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NUMERICAL INVESTIGATION OF THE PERFORMANCE OF INTERIOR PERMANENT
MAGNET SYNCHRONOUS MOTOR DRIVE
SHAHIDA PERVIN
THESIS SUBMITTED IN FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE IN MATHEMATICS
FACULTY OF SCIENCEUNIVERSITY OF MALAYA
KUALA LUMPUR
2014
UNIVERSITI MALAYA
ORIGINAL LITERARY WORK DECLARATION
Name of Candidate: Shahida Pervin (I.C/Passport No: WS587463)
Registration/Matric No: SGP 110002
Name of Degree: Master of Science
Title of Project Paper/Research Report/Dissertation/Thesis (“this Work”): Numerical Analysis Based Performance Investigation of Interior Permanent Magnet Synchronous Motor Drive
Field of Study: Numerical Analysis of Motor Drives
I do solemnly and sincerely declare that:
(1) I am the sole author/writer of this Work; (2) This Work is original; (3) Any use of any work in which copyright exists was done by way of fair dealing and for
permitted purposes and any excerpt or extract from, or reference to or reproduction of any copyright work has been disclosed expressly and sufficiently and the title of the Work and its authorship have been acknowledged in this Work;
(4) I do not have any actual knowledge nor ought I reasonably to know that the making of this work constitutes an infringement of any copyright work;
(5) I hereby assign all and every rights in the copyright to this Work to the University of Malaya (“UM”), who henceforth shall be owner of the copyright in this Work and that any reproduction or use in any form or by any means whatsoever is prohibited without the written consent of UM having been first had and obtained;
(6) I am fully aware that if in the course of making this Work I have infringed any copyright whether intentionally or otherwise, I may be subject to legal action or any other action as may be determined by UM.
Candidate’s Signature Date
Subscribed and solemnly declared before,
Witness’s Signature Date
Name:
Designation:
ABSTRACT
The interior permanent magnet synchronous motor (IPMSM) is getting popular in
industrial drives because of its' advantageous features such as high power to weight
ratio, high efficiency, high power factor, low noise, and robustness. The vector control
technique is widely used for high performance IPMSM drive as the motor torque and
flux can be controlled separately. Fast and accurate response, quick recovery of speed
from any disturbances and insensitivity to parameter variations are some of the
important characteristics of high performance drive system used in electric vehicles,
robotics, rolling mills, traction and spindle drives.
Despite many advantageous features of IPMSM, precise control of this motor at
high-speed conditions especially, above the rated speed remains an engineering
challenge. At high speeds the voltage, current and power capabilities of the motor
exceed the rated limits. Consequently, the nonlinearity due to magnetic saturation of the
rotor core and hence the variation of d- and q- axes flux linkages will be significant.
Thus, it severely affects the performance of the drive at high speeds. The IPMSM drive
can be operated above the rated speed using the field-weakening (FW) technique. In
IPMSM the flux/field can only be weakened by the demagnetizing effect of d-axis
armature reaction current, id. Recently, researchers developed FW control algorithms
but often ignored the high precision computation of the algorithm. Mostly they simplify
the equations of flux control by ignoring the stator resistance and depend on
MATLAB/Simulink library. This results in improper flux weakening operation of
IPMSM. However, the proper flux computation is a crucial issue for motor control
particularly, at high speed condition. Therefore, there is a need to investigate the other
computational methods.
iii
In this study, accurate flux estimation for proper FW operation of IPMSM is
developed by incorporating the stator resistance of the motor. The Newton-Raphson
method (NRM) based numerical computation is used for high precision computation of
flux component of stator current, id to enhance the performance of the IPMSM drive
over wide speed range. The performance of the proposed NRM based computation of id
for IPMSM drive is evaluated in simulation using MATLAB/Simulink at different
operating conditions. The performance of the IPMSM drive with the proposed NRM
method is also compared with the conventional simplified computation of id. It has been
found that the IPMSM drive with proposed calculation of id provides better response as
compared to the conventional calculation of id. Thus, the proposed method could be a
potential candidate for real-time field weakening operation of IPM motor.
In the next step, fourth order Runge-Kutta method (RKM) is used to solve the motor
differential equations and the performance of the IPMSM drive is tested and compared
with the MATLAB/Simulink library built-in motor model. The results found that there
is no significant difference between RKM based calculation and MATLAB/Simulink
library. Thus, the MATLAB/Simulink library motor model can be used for motor drives
simulation with sufficient accuracy.
iv
ABSTRAK
Pemacu motor segerak magnet dalaman kekal (PMSMDK) semakin popular dalam
industri pemacu kerana kelebihan cirinya seperti berkuasa tinggi berbanding nisbah,
berkecekapan tinggi, faktor kuasa tinggi, bunyi rendah dan kekukuhan. Teknik kawalan
vektor digunakan secara meluas untuk pemacu PMSMDK berprestasi tinggi kerana tork
motor dan fluks boleh dikawal secara berasingan. Tindak balas cepat dan tepat, kelajuan
dipulihkan segera dari sebarang gangguan dan tidak sensitif kepada variasi parameter
adalah beberapa ciri penting sistem pemacu berprestasi tinggi yang digunakan dalam
kenderaan elektrik, robotik, mesin penggelek, peranti tarikan dan pengumpal.
Walaupun banyak ciri berfaedah PMSMDK, kawalan persis motor ini pada kelajuan
tinggi terutamanya di atas laju kadaran kekal sebagai cabaran kejuruteraan. Pada
kelajuan tinggi, voltan, arus dan keupayaan tenaga motor melebihi had laju kadaran.
Oleh itu, , ketaklinearan disebabkan penepuan magnet teras rotor dan dengan itu variasi
hubungan paksi-d dan -q flux menjadi penting. Oleh itu, ianya memberi kesan kepada
prestasi pemacu pada kelajuan tinggi. Pemacu PMSMDK boleh beroperasi di atas laju
kadaran menggunakan teknik medan lemah (ML). Dalam IPMSM fluks/medan hanya
menjadi lemah oleh kesan penyahmagnetan paksi-d arus tindakbalas armatur, id.
Baru-baru ini, penyelidik membangunkan algoritma kawalan ML tetapi sering
mengabaikan algoritma pengiraan kejituan tinggi. Kebanyakan mereka meringkaskan
persamaan kawalan fluks dengan mengabaikan rintangan pemegun dan bergantung
kepada perpustakaan MATLAB/Simulink. Keputusan ini menyebabkan operasi
PMSMDK medan lemah tidak wajar. Walau bagaimanapun, pengiraan fluks yang betul
adalah isu penting untuk kawalan motor, terutamanya pada keadaan kelajuan tinggi.
Oleh itu, ianya perlu untuk mengkaji kaedah pengiraan lain.
v
Dalam kajian ini, anggaran fluks tepat untuk operasi ML PMSMDK dibangunkan
dengan menggabungkan motor rintangan pemegun Kaedah Newton-Raphson (KNR)
berasaskan pengiraan berangka digunakan untuk membuat pengiraan ketepatan tinggi
komponen fluks arus pemegun, id untuk meningkatkan prestasi pemacu PMSMDK
melangkaui julat kelajuan. Prestasi pengiraan KNR berdasarkan cadangan id PMSMDK
dinilai dalam simulasi menggunakan MATLAB/Simulink pada keadaan operasi yang
berbeza. Prestasi pemacu PMSMDK dengan KNR yang dicadangkan juga berbanding
dengan pengiraan id konvensional yang dipermudahkan. Didapati pemacu PMSMDK
dengan pengiraan id yang dicadangkan member respon lebih baik berbanding dengan
pengiraan id konvensional. Oleh itu, kaedah yang dicadangkan itu boleh menjadi bahan
berpotensi untuk masa sebenar operasi motor MDK medan lemah.
Seterusnya, kaedah Runge-Kutta (KRK) peringkat empat digunakan untuk
menyelesaikan persamaan pembezaan motor dan prestasi pemacu PMSMDK diuji dan
dibandingkan dengan perpustakaan model motor MATLAB/Simulink terbina dalaman.
Keputusan didapati bahawa terdapat perbezaan yang signifikan antara pengiraan
berdasarkan KRK dan perpustakaan MATLAB/Simulink. Oleh itu, perpustakaan model
motor MATLAB/Simulink boleh digunakan untuk simulasi pemacu motor dengan
ketepatan yang mencukupi.
vi
Acknowledgements
I would like to express my most sincere gratitude and appreciation to my supervisor
Dr. Zailan bin Siri for his guidance, advice and encouragement throughout of this study.
I wish to thank Prof. Dr. Angelina Chin Yan Mui (Postgraduate Coordinator, Institute of
Mathematical Sciences) for her help although out my study at University of Malaya. I
also offer my sincere thanks to Professor Dr. M. Nasir Uddin who provides me all the
support on the technical side of motor drives.
I would like to acknowledge the assistance from the Institute of Mathematical
Sciences, University of Malaya, all Faculty members, graduate fellows and staff
members.
