Practical Control of Electric Machines for EV/HEVs: Design, Analysis, and Implementation [1 ed.] 3031381602, 9783031381607

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Lecture Notes in Electrical Engineering 1064

Shuiwen Shen Qiong-zhong Chen

Practical Control of Electric Machines for EV/HEVs Design, Analysis, and Implementation

Lecture Notes in Electrical Engineering Volume 1064

Series Editors Leopoldo Angrisani, Department of Electrical and Information Technologies Engineering, University of Napoli Federico II, Napoli, Italy Marco Arteaga, Departament de Control y Robótica, Universidad Nacional Autónoma de México, Coyoacán, Mexico Samarjit Chakraborty, Fakultät für Elektrotechnik und Informationstechnik, TU München, München, Germany Jiming Chen, Zhejiang University, Hangzhou, Zhejiang, China Shanben Chen, School of Materials Science and Engineering, Shanghai Jiao Tong University, Shanghai, China Tan Kay Chen, Department of Electrical and Computer Engineering, National University of Singapore, Singapore, Singapore Rüdiger Dillmann, University of Karlsruhe (TH) IAIM, Karlsruhe, Baden-Württemberg, Germany Haibin Duan, Beijing University of Aeronautics and Astronautics, Beijing, China Gianluigi Ferrari, Dipartimento di Ingegneria dell’Informazione, Sede Scientifica Università degli Studi di Parma, Parma, Italy Manuel Ferre, Centre for Automation and Robotics CAR (UPM-CSIC), Universidad Politécnica de Madrid, Madrid, Spain Faryar Jabbari, Department of Mechanical and Aerospace Engineering, University of California, Irvine, CA, USA Limin Jia, State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University, Beijing, China Janusz Kacprzyk, Intelligent Systems Laboratory, Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland Alaa Khamis, Department of Mechatronics Engineering, German University in Egypt El Tagamoa El Khames, New Cairo City, Egypt Torsten Kroeger, Intrinsic Innovation, Mountain View, CA, USA Yong Li, College of Electrical and Information Engineering, Hunan University, Changsha, Hunan, China Qilian Liang, Department of Electrical Engineering, University of Texas at Arlington, Arlington, TX, USA Ferran Martín, Departament d’Enginyeria Electrònica, Universitat Autònoma de Barcelona, Bellaterra, Barcelona, Spain Tan Cher Ming, College of Engineering, Nanyang Technological University, Singapore, Singapore Wolfgang Minker, Institute of Information Technology, University of Ulm, Ulm, Germany Pradeep Misra, Department of Electrical Engineering, Wright State University, Dayton, OH, USA Subhas Mukhopadhyay, School of Engineering, Macquarie University, NSW, Australia Cun-Zheng Ning, Department of Electrical Engineering, Arizona State University, Tempe, AZ, USA Toyoaki Nishida, Department of Intelligence Science and Technology, Kyoto University, Kyoto, Japan Luca Oneto, Department of Informatics, Bioengineering, Robotics and Systems Engineering, University of Genova, Genova, Genova, Italy Bijaya Ketan Panigrahi, Department of Electrical Engineering, Indian Institute of Technology Delhi, New Delhi, Delhi, India Federica Pascucci, Department di Ingegneria, Università degli Studi Roma Tre, Roma, Italy Yong Qin, State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University, Beijing, China Gan Woon Seng, School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore, Singapore Joachim Speidel, Institute of Telecommunications, University of Stuttgart, Stuttgart, Germany Germano Veiga, FEUP Campus, INESC Porto, Porto, Portugal Haitao Wu, Academy of Opto-electronics, Chinese Academy of Sciences, Haidian District Beijing, China Walter Zamboni, Department of Computer Engineering, Electrical Engineering and Applied Mathematics, DIEM—Università degli studi di Salerno, Fisciano, Salerno, Italy Junjie James Zhang, Charlotte, NC, USA Kay Chen Tan, Department of Computing, Hong Kong Polytechnic University, Kowloon Tong, Hong Kong

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Shuiwen Shen · Qiong-zhong Chen

Practical Control of Electric Machines for EV/HEVs Design, Analysis, and Implementation

Shuiwen Shen Hozon New Energy Automobile Company Limited Shanghai, China

Qiong-zhong Chen GKN Aerospace Services Limited Global Technology Centres Bristol, UK

ISSN 1876-1100 ISSN 1876-1119 (electronic) Lecture Notes in Electrical Engineering ISBN 978-3-031-38160-7 ISBN 978-3-031-38161-4 (eBook) https://doi.org/10.1007/978-3-031-38161-4 © Springer Nature Switzerland AG 2024 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

The main motivation of this book is to give some practicality for electric machine control for electric vehicles (EVs) and hybrid electric vehicles (HEVs). The ideas which form the core part of the book are originated from the development of the application software for the mass-production motor control unit (MCU). There are some key features for an electric machine used as a device to drive the vehicles. Firstly, in automotive applications, the working condition changes rapidly and massively along with the vehicle driving conditions, which could be accelerating at one instant and braking at the next instant. This will inevitably require a good dynamics of the electric motor torque. Secondly, optimization of the vehicle powertrain system for a better efficiency without compromising the performance at all driving conditions is one of the key control targets. With the predefined vehicle configuration, the powertrain shall be able to meet the requirements from different driving modes. The optimization objective will need to cover multiple aspects, including efficiency, dynamics, and also the powertrain capability such as power boosting at high speeds. Thirdly, the design of the electric machine control must have adaptation ability for varying parameters. Electric machine control is decisively sensitive to the machine and environment parameters. The runtime temperature change, the machine aging, or the slight manufacturing difference from even the same production line will lead to variation of the machine parameters. The control strategies shall be capable of detecting and adapting to the variation. Furthermore, although not covered within the scope of this work, it is noteworthy that meeting the functional safety ISO 26262 is the general requirement for the control software design of an EV/HEV motor drive. This implies not only a stringent development process but also a monitoring and fault-handling concept for fault mitigation as per the safety goals. As such, the vehicle shall be kept in or returned to a safe state in case of fault occurrence.

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Having the above-mentioned specifications in mind, this book tries to address the specialties in electric machine control for vehicle applications. In practice, both the Alternating Current Induction Machine (ACIM) and Permanent Magnetic Synchronous Machine (PMSM) are applicable to EV/HEVs. Even if the latter one has a wider application, the ACIM is selected as the benchmark for electric machine control design simply because from the control perspective, an ACIM has more degrees of freedom given both the stator and rotor variables. Its vector control and optimization strategies can be ported across to other AC motor types without losing much generality. In this book, the technologies addressing the above specifications are demonstrated by means of design, analysis, and examples with Matlab scripts and practical control software architectures, which are built up step by step throughout each chapter. Please bear in mind that the example given in this book is a machine not used by any OEM for the IP protection reason. Especially the control strategies presented throughout the book are also developed for illustration of the practicality. Following a review of the currently available or potential motor types in the vehicle markets, this book tries to build up the in-depth control technologies by starting with the basic theory of the ACIM. It develops gradually from the steady state to the dynamic models. The motor characteristics together with the performance analysis are then given in the wake of either the equivalent circuit models or the steady-state complex space model. Based on the latter one, the optimization methods on efficiency or capability are presented. The dynamic analysis of the motor model, using the example motor parameters, clarifies the fast and slow modes of the ACIM. This benefits for determining the fast torque delivery path. Chapters 2–4 are particularly suitable for university undergraduate or graduate study, or junior engineers. The provided MATLAB scripts are intended to facilitate the understanding of the motor concept and the dynamics. The following Chaps. from 5 to 8 focus on the machine control and the parameter adaptation. The scalar control is not of particular interest in vehicle applications but could be interesting for fast prototyping thanks to a simple control structure. A strong emphasis is on the vector control, which is incorporated with various optimization methods. This part is elaborated using a practical and rather detailed control software architecture example, which will therefore be helpful with the learning process of practical design. Torque delivery is desired to be fast, efficient, and capable of pushing the motor drive to the limit of power delivery—subject to the thermal and voltage limits. Given the various possibilities of control over the field-weakening region(s), different strategies are extensively discussed as a separate chapter. Plus, the parameter adaptation and sensorless control are presented to complement parameter inaccuracy or runtime variation. These chapters are considered for an enhanced understanding of AC electric machine control.

Preface

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Control signals shall be fed to the voltage source inverter (VSI) by means of pulse width modulation (PWM). The last chapter of this book presents the commonly used continuous PWM schemes. The efficiency of voltage usage, nonlinear region behavior, and harmonics of these schemes are compared. This chapter aims to complete the process of AC electric machine control. Cambridge, UK Bristol, UK

Shuiwen Shen Qiong-zhong Chen

Acknowledgements

Writing of this book can be traced back to the year of 2017. In the past nearly 6 years, this work was vastly disrupted by the pandemic. Nevertheless, there were two extremely intensive periods in preparing the manuscript. The authors feel very grateful for all the kind support they have received to make this book possible. In particular, the authors are thankful to Springer for having been patient. Without the patience and encouragement, this book would not have been completed. Shuiwen Shen would like to thank his family for the relentless support. Qiong-zhong Chen wants to give special thanks to his family for tolerating his unavailability.

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Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Background on EV/HEV Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 EV/HEV Electric Machines and Comparison . . . . . . . . . . . . . . . . . . . 1.2.1 Induction Motor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Interior Permanent Magnet Motor . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Surface-Mounted Permanent Magnet Motor . . . . . . . . . . . . . . 1.2.4 Switched Reluctance Motor . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.5 Wound Field Synchronous Motor . . . . . . . . . . . . . . . . . . . . . . . 1.2.6 Synchronous Reluctance Motor . . . . . . . . . . . . . . . . . . . . . . . . 1.2.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Selection Criteria of EV/HEV Traction Motors . . . . . . . . . . . . . . . . . 1.4 Requirements from Motor Control Perspective . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 4 6 8 10 12 14 15 17 20 21

2 Equivalent Circuit Modelling and Analysis . . . . . . . . . . . . . . . . . . . . . . . 2.1 Induction Machine Working Principle . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Rotating Stator Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Electromagnetic Force and Torque . . . . . . . . . . . . . . . . . . . . . . 2.2 Equivalent Circuit Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 System Performance Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Slip to Torque Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Thevenin’s Theorem Application to Torque Profile Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Speed Torque Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.4 Electric Machine Operation Analysis . . . . . . . . . . . . . . . . . . . 2.3.5 Phasor Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.6 The System Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25 25 25 31 34 40 41 51 56 60 63 67 71

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3 Rotating Frame Modelling and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Vector Representation of Rotating Fields . . . . . . . . . . . . . . . . . . . . . . . 3.2 Complex Space Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Three Phase Frame Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Fixed Frame Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Rotating Frame Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Steady-State Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Maximum Torque per Ampere and Maximum Torque per Loss . . . . 3.8 Maximum Torque per Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9 Speed-Torque Characteristics and System Operations . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73 73 79 88 90 96 99 107 112 119 128

4 Induction Machine Dynamics Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Rotor Field Oriented Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 System Dynamics Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Equivalent Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

129 129 134 145 153

5 Scalar Torque Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Equivalent Circuits for Scalar Control . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Improved -model Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Electromagnetic Torque Analysis . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Analytical Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

155 155 157 158 163 169 170 173

6 Vector Torque Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Principle of Vector Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Vector Control Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Vector Control Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Fast Torque Delivery Control . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Current Feedback Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.3 Decoupling Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.4 Current Feed-Forward Control . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Vector Control Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Steady-State Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

175 175 179 184 185 188 189 189 190 192 201

7 Flux-Weakening Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Flux-Weakening Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 I-Limit Dominated Flux-Weakening Control . . . . . . . . . . . . . . . . . . . . 7.3 Q-Limit Dominated Flux-Weakening Control . . . . . . . . . . . . . . . . . . . 7.4 Q/I-Limit Dominated Flux-Weakening Control . . . . . . . . . . . . . . . . . 7.5 Explicit Flux Limitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

203 203 206 214 223 234 249

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8 Rotor Field Oriented Control and Senseless Control . . . . . . . . . . . . . . . 8.1 Issue with Indirect Flux Oriented Control . . . . . . . . . . . . . . . . . . . . . . 8.2 Direct Versus Indirect Flux Oriented Control . . . . . . . . . . . . . . . . . . . 8.3 Full State Versus Reduced State Observer . . . . . . . . . . . . . . . . . . . . . . 8.3.1 Full State Observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.2 Reduced State Observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Parameter Identification by Adaptive Law . . . . . . . . . . . . . . . . . . . . . . 8.4.1 Rotor and Stator Resistance Estimation . . . . . . . . . . . . . . . . . . 8.4.2 Sensorless Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Model Reference-Based Sensorless Control . . . . . . . . . . . . . . . . . . . . 8.5.1 Rotor Flux-Based Model Reference . . . . . . . . . . . . . . . . . . . . . 8.5.2 Induced Voltage-Based Model Reference . . . . . . . . . . . . . . . . 8.5.3 Phase Alignment Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

251 251 258 259 260 264 268 270 275 278 278 282 284 287

9 Inverter PWM Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Sinusoidal PWM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Third Harmonic Injection PWM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Space Vector PWM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.1 Implementation of SVPWM . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.2 Linear and Nonlinear Boundaries . . . . . . . . . . . . . . . . . . . . . . . 9.5 Comparative Study of Harmonics and Nonlinearity . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

289 289 292 298 301 303 310 313 318

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319

Acronyms

AC ACIM CPSR DC DFOC DOF DTC EMF EV FMEA FOC FOM FTA HEV HSM IFOC INV IPM MCU MI MMF MTPA MTPF MTPL NVH NZE OEM

Alternating Current Alternating Current Induction Machine Constant Power Speed Range Direct Current Direct Field-Oriented Control Degree of Freedom Direct Torque Control Electromotive Force Electric Vehicle Failure Mode and Effects Analysis Field-Oriented Control Field-Oriented Model Fault Tree Analysis Hybrid and Electric Vehicle Hybrid Synchronous Machine Indirect Field-Oriented Control Inverter Interior Permanent Magnet Motor Control Unit Modulation Index Magneto-Motive Force Maximum Torque per Ampere Maximum Torque per Flux Maximum Torque per Loss Noise, Vibration, and Harshness Net Zero Emissions Original Equipment Manufacturer

xv

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Acronyms

PHEV PI PID PM PMSM PWM SPM SPWM SQIM SRM SVPWM SynRM THIPWM WFSM WLTP WRSM

Plugged-in Hybrid Electric Vehicle Proportional and Integral control Proportional, Integral, and Derivative control Permanent Magnet Permanent Magnet Synchronous Machine Pulse Width Modulation Surface-Mounted Permanent Magnet Sinusoidal Pulse Width Modulation Squirrel Cage Induction Motor Switched Reluctance Motor Space Vector Pulse Width Modulation Synchronous Reluctance Motor Third Harmonic Injection Pulse Width Modulation Wound Field Synchronous Motor World Harmonized Light Vehicle Test Procedure Wound Rotor Synchronous Motor

Abbreviations abc cnst cont dec dq err ie img inc pu rel rms rpm st wrt v/f αβ

Three-phase fixed frame in stator Constant Continuous Decrease Direct and quadrature rotating frame synchronized with rotor flux Error Id est Imaginary part of a complex variable Increase Per unit Real part of a complex variable Root mean square Revolutions per minute Subject to With respect to Voltage-frequency control or Volts Hertz control Two-phase fixed frame in stator

Acronyms

Symbols E Es Er f f cw fr fs fsrt i ia ib ic id iq iα iβ I IDC IAC Ia Ib Ic Id Il Ip Ipeak Iq Imax aC Imax iC Imax fC Imax pC Imax qC Imax IM Ir Ir∗ Is Isrt Isd Isq Isrt Ismax

EMF (V) Back EMF at the stator side (V) Back EMF at the rotor side (V) Frequency in general (Hz) Carrier wave frequency (Hz) Rotor frequency (Hz) Stator frequency (Hz) Rated stator frequency (Hz) Instantaneous current (–) Instantaneous Phase A current (A) Instantaneous Phase B current (A) Instantaneous Phase C current (A) Instantaneous direct-axis current (A) Instantaneous quadrature-axis current (A) Instantaneous α Phase current (A) Instantaneous β Phase current (A) Current in general (A) DC current (A) AC current (A) Phase A current (A) Phase B current (A) Phase C current (A) Field or direct-axis current (A) Line current (A) Phase current (A) Absolute maximum current (A) Torque or quadrature-axis current (A) Maximum current, or current limit(A) Imax when I-limit dominates for ψ  I-limits (A) Imax when I-limit dominates for Q I-limits (A)  Imax that ψ I-limit peak torque equals Q-limit’s (A)  Imax when ψ I-limits joint at MTPF (A)  Imax when Q-limit dominates for Q I-limits (A) Magnetizing current (A) Rotor current (A) Equivalent rotor current (A) Stator current (A) Rated stator current (A) Stator field or direct-axis current (A) Stator torque or quadrature-axis current (A) Rated stator current limit (A) Maximum stator current (A)

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xviii

Islim dmd Isd dmd Isq j kT Lδ Lˆ δ LM Lm Lˆ m Lr Lrm Lrl L∗rl Lrr LRR Ls Lsm Lsl Lss LSS Np Ns Nr P P Pag Pc Pcls Pclr Pe Pm Pmrt Pml re rv Rc Req Rˆ eq Rs Rˆ s Rr R∗r Rˆ r

Acronyms

Stator current limit (A) Stator field current demand (A) Stator torque current demand (A) Imaginary unit (–) Torque gain (–) Total leakage inductance at stator side (H) Total leakage inductance at stator side—Estimated (H) Magnetizing inductance (H) Mutual inductance (H) Mutual inductance-Estimated (H) Rotor inductance (H) Rotor mutual inductance (H) Rotor leakage inductance (H) Equivalent rotor leakage inductance (H) Rotor self-inductance (H) Rotor self-inductance in two-phase model, LRR = 1.5Lrr (H) Stator inductance (H) Stator mutual inductance (H) Stator leakage inductance (H) Stator self-inductance (H) Stator self-inductance in two-phase model, LSS = 1.5Lss (H) Number of pole-pairs (–) Turns of stator winding (–) Turns of equivalent rotor winding (–) Number of poles (–) Power in general (W) Air gap power (W) Core loss power (W) Stator copper loss power (W) Rotor copper loss power (W) Electric input power (W) Mechanical output power (W) Rated mechanical output power (W) Mechanical loss power (W) Effective turn ratio between rotor and stator (–) Voltage factor in Thevenion transform (–) Core loss equivalent resistance (Ω) Total equivalent resistance at stator side (Ω) Total equivalent resistance at stator side—Estimated (Ω) Stator winding resistance (Ω) Stator winding resistance—Estimated (Ω) Rotor winding resistance (Ω) Equivalent rotor winding resistance (Ω) Rotor winding resistance—Estimated (Ω)

Acronyms

s s Te Tecn pk Te Tem cn Tem cap Tem pk Tem rt Tem st Tem dmd Tem max Tem va vb vc vd vq Vs vsd vsq vα vβ V VDC VAC Va Vb Vc Vd Vl Vmax Vp Vq Vs Vsrms Vsd Vsq Vsrt Xm XM Xr Xrl Xrl∗

Slip of induction machines (–) The complex factor of s = jω for Laplace transform (–) Electric machine torque (Nm) Continuous electric machine torque (Nm) Peak electric machine torque (Nm) Electric machine torque, same as Te (Nm) Continuous electric machine torque (Nm) Torque capability at a given speed (Nm) Peak electric machine torque (Nm) Rated machine torque (Nm) Electric machine starting torque (Nm) Electric machine torque demand (Nm) pk Maximum electric machine torque, same as Te (Nm) Instantaneous Phase A voltage (V) Instantaneous Phase B voltage (V) Instantaneous Phase C voltage (V) Instantaneous direct-axis voltage (V) Instantaneous quadrature-axis voltage (V) Instantaneous phase voltage (V) Instantaneous stator direct-axis voltage (V) Instantaneous stator quadrature-axis voltage (V) Instantaneous α phase voltage (V) Instantaneous β phase voltage (V) Voltage in general (V) DC voltage (V) AC voltage (V) Phase A voltage (V) Phase B voltage (V) Phase C voltage (V) Direct-axis voltage (V) Line voltage (V) Maximum voltage, or voltage limit (V) Phase to Neutral voltage (V) Quadrature-axis voltage (V) Phase voltage (V) Phase voltage in rms (V) Stator direct-axis voltage (V) Stator quadrature-axis voltage (V) Rated phase voltage (V) Reactance of mutual inductance (Ω) Reactance of magnetizing inductance (Ω) Reactance of rotor inductance (Ω) Reactance of rotor leakage inductance (Ω) Reactance of equivalent rotor leakage inductance (Ω)

xix

xx

Xrm Xsm Xs Xsl Z Zr Zs κr κˆ r ω ωc ωb ωe ωf ωmax ωr ωrm ωre ωs ωsm ωsp ωv μ φ  m max rms max M r r∗ ˆr  rd rq s sd sq τM τr τˆr τT τˆT η

Acronyms

Reactance of rotor mutual inductance (Ω) Reactance of stator mutual inductance (Ω) Reactance of stator inductance (Ω) Reactance of stator leakage inductance (Ω) Impedance in general, or total impedance of the machine (Ω) Rotor impedance (Ω) Stator impedance (Ω) Rotor gain (–) Rotor gain-Estimated (–) Angular speed or frequency in general (rad/s) Critical speed of the flux-weakening strategy of this book (rad/s) Base speed above which flux-weakening starts (rad/s) Phase voltage or current electric frequency (rad/s) Full flux-weakening speed (rad/s) Maximum machine speed, or speed limit (rad/s) General rotor speed, can be ωrm or ωre (rad/s) Rotor mechanical speed (rad/s) Rotor electrical speed (rad/s) Synchronous speed in electrical scale. It is the same as ωe (rad/s) Synchronous speed in mechanical scale (rad/s) Slip speed of induction machines (rad/s) Speed where V-limit becomes effective (rad/s) Magnetic permeability (H/m) Flux in general (Wb) Flux linkage in general (Wb) Mutual flux linkage (Wb) Maximum flux linkage, or flux limit (Wb) Maximum flux linkage, or flux limit, in rms (Wb) Magnetizing flux linkage (Wb) Rotor flux linkage (Wb) Rotor flux linkage mapped to stator side (Wb) Rotor flux linkage—observed (Wb) Rotor flux linkage in field direction (Wb) Rotor flux linkage in quadrature-axis direction (Wb) Stator flux linkage (Wb) Stator flux linkage in field direction (Wb) Stator flux linkage in quadrature-axis direction (Wb) Magnetizing time constant (s) Rotor time constant (s) Rotor time constant—Estimated (s) Torque time constant (s) Torque time constant—Estimated (s) Duty cycle (–)

Acronyms

xxi

Typical Parameter Values See Table 1.

Table 1 Typical parameters of electric machine Symbol

Typical Value

Description

Ipeak

600

Absolute maximum current (A)

LM

3.0 × 10−3

Magnetizing inductance (H)

Lm

3.7 × 10−3

Mutual inductance (H)

Lr

3.2 × 10−3

Rotor inductance (H)

Lrl

0.18 × 10−3

Rotor leakage inductance (H)

L∗rl

0.27 × 10−3

Equivalent rotor leakage inductance (H)

Ls

4.8 × 10−3

Stator inductance (H)

Lsl

0.25 × 10−3

Stator leakage inductance (H)

Np

2

Pole pairs (–)

Qmax

0.30

Equivalent permissible flux linkage at stator (Wb)

re

0.82

Effective turn ratio (–)

Rr

38 × 10−3

Rotor winding resistance (Ω)

R∗r

57 × 10−3

Equivalent rotor winding resistance (Ω)

Rs

24 × 10−3

Stator winding resistance (Ω)

VDC

330

DC voltage (V)

Vs

190

Phase voltage, Y connection (V)

Vsrms

134

Phase voltage in rms, Y connection (V)

max

0.28

Maximum permissible flux linkage (Wb)

rms max

0.2

Maximum permissible flux linkage in rms (Wb)

Chapter 1

Introduction

1.1 Background on EV/HEV Transition Driven by the environmental concerns, it is now commonly acknowledged that conventional fossil fuel-powered vehicles will gradually phase out and be vastly replaced by electrified vehicles. In order to boost this transition, many countries have strengthened their policy support covering from the development of hybrid electric vehicles (HEVs) and pure electric vehicles (EVs) down to their deployment into the market. This has become an ongoing profound revolution in the car industry, not only because of the transition of technologies and the reshuffling of the downstream powertrain supply chains, but also due to many more niche competitors joining this race. It can be predicted that the future of the car industry will be very much reshaped in the coming dozens of years if not shorter. As a result, market shares of EVs and HEVs have increased drastically over the past years: the annual sales have mounted to 9% of the global car market in 2021, around four times as high as in 2019 [3]. The projected uptake of electric vehicles in the period to 2030 has been assessed by approaches based on scenarios, among which the Net Zero Emissions (NZE) by 2050 Scenario predicts an outlook of the share of the global EV sales and stock to reach around 60% and 20% in 2030 respectively. More precisely, this statistics covers the pure EVs and the plugged-in HEVs (PHEVs). As of now, the projection from the Stated Policies Scenario is lagging that from the NZE Scenario. However, several major automakers have announced plans accelerating their transition to a fully electric future. Benefiting from this, the number of available car models in the market has increased to over 450 as of 2021, more than twice the number in 2018 [3]. EV application is greatly affected by the availability of the charging infrastructure and the so-called range anxiety of the end users. Nevertheless, large battery capacity, high powertrain efficiency and light vehicle payload are three major contributors to improve the EV range on one full battery charge. By far, the predominant energy storage technology, in particular for the light-duty passenger vehicles, is the Lithium-ion © Springer Nature Switzerland AG 2024 S. Shen and Q.-z. Chen, Practical Control of Electric Machines for EV/HEVs, Lecture Notes in Electrical Engineering 1064, https://doi.org/10.1007/978-3-031-38161-4_1

1

2

1 Introduction

battery. There has been steady progress in regards to cost reduction and energy density increase [3, 20]. All these make the large-capacity battery packs more available and affordable, extending the mile range even further. Motor drives, as a key component to the powertrain, have been attracting lots of attention over decades, not only in automotive industry, but across many other sectors. In most automotive applications, motors are connected via a transmission unit before applying torque to the wheels, but they can drive the wheels directly in a few cases. The latter ones are then referred to as direct-drive in-wheel motors. Potential motor drives serving for traction or auxiliary purpose need to be optimized for case-specific applications. Without a single exception, they all have pros and cons, and there will not be a one-size-fits-all approach. Factors including cost, efficiency, power/torque to weight ratio, power/torque density over volume, reliability and fault tolerance, maintainability, NVH (noise, vibration and harshness) and so forth, all need to be carefully assessed and balanced. Drivability, electric range, cost and safety, beyond the EV support infrastructure, are probably the utmost important factors that impact the acceptance of the electric vehicles at the user end. These have to be reflected back on the requirements of the traction motors. EV market reveals the three commercially applied and currently available electric motor types in mass production: permanent magnet (PM) motors, AC induction motors (ACIMs or equally IMs) and wound rotor synchronous motors (WRSMs) [8, 11]. Besides, switched reluctance motors (SRMs) equipped EVs have been seen in Motor Shows or demonstration workshops, though few commercial deployment has been found. They are considered as a potentially viable solution. Currently PM motors, or to be more accurate, the interior permanent magnet (IPM) motors, reign the market share of the EV applications, followed by the ACIMs in the second position. However, the current status and trend on deployment of PM motors can be diversified depending on regional markets. On the one hand, as one of the major electric vehicle markets, China tends to adopt more rare-earth PM motors in electric automotive applications. This is rather sensible as China is rich in rare earth resources and predominates the global production. On the other hand, other countries are more motivated in searching for alternatives in lieu of PM motors. This can be seen via the fact that there are more permanent magnet-free car models being launched by European or US car makers than by their Chinese counterparts.

1.2 EV/HEV Electric Machines and Comparison The electric motor technology keeps evolving since the first DC motor was invented in 1800s. Not long after, human being started investigating the electric vehicle, or more accurately, the electric-powered carriage. It then staggered for nearly a century until Ford presented the first commercially successful electric car model, Ford Model T, in 1908. The first electric car was powered by a DC motor, then the motor technology underwent numerous milestones, such as the invention of AC induction motors, the reluctance motors, the AC variable-frequency drives and the PM motors and so on.

1.2 EV/HEV Electric Machines and Comparison

3

Electric Motor Types

AC Motors Synchronous Motors

Permanent Magnet Motors

Surface Permanent Magnet Synchronous Motors

Asynchronous Motors

Reluctance Motors

Variable Reluctance Synchronous Motors

Externally Excited Synchronous Motors

Inducon Motors

Wound Rotor Synchronous Motors

Interior Permanent Magnet Synchronous Motors

DC Motors Switched Reluctance Motors

Fig. 1.1 Classification of EV traction motors

Motor technology evolution always came with leap-forward progress on materials, methodology and the achievements on controllers and power electronics. This section reviews the motor types that are commercially available in the current EV markets or the potential alternative candidates. The authors’ purpose is to highlight the pros and cons of these motor types used as automotive traction motors based on experience and a literature review, rather than to elaborate the motor operating principle. Figure 1.1 shows where these motor types fit in the hierarchical classification of electric motors. SRMs are categorized within the group of DC motors, as they do carry unidirectional phase currents. However, keep in mind that their operation is polarity independent due to the nature of reluctance torque. They can thus fit in either group. Conventional brushed or brushless DC motors are out of this scope. They prevailed in the EV and HEV models in 1990s [8], but have phased out since more than a decade ago except for non-traction motor applications, e.g. wipers, window lifts and pumps. Except for induction motors (IMs), all other motor types that are captured in this chart are synchronous motors. Being synchronous or asynchronous refers to the rotor speed in comparison to the excitation frequency of stator phase windings. If the rotor speed in the electrical measure is the same as the stator supply frequency, the motor is then called a synchronous motor; otherwise, it is an asynchronous motor. These motors are further classified according to the torque production mechanism. An IPM motor takes advantage of both magnetic torque1 and reluctance torque, yielding a major magnetic torque assisted by a reluctance torque in general, and therefore it is

1

It is also referred to as alignment torque.

4

1 Introduction

usually grouped within the permanent magnet motors. Its difference to a (permanent) magnet-assisted synchronous reluctance motor, however, needs to be brought into attention. It is worth noting that the following comparison of the motor characteristics will be exclusively for inner-rotor radial-flux topologies. Other motor types, such as inwheel motors and axial-flux motors are not included within this scope of this work. Axial-flux PM motors are an emerging technology [11]. They seek good potential applications due to its pancake structure and a claimed high power density, and therefore can fit well in scenarios such as the integration with the engine and the drivetrain for HEVs. In-wheel motors, as the name indicates, are installed directly inside the driving wheels and have a more compact integration.

1.2.1 Induction Motor IMs have been dominant in industrial applications for decades. Under this category, there are wound rotor, doubly-fed, and squirrel cage induction motors (SQIMs). Wound rotor induction motors are less common and are only used in applications where access to the rotor for connection to an external resistor is beneficial in order to improve the start-up performance [51]. Doubly-fed induction machines have been a prevailing generator technology in wind energy sector thanks to the less demanding requirement of power converters [38]. However, these two motors are not suitable for automotive applications as they are not likely to meet the requirements on power density and efficiency [1]. As of today, SQIMs are the only IM motor type applied in EVs. The cross-sectional view of a SQIM can be see in Fig. 1.2. SQIMs are magnet-free. The rotor of a SQIM consists of stacks of laminated iron with embedded short-circuited conductor bars made in the shape of a cage. The stator is wound by distributed windings which are fed with polyphase AC currents to create a rotating electromagnetic field. An EMF and therefore current are induced in the rotor conductor due to its relative movement with respect to the rotating stator field.

Fig. 1.2 Cross-sectional view of squirrel cage induction motor (SQIM)

A

L

E

X Stator windings

Rotor conductor bars

Stator core

Rotor core

L O U

I

S

1.2 EV/HEV Electric Machines and Comparison

5

The stator field then exerts a Lorenz force on the rotor conductor bars and causes the rotor to spin. In order to generate torque, there shall always exist a slip between the rotor speed and the speed of the stator field, and therefore induction motors are known as asynchronous motors [4, 51]. The torque production of an induction motor will be detailed in Chaps. 2 and 3 from the perspective of both steady-state and dynamic models. SQIMs are relatively cheap, extremely reliable, highly fault-tolerant, suitable for operating in harsh environment and having a rugged structure that makes it easy for construction. In general, SQIMs are known to be less efficient and have lower power/torque density than the equivalent PM counterparts. Nevertheless, measures such as the adoption of low-loss ferromagnetic materials and reduced-resistance rotor cage can significantly improve their efficiency, leading to only marginal difference compared to the PM motors. By referencing a few comparative studies of the motor efficiency maps published in [15, 39, 56], the difference in the peak efficiency between the IM and the IPM motor is no greater than 1%, or those two may be even similar to each other. When assessed using a drive cycle such as the World Harmonized Light Vehicle Test Procedure (WLTP) cycle, a difference of 1.15% in average efficiency is reported in [14] as an example. In automotive applications, these improvements include a copper cage in replacement of the traditional aluminum cage in order to increase the conductivity, and also a more efficient rotor cooling method [40, 50]. Although adoption of a copper bar cage adds cost to both the material and manufacturing process, it is paid off via efficiency gain [5]. Being less efficient than PM motors is probably too general to be accurate. As a matter of fact, IMs and PM motors compete in different operating regions, making them an alternative solution as well as a compensation to each other. On the one hand, IMs have less favorable efficiency range in high load conditions. The rotor conductor of an induction motor carries current and poses a resistive loss. Due to a lower power factor, not only the power rating of the inverter is negatively impacted, but the reactive load again leads indirectly to an increase of stator copper loss. Consequently, conducting loss of SQIMs is higher than that of PM motors, where the rotor excitation is provided by permanent magnets. On the other hand, IMs have good flexibility in regulating the field strength, as opposed to the PM motors. With this, the iron loss and conductor loss can be leveraged for the benefit of low-torque or high-speed operation. Due to the absence of magnets, IMs can operate in freewheeling state without yielding drag torque. This is an advantageous feature for a multi-motor EV architecture. Indeed, the power boost mode is not always necessary, but having a motor standby can optimize the system efficiency in many operating scenarios. In short, though IMs do not compete with the PM motors regarding efficiency in general, their benefit however resides in the low-load or high-speed regions [56]. When evaluated via an automotive drive cycle, the induction motor approach may still be a favored approach as it could outperform for overall efficiency. When it comes to selection of a motor drive technology, cost and efficiency are two major drives. As aforementioned, the PM motors are predominant in the EV markets and this trend is ongoing. However, their deployment is highly prone to the price variation of PM materials, and meanwhile, as a cost-effective and mature

6

1 Introduction

technology, IMs are an excellent compensator to the PM motors. SQIMs have been used in various commercially successful car models. Particularly in the dual motor configuration in high performance EVs, where a combination of front and rear motors is used for optimal performance, IMs remain as a viable and popular selection. These applications include the Tesla Ludicrous motor for the Tesla Model S/X models, Audi e-Tron and Mercedes-Benz EQC to name a few [14, 50]. Compared to the PM motors, control of an induction motor offers more degrees of freedom (DOFs) given both the controllable stator and rotor variables. Its vector control and optimization analysis ports similarity to the control of other AC motor types, and thus most can be deployed across. Bear in mind that this book revolves around the ACIM control for benchmark EV applications.

1.2.2 Interior Permanent Magnet Motor IPM motors have magnets embedded inside the rotor core, which very often has a multi-layer V-shaped recess structure in order to optimize the torque and safety performance [14, 17, 36, 37]. As an indicator, Fig. 1.3 shows the cross-sectional view of a single-layer V-shaped IPM motor with four pole pairs. The V-shaped slots have far smaller permeability than the rotor iron, and result in variable reluctance and thus inductance distribution around the rotor seen circumferentially. The stator is analogous to that of an IM, wound by distributed windings supplied with polyphase AC currents. Torque can be generated by two components: • magnetic torque, which results from the interaction between the magnet field and the rotating electromagnetic field from the stator windings; • reluctance torque, attributing to the magnetic saliency structure that tends to drive the rotor to move in a way where reluctance is minimized. Saliency ratio is defined as the ratio of the q-axis inductance L q over the d-axis inductance L d from the vector perspective. The higher the saliency ratio, the more

Fig. 1.3 Cross-sectional view of interior permanent magnet (IPM) motor

Permanent magnets

1.2 EV/HEV Electric Machines and Comparison

7

reluctance torque it can offer. Because of this, IPM motors are sometimes referred to as IPM-SynRM motors, literally interior permanent magnet—synchronous reluctance motors, as is named for the Tesla Plaid motors. The electromagnetic torque of a three-phase IPM can be expressed as: Tem =

3 N p [ M Iq + (L d − L q )Id Iq ] 2

(1.1)

where N p denotes the number of pole pairs and  M is the flux linkage due to the permanent magnet. The quantities are expressed in the amplitude-invariant scale. IPM motors offer the flexibility to regulate the sharing of magnetic torque and reluctance torque, which, as a combination, can be optimized for loss minimization for instance. One commonly-used optimization method is the maximum torque per Ampere (MTPA) strategy where the orthogonal d, q-axis currents, Id and Iq respectively, follow a trajectory so that the requested torque is delivered with minimum current amplitude, and therefore it ensures a minimum DC copper loss on the stator windings. At high speeds, a field weakening Id will be mandatory in order to fit the operation within the DC-bus voltage constraint boundary. The magnetic saliency structure gives benefit that a component of d-axis stator current that weakens the magnetic field will however boost the reluctance torque to some extent. This feature makes IPM motor an intrinsic solution to avoid major stator iron loss in the high-speed regime, in comparison with the surface-mounted PM (SPM) motors. As a rare-earth PM motor, an IPM motor is more like a successor to the SPM motor in the automotive applications. It boasts of numerous advantages including high efficiency, high torque and power density with respect to both weight and volume, and wide constant power speed range (CPSR) etc. Compared to its SPM counterpart, an IPM motor has nearly no magnet loss [36, 46], partly because the magnet-burying structure diverts the stator field from running through the permanent magnets and partly due to the reluctance torque feature. Therefore, it tends to have better efficiency at medium to high speeds than SPM motors. The reluctance feature also extends the range of operating speed with constant power. As a benefit of having magnets buried inside the rotor, IPM motors usually have stronger mechanical structure for high speed applications than the SPM motors. Because of all these, IPM motors are more favoured in the EV applications than SPM motors. Major downsides of the IPMs are the cost, risk of demagnetization and the possible high voltage hazard from back EMF at high speeds, which are common for all PM motor types. High cost of IPM motors comes from not only the volatile and pricey rare-earth materials, but also from the more complicated manufacturing process—the fragile die-cast rotor lamination and the insertion of magnets. The material cost could be 20–30% higher than their IM counterparts and even much higher than the SRMs [54]. This surplus in cost however may be paid off over a product life cycle given the benefit of its high efficiency. Besides the price, environmental concerns and the geopolitical unbalance in the production of rare-earth materials are also major drives for research units or enterprises to search for potential alternative motor technologies.

8

1 Introduction

An undesirable field weakening current, resulting from either short circuit transients or aggressive field weakening control, may irreversibly demagnetize the magnets and change the motor finger prints as a consequence. Once it happens, the motor performance degrades and vehicle safety standard will be violated due to the resultant torque inaccuracy. Demagnetization usually occurs under scenarios of short circuiting or magnet overheating. The latter is rather critical for NdFeB magnets, which are considered as highest-performance magnets, but have a comparatively low operating temperature depending on the material grades. For automotive traction motors, this temperature is typically between 150 ◦ C and 220 ◦ C [43, 44]. It is worth noting that active short circuit, as a safety protection measure for the back EMF at high speeds, can also lead to partial demagnetization of the magnets. Back EMF, as a side effect from the magnets, is exerted on the stator windings while the rotor spins. As the magnet strength is uncontrollable, the back EMF, if not well controlled, may surpass the inverter voltage rating and lead to damages of power devices. Therefore, when there is critical fault occurrence at high speeds, the motor drive should not be shut down directly, but will be swopped to an active short circuit mode [53]. Otherwise, it ends up with an unconditional generating condition, which may blow either the battery or the power devices such as the power switches or the DC-link capacitor. Other drawbacks of IPM motor are minor but not negligible, in particular the drag torque and the persistent iron loss due to the uncontrollable magnet field. They are present as long as the rotor spins no matter whether the stator windings are energized or not. This impacts the selection of motor technologies in the vehicle architectural design when using multiple motor drives. Reluctance torque usually comes with torque ripple. Higher saliency ratio usually means more torque ripple. Trade-off between magnetic torque and reluctance torque shall be optimized from the motor design phase till the motor control strategies. With regard to the mechanical strength, although there is no peeling-off issue of the magnets like the surface mounting topology, the bridges of the V-shaped slots are the structural weaknesses. Bridge thickness needs to compromise between minimizing the magnet flux leakage on one hand and ensuring enough tolerance of structural stress on the other hand. A retaining rotor sleeve may be considered in some applications.

1.2.3 Surface-Mounted Permanent Magnet Motor A SPM motor has permanent magnets mounted on and evenly distributed around the rotor surface, as shown in Fig. 1.4 for an example SPM motor with two pole pairs. The stator can be wound by either distributed windings or concentrated windings. In case of concentrated winding, it is essentially identical to a brushless direct current (BLDC) motor despite their different control strategies, where a SPM stator is fed by polyphase sinudoidal AC currents in opposition to the staircase or rectangular DC currents for a BLDC motor. Alternating currents flowing into the stator windings

1.2 EV/HEV Electric Machines and Comparison

9

Fig. 1.4 Cross-sectional view of surface-mounted permanent magnet (SPM) motor

create a rotating electromagnetic field that interacts with the rotor magnetic field, and as such, it attracts or expels the rotor to run synchronously along with it. Ignoring the magnetic saturation effects, a SPM motor has nearly no reluctance saliency meaning an even distribution of inductance circumferentially, and therefore it delivers pure magnetic torque. The SPM torque expression can be simplified as: Tem =

3 N p  M Iq 2

(1.2)

Selection of magnet remanence depends on a bunch of factors such as peak torque, top speed and the DC supply voltage to the inverter. A strong magnet field makes it easy for delivering a high torque right from standstill, but contrarily, it limits the CPSR and will induce a higher back-EMF on the stator windings. As the rotor excitation field is provided by magnets, there exists no rotor conductor loss due to the lack of armature on the rotor. On the one hand, SPM motors exhibit numerous notable advantages such as good dynamic performance, outstanding efficiency, very low torque ripple, and high torque and power density with respect to both weight and volume. Compared to the IPM motors, SPM motors are easy to be constructed. Control of SPM motors are also relatively simple and flexible. Usually q-axis current is the one variable to be controlled, while d-axis current is only required upon the field-weakening regime. On the other hand, SPM motors have the common drawbacks of PM motors such as cost, demagnetization risk and the handling of back-EMF, as stated in Sect. 1.2.2. They also present some unique disadvantages. To start with, it has a relatively low mechanical strength that limits the motor speed. The surface-mounted magnets face risk of being peeled off due to shock or vibration, or by the centrifugal forces as speed goes high. This risk gets further aggravated with the increase of rotor temperature. High speed application cannot be obtained without magnet retention measures, which are usually done by adopting a metallic or carbon fiber rotor sleeve. However, metallic

10

1 Introduction

sleeve increases eddy current loss while a carbon fiber sleeve abates heat dissipation and thus increases demagnetization risk [17]. Secondly, the most efficient operating range of a SPM motor remains at low to medium speeds, while at high speeds, its efficiency tends to be inferior to its counterpart motor types such as IPMs, IMs and SRMs [27, 36, 37]. The permanent magnet material NdFeB has good electrical conductivity. As nearly all airgap flux passes through the magnets, it induces eddy current in the magnets and leads to magnet loss. Magnet loss is one of the main loss components in the SPM motors, different from their IPM counterparts, where magnet loss is rather minor. A strong magnet is not always desirable as its magnetic strength is uncontrollable and there lacks flexibility in regulating the airgap field strength. As the rotor speeds up, the stator iron loss becomes significant even at low loads. This brings down its efficiency in the high-speed low-torque operating regions, which occur frequently in a highway cruise scenario in automotive applications. The uncontrolled magnet strength leads to another side effect—drag torque. When the stator iron is magnetized and demagnetized, back and forth repeatedly by the rotating magnet field, the energy enclosed by the hysteresis loop due to the iron’s hysteresis effect opposes the rotor from rotating. This drag torque and therefore loss exists as long as the rotor spins, even if the stator is open-circuited. Therefore, there is no truly freewheeling mode in permanent magnet motors, but some energy is required to combat this loss in order to run at zero torque output. Due to this, vehicle equipped with SPM motors cannot usually be towed directly unless they can be decoupled by a clutch. Otherwise, the joule heating from the induced iron loss may impair the insulation coating of the winding coils without coolant being activated. SPM motors are still found in EV applications, but they are now quite limited compared to their IPM counterparts. They are however favored in electric aircraft applications, where the load profile is predefined and the propulsion system is required to work at missions like low speed taxing, take off or at constant loads [1, 10, 46].

1.2.4 Switched Reluctance Motor Switched reluctance motors have a double saliency structure on the stator and rotor core. The stator poles are wound with concentrated windings, while the rotor has neither magnets nor windings. Figure 1.5 shows the cross-sectional view of a three-phase SRM with a 12/8 topology. SRMs are most commonly driven by an asymmetric halfbridge converter. When a stator phase is energized, the rotor tends to move towards a pole alignment position to minimize the reluctance along the magnetic path. Stator phases are sequentially energized in accordance with the rotor positions, and the phase current gives a switched, unidirectional pulse waveform in comparison with the sinusoidal waveform as in other AC motors. As such, the motor delivers a summation of reluctance torque from each phase nearly independently when ignoring the minor mutual coupling effects [24, 28]. SRMs has a highly nonlinear characteristic

1.2 EV/HEV Electric Machines and Comparison

11

Fig. 1.5 Cross-sectional view of switched reluctance motor (SRM)

Concentrated stator windings

of torque depending on phase current and position. The torque from each phase can be expressed as:  Tem =

∂

(i, θ )di ∂θ

When no saturation occurs, this torque expression can be simplified as Tem =

1 2 ∂ L(θ ) i 2 ∂θ

(1.3)

where i is the phase current, θ the rotor position, and L(θ ) the phase inductance depending on the rotor position. The reluctance torque feature then becomes more straightforward. As a magnet-free solution, SRMs have been attracting lots of attention for automotive applications. Even though they are rarely seen in commercially mass production, SRM-based EV/HEVs do appear quite often in demonstrations or car shows, such as the Land Rover’s 110 Defender model [16]. SRMs have many advantages with the most outstanding ones including low cost, intrinsic high fault tolerance, and suitability for hostile environment. Having no magnets and with a rugged structure, SRMs are the cheapest of the aforementioned motor types considering a combined cost of material and manufacturing [54]. Phases are independent in SRMs, meaning that short or open-circuit fault in one phase will not affect other phases and the motor as a whole can still deliver a pro-rata amount of torque, even though with aggravated ripples. Cooling in a SRM is easy and the least demanding as dominant losses are generated on the stator side, and the rotor has neither windings nor magnets that could lead to hazards from a temperature increase. Unlike the IMs and PM motors, whose motor characteristics are highly correlated to the rotor cage temperature or magnet temperature, a SRM does not have temperature effect on its torque behaviour. This makes its control less error prone. As another main driver that makes SRMs an attractive candidate for replacing the PM motors,

12

1 Introduction

efficiency of SRMs is relatively good, especially in the medium and high speed ranges where they can be very competitive compared to either IM or PM motors [23, 49]. Besides, SRMs have relatively good power density among all motor types [11]. Disadvantages of SRMs are equally distinctive, among which the NVH behavior is the major one that affects their drivability. Due to the double saliency structure, SRMs have a highly nonlinear electromagnetic characteristic, which intrinsically leads to massive torque ripple and significant noise if not well controlled. In general, the larger the saliency ratio, the higher the power and torque density. The downside is however the aggravated NVH behavior. Torque ripple can be effectively reduced via an optimal motor design together with torque ripple control strategies such as direct instantaneous torque control or torque sharing functions [30, 57]. Bear in mind that torque ripple reduction method usually comes with compromise in efficiency [42]. Even so, noise remains a bigger issue for their commercial application. It is acknowledged that acoustic noise of SRMs is mainly induced by the deformation of stator structure caused by the pulsating radial force [9]. Efforts on noise suppression include motor geometric optimization, design of a stiff structure in order to shift its vibration modes, and acoustic insulation in the motor packaging [13]. Control methods on radial force shaping also aim to suppress the low-order vibration modes to reduce the highly-sensitive audible noise, but a wider range noise reduction still seems challenging [6, 19]. In general, despite the maturity level of the technology, it remains as a trade-off between NVH and efficiency. That being said, the control of SRMs is much more complicated than that of the AC machines. As of now, the drivability and technical maturity level are two major reasons that hinder a wider application of SRMs in EVs. Nevertheless, lots of research interest in seeing them as a promising magnet-free alternative solution can still be anticipated [35].

1.2.5 Wound Field Synchronous Motor A wound field synchronous motor (WFSM) is also referred to as wound rotor synchronous motor (WRSM), whose cross-sectional view can be seen in Fig. 1.6 as an example. It has a stator analogous to that of induction motors with the phase windings fed by AC voltages. The rotor however has excitation windings in order to yield a controllable electromagnet field. Current is supplied to the rotating rotor windings via the contact of a pair of brushes and slip rings. Torque is then yielded as a result of the interaction between the stator and rotor fields. The electromagnetic torque can be expressed as: 3 (1.4) Tem = N p (sd Isq − sq Isd ) 2 where both the stator d, q-axis flux linkage sd and sq are a function depending on not only the stator d, q-axis current Isd and Isq , but also the rotor field current I f . The field current is controlled separately by a field exciter [32]. Therefore it offers three current-control variables—one more DOF compared to the PM motors. This

1.2 EV/HEV Electric Machines and Comparison

13

Fig. 1.6 Cross-sectional view of wound field synchronous motor (WFSM)

Rotor windings

gives possibility to optimize the field excitation in the field weakening region and can therefore lead to reduction on the iron loss particularly and even the stator copper loss. The stator can operate with a high power factor, as there is no need for the stator to provide field weakening current. The consequent reduction in current magnitude and apparent power can potentially downsize the power rating of the inverter. Downsides of the external rotor excitation are the rotor winding resistance loss and the added complexity in control which needs to take into account the disturbance from the strong mutual coupling between the rotor and stator flux [21, 31]. That said, WFSMs can be more efficient than PM motors in the regions of low loads or high speeds, but will be less otherwise [14, 39]. Major shortcoming of WFSMs is the usage of brushes and slip rings. They not only decrease the power density of the motor drive, but more importantly, lead to concerns on reliability, early wear and necessity of more frequent maintenance of the parts, and even the potential risk due to the conductive dust from the process. All these tend to make WFSMs less viable for a harsh automotive application environment. Brushless excitation strategies, such as capacitive power coupler (CPC), inductive power transfer systems or rotating transformers, have been studied in order to replace the rings and brushes [26, 32]. However they come with a price of oversizing and cost, and the technical maturity level for commercialization is obscure for the time being. Other than that, as the WFSM is a magnet-free motor, there is no concern on either demagnetization or the handling of back EMF, which is an apparent advantage over its PM counterparts. WFSM can also survive a higher operating temperature due to the lack of PM materials. As automotive traction motors, WFSMs can be found in commercialized vehicles such as the Renault Zoe, BMW iX3 and the newly launched BMW iX M6 [25]. Nevertheless, there has not been a wide application mainly due to the concern on the durability, reliability and maintenance of the brushes and slip rings.

14

1 Introduction

1.2.6 Synchronous Reluctance Motor A synchronous reluctance motor (SynRM) has a similar structure to an IPM motor, except that there is no PMs embedded in the rotor recesses and there are more layers of flux barriers in order to increase the saliency ratio. Figure 1.7 shows the cross section of a SynRM with 2 pole pairs and 36 stator slots. The flux barriers guide the flux through the desired iron path in d-axis, while block it in the other axis otherwise. When the stator windings are fed by polyphase AC currents, the rotor is driven solely by the reluctance torque and runs in synchronism with the stator field. Torque yielding in a three-phase SynRM can be written as: Tem =

3 N p (L d − L q )Id Iq 2

(1.5)

It is commonly acknowledged that SynRMs need to be designed with a high saliency ratio of L d /L q in order to increase peak torque and the power factor [18]. The saliency ratio can be increased by designing appropriate geometry, number and placement of the barriers [7, 44, 47, 48, 52]. However, it is limited by the mechanical constraints due to the aggravated structural weakness. With more barriers, finer ribs and narrower bridges, the structure becomes too delicate to withstand the mechanical stress in high-speed applications. On the electromagnetic aspect, this geometry can easily lead to magnetic saturation along its desired d-axis path. The resultant magnetic saturation also limits the torque overload capability. The power factor of a SynRM is directly linked its saliency ratio [47]. Because of the constraint on the saliency ratio, the power factor of SynRMs is notoriously small—significantly smaller than their IM counterparts as well [34, 48, 52]. Consequence is that a larger power-rating inverter is usually required. Belonging to the family of reluctance motors, SynRMs inherit the same common drawbacks of high

Fig. 1.7 Cross-sectional view of synchronous reluctance motor (SynRM)

1.2 EV/HEV Electric Machines and Comparison

15

torque ripple and noise [12, 59]. However, this is less of a concern, as appropriate motor design including flux barriers and stator slots may mitigate these effects [2, 33]. SynRMs have received fairly amount of attention as a PM-free candidate for EV applications, but there seems no applications in EVs as of today. Their saliency ratio tends to be much smaller than their SRM counterparts, which limits their torque performance. This may explain why there have been more efforts on the study of SRMs than that of SynRMs for EV applications. Due to the limited saliency ratio, they tend to have a low power density and will be comparatively more suitable for low-power industrial applications rather than EV traction [34]. There has been efforts made in order to improve the power factor and efficiency. This ends up with a variant of SynRMs—PM-assisted SynRMs (PMaSynRMs), where ferrite PMs or rare-earth PMs are inserted in the slots pointing to the negative q-axis to produce an assistive component of magnet torque. It is claimed that both the power factor, efficiency and power density can be significantly improved, which is sensible as it basically turns into a variant of an IPM as well, even though the used amount of PMs may be reduced. A bunch of study and EV-application attempts have been carried out, see in [2, 22, 35, 41, 59].

1.2.7 Summary It is worth re-clarifying that only the motor types that are commercially available or the widely recognized potential candidates for HEV/EV applications are reviewed in this section. This review covers merely the radial-flux, inner-rotor topologies in order to have a relatively fair comparison. The axial-flux motors and outer-rotor inwheel motors as emerging technologies are predicted to make a boost in the coming decade due to the compact pancake structure or the easy integration into a vehicle powertrain. However, they are not included within this scope. This study is focused on the automotive traction applications, so the relevant features are particularly highlighted. Some motors may have appealing features in certain applications, but if those features are not equally beneficial for an EV, they may still be slightly penalized. For instance, SPM motors have outstanding peak efficiency, but these high-efficiency regions are less favorable for an EV than that of IPM motors. Taking this into consideration, the scoring of SPM motors is then slightly penalized, even though the efficiency range of a SPM motor may be endorsed in applications such as electric aircrafts.

16

1 Introduction

IPM and SPM motors have been long well known for high efficiency and torque/power density. The high cost of PM motors are expected to be paid back over a product life cycle. The volatility in price and supply however sparks lots of interest in searching for rare-earth PM-free alternative solutions. Activities in this regard have been particularly advocated in western countries. This tendency will continue in the coming future. Moreover, for the dual-motor EV configuration, where increased vehicle performance is targeted, it is preferential to have at least one PMfree motor equipped so that the motor can be fully disengaged in a freewheeling mode without having to employ a clutch. This however cannot be achieved by a PM motor, as the drag torque and back EMF will be constantly present. IMs are dominant in the fixed/adjustable-speed industrial applications. They are probably the most mature motor technology that has been used in the EVs in order to maintain a good compromise. Automotive IMs are designed with a copper cage and appropriate rotor cooling so that the efficiency is significantly improved. These improvements make IMs competitive with the PM motors in certain frequent operating regimes of a passenger vehicle. The option of IMs well balances among materials (PM-free), efficiency, cost and immunity of potential hazards. WFSMs have the benefits of being able to regulate the rotor field which makes them highly efficient and capable of operating with a high power factor for downsizing the inverter. However, it is not insensible to say that WFSMs will still have limited EV applications in the foreseeable future until the brushes and slip ring can be replaced by a viable and mature brushless solution. SRMs have sought lots of progress on torque ripple control, but the acoustic noise still remains a major concern. They have a wide band of competitive efficiency. However, the NVH-suppression control methods will almost certainly impose penalty on the efficiency. These concerns need to be tackled before a brighter future of SRMs in the commercial passenger EVs. SynRMs have limited saliency ratio resulting in deficient torque overload capability, a low power factor and low power density. There seems only dim light for the EV applications as is. The variants of SynRMs with the PM assistance however may find some controversial breakthrough, as on one hand, they significantly offset the downsides of the SynRMs and on the other hand, they still require a certain amount of PM materials. Based on the findings from a list of publications, the authors extract and compile the evaluation for each of the aforementioned motor types, see in Fig. 1.8 [8, 14, 27, 29, 37, 39, 44, 45, 49, 50, 56, 58, 59].

1.3 Selection Criteria of EV/HEV Traction Motors

17

Cost compeveness 5

Technical maturity

Efficiency

4.5 4 3.5 3

Cooling complexity

Power & torque density

2.5 2 1.5 1 0.5

Environment tolerance

NVH

0

Fault tolerance

Controllability

CPSR

Reliability Power factor

IPM

SPM

IM

WFSM

SRM

SynRM

Fig. 1.8 Comparison of motor types

1.3 Selection Criteria of EV/HEV Traction Motors Selection of electric powertrain candidates for EV applications depends on multiple factors. There is no one-size-fits-all solution. From the perspective of OEMs, this assessment will generally include requirements or considerations as below: • • • • • • • •

Cost Efficiency Power/torque density over weight Power/torque density over volume NVH Fault tolerance Reliability and maintainability Technical maturity or risks.

Criteria-based decision method such as Pugh Matrix or an objective function can then be defined with weighting coefficients associated to each of these aspects as indication of importance. The comparison shall start with an assumption of the positioning of the vehicle in the market, EV or HEV powertrain topologies, mechanical integration constraints, cost and power and so forth. These different application

18

1 Introduction

scenarios may come up with significantly different weighting coefficients and the assessment matrix. Some factors can be quantized while most other aspects will be rather subjective. For one example, PM motors are known to have superior efficiency to other motor types, but they do not necessarily outperform others when assessed using drive cycles predefined by the authorities. The drive cycles are usually designed for more economic driving behaviour, but the EVs and HEVs may put more values on performance. Therefore when coming to assessment of the efficiency, aspects including peak efficiency, high efficiency range, and the evaluation results from standardized test cycles all need to be weighted. Another representative example is the assessment of acoustic noise, which directly links to the vehicle drivability. The maximum allowed decibels (dB) are defined by the ISO 362-1. However, the noise level is not always the lower the better, but in reality, a certain level of noise is required to ensure vehicle safety for both the drivers and the pedestrians. Furthermore, the NVH behavior represents vibration modes at different frequency components that may be tolerable to one but uncomfortable to the other. Therefore, it is always important to select a reference and set boundaries to the evaluation algorithm when comparing multiple candidates. As there is no one unique solution for all applications, even for the same motor type, the design itself needs to go through numerous iteration loops before it is customized with a combination of most optimal architecture, topology and geometric dimensions etc. Cost and efficiency usually receive highest weighting coefficients. Cost affects the acceptance of the products as the end users tend to choose a cost-effective solution that fits for purpose. Cost breakdown comprises of both the material cost and the development cost spread over a predicted volume of mass production. Efficiency is directly linked to the range of the vehicle. A highly efficient motor drive not only alleviates the range anxiety but also leads to an early payback of the cost. Efficiency is usually evaluated over a standard drive cycle such as the WLTP. Figure 1.9 emulates a WLTP drive cycle with a general EV vehicle curb weight, transmission ratio and other geometric dimension as an example (Used in the example is the curb weight of 1620 kg, the transmission ratio of 9.69:1 and the tyre size of 225/45R18). The red circle marker indicates repetitions of the operating points, while the blue means no repetition. As such, more darkness of the red color represents more densely repeated operating points in the drive cycle. It can be seen that most often the vehicle will operate in the low-load situation, which actually favors the PM-free motors. Power to weight ratio, i.e. specific power, is a factor to affect the payload of the vehicle. The lighter the motor drive, the better. Integration of the electric motor drive into the vehicle powertrain largely depends on the dimension and volume of the motor drive itself. This can be even more important for the HEV applications. The hybrid architecture, either serial or parallel, requires the electric motor drive to be integrated with the engine and the transmission line within the confined space. Some motor topologies, such as outer-rotor motors or axial flux motors, may fit particularly well, such as HEV architecture P1 or P2 topology or the in-wheel applications [55]. NVH can in a way be referred to as drivability and can draw a red line regarding the acceptance of the technology. In case of standard not being met, it can hardly be further considered. Electric vehicles are generally much quieter than conventional

1.3 Selection Criteria of EV/HEV Traction Motors

19

(a)

Dense operang points Sparse operang points

(b) Fig. 1.9 WLTP Class 3 drive cycle: a Vehicle speed, b Motor torque-speed projection

20

1 Introduction

internal combustion engine-based vehicles. Nevertheless, NVH characteristic is one of the finger prints of a motor drive, which can be attributed to motor types, design and control. Electric motor topologies, such as switched reluctance motors and also synchronous reluctance motors, are intrinsically more noisy and more pulsating than the PM and IM motors. Therefore they require a customized system design approach and more complex motor control techniques to suppress the NVH. Fault tolerance may not be a mandatory feature in passenger vehicles compared to the aerospace applications. In case of critical fault occurrence, the safety state of the controller shall be capable of disabling the power delivery and putting the vehicle in an agreed safe state, such as standstill or freewheeling. This is very much unlike the applications in aerospace, where loss of power can be classified as severe as a catastrophe. However, high fault tolerance is a nice-to-have feature, at least beneficial to ensure a limp home mode. Motor with rugged structures tend to have higher fault tolerance, and less costly in maintenance. High-maintenance motors and components such as the brushes and slip ring in a WFSR cause more uncertainty on the product. Another important factor that manufacturers will consider for commercialization is the technical maturity level. It is quite often used to assess the continuation of development of the motor technologies. This usually occurs when an alternative technology is investigated to replace the current existing technologies. Nowadays, many technologies have been studied to replace the rare-earth PM motors. Among these, IMs are the most mature technology with regard to the vehicle application, where both acceptable accoustic noise and immunity to a hostile environment seem to be prerequisites. As a summary, this section briefs the general methodologies when assessing an EV traction motor solution. Assessment chart of the motor types as shown in Fig. 1.8 can serve as a reference, but when it comes to the selection of the technology using the criteria-based decision methods, the scoring indeed needs to be calibrated and some aspects can be rather subjective, depending on the case-specific applications.

1.4 Requirements from Motor Control Perspective The ideas which form the core part of the book is originated at the development of the application software for the mass-production motor control units (MCUs). There are some key features for electric machines used as a device to drive the vehicles. Firstly, unlike most industrial applications, where the electric machine works more or less around the rated load, the working condition (or working load) changes rapidly and massively along with the vehicle driving conditions, which could be accelerating at one instant and braking at the next instant. This will inevitably require a good dynamics of the electric motor torque. Secondly, the design of the electric machine control must be able to work well with all the electric machines from the same production lines, which could have a difference in characteristics even if it is only slight. Therefore, the control strategy shall have an adaptation ability against the possible varying parts. This adaptation

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ability will be extended to conditions when there is a fault within the electric drive system. Under such a faulty condition, it is required that the vehicle shall be operative at least at the minimum level not to leave the user of the vehicle in the risk condition. Next, the optimization of the vehicle powertrain system to have a better efficiency without compromising the performance at all driving conditions is one of the key control targets. Equally, this is to request the optimization of the electric machine operations under the parameters uncertainties, which may be varying the machine load, environment conditions (e.g. temperature), and the machine aging. Furthermore, it is a general requirement that the control of the electric machine must meet the ISO 26262 for the functional safety for vehicle applications. This implies not only a stringent development process, but a safety design to monitor the electric machine torque and to ensure no un-intended torque is generated as well.

