115,99 €
Mehr erfahren.
- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Serie: Wiley-IEEE Press Book Series on Control Systems Theory and Applications
- Sprache: Englisch
A comprehensive resource providing basic principles and state-of-the art developments in sensorless control technologies for permanent magnet synchronous machine drives Sensorless Control of Permanent Magnet Synchronous Machine Drives highlights the global research achievements over the last three decades and the sensorless techniques developed by the authors and their colleagues, and covers sensorless control techniques of permanent magnet machines, discussing issues and solutions. Many worked application examples are included to aid in practical understanding of concepts. Written by two pioneering authors in the field, Sensorless Control of Permanent Magnet Synchronous Machine Drives covers sample topics such as: * Permanent magnet brushless AC and DC drives * Single three-phase, dual three-phase, and open winding machines * Modern control theory based sensorless methods, covering model reference adaptive system, sliding mode observer, extended Kalman filter, and model predictive control * Flux-linkage and back-EMF based methods for non-salient machines, and active flux-linkage and extended back-EMF methods for salient machines * Pulsating and rotating high frequency sinusoidal and square wave signal injection methods with current or voltage response, at different reference frames, and selection of amplitude and frequency for injection signal * Sensorless control techniques based on detecting third harmonic or zero-crossings of back-EMF waveforms * Parasitic effects in fundamental and high frequency models, impacts on position estimation, and compensation schemes, covering cross-coupling magnetic saturation, load effect, machine saliency and multiple saliencies, inverter non-linearities, voltage and current harmonics, parameter asymmetries, and parameter mismatches * Techniques for rotor initial position estimation, magnetic polarity detection, and transition between low and high speeds Describing basic principles, examples, challenges, and practical solutions, Sensorless Control of Permanent Magnet Synchronous Machine Drives is a highly comprehensive resource on the subject for professionals working on electrical machines and drives, particularly permanent magnet machines, and researchers working on electric vehicles, wind power generators, household appliances, and industrial automation.
Sie lesen das E-Book in den Legimi-Apps auf:
Seitenzahl: 568
Veröffentlichungsjahr: 2023
Ähnliche
IEEE Press
445 Hoes Lane
Piscataway, NJ 08854
IEEE Press Editorial BoardSarah Spurgeon, Editor in Chief
Jón Atli Benediktsson
Behzad Razavi
Jeffrey Reed
Anjan Bose
Jim Lyke
Diomidis Spinellis
James Duncan
Hai Li
Adam Drobot
Amin Moeness
Brian Johnson
Tom Robertazzi
Desineni Subbaram Naidu
Ahmet Murat Tekalp
Sensorless Control of Permanent Magnet Synchronous Machine Drives
Zi Qiang Zhu
and
Xi Meng Wu
University of Sheffield, UK
IEEE Press Series on Control Systems Theory and Applications
Maria Domenica Di Benedetto, Series Editor
Copyright © 2024 by The Institute of Electrical and Electronics Engineers, Inc. All rights reserved.
Published by John Wiley & Sons, Inc., Hoboken, New Jersey.Published simultaneously in Canada.
No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per‐copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750‐8400, fax (978) 750‐4470, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748‐6011, fax (201) 748‐6008, or online at http://www.wiley.com/go/permission.
Trademarks: Wiley and the Wiley logo are trademarks or registered trademarks of John Wiley & Sons, Inc. and/or its affiliates in the United States and other countries and may not be used without written permission. All other trademarks are the property of their respective owners. John Wiley & Sons, Inc. is not associated with any product or vendor mentioned in this book.
Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Further, readers should be aware that websites listed in this work may have changed or disappeared between when this work was written and when it is read. Neither the publisher nor authors shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages.
For general information on our other products and services or for technical support, please contact our Customer Care Department within the United States at (800) 762‐2974, outside the United States at (317) 572‐3993 or fax (317) 572‐4002.
Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic formats. For more information about Wiley products, visit our web site at www.wiley.com.