Finally, I express my sincere appreciation to my husband, daughter, as well as other
family members, relatives and friends without whose support and encouragement it
would not have been possible to complete this study.
vii
DEDICATED TO:
My husband Prof. Dr. M. Nasir Uddin
and,
My daughter Miss Shirley Naima Uddin
viii
TABLE OF CONTENTS
Page
ABSTRACT iii
ABSTRAK MALAYA v
ACKNOWLEDGEMENTS vii
LIST OF FIGURES xiii
LIST OF SYMBOLS xv
LIST OF ACRONYMS xvii
1 Introduction 1
1.1 Background of Motor
1.1.1 Direct Current (DC) Motor
1.1.2 Alternating Current (AC) Motor
1.2 Background of the Permanent Magnet Synchronous Motor (PMSM)
ix
1.2.1 Classifications of PMSM
1.3 Motivation and Objectives
1.4 Methodology
1.5 Introduction to Numerical Analysis Methods
1.5.1 Newton-Raphson Method
1.5.2 Runge-Kutta Method
1.6 Organizations of the Thesis
2 Literature Review 15
2.1 Mathematical Model of Interior Permanent Magnet Synchronous Motor
(IPMSM)
2.2 Vector Control Strategy for IPMSM Drives
2.2.1 Speed Controller
2.2.2 Current Controller and Voltage Source Inverter
2.3 Flux Control Methods for IPMSM Drives
2.4 Application of Numerical Analysis for Motor Drives
2.5 Conclusion
x
3 Flux Control Techniques for IPMSM Drive 26
3.1 Introduction
3.2 Conventional Flux Control of IPMSM
3.2.1 Conventional Flux Control below Rated Speed – Maximum Torque
per Ampere (MTPA) Control
3.2.2 Conventional Flux Control above Rated Speed – Field Weakening
(FW) Control
3.3 Proposed Flux Control Techniques
3.3.1 Proposed Flux Control below Rated Speed
3.3.2 Proposed Flux Control above Rated Speed
3.4 Simulation Results and Discussions
3.5 Experimental Test of the Proposed Field Weakening
3.6 Conclusion
4 Application of Runge-Kutta Method for Solution of IPMSM Model 46
4.1 Introduction
4.2 Modelling of IPMSM
4.3 Simulation Results of the RKM and MATLAB/Simulink Based Motor
Model
4.4 Conclusion
xi
5 Conclusion 51
5.1 Summary of the Thesis
5.2 Major Achievements of the Current Work
5.3 Further Scope of the Work
List of Papers Published from this Work 55
REFERENCES 56
APPENDICES 61
xii
List of Figures
1.1 Structure of a simple 2-pole PM dc motor ................................................ 3
1.2 Squirrel-cage induction motor. .............................................................. 4
1.3 Cross section of a 4-pole: (a) surface mounted, (b) inset and (c) IPM type motor.. 7
2.1 Equivalent circuit model of the IPMSM: (a) d-axis, (b) q-axis.…………. 18
2.2 Basic vector diagram of IPMSM: (a) general; (b) modified with id=0………… 20
2.3 Block diagram of the closed loop vector control of IPMSM drive……....….. 21
3.1 Typical torque-speed characteristic curve of a motor drive……………….... 32
3.2 Locus of stator current Ia at different speeds staring from 188 rad/s to 363 rad/s in a
step of 25 rad/s. ………………………………………………………........... 32
Fig. 3.3: Variation of id with speed, r…………………………………. 36
Fig. 3.4: Comparison of curve fitting polynomial with actual id. . ……… 36
Fig. 3.5. (a) Simulink block diagram for overall control system of IPMSM drive, (b)
Reference current generator subsystem. ………………………………. 38
Fig. 3.6: Simulated transient responses of the drive for step change of speed at rated
load using the conventional computation of id [12]; (a) speed, (b) iq*, (c) id
*, (d) stator
phase current, ia(actual and command)……… ……………………….……….... 39
3.7: Simulated transient responses of the IPMSM drive for step change of speed at rated
load using the proposed NRM based id computation; ; (a) speed, (b) iq*, (c) id
*, (d) stator
phase current, ia (actual and command)………………………........... 40
xiii
Fig. 3.8: Comparison of speed error (= r* - r ): (a) conventional (b) NRM based
computation of id. ....................................................................................... 41
Fig. 3.9: Simulated responses of the IPMSM drive at very high speed (320 rad/s)
condition at rated load using the proposed NRM based id computation. ………...... 43
Fig. 3.10: Simulated responses of the proposed IPMSM drive for a step increase in
power in field weakening region……………………………………………….... 44
Fig. 3.11 Experimental responses of the IPMSM drive for a step increase in speed using
the proposed NRM based computation of id: (a) speed, (b) id. …………....... 45
Fig. 4.1 Simulink block diagram to get 3-phase currents from id and iq............... 49
Fig. 4.2 MATLAB/Simulink based built-in IPMSM model.............................. 49
Fig. 4.3 RKM (ode4) based calculation of IPMSM model: (a) ‘a’ phase stator current
(A), (b) steady-state 3-phase currents (A)................................................................ 50
Fig. 4.4 Result based on MATLAB/Simulink based built-in IPMSM model: (a) ‘a’
phase stator current (A), (b) steady-state 3-phase currents (A).............................. 50
xiv
List of Symbols
va, vb and vc a, b and c, phase voltages, respectively
ia, ib and ic Actual a, b and c, phase currents, respectively
ia*, ib
* and ic* Command a, b and c, phase currents, respectively
vd d-axis voltage
vq q-axis voltage
id d-axis current
iq q-axis current
Rs stator resistance per phase
Ld d-axis inductance
Lq q-axis inductance
s stator angular frequency (=Pr) or electrical frequency
r actual rotor speed (=dr/dt)
r* motor command speed
r error between actual and command speeds
r rotor position
P number of pole pairs
Te electromagnetic developed torque
TL load torque
Jm rotor inertia constant
Bm friction damping coefficient
m magnet flux linkage
Vm maximum stator phase voltage
Vm maximum stator phase voltage neglecting Rs
xv
Im maximum stator phase current
VB dc bus voltage for the inverter
Ts sampling period
xvi
List of Acronyms
AC Alternating current
DC Direct current
FW Flux-weakening
HPD High performance drive
IGBT Insulated gate bipolar transistor
IM Induction motor
IPMSM Interior permanent magnet synchronous motor
MTPA Maximum torque per ampere
NRM Newton-Raphson method
PI Proportional-integral
PID Proportional-integral-derivative
PM Permanent magnet
PMSM Permanent magnet synchronous motor
PWM Pulse width modulation
RKM Runge-Kutta method
VSI Voltage source inverter
xvii
Chapter 1
Introduction
1.1 Background of Motor
Advancement of modern societies is highly dependent on the development of
faster, efficient, and environmentally friendly motor drives because the motors consume
more than 50% of all the electrical power generated on the earth [1]. The motors are
used in all motions and locomotion of all devices such as electric vehicles, robotics,
rolling mills, machine tools, etc. This may be the most important factor behind today’s
high demand for more efficient motor control systems. Thus, the sophisticated and
precise computation of mathematical model of the motor is very important for high
performance motor drives. The development of electric motor has given us the most
efficient and effective means to do work known to the history. The electric motor
converts electrical energy into mechanical energy and the generator converts
mechanical energy in to electrical energy. Generally, there are two windings in a motor,
one receives electric energy and other gets electric energy by induction. Both of them
produce magnetic fluxes and due to the interaction of two fluxes rotating motion is
1
produced in the rotor. Over the years, electric motors have changed substantially in
design; however the basic principles have remained the same. In general electric motors
can be classified in to two major groups such as direct current (DC) and alternating
current (AC) motors.
1.1.1 Direct Current (DC) Motor
The modern DC motor was invented in 1873 by Belgian-born electrical engineer
Mr. Gramme. A simple permanent magnet DC motor is shown in Fig. 1.1 in which the
electric power in supplied from a battery to one winding and the main flux is supplied
by permanent magnets. Torque is produced according to Ampere’s Law, “If a current-
carrying conductor placed in a magnetic field, it experiences a force”.The name DC
motor comes from the DC electric power used to supply the motor. The DC motor was
in wide spread use in street railways, mining and industrial applications by the year
1900. However, the disadvantage of DC motors such as excessive wear in the electro-
mechanical commutator, low efficiency, fire hazards due to sparking, limited speed and
the extra room requirement to house the commutator, high cost of maintenance, became
evident and leads to further investigation in order to overcome these DC motor
disadvantages [2].
2
Fig. 1.1 Structure of a simple 2-pole PM DC motor.
1.1.2 Alternating Current (AC) Motor
Most AC motors being used today are so-called “induction motors” which are
the workhorse of the industry. In an induction motor (IM) a 3-phase AC supply is
applied to the stator and voltage is induced in the rotor according to Faraday’s law of
electromagnetic induction. The most common type of IM has a squirrel cage rotor in
which aluminum conductors or bars are shorted together at both ends of the rotor by
cast aluminum end rings as shown in Fig. 1.2. Due to the interaction of stator and rotor
fluxes rotating motion is produced. The rotor is connected to the motor shaft, so the
shaft will also rotate and drive the load. When a 3-phase supply is applied to the stator
winding a rotating magnetic flux/field is produced. The mechanical speed of the rotor is
always lower than the speed of the rotating magnetic flux (synchronous speed) by the so
called slip speed. The main advantages of IM are its lower cost, maintenance free
operation and greater reliability especially, in harsh industrial environment [2]. On the
contrary, IMs require very complex control scheme because of their nonlinear
relationship between the torque generating and magnetizing currents. In order to
3
overcome this problem, researcher developed another type of motor which is known as
synchronous motor.
Fig.1.2 Squirrel-cage induction motor.
1.1.3 Synchronous Motor
Synchronous motors utilize the same type of stator winding structure as IM and
which is either a wound DC field or permanent magnet rotor [2]. Synchronous motors
run at the synchronous speed which is the same as the supply frequency. In a
synchronous motor 3-phase AC supply is applied to the stator winding and a DC supply
is applied to the rotor winding which also have shorted squirrel cage bar. Initially, the
DC supply is not applied to the rotor and the motor starts like an IM when a 3-phase
supply is applied to the stator. When the speed of the motor is near synchronous speed
then the DC supply is applied to the rotor and the rotor magnetic field catches the stator
4
magnetic field and gets magnetically locked. Thus, the rotor rotates at synchronous
speed.
1.2 Background of the Permanent Magnet Synchronous
Motor (PMSM)
A permanent magnet synchronous motor (PMSM) is exactly similar to a
conventional synchronous motor with the exception that the field winding and DC
power supply are replaced by the permanent magnets (PMs). The PMSM has
advantages of high torque to current ratio, large power to weight ratio, high efficiency,
high power factor, low noise and robustness etc. Current improvement of PM motors is
directly related to the recent achievement in high energy permanent magnet materials.