References 1. Agamloh E, Von Jouanne A, Yokochi A (2020) An overview of electric machine trends in modern electric vehicles. Machines 8(2):20 2. Bianchi N, Bolognani S, Bon D, Dai Pre M (2009) Rotor flux-barrier design for torque ripple reduction in synchronous reluctance and pm-assisted synchronous reluctance motors. IEEE Trans Ind Appl 45(3):921–928 3. Bibra EM, Connelly E, Dhir S, Drtil M, Henriot P, Hwang I, Le Marois JB, McBain S, Paoli L, Teter J (2022) Global ev outlook 2022: securing supplies for an electric future 4. Bose BK et al (2002) Modern power electronics and AC drives, vol 123. Prentice hall, Upper Saddle River, NJ 5. Burwell M, Carosa P, Kirtley J, Rippel W, Sanner J, Seger D (2014) Improving the high speed efficiency of xev induction motors 6. Callegaro AD, Bilgin B, Emadi A (2019) Radial force shaping for acoustic noise reduction in switched reluctance machines. IEEE Trans Power Electron 34(10):9866–9878 7. Credo A, Fabri G, Villani M, Popescu M (2020) Adopting the topology optimization in the design of high-speed synchronous reluctance motors for electric vehicles. IEEE Trans Ind Appl 56(5):5429–5438 8. De Santiago J, Bernhoff H, Ekergård B, Eriksson S, Ferhatovic S, Waters R, Leijon M (2011) Electrical motor drivelines in commercial all-electric vehicles: A review. IEEE Trans Veh Technol 61(2):475–484 9. Dos Santos FL, Anthonis J, Naclerio F, Gyselinck JJ, Van der Auweraer H, Góes LC (2013) Multiphysics NVH modeling: Simulation of a switched reluctance motor for an electric vehicle. IEEE Trans Ind Electron 61(1):469–476 10. Dubois A, Van Der Geest M, Bevirt J, Christie R, Borer NK, Clarke SC (2016) Design of an electric propulsion system for sceptor’s outboard nacelle. In: 16th AIAA aviation technology, integration, and operations conference, p 3925 11. Edmondson J (2022) Electric motors for electric vehicles: technologies and market outlook https://fpc-event.co.uk/wp-content/uploads/2022/03/james.edmondson.electric_ machines-1.pdf 12. Fratta A, Troglia G, Vagati A, Villata F (1993) Evaluation of torque ripple in high performance synchronous reluctance machines. In: Conference record of the 1993 IEEE industry applications conference twenty-eighth IAS annual meeting. IEEE, pp 163–170

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13. Gan C, Wu J, Sun Q, Kong W, Li H, Hu Y (2018) A review on machine topologies and control techniques for low-noise switched reluctance motors in electric vehicle applications. IEEE Access 6:31,430–31,443 14. Goss J (2019) Performance analysis of electric motor technologies for an electric vehicle powertrain. Wrexham, UK, Motor Design Ltd, White Paper 15. Gundogdu T, Zhu ZQ, Chan CC (2022) Comparative study of permanent magnet, conventional, and advanced induction machines for traction applications. World Electric Veh J 13(8):137 16. Han S, Diao K, Sun X (2021) Overview of multi-phase switched reluctance motor drives for electric vehicles. Adv Mech Eng 13(9):16878140211045,195 17. He T, Zhu Z, Eastham F, Wang Y, Bin H, Wu D, Gong L, Chen J (2022) Permanent magnet machines for high-speed applications. World Electric Veh J 13(1):18 18. Heidari H, Rassõlkin A, Kallaste A, Vaimann T, Andriushchenko E, Belahcen A, Lukichev DV (2021) A review of synchronous reluctance motor-drive advancements. Sustainability 13(2):729 19. Hofmann A, Al-Dajani A, Bösing M, De Doncker RW (2013) Direct instantaneous force control: A method to eliminate mode-0-borne noise in switched reluctance machines. In: 2013 international electric machines & drives conference. IEEE, pp 1009–1016 20. Houache MS, Yim CH, Karkar Z, Abu-Lebdeh Y (2022) On the current and future outlook of battery chemistries for electric vehicles-mini review. Batteries 8(7):70 21. Hwang D, Gu BG (2020) Field current control strategy for wound-rotor synchronous motors considering coupled stator flux linkage. IEEE Access 8:111,811–111,821 22. Kim H, Park Y, Liu HC, Han PW, Lee J (2020) Study on line-start permanent magnet assistance synchronous reluctance motor for improving efficiency and power factor. Energies 13(2):384 23. Kiyota K, Kakishima T, Chiba A (2014) Comparison of test result and design stage prediction of switched reluctance motor competitive with 60-kw rare-earth pm motor. IEEE Trans Ind Electron 61(10):5712–5721 24. Krishnan R (2017) Switched reluctance motor drives: modeling, simulation, analysis, design, and applications. CRC Press 25. Lee CH, Hua W, Long T, Jiang C, Iyer LV (2021) A critical review of emerging technologies for electric and hybrid vehicles. IEEE Open J Veh Technol 2:471–485 26. Ludois DC, Brown I (2017) Brushless and permanent magnet free wound field synchronous motors for ev traction. Tech. rep., Univ. of Wisconsin, Madison, WI (United States) 27. Mahmoudi A, Soong WL, Pellegrino G, Armando E (2015) Efficiency maps of electrical machines. In: 2015 IEEE energy conversion congress and exposition (ECCE). IEEE, pp 2791– 2799 28. Miller TJE (2001) Electronic control of switched reluctance machines. Elsevier 29. Motor XP (2020) Performance analysis of the tesla model 3 electric motor using motorxp-pm 30. Neuhaus CR, Fuengwarodsakul NH, De Doncker RW (2006) Predictive PWM-based direct instantaneous torque control of switched reluctance drives. In: 2006 37th IEEE power electronics specialists conference. IEEE, pp 1–7 31. Nie Y, Brown IP, Ludois DC (2017) Deadbeat-direct torque and flux control for wound field synchronous machines. IEEE Trans Ind Electron 65(3):2069–2079 32. Nøland JK, Nuzzo S, Tessarolo A, Alves EF (2019) Excitation system technologies for woundfield synchronous machines: survey of solutions and evolving trends. IEEE Access 7:109,699– 109,718 33. Oprea C, Dziechciarz A, Martis C (2015) Comparative analysis of different synchronous reluctance motor topologies. In: 2015 IEEE 15th international conference on environment and electrical engineering (EEEIC). IEEE, pp 1904–1909 34. Ozcelik NG, Dogru UE, Imeryuz M, Ergene LT (2019) Synchronous reluctance motor vs. induction motor at low-power industrial applications: design and comparison. Energies 12(11):2190 35. Pavel CC, Marmier A, Alves Dias P, Blagoeva D, Tzimas E, Schüler D, Schleicher T, Jenseit W, Degreif S, Buchert M (2016) Substitution of critical raw materials in low-carbon technologies: lighting, wind turbines and electric vehicles. European Commission, Oko-Institut eV, Luxembourg

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Chapter 2

Equivalent Circuit Modelling and Analysis

2.1 Induction Machine Working Principle The fundamental principle of induction machines is introduced in this section, which is further broken down into two parts stating respectively • The rotating stator flux generated by alternating current; • The production of electromagnetic torque resulting from the interaction between the stator field and the rotor field.

2.1.1 Rotating Stator Flux When an induction machine is supplied with the three-phase AC current with a 120° phase shift wrt each other, it will produce a rotating magnetic field. Due to the relative movement between the stator and the rotor, this rotating field will induce an alternating current in the rotor, which generates another rotating magnetic field as a return. The electromagnetic torque is a result of the interaction between these two rotating fields. As shown in Fig. 2.1, a three-phase winding1 of AA’, BB’ and CC’ are configured in a Y connection,2 where the respective phase-end A’, B’, and C’ are joint together to form a neutral point. Though demonstrated by three concentrated coil windings, they are usually distributed in reality. The three phases AA’, BB’ and CC’ are displaced with 120° apart from each other in space. When fed with three-phase AC current 1

Connection of the illustrated three-phase winding gives rise to a single pole-pair electric machine. However, three-phase motors are commonly configured with multiple pole pairs. 2 Three phases can be connected in either a Y or Δ manner. Y connection is more widely used in automotive and is adopted in this book, but Δ connection also has its applications as it can produce a rather high power and starting torque. © Springer Nature Switzerland AG 2024 S. Shen and Q.-z. Chen, Practical Control of Electric Machines for EV/HEVs, Lecture Notes in Electrical Engineering 1064, https://doi.org/10.1007/978-3-031-38161-4_2

25

26

2 Equivalent Circuit Modelling and Analysis B-axis

B-axis

A’

B

B

C

B’ A-axis B’

C-axis

C’ A

A’ C’

A

Phase A magnetic axis

C

C-axis

Current

Time

Fig. 2.1 Induction motor working principle

with 120° out of phase to each other in time, it generates a rotating magnetic field across the stator, rotor and air gap. This electromagnetic field at different timing instants is depicted in Figs. 2.2, 2.3, 2.4 and 2.5. Combination of 120° difference of the three phases in both space displacement and time sequence respectively is the key for producing a resultant rotating field with a constant speed at steady state. Prior to deriving the motor principle, it is necessary to get aligned on the polarity convention of the motor speed and the phase current. Here, the positive rotating direction corresponds to the motor spinning anticlockwise (when viewed from the opposite end towards the output shaft). A phase current is said to be positive if it flows from the power supply into the phase terminal. When supplied with a positive phase current, the phase-induced magnetic field points to the positive direction along the corresponding phase magnetic axis, as indicated in the figure. In Fig. 2.2, Phase A is fed with a current of I , while Phase B and Phase C with a current of − 21 I respectively. The direction of the current flow as well as current strength is indicated with red arrows in the simplified phase connection diagram. That being said, phase current flows from terminal A to the neutral point A’ as in the example. In the simplified motor diagram, the cross and dot symbols are used to denote the direction of the phase current flow. A cross sign means the current flowing in an inward direction, while with a dot sign the current flows outwards. Given the schematic winding configuration in Fig. 2.1, the generated magnetic field has the direction pointing to the right, which is the positive direction of the phase A magnetic axis. Compared to Fig. 2.2, the three-phase currents in Fig. 2.3 have moved forward by 120° in phase, resulting in the current strength of I in Phase B while − 21 I in both Phase C and Phase A. The generated magnetic field in this case points towards the

2.1 Induction Machine Working Principle

27

A’

B

C

B

N

B’

S

B’

C’

A

A’

Ia

C’ C A

Current

A

B

C

Time

Case 1

Fig. 2.2 Magnetic field at θe = 0° A’

B

C

B B’ C’

B’

C’

A’

A Ia

C

A Current A

B

C

Time

Case 2

Fig. 2.3 Magnetic field at θe = 120°

up-left direction. That is, this magnetic field has rotated 120° anticlockwise3 in space relative to that in Fig. 2.2. In Fig. 2.4, the phase currents have further advanced by another 120° in phase. This time, Phase C has current of I while the currents of Phase A and Phase B are − 21 I . Consequently, the resultant magnetic field rotates by 120° anticlockwise in space from the direction in Fig. 2.3. 3

Clockwise or anticlockwise rotation of the magnetic field depends on the configuration of phase winding arrangement. If two phases are swopped in space, or if the phase current excitation sequence is changed to ACB instead of ABC, the magnetic field will rotates in a clockwise direction.

28

2 Equivalent Circuit Modelling and Analysis A’

B

C

B B’

C’

B’

C’

A’

A Ia

C

A Current A

B

C

Time

Case 3

Fig. 2.4 Magnetic field at θe = 240° A’

B

C

B

N

B’

S

B’

C’

A’

A Ia

C’ C

A Current A

B

C

Time

Case 4

Fig. 2.5 Magnetic field at θe = 360°

Nevertheless, the magnetic field presented in Fig. 2.5 is reverted back to the same state as in Fig. 2.2 due to the period of 360° in electrical scale for the phase currents. As a summary, assuming the angular frequency of the three-phase current is ωe , the magnetic field will rotate at the same speed as ωe for a single pole-pair induction machine.4 The speed of the rotating magnetic field is referred to as the synchronous speed (or virtual mechanical speed) ωsm . For multiple pole-pair motors, denoting

4

This principle is generic for AC machines.

2.1 Induction Machine Working Principle

C1

180° A2’

B1

B2

B1’

29

C1 B1’ B2

C1’

180° A2’

B1

B2

B1’

B1’ B2

C1’

Ia

360°

A1

A2

B2’ C2’ A2’ A2 A1’

0°

C2 C1’

B2’

C2’ B1

A1

A2

C2’

0°

C2 C1’

B2’

C2’

C1

C2

A1’

360°

A1

B1

540°

Ia

A1

C1

C2

A1’

B2’

A2’ A2 A1’

540° Current

Current A

B

A

C

B

C

Time

Time

Case 2

Case 1

(a) Magnetic field at θ e = 0° C1

180° A2’

C1

B1

B2

B1’

(b) Magnetic field at θ e = 120°

B1’ B2

C1’

180° A2’

B1

B2

B1’

B1’ B2

C1’

Ia

360°

A1

A2

C2’

0°

C2 C1’

B2’

C2’ B1

360°

A1

A2

A1

C2’

0°

C2 C1’

B2’

C2’

C1

C2

A1’

B2’ A2’ A2 A1’

B1

Ia

A1

C1

C2

A1’

B2’

A2’ A2 A1’

540°

540°

Current

Current A

B

A

C

B

C

Time

Time

Case 4

Case 3

(c) Magnetic field at θ e = 240°

(d) Magnetic field at θ e = 360°

Fig. 2.6 Multi pole-pair magnetic field working principle

the pole-pair number by N p , the relationship between the stator electric speed (or frequency) ωe and the synchronous speed arises as ωe = N p ωsm This relationship is illustrated in Fig. 2.6 for a two pole-pair motor. It is clear that indeed rotation of the magnetic field will complete one mechanical cycle only when the current passes two electric cycles. For the sake of convenience at the later-stage analysis, an equivalent rotor electric speed ωr e is introduced, satisfying ωr e = N p ωr m

(2.1)

where ωr m 5 is the rotor mechanical rotating speed. As the induced rotor current and therefore electromagnetic torque result from the relative movement between the magnetic field and the rotor bar. There always exists a slip between the rotor mechanical speed and the synchronous speed in order for yielding torque. Mathematically, the three phase winding currents can be expressed as 5

The notation of ωr is also often used in this book for simplicity sake.

30

2 Equivalent Circuit Modelling and Analysis

i a = I cos (ωe t)   2π i b = I cos ωe t − 3   2π i c = I cos ωe t + 3

(2.2) (2.3) (2.4)

By neglecting the effect of stator slots and other space harmonics due to non-ideal winding distribution, each phase winding will produce a sinusoidally distributed MMF. At a given angular position θ in space, the respective instantaneous MMF is given by mm f a (θ ) = N i a cos (θ )   2π mm f b (θ ) = N i b cos θ − 3   2π mm f c (θ ) = N i c cos θ + 3

(2.5) (2.6) (2.7)

where N is the number of turns in one phase winding. It is clear that the currents are shifted in phase by 120°and MMF waves are shifted in space by 120°. The resultant MMF at angle θ is the sum of the MMF of each phase winding as mm f (θ ) = mm f a (θ ) + mm f b (θ ) +  )  mm f c (θ  + N i c cos θ + = N i a cos (θ ) + N i b cos θ − 2π 3

2π 3



(2.8)

Substituting Eqs. (2.2)–(2.4) into Eq. (2.8), yields  mm f (θ, t) = N I cos (ωe t) cos (θ )   + cos ωe t − 2π cos (θ − 3   cos (θ + + cos ωe t + 2π 3

2π ) 3 

(2.9)

2π ) 3

Bearing in mind that cos(x) cos(y) =

1 [cos(x − y) + cos(x + y)] 2

we then have (Fig. 2.7) cos (ωe t) cos (θ )    cos ωe t − 2π cos θ − 3    cos ωe t + 2π cos θ + 3

= 21 [cos(θ − ωe t) + cos (θ + ωe t)]  = 21 [cos(θ − ωe t) + cos θ + ωe t + 3   2π = 21 [cos(θ − ωe t) + cos θ + ωe t − 3  2π



2π ] 3  2π ] 3

(2.10)

2.1 Induction Machine Working Principle

31

mmf cos(we*t) * cos( )

we*t=180

-180

-90

0

90

180

we*t=0

A’ cos(

+ we*t) mmf -

-180

-90

cos(

0

- we*t) mmf +

90

mmf + B

C

mmf

180

mmf C’

B’ A

Fig. 2.7 Magnetic field decomposition for one phase

Equation (2.10) indicates that the MMF wave of one phase can be decomposed into one MMF component of mm f + = N2I cos(θ − ωe t) rotating anticlockwise, and } rotating clockwise. Furthe other of mm f − = N2I cos(θ + ωe t − φi ), φi = {0, ± 2π 3 thermore, the mm f + components from the three phases are in phase, and thus enhance each other. On the other hand, the mm f − components are out of phase, and therefore, are cancelled out by each other, i.e.,     2π 2π + cos θ + ωe t − =0 cos (θ + ωe t) + cos θ + ωe t + 3 3 The resultant MMF then arises as mm f (θ, t) =

3 N I cos(θ − ωe t) 2

(2.11)

This explains the rotating nature of the stator magnetic field mathematically.

2.1.2 Electromagnetic Force and Torque As aforementioned, injection of three-phase AC current into the stator windings generates a rotating magnetic field. This is equivalent to a permanent magnetic pole pair of N and S rotating at a synchronous speed of ωs as illustrated in Figs. 2.8 and 2.9. When a rotor, with buried aluminum or copper bars that are short-circuited at two ends with each other, rotates at a speed ωr e different from the electrical speed ωs of

32

2 Equivalent Circuit Modelling and Analysis

Fig. 2.8 Electro-magnetic motoring torque

A’ s >

C

B

re

Rotor bars cutting the stator flux induce a current Moving rotor bars with a current produce an electro-magnetic force

C’

B’

A pair of force generates a motoring torque

A s

mmf

Fleming’s righthand rule

s

s

Fleming’s lefthand rule

N

N F

re

re

re

re

-

-

re

s

s

re

F S

S s

Fig. 2.9 Electro-magnetic brake torque

s

A’

B

C

s
ωr e , while Fig. 2.9 is the case with ωs < ωr e . The effect of the change of current direction to the induction machine will be further clarified in the upcoming parts of this section. Subsequently, with the current i flowing in the rotor bars, the stator magnetic field will exert a magnetic force on the bars as F = (i × B)L Due to the fact that the upper bars and low bars of the rotor have opposite current directions, the force in the low bars has opposite direction to the that of the upper bars. These pairs of forces form a rotating torque Tem as Tem = 2r (i × B)L

(2.13)

where r is the equivalent rotor radius. In short, the working principle of induction machines can be summarized as • The alternating current in the stator windings generates a rotating magnetic field; • The relative speed between stator and rotor induces a current in the rotor conductors, and in turn produces a torque between the stator and the rotor. A flowchart is given in Fig. 2.10 in this regard. In Fig. 2.8, the stator magnetic field rotates faster than rotor. Rotor bars cut the flux from behind and induce a torque to speed the rotor up. It is a motoring torque that tends to synchronize the rotor speed with the stator flux speed.6 By contrast, in Fig. 2.9, the rotor speed exceeds the stator flux speed, resulting in cutting the flux from a reverse direction due to the relative movement. As a consequence, the induced torque will slow down the rotor. This is the braking torque to the rotor or is also referred to as the regenerating torque. Likewise, the braking torque also tries to synchronize the rotor speed with the stator field speed. In an ideal freewheeling scenario, the rotor speed is synchronized with the stator flux speed, there will then be no current induced and consequently no torque is yielded. Thus, a key factor in 6

Hereinafter, the stator speed is referred to as the stator flux rotating speed in this book.

34

Alternating current

2 Equivalent Circuit Modelling and Analysis

Rotating flux

Different Speed ? s > re

Different Speed ? s < re

No

Rotor rotates

No current

Yes

Yes

Cut flux

Cut flux

Produces EMF

Produces EMF

Induces current

No Stator flux and rotor rotate at the same speed

Generates a motoring torque

Induces Current

Generates a braking torque

No torque

Rotor speeds up Rotor slows down Rotor holds the speed

Fig. 2.10 Electro-magnetic torque flow diagram

the working principle of induction machines is the relative movement between the rotor and the stator. The relative movement is referred to as slip, which is defined as s=

ωs − ωr e ωs

(2.14)

Here, ωr e standing for the electrical speed of the rotor. Literally, slip describes the relative movement between the stator and rotor as a fraction of the stator electrical speed. Bear in mind that the above expression of the slip is in the electrical speed domain. The slip can be equally specified in the mechanical scale by replacing ωs with ωsm , and ωr e with ωr m (or ωr ). Nevertheless, the slip speed (being ωs − ωr e ) is normally defined in the electrical speed domain only. Combining Eqs. (2.12) and (2.13) results in the expression of the overall electric machine torque as Tem = r B 2 L 2 v/R ∝ r 2 B2 L2 (2.15) ∝ r4 L2 Here, it is assumed that number of winding turns is proportional to the perimeter of the stator. Equation (2.15) is helpful from electric machine design point of view. It suggests that, for instance, in order to increase the electric machine torque, it is more effective to increase the machine radius rather than the motor length.

2.2 Equivalent Circuit Model An equivalent circuit model of an induction motor is a simple but important tool for performance analysis. In particular, this tool helps in predicting steady-state characteristics of a motor. The equivalent circuit model is normally built upon a

2.2 Equivalent Circuit Model

35

Fig. 2.11 Rotating transformer circuit

Rs Is

Lrl

Lsl Iψ

Ir*

Es

Rr

Rc

Ic

Er

LM

IM

Vs

Ir

Ideal rotating transformer

single phase as shown in Fig. 2.11. The idea of equivalent circuit model comes form the fact that an induction motor is actually a rotating transformer, where the equivalent circuit model is widely used in its performance analysis [1, 10]. Figure 2.11 depicts a per-phase equivalent circuit model of an induction machine. The primary side of the transformer equivalent circuit model stands for the stator winding, while the secondary side represents the equivalent rotor circuit. The primary voltage Vs is considered as voltage supply to phase terminals. Notations in this figure and in later-on transformed equivalent models are respectively defined as Er : Es : Ic : IM : Ir : Ir∗ : Is : LM: L rl : ∗ : L rl L sl : Rc : Rr : Rr∗ : Rs : Vs :

Back EMF at rotor side; Back EMF at stator side; Core loss current; Magnetizing current Rotor current; Equivalent rotor current referred to the stator side; Stator current; Magnetizing inductance; Rotor leakage inductance; Equivalent rotor leakage inductance referred to the stator side; Stator leakage inductance; Core loss resistance; Rotor winding resistance; Equivalent rotor winding resistance referred to the stator side; Stator winding resistance; Stator phase voltage.

Note that in an equivalent circuit model, the quantities are quoted in rms values instead of peak values. Nevertheless, the model in this form is not readily analyzable. In order for identification of the model parameters, further conversion and simplification of the model are required. The process is broken down into the following steps and elaborated correspondingly.

36

2 Equivalent Circuit Modelling and Analysis

Fig. 2.12 Rotor side equivalency +

-

jXrl=jsXrl0

jXrl0

Ir

Ir

Er = sEr0

Rr

+

-

Er0

Rr s

Step1: Rotor Model Normalization On the left-hand side of Fig. 2.12 is a rotor model, which is the same as the secondary circuit shown in Fig. 2.11. This simple rotor model is composed of a resistance Rr and a leakage inductance L rl , which are powered by the induced EMF Er . The EMF Er depends on the stator flux and the slip speed between the rotor and stator. If the stator flux7 is Φs = Φs0 sin(ωs t) and the rotor is stationary, then the induced EMF at the rotor will be s er = Nr dΦ dt = Nr Φs0 ωs cos(ωs t) = Er 0 cos(ωs t)

where Nr is the equivalent turns of the rotor, and Er 0 the magnitude of the induced EMF when the rotor is locked (i.e., stationary). However, when the rotor rotates at a speed of ωr e ,8 the stator flux viewed by the rotor becomes Φsr = Φs0 sin[(ωs − ωr e )t] = Φs0 sin(sωs t) where ωsp = ωs − ωr e denoting the slip speed. Accordingly, the induced rotor EMF arises as dΦ r er (s) = Nr dts = s Nr Φs0 ωs cos(sωs t) = Er cos(sωs t) = s Er 0 cos(sωs t) Bear in mind that Eq. (2.14) is applied in the above expressions. It can be seen that the induced EMF at locked-rotor scenario is indeed one particular situation when s = 1. Consequently, the magnitude Er of EMF at the slip s is s times of that when the rotor is locked. That is, Er can be normalized to Er 0 by a factor of s. Although 7

Flux is the result of MMF and the magnetic reluctance of the corresponding magnetic path. Here ωr e is referred to as the electrical rotating speed of the rotor. ωr e = N p ωr as defined by Eq. (2.1).

8

2.2 Equivalent Circuit Model

37

the above analysis is given at the magnitude level for EMF, the same equation still applies if converted to a rms manner which can be more common for investigating the steady states of AC circuits. Hereinafter in this section, rms and magnitude of the signals may be used exchangeably unless otherwise specified for an exception. When the rotor is locked, the induced EMF on the rotor bars has the same electric frequency of ωs as the stator. When the rotor rotates, its induced EMF is with an electric frequency of sωs which exhibits as a slip frequency compared to that of the stator. However, the overall rotating speed (in space) of the rotor flux is the sum of rotor electric rotating speed and the electric frequency of the rotor flux as ωrflux = ωr e + sωs = ωr e + (ωs − ωr e ) = ωs

(2.16)

Thus from both cases, it can be generalized that the rotor flux always rotates at the same speed as that of the stator flux. Though the rotor resistance remains unchanged irrespective of the electric frequency, the reactance of rotor leakage inductance is dependent on the electric frequency and is given by X rl (s) = sωs L rl = s X rl0 Thus, the reactance of rotor leakage inductance X rl at the slip s is normalized to X rl0 , which is the locked-rotor reactance. The principle of equivalent circuit on the rotor side is based on the assumption that the rotor current Ir remains unchanged before and after the conversion. As such, Ir = = = =

Er Zr Er Rr + j X rl s Er 0 Rr + js X rl0 Er 0 Rr /s+ j X rl0

When Er is normalized to Er 0 and X rl to X rl0 , Rr needs to be scaled to Rr /s in order to keep Ir the same prior to and after the transform. This equivalent process is illustrated in Fig. 2.12. With this conversion, electric frequency of the equivalent rotor model gets aligned with that of the stator, i.e. the frequency of ωs or ωe . Step2: Mapping the Rotor Model to Stator Side The principle of mapping the rotor model to the stator side is to maintain the following conditions during the conversion • The current Ir flowing to the rotor when referred to the stator side shall be unchanged; • The back EMF at the stator side due to the interaction between the stator and rotor shall be invariant.

38

2 Equivalent Circuit Modelling and Analysis

The above two conditions also imply that the power injected into the rotor from the stator remains the same. Figure 2.13 manifests this conversion process. After the conversion, the equivalent reactance X rl0 and the equivalent resistance Rr /s are to be identified. We define the parameter re as the effective turns ratio of the rotating transformer, given by re = Er 0 /E s = Nr /Ns Mapping the rotor model to the stator side is actually an impedance matching process. Let Z r∗ stand for the impedance of the rotor but measured at the stator side, we have the following expression Z r∗ = E s /Ir∗ Here Ir∗ is the current flowing to the rotor referred to the stator side, or is considered as an equivalent rotor current viewed from the stator. Due to the power invariance during the conversion, it yields E s = Er 0 /re Ir∗ = Ir re The rotor impedance measured at the rotor side is Z r = Er 0 /Ir Combining the above equations results in Z r∗ = = = =

Zr re2 Rr /s+ j X rl0 re2 Rr /s + j Xrrl0 2 re2 e ∗ Rr ∗ + j X rl0 s

Therefore, when the rotor is mapped to the stator side, its resistance Rr∗ and inductance ∗ are multiplied by a factor of 1/re2 . The converted equivalent circuit is given in L rl the lower diagram in Fig. 2.13. The equivalent circuit is now obtained. As per the modelling conversion process, ∗ referred to the stator side are derived from Rr and L rl . In the equivalent Rr∗ and L rl practice, these parameters are measured via a combination of DC test, blocked-rotor test and no-load test [4, 6, 10, 12]. Alternatively, parameter identification methods have also been extensively studied such as in Refs. [7, 11]. As such, these parameters are mapped to the stator side. As the parameter measurement is viewed from the stator perspective, extra caution is thus required in order to avoid confusion. In summary, the equivalent model by far is already feasible for performance analysis of an induction machine. This is a linear model for a given slip s, ignoring the

2.2 Equivalent Circuit Model

39

Fig. 2.13 Mapping the rotor model to the stator side

Rs

Xrl0

Xsl Iψ

Is

Ir*

Es

Rs

Rr/s

Rc Ic

Er0

XM IM

Vs

Ir

re

* Xrl0

Xsl Iψ

Is

Ir* Rc Ic

Es

* Rr/s

XM IM

Vs

Ir*

parameter variation along the changing environmental conditions such as temperature rise during machine operation. Though some analysis later on is indeed based on this model, this model will still be further simplified in the next step. Step3: Equivalent Model Simplification The purpose of further simplification is to decouple the effects of the core loss resistance Rc and the magnetizing inductance L M from the rest of the equivalent model. Let Z M denote the impedance in the magnetizing branch that comprises of the magnetizing inductance and the core loss resistance, i.e., Z M = jX M  Rc . The rest of stator impedance is denoted by Z s with Z s = Rs + jX sl . Let Z be the total impedance viewed from the stator. The core loss and magnetizing combined current IΨ is given by s Zs IΨ = Vs −I ZM Vs = Z M − VZs ZZMs As long as the magnitude of Z s /Z M is sufficiently small, which is generally true for most induction machines, the second term in the equation can be neglected. Therefore, the component Z M can be transferred to the leftmost to the voltage input terminal. Moreover, the term of core loss is very often omitted when efficiency is not a focus of the analysis, and thus Z M can be further simplified as Z M = X M . With this approximation, the equivalent circuit model is transformed from a ‘T’-shaped model into a ‘ ’-shaped model. The whole step of simplification introduces a small amount of errors to the model, and thus the final model is also referred to as approximately equivalent circuit model.

40

2 Equivalent Circuit Modelling and Analysis

Fig. 2.14 Equivalent circuit

* Lrl

Lsl

Rs

Iψ

Is

Ir*

Rc Ic

* Rr/s

XM IM

Vs

Rs

* Lrl Ir*

Is

IM

* Rr/s

XM

Vs

Lsl

The bottom diagram in Fig. 2.14 shows the final equivalent circuit model after the above mentioned three steps of conversion and simplification.

2.3 System Performance Analysis In this section, the performance analysis of induction motors is carried out, covering the following aspects: • • • • •

The slip to torque characteristics; The speed to torque characteristics; Operation regions; Phasor diagram; System efficiency.

For convenience sake, the following components of system power are denoted respectively by Pag : Air gap power; Pc : Core loss power, including hysteresis and eddy current losses of both the stator and rotor; Pclr : Rotor copper loss; Pcls : Stator copper loss; Pm : Mechanical Power; Pml : Mechanical loss, such as bearing and air drag losses; Pnet : Net power output; Pe : Electric power input, sometimes Ps is used.

2.3 System Performance Analysis

41

Table 2.1 Typical parameters of the induction machine Symbol Typical value ISO unit Description LM ∗ L rl L sl Np Rr∗ Rs VDC Vs rms Ψmax

3.0×10−3 0.27×10−3 0.25×10−3 2 57×10−3 24×10−3 330 134 0.2

[H] [H] [H] [-] [Ω] [Ω] [V] [V] [Wb]

Magnetizing inductance Equivalent rotor leakage inductance Stator leakage inductance Number of Pole pairs Equivalent rotor resistance Stator winding resistance DC voltage Phase voltage in rms value, Y connection Maximum permissible flux linkage in rms value

Furthermore, the electric machine parameters utilized in this analysis are summarized in Table 2.1. By default, the stator supply frequency is 100 Hz and the rms phase voltage is set to 134V unless otherwise specified.

2.3.1 Slip to Torque Characteristics It is worth noting that the AC electrical quantities used in this section are described in a rms manner, even though the symbols by themselves do not have the indication of ‘rms’ for the sake of conciseness. For instance, I and voltage Vs are actually referred to as I rms and Vs rms . The air gap power Pag is the power stored in the air gap coupling the stator and rotor. On the rotor side, this power is consumed by the equivalent resistor Rr∗ /s. Thus, in a per-phase scale, (2.17) Pagph = I 2 Rr∗ /s The rotor conductor loss can be written as r, ph

Pcl

= I 2 Rr∗

(2.18)

ph

This gives rise to the mechanical power Pm generated by one phase as ph

ph

r, ph

Pm = Pag − Pcl ph = (1 − s)Pag 1−s = I 2 Rr∗ s

(2.19)

42

2 Equivalent Circuit Modelling and Analysis

In the above I is the phase current without the magnetizing part. It is actually the equivalent rotor current Ir∗ viewed by the stator side.9 For a three-phase motor, summation of the corresponding power becomes Pag = 3I 2 Rr∗ /s Pclr = Pag s Pm = Pag (1 − s) The electromagnetic torque Tem is therefore given by

Tem =

Pm Pm = Np ωr m ωr e Pag (1 − s) = Np ωs (1 − s) Pag = Np ωs R∗ = 3N p I 2 r sωs

(2.20) (2.21)

where ωr m is the mechanical rotor speed while ωr e the rotor speed in the electrical scale. Equation (2.20) indicates the air gap power is actually the result of the machine torque multiplied by the synchronized speed as Pag = Tem ωsm

(2.22)

The expression (2.20) elucidates the fact that the mechanical torque is actually determined by the useful work of the air gap and stator supply frequency. The mechanical power expression in Eq. (2.19) suggests that the separation of term Rr∗ /s into the two as shown in Fig. 2.15 has more sensible corresponding components of Rr∗ and Rr∗ 1−s s corresponds physical meaning. Rr∗ represents the rotor resistance loss, while Rr∗ 1−s s to the desirable power output. Accordingly, the current I is given by I =

Vs (Rs +

Rr∗ /s)2

+ (X sl + X rl∗ )2

(2.23)

Combining Eqs. (2.21) and (2.23) yields the overall expression for Tem as Tem = 3N p

Vs2 Rr∗ /s ωs (Rs + Rr∗ /s)2 + (X sl + X rl∗ )2

(2.24)

The notations of I and Ir∗ as the rotor current at the stator side will be used exchangeably in this section. It is also called the torque current later on.

9

2.3 System Performance Analysis

43

Fig. 2.15 Equivalent circuit model

Rs

* Lrl

Rr*

Ir* (I)

Is

IM

* Rr(1-s)/s

XM

Vs

Lsl

Investigating Figs. 2.11, 2.12 and 2.13, when s < 0 (i.e., ωr e > ωs ), the direction of the induced voltage Er and current Ir is reversed, implying that the rotor will then work as a prime mover to supply power back to the stator side. As a result, the motoring and generation characteristics are not completely symmetric. As a matter of fact, the peak generating torque is relatively larger than that of motoring. Nevertheless, the slip to achieve the peak torque in generating and motoring is symmetric (same slip in terms of absolute values). The following discussions are made for the electromagnetic torque. D1: s = 1, i.e. the rotor starts at standstill or is locked , ωs > 0 but ωr e = 0. In this case, V2 Rr∗ Tst = 3N p s ωs (Rs + Rr∗ )2 + (X sl + X rl∗ )2 ∗ The starting torque Tst depends not only on the design parameters of N p , L rl , ∗ L sl , Rr and Rs , but also on the operating parameters of Vs and ωs . In order to have a high starting torque, it is desirable to have a low stator supply frequency and a high supply phase voltage. D2: s = 0, i.e. the rotor is synchronized with the stator. Repairing Eq. (2.24) yields

Tem = 3N p

Vs2 Rr∗ s ωs (Rs s + Rr∗ )2 + (X sl + X rl∗ )2 s 2

It is now clear that synchronizing torque Tem = 0 when synchronized. D3: s > 0, but s is sufficiently small. The expression of Tem can be approximated as V 2s Tem = 3N p s ∗ ωs Rr Therefore, at the low slip level, the torque Tem is linearly proportional to the slip s approximately. D4: s < 0, but s is sufficiently small. This is similar to case D3 except that the torque is negative. D5: s < 1, but s is very close to 1. As per Eq. (2.24), the term Rr∗ /s will increase when s decreases. Tem is more sensitive to term Rr∗ /s in the numerator than in

44

2 Equivalent Circuit Modelling and Analysis

the denominator. Thus, the overall Tem increases with the decrease of s when s < 1 and s is close to 1. D6: s > 1, but s is very close to 1. The analysis is similar to D5. Tem decreases with the increase of s when s > 1 and s is close to 1. Recall the cases form D1 to D6, Tem = 0 when s = 0, and Tem = Temst when s = 1. The change of Tem is positively correlated with the slip near s = 0, while a negative correlation between Tem and the slip occurs near s = 1. This implies that Tem must have at least one local maximum in the range 0 < s < 1. To identify Temmax , Eq. (2.24) is revised as V2 Rr∗ (2.25) Tem = 3N p s ωs (Rs s + Rr∗ )2 /s + (X sl + X rl∗ )2 s Tem reaches its maximum when the denominator term Ξ = (Rs s + Rr∗ )2 /s + (X sl + X rl∗ )2 s is at its minimum. It is well known that minimum of Ξ is obtained when ∂Ξ /∂s = 0. We therefore have ∂Ξ (Rs s + Rr∗ )Rs (Rs s + Rr∗ )2 + 2 =− + (X sl + X rl∗ )2 = 0 ∂s s2 s

(2.26)

Multiplying both the denominator and numerator of the above equation by s 2 (as s = 0 is not the solution) and tidying it up, it results in s∗ = 

Rr∗ Rs2 + (X sl + X rl∗ )2

(2.27)

∗ or using L sl and L rl in the expression by instead, the above equation then arises as

s∗ = 

Rr∗ ∗ 2 Rs2 + ωs2 (L sl + L rl )

Thus, the maximum machine torque is arrived at when s = s ∗ . Substituting Eq. (2.27) into (2.24), yields the maximum torque at motoring as Temmax =

1.5N p Vs2  2 ωs Rs + (X sl + X rl∗ )2 + Rs

(2.28)

It is worth noting that the condition in Eq. (2.26) is valid not only for the maximum motoring torque, but also for the peak torque in general including both motoring and generating. In case that the slip s < 0, the maximum generating torque, which is referred to as the minimum machine torque here considering the polarity, is obtained as

2.3 System Performance Analysis

45

Temmin = −

1.5N p Vs2  ωs Rs2 + (X sl + X rl∗ )2 − Rs

subjected to s∗ = − 

(2.29)

Rr∗ Rs2 + (X sl + X rl∗ )2

Apparently, the difference in the denominators in Eqs. (2.29) and (2.28) implies that the amplitude of the peak generating torque is comparatively larger than that in motoring. This supports the claim made earlier regarding the asymmetric behavior between motoring and generating. The maximum torque equation can also be re-organized in the following format, boasting an advantage that will be clear later on  Temmax =

Vs ωs

2

1.5N p  ∗ 2 Rs /ωs + (Rs /ωs )2 + (L sl + L rl )

The following comments are made C1: When electric frequency ωs is sufficiently high, Rs /ωs can be neglected when ∗ terms. This yields the maximum torque as compared with L sl and L rl  Temmax

=

Vs ωs

at the slip speed of ∗ = ωsp

2

1.5N p ∗ L sl + L rl

Rr∗ ∗ L sl + L rl

(2.30)

(2.31)

This is the fundamental principle of variable frequency control. The torque remains constant when the voltage-frequency ratio V /F is controlled to be constant.10 C2: The maximum machine torque is determined by the air gap flux linkage Vs /ωs ∗ (or magnetic field strength), the stator and rotor inductance L sl and L rl . The mutual flux linkage is given by Ψ M = E s /ωs ≈ Vs /ωs

(2.32)

where Ψ M = L M I M . Bear in mind that inductance L is determined by the area A of the cross section of the magnetic conducting materials, the length l of

10

This is only valid in the relatively high frequency range. Extra voltage is required to compensate the stator resistance loss (and etc.) to maintain the desired magnetic strength, and the therefore constant machine torque.

46

2 Equivalent Circuit Modelling and Analysis

Fig. 2.16 Slip to torque curve [ f s = 100 Hz]

Slip to torque curve

300

Torque [Nm]

200 100 0 -100 -200 -300 -1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Slip [-]

the magnetic path and the turns N of the winding according to the following equation L = μN 2 A/l

C3:

C4: C5: C6:

This once again indicates the increase the machine radius is more effective than the increase of the machine length by assuming that N is proportional to the machine perimeter while A is proportional to the machine length. Maximum torque Temmax does not depend on the rotor resistance Rr∗ , but the slip ∗ at which the maximum torque is arrived does. That being said, the speed ωsp slip to torque profile will vary for different Rr∗ . The higher the Rr∗ , the bigger ∗ . the ωsp ∗ and L M are fixed, the maximum Once the design parameters of Rr∗ , L sl , L rl max torque Tem is governed by the operating parameters Vs and ωs . Maximum torque is proportional to the square term of the input voltage Vs as per Eq. (2.30), provided magnetic field is not saturated. Maximum torque is limited by the system permissible flux linkage Ψmax .11 Equation (2.30) can be rewritten as 1.5N

Temmax = (Ψ M )2 L sl +Lp∗

rl

=

1.5N (L M I M )2 L sl +Lp∗ rl

The flux linkage is subjected to saturation. When the phase voltage is further raised, the torque will not be increased any more, but more heat is generated. A torque-slip characteristics diagram is given in Fig. 2.16. The corresponding Matlab script is given as in Listing 2.1. This diagram validates the following aspects: • At the low s range, the machine torque is almost linearly proportional to the slip; • The machine is motoring when s > 0 and generating when s < 0; However, the torque-slip behavior is not strictly odd symmetrical. 11

In this case, the maximum torque is achieved with the optimal slip under the constant phase voltage.

2.3 System Performance Analysis

47

• The peak torque Temmax = 246 Nm is arrived at with s ∗ = 17.4%. According to Eqs. (2.30) and (2.31), these will be 265 Nm and 17.5% respectively. The difference is caused by neglecting the term Rs /ωs . Rs /ωs is 0.038 × 10−3 H in this example, about 7% of L sl + L sr being 0.52 × 10−3 H.

Listing 2.1 Slip to torque characteristics calculation 1 2 3 4

Freq ws Vdc vs

= = = =

100; 2* pi * Freq ; 330; Vdc / sqrt (6) ;

5 6

% For Y c o n n e c t i o n and SPWM % Vpn = Vdc * sqrt (3) /3 % Vpn_rms = Vpn / sqrt (2)

7 8

Motor_Parameters_Eqv ;

9 10 11 12 13 14 15 16 17 18 19 20

S l i p P l o t = 1; % Plot s e t t i n g % % P r e p a i r the f i g u r e figname = ' SlipSpdVsTqCalc '; h _ s l i p S p d T q f i g = f i n d o b j (0 , ' Name ' , [ figname , ' _ R e s u l t s ' ]) ; if i s e m p t y ( h _ s l i p S p d T q f i g ) h _ s l i p S p d T q f i g = f i g u r e ( ' P o s i t i o n ' ,[50 50 452*1 .8 257*1 .8 ] , ... ' Name ' , [ figname , ' _ R e s u l t s ' ] , ... ' N u m b e r T i t l e ' , ' off ' ) ; end figure ( h_slipSpdTqfig ); set ( h _ s l i p S p d T q f i g , ' D e f a u l t A x e s F o n t S i z e ' ,14 , ' D e f a u l t A x e s F o n t N a m e ' , ' Times ... New Roman ' ) ;

21 22 23 24

% % C a l c u l a t i o n bas e d on e q u i v a l e n t c i r c u i t X_term = ( Lsl + Lrl ) * ws ; X _ s q u a r e = X _ t e r m ^2;

25 26 27 28

W_RE = [ 0 : 1 : 2 * ws ]; S = ( ws - W_RE ) / ws ; WSP = S. * ws ;

% Rotor electric speed vector % Slip vector % Slip speed vector

29 30 31 32 33 34

% This part R_term = R_square = I_term = Te =

contains vector related calculation ( Rs. * S + Rr ) ; R _ t e r m . ^2; vs ^2 . * S. /( R _ s q u a r e + X _ s q u a r e . * S. ^2) ; 3* P o l e _ p a i r . * I _ t e r m * Rr / ws ;

35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54

% % Plot the r e s u l t s Tmax = abs ( floor ( min ( Te ) /50) *50) ; if S l i p P l o t plot (S , Te , ' L i n e W i d t h ' ,2) , hold on Smax = 1; x l a b e l ( ' Slip [ -] ' ) ; title ( ' S l i p to t o r q u e c u r v e ' ) ; else plot ( WSP , Te , ' L i n e W i d t h ' ,2) , hold on Smax = min (400 , abs ( f l o o r ( max ( WSP ) / 1 0 0 ) * 1 0 0 ) ) ; x l a b e l ( ' Slip Speed [ rad / s ] ' ) ; title ( ' S l i p s p e e d to t o r q u e c u r v e ' ) ; end plot ([ - Smax , Smax ] ,[0 ,0] , ' k ' , ' L i n e W i d t h ' ,1) ; plot ([ 0 ,0] ,[ - Tmax , Tmax ] , ' k ' , ' L i n e W i d t h ' ,1) ; y l a b e l ( ' T o r q u e [ Nm ] ' ) ; set ( gca , ' XLim ' ,[ - Smax Smax ]) ; set ( gca , ' YLim ' ,[ - Tmax Tmax ]) ; grid on ; hold off ;

48

2 Equivalent Circuit Modelling and Analysis

Fig. 2.17 V/F voltage boost curve

V/F volt drop compensation

4

Boost Ratio [-]

3.5 3 2.5 2 1.5 1 0

10

20

30

40

50

60

70

80

90

100

Freq [Hz]

For V/F control, in order to maintain a constant peak torque, compensation of the dropped Rs effect is required as below   Vs / f s = γ Vs rt / f s rt

(2.33)

where Vs rt and f s rt are respectively the rated phase voltage and frequency, and γ is a boost ratio to compensate the voltage drop across Rs .12 Introducing κr as κr =

Rs ∗ ωs (L sl + L rl )



then γ =

κr +

κr2 + 1

(2.34)

Clearly, γ ≈ 1 if κr is adequately small, which is the case when frequency ωs is sufficiently large. On the other hand, γ increases in line with κr , and this scaling factor becomes non-negligible when ωs is low. Figure 2.17 gives the change of γ with respect to f s . When the frequency f s = 1, the boost ratio γ is upto 3.84; at high frequencies, γ is however closely approaching 1. The Matlab script for Fig. 2.17 is provided in Listing 2.2. Listing 2.2 Voltage boost profile calculation 1 2 3 4 5 6 7

% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Freq = 100; ws = 2* pi * Freq ; Vdc = 330; vs = Vdc / sqrt (6) ; % For Y c o n n e c t i o n and SPWM % Vpn = Vdc * sqrt (3) /3 % Vpn_rms = Vpn / sqrt (2)

8 9 10

F r e q _ R a t e = Freq ; Vs_Rate = vs ;

% Rated Freqency ; % Rate p h a s e v o l t a g e

11

12

This method of boosting the phase voltage has an issue when the stator frequency is at or very close to zero. At that range, offset compensation is more effective than scaling compensation.

2.3 System Performance Analysis 12 13 14 15 16 17 18 19 20 21 22 23 24

49

Motor_Parameters_Eqv ; % % P r e p a i r the f i g u r e figname = ' VFCompensation '; h _ V F C o m p = f i n d o b j (0 , ' Name ' , [ figname , ' _ R e s u l t s ' ]) ; if i s e m p t y ( h _ V F C o m p ) h _ V F C o m p = f i g u r e ( ' P o s i t i o n ' , ... [50 50 4 5 2 * 1 .8 2 5 7 * 1 .5 ] , ... ' Name ' , [ figname , ' _ R e s u l t s ' ] , ... ' N u m b e r T i t l e ' , ' off ' ) ; end figure ( h_VFComp ); set ( h_VFComp , ' D e f a u l t A x e s F o n t S i z e ' ,14 , ... ' D e f a u l t A x e s F o n t N a m e ' , ' T i m e s New R o m a n ' ) ;

25 26 27 28

%% Factor Calculation F R E Q = [1 2 : 2 : 1 0 15 20 30 40 6 0 : 5 : 1 0 0 ] ; % F r e q u e n c y v e c t o r WS = 2* pi * FREQ ; % F r e q u e n c y v e c t o r in ISO

29 30 31 32

%% Voltage compensation Kr = Rs. /(( Lsl + Lrl ) . * WS ) ; r_boost = sqrt ( Kr + sqrt ( Kr. ^2+1) ) ;

% K factor % r_factor

33 34 35 36 37

plot ( FREQ , r_boost , ' L i n e W i d t h ' ,2) ; grid on ; hold on ; t i t le ( 'V / F volt drop c o m p e n s a t i o n ' ) ; y l a b e l ( ' Boost Ratio [ -] ' ) ; x l a b e l ( ' Freq [ Hz ] ' ) ;

Figure 2.18 shows the impact to the V /F strategy with/out the voltage boost compensation at the stator frequency f s of 25, 50, 75 Hz respectively. The upper diagram represents the results without the voltage boost, while the bottom diagram gives the results with the compensation. It is clear that • Without the voltage boost, the peak machine torque drops considerably at low frequencies. The lower the frequency, the more significantly the torque drops; • The compensation to the drop in peak torque varies with frequencies. More compensation is applied to lower supply frequencies, ending up with the peak torque having increased by 21% with f s = 50 Hz and 36% with f s = 25 Hz for instance. Nevertheless, according to Eq. (2.31), the slip speed at which the machine torque reaches its peak value is invariant irrespective of the stator frequency. In case that the slip speed ωsp is small, the torque-slip speed profile is approximately linear and the machine torque can be simplified as em (ωsp ) = 1.5N p T Rr∗



Vs ωs

2 ωsp

(2.35)

Therefore, the slip-speed to torque characteristics is more or less independent of the stator frequency. This feature is greatly beneficial for both performance analysis and control of the electric machines. Figure 2.19 illustrates and validates this analysis. It can be seen that the slip speed to torque relationship at the respective frequency of 25, 50, 70 Hz remains nearly unchanged. The use of the slip speed to

50

2 Equivalent Circuit Modelling and Analysis

Fig. 2.18 V/F torque curves Slip speed to torque curve

200

250

100

200

0

fs=25 fs=50 fs=75

150

-100

100

-200

50

-300 -400

V/F machine torque - With voltage boost

300

Torque [Nm]

Torque [Nm]

300

0

-300

-200

-100

0

100

200

Slip Speed [rad/s]

(a) fs = 100 Hz

300

400

0

50

100

150

200

250

300

350

400

Slip speed [rad/s]

(b) fs = 25, 50, 75 Hz

Fig. 2.19 Slip speed to torque curves

regulate the machine torque is the fundamental principle of the scalar control of ACIMs, which will be elaborated in Chap. 5. Though in vector torque control of ACIMs, machine flux and torque are regulated seemingly irrespective of the slip speed, the slip speed is however indirectly used to control the machine torque, see Chap. 6 for further evidence.

2.3 System Performance Analysis

51

2.3.2 Thevenin’s Theorem Application to Torque Profile Analysis The slip or slip speed dependent torque analysis is based on the approximated equivalent circuit model as shown in Fig. 2.14. It is understandable that this approximation introduces certain errors in the torque analysis. As shown in Fig. 2.20, the blue curves represent the slip-to-torque results of the simplified equivalent circuit model, while the black curves are those of the nonsimplified model (assuming Rc = ∞). For instance, at f s = 100 Hz, the maximum torque Temmax is 246 Nm for the simplified model but 222 Nm for the non-simplified model. Thus, there is a substantial error of 26 Nm due to the simplification, accounting for 12% of the maximum torque from the detailed model. Fortunately, the shape of the torque-slip profiles remains consistent with minor change. In particular, the critical slip s ∗ , where the maximum torque is arrived at, remains unchanged. This conclusion is true regardless of the operating frequency, be it 50 Hz or 100 Hz. In order to improve model accuracy with the maximum torque analysis, Thevenin’s theorem is introduced and applied [2]. Theorem 2.1 Thevenin’s theorem: Any linear electrical network containing only voltage sources and impedances can be replaced at terminals by an equivalent combination of a voltage source Vth in series with a impedance Z th . Thevenin’s theorem is illustrated by Fig. 2.21, where Vs is the voltage source; Z 1 , Z 2 and Z are impedances. Vz and Iz are respectively the voltage and current applied to Z . Z 1 and Z 2 must be linear components, but Z can be any single component or electric networks. By applying Thevenin’s theorem, the partitioned electric network consisting of Vs , Z 1 and Z 2 is reduced to an equivalent voltage source Vth in series with an equivalent impendance Z th . Mapping to the unsimplified equivalent circuit model of induction motors as seen in Fig. 2.13, these impedances are correspondingly

Slip to torque curve

300

Slip to torque curve Simple model Full model

Simple model Full model

300

200 100

Torque [Nm]

Torque [Nm]

200

0 -100

100 0 -100 -200

-200 -300 -300 -1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

Slip [-]

(a) fs = 100 Hz

Fig. 2.20 Slip to torque curve comparison

1

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

Slip [-]

(b) fs = 50 Hz

0.4

0.6

0.8

1

52

2 Equivalent Circuit Modelling and Analysis

Fig. 2.21 Thevenin’s theorem illustration

Z 1 = Rs + jX sl Z 2 = Rc + jX M ∗ Z = Rr∗ /s + jX rl0 ∗ We recap that the stationary or blocked rotor leakage reactance X rl0 = ωs L rl . X rl0 is the reactance referred to the stator side, which has been elaborated in Sect. 2.2. Besides, it is assumed that the core loss resistance Rc = ∞ in this subsection.

Proof The proof of Thevenin’s theorem concentrates on keeping Vz the same before and after applying the thevenin’s transformation. Iz stays the same as long as Vz remains unchanged. With the equivalent model Vz =

Z Vth Z + Z th

On the other hand, Vz is derived by the following equations from the original model prior to transformation. I = and

Z + Z2 Vs Vs = Z ||Z 2 + Z 1 Z Z 2 + Z 1 (Z + Z 2 ) Vz = Vs − Z 1 I = = =

Z Z2 V Z Z 2 +Z 1 (Z +Z 2 ) s Z Z2 V Z (Z 1 +Z 2 )+Z 1 Z 2 s Z2 Z Vs Z Z Z + 1 2 Z 1 +Z 2 Z 1 +Z 2

Comparing these two Vs expression, it concludes Vth =

Z2 Vs Z1 + Z2

2.3 System Performance Analysis

53

and Z th = Z 1 ||Z 2 =

Z1 Z2 Z1 + Z2 

With Thevenin transformation, it is clear that: (1) The equivalent impedance Z th is the resultant as if the impedances Z 1 and Z 2 are connected in parallel; (2) The equivalent terminal voltage Vth is equal to the Vs fraction across Z 2 as if Z 1 and Z 2 are in series. Figure 2.22 shows the application of Thevenin’s theorem to the equivalent circuit analysis. The equivalent stator resistance Rs∗ and leakage inductance L ∗sl are determined by X 2M Rs∗ = 2 Rs (2.36) Rs + (X sl + X M )2 L ∗sl =

Rs2 L M /L sl + (X sl + X M )X M L sl Rs2 + (X sl + X M )2

(2.37)

and the terminal voltage Vs∗ becomes Vs∗ = 

XM Rs2 + (X sl + X M )2

Vs

(2.38)

Thevenin transformation is an equivalent conversion, and hence there is no associated accuracy loss. Torque profile of the Thevenin model will be exactly the same as those from the unsimplified model as shown in Fig. 2.20 (by black curves). Compared to the ‘ ’-shaped model in Fig. 2.15, the model is exactly in the same formation, except that the parameters Rs and L sl are updated to Rs∗ and L ∗sl , and phase voltage is amended to Vs∗ . Therefore, the analysis made in the Sect. 2.3.1 holds true from a Fig. 2.22 Thevenin’s theorem application to the equivalent circuit model

Rs

* Lrl

Lsl

Ir*

Is

* Rr/s

XM

Vs IM

Rs*

* Lsl

* Lrl Ir* * Rr/s

Vs*

54

2 Equivalent Circuit Modelling and Analysis

Fig. 2.23 Thevenin factors

1.2 R-factor L-factor V-factor

Coefficient gain [-]

1.1 1 0.9 0.8 0.7 0.6 10

20

30

40

50

60

70

80

90

100

Frequency [Hz]

qualitative point of view. Furthermore, the factors of Rs∗ /Rs , L ∗sl /L sl and Vs∗ /Vs are frequency dependent, and are given in Fig. 2.23. The voltage factor Vs∗ /Vs (or V-factor) and resistance factor Rs∗ /Rs (or R-factor) are almost constant, being 0.92 and 0.85 respectively. By contrast, the inductance factor L ∗sl /L sl (or L-factor) is more sensitive at low frequencies, but then stabilizes to around 0.925 at high frequencies. According to the R & L factors, the optimal slip from the Thevenin model and the simplified model as determined by Eq. (2.27) should be different. However, since this difference is rather small, it is unnoticeable as shown in Fig. 2.20. The voltage factor is therefore the main cause leading to a positive estimation error of the peak torque with the simplified model. 1 = 1.18 0.922 It indicates that the V-factor alone introduces a 18% torque error. As per Eq. (2.30), the torque ratio between the simple and Thevenin models is given as  2 ∗ ∗ L sl + L rl Vs T rat = ∗ ∗ Vs L sl + L rl In particular, T rat = 1.12 for a given case of f s = 100 Hz. This matches the previous analysis in Fig. 2.20 and explains the torque error of the simplified model. Thereby, the maximum torque expression in Eq. (2.30) shall be updated as  Temmax

=

Vs∗ ωs

2

1.5N p ∗ L ∗sl + L rl

(2.39)

Once the machine parameters are fixed, the maximum torque is not determined by Vs∗ (or subsequently by Vs ), but rather by Vs∗ /ωs (or in turn by Vs /ωs ). This ratio is not arbitrary, but it is decided by the maximum permissible flux Ψmax of the machine. Therefore, as long as Vs /ωs is designed to have the value of Ψmax at the air gap, the torque of this updated simplified model could be sufficiently accurate. In general,

2.3 System Performance Analysis

55

V /ω represents the air-gap flux linkage. In practice, Vs∗ /ωs is a more realistic measure than Vs /ωs . In either way, these V /ω ratios are commonly used to regulate the air-gap flux linkage by the V /F control. Therefore, in order to get the targeted air-gap flux linkage, if Vs /ωs is adopted, the V-factor shall be taken into account for boosting the terminal voltage. On the same basis, when the flux limit is a control criteria, the V-factor is then a useful indicator of the permissible phase voltage (or permissible V/F ratio). However, in order to reach the maximum torque, apart form the voltage boost as given by Fig. 2.17 (which is mainly for the system to have desired current), further boost by the inverse of V-factor as in Fig. 2.23 is required (which is mainly for the system to have the desired flux). Nevertheless, it is worth noting that determining the peak torque of the machine is ultimately a fine calibration process. The permissible flux linkage at the air gap is usually determined by the rated voltage of the machine at the rated frequency as (2.40) Ψmax = rv Vs rt / f s rt in which the V-factor rv is defined as Vs∗ /Vs from Eq. (2.38), and rv = 0.92 is adopted in this chapter. For the given example machine, the maximum flux linkage is Ψmax = 0.2 Wb.13 The peak torque of the machine is limited by this permissible flux. Nonetheless, it is noteworthy that short-time overload to surpass this flux linkage limit for transient torque boosting is possible. Bear in mind that Ψmax defined in Eq. (2.40) does not yet result in full neutralization (or utilization) of air-gap flux in terms of deriving maximum possible machine torque. This is analyzed in the following. Given the flux linkage limit is such an important factor, further analysis is worthwhile. According to Fig. 2.21 and the proof process of Thevenin theorem, the magnetizing voltage VM is given by Z VM = Vz = Vth Z + Z th Based on this equation, the normalized magnetizing voltage (solid red line) with reference to the phase voltage and the air-gap flux linkage (dash-dotted blue line) at stator frequency of f s = 100 Hz are given in Fig. 2.24. Also given in the figure is the normalized machine torque (dashed black line) for comparison. It is evident that • The maximum flux linkage is derived at s = 0. At s = 0, the equivalent load resistance on the rotor side becomes infinitely large, leading to VM ≈ Vs rt ; • The minimum flux is at s = 1. The magnetizing inductance is shunted by the rotor impedance. This results in the minimum voltage division on the magnetizing branch; √ • The magnetizing voltage at s = s ∗ is VM = Vs rt / 2. This observation will be further proved in Chap. 3 when MTPF is investigated. Thus, the maximum air-gap flux linkage at rated voltage and frequency only appears at no-load operation. As long as torque is generated, the air-gap flux linkage 13

This is a rms value, which shall not be confused with the peak value used in the vector torque control in Chap. 6.

56

2 Equivalent Circuit Modelling and Analysis 0.2

Torque [-] / EMF [-]

1

0.8

0.18

0.6

0.16

0.4

0.14

0.2

0.12

Flux linkage [Wb]

Fig. 2.24 Magnetizing voltage and air-gap flux characteristics

0.1

0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Slip [-]

reduces sharply. At the maximum machine torque, the air-gap flux linkage is only around 70% of its peak value. Since there is a significant flux reservation (around the maximum torque), the phase voltage can be further boosted to neutralize the air-gap flux and to promote the maximum machine torque. That being said, the maximum torque could be doubled at the most.

2.3.3 Speed Torque Characteristics In Sect. 2.3.1, the advantage of using the slip speed to torque characteristics instead of the slip to torque characteristics is demonstrated. Dependency of the torque on the rotor speed will be analyzed in this subsection. For a given stator frequency ωs , torque Tem is a function of slip s as per Eq. (2.25). Furthermore, the slip s is uniquely defined by ωr according to Eqs. (2.1) and (2.14). Hence, torque Tem is a function of rotor speed ωr . Figure 2.25 gives a profile of torque versus rotor speed based on the parameters as in Table 2.1. Investigating Fig. 2.10 with Fig. 2.25 together, the following comments are made: Braking.

Motoring. Synchronization. Generation.

ωr < 0, this is when the direction of rotor speed is opposite to the stator speed. The machine torque brakes the rotor and tends to bring the rotor speed back to the same direction as that of the stator. 0 < ωr < ωsm , this is when the synchronous stator speed exceeds the rotor speed. The machine torque accelerates the rotor. ωr = ωsm , this is when the rotor is synchronized with the stator. At this case, no motor torque is yielded. ωr > ωsm , this is when the rotor speed exceeds the synchronous stator speed. The machine torque changes its polarity into negative and tends to decelerate the rotor. The power flows from the rotor side to the stator side. It is the generating case.

2.3 System Performance Analysis

57

We

Wr

Peak Motoring Starting Torque We

We

Wr

Wr

s=2

Peak Generation

Slip s

s=1

Motoring

Braking

s=0

s=-1 Generation

Fig. 2.25 Rotor speed to torque curve at f s = 100 Hz

Combining Eqs. (2.28) and (2.33) results in Eq. (2.30). This seemingly suggests that this theoretical peak torque of electric machine can always be achievable regardless of motor speed. This however is untrue, as otherwise the phase voltage will eventually exceed its limit with a rising motor speed. Once Vs grows along with the stator frequency upto the physical limit of Vs max , it needs to be capped. As a consequence, the peak torque and the system flux linkage must decrease. This phenomenon is called flux weakening. The achievable maximum flux linkage is subjected to the voltage limit, arising as

Φ M (ωs ) =

if Ψ Mmax ωs < rv Vs max Ψ Mmax , max rv Vs /ωs , Otherwise

(2.41)

Hence, once Vs reaches the limit, the flux linkage becomes inversely proportional to the stator frequency ωs . Similarly Temmax (ωs ) =

⎧ ⎨ ⎩

1.5N p ∗ L sl +L rl 1.5N p ∗ L sl +L rl

 

2

, i f Vs < Vs max  2 rv Vs max , i f Vs = Vs max ωs rv Vs ωs

(2.42)

However, caution must be taken here. Once in the flux weakening region, the peak machine torque is not inversely proportional to the stator frequency, but to the

58

2 Equivalent Circuit Modelling and Analysis

Motor torque

250

Torque [Nm]

200

150

100

50

0 0

100

200

300

400

500

600

Rotor speed [rad/s] Fig. 2.26 Peak torque profile, f s = 100 Hz, Vs = 134 V

square of the frequency; Nevertheless, the peak power is inversely proportional to the stator frequency. That being said, in the flux weakening region, 2 = cnst Temmax ωsm

(2.43)

Figure 2.26 is given to validate these analyses, demonstrating that: (1) With the V/F control strategy and the voltage boost at low frequencies, the peak torque can be regulated to keep constant, as long as the stator frequency is below the rated frequency or equivalently, the rotor speed is below the base speed; (2) The peak torque will drop at a rate of (ωb /ωr )2 above the base speed. The blue curve in this figure is the peak machine torque at different stator speeds, governed by Eqs. (2.41) and (2.42), while the black curves are a family of torque profiles versus rotor speed at each given stator speed. It is worth noting that the peak torque curve is well matched with the envelope curve of rotor speed to torque profiles below the based speed. However, the envelope curve is slightly above the peak torque curve beyond the based speed. As a consequence, the maximum machine torque could be higher than the peak torque defined by Eq. (2.42) in the field weakening region, in particular where the stator speed gets over but still near the base speed. Matlab script as in Listing 2.3 is provided for Fig. 2.26. To summarize, the torque profile versus slip speed is helpful for understanding the scalar control of induction machines, while the torque curves versus rotor speed enhance the understanding of operation of induction machines in general.