Library of Congress Cataloging‐in‐Publication Data Applied for:
Hardback ISBN: 9781394194353
Cover Design: WileyCover Image: © Bosca78/Getty Images
About the Authors
Professor Zi Qiang Zhu received the BEng and MSc degrees from Zhejiang University, Hangzhou, China, in 1982 and 1984, respectively, and the Ph.D. degree from The University of Sheffield, Sheffield, U.K., in 1991, all in electrical engineering.
After working as Lecturer/Assistant Lecturer at Zhejiang University from 1984 to 1988, he has been with The University of Sheffield since 1988, initially as Visiting Research Fellow sponsored by the British Council (1988–1989), then Research Associate working with Philips (1989‐1992), Senior Research Scientist/Officer––an established university post (1992–2000), Professor (2000–present), Royal Academy of Engineering/Siemens Research Chair (2014–present), and Head of the Electrical Machines and Drives Research Group (2008–present). As the Founding Director, he has helped several industries in establishing their research centers, most notably, Siemens Gamesa Renewable Energy Research Center at Sheffield (2009–present) and Midea Electrical Machines and Control Research Centers at Shanghai and Sheffield (2010–present).
His research interests include design and control of permanent magnet brushless machines and drives for applications ranging from electrified transportation (electric vehicles, fast trains, and more electric aircraft) to domestic appliances to wind power generation. He has published more than 1400 papers, including more than 550 for IEEE Transactions and IET Proceedings.
He is a Fellow of the Royal Academy of Engineering, UK; Institute of Electrical and Electronics Engineers (IEEE), USA; Institution of Engineering and Technology (IET), UK; and Chinese Society for Electrical Engineering (CSEE), China. He is the recipient of the 2019 IEEE Industry Applications Society Outstanding Achievement Award and the 2021 IEEE Nikola Tesla Award.
Dr Xi Meng Wu received the BEng degree in electrical engineering and automation from Hefei University of Technology, Hefei, China, in 2011, and MSc and PhD degrees in electrical engineering from The University of Sheffield, Sheffield, UK in 2016 and 2020, respectively. Since 2020, he has been a Postgraduate Research Associate at the University of Sheffield associated with the Sheffield Siemens Gamesa Renewable Energy Research Centre. His research interests include control of permanent magnet synchronous machines.
Preface
Permanent magnet (PM) synchronous machine (PMSM) drives, including PM brushless ac and dc drives, exhibit many advantages such as high efficiency and high torque density. A high‐performance PM brushless ac or dc drive needs accurate rotor position information. This is usually obtained by using a hardware rotor position sensor, such as a resolver, encoder, or Hall sensor. However, these sensors increase drive size and cost and reduce reliability, particularly in harsh environments. Therefore, it is desirable to replace hardware rotor position sensors with software‐based rotor position sensorless techniques.
This book aims to comprehensively describe sensorless control techniques of PMSM drives. We have strived to highlight the global research achievements and also many new techniques developed at the University of Sheffield. The basic principles and state‐of‐the‐art rotor position sensorless control techniques are explained, together with their challenges and practical solutions. The scope is very broad, and readers may find the summary diagram in section 1.7 useful.
Thirty years ago, sensorless control of PMSM drives had limited application and was mostly used for driving ventilation fans with brushless dc drives. However, over the last 10 years, the field has rapidly expanded. Today, numerous commercial products are using sensorless control techniques for a wide variety of applications, for example wind power generators, automotive compressors, water and oil cooling pumps, electric bicycles, drones, general purpose variable frequency drives, and household appliances (e.g. air‐conditioning and refrigerator compressors and fans, washing machines, dishwasher pumps, heat circulating pumps, and vacuum cleaners), as well as fault‐tolerance drives in electrified transportation and aerospace applications. Despite these successes, the use of sensorless control for applications that require high torque for rapid starting remains challenging. As the technology continues to evolve and improve, it is likely that sensorless control will find even broader applications, offering a reliable, cost‐effective, and efficient solution for a wide range of industrial and commercial needs.
Beginning in 1998, I have been working with my PhD students on various topics of sensorless control of PMSM drives. I would like to take this opportunity to thank them for their contributions, including J. Ede, J. X. Shen, Y. F. Shi, Y. Liu, Y. Li, L. M. Gong, J. M. Liu, T. C. Lin, A. H. Almarhoon, P. L. Xu, H. L. Zhan, X. M. Wu, L. Yang, B. Shuang, and T. Y. Liu. In particular, Dr. Xi Meng Wu, currently a post‐doctoral researcher at the University of Sheffield and the co‐author of this book, has made significant contributions to sensorless control, especially the novel techniques for initial rotor position detection.