Ferrite and rare earth/cobalt alloys such as neodymium-boron-iron (Nd-B-Fe),
aluminum-nickel-cobalt (Al-Ni-Co), samarium-cobalt (Sm-Co) are widely used as
magnetic materials. Rare earth/cobalt alloys have a high residual induction and coercive
force than the ferrite materials. But the cost of the materials is also high. So rare earth
magnets are usually used for high performance motor drives as high torque to inertia
ratio is attractive. Depending on the position of PM, there are different types of PMSM
which are discussed in the following subsection.
1.2.1 Classifications of PMSM
The performance of a PMSM drive varies with the magnet material, placement
of the magnet in the rotor, configuration of the rotor, the number of poles, and the
5
presence of dampers on the rotor [3]. Depending on magnet configuration, the PMSM
can be classified into following three types:
(a) Surface mounted PM motor: In this type of PM motor, the PMs are
typically glued or banded with a non-conducting material to the surface
of the rotor core as shown in Fig. 1.2(a). This type of motor is not
suitable for high speed.
(b) Inset type PM motor: In this type of PM motor, the PMs are typically
glued directly or banded with a non-conducting material inside the rotor
core as shown in Fig. 1.2(b). This type of motor is not suitable for high
speed.
(c) Interior type PM (IPM) synchronous motor (IPMSM): In an IPMSM, the
PMs are buried inside the rotor core as shown in Fig. 1.2(c). This is the
most recently developed method of mounting the magnets. Interior
magnet designs offer q-axis inductance (Lq) larger than the d-axis
inductance (Ld). The saliency makes possible a degree of flux
weakening, enabling operation above nominal speed at constant voltage
and should also help to reduce the harmonic losses in the motor. The
IPMSM has the advantages of smooth rotor surface, mechanical
robustness and a smaller air gap as compared to other types of PM
motors. Thus, this motor is more suitable for high speed operation and
hence, considered in this thesis. The parameters of the particular
laboratory 1 hp motor used in this thesis are given in Appendix A.
6
d-axisq-axis
(a)
(b)
(c)
Fig. 1.2: Cross section of a 4-pole: (a) surface mounted, (b) inset and (c) IPM type motor.
7
1.3 Motivation and Objectives
The permanent magnet synchronous motors (PMSM) get rapid industrial
acceptance because of their advantageous features such as high torque to current ratio,
high power to weight ratio, high efficiency, high power factor, low noise, and
robustness [6]. The advances in PM material quality and power electronics have boosted
the use of PMSM in electric motor drives for high performance applications such as
automotive, aerospace, rolling mills, etc. Among different types of PM synchronous
motors, the IPMSM shows excellent properties such as mechanically robust rotor, small
effective air gap, and rotor physical non-saliency [7]. The vector control technique is
widely used for high performance control of an IPMSM drive as the motor torque and
flux can be controlled separately in vector control [8]. Thus, the motor acts like a DC
motor while maintaining the general advantages of AC motor over DC motors.
Precise control of high performance IPMSM over wide speed range is an
engineering challenge. Fast and accurate speed response, quick recovery of speed from
any disturbances and insensitivity to parameter variations are some of the important
characteristics of high performance drive (HPD) system used in electric vehicles,
robotics, rolling mills, traction and spindle drives [6]. Some of the drive systems like
spindle drive and traction also need constant power operation [9]. The IPMSM drive
can be operated in constant power mode (above the rated speed) using the field-
weakening technique. In an IPMSM the direct control of field flux is not possible.
However, the field can be weakened by the demagnetizing effect of d-axis armature
reaction current, id [9]. Recently, the researchers [9-31] continue their efforts to develop
the IPMSM drive system incorporating the flux-weakening mode. The researchers
develop sophisticated control algorithms but often ignored the high precision
computation of the algorithm [9-31].
8
Electrical engineers often develop the mathematical model for voltage, current
and flux calculation of IPMSM drive but they simplify the equation for easier
computation. However, with the simplified equation the performance of the drive
deteriorates, which may not be suitable for HPD applications. Thus, the main objective
of the study is to investigate the numerical iterative computation methods (i.e., Newton-
Raphson and fourth order Runge-Kutta Methods) for high precision computation of
stator voltage/current and flux from mathematical model of IPMSM to improve the
performance of the drive. The particular objectives of this thesis are as follows:
to develop more accurate analytical model for flux control of IPMSM drive
based on d-axis stator current, id.
to incorporate the Newton-Raphson method (NRM) based iterative computation
method to calculate id for the closed loop vector control of IPMSM drive in
order to achieve high performance of the drive over wide speed range.
to investigate the performance of the IPMSM drive incorporating the proposed
NRM based calculation of id and it’s comparison with the conventional
simplified calculation of id.
to incorporate Runge-Kutta method (RKM) based iterative computation method
to solve the differential equation of IPMSM mathematical model and compare
the stator currents obtained from RKM with the MATLAB/Simulink library
model.
1.4 Research Methodology
The closed loop vector control of IPMSM drive is familiarized. For vector control
scheme the mathematical model of speed and current control algorithms are
familiarized. Then, the conventional flux control algorithm (i.e., computation of d-axis
9
stator current, id) is revisited. The performance (e.g. speed, current, etc.) of the IPMSM
drive with conventional flux control algorithm is investigated. After that a more
accurate analytical model for flux control of IPMSM is developed. The NRM is used to
compute the flux component of stator current, id. The performance of the IPMSM drive
is investigated with the proposed NRM based computation of id. These speed response
of the IPMSM drive with the proposed NRM based computation of id are compared with
the conventional computation of id in term of settling time, speed overshoot/undershoot,
steady-state error, etc.
Again, the fourth- order RKM (ode4) is applied to solve the set of first order
differential equations representing the mathematical model of IPMSM. The calculation
of motor stator currents using the ode4 are compared with those obtained from
MATLAB/Simulink library motor model.
1.5 Background of Numerical Computation Methodology
Various numerical methods e.g. Netwon-Rahpson and Runge-Kutta methods are
used for computation to find the solutions of different types of equations [4,5].
Electrical engineers develop mathematical model of electric motors for controlling
and/or efficiency optimization [5]. However, sophisticated computation of motor
mathematical model is often ignored by electrical engineers. Traditionally, they depend
on MATLAB/Simulink library and simplified equation for computation. Some
numerical methods used in this thesis are discussed below.
10
1.5.1 Newton-Raphson Method
In order to get more accurate value of id for proper flux control Newton-Raphson
Method (NRM) based numerical computation of id is proposed in this work.
The NRM is one of the most powerful, popular numerical methods and easy for
computer program to solve some root finding equations. As compared to other
numerical methods (e.g. Secant and Bisection) the NRM has some advantages such as,
it needs only one initial value, it's convergence is fast and it is easy to implement.
Therefore, in this thesis, the NRM will be used to solve some motor equations,
particularly, for high precision computation of flux for IPMSM. The NRM method is
illustrated below.
Let the function, f(x)=0, then according to NRM the roots of this equation can
be found using the iterative formulae,
xn+1=xn−f ( xn )
f '( xn ) , (1.1)
where, f΄(x) is the derivative of f(x).
Thus, the root of f(x) can be solved using the following steps.
1. Select: initial approximation, x1 = p0, tolerance (TOL), and maximum number of
iteration N0.
2. Do the following iterations using ‘for’ loop
For i=1:N0
calculate, x2=x1−
f ( x1)
f '( x1 )
3. If |x2-x1|<TOL, then the root of f(x) is: x2,
4. stop; otherwise,
5. set x1=x2; and repeat step 2 until step 4.
11
6. End of calculation.
1.5.2 Runge-Kutta Methods
In numerical analysis, the Runge–Kutta Methods (RKMs) are an important family of
implicit and explicit iterative methods for the approximation of solutions of ordinary
differential equations. These techniques were developed around 1900 by the German
mathematicians C. Runge and M.W. Kutta [5]. As compared to other methods, the
fourth order RKM (which is known as ode4) is well-known and very stable method to
solve for ordinary differential equations (ode) and this method is easy to implement in
computer program. Therefore, in this thesis, the fourth-order Runge-Kutta (RK4)
method will be used to solve some first ode representing the mathematical model of
IPMSM for high precision computation of its stator current. The RKM method is
illustrated below.
A first -order ordinary differential equation has the first derivative as its highest
derivative and it usually can be cast in the following form:
y'(x) = f (x,y) with initial condition y=y0 at x=x0 (1.2)
where, x is the independent variable, y(x) is the dependent variable and y'(x) is the first
derivative of y(x). The dependent variable at x+h, i.e. y(x+h), of equation (1.2), is given
by:
y(x+h) = y(x) + (k1 +2k2 + 2k3 + k4)/6 (1.3)
12
where,
k1 = hf(x0,y0)),
k2 = hf(x0+h/2,y0+k1/2),
k3 = hf(x0+h/2,y0+k2/2),
k4 = hf(x0+h,y0+k3),
and h is the fixed step-size.
The algorithm to solve Eqn. (1.3) using RK4 is:
To approximate the solution of the initial value problem y'(x) = f(x,y), a<x <b, y(0)=y0
at N+1 equally spaced numbers in the interval [a,b]:
INPUT end points a,b; integer N; initial condition y0:
OUTPUT approximation of y at the N+1 values of x.
Step 1: Set h=(b-a)/N;
x=a;
y=y0;
OUTPUT (x,y)
Step 2: For i=1,2,..,.N do Steps 3-5
Step 3: Set k1=hf(x,y);
k2=hf(x+h/2, y+k1/2);
k3=hf(x+h/2, y+k2/2);
k4=hf(x+h, y+k3);
Step 4: y=y+(k1+2k2+2k3+k4)/6; (Compute yi.)
x=a+ih. (Compute xi.)
Step 5: OUTPUT (x,y)
13
Step 6 STOP.