2.3 System Performance Analysis

59

Listing 2.3 Speed torque profile calculation 1 2 3 4

Freq ws Vdc vs

= = = =

100; 2* pi * Freq ; 330; Vdc / sqrt (6) ;

5 6

% For Y c o n n e c t i o n and SPWM % Vpn = Vdc * sqrt (3) /3 % Vpn_rms = Vpn / sqrt (2)

7 8 9

F r e q _ R a t e = Freq ; Vs_Rate = vs ;

% Rated F r e q e n c y ; % Rate p h a s e v o l t a g e

10 11

Motor_Parameters_Eqv ;

12 13 14 15 16

% % T h e v e n i o n p a r a m e t e r c a l c at r a t i n g f r e q u e n c y Xsl = Lsl * ws ; Xm = LM * ws ; X_SQR = Rs ^2+( Xm + Xsl ) . ^2;

17 18

r_v

= Xm / sqrt ( X_SQR ) ;

19 20 21 22 23 24 25 26 27 28 29 30 31

% % P r e p a i r the f i g u r e figname = ' SpdVsTqCalc '; h _ s p d T q f i g = f i n d o b j (0 , ' Name ' , [ figname , ' _ R e s u l t s ' ]) ; if i s e m p t y ( h _ s p d T q f i g ) h _ s p d T q f i g = f i g u r e ( ' P o s i t i o n ' , ... [50 50 452*1 .8 257*1 .8 ] , ... ' Name ' , [ figname , ' _ R e s u l t s ' ] , ... ' N u m b e r T i t l e ' , ' off ') ; end figure ( h_spdTqfig ); set ( h_spdTqfig , ' D e f a u l t A x e s F o n t S i z e ' ,14 , ... ' D e f a u l t A x e s F o n t N a m e ' , ' Times New Roman ' ) ;

32 33 34 35

% % F r e q u e n c y and v o l t a g e s e t u p FREQ = [1: 2: 200]; WS = 2* pi * FREQ ;

36 37 38 39 40 41 42

%% Voltage Boost Kr = Rs. /(( Lsl + Lrl ) . * WS ) ; % K factor r_boost = sqrt ( Kr + sqrt ( Kr. ^2+1) ) ; % r_factor % VS = min ( V s _ R a t e / F r e q _ R a t e . * FREQ. * r_boost , V s _ R a t e ) ; VS = min ( V s _ R a t e / F r e q _ R a t e . * FREQ. * r_boost , V s _ R a t e ) * r_v ;

43 44 45 46 47

for II =1: l e n g t h ( F R E Q ) ws = WS ( II ) ; vs = VS ( II ) ;

48 49 50 51

% % C a l c u l a t i o n based on e q u i v a l e n t c i r c u i t X_term = ( Lsl + Lrl ) * ws ; X _ s q u a r e = X _ t e r m ^2;

52 53 54

W_RE = [0:1: ws ws ]; S = ( ws - W_RE ) / ws ;

% Rotor e l e c t r i c s p e e d v e c t o r % Slip vector

55 56 57

% This part c o n t a i n s v e c t o r r e l a t e d c a l c u l a t i o n R_term = ( Rs. * abs ( S ) + Rr ) ;

60 58 59 60

R_square I_term Te

2 Equivalent Circuit Modelling and Analysis = R _ t e r m . ^2; = vs ^2 . * S. /( R _ s q u a r e + X _ s q u a r e . * S. ^2) ; = 3* P o l e _ p a i r . * I _ t e r m * Rr / ws ;

61 62 63

W_RM = W_RE / P o l e _ p a i r ; plot ( W_RM ,Te , 'k ' , ' L i n e W i d t h ' ,0 .5 ) ; hold on ;

64 65

end % II

66 67 68 69 70 71

% % Max T o r q u e C a l c W s _ R a t e = F r e q _ R a t e *2* pi ; V W _ R a t e = V s _ R a t e * r_v / W s _ R a t e ; L_lk = Lsl + Lrl ; T_max = 1 .5 * P o l e _ p a i r * V W _ R a t e ^2/ L_lk ;

72 73 74 75 76

% % Max T o r q u e P r o f i l e Calc T p e a k = T _ m a x . * min (( F r e q _ R a t e . / F R E Q . / r _ b o o s t ) . ^2 ,1) ; Wsp = Rr / L_lk ; WR = max ( WS - Wsp ,0) ;

77 78 79 80 81 82 83 84 85 86 87

% % Plot the r e s u l t s % C o n v e r t from elec to mech speed WR_m = WR / P o l e _ p a i r ; plot ( WR_m , Tpeak , 'b ' , ' L i n e W i d t h ' ,2) ; title ( ' M o t o r t o r q u e ' ) ; y l a b e l ( ' T o r q u e [ Nm ] ' ) ; x l a b e l ( ' Rotor speed [ rad / s ] ' ) ; Tmax = ceil ( T_max /50) *50; axis ([0 ws / P o l e _ p a i r 0 Tmax ]) ; grid on ; hold off ;

2.3.4 Electric Machine Operation Analysis Based on Fig. 2.26, it is obvious that the machine could be operated in two ranges. One is referred to as the constant torque range, which is when the rotor speed is below the base speed ωb . The other range is above the base speed, where the peak power is inversely proportional to the speed. However, it is well known that the electric machine can also be operated in a constant power range during flux weakening. To see how this works, Fig. 2.27 is given, where the blue solid curve is the peak torque while the blue dashed curve the continuous torque, red sold and dashed curves are respectively the peak and continuous power. The peak torque is a short time machine torque. Otherwise, temperature of the inverter and motor will rise quickly under such operation, making it unsustainable. In principle, a machine can operate anywhere within the peak torque envelope. Nonetheless, there exists one torque level that can be maintained in compliance with the thermal constraint. This torque level is the so-called continuous torque or steady-state torque, which is illustrated in Fig. 2.27. In the flux weakening region, the continuous torque is controlled by regulating the slip speed so that the machine power maintains constant. This is known as constant power operation. However, this operation

2.3 System Performance Analysis

61

Peak Torque

Peak Power

Cont Torque

Cont Power

Constant Torque Zone

ωb

Constant Power Zone

ωc

Power decrease Zone

Fig. 2.27 Machine operation zones

is rather a controlled machine behaviour. The constant power is sustained upto the critical speed ωc . Above ωc , the machine goes into the intrinsic power decrease mode. As depicted in Fig. 2.27, the electric machine has the following operation regions: • Constant torque region. Tem = cnst in this region. This is when ωr < ωb ; • Constant power region. Pem = cnst in this region. This is when ωb < ωr < ωc ; • Power decrease region. Pem ωsm = cnst in this region. This is when ωr > ωc and the continuous torque overlaps with the peak torque. While the characteristics of the machine torque and power at each region are clear, the characteristics of current, voltage, flux linkage and slip are not equally evident. This subsection focuses on the analysis in this regard. Recall Eq. (2.21) I2 Tem = 3N p Rr∗ sω s 2 = 3N p Rr∗ ωIsp It implies that as long as the slip speed ωsp stays the same, the demand for current is invariant for the same torque at different operating frequencies f s (or ωe ), regardless of the operating range, be it constant torque or flux weakening range. Bear in mind that the current I here is the rotor current referred to the stator circuitry. On the other hand, the magnetizing current I M , which is determined by Vs∗ and ωs , will however vary with the frequency f s during flux-weakening region.

62 400

Torque Current

350

Torque [Nm] / Current [A]

Fig. 2.28 Torque and rotor current profile at f s = 100 Hz and Vs = 134 V

2 Equivalent Circuit Modelling and Analysis

300 250 200 150 100 50 0 0

50

100

150

200

250

300

350

Rotor Speed [rad/s]

Figure 2.28 shows the rotor current in relation to the rotor speed. Also shown is the machine torque Tem for comparison. Key points can be summarized as • When the rotor is synchronized with the stator, the rotor current I = 0; • When slip speed is small, the rotor current I depends almost linearly on the slip speed. Since Tem has a similar characteristic, it is true that Tem relies linearly on I approximately. Thus, it is reasonable to call rotor current I (viewed at the stator side) as the torque current; • Once the slip speed becomes greater across the threshold of the breakdown torque, Te starts decreasing but I keeps increasing though it becomes saturated when the slip speed is really large. This implies that the system efficiency and the power factor drop significantly. Hence, it is more preferable to operate the electric machine in the range with small slip speed. This is because not only there is benefit of approximate linearity between the power, current and the slip speed, but the system efficiency and power factor are better in this range as well. According to Eq. (2.35), there are different means to regulate the machine torque M1: Regulating the slip speed ωsp only. By this method, Vs / f s is normally constant or the flux linkage Ψ M is kept constant. The machine torque relies almost linearly on ωsp . As just-analyzed, the torque current I is also linearly dependent on ωsp . The fundamental of this method is to control the machine torque by regulating the torque current I ; M2: Regulating the phase voltage Vs only. By this method, the slip speed ωsp remains constant. The varying Vs results in changing both the magnetizing flux linkage Ψ M and the torque current I at the same time.14 Hence, machine torque is very sensitive wrt phase voltage. The essence of this method is to control the machine torque by regulating both the magnetizing current I M and the torque current I ; 14

By the voltage control alone, the air-gap flux and torque current are coupled to derive the demanded machine torque. By contrast, the vector control decouples the air-gap flux and torque current to regulate torque.

2.3 System Performance Analysis

63

M3: Regulating both the voltage and the slip speed. This is the combination of M1 and M2. That being said, the general principle of governing the machine torque is by regulating either or both of the torque current I (or Ir∗ ) and the magnetizing current I M . Scalar torque control presented in Chap. 5 regulates I only, which is the reason why it is called scalar control. While, vector control stated in Chap. 6 controls both I M and Ir∗ simultaneously and independently in a decoupled way. As shown in Fig. 2.27, with the method M1, the constant torque results in a constant slip speed ωsp . However, the slip s decreases, as the rotor speed ωr goes up. On the other hand, the constant power requires constantly increasing the slip speed ωsp , but results in a constant slip s. Referring to Eq. (2.21), constant current I and constant slip speed ωsp leads to a constant torque Tem while constant current I and constant slip s to a constant power Tem ωs /N p . The characteristics of voltage, flux linkage, current, slip speed, slip and power of the different operating regions for the induction machine are summarized in Fig. 2.2915 and Table 2.2. One immediate comment is added that the flux linkage in the constant torque region is only 0.14 Wb, significantly away from its peak 0.20 Wb. This is due to the fact that the peak flux linkage does√not occur at the operating point of peak torque, where s = s ∗ leading to VM = Vs / 2.

2.3.5 Phasor Diagram Based on the equivalent model as in the upper part of Fig. 2.14, a phasor diagram16 is given in Fig. 2.30. For the sake of simplicity, the rotor part of the phasor diagram is mirrored to the lower part of the figure, which leaves the stator side of the phasor diagram taking the upper part. All the variables used in this figure are in the rms manner. The light blue lines represent the current vectors, while the light red lines are for the voltage vectors and cyan line is for flux linkage. For the convenience of the analysis, the following variables are defined Vrl : Vsl : Vrr : Vsr : θ: θs : θr : δ: 15

Rotor leak voltage Vrl = Ir∗ X rl∗ ; Stator leak voltage Vsl = Is X sl ; Rotor slip and resistance voltage Vrr = Is Rs /s; Stator resistance voltage Vsr = Is Rs ; Phase angle between phase current Is and magnetizing flux linkage Ψ M ; Phase angle between phase voltage Vs and phase current Is ; Phase angle between rotor voltage VM and rotor current Ir∗ ; Phase angle between magnetizing flux linkage Ψ M and rotor current Is .

Due to the numerical errors, the continuous current is not truly constant in the constant power region. 16 The use of the equivalent model as in Fig. 2.15 for phase diagram analysis is not recommended.

64

2 Equivalent Circuit Modelling and Analysis Constant Torque

Constant Power

Constant P*We

Peak Torque

Cont Torque

Cont Power

Constant Peak Current

Constant Voltage

Peak Current Phase Voltage

Cont Current

Constant Cont Current

Constant Flux

Flux Weakening

Normalized peak slip speed

Flux Normalized cont slip speed

Constant cont slip speed Constant cont slip

Fig. 2.29 Operation zone characteristics

The machine torque17 is a result of the rotor current Ir∗ rotating in the magnetic field Ψ M .18 Tem = 3N p (Ψ M × Ir ∗ ) (2.44) = 3N p Ψ M Ir∗ sin(δ) When s is really small, i.e. when the rotor is close to synchronization with the stator, the phase angle θr will become very small. Under such conditions, torque can be approximated as Tem = 3N p Ψ M Ir∗ . Also, if Ic is negligible, which is the case when Is and Ir∗ both are relatively large and which is normally true, then Is sin(θ ) = Ir∗ sin(δ)

(2.45)

When using the magnitude instead of the rms value, the torque is given as Tem = 3/2N p Ψ m Ir∗ sin(δ). 18 × is the cross-product operation for vectors. 17

2.3 System Performance Analysis

65

Table 2.2 Operation region characteristics summary Operation region Characteristics Constant torque

Tem = cnst

Supporting equations  2 1.5N p max = rv Vs Tem ωs L +L ∗

cn = cnst Tem

Controlled

Pem ∝ ω cn ∝ ω Pem Vs ∝ ω Ψ = cnst

P = Tω

pk

pk

pk

rl

Vs /ωs = cnst √ Ψ = Vs /(ωs 2)

ωsp = cnst

∗ = ωsp

cn = cnst ωsp

Tem ∝

s pk ∝ 1/ω s cn ∝ 1/ω

Constant power

sl

Rr∗ ∗ L sl +L rl



rv Vs ωs

2

ωsp

cn = cnst ⇒ ω cn = cnst Tem sp ωsp = sωs 2

I pk = cnst

Tem = 3N p Rr∗ ωIsp

I cn = cnst

Tem = cnst and ωsp = cnst ⇒ I = cnst

pk

Tem ∝

1 ω2

max = Tem

2 1.5N p rv2 Vmax ∗ ωs2 L sl +L rl

cn ∝ 1/ω Tem

Controlled

Pem ∝ 1/ω cn = cnst Pem Vs = cnst Ψ ∝ 1/ω

P = Tω

pk

pk

Vs = Vs max √ Ψ = Vs /(ωs 2)

ωsp = cnst

∗ = ωsp

cn ∝ ω ωsp

Tem ∝

s pk ∝ 1/ω s cn = cnst I pk ∝ 1/ω

I cn = cnst

Rr∗ ∗ L sl +L rl 2 rv2 Vmax ωsp ωs2

cn ∝ ω T ω = cnst ⇒ ωsp ωsp = sωs

T ∝ Ψ Ir∗ pk Tem ∝ ω12 and Ψ ∝ ⇒ I pk ∝ ω1 I 2 R∗

1 ω

I 2 R∗

T ∝ sωr ⇒ P ∝ s r P = cnst and s = cnst ⇒ I cn = cnst

(continued)

66 Table 2.2 (continued) Operation region Constant Pω

2 Equivalent Circuit Modelling and Analysis

Characteristics pk

Tem ∝ cn ∝ Tem pk

Supporting equations max = Tem

1 ω2 1 ω2

2 1.5N p rv2 Vmax ∗ ωs2 L sl +L rl

pk

cn = T Tem em

Pem ∝ 1/ω

P = Tω

cn ∝ 1/ω Pem Vs = cnst Ψ ∝ 1/ω

cn Pem = Pem max Vs = Vs

pk

pk

Rr∗ ∗ L sl +L rl

ωsp = cnst

∗ = ωsp

cn = cnst ωsp s pk ∝ 1/ω s cn ∝ 1/ω I pk ∝ 1/ω I cn ∝ 1/ω

cn ωsp = ωsp ωsp = sωs s pk = s cn T ∝ΨI I pk = I cn

pk

Fig. 2.30 Phasor diagram

VM

Isr s

Ic IM

Ism

ψM

r

-VM

Therefore, Eq. (2.44) can be rewritten as Tem = 3N p Ψ M Is sin(θ )

(2.46)

Furthermore, Is can be decomposed into the components Ism and Isr . in particular, Ism = I M and Isr = Ir∗ when Ic and Vrl are neglected. Bearing in mind that Ψ M = L M I M , this leads to the following expression of the machine torque Tem

2.3 System Performance Analysis

67

Tem ≈ 3N p L M I M Ir∗ ≈ 3N p L M Ism Isr

(2.47)

Thus, at relatively high speed and at relatively high machine torque, the machine torque can be well regulated by the appropriate Is components, namely the magnetizing field component Ism and rotor current component Isr . This is the fundamental principle for vector torque control. The power injected into the machine system is given by Ps = Vs Is cos(θs ) Thus, cos(θs ) is the so-called power factor. By investigation of Fig. 2.30, it is clear that the power factor angle is affected by θr : Large θr leads to a large ( π2 − θ ), which forms a part of θs . θr is determined by both the slip s and the stator frequency ωs . A small θr requires either or both of a small slip and a low frequency; Im : A large I M results in a large ( π2 − θ ). I M is mainly determined by Vs . Thus, unnecessarily large Ψ M will reduce the power factor; Vsl : Large Vsl (relative to Vs ) brings in a large θs . This effect gets aggravated with a high supply frequency. In general, a higher power factor requires: (1) Reduction of the leakage inductance for both the stator and rotor; (2) Decrease in the magnetizing current; and (3) Operation of the machine at the small slip range, which makes the high frequency operation preferable.

2.3.6 The System Efficiency The electric machine has various losses as below Copper loss19 Iron loss: Stray loss:

Mechanical loss:

It is the loss to the stator and the rotor resistance; It consists of the eddy-current and hysteresis losses at both the stator and rotor. It is also referred to as core loss; It consists of the losses from the non-uniform and/or nonsinusoidal distribution and distortion of the current and flux, which generate high order harmonics that cause extra copper, eddy-current and hysteresis losses; Such as bearing and air drag losses.

The analysis of the cooper, iron and stray losses is the topic of this subsection. As shown in Fig. 2.31, the mechanical power Pm and the net power Pnet are results of the input power less the losses, given by

68

2 Equivalent Circuit Modelling and Analysis P Air gap Ps = Tind

Ps = Tem

sm

P Mechanical sm

Ps = Tem

rm

PNet

P s = V s Is Input Power

P Bearing P Stator Copper loss

P Rotor

P Core

Copper loss loss Stator and Rotor Hysteresis Eddy Current

PStray

Air drag

Loss Stator and Rotor Harmonics Leak flux Distortion Nonlinear

Fig. 2.31 Power flow diagram per phase

Pm = Ps − Pcls − Pclr − Pc − Pstr y Pnet = Pm − Pml

(2.48) (2.49)

where Ps : Pc : Pclr : Pcls : Pm : Pml : Pnet :

Electric power input; Core loss; Rotor copper loss; Stator copper loss; Mechanical Power; Mechanical loss; Net power output.

The electric efficiency ηe and overall efficiency η are defined as ηe = Pm /Ps η = Pnet /Ps

(2.50) (2.51)

The input power is determined by the phase voltage, current as well as the power factor20 Ps = 3 Vs , Is

(2.52) = 3Vs Is cos(θs ) The calculation of the copper losses is quite obvious and is given by Pcls = 3Is2 Rs Pclr 20

=

3Ir∗ 2 Rr∗

·, · is the inner product operation for vectors.

(2.53) (2.54)

2.3 System Performance Analysis

69

B

ψ

Eddy Current Loss V 2/R ψ V

H Hysteresis Shaded area is the loss Shaded area

ψ

2

Fig. 2.32 Hysteresis and Eddy current loss

However, the calculation of the iron losses is not straightforward, see Fig. 2.32. Hysteresis loss is determined by the surface area spanned by the B − H loop, which is again determined by both the strength of the magnetic field and the electric frequency. It is also determined by the iron material properties and the structure geometric parameters such as volume and size. In reality, it is difficult to quantify the area of the hysteresis loop due to its nonlinearity and multivalued characteristics. Based on extensive experiments, Charles Steinmetz found that the hysteresis loss n , with exponent n being in the range [1.5, 2.5] is approximately proportional to B M depending on the core material [5, 8, 10]. Here, B M is the magnetizing flux density. Though not accurate enough, we round the value of n to 2 for qualitative analysis. On the other hand, eddy current loss is proportional to the square of the induced voltage, which is the function of the flux and the electric frequency. Also, the thickness of the laminated silicon steel plays a significant part in the loss. The thinner the steel, the lower the eddy current. Pc = ke (Ψ M ωs Δe )2 + kh Ψ M2 ωs

(2.55)

in which ke is the eddy current lost constant, determined by the size and conductivity of the silicon steel; Δe the thickness of the silicon steel; kh the hysteresis loss constant, depending on the volume and reluctance of the iron. To simplify the analysis, the following expression of the system efficiency is adopted. This simplification is assuming Pnet ≈ Pm (focusing on the electromagnetic domain losses), and ignoring Pstr y . Thus, it is only valid for qualitative analysis, but not for quantitative analysis. ηe ≈

Tind ωsm −(Pcls +Pclr +Pc ) Tind ωsm

= 1− = 1−





Is2 Rs +Ir∗2 Rr∗ Tind ωsm

+

ke (Ψ M ωs Δe )2 +kh Ψ M2 ωs Tind ωsm

Is2 Rs +Ir∗2 Rr∗ Tind ωsm

+

ke (Ψ M Δe )2 ωs +kh Ψ M2 Tind /N p





where Tind is the indicated (or virtual) electric torque viewed from the phase input side, Tind > Tem for motoring, Tind < Tem otherwise. To simplify the analysis further,

Power

2 Equivalent Circuit Modelling and Analysis

Power

70

Pcl Pag

Pc Pag PEdy Pag Phys Pag

Ploss Pag

Pc Pag

Ploss Pag

Pcl Pag

Torque

Speed

Fig. 2.33 Optimal efficiency over the torque and speed range

it is assumed that Tind = Tem without loss of generality. It is now in the position to investigate the machine efficiency over two particular cases: C1: Tem = cnst and speed increases continuously; Tem = cnst implies Ir∗ = cnst as long as the torque limit is not reached by the operation method M1; Method M1 means Ψ M = cnst as long as flux weakening is not required; Current Is is complicated. At really low speeds, a small Rr∗ /s causes a relatively large phase delay θr , which will increase Is as per the phasor diagram shown in Fig. 2.30. In general, Ir∗ = cnst and Ψ M = cnst lead to Is = cnst. Copper loss Pcl , expressed as loss percentage, can then be approximated as Pcl 1 ≈ Pag ωsm



2 ∗2 Is,cnst Rs + Ir,cnst Rr∗ Tem,cnst



Similarly, core loss Pc is further simplified as Pc 2 = Ψ M,cnst Np Pag



ke Δ2e kh ωs + Tem,cnst Tem,cnst



It is clear that, copper loss, relative to air gap electric power, decreases with the speed while the core loss increases linearly with the speed. There must exist one speed where the sum of the copper loss and core loss is minimized or the system efficiency is maximized. This is illustrated by Fig. 2.33. C2: ωs = cnst and torque increases continuously. Ir∗ /Tem = cnst as long as the torque limit is not reached by the operation method M1; Method M1 means Ψ M = cnst as long as the flux weakening is not required; Is comprises of both rotor Ir∗ and magnetizing current I M .

References

71

Copper loss can then be repaired as I ∗Np Pcl ≈ r Pag Tem



∗ Rs Is ∗ Rr I + I s r ωs,cnst Ir∗ ωs,cnst



Since both Ir∗ /Tem and Is /Ir∗ are nearly constant, and Is and Ir∗ increase with Tem , the copper loss is almost linearly dependent on the machine torque. Meanwhile, core loss Pc is tidied up as ke (Ψ M Δe )2 ωs,cnst + kh Ψ M2 Pc = Pag Tem /N p Due to Ψ M and ωs are both constant, the core loss is inversely proportional to the machine torque. Once again, There will exist a torque level where the system efficiency is maximized as illustrated in Fig. 2.33. Therefore, there will be a best efficiency of the electric machine that will be obtained at an optimal operating point of speed and torque. This speed and torque can be neither too high nor too low. As a validation of this analysis, we refer to the EV traction induction machine efficiency maps reported in [3, 9]. It serves as an indication that: • • • • •

At extremely low speed, the efficiency is low; At extremely high speed, the efficiency is also low; At extremely low torque, the efficiency is low; At extremely high torque, the efficiency is also low; Optimal efficiency is in general arrived at in the range of both rather medium speed and torque.21

Nevertheless, both the copper and core loss depend on the magnetizing current I M . In particular, the field strength plays a critical role in core loss. Thus, if no high torque is requested, it is beneficial to have a weaker electromagnetic field from the system efficiency point of view. However, by doing so, the trade-off is the sacrifice of system dynamic performance. This will be analyzed in the next chapter.

References 1. Bose BK et al (2002) Modern power electronics and AC drives, vol 123. Prentice hall Upper Saddle River, NJ 2. Brittain J (1990) Thevenin’s theorem. IEEE Spectr 27(3):42 3. Gundogdu T, Zhu ZQ, Chan CC (2022) Comparative study of permanent magnet, conventional, and advanced induction machines for traction applications. World Electric Veh J 13(8):137–154 4. Lin WM, Su TJ, Wu RC (2011) Parameter identification of induction machine with a starting no-load low-voltage test. IEEE Trans Ind Electron 59(1):352–360 21

IM motors have good efficiency at high speeds, attributed to a low core loss compared to the PM motors.

72

2 Equivalent Circuit Modelling and Analysis

5. Melkebeek JA (2018) Electrical machines and drives. Springer 6. Van der Merwe C, Van der Merwe F (1995) A study of methods to measure the parameters of single-phase induction motors. IEEE Trans Energy Convers 10(2):248–253 7. Moons C, De Moor B (1995) Parameter identification of induction motor drives. Automatica 31(8):1137–1147 8. Nam KH (2018) AC motor control and electrical vehicle applications. CRC Press 9. Popescu M, Goss J, Staton DA, Hawkins D, Chong YC, Boglietti A (2018) Electrical vehicles– practical solutions for power traction motor systems. IEEE Trans Ind Appl 54(3):2751–2762 10. Sen PC (2021) Principles of electric machines and power electronics. Wiley 11. Toliyat HA, Levi E, Raina M (2003) A review of RFO induction motor parameter estimation techniques. IEEE Trans Energy Convers 18(2):271–283 12. Trzynadlowski AM (2000) Control of induction motors. Elsevier

Chapter 3

Rotating Frame Modelling and Analysis

3.1 Vector Representation of Rotating Fields The equivalent model in Chap. 2 is only valid for steady-state characteristics analysis. However, the transient and/or dynamic performance is equally important. The dynamic modelling of three-phase electric machines is somewhat complicated as the stator and rotor voltage, current and flux are distributed field variables which are also time-varying. The dynamics model is to look at the interaction between these both space and time-dependent variables. For this end, the vector representation of sinusoidal variables either in time or in space is adopted as the main tool in this chapter for the modelling and analysis of electric machines. As shown in Fig. 3.1, a sinusoidal signal with frequency ω can be modelled as a vector in a complex space rotating with a speed of ω. The projection of this vector to the real axis produces a cosine signal, while the projection to the imaginary axis is a sine signal, according to the Euler equation ejθ = cos θ + j sin θ Thus, the superposition and multiplication of sinusoidally distributed functions become simplified. Figure 3.2 illustrates the phase current waveform in the time domain, and the spatial distribution waveforms of current density and magnetic field density for Phase A. It is noteworthy that the stator is represented by concentrated windings, which are usually distributed in reality. We also recap the convention that a positive phase current is referred to a current flowing from the external into the phase terminal and a positive magnetic flux is when it points towards the positive direction along the corresponding phase magnetic axis.1 That being said, the current flow 1

For the flux density distribution expressed along the stator circumference, flux signals are positive when flowing towards the motor coils, while negative when exiting, see Fig. 3.2. © Springer Nature Switzerland AG 2024 S. Shen and Q.-z. Chen, Practical Control of Electric Machines for EV/HEVs, Lecture Notes in Electrical Engineering 1064, https://doi.org/10.1007/978-3-031-38161-4_3

73

74

3 Rotating Frame Modelling and Analysis Img sin( ) sin( )

Real

cos( )

t

Time

t

t

=

e j = cos( ) + j sin( )

cos( )

Time

Fig. 3.1 Euler representation of a rotating vector in space Current

A

Flux density Field intensity

A’

t

t

C B

C

B

t

a

Time

A

B’

C’

B

A’

C

B’

A

Space

C’ t

A

Case

Current density

B

t

A’

C B B’ C’

A’

Ia

A Ia

t A

B’

C’

B

A’

C

B’

A

Space

C’ C

A

t

Fig. 3.2 Current and flux density distribution diagram for Phase A

orientation is perpendicular to the plane of the paper, while flux is in the radial direction viewed from the centre axis of the motor cross section. In the bottom-center schematic diagram of the current density distribution, which is illustrated as the wave along the stator circumference, the density distribution wave with arrows pointing inwards radially towards the circumference indicates a positive current flow (current flows into the plane of the paper); vice versa, the wave with arrows pointing outwards from the circumference represents a negative current flow (current flows out from the plane of the paper). Bear in mind that this convention of the representation will be adopted throughout this chapter.

3.1 Vector Representation of Rotating Fields

75

In the bottom-left diagram, the current flows into the phase winding. It then arises as a positive current vector pointing to the positive direction along the winding axis (A-axis). Equally, this is the direction of the induced magnetic field and the MMF. The phase current is a sinusoidal function in time, while the current density distribution wave and magnetic field density distribution wave are sinusoidal circumferentially in space. The current and flux density at their respective peak values in space are shown in the figure when the current takes the peak value in time. At different time instant other than the one indicated in Fig. 3.2, the phase current will change, so do the values of the current and flux densities, but the distribution shapes of these densities remain unchanged except the amplitude. Furthermore, the flux (or MMF) density waveform, since it is a spatial integration of current density, shifts from the current density waveform by 90◦ in space abiding by the right-hand rule and it is also a sinusoidal waveform. Caution must be taken on the flattened view of the flux density waveform as in the upper-right diagram, which is derived by unrolling the anticlockwise stator circumference from its bottom point and having the radial distribution waveform projected from the perspective of an orthogonal coordinate system. However, the vector current Ia for phase A, representing the current density distribution, is defined as a vector pointing to the right according to the aforementioned convention. This will lead to the vector a of the magnetic field density2 being as a = κa Ia where κa = Na /R is a structure parameter that is largely determined by the turns Na of the winding and by the reluctance R of the flux path. The phase current Ia is given by Ia = rel(ejωs t )Is in which Is is the amplitude of the phase current vector and ωs the electric frequency. This convention of the vector definition greatly simplifies the modelling for three phase electric machine dynamics. The waveforms of the current and magnetic field density of the three phase windings are the vector sum of each individual. This offers an apparent benefit by representing the distributed field with a vector. Bearing in mind that current and flux vectors have 120◦ difference in space, which is how three phase windings are configured, the phase current vector therefore becomes Is =

2

 2 2 2 Ia + Ib e+j 3 π + Ic e−j 3 π 3

(3.1)

MMF and flux waveform is more step-wised sinusoidal, which will introduce high order harmonics. Also, since the phase flux is coupled, this simple expression is only true when there is no flux generated by other phases and by the rotor.

76

3 Rotating Frame Modelling and Analysis

The factor 2/3 in above expression is to normalize the resultant electric density field in an amplitude-invariant manner. The purpose of such a normalization will become clear at the later part of this section. The phase currents are further given by Ia = rel(ejωs t )Is 2 Ib = rel(ej(ωs t− 3 π) )Is 2 Ic = rel(ej(ωs t+ 3 π) )Is

(3.2)

As already demonstrated in Chap. 2, the three-phase winding fed with the balanced sinusoidal AC current generates a rotating electric field and in turn a rotating magnetic field. This is demonstrated again by using the vector representation approach. Combining Eqs. (3.1) and (3.2) yields Is = Since

     2   jωs t  j0 2 2 2 2 Is rel e e + rel ejωs t e−j 3 π e+j 3 π + rel ejωs t e+j 3 π e−j 3 π 3 rel(ejωs t e−j 3 π ) = − 21 cos(ωs t) + 2 rel(ejωs t e+j 3 π ) = − 21 cos(ωs t) − 2

√ 3 √2 3 2

sin(ωs t) sin(ωs t)

the expression in the bracket can be further decomposed into real and imaginary parts as √

♣r el = cos(ωs t) − 21 [− 21 (cos(ωs t) + 23 sin(ωs t)] − 21 [− 21 (cos(ωs t) − = 23 cos(ωs t) √ √ √ + 23 [− 21 (cos(ωs t) + 23 sin(ωs t)] − 23 [− 21 (cos(ωs t) − ♣img = = 23 sin(ωs t)

√

3 2

sin(ωs t)]

√ 3 2

sin(ωs t)]

Therefore, Eq. (3.1) is nicely reduced to as below Is = Is ejωs t

(3.3)

It is indeed a rotating vector in the complex space as shown in Fig. 3.1. The need for the normalization factor 2/3 is now also appreciated. Likewise, the stator voltage is the vector sum of each phase as Vs = in which

 2 2 2 Va + Vb e+j 3 π + Vc e−j 3 π 3 Va = rel(ej(ωs t+θv ) )Vs 2 Vb = rel(ej(ωs t+θv − 3 π) )Vs 2 Vc = rel(ej(ωs t+θv + 3 π) )Vs

This rotating voltage vector is then expressed as

(3.4)

3.1 Vector Representation of Rotating Fields B

77 B

B

Ib

Ic

Ic Ia

A

Ia

A

Ia

C

A Ic

Ib

C

Case1

C

Case3

B

Ib

Case5

B Ic Ib

Ib Ia

A

Ia

A Ic

C

C

Case2

Case4

Current

A

C

B

Time

Case1

Case2

Case3

Case4

Case5

Fig. 3.3 Electricity space vectors

Vs = Vs ej(ωs t+θv )

(3.5)

The phase angle θv is present because the vectors Vs and Is are rarely in phase. This is also true for the rotor voltage Vr , current Ir , flux r vectors, and the stator flux s vector, though they will have different phase angles in the complex space wrt the stator current vector Is . Besides, this is also true for the stator and rotor flux linkage  s and  r , which will be the topic of the next section. Nonetheless, caution must be taken as below: • Vs , Is etc. in this chapter represents magnitude of a vector, while these are in rms values in Chap. 2 for equivalent modelling and analysis; • The current density field and the magnetic field has 90◦ difference in space. However, due to the current vector convention definition,  = κI is then held. This process of determining a rotating vector in space is also illustrated in Fig. 3.3. The current density field vector of the stator windings is obtained circumferentially at the electric angle θe being 0◦ , 60◦ , 120◦ , 180◦ and 240◦ , respectively, which corresponds to from Case 1 to Case 5 in the figure. According to this figure, the following observations can be summarized • The winding axes to accommodate current vectors of phase A, B, C are displaced with 120◦ apart in space; • The current density field vectors or current vectors Ia , Ib and Ic are aligned with these winding axes respectively as per the convention on current polarity; • Vectors Ia , Ib and Ic vary their magnitudes over time, but the directions stay on the corresponding winding axes, being either positive or negative. Thus, Ia , Ib and Ic do not rotate; • However, the vector sum Is of Ia , Ib and Ic rotates at speed ωs over the space.

78

3 Rotating Frame Modelling and Analysis Flux density Field intensity

Current A’

t

A

B

t

C

C

B

žs Time

¹

B’

A

C’

B

A’

C

B’

A

C

B’

A Space

Space

C’ A

Case

(B-axis)

Current density

B

A’ C

A’

B

Ic

B’

C’

A Ia

Ia

B’

Ib

Is

¹

(A-axis)

C’ C

A

A

C’

B

A’

t

(C-axis)

Fig. 3.4 Current and flux density distribution diagram for 3 phases

Therefore, this design of three phase windings to generate the rotating current and magnetic field is truly remarkable and still inspiring nowadays. It is worth noting that according to the Euler projection principle, the projection of a rotating vector onto axes of Phase A, B and C, yields three sinusoidal waveforms with 120◦ electric phase difference to each other as a result of the 120◦ spatial displacement between the phases. This is exactly what happens in the stator windings as demonstrated by the cases in Fig. 3.3. Vice versa, the vector sum of three sinusoidal waveforms having 120◦ electric phase shift to each other shall generate a rotating vector in space for the three phase windings with 120◦ apart in spatial displacement. This is generally true that for any N > 2 sinusoidal waveforms having 360/N phase difference in time and along the axes with 360/N phase difference in space will create a rotating vector in space.3 This is the nature and beauty of the vector and linear space analysis of mathematics. With this fact as a result of vector analysis, the current and magnetic fields of three phase windings are given in Fig. 3.4 for time t = 0. Comparing to that of the single phase A in Fig. 3.2, it is evident that: • The fields of Phase A are similar to the resultant counterparts in the three-phase diagram. This is because the resultant current and flux vectors of the three phases are now aligned with Phase A axis at electric angle θe = 0; • However, the three-phase fields are stronger than those from Phase A as a single phase excitation, since Phase B and C also contribute to the three-phase fields in the meantime; • Also, as time advances, Phase A fields change the strength while the resultant three-phase fields rotate in the space.

3

This is the basis for 5 or 7-phase machines.

3.2 Complex Space Model

79

3.2 Complex Space Model The stator and rotor dynamic models are coupled through the stator and rotor flux linkage [9]. The stator flux linkage  s consists of the stator self flux linkage  ss and the coupling flux linkage  sr , while the rotor flux linkage r is composed of the rotor self flux linkage  rr and the coupling flux linkage  r s  s =  ss +  sr

(3.6)

 r =  rr +  r s

(3.7)

The self flux linkage  ss is a vector sum of the stator self flux linkage of three individual phases as  ss =

 2 2 2  sa +  sb e+j 3 π +  sc e−j 3 π 3

(3.8)

In a per-phase scale, the stator self flux linkage of Phase A is a vector sum of the flux of the three phases passing through Phase A winding as   2 2  sa = rel a + b e+j 3 π + c e−j 3 π Ns   2 2 = rel (κss + κsl )Ia + κss Ib e+j 3 π + κss Ic e−j 3 π Ns

(3.9)

in which L ss = Ns κss is the stator self inductance per phase and L sl = Ns κsl the stator leakage inductance per phase. According to Eqs. (3.1) and (3.3),  sa is reduced to  sa = L sl Ia +

3 L ss rel(Is ) 2

or  sa = L s Ia

(3.10)

where stator self inductance L s is defined as L s = 23 L ss + L sl , and Ia = rel(Is ) bearing in mind that Is is normalised. Similarly  sb = L s Ib

(3.11)

 sc = L s Ic

(3.12)

Substituting Eqs. (3.10), (3.11) and (3.12) into Eq. (3.8), results in a rotating flux linkage vector given by (3.13)  ss = L s Is or  ss = Ψss ejωs t

(3.14)

80 Fig. 3.5 Stator related flux linkages

3 Rotating Frame Modelling and Analysis B

B

Ψsc

c

a

Ψsa

Ψsa Ψss A

A

Ψsb

b

C

C

It seems that the resultant flux linkage vector expression is similar to those of the current and voltage vectors, and that the above complicated process of deriving flux linkage is unnecessary. However, the stator self inductance L s in the vector expression is not the same as the self inductance of each phase winding. This is an important discovery through the flux linkage determination. Note that L SS is the self-inductance of the stator without leakage effect. Ls =

3 L ss + L sl = L SS + L sl 2

(3.15)

This process of determining the stator self flux linkage and self inductance is illustrated as in Fig. 3.5. Going through the same process as above, the rotor self linkage Ψrr is derived as (3.16)  rr = L r Ir or

 rr = Ψrr ej(ωs t+θr )

(3.17)

where θr is the phase angle between the stator Is and the rotor Ir current vectors. L r is the rotor inductance, and it is determined by self L rr and leakage L rl inductance per phase.4 Note that L R R represents the rotor self inductance without leakage. Lr =

3 L rr + L rl = L R R + L rl 2

(3.18)

The coupling terms  sr and  r s will have different magnitudes and take different directions in the vector space, but the coupling (or mutual) inductance shall be the same. In order to have a more general conclusion, a two arbitrary rotating current Ix and I y in the vector space, which have a spatial phase difference of θ , are taken as IxX = IX ejωt

4

I yX = IY ej(ωt−θ)

Rotor does not necessarily have three phases in reality. However, it is equivalent to have so for the sake of ACIMs modelling and analysis.

3.2 Complex Space Model

81

Fig. 3.6 Coupling flux linkages Iy

IX

xy

X

Y

IX

Ψxy

X

Y

X

Y

X

X

Y

or IYy = IY ejωt

IYx = IX ej(ωt+θ)

When the current IY is described in a way of I yX , it is already related to the vector IX . In particular, the term I y = IY cos(θ )ejωt is its projection on the vector IX . Similarly, IYx describes the current IX in relation to the current IY , and Ix = IX cos(θ )ejωt is the corresponding projection. These are demonstrated in Fig. 3.6. Based on this geometric illustration, the coupling flux linkage  x y at Phase X from Phase Y is then governed by  x y = N x x y = N x κx y I y = L x y Iy in which L x y = N x N y /R is the mutual inductance. As illustrated in Fig. 3.6, applying the same process yields the following  yx = L yx Ix but the mutual inductance L yx is the same as L x y as it represents the same flux path. In particular, when the flux path is universal in the rotating vector space, then mutual inductance does not depend on actual vector position in the space, which is generally true for induction machines, but not for permanent magnet synchronous machines particularly for interior mounted configurations. Therefore, the coupling term  sr and  r s are governed by (3.19)  sr = L m Ir  r s = L m Is

(3.20)

82

3 Rotating Frame Modelling and Analysis

where L m is the stator and rotor mutual inductance, defined as Lm =

3 L mm 2

(3.21)

L mm is the stator and rotor mutual inductance per phase, given as L mm = √ Nr Ns /R = L ss L rr Comparing to Eqs. (3.15) and (3.18), the difference is clearly shown, which is not unforeseen since the leakage term of neither L sl nor L rl has functions related to the coupling flux linkage whatsoever. Nevertheless, the flux linkages among the stator and rotor phases such as A = L mm Ira  sr B  sr = L mm Ir b  Csr = L mm Ir c

 rAs = L mm Isa  rBs = L mm Isb  rCs = L mm Isc

are not rotating vectors anymore. They are fixed in space but vary their amplitudes over time. Bear in mind that the above is only the coupling flux linkage of the same phase. As a summary, the stator and the rotor flux linkage  s and  r are governed by the corresponding current vectors of Is and Ir as  s = L s Is + L m Ir

(3.22)

 r = L m Is + L r Ir

(3.23)

The electric torque is a result of an interaction between the stator flux field (or more accurately air gap flux field) and the rotor MMF field which is the integration of rotor current density field,5 as long as there is a slip between the rotor and the stator. These fields are sinusoidal waves (or dominated by sinusoidal wave at fundamental frequency) travelling in space at speed ωs , and are illustrated in Fig. 3.7. Bear in mind that the pole pair N p = 1 is adopted in the example for simplification, thus the rotor electric speed ωr e equals its mechanical speed ωr . The following comments are made based on the diagrams. C1: Alternating current at the phase windings produces a rotating air gap flux at the speed of ωs , which is the stator electric frequency; C2: The air gap flux is a spatial integration of current density waveform. Thus, it has a 90◦ phase difference in space;

5

The moving electric chargers in the magnetic field will generate a reactive force, according to the Lorentz law.

3.2 Complex Space Model

83

Current B

A

B

C B’ C’

Time

A

A’

Ia

C Current density A’ C B

2π

π

0

Is

s

A

B’

C’

B

A’

C’

C

A Space Stator View

B’

s

A

Flux density – Air Gap Current density A’ s

C B

A

m

2π

π

0 s

s

B’

C’

B

A’

C

B’

A Space Stator View

C’

A

Induced Voltage Wave Flux density – Air Gap

s

A’ C B r

s

>

Er

r dψ dt

0

Cutting Direction

2π

π

Space Rotor View

mωSP

B’

B v

r

C’ sp

A Er

Rotor Current Density Induced Voltage Wave r

A’ C B B’

2π

π

0

r

Space Rotor View

r

r

C’ A

sp

MMF intensity- Rotor Flux density- Air Gap A’ C

B B’

2π

π

0

r

Space Rotor View

r

90°+ r C’

A

s

Tem

Fig. 3.7 Torque production and field waves

sp

mmf r

m

sin r

r

84

3 Rotating Frame Modelling and Analysis

Cutting speed v

v v = r

Fig. 3.8 Stator flux and rotor induced voltage waveform

C3: The rotor rotates at a speed lower than the stator synchronous speed during motoring. This slip speed causes the rotor bar cutting the air gap flux, and induces an EMF at the rotor bars. The induced voltage has the same spatial phase as the air gap flux, which is illustrated in Fig. 3.8, and briefed as below. Define the rotor position θ = 0 for the spatial position at A. The flux density distribution at any position θ along the rotor circumference at the cross section can be written as: B(θ ) = −Bm sin θ where Bm denotes the amplitude of the flux density wave. For any rotor bar, its motion induced voltage is determined by the flux density at that spatial position as well as the velocity with which it cuts the flux field. The induced voltage is expressed in the vector format as Er (θ ) = lv × B(θ ) or in the scalar form given the polarity convention, as Er (θ ) = −ωsp ψm sin θ where ψm = lr Bm , l denotes the shaft length and r the radius of the rotor. According to the Fleming’s right-hand rule, the induced current direction is given in Fig. 3.8. The polarity of the voltage distribution is the same as that for the current density (positive for causing inward flow and negative otherwise). When motoring, it is now clear that the induced voltage has the same spatial phase as the air gap flux. Otherwise for the generating scenario, the rotor speed is ahead of the stator flux, leading to ωsp < 0. Consequently, their spatial phase difference is 180◦ ; C4: The induced rotor voltage has an electric frequency of ωsp or has a rotating speed ωsp relative to the rotor, which is also rotating at a speed ωr . Thus, its traveling speed at spatial space is ωs . This is true for the rotor current and flux vectors;

3.2 Complex Space Model

85 Wf

Fig. 3.9 Energy balancing Pe Electric Power

Magnetic Energy

- Pm Mechanical Power

C5: The rotor current field lags its voltage field by the rotor power factor angle θr . This angle is usually small as the rotor leakage inductance is quite small compared to the rotor resistance at low slip speed; C6: The rotor MMF wave generated by the rotor current field is in a step-wised sinusoidal waveform, but its dominant component is the sinusoidal wave at fundamental frequency; C7: The direction of vectors agrees with those in Fig. 2.30, except for the rotor current and voltage vectors, whose phases are inverted. This is because the rotor vectors are mirrored to the stator side in the equivalent model; C8: The machine torque Tem is a consequence of the interaction between the rotor MMF and air-gap flux. This torque is maximized when the phase angle between these two vectors is 90◦ . Deviation from 90◦ leads to a reduction of the torque. Tem is the cross product of these two vectors as 3 3 Tem = k N p  m × Ir = k N p Ψm Ir sin θr 2 2

(3.24)

where the parameter k represents the conversion ratio from Ir to the fundamental component of MMF. Another way of determining the machine torque is through electric and mechanical energy conversion. As a matter of fact, the electric machine is a device to convert either from electric energy to mechanical energy or vice versa, using magnetic field as a energy storage medium. This is illustrated as in Fig. 3.9. Here, in order to avoid confusion, let us consider that the electric power flowing into the magnetic field is positive, while the mechanical power flowing out from the magnetic field is positive. Both the electric and mechanical power contribute to the energy stored in the magnetic field, or the variation of the magnetic energy is a result of the electric and mechanical power flowing to and from the magnetic energy pool given by dW f = Pe − Pm dt This equation can be rewritten for the mechanical power as Pm = Pe −

dW f dt

86

3 Rotating Frame Modelling and Analysis Ψ

Ψ Wf=Wf’ = 0.5LI 2

Ψ=Li

W = Wf+Wf’ = ΨI

Ψ(i)

Ψ

Linear

Ψ

Nonlinear Wf Energy

Wf Energy

Wf’ Co-Energy

Wf’ Co-Energy I

i

I

i

Fig. 3.10 Energy and co-energy of magnetic field

where W f is the magnetic field energy. The mechanical power Pm can also be expressed as the production of mechanical torque and rotor speed such that Pm = Tem

dΘ dt

in which Θ is the rotor angular position. Combining the above two equations, and bearing in mind that ∂ Pe /∂Θ = 0 as electric power contains no mechanical rotating angle factor, it yields ∂W f (3.25) Tem = − ∂Θ 

Nevertheless, the concept of co-energy W f is commonly introduced to derive  electromagnetic torque [5, 7, 13]. Field energy is defined as W f = idψ, while   co-energy as W f = ψdi, see Fig. 3.10. According to the principle of integration  by parts, W f = W − W f , where W = I and Ψ and I are the terminal values for the respective ψ and i. Given this relation, the above expressions for Pm and Tem become   dW f ∂W f and Tem = Pm = Pe + dt ∂Θ The benefit of avoiding the negative sign when deriving Tem is clearly shown, but the field energy W f is adopted in this book due to its real physical meaning. As a matter  of fact, W f = W f = 0.5W when the system works in the linear region.6 According to the previous analysis and as shown in Fig. 3.11, the stator phase windings can be modelled with a rotating current Is with its inductance L s . Similarly, Ir and L r are for the rotor model. The stator and rotor has coupling inductance L m . 6



  For a given relative position, ψ = L(i)i. In the linear region, L = cnst. Thus, idψ = i Ldi =  ψdi, namely, W f = W f . However, this relation does not hold in the nonlinear region. Further

more, since the inductance L is a nonlinear function of i, the co-energy W f does have computational advantage over the field energy W f .

3.2 Complex Space Model

87

Fig. 3.11 Machine torque principle s

I s Ψs e

s

I r Ψr

The current vectors of the stator and rotor have a phase difference of electric angle θe , which depends directly on the rotor angular position Θ. Particularly, the default rotor motion tends to eliminate this electric angle, and the electric angle and mechanical angle are related by the pole-pairs N p , thus ∂θe = −N p ∂Θ For a linear system (or close to linear), the magnetic energy is governed by W f = 0.5L s Is2 + 0.5L s Ir2 + L m Is Ir cos(θe ) Thus Tem = −

∂ W f ∂θe ∂W f =− = −N p L m Is Ir sin(θe ) ∂Θ ∂θe ∂Θ

Negative sign in the above equation implies that in motoring, the induced rotor current is lagging the magnetizing current. As such θe < 0. An immediate remark can be made that in generating, the induced current will lead the magnetizing current with θe > 0. As a consequence, the torque Tem can be expressed by a more simple form as the vector cross production Tem = −N p L m (Is × Ir ) = N p L m (Ir × Is ) Combining with conversion factor from 2 phases to 3 phases, the machine torque is finally given as (3.26) Tem = 1.5N p L m (Ir × Is ) This is the version that will be used throughout this book unless otherwise specified. This then boils down to an overall induction machine model in complex vector space as

88

3 Rotating Frame Modelling and Analysis

Vs = Rs Is + 0

= Rr Ir +

d s dt d r dt

 s = L s Is + L m Ir

(3.27)

 r = L m Is + L r Ir Tem = 1.5N p L m (Ir × Is ) in which the stator and rotor self as well as their mutual inductance L s , L r and L m are given by Eqs. (3.15), (3.18) and (3.21), respectively. Stator voltage and current vectors Vs and Is are governed by Eqs. (3.4) and (3.1), while the rotor current vector is similar to the stator current vector. This model of induction machines can be readily mapped to the abc, αβ and dq frames, which is the focus of the next three sections.

3.3 Three Phase Frame Model Abc frame refers to a static frame in space that its three axes a, b and c are aligned with the magnetic axes of the windings of Phase A, B and C respectively. When projecting a rotating vector such as stator current Is onto the abc axes, the corresponding projected components are not rotating anymore but each projected length varies wrt the spatial position of that vector. This decomposition of a spatial vector is illustrated in Fig. 3.12. Bearing in mind that when mapping to the abc axes, the stator and rotor vectors are mapped to different frames: the stator vectors to the fixed abc frame while the rotor vectors to the rotating frame ra − r b − r c at a speed of ωr e . The model in the abc frame comes directly from the physics of the interaction between the electric and magnetic fields, as illustrated in Fig. 3.13.

Fig. 3.12 Vector projected to abc frame

b

i sc

Is i sa

i sb

c

a

3.3 Three Phase Frame Model

89

Fig. 3.13 Three phase model

It is more convenient to give the model in abc frame in matrix format as below ⎡ ⎤ ⎡ ⎤⎡ ⎤ va Rs 0 0 i sa ⎣vb ⎦ = ⎣ 0 Rs 0 ⎦ ⎣i sb ⎦ + vc i sc 0 0 Rs

 

  Vsabc Isabc ⎤⎡ ⎤ ⎡ ⎤ ⎡ ira 0 Rr 0 0 ⎣0⎦ = ⎣ 0 Rr 0 ⎦ ⎣ir b ⎦ + 0 ir c 0 0 Rr

  Irabc

d dt

d dt

⎡ ⎤ ψsa ⎣ψsb ⎦ ψsc

(3.28)

⎡ ⎤ ψra ⎣ψr b ⎦ ψr c

(3.29)

where Vsabc is the stator phase voltage vector, Isabc the stator phase current vector,  sabc the stator phase flux linkage vector. Similarly, Irabc and  rabc are the respective rotor current and flux linkage vectors. The stator and rotor flux linkage can be further expressed as ⎤ ⎤⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ i sa ira cos(θr + 120) cos(θr − 120) ψsa cos(θr ) L s0 L sm L sm ⎣ψsb ⎦ = ⎣ L sm L s0 L sm ⎦ ⎣i sb ⎦ + L mm ⎣cos(θr − 120) cos(θr ) cos(θr + 120)⎦ ⎣ir b ⎦ cos(θr + 120) cos(θr − 120) ψsc L sm L sm L s0 i sc cos(θr ) ir c    

 

 sabc L sabc L mabc ⎡

⎡ ⎤ ⎡(3.30) ⎤ ⎤⎡ ⎤ ⎡ ⎤ cos(θr + 120) cos(θr − 120) Lr0 Lrm Lrm ira i sa ψra cos(θr ) ⎣ψr b ⎦ = ⎣ L r m L r 0 L r m ⎦ ⎣ir b ⎦ + L mm ⎣cos(θr − 120) cos(θr ) cos(θr + 120)⎦ ⎣i sb ⎦ cos(θr + 120) cos(θr − 120) ψr c Lrm Lrm Lr0 ir c cos(θr ) i sc    

 

L rabc L mabc  rabc ⎡

(3.31) where L s0 = L ss + L sl is the stator self inductance per phase, and L sm = − 21 L ss the mutual inductance between stator phases. Likewise, L r 0 = L rr + L rl and L r m = − 21 L rr are the respective rotor self inductance per phase and mutual inductance between phases.

90

3 Rotating Frame Modelling and Analysis

With these definitions, the induction machine model in matrix format can be neatly expressed as Vsabc = Rs Isabc + = Rr Irabc +

0

d sabc dt d abc r

dt

(3.32)

 sabc = L sabc Isabc + L mabc Irabc  rabc = L mabc Isabc + L rabc Irabc

3.4 Fixed Frame Model αβ frame refers to a frame that α axis is aligned with the magnetic axis of Phase A winding while β axis is orthogonal to α axis. αβ is a fixed frame in space. The rotating vectors of Vs , Is ,  s , Ir and  r can be mapped into the αβ frame. For instance (3.33) Vs = vsα + jvsβ When Vs rotates in space, both vsα and vsβ will be sinusoidal wrt time along α and β axes. Similarly (3.34) Is = i sα + ji sβ Ir = ir α + jirβ

(3.35)

 s = Ψsα + jΨsβ

(3.36)

 r = Ψr α + jΨrβ

(3.37)

The system dynamics is still governed by Eq. (3.32), but now the derivative of stator and rotor flux  s and  r can be expanded based on the αβ frame. Before moving to further discussion in this section, the derivative of a rotating vector in a fixed frame and a rotating (co-moving) frame is introduced as shown in Fig. 3.14. In the left-hand side of the diagram, the vector  rotates at speed ω in the fixed αβ frame  = ej(ωt+θ) = [cos(ωt + θ ) + j sin(ωt + θ )] = Ψα + jΨβ The time derivative of  is then given by dΨα dΨβ d = +j dt dt dt

3.4 Fixed Frame Model

91

Fig. 3.14 Derivative of a vector

For the same vector in a rotating dq frame, which is shown in the right-hand side of the diagram,  = ej(ωt+θ) = [ cos(θ ) + j sin(θ )]ejωt = [Ψd + jΨq ]ejωt The time derivative of  becomes     dΨq dΨd d = +j ejωt + jω Ψd + jΨq ejωt dt dt dt     dΨq dΨd − ωΨq + j + ωΨd ejωt = dt dt The term ejωt represents that the dq frame rotates in space at the speed of ω. Therefore, the time derivative in the rotating dq frame is      dΨq dΨd d  + j − ωΨ + ωΨ = q d dt dq dt dt or

 d dq d αβ  + jω dq =  dt dq dt

(3.38)

in which  αβ = Ψα + jΨβ and  dq = Ψd + jΨq . On the right-hand side of the equation, the first term d dq /dt is the rate of change of  dq as viewed in the rotating dq frame; the second term is indeed the vector product of the angular speed ω and  dq , representing the rotating effect of the frame [4]. It is clear that both the original vector itself and its derivative are fixed vectors in the rotating frame. Nevertheless, the derivative of the same vector wrt different frames that are rotating will introduce an extra rotating term in its derivative. For the case of the induction machine modelling, the flux linkage  s rotates at a speed of

92

3 Rotating Frame Modelling and Analysis

ωs wrt a fixed αβ frame, while  r rotates at a speed of ωsp wrt a frame fixed to an imaginary rotor,7 which by itself in turn rotates around the αβ frame at a speed of ωr e . Take this fact into account, Eq. (3.32) of induction machine in the αβ frame in complex vector space becomes Vs 0 s r

= = = =

s Rs Is + d dt d r Rr Ir + dt − jωr e  r L s Is + L m Ir L m Is + L r Ir

(3.39)

Bearing in mind that the negative sign before ωr e is due to that the αβ frame rotates at a speed of −ωr e wrt the rotor. Or, in matrix format it becomes     vα Rs 0 = vβ 0 Rs

  αβ Vs

  i sα + i sβ

  αβ Is

     0 ir α Rr 0 + = irβ 0 Rr 0

  αβ Ir

 d dt

  Ψsα Ψsβ

(3.40)

   Ψr α +Ψrβ + ωr e Ψrβ −Ψr α

(3.41)

d dt

in which the flux linkage  s ,  r are further given by         Ψsα Lm 0 Ls 0 i sα ir α + = Ψsβ 0 L s i sβ 0 L m irβ

   

  αβ αβ  αβ L Lm s s

(3.42)



       Ψr α Lr 0 i sα ir α Lm 0 + = Ψrβ irβ 0 L m i sβ 0 Lr

   

  αβ αβ  rαβ Lm Lr

(3.43)

Note that the variables in the αβ frame are normalized, and L s , L r and L m are given by Eqs. (3.15), (3.18) and (3.21) respectively. The model in the αβ frame can also be derived from the transformation of the model in abc frame as given in Sect. 3.3. This is the so-called Clarke transformation [10], which is illustrated in Fig. 3.15 and is defined as

7

The frame for the rotor model does not rotate at the same speed as the rotor, but the same as as the electrical speed of the rotor. The relation between the mechanical and electrical speeds is as ωr e = N P ωr m .

3.4 Fixed Frame Model

93

Fig. 3.15 Clarke transformation

Tαβ

⎤ ⎡ 1 √ −1/2 −1/2 √ 2⎣ = 3/2 − 3/2⎦ 0 3 1/2 1/2 1/2

(3.44)

The Clarke Transformation transfers the three-phase model into two-phase model.8 The inverse of Clarke transformation is also given as below in the meanwhile ⎡

−1 Tαβ

1 √0 ⎣ = −1/2 √3/2 −1/2 − 3/2

⎤ 1 1⎦ 1

(3.45)

Comparing the above two equation, an immediate conclusion is that the inverse of normalizing factor is not required for the inverse Clarke transformation. This is due to the fact that superposition of three vectors in two dimensional space results in an enlarged vector. The normalization is therefore required. On the other hand, projection of this vector from the two-phase orthogonal reference frame back to three individual subspaces, the scaling property remains. As such, no normalization factor is needed for inverse Clarke transformation. Furthermore, as shown in Fig. 3.13, the rotor model rotates with a speed of ωr e , which is also required to be transferred to the fixed αβ frame by the following   cos(θr ) − sin(θr ) Tθr = sin(θr ) cos(θr ) and subsequently, its inverse is given by Tθ−1 r

8

 cos(θr ) sin(θr ) = − sin(θr ) cos(θr ) 

Zero sequence component is not taken into account.

94

3 Rotating Frame Modelling and Analysis

Caution must be taken here as Tθr defines a clockwise coordinator transformation. With Clarke transformation Tαβ , the relationship between stator flux linkage in the abc and αβ frame is given as below = Tαβ  sabc  αβ s or

−1 αβ s  sabc = Tαβ

This transformation is also valid for the current, voltage vectors of the stator. However, the transformation of a rotor vector component from the abc frame to the αβ frame requires one more step of conversion since the rotor is rotating. The conversion for the rotor flux linkage is given as  rαβ = Tαβ Tθr  rabc or

−1 αβ Tαβ r  rabc = Tθ−1 r

This relation is also held for other rotor vectors such as current and voltage. Reference to the third equation in Eq. (3.32), it can be rewritten as −1 αβ −1 αβ −1 −1 αβ  s = L sabc Tαβ Is + L mabc Tαβ Tθr Ir Tαβ

or

Similarly

−1 αβ −1 −1 αβ = Tαβ L sabc Tαβ Is + Tαβ L mabc Tαβ Tθr Ir  αβ s

−1 αβ −1 −1 αβ Is + Tθr Tαβ L rabc Tαβ Tθr Ir  rαβ = Tθr Tαβ L mabc Tαβ

Therefore −1 = Tαβ L sabc Tαβ L αβ s

L rαβ L αβ m

= =

(3.46)

−1 −1 Tθr Tαβ L rabc Tαβ Tθr abc −1 −1 Tαβ L m Tαβ Tθr αβ

αβ

(3.47) (3.48) αβ

Though the rather tedious process of deriving L s , L r and L m based on the above equation is omitted, the result is exactly the same as given by Eqs. (3.42) and (3.43). According to the abc frame model as in Eq. (3.32), and with the above transformation definition, the following expression is derived

3.4 Fixed Frame Model

95

Vsabc αβ

−1 ⇒ Tαβ Vs

=

⇒

=

⇒ ⇒

αβ Vs αβ Vs αβ Vs

d sabc dt dT −1  αβ −1 αβ Rs Tαβ Is + αβdt s dT −1  αβ −1 αβ Tαβ Rs Tαβ Is + Tαβ αβdt s αβ −1 αβ −1 d s Rs Tαβ Tαβ Is + Tαβ Tαβ dt d αβ αβ Rs Is + dts

= Rs Isabc +

= =

The stator model in αβ frame has the same expression as that in abc frame, though one is looked at in a three-phase space and the other in a two-phase space. This is however not true for the rotor model, which is demonstrated in the following d rabc dt dT −1 T −1  αβ −1 −1 αβ Rr Tαβ Tθr Ir + αβ dtθr r −1 αβ αβ −1 −1 dTθr  r Tαβ Rr Tθ−1 Ir + Tαβ dt r dTθ−1  rαβ −1 αβ r Rr Tθr Ir + dt αβ dTθ−1 αβ −1 d r r Tθ−1 R I + T +  rαβ r r θ dt dt r r αβ αβ −1 d r Rr Ir + Tθr Tθr dt + Tθr Er ωr e Tθ−1  rαβ r αβ d αβ Rr Ir + dtr + ωr e Er  rαβ

0 = Rr Irabc + ⇒0= ⇒0= ⇒0= ⇒0= ⇒0= ⇒0=



in which

0 1 Er = −1 0 dTθ−1 r dt

= Er Tθ−1 r

 (3.49)

dθr = ωr e Er Tθ−1 r dt

This is attributed to the fact that the rotor rotates with an electric rotating speed of ωr e . Tidying up the above process, the matrix format model of induction machine arises as d αβ αβ αβ Vs = Rs Is + dts d αβ αβ 0 = Rr Ir + dtr + ωr e Er  rαβ (3.50) αβ αβ αβ αβ  αβ = L s Is + L m Ir s αβ αβ αβ αβ  rαβ = L m Is + L r Ir These expressions are exactly the same as those given by equations from (3.40) to (3.43) when describing the vector model in complex space. The parameters and structure of the model are illustrated in Fig. 3.16. An immediate comment for this model is that signals of current, voltage and flux linkage in αβ frame are generally in sinusoidal waveforms. Hence, it is still far too difficult to analyze.