I would like to acknowledge the pioneering work on sensorless control of PMSM drives by Professor Robert D. Lorenz at University of Wisconsin‐Madison, USA, Professor Seung‐Ki Sul at Seoul National University, South Korea, and Professor Fernando Briz and Professor David Reigosa at University of Oviedo, Spain, as well as Professor Nobuyuki Matsui, Dr. Zhiqian Chen, Professor Shigeo Morimoto, Professor Ion Boldea, and Professor Kaushik Rajashekara, and others.
I dedicate my thanks to our sponsors, including the UK government, particularly the UK EPSRC Prosperity Partnership (“A New Partnership in Offshore Wind” under Grant No. EP/R004900/1), and our partners in industry such as Siemens Gamesa Renewable Energy Ltd. on wind power, Midea group on household appliances, Nissan and Toyota group on electric vehicles.
My co‐author and I sincerely hope this book will be useful to industrial engineers and researchers, university professionals, post‐doctoral researchers, and students alike.
Professor Zi Qiang Zhu
FREng, FIEEE, FIET, FCSEE
University of Sheffield, UK
List of Abbreviations
ac
Alternating current
AD
Analog to digital
ADC
Analog‐to‐digital converter
AF
Active flux
ANF
Adaptive notch filter
BDS
Boundary selection strategy
BLAC
Brushless ac
BLDC
Brushless dc
BPF
Band‐pass filter
CCS‐MPC
Continuous‐control‐set model predictive control
CM
Current model
CMV
Common mode voltage
CPU
Central processing unit
dc
Direct current
DEA
Differential evolution algorithm
DFT
Discrete Fourier transform
DQZ
Direct‐quadrature‐zero
DSP
Digital signal processor
DTC
Direct torque control
DTP‐PMSM
Dual three‐phase permanent magnet synchronous machine
EEMF
Extended electromotive force
EKF
Extended Kalman filter
EMF
Electromotive force
FCS‐MPC
Finite‐control‐set model predictive control
FE
Finite element
FEA
Finite element analysis
FO
Flux‐linkage observer
FOC
Field oriented control
HF
High frequency
HPF
High‐pass filter
INFORM
Indirect flux detection by online reactance measurement
IPM
Interior PM
IPMSM
Interior permanent magnet synchronous machine
LDF
Lower‐diode freewheeling
LMS
Least‐mean‐squares
LPF
Low‐pass filter
LUT
Look‐up table
MMF
Magneto‐motive force
MPC
Model predictive control
MRAS
Model reference adaptive system
MTPA
Maximum torque per ampere
OW‐PMSM
Open‐winding permanent magnet synchronous machine
PI
Proportional integral
PLL
Phase‐locked loop
PM
Permanent magnet
PMSM
Permanent magnet synchronous machine
PO
Position observer
PWM
Pulse width modulation
QSG
Quadrature signal generator
RSA
Reliable selection area
RVD
Resistance voltage divider
SMO
Sliding mode observer
SNR
Signal‐to‐noise ratio
SOA
Safe operating area
SOGI
Second order generalized integrator
SPM
Surface‐mounted PM
SPMSM
Surface‐mounted permanent magnet synchronous machine
SSOA
Sensorless safe operation area
STP‐PMSM
Single three‐phase permanent magnet synchronous machine
SVPWM
Space vector pulse width modulation
THD
Total harmonic distortion
UDF
Upper‐diode freewheeling
VC
Vector control
VM
Voltage model
VSD
Vector space decomposition
VSI
Voltage source inverter
ZCD