1.6 Organizations of the Thesis
The thesis consists of five Chapters. Chapter 1 presents the background about different
types of motors, particularly, IPMSM. Introduction about various numerical analysis
methods (e.g., NRM and RKMs) associated with the MATLAB programming. Then,
the motivation and the specific objectives of the current work are presented.
Chapter 2 provides a literature review on the IPMSM, its control techniques and
numerical analysis methods. Particularly, the mathematical model of IPMSM, vector
control strategy, speed and current control techniques are presented. Literature search
on flux control techniques for IPMSM and application of numerical analysis for motor
drives are also presented in this chapter.
In Chapter 3, the conventional and the proposed flux control techniques are
presented. For the proposed flux control, a more accurate analytical model of d-axis
stator current id incorporating stator resistance is developed. The NRM based
computation of id is also given in this chapter. The effectiveness of the proposed flux
control technique for IPMSM drive is investigated in both simulations and real-time.
Chapter 4 presents the application of 4 th order RKM to solve for d-q axis
components of stator current from the differential equations representing the
mathematical model of IPMSM. Then the a-b-c phase currents are calculated from d-q
axis currents. These currents are compared with the stator phase currents obtained from
MATLAB/Simulink built-in motor model.
Chapter 5 concludes the thesis indicating the achievements and limitations.
Finally, the future scope of the current work is also summarized.
14
Chapter 2
Literature Review
2.1 Mathematical Model of Interior Permanent Magnet
Synchronous Motor (IPMSM)
The IPMSM is similar to the conventional wire-wound excited synchronous
motor with the exception that the excitation is provided by the permanent magnets
instead of a wire-wound DC rotor field. The mathematical model of an IPMSM in the
d-, q-axis is given as [12],
vd=Ld
did
dt+Rs id−Pωr Lq iq (2.1)
vq=Lq
diq
dt+Rs iq+Pωr Ld id+Pωr ψm (2.2)
15
The electromagnetic developed torque in terms of electrical parameters is given by,
T e=3 P2
(ψm iq+(Ld−Lq ) id iq ) (2.3)
While, the electromagnetic developed torque in terms of mechanical parameters is given
by,
T e=T L+J m pωr+Bm ωr (2.4)
where, vd and vq are the d,q-axis voltages, Ld and Lq are the d,q-axis inductances, id and
iq are the d,q-axis stator currents, respectively; Rs is the stator resistance per phase, m is
the constant flux linkage due to rotor permanent magnet, r is the angular rotor speed, P
is the number of pole pairs of the motor, p is the differential operator (d/dt), and Te is the
developed electromagnetic torque, TL is the load torque, Bm is the viscous coefficient
and Jm is the inertia constant. The first term of (2.3) represents the magnet torque due to
the rotor permanent magnet flux m and the second term represents the reluctance torque
due to the complex interaction of d,q-axis currents and inductances of the IPMSM. The
complexity of the control arises due to the nonlinear nature of the torque equation (2.3)
because, Ld, Lq, id and iq are not constants. All these quantities vary during dynamic
operating conditions [9]. To make the torque equation linear and the control task easier,
usually id is set to zero [32]. However, in an actual IPMSM drive, it is inappropriate, the
assumption of id = 0 leads to erroneous results. If id = 0, the reluctance torque will be
zero and hence, the motor operates at below optimum torque condition. In this thesis id
is not considered zero. The value of id is calculated from iq maintaining the armature
voltage and current within the capacity of the motor and the inverter. This improves the
performance of the drive system as compared to the id = 0 technique as it can utilize the
16
PrLqiq+-Rs Ldid
vd
PrLdid+ -Rs Lq
vq
-+Prm
iq
reluctance torque below rated speed and operate the motor above rated speed by flux
weakening. The motor parameters used in the simulation are given in Appendix-A.
According to (2.1) and (2.2) the d,q-axis equivalent circuit models are shown in Fig. 2.1
(a) and (b), respectively.
(a)
(b)
Fig. 2.1. Equivalent circuit model of the IPMSM: (a) d-axis, (b) q-axis.
2.2 Vector Control Strategy for IPMSM Drive
The vector control means the control of both magnitude and phase angle of either
the motor voltage or current or both instead of their magnitudes only. As mentioned
earlier, the vector control technique is one of the most effective techniques for in high
performance AC motors drives. In the case of the permanent magnet synchronous motor
(PMSM), the torque eqn. (2.3) has two terms: the first term represents the magnet
17
torque produced by the permanent magnet flux m and the torque producing current
component iq; the second term represents the reluctance torque produced by the
complex interaction of inductances Ld and Lq and also the currents id and iq. The
excitation voltage due to permanent magnets, and the values of the inductances Ld and
Lq undergo significant variations in an interior type permanent magnet motor under
different steady state and dynamic loading conditions [9,33]. Thus, the complexity of
the control of the IPMSM drive arises due to the nonlinear nature of the torque eqn.
(2.3). In order to operate the motor in a vector control scheme avoiding the complexity,
id is set to zero. Then the torque equation becomes linear and is given by,
T e =
3 P2
ψm iq = K t iq (2.5)
where, the constant K t =
3 P2
ψm. Using phasor notations and taking the d-axis as a
reference phasor, the steady-state phase voltage Va can be derived from the steady-state
d -q axis voltage eqn. (2.1) and (2.2) as,
V a= vd + j vq
= R s Ia − ωs Lq iq + j ωs Ld id + j ωs ψm (2.6)
where, the phase current, I a= id + j iq (2.7)
In the case of the IPM motor, the d-axis current is negative and it demagnetizes the
main flux provided by the permanent magnets.
18
(a)
d-axis
q-axis
mid
iqIa
Lqiq
Ldido
-s Lqiq
jsm-jsLdidIaRs
Va vq
vd
d-axis
q-axis
m
Ia
Lqiqo
-sLqiq jsmIaRs
Va vq
vd
According to eqns. (2.6) and (2.7) the basic vector diagram of IPMSM is shown in Fig.
2.2 (a). The vector control scheme can be clearly understood by this vector diagram. It
is shown in the vector diagram that the stator current, Ia, can be controlled by
controlling the d- and q- axis current components. In the vector control scheme, when id
is set to zero then all the flux linkages are oriented in the d-axis as shown in Fig. 2.2 (b).
After setting id = 0, eqn. (2.3) shows that the torque is a function of only the quadrature
axis current component, iq and hence a constant torque can be obtained by ensuring iq
constant.
(b)
Fig. 2.2 Basic vector diagram of IPMSM: (a) general; (b) modified with id = 0.
19
+ r*
Current Controller
Vector rotator
SpeedController
VB
d/dt
ia* ib* ic*
r
iq* id*
ia ib
r-
ic
IPMSM Position sensorPWM Inverter
Base drive circuit
r
f(iq*, r)
The closed-loop vector control of voltage source inverter (VSI) fed IPMSM drive is
shown in Fig. 2.3. In this figure VB is the DC bus voltage for the inverter, which
supplies AC voltage to the motor with variable frequency and magnitude. The IPMSM
drive consists of the current controller and the speed controller. The speed controller
generates the torque command and hence the q-axis current command iq* from the error
between the command speed and the actual speed. As mentioned earlier, in the vector
control scheme traditionally, the d-axis command current id* is set to zero to simplify the
nonlinear dynamic model. If the flux control is needed the id should be calculated based
on some flux control algorithm
Fig. 2.3. Block diagram of the closed loop vector control of IPMSM drive.
20
of the IPMSM. The command phase currents ia*, ib
* and ic* are generated from the d,q
axis command currents using inverse Park’s transformation given in Eqn. (2.8) [34].
The current controller forces the load current to follow the command current as closely
as possible and hence forces the motor to follow the command speed due to the
feedback control. Therefore, in order to operate the motor in a vector control scheme the
feedback quantities will be the rotor angular position and the actual motor currents. In
the control scheme, the torque is maintained constant up to the rated speed, which is
also called the constant flux or the constant voltage to frequency ratio (V/f) control
technique.
[ia¿ ¿ ] [ib¿ ¿ ] ¿¿
¿¿ (2.8)
The (V/f) is maintained constant by using PWM operation of the VSI. The designs of
the speed controller, current controller and voltage source inverter to perform their
specific functions are given in the following sub-sections.
2.2.1 Speed Controller
The speed controller processes the speed error (r) between command and actual
speeds and generates the command q-axis current (iq*). The small change in speed r
produces a corresponding change in torque Te and taking the load torque TL as a
constant, the motor dynamic Eqn.(2.4) becomes,
21
ΔT e = J m
d ( Δωr)dt
+ Bm Δωr (2.9)
Integrating Eqn.(2.9) gives us the total change of torque as,
T e =K t iq = J m Δωr + Bm∫ 0
tΔωr (τ ) dτ
(2.10)
According to Eqn. (2.10), a proportional-integral (PI) algorithm can be used for the
speed controller which may be written as,
iq¿ = K p Δωr + K i∫0
tΔωr ( τ ) dτ
(2.11)
r = r*-r (2.12)
where, Kp is the proportional constant, Ki is the integral constant and r is the speed
error between the command speed, r* and the actual motor speed, r.
For discrete-time representation, Eqn. (2.11) can be differentiated and written in
discrete-time domain, respectively as,
diq¿
dt= K p
dΔωr
dt+ K i Δωr
(2.13)
iq¿ (k ) = iq¿ (k−1 ) + K p [Δωr ( k ) − Δωr (k − 1 ) ] +K iT s Δωr (k ) (2.14)
where, iq* (k) is the present sample of command torque, iq
*(k-1) is the past sample of
command torque, r(k) is the present sample of speed error and r(k-1) is the past
sample of speed error and Ts is the sampling period.
22
23
2.2.2 Current Controller and Voltage Source Inverter
The current controller is used to force the motor currents to follow the command
currents and hence forces the motor to follow the command speed due to the feedback
control. The outputs of the current controller are the pulse width modulated (PWM)
signals for the insulated gate bipolar resistor (IGBT) inverter switches. The voltage
source inverter (VSI) converts fixed DC voltage to a variable AC voltage for the motor
so that it can follow command speed with the required load. The current control
principle and PWM generation for the VSI can be found in [35].
