the transient and steady state responses of the current-fed resonant load model were
demonstrated to emphasize the importance of a constant current at the input. A ripple-free DC
current was applied at the input of the inverter and a phase angle was used to operate the inverter
according to the load requirement. The resonant load arrangement formed a resonant frequency
which was tracked and the inverter bridge was operated in all modes i.e., capacitive, inductive and
resistive mode. The analytical analysis of a few kilowatt load model was verified through
simulation results; the outcomes showed a good agreement due to the provision of a smooth DC
current at the input.
The phase angle control scheme employed in line commutated rectifier and forced commutated
rectifier with typical topologies were described. Different modulation patterns were addressed. A
space-vector pulse width modulation based PI-control was selected to control the DC-link current.
A current-source converter feeding parallel resonant load with SV-PWM based control strategy
was modeled in Simulink. A constant current was supplied to the load circuit through this strategy.
An attractive response of the control technique was received for reference tracking. A dynamic
load model was also established by insertion of different loads instantly in the on-line condition.
The control approach maintained an uninterrupted constant current during dynamic load situation
and also regulated the power of the system effectively.
The state space model of the current-source converter feeding an induction heating load was
formed based on the linearized model given in chapter 4. The linear quadratic regulator i.e. a well-
known optimal control technique is studied and its drawbacks were discussed. The generalized
predictive control cost function and system constraints were expressed. It is obvious that model
101
predictive control outperforms over linear quadratic regulator in terms of ill-conditioning problems
and constraints handling. Hence, generalized predictive control was focused in this work. The GPC
was employed to control the active and reactive power flow of the current source converter. The
DC-link current and input quadrature current was used to regulate the active and reactive power,
respectively. Through this feedback control loop both currents were maintained at their set values.
The predictive control results were shown with constraints and without constraints. The GPC has
given desired reference tracking and constraints handling for the multivariable system. The results
have verified the effectiveness of the application of presented control algorithm for the smooth
regulation of power within defined constraints. Significance of the constraints based control
algorithm compared to a control law without constraints has also been observed.
In conclusion, both control schemes, SVPWM-PI technique and generalized predictive control,
were presented to track the DC current reference and power regulation by means of DC-link
current. These control strategies have offered a constant current flow into the work piece which
is essential to produce a good quality product with a reduced cost. Both control techniques have
tracked the desired reference effectively. The PI control method has a benefit of simplicity and
easy implementation but cannot meet the advanced control requirements. However, MPC has a
potential both in reference tracking and constraints handling in an efficient manner.
8.2 Future Works
In the extension of this work, following topics are proposed for future possible research:
This work recommends the application of GPC technique to the actual industrial process
and analysis of the results accordingly.
MPC is a MIMO based control method, so integration of power and temperature can be
achieved within a single module through multimodal approach.
Some heating process needs to stay about few minutes to an hour at final temperature,
operator does it by adjusting the power manually but it cannot be accurate. A hybrid model
which works in this mode would improve the efficiency and quality of the heated product.
DC link current can be predicted by MPC in advance to maintain it at a constant value,
hence, the size of the DC reactor could be reduced to minimize the energy loss.
102
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Appendix-A
Various coordinates’ matrices are described in different reference frames i.e. abc, αβ and dq in
chapter 4 and their simulation models are presented in this appendix.
A.1 abc-Reference Frames into dq-Coordinates
Simulink model of the abc-reference frames into αβ-coordinates is shown in Figure A.1.1
Figure A.1.1: Simulation of abc-reference frame into αβ- transformation.
The Figure A.1.1 is further extended into dq-reference frame and is shown in Figure A.1.2.
Figure A.1.2: Simulation of αβ -reference frame into dq- transformation.
122
Direct transformation from abc-reference frames into dq-coordinates is shown in Figure A1.3.
Figure A.1.3: Simulation of abc -reference frame into dq- transformation.
123
A.2 dq-Coordinates into abc-Reference Frame
The inverse transformation from dq-coordinates into abc-reference frame is also possible by
converting dq into αβ -reference frame and then into abc- frame. Figure A.2.1 depicts the dq-
transformation into αβ.
Figure A.2.1: Simulation of dq-reference frame into αβ - transformation.
Similarly, the αβ - reference frame is further extended into abc-reference frame and is shown in
Figure A.2.2.
Figure A.2.2: Simulation of αβ -reference frame into abc- transformation.
The direct transformation from dq-coordinates into abc-reference frame is obtained as given in
Figure A.2.3.
124
Figure A.2.3: Simulation of dq -reference frame into abc- transformation.
125
Appendix-B
B.1 Parallel Resonant Load Model
The state space model of the inverter with resonant circuit is described in chapter 3. The
simulation model of the parallel resonant load is shown in Figure B.1.1.
Figure B.1.1: Simulation model of the parallel resonant load.
126
B.2 CSC with Equivalent Load Model
Current-source converter based equivalent resonant load in dq- reference frame is illustrated
in chapter 4. In this appendix, Simulink model of the converter with load is presented. The
non-linear and coupled mathematical model of the system is initially developed and is shown
in Figure B.2.1.