96

3 Rotating Frame Modelling and Analysis

Fig. 3.16 αβ model

3.5 Rotating Frame Model dq frame refers to a frame that rotates at a speed of ωs in space.9 Normally, d axis is aligned with the rotor flux, which is called rotor flux oriented model, but d axis can also be aligned with the stator flux or air gap flux. When mapping the vectors that rotate at the same speed as the dq frame, these vectors become stationary in the dq frame. This is one of the advantages of adopting dq frame as it makes both performance analysis and system control convenient (Fig. 3.17). The rotating voltage vector of Vs in the dq frame is given by Vs = vsd + jvsq

(3.51)

Similarly, Is ,  s , Ir and  r are as Is = i sd + ji sq

(3.52)

Ir = ir d + jirq

(3.53)

 s = Ψsd + jΨsq

(3.54)

Fig. 3.17 Park transformation 9

There are frames which have rotating speed different from the stator speed ωs .

3.5 Rotating Frame Model

97

 r = Ψr d + jΨrq

(3.55)

As already proven and shown by Eq. (3.38) in last section, the time derivative of a vector is the sum of its derivative as observed in the rotating frame and the vector product of the frame’s rotating speed and the vector components expressed in that rotating frame. For the case of induction motor modelling, we designate a rotating dq frame with the synchronous angular speed of ωs . This frame then has a relative speed of ωs wrt the stator, and a speed of ωsp relative to the rotor. Hence, the motor equations in the dq frame turn into Vs 0 s r

= = = =

s Rs Is + d + jωs  s dt d r Rr Ir + dt + jωsp  r L s Is + L m Ir L m Is + L r Ir

(3.56)

Here, the stator and rotor vectors are represented using the dq-axis components. In a matrix format, the above set of equations arises as      vd i sd Rs 0 = + vq 0 Rs i sq



  dq dq Is Vs      0 ir d Rr 0 = + irq 0 Rr 0

  dq Ir

 d dt

 d dt

   Ψsd −Ψsq + ωs Ψsq +Ψsd

(3.57)

   Ψr d −Ψrq + ωsp Ψrq +Ψr d

(3.58)

where the flux linkage  s ,  r are further given by         Ψsd Lm 0 i sd ir d Ls 0 + = Ψsq 0 L s i sq 0 L m irq

   

  dq dq dq s Ls Lm

(3.59)



       Ψr d Lr 0 i sd ir d Lm 0 + = Ψrq irq 0 L m i sq 0 Lr

   

  dq dq  rdq Lm Lr

(3.60)

Note that L s , L r and L m are given by Eqs. (3.15), (3.18) and (3.21) respectively, which are the same as those in the αβ frame. In general, the stator and rotor equations in any frame with a rotating speed of ωk are given as s + jωk  s Vs = Rs Is + d dt d r 0 = Rr Ir + dt + jωks  r

98

3 Rotating Frame Modelling and Analysis

Fig. 3.18 dq model

in which ωks = ωk − ωr e . As already shown in last section, the αβ frame model can be derived from the abc frame model by coordination transformation. The dq frame model can be obtained in the same way by transforming the αβ frame model. This is the so-called Park transformation. It is illustrated in Fig. 3.18, and defined as 

 cos(θ ) sin(θ ) − sin(θ ) cos(θ )

Tθ =

The inverse Park transformation is as   cos(θ ) − sin(θ ) Tθ−1 = sin(θ ) cos(θ )

(3.61)

(3.62)

Tθ is an anticlosewise coordinator transformation. Compared with the definition of Tθr in last section, the difference is evident. With this definition, the model in the αβ frame as given by Eq. (3.50) can be readily transferred to the dq frame as below αβ

⇒ Tθ−1 Vs = dq

in which

d αβ s dt dq dT −1  dq Rs Tθ−1 Is + θ dt s dq dT −1 dq Tθ Rs Tθ−1 Is + Tθ Tθ−1 ddts Tθ  sdq dtθ dq dq −1 Rs Tθ Tθ−1 Is + ddts + Tθ ωs E ∗ Tαβ  sdq dq dq Rs Is + ddts + ωs E ∗  sdq αβ

= R s Is

Vs ⇒

Vs

dq

=

⇒

dq Vs

=

⇒

dq Vs

=

+

  0 −1 E = 1 0 ∗

dθ dTθ−1 = E ∗ Tθ−1 = ωs E ∗ Tθ−1 dt dt

(3.63)

3.6 Steady-State Analysis

99

and E ∗ = −Er where Er is defined by Eq. (3.49). Similarly d rαβ + ωr e Er  rαβ dt dq −1 dT  dq Rr Tθ−1 Ir + θ dt r + ωr e Er Tθ−1  rdq dq dT −1 dq Tθ−1 Rr Ir + Tθ−1 ddtr + dtθ  rdq + Tθ−1 ωr e Er  rdq dq dq Rr Ir + ddtr + ωs E ∗  rdq + ωr e Er  rdq dq dq Rr Ir + ddtr + ωsp E ∗  rdq αβ

0 = Rr Ir ⇒0= ⇒0= ⇒0= ⇒0=

+

Hence, the matrix format model of an induction machine in the dq frame is given as dq

Vs

dq d s + ωs E ∗  sdq dt dq dq Rr Ir + ddtr + ωsp E ∗  rdq dq dq dq dq L s Is + L m Ir dq dq dq dq L m Is + L r Ir dq

= R s Is +

0 =  sdq =  rdq =

(3.64)

Note that this induction motor model shall be the same as given by equations from (3.57) to (3.60). The parameters and structure of the model are illustrated in Fig. 3.18. Furthermore, inductances L s , L r and L m shall have no difference regardless of αβ or dq frame. The induction model in the dq frame as represented either by Eq. (3.56), or by equations from (3.57) to (3.60), is by far the most convenient model for both performance analysis and system control. Hence, hereinafter, the dq frame model will be adopted unless stated explicitly otherwise.

3.6 Steady-State Analysis The set of parameters used for system analysis in this section as well as for system control in later sections is summarized in Table 3.1, which is an updated version from Table 2.1 to account for the characteristics of the system model in the dq frame. Though the dq frame is defined with a rotating speed of ωs , its direction requires further clarification, which is given in three scenarios as below. F1: d axis is aligned with the rotor flux. Thus Ψr d = Ψr and Ψrq = 0. This is the so-called rotor field oriented model. This model is commonly used for both analysis and control; F2: d axis is aligned with the stator flux. Thus Ψsd = Ψs and Ψsq = 0. This is the so-called stator field oriented model; F3: d axis is aligned with the air-gap flux. This is the so-called air-gap field oriented model.

100

3 Rotating Frame Modelling and Analysis

Table 3.1 Typical parameters of electric machine for dq frame model Symbol Typical value ISO Unit Description KT Lm Lr Ls L rl L sl Lδ Q max re Req Rr Rs VDC Vs δ κr Ψmax τT τr

3.47 3.7 ×10−3 3.2 ×10−3 4.8 ×10−3 0.18×10−3 0.25×10−3 0.50×10−3 0.30 0.82 82×10−3 38×10−3 24×10−3 330 190 0.11 1.16 0.28 6.8 83.7

[–] [H] [H] [H] [H] [H] [H] [Wb] [–] [Ω] [Ω] [Ω] [V] [V] [–] [–] [Wb] [ms] [ms]

Induction motor torque constant Mutual inductance Rotor self inductance Stator self inductance Rotor leakage inductance Stator leakage inductance Total leakage at stator side Equivalent permissible flux linkage at stator Effective turn ratio Overall equivalent resistance Rotor winding resistance Stator winding resistance DC voltage Phase voltage, Y connection Leakage factor Coupling factor Maximum permissible flux linkage Stator or torque time constant Rotor time constant

The rotor field oriented model is utilized in this section. Thus, the dq frame model as given by Eqs. (3.57) and (3.58) is reduced to as vd vq 0 0

= = = =

Rs i sd + dΨdtsd − ωs Ψsq dΨ Rs i sq + dtsq + ωs Ψsd r Rr ir d + dΨ dt Rr irq + ωsp Ψr

(3.65)

while the flux linkage Eqs. (3.59) and (3.60) become Ψsd Ψsq Ψr 0

= = = =

L s i sd + L m ir d L s i sq + L m irq L m i sd + L r ir d L m i sq + L r irq

(3.66)

The steady-sate analysis refers to as analyzing the induction machine performance by setting d x / dt = 0, x = (s, r ) in the dynamic model. Equation (3.65) is further reduced to as

3.6 Steady-State Analysis

101

vd vq 0 0

= = = =

Rs i sd − ωs Ψsq Rs i sq + ωs Ψsd Rr ir d Rr irq + ωsp Ψr

(3.67)

where ωsp is the slip speed defined as ωsp = ωs − ωr e . According to the 3rd equation in (3.67), the rotor field oriented model makes ir d = 0

(3.68)

which in turn leads to the following as per Eq. (3.66) Ψsd = L s i sd

(3.69)

Ψr = L m i sd

(3.70)

Thus, it concludes that the rotor flux is determined by the stator current i sd . This is why d is called field component. The 4th equation in (3.66) results in ir = irq = −

Lm i sq Lr

(3.71)

which implies that the rotor current irq intends to cancel out the magnetic field generated by the stator in q direction. Combining Eq. (3.71) with Eq. (3.66), and introducing a leakage factor δ as δ =1−

L 2m Ls Lr

(3.72)

the total leakage at the stator side is then given as L δ = δL s

(3.73)

With the definitions in Eqs. (3.15) and (3.18), the total leakage can be approximated as L δ ≈ L sl LLRrR + L rl LLSSr ∗ ≈ L sl + L rl In the last step of approximation, L R R is used to estimate L r . Refer to Fig. 2.15, then the physical meaning of L δ is obvious. However, an immediate note is added here that the total leakage definition in the dq frame is slightly different form that in the equivalent circuit model. This slight difference will introduce a slight error in machine torque, which will be demonstrated in the later part of this section.

102

3 Rotating Frame Modelling and Analysis

Fig. 3.19 Phasor diagram

With the total leakage definition, the stator flux linkage in the q direction is given by Ψsq = L δ i sq

(3.74)

Compared to the stator flux linkage in the d direction, the magnitude of Ψsq is usually very small. However, when the stator speed is high, the back EMF caused by Ψsq is significant and could then dominate the stator vd . The phasor diagram is given in Fig. 3.19 based on the steady-state analysis by far. This phasor diagram agrees with what is illustrated in Fig. 3.7, but is different from the phasor diagram given in Fig. 2.30 in regard of the rotor current direction since the rotor current is inverted in the equivalent circuit. According to Eq. (3.26), the machine torque is expanded as an interaction between the electric and magnetic fields of the stator and rotor. Tem = 23 N p L m (ir d i sq − irq i sd )

(3.75)

Based on Eqs. (3.59) and (3.60), it is not difficult to prove the following expressions of the machine torque Tem = 23 N p (Ψsd i sq − Ψsq i sd ) = 23 N p LLmr (Ψr d i sq − Ψrq i sd )

(3.76)

in which the coefficient of 3/2 is to account for the conversion from the 2 phases to 3 phases. The model in dq as well as in αβ frame is a 2-phase model while it is actually a 3-phase model in abc frame. Based on the previous steady-state analysis, this machine torque can be further expressed as

3.6 Steady-State Analysis

103

Tem = − 23 N p L m ir i sd L2 = 23 N p Lmr i sq i sd

(3.77)

Bearing in mind that when the leakage inductances of the stator and rotor are neglected, L s = L 2m /L r , according to Eqs. (3.18), (3.15) and (3.21). Therefore, the induction machine torque can be governed by the stator current components alone. This is a useful observation for induction machine control, as it is very difficult to obtain any rotor electrical signals. Combining Eq. (3.70) into the above yields Tem =

Lm 3 Np i sq Ψr 2 Lr

(3.78)

That being said, when the rotor flux remains constant (by regulating direct-axis current i sd ), the torque is then determined by quadrature-axis current i sq . Thus, i sq is called torque current and q is named as quadrature component. Substituting Eq. (3.70) into (3.67) results in ωsp = sωs =

Rr i sq L r i sd

(3.79)

The following comments are made based on the above expression: C8: The slip speed is controlled by stator current. It requires delicate control of the stator current components over the field and quadrature direction. The slip speed is largely affected by both rotor inductance and resistance, which are time-varying; C9: If this relation is not satisfied, then Ψrq will be present. Consequently, it results in the direct axis of the dq frame deviating away from the rotor field direction; C10: If field current i sd remains constant, then the quadrature current i sq governs the slip speed completely. Therefore, the machine torque is determined by the rotor slip speed, though implicitly. This analysis agrees with what is presented in Chap. 2. Substituting Eq. (3.79) into (3.77), the torque is then given by Tem =

L2 3 2 N p m sωs i sd 2 Rr

(3.80)

It is immediately clear how the slip speed governs the machine torque. In particular, Tem = 0 if ωsp = 0 or s = 0. As a quick summary, the machine torque is regulated by stator current in the field and quadrature directions. The stator field current i sd determines the rotor flux, while the quadrature current i sq regulates the machine torque. Hence, the task now is to investigate how the stator current is driven by the stator voltage.

104

3 Rotating Frame Modelling and Analysis

Fig. 3.20 Signal flow diagram

Substituting Eqs. (3.69) and (3.74) into Eq. (3.67) yields vd = Rs i sd − ωs L δ i sq vq = Rs i sq + ωs L s i sd

(3.81)

Accordingly, i sd and i sq are largely controlled by vq and vd respectively, although i sd and i sq are coupled. It is also clear why it is called the vector control since it requires not only the precise magnitude of Vs but its precise phase as well in order to achieve the desirable i sd and i sq . Solving Eq. (3.81) results in Rs vd + ωs L δ vq Rs2 + ωs2 L s L δ Rs vq − ωs L s vd = 2 Rs + ωs2 L s L δ

i sd = i sq

(3.82)

The machine torque can now be completely determined by Vs through Eqs. (3.82) and (3.77), and so can the slip speed. A signal pathway flow diagram is given in Fig. 3.20, which summarizes the steadystate analysis in this section by far as below. S1: The stator field and quadrature current components i sd and i sq are respectively regulated by the stator quadrature and field voltage components vq and vd , according to Eq. (3.82); S2: The stator flux in d-axis is determined by the stator field current i sd , and so is the rotor flux, according to Eqs. (3.69) and (3.70); S3: The stator flux in q-axis is determined by the stator quadrature current i sq , according to Eq. (3.74); S4: The rotor current Ir is determined by the stator quadrature current i sq , according to Eq. (3.71); S4: The rotor slip speed ωsp is governed by the stator field and quadrature current i sd and i sq , according to Eq. (3.79); S5: The machine torque Tem is governed by both the stator field and quadrature current i sd and i sq , according to Eq. (3.77).

3.6 Steady-State Analysis Slip speed to torque curve

300 200

200

100

100

0

0

-100

-100

-200

-200

-300 -400

-300

-200

-100

0

100

Slip speed to torque curve

300

Torque [Nm]

Torque [Nm]

105

200

300

Equivalent model dq frame model

-300 -400

400

-300

-200

-100

0

100

200

300

400

Slip Speed [rad/s]

Slip Speed [rad/s]

(a) steady-sate results

(b) Comparison

Fig. 3.21 Slip speed to torque curve

To envisage the fact that machine torque depends on the slip speed which is hidden by the rotating frame model, Fig. 3.21 is provided. This figure plots the machine torque as a function of the slip speed. Substituting Eq. (3.79) into (3.81), we get 2 = i sd

Vs2 (Rs − sδ X s X r /Rr )2 + (s Rs X r /Rr + X s )2

The machine torque can then be derived by Eq. (3.80) as Tem =

1.5N p L 2m Rr ωs Vs2 (Rs Rr /s − δ X s X r )2 + (Rs X r + X s Rr /s)2

When both Vs and ωs are constant, the torque reaches the maximum when the term Ξ = (Rs Rr /s − δ X s X r )2 + (Rs X r + X s Rr /s)2 

is minimum at ∂Ξ = 0, ∂s

Rr s = Xr ∗

Rs2 + X s2 2 δ X s2 + Rs2

At sufficiently high speed ωs so that Rs is negligible, the above equation is repaired as Rr r) = Rr (LδLs /L ωsp = δL r s (3.83) Rr∗ Rr∗ ≈ L δ ≈ L sl +L ∗ rl

This equation is more or less the same as Eq. (2.31), validating the equivalence of the equivalent circuit model and the dq frame model during steady-state. The optimal slip s ∗ leads to the peak torque as Temmax =

1.5N p (L 2m /L r )Vs2   2 Rs X s (1 − δ) + (Rs2 + X s2 )(Rs2 + δ X s2 ) 

(3.84)

106

3 Rotating Frame Modelling and Analysis

With the fact that δ 1, X s Rs (at high frequencies), and that L s ≈ L 2m /L r , the expression of the peak torque can be reduced as Temmax ≈ ≈ ≈

1.5N p L s Vs2   √ 2 Rs X s +X s Rs2 +δ X s2



Vs2 2ωs2



1.5N p √

2  rms R2s /ωs + (Rs /ωs ) +L δ 1.5N p Vs ∗ ωs L sl +L rl

Given that the phase voltage Vs in the above equation takes its magnitude value (due to the magnitude-invariant transform), while the phase voltage Vs in Sect. 2.2 takes a rms value, its similarity to Eq. (2.30) is evidently demonstrated. Figure 3.21a is produced using the following Matlab scripts with parameters as in Table 2.1. Compared to the results in Fig. 2.19, the profile of the torque curve is very similar, which proves the steady-state analysis in this section. However, the peak torque is slightly increased with the dq frame model. This is due to a marginal difference in the total inductance leakage with the dq frame model compared to the equivalent circuit model. Figure 3.21b is given for a direct comparison. This analysis proves that in steady-state dq frame model and equivalent circuit model are equivalent. Listing 3.1 Slip speed to torque profile calculation 1 2 3 4

Freq ws Vdc vs

= = = =

100; 2* pi * Freq ; 330; Vdc / sqrt (3) ;

5 6

% For Y c o n n e c t i o n and SPWM % Vpn = Vdc * sqrt (3) /3 % Vpn_rms = Vpn / sqrt (2)

7 8 9 10 11 12 13 14 15 16 17

% % P r e p a i r the f i g u r e figname = ' SlipSpdVsTqCalcDQ '; h _ s l i p S p d T q f i g = f i n d o b j (0 , ' Name ' , [ figname , ' _ R e s u l t s ' ]) ; if i s e m p t y ( h _ s l i p S p d T q f i g ) h _ s l i p S p d T q f i g = f i g u r e ( ' P o s i t i o n ' ,[50 50 452*1 .8 257*1 .8 ] , ... ' Name ' , [ figname , ' _ R e s u l t s ' ] , ... ' N u m b e r T i t l e ' , ' off ' ) ; end figure ( h_slipSpdTqfig ); set ( h _ s l i p S p d T q f i g , ' D e f a u l t A x e s F o n t S i z e ' ,14 , ' D e f a u l t A x e s F o n t N a m e ' , ' Times ... New Roman ' ) ;

18 19 20 21

% % C a l c u l a t i o n bas e d on e q u i v a l e n t c i r c u i t X_term = ( Lsl + Lrl ) * ws ; X _ s q u a r e = X _ t e r m ^2;

22 23 24 25 26

% W_RE = [0:1: ws ws ]; W_RE = [ 0 : 1 : 2 * ws ]; S = ( ws - W_RE ) / ws ; WSP = S. * ws ;

% % % %

Rotor e l e c t r i c s p e e d v e c t o r Rotor electric speed vector Slip vector Slip speed vector

27 28 29 30 31

% This part c a l c u l a t e s the t e r m s for Isd Delta = 1 - Lm ^2/( Ls * Lr ) ; Xr = ws * Lr ; Xs = ws * Ls ;

32 33

Xr_term

= Xr / Rr. * S ;

3.7 Maximum Torque per Ampere and Maximum Torque per Loss 34 35 36 37

Xd_term Xq_term Xsum_squr Isd_squr

= = = =

107

Rs - D e l t a * Xs. * X r _ t e r m ; Rs. * X r _ t e r m + Xs ; X d _ t e r m . ^2 + X q _ t e r m . ^2; vs ^2 . / X s u m _ s q u r ;

38 39 40 41 42

% This part c a l c u l a t e s Te % Te = 1 .5 * Np * Lm ^2/ Rr * s * ws * isd ^2 kt = 3/2* P o l e _ p a i r * Lm ^2/ Rr ; Te = kt. * WSP. * I s d _ s q u r ;

43 44 45 46 47 48 49 50 51 52 53 54

% % Plot the r e s u l t s Tmax = abs ( floor ( min ( Te ) /50) *50) ; plot ( WSP , Te , ' L i n e W i d t h ' ,2) , hold on plot ([ -400 ,400] ,[0 ,0] , ' k ' , ' L i n e W i d t h ' ,1) ; plot ([ 0 ,0] ,[ - Tmax , Tmax ] , ' k ' , ' L i n e W i d t h ' ,1) ; t i t le ( ' S l i p s p e e d to t o r q u e c u r v e ' ) ; y l a b e l ( ' T o r q u e [ Nm ] ' ) ; x l a b e l ( ' Slip Speed [ rad / s ] ' ) ; Smax = 400; axis ([ - Smax Smax - Tmax Tmax ]) ; grid on ; hold off ;

Note that the steady-state analysis is helpful in • Providing feed-forward control strategy; • Determining the conditions to optimize the machine operation; • Analyzing the machine speed to torque characteristics. For the feed-forward control, Eq. (3.81) shall be utilized to determine targeted Vs . When the parameters ωs , δ and L s are sufficiently accurate, the machine torque will stabilize around the desired toque after a transient phase. This will be detailed and analyzed in Chap. 6. The next sections will present some bases for machine operation optimization and investigate the machine speed-torque characteristics.

3.7 Maximum Torque per Ampere and Maximum Torque per Loss The machine torque is determined by appropriate operation of the phase current Is , according to Eq. (3.77). Nevertheless, constraint of maximum machine current Imax depending on the thermal characteristics of the phase windings and the switch devices must be applied. 2 2 2 + i sq ≤ Imax (3.85) i sd Maximizing the machine torque over phase current, which is normally called Maximum Torque per Ampere (MTPA) under the current constraint, is investigated in this section [2, 15]. Define K T as KT = N p

3 Lm 2 Lr

(3.86)

108

3 Rotating Frame Modelling and Analysis MPTA

150

e rqu r To nt5o0u Co

0

1 00

25

50

20

MTPA curve

50 25

0 25 50

-50

nt rre r Cu ntou Co

50

-100

100

25

Quadrature Current Iq [A]

100

20

-150 -150

0

-100

-50

0

50

100

150

Field Current Id [A]

Fig. 3.22 MTPA curves

which is a torque coefficient. Then, Eq. (3.77) can be rewritten as Tem = K T L m i sq i sd MTPA is an optimization problem as below Maximizing f (x, y) : f (x, y) = Tem  2 + i 2 = cnst Subject to c(x, y) : c(x, y) = Is = i sq sd where Is is the magnitude of phase current. Therefore, for a fixed Is Temmax =

1 K T L m Is2 2

(3.87)

√

is arrived at when i sq = ±i sd = ±

2 Is 2

(3.88)

An example of MTPA is given in Fig. 3.22, which deploys the parameters as in Table 2.1. The magenta solid lines are constant torque contours while the dash blue lines are current contours. Torque levels of 25, 50, 100, 200 Nm are marked. MTPA line is represented in cyan color. The following are concluded based on observation of this diagram:

3.7 Maximum Torque per Ampere and Maximum Torque per Loss

109

O1: Current contours are circular curves; O2: Torque contours are symmetrically hyperbolic curves wrt i sd = ±i sq . Note that negative torque contours are not shown in the figure; O3: Geometrically, MTPA is the trace of the points at which torque and current contours are tangent to each other. It is evident that i sd = i sq . Take Tem = 50 Nm as an example. When i sd = i sq = 62.7 A, phase current of Is = 88.6 A will be sufficient to produce this torque. Alternatively, the same torque can be obtained by a combination of i sd = 120.9 A and i sq = 32.5 A, which consequently results in a larger phase current of Is = 125.2 A. These two phase current operation points are also indicated by green diamond markers in the figure. Furthermore, the current contour of Is = 125.2 A may yield a machine torque as high as 100 Nm, if i sd and i sq are regulated appropriately. Apparently, for generating the same amount of the demanded torque, the minimum current is reached on MTPA. That said, torque is maximized for a given current along the MTPA trace. Operating an electric machine along the MTPA curve ends up with the minimum phase current, which generates less heat from the stator dc copper loss point of view, and improves the system efficiency in general. Nonetheless, system losses comprise of more forms than just the stator dc copper loss and have different dependencies on the field and quadrature current components. Therefore, the system efficiency may not be optimal along the MTPA curve [1, 12, 14]. The electric machine model as given in Eq. (3.65), only the dc cooper losses are captured, which are represented by 2 2 + i sq ) + Rr ir2 Pc = Rs (i sd 2 2 = Rs (i sd + i sq ) + Rr

L 2m 2 i L r2 sq

Introducing Rreq = Rr

L 2m L r2

(3.89)

which is the rotor resistance seen on the stator side, the electric machine loss in the dq model can then be expressed as 2 2 + (Rs + Rreq )i sq Ploss = Rs i sd

(3.90)

To obtain the desired torque, based on the torque equation of (3.77), there are infinite combinations of i sq and i sq . One of the targets could be that 1. The combination of i sq and i sq gives rise to the required torque Tem ; 2. The combination of i sq and i sq minimizes the machine loss given by Eq. (3.90). This is the so-call Maximum Torque per Loss (MTPL) principle. It can also be expressed mathematically as

110

3 Rotating Frame Modelling and Analysis MPTL

150

0 200

50

Torq u Conto1 e u0r0

25

50

50 25

0

25 50

10 0

25

-50 200

10

-100

s os rL we u r Po onto C

50

Quadrature Current Iq [A]

MTPL curve

10

100

0

-150 -250 -200 -150 -100

-50

0

50

100

150

200

250

Field Current Id [A]

Fig. 3.23 MTPL curves

Maximizing f (x, y) : f (x, y) = Tem Subject to c(x, y) : c(x, y) = Ploss = cnst where x = i sd and y = i sq . Using the well-known Lagrange-Hamilton principle [6], the above constrained optimization problem is converted to unconstrained optimization problem as maximizing F(x, y) = f (x, y) + λ[c(x, y) − cnst] which has a maximum (or minimum) solution at ∂F ∂x ∂F ∂y

This results in

= K T L m i sq + 2λRs i sd =0 eq = K T L m i sd + 2λ(Rs + Rr )i sq = 0  i sq = ±

Rs eq i sd Rs + Rr

(3.91)

An example of MTPL is given in Fig. 3.23, in which the magenta solid lines are constant torque contours while the dash blue lines are power loss contours. Torque levels of 25, 50, 100, 200 Nm are marked. MPTL line is represented in cyan color. The following are concluded based on observation: O4: Power loss contours are ellipse curves, and the power loss increases with the current magnitude;

3.7 Maximum Torque per Ampere and Maximum Torque per Loss

111

O5: Torque contours are symmetrically hyperbolic curves wrt i sd = ±i sq . Note that negative torque contours are not shown in the figure; O6: Geometrically, MTPL is the trace of the points at which torque and power loss contours are tangent to each other. Take Tem = 50 Nm as an example. When the stator current (i sd , i sq ) is controlled at (83.2A, 47.2A), the corresponding power loss Ploss reaches 0.32 kW. Alternatively, the same amount of torque can be achieved by requesting the current at (159.8A, 24.6A), which corresponds to a loss of 0.36kW . On the other hand, with this power loss, the torque could be increased to as high as 100 Nm if the stator current (i sd ,i sq ) is regulated to minimize the power loss. Clearly, the power loss is minimized for a given demanded torque with MTPL. Equally, the torque is maximized for a given power loss along the MTPL trace. By far, only the DC copper loss is considered. However, this principle of electric machine power loss minimization can be extended to include the iron loss (and more) whenever it is necessary and efficient to do so. According to Eq. (2.55), the iron loss is a function of the strength and frequency of the air-gap magnetic field [13]. In particular, the air-gap magnetic field strength depends on the field current i sd . To capture this effect and to simply the power loss optimization analysis, an equivalent eq core resistance Rc is introduced as Rceq (ωs ) = (ke Δe ωs + kh )ωs L 2m

(3.92)

Then the power loss given in Eq. (3.90) is updated to 2 2 + (Rs + Rreq )i sq Ploss (ωs ) = [Rs + Rceq (ωs )]i sd

(3.93)

and the optimal solution of MTPL becomes  i sq = ±

eq

Rs + Rc (ωs ) eq i sd Rs + Rr

(3.94)

The following comments are made for this optimal power loss solution C11: The optimal power loss solution for a given demanded torque depends on the electric machine frequency ωs ; C12: At low frequency, copper loss dominates. Optimal solution tends to approximate the solution given in Eq. (3.90), ie to rotate more towards d axis. Therefore, high field current is beneficial; C13: On the other hand, at high frequency, iron loss dominates. Optimal solution rotates more towards q axis, thus relatively low field current is preferable. By operating an electric machine along the MTPL curve, the system will benefit a lower power loss. However, this will require constant change of the field current i sd whenever there is a torque request variation, which can subsequently lead to a

112

3 Rotating Frame Modelling and Analysis

deteriorated dynamics response. There is a trade-off between system dynamics and efficiency performance.

3.8 Maximum Torque per Flux An electric machine shall be operated under the constraints that the back EMF is within the DC power supply limit and that the air-gap flux is not over the magnetic material flux saturation limit. Referring to Eq. (3.56) and bearing in mind that the stator and rotor induction are respectively defined by Eqs. (3.15) and (3.18), the expressions of the stator and rotor flux can be repaired as √ √ √  s = L sl Is + √ L SS ( √L SS Is + √L R R Ir )  r = L sl Ir + L R R ( L SS Is + L R R Ir ) in which L SS = 1.5L ss and L R R = 1.5L rr . The second term in above equation is the air-gap flux linkage, which is √ √ √ sm = L SS ( L SS Is + L R R Ir ) = L SS Is + L m Ir

(3.95)

when seen by the stator, and √ √ √ r m = L R R ( L SS Is + L R R Ir ) = L m Is + L R R Ir when seen on the rotor side. The stator side flux linkage sm is commonly used to represent the air-gap flux linkage m . In the rotor field oriented model, the air-gap flux becomes md = L SS i sd mq = (L SS − L m /L r )i sq Comparing to the stator flux as given in Eqs. (3.69) and (3.74), the difference is caused by the stator self-leakage in the field direction and the rotor self-leakage in quadrature direction, which is negligible. Thus, the use of stator flux instead of airgap flux as a measure for magnetic field saturation and back EMF analysis becomes creditable. When the operating frequency is relatively high, the voltage drop due to the ohmic resistance in Eq. (3.81) can be neglected. The voltage constraint of 2 2 2 + vsq ≤ Vmax vsd

is then reduced to

(3.96)

3.8 Maximum Torque per Flux

113

2 (ωs L δ i sq )2 + (ωs L s i sd )2 ≤ Vmax

or10 2 (L δ i sq )2 + (L s i sd )2 ≤ min[(Vmax /ωs )2 , Ψmax /rv2 ]

(3.97)

Equation (3.97) combines the voltage and flux limits. At low speed, flux limit becomes effective, while at high speed, voltage limit dominates. Maximizing machine torque under constant flux is the well-known Maximum Torque per Flux (MTPF) method, and investigation of this method is conducted in this section [3, 8, 11]. This requirement is captured mathematically as below: Maximizing f (x, y) : f (x, y) = Tem Subject to c(x, y) : c(x, y) = Ψm2 = cnst where x = i sd and y = i sq . Using the same Lagrange-Hamilton principle as used in the previous section, the converted unconstrained optimization problem becomes F(x, y) = f (x, y) + λ[c(x, y) − cnst] which has a maximum (or minimum) solution at ∂F ∂x ∂F ∂y

= K T L m i sq + 2λL 2s i sd = 0 = K T L m i sd + 2λL 2δ i sq = 0

This results in

1 i sq = ± i sd δ

(3.98)

When the voltage limits are active, the optimal current is as i sd

√ 2 Vmax = 2 L s ωs

The machine torque then becomes Tem =

2 K T L m Vmax 2δL 2s ωs2

(3.99)

An example of MTPF is given in Fig. 3.24, where the magenta solid lines are constant torque contours while the dash blue lines are constant flux (or constant voltage) contours. Torque levels of 25, 50, 100, 200 Nm are marked. MPTF line is represented in cyan color. Followings are concluded based on observation: Back EMF of ωs Ψmax is viewed at the magnetizing inductance terminal, not the phase winding terminal. Thus, rv as defined in Sect. 2.3.2 is required for this conversion.

10

114

3 Rotating Frame Modelling and Analysis MPTF

500

MTPF curve

400

0 10 50

100

25 25 50 100 200

200 100 50 25

25

10

-200

Flux Conto ur

0 -100

Torq u Conto e ur

0

200

50

0

Quadrature Current Iq [A]

20

Speed Increase Direction

300

-300 20 0

-400 -500 -150

-100

-50

0

50

100

150

Field Current Id [A]

Fig. 3.24 MTPF curves

O7: Flux contours are ellipse curves and flux contour reduces with the machine operating frequency (or speed); O8: Torque contours are symmetrically hyperbolic curves wrt i sd = ±i sq . Note that negative torque contours are not shown in the figure; O9: Geometrically, MTPF is the trace of the points at which torque and flux contours are tangent to each other. Once again, take Tem = 50 Nm as an example. When the stator current (i sd , i sq ) is controlled at (20.4A, 192.2A), the resulted equivalent flux linkage Q = Vmax /ωs is 0.019 Wb. Alternatively, the same torque can be achieved by setting the current to be (39.2A, 100.3A), which corresponds to an flux linkage of 0.037 Wb. However, with this equivalent flux linkage level, the torque could be increased to as high as 100 Nm if stator current (i sd , i sq ) is regulated to minimize the flux. It is clear that the flux is minimized for a given demanded torque with the MTPF. Equally, the torque is maximized for a given flux along the MTPF trace. Substituting Eq. (3.79) into (3.98) results in the slip speed at MTPF as ωsp =

Rr δL r

This is exactly the same as Eq. (3.83). Thus, MTPF yields the maximum torque when phase voltage is fixed. All the analyses in Chap. 2 shall still be valid, at least from the stead-state point of view. For instance, the slip-speed to torque profile (such as Fig. 2.19) is derived by varying the slip speed (or slip). According to Eq. (3.79), the slip speeds are constant along the radial lines from the origin. As a mater of fact, ∗ , where maximum torque is reached. the MTPF line also represents the slip speed ωsp Figure 3.25 indicates how the slip speed changes on constant voltage contour.

3.8 Maximum Torque per Flux

115

Fig. 3.25 Slip on Q-limit

The constant voltage contour at rated voltage and rated frequency is defined as Q-limit in this book. Along the Q-limit, the slip speed ωsp increases from 0 at d-axis, through Rr /L r at MTPA and Rr /(δL r ) at MTPF and ωs at s = 1, to ∞ at q-axis. Between s = 1 line and q-axis, rotor speed is reversed. As a result of monotonous increase of slip speed, the machine torque also varies. Obviously, the torque Tem grows continuously in the region between d-axis and the MTPF line along the Qlimit ellipse, but it is not obvious any more after that. In order to see how the torque varies on the Q-limit contour, Fig. 3.26 is given, in which • The solid blue line is the Q-limit torque profile; • The dashed black line is the torque profile based on steady-state analysis as in Fig. 3.21, re-produced here for comparison; • The solid pink line is the torque profile wrt the I-limit circle at Imax = 160 A. At this current level and at rated frequency of 100 Hz, it breaks the Q-limit at some part of operation. • The dashed pink line is the torque profile wrt the I-limit circle at Imax = 60 A. At this current level and at rated frequency, it is just within the Q-limit completely. Comparing the Q-limit to steady-state results, we could conclude that: (1) the Q-limit torque is higher than the steady-state torque. This is because the resistance terms are neglected; (2) Q-limit torque has a very similar profile to that of the steadystate torque, and to that of the equivalent circuit model in Chap. 2; (3) Optimal slip speed is arrived at on the MTPF line. This proves that the Q-limit, steady-state and equivalent circuit analyses are consistent. Nevertheless, the use of Q-limit may still

116

3 Rotating Frame Modelling and Analysis 250 Q-limit Steady-State I=60A I=160A

Torque [Nm]

200 150 100 50 0 0

100

200

300

400

500

600

Slip speed [rad/s] Fig. 3.26 Slip speed to torque curves

result in violation of the flux-limit at the machine free rotating condition (s = 0) due to the negligence of resistances. On the other hand, MTPA results in the maximum torque when phase current is constant. The torque profiles on constant I-circles are also given in Fig. 3.26. Comparing Q-limit and I-limit torque profiles, we could conclude that: (1) Torque profiles of Q-limit and I-limit are very different; (2) Q-limit torque reaches maximum ∗ = Rr /(δL r ); (3) I-limit torque reaches maximum on on MTPF with slip speed ωsp ∗ = Rr /L r , which is a much lower slip speed; (4) The MTPA with slip speed ωsp torque profiles for different I-circles are the same, but the maximum torque Temmax is 2 . proportional to Imax As an extension to Fig. 3.25 for further illustration on the shaded region of ωr < 0, Fig. 3.27 is given. In the figure, blue lines represent the constant slip speed ωsp . The marked numbers show the slip speed in rad/s. Similarly, black ellipses are constant stator speed (or operating frequency) contours. The thick red line is the locus that ωs = ωsp

or,

ωr = 0

above which, it implies reverse rotating of the electric machine. Therefore, ωs = ωsp is an operation limit. Operating on ωs = ωsp line would result in the maximum torque at standstill. However, the phase current is so high along this curve, making this limit impractical. Nevertheless, Fig. 3.27 does suggest that the pair of (ωs , ωsp ) could be employed as a coordinator to interpret and analyze the electric machine performance at least from the mathematics point of view, since the combination of ωs and ωsp (or the slip s) can span the whole space for the electric machine operation, the same as what the combination of Id and Iq can do. This is one of the foundations for the scalar control as will be discussed in Chap. 5. By comparing Figs. 3.25 and 3.26, we can observe an interesting fact that the slip speed variation along Q-limit and I-circle is very nonlinear. For slip speed From 0

3.8 Maximum Torque per Flux

117

Fig. 3.27 Speed contours [Zero rotor speed curve]

rad/s to 118 rad/s, it spans from d-axis to the MTPF line. However, from 118 rad/s to 600 rad/s, it only spreads from the MTPF line to the s = 1 line in Fig. 3.25, which is a very narrow area. Therefore, the slip speed is much more sensitive at MTPF than that at MTPA. As a summary of MPTA, MPTL and MTPF characteristics, and of the use of these characteristics for the electric machine operation, Fig. 3.28 is presented. The electric machine operation must satisfy the following constraints: • Current limit (or I-Limit) Imax : This limit is presented by Eq. (3.85). The electric machine must be under the current limit. Otherwise, the switch devices and/or stator windings can be damaged; • Voltage limit (or V-Limit) Vmax : This limit is presented by Eq. (3.97) and is an actual physical limit. On this limit, the system has a strong tendency for oscillation, due to the coupling nature between the field and quadrature channels as given in Eq. (3.65). Thus, some headroom from the voltage limit shall be reserved; • Flux limit (or Ψ -Limit) Ψmax : This limit is presented by Eq. (3.97). The electric machine shall be under the flux limit. Otherwise, excessive heat will be generated.

3 Rotating Frame Modelling and Analysis

Voltage Limit Contour

118

MTPA Peak Torque

T5 P1

P2

P3

T4

T3 T2

T1

P4

MTPL 1 2

Base Speed

Full Flux Weakening Speed

3

MTPF

4

Fig. 3.28 Operation curves

The current limit is indicated in this figure by a black circle. The constraint contours of voltage limit, which vary with the operating frequency (or speed) ωs , are given by a family of blue ellipses. The cyan ellipses are the power loss contours. These ellipses as well as MTPL line change according to the operating frequency. The machine torque contours are presented by magenta hyperbolic contours (Only in the quadrant of Id > 0 and Iq > 0). ωx (x = 1, 2, 3, 4) are the different operating speeds and ω1 > ω2 > ω3 > ω4 , while Px , (x = 1, 2, 3, 4) the different power losses and P1 > P2 > P3 > P4 , and Tx , (x = 1, 2, 3, 4, 5) the different torques and T1 > T2 > T3 > T4 > T5 . A summary of the MTP‘X’ characteristics and the relevant electric machine operations are as below: S6:

S7:

The maximum machine torque Temmax is obtained by MTPA subject to the current limit. This maximum torque can be maintained as long as the operating frequency ωs < ωb , the base speed; The voltage limit ellipses vary with operating speed ωs . The encircled area of the limit ellipse tightens with the increase of ωs . At the base speed of ωb (ωb = ω4 in the figure), voltage limit ellipse joints the current limit circle at the MTPA line. When the speed gets sufficiently high, voltage limit ellipse

3.9 Speed-Torque Characteristics and System Operations

S8:

S9:

S10:

S11: S12: S13:

S14:

119

will touch the current limit circle at the MTPF line. This speed level is called full flux weakening speed ω f (ω f = ω1 in this figure); Below the base speed (i.e. ωs < ωb ), the current limit and flux limit dominate the system. It is assumed that the flux limit is a much more relaxed limit compared to the current limit. Thus it is not shown in this figure. Above the base speed(i.e. ωs > ωb ), voltage limit becomes effective. Nevertheless, both the current and voltage limit constraints must be satisfied in all cases. For instance, at base speed ωb , a larger i sd is prohibited by the voltage limit on top of the current limit constraint; Below the base speed ωb , the constant magnetizing flux linkage Ψm can be maintained since Vs = Ψm ωs < Vmax . Above the base speed ωb , Ψm = Vmax /ωs . Therefore, the air gap magnetic field strength has to be continuously reduced along with the operating frequency. This is the flux weakening; At the speed range of ωb < ωs < ω f , the electric machine can be operated max along the current limit circle toward MTPF line. The maximum torque Tem now is a function of and reduces with the operating speed ωs . Along the current limit circle, the torque decrease rate is minimized. For instance, if the system follows along the MPTA line to confine with the voltage limit with the increase of speed, the max toque will reduce at a rate of the square of the phase current; When ωs > ω f , in order to maximize the machine torque Tem , current i sd and i sq shall be regulated to stay along the MTPF line; The maximum torque curve is determined by, and is made up by parts of, the MTPA line, the current limit circle and the MTPF line; In general, the maximum torque curve is not aligned with the MTPL line at which the system efficiency is optimized. This said, the maximum torque is obtained with sacrifice of the system efficiency; The MTPL line will rotate towards the q axis with the increase of the operating speed as the iron loss depends on the operating frequency. This is to say that high speed flux weakening can also benefit the system efficiency.

3.9 Speed-Torque Characteristics and System Operations The speed to torque characteristics were analyzed based on the equivalent circuit model in Sect. 2.3. These characteristics will be analyzed again in this section, but based on the steady-state model in the dq frame. The focus of this analysis is to characterize the constant torque, constant power and constant speed power operation ranges of the electric machine. Figure 3.29 illustrates the possible operations of the electric machine in the dq frame. In this figure, • The black circle represents the constant current limit. The slip s = 0 when i sq = 0, and s = ∞ when i sd = 0. The slip increases with i sq along the current limit circle.

120

•

•

•

•

3 Rotating Frame Modelling and Analysis

The radial lines from the origin such as the lines of MTPF, MTPA and MTPL represent different constant slip speeds. Slip speed ωsp is 0 along the d-axis, ωsp = Rr /L r along the MTPA line and ωsp = Rr /(δL r ) along the MTPF line. Along the current circle, the slip (as well as the slip speed) varies with strong nonlinearity. The slip s spans from s = 0 on the d-axis to about s = 0.1 on the MTPF line. However, it increases abruptly from 0.1 to ∞ just from the MTPF line to the q-axis, taking up a very narrow region of the current limit circle. The family of blue ellipses represent the current limit profiles induced by the voltage limit in accordance with different operating speeds. The profile passes through Point A when the operating speed ωs reaches the base speed ωb , while it passes through Point B when ωs = ω f . The higher the operating frequency (or speed), the narrower the permissible current range. The magenta hyperbolic contours represent constant torque at different levels. ∗ is the torque contour tangent to V-limit ellipse at speed ωb . Note that current T peak ∗ is not durable but goes beyond the I-limit at this operating point. Therefore, T peak only available in a very short period. T peak is the torque contour tangent to I-limit circle. At this level, the inverter and motor temperature will rise quickly. This peak torque is not intended to sustain a ∗ . long time, although it can run longer than T peak Tcont is a torque level that can be sustained and is therefore called continuous torque. This continuous torque level depends on the thermal system behavior; MTPF, MTPA and MTPL are illustrated. Point A on MTPA is the only point permitted by I-limit for T peak operation. There are infinite points within I-limit for Tcont operation, but Point C on MTPL can minimize the losses. However, bear in mind that MTPL line is variant wrt operating speed. Point D on MTPA can minimize the required current. Point B is the joint of the I-limit and V-limit at the operating speed ωs = ω f . It is critical since this point separates the operation principle to obtain maximum torque under the voltage and current limits. Before this point when ωs < ω f , the maximum torque is attained along the I-limit curve. However, after ωs > ω f , the maximum torque is obtained on the MTPF line.

Adopting the optimization approach, the solution to maximum torque under both the V- & I-limits are summarized as below: Range 1:

Working on Point A when ωs < ωb . Tem = cnst

thus

Pem = Tem ωsm

Therefore it is a constant torque operation. The power11 will increase with the operating speed linearly; 11

Pem here (for simplifying the analysis) is the net power flowing into (or out of) the rotor, which is actually Pag according to Eq. (2.22). The mechanical power is as Pm = Tem ωr . The losses associated with the rotor define the difference between Pem and Pm . ωr = (1 − s)ωsm and ωs = N p ωsm . Thus as long as s is small (which is true for most operation of an ACIM), there is not much difference between the two.

3.9 Speed-Torque Characteristics and System Operations

121

Fig. 3.29 Electric machine operations

Range 2: Working on the I-limit circle alone when ωb < ωs < ω f . The profiles of the torque and power wrt speed are not clear yet. Further analysis is provided in this section. Range 3: Working on the MTPF line alone when ωs > ω f . Along MTPF, i sq depends linearly on i sd according to Eq. (3.98), and the machine torque is governed by Eq. (3.99). Recall and tidy up this torque expression as Tem ωs2 =

2 K T L m Vmax 2δL 2s

Clearly, Pω = cnst in this range. Note that Pω = cnst not only along the MTPF line, but along the MTPA and MTPL lines as well. As a matter of fact, any line of i sq = ki sd , with k being a factor, will lead to Pω = cnst. 1 The solutions are indicated in Fig. 3.29 as Operation . In order to analyse the speed-torque characteristics in Range 2, Fig. 3.30 is given, in which further three different cases are presented for investigation. Prior to this, the base speed and full flux weakening speed are determined first. Both require the knowledge of the V- & I-limits, whose curves are governed by the following equations: 2 2 2 + i sq = Imax C I lim (i sd , i sq ) −→ i sd 2 2 CV lim (i sd , i sq ) −→ i sd + δ 2 i sq =

2 Vmax L 2s ωs2

122

3 Rotating Frame Modelling and Analysis

Fig. 3.30 Constant power operation

The base speed ωb is when the V-limit eclipse joints the I-limit circle at the MTPA line. Thus, to satisfy both CV lim (i sd , i sq ) and MTPA equations, it yields Vmax ωb = L s Imax



2 1 + δ2

(3.100)

Similarly, the full flux weakening speed ω f is when the V-limit joints the I-limit at the MTPF line. Solving C I lim (i sd , i sq ) and MTPF equations, ω f arises as Vmax ωf = L s Imax



1 + δ2 2δ 2

(3.101)

B the system moves along a constant i sq line to the left-hand side of For Case , the figure. √ 2Imax = cnst i sq = 2

However, i sd reduces along the line, leading to a reduction of the torque. The operation is away from and under the I-limit, thus only the V-limit from CV lim (i sd , i sq ) becomes effective. The resulted power becomes  2 /(2L 2 ) − δ 2 I 2 ω2 /4 Pem = K T L m Imax Vmax s max s B the power decreases with the operating speed. Clearly, with Case , 1 the system moves along the I-limit circle towards the left-hand side For Case , of the figure. Current i sd and i sq must satisfy both the V- & I-limits defined by C I lim (i sd , i sq ) and CV lim (i sd , i sq ) respectively. Machine torque then becomes

3.9 Speed-Torque Characteristics and System Operations

123



 2 2 /(L 2 ω2 ) 2 /(L 2 ω2 ) − δ 2 I 2 Imax − Vmax Vmax s s s s max Tem (ωs ) = K T L m 1 − δ2 1 − δ2

    i sd i sq Based on the torque contours, it is obvious that torque will decrease when the system moves along the I-limit circle via the designated path. However, speed will increase in the meantime. It is then not obvious regarding how the power changes with the operating speed. Nevertheless, by substituting Eqs. (3.100) and (3.101) into the above equation respectively, we can find that the power at the base speed and full flux weakening speed are the same. Tem (ωb )ωb = Tem (ω f )ω f

(3.102)

or PA = PB Points A and B are indicated in both Figs. 3.29 and 3.30. Furthermore, as a matter of fact, it can be generalized that for any constant I-limit circle that joints MTPA and MTPF at Point X and Point Y, the power at these two points will be the same, namely Px = Py . To see how the power varies with the operating speed, ∂ Pem /∂ωs could be derived and analyzed. This process is tedious and omitted. Some useful conclusions are given as below ∂ Pem  ω =ω > 0 ∂ωs s b and

∂ Pem  ω =ω < 0 ∂ωs s f

This implies that: • • • •

The power along the I-limit circle is not constant; The power near MTPA increases with speed; The power near MTPF decreases with speed; There exist at least one maximum power point along the I-limit circle.

B and that the power Combining the facts that the power decreases in Case  1 near MTPA, there must exist a case between Case  1 and Case increases in Case  B so that the power remains constant in spite of speed variation.  Let us introduce an unknown constant K such that

i sq i sd ωs = K Then Power Pem will then become constant. This condition shall also be held at ωb on the I-limit, yielding

124

3 Rotating Frame Modelling and Analysis Current Curves

220 200 180 Constant Power

Current Id / Iq [A]

ωb

Voltage Limit Contour

ωf

A

MTPA

160 140 120 100 80 60 40

MTPF

20 200

400

600

800

1000

1200

1400

Operating Speed [rad/s]

(a) Operation profile

(b) isd and isq

Fig. 3.31 Constant power curve

2 K = Imax ωb /2

Therefore, if i sd and i sq satisfy the following conditions, it will result in a constant power operation. 2 ωb 2i sd i sq ωs = Imax 2 (3.103) V 2 2 i sd + δi sq = 2max2 L s ωs The solution of the above equation is given by

2 = i sd

2 i sq =

2 Vmax +

2 Vmax −

 

4 − I 4 δ 2 ω2 ω2 L 4 Vmax max b s s

2L 2s ωs2

(3.104)

4 − I 4 δ 2 ω2 ω2 L 4 Vmax max b s s

2δ 2 L 2s ωs2

Unsurprisingly, ωb and ω f as defined by Eqs. (3.100) and (3.101) satisfy the above equation since this solution starts from Point A and ends up at Point B. An example of constant power operation is given in Fig. 3.31a. Basically, a constant power contour in red color is added on top of the information given in Fig. 3.28. This figure looks less busy as power loss contours are removed and less levels of torque and V-limit contours are presented. It is clear that the constant power contour joints V- & I-limits at MTPA when ωb = ωs and at MTPF when ωb = ω f . Compared 2 in Fig. 3.30, this constant power to the illustrated constant power contours as Case  curve is indeed under the I-limit circle. Given in Fig. 3.31b are the corresponding current i sd and i sd . Nevertheless, the maximum torque in Range2 is along the I-limit circle and it is not constant power operation. Constant power operation in Range 2

3.9 Speed-Torque Characteristics and System Operations

125

is available, but it is a controlled characteristic. The speed-torque characteristics of different operation range are re-called as Range 1:

Constant torque range. Tem = cnst for ωs ≤ ωb

Range 2:

Constant power range (Controlled behaviour). Pem = cnst for ωb < ωs ≤ ω f

Range 3:

Constant power speed range. Pem ωs = cnst for ωs > ω f

This analysis agrees with what was presented in Sect. 2.3. The detailed profiles for torque, power, current, and flux linkage are given in Fig. 3.32. With the above analysis of speed-torque characteristics, and along with the operation cases given in Fig. 3.29, Fig. 3.32 presents the induction machine operating characteristics. Note that definitions of the illustrated operation cases are the same for Figs. 3.29 and 3.32 as 0 Case : 1 Case :

2 Case :

3 Case :

4 Case :

5 Case :

The system is operated on MTPF with the V-limit constraint. The very ∗ applies; short period of T peak The system is operated on MTPA under the I-limit constraint, and then along the I-limit circle for applying the V-limit constraint when ωs > ωb , and finally along the MTPF line for ωs > ω f ; The system is operated on MTPA under the I-limit constraint, and then along the constant power curve for applying the V-limit constraint when ωs > ωb , and finally along the MTPF line for ωs > ω f ; The system is operated on MTPA, and then along the constant torque curve for applying the V-limit constraint at high speed, and finally along the MTPF line once jointed; The system is operated on MTPA, and then along the constant power curve for applying the V-limit constraint at high speed, and finally along the MTPF line once jointed; The system is operated on MTPL, and then along the constant Iq curve for applying the V-limit constraint at high speed, and finally along the MTPF line once jointed.

A summary of the characteristics of the above-mentioned operations is as below S15:

∗ 0 provides the very high torque T peak , but only for a very Operation case  limited period. Once ωs > ωb , flux weakening applies and results in

126 Fig. 3.32 Speed torque characteristics

3 Rotating Frame Modelling and Analysis

3.9 Speed-Torque Characteristics and System Operations

Ψm ∝ 1/ωs Pem ∝ 1/ωs S16:

S17:

127

Is ∝ 1/ωs Tem ∝ 1/ωs2

This is Pω = cnst operation; 1 it is a constant torque operation as long as ωs < ωb . For operation case , The peak torque T peak is still not sustaining, but it can hold for a longer ∗ . For range ωb < ωs < ω f , it is neither constant torque nor time than T peak constant power operation based on the aforementioned analysis. However, along the I-limit circle, it results in a maximum torque operation under the constraint of both V- & I-limits. The maximum torque operation joints the MTPF line once ωs ≥ ω f ; 2 differs from case  1 at range ωb < ωs < ω f . It is a constant Operation case  power operation in this range: Pem = cnst

S18:

S19:

S20:

Bear in mind that the magnitude of the phase current is not constant. 3 The system starts at Point D on the MTPA line, which For operation case . is within both the I-limit and V-limit. The design of this torque Tcont is to make it sustainable (though the actual level Tcont depends on the thermal system behavior). It is a constant torque, constant current, and constant flux operation. The system continues along the constant torque curve at the boundary of the V-limit constraint till it reaches the MTPA operation. The phase current increases in this range while the flux decreases; 4 differs from case  3 in the medium speed range. The Operation case  constant power operation is demonstrated in here. Unlike constant power 2 it is a more sustainable case; operation in case , 5 illustrates a more efficient operation. The system operates on MTPL Case  as long as the V- & I-limits permit. It is again a constant torque, current and flux operation. When V-limit becomes active, it then transitions to a constant Id operation.

Comparing to the results shown in Fig. 2.29, taking continuous torque and continuous power operation as an example, attention is paid to some differences as below during the constant power range: • Current is constant based on the equivalent model analysis, but it is not strictly true for the dq model; • Flux is inversely proportional to speed on the former analysis, while the flux is reduced but not inversely proportional to speed for the dq model analysis. Over simplification on the magnetizing current of the equivalent model as given in Fig. 2.14 has significant contribution to the above discoveries. It is one of the differences between the equivalent model and the dq model in steady-state analysis.

128

3 Rotating Frame Modelling and Analysis

References 1. Abootorabi Zarchi H, Mosaddegh Hesar H, Ayaz Khoshhava M (2019) Online maximum torque per power losses strategy for indirect rotor flux-oriented control-based induction motor drives. IET Electric Power Appl 13(2):259–265 2. Bozhko S, Dymko S, Kovbasa S, Peresada SM (2016) Maximum Torque-per-Amp control for traction IM drives: theory and experimental results. IEEE Trans Ind Appl 53(1):181–193 3. Faiz J, Mohseni-Zonoozi SH (2003) A novel technique for estimation and control of stator flux of a salient-pole PMSM in DTC method based on MTPF. IEEE Trans Ind Electron 50(2):262– 271 4. Holton JR (1973) An introduction to dynamic meteorology. Am J Phys 41(5):752–754 5. Jokinen T, Hrabovcova V, Pyrhonen J (2013) Design of rotating electrical machines. Wiley 6. Kirk DE (2004) Optimal control theory: an introduction. Courier Corporation 7. Mawardi OB (1957) On the concept of coenergy. J Franklin Inst 264(4):313–332 8. Miguel-Espinar C, Heredero-Peris D, Gross G, Llonch-Masachs M, Montesinos-Miracle D (2020) Maximum torque per voltage flux-weakening strategy with speed limiter for PMSM drives. IEEE Trans Ind Electron 68(10):9254–9264 9. Novotny DW, Lipo TA (1996) Vector control and dynamics of AC drives, vol 41. Oxford University Press 10. O’Rourke CJ, Qasim MM, Overlin MR, Kirtley JL (2019) A geometric interpretation of reference frames and transformations: dq0, clarke, and park. IEEE Trans Energy Convers 34(4):2070–2083 11. Pellegrino G, Bojoi RI, Guglielmi P (2011) Unified direct-flux vector control for ac motor drives. IEEE Trans Ind Appl 47(5):2093–2102 12. Rashad E, Radwan T, Rahman M (2004) A maximum torque per ampere vector control strategy for synchronous reluctance motors considering saturation and iron losses. In: Conference record of the 2004 IEEE industry applications conference, 2004. 39th IAS annual meeting, vol 4. IEEE, pp 2411–2417 13. Sen PC (2021) Principles of electric machines and power electronics. Wiley 14. Uddin MN, Nam SW (2008) New online loss-minimization-based control of an induction motor drive. IEEE Trans Power Electron 23(2):926–933 15. Wasynczuk O, Sudhoff S, Corzine K, Tichenor JL, Krause P, Hansen I, Taylor L (1998) A maximum torque per ampere control strategy for induction motor drives. IEEE Trans Energy Convers 13(2):163–169

Chapter 4

Induction Machine Dynamics Analysis

4.1 Rotor Field Oriented Model The system equation as given in Eq. (3.56) in Chap. 3 is not suitable for dynamic analysis, further transformation is required. There are different choices on selection of the state variables, but this only affects the way how the system is expressed and shall not affect the result of the system dynamics. The set of state variables can be either of the following options: V1: V2: V3: V4:

Taking (Is , Ir ) as state variables.  s ,  r ) as state variables [5]. Taking ( Taking (Is ,  r ) as state variables [1, 3, 6].  s , Ir ) as state variables. Taking (

Selection of V1 or V2 benefits from simple transformation and easy understanding of the system. The drawback is that only the state Is is measurable. For Selection V4, not only complicated transformation of the model equations is required, but all the state variables are unmeasured as well. Selection V3 has the advantage of being rotor field-oriented, that is, the d-axis of the dq frame is aligned with the rotor flux. Thus, Selection V3 is adopted in this book. Based on Eq. (3.56), the rotor current Ir can be written as Ir =

 r − L m Is Lr

The rotor dynamics equation then becomes r d Rr = (L m Is −  r ) − jsωs  r dt Lr

© Springer Nature Switzerland AG 2024 S. Shen and Q.-z. Chen, Practical Control of Electric Machines for EV/HEVs, Lecture Notes in Electrical Engineering 1064, https://doi.org/10.1007/978-3-031-38161-4_4

129

130

4 Induction Machine Dynamics Analysis

Denoting the rotor time constant by τr , this dynamics expression arises as τr

r d r +  r = L m Is − jτr (ωs − ωr e ) dt

(4.1)

where τr = L r /Rr . Given the leakage factor defined in Eq. (3.72), the stator flux linkage  s can be rewritten as Lm r  s = δL s Is + Lr Substituting the above equation into the first equation in (3.56) yields δL s

  r L m d dIs Lm + = Vs − Rs Is − jωs δL s Is + r dt L r dt Lr

Tidying up and combining with the rotor dynamics Eq. (4.1), we get the stator dynamics equation as 1 dIs Rs + Rr L 2m /L r2 1 Lm = Vs − Is − jωs Is + dt δL s δL s δL s L r



 Rr − jωr e  r Lr

Here we define the following parameters: the rotor to stator coupling factor κr , the overall equivalent resistance Req , the leakage inductance L δ , and the torque time constant τT respectively as κr = L m /Lr Req = Rs + κr2 Rr L δ = δL s τT = L δ /Req The expression of stator dynamics then becomes τT

dIs 1 κr + Is = Vs − jτT ωs Is − (jτr ωr e − 1)  r dt Req Req τr

(4.2)

Equations (4.1) and (4.2) define and govern the system dynamics. The structure of this dynamic model is given in Fig. 4.1, where striped connectors indicate vector signal flows (eg Vs , Is and  r ), while thin lines imply scalar signal flows (eg ωs , ωr , Tem ). Blocks symbolized by a filter curve pattern represent a general first-order system as dx +x =u τ dt with u being a general input, x a general state variable, and τ a time constant. Also, bear in mind that the symbol | × | denotes cross product of vectors. It can be seen

4.1 Rotor Field Oriented Model Kr/Req

131

r

+

-

Is Vs 1/Req

+

-

Is

T

Lm

j Stator Model

+

|X|

j Rotor Model

Tem

r

-

T

X

r

r

KT

r

X

X sp

s

+

-

rm

Np

re

Fig. 4.1 Dynamic model structure [6]

that the stator dynamics is dominated by the time constant of τT , while the rotor dynamics by the time constant of τr . We recap the parameters given in Table 3.1, and τT and τr are τT = 6.8 ms τr = 83.7 ms respectively. Clearly, the time constant of rotor dynamics is in the order of 10 times higher as compared to that of the stator dynamics (or torque dynamics). Therefore, if the induction motor torque control is done by regulating the rotor flux (linkage),1 it will take a much longer time to achieve the desired torque. On the contrary, if the rotor flux can be kept constant, while having torque controlled via regulating the stator current, then the torque response will become much faster. As it will be shown in the later part of this book, the vector torque control strategy achieves a fast torque response (in about less than 10 ms) by regulating the quadrature current independently and by keeping the field current constant in the meantime. This is basically controlling the phase voltage to have a specific direction wrt the field flux. However, scalar control strategy has no such capability, but rather results in varying both the stator current and rotor flux in the meantime. Thus, torque response with the scalar control method is significantly more sluggish, with the time constant being as slow as 100 ms in the given example. In the above, Eqs. (4.2) and (4.1) are given in a complex vector format, which can be expanded as below

1

We use the term ‘flux’ for ‘flux linkage’ in this chapter if it does not necessarily have to be mentioned by ‘flux linkage’ explicitly.