Zero‐crossing detection
ZCP
Zero‐crossing point
ZSC
Zero sequence current
ZSV
Zero sequence voltage
ZVC
Zero vector current
ZVCD
Zero vector current derivative
List of Symbols
Symbol
Description
Unit
A
ac
Amplitude of injected ac current reference
A
A
d
,2
f
,
A
q
,2
f
Amplitudes of second order harmonic of
d
‐ and
q
‐axis estimated back‐EMFs
V
A
mp
Amplitude of third harmonic flux‐linkage
Wb
Amp
_
i
A
h
,
Amp
_
i
B
h
,
Amp
_
i
C
h
Amplitudes of three‐phase HF currents
A
B
Viscous damping factor
Ns/m
B
3
Amplitude of third harmonic component of excitation flux density
T
D
Duty ratio
E
A
(
s
),
E
B
(
s
),
E
C
(
s
)
Three‐phase back‐EMFs after Laplace transform
V
e
A
,
e
B
,
e
C
Three‐phase back‐EMFs
V
e
A5
,
e
B5
,
e
C5
Three‐phase fifth order harmonic back‐EMFs
V
e
A7
,
e
B7
,
e
C7
Three‐phase seventh order harmonic back‐EMFs
V
E
a
,
d
,
E
a
,
q
dq
‐axis back‐EMFs of active flux model
V
e
c
Back‐EMF of current model
V
,
Estimated
d
‐ and
q
‐axis back‐EMFs
V
,
dc components of
d
‐ and
q
‐axis estimated back‐EMFs
V
,
Second order harmonic of
d
‐ and
q
‐axis estimated back‐EMFs
V
E
ex
Extended back‐EMF
V
,
Estimated
d
‐ and
q
‐axis extended back‐EMFs
V
E
ex
_
imp
Improved extended back‐EMF
V
E
ex
1
Equivalent extended back‐EMF of DTP‐PMSM
V
e
ex
,
α
,
e
ex
,
β
α
‐ and
β
‐axis extended back‐EMFs
V
e
H
,
e
L
,
e
O
High‐level, low‐level, and floating phase back‐EMFs
V
E
m
Peak value of phase back‐EMF
V
E
m
3
Amplitude of third harmonic back‐EMF
V
e
α
,
e
β
α
‐ and
β
‐axis back‐EMFs
V
e
0
Zero sequence back‐EMF
V
Estimated zero sequence back‐EMF
V
E
3
Amplitude of third harmonic back‐EMF
V
e
3_
set
1
,
e
3_
set
2
Third harmonic back‐EMFs in two sets
V
e
3_
set
1
,
e
9_
set
1
,
e
15_
set
1
Third, ninth, and fifteenth harmonic back‐EMFs of first set of DTP‐PMSM
V
e
3_
set
2
,
e
9_
set
2
,
e
15_
set
2
Third, ninth, and fifteenth harmonic back‐EMFs of second set of DTP‐PMSM
V
e
9
Ninth order harmonic back‐EMF
V
e
15
Fifteenth order harmonic back‐EMF
V
f
e
Electrical rotor frequency
Hz
f
h
Frequency of injected high‐frequency voltage signal
Hz
f
(Δ
θ
)
Position error signal
I
*
Amplitude of extra injected current signal
A
I
A
(
s
),
I
B
(
s
),
I
C
(
s
)
Three‐phase currents after Laplace transform
A
i
A
,
i
B
,
i
C
Three‐phase stator currents
A
, ,
Three‐phase primary current response peak values
A
, ,
Three‐phase secondary current response peak values
A
i
ABC
h
Three‐phase high‐frequency current responses
A
I
AD
_
error
Disturbance current vector due to current measurement error
A
I
d
,
I
q
Amplitudes of
dq
‐axis currents
A
i
d
,
i
q
d
‐ and
q
‐axis currents
A
,
Estimated
d
‐ and
q
‐axis currents
A
i
dc
dc‐link current
A
dc‐link current reference
A
,
Estimated
d
‐ and
q
‐axis ac current components
A
i
d
,
CM
,
i
q
,
CM
d
‐ and
q
‐axis currents of current model
A
,
Estimated
d
‐ and
q
‐axis dc current components
A
i
df
,
i
qf
Fundamental
d
‐ and
q
‐axis currents
A
,
Amplitudes of estimated
dq
‐axis high‐frequency currents
A
i
dh
,
i
qh
d
‐and
q
‐axis high‐frequency currents
A
,
Estimated
d
‐ and
q
‐axis high‐frequency currents
A
,
Virtual