2.3 Flux Control Methods for IPMSM Drives
In an IPMSM the flux cannot be controlled directly as the flux is supplied the
permanent magnets which are buried inside the rotor. The flux can only be controlled by
the armature reaction of the d-axis component of stator current, id. Traditionally, for
vector control of IPMSM drive id is set to zero so that the torque equation becomes
linear with iq and hence, the control task becomes easier. With id=0, motor cannot run
properly above rated speed as the flux remains fixed. If the flux is fixed, the voltage and
current exceed the rated limits of the motor with the speed since the voltage is
proportional to speed and flux. So, the flux control using id is needed to operate the
IPMSM above rated speed. Above the rated speed, the air gap flux is reduced by
demagnetizing effect of id so that the voltage and current don’t exceed the rated limits
although the speed exceeds the rated value. Moreover, with id 0, more torque can be
produced by the motor as the reluctance torque is utilized, which can be seen from Eqn.
(2.3).
24
Researchers reported some field weakening operations of IPMSM but the
computation of id is based on simplified equation which results in inappropriate control
of the flux and hence, non-optimum performance of the drive, particularly, above the
rated speed condition [14-31]. In this thesis, the Newton-Raphson method is used for
high precision computation of id without simplifying the equation. The flux control
techniques of IPMSM are discussed in detail in the next chapter.
2.4 Application of Numerical Analysis for Motor Drives
Traditionally the researchers in motor drives area depend on Matlab/Simulink
library as a dependable platform for computation. In order to reduce the computational
burden especially in real-time, sometimes they also simplify the equations even in
simulation so that they can resemble the simulation performance with the experimental
results. Recently, researchers looked into the application of numerical analysis methods
for modelling in motor drives [36-38]. In [36] authors used numerical analysis
technique for ultrasonic motor. Phyu et. al. used numerical modeling and analysis of
spindle motor for disk drive applications [37]. Han et. al. used numerical method for
brushless dc motor modeling [38]. None of the works are reported on the modeling of
IPMSM or its control. Moreover, the application of numerical analysis methods for
modeling of motor and/or control techniques are at its initial stage. There is a big scope
to investigate the application of numerical analysis methods for these applications
instead of readily available Matlab/Simulink or any other software.
25
2.5 Conclusion
In this Chapter first, mathematical model of IPMSM and vector control strategy
have been provided. Then, the speed and current control techniques have been briefly
described. A literature search on flux control techniques for IPMSM has been provided.
Finally, a literature search on application of numerical analysis for motor drives has
been provided.
26
Chapter 3
Flux Control Techniques for IPMSM
Drive
3.1 Introduction
The high performance motor drives used in robotics, rolling mills, machine tools,
electric vehicles, spindle drives, traction drives, etc. require low to high speed operation,
fast and accurate speed response, quick recovery of speed from any disturbances and
insensitivity of speed response to parameter variations. Some of the drive systems like
machine tool spindles and traction drives also need above rated speed (constant power)
operation. Despite many advantageous features, the precise torque and speed control of
IPMSM at high speed, particularly, above the rated speed, has always been a challenge
for researchers due to the nonlinearity present in the electromagnetic developed torque
because of magnetic saturation of the rotor core [33]. At high speeds, the voltage,
current and power capabilities of the motor exceed the rated limits. Consequently, the
27
magnetic saturation of the rotor core and shifting the operating point of permanent
magnet loading, and hence the variation of d, q axis flux linkages will be significant [9].
Thus, it severely affects the performance of the IPMSM drive at high speeds. For
IPMSM the direct flux weakening is not possible as the main flux is supplied by the
permanent magnets embedded in the rotor core. The IPMSM can be operated above the
rated speed by utilizing the demagnetizing effect of d-axis armature reaction current, id,
which makes the flux weakening operation [14-31]. Moreover, below the rated speed
with proper control of id the reluctance torque can also be utilized [6]. So, without
controlling the flux, the additional advantages of IPMSM over other PM motors cannot
be achieved. Usually, for vector control of IPMSM id is set to zero in order to linearize
the IPMSM model and to make the control task simpler [35]. Researchers reported
some field weakening operations of IPMSM but the computation of id is based on
simplified equation which results in inappropriate control of the flux and hence, non-
optimum performance of the drive, particularly, above the rated speed condition [14-
31]. In this work, the Newton-Raphson method is used for high precision computation
of id without simplifying the equation [39]. In order to control the net air-gap flux of
IPMSM, the d-axis stator current id is controlled according to the maximum torque per
ampere (MTPA) operation in constant torque region (below the rated speed) and the
flux-weakening (FW) operation in constant power region (above the rated speed).
3.2 Conventional Flux Control of IPMSM
As per phasor diagram Fig. 2.2(a)) the net air-gap flux linkage, o can be defined as,
ψo = √ψd2+ψq
2 (3.1)
where, d and q are d,q axis flux linkages, respectively, which can be defined as,
28
ψd = Ld id+ψm (3.2)
ψq = Lq iq (3.3)
Since id is negative for an IPMSM, it can be seen from Eqns. (3.1) and (3.2) that air-gap
flux linkage can be reduced by varying id. This is called demagnetizing effect of id.
Traditionally, id is set to zero to make the control task easier [29]. With id=0, motor
cannot run properly above rated speed as the flux can’t be controlled. So, the flux
control using id is needed to operate the IPMSM above rated speed within the rated
voltage and current capacities of the motor. Moreover, with id 0, more torque can be
produced by the motor as the reluctance torque is utilized.
3.2.1 Conventional Flux Control below Rated Speed – Maximum
Torque per Ampere (MTPA) Control
Referring to the phasor diagram of Fig. 2.2(a), the stator phase voltage (V a ) and
current ( I a ) can be related to the d-q axis voltages and currents as,
V a = vd + jvq (3.4)
I a = id + jiq (3.5)
The maximum value of stator phase voltage and current are considered as, Vm and Im,
respectively. Below the rated speed, with the assumption of keeping the absolute value
of stator current ( I a ) constant, id can be calculated in terms of iq for MTPA control. This
is obtained by differentiating electromagnetic developed torque equation (2.3) and
setting it to zero to get the maximum torque as [12],
29
dT e
diq= d
diq( 3 P
2 [ψmiq + (Ld − Lq ) id iq] )=0
(3.6)
and, I a =√( id )2+( iq )
2 (3.7)
After solving id can be obtained as,
id =ψm
2(Lq−Ld )− √ ψm
2
4 (Lq−Ld )2+ iq
2
(3.8)
In real-time, implementatation of the drive system becomes complex, and it
overburdens the DSP with expressions in equation (3.8). In order to solve this problem,
usually a simpler relationship between d- and q-axis currents, which is obtained by
expanding the square root term of equation (3.8) using Maclaurin series expansion as
(3.9). The values of the motor parameters in (3.8) are given in Appendix-A. In this case
only first two terms of series expansion are considered for simplicity.
(3.9)
This is the equation which is used to get id below rated speed.
3.2.2 Conventional Flux Control above Rated Speed – Field
Weakening (FW) Control
As the voltage is proportional to speed and flux, above rated speed the voltage and
hence the currents exceed the rated limit if the flux remains constant. In order to operate
the motor within the rated capacities the flux must be weakened. As discussed earlier,
for an IPMSM the flux can be weakened by the demagnetizing effect of id.
Conventionally, above the rated speed id is obtained from Eqns. (2.1), (2.2), and (3.4)
30
neglecting the stator resistance (Rs) and assuming the stator voltage constant at its
maximum value (Vm) as follows [12].
Neglecting Rs, from Eqn. (2.1) we get,
vdo = −Pωr iq Lq (3.10)
Neglecting Rs, from Eqn. (2.2) one can get,
vqo = Pωr Ld id+Pωr ψm (3.11)
From Eqn. (3.4),
V m' =√(vdo )
2+( vqo)2
(3.12)
From Eqns. (3.10)-(3.12) id can be obtained as,
id=−ψm
Ld+
1Ld √ V
m2'
P2 ωr2 −Lq
2 iq2
(3.13)
This equation is used to calculate id above the rated speed. Further to reduce the
computational burden, Maclaurin series expansion (only first one term is considered)
can be applied to Eqn. (3.13), which yields [30]:
id ≈ −10 . 328+2356 .27ωr
(1−316 .57 x10−9 ωr2 iq
2 ) (3.14)
Above the rated speed, the voltage remains constant as the magnet flux is weakened by
the d-axis armature reaction current, id and hence, the power remains constant in this
region. The typical torque-speed characteristic curve of a motor drive is shown in
Fig.3.1. Equation (3.13) represents an ellipse in the d-q plane, which indicates that an
increase in rotor speed results in smaller ranges for the current vector as shown in
Fig.3.2. By appropriately controlling id, the amplitude of the terminal voltage is adjusted 31
Speed
Torque
r,rated r,max
MTPA Region FW Region
Constant torque region Constant power region
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-18
-16
-14
-12
-10
-8
-6
-4
-2
0
2
Ia 188213238
Current iq, A
Current id, A
to Vm. Above the base speed i.e., in constant power region, the voltage remains constant
as the magnet flux is weakened by the armature reaction of id in order to decrease the
total air gap flux. The flux-weakening control not only extends the operating limits of
IPMSM drive but also relieves the current controllers from saturation that occurs at high
speeds. Saturation of the current controllers occur at high speeds for a given torque,
when the motor terminal voltage approaches Vm, which may cause instability of the
drive for id=0 control.
Fig. 3.1 Typical torque-speed characteristic curve of a motor drive.
Fig. 3.2: Locus of stator current Ia at different speeds staring from 188 rad/s to 363 rad/s
in a step of 25 rad/s.