132

4 Induction Machine Dynamics Analysis

Ψrq vsd di sd κr Ψr d 1 = +ωs i sq + κr ωr e + − i sd dt Lδ Lδ τr L δ τT vsq di sq Ψr d κr Ψrq 1 = −ωs i sd − κr ωr e + − i sq dt Lδ Lδ τr L δ τT          Back EMF

Coupling

(4.3)

Dynamics

and dΨr d = +(ωs − ωr e )Ψrq +(L m i sd − Ψr d )/τr dt dΨrq = −(ωs − ωr e )Ψr d +(L m i sq − Ψrq )/τr       dt Back EMF

(4.4)

Dynamics

The second and third terms in the right-hand side of Eq. (4.3) demonstrate the back EMF induced current. The fifth term relates to the stator current. The dynamics of the stator and rotor are coupled as shown in Fig. 4.1 [6]. The stator injects a flux via the mutual inductance L m to the rotor. In response, the rotor feedbacks its EMF to the stator (in a complex way) by the same mutual inductance L m . Similar to Sect. 3.6, the analysis can be further simplified using the rotor fieldoriented modelling approach such that 

Ψr d = Ψr Ψrq = 0

Then, the expression of the rotor dynamics as in Eq. (4.4) reduces to dΨr = (L m i sd − Ψr )/τr dt

(4.5)

In the meanwhile, the second expression in Eq. (4.4) turns into an algebraic constraint as τr (ωs − ωr e )Ψr = L m i sq or ωsp =

L m i sq τr Ψr

(4.6)

with ωsp = ωs − ωr e . This is equivalent to specify that the operating frequency ωs has to be regulated according to the rotor speed in order to align the d-axis with the rotor flux. Note that this equation is exactly the same as Eq. (3.79) in steady sate. However, they are different during transient since Ψr = L m i sd . The state dynamics becomes

4.1 Rotor Field Oriented Model Kr/Req

133

r

-

+

r

Vs +

1/Req

-

Is

T

j

Isd Isq

r

Lm

sp =

T

j

r

X

I sq Lm r

r r

Stator Model

X

Rotor Model s

rm

Np

sp

Isq

X

KT

Tem

+ + re

Fig. 4.2 Field-oriented model structure

vsd di sd κr Ψr i sd = + ωsp (i sq , Ψr ) + ωr e i sq + − dt Lδ τr L δ τT

vsq di sq Ψr i sq = − ωsp (i sq , Ψr ) + ωr e i sd −κr ωr e − dt Lδ L δ τT

(4.7)

in which the slip speed is a function of quadrature current and rotor flux linkage. Even if taking ωr e as an operating parameter, it is still clear that the system represented by Eqs. (4.5), (4.6) and (4.7) is nonlinear. The nonlinearity lies in the slip speed part in the equation. Its stability and dynamics are not so straightforward any more. If further accounting for the rotor mechanical dynamics, ωr e has to be treated as a state variable. The stability and dynamics analysis become even more complicated. In most applications, the rotor mechanical dynamics is slow and stable. This is particularly true for the electric vehicle applications as the rotor is rigidly connected to the vehicle, which has a massive inertia. In these cases, it is not unreasonable to consider ωr e as a parameter rather than a stable variable, and consequently this concept is adopted in this book. Based on Eqs. (3.78) and (3.86), the machine torque in the rotor field-oriented model can be simplified as Tem = K T Ψr i sq

(4.8)

Before moving forward to further dynamic analysis, provided is Fig. 4.2, which is an updated version of Fig. 4.1 to reflect the rotor field-oriented modelling. It is explicitly indicated that the field current i sd governs the rotor flux but through the path of the rather slow rotor dynamics, and that the quadrature current i sq plays a direct role in producing the machine torque. Also, there is an extra feedback path in accordance with the requirement of rotor field orientation. The dynamic slip speed depends on the rotor flux, which in turn determines the stator winding operating frequency. Note that this extra feedback loop is closed by system control, but not an intrinsic nature of the induction motor itself. Hence, this extra feedback loop will be excluded for the system dynamic analysis in this chapter.

134

4 Induction Machine Dynamics Analysis

The following comments on the machine torque implementation are made according to Fig. 4.2, the dynamic Eqs. (4.5) and (4.7), and the FOC torque equation (4.8): M1: Torque Tem can be achieved by varying i sq while keeping rotor flux linkage Ψr unchanged. Invariation of the rotor flux linkage is equivalent to fixing the field current i sd . This is the so-called fast torque path (time constant τT = 6.75 ms for the the example used in this book); M2: Torque Tem can be equally achieved by varying Ψr which is equivalent to varying i sd , while keeping i sq unchanged. This is the so-called slow torque path (time constant of τr = 83.75 ms for the given example machine); M3: Torque Tem can be further achieved by varying both i sq and Ψr (or i sd ). This is generally the case during flux weakening range. It is noteworthy that the strategies summarized in Sect. 2.3.4 regarding regulation of slip speed and phase voltage for controlling torque are not equivalent to those listed in the above. Regulation of slip speed to govern the machine torque could be implemented in different ways. If rotor flux is kept constant, slip speed will then depend linearly on the quadrature current i sq . Under such condition, the regulation of slip speed becomes similar to the above-mentioned method M1. However, regulation of the magnitude of the phase voltage, which is the method M2 in Sect. 2.3, will generally result in variation of both the field and quadrature current. However, both M1 and M2 in this section imply a regulation of not only the magnitude but also the direction (or the phase angle) of phase voltage. This is one of the advantages with the utilization of the dq-frame model.

4.2 System Dynamics Analysis For system stability and dynamics analysis, the 4-state (i sd , i sq , Ψr d and Ψrq ) equation as given by (4.3) and (4.4), rather than the 3-state equation (i sd , i sq and Ψr ) as given by (4.7) and (4.5), is explored. The matrix format of the 4-state dynamics equation is as below, taking ωs and ωr e as parameters ⎤⎡ ⎤ ⎤ ⎡ 1 κr ωr e κr − τT ωs i sd i sd τr L δ Lδ κr ⎥⎢ ⎥ ⎢−ω − 1 − κr ωr e ⎥ d ⎢ i s ⎥ ⎢ i sq ⎥ τT Lδ τr L δ ⎢ sq ⎥ = ⎢ + ⎥ ⎢ Lm 1 ⎦ ⎣ Ψ 0 − τr ωs − ωr e ⎦ ⎣Ψr d ⎦ ⎣ τr dt rd Ψrq Ψrq 0 Lτrm ωr e − ωs − τ1r          x˙ A x ⎡

⎡

1 Lδ

0

⎤

  ⎢ 0 1 ⎥ vsd ⎢ Lδ ⎥ ⎣ 0 0 ⎦ vsq    (4.9) 0 0    B u

The dynamics and stability of the open-loop system are now determined by the eigenvalues and eigenvectors of its characteristic equation of matrix A as det(λI − A) = 0

4.2 System Dynamics Analysis

135

f=100Hz

Rotor mode

f=0Hz

Stator frequency when decoupled

Rotor frequency when decoupled

f=100Hz

f=0Hz

Stator mode

Fig. 4.3 Root locus for the 3-state model of different operating frequencies

in which det(·) is the matrix determinant calculator and the solution of det(λ) = 0 is the eigenvalues of the system. Figure 4.4 shows the root locus of the 4-state model. The damping ratio as well as the time constant of this model is also shown in Fig. 4.5. For comparison purpose, the root locus for the 3-state model is given in Fig. 4.3. All these results are conducted with the slip of s = 0.01, and with the operating frequency from 0 to 100 Hz. Bear in mind that the root locus of the 3-state model is based on the equations of (4.5) and (4.7), in which the constraint of Eq. (4.6) must be satisfied by means of feedback control. The results in Fig. 4.3 are derived by relaxing the freedom between ωs and ωr e . It clearly indicates in Fig. 4.4 that the system has two modes: one mode with high frequency being associated with the stator dynamics, and the other associated with the rotor dynamics. These two modes are coupled, and this coupling grows with the operating frequency. Comparing the root locus of the 3-state and 4-state models, the difference is immediately shown. The 4-state model shows that the stator mode is well damped, while the rotor mode is very lightly damped at high operating frequency. On the contrary, the 3-state model demonstrates a completely different feature. The rotor dynamics is well damped, while the stator dynamics becomes under-damped at high operating frequency. Furthermore, there is much less overall damping for the 4-state model than the 3-state model at the same frequency when it is high. Also, the overall system frequency (determined by the low frequency of the associated modes) of the 4-state model is much lower than that of the 3-state model at high frequency. The key of these differences is the introduction of the slip speed feedback path which is assumed to be infinitely fast, and these differences indicate that the 3-state model is not suitable for dynamics analysis.

136

4 Induction Machine Dynamics Analysis

f=100Hz

f=0Hz

f=100Hz

Stator mode

f=35Hz

Rotor mode

Stator frequency when decoupled

Stator frequency limit at high operating frequency

f=0Hz

f=23Hz

Rotor frequency when decoupled

Rotor frequency limit at high operating frequency

Fig. 4.4 Root locus for the 4-state model of different operating frequencies

Rotor mode Time constant Local maximum damping ratio at f=23Hz Stator mode Time constant

Fig. 4.5 System damping and time constant for different operating frequencies for the 4-state model

Listing 4.1 Dynamics analysis for 4-state model 1 2 3 4 5 6 7

Freq ws Vdc vs

= = = =

100; 2* pi * Freq ; 330; Vdc / sqrt (3) ;

% For Y c o n n e c t i o n and SPWM % Vpn = Vdc * sqrt (3) /3 % Vpn_rms = Vpn / sqrt (2)

8 9 10

% % D e f i n e some c o m m o n p a r a m e t e r s k_r = Lm / Lr ;

4.2 System Dynamics Analysis 11 12 13 14 15

137

Req =( Rs + Rr *( k_r ) ^2) ; = 1 -( Lm ^2/( Ls * Lr ) ) ; tau _r = Lr / Rr ; tau _s = * Ls / Req ; L_lk = * Ls ;



16 17 18

% % Set slip ratio s = 0 .01 ;

19 20 21 22 23 24 25 26 27 28

% % I n i t i a l i z e m a t r i x AA A11 = -1/ ta u_ s ; A13 = k_r / tau_r / L_lk ; A21 = -1; A23 = - k_r *(1 - s ) / L_lk ; A31 = Lm / t au _r ; A33 = -1/ ta u_ r ; A41 = 0; A43 = -s ;

A12 A14 A22 A24 A32 A34 A42 A44

= = = = = = = =

1; k_r *(1 - s ) / L_lk ; -1/ tau_s ; k_r / tau_r / L_lk ; 0; s; Lm / tau_r ; -1/ tau_r ;

29 30 31 32 33

AA

= [ A11 A21 A31 A41

A12 A22 A32 A42

A13 A23 A33 A43

A14 ; ... A24 ; ... A34 ; ... A44 ];

34 35 36 37 38 39 40 41

%% Initialize FREQ = [ 0 : 1 : 1 0 0 ] ; N = l e n g t h ( FREQ ) ; EIGA = zeros (N ,4) ; WnF = z er os ( N ,4) ; DAMP = zeros (N ,4) ; TAU = z er os ( N ,4) ;

42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62

f i g u r e (2) hold on for I =1: N Freq = FREQ ( I ) ; ws = 2* pi * Freq ; AA (1 ,2) = A12 * ws ; AA (1 ,4) = A14 * ws ; AA (2 ,1) = A21 * ws ; AA (2 ,3) = A23 * ws ; AA (3 ,4) = A34 * ws ; AA (4 ,3) = A43 * ws ; eigA = eig ( AA ) ; EIGA (I ,:) = eigA ; [ Wn , Damp ] = damp ( AA ) ; WnF (I ,:) = Wn ; DAMP (I ,:) = Damp ; TAU (I ,:) = 1 . /( Wn. * Damp ) ; M K _ s i z e = ceil (10* I / N ) +2; plot ( real ( eigA ) , imag ( eigA ) , 'o ' , ' M a r k e r S i z e ' , M K _ s i z e ) ; end

63 64 65 66

% % Plot d e c o u p l e d e i g v e n value at Freq =0; Freq = 0; ws = 2* pi * Freq ;

138 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82

4 Induction Machine Dynamics Analysis

AA (1 ,2) = A12 * ws ; AA (1 ,4) = A14 * ws ; AA (2 ,1) = A21 * ws ; AA (2 ,3) = A23 * ws ; AA (3 ,4) = A34 * ws ; AA (4 ,3) = A43 * ws ; AA (1 ,3) = 0; % D e c o u p l e the s y s t e m AA (2 ,4) = 0; % D e c o u p l e the s y s t e m eigA = eig ( AA ) ; plot ( real ( eigA ) , imag ( eigA ) , 'd ' , ' M a r k e r S i z e ' ,8 , ... ' M a r k e r E d g e C o l o r ' , 'k ' , ' M a r k e r F a c e C o l o r ' , 'r ' ) ; grid on ; hold off ; tit le ( ' E i g e n v a l u e s - dq frame model ' ) ; y l a b e l ( ' I m a g i n a r y [ -] ' ) ; x l a b e l ( ' Real [ -] ' ) ; set ( gca , ' Box ' , ' on ' )

83 84 85 86 87 88 89 90 91

% % Plot d a m p i n g and time c o n s t a n t f i g u r e (1) s u b p l o t (1 ,2 ,1) ; plot ( FREQ , min ( DAMP ') ', ' -d ' , ' M a r k e r S i z e ' ,4) ; y l a b e l ( ' D a m p i n g r a t i o [ -] ' ) ; x l a b e l ( ' O p e r a t i n g f r e q u e n c y [ Hz ] ' ) ; grid on

92 93 94 95 96 97

s u b p l o t (1 ,2 ,2) ; plot ( FREQ , [ max ( TAU ') ', min ( TAU ') ']*1 e3 , ' -d ' , ' M a r k e r S i z e ' ,3) ; y l a b e l ( ' Time c o n s t a n t [ ms ] ' ) ; x l a b e l ( ' O p e r a t i n g f r e q u e n c y [ Hz ] ' ) ; grid on

Based on the results of the dynamics analysis as presented in Figs. 4.4 and 4.5, which are generated by the Matlab scripts as shown in List 4.1, the following observations are made: O01: The system has two modes, the high frequency mode is associated with the stator dynamics while the low one with the rotor dynamics; O02: The stator and rotor dynamics are coupled. This coupling is illustrated as in Fig. 4.1. With the increase of operating frequency, the stator mode dynamics becomes slow whereas the rotor dynamics turns fast. This effect is more obvious in the high operating frequency range due to a stronger coupling between the rotor and stator at high rotor speed as shown in Fig. 4.1; O03: As a comparison to the coupled system, the eigenvalues of the decoupled system are given in Fig. 4.4 indicated by red diamond markers. These eigenvalues correspond to the time constants of τT and τr respectively. The system is decoupled by setting κr = 0 and ωs = 0. κr = 0 decouples the rotor dynamics from the stator dynamics, while ωs = 0 decouples the interaction between the field (direct axis) and the quadrature components of the stator;

4.2 System Dynamics Analysis

139

f=80Hz

s=1 s=0

s=1

s=1

s=1

s=0 s=0

f=20Hz

s=0

Rotor mode Stator mode

Fig. 4.6 Root locus for different slips at f s = 20 Hz and f s = 80 Hz

O04: The bandwidth (of frequency) of the coupled system is determined by the dynamics of the slow mode, which is the rotor mode for the induction motor case. As shown in Fig. 4.5, the time constant of the rotor decreases so drastically in the low operating frequency range, and it then stabilizes at around τr = 21.3 ms. The coupled system will have a rather slow response because of this; O05: Decoupling of the system will lead to independence of the slow and fast modes. One benefit is to improve the system torque response by regulating the fast mode only. Furthermore, the vector control, which will be introduced in Sect. 6, is sort of indirect decoupled control. It governs the rotor dynamics by regulating the field current, while it controls the stator dynamics by regulating the quadrature current; O06: The stability of the coupled system is determined by the mode with the lowest damping. This is, once again, the rotor mode. The overall system damping ratio is shown in Fig. 4.5. The system is well-damped at low operating frequencies, but it becomes much less-damped at high frequencies. The damping ratio ξ is less than 10% when f s > 80 Hz. Hence, the system will be very sensitive to external disturbance and un-modelled errors. It is therefore required to improve the system stability at high motor speed by means of feedback control for instance. The above analysis indicates that the system dynamics varies with the operation frequency, or equivalently, with the machine rotating speed. Nevertheless, the above analysis is based on a fixed slip speed ratio. Analysis with varying slip speed is presented in the following.

140

4 Induction Machine Dynamics Analysis

Fig. 4.7 System damping and time constant for different slips at f s = 20 Hz Rotor mode Time constant

Stator mode Time constant

Fig. 4.8 System damping and time constant for different slips at f s = 80 Hz

Rotor mode Time constant

Stator mode Time constant

Figure 4.6 shows the root locus of the 4-state model with the slip speed ratio varying from s = 0 to s = 1. For comparison purpose, the root loci at both low frequency ( f s = 20 Hz) and high frequency ( f s = 80 Hz) are presented in the same figure. The damping ratio as well as the time constant of f s = 20 Hz is given in Fig. 4.7, while those of f s = 80 Hz are given in Fig. 4.8. When close attention is paid, one can see that the results of the damping ratio and time constants given in these two figures indeed agree with those in Fig. 4.5 for conditions of s = 0.01, and f s being 20 Hz and 80 Hz respectively. Further observations are as below: O07: The system dynamics clearly depends on the rotor slip speed (or slip ratio). The system tends to be more stable at low slip. The best stability is arrived at when rotor is synchronized with the stator. Therefore, operating the system at unnecessary high slip speed shall be avoided, as it not only greatly reduces the system stability but also sacrifices the machine efficiency in the meantime; O08: At high rotor speed, operating the machine at s = 1 is risky. For instance, the damping ratio is less than 1% when f s = 80 Hz as shown in Fig. 4.8;

4.2 System Dynamics Analysis

141

Motoring

f=80Hz

s=-1

s=-1

s=1

s=0

Rotor mode

Generation

f=20Hz s=1 s=-1 s=0

s=-1

Motoring

s=0

s=1

s=0

Stator mode

Generation Motoring

Generation s=1

Fig. 4.9 Root locus for different slips at f s = 20 Hz and f s = 80 Hz (including generation)

O09: The system response time or the frequency bandwidth depends on the slip speed as well. The system becomes slow at high slip, and this is another reason to avoid unnecessary high slip speed. Nevertheless, it is interesting to see that the time constant at both the high and low operating frequencies tends to be the same (or very close to each other) when s = 1. This is true for both the stator and rotor modes; O10: Tendency of the system being fast and stable at low slip is evident for both the high and low operating frequencies. Combining with the fact that the torque decreases at high slip as shown in Figs. 2.19 and 3.21. It is highly recommended that the system shall be operated within the slip speed range where torque is monotonously (and almost linearly) increasing wrt the slip speed. Based on these observations, it is generally true that the system is more stable and responsive when delivering low torque, but less stable and sluggish for delivering high torque. However, these observations so far are validated only for motoring torque (s > 0). Figure 4.9 is therefore given as an extended version of Fig. 4.6 to include the scenario of s < 0, which corresponds to the generation cases. Similarly, Fig. 4.10 shows the system damping and time constant including the generation scenario. Keep in mind that in Fig. 4.9, the trace with blue circle markers illustrates the root locus of the system at motoring state, while the red circles are for generating state. Further observations are summarized as below: O11: The system is much more stable during generation than motoring for the same operating frequency. This finding is evidenced by Figs. 4.9 and 4.10;

4 Induction Machine Dynamics Analysis

Generation

Motoring

Generation

Motoring

Time constant [ms]

142

Generation

Motoring

Time constant [ms]

f=20Hz

Generation

Motoring

f=80Hz

Fig. 4.10 System damping and time constant for different slips at f s = 20 Hz and f s = 80 Hz (Including Generation)

O12: The system damping ratio only slightly depends on the level of generation, by contrast to the way higher sensitivity in motoring. This is particularly true for high frequency operation; O13: Likewise, the system bandwidth only slightly depends on the level of generation, in particular at high frequency operation. This is a distinctive characteristics compared to the motoring. It can be concluded that the system dynamics is substantially better in generation than in motoring. Combining with the results given in Fig. 3.21, the system during generation has the benefits of: (1) being capable of delivering a higher torque; (2) being much more stable; and (3) featuring a much quicker response.

4.2 System Dynamics Analysis

143 Eigenvalues

dq frame model

Rr = 57m Rr = 47m Rr = 38m

Rotor mode

f=0Hz

Stator mode

f=100Hz

Fig. 4.11 Root locus of different Rr from 0 to 100 Hz Eigenvalues

dq frame model

Rotor mode

f=0Hz

Stator mode

f=100Hz

Fig. 4.12 Root locus of different Rs from 0 to 100 Hz

Other parameters that can and almost definitely will vary during operation are the rotor and stator resistances, which are temperature dependent. The root loci of the system with varying Rr and Rs at slip s = 0.01 are illustrated in Figs. 4.11 and 4.12, respectively. These results are derived simply by repeating the simulation test as in Fig. 4.4, but of course with different parameter settings for the rotor and stator resistances.

144

4 Induction Machine Dynamics Analysis

Figure 4.11 presents the root locus for Rr being 1.0, 1.25 and 1.5 times of the original Rr . The rotor mode dynamics is slightly affected by this variation, in particular at the frequency range around 20–40 Hz. The stator mode dynamics, on the other hand, is greatly dependent on Rr . Since the overall system performance is determined by the rotor mode, the system dynamic is more or less insensitive to rotor resistance variation. Figure 4.12 presents the root locus for Rs being 1.0, 1.25 and 1.5 times of the original Rs . Unlike the case of Rr variation, this time, the rotor mode dynamics is significantly impacted. The increase of Rs leads to a more stable and quicker system. These results suggest that the rotor mode is mostly affected by Rs while the stator mode by Rr . At a first glance of Eq. (4.9), impact of Rr on the stator mode of dynamics is understandable since Rr does affect both the stator and rotor time constants, respectively τT and τr . However, variation of Rs affects the stator time constant τT only. Therefore, the significant dynamical performance shift of the rotor due to the stator resistance must come from the coupling between the stator and rotor. As a closure of this section, the induction motor dynamics is measured by the roots of the rotor mode and the characteristic equation is illustrated in Fig. 4.13. This Figure combines together the effects of the frequency variation and slip speed variation as presented separately in Figs. 4.4 and 4.6. It then gives a global view of the system dynamics. It is clear that:

Eigenvalues s=0

s=0.5

f=100Hz

4-state dq frame model s=1

s=0.75 f=100Hz

f=100Hz f=80Hz

80Hz

80Hz f=60Hz

60Hz

60Hz 40Hz

40Hz

20Hz 20Hz

f=0Hz

20Hz

20Hz

40Hz

40Hz 60Hz

f=60Hz

80Hz

f=100Hz s=0 s=0.5

f=80Hz f=100Hz s=0.75

Fig. 4.13 Root locus of 4-state dq model

60Hz 80Hz

f=100Hz s=1

4.3 Equivalent Circuit

145

O14: The system time constant2 (or the rotor mode time constant) is mostly affected by the slip speed. High operating frequency offers a wide range of time constant. The time constant is invariant at s = 1 regardless of the operating frequency; O15: The system damped frequency,3 on the other hand, is mostly impacted by the operating frequency, and is almost independent form the slip speed; O16: Both the operating frequency and slip speed contribute to the system stability, measured by the damping ratio. The system is less damped at high frequency and high slip speed. When operating at these conditions, the system will be more exposed to external noise and disturbance. Nonetheless, the most important discovery of the induction motor dynamic analysis is that the system has a fast stator mode and a slow rotor mode. Therefore, in order to get a prompt torque response, the stator mode variable (e.g. the quadrature current i sq ), rather than the rotor variable (e.g. the rotor flux Ψr ), shall be regulated to deliver the demanded torque. This is the fundamental of the vector torque control, as will be elaborated in Chap. 6.

4.3 Equivalent Circuit It is well-known that the equivalent circuit model developed in Sect. 2.2 works well for steady-state analysis, but it is not suitable for dynamic analysis [8]. The comparison of the results given in Figs. 2.19 and 3.21 proves the effectiveness of the equivalent circuit model from the steady-state point of view. It is the task of this section to show the deficiency of the equivalent model, to improve the model for dynamic analysis. Difference in dynamics will appear when comparing the equivalent circuit model in vector format with that of the rotating dq-frame model as given in Eq. (3.56). The former needs to be developed first. Figure 2.13, which presents the equivalent circuit model, is re-drawn as Fig. 4.14 by ignoring the core loss resistance Rc . As per Fig. 4.14, and for the given default direction of Is and Ir , the stator and rotor flux linkage are given by  s = L sl Is + L M (Is − Ir )  r = L M (Is − Ir ) − L rl Ir It is noteworthy that confusion shall be avoided for L M and L m . The former, given by per phase; the later L M = L ss , is the magnetizing inductance√ √is the mutual inductance of the machine and is governed by L m = L SS L R R = 1.5 L ss L rr . They are indeed different. Also, bear in mind that Vs , Is , Ir ,  s and  r are vectors in a per-phase scale as the equivalent circuit model is at a single phase level. They have their own 2 3

Represented by the real part of the eigenvalue. Represented by the imaginary part of the eigenvalue.

146

4 Induction Machine Dynamics Analysis Rs

Ir

Is

IM

Lrl

Rr/s

LM

Vs

Lsl

Fig. 4.14 Equivalent circuit model

magnitudes and phases in space. Assuming L ss = L rr for the time being for the sake of envisaging the dynamic deficiency, it yields  s = L s Is − L M Ir  r = L M Is − L r Ir in which L s = L ss + L sl and L r = L rr + L sl are respectively the stator and rotor inductance per phase. Compared to Eq. (3.56), these expressions are really similar apart from that • The equivalent circuit model is at a single phase level, while the dq-frame model combines the effects from all three phases; • The rotor current is reversed in the equivalent circuit model. This is one of the fundamental differences between the equivalent circuit and dq-frame model. s As shown in Eq. (3.56), the derivative of  s is composed of the terms of d dt and ωs  s , as a result of the rotating frame. The former represents the changes in magnitude, while the latter reflects the changes in phase, according to Eq. (3.38). This is due to the fact that dq is a rotating frame, and that all the variables rotates with the dq frame. However, for the equivalent circuit model, the variables for the single phase are sinusoidal signals in time with high-order harmonics, but they are not treated as rotating in space. The voltage signal across the inductor L sl and L M contains all the derivatives of  s in the equivalent circuit model. This yields the single phase stator model as s d Vs = Rs Is + dt

Similarly for the rotor model as r d Rr = Ir dt s

4.3 Equivalent Circuit

147

This equivalent circuit model is indeed analogous to the αβ frame model as given in Eq. (3.39), but the difference in the rotor model is also immediately shown. • The back EMF term of ωr e r has not been considered in the equivalent circuit model. back EMF reflects the coupling between the mechanical and electric systems. It is not straightforwardly represented by a pure electric component; • The term Rr /s instead of Rr appears in the equivalent model. This, combined with the missing of the back EMF effect, makes a fundamental difference in rotor dynamic. Caution must be taken that: (1) The single phase variables adopted here are runtime varying vector signals, not just the scalar rms values as in Chap. 2; However, (2) these are not rotating variables in space, which make the equivalent model not directly comparable to the dq-frame model. In order to make a direct comparison, the following developments to the equivalent model are necessary • The model is updated to capture three-phase behaviours; • Vs , Is , Ir ,  s and  r are now the variables to account for three phases. Based on the previous analysis, these variables turn into rotating vectors in space; • Also needed to be updated is the magnetic inductance parameter, required to include a two-phase to three-phase conversion coefficient. It now becomes L M = L SS ;4 Going through the same coordinate transformation as in Sect. 3.5, the equivalent circuit model in the dq frame evolves as dΨsd − ωs Ψsq dt dΨsq + ωs Ψsd vq = Rs i sq + dt dΨr d + sωs Ψrq 0 = Rr ir d − s dt dΨr d − sωs Ψr d 0 = Rr irq − s dt

vd = Rs i sd +

(4.10)

in which the flux linkages are given as Ψsd = L s i sd − L M ir d Ψsq = L s i sq − L M irq Ψr d = L M i sd − L r ir d Ψrq = L M i sq − L r irq

(4.11)

The magnetizing inductance L M of equivalent circuit model is for one phase. However, L M of αβ and dq-frame model is after the Clark transformation which takes combined effect of three phases into account.

4

148

4 Induction Machine Dynamics Analysis

Comparing to the induction machine model in the dq frame as given by Eqs. (3.57) and (3.58), the difference (in Red) is evident. Compared to the dq-frame model, the rotor mode becomes significantly faster with the equivalent circuit model, which makes it not suitable for dynamic analysis. As just-mentioned above that the equivalent model is more analogue to the αβ-frame model, but this equivalent model still requires further modification in order for dynamic analysis. Referring back to Eq. (3.39), the stator flux linkage equation is repaired as  s = L s Is + L m Ir √ = L sl Is + L SS Is + L SS L R R Ir = L sl Is + L M (Is + re Ir ) so that it can be readily represented in the equivalent circuit. Here, re = is the effective turn ratio of the rotating transformer. By defining

√

L R R /L SS

Ir∗ = re Ir it then yields the stator equation as below Vs = Rs Is + L sl

dIs d(Is + Ir∗ ) + LM dt dt

The last two terms in the above equation are due to the inductors in the equivalent circuit but with different current passing through. Similarly, the rotor flux can also be tidied up as  r = L m Is + L r Ir √ = L SS L R R Is + (L rl + L R R )Ir = re L SS (Is + re Ir ) + L rl Ir ∗ ∗ = re [L M (Is + Ir∗ ) + L rl Ir ]

in which

∗ = L rl /re2 L rl

Defining

 r∗ =  r /re

results in an updated expression of the rotor model as ∗ 0 = Rr∗ Ir∗ + L rl

where

dIr∗ d(Is + Ir∗ ) + LM − jωr e r∗ dt dt Rr∗ = Rr /re2

4.3 Equivalent Circuit Rs

149 Lsl

Ir*

Is

Rr*

*

IM LM

Vs

Rs

Lsl

Ir*

Is

IM LM

Vs

* Lrl

* Lrl

*

+

*

+

-

Rr*

*

-

Fig. 4.15 Dynamically equivalent circuit model

Once again, the two middle terms in the right-hand side of the equation are represented by the inductors in the equivalent circuit. However, the last term regarding the back EMF has no corresponding electric component, which has to be modelled algebraically. Figure 4.15 gives an equivalent circuit model, according to the above conversion of the αβ-frame equations. This model is capable of capturing the dynamics of induction machines. In the bottom diagram, the rotor current is reversed, aligned with the convention of the equivalent circuit model. Comparing to the equivalent model as in Fig. 4.14, there are two main differences 1. The parameters Rr and L rl of rotor model are scaled by 1/re2 , while the variable  r by 1/re and Ir by re . All these scalings in the dynamic model allow the stator and rotor models to share the common magnetizing inductor; 2. The variables in the equivalent circuit model are scalar signals in an rms scale, while they are complex vector signals in the dynamically equivalent circuit model; 3. The back EMF term is directly added to the rotor model of the dynamically equivalent circuit; 4. As a result of introducing the back EMF in the equivalent circuit, the scaling of Rr by 1/s in the original model is not required.

150

4 Induction Machine Dynamics Analysis

Fig. 4.16 dq model equivalent circuit

To some extent, the term Ir Rr /s in the original equivalent circuit models represents the voltage drop over the rotor resistive load, or the power injected into or flowing out of the rotor when multiplied by the current Ir . When s = 1, the rotor is stationary. represents The voltage drop across the rotor is reduced to Ir Rr . Therefore, Ir Rr 1−s s is obligatory in the original equivalent the rotor back EMF. The term Ir Rr 1−s s circuit to correctly capture the machine torque in steady-state. However, it is also this term that makes the equivalent model fundamentally incorrect during transient as it substantially changes the time constant of the rotor mode. According to Eqs. (3.78) and (3.71), the machine torque5 can also be expressed as 3 (4.12) Tem = −N p Ir Ψr 2 Bearing mind that the rotor current is reversed in the equivalent circuit model and that Ir Ψr = Ir∗ Ψr∗ , then Tem ∝ Ir Ψr = Ir∗ Ψr∗ With this interpretation, the updated equivalent circuit model as given in Fig. 4.15 is quite similar to that of DC motors. To conclude this section, the equivalent circuits for the dq-frame model are given in Fig. 4.16. The circuits in the left-hand side of the figure represents the general dq-frame model, while the circuits in the right-hand side is for the rotor fieldoriented model. Please note that for the FOM, current direction takes the equivalent circuit convention, so that it makes the model more readable. Also, the fundamental difference of the equivalent models between Figs. 4.15 and 4.16 is that the voltages 5

The machine torque has many expressions as a result of cross product of vectors of stator current, rotor current, stator flux, rotor flux or air gap flux etc. This is not in the scope of this book.

4.3 Equivalent Circuit

151

injected into the circuits in the former model are sinusoidal AC signals, while the latter are DC signals. This explains why there is higher back EMF caused by the rotating flux in the dq equivalent circuits. Furthermore, the gray color shadows the parts of the model which are negligible. • With DC injected signal of Vsd , the impedance of magnetic L m is tiny, compared ∗ and Rr∗ . Therefore, most direct current flows through to that of the rotor path L rl L M . During steady-state, since the impedance of the magnetizing path drops to ∗ and Rr∗ , namely Ir∗d = 0; X M = 0, and consequently no current flows through L rl • Ψrq = 0, which has to be satisfied for FOM, requires the stator and rotor fluxes cancel each other in the quadrature direction. This is equivalent to requesting ∗ ∗ (or Isq = Irq with default direction at the right-hand side of the figure) Isq = −Irq by means of an appropriate slip speed control as explained in Sect. 3.6 as per Eq. (3.79). Therefore, both stator and rotor pump the same amount but in opposite direction of currents into the magnetizing path in q-axis, making its net effect of L m negligible; • Though L M has no effect in q-axis, the stator still has a leak flux linkage Ψsq . The introduced EMF ωs Ψsq still requires to be compensated by Vsd . In general, the leak flux linkage and its induced voltage are small; • The stator flux linkage Ψsd and rotor flux linkage Ψr∗ are almost the same, since they are dominated by L M I M , which is shared by both the stator and rotor. It is therefore reasonable to consider the EMFs of ωs Ψsd and ωs Ψr∗ are counterbalanced, making the back EMF due to the rotor rotation the only main term to be compensated by Vsq ; With the above interpretation of negligible components, the physical meaning of the equivalent circuit becomes clear. • The d-axis circuit models the generation of the magnetic field and its associated dynamics. The magnetic field is regulated by Vsd . In order to get an accurate field strength, the back EMF ωs Ψsq , which can be significant at high frequency and high quadrature current, has to be appropriately compensated. So does the voltage drop across Rs ;6 • The magnetic field generation part of the model has a time constant of τM =

Ls Rs

It is 198 ms for the given example of the induction machine, and this is a slow mode. Increasing the voltage Vsd will result in a fast change of the rotor current Ir∗d ,7 but not necessarily a fast change of the magnetic current I M = Isd − Ir∗d . Therefore, 6

The voltage drop at L sl is only during transient. Time constant for rotor current in the field direction is the same as that of the torque or the quadrature current.

7

152

4 Induction Machine Dynamics Analysis

in order to achieve prompt response of the machine toque, the magnetic field shall remain invariant; • The q-axis circuit models the torque current dynamics. The torque current Iq 8 is regulated by Vsq . The quadrature voltage shall firstly compensate the rotor back EMF, which represents the power flowing into or out of the rotor. To see this, the mechanical power at rotor is as Pem = Tem ωr m = N p Ψr∗ Ir∗ ωr m = Ψr∗ ωr e Ir∗ This is exactly the same electric power based on the equivalent model. Precise compensation of the rotor back EMF is important for improving the torque accuracy, as any uncompensated EMF will lead to deviation of the torque current from the desirable set point. The voltage drop over Rs and Rr∗ is less significant due to its relatively small magnitudes. • The time constant of torque current change is governed by τT =

∗ L sl + L rl Rs + Rr∗

This equation is exactly the same as what is given in Eq. (4.2). The time constant is 6.75 ms for this given sample. Comparison to the magnetizing time constant indicates that the torque current is a fast-changing path. Thus, quick response of machine torque can be obtained by regulating the torque current Iq ; • At a high rotor speed ωr e such that Ψr∗ ωr e ≥ Vsq Ψr∗ has to be reduced. Otherwise uncontrolled generation will occur. This is when the flux weakening comes into force. • when ωr e > ωs , the Vsq shall be regulated to for a reversed torque current Iq at the desirable level. This turns into the controlled generation. In summary, the equivalent dq model captures the steady-state and dynamic characteristics as analyzed in the previous sections. It is a simple and convenient analytic tool. In particular, this model resembles many features of DC motor models.

8

∗ . Iq = Isq = Irq

References

153

References 1. Briz F, Degner MW, Lorenz RD (2000) Analysis and design of current regulators using complex vectors. IEEE Trans Ind Appl 36(3):817–825 2. Clark RN (1996) Control system dynamics. Cambridge University Press 3. Del Blanco FB, Degner MW, Lorenz RD (1999) Dynamic analysis of current regulators for ac motors using complex vectors. IEEE Trans Ind Appl 35(6):1424–1432 4. Franklin GF, Powell JD, Emami-Naeini A, Powell JD (2002) Feedback control of dynamic systems, vol 4. Prentice Hall, Upper Saddle River 5. Holtz J (1996) On the spatial propagation of transient magnetic fields in ac machines. IEEE Trans Ind Appl 32(4):927–937 6. Holtz J (2002) Sensorless control of induction motor drives. Proc IEEE 90(8):1359–1394 7. Holtz J, Quan J, Schmittt G, Pontt J, Rodriguez J, Newman P, Miranda H (2003) Design of fast and robust current regulators for high power drives based on complex state variables. In: 38th IAS annual meeting on conference record of the industry applications conference, IEEE, vol 3 (2003), pp 1997–2004 8. Novotny DW, Lipo TA (1996) Vector control and dynamics of AC drives, vol 41. Oxford University Press 9. Sen PC (2021) Principles of electric machines and power electronics. Wiley

Chapter 5

Scalar Torque Control

5.1 Introduction Scalar control of an induction machine is generally referred to as scalar speed control, which adjusts the supply frequency while keeping a constant ratio of voltage magnitude versus supply frequency in order to regulate the speed. Therefore, scalar control is also called Volts Hertz (V/F) control. The scalar torque control mentioned here, however, is the so called constant Volts Hertz (V/F) and slip speed control. It assumes that the torque control can be decoupled by controlling separately the stator flux linkage and the slip speed, although not from the vector decomposition point of view. The control method is based on the steady states of an induction machine, in particular, the torque-slip characteristics as per the equivalent circuit models. In Sect. 4.3, it has been demonstrated via comparison that the equivalent circuit model is a simple and useful analysis tool, but the drawback is the lack or even misrepresentation of the motor dynamics such as phase delay. Due to this, the dynamic coupling effects are omitted, which consequently results in sluggish torque response. In general, the scalar torque control is only suitable for applications when fast dynamics is not a requirement. However, due to its simplest control structure and the cheap while stable implementation, it is still beneficial for prototyping or for the auxiliary applications where fast dynamics and a high level of torque accuracy are not required. V/F control is to control the stator flux by imposing a constant relationship between the amplitude of the stator voltage and the stator frequency. As a result, the stator flux is kept approximately constant. The advantage of keeping a constant Vs / f s is the capability of maintaining an envelope of constant peak torque below the rated speed, although above that speed, the peak torque decreases quadratically. In industrial applications, V/F control by itself is usually used in the open-loop speed control when high precision is not required. Automotive application as a traction drive however is in torque control mode and thus the conventional V/F control is obsolete. It has to come together with the slip speed control to be able to operate in the torque control mode. © Springer Nature Switzerland AG 2024 S. Shen and Q.-z. Chen, Practical Control of Electric Machines for EV/HEVs, Lecture Notes in Electrical Engineering 1064, https://doi.org/10.1007/978-3-031-38161-4_5

155

156

5 Scalar Torque Control Torque

Unstable region

Stable region Tmax Stable

Unstable

0 Slip ratio 1

Load torque curve

ωs Rotor speed 0

Fig. 5.1 Examplar torque speed characteristics of an ACIM

For an induction machine, slip is always needed in order to produce torque. In the scalar torque control, the slip speed is used to adjust the output of machine torque. The machine torque versus rotor speed characteristics of an induction machine shows that the operating region is stable only within a small range of the slip speed, see Fig. 5.1. In this figure, the blue curve shows an example of the machine torque versus rotor speed (or the slip that has the coordinate axis pointing to the opposite direction of that of the speed), and the red curve represents the load. Intersections of the machine torque curve and the load curve are two presumed operating points: one is stable, but the other is not. Assume that there is a small signal variation to the speed, the electromagnetic torque at the unstable operating point will then change in a way either to continuously decelerate till stall or to continuously accelerate the motor; at the stable operating point, any variation to the speed will cause the machine torque to counteract the load and thus stabilize the speed. The shadowed region in the figure is the stable operating region. Only if the slip speed is located within this region, the system will be stable. The stability issue is more of a concern in the applications with open-loop speed control. Nonetheless, it has been demonstrated in Figs. 4.7 and 4.8 in the chapter of dynamics analysis that higher slip ratio is directly linked to a slower system time constant and a more deteriorated damping ratio. On top of that, the unnecessary high slip ratio also reduces the system efficiency, which is rather straightfoward, as a higher current is required as a result of the reduced effective power load Rr∗ (1 − s)/s, see in Figs. 2.15 and 2.28. Due to all these, the operating range of an induction is limited in the shadowed region as shown in Fig. 5.1. In this chapter, we first recall the equivalent electric circuits of an ACIM, as stated in Chap. 2. Following that, a so called improved ‘’-model equivalent circuit, which has inherent better torque accuracy estimation compared to the ‘T-model’, is derived for the analytical scalar torque control method. Sensitivity of the motor parameters is discussed and the implementation is elaborated afterward.

5.2 Equivalent Circuits for Scalar Control

157

5.2 Equivalent Circuits for Scalar Control Equivalent circuit model of an induction machine has been extensively discussed in this book. In particalar, a 3-stage simplification of the equivalent electric circuit of the motor in a per-phase scale has been illustrated in Chap. 2.2. Bear in mind that the scalar torque control method is to maintain a constant Vs / f s and then use the slip speed ωsp to regulate the torque. That being said, the Vs / f s and ωsp are desired to be decoupled. Having this in mind, we therefore investigate a way to improve the model accuracy based on the previous analysis, while having that decoupled relationship in the meanwhile. Let us recall the ‘T-model’ equivalent circuit, in which the iron loss and AC loss are omitted, as shown in Fig. 5.2a. Previous analysis assumes that the magnetizing impedance is large enough satisfying  ωs L M 

Rs 2 + (ωs L sl )2

(5.1)

Then the shunt magnetization branch can be moved from the middle of the T-model circuit to the input terminal on the stator side. This forms a ‘’-shaped model, see Fig. 5.2b. As such, maintaining a constant magnetizing flux is done by imposing a constant relation between the stator voltage per supply frequency. To differ from a ‘-model’ developed later in this section, we therefore denote this equivalent circuit model as a ‘-like model’ or the simplified -model. With this simplifiction, we recap the electromagnetic torque expression in Eq. (2.24) with minor adjustment as Tem = 3N p

Vs2 Rr∗ s ωs (Rs s + Rr∗ )2 + (X sl + X rl∗ )2 s 2

(5.2)

In order to be able to use this expression for torque control, a fundamental basis is that the simplification from the T-model to the -like model is valid and sensibly accurate. This assumption is true only if the magnetizing reactance is sufficiently larger than the total impedance of the stator resistance and leak inductance. At high frequency, the voltage drop caused by the stator resistance is negligible, as the impedance caused by

Rs

L*rl

Lsl

Is

L*rl

R*r/s

IM

IM

(a)

LM

Vs

Lsl

I*r

Is R*r/s

LM

Vs

Rs

I*r

(b)

Fig. 5.2 Induction motor equivalent circuit (Sect. 2.2): a T-model equivalent circuit; b Simplified -model equivalent circuit

158

5 Scalar Torque Control

L M is much larger than Rs . However, the contribution of the stator leakage inductance L sl is not negligible irrespective of frequency. Take the motor parameters given in Table 2.1 for instance, this leak inductance is around 10% of the mutual inductance L M . By moving the L M branch to the terminal on the stator side, the voltages over both Rs and L sl are neglected and it implies that the voltage across the magnetizing branch approximates the stator voltage. Hence, the -like model will introduce significant error in the torque estimation. In Sect. 2.3.2, the Thevenin’s theorem is applied to derive a ‘-model’ without accuracy sacrifice. However, the model by itself does not decouple the V/F and the slip speed. Indeed, the corrected stator resistance and inductance, as a result of the Thevenin’s transform, couple the stator frequency and the slip ratio in the torque Eq. 2.25. Hence, Thevenin’s model is beneficial for motor performance analysis, but is not immediately convenient for applying the scalar torque control.

5.2.1 Improved -model Circuit In this section, an equivalent transformation is investigated so that the -like model can be improved to obtain better accuracy with regard to the torque-speed characteristics. The idea is to find an equivalent transform to move the stator leak inductance to the middle branch as shown in Fig. 5.2a, so that when the converted magnetizing branch is relocated to the input terminal of the stator circuit, the resulted approximation error can be substantially reduced. It is reasonable to assume that the circuit after the equivalent transform takes the form as in Fig. 5.3. By equivalent transform, it implies that the input power, air gap power, supply frequency and rotor slip shall all be kept unchanged before and after the transformation. This further signifies that the stator voltage, stator current and stator flux linkage shall be kept invariant regardless of the transformation as well [8]. Here, we denote the following symbols as •  L M is the transformed magnetizing inductance that collects both L sl and L M . That being said, the  L M is equivalent to the per-phase stator inductance as  L M = L sl + L M

Fig. 5.3 Transformed T-model equivalent circuit

LL

Is

IR

IM

RR/s

LM

Vs

Rs ( RS )

5.2 Equivalent Circuits for Scalar Control

159

Subsequently, the transformation coefficient is defined by ζ =

 LM LM

(5.3)

•  L L is the transformed total leak inductance; • Rs is the stator resistance, which keeps unchanged after the transformation; R is the transformed rotor resistance; • R • Is is the stator current vector, with Is = |Is |; I R = | I R |; • I R is the transformed rotor current vector, with  I M = | I M |; • I M is the transformed magnetizing current vector, with  • s is the same slip ratio as prior to the transformation. By applying the Kirchhoff laws on the stator side, the steady-state stator voltage is given by Vs = Rs Is + jωs  s The stator flux linkage keeps invariant with the transform and can be written as  s = L ls Is + L M I M IM = L M

(5.4) (5.5)

where Eqs. (5.4) and (5.5) represent the stator flux linkage before and after the transformation respectively. Likewise, the stator current stays invariant after the transformation. Applying the Kirchhoff current laws, we have I M = Is − Ir∗  I M = Is −  IR

(5.6) (5.7)

Substitute Eqs. (5.6)–(5.7) into (5.4)–(5.5) and tidy up, then the tranformed rotor current arises as I∗  IR = r ζ

(5.8)

The equivalent transformation applies invariant air-gap power such that R R s R∗ = Ir∗ 2 r s

ph Pag = I R2

(5.9)

Substituting Eq. (5.8) into (5.9) yields R = ζ 2 Rr∗ R

(5.10)

160

5 Scalar Torque Control

On the rotor side, applying the Kirchhoff laws, it yields   R + R R jωs  IR = 0 s where

(5.11)

R =   IR −  IM L L L M

Likewise, the rotor side Kirchhoff laws before the transform are given by jωs  r∗ + Rr∗ Ir∗ = 0 where

(5.12)

 r∗ = L lr∗ Ir∗ − L M I M

Substitute the Eqs. (5.8) and (5.10) into (5.11), and compare to Eq. (5.12), then the rotor flux linkage is updated as  R = ζ  r∗  (5.13) or equivalently

   IR −  I M = ζ L lr∗ Ir∗ − L M I M L M L L

(5.14)

By substituting  L M = ζ L M and Eqs. (5.6)–(5.8) into (5.14), the transformed total leakage inductance can therefore be obtained as  L L = ζ L ls + ζ 2 L lr∗

(5.15)

By this far, all parameters after the transform have been derived. To sum up, the conversion parameters are rewritten in the following form: ⎧ s = Rs R ⎪ ⎪ ⎪ ⎪ L M = L sl + L M ⎨ R = ζ 2 Rr∗ (5.16) R ⎪ 2 ∗ ⎪  = ζ L + ζ L L ⎪ ls lr ⎪ ⎩ L ζ = L M /L M With this transform, the contribution of the stator leakage inductance is taken into account in the stator flux path. Compared to Eq. (5.1), the following condition becomes therefore more valid: L M = ωs (L M + L sl )  Rs ωs 

(5.17)

Indeed, by rearranging Eq. (5.1), the former -like model takes the prerequisite of

5.2 Equivalent Circuits for Scalar Control

161

Fig. 5.4 Improved -model equivalent circuit

Rs ( RS )

IR

Is

RR/s

LM

Vs

LL

IM

 ωs L 2M − L 2sl  Rs The benefit after the transform is then immediately evident. With Eq. (5.17) being satisfied, the middle branch ‘ L M ’ can be moved to the input terminal of the stator, see Fig. 5.4, while keeping all other parts unchanged. This step introduces only an error as a second-order term of the stator resistance, and thus the inaccuracy is rather insignificant, which will be proven later in this chapter. The equivalent circuit then reduces to a simple series impedance load viewed from the stator terminal. One distinct benefit is that the inductive and resistive loads are decoupled in such a way that cannot be achieved by the Thevenin’s transform. We then denote this transformed equivalent circuit model by the ‘improved -model’. With the improved -model, All the torque expression and analysis in Chap. 2 shall still hold valid, except that the parameters now need to be updated accordingly. Consequently, the electromagnetic torque in Eq. (5.2) can be rewritten as Tem = 3N p

R s Vs2 R R )2 +  ωs (Rs s + R X 2L s 2

(5.18)

where  X L = ωs  LL. In order to validate the accuracy of the improved -model, Fig. 5.5 is presented to show the comparison of the slip-to-torque curves with the simplified -like model (indicated by ‘Simple model’ in the graph) and the full model as illustrated in Chap. 2. The example motor parameters listed in Table 2.1 are used for this analysis. This figure is to be compared with Fig. 2.20. As a matter of fact, the errors from the improved -model compared to the full model at high frequency is too invisible to be displayed without zooming in. For this reason, a lower frequency of f s = 25 Hz is showcased in Fig. 5.5b to highlight the difference. Even so, this difference is still rather minor, and it mainly lies within the generating region.1 For quantitative analysis of the torque accuracy, we define the following relative error on the L 1 norm  N N



∗ |ei | |Tem (si )|, (5.19) e1 = i=0

1

i=0

Asymmetry between motoring and generating is explained in Chap. 2.

162

5 Scalar Torque Control Slip speed to torque curve

300

Slip speed to torque curve

400

200

Torque [Nm]

Torque [Nm]

200 100 0 -100

0

-200 -200 -300 -1

-0.5

0

0.5

1

-400 -1

-0.5

0

Slip [-]

0.5

1

Slip [-]

(a) fs = 100 Hz

(b) fs = 25 Hz

Fig. 5.5 Slip to torque curve comparison 50

16 14

40

Torque Error [%]

Torque Error [%]

12 10 8 6 4

30 20 10

2 0

10

20

30

40

50

60

70

80

90

0

100

10

20

30

40

50

60

70

80

90

100

Stator frequency [Hz]

(a)

(b)

Fig. 5.6 Torque errors compared to the full model: a Motoring torque errors; b Generating torque errors

with

∗ (si ) ei = Tem (si ) − Tem

where si is the discretized slip ratio; N is the number of discretized steps; Tem (si ) and ∗ (si ) are respectively the electromagnetic torque from the improved -model and Tem the full model at the slip ratio of si . Likewise, the same relative error norm is defined for the simplified -like model. Figure 5.6 shows respectively the quantitative torque inaccuracy for the simple model and the improved -model with reference to the full model. Errors in the quadrants of motoring and generating are separated in Fig. 5.6a and b respectively so that they can be better evaluated. The following observations can be made: 01. The torque errors decrease monotonously with the increase of the stator supply frequency. This is in agreement with the basis of approximation. Indeed, the higher the frequency, the higher ratio the impedance versus the stator resistance. Thus, the approximation error of relocating the middle  L M branch for a -model circuit tends to diminish;

5.2 Equivalent Circuits for Scalar Control

163

02. The torque accuracy from the improved -model has increased substantially compared to the simplified -like model. The errors reduce vastly from above 10% to generally below 2% in motoring mode, or even below 0.5% above 40Hz. Similar improvement can also be found in the generating mode; 03. In generating mode, the simplified model yields significant errors at low frequency below 10 Hz, and thus within this frequency range, the model can mainly work for qualitative analysis. On the other hand, at particularly low frequency such as f s = 5 Hz, the improved -model yields around 13% average error, which then decreases down to 5% at 10 Hz, and further down quickly to below 2% above 20 Hz and then even much lower at higher frequencies; 04. When the supply frequency is sufficiently high, the torque error from the simplified model stabilizes at around 11%. Further increase of frequency does not tend to reduce this error meaningfully any more. At this point, the stator resistance impact from moving the middle branch to the stator input terminal has disappeared; however, the influence from the implicit relocation2 of stator leak inductance remains, despite the increase of the frequency. By contrast, the torque error from the improved -model almost diminishes to null at and above the frequency of 100 Hz, evidencing that there is no error attributed to the stator leak inductance. As a brief summary, the improved -model has substantially reduced the approximation error, which results in much better accuracy than that of the simplified -like model. Meanwhile, it keeps the terms of reactance and resistance decoupled, and thus, it is consistent with the forms of the torque equations elaborated in Chap. 2. The improved -model can be readily used to derive for the scalar torque control.

5.2.2 Electromagnetic Torque Analysis Bear in mind that one basis of the scalar torque control is to keep a constant ratio of voltage versus frequency. Consequently, the maximum torque stays constant despite the frequency as long as the voltage is within the achievable constraint. This constant maximum torque is characterized by the peak torque of the induction motor with the rated voltage Vsrt and the rated stator frequency f srt . Hence, with the updated model parameters, the peak torque based on Eq. (2.30) emerges as Tepk =

2

Vsrt 2π f srt

2

1.5N p  LL

Relocation of the middle branch to the stator terminal is equivalently an implicit relocation of the stator resistance and leak inductance.

164

5 Scalar Torque Control

By substituting the conversion relationship of  L L , as given in Eq. (5.16), into the above equation, the peak torque is rewritten as

Vsrt 2π f srt

Tepk =

2

1.5N p ζ L ls + ζ 2 L lr∗

(5.20)

By comparing Eq. (5.20) with Eq. (2.30), the difference is obvious. Given that ζ is strictly greater than 1, it further proves that the torque is over estimated by the simplified -model equivalent circuit model as stated in Sect. 2.2. The peak torque of a motor can be delivered for only a short period of time, subjected to the thermal characteristics of the system. This peak torque can be achieved as long as V rt Vs = srt fs fs

subjected to

  f s ∈ 0, f srt

(5.21)

Because of this, scalar control is also called V/F control. Nonetheless, at low frequency, the voltage drop across the stator resistance is not negligible. This voltage drop will bring down the peak torque if not compensated. Section 2.3.1 has discussed the voltage boost factor for compensating this effect. Here, we briefly recall the voltage boost factor and update it accordingly for the improved ‘-model’ circuit model as   γ =

κr +

κr2 + 1

with κr = Rs /(ωs  L L ). The applied stator voltage becomes

V rt Vs = min γ f s srt , Vsrt fs

(5.22)

Figure 5.7 shows the stator voltage in the rms value w/out the voltage boost factor. The voltage with the boost factor is almost linear before the clamping at its rated

Fig. 5.7 Stator voltage versus stator frequency

Voltage boost compensation

140

Stator voltage [V]

120 100 80 60 40 20 0

with boost factor without boost factor

0

20

40

60

80

100

Stator frequency [Hz]

120

140

160

5.2 Equivalent Circuits for Scalar Control Fig. 5.8 Torque-speed envelopes of the respective simplified and improved ‘’ circuit models

165 Torque speed envelope

300

Torque [Nm]

250 200 150 100 50

0

200

400

600

800

1000

1200

Rotor speed [rad/s]

value. The difference between the two curves shows an almost perfectly constant offset, meaning that in implementation, the stator voltage can be approximated by adopting this offset on top of a strictly constant ‘Vs / f s ’ ratio as below Vs = min

fs

Vsrt + ΔVs , Vsrt f srt

(5.23)

After applying the voltage with the boost factor from either Eq. (5.22) or (5.23), Fig. 5.8 is presented to compare the maximum torque-speed envelopes of the simplified -like model and the improved -model. In comparison to the simplified model, the peak torque from the newly-developed model drops from 265.2 Nm to 234.7 Nm. Once again, this evidences that the simplified model underestimates the voltage drop over the impedances on both the stator and the rotor branches, and as a consequence, it overestimates the motor torque. This figure shows that both the peak torque before the rated speed and the achievable maximum torque within the power-decrease region are significantly overestimated by the simplified model. Here, the torque-speed envelope from the improved -model is used as a reference, whose accuracy is well validated in Figs. 5.5 and 5.6. Let us assume an ideal scenario without the presence of stator resistance, i.e., Rs = 0 . the electromagnetc torque in Eq. (5.18) can be rewritten as Tem = 3N p

Vs ωs

2

R (sωs ) R 2R +  L 2L (sωs )2 R

(5.24)

Provided Vs /ωs is maintained constant, the electromagnetic torque arises as a function dependent solely on the slip speed, i.e., sωs or ωsp , irrespective of both Vs and ωs . As per Eq. (2.27), the torque reaches its maximum when sωs =

R R  LL

(5.25)

166

5 Scalar Torque Control Slip speed at max torque

120

Torque speed envelope

250

110

Torque [Nm]

Slip speed [rad/s]

200 100 90 80

150

100 70 60

5

50

100

150

50

200

(a)

0

200

400

600

800

1000

1200

Rotor speed [rad/s]

Stator frequency [Hz]

(b)

Fig. 5.9 Effect of the stator resistance: a slip speed at which max torque is achieved; b change of max torque against stator frequency

In this ideal situation, the maximum torque is always acquired at the constant slip speed of sωs = 113.8rad/s for the example motor. While in the presence of the stator resistance, the condition for the slip speed where the maximum torque is reached can be rewritten as R R sωs =  (5.26) (Rs /ωs )2 +  L 2L It is clear that with the decrease of the stator frequency, the slip speed where the maximum torque is arrived decreases. Figure 5.9a shows this slip speed with the change of the stator frequency. When the stator frequency is sufficiently high, both the approximation of Eq. (5.26) to (5.25) and the approximation of Eq. (5.2) to (5.24) do not bring in much error. Having said that, the machine torque becomes a function purely depending on the slip speed, provided a constant Vs / f s is kept. In fact, from the stator frequency of f s = 45Hz onwards for instance, the shifting in slip speed compared to the ideal non stator resistance case drops below 1%. Given that the voltage boost factor is constantly greater than 1, the peak torque can be held constant up to a stator frequency slightly smaller than the rated frequency, as shown in Fig. 5.9b. Above this frequency, the stator voltage is clamped at its rated value and the maximum torque starts to decrease quadratically against the stator frequency (and therefore the rotor speed) as shown in both Figs. 5.9b and 5.8. This is the so-called natural characteristics of an electric machine. Figure 5.10 shows the torque-slip characteristics for the frequency cases below the rated stator frequency of 100 Hz. At no presence of the stator resistance, the family of the torque curves overlap completely. However, when the stator resistance is present, the family of curves shift nonlinearly a little depending on the stator frequency. Meanwhile, the peak torque starts to drop from the rated frequency onwards. Combining Figs. 5.9 and 5.10, it can be seen that using the model with omitted stator resistance for torque control will lead to significant overestimation of the required slip speed. There are two consequences attributed to this. One is the introduced torque

167

250

250

200

200

Motor torque [Nm]

Motor torque [Nm]

5.2 Equivalent Circuits for Scalar Control

150 100

150 100

50 0

50

0

20

40

60

80

0

100

0

20

40

Slip speed [rad/s]

60

80

100

Slip speed [rad/s]

(a)

(b)

Fig. 5.10 Torque versus slip speed: a without stator resistance; b with stator resistance

20

30

40

250

150

3040

Torque [Nm]

50

200 20

10

10

100

2

2 2

50

0.5

0

0

200

0.1

0.3 0.05

400

0.5

0.4 0.2 0.01

600

0.1

0.4

0.3

800

0.05

0.2 0.01

1000

Rotor speed [rad/s]

Fig. 5.11 Torque errors of Eq. (5.24) with neglected stator resistance compared to the full model

inaccuracy and the other is the decrease of motor efficiency, in particular, if it crosses over the breakdown torque and enters the unstable operating region, see Fig. 5.1. In an open-loop speed control scenario, the latter case may cause unstable operating and finally lead to a motor stall. In order to quantize the torque inaccuracy, Fig. 5.11 shows the contour of the torque errors of the improved -model with stator resistance being omitted, i.e., the model on Eq. (5.24). The errors are referenced to the full model,3 stated in Chap. 2. It can be seen that the torque is constantly overestimated. The worst estimation occurs in the low-speed high-torque region. On the contrary, at low torque or high speed operation, the omission of the stator resistance does not bring in much error. With that being said, Eq. (5.24) can be used for the scalar torque control for fast prototyping, but it still needs to be improved. 3

The full model can be either from direct calculation of the equivalent circuit in a sense of the full representation, or equivalently, from the Thevenin’s transform.

168

5 Scalar Torque Control

0.5

2

0.2

0.

05

0.3

0.01

0.1

2

50

0

0.

0.

0.4 20

0.5

10

100

05

1

10

Torque [Nm]

30

150

0.3

200

0. 2

0.4

250

1

0.0

0.001

0.001

0

200

400

600

800

1000

Rotor speed [rad/s]

(a)

0.2

250

0.1

1

150

5

0.05

0.2

10

100

0.0

3

20

Torque [Nm]

0.0

5

200

5

0.01

10

0.

01

0.03

0.1

1

20

50

0

0

200

400

600

800

1000

Rotor speed [rad/s]

(b) Fig. 5.12 Torque errors of Eq. (5.27) with the presence of stator resistance compared to the full model: a absolute error; b relative error

We therefore recap the electromagnetic torque equation of the improved -model in (5.18) and rewrite it in the following form Tem = 3N p

Vs ωs

2

R ωs s R R )2 +  (Rs s + R L 2L ωs2 s 2

(5.27)

Instead of using the slip speed ωsp in the control, this equation indeed suggests to use the slip ratio for torque regulation while assuming the stator frequency as known runtime information. As ωs is not decoupled and is dependent on the rotor speed and the slip speed, iteration will be needed in implementation. By rearrangement, the equation becomes actually a second-order equation of the slip ratio s, whose

169

250

250

200

200

150

Motor torque [Nm]

Motor torque [Nm]

5.2 Equivalent Circuits for Scalar Control

Increase of rotor resistance

100

150 100

50 0

Increase of stator resistance

50

0

20

40

60

80

100

120

140

0

160

Slip speed [rad/s]

(a)

0

20

40

60

80

100

Slip speed [rad/s]

(b)

Fig. 5.13 Change of torque curves with respect to thermal effects at f s = 50Hz: a change of rotor resistance; b change of stator resistance

solution is simply analytical. In comparison to the above presented torque error contour with the stator resistance being omitted, Fig. 5.12 presents the torque error contour with the stator resistance included in the model, i.e., Eq. (5.27). Again, the errors are referenced to the full model, but are expressed in both the absolute and relative scales. It is evident that the torque accuracy, compared to the full model, is nearly perfect except for a very limited collection of operating points at extremely low operating frequency with low-medium torque. This can therefore reduce the work load from calibration drastically.