d
‐ and
q
‐axis high‐frequency currents
A
References of
d
‐ and
q
‐axis currents
A
Predicted
d
‐ and
q
‐axis currents
A
Δ
I
error
Error between real and recorded currents
A
Extra injected current signal
A
i
H
,
i
L
,
i
O
High‐level, low‐level, and floating phase currents
A
I
m
Peak value of phase current
A
I
max
Maximum current response peak value
A
I
mean
Average current response peak value
A
I
n
Amplitude of negative sequence current response
A
i
n
Negative sequence current response
A
Estimated negative sequence current response
A
Amplitude of negative sequence current response of square‐wave injection
A
,
Estimated
d
‐ and
q
‐axis negative sequence current responses
A
I
neg
_
α
,
I
neg
_
β
α
‐ and
β
‐axis components of negative sequence HF current
A
I
p
Amplitude of positive sequence current response
A
i
p
Positive sequence current response
A
Amplitude of positive sequence current response of square‐wave injection
A
I
pos
_
α
,
I
pos
_
β
α
‐ and
β
‐axis components of positive sequence HF current
A
I
q
_
MAX
Maximum
q
‐axis current
A
I
qu
Quantum current of analog‐to‐digital converter
A
I
real
Real current
A
I
record
Recorded current
A
I
s
Amplitude of stator current
A
i
s
Stator currents
A
Δ
I
th
Threshold current value
A
i
X
,
i
Y
,
i
Z
Phase currents of second winding set of DTP‐PMSM
A
i
X
f
Fundamental current in arbitrary phase
A
i
X
h
High‐frequency current in arbitrary phase
A
i
z
1
z
2
Stator current in
z
1
z
2
subspaces
A
i
α
,
i
β
α
‐ and
β
‐axis currents
A
,
α
‐ and
β
‐axis estimated currents
A
Δ
i
α
, Δ
i
β
α
‐ and
β
‐axis current estimation errors
A
I
αβh
Amplitudes of
α
‐and
β
‐axis high‐frequency currents
A
i
αβh
α
‐ and
β
‐axis high‐frequency currents
A
α
‐ and
β
‐axis high‐frequency currents before compensating positive current
A
α
‐ and
β
‐axis high‐frequency currents after compensating for positive current
A
I
0
Amplitude of dc component of three‐phase current responses
A
i
0
Zero sequence current
A
I
2
Amplitude of second order harmonic component of three‐phase current responses
A
i
2
nd
Secondary positive sequence harmonics in HF current response
A
J
Inertia
kg.m
2
k
c
Compensation factor for cross‐coupling inductances
mH/A
K
i
Integration gain of PI controller
Deviation factor of
q
‐axis inductance
rad/A
Deviation factor of
q
‐axis self‐inductance in Set 1
rad/A
Deviation factor of
q
‐axis mutual-inductance between two sets
rad/A
K
p
Proportional gain of PI controller
k
p
3
,
k
d
3
,
k
s
3
Coil pitch factor, distribution factor, and skew factor for third harmonic
K
R
Equivalent gain of resistance voltage divider
k
r
Compensation factor for cross‐coupling error angle
rad/A
,
Deviation factors of resistance and
q
‐axis equivalent inductance
rad/A
k
w
3
Winding factor for third harmonic
K
ω
Slope of back‐EMF envelope around ZCP
V
L
Phase self‐inductance of BLDC
mH
Δ
L
Asymmetric inductance
mH
L
AA
,
L
BB
,
L
CC
Three‐phase self‐inductances
mH
Δ
L
AB
, Δ
L
BC
, Δ
L
CA
Three‐phase inductance asymmetric errors of BLDC
mH
L
c
Second order harmonic amplitude of sine inductance term in self‐inductance
mH
L