Figure B.2.1: Simulation model of the non-linear model of CSC with equivalent load.
The linearized model of the system is then developed and simulation model of the circuit is
depicted in Figure B.2.2.
127
Figure B.2.2: Simulation model of the linearized model of CSC with equivalent load.
128
Appendix-C
C.1 Switching Devices
The conversion from AC to DC and again into AC is accomplished by switching devices, i.e.
thyristors, MOSFETS or IGBTs provided that a control signal is applied to them. Power and
frequency are the major deciding factors for the switching devices to be used in an induction
furnace. For low-frequency and high-power applications thyristors are commonly used as
compared to transistors. A thyristor requires a definite time to turn it into off state. At a high
frequency there is a probability of short circuiting of legs in a full bridge inverter if appropriate
delay time is not provided. An IGBT, on the other hand, requires a very small delay time and
can be switched very fast at high frequencies to reduce the overall power loss and increase the
efficiency of a system. The selection of these devices, in power-conversion circuits, depends
upon the type of the application. The solid-state devices, in converters, are used with different
frequency ranges. Typically SCRs are used below 1 kHz, IGBTs below 100 kHz and
MOSFETs above 100 KHz. The frequency and power ranges of these switching devices are
presented in Table C.1. A brief overview of the switching devices is given below and their
symbols are depicted in Figure C.1.
C.1.1 Thyristor
A thyristor has the capability to control a large amount of power, and it can be used in
applications requiring from a few amperes to several thousand amperes as discussed earlier.
Usually thyristors are used in controlled rectification; however, they can also be connected to
generate alternating current in an inverter configuration. On applying a sufficient positive
voltage at the anode with respect to its cathode (generally above 1-3 V, which is the forward
voltage drop across the conducting device), an SCR is turned-on by applying a positive gate
pulse with respect to cathode. It remains in the on state until its forward (anode) current does
not fall below a value known as ‘latching current’ (this can be done, i.e. the device is turned
off by the application of a negative anode to cathode voltage or negative anode current). In
the on state, the SCR can be modelled as a forward-biased diode junction in series with a low
value resistance. In a rectification circuit, inputs are AC voltages and zero crossing is
inheritance; however, in DC conversion forced commutation is required to turn it off. In
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induction-heating applications an SCR is also used in an inverter where zero crossing is
achieved by output circuit resonance. The disadvantage of an SCR, in an inverter circuit, is its
low switching speed and a longer turn-off time (associated with minority carriers’ stored
charge, which needs to be removed). For high-power applications SCRs are commonly used
but their turn-off time is significantly large as this time has direct relation with the magnitude
of current. A large current passing through a thyristor needs more turn-off time and vice versa.
C.1.2 Transistors
Two common controlled switches MOSFET (Metal oxide semiconductor field effect
transistor) and IGBT (Insulated gate bipolar transistor) are used in heat-treatment applications.
MOSFET is a very popular switching device for those applications that require low voltage
and low current with fast switching. IGBT was developed through merging of two
technologies (i.e. bipolar junction transistor and metal oxide semiconductor field effect
transistor) to achieve higher power capability at high frequency. In recent years, IGBTs are
replacing thyristors in small to medium furnaces. However, thyristors are still being used in
large induction furnaces due to their high power capability. Transistors may operate at the
resonant frequency and their turn-off time can be ignored. This is the reason for the popularity
of IGBTs’ usage in modern induction furnaces. At resonance, maximum power can be
delivered from constant source to the charge material.
Figure C.1.1: a) Silicon controlled rectifier, b) Metal oxide semiconductor field effect transistor,
c) Insulated gate bipolar transistor [15].
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Table C.1: Frequency and power ranges of switching devices in induction heating inverters.
SCR MOSFET IGBT
Very high power
(>1000 kW)
Low to high power
(1-1000 kW)
Medium to high power
(10-1000 kW)
Low frequency
(0.5-1 kHz)
High to very high frequency
(100-600 kHz)
Medium to high frequency
(1-100 kHz)
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Research Papers
1. M. Nawaz and M. A. Saqib, “Power-control strategy of a current source converter for
high-power induction melting,” Pakistan Journal of Engineering and Applied
Sciences, Vol. 18, No. 1, pp.11-20, 2016.
2. M. Nawaz, M. A. Saqib and S. A. R. Kashif, “Model predictive control in induction
heating powered by solar cells”, International Conference on Energy for
Environmental and Economic Sustainability (ICEEES2016), Lahore Pakistan,
October 20-23, 2016.
3. M. Nawaz, M. A. Saqib and S. A. R. Kashif, “Orthonormal functions based model
predictive control of a current-source resonant inverter”, International Conference on
Electrical Engineering, UET Lahore, Pakistan, March 2-4, 2017.
4. M. Nawaz, M. A. Saqib and S. A. R. Kashif, “Model predictive control strategy for a
solar based series resonant inverter in domestic heating”, Electronics Letters, Vol. 53,
No. 8, pp. 556-558, 2017.
5. M. Nawaz, M. A. Saqib and S. A. R. Kashif, “Constrained model predictive control
for an induction heating load”, Transactions of the Institute of Measurement and
Control, in review.