5.2.3 Sensitivity Analysis The thermal effects shall also be taken into account. Due to the iron loss and conductor loss in the rotor, and also the likely heat exchange between the stator and rotor, the rotor bars will almost inevitably heat up during operating, particularly with high torque or power demand. As a consequence, the temperature-correlated rotor resistance will vary depending on the operating condition. This directly impacts the slip speed at which the maximum torque is reached, as defined in Eqs. (5.25) and (5.26). Figure 5.13a shows the sensitivity of the motor torque with respect to the varying rotor resistance. It is evident that increase of the rotor resistance will cause an increase in slip speed to deliver the same amount of torque. This temperature rise effect will widen the stable operating region; however, the actual torque output will be degraded accordingly. The figure shows the change of the machine torque curves with respect to the rotor resistance at f s = 50Hz. In case that the rotor resistance increases by 50%, the machine torque may drop by more than 20%. The change of the stator resistance, on the other hand, does not have as significant impact as that of the rotor resistance. Figure 5.13b shows the change of the machine torque with respect to the stator resistance at f s = 50Hz as well. The effects are only

170

5 Scalar Torque Control 250 200

150

Motor torque [Nm]

Motor torque [Nm]

200

100

50

0

150 100 50

0

5

10

15

20

25

0

30

Slip speed [rad/s]

(a)

0

5

10

15

20

25

30

Slip speed [rad/s]

(b)

Fig. 5.14 Change of torques curve with respect to thermal effects at f s = 5Hz: a change of rotor resistance; b change of stator resistance

marginal. Nevertheless, as the comparison between Figs. 5.11 and 5.12 implies, the impact of the stator resistance mainly lies in the low-frequency operating region. It is therefore sensible to assume that at low frequency, the impact from the change of the stator resistance will be amplified. Hence, the variation of stator resistance cannot be neglected as well, in order to obtain satisfactory torque accuracy in the low speed region. This assumption is evidenced by Fig. 5.14, which provides the sensitivity of the rotor and stator resistance parameters at low frequency of f s = 5Hz.

5.3 Analytical Implementation In the scalar torque control scheme, the V/F relation is defined by Eq. (5.22) or (5.23). As mentioned in Sect. 5.2.2, the machine torque is dependent solely on the slip speed regardless of stator frequency, assuming that the stator frequency is sufficiently high. The electromagnetic torque can therefore be approximated using Eq. (5.24), which is purely analytical. This approximation decouples the two terms of Vs /ωs and the slip speed sωs . While keeping the former invariant, the slip speed is derived accordingly. It then goes through a limiter constraint and adds to the measured electrical rotor speed to provide the synchronous electrical frequency, i.e., the stator supply frequency. Subsequently, this information is used to calculate the stator voltage as per the V/F relation. Using the PWM schemes, as will be elaborated in Chap. 9, the duty cycles are calculated to control the power switches. A corresponding control block diagram is given in Fig. 5.15. Indeed, Eq. (5.24) turns into a simple second-order equation in terms of the slip speed sωs or ωsp , and the solution of the targeted ωsp is rather straightforward. Further simplification of the equation into a first-order linear equation is possible, referencing Chap. 2; However, the introduced error increases accordingly.

5.3 Analytical Implementation

171 Inverter

Ba ery

ACIM

ωm

PWM Vs

Np

Eqn. (5.22) or Eqn. (5.23)

ωre ωs

Limiter Tem reference

Eqn. (5.24)

ωsp

ω*sp

Fig. 5.15 Simplified induction motor scalar torque control block diagram

As aforementioned, the phase voltage needs to be scaled with a voltage boost factor to account for the voltage drop across the stator resistance. Therefore, Vs /ωs is not strictly constant. Runtime information of both the voltage and supply frequency is required. In particular, above the rated frequency, the voltage is capped by the power supply. Vs /ωs cannot be maintained close to a constant any more, but becomes inversely proportional to the supply frequency. The torque-slip speed characteristics change vastly in such a way that the torque drops quadratically with the increase of frequency. The control structure as given in Fig. 5.15 gives a simple computational scheme for regulating the torque; however, it suffers from the torque accuracy loss due to the omission of the stator resistance. Meanwhile, the runtime varying motor parameters due to thermal effects are not taken into account either, which can be rather influential on both torque accuracy and motor efficiency. In particular, once the rotor is heated up, the change of the rotor resistance will substantially deviate the torque speed characteristics, as can be seen from the sensitivity analysis in Sect. 5.2.3. To take these effects into account, a control structure based on the improved -model without negligence of the stator resistance is provided in Fig. 5.16. This control method is proven accurate over almost the whole operating region, see in Fig. 5.12. It is acknowledged that, as the supply frequency and the slip ratio are not decoupled, iterations in runtime calculation are expected in order to have the desired slip converging for the reference torque. The stator winding temperature can be directly measured. This temperature is then used to compensate for the stator resistance variation. On the other hand, in regard to the temperature-dependent rotor resistance, it is generally impractical to have a rotor temperature sensor installed in the vehicle application. Thus the rotor temperature usually needs to be identified via an observer [1, 3].

172

5 Scalar Torque Control Inverter

Ba ery

ACIM

ωm PWM Vs

Np

Eqn. (5.22) or Eqn. (5.23)

ωre ωs

Limiter Tem reference

Eqn. (5.27)

s

x

ωsp

ω*sp

Rs & R r

Fig. 5.16 Improved -model induction motor scalar torque control block diagram

It is noteworthy that in battery electric vehicle applications, power is supplied by a battery pack. The stator voltage is provided by modulating the DC bus voltage, which depends mainly on the state of charge (SoC) of the battery, but also the operating points, power cable connection and operating temperature etc. The DC-bus voltage is normally sensored as an input for the control of the motor and also for safety monitoring. The stator voltage will be clamped by the runtime DC voltage, meaning that the constraint to Vsrt in Eqs. (5.22) and (5.23) shall also depends on the DC bus voltage. Consequently, the voltage constraint in Eq. (5.22) or (5.23) will change during operation. In summary, this chapter investigates an improved equivalent circuit model to implement the scalar torque control, or the so-called V/F and slip speed torque control. Accordingly, the brief control structures using the analytical expressions are discussed. The torque accuracy of the improved -model is proven accurate; however, the inaccuracy in the small vicinity of very low-speed and low-medium torque operating points, as shown in Fig. 5.12, needs to be compensated by calibration. It is worth mentioning that prompt torque response is not of the intention for scalar control and that there exist other implementations of the scalar torque control, for instance, using the full equivalent circuit model or the model from the Thevenin’s transform. Consequently, they may require more complex calculation or even numerical methods, such as lookup tables.

References

173

References 1. Beguenane R, Benbouzid MEH (1999) Induction motors thermal monitoring by means of rotor resistance identification. IEEE Trans Energy Convers 14(3):566–570 2. Bose BK et al (2002) Modern power electronics and AC drives, vol 123. Prentice hall Upper Saddle River, NJ 3. Kral C, Habetler TG, Harley RG, Pirker F, Pascoli G, Oberguggenberger H, Fenz CJ (2004) Rotor temperature estimation of squirrel-cage induction motors by means of a combined scheme of parameter estimation and a thermal equivalent model. IEEE Trans Ind Appl 40(4):1049–1057 4. Leonhard W (2001) Control of electrical drives. Springer Science & Business Media 5. Munoz-Garcia A, Lipo TA, Novotny DW (1998) A new induction motor v/f control method capable of high-performance regulation at low speeds. IEEE Trans Ind Appl 34(4):813–821 6. Nam KH (2018) AC motor control and electrical vehicle applications. CRC Press 7. Sen PC (2021) Principles of electric machines and power electronics. John Wiley & Sons 8. Trzynadlowski AM (2000) Control of induction motors. Elsevier

Chapter 6

Vector Torque Control

6.1 Principle of Vector Control As already demonstrated in Chap. 5, and explained in Chap. 2 that the scalar control is to regulate the operating frequency based on the rotor speed and the required slip speed (which in turn relies on the demanded torque), and to regulate the stator phase voltage based on the operating frequency (with the compensation of the voltage loss in stator windings, particularly at low frequencies). This simple control has an intrinsic drawback of slow torque response. This process of determining the phase voltage Vs is illustrated as in Fig. 6.1. The name of the scalar control comes from the fact that the electric machine torque is derived by regulating magnitude of the three-phase winding voltage (as well as its frequency), but no request of regulating the direction (or its phase) of the winding voltage. This is basically treating the phase voltage as a scalar variable, but not a vector variable. It will become clear in the later part of this section that the lack of the control of the voltage phase makes undesirable transient performance from the scalar control. The rather simple scalar torque control could deliver the demanded torque with accuracy at steady state as long as the equivalent model developed in Chaps. 2 or 5 is sufficiently accurate and there are no significant disturbances to the system. This is because the equivalent model is capable of representing the system torque characteristics during steady state, but is unable to correctly capture the system dynamics performance, as proved in Chap. 3. Unlike the scalar control, the vector control regulates both the magnitude and the phase of the winding voltage, in order to yield the desired torque during both steady and transient states. The key signal flow for the vector control is illustrated as in Fig. 6.2. Bear in mind that Fig. 6.2 implies a dq-frame model by default. In general, when flux-weakening is not required, it is preferable to fix the magnetic field at its maximum level. With this setting, the machine torque will be regulated by the quadrature current only. This not only simplifies the control, but more importantly results in a fast torque delivery. We hereby refer to Sect. 4.2 for the details of the fast © Springer Nature Switzerland AG 2024 S. Shen and Q.-z. Chen, Practical Control of Electric Machines for EV/HEVs, Lecture Notes in Electrical Engineering 1064, https://doi.org/10.1007/978-3-031-38161-4_6

175

176

6 Vector Torque Control

Fig. 6.1 Scalar control signal flow Fig. 6.2 Vector control signal flow

Tem

ω sp (s)

ωs

Vs

ω re

Ψr= cnst

Isd= cnst

Vsd

Tem

Isq

Vsq

ω sp (s)

ωs

ω re

torque delivery principle. Once the magnetic field is fixed, the quadrature current is driven by the desired torque according to Eq. (3.78). It is similar for the slip (or the slip speed more precisely) according to Eq. (3.79). The operating frequency ωs is then determined by the rotor speed and the slip speed. This method of determining operating frequency is called Indirect Field Oriented Control (IFOC) [10, 13]. At last, the phase voltage is yielded according to Eq. (3.81). Clearly, by this approach, it is not a scalar variable Vs but a vector variable Vs being determined with both required magnitude and phase in the dq frame. The principle of vector control is demonstrated in Fig. 6.3, which is quite similar to Fig. 3.29, but extended (at the left-hand side) to include Vd -Vq coordinates. In addition to the torque contour, the slip speed contour, the working current points and the required phase voltages are also presented. As indicated in the figure, there are infinite possible combinations of Id and Iq to deliver torque T2 . The pink lines in the left-hand side of the figure show the required phase voltages corresponding to the demanded torque levels. V2 corresponds to the case (Idm , Iq2 ). However, for torque level T1 , there is only one feasible operation of (Idm , Iq1 ) with the resulted phase voltage V1 . The field current Isd = Idm is the same for both torque T1 and T2 . This design allows the rotor flux to remain unchanged from T = 0 to T = T max as long as the flux-weakening is not required. For instance, increasing torque from T = T1 to T = T2 requires only regulating the quadrature current Isq .1 This is the quickest way to deliver torque, but sacrificing some efficiency. For vector control, the required phase voltage will switch from Vs = V2 over to Vs = V1 as illustrated in Fig. 6.3. It is clear that the phase voltage is a vector variable, and both the phase and magnitude have to be precisely controlled in order to obtain the desired field and torque current. As a matter of fact, the phase voltage is regulated via its field and quadrature components, instead of its magnitude and phase angle. In general, vector control has the following features • The current, not the voltage, is the controlled variable. The voltage is adopted to regulate the current to have the desired field and quadrature components; • There is one degree of freedom in determining the magnetic and torque current. This is an optimization topic. The MPTA, MPTL and MPTF, as illustrated in Sects. 3.7 and 3.8, are examples of optimization results [1, 2, 11, 14]; 1

This is why sometimes it is called the torque current.

6.1 Principle of Vector Control ω1sp

177

ω2sp

V1

Vq Iq

V2

Vq = ω sψ m S=∞

T1

MTPA

T2

S1

Iq1

ω1sp

ω2sp

Iq2

S2

S Inc

-Vd

Vd1

Vd2

T1=Tpeak

S=0

Ψm = Ls Im d

T2 Id

Imax

Fig. 6.3 Vector control principle

• The feedback compensation to the current deviation from the desired value is required as the model and parameter uncertainties and external disturbances are present. Bear in mind that the slip speed (in green) and torque (in pick) contours in Fig. 6.3 are given in both the Id -Iq (right-hand side) as well as the Vd -Vq (left-hand side) coordinates. It could be concluded that the same torque can be delivered by different phase voltage vectors, but will lead to the different slip speeds accordingly. If the slip speed of the motor is fixed, then there will be an unique solution of the phase voltage for a given torque. Thus, for vector control, the motor slip is still an important control variable, though it is not regulated explicitly. Because of this, regulating the motor slip (no matter explicitly or implicitly) represents a link between the scalar and vector control. The principle of scalar control is presented in Fig. 6.4 for comparison. For scalar control, the magnetic field is regulated by V/F, i.e., different voltage to frequency ratio manages the steady-state air-gap flux. This is equivalent to regulating the filed current Id although indirectly. In the meantime, the torque is regulated by the desired slip, which is equivalent to requesting control of the torque current Iq , again indirectly. Therefore, the scalar and vector control are the same from the steady-state point of view.2 The difference is at how the desired magnetic field (or desired field current) and the targeted slip (or the targeted torque current) are reached through regulating the phase voltage.

2

The equivalency of the scalar and vector control at steady state is detailed in Sects. 3.6 and 4.3.

178

6 Vector Torque Control ω2sp

ω1sp

‘1 V V1 V2

T1

Vq Iq Vq = ω sψ m V1 (ω s1)

S=∞

MTPA ω1sp

V2 (ω s2)

T2

S1

Iq1

ω2sp S2

T1=Tpeak

Iq2 S Inc

-Vd

Vd1

Vd2

Ψm = Ls Idm

S=0

Imax

T2 Id

Fig. 6.4 Scalar control principle

With reference to Fig. 6.4, take the torque increase from T2 to T1 as an example. The constant voltage contours both in Id -Iq and Vd -Vq coordinates are supplemented to enhance the understanding of the transient behaviours. At T = T2 , the phase voltage 2 at must be V2 at steady sate, as it is the only solution giving rise to the desired slip ωsp T = T2 . However, since the scalar control has no ability to regulate the phase voltage direction, it can only rise up the phase voltage magnitude from |V2 | to |V1 | in response to the increased torque request. As such, the phase voltage becomes Vs = V1 initially. However, at Vs = V1 ,3 the system slip is not the same as required yet. Thus, the 1 . resulted phase current will vary continuously until it reaches the equilibrium of ωsp During this process, the phase voltage is more or less constant in magnitude. On the other hand, as a result of the rapid increase of the stator frequency, the phase voltage is not aligned with the rotor flux anymore. Thus, the phase voltage requires to have an appropriate rotation (relative to the rotor flux) as illustrated in the figure. Nevertheless, the magnetizing current varies throughout. As already discussed in Sect. 4.3, the mode associated to the magnetic field is slow. In principle, the scalar control takes a relatively long time to stabilize. Furthermore, with the scalar control, there is no current feedback loop to compensate any system uncertainties and disturbances, to break the system coupling, and to enhance the system damping. Consequently, the torque accuracy is not guaranteed by the scalar control, and the system will be exposed to oscillation during the transition.

3

V1 is not the steady-state solution of the system, but rather a transient solution.

6.2 Vector Control Architecture

179

The main benefits with vector control, relative to scalar control, are • A fast torque response (in the order of 10 ms); • An accurate torque delivery4 ; • The ability to minimize the machine losses.

6.2 Vector Control Architecture Vector control is the most common approach for vehicle traction machine control. It is also the main stream of this book. Thus, the vector control architecture, which is one critical step in the control system design, is introduced in this section. The control architecture shall cover the following: • The functions of the vector control; • The key interactions among the functions; • The key interactions with the external world. Figure 6.5 presents an control architecture designed in this book, with functions as given in Tables 6.1, 6.2, 6.3, 6.4, 6.5, 6.6, 6.7 and 6.8. Keep in mind that this is not a full list but a general list of the functions that are required, and also only some of the functions in the following list are discussed in this book.

Power Limit

Current Limit

Voltage Limit

MinDeadTime Limit

Temperature Limit

Flux Limit

OverVolt Control

Max/Min Duty Limit

DC BUS

Fast Channel

Fast Channel

Tem

Torque Filter

Current Arbitration Idmd d

Tem

Torque Arbitration

ωr

Iqdmd

err

Id

Feedback Control

Iqerr

Inv Park fb

vd

Phase Voltage Arbitration

vfb q PID

Quadrant Limit

Optimization

Torque Capability

MTPL

dq

vd

Modulation vα

PWMa

Inveter

PWMb

vβ

vq

PWMc

αβ

Gain Schedule

ACIM ff

Decouple & BackEMF Control Faults

Torque fault

Fault De-rating LimpHome

Torque Safety

vd vqff

DC Voltage Ripple Rejection

MTPA

Harmonics Injection

Harmonics Compensation

MTPF

emf

vd

emf

vq

loss

vd

loss

vq

Motor Model Temp

Thermal Control

Active Damping

Parameter Estimation

Fast Torque Deliverry

Rr Lm ψr ωs

Flux Observer

Flux Weakening

θ

ωs

Speed Control

Id Speed Limit

Fast Discharge

raw Id

Iα

dq

raw

αβ Park ω re

Ia Ib Ic

αβ

Iβ

Iq

Iq

ω rm

Slow Control Loop

Current Filter

abc Clark

Speed Filter Speed/Position Calculation

Position/Speed

Fast Control Loop

Fig. 6.5 Control architecture

4

The vector controlled electric machines are sometimes used as a torque sensor due to its high accuracy.

180

6 Vector Torque Control

Table 6.1 Torque management-related functions ID Title Description F01

Torque arbitration

F02

Torque filter

F03

Quadrant limit

F04

Torque capability

F05

Torque de-rating

F06

Torque safety

F07

Thermal control

F08

Active damping

F09

Speed control

F10

Speed limit

It manages the torque requests either internally or externally It avoids unnecessary sudden change of the demanding torque. The time constant can be a function of the machine speed, machine torque, and machine temperature etc. It permits the machine to operate only at pre-defined quadrants. For instance, when the forward gear is selected, the machine will only work in its first and fourth quadrant for the electric vehicles It dynamically defines the maximum machine torque as a function of its speed. The maximum torque shall be reduced to protect the machine when temperature increases beyond a threshold. Similarly, It derates the capability when a fault is present It degrades the torque capability once a fault is detected. The level of reduction can depend on the severity of the fault. No torque is permitted for critical faults It reduces the torque capability to a pre-defined limp-home level once a torque safety issue is identified It degrades the torque capability when the machine or inverter temperature is high. No torque can be requested at a critical level It adds the damping to the machine by means of active torque control. This function helps electric vehicle to reduce oscillation at kick-off and during regenerative braking This controls the machine speed to align with the demanded speed by regulating the machine torque. The use of speed control function benefits the gear shifting (if any) as it has a higher bandwidth than other automotive torque sources in a vehicle It limits the machine torque to protect the machine from over the pre-defined speed

These functions cover • The machine torque determination and management; • The machine vector current determination to optimize the machine operations; • The machine current control, in particular the feedback control to improve the accuracy under disturbances and uncertainties; • System protection includes (but not limited to) over-current, over-voltage, overspeed, over-load, over-temperature, and short-circuit protections;

6.2 Vector Control Architecture

181

Table 6.2 Current determination-related functions ID Title Description F11

Current arbitration

F12

Flux weakening

F13

Optimization

F14

MTPA

F15

MTPL

F16

MTPF

F17

Fast torque

F18

Fast discharge

It manages the field and quadrature current requests from various request sources It decreases the filed current demands to reduce the strength of the magnetic filed at high machine speed This optimizes the machine operation considering the efficiency, torque response time, flux, power factor, phase current and rotor temperature etc. This optimizes the machine operation so that the torque is delivered with a minimum phase current This optimizes the machine operation so that the torque is delivered with a minimum loss This optimizes the machine operation so that the torque is delivered with a minimum level of flux It targets to deliver the torque with no or minimum variation of the magnetic field This is to drain the energy in the DC-link capacitor by setting torque current to zero while magnetizing the filed continuously

Table 6.3 Current control and voltage generation-related functions ID Title Description F19

Voltage arbitration

F20

Feedback control

F21

Gain schedule

F22

Decoupling control

It manages voltage requests from various request sources This is to regulate the field and quadrature voltage based on the error between the reference and actual phase currenta It determines the control parameters according to the machine operating conditions such as torque, speed, temperature, as well as the control errors This is to reduce and/or remove the coupling between the stator and rotor modes, and between the field and quadrature sub-modes

a PID control is employed in this book, but other linear or nonlinear control strategies can also be applied

• System modelling, and variable and parameter estimation. This allows the online model-based observation and control; • Harmonics compensation by feedforward and feedback approaches; • Fault-tolerance and system safety. It is not the intention of this book to cover all the topics as above, but only the system model observation, current determination and current control are further

182

6 Vector Torque Control

Table 6.4 Coordinates transformation and modulation-related functions ID Title Description F23 F24 F25

F26

F27

Clark transformation It converts the 3-phase signals to 2-phase signals in the fixed αβ frame Park transformation It converts 2-phase signals from the fixed αβ frame to the rotating dq frame Inverse Park It is inverse Park transformation, which converts the 2-phase signals back to the fixed αβ frame from the rotating dq frame Modulation This controls the on/off states of the switches in the inverter, based on the required phase voltage. SVPWM is one of the most well-known approach, which is elaborated in Chap. 9 DC ripple This is to update the PWM duty ratio requirement to account for the DC bus voltage variation

Table 6.5 System protection-related functions ID Title Description F28

Power limit

F29

Temperature limit

F30

Current limit

F31

Flux limit

F32

Voltage limit

F33

Voltage control

F34

Dead-time limit

F35

Duty ratio limit

It limits the system power to a pre-defined level. This level could be dynamic to account for temperature variation and system faults It controls the system to operate under the pre-defined temperature level of the stator, rotor and inverter It limits the system current to a pre-defined level to protect the inverter switch devices and the stator windings. This level could be dynamic to account for temperature variation and system faults It limits the system flux to a pre-defined level by controlling the field current. This level could be dynamic to account for temperature variation and system faults It limits the system voltage to a pre-defined level. This level could be dynamic to account for voltage source variation and system faults It controls the system voltage within the permissible range. This control will regulate the phase current in both field and quadrature directions whenever necessary This is to ensure the minimum dead time between the upper and bottom switch devices on a same phase leg to protect from direct shoot through of the DC supply This is to ensure the duty ratio of switch devices within a safe range to protect them from possible short circuit

6.2 Vector Control Architecture

183

Table 6.6 Torque ripple reduction-related functions ID Title Description F36

Harmonics compensation

F37

Harmonics injection

This is to select, manipulate and compensate the pre-defined order of harmonics. This is a feedback approach for harmonics reduction. It normally uses the raw current signal, but not the filtered signal This is to inject harmonics with pre-defined magnitude and phase to counteract the harmonics in the system. This is a feedforward approach for harmonics reduction

Table 6.7 Model observation-related functions ID Title Description F38

F39 F40

Motor model

It models ACIMs based on the estimated parameters and variables. This is important for model-based decoupling control and for senseless motor control Parameter estimation This is to estimate the time-varying parameters such as rotor and stator resistance (and inductance) Flux observer This is to observe the direction and/or magnitude of the rotor flux. It is critical for the FOC

Table 6.8 Signal process-related functions ID Title Description F41

Torque filter

42 F43

Current filter Speed filter

F44

Speed calculation

This is to smoothen the demanded torque. The time constant could depend on the temperature, speed, and torque This is to reject the noise in the current signal This is to reject the noise in the speed signal. The time constant could be dynamic It calculates the rotor speed based on the encoder or resolver inputs. It also converts the mechanical speed to electric speed when necessary

investigated in this chapter. Bear in mind that system optimization is covered by Sects. 3.7 and 3.8, at least partially. As indicated in Fig. 6.5 , there are three loops presented: (1) Current feedback loop; (2) Voltage feedback loop; and (3) Speed feedback loop. The current loop is the main control loop, while voltage and speed loops are supplementary loops. The current feedback loop is to compensate the disturbances, model and parameter uncertainties and variations. This is one of the key features of the vector control. Voltage loop intends to ensure the voltage limit is not broken by updating the phase current, which becomes effective normally during flux-weakening. The current loop can fight with

184

6 Vector Torque Control

the voltage loop, which gives rise to a stability issue. The speed feedback loop have two functions. One function is for motor speed control which is optional, while the other is rotor field orientation. Incorrect orientation will introduce a significant torque error as a result of misleading slip speed. The vector control works across different bandwidths. The torque management related and system performance optimization functions have a low task rate at 1 kHz or so. On the other hand, the current control, modulation, system observation, field orientation, and coordinate transformation are at fast task rate (e.g. 20 kHz, depending on switching frequency) functions. The boundary between the low and high frequency functions is illustrated in Fig. 6.5. Clearly, the current, voltage and FOC speed loops are the fast control loops, while motor speed control loop is a slow loop. The high frequency loop reacts quickly to external disturbances. Therefore, it not only has better transient performance, but makes the system more stable as well. As such, if the active damping is moved into the high frequency domain, its effectiveness with oscillation suppression will be greatly improved. However, this will impose an increase on computational load. As indicated in Eqs. (4.3) and (4.4), the system is nonlinear since the rotor speed ωr e is a variable, not a parameter. However, the bandwidth associated with rotor mechanical dynamics is less than 20 Hz for automotive powertrain application. This dynamics is so low compared to the fast vector control loop, the rotor speed can be considered as a parameter, which will then make the induction motor model become linear. One benefit is that the frequency decomposition design can be applied. This is a great advantage for harmonics suppression. Bear in mind that the torque-related function and flux-weakening uses the mechanical rotor speed ωr m , while the model observation, model-based control, and FOC uses the electrical rotor speed ωr e .

6.3 Vector Control Design The current determination and current control are discussed in this section [3, 5, 8]. This section is based on the FOM as developed in Chap. 3. The model given in Eq. (4.3) and Eq. (4.4) is aligned with the rotor flux. For the completeness of this section, these equations are recapped as below di sd dt di sq dt dΨr dt

or

= − τ1T i sd +ωs i sq + τrκLr δ Ψr + L1δ vsd = − τ1T i sq −ωs i sd − κrLωδr e Ψr + L1δ vsq = − τ1r Ψr + Lτrm i sd

τT didtsd + i sd = + RLeqδ ωs i sq +

κr Req τr

di sq dt dΨr τr dt

κr ωr e Ψr Req

τT

+ i sq = − RLeqδ ωs i sd − + Ψr =

L m i sd

Ψr + R1eq vsd + R1eq vsq

(6.1)

6.3 Vector Control Design

185

in which i sd , i sq and Ψr are state-variables; vsd and vsq are control inputs. ωs , ωr e , τT , τr are time-varying parameters, while parameters L δ and κr are treated as constants. This model for the induction machine is also graphically represented as in Figs. 4.2 and 4.16. Combining Eqs. (3.78) and (3.86) results in the following expression for the torque Tem = k T i sq Ψr

(6.2)

In the above expression, ωs is not arbitrary, but it must satisfy Eq. (3.79). Thus ωs = ωr e + Rr κr

i sq Ψr

(6.3)

The angular position θ required by Park and inverse Park transformation is then governed by  θ=

ωs dt

(6.4)

Based on Eqs. (6.3) and (6.4) to yield rotor flux orientation is the so-called Indirect Field Oriented Control (IFOC), which is adopted in this chapter. In contrast to the IFOC, Direct Field Oriented Control (DFOC) aligns the dq frame directly to the rotor flux Ψr , which requires the observation of Ψr since it is not measurable [10, 13]. DFOC is discussed in Chap. 8. One way to improve the accuracy of the delivered torque is to feedback the measured torque and then adjust the phase current according to the torque error. However, this method requires a torque sensor, which is very expensive. Alternatively, as shown in Eq. (6.2), torque is determined by the stator quadrature current i sq and the rotor flux linkage Ψr which is in turn determined by stator field current i sd . Therefore, if the parameters L m and L r are known and accurate, then the stator current is a measure of machine torque. Fortunately, L s , L r and L δ can be detected based on the machine no-load and lock tests,5 and they are not dependent on the stator and rotor temperature (in normal machine operation range) as well. All these make it possible that the stator current can be treated as an accurate torque sensor. Consequently, the feedback of the stator magnetic and torque current to close the torque loop (or equally the current loop) is a practical way to improve the machine torque accuracy. This is the method utilized in this book.

6.3.1 Fast Torque Delivery Control To determine the desired (or demanded) phase current i sd , i sq , there are different strategies as given in Table 6.2. The fast torque delivery strategy is adopted here. 5

L m can be derived from L s , L r and L δ .

186

6 Vector Torque Control

Fig. 6.6 Fast torque strategy

Iq

Current limit MTPA Iqlim

Iraw q Iqlim(V) dmd

Tmax

Iq

T1 T2

Tem dmd

Id

Id Imax

The field current i sd shall be fixed so that the rotor flux remains constant during transient torque control, which is the key to make a fast torque response as analyzed in Chap. 4. The field current i sd is determined as follows √ i sddmd

=

2 Imax 2

(6.5)

in which i sddmd is the magnetic current demand, and Imax the current limit. which relies on the stator winding and inverter switch device characteristics. For a practical control, Imax is a calibration map depending on stator and rotor temperature. Nevertheless, it is assumed to be a constant here as any variation introduced by temperature is slow. In reference to Fig. 6.6, this determination of the desired magnetic current is based on the MPTA principle. This is the only permissible design point so that the machine torque can span from Tem = 0, to Tem = Temmax when i sd = cnst. However, this inevitably leads to a large phase current and a compromised power factor. Therefore, from practical engineering point of view, the field current can be relaxed at low torque demand to optimize the system efficiency and power factor if necessary, which is discussed later in this chapter. Once the magnetic current is determined, the torque current is the only degree of freedom left to deliver the required torque. There are different ways to determine the i sq . One is based on the steady-state rotor flux (which is an algebraic function of i sd ), the other on the transient rotor flux (which is a dynamic function of i sd ). The latter has a better response time. T dmd (6.6) i sqraw = em k T Ψr where i sqraw is the raw torque current demand and Ψr = L m i sd

6.3 Vector Control Design

187

at steady state. Otherwise, for the transient Ψr , the rotor model as in Eq. (6.1) shall be used to derive the rotor flux, which will have certain model errors attributed to the lumped parameter nature of the model and the online parameter variation. Nevertheless, transient rotor flux will converge to the steady-state value. Due to the fact that Isd is fixed for fast torque control, the use of the steady-state rotor flux makes more sense. However, the desired torque current must meet the constraint from the system current and voltage limits. By applying both the voltage and current limit, it yields the torque current limit as

Isqlim

   ⎫  2 ⎬ 

 Vmax 1 2 Imax , max = min − i sddmd , 0 ⎩ 2 ⎭ δ L s ωs ⎧ ⎨ √2

(6.7)

In the above equation, when max ωs,F T =

Vmax L s i sddmd

the torque current limit reduces to Isqlim = 0, implying that no torque is yielded at and beyond this speed with fast torque strategy. Finally, the demanded torque current is as ⎧

 ⎨min i sqraw , +Isqlim Temdmd > 0 dmd (6.8) i sq = ⎩max i raw , −I lim  Otherwise sq sq This method only ensures that the current request will fall under the system voltage limit, but there is no feedback loop yet to regulate the request if the system does go over the limit in any case. With the feedback current control, the integral term will contribute to extra voltage request, which may make the system potentially break the voltage limit. Thus, voltage limit feedback control is essential to ensure that the final voltage request is well within the limit. This is particularly helpful during flux-weakening operation. It is therefore introduced in the flux-weakening control Sect. 7.1. As illustrated in Fig. 6.6, i sqraw > Isqlim (V ) when Temdmd = T1 As a result of Eq. (6.8), i sqdmd = Isqlim (v) This implies that the desired torque T1 can not be implemented by fast torque strategy. The delivered torque becomes T2 . Nevertheless, some part of T1 torque contour is still under both the current and voltage limit. T1 is reachable by reducing the filed current i sd for instance. Thus, fast torque strategy reduces the machine torque

188

6 Vector Torque Control

capability when the rotor speed is high such that voltage limit becomes effective. This can be improved by flux-weakening control, but with sacrifice on the torque response time.

6.3.2 Current Feedback Control According to Eq. (6.1), the machine phase current i sd and i sq are governed by phase voltage vsd and vsq . There are two different ways to regulate the magnetic and torque currents, namely, the feed-forward and feedback methods. The feedback method of regulation is discussed in this subsection, while the feed-forward in next subsection. It is well known that feedback control can greatly reduce the system sensitivities to the parameter variations, model uncertainties and external disturbances. The Proportional and Integral (PI) control is adopted, and is given as below. gn

vsdfb = Pd i derr + Idterm vsqfb = Pqgn i qerr + Iqterm dIdterm gn = Id i derr dt dIqterm = Iqgn i qerr dt gn

(6.9) (6.10) (6.11) (6.12)

gn

in which Pd and Pq are the proportional gains for the magnetic and torque current gn gn control, while Id and Iq the integral gains. The integral term Idterm in Eq. (6.9) for the magnetic current control and Iqterm in Eq. (6.10) for the torque current control are further expressed in Eqs. (6.11) and (6.12). i derr and i qerr are respectively the errors between the desired and actual phase current in the field and quadrature directions, which are defined as i derr = i sddmd − i sdact i qerr

=

i sqdmd

−

i sqact

(6.13) (6.14)

where i sdact and i sqact are actual magnetic and torque current, which are converted from the phase current i a , i b and i c through the Clark and Park transformation. In general, these measured current signals will be filtered to reject the high-frequency components of noises. Nevertheless, i sdact and i sqact are the same as i sd and i sq when the current filter is ignored.

6.3 Vector Control Design

189

6.3.3 Decoupling Control As shown in Chap. 4, the coupling between the stator and rotor modes and between the field and quadrature sub-modes make the dynamics of the induction motor complicated. In this subsection, these couplings are compensated (at least partially) by appropriate vsd and vsq control. One version of decoupled control is the use of feedback current i sdact and i sqact , which is defined as dp vsd = −ωs Lˆ δ i sqact − κˆr Ψˆ r /τˆr

+ωs Lˆ δ i sdact

(6.15)

+ κˆr ωr e Ψˆ r

(6.16)

vsd = −ωs Lˆ δ i sqdmd − κˆr Ψˆ r /τˆr

(6.17)

dp vsq = +ωs Lˆ δ i sddmd + κˆr ωr e Ψˆ r

(6.18)

dp vsq

=

Another is a feed-forward version as dp

Both methods shall lead to the same steady-state results, but their closed-loop system dynamic performances will be different. The use of the demanded magnetic and torque current does avoid to add extra feedback loops in the system, and therefore simplifies the closed system dynamics. This is the method employed in this book. For both cases, Ψˆ is the observed rotor flux linkage. A simple version of rotor flux observer is given as dΨˆ r = −Ψˆ r + Lˆm i sddmd (6.19) τr dt Nevertheless, the use of un-filtered i sdact as an input variable to rotor flux observer is more realistic, but for fast torque control, the magnetic current remains constant. Therefore, the use of either the desired or actual magnetic current does not make much difference. In the above equations, Lˆ δ , Lˆ m , κˆr and τˆr are observed machine parameters by either offline measurement or online estimation.

6.3.4 Current Feed-Forward Control Based on Eq. (6.1), in order to obtain accurate magnetic and torque current control, the voltage loss in stator and rotor resistances shall be compensated. The use of the feedback control law as given in Sect. 6.3.2 is capable of completing the task. However, this often leads to a relatively larger gain than necessary if the loss is significant. High gain will risk the system stability, which shall be avoided if possible. Another way of compensation is to use the feed-forward approach to include the voltage loss explicitly. The feed-forward approach normally can not perform the complete compensation since the used parameters usually are not accurate enough

190

6 Vector Torque Control

and may also be time-varying, but it will greatly reduce the need of a high gain for the feedback loop, leaving the feedback control to focus on the compensation of the external disturbances and system uncertainties (such as unmodelled errors and parameter variations). There are also two feed-forward versions of resistance voltage loss compensation, one is using the actual currents as R = Rˆ eq i sdact vsd R = Rˆ eq i sqact vsq

(6.20) (6.21)

Another is the use of the demanded currents R = Rˆ eq i sddmd vsd R = Rˆ eq i sqdmd vsq

(6.22) (6.23)

in which Rˆ eq are the observed equivalent resistance at the stator side. Similar to the decoupling control, the use of the demanded current approach is chosen for simplifying the closed-loop system dynamics. In this book, both the decoupling control and resistance voltage loss compensation are adopting the feed-forward approach. Therefore, dp

R vsdff = vsd + vsd ff R dp vsq = vsq + vsq

(6.24) (6.25)

Substituting Eqs. (6.17), (6.18), and Eqs. (6.22), (6.23) into Eqs. (6.24) and (6.25), it yields vsdff = Rˆ eq i sddmd − ωs Lˆ δ i sqdmd − κˆr Ψˆ r /τˆr vsqff = Rˆ eq i sqdmd + ωs Lˆ δ i sddmd + κˆr ωr e Ψˆ r

6.4 Vector Control Performance The fast torque control strategy is summarized as in Fig. 6.7, which includes • The current determination. See Fig. 6.6, Eqs. (6.5), (6.6), (6.7) and (6.8) for details; • Current feedback control. See Eqs. (6.9), (6.10), (6.11), (6.12) for details; • Current feed-forward control (consisting of decoupling control and Resistance voltage loss compensation), see Eqs. (6.17), (6.18), (6.22), (6.23) for details; • Rotor flux orientation. See Eq. (6.3) for details. Bear in mind that the observed parameters shall be used for the flux oriented control; • Rotor flux observation. See Eq. (6.19) for details;

6.4 Vector Control Performance

191

Vlim

dmd

Tem

Current Determination Vlim Ilim

iqdmd Tem

dmd isd

-

dmd isq

act

act

Speed Filter

Np

vsd

+ +

vsq

ff vsq

idmd sq ω re

idmd sd

|

ff vsd

isq

ω rm

|

fb vsq

PIq dmd isd dmd isq

Current Filter

|vs |

fb vsd

PId -

iddmd

isd

-

Pv

Ilim

ωs

Lm

ωs

τr

Vector Control

Fig. 6.7 Fast torque control

• Voltage limit control. This control uses a single-sided proportional controller Pv, where only the voltage over the limit is fed back for regulation; • Current and speed signal processing. The vector control system as represented in Fig. 6.7 and the machine model as given in Fig. 4.2 form a closed-loop system as indicated in Fig. 6.8, which is the subject under investigation in this section. With this closed system diagram, it is also clear that the vector control regulates the phase voltage in the field and quadrature direction independently, and that the stator frequency is not arbitrary. Bear in mind that the coordinate transformation and inverter modulation are omitted in this closeloop system study. Since the modulation and coordinate transformation is at the fast frequency loop, the time delays caused by these functions are negligible compared to the time constant of the induction machine model. Another noteworthy point is the rotor speed filter. In general, a high bandwidth filter is designed for rotor model observation and for rotor flux orientation, but low bandwidth filters for the machine speed control, flux-weakening reference current generation, gain schedule etc. These filters do not affect the steady-state performance of the closed-loop system. Also, the influence of the high bandwidth filter on the dynamic performance could be negligible for the same reason as just-mentioned. Thus, the current filter is ignored. As indicated in Fig. 6.8, the closed-loop system has two inputs of Temdmd and ωr m , and one output of Tem . The input ωr m is often considered as a slow time-varying parameter rather than an input variable.

192

6 Vector Torque Control Vlim

Current Determination Vlim Ilim

dmd

Tem

iqdmd Tem

dmd isd

-

dmd isq

act

Np

vsd

+

vsq

+

ff vsq

idmd sq ω re

Speed Filter

|

ff vsd

dmd isd dmd isq

act isq

|

fb vsq

PIq

isd

Current Filter

|vs |

fb vsd

PId -

iddmd

ω rm

-

Pv

Ilim

ωs

idmd sd

τr

Lm

Vector Control

Kr/Reqτ r

-

re

Vs

1/Req

+

-

τT

Is

I sd

ψr

Lm

τr

iτT ωs Stator Model

X

Np

+

iτr

ψr

Rotor Model

I sq

X

X

ω re

KT

T em

Fig. 6.8 Vector controlled system

6.4.1 Steady-State Analysis At steady state, the system represented in Eq. (6.1) reduces to as vsd = Req i sd − L δ ωs i sq − κr Ψr /τr vsq = Req i sq + L δ ωs i sd + κr ωr e Ψr

(6.26)

Ψr = L m i sd Substituting the third equation into the the first and second equations in the above set, it yields vsd = Rs i sd − L δ ωs i sq vsq = Rs i sq + L s ωs i sd This is exactly the same as Eq. (3.81). Thus, the feed-forward control, including the decoupling and resistance compensation terms, as presented in Sect. 6.3, can be simplified as

6.4 Vector Control Performance

193

Vlim

Current Determination Vlim Ilim

dmd

Tem

-

Pv

Ilim

iqdmd Tem

dmd isd

-

dmd isq

act

act

dmd isd

ff vsd

dmd isq

ff vsq

isq

dmd isd idmd sq ω re

Speed Filter

ω rm

Np

|

vsd

+

fb vsq

PIq

isd

Current Filter

| fb vsd

PId -

iddmd

|vs|

vsq

+

ωs

Vector Control

Kr/Reqτ r

-

re

Vs

+

1/Req

-

τT

Is

I sd

Stator Model

X

Np ψr

Lm

τr

iτT ωs

+

iτr

ψr

Rotor Model

I sq

X

X

ω re

KT

T em

Fig. 6.9 Simplified close loop system

vsdff = Rˆ s i sddmd − Lˆ δ ωs i sqdmd vsqff = Rˆ s i sqdmd + Lˆ s ωs i sddmd

(6.27)

in which Rˆ s , Lˆ s and Lˆ δ are observed parameters. The benefit of this simplified feedforward control is that no rotor flux observation is required. This simple version is presented in Fig. 6.9. Combining the feedback and feed-forward control (simplified in this subsection) together, it yields the overall phase voltage as gn vsd = Pd i derr + Idterm + Rˆ s i sddmd − ωs Lˆ δ i sqdmd gn vsq = Pq i qerr + Iqterm + Rˆ s i sqdmd + ωs Lˆ s i sddmd

(6.28)

Equations (6.26) and (6.28) must be the same during steady state, which leads to the following balance for vsd     gn 0 = Pd i derr + Idterm + Rˆ s i sddmd − Rs i sd − ωs Lˆ δ i sqdmd − L δ i sq     gn = Pd + Rˆ s i derr + Idterm + i sddmd Rserr − ωs Lˆ δ i qerr + i sqdmd L δerr

194

6 Vector Torque Control

where the parameter errors are defined as Rserr = Rˆ s − Rs L δerr = Lˆ δ − L δ and i derr and i qerr are defined as in Eqs. (6.13) and (6.14) respectively. The leak inductance L δ shall be quite accurate as it is not sensitive to temperature (or other environment changes) as long as no flux saturation takes place.6 Consequently, the L δ estimation error could be ignored though the life-time and manufacturing variations do exist. On the other hand, Rs (likewise Rr ) is temperature dependent, and is therefore time-varying. There will be sufficient estimation errors when off-line measured data is applied for deriving these parameters. Therefore, it is reasonable to assume that L δerr = 0 Rserr = 0 By replacing Rˆ s with Rs and Lˆ δ with L δ , which makes no significant difference in terms of steady-state error but makes the analysis simpler, it yields

 gn Pd + Rs i derr + Idterm + i sddmd Reqerr − ωs L δ i qerr = 0

(6.29)

Similarly, the vsq balancing results in

 Pqgn + Rs i qerr + Iqterm + i sqdmd Reqerr + ωs L s i derr = 0

(6.30)

The following discussions are made on the close-loop steady-state error analysis. Discussion1: Proportional + Feed-forward control If there is only proportional control, the integral control terms will be dropped, i.e., Idterm = 0, and Iqterm = 0 in the above steady-state equation, leading to a simple version as below

gn  Pd + Rs i derr + i sddmd Rserr − ωs L δ i qerr = 0 (6.31) 

gn Pq + Rs i qerr + i sqdmd Rserr + ωs L s i derr = 0

6

For the similar reason, the estimation errors with stator, rotor and mutual L s , L r and L m are negligible.

6.4 Vector Control Performance

195

Consequently, the state-state error is given by i derr = −

gn (Pq +Rs )i sddmd +L δ ωs i sqdmd gn gn (Pd +Rs )(Pq +Rs )+δL 2s ωs2

Rserr

i qerr = +

gn (Pd +Rs )i sqdmd −L s ωs i sddmd gn gn (Pd +Rs )(Pq +Rs )+δL 2s ωs2

Rserr

(6.32)

Based on Eq. (6.32), it is clear that C1: Both the magnetic and torque currents have control errors. The estimation error of stator resistance is the main cause. Stator resistance is normally measured off-line as a calibration parameter. However, the stator resistance is temperature dependent. It will increase when the temperature rises; C2: Any means of increasing stator resistance accuracy improve the current control accuracy, and thus improve the torque accuracy. One way of doing this is to measure the stator resistance at different temperature, and use map-based method to adjust the stator resistance according to the stator winding temperature. This method however will greatly increase the work load on testing. C3: Another way is to reduce the sensitivity of steady-state error, associated with gn the stator resistance estimation error, by increasing the proportional gain Pd gn and Pq ; however, increasing proportional gain can risk the system stability; C4: The current error also depends on the current demands. Relatively large current error is caused by a large current demand. This is particularly true for for the field current error. Also, there is cross coupling between the current demand in one direction and the current error in the other direction. That is, the magnetic current error i derr depends crossly on the torque current demand i qdmd , and vice versa for the toque current error i qerr . Torque current error i qerr is reduced with high torque demand, benefiting from this cross coupling. However, it leads to an even larger magnetic current error. For a given torque, both i sddmd and i sqdmd are fixed according to Eqs. (6.5) and (6.6). Based on Eq. (6.2), the torque error can be approximated as Temerr = k T L m (i sddmd i sqerr + i sqdmd i sderr )

(6.33)

Clearly, the torque error is dominated by the resistance estimation error Rserr . In general Rserr < 0, thus, Eq. (6.31) can be repaired as ax − by = m cx + dy = n where x and y are defined as x = i derr ,

y = i qerr

196

6 Vector Torque Control

d0

y Iq Error

d inc

d1

a2 a1

d2

a0

err

Tem = 0

x Id Error

err

Tem,1

err

Tem,2

err

Tem,3

s8

s7 s6

s3

s5 s4

s0

s1

s2

Fig. 6.10 Steady state error analysis

Coefficients a, b, c, d, m and n are given by a b c d m n

= = = = = =

gn

Pd + Rs ωs L δ ωs L s gn Pq + Rs −i sddmd Rserr −i sqdmd Rserr gn

Here, normally m > 0 and n > 0. Having these in mind, how the control gains Pd gn and Pq affect the current errors i derr and i qerr , which in turn affect the torque error err Tem , is illustrated geometrically in Fig. 6.10. In Fig. 6.10, the parallel dotted thin lines represent the constant torque error contours, in which err err err < Tem,2 < Tem,1 |Tem,2 | > |Tem,1 | |Tem,3

With these assumptions and definitions, the blue lines illustrate the equation gn ax − by = m for different Pd gains, and a2 > a1 > a0

6.4 Vector Control Performance

represents

197

gn

gn

gn

Pd,2 > Pd,1 > Pd,0 = 0 It is clear that the line ax − by = m intersects Y axis of x = 0 at an invariant point gn of −m/b, regardless of the variation of coefficient a (or gain Pd ). Similarly, lines cx + dy = n with d2 > d1 > d0 gn

represent different torque current control gain Pq that has the following relation gn

gn

gn

Pq,2 > Pq,1 > Pq,0 = 0 Also, these lines have an invariant point of n/c. Nevertheless, the confluences such as Si , i = (0, 1, . . . , 8) of line ax − by = m and cx + dy = n are the solutions of i sd and i sq errors. According to Fig. 6.10, it is obvious that, • The resulted current errors are such that i derr > 0

and

i qerr < 0

In general, the large current (in magnitude) errors i derr and i qerr result in large torque error as long as both the current errors have the same sign. The current error could cancel out (at least partially) each other if their polarities are different. gn gn • When Pq is fixed, increasing gain Pd leads to the line ax − by = m rotating gn err anticlockwise. Thus, current error i d is inversely proportional to Pd ; gn • Similarly, when Pd is fixed, the line cx + dy = m also rotates anticlockwise, resulting in a reduced current error i qerr (in magnitude) with increased control gain gn Pq ; gn

In general, the magnetic current error i derr is quite sensitive to its control gain Pd , gn and i qerr to Pq . gn i derr = 0 when Pd = ∞ and i qerr = 0 when Pqgn = ∞ Thus

gn

Temerr = 0 when Pd

= ∞ and Pqgn = ∞

However, this way of reducing the torque error to zero is not preferable since the system stability becomes questionable. As indicated in the figure, there are infinite moderate control gain combinations resulting in the cancellation between the magnetic and torque current errors.

198

6 Vector Torque Control

Nevertheless, these control gains also have cross-term effects on the current components at the other direction. Take the torque current error i qerr for an example. gn Increasing gain Pq leads to the decrease of i qerr (in magnitude), but the increase of gn i derr on the other hand, as illustrated in Fig. 6.10. This is not true for gain Pd , which causes a reduction on both current errors (in magnitude). It is interesting to see that gn there is a gain Pd , which makes line ax − by = m orthogonal to the torque gn error contours. At this level of Pd , the torque error will be mostly sensitive to gn the torque gain Pq . gn gn By all means, the torque error can be regulated by the control gains Pd and Pq , which make the system less sensitive or insensitive to stator resistance estimation error. This requires delicate gain tuning. Also the gains that lead to a minimised torque error are also torque demand dependent, which makes the gain calibration rather complicated. The torque errors for Temdmd being respectively 100 Nm and 200 Nm with different control gains are given in Fig. 6.11, as a validation of the above analysis. The following could be concluded by observation of the results • The control gains that result in zero torque error are also torque demand dependent. The low torque demand requires more control efforts; • Inappropriate large gains (as shown in Fig. 6.11b) can even lead to the increased torque error; • There is a linear combination of magnetic and torque control gains, which gives rise to zero torque error. • There is also another linear combination of magnetic and torque control gains, which gives rise to the quickest torque error descending. However, a note is immediately added here that zero torque error in the above analysis is obtained by assuming accurate measurement of the current. As in Eq. (6.14), the actual current i sxact , x = (d, q) are subject to measurement errors, which is out of the scope of this book.

dmd = 100Nm (a) Tem

Fig. 6.11 Torque error as function of control gains

dmd = 200Nm (b) Tem

6.4 Vector Control Performance

199

Discussion2: PI + Feed-forward control With the integral control, according to Eqs. (6.11) and (6.12), it will achieve the zero steady-state current error as, i derr = 0 and i qerr = 0

(6.34)

Combining this, the steady-state error Eqs. (6.29) and (6.30) are reduced to as Idterm + i sddmd Rserr = 0 Iqterm + i sqdmd Rserr = 0

(6.35)

The system is now fully capable of compensating the parameter uncertainties during steady state. It is already assumed that parameters L δ , L m , L s and L r are quite accurate. This makes the integral control applied in this chapter counteract the stator resistance variation on the stator temperature mainly. Since temperature time constant is rather slow, it would be a reasonable to claim that the working load for the integral control is not that high during both steady and transient states. Nevertheless, even though the rotor flux estimation error Ψr err = Ψˆ r − Ψr does not appear in Eqs. (6.29) and (6.30), which could be the results of the mutual inductance estimation error L merr = Lˆ m − L m , the rotor resistance estimation error Rrerr = Rˆ r − Rr , the rotor inductance estimation error L rerr = Lˆ r − L r , and any other unmodelled errors, the current integral control shall still have the ability to compensate the steady-state part of it. To see this, let the first expression of Eq. (6.26) be given as vsd = f (i sd , i sq , Ψr ) = Req i sd − L δ ωs i sq − κr Ψr /τr In this way, it is now assumed i sd , i sq and Ψr are the interested variables. Req and τr are subjected to variations, but these are not in the consideration of analyzing Ψr err . Conducting Taylor expansion around the nominal states i sddmd , i sqdmd and Ψˆ r , then f (i sd , i sq , Ψr ) = f (i sddmd , i sqdmd , Ψˆ r ) + Req i sderr − L δ ωs i sqerr − κr Ψr err /τr The first term in the right-hand side of the above expression is exactly the feedforward compensation as detailed in Sect. 6.3.4, which leaves the rest to be compensated by the feedback loop. Thus gn

Pd i derr + Idterm + Req i sderr − L δ ωs i sqerr − κr Ψr err /τr = 0 According to Eq. (6.34) by integral control, it yields Idterm = κr Ψr err /τr

200

6 Vector Torque Control 4

4

3.5

3.5 3

Torque error [Nm]

Torque error [Nm]

3 2.5 2 1.5

2.5 2 1.5

1

1

0.5

0.5 0

0 0

50

100

150

200

0

50

100

150

200

Torque demand [Nm]

Torque demand [Nm]

(a) Current error = 1%

(b) Current error = 0.1 A

Fig. 6.12 Torque error as a function of demanded torque

Similarly, Iqterm = −κr ωr e Ψr err It confirms that the integral control can compensate the rotor flux error. However, based on Eq. (6.19), the steady-state error of Ψr err is contributed mainly by L m , which will become inaccurate when flux saturation occurs. In the meanwhile, the transient error caused by τr where Rr plays a significant part. Therefore, the transient flux error compensation is a major concern. Furthermore, another fundamental error is that the field direction adopted by the control system is not accurately aligned to the rotor flux. Even if the values of i sd and i sq are regulated to the desired value precisely, but the machine torque will not be as the demanded due to this directional error. All these will be discussed separately in Chap. 8, Same as the proportional control, the current measurement errors can not be neutralized by integral control either. This is the intrinsic drawback of the feedback control, requiring the accuracy of the core feedback signals. Assume that there is no error with rotor flux and rotor flux direction, and that the parameter uncertainties are compensated by integral control, then the torque error as given in Eq. (6.33) is also driven by current measurement error. Given current measurement error of 0.2 A or 1%, whichever is larger, the resulted torque error is shown in Fig. 6.12a. It is clear that the torque error relies on the torque demand. If the current measurement error could be improved to 0.1 A, the torque error result is then given in Fig. 6.12b for comparison. It can be seen that the torque error is drastically reduced as a result of increased current measurement accuracy. Consequently, any efforts on improving the current sensor quality are encouraged. Discussion3: PI only When there is no feed-forward control, including the decoupling control and the stator resistance voltage drop compensation), Eqs. (6.29) and (6.30) are updated as

References

201 gn

Pd i derr + Idterm + Rs i sd − L δ ωs i sq = 0 gn

Pq i qerr + Iqterm + Rs i sq + L s ωs i sd = 0 Equation (6.34) is a result of the integral control, which still holds. Thus Idterm + Rs i sd − L δ ωs i sq = 0 Iqterm + Rs i sq + L s ωs i sd = 0

(6.36)

Compared to Eq. (6.35), it is clear that • With the feed-forward strategy, the integral control is only required to compensate the effects caused by the stator resistance error; • However, without the feed-forward strategy, the integral control is required to compensate the voltage drops caused by the stator resistance and the back EMF introduced by the coupling between the field and quadrature components. Thus, without feed-forward compensation, the working load for the integral control is greatly increased. This occurs not only during steady sate, as terms Rs i sd − L δ ωs i sq and Rs i sq + L s ωs i sd can be 10 times (or even more) greater than i sddmd Rserr and i sqdmd Rserr , but also during transient as the coupling between the field and quadrature components is fast from dynamics point of view. The integral control is well capable of compensating a slow changing uncertainty (such as the case for resistance variation), but is not able to counteract the fast change (such as the case for the back EMF variation). This is because the integral term is lagging and requires time to adapt to uncertainties. Any fast-changing uncertainties for integral control to compensate is inappropriate. Therefore, it is strongly recommended to have feed-forward strategy included in the vector torque control.

References 1. Abootorabi Zarchi H, Mosaddegh Hesar H, Ayaz Khoshhava M (2019) Online maximum torque per power losses strategy for indirect rotor flux-oriented control-based induction motor drives. IET Electr Power Appl 13(2):259–265 2. Bozhko S, Dymko S, Kovbasa S, Peresada SM (2016) Maximum torque-per-amp control for traction im drives: theory and experimental results. IEEE Trans Ind Appl 53(1):181–193 3. Briz F, Degner MW, Lorenz RD (2000) Analysis and design of current regulators using complex vectors. IEEE Trans Ind Appl 36(3):817–825 4. Clark RN (1996) Control system dynamics. Cambridge University Press 5. Del Blanco FB, Degner MW, Lorenz RD (1999) Dynamic analysis of current regulators for ac motors using complex vectors. IEEE Trans Ind Appl 35(6):1424–1432 6. Franklin GF, Powell JD, Emami-Naeini A, Powell JD (2002) Feedback control of dynamic systems, vol 4. Prentice hall Upper Saddle River 7. Holtz J (1996) On the spatial propagation of transient magnetic fields in ac machines. IEEE Trans Ind Appl 32(4):927–937

202

6 Vector Torque Control

8. Holtz J, Quan J, Schmittt G, Pontt J, Rodriguez J, Newman P, Miranda H (2003) Design of fast and robust current regulators for high power drives based on complex state variables. In: 38th IAS Annual Meeting on Conference Record of the Industry Applications Conference, 2003., IEEE, vol 3, pp 1997–2004 9. Idris NRN, Yatim AHM (2002) An improved stator flux estimation in steady-state operation for direct torque control of induction machines. IEEE Trans Ind Appl 38(1):110–116 10. Novotny DW, Lipo TA (1996) Vector control and dynamics of AC drives, vol 41. Oxford University Press 11. Pellegrino G, Bojoi RI, Guglielmi P (2011) Unified direct-flux vector control for ac motor drives. IEEE Trans Ind Appl 47(5):2093–2102 12. Sen PC (2021) Principles of electric machines and power electronics. Wiley 13. Trzynadlowski AM (2000) Control of induction motors. Elsevier 14. Vukosavic SN, Levi E (2003) A method for transient torque response improvement in optimum efficiency induction motor drives. IEEE Trans Energy Convers 18(4):484–493

Chapter 7

Flux-Weakening Control

7.1 Flux-Weakening Principle As explained in Sect. 6.3, the fast torque control strategy is not able to neutralize the machine torque capacity at high speed, which is illustrated in Fig. 6.6. It is the topic of this chapter to improve the fast torque control strategy. At high operating frequency, the machine control is subject to current limit (or I-limit), voltage limit (or V-limit), flux saturation limit (Ψ -limit, or Q-limit at rated frequency), and thermal limit (or T-limit). The flux-weakening strategy relies on which limit dominates the machine performance [3, 7, 8]. At comparatively low speed, I-limit and Q-limit play a major role in the control law design. pk The machine peak torque Tem is managed by I-limit. In other words, the I-limit constraint is tighter than the Q-limit so that I-limit becomes effective in determining the machine peak torque. This is illustrated in Fig. 7.1. In this chapter, it is assumed that Imax = 200 A and Ψmax = 0.84 Wb. dmd = T1 at operating frequency As shown in Fig. 7.1a, when the toque request Tem 3 Accordof ωs = ω1 , V-limit (or voltage limit) intersects the line i sd = Idlim 1 at . 3 ingly,  is also on the torque contour T1 for the given case. Therefore, at operating frequency ω1 ,2 T1 is the maximum torque that the fast torque strategy can deliver. Nevertheless, there exist many feasible operations under I/V-limits3 for the torque 1 and  2 are two interesting contour T1 . Compared to others, operating points  examples. For both operations, the field current i sd is reduced (greatly), compared to 3 Therefore, the resulted air-gap flux becomes weakened. This Idlim for operation . is why it is called flux weakening. However, for both the cases, the operations are well under the I/V-limits, and they are capable of delivering more torque than T1 if required. Their improvement over the fast torque control strategy is clearly demonIdlim is the same as Iddmd for fast torque control strategy. ω1 > ωb . ωb is the base speed as defined in Sect. 3.9. 3 Q-limit in this case is not a limiting factor. 1 2

© Springer Nature Switzerland AG 2024 S. Shen and Q.-z. Chen, Practical Control of Electric Machines for EV/HEVs, Lecture Notes in Electrical Engineering 1064, https://doi.org/10.1007/978-3-031-38161-4_7

203

204

7 Flux-Weakening Control Iq

Iq

MTPF

Iqlim

I-limit

MTPA

1

MTPF

MTPA

dmd Ilim q Iq2

2

3

Tmax T1

Idmd q2

Tmax T1 Tf

4

dmd

Iq3

Strategy 2

Strategy 1

Id

lim Id

T3 lim

dmd Id2 Id

Imax

(a) Principle

Id

Imax

(b) Strategy

Fig. 7.1 Flux Weakening torque strategy [I/V-limits]

strated, and they are named respectively as flux-weakening Strategy1 (as indicated by the shaded area in Fig. 7.1a) and Strategy2 (as indicated by the shaded area in Fig. 7.1b). The improvement gets even clearer when investigating operating point 4 at an increased speed ω2 (ω2 > ω1 ). As a consequence of the constraint of V limit, the torque T1 is unavailable any more with the fast torque control strategy, but it is still achievable under the flux-weakening strategies as illustrated. Nonetheless, flux-weakening strategies also have downsides. Particularly, for flux-weakening Strategy1, any torque demand change along the I-limit will be accompanied by a flux change, which is rather slow. In Fig. 7.1, ωb , ω f are the base and full flux-weakening speed, which is defined by Eqs. (3.100) and (3.101) respectively. ωc is the critical speed for flux-weakening Strategy2, where the torque current i sq from the MTPF has the same value as what the MPTA joints the I-limit. For the V-limit, it is defined as 2 2 + δ 2 i sq = CV lim (i sd , i sq ) −→ i sd

2 Vmax L 2s ωs2

√

with i sd = δi sq and i sq =

2 Imax 2

it yields ωc =

Vmax δL s Imax

(7.1)

Interestingly, at ωc , the I/V-limits joint at the q-axis. There are more speeds defined in Fig. 7.1, which are summarized in Table 7.1. Bear in mind that these speeds depend on the current limit Imax , being set to Imax = 200 A for Table 7.1. Also, note that these are stator operating frequency rather than rotor rotating speed.

7.1 Flux-Weakening Principle

205

Table 7.1 Machine speed definition summary Definition Equation Initial speed ω0

ω0 =

Base speed ωb

ωb =

Flux-weakening speed ω f 0 Flux-weakening speed ω f Critical speed ωc

Vmax L s Imax



2 ω 1+δ 2 0

√

ω f 0 = 2ω0  2 ω f = 1+δ ω0 2δ 2 ωc = 1δ ω0 √ ωc0 = δ2 ω0

Critical speed ωc0

Description

Value [rad/s]

V and I limits joint at d-axis

200.5

V and I limits joint at MTPA

282.0

V-limit joints d-axis at

I√ max 2

283.6

V and I limits joint at MTPF

1342.1

V and I limits joint at q-axis

1887.4

V-limit joints q-axis at

I√ max 2

2669.3

300 Id-limit line Iq-limit line I-limit circle MTPF line MTPA line

Machine Torqe [Nm]

250

200

150

100

50

0 0

200

400

600

800

1000

1200

1400

Machine Speed [rad/s] Fig. 7.2 Torque profiles of flux-weakening [I/V-limit dominated]

With these definitions and with the current limit of Imax = 200 A and Ψmax = 0.84 Wb, the speed-torque profiles of different strategies are given in Fig. 7.2 [1, 6, 11]. Note that the machine speed used in this figure is simplified as ωs /N p . • Fast torque strategy (in green line). The field current is fixed at i sd = Idlim . The torque current i sq is limited according to the I-limit when ωs < ωb , but to the max to 0 Nm when the V-limits once ωs > ωb . The torque reduces rapidly from Tem stator speed changes from ωb to ω f 0 , which is a tiny speed range (see Table 7.1 for details). This is a result of a quick deduction of i sq complying with the varying

206

•

•

•

•

7 Flux-Weakening Control

V-limits, which is a function of the stator speed. Thus, the fast torque strategy must be improved over the speed range of ωs > ωb ; I-limit strategy (in blue line). As the operating frequency goes up, the phase current sweeps over the I-limit circle from right to left. The torque rises from 0 Nm at max at ωb , which is where the I-limit circle connects the MTPA line. speed of ω0 to Tem After that, the machine torque shrinks gradually to 0 Nm at speed ωc . This strategy delivers the most torque during ωb < ωs < ω f . Also, this strategy performs fluxweakening once ωs > ωb ; Constant Iq strategy (in black line). The torque current is fixed at i sq = Iqlim . max when ωs > ωb at a speed proportional to 1/ωs The torque descends from Tem approximately. It reaches 0 Nm torque at ωc0 ; It delivers the second most torque during ωb < ωs < ω f . The field current i sd is more or less inversely proportional to the stator frequency ωs , which makes it one of the favorable flux-weakening approaches; MTPA strategy (in cyan line). This strategy has the same torque as the fast torque and constant Iq strategy as long as ωs < ωb , due to the torque is limited by I-limit. The torque of this strategy also drops very quickly after ωs > ωb . It does not reach 0 Nm torque as quickly as the fast torque strategy, because it has flux-weakening function though moderately; MTPF strategy (in red line). The torque for ωs < ω f is limited by the I-limit at 2δ max max MTPF, which is 1+δ 2 Tem . It is very different from Tem , making operating MTPF alone infeasible at low speed if the I-limit dominates the system performance. However, once ωs > ω f , the largest torque is derived by the MTPF strategy. This is the best flux-weakening strategy in terms of torque performance. The constant Iq strategy meets MTPF at ωc .