D
,
L
Q
Equivalent
d
‐ and
q
‐axis inductances of DTP‐PMSM
mH
L
d
,
L
q
d
‐ and
q
‐axis self‐inductances
mH
,
Nominal values of
d
‐ and
q
‐axis inductances
mH
Δ
L
d
, Δ
L
q
Mismatch values of
d
‐ and
q
‐axis inductances
mH
L
dh
,
L
qh
dq
‐axis incremental self‐inductances
mH
L
dq
,
L
qd
Cross‐coupling
dq
‐axis inductances
mH
L
dqh
,
L
qdh
Cross‐coupling
dq
‐axis incremental inductances
mH
L
d
1
,
L
d
2
,
L
q
1
,
L
q
2
d‐
and
q
‐axis self‐inductances of two winding sets
mH
L
eq
Equivalent inductance
mH
L
H
,
L
L
High‐ and low‐level inductances
mH
L
Δ
h
Amplitude of
h
th spatial inductance
mH
L
lm
Boundary inductance
mH
L
ls
Leakage self‐inductance
mH
L
MAX
,
L
MIN
Maximum and minimum inductances
mH
L
n
Negative sequence inductance
mH
L
p
Positive sequence inductance
mH
L
qds
Approximated cross‐coupling inductances
mH
L
sa
Average between
d
‐ and
q
‐axis incremental inductances
mH
L
sd
Difference of
d
‐ and
q
‐axis incremental inductances
mH
L
sj
,
L
sk
j
th and
k
th order self‐inductances
mH
L
s
0
Average value of self‐inductance
Wb
L
s
2
Amplitude of second order harmonic component of self‐inductance
Wb
L
XX
,
L
YY
,
L
ZZ
Three‐phase self‐inductances of first winding set in DTP‐PMSM
mH
L
αα
,
L
ββ
α
‐ and
β
‐axis self‐inductances
mH
L
0
Zero sequence inductance
mH
M
Phase mutual-inductance of BLDC
mH
M
AB
,
M
BA
,
M
AC
,
M
CA
,
M
BC
,
M
CB
Three‐phase mutual-inductances
mH
M
c
Second order harmonic amplitudes of sine inductance terms in mutual-inductances
mH
M
d
12
,
M
d
21
,
M
q
12
,
M
q
21
d‐
and
q
‐axis mutual-inductances between two winding sets
mH
Δ
M
q
21
, Δ
M
q
12
Deviation values of
q
‐axis inductances between two sets
mH
M
sj
,
M
sk
j
th and
k
th order mutual-inductances
mH
M
s
0
Average value of mutual-inductance
Wb
M
s
2
Second order harmonic of mutual-inductance
Wb
M
XY
,
M
YX
,
M
YZ
,
M
ZY
,
M
ZX
,
M
XZ
Three phase mutual-inductances of second winding set in DTP‐PMSM
mH
M
αβ
,
M
βα
α
‐ and
β
‐axis mutual-inductances
mH
N
s
Number of sample points
P
(
k
)
Covariance matrix of EKF
P
Number of pole pairs
p
Derivative operator
Q
(
k
),
R
(
k
)
Covariances of process noise and measurement noise of EKF
R
Phase resistance of BLDC
Ω
Δ
R
A
, Δ
R
B
, Δ
R
C
Asymmetric resistances components
Ω
Δ
R
ave
dc offset due to resistance asymmetry
Ω
R
d
,
R
q
d
‐ and
q
‐axis resistances
Ω
R
dc
dc‐link resistance
Ω
R
dq
dq
‐axis mutual-resistance
Ω
R
eq
Equivalent resistance
Ω
R
N
Resistance of auxiliary resistor network
Ω
R
s
Phase resistance
Ω
Nominal value of phase resistance
Ω
Balanced part of three‐phase resistances
Ω
Δ
R
s
Mismatch value of phase resistance
Ω
R
X
h
Equivalent HF resistance of inverter in arbitrary phase
Ω
R
1
,
R
2
Nominal values of low and high side resistances of resistance voltage divider
Ω
,
Actual values of low and high side resistances of resistance voltage divider
Ω
S
Sliding mode surface
S
A
, S
B
, S
C
Switching states of three legs of VSI
t
Time
s
Δ
T
Period of injected square‐wave voltage signal
s
Δ
t
Time step
s
T
c
Time constant of LPF
s
t
d
Time interval of half cycle between two zero‐crossing points