32
3.3 Proposed Flux Control Techniques
In section 3.2 the current id is calculated based on Maclaurin series in order to
reduce the computational burden so that it becomes suitable for real-time
implementation. However, some approximation is taken in Maclaurin series to compute
id, which results in an improper value of the flux. Moreover, in order to simplify the
calculation most of the reported works neglected the stator resistance to calculate id in
the field weakening region [5-27]. Thus, the motor performance gets affected especially
at high speed condition. Therefore, in this thesis first, a more accurate analytical model
of id is developed in the FW region including the stator resistance. Then, the NRM
based numerical computation of id is used so that the flux calculation becomes more
accurate and hence, the motor runs smoothly above rated speed conditions.
33
3.3.1 Proposed Flux Control below Rated Speed
Below the rated speed for NRM based computation of id, Eqn. (3.8) can be
rewritten as,
f ( id)= {id2−
ψm id
(Lq−Ld)− iq
2}=0 (3.15)
According to NRM,
id (n+1 )= id (n )−f ( id )
f ' ( id )|id=id(n)
(3.16)
The Matlab program to solve for id below rated speed using (3.16) is shown in
Appendix-B. Below the rated speed the calculation of id does not show that much
difference between NRM based computation using Eqn. (3.16) and the conventional
computation using Eqn. (3.14). That is why, further analysis for id calculation below the
rated speed is not provided. The flux calculation above the rated speed is more
important than below the rated speed and hence, the numerical computation of id above
the rated speed is provided below.
3.3.2 Proposed Flux Control above Rated Speed
Above the rated speed for NRM based computation of id without neglecting stator
resistance, in steady-state the FW method can be re-evaluated from Eqn. (2.1), (2.2) and
(3.4) as follows [40]:
vd = R sid−Pωr iq Lq (3.17)
vq = R s iq+Pωr id Ld+Pωr ψm (3.18)
V m =√(vd )2+( vq )
2 (3.19)
⇒(Pωr ( Ld id+ψm)+R s iq )2+(R s id−Pωr Lq iq)
2=V m2
(3.20)
34
So, f ( id )= (Pωr (Ld id+ψ f )+Rs iq)2+(R s id−Pωr Lq iq )
2−V m2 =0 (3.21)
where, Vm is the maximum stator phase voltage without neglecting stator resistance.
From Eqns. (3.16) and (3.21), for certain values of iq and r, id can iteratively calculated
as,
(3.22)
Based on the iteration the value of id is calculated if the consecutive values converges
with the precision of =10-5. That means if |id (n+1 )− id (n )|≤ε then id=id(n+1). The
Matlab program to solve for id above rated speed using (3.22) is shown in Appendix-C.
First the value of id is calculated offline at each speed using the NRM and then a
polynomial of id as a function of speed is developed, which is used for flux weakening
algorithm.
Above the rated speed the variations of the proposed id with different speeds are
shown in Fig. 3.3. The values of id calculated based on simplified Eqn. (3.14) are also
shown in this figure. It is found that there is a significant difference between the two
values that causes the enhancement in the performance. It is also found that id calculated
based on Eqn. (3.14) is positive for speeds above but near rated speed (188.5 rad/s =
1800 rev/min(rpm)), which is not making the field weakening operation. Whereas, the
NRM based numerical computation of id incorporating the stator resistance (Rs) using
Eqn. (3.22) is always negative and hence, ensuring the proper field weakening
operation. Using the cubic polynomial curve fitting method the equation of id (computed
based on NRM) as a function of speed is determined as,
id = 0.0004r3 - 0.0129r
2 + 15.829r (3.23)
35
180 200 220 240 260 280 300 320 340-4
-3
-2
-1
0
1
2
id based on simplified equation (3.14)
id based on NRM incorporating Rs
Speed, rad/s
id, A
180 200 220 240 260 280 300 320 340 360-4
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
O - id calculated based on NRM incorporating Rs
id based on polynomial (3.23) id
Speed, rad/svd
id, A
This equation is used to calculate id for field weakening operation of IPMSM
above rated speed. The curve fitting for higher degree polynomial exhibits bad
conditioned. Moreover, unnecessary it will increase the computational burden. The
curve fitting equation was checked with the earlier data and the plot is shown in Fig.
3.4. It is found that the actual id matches almost perfectly with the values obtained from
polynomial (3.23). Thus, Eqn. (3.23) is used to calculate id above rated speed.
Fig. 3.3: Variation of id with speed, r.
Fig. 3.4: Comparison of curve fitting polynomial with actual id.
3.4 Simulation Results and Discussions
A closed loop vector of IPMSM drive incorporating the proposed control
algorithms mentioned earlier is developed in Matlab/Simulink [41]. The Simulink block
36
diagram for the proposed IPMSM drive and the reference current generator subsystem
are shown in Fig. 3.5(a) and 3.5(b), respectively. The flux control algorithm blocks
(MTPA and Flux_weaken), which are the main concern of this thesis, are shown in Fig.
3.5(b). The inputs of the command current generator subsystem shown in Fig. 3.5(b) are
command speed and the feedback actual speed of the motor. The outputs are the
command currents IAC, IBC and ICC. The speed controller is shown as proportional-
integral-derivative (PID) but the derivative part is considered zero so that it acts like PI
controller. The other Simulink subsystems of Fig 3.5(a) are given in Appendix-D. The
performance of the proposed drive system is investigated in simulation at different
operating conditions. Sample simulation results are presented in Figs. (3.6)-(3.10).
The simulated transient responses of the IPMSM drive for a step change of
command speed are shown in Figs. 3.6 and 3.7 for the conventional and proposed
calculation of id [8], respectively. It is clearly seen from Fig. 3.7(a) that the proposed
NRM based numerical computation of id ensures proper flux weakening operation and
hence there is no overshoot in the speed response when the speed increase from
188.5rad/s (=1800 revolution/min (rpm)) to 250 rad/s (=2387 rpm). Whereas, it is seen
from Fig. 3.6(a) that the speed response suffers from big overshoot due to the simplified
calculation of id. Therefore, the incorporation of the stator resistance to calculate id
based on NRM is justified. The torque component of the current (iq) decreases above
rated speed since the torque decreases in the field weakening region (Figs. 3.6(b) and
3.7(b)). For field weakening operation id is becoming more negative when the speed
goes above rated speed, which is shown in Fig. 3.7(c). It is seen from Fig. 3.6(d) that the
actual motor current cannot follow the command current at 250rad/s for conventional id
control as the current controller gets saturated due to improper flux control. Whereas, it
is seen from 3.7(d) that for the proposed id control actual motor current can follow the
command current in steady-state without any error.
37
iq
To Workspace9
ib
To Workspace8
ic
To Workspace7
t
To Workspace6
wrc
To Workspace5
ICC
To Workspace4
IBC
To Workspace3
IAC
To Workspace2
NA
To Workspace11
id
To Workspace10
ia
To Workspace1
wr
To Workspace
Switch1
Scope
wrc
cos_the
wr
sin_the
IAC
IBC
ICC
Ref_curr
iq
id
wr
th_e
Motor_out
NA
NB
NC
v qs
v ds
Inverter
IAC
ia
IBC
ib
ICC
ic
NA
NB
NC
Hys_cnt
sin(u)
Fcn2
cos(u)
Fcn1
wr
v q
v d
iq
Curr_trans
cos_the
v qs
sin_the
v dsv d
Coor_trans
250
Constant6
188.5
Constant4
0
Clock1
0
Clock
cos_the
iq
id
sin_the
ia
ib
ic
Actual_curr
3ICC
2IBC
1IAC
costheta
iq
sintheta
id
iac
ibc
icc
dq-abc trans
Switch
Sum1
PID
PID Controller(with Approximate
Derivative)
iqr* idr*
MTPA
iqr*
wr
idr*
Flux_weaken
-K-
(1/K)T*
4sin_the
3wr
2cos_the1wrc
Fig. 3.5. (a) Simulink block diagram for overall control system of IPMSM drive, (b)
Reference current generator subsystem.
38
(a)
(b)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
50
100
150
200
250
300
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
5
10
15
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-20
-18
-16
-14
-12
-10
-8
-6
-4
-2
0
Current id*, ACurrent iq*, A
Speed, rad/s
(a)
(b)
(c)
0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75-8
-6
-4
-2
0
2
4
6 (d)
Current ia, A
Time, s.
Actual motor speed
Reference speed
Fig. 3.6: Simulated transient responses of the drive for step change of speed at rated
load using the conventional computation of id [12]; (a) speed, (b) iq*, (c) id
*, (d) stator
phase current, ia (actual and command).
39
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-8
-7
-6
-5
-4
-3
-2
-1
0
id*, A
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
5
10
15
(b)
Speed, rad/s
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
50
100
150
200
250
300
Actual speed
(a)
Reference speed
(c)iq*, A
0.4 0.45 0.5 0.55 0.6 0.65 0.7
-6
-4
-2
0
2
4
6
Time, s
ia, A
(d)
Fig. 3.7: Simulated transient responses of the IPMSM drive for step change of speed at
rated load using the proposed NRM based id computation; ; (a) speed, (b) iq*, (c) id
*, (d)
stator phase current, ia (actual and command).
40
Fig. 3.8 shows the comparative speed errors for conventional and proposed NRM based
computation of id. The performance comparison of the IPMSM drive with conventional
and proposed id computation is summarized in Table-I. It is found that the conventional
drive has 10.8 % overshoot and the proposed drive has no overshoot at high speed (250
rad/s) condition. Moreover, the proposed drive has less settling time than the
conventional one.
Fig. 3.8: Comparison of speed error (= r* - r ): (a) conventional (b) NRM based
computation of id.