7.2 I-Limit Dominated Flux-Weakening Control Bear in mind that the flux saturation limit (Q-limit) is fully relaxed (by setting Ψmax = 0.84 Wb) in this section in order to investigate the flux-weakening performance when I-limit is dominant. However, more realistic flux limit is adopted in Sect. 7.3 and thereafter. Upon the investigation of the torque profiles in Fig. 7.2 and the fluxweakening principle as given in Fig. 7.1, the following could be concluded when the I-limit dominates the peak torque performance. • Maximum torque is reached by the fast torque strategy when ωs < ωb , I-limit when ωb < ωs < ω f , MTPF when ωs > ω f . Flux-weakening strategy1 is a combination of I-limit and MTPF; • The second maximum torque profile consists of the fast torque strategy when ωs < ωb , constant Iq strategy when ωb < ωs < ωc , and MTPF when ωs > ωc . Flux-weakening strategy2 is referred to as the composition of Iq -limit and MTPF; • The Flux-weakening strategy2 delivers slightly lower torque than strategy1 during ωb < ωs < ωc . However, in this same speed range, strategy2 has a great

7.2 I-Limit Dominated Flux-Weakening Control

207

advantage that the torque current is not required to rise further up from Iqlim , which is defined at the peak torque operating condition (it is MPTA at I-limit for this case). The flux-weakening Strategy2 is selected in this book, but an immediate comment is added that flux-weakening strategy1 by all means is well-applied in automotive section, where the torque capability is a major concern. The torque profile and the corresponding torque and field current are summarized in Fig. 7.3 As shown in Fig. 7.3b, it is clear that the torque current is greatly increased for Strategy1 just in order for elevating the machine torque very gently, which implies a great sacrifice of the system efficiency. Nevertheless, The field current profiles of these two strategies are very similar. Thus, from the flux-weakening point of view, the Strategy1 and Strategy2 are more or less the same. The main difference is how the permissible torque current i sq is determined. By the way, Fig. 7.3c is commonly used as a map in flux-weakening control to determine the field current i sd based on the machine speed. Flux-weakening control is a common approach for neutralizing the machine capability in the high speed range, for either ACIMs or PMSMs. To maintain the fast torque capability as much as possible is one design criterion for flux-weakening control. One evident advantage of electric machines over internal combustion engines is its fast torque ability. Electric machines can deliver the demanded torque around or less than 10 ms, while internal combustion engines require 100 ms to do so. It is the same reason to design a fast torque strategy as in Sect. 6.3 of Chap. 6. By doing so, it requests a large field current regardless of the torque demand, which trades off the system efficiency undoubtedly. For flux-weakening control, in order to have a fast torque response, the field current shall be designed with slow variation. If the demanded flux-weakening has sufficiently low bandwidth than that of the rotor dynamics, it can be well considered as a constant. As such, the fast torque capability is well maintained over the flux-weakening region. Nevertheless, the mechanical dynamics of an electric machine for a driving system is rather slow. This is indeed true for EV/HEVs as the electric machine is rigidly coupled to the vehicle with a large inertia. Thus, the rotor speed is treated as a parameter rather than a state for machine model-based control and for electric machine dynamics analysis. Therefore, Determining the flux-weakening based on the rotor speed resumes the fast torque capability (of the vector control) due to the slow changing nature of the rotor speed. This is, to some extent, decoupling the field control from the torque control in the frequency domain. Prior to a review and summary of the flux-weakening strategy2, as given in Fig. 7.1, the following definitions are given

208

7 Flux-Weakening Control

Fig. 7.3 Torque and current profiles [I/V-limit]

300 Flux-Weakening Strategy 1 Flux-Weakening Strategy 2

Machine Torqe [Nm]

250

200

150

100

50

0 0

200

400

600

800

1000

1200

1400

Machine Speed [rad/s]

(a) Torque profile 250 Flux-Weakening Strategy 1 Flux-Weakening Strategy 2

Torqe current [A]

200

150

100

50

0 0

200

400

600

800

1000

1200

1400

Machine Speed [rad/s]

(b) Torque current profile 200 Flux-Weakening Strategy 1 Flux-Weakening Strategy 2

180 160

Filed current [A]

140 120 100 80 60 40 20 0 0

200

400

600

800

1000

Machine Speed [rad/s]

(c) Field current profile

1200

1400

7.2 I-Limit Dominated Flux-Weakening Control

209

• Type1 flux-weakening is defined as weakening the flux along MTPF; • Type2 flux-weakening is defined as weakening the flux along the constant torque current limit. The field current i sd is determined based on a reference stator speed ωˆ s as follows

dmd i sd

⎧ √ 2 ⎪ ⎪ ⎨ Imax 2 2 ω2 b = Imax 1+δ − 2 ωˆ s2 ⎪ ⎪ ⎩I √ δ ω f max

 ωs ≤ ωb δ2 2

1+δ 2 ωˆ s

ωb <  ωs ≤ ωc

(7.2)

ωˆ s > ωc

in which the reference speed ωˆ s is determined by cap (ωr e ) ωˆ s = ωr e + ωsp

(7.3)

By comparing to Eq. (6.3), the intention of adopting the reference speed ωˆ s becomes clear. In Eq. (6.3), the required slip speed is used to determine operating frequency. However, the demanded torque (or torque current) is a fast changing signal. This in return requires a fast implementation of the rotor flux, which is contradictory to the flux-weakening design target. Instead of the required slip speed req cap ωsp = Rˆ r κr i sq /Ψˆ r , the slip speed ωsp that delivers the maximum torque at the given speed is applied in Eq. (7.3), making ωˆ s independent from the torque demand. Also, if ωˆ s is directly and totally determined by ωr e as ωˆ s = ωr e , it is simple but has a risk of breaking the voltage limit caused by a slightly lagging flux-weakening. Neverthedmd by Eq. (7.2) can be equally represented using a map. less, the determination of i sd Map-based design is more readily usable for coding (e.g. graphic Simulink coding or other model-based design) and for calibration, which is commonly used in automodmd determination resembles Fig. 7.3c. Please tive industry. The map representing i sd note that Imax in Eq. (7.2) is not strictly a constant. When temperature rises, Imax shall be adjusted accordingly to avoid potential damage to the system. Therefore, a map-based design approach shall have a control factor on updating the field current demands based on the system temperature. Equation (6.6) is still valid during flux-weakening for demanding the torque curdmd dmd , since the voltage limit is already applied in i sd determination. Therefore, rent i sq Eq. (6.8) is not necessary any more. Nevertheless, as shown in Fig. 7.3a, the machine torque capability is reduced at high speed. The torque demand must comply to this limit.  raw cap dmd = min Tem , Tem (ωr e ) (7.4) Tem cap

where Tem represents the machine torque capability as shown in Fig. 7.3a, and is given by the following

210

7 Flux-Weakening Control

cap Tem

=

⎧ KT Lm 2 ⎪ I ⎪ ⎨ 2 max

2 Vmax K √T L m  ωs2 2L s ⎪ 2 ⎪ ⎩ 1 K T L m Vmax 2 Ls Lδ  ωs2

 ωs ≤ ωb −

1 2 2 L I 2 δ max

ωb <  ωs ≤ ωc

(7.5)

 ωs > ωc

cap

Bear in mind that Tem is also updated according to the system temperature, and that cap the map-based representation of Tem can also be equally applied. For the completeness of this section, Eq. (6.6) is recalled as below raw = i sq

dmd Tem k T L m i sd

However, the voltage limit control is essential for flux-weakening, in particular when the torque demand is close to or at the torque capability for the given machine speed. Even with a perfect machine model, the voltage is on the boundary of limit already. The current feedback control and resistance compensation control will require extra phase voltage. Thus, it is very likely that the resulted voltage may exceed the limit. Unlike exceeding the current limit takes time to become effective as it is ultimately a thermal consequence, the over voltage can have an immediate effect. This is why an internal voltage loop is introduced as shown in Fig. 7.4. The designed voltage feedback loop is an one-sided limit control using only the proportional term. Pv denotes the proportional voltage controller. The integral term is not adopted due to its slow reaction nature which takes time to adapt to the uncertainties and disturbances. This nature is somehow against the requirement of the limit control, where the controller becomes active sporadically and intermittently leaving no sufficient time for the integral part to learn the uncertainties and disturbances.4 The voltage error Vserr is defined as

Vserr

=

kv Vmax − 0,



2 2 , k V vsd + vsq v max
δi sq for motoring and i sd > −δi sq for generation

in the i d -i q coordinates. In order to have a smooth operation of voltage limit control, the direction of Imod is further refined as E I mod = αe M T P F + (1 − α)eGrad V

(7.8)

in which E I mod , e M T P F and eGrad V represent the directions of the vectors in the Id -Iq coordinates for Imod , MPTF line, and gradient of voltage contour, respectively. Particularly, e M T P F and eGrad V are unit vectors. α is a control parameter taking value from 0 to 1 depending on the distance d I F of the demanded current Idmd (or actual current Iact s ) to the MPTF line. α can be linearly determined as

α=

1 − d I F /dth d I F < dth 0 Otherwise

214

7 Flux-Weakening Control

where dth is a threshold distance which can be calibrated. Nevertheless, there are more nonlinear choices, which will not be detailed in this book. Once E I mod is normalized as 1 e I mod = E I mod |E I mod | then mod = |edI mod |Imod i sd

(7.9)

q |e I mod |Imod

(7.10)

mod i sq

=

q

in which edI mod and e I mod are the projection of e I mod to the respective d and q axes. To conclude this section, the flux-weakening strategy as shown in Fig. 7.1b is described as below • For the operating speed ω2 > ωb , flux-weakening is required; • At ω2 , the torque capability with flux-weakening Strategy2 is T1 ; dmd is determined by ωs (or ωr ), geometrically according • The field current demand Id2 to Fig. 7.1b, mathematically by Eq. (7.2), and graphically as in Fig. 7.3c; dmd dmd = T1 , the torque current demand is Iq2 (black color in • For the torque demand Tem the figure), geometrically according to Fig. 7.1b, and mathematically by Eqs. (7.4) dmd = Iqlim ; and (6.6). At torque capability level of T1 , Iq2 dmd • For the torque demand T3 < T1 , the field current demand is still Id2 as long as dmd the speed ωs remains unchanged. The torque current becomes Iq3 , geometrically according to Fig. 7.1b, and mathematically by Eq. (6.6).

7.3 Q-Limit Dominated Flux-Weakening Control If sufficient care is taken, one can find that the speed to torque profile as given in Fig. 7.3a is very different from what is presented in Fig. 2.26. In particular, the base speed in Fig. 2.26 is about 260 rad/s, but it is 141(= 282/N p ) rad/s in Fig. 7.3a. This is because • For the torque profiles as in Fig. 7.3a, the peak torque is derived by the I-limit on MTPA, and flux-weakening starts when the V-limit joints the I-limit at MTPA. The Q-limit (which is relaxed in the previous section) is not a limiting factor in this case; • For the torque profiles as in Fig. 2.26, the peak torque is derived by the Q-limit on MTPF, and flux-weakening starts when the V-limit becomes the same as the Qlimit. The I-limit (which is relaxed in that case) becomes irrelevant when deriving the peak torque in such a case. Along the Q-limit line yields the optimal slip at MTPF as defined in Eq. (2.27).

7.3 Q-Limit Dominated Flux-Weakening Control

215

In general, I-limit is more a short-period limit. The thermal limit plays a more practical role. Firstly, I-limit can be quite large. The current limit depending on switch devices and stator windings are as high as 600 A for the given example machine. When the temperature of either the component or the system increases, the I-limit will be reduced. Thus, I-limit is a dynamic limit. As a result, system performance will change dramatically, depending on whether it is determined by I-limit, partially or completely. This process is illustrated in Fig. 7.6. Keep in mind that V-limit is also a dynamic limit, which develops with the operating speed ωs . However, Q-limit is a fixed limit once the machine design is frozen. Therefore, unsurprisingly, the Qlimit does not change throughout Fig. 7.6a–e. By contrast, the I-limit varies from one figure to another, while the V-limits are a family of curves in each figure. There is the largest I-limit in Fig. 7.6a, which gradually reduces from Fig. 7.6b–d, and becomes the narrowest in Fig. 7.6e. Also, the base speed ωb , flux-weakening speed ω f , critical speed ωc , and the corresponding V-limits are indicated. ωb and ωc collapse to ω f in Fig. 7.6a and b. Referring to Fig. 4.15, which represents an equivalent dq-frame model, the back EMF VM = Ψ M ωs generated by the air-gap flux is at the terminal of the magnetizing inductances L M , but not the terminal of the phase windings. Therefore, the use of Vs /ωs to estimate Ψ M is inaccurate. The similarity from Figs. 4.15 to 2.22 suggests that application of Thevenin’s theorem is not accidental. Q-limit is defined as Q max = Vsrt /ωsrt = Ψmax /rv Iq

Iq

MTPF

(7.11)

MTPF

Iqlim

Iqlim MTPA

lim

Id (a) Q-limit fully dominates

Iq

Iq

MTPF

MTPA

Tmax

Tmax

Tem

Tem

lim Imax Id (b) Q-limit dominates

Imax

Iq

MTPF

MTPA

MTPF

MTPA

MTPA

Iqlim Iqlim Iqlim

Tmax Tmax Tem

Tmax

Tem

Tem lim

Imax Id (c) Q/I limit jointly dominates

Fig. 7.6 Q/I limit effects

Id

lim

Imax Id (d) I-limit dominates

Id

lim

Imax Id (e) I-limit fully dominates

Id

216

7 Flux-Weakening Control

in which rv is the voltage ratio between VM and Vs as in Sect. 2.3.2, while the permissible flux Ψmax of the system is measured at the rated voltage and rated frequency. As a mater fact, the above equation is the inverse of Eq. (2.40). Equation (2.40) defines how to measure the permissible flux, while Eq. (7.11) determines Q-limit based on the known Ψ limit. The information given in Fig. 7.6a–e are interpreted as below: • In Fig. 7.6a, the system performance is determined by the Q-limit completely. The peak torque is reached where the Q-limit joints the MTPF line as5 max Tem =

1.5N p 2 1 KT Lm 2 Q max ≈ Q max 2 Ls Lδ 2L δ

(7.12)

Bear in mind that Q max is in magnitude rather than rms value, which is similar to Eq. (2.39) . The base speed, flux-weakening speed, and critical speed in this case are the same as (7.13) ωb = Vmax /Q max • In Fig. 7.6b, the system performance is also determined completely by the Q-limit. However, the I-limit just touches the Q-limit at MTPF. Thus,  qC Imax

=

1 + δ 2 Q max 2δ 2 Ls

(7.14)

Consequently, the base, flux-weakening and critical speed can also be expressed as a function of Imax as  ωc = ωb = ω f =

1 + δ 2 Vmax qC 2δ 2 L s Imax

qC

When Imax ≥ Imax , the system performance is determined by the Q-limit; • In Fig. 7.6c, the maximum torque is jointly determined by both the Q/I-limits. At the point where the I-limit intersects the Q-limit, the field and torque current are as  2 2 L s Imax −Q 2max QI = i sq 2 2  L s −L δ (7.15) 2 2 2 Q max −L δ Imax QI i sd = 2 2 L −L s

Thus max = Tem

5

L 2m = L SS L R R ≈ L s L r .

KT Lm L 2s − L 2δ

δ

 2 2 ) (L 2s Imax − Q 2max )(Q 2max − L 2δ Imax

(7.16)

7.3 Q-Limit Dominated Flux-Weakening Control

217

Equation (7.13) is still held valid for the base speed calculation, but the fluxweakening speed and critical speed are determined differently and respectively as  1 + δ 2 Vmax (7.17) ωf = 2δ 2 L s Imax √  2 1/δ 2 − 1 Vmax ωc = 2 2 2 1 − Q max /(L s Imax ) L s Imax

(7.18)

Equation (7.17) is actually the same as Eq. (3.101), it is re-given here for the sake of completeness. It is clear that now both ω f and ωc vary with Imax , but ωb is independent from the I-limit. Keep in mind that the flux-weakening speed ω f as given in Table 7.1 is the same as in Eq. (7.17) in terms of expression. The difference in value comes from the difference in the I-limit; • In Fig. 7.6d, the system performance starts to be determined by I-limit alone. Qlimit is just touching the I-limit at MTPA. The maximum torque is determined as in Eq. (3.87). For the completeness of this section, it is recapped as max = Tem

1 2 K T L m Imax 2

The Q/I-limits have the following relationship  iC Imax

=

2 Q max 1 + δ2 L s

(7.19)

Equation (7.13) for ωb and Eq. (7.17) for ω f are still valid. However ω f will have a different value since Imax becomes different. The critical speed ωc is determined by 1 Vmax (7.20) ωc = δ L s Imax This expression of ωc is the same as in Table 7.1 and as Eq. (7.1). However, Imax is a dynamically changing parameter in this section. The flux-weakening control strategies developed in the previous section are applicable; • In Fig. 7.6e, the system performance is completely determined by the I-limit. Q max is not a limiting factor any more, since Imax is a more tightened limit. The peak torque is governed by Eq. (3.87) as in Fig. 7.6d. Equation (7.17) for ω f and Eq. (7.20) for ωc are still valid. However, the base speed is now determined by Imax rather than Q max as  ωb =

2 Vmax 1 + δ 2 L s Imax

(7.21)

The flux-weakening control strategies developed in last subsection are for this case.

218

7 Flux-Weakening Control

Table 7.2 Machine performance summary max [Nm] Dominant limits Imax [A] Tem Q-limit

Imax > 411

Q-limit Q/I-limits I-limit I-limit

Imax Imax Imax iC Imax

qC

= 411 < 411 = 86 < 86

ωb [rad/s]

ω f [rad/s]

ωc [rad/s]

226

653

653

653

226 Varying 48 Varying

653 653 653 Varying

653 Varying 3108 Varying

653 Varying 4372 Varying

Based on the above analysis, the Q-limit dominates the system performance when qC Imax ≥ Imax . On the other hand, the I-limit governs the system performance when iC . the system performance is jointly determined by the Q/I-limits if Imax Imax ≤ Imax is in-between. With the parameters in Table 3.1, specifically Q max = 0.30 Wb, the qC iC are as respective Imax and Imax qC

Imax = 411 A iC = 86 A Imax qC

Practically, instant I-limit could be well over Imax = 411 A. Therefore, the Q-limit that dominates flux-weakening control shall require more attention. Table 7.2 summarizes the Q/I-limits in determining the machine performance. It is clear that the maximum torque decreases dramatically from the Q-limit dominance to the I-limit dominance. The maximum current ratio between the Q-limit and the I-limit is as qC iC r IQI = Imax  /Imax  = 21 LL δs + LL δs ≈

1 Ls 2 Lδ

For the given example machine, r IQI is about 4.7. According to Table 7.2, torque ratio is 233/48 = 4.7. Thus, in case that both the Q/I-limits are active, the maximum torque is reduced almost linearly with respect to the I-limit. Once the I-limit becomes dominant, the maximum torque is inversely proportional to the square of Imax according to Eq. (3.87), which is a much faster rate than that in the Q/I-limits dominating region. Table 7.2 clearly indicates that: (1), Once I-limit becomes active, even if it determines the system performance jointly with Q-limit, the maximum torque, ω f and ωc are functions of the I-limit. The maximum torque decreases almost linearly with the descending I-limit. On the other hand, ω f and ωc increase significantly; (2) Base speed ωb only goes up when I-limit starts dominating the performance; (3) ωb , ω f and ωc are inversely proportional to I-limit. These trends are shown in Fig. 7.7. One interesting fact is that system performance remains invariant as long as Q-limit stays dominant. This is very beneficial for system control.

7.3 Q-Limit Dominated Flux-Weakening Control

219

Maximum torque

250

Flux-weakening speed

9000

Base Speed Flux Speed Critical Speed

8000 7000

Stator Speed [rad/s]

Torque [Nm]

200

150

100

6000 5000 4000 3000 2000

50

1000 0

0 0

50

100

150

200

250

300

350

400

450

500

0

50

I-limit [A]

(a) Maximum torque

100

150

200

250

300

350

400

450

500

I-limit [A]

(b) Flux weakening speeds

Fig. 7.7 Flux weakening curves as a function of the I-limits

The flux-weakening control strategy when Q-limit dominates is demonstrated as in Fig. 7.8. Based on the stator operating frequency (or speed), the strategy is partitioned into two regions • Fast torque control region. When ωs < ωb , only Q-limit is active in determining the feasible operating region. i sd is fixed to Idlim , which is in turn determined by dmd ; the Q-limit at MTPF. i sq is then left to deliver the demanded torque Tem • Flux-weakening region. This is when ωs > ωb . V-limit overtakes Q-limit to become a dominating factor. i sd now is derived by the V-limit, which in turn is a function of stator frequency, along the MTPF line. Similar to the fast torque region, i sq is applied for the machine torque demand. By this strategy, the flux-weakening and torque control are decoupled. Field current i sd controls the system under the Q/V-limits, while torque current i sq regulates the machine torque. In order to add more properties to this decoupling, it is preferable to make the torque path more transient and dynamic, while making the flux-weakening path more static or quasi-static. One way to do so, which is already adopted in the previous section, is to let the desired flux or field current depend only on rotor speed due to its slowly changing nature. This is more readily usable with Q-limit dominated flux-weakening, since slip speed is a constant along the MTPF line. This strategy is illustrated in Fig. 7.8. Since the operating speed ωs > ωb , the electric machine is operated in the flux-weakening region. the field current demand Iddmd as shown in the figure is determined by the speed dependent V-limit (ωs ). The torque demand Tem in this figure is well within the machine torque capability at the speed ωs . Thus, the intersection of the line Id = Iddmd with the torque contour Tem gives rise to the demanded torque current Iqdmd . When the torque demand Tem varies, only Iqdmd varies as a response, but not Iddmd . It is noteworthy that T peak is the peak max (ωs ) at the operating torque for the given Q-limit, but not the capability torque Tem speed ωs which will be below T peak obviously.

220

7 Flux-Weakening Control

Iq

MTPF

Iqlim MTPA Idmd q

Tmax

Tem lim Idmd Id d

Imax

Id

Fig. 7.8 Flux weakening torque strategies [Q/V-limit]

If sufficient attention is paid, one could see that varying Iqdmd (with fixed Iddmd ) is actually to set an appropriate slip speed for the demanded torque. This is somehow analogous to the scalar control as described in Chap. 5. The only difference is that the scalar control regulates the required slip speed along the V-limit contour, while the vector control along the Id = Iddmd line. Thus, the scalar control will result in a slight change of the rotor flux, but the vector control can maintain it as a constant. The torque profile, together with the associated field and torque current profiles of the above-mentioned strategy for the given example machine, is illustrated in Fig. 7.9. Compared to the torque profile as given in Fig. 2.26, they are very similar. This indicates that the peak torque analysis in Chap. 2 is the case that the Q-limit plays a dominating role. Unlike Fig. 7.3, where slip speed is not considered in determining the machine speed, it is taken into account in this figure by instead. Two comments are immediately added as • The torque and field current profiles as in Fig. 7.9 are commonly used as lookup tables for flux-weakening control strategy implementation; max ∝ 1/ωs2 . Consequently, the power in the • Within the flux-weakening region, Tem same region is inversely proportional to the stator electric speed ωs . The peak torque of a Q-limit dominated system is determined by the intersection of either the MTPF line with Q-limit when the flux-weakening is not required, or the MTPF line with V-limit when it is required. As a matter of fact, the peak torque, as given below according to Eq. (3.98), is arrived at on the MTPF line.

7.3 Q-Limit Dominated Flux-Weakening Control Fig. 7.9 Torque and current profiles [Q/V-limit]

221

250

Torqe [Nm]

200

150

100

50

0 0

100

200

300

400

500

600

Machine Speed [rad/s]

(a) Torque profile 450 Field current Torque curent

400 350

Current [A]

300 250 200 150 100 50 0 0

100

200

300

400

500

600

Machine Speed [rad/s]

(b) Current profiles

max Tem =

⎧ ⎨ 1 K T L m Q 2max ωs < ωb 2 Ls Lδ ⎩ 1 KT Lm 2 Ls Lδ

2 Vmax ωs2

(7.22)

Otherwise

max max Now it becomes clear that Tem ∝ 1/ωs2 or Pem ∝ 1/ωs during flux-weakening. The Q-limit dominated flux-weakening control as given in Fig. 7.10 is summarized below. Compared to the I-limit dominated strategy as in Fig. 7.4, the main differences are the torque capability and the reference current determination. Equation (7.4) to limit the raw torque demand to the machine torque capability is valid, but for the completeness reason is recalled as below

 raw cap dmd = min Tem , Tem (ωr e ) Tem

222

7 Flux-Weakening Control cap

in which Tem is governed by Eq. (7.22), and is illustrated in Fig. 7.9a. Equation (7.2) for determining the field current i sd is hereby updated as dmd i sd =

⎧√ ⎨ 2 Q max  ωs ≤ ωb 2 Ls ⎩

√ 2 Vmax 2 Ls  ωs

(7.23)

Otherwise

 ωs is now defined as ωˆ s = ωr e +

Rr Lr δ

(7.24)

The second term in the above equation is the optimal slip speed from the maximum torque point of view, according to Eqs. (3.79) and (3.98). Based on the above equation, dmd depends only on ωr e (or ωr m in reality), which is a slowly changing state. Thus, i sd dmd is a quasi-static demand. i sd Similarly, Eq. (6.6) is still valid for determining the torque current demand based on the torque and field current demand, and is re-given as below for the sake of completeness

Vlim

|vs|

Current Determination

raw

Torque Capability

Tem

-

Pv

Ilim dmd

Tem

cap Tem

dmd isd

-

ωr

Tem

dmd isq

fb vsd

PId

dmd isq

|

vsd

+

fb vsq

PIq

-

|

vsq

+

idmd sd

ff vsd

act

dmd isq

ff vsq

act

dmd isd idmd sq

ωr

dmd isd

Current Filter

Speed Filter

ω rm

isd

isq ω rm

Np

ωs

ω re

Vector Control with flux-weakening

Kr/Reqτ r

-

re

+ Np ψr

Vs 1/Req

+

-

τT

Is

I sd

Lm

iτT

ωs Stator Model

X

τr

iτr

ψr

Rotor Model

Fig. 7.10 Flux-weakening close loop system [Q/V-limit]

I sq

X

X

ω re

KT

T em

7.4 Q/I-Limit Dominated Flux-Weakening Control raw i sq =

223

dmd Tem k T L m i sd

As for the internal feedback loop for voltage limit control as given Fig. 7.10, the strategy employed here is the same as in Sect. 7.2.

7.4 Q/I-Limit Dominated Flux-Weakening Control As explained in the above section that maximum current is a dynamic limiting factor. In particular, I-limit depends on the component as well as system temperature. It is part of the electric machine thermal control. Generally, thermal control regulates the machine torque capability according to the system temperature. This can be either a model-based or a map-based approach, but deduction of the torque capability is a result of the reduced I-limit as shown in Fig. 7.7a. The flux-weakening strategy in last section is widely used in automotive industry as well as other sectors. However, the operating point where the Q-limit joints the MTPF line is usually not sustainable, when the I-limit is reduced below the Q-limit along the MTPF line, as illustrated by Fig. 7.11. This is generally the case when temperature goes up. In other words, the peak torque determined by the Q-limit on MTPF is not always available. A typical duration is about 30 s, depending on the thermal system design, integration and control. Nevertheless, with a given thermal system, there must exist a balance between the heat generated by the machine as

Iq

MTPF

MTPA

Ilim q Idmd q

Tmax

Tem

Idmd d

lim

Id

Fig. 7.11 Flux weakening torque strategies [Q/I-limit]

Imax

Id

224

7 Flux-Weakening Control

well as inverter and the heat dissipated by the cooling system, which is subsequently dumped to the surrounding environment. This balance suggests a sustaining torque, which is the well-known continuous torque of the electric machine. This balance also implies a sustaining level of I-limit. By the way, the peak and continuous torque concepts are introduced in Sect. 2.3. The task of this section is to • Design a control strategy when both Q/I-limits are active, which is the case when the I-limit reduces significantly with an increased temperature; • Analyze the system performance under the Q/I-limits-constrained flux-weakening strategy. Figure 7.11 shows a case of the flux-weakening strategy with active Q/I-limits. Although there is an infinite combination of Q/I-limitations, the control strategy is more or less the same, and is similar to that of the I-limit dominated flux-weakening Strategy2. It consists of the following three operating regions. • Constant torque region at ωs < ωb . Field current i sd is fixed at Idlim , which is determined as Eq. (7.15) and is a function of the I-limit Imax . Different torque requests are implemented by different torque current i sq . To account for and apply the I-limit, i sq must satisfy i sq < Iqlim . • Nearly constant power region when ωb < ωs < ωc . In this region, the V-limit takes over the Q-limit in determining the system performance, and consequently fluxweakening starts. Field current i sd is now a function of rotor speed in order to maintain a fast torque response. This leaves the torque current i sq to deliver the torque demand. Bear in mind that, along the Iqlim line, it does not produce constant power operation. The constant power operation is between the I-limit circle and the Iqlim line, as detailed in Sect. 3.9. Nevertheless, results given in Fig. 7.3a indicate that the Iqlim line is close to constant power operation; • Constant speed power region if ωs > ωc . The system performance is completely determined by the V-limit along MTPF. The field current i sd is driven by the V-limit, while the torque current derived by torque demand. The flux-weakening strategy in this region has no difference with that presented in Sect. 7.3. The torque capability governed by Eq. (7.22) is still held for the case of ωs > ωb , but the speed condition shall be updated as ωs > ωc . Comparing to the Q-limit dominated flux-weakening control, it is interesting to see the following facts • The peak torque is reduced. This is expected as a results of the reduced I-limit; • The field current is significantly increased at the peak torque, compared to the Q-limit flux-weakening operations. This is because in order to maximize the peak torque, the operating point is shifted away from the MTPF line towards the MTPA line, depending on the level of the I-limit. One extreme case is on the MTPA line when the I-limit becomes dominant; • The nearly constant power operation is introduced as a result of the interaction of Q/I-limits. Moreover, this nearly constant power operation region is extended when the I-limit is further reduced.

7.4 Q/I-Limit Dominated Flux-Weakening Control

225

Listing 7.1 Performance profiles calculation 1 2 3 4

Freq ws Vdc vs

= = = =

100; 2* pi * Freq ; 330; Vdc / sqrt (3) ;

5 6

% For Y c o n n e c t i o n and SPWM % Vpn = Vdc * sqrt (3) /3 % Vpn_rms = Vpn / sqrt (2)

7 8

Qmax = 0 .28 /0 .96 ;

9

10

11 12 13 14 15 16 17 18 19 20 21 22

% Max flux at Phase t e r m i n a l % r_v =0 .96 rather than 0 .92 to a c c o u n t for voltage % Loss at L_sl and Rs w h i c h is d r o p e d at SS analysis % Maximum voltage % This level , Q - limit d o m i n a t e s

... ...

Vmax = vs ; Imax = 500; Motor_Parameters_DQ ; % % D e f i n e some common p a r a m e t e r s Np = Pole_pair ; k_r = Lm / Lr ; k_T = 1 .5 * Np * k_r ; Req =( Rs + Rr *( k_r ) ^2) ;  = 1 -( Lm ^2/( Ls * Lr ) ) ; tau_r = Lr / Rr ; tau_s =  * Ls / Req ; L_lk =  * Ls ;

23 24 25 26

%% Common calculation Imax_qC = Qmax / sqrt (2) * sqrt (1/ Ls ^2+1/ L_lk ^2) ; Imax_iC = Qmax * sqrt (2) / sqrt ( Ls ^2+ L_lk ^2) ;

27 28 29

T _ p e a k _ q C = 0 .5 * k_T * Lm / Ls * Qmax ^2/ L_lk ; T _ p e a k _ i C = 0 .5 * k_T * Lm * I m a x _ i C ^2;

30 31 32 33

wb_qc wf_ic wc_ic

= Vmax / Qmax ; = ( Vmax /( Ls * I m a x _ i C ) ) * sqrt ((1+  ^2) /2/  ^2) ; = ( Vmax /( Ls * I m a x _ i C ) ) /  ;

34 35 36 37 38 39 40 41 42 43 44 45

%% Prepare figure close all ; % c l o s e all f i g u r e figname = ' TqCurve '; h _ T q f i g = f i n d o b j (0 , ' Name ' , [ figname , ' _ R e s u l t s ' ]) ; if i s e m p t y ( h _ T q f i g ) h _ T q f i g = f i g u r e ( ' P o s i t i o n ' ,[50 50 452*1 .8 257*1 .8 ] , ... ' Name ' , [ figname , ' _ R e s u l t s ' ] , ... ' N u m b e r T i t l e ' , ' off ' ) ; end f i g u r e ( h _ T q f i g ) ; box on , hold on ; set ( h_Tqfig , ' D e f a u l t A x e s F o n t S i z e ' ,14 , ' D e f a u l t A x e s F o n t N a m e ' , ' Times ... New Roman ' ) ;

46 47 48 49 50 51 52 53 54 55

56

figname = ' IdCurve '; h _ I d f i g = f i n d o b j (0 , ' Name ' , [ figname , ' _ R e s u l t s ' ]) ; if i s e m p t y ( h _ I d f i g ) h _ I d f i g = f i g u r e ( ' P o s i t i o n ' ,[50 50 452*1 .8 257*1 .8 ] , ... ' Name ' , [ figname , ' _ R e s u l t s ' ] , ... ' N u m b e r T i t l e ' , ' off ' ) ; end f i g u r e ( h _ I d f i g ) ; box on , hold on ; set ( h_Idfig , ' D e f a u l t A x e s F o n t S i z e ' ,14 , ' D e f a u l t A x e s F o n t N a m e ' , ' Times ... New Roman ' ) ;

226 57 58 59 60 61 62 63 64 65

7 Flux-Weakening Control

figname = ' IqCurve '; h _ I q f i g = f i n d o b j (0 , ' Name ' , [ figname , ' _ R e s u l t s ' ]) ; if i s e m p t y ( h _ I q f i g ) h _ I q f i g = f i g u r e ( ' P o s i t i o n ' ,[50 50 452*1 .8 257*1 .8 ] , ... ' Name ' , [ figname , ' _ R e s u l t s ' ] , ... ' N u m b e r T i t l e ' , ' off ' ) ; end f i g u r e ( h _ I q f i g ) ; box on , hold on ; set ( h_Iqfig , ' D e f a u l t A x e s F o n t S i z e ' ,14 , ' D e f a u l t A x e s F o n t N a m e ' , ' Times ... New Roman ' ) ;

66 67 68

69

70

% % Setup Imax % IMAX = [(1 .5 * I m a x _ q C ) : -50: I m a x _ q C I m a x _ q C : -10: I m a x _ i C I m a x _ i C : -5: I m a x _ i C /2]; IMAX = [ I m a x _ i C / 2 : 1 0 : Imax_iC -10 I m a x _ i C :25: I m a x _ q C ... I m a x _ q C : 1 0 0 : 1 .5 * I m a x _ q C ]; % IMAX = [ I m a x _ i C I m a x _ i C ];

...

71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96

% % Main C a l c u l a t i o n for II = 1: length ( IMAX ) Imax = IMAX ( II ) ; if ( Imax ≤ I m a x _ i C ) Iq = Imax / sqrt (2) ; Id = Imax / sqrt (2) ; w0 = Vmax /( Ls * Imax ) ; wb = sqrt (2/(1+  ^2) ) * w0 ; wc = 1/  * w0 ; e l s e i f ( Imax ≥ I m a x _ q C ) Iq = Qmax / L_lk / sqrt (2) ; Id = Qmax / Ls / sqrt (2) ; wb = Vmax / Qmax ; wc = wb ; else w0 = Vmax /( Ls * Imax ) ; L0s = Ls ^2 - L_lk ^2; Iq = sqrt (( Ls ^2* Imax ^2 - Qmax ^2) / L0s ) ; Id = sqrt (( Qmax ^2 - L_lk ^2* Imax ^2) / L0s ) ; wb = Vmax / Qmax ; wc = sqrt ( L0s /(2*(1 -( Qmax / Ls / Imax ) ^2) ) ) / L_lk * w0 ; end Wsp0 = Rr / Lr * Iq / Id ; Tp = k_T * Lm * Iq * Id ; WS = [ max ( Wsp0 ,2) :2: wb wb : wc wc :10: wc_ic + 1 0 0 0 ] ;

97 98 99 100 101 102

% % PEAK t o r q u e p r o f i l e TP_MAX = Tp * ones ( size ( WS ) ) ; ID_MAX = Id * ones ( size ( WS ) ) ; IQ_MAX = Iq * ones ( size ( WS ) ) ; % WP_MAX = Rr / Lr * Iq / Id ;

103 104 105 106 107 108 109

% % MTPF t o r q u e p r o f i l e Qlim = min ( Qmax , Vmax. / WS ) ; I Q _ M T P F = Qlim / L_lk / sqrt (2) ; I D _ M T P F = Qlim / Ls / sqrt (2) ; T P _ M T P F = k_T * Lm * I D _ M T P F . * I Q _ M T P F ; % W P _ M T P F = Rr / Lr * I Q _ M T P F . / I D _ M T P F ;

110 111 112 113 114 115

% % Iq = I q _ l i m t o r q u e p r o f i l e Iq_lim = Iq ; T e r m _ I d S Q = Qlim. ^2 - L_lk ^2* Iq_lim ^2; I D _ I q C n s t = sqrt ( T e r m _ I d S Q / Ls ^2) ; I Q _ I q C n s t = I q _ l i m * ones ( size ( WS ) ) ;

T e r m _ I d S Q ( Term_IdSQ wb & WS ≤ wc ) + T P _ M T P F . *( WS > wc ) ; ID = I D _ M A X . *( WS ≤ wb ) + I D _ I q C n s t . *( WS > wb & WS ≤ wc ) + I D _ M T P F . *( WS > wc ) ; IQ = I Q _ M A X . *( WS ≤ wb ) + I Q _ I q C n s t . *( WS > wb & WS ≤ wc ) + I Q _ M T P F . *( WS > wc ) ; WP = Rr / Lr * IQ. / ID ;

119

120

121

122

... ... ...

123 124 125 126 127 128

% % Plot WR = max ( WS - WP ,0) /2; figure ( h_Tqfig ); plot ( WR , TP ) ; if ( Imax == I m a x _ i C ) plot ( WR , TP , ' b ' , ' L i n e W i d t h ' ,1 .5 ) ; if ( abs ( Tp - T _ p e a k _ q C /2) ωc as it operates on the MTPF line. All these indicate that the Q/I-limits dominated flux-weakening strategy resembles the analysis in Chap. 2, but the torque response time is very short as a result of vector control. The flux-weakening strategy with joint determination by the Q/I-limits will be very similar to that when the I-limit is dominating, which is detailed in Sect. 7.2. The control architecture is given in Fig. 7.16, and the control strategy is summarized as below. Use of Eq. (7.4) to limit the raw torque is valid, but the torque capability requires to be updated to reflect the I-limit effect.  raw cap dmd = min Tem , Tem (ωr e , Imax ) Tem cap

where Tem is determined by Eq. (7.25) and is illustrated as in Fig. 7.12.

cap Tem

⎧  KT Lm ⎪ 2 )  ⎪ ωs ≤ ωb (L 2 I 2 − Q 2max )(Q 2max − L 2δ Imax 2 2 ⎪ ⎨ L s −L δ  s max K L T m 2 2 ) ω ωc 2 Ls Lδ  ω2 s

(7.25)

232

7 Flux-Weakening Control

where Qˆ max is defined as 2

V Qˆ 2max = (1 − δ 2 ) max + δ 2 Q 2max  ωs2

(7.26) qC

iC  ωs is defined by Eq. (7.3). Imax is a dynamic limit satisfying Imax < Imax < Imax .7 The base speed ωb and critical speed ωc are defined in Eqs. (7.13) and (7.18) respectively. Nevertheless, it is strongly recommended to use the map-based approach as demonstrated in Fig. 7.12 rather than the model-based approach for representing torque capacity. I-limit variation is a result of thermal control, which is not within the scope of this book, but the I-limit shall be reduced once the system temperature is sufficiently high. Hence, a rather simple thermal strategy for I-limit regulation is illustrated in Fig. 7.17, and is specified as below: Step1. Determining the permissible current as a result of the inverter or Motor Control Unit (MCU) thermal control.

max Iinv

⎧ pk bC ⎪ I Tinv < Tinv ⎪ ⎨ inv max −T pk Tinv inv bC max = Iinv T max −T bC Tinv ≤ Tinv < Tinv inv inv ⎪ ⎪ ⎩ 0 Otherwise

(7.27)

bC max and Tinv are calibration parameters. Permissible phase current starts where Tinv bC , and reduces to a level with no declining (and thus torque is reducing) at Tinv = Tinv max . current (thus no torque) being permitted at Tinv = Tinv Step2. Determining the permissible current as a result of the stator or electric machine thermal control.

Ista

max

Imax inv

pk Ista

Iinv

qC Imax

qC Imax

tC Imax

tC Imax

iC Imax

iC Imax

pk

bC qC tC iC max Tsta Tsta Tsta Tsta Tsta

Tsta

bC iC tC Tinv TqC inv Tinv Tinv

max Tinv

Tinv

Fig. 7.17 Thermal control

7

Flux-weakening specified as in Sect. 7.2 (covering both Type1 and Type2) shall be used when qC iC , and Sect. 7.3 (Type1 only) when I Imax < Imax max > Imax .

7.4 Q/I-Limit Dominated Flux-Weakening Control

max Ista

⎧ pk bC ⎪ I Tsta < Tsta ⎪ ⎪ sta ⎨ max −T pk Tsta sta max bC = Ista T max bC Tsta ≤ Tsta < Tinv sta −Tsta ⎪ ⎪ ⎪ ⎩0 Otherwise

233

(7.28)

bC max where Tsta and Tsta are once again calibration parameters. Similarly, Permissible bC max , and diminishes completely at Tsta = Tsta . current starts declining at Tsta = Tsta Step3. Determining the permissible current combining the results of Step1 and Step2. max max , Ista ) (7.29) Imax = min(Iinv

The parameters and variables used in the thermal control are given as below max Iinv : pk Iinv :

Imax : iC : Imax qC Imax : tC : Imax max Ista : pk Ista : Tinv : bC : Tinv iC : Tinv qC Tinv : tC : Tinv max Tinv : Tsta : bC : Tsta iC : Tsta qC Tsta : tC : Tsta max Tsta :

Inverter permissible current; Maximum inverter permissible current; Permissible current, or I-limit; Permissible current where Q/I-limits joint at MTPA; Permissible current where Q/I-limits joint at MTPF; Permissible current corresponding to continuous torque; Stator permissible current; Maximum stator permissible current; Inverter temperature; Base inverter temperature when permissible current starts declining; iC Inverter temperature corresponding to Imax ; qC Inverter temperature corresponding to Imax ; tC Inverter temperature corresponding to Imax ; Maximum allowable inverter temperature; Stator and machine temperature; Base stator temperature when permissible current starts declining; iC Stator temperature corresponding to Imax ; qC Stator temperature corresponding to Imax ; tC Stator temperature corresponding to Imax ; Maximum allowable stator temperature.

Equation (7.2) for determining field current i sd is hereby updated hereby as

dmd i sd

⎧ 2 Q 2max −L 2δ Imax ⎪ ⎪  ωs ≤ ωb ⎪ L 2s −L 2δ ⎪ ⎪ ⎨ = 2 2 2 ⎪ Qˆ max 2−L δ2Imax ωb <  ωs ≤ ωc ⎪ ⎪ L −L ⎪ ⎪√ V s δ ⎩ 2 max Otherwise 2 Ls  ωs

(7.30)

ωs in Eq. (7.3). Based on the above equawhere Qˆ max is defined in Eq. (7.26) and  dmd tion, i sd depends on ωr e (or ωr m in reality), which changes slowly. For the same

234

7 Flux-Weakening Control

reason as for the machine torque capability determination, the map-based approach as illustrated in Fig. 7.13 is recommended. Similarly, Eq. (6.6) is still valid for determining the torque current demand based on the torque and field current demand, and is re-given as below for completeness raw = i sq

dmd Tem k T L m i sd

dmd It is worth mentioning that the I-limit is satisfied implicitly when Tem is limited by cap dmd Tem and the field current i sd is determined according to Eq. (7.30). The internal feedback loop for the voltage limit control is given in Fig. 7.10. The strategy adopted here is the same as is designed in Sect. 7.2. A summary of the Q/I limits dominated flux-weakening strategies is given in the following

• This strategy is very similar to the conventional V/F and slip speed scalar control (see in Chap. 5), from the resulted steady-state machine performance point of view. The similarity of the torque profiles between those in Figs. 2.29 and 7.12 is one evidence; • The field current or air-gap flux is maximized only when the Q-limit meets the I-limit, which is the case when ωs ≤ ωb . As the speed rises, the field current will be reduced. This implies that the air-gap flux is not fully utilized. The air-gap flux neutralization strategy is investigated in next subsection. As a result, the peak torque of the machine could be significantly increased, compared to conventional V/F and scalar control.

7.5 Explicit Flux Limitation In this section, the explicit application Ψ -limit and its resulting system performance are investigated. Using the given example machine, the permissible air-gap flux linkage is Ψmax = 0.28 Wb, which corresponds to the machine being operated at the rated voltage and rated frequency with the no-load rotor condition. The air-gap flux linage is generated by the magnetizing inductance. Therefore, the flux linkage limit is essential to the field current limit as lim = Ψmax /L M ≈ Q max /L s Isd

(7.31)

Bear in mind that L M in the above equation is the dq frame magnetizing inductance, lim = 63 A for the and is defined as L M = L SS . The field current limit is set as Isd example machine. The principles of torque strategy and flux-weakening strategy are illustrated in Fig. 7.18, where the direct flux limitation will be applied. In Fig. 7.18,

7.5 Explicit Flux Limitation

235

Iq

MTPF

MTPA

Ilim q Idmd q

Tmax

ψ -limit

Idmd d

lim

Id

Tem

Imax

Id

Fig. 7.18 Flux weakening torque strategies [Ψ -limit]

lim • Red line Ψ -limit is the flux limit, which requires i sd ≤ Isd ; 2 2 2 + i sq ≤ Imax ; • Black circle is the current limit or the I-limit, which requires i sd 2 2 2 2 2 • Blue ellipses are the voltage limits, which require L s i sd + L δ i sq < Vmax /ωs2 . Thereby, they are dynamic limits from the current limitation point of view; • Green ellipse is the Q-limit, which is actually a voltage limit at rated voltage and rated frequency.

The Q-limit takes the format of flux-linkage, and is often treated as a tighter flux limit such as the one used by the scalar torque control. Referring to Fig. 2.24, at the maximum torque, the flux deviates significantly away form the maximum flux. Referring √ to Fig. 7.8, the peak torque under the Q-limit is arrived at when i sd = Q max /( 2L s ). All these indicate that the air-gap flux is not neutralized at maximum torque operation. Now, let the Q-limit be relaxed. The V-limit does not take effect at sufficiently low speed, and the maximum torque is therefore derived at where the Ψ -limit joints the I-limit, as demonstrated in Fig. 7.18. It yields the torque current as below 1 2 2 FI = L s Imax − Q 2max (7.32) i sq Ls Thus max = Tem

K T L m Q max  2 2 L s Imax − Q 2max L 2s

(7.33)

236

7 Flux-Weakening Control

Fig. 7.19 Peak torque of Ψ /Q-limit

Iq

MTPF

MTPA Ilim q

ψ - limit

Tmax

lim

Id

fC Imax

IqC max

Id

Comparing to the maximum torque of Q-limit as in Eq. (7.12), it could be confC , cluded that the Ψ -limit will result in a higher maximum torque once Imax > Imax fC where Imax is defined as fC Imax

Q max = Ls



 1 1 lim 1 + 2 = Isd 1 + 2 4δ 4δ

(7.34)

fC For the given example machine in this book, Imax = 295 A, which is a much lower qC qC fC value than Imax = 411 A. Imax and Imax are illustrated in Fig. 7.19 conceptually. Similar to the Q/I-limits dominated case, the torque and flux-weakening strategy is designed as lim • Fast torque region when ωs ≤ ωb . i sd is fixed at i sd = Isd and i sq is left for torque regulation; • Nearly constant power flux-weakening region when ωb < ωs ≤ ωc . i sd is reduced FI . i sq is left for torque regulation; according to the V-limit along i sq = i sq • MTPF flux-weakening region when ωs > ωc . i sd is now reduced according to the V-limit on the MTPF line and i sq is still left for torque regulation.

This strategy is demonstrated in Fig. 7.18. The base speed ωb now becomes when the V-limit joints Ψ -limit at I-limit, and is defined as ωb = 

1 (1 − δ 2 ) +

max 2 δ 2 ( LQs Imax )

Vmax Q max

(7.35)

Unlike the fixed base speed as defined in Eq. (7.13) for the Q-limit dominated case, it is also dependent on the I-limit now, and is significantly lower in general. Definition of the critical speed ωc is similar to that of the Q/I-limits dominated cases. Nevertheless, Eq. (7.18) needs to be updated as below.

7.5 Explicit Flux Limitation

237

ωc =

 2δ

√ 2 L s Imax Q max

2

−1

Vmax Q max

(7.36)

The commonly adopted flux-weakening concept is referred to reduction of the airgap flux linkage at high speed so that there remains sufficient voltage to deliver a torque current component. This flux weakening is introduced by the dynamic Vlimit. However, there is another type of flux weakening caused by the dynamic I-limit, which is the result of thermal control. Once the I-limit becomes adequately small. The air-gap flux shall be reduced in order to maximize the machine torque. The I-limit introduced flux weakening is illustrated in Fig. 7.20 and is described as below. Keep in mind that from Fig. 7.20a–e, the I-limit is reduced monotonously. • The maximum torque performance is jointly determined by the Ψ /I-limits in Fig. 7.20a, but the I-limit joints the Ψ -limit above the MTPF line, inferring a high pC I-limit of Imax > Imax . The maximum torque is managed by Eq. (7.33). One special speed is defined as Eq. (7.35), but has a very different meaning. For completeness, this special speed ωv is re-define as ωv =

1 Vmax  Q max    max 2 1 − δ 2 + LQδ Imax

(7.37)

When ωs > ωv , the V-limit comes into force, but flux weakening is not required. As a result, i sq shall be reduced. This operation yields maximum machine torque under the V-limit. However, the flux weakening starts when ωs > ωb , as illustrated in the figure. The base and critical speeds are the same in this case and are governed by Eq. (7.40). • The maximum torque performance is jointly determined by the Ψ /I-limits in Fig. 7.20b, but the I-limit joints the Ψ -limit at the MTPF line. Thus  pC = Imax

1 Q max δ2 L s

(7.38)

KT Lm 2 Q L s L δ max

(7.39)

1+

The resulting maximum torque is as max = Tem

Compared to Eq. (7.12), which defines the Q-limit dominated torque performance, the maximum torque by the√Ψ -limit is doubled, as the permissible current Imax is increased by a factor of 2 (reference to Eqs. (7.14) and (7.38) for

238

7 Flux-Weakening Control Iq

Iq

Imax

MTPF

Iqlim

Imax

MTPF

Iqlim MTPA

MTPA

Tmax

Tmax

Q-limit

Iq

ψ - limit

Q-limit

ψ - limit

Ilim d (a) ψ /I-limit joint dominates

Id

Iq

MTPF

MTPF

MTPA

MTPA

Ilim q

Id

lim

Id (b) ψ /I-limit joint dominates

Ilim q

Tmax

Iq

Id

lim Imax Id (d) I-limit start dominating

Iq

MTPF

ψ - limit

Q-limit

ψ-limit lim Imax Id (c) ψ/I-limit joint dominates

Tmax

Id

MTPF

MTPA

MTPA

Imax ψ - limit

Imax

ψ - limit

Ilim q

Ilim q

Tmax Tmax lim

Id (e) I-limit fully dominates

Id

Id

lim

Id

(f) I-limit fully dominates

Fig. 7.20 Ψ /I-limit effects

details). This is an important case since it separates the principle of flux-weakening control.8 The base and critical speeds are the same in this case and are defined as 1 Vmax ωb = √ 2 Q max

(7.40) pC

The flux weakening is along the MTPF line when Imax ≥ Imax . By contrast, the system shall operate along i sq = cnst to perform flux-weakening firstly, and then along the MTPF line once ωs > ωc .

8

7.5 Explicit Flux Limitation

239

• The maximum torque performance is jointly determined by the Ψ /I-limits in Fig. 7.20c. This is the same case as in Fig. 7.18. The maximum torque is managed by Eq. (7.33), which is I-limit dependent. The base and critical speeds are respectively defined as in Eqs. (7.35) and (7.36); • The maximum torque performance starts to be determined by the I-limit in Fig. 7.20d. For this case, the I-limit and the Ψ -limit meet at MTPA. Therefore aC = Imax

√

lim 2Isd =

√

2Q max /L s

(7.41)

The control strategy still consists of constant torque, constant power and constant speed power regions as indicated in the figure. The base and critical speed are respectively managed by Vmax 1 (7.42) ωb = √ 1 + δ 2 Q max and

1 Vmax ωc = √ 2δ Q max

(7.43)

• The maximum torque performance is completely determined by the I-limit in Fig. 7.20e. This figure demonstrates a special case where the I-limit of kC = Q max /L s Imax

(7.44)

is tangent to the Ψ -limit at the d-axis. Thereby, without flux-weakening, it would lead to i sq = 0, thus the machine torque capability would completely diminish to null. With flux-weakening, the maximum torque is governed by Eq. (3.87). It is recalled as below for the completeness of this section, max = Tem

1 2 K T L m Imax 2

It is clear that now the maximum torque depends on the I-limit only. The base and critical speeds are governed by Eqs. (7.21) and (7.20) respectively. For completeness, they are re-given by the following,  ωb =

2 Vmax 2 1 + δ L s Imax

and ωc =

1 Vmax δ L s Imax

The system starts flux weakening initially due to the I-limit. When ωs > ωb , further flux-weakening introduced by the V-limit will be required. The torque and field current limits are managed by

240

7 Flux-Weakening Control

Table 7.3 Machine performance summary max [Nm] De f inition Imax [A] Tem

ωv [rad/s]

ωb [rad/s]

ωc [rad/s]

467 Varying

454 Varying

462 462

462 462

Imax = 581 Imax < 581 Imax = 295

452 Varying 226

462 Varying 584

462 Varying 584

462 Varying 924

Imax < 295 aC = 86 Imax Imax < 86

Varying 48 Varying

Varying 650 Varying

Varying 650 Varying

Varying 4347 Varying

I-peak Torque Imax = 600 Ψ /I-limits Imax > 581 MTPF Torque Ψ /I-limits Q-limit Torque Ψ /I-limits MTPA Torque I-limit

pC

√ lim lim Isd = Isq = Imax / 2 • The case demonstrated in Fig. 7.20f is similar to that of Fig. 7.20e, but with a further reduced I-limit. Consequently, the maximum torque is managed by Eq. (3.87), and the base and critical speed are governed by Eqs. (7.21) and (7.20) respectively. According to the analysis on Fig. 7.20, it is clear that for a given Ψ -limit, the system performance depends on the maximum current Imax , which is a dynamic limit varying with the machine and inverter temperature. The system performance variation is summarized in Table 7.3 and is illustrated in Fig. 7.21. In Table 7.3, the following are some important discoveries. • The peak torque could reach 467 Nm for the same flux limit, which is more than double from the conventional control. This is attributed to the benefits that the vector control not only decouples the flux control from the torque control, but also has precise regulation on the field current and the air-gap flux. Conversely, torque and flux are coupled in the conventional scalar and V/F control which is demonstrated in Fig. 2.24. Therefore, flux is indirectly and inaccurately controlled. Because of this, vector control can fully neutralize the air-gap flux limit; • For the same peak torque of 226 Nm as that of the Q-limit control , it only requires 295 A of the phase current. It is significantly reduced, compared to 411 A as given in Table 7.2. This is because once the Q-limit is relaxed, the system will allow to be operated more towards the MTPA line, thus less current is required for the same amount of requested torque; • The principles of flux and torque control are very different for the Q-limit and the Ψ -limit approach, when comparing the results in Tables 7.2 and 7.3. This implies another benefit from the vector control that it can optimize the direction of the phase current to deliver the desired torque such that the system loss could be reduced.

7.5 Explicit Flux Limitation Maximum torque

500

Flux-weakening speed

10000

450

9000

400

8000

350

7000

Stator Speed [rad/s]

Torque [Nm]

241

300 250 200 150

V-limit Speed Base Speed Critical Speed

6000 5000 4000 3000

100

2000

50

1000 0

0 0

100

200

300

400

500

0

600

100

200

(a) Maximum torque

400

500

600

(b) Flux weakening speeds

Maximum field current

70

300

I-limit [A]

I-limit [A]

Maximum torque current

600

60

500

50

Currents [A]

Currents [A]

400 40

30

300

200 20 100

10

0

0 0

100

200

300

400

500

600

0

100

200

300

I-limit [A]

I-limit [A]

(c) Idlim

(d) Iqlim

400

500

600

Fig. 7.21 Flux weakening curves as a function of I-limit

The investigation of the results is given in Fig. 7.21 and interesting observations are as below • The maximum torque is almost linearly dependent on the maximum current as aC long as Imax > Imax . Thus, with a fixed flux limit, the peak torque performance could be increased by lifting up the permissible current (at least for a short period); • When the I-limit decreases linearly and monotonously, the base speed ωb varies aC . On the other hand, the critical speed rises up quick, rather slowly till Imax < Imax aC . In general, ωb and ωc are very nonand increases even faster when Imax < Imax linear depending on the I-limit. aC , flux weakening is also required to improve the torque perfor• For Imax < Imax mance. This is clearly indicated in Fig. 7.21c.

242

7 Flux-Weakening Control

Listing 7.2 Performance profiles calculation 1 2 3 4

Freq ws Vdc vs

= = = =

100; 2* pi * Freq ; 330; Vdc / sqrt (3) ;

5 6

% For Y c o n n e c t i o n and SPWM % Vpn = Vdc * sqrt (3) /3 % Vpn_rms = Vpn / sqrt (2)

7 8

Qmax = 0 .28 /0 .96 ;

9

10

11 12

Vmax Ipk

= vs ; = 600;

% Max flux at Phase t e r m i n a l % r_v =0 .96 rather than 0 .92 to a c c o u n t for voltage % Loss at L_sl and Rs w h i c h is d r o p e d at SS analysis % Maximum voltage % This level , Q - limit d o m i n a t e s

... ...

13 14 15 16 17 18 19 20 21 22 23 24

Motor_Parameters_DQ ; Idmax = Qmax / Ls ; % M a x i m u m flux limit % % D e f i n e some common p a r a m e t e r s Np = Pole_pair ; k_r = Lm / Lr ; k_T = 1 .5 * Np * k_r ; Req =( Rs + Rr *( k_r ) ^2) ;  = 1 -( Lm ^2/( Ls * Lr ) ) ; tau_r = Lr / Rr ; tau_s =  * Ls / Req ; L_lk =  * Ls ;

25 26 27 28 29 30

%% Common calculation Imax_pC = Qmax / Ls * sqrt (1+1/  ^2) ; Imax_fC = Qmax / Ls * sqrt ( 1 + 1 / ( 4 *  ^2) ) ; Imax_aC = Qmax / Ls * sqrt (2) ; Imax_kC = Qmax / Ls ;

31 32 33 34

T _ p e a k _ p C = 1 .0 * k_T * Lm /( Ls * L_lk ) * Qmax ^2; T _ p e a k _ f C = 0 .5 * k_T * Lm /( Ls * L_lk ) * Qmax ^2; T _ p e a k _ a C = 0 .5 * k_T * Lm * I m a x _ a C ^2;

35 36 37 38

wb_pc wb_ac wc_ac

= Vmax / Qmax / sqrt (2) ; = Vmax / Qmax / sqrt (1+  ^2) ; = Vmax / Qmax / sqrt (2) /  ;

39 40 41 42 43 44 45 46 47 48 49 50

%% Prepare figure close all ; % c l o s e all f i g u r e figname = ' TqCurve '; h _ T q f i g = f i n d o b j (0 , ' Name ' , [ figname , ' _ R e s u l t s ' ]) ; if i s e m p t y ( h _ T q f i g ) h _ T q f i g = f i g u r e ( ' P o s i t i o n ' ,[50 50 452*1 .8 257*1 .8 ] , ... ' Name ' , [ figname , ' _ R e s u l t s ' ] , ... ' N u m b e r T i t l e ' , ' off ' ) ; end f i g u r e ( h _ T q f i g ) ; box on , hold on ; set ( h_Tqfig , ' D e f a u l t A x e s F o n t S i z e ' ,14 , ' D e f a u l t A x e s F o n t N a m e ' , ' Times ... New Roman ' ) ;

51 52 53 54 55 56 57

figname = ' IdCurve '; h _ I d f i g = f i n d o b j (0 , ' Name ' , [ figname , ' _ R e s u l t s ' ]) ; if i s e m p t y ( h _ I d f i g ) h _ I d f i g = f i g u r e ( ' P o s i t i o n ' ,[50 50 452*1 .8 ' Name ' , [ figname , ' _ R e s u l t s ' ] , ... ' N u m b e r T i t l e ' , ' off ' ) ;

257*1 .8 ] , ...

7.5 Explicit Flux Limitation 58 59 60

243

end f i g u r e ( h _ I d f i g ) ; box on , hold on ; set ( h_Idfig , ' D e f a u l t A x e s F o n t S i z e ' ,14 , ' D e f a u l t A x e s F o n t N a m e ' , ' Times New Roman ' ) ;

...

61 62 63 64 65 66 67 68 69 70

figname = ' IqCurve '; h _ I q f i g = f i n d o b j (0 , ' Name ' , [ figname , ' _ R e s u l t s ' ]) ; if i s e m p t y ( h _ I q f i g ) h _ I q f i g = f i g u r e ( ' P o s i t i o n ' ,[50 50 452*1 .8 257*1 .8 ] , ... ' Name ' , [ figname , ' _ R e s u l t s ' ] , ... ' N u m b e r T i t l e ' , ' off ' ) ; end f i g u r e ( h _ I q f i g ) ; box on , hold on ; set ( h_Iqfig , ' D e f a u l t A x e s F o n t S i z e ' ,14 , ' D e f a u l t A x e s F o n t N a m e ' , ' Times ... New Roman ' ) ;

71 72 73

74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98

% % Setup Imax IMAX = [ I m a x _ k C / 2 : 1 0 : Imax_aC -10 I m a x _ a C :25: Imax_fC -25 I m a x _ f C :50: Imax_pC -50 I m a x _ p C : 1 0 0 : Ipk Ipk ]; % % Main C a l c u l a t i o n for II = 1: length ( IMAX ) Imax = IMAX ( II ) ; if ( Imax > I m a x _ p C ) Id = Idmax ; Iq = sqrt ( Imax ^2 - Id ^2) ; wv = Vmax / sqrt ( Ls ^2* Id ^2+ L_lk ^2* Iq ^2) ; wb = Vmax /( sqrt (2) * Ls * Id ) ; wc = wb ; e l s e i f ( Imax ≥ I m a x _ a C ) Id = Idmax ; Iq = sqrt ( Imax ^2 - Id ^2) ; wv = Vmax / sqrt ( Ls ^2* Id ^2+ L_lk ^2* Iq ^2) ; wb = wv ; wc = Vmax /( sqrt (2) * L_lk * Iq ) ; else Id = Imax / sqrt (2) ; Iq = Id ; wv = Vmax / sqrt ( Ls ^2* Id ^2+ L_lk ^2* Iq ^2) ; wb = wv ; wc = Vmax /( sqrt (2) * L_lk * Iq ) ; end Wsp0 = Rr / Lr * Iq / Id ; Tp = k_T * Lm * Iq * Id ; WS = [ max ( Wsp0 ,2) :2: wv wv : wc wc :10: wc_ac + 1 0 0 0 ] ;

...