s
t
dd
Turn‐off delay of power device
s
t
dt
Deadtime
s
t
du
Turn‐on delay of power device
s
T
inj
Period of extra injected current signal
s
T
i
1
,
T
i
2
Periods of first and second injected HF voltage signals
s
T
L
Load torque
Nm
T
m
_
BLAC
Electromagnetic torque of a BLAC machine
Nm
T
m
_
BLDC
Electromagnetic torque of a BLDC machine
Nm
T
opt
Optimal duration of voltage pulse
s
T
P
Duration of voltage pulse
s
t
period
Fundamental period
s
T
P
_
MAX
,
T
P
_
MIN
Maximum and minimum durations of voltage pulse
s
t
r
Remainder of division of time by injection period
s
T
s
Sampling period
s
t
φ
[
n
]
n
th time delay for commutation instant
s
t
θ
[
n
]
n
th ZCP time interval
s
T
0
Duration of zero voltage vector
s
T
1
Duration of voltage vector 1
s
T
2
Duration of voltage vector 2
s
t
23
,
t
34
,
t
45
,
t
25
,
t
52
,
t
56
,
t
61
,
t
12
Period between sectors
s
u
VA
,
u
VB
,
u
VC
Three‐phase vertical error correction common‐mode bias
V
u
1
,
u
2
,
u
3
,
u
4
,
u
5
,
u
6
Zero‐crossing thresholds
V
Δ
V
Average terminal voltage error
V
v
A
,
v
B
,
v
C
Three‐phase stator voltages
V
Δ
v
AB
, Δ
v
BC
, Δ
v
CA
Three‐phase horizontal voltage shifts
V
V
AD
Maximum sampling voltage
V
V
AG
(
s
),
V
BG
(
s
),
V
CG
(
s
)
Three‐phase terminal voltages after Laplace transform
V
v
AG
,
v
BG
,
v
CG
Voltage between phase terminal and ground
V
v
A
h
,
v
B
h
,
v
C
h
Injected three‐phase HF voltages
V
v
AN
,
v
BN
,
v
BN
Phase voltages of a PM machine in ABC reference frame
V
Acquired phase B terminal voltage
V
V
c
Amplitude of equivalent voltage source
V
v
d
,
v
q
d
‐ and
q
‐axis voltages
V
,
Estimated
d
‐ and
q
‐axis voltages
V
V
dc
dc‐link voltage
V
Δ
V
dc
dc‐link voltage variation
V
Estimated
d
‐axis feed‐forward voltage
V
v
dh
,
v
qh
d‐
and
q
‐axis high‐frequency voltages
V
,
Estimated
d
‐ and
q
‐axis high‐frequency voltages
V
,
Virtual
d
‐ and
q
‐axis high‐frequency voltages
V
v
dh
1
,
v
dh
2
First and second injected HF voltage signals
V
,
Voltages of voltage model
V
Δ
v
f
Fundamental disturbance voltage
V
Fundamental voltage reference solution
V
V
h
Amplitude of injected high‐frequency voltage
V
Δ
V
H
(
s
)
Voltage shift caused by asymmetric parameters after Laplace transform
V
Δ
v
H
Horizontal voltage shift
V
Δ
v
h
High‐frequency disturbance voltage
V
High‐frequency voltage signal reference
V
v
HG
,
v
LG
High‐level and low‐level terminal voltages
V
Δ
v
HL
Voltage error caused by asymmetric parameters
V
V
h
1
,
V
h
2
Amplitudes of first and second injected HF voltage signals
V
v
(
k
)
Measurement noise
V
v
N
Neutral voltage
V
V
NG
(
s
)
Zero sequence voltage after Laplace transform
V
V
N1N2
Amplitude of zero sequence HF voltage of DTP‐PMSM
V
v
N1N2
Zero sequence HF voltage of DTP‐PMSM
V
V
P
Magnitude of voltage pulse
V
V
P
_
MAX
,
V
P
_
MIN
Maximum and minimum magnitudes of voltage pulse
V
V
RL
Voltage drop on winding resistance and inductance
V
Amplitude of voltage reference
V
Voltage reference vector
V
v
SM
Voltage between network central point and capacitor mid‐point
V
Upper envelope of measured virtual third harmonic voltage
V
v
SN