Table-I: Performance comparison of conventional and proposed computation of id
Conventional id computation
Proposed NRM based id computation
188.5 rad/s 250 rad/s 188.5 rad/s 250 rad/s
% Speed overshoot ((/ r
*)*100%)4.51 10.8 4.51 0
Settling time (s) 0.2 s 0.2 s 0.2 s 0.1 s
41
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-50
0
50
100
150
200
Time, s
(b)
Spe
ed e
rror
(r
ad/s
) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-50
0
50
100
150
200
Time, s
Spe
ed e
rror
(r
ad/s
)
(a)
Fig. 3.9 shows the response of the IPMSM drive at very high speed (320 rad/s
3056rpm) using the proposed NRM based id computation. The motor can follow this
high command speed only due to the proper calculation of id. With conventional
simplified computation of id, it was not possible to run the motor at such high speed
condition. Thus, the proposed technique extends the operating speed region of the
motor. As the speed increases, the frequency of the stator current increases, which is
shown in Fig. 3.9(c). The 3-phase currents shown in Fig. 3.9(d) indicate the balanced
operation of the motor.
A load disturbance is also applied to the motor and the corresponding responses are
shown in Fig. 3.10. The load is increased at t=0.8 s while the motor is running at 250
rad/s. The d,q axis currents adjusted with the new load condition as the id decreases and
iq increases further at t=0.8s. It is found that the motor can handle the load disturbance
while running above rated speed condition. Therefore, the proposed NRM based
numerical computation of id provides proper field weakening operation and hence the
drive can handle the uncertainties such as sudden change in reference speed and load
while running at high speed (above rated speed) condition. It has been found from the
results that the IPMSM drive with proposed calculation of id provides better response as
compared to the conventional calculation of id. Thus, the proposed method could be a
potential candidate for real-time field weakening operation of IPM motor.
42
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
50
100
150
200
250
300
350
0.605 0.61 0.615 0.62 0.625-3
-2
-1
0
1
2
30 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-8
-6
-4
-2
0
2
4
6
8
10
Time, s
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-30
-25
-20
-15
-10
-5
0
5
10
15
iq*
id*
Actual speed
Reference speed
(a)
(b)
(c)
(d)
Speed, rad/sd/dt
iq*, id* , A ia, AEncoder
ia, ib, ic, A
Fig. 3.9: Simulated responses of the IPMSM drive at very high speed (320 rad/s)
condition at rated load using the proposed NRM based id computation: (a) speed, (b)
command d-q axis currents id* and iq
*, (c)’a’ phase stator current, (d) steady-state 3-
phase stator currents of the motor.
43
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
50
100
150
200
250
300
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-12
-10
-8
-6
-4
-2
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
5
10
15
(a)
(b)
(c)
Current iq*, ACurrent id*, A
Speed, rad/s
Time, s
Fig. 3.10: Simulated responses of the proposed IPMSM drive for a step increase in
power in field weakening region: (a) speed, (b) command d-axis current id*, and (c)
command q-axis current iq*.
44
X-scale: 500ms/div
Y-scale: 40 rad/sec/div
100
200 250
188.5Y-scale: 25 rad/s/divX-scale: 0.5 s/div
Actual speed
Command speed
(b)
id, A
3.5 Experimental Test of the Proposed Field Weakening
The proposed id control method was tested in real-time at “Lakehead University
Power Electronics and Drives Laboratory” [26]. A sample experimental result is given
below. The experimental speed and the corresponding d-axis current responses for the
proposed NRM based IPMSM drive at a step change in command speed from 188.5
rad/s to 250 rad/s are shown in Fig. 3.11. It is found that the motor follows the above
rated command speed (250 rad/s) smoothly without any steady-state error, overshoot
and undershoot. It is seen from Fig. 3.11(b) that the d-axis current becomes more
negative for proper field weakening operation when the speed increases above rated
speed. Thus, the model developed for id and it’s computation using NRM are also
verified in real-time.
Fig. 3.11 Experimental responses of the IPMSM drive for a step increase in speed using
the proposed NRM based computation of id: (a) speed, (b) id.
45
(a)
3.6 Conclusion
In this Chapter at the beginning the necessity of flux control technique has been
discussed. Then, the conventional flux control technique and the proposed flux control
algorithms have been discussed in detail. For the proposed flux control, a more accurate
analytical model of d-axis stator current id incorporating stator resistance has been
developed. The NRM based computation of id has also been presented. The simulation
results of the proposed flux control technique for IPMSM drive at different operating
conditions have been presented. The effectiveness of the proposed flux control
technique has also been verified in real-time.
46
Chapter 4
Application of Runge-Kutta Method for
Solution of IPMSM Mathematical Model
4.1 Introduction
Traditionally, the MATLAB/Simulink library built-in motor model is used for
drives simulation [12,30]. Researchers usually overlooked about the accuracy of the
simulation performance of the drive caused by MATLAB/Simulink built-in motor
model. Therefore, it is necessary to investigate the use of numerical methods for the
solution of motor mathematical model. In this chapter the fourth order Runge-Kutta
method (RK4) is applied to solve the 1st order differential equations representing the
mathematical model of IPMSM. Then the motor currents obtained from RK4 are
compared with those of obtained from MATLAB/Simulink built-in motor model. The
limitations of the built-in motor model is also discussed in this Chapter.
47
4.2 Modelling of IPMSM
In order to apply RK4 the current based mathematical models of IPMSM are used,
which are given by,
Ld
did
dt=vd−R s id+Pωr Lq iq
(4.1)
Lq
diq
dt=vq−R s iq−Pωr Ld id−Pωr ψm (4.2)
For the solutions of (4.1) and (4.2) the values of vq and vd are considered as +150V and -
150V, respectively, which are corresponding to the rated values of stator voltages. After
putting all the motor parameters and rated speed values Eqns. (4.1) and (4.2) becomes,
did
dt=−3534 .401−45 .48 id+7066.83 iq=f 1( t , i d , iq ) (4.3)
diq
dt=397 . 78−24 .26 iq−201. 03 id= f 2( t , i d , iq ) (4.4)
Utilizing 4th order RKM id and iq are solved from these equations. The corresponding
Matlab codes are shown in Appendix E.
4.3 Simulation Results of the RKM and MATLAB/Simulink
Based Motor Model
The 3-phase stator currents (ia*, ib
* and ic*) are calculated from the d-q axis currents,
id and iq, obtained from Rk4 using inverse Parks’ transformation as given in Eqn. (2.8).
The Simulink block diagram to implement Eqn. (2.8) is shown in Fig. 4.1. Then the
stator ‘a’ phase current obtained from RK4 is compared with the one obtained from the
Simulink built-in machine model, which is shown in Fig. 4.2 [31].
48
te
torque
wr
speed
VDC
g
A
B
C
+
-
Universal Bridge
z
1
2.25
Load (Nm)
Tm
mABC
I nterior Permanent MagnetSynchronous Machine
Iabc*
Iabc
Pulses
Current RegulatorCommand currents from speed controller
Fig. 4.1 Simulink block diagram to get 3-phase currents from id and iq.
Fig. 4.2 MATLAB/Simulink based built-in IPMSM model.
The comparative results are shown in Figs. 4.3 and 4.4 for RKM and
MATLAB/Simulink, respectively. It is found that the steady-state current responses are
almost similar. There is a little bit difference in transient state (beginning part) as Fig.
4.3 was obtained in open loop just for the sake of testing the RKM and Fig. 4.4 was
obtained under closed loop condition. If the RKM is applied in closed loop condition,
49
iq
3icc
2 ibc
1 iac
id
To Workspace4
iq
To Workspace2
Sum7
Sum6
Sum3
Sum2
Product6
Product5
Product1
Product
1/2
Gain3
1
Gain1
-K-
1.732
4id
3sintheta
2iq
1costheta
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-12
-10
-8
-6
-4
-2
0
2
4
6
Time, s
(a)
ia, A
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-8
-6
-4
-2
0
2
4
6
8
10
12
(a)
ia, A
Time, s
0.4 0.41 0.42 0.43 0.44 0.45 0.46-5
-4
-3
-2
-1
0
1
2
3
4
5
(b)ia, ib, ic, A
Time, s0.4 0.41 0.42 0.43 0.44 0.45 0.46
-5
-4
-3
-2
-1
0
1
2
3
4
5
Time, s
(b)
ia, ib, ic, A
the transient response will also be same. Therefore, the machine model available in
MATLAB/Simulink can be used without any hesitation.
50
Fig. 4.3 RKM (ode4) based calculation of IPMSM model: (a) ‘a’ phase stator current (A), (b) steady-state 3-phase currents (A).
Fig. 4.4 Result based on MATLAB/Simulink based built-in IPMSM model: (a) ‘a’ phase stator current (A), (b) steady-state 3-phase currents (A).
4.4 Conclusion
The RKM based solution of IPMSM mathematical model has been presented in this
Chapter. From comparative results it is found that the MATLAB/SIMULINK library
built-in motor model can be used for general purpose as it provides acceptable results.
The built-in motor model can save lot of time for the researcher to simulate the drive's
performance. However, if more flexibility is required to change/include the motor
parameters (e.g., the effect of magnetic saturation, core loss, etc.) values the
mathematical model of the motor should be developed in software as these options are
not available in Simulink library built-in machine model.
51
Chapter 5
Conclusions
5.1 Summary of the Thesis
Based on the review of various electric motors in Chapter 1, it is concluded that the
interior permanent magnet synchronous motor (IPMSM) can be used to meet the criteria
of variable speed high performance electric motor drives. Problems of controlling the
IPMSM above rated speed have been identified. It has been found that the high
precision computation of flux is needed to run the IPMSM properly above rated speed.
The numerical analysis techniques such as Newton-Raphson and Runge-Kutta methods
have been proposed for performance investigation of the IPMSM.
The existing literature about mathematical model of the IPMSM, vector control
technique, speed controller, current controller and flux control techniques have been
reviewed in Chapter 2.
In Chapter 3 the flux control techniques of the IPMSM to extend the operating
speed range has been discussed. The techniques include the maximum torque per
ampere operation in the constant torque region (i.e., below the base speed) and the flux-
weakening operation in the constant power region (i.e., above the base speed). The flux
52
of IPMSM can be controlled by demagnetizing effect of d-axis stator current, id. A new
analytical model of id incorporating the stator resistance has been developed in this
Chapter. Then the NRM has been applied for high precision computation of id. The
performance of the proposed NRM based id control technique has been investigated in
simulation. A comparison has also been made between the proposed and the
conventional id computation based IPMSM drive. The proposed id computation
technique has been found superior to the conventional id control technique over a wide
speed range. Finally, the performance of the proposed id computation based IPMSM
drive has been verified in real-time.
In Chapter 4 the fourth order Runge-Kutta method (RK4) has been applied to solve
the ordinary differential equation representing the mathematical model of IPMSM. The
stator currents of the IPMSM have been computed based on RK4 and compared with
those of MATLAB/Simulink built-in motor model. It has been found that the current are
almost similar. However, it has been proposed to solve the mathematical model using
RK4 if greater flexibility such as to incorporate magnetic saturation, parameter variation
and iron loss resistance is needed to investigate the performance of the drive in
simulation.
5.2 Major Achievement of the Thesis
The achievement of the current work can be summarized as follows:
(1) In this work, the Newton-Raphson based numerical method has been used for high
precision computation of flux component of stator current, id.
53
(2) A new analytical model of id incorporating the stator resistance has been developed.
The calculated id provides proper flux weakening operation of IPMSM drive to
operate the motor above the rated speed over a wide speed range.
(3) The performance of the drive system with the proposed NRM based computation of
id has been investigated in simulation using Matlab/Simulink at different operating
conditions. The effectiveness of the proposed NRM based id computation has also
been tested in real-time.
(4) The performance of the IPMSM drive with the proposed NRM based computation of
id has also been compared with the conventional [12] simplified computation of id.
The IPMSM drive with proposed calculation of id provides better speed response as
compared to the conventional calculation of id. Thus, the proposed method could be a
potential candidate for industrial field weakening operation of IPMSM.
5) The fourth order Runge-Kutta method has been used to solve the mathematical model
of IPMSM and then the performance has been compared with the Simulink built-in
motor model. Through this investigation researcher can find the direction if more
flexibility in motor model is required for simulating the motor performance at
different operating conditions.
5.3 Future Scope of the Current Work
Other numerical methods may be applied for flux control and the results can be
compared with the current work.
Mathematical model of intelligent (e.g., fuzzy logic, neural network, etc.) flux
control scheme can be developed based on motor operating conditions (e.g.,
voltage, current and load) so that the model will be independent of motor
parameters (e.g., inductance and resistance). Then, the performance can be
54
compared with the current work.
Fourth order Runge-Kutta Method may be applied to solve the mathematical
model of IPMSM incorporating magnetic saturation and iron loss. Higher order
Runge-Kutta Methods may be also applied to investigate the solution of IPMSM
mathematical model and then it can be compared with the MATLAB/Simulink
motor model.
55
56
List of Papers Published from this Work
1. S. Pervin, Z. Siri and M. Nasir Uddin, “Newton-Raphson Based Computation of id
in the Field Weakening Region of IPM Motor Incorporating the Stator Resistance to
Improve the Performance”,Proceedings of 47thIEEE (Institute of
Electrical and Electronic Engineers) Industry Applications Society
(IAS) Annual Meeting, Oct. 2012, Las Vegas, USA, pp.1-6.
2. S. Pervin, M. Nasir Uddin and Z. Siri, “Improved Dynamic
Performance of IPMSM over Wide Speed Range Based on
Numerical Computation of id in the Field Weakening Region”,
Journal of International Review of Modeling and Simulations
(IREMOS), Praise Worthy Prize publishing house, Italy, vol.5, no. 5,
Oct. 2012, pp. 2042-2048. (SCOPUS cited)
3. S. Pervin, “Numerical Methods Based Computation and Analysis of Interior
Permanent Magnet Synchronous Motor Drive”, Colloquiums series, Institute of
Mathematical Sciences, University of Malaya, 27 June 2012.
57
58
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63
APPENDIX-A: IPMSM PARAMETERS
Appendix-B: (Application of NRM to solve for id below rated speed)
% Name: Shahida Pervin% Matric #: SGP 110002% Supervisor: Dr. Zailan Siri% Newton-Rephson Method for PM motor flux calculation below rated speed% id calculation if w,wrated; f(id)= id^2-xf*id/(2*(Ld-Lq))-iq^2Ld=0.04244;Lq=0.07957;iq= 2.45;xf=0.314;id1=0;for i=1:10id2=id1- (id1^2-(xf*id1)/(2*(Lq-Ld))-iq^2)/(2*id1-xf/(2*(Lq-Ld)));if abs(id2-id1)<=0.00001 output=id2endid1=id2;end
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Motor rated power 1 hp
Rated voltage 208 V
Rated current 3 A
Rated frequency and phase 60 Hz, 3-phase
Pole numbers 4 (pole-pair, P=2)
d-axis Inductance, Ld 42.44 mH
q-axis Inductance, Lq 79.57 mH
Damping coefficient, Bm .0008 Nm/rad/sec.
Motor inertia, Jm .003 kg-m
Stator resistance, Rs 1.93 Ohm
Magnet flux linkage, m 0.314 V/(rad/s)
Magnet type Samarium cobalt
Appendix-C: (Application of NRM to solve for id above rated speed)
% Name: Shahida Pervin% Matric #: SGP 110002% Supervisor: Dr. Zailan Siri% Newton-Rephson Method for PM motor flux calculation above rated speed% id calculation if w>wrated;clearP=2;R=1.93;Ld=0.04244;Lq=0.07957;xf=0.313;Vm=240;wr=300;iq=424/(wr*.939);%wr=188:12.5:363;%NN=length(wr);%for ii=1:NNid1=0; for i=1:10id11=(P*wr*(Ld*id1+xf)+R*iq)^2+(R*id1-P*wr*Lq*iq)^2-Vm^2;
id22=2*R*(R*id1-P*wr*Lq*iq)+2*P*wr*Ld*(P*wr*(Ld*id1+xf)+R*iq); id2=id1-id11/id22; if abs(id2-id1)<=0.00001 output=id2;endid1=id2;endid=output%id(ii)=output;%end%plot(id,wr)
65
Appendix-D: Simulink Subsystems
The Simulink block diagram for the proposed IPMSM drive and the reference
current generator subsystem were shown in Fig. 3.5(a) and 3.5(b), respectively. The
details of the other subsystem blocks for the Simulink schematic shown in Fig. 3.5 (a)
are presented in this appendix. The inputs of the command current generator subsystem
shown in Fig. 3.5(b) are command speed and the feedback actual speed of the motor.
The outputs are the command currents IAC, IBC and ICC. The command currents act as
inputs to the current controller subsystems (D.1). In current controller subsystems, the
relays are used to represent the hysteresis band. The actual motor currents (ia, ib, ic) also
act as inputs to this subsystem. The outputs of this subsystem are the logic signals NA,
NB and NC, which are used to turn the inverter switches on and off. The inputs of the
inverter subsystem (D.2) are logic signals NA, NB, NC and the dc bus voltage VB. The
outputs of this subsystem are the d-q axis voltages in the stationary reference frame. The
coordinate transformation subsystem (D.3) transforms d-q axis voltages from stationary
flame to the synchronously rotating rotor reference frame. The d-q axis currents
generation subsystem (D.4) generates d-q axis currents in the rotating reference frame
from the synchronously rotating d-q axis voltages and motor parameters. The actual
current generation subsystem was shown in Fig. 4.1, which generates actual currents ia,
ib and ic from the synchronously rotating d-q axis currents and the rotor position angle.
The motor output subsystem (D.5) generates the rotor position angle, th_e, and the
motor speed, r, from the synchronously rotating d-q axis currents and the motor
parameters using the motor dynamics.
66
D.1 Hysteresis Current Controller Subsystem
67
D.2 Inverter Subsystem
68
D.3 Coordinate Transformation Subsystem
69
D.4 d-q axis Current Generation Subsystem
70
D.5 Motor Output Subsystem
71
Appendix-E: Application of RK4 (ode4) to solve for id and iq from IPMSM Mathematical Model
% Name: Shahida Pervin% Matric #: SGP 110002% Supervisor: Dr. Zailan Siri% 4th order Runge-Kutta Method to solve for id and iq% Solve the 1st order differential Eqn. using ode4; % diq/dt=397.78-24.26iq-201.03id =f(t,iq,id);% did/dt=-3534.401-45.48id+706.83*iq = f(t,iq,id);% iq(n+1)=iq(n)+(k1+2k2+2k3+k4)h/6; iq(0)=5, t(0,1), h=0.0001, N=10000;% k1=hf(x0,t0); k2=h*f(iq0+k1/2, t0+h/2); k3=h*f(iq0+k2/2, t0+h/2);% k4=h*f(iq0+k3, t0+h);% t0=0;% iq0=10;
cleara=0;b=0.1;t=a;h=.0001;N=(b-a)/h;iq(1)=10;id(1)=-20;for i=1:Nk1(i)=h*(397.78-24.26*iq(i)-201.03*id(i));k2(i)=h*(397.78-24.26*(iq(i)+k1(i)/2)-201.03*id(i));k3(i)=h*(397.78-24.26*(iq(i)+k2(i)/2)-201.03*id(i));k4(i)=h*(397.78-24.26*(iq(i)+k3(i))-201.03*id(i));iq(i+1)=iq(i)+(k1(i)+2*k2(i)+2*k3(i)+k4(i))/6; k11(i)=h*(-3534.401-45.48*id(i)+706.83*iq(i));k12(i)=h*(-3534.401-45.48*(id(i)+k11(i)/2)+706.83*iq(i));k13(i)=h*(-3534.401-45.48*(id(i)+k12(i)/2)+706.83*iq(i));k14(i)=h*(-3534.401-45.48*(id(i)+k13(i))+706.83*id(i));id(i+1)=id(i)+(k1(i)+2*k2(i)+2*k3(i)+k4(i))/6;t=t+h;endt=a:h:b;plot(t,iq)hold onplot(t,id),grid
72