99 100 101 102 103 104

% % PEAK t o r q u e p r o f i l e TP_MAX = Tp * ones ( size ( WS ) ) ; ID_MAX = Id * ones ( size ( WS ) ) ; IQ_MAX = Iq * ones ( size ( WS ) ) ; % WP_MAX = Rr / Lr * Iq / Id ;

105 106 107 108 109 110 111

% % MTPF t o r q u e p r o f i l e Qlim = min (2* Qmax , Vmax. / WS ) ; I Q _ M T P F = Qlim / L_lk / sqrt (2) ; I D _ M T P F = Qlim / Ls / sqrt (2) ; T P _ M T P F = k_T * Lm * I D _ M T P F . * I Q _ M T P F ; % W P _ M T P F = Rr / Lr * I Q _ M T P F . / I D _ M T P F ;

112 113 114 115 116

% % Iq = I q _ l i m t o r q u e p r o f i l e Iq_lim = Iq ; T e r m _ I d S Q = Qlim. ^2 - L_lk ^2* Iq_lim ^2; I D _ I q C n s t = sqrt ( T e r m _ I d S Q / Ls ^2) ;

T e r m _ I d S Q ( Term_IdSQ wb IQ = I Q _ M A X . *( WS ≤ wv ) + I Q _ I q C n s t . *( WS > wb WP = Rr / Lr * IQ. / ID ;

129

130

131

132

T P _ I d C n s t . *( WS > wv & WS ≤ wb ) + & WS ≤ wc ) + T P _ M T P F . *( WS > wc ) ; I D _ I d C n s t . *( WS > wv & WS ≤ wb ) + & WS ≤ wc ) + I D _ M T P F . *( WS > wc ) ; I Q _ I d C n s t . *( WS > wv & WS ≤ wb ) + & WS ≤ wc ) + I Q _ M T P F . *( WS > wc ) ;

... ... ...

133 134 135 136 137 138 139

% % Plot WR = max ( WS - WP ,0) /2; figure ( h_Tqfig ); if ( Imax == I m a x _ p C ) if ( Imax == I m a x _ f C ) if ( Imax == I m a x _ a C )

plot ( WR , TP ) ; plot ( WR , TP , ' m ' , ' L i n e W i d t h ' ,1 .5 ) ; plot ( WR , TP , ' g ' , ' L i n e W i d t h ' ,1 .5 ) ; plot ( WR , TP , ' b ' , ' L i n e W i d t h ' ,1 .5 ) ;

end end end

figure ( h_Idfig ); if ( Imax == I m a x _ p C ) if ( Imax == I m a x _ f C ) if ( Imax == I m a x _ a C )

plot ( WR , ID ) ; plot ( WR , ID , ' m ' , ' L i n e W i d t h ' ,1 .5 ) ; plot ( WR , ID , ' g ' , ' L i n e W i d t h ' ,1 .5 ) ; plot ( WR , ID , ' b ' , ' L i n e W i d t h ' ,1 .5 ) ;

end end end

figure ( h_Iqfig ); if ( Imax == I m a x _ p C ) if ( Imax == I m a x _ f C ) if ( Imax == I m a x _ a C )

plot ( WR , IQ ) ; plot ( WR , IQ , ' m ' , ' L i n e W i d t h ' ,1 .5 ) ; plot ( WR , IQ , ' g ' , ' L i n e W i d t h ' ,1 .5 ) ; plot ( WR , IQ , ' b ' , ' L i n e W i d t h ' ,1 .5 ) ;

end end end

140 141 142 143 144 145 146 147 148 149 150 151

end

152 153 154 155 156 157

% % C a l c l u a t e Q - limit data Vlim = min ( Qmax , Vmax. / WS ) ; IQ_Qlim = Vlim / L_lk / sqrt (2) ; ID_Qlim = * I Q _ Q l i m ; TP_Qlim = k_T * Lm *  * I Q _ Q l i m . ^2;

158 159 160 161 162 163 164 165 166 167 168 169 170 171

% % Tide up figure figure ( h_Tqfig ); plot ( WR , TP_Qlim , ' r ' , ' L i n e W i d t h ' ,1 .5 ) ; plot ( WR , TP , ' k ' , ' L i n e W i d t h ' ,1 .5 ) ; set ( gca , ' Xlim ' ,[0 1 0 0 0 ] ) ; set ( gca , ' Ylim ' ,[0 500]) ; y l a b e l ( ' T o r q u e [ Nm ] ' ) ; x l a b e l ( ' M a c h i n e S p e e d [ rad / s ] ' ) ; title ( ' T o r q u e c u r v e s ' ) ; grid on , hold off ; % R e s o l u t i o n is 500 dpi % print ( ' F l u x W e a k e n i n g _ F I l i m _ T q ' , ' - depsc2 ' , ' - r500 ') ; % print ( ' F l u x W e a k e n i n g _ F I l i m _ T q ' , ' - djpeg ' , ' - r500 ') ;

172 173 174

figure ( h_Idfig ); plot ( WR , ID_Qlim , ' r ' , ' L i n e W i d t h ' ,1 .5 ) ;

7.5 Explicit Flux Limitation 175 176 177 178 179 180 181 182 183 184

245

plot ( WR , ID , ' k ' , ' L i n e W i d t h ' ,1 .5 ) ; set ( gca , ' Xlim ' ,[0 1 0 0 0 ] ) ; set ( gca , ' Ylim ' ,[0 70]) ; ylabel ( ' Current [A] '); x l a b e l ( ' M a c h i n e S p e e d [ rad / s ] ' ) ; title ( ' F i e l d c u r r e n t c u r v e s ' ) ; grid on , hold off ; % R e s o l u t i o n is 500 dpi % print ( ' F l u x W e a k e n i n g _ F I l i m _ I d ' , ' - depsc2 ' , ' - r500 ') ; % print ( ' F l u x W e a k e n i n g _ F I l i m _ I d ' , ' - djpeg ' , ' - r500 ') ;

185

187 188 189 190 191 192 193 194 195 196 197

figure ( h_Iqfig ); plot ( WR , IQ_Qlim , ' r ' , ' L i n e W i d t h ' ,1 .5 ) ; plot ( WR , IQ , ' k ' , ' L i n e W i d t h ' ,1 .5 ) ; set ( gca , ' Xlim ' ,[0 1 0 0 0 ] ) ; set ( gca , ' Ylim ' ,[0 600]) ; ylabel ( ' Current [A] '); x l a b e l ( ' M a c h i n e S p e e d [ rad / s ] ' ) ; title ( ' T o r q u e c u r r e n t c u r v e s ' ) ; grid on , hold off ; % R e s o l u t i o n is 500 dpi % print ( ' F l u x W e a k e n i n g _ F I l i m _ I q ' , ' - depsc2 ' , ' - r500 ') ; % print ( ' F l u x W e a k e n i n g _ F I l i m _ I q ' , ' - djpeg ' , ' - r500 ') ;

Figure 7.22 presents the torque profiles with different I-limits. Similarly, Fig. 7.23 gives the field current profiles, and Fig. 7.24 the torque current profiles. These profiles are made using the Matlab scripts provided in Listing 7.2. Detailed interpretations of these results are omitted. Section 7.4 shall be referred to as these results are similar from the profile pattern point of view. However, the key interesting facts are listed below. Torque curves

500 450 400 350

Torque [Nm]

186

300 250 200 150 100 50 0 0

100

200

300

400

500

600

Machine Speed [rad/s]

Fig. 7.22 Torque profiles [Ψ /I-limit]

700

800

900

1000

246

7 Flux-Weakening Control Field current curves

70

60

Current [A]

50

40

30

20

10

0 0

100

200

300

400

500

600

700

800

900

1000

Machine Speed [rad/s]

Fig. 7.23 Field current profiles [Ψ /I-limit]

• Color code used in Figs. 7.22, 7.23 and 7.24 are the same. The thick black curves are for the system at the maximum I-limit (I peak = 600 A as given in Table 1). The maximum I-limit is only available for a very short period. The thick pink curves are for the case when the I-limit and the Ψ -limit meet at MTPF as illustrated in Fig. 7.20b. The thick red curves correspond to the thick black curves in Figs. 7.12, 7.13 and 7.14, representing the maximum torque capability under the Q-limit. It is re-plotted here for comparison. The thick green curves represent the result that is mimic to the Q-limit torque profile. It is better interpreted in Fig. 7.19. Finally, The thick blue curves are for the case when the I-limit starts dominating as shown in Fig. 7.20d; • Torque capability under the Q-limit (in red) is very moderate. This clearly indicates the benefit of vector control in term of torque performance. Under the Ψ -limit, the peak torque is restricted to the I-limit only; • Comparing the Q-limit torque capability (in red) to the mocked one (in green) with the Ψ -limit, the difference is hardly noticeable, but the mocked one has a Type2 flux-weakening though the speed range for the constant Isq is narrow. Nevertheless, the corresponding field and torque current are very different as shown in Figs. 7.23 and 7.24; • Comparing the results for the cases when the I-limit starts dominating, the Ψ and Q-limits are very similar in all aspects (such as torque, field current, torque current, base speed, critical speed and etc). This is simply because the I-limit becomes critical while the Ψ and Q-limits become ineffective; • Comparing the resulting field current as given in Figs. 7.21c and 7.15a, it is clear that the field control becomes more regular under the Ψ -limit scheme. It now depends on the I-limit monotonously. Nevertheless, clearly indicated in Fig. 7.23

7.5 Explicit Flux Limitation

247 Torque current curves

600

500

Current [A]

400

300

200

100

0 0

100

200

300

400

500

600

700

800

900

1000

Machine Speed [rad/s]

Fig. 7.24 Torque current profiles [Ψ /I-limit]

are the three types of flux weakening: Type1 flux-weakening along the MTPF line when ωs > ωc ; Type2 flux-weakening along the constant i sq when ωb < ωs < ωc ; aC and Type3 flux-weakening along the MTPA line when Imax < Imax . When comparing the maximum field currents in Fig. 7.23 with those in Fig. 7.13, one can immediately discover that the difference between the two is hardly distinguishable. Maximum field current for Q-limit flux-weakening is as √

1 1+

δ2

Ψmax Ψmax ≈ rv L s Ls

given δ = 0.11 and rv = 0.95. The right-hand side term represents the maximum current for the Ψ -limit flux-weakening. By these results, it seems to suggest that the machine torque would be unlimited if only with the Ψ -limit. This is not completely true, since even without the I-limit, there is an equivalent slip speed and operating frequency limit as given in Fig. 3.27. This limit would put a cap on the machine torque capability. Nevertheless, this limit is far too high comparing to the current limit. Thus, it is impractical, and is therefore neglected without affecting the system performance. The flux-weakening strategy with joint determination by the Ψ /I-limits, will be very similar to that when Q/I-limits are dominating, as detailed in Sect. 7.4. The control architecture and the closed-loop system are given in Fig. 7.25, and the control strategy is summarized as below. For simplifying the design of the strategy, it is pC assumed that Imax = Imax = 581 A. This is the current level where the Ψ /I limits meet at MTPF.

248

7 Flux-Weakening Control Vlim

Ttemp

|vs|

Current Determination raw

Torque Capability cap Tem

dmd isd

dmd

Tem

idmd sq

ψ lim

Tem

-

Pv

Ilim

Tem

ωr

-

dmd isq

PId -

| fb vsd

vsd

+

fb vsq

PIq

|

vsq

+

idmd sd

ωr

Current Filter

act

isd

ff vsd

dmd isq

ff vsq

dmd isd idmd sq

act

Speed Filter

ω rm

dmd isd

isq ω rm

Np

ωs

ω re

Vector Control with flux-weakening

Kr/Reqτ r

-

re

+

Np ψr

Vs 1/Req

+

-

τT

Is

I sd

Lm

τr

iτT ωs Stator Model

X

iτr

ψr

Rotor Model

I sq

X

ω re

T em X

KT

Fig. 7.25 Flux-weakening control closed loop system [Ψ /I-limit]

Use of Eq. (7.4) to limit the raw torque is still valid. However, similar to Sect. 7.4, the torque capability needs to be updated to reflect the I-limit effect.  raw cap dmd = min Tem , Tem (ωr e , Imax ) Tem cap

in which, Tem is determined by Eq. (7.22), and is illustrated as in Fig. 7.22.

cap Tem

⎧K L Q  T m max 2 L 2s Imax − Q 2max  ωs ≤ ωb ⎪ ⎪ ⎨ L 2s K L 2 2 ) ω ωc 2 Ls Lδ  ω2

(7.45)

s

in which Qˆ max is defined as Eq. (7.26), and  ωs as Eq. (7.3). Imax is a dynamic limit pC aC satisfying Imax ≤ Imax ≤ Imax . The base speed ωb and critical speed ωc are defined as in Eqs. (7.35) and (7.36) respectively. Equation (7.5) shall be applied when Imax < cap aC . Given the complexity in formulating Tem , the advantage of using the map-based Imax approach as in Fig. 7.22 rather than the model-based approach becomes clear. I-limit variation is a result of thermal control as illustrated in Fig. 7.17, and the simple thermal control strategy as given in Sect. 7.4 is utilized.

References

249

aC Equation (7.2) for determining the field current i sd is only valid when Imax < Imax , aC and it is updated as below to account for the case when Imax ≥ Imax

dmd i sd =

⎧Q max ⎪ ⎪ L ⎪ ⎨ √s

 ωs ≤ ωb

2 Qˆ 2max −L 2δ Imax Ls

⎪ ⎪ ⎪ ⎩ √2

Vmax 2 Ls  ωs

ωb <  ωs ≤ ωc

(7.46)

Otherwise

where Qˆ max and  ωs are the same as the above-defined for the torque capability dmd depends on ωr e (or ωr m in reality), determination. Based on the above expression, i sd which changes slowly. For the same reason as in the machine torque capability determination, the map-based approach as illustrated in Fig. 7.23 is recommended. Similarly, Eq. (6.6) is still valid for determining the requested torque current based on the torque and field current demand, and is re-given below for completeness sake raw = i sq

dmd Tem k T L m i sd cap

dmd The Ψ /I-limits are satisfied implicitly when Tem is limited by Tem and the field dmd current i sd . The internal feedback loop for voltage limit control is the same as given in Fig. 7.10. Likewise, the same strategy that is designed in Sect. 7.2 is also adopted here. A quick summary of the Ψ /I-limits dominated flux-weakening control is as below

• The use of direct flux limit demonstrates the great benefit of vector control. It increases the torque capability significantly as it is capable of neutralizing the air-gap flux; • The flux-weakening control highly depends on the current limit. Only Type1 fluxpC weakening is required when Imax > Imax , Type1 and Type2 are applied when aC Imax > Imax , and all the three types of flux-weakening are employed once Imax < aC ; Imax • I-limit is a dynamics limit, which varies with the system temperature; • Map-based representation of the torque capability and the desired field current is strongly recommended.

References 1. Abu-Rub H, Schmirgel H, Holtz J (2007) Maximum torque production in rotor field oriented control of an induction motor at field weakening. In: 2007 IEEE international symposium on industrial electronics. IEEE, pp 1159–1164 2. Del Blanco FB, Degner MW, Lorenz RD (1999) Dynamic analysis of current regulators for ac motors using complex vectors. IEEE Trans Ind Appl 35(6):1424–1432

250

7 Flux-Weakening Control

3. Dong Z, Yu Y, Li W, Wang B, Xu D (2017) Flux-weakening control for induction motor in voltage extension region: torque analysis and dynamic performance improvement. IEEE Trans Ind Electron 65(5):3740–3751 4. Holtz J, Quan J, Schmittt G, Pontt J, Rodriguez J, Newman P, Miranda H (2003) Design of fast and robust current regulators for high power drives based on complex state variables. In: 38th IAS annual meeting on conference record of the industry applications conference, 2003. IEEE, vol 3, pp 1997–2004 5. Kaboli S, Zolghadri MR, Vahdati-Khajeh E (2007) A fast flux search controller for dtc-based induction motor drives. IEEE Trans Ind Electron 54(5):2407–2416 6. Kim SH, Sul SK (1995) Maximum torque control of an induction machine in the field weakening region. IEEE Trans Ind Appl 31(4):787–794 7. Lin PY, Lai YS (2010) Novel voltage trajectory control for field-weakening operation of induction motor drives. IEEE Trans Ind Appl 47(1):122–127 8. Miguel-Espinar C, Heredero-Peris D, Gross G, Llonch-Masachs M, Montesinos-Miracle D (2020) Maximum torque per voltage flux-weakening strategy with speed limiter for pmsm drives. IEEE Trans Ind Electron 68(10):9254–9264 9. Novotny DW, Lipo TA (1996) Vector control and dynamics of AC drives, vol 41. Oxford University Press 10. Pellegrino G, Bojoi RI, Guglielmi P (2011) Unified direct-flux vector control for ac motor drives. IEEE Trans Ind Appl 47(5):2093–2102 11. Xu X, Novotny DW (1992) Selection of the flux reference for induction machine drives in the field weakening region. IEEE Trans Ind Appl 28(6):1353–1358

Chapter 8

Rotor Field Oriented Control and Senseless Control

8.1 Issue with Indirect Flux Oriented Control The indirect flux oriented control as presented in Sect. 6.3 has a fundamental issue of relying on accurate knowledge of machine parameters such as rotor resistance Rr , rotor inductance L r and mutual inductance L m [9, 10]. Moreover, Rr depends on rotor temperature, and L r and L m vary with both or either of rotor and stator current given the flux saturation. All these lead to deviation of the d-axis away from the rotor flux direction instantaneously, which could result in a significant torque error. The difference between the desired and actual slip speed is the root cause. According to Eq. (6.3), the desired slip speed to deliver the requested torque and to make the d-axis align with the rotor flux shall be des dmd = Rr κr (i sq /Ψr ) ωsp

However, the actual slip speed is governed by act = Rˆ r κˆr (i sq /Ψˆ r ) ωsp

Where Rˆ r , κˆr and Ψˆ r are estimations of Rr , κr and Ψr . Let us assume • κr estimation is adequately accurate. This is generally true when L r and L m are measured parameters and when the system does not operate within the flux saturation zone; • Ψr estimation is rather accurate. Ψr is estimated by Ψr = L m i sd . Since it is the design principle (as detailed in Chap. 6) to maintain a steady (or at least quasisteady) rotor flux, the use of i sd to estimate Ψr is reasonable.

© Springer Nature Switzerland AG 2024 S. Shen and Q.-z. Chen, Practical Control of Electric Machines for EV/HEVs, Lecture Notes in Electrical Engineering 1064, https://doi.org/10.1007/978-3-031-38161-4_8

251

252 Fig. 8.1 Torque error principle

8 Rotor Field Oriented Control and Senseless Control

Iq

MTPF

Iqlim

( ωdes sp ) act ω sp des ω sp

δθ

act Tem Tdmd em

Iqdmd δθ

θ2 θ1

lim

Id

δθ

Id

Thus, the slip speed can be approximated with sufficient accuracy, by the following expression Rr i sq ωsp = (8.1) L r i sd and the key estimation error in slip speed results from the variation of rotor resistance Rr due to the changing rotor temperature. Figure 8.1 illustrates the error associated with the machine torque because of the dmd dmd , both i sd rotor resistance variation. In order to deliver the torque demand Tem dmd lim dmd (i sd = i sd ) and i sq require to be appropriately determined, and both shall be well des regulated, see Chap. 6 for more details. Meanwhile, the desired slip speed ωsp needs to be well governed by Eq. (8.1). With the feedback control strategies as presented in Chap. 6, it is reasonable to assume sufficient accuracy in the stator current Is in terms of magnitude. However, the estimated slip speed may not be accurate due to the above-mentioned reason. The actual slip speed is based on the estimated resistance (or nominal resistance) Rˆ r , while the desired one relies on the actual runtime resistance Rr . Hence, there exists a significant error in the slip speed when the rotor is sufficiently hot. The following analysis and remarks are made according to Fig. 8.1. des • The black line ωsp represents the desired slip speed in the original Id -Iq coordinate system (with black axes) and is managed by des = ωsp

Rr i sq L r i sd

8.1 Issue with Indirect Flux Oriented Control

253

act • The orange line ωsp represents the actual slip speed in the original Id -Iq coordinate system and is managed by Rˆ r i sq act = ωsp L r i sd

• The difference between the desired and actual slip speeds is denoted by δθ . As a result of this difference, the original stator field current is not aligned with the rotor flux. Equivalently, the dq-frame model is no longer rotor field oriented; • The new Id -Iq coordinate system (in orange) becomes rotor field oriented by instead, and it rotates clockwise (for the given example) by an angle of δθ . Based on the assumption that i sd and i sq are well regulated (but in the original coordinate), the magnitude of Is shall be maintained. This is to say that the desired and actual phase current are well on a circle as indicated in Fig. 8.1; • Due to the misalignment of rotation, the desired slip speed line in the original coordinate now represents the actual one in the new coordinate, while the desired slip line (in the new coordinate) is the result of clockwise rotation with the angle δθ from the old one, as shown in the figure; • The demanded torque is governed by dmd lim dmd = k T L m Isd i sq Tem

while the actual torque is derived by act act act = k T L m i sd i sq Tem

There is an apparent difference caused by the dq model being unaligned with the rotor flux; • It could be confusing, but torque control does use desired slip speed line (in the original Id -Iq coordinate system) to determine the demanded field and torque currents. Due to misalignment of stator current and rotor flux, the actual field and torque currents are however governed by the actual slip speed line which is away from the desired one. Let us define the magnitude of phase current Is as 2 2 + i sd Is2 = i sq

Due to the current feedback control, it is considered to be invariant regardless of the coordinate misalignment. Geometrically, dmd = k T L m Is2 sin(θ1 ) cos(θ1 ) = 0.5k T L m Is2 sin(2θ1 ) Tem

and act = 0.5k T L m Is2 sin(2θ2 ) Tem

254

8 Rotor Field Oriented Control and Senseless Control

where θ1 and θ2 are the respective angles of the desired and actual slip speed lines with reference to the d-axis in the new Id -Iq coordinate system. Since the angles θ1 and θ2 are respectively determined by the desired and actual slip speeds as tan(θ1 ) =

i sq i sd

and tan(θ2 ) =

∗ i sq

i sd

∗ des where i sq is the torque current at ωsp when i sd is fixed with Idlim , satisfying ∗ ˆ i sq Rr = i sq Rr

It is clear that the torque deviation is introduced by the error in Rr , which is in turn introduced by the changing rotor temperature Tr ot . The sensitivity of rotor flux direction to rotor resistance is defined as ∂θ/∂ Rr ,1 and torque to rotor resistance as ∂ Tem /∂ Rr , which are given respectively as below

and

cos2 (θ ) i sq ∂θ = ∂ Rr Rˆ r i sd

(8.2)

∂ Tem k T L m i sq = [Is cos(θ )]2 cos(2θ ) ˆ ∂ Rr i Rr sd

(8.3)

Given the fact that Is cos(θ ) = i sd and cos(2θ ) =

2 2 i sd − i sq

Is2

These sensitivity functions are further reduced as 2 1/i sd ∂θ Tem = 2 2 ∂ Rr k T L m Rˆ r 1 + i sq /i sd

and

2 2 /i sd ∂ Tem Tem 1 − i sq = 2 /i 2 ∂ Rr Rˆ r 1 + i sq sd

It is now evident that both the sensitivity functions depends on the torque entirely when inductance variations are neglected and the field current is fixed. The above equations can be further normalized as ∂θ/(i sq /i sd ) 1 = 2 /i 2 1 + i sq ∂ Rr / Rˆ r sd 1

The angle of the actual slip line can be approximated as tan(θ) =

(8.4)

Rr i sq Rˆ r i sd

.

8.1 Issue with Indirect Flux Oriented Control Sensitivities to resistance change

1

Sensitivities to resistance change

1

0.8

0.8

0.6

0.6

0.4

0.4

Sensitivity [-]

Sensitivity [-]

255

0.2 0 -0.2

0.2 0 -0.2

-0.4

-0.4

-0.6

-0.6

-0.8

-0.8

-1

-1 0

20

40

60

80

100

120

140

160

180

200

0

50

Machine Torque [Nm]

100

150

200

250

300

350

400

Torque Current [A]

(a) Machine torque as a dependent variable

(b) Torque current as a dependent variable

Fig. 8.2 Sensitivity functions of slip speed angle and torque

and

2 2 1 − i sq /i sd ∂ Tem /Tem = 2 /i 2 1 + i sq ∂ Rr / Rˆ r sd

(8.5)

Note that the variation of the slip speed angle expressed using Id -Iq coordinates is only approximately normalized here by a given quantity of tan θ instead of θ . Using the typical electric machine parameters in this book, the sensitivity functions are presented in Fig. 8.2, in which the solid blue and dash black curves are respectively the sensitivity of Tem to Rr and θ to Rr . It clearly indicates that • The machine torque is sensitive in the range of low or high torque; • The machine torque increases with the rotor resistance (or temperature) when Tem < 50 Nm, but decreases otherwise; • The machine torque is less (or even not) sensitive around Tem = 50 Nm. However, the slip speed angle sensitivity has different trend. It is more sensitive at low machine torque, while less sensitive at high torque. A close look at the torque sensitivity function in the torque current domain as given in Fig. 8.2b indicates that the torque sensitivity diminishes at i sd = i sq = 61 A or when torque is delivered on the MTPA line. Therefore, machine torque is insensitive to the variation of rotor flux direction (or rotor resistance) when the electric machine is operated along or around MTPA. The more the system shifts away from MTPA, the more sensitive the system is to the change of rotor flux direction. The torque error is positive when operating below MTPA, but negative on the other hand. These are illustrated in Fig. 8.3. Based on the above analysis, the blue and red arrows in Fig. 8.3 are employed to demonstrate the torque variation caused by the change of rotor flux direction (or rotor resistance). Two widely-adopted electric machine operations are shown, one being vector control that operates the machine along i sd = Idlim as long as no flux

256 Fig. 8.3 Torque sensitivity illustration

8 Rotor Field Oriented Control and Senseless Control

Iq

MTPF

Iqlim

Slip speed 2 em

T

MTPA Slip rotating direction

Tem1

Torque Invariant points

lim

Id

Id

weakening is required, the other being slip speed torque control that operates the machine along the Q-limit. • The blue arrows show that machine torque will decrease as a result of rotor flux direction error. The length of the arrows indicates the sensitivity of the torque, qualitatively; • The red arrows show that machine torque will increase. Likewise, the length of the arrows indicates the sensitivity of the torque, qualitatively; 1 in • At and around the MTPA line, machine torque contour (such as Tem = Tem the figure) is tangent to phase current contour (which is a circle). Thus, once the magnitude of phase current can be guaranteed by feedback control, the machine torque will be insensitive; • By contrast, when the system is operating away from MTPA, constant torque 2 in the figure) is not tangent to constant phase current contour (such as Tem = Tem contour (which is what current feedback could ensure) any more. Therefore, along the constant current contour, the machine torque will change (more sharply). For 2 , the machine torque will be reduced as a result of variation the case of Tem = Tem of rotor flux direction. These somewhat agree with the results given in Fig. 8.2 and therefore validate the sensitivity analysis. To conclude this section, illustration of the scalar control torque error due to rotor resistance variation is given in Fig. 8.4. For scalar control, the torque is regulated based on the slip speed to torque characteristics. In the meanwhile, magnetic field is intended to be kept constant by the V/F principle. Though the maximum torque remains constant regardless of the variation of the rotor resistance, the slip speed at which the machine torque is maximized does depend on the rotor resistance.

8.1 Issue with Indirect Flux Oriented Control

257

Iq Tem

MTPF

ω*

ω des sp

Tdmd em

ω act sp

ω* Rr Rr

act

Tem

ω sp

Id des ω act sp ω sp

Fig. 8.4 Torque depends on temperature principle

This fact makes the machine torque sensitive to rotor temperature. Without appropriate feedback strategies, this error could become significant at high rotor temperature. The orange and black curves in the right-hand side of Fig. 8.4 represent the slip speed to torque characteristics of the electric machine, which normally does not vary (or vary little) with the slip speed as long as flux weakening is not required and the key parameters are not altered. These curves reflect how the torque changes along the Q-limit at the left-hand side of the figure. • Along the Q-limit, i sd decreases and i sd increases gradually; • According to Eq. (8.1), slip speeds are radial lines in the Id -Iq coordinate system; • If Rr and L r remain unchanged, the same radial lines represent the same slip speeds. These lines intersect the Q-limit at different torque levels, generating the slip speed to torque curves as shown in the right-hand side of the figure; • However, Rr does vary with rotor temperature. Consequently, the same radial line represents different slip speeds for different Rr . Since the same radial line always corresponds to a same torque at the Q-limit, the same torque is therefore related to different slip speeds if Rr changes. This is again illustrated on the right-hand side of the figure; • With the nominal slip speed to torque characteristics (orange curve on the right), act dmd for the demanded torque Tem is expected. However, due to the change of Rr , ωsp the slip speed to torque curve is altered to the black one, and hence the resulting act dmd becomes significantly lower. To deliver the torque Tem , an updated torque Tem des ωsp is then desired. The above analysis shows that even if with accurate feedback current, torque error is still present due to inappropriate slip speed, which leads to variation of rotor flux direction. This torque error can be significant in the range of either low or high torque demand. Methods to improve the torque accuracy are required, and are elaborated in the following sections.

258

8 Rotor Field Oriented Control and Senseless Control

8.2 Direct Versus Indirect Flux Oriented Control Indirect flux oriented control, as indicated in Fig. 8.5, employs the slip speed (as defined by Eq. (8.1) as a means to regulate the field direction of the dq frame to align with the rotor flux [9, 10]. The principle of the regulation is well explained in Chap. 6. This simple indirect way of field oriented control depends heavily on the accuracy of the slip speed determination, which however requires precise knowledge of the rotor time constant τr = L r /Rr . Unfortunately, the rotor time constant is not precisely known and is time varying, which will lead to the misalignment of the d-axis with the rotor flux as analyzed previously. That being said, the stator flux can not be regulated to have the correct angular position relative to the rotor flux. This is an obvious drawback of the indirect flux oriented control. This drawback can be improved if the rotor time constant can be estimated online with necessary accuracy and with sufficient bandwidth. Direct flux oriented control (Fig. 8.6), on the other hand, directly aligns the d-axis to the rotor flux (or controls the stator flux to have the required angular position in relation to the rotor flux) [9, 10]. As expected, this method requires the the rotor flux to be known, which is generally untrue. Thus, a rotor flux observer is essential. It is worth noticing that rotor flux observer is based on signals from the αβ coordinates rather than the dq coordinates [6]. The use of signals from the dq coordinates to estimate rotor flux is non-causal since the d-axis is determined by the direction of rotor flux. Nevertheless, the difference between the IFOC and the DFOC is merely at how the stator electric frequency is determined. The rest of the control strategies are more or less the same. Therefore, with no surprise, the focus of this chapter is on the preservation of rotor flux  r and the calculation of stator frequency or speed ωs .

Ttemp

Vlim

|vs| Pv

Ilim Current Determination raw

Torque Capability cap Tem

dmd

Tem

idmd sq

Tem

ωr

-

PId

| fb vsd

|

vsd

+

dq

Modulation

vsα

PWMa

Inveter

PWMb

ψ lim

Tem

dmd isd

-

dmd isq

-

PIq

fb vsq

+

vsq

vsβ

PWMc

αβ

idmd sd

ωr

dmd isd

ff vsd

dmd isq

ff vsq

dmd isd idmd sq

ω rm

Np

ω re

ACIM

ωs

θ

act

isd

dq

act

isq

Fig. 8.5 Indirect field oriented control architecture

αβ

isa isb

isβ

αβ

IFOC = Indirect Field Oriented Control

isα

abc

8.3 Full State Versus Reduced State Observer Ttemp

Vlim

|vs|

Current Determination

raw

dmd isd

dmd

Tem

idmd sq

Tem

ωr

-

| fb vsd

PId

|

vsd

+

dq

Modulation

vsα

PWMa

Inveter

PWMb

ψ lim

Torque Capability cap Tem

-

Pv

Ilim

Tem

259

dmd isq

-

fb vsq

PIq

+

vsq

vsβ

PWMc

αβ

idmd sd

ωr

ω rm

Np

dmd isd

ff vsd

dmd isq

ff vsq

ω re

ACIM

ψ r Observe ω s Calculator

ωs

θ

act

isd

dq

act isq

isα

αβ

αβ

isa isb

isβ abc

DFOC = Direct Field Oriented Control

Fig. 8.6 Direct field oriented control architecture

8.3 Full State Versus Reduced State Observer The estimation of  r is based on the αβ-frame model as given by Eqs. (3.40), (3.41), (3.42) and (3.43). When taking Is and  r as state variables, the αβ-frame model becomes di sα = − τ1T i sα + τrκLr δ Ψr α + κrLωδr e Ψrβ + L1δ vsd dt di sβ dt dΨr α dt dΨrβ dt

κr ωr e Ψr α Lδ

−

= − τ1r Ψr α +

κr Ψ τr L δ rβ Lm i τr sα

= − τ1r Ψrβ +

Lm i τr sβ

− ωr e Ψr α

= − τ1T i sβ +

+

1 v L δ sq

+ ωr e Ψrβ

(8.6)

The observer design is to detect Ψr α and Ψrβ on the basis that ir α and irβ are measurable. The model-based Ψr α and Ψrβ observation could be interpreted by the following         d Ψr α i sα 1/τr −ωr e Ψr α L m /τr 0 + = +ωr e 1/τr Ψrβ 0 L m /τr i sβ dt Ψrβ          Derivative Plant model Input This equation represents the rotor flux dynamics. If the plant model and the input gains are precisely known, the rotor flux can be estimated with sufficient accuracy based on the stator current measurement according to the rotor flux dynamics alone. However, this assumption is not true. Estimation of the rotor flux based on the rotor flux dynamics is not reliable.            d i sα κr 1/τr ωr e i sα vsα Ψr α 1/τT 0 1/L δ 0 + = + 0 1/τT i sβ 0 1/L δ vsβ dt i sβ L δ −ωr e 1/τr Ψrβ             Derivative Plant model Input Disturbuance

260

8 Rotor Field Oriented Control and Senseless Control

The above expression represents the stator dynamic model, where the stator voltages are controlled inputs, while the rotor flux is treated as disturbance. The difference ΔIs between the measured Is and estimated Iˆs based on the stator model is contributed by the unknown rotor flux  r as well as the parameter uncertainties. In particular, the rotor flux  r plays a major role in generating stator current estimation error ΔIs . It is therefore possible to estimate the rotor flux  r based on the stator current error ΔIs information. The rotor time constant τr can be approximated with the same principle. Tidying up Eq. (8.6) into the matrix format, it yields ⎤ ⎡ 1 −τ 0 i sα ⎢ 0T − 1 ⎥ d ⎢ τT ⎢ i sβ ⎥ = ⎢ ⎢ L dt ⎣Ψr α ⎦ ⎣ τrm 0 Ψrβ 0 Lτrm     x˙ ⎡

⎤⎡

⎤ ⎡ ⎤ i sα 1/L δ 0   ⎢ i sβ ⎥ ⎢ 0 1/L δ ⎥ vsα ⎢ ⎥+⎢ ⎥ ⎣Ψr α ⎦ ⎣ 0 0 ⎦ vsβ    Ψrβ 0 0        x B u

κr ωr e κr L δ τr Lδ κr ωr e − L δ Lκδrτr ⎥ ⎥ ⎥ − τ1r ωr e ⎦ −ωr e − τ1r

 A

(8.7)

The plant version of the model can be denoted by x˙ = A(τT , τr )x + Bu

(8.8)

and correspondingly, the observer version of the model is written by ˆ τˆT , τˆr )ˆx + Bu x˙ˆ = A(

(8.9)

The following comments are made • Resistances are the main varying parameters. Thus, the time constants τT , τr are adopted here to indicate the difference between the characteristic matrices A (plant ˆ (observer version); model version) and A • The electric speed ωr e is measurable in this book. However, it can also be treated as a varying parameter of A, which is the case for sensorless induction machine control; • The input matrix B is assumed invariant for both the plant model and the observer. Also the control input array u is assumed measurable or known.

8.3.1 Full State Observer In general, to estimate the full state x, it is required that some linear combinations of the state variables are know. Let us define vector y as the known output having the following format y = Cx, yˆ = Cˆx (8.10)

8.3 Full State Versus Reduced State Observer

261

Theorem 8.1 Identical state matrix observation theorem. x˙ = Ax + Bu is the physical model, and x˙ˆ = Aˆx + Bu is the observer model. Then xˆ approaches x if state matrix A is stable. Proof Let ex be the observation error as ex = xˆ − x Then the subtraction of the above two state equations results in e˙ x = Aex Once A is a stable matrix, then ex (t) = 0 when t = ∞  However, though the observation error ex converges to zero, this process can be slow depending on the eigenvalues of A. It is a general requirement that observer dynamics shall be faster than the physical model. Having said that, the observation error shall converge to zero quickly. The following feedback control loop is introduced for this purpose x˙ˆ = Aˆx + Bu − G(ˆy − y) in which G is the observer matrix, containing feedback gains that requires tuning. With the definition of the output y in Eq. (8.10), the above observer becomes x˙ˆ = Aˆx + Bu − L(ˆx − x) and error dynamics becomes

e˙ˆ x = (A − L)ex

Now the error dynamics can be regulated by the observer feedback gain matrix L = GC. Fast convergence of observation can be achieved by a large feedback gain, but caution must be taken for the stability and sensitivity of the system. Theorem 8.2 Full state observation theorem.

262

8 Rotor Field Oriented Control and Senseless Control

Let the observer system be represented as ˆ x + Bu x˙ˆ = Aˆ

(8.11)

ˆ = A, then introduce the output error feedback with parameter uncertainties, i.e. A with the gain matrix G ˆ x + Bu − G(ˆy − y) x˙ˆ = Aˆ (8.12) to regulate the observer model to track the physical output. This mechanism can estimate state variables with sufficient accuracy if the control matrix G is properly designed. Proof

Introducing

e˙ x = x˙ˆ − x˙ ˆ x + Bu − G(ˆy − y) − Ax − Bu = Aˆ ˆ x − Ax) − L(ˆx − x) = (Aˆ ˆ − A)ˆx − L(ˆx − x) = A(ˆx − x) + (A ˆ −A ΔA = A

Results in that e˙ x = (A − L)ex + ΔAˆx

(8.13)

From the dynamics point of view, if A − L is a stable matrix, then the observation error will be stable. From the steady-state point of view, the steady-state error ex does rely on the deviation of matrix A as well as the observed state xˆ as (A − L)ex = −ΔAˆx

(8.14)

Take the singular value decomposition of A − L as A − L = UΣVT where Σ is a diagonal matrix containing all the eigenvalues of A − L, while U and V are orthogonal matrices consisting of the eigenvectors of A − L. Substituting the above equation into Eq. (8.14) yields ex = −VΣ −1 UT ΔAˆx The norm of the error vector becomes ex 22 = (VΣ −1 UT ΔAˆx)T (VΣ −1 UT ΔAˆx) = xˆ T ΔAT UΣ −1 VT VΣ −1 UT ΔAˆx = xˆ T ΔAT UΣ −2 UT ΔAˆx

8.3 Full State Versus Reduced State Observer Fig. 8.7 Full state observer architecture

263

u

y

Plant Model

Observer Model

y

x

-

C

G

Δy

Assuming di is the ith element of the disturbance vector ΔAˆx, λi is the ith eigenvalue of the matrix A − L, and n is the number of the elements. Bear in mind that U only performs rotating operation to vector ΔAˆx; it is the matrix Σ that plays the stretching and shrinking role. Then n  di2 /λi2 ex 22 = i=1

It is now clear that how the increase of the eigenvalues of A − L makes the estimation error insensitive to parameter uncertainties. The high gain of L is necessary to reject from parameter variation into an estimation error.  It is worth noting that since not all the states are measurable, the output feedback gain G is not possible to place all the eigenvalues of matrix A − L as required. Thus, the above approach of observer design certainly has limits. The full state observation of an induction machine is illustrated in Fig. 8.7, where • • • •

u is the control inputs [vsα vsβ ]T ; y is the plant outputs of [i sα i sβ ]T ; xˆ is the observed states [iˆsα iˆsβ Ψˆ r α Ψˆ rβ ]T , with Ψˆ r α and Ψˆ rβ being of interest; yˆ is the observed outputs [iˆsα iˆsβ ]T .

By defining the stator current vector Is = [i sα i sβ ]T , rotor flux vector  r = [Ψr α Ψrβ ]T , and stator voltage vector Vs = [vsα vsβ ]T , Eq. (8.7) can be revised in the block matrix format as        Is AI I AI Ψ Is BI V   d Vs = + (8.15) dt  AΨ I AΨ Ψ r 0 r where AI I

  1  κr − τT 0 L δ τr AI Ψ = = − κrLωδr e 0 − τ1T 

AΨ I =

Lm τr

0

0 Lm τr





AΨ Ψ

κr ωr e  Lδ κr L δ τr

− τ1r ωr e = −ωr e − τ1r



264

8 Rotor Field Oriented Control and Senseless Control

and

 BI V =

1 Lδ

0

0



1 Lδ

Similarly, the observed version of the plant model is simplified as  d dt

      ˆ IΨ ˆ II A Iˆs Iˆs A BI V   Vs = ˆ + ˆ ΨΨ 0 AΨ I A ˆ r ˆ r

The output y in this case is

(8.16)

    Is = Is y= 10 r

According to Eq. (8.12), the observer model is given by  d dt

 Iˆs = ˆ r

 

=

      Iˆs BI V   LI I  Vs + + Is − Iˆs 0 LΨ I ˆ r (8.17)       ˆ IΨ Iˆs A BI V   LI I   Vs + Is + ˆ ΨΨ 0 LΨ I A ˆ r

ˆ IΨ ˆ II A A ˆΨI A ˆ ΨΨ A

ˆ I I − LI I A ˆ AΨ I − LΨ I



Then the full state rotor flux observer relies on the design and tuning of the observer gain L I I and LΨ I so that • The closed-loop observer is stable; • The closed-loop observer rejects parameter variation online. The full model of the closed-loop observer is given in the following diˆsα dt diˆsβ dt dΨˆ r α dt dΨˆ rβ dt

= − τˆ1T iˆsα +

κr ˆ Ψ τˆr L δ r α

+

κr ωr e ˆ Ψrβ Lδ

+

1 v L δ sd

− lsα (iˆsα − i sα )

= − τ1ˆT iˆsβ +

κr ˆ Ψ τˆr L δ rβ

−

κr ωr e ˆ Ψr α Lδ

+

1 v L δ sq

− lsβ (iˆsβ − i sβ )

= − τˆ1r Ψˆ r α +

Lm ˆ i τˆr sα

+ ωr e Ψˆ rβ − lr α (iˆsα − i sα )

= − τˆ1r Ψ˜ rβ +

Lm ˆ i τˆr sβ

− ωr e Ψˆ r α − lrβ (iˆsβ − i sβ )

.

(8.18)

8.3.2 Reduced State Observer The full state observer design has advantages, for instance, even though Is is measurable, Iˆs will filter out the noise automatically, giving rise to a rather noise-free rotor flux estimation. However, the full state observer demands more computational resource and is more complicated than the reduced-order observer, which estimates the non-measurable states only. The reduced-order observer will be introduced in this section.

8.3 Full State Versus Reduced State Observer

265

The system given in Eq. (8.11) is partitioned as 

      x˙ m Amm Amn xm Bm   u = + x˙ n Anm Ann xn Bn

(8.19)

where xm represents the measurable states, while xn the non-measurable. For the case of an induction motor xm = [i sα i sβ ]T xn = [Ψr α Ψrβ ]T Amm , Amn , Anm and Ann are the associated matrices as defined in Eq. (8.15). In particular, Amm = A I I Amn = A I Ψ Anm = AΨ I Ann = AΨ Ψ Bm and Bn are the corresponding input matrices as Bm = B I V Bn = 0 and u = vs is the input. The output y as defined by Eq. (8.10) is simplified as y = xm

(8.20)

Thus y˙ = x˙ m = Amm xm + Amn xn + Bm u or

y˙ − Amm y − Bm u = Amn xn       known part of model unknown

(8.21)

The left-hand side of the equation could be used to measure xn based on the output y. On the other hand, the right-hand side represents the observed xn according to the model. The difference between these two shall be adopted to correct the observed model.   x˙ˆ n = Ann xˆ n + Anm y + L (˙y − Amm y − Bm u) − Amn xˆ n (8.22)   = Ann xˆ n + Anm y + L Amn xn − Amn xˆ n Defining the observed error exn as exn = xˆ n − xn Combining Eqs. (8.19) and (8.22), the observed error dynamics is as e˙ xn = (Ann − LAmn )exn

(8.23)

266

8 Rotor Field Oriented Control and Senseless Control

u

y ( xm)

Plant Model

L

Anm-LAmm

+

-LBm

xT

xT

+

xn

Ann-LAmn

Fig. 8.8 Reduced-order observer architecture

With appropriate design of L, the system will have the desired dynamics (i.e., sufficiently faster than the physical dynamics of the induction motor) and stability. The use of Eq. (8.21) contains a lot of noise due to the presence of y˙ , Eq. (8.22) is repaired as x˙ˆ n − L˙y = (Ann − LAmn )ˆxn + (Anm − LAmm )y − LBm u Introducing a temporary vector xˆ T as xˆ T = xˆ n − Ly The observer model then becomes x˙ˆ T = (Ann − LAmn )ˆxn + (Anm − LAmm )y − LBm u and is illustrated as in Fig. 8.8. For the case of induction motors, the reduced-order state observer is derived as follows. The αβ-frame model in the complex space is employed here for simplicity reason.   I˙s = − τ1T Is + Lκrδ τ1r − jωr e  r + L1δ Vs   (8.24) ˙ r = − τ1r + jωr e  r + Lτrm Is Since the stator current Is is measurable, the task turns into observing the rotor flux  r . According the the first equation in (8.24), the rotor flux  r could be indirectly measured by the stator current Is as κr Lδ



 1 1 1 − jωr e  r = I˙s + Is − Vs τr τT Lδ

8.3 Full State Versus Reduced State Observer

or r =

267

I˙s + τ1T Is − L1δ Vs   κr 1 − jωr e L δ τr

(8.25)

Unfortunately, parameters τT and τr are both varying online as a result of the temperature going up and down, and the derivative of Is introduces a lot of noises. Therefore, the use of Is to identify  r based on the above equation is never practical. It is necessary to modify the second equation in (8.24) so that it could estimate  r with accuracy. This is done by introducing the feedback of the error eΨ r between the indirectly measured  r and observed ˆ r as ˙ ˆ r = −



    1 Lm κr 1 + jωr e ˆ r + Is + L − jωr e  r − ˆ r τˆr τˆr L δ τˆr

(8.26)

Where L is observer gain vector. Substituting Eq. (8.25) into the above equation and tidying up, it yields ˙ ˆ r − LI˙s = −



1 τˆr

 1+L

κr Lδ



    κr Lm L L + jωr e 1 − L Is − + Vs ˆ r + Lδ τˆr τˆT Lδ

The dynamics of the resulting observation error becomes ˙ e˙ r = ˆ r − ˙ r      = τ1r + jωr e  r − Lτrm Is − τˆ1r + jωr e ˆ r −    −L Lκrδ τˆ1r − jωr e ˆ r −  r

Lm I τˆr s



which is the combination of Eqs. (8.24) and (8.26). The resulting error dynamics is as       κr κr 1 ΔRr  1+L + jωr e 1 − L eΨ r + L m Is −  r (8.27) e˙ Ψ r = − τˆr Lδ Lδ Lr in which

ΔRr 1 1 = − Lr τˆr τr

and ΔRr is the variation of the rotor resistance. It is clear once again that: • With the existence of parameter uncertainties, the steady-sate error ess Ψ r  = 0; • With the feedback gain L, the effect of parameter uncertainties can be regulated; • More interestingly, the steady-sate error ess Ψ r contains the parameter deviation information. Therefore, it could be used to estimate varying parameters such as the rotor resistance Rr .

268

8 Rotor Field Oriented Control and Senseless Control

The parameter estimation based on ess Ψ r is impractical. The method to estimate the rotor resistance and the rotor speed is briefed in the next section.

8.4 Parameter Identification by Adaptive Law The general principle for parameter identification by an adaptive law is illustrated as in Fig. 8.9. Parameters in Φ are updated according to the output error Δy, which is the difference between the measured output y and the estimated one yˆ . For the case of an induction motor, the output is Is , and parameters for identification are Rs and Rr . The idea behind is to design an adaptive law such that the plant model converges to the real plant. As such, the parameters used in the plant model shall track the parameters of the the real plant. This will be true for linear or monotonous systems. Theorem 8.3 Adaptive parameter identification theorem. For a bilinear system shown in Fig. 8.10 e˙ x = Aex + Bv∗ , e y = CT ex

(8.28)

where ex represents the state error between the real and estimated plants, while e y is a reduced-order output of ex in general. The input v∗ = vT Φ is a bilinear function of the parameter Φ and the reference input v. The following adaptive law is able to identify the parameter(s) Φ Φ˙ = −γ ve y (8.29) where γ is a control parameter to tune the converging speed of the parameters. Bear in mind that the reference input v is very different from the input u as shown in

R Tem

Controller

u Vs

y Plant

Is

P0 ωr Plant Model

Φ Rs /Rr Fig. 8.9 Parameter identification structure

y Is

-

Δy eis Adaptive Law

8.4 Parameter Identification by Adaptive Law Fig. 8.10 Adaptive law principle for parameter identification

269

v

x v*

System Model

ey

Φ

x

-rv

Fig. 8.9. In this case, v represents the estimated state xˆ . The adaptive law uses the error between the measured and estimated outputs to correct the estimated parameter, which implies that the plant output and the plant parameter must be interrelated to some degree. Proof The system is assumed to be that • A is a stable matrix; • The output e y depends on v∗ monotonously. This is true since both A and B are constant matrices, indicating a linear system for the input v∗ . A stable matrix A implies AT P + PT A = −Q where P is a symmetric positive definite matrix (PT = P), while Q a positive matrix. Furthermore, P is not arbitrary, but requires PT B = C Now, introducing an energy-like function V V (ex , Φ) =

1 T 11 T ex Pex + Φ Φ 2 2γ

the derivative of V becomes2 1 V˙ = eTx P˙ex + Φ˙ T Φ γ By substituting the system model equation into the above energy-like function, it yields3   V˙ = eTx P Aex + BvT Φ + γ1 Φ˙ T Φ     = 21 eTx AT P + PT A ex + eTx PT BvT + γ1 Φ˙T Φ 2 eT P˙ ˙ ˙ Tx Pex and Φ˙ T Φ = Φ T Φ. x ex = e 3 For x T ˙ = Ax, then d(x Px)/ dt = x˙ T Px + xT P˙x

= xT (AT P + PT A)x.

270

8 Rotor Field Oriented Control and Senseless Control

if the update law of the parameter set Φ is designed such that 1 ˙T Φ + eTx PT BvT = 0 γ or

Φ˙ = −γ v(PT B)T ex = −γ ve y

Then

1 V˙ = − eTx Qex < 0 2

According to the Lyapunov principle, the system as described in Fig. 8.10 will be asymptotically stable, namely ex = 0 and e y = 0 when t → ∞ The estimated parameter Φˆ must approach Φ (the real parameter) when ex = 0  It is worth noting that PT B = C implies e y = BT Pex When B is full rank, which is normally the case, the matrix P is merely to select those error states ex affecting the error output e y . For an ACIM, it will be the stator current error ei , leaving the rotor flux  r as an internal state. Therefore, an extra requirement for the above theorem is to have a stable internal dynamics. Fortunately, this is the case for an ACIM.

8.4.1 Rotor and Stator Resistance Estimation Unlike the αβ-frame model, which is employed for the rotor flux observation as in Sect. 8.3, it is more convenient to use the dq-frame model as given in Eqs. (4.9) and (6.1) for the rotor resistance estimation based on the adaptive parameter identification theorem. This is because the estimation of Rr does not affect causality of the system. The dq-frame model is recapped in the matrix format for the completeness of this section as below.        Is AI I AI Ψ Is BI V   d Vs = + dt  AΨ I AΨ Ψ r 0 r

8.4 Parameter Identification by Adaptive Law

where AI I =

271

  1  κr − τT ωs L δ τr A = IΨ − κrLωδr e ωs − τ1T 

AΨ I =

Lm τr

0

0



κr ωr e  Lδ κr L δ τr

 AΨ Ψ =

Lm τr

− τ1r ωsp −ωsp − τ1r



in which the slip speed ωsp = ωs − ωr e , and  BI V =

1 Lδ

0

0



1 Lδ

Or it can be rewritten in the full form as ⎤⎡ ⎤ ⎡ ⎤ ⎡ 1 − τT ωs τrκLr δ κrLωδr e i sd i sd κr ωr e κr ⎥ ⎢ 1 ⎥ ⎢ −ω − − d ⎢ i i sd ⎥ s ⎥ ⎢ sd τ L τ L T δ r δ ⎢ ⎥=⎢ L ⎥+ ⎥⎢ 1 m ⎣ ⎦ ⎣ 0 − τr ωsp ⎦ Ψr d ⎦ ⎣ τr dt Ψr d Ψrq Ψrq 0 Lτrm −ωsp − τ1r          x˙ A x

⎡

1 Lδ

0

⎤

  ⎢ 0 1 ⎥ vsd ⎢ Lδ ⎥ ⎣ 0 0 ⎦ vsq    0 0    B u

(8.30)

The characteristics matrix A clearly depends on the respective rotor and stator resistance Rr and Req or the respective time constants τr and τT more precisely. We define A(Rr , Req ) as A(Rr , Req ) = A0 + Req Aeq + Rr Ar ⎡

where A0

0 ⎢−ωs =⎢ ⎣ 0 0

⎤ ωs 0 κrLωδr e 0 − κrLωδr e 0 ⎥ ⎥ 0 0 ωsp ⎦ 0 −ωsp 0

⎡

Aeq

− L1δ 0 ⎢ 0 −1 Lδ =⎢ ⎣ 0 0 0 0

0 0 0 0 ⎡

Ar =

1 Lr

Ar∗ =

1 Lr

⎤ 0 0⎥ ⎥ 0⎦ 0

0 ⎢0 ⎢ ⎣L m 0

⎤ 0 Lκrδ 0 0 0 Lκrδ ⎥ ⎥ 0 −1 0 ⎦ L m 0 −1

Similarly, the observed version of the plant model is simplified as

272

8 Rotor Field Oriented Control and Senseless Control

ˆ Rˆ r , Rˆ eq )ˆx + Bvs x˙ˆ = A( where

(8.31)

ˆ Rˆ r , Rˆ eq ) = A0 + Rˆ eq Aeq + Rˆ r Ar A(

The output y in this case is

  i y = Is = sd i sq

By defining ex = xˆ − x, and combining Eqs. (8.30) and (8.31), it results in   e˙ x = Aex + ΔReq Aeq + ΔRr Ar xˆ

(8.32)

It is a bilinear system that is similar to what is given in Eq. (8.28). Thus, the adaptive parameter identification theorem is applicable. Let us define     ΔReq 1/k Re , KR = eR = ΔRr 1/k Rr where K R is the control gains, which will be clear later on, and introduce an energylike function V as   2 1 T 1 ΔReq ΔRr2 + (8.33) V (ex , e R ) = ex Pex + 2 2γ k Re k Rr Its derivative then becomes    1  1 V˙ = eTx AT P + PT A ex + eTx P ΔReq Aeq + ΔRr Ar xˆ + 2 γ



ΔReq Δ R˙ eq ΔRr Δ R˙ r + k Re k Rr



According to the adaptive parameter identification theorem, the design of the parameter updating laws as below yields a globally asymptotically stable system as long as A is stable. Δ R˙ eq = −γ k Re eTx PAeq xˆ Δ R˙ r = −γ k Rr eTx PAr xˆ The following notes are immediately added for the above estimation of the stator and rotor resistance: • The update laws are not unique by all means. For instance, with the following laws Δ R˙ eq < −γ k Re eTx PAeq xˆ Δ R˙ r < −γ k Rr eTx PAr xˆ

8.4 Parameter Identification by Adaptive Law

273

it yields 1 V˙ < − eTx Qex < 0 2 implying that the system is not only stable, but has a faster converging speed in the meanwhile; • All adaptive laws form a feasible space. In this space, the design of the matrix P can have different converging characteristics, being a similar effect to that of the control gain K R . Thus, for the simplification reason, letting P be a positive diagonal matrix will not lose the generality of the system: ⎡

1 ⎢0 P=⎢ ⎣0 0

0 k2 0 0

0 0 k3 0

⎤ 0 0⎥ ⎥ 0⎦ k4

As such, eTx P reduces to eTx P = [eid k2 eiq k3 eΨ d k4 eΨ q ] where k2 , k3 and k4 are appropriate gains requiring careful calibrations; • Since  r is not measurable, the use of eΨ d and eΨ q is physically infeasible. It is reasonable to have k3 = k4 = 0. This action makes the rotor dynamics become internal. Nevertheless, the overall system stability is still ensured. This further reduces eTx P to eTx P = [eid k2 eiq 0 0] Since γ is a redundant parameter in the original adaptation law, it is now assigned to k2 ; • The derivatives of ΔReq and ΔRr are the same as those of Req and Rr respectively. Therefore, a simple and practical adaptation law is derived as  k Re  ˆ R˙ eq = eid i sd + γ eiq iˆsq Lδ or

Similarly

or

   k Re ˆ ˆ i sd i sd − i sd + γ iˆsq iˆsq − i sq R˙ eq = Lδ

(8.34)

 k Rr κr  R˙ r = − eid Ψˆ r d + γ eiq Ψˆ rq Lδ Lr     k Rr κr  R˙ r = − Ψˆ r d iˆsd − i sd + γ Ψˆ rq iˆsq − i sq Lδ Lr

(8.35)

274

8 Rotor Field Oriented Control and Senseless Control

Let us recall Eq. (8.33), with adaptive laws (8.34) and (8.35), we have V˙ < 0 Therefore, as t → ∞, eid → 0, eiq → 0, ΔReq → 0, ΔRr → 0, According to Eqs. (8.34) and (8.35), Rˆ eq = Req ,

Rˆ r = Rr when eid = 0, eiq = 0

Bear in mind that with k3 = k4 = 0, this specially designed Laypunov function does not guarantee eΨ d = eΨ q = 0 by itself. Nevertheless, Eq. (8.32) reduces to e˙ x = Aex when e R = 0. A stable A will ensure ex = 0 at last. This indicates that the Lyapunov stability is only a sufficient condition. That being said, satisfying the Lyapunov condition will lead to a stable system, but the system can still be stable if the Lyapunov condition is not satisfied. The adaptive mechanism as given in Eqs. (8.34) and (8.35) looks like an integral law of the classical control. This is even true when combined with Eq. (8.32). The terms of iˆsd and iˆsq in the Rs (Req actually) estimation law and Ψˆ r d and Ψˆ rq in the Rr estimation law make a fundamental difference. Without these terms, they are indeed the integral control laws, having the following solutions for the closed-loop system eid = 0, eiq = 0 based on the integral control-like laws of Rs and Rr estimation ΔReq Aeq + ΔRr Ar = 0 and based on the integral control-like laws for the overall error dynamics ΔReq = 0, ΔRr = 0 In particular, the different signs of Aeq (the A I I part) and Ar (the A I Ψ part) make the trivial solution of ΔReq = 0, ΔRr = 0 become the integral function to some extent, but it is carefully designed to ensure that the estimation error decays to zero asymptotically.

8.4 Parameter Identification by Adaptive Law Ttemp

Vlim

Current Determination

raw

dmd isd

dmd

Tem

idmd sq

ψ lim

Torque Capability cap Tem

-

Pv

Ilim

Tem

275

Tem

ωr

-

|vs | | fb vsd

PId

|

vsd

+

dq

Modulation

vsα

PWMa

Inveter

PWMb

dmd isq

fb vsq

PIq

-

+

vsq

αβ

vsβ

PWMc

idmd sd ff vsd ff vsq

ωr

dmd isd

dmd isq

ω rm

Np

ACIM

dmd isd idmd sq ω re

ωs Rs /Rr Estimation

Rr

θ

Rs act

isd

dq

act isq

isα

αβ

αβ

isa isb

isβ abc

IFOC = Indirect Field Oriented Control

Fig. 8.11 Rr estimation based overall control architecture

Finally, the replacement of iˆsd and iˆsq by the respective i sd and i sq in Eq. (8.34) is possible, but not recommended since this will introduce a lot of noise. The replacement of the state variables iˆsd and iˆsq by the respective input variables vsd and vsq 4 is possible, but it is out of the scope of this book. Figure 8.11 presents the overall control architecture of the IFOC with an improved accuracy of the rotor and stator resistance, which gives rise to a more accurate calculation of the rotor flux direction. Improvement of the accuracy is attributed to the use of the feedback of the stator current error between the estimated and measured values.

8.4.2 Sensorless Control The adaptive parameter identification theorem could also be applied for the rotor speed estimation [11]. This is the so-called sensorless control due to the fact that the rotor speed sensor is no longer required. However, • The sensorless control is less practical than the case with speed sensors. A speed sensor has more use for the system state check in terms of diagnosis and safety monitoring. Thus, removal of the speed sensor is not recommended in reality; • The use of the error between the observed and measured current is the key. Therefore, when the number of the parameters required for estimation exceeds the number of the states (or the outputs) that could be measured, there will be an overestimation issue. The accuracy of the estimation can not be guaranteed anymore. It is therefore assumed that the rotor and stator resistance Rr and Rs are known. Also, since ωr is a fundamental signal for the rotor flux direction determination, the 4

Equation (8.29) does suggest the use of the input variable(s) instead of the state variable(s).

276

8 Rotor Field Oriented Control and Senseless Control

estimation of the rotor speed is better performed by using the αβ-frame model as given in Eq. (8.7) as x˙ = A(ωr e )x + Bu For the speed identification purpose, the matrix A(ωr e ) is repaired as ˆ r e ) = A0 + ωr e Aw A(ω where A0 is constant term, while Aw is the first-order term from the rotor speed perspective. They are given by ⎤ − τ1T 0 Lκδrτr 0 ⎢ 0 − 1 0 κr ⎥ ⎢ τT L δ τr ⎥ A0 = ⎢ L m ⎥ ⎣ τr 0 − τ1r 0 ⎦ 0 Lτrm 0 − τ1r ⎡

⎡

0 ⎢0 Aw = ⎢ ⎣0 0

⎤ 0 0 Lκrδ 0 − Lκrδ 0 ⎥ ⎥ 0 0 1 ⎦ 0 −1 0

Going through the same process as in the last section, the plant model is given as ˆ ωˆ r e )ˆx + Bu x˙ˆ = A( and the error dynamics as e˙ x = Aex + Δωr e Aw xˆ

(8.36)

The energy-related function V is defined as V (ex , Δωr e ) =

1 T Δωr2e ex Pex + 2 2γ

(8.37)

Substituting the error dynamics into the derivation of the energy-related function, it yields  1 1  V˙ = eTx AT P + PT A ex + eTx PΔωr e Aw xˆ + Δωr e Δω˙ r e 2 γ In order to have the global asymptotic stability, The adaptive law for the speed ωr e identification shall be as follows Δω˙ r e = −γ eTx PAw xˆ

8.4 Parameter Identification by Adaptive Law

277

Similar to the adaptive law for the resistance identification, the symmetric positive definite matrix P is designed in a way to ignore the unavailable error states eΨ d and eΨ q ⎤ ⎡ 1 0 0 0 ⎢ 0 k 0 0 ⎥ ⎥ P=⎢ ⎣ 0 0 0 0 ⎦ 0 0 0 0 With such a deign, ⎡

0 ⎢   0 eTx PAw xˆ = eiα keiβ 0 0 ⎢ ⎣0 0   κr = L δ eiα Ψˆ rβ − keiβ Ψˆ r α

⎤⎡ ˆ ⎤ i sα 0 0 Lκrδ ⎢ˆ ⎥ κr ⎥ 0 − L δ 0 ⎥ ⎢ i sq ⎥ ⎥ ⎢ 0 0 1 ⎦ ⎣ Ψˆ r α ⎦ 0 −1 0 Ψˆ rβ

Finally, by replacing Δω˙ r e with ω˙ r e (as they are the same), the adaptive law of the rotor speed estimation is achieved. ω˙ r e = −γ

 κr  eiα Ψˆ rβ − keiβ Ψˆ r α Lδ

(8.38)

It is general the case in many designs that the control gain is set as γ = 1 in Eqs. (8.34) and (8.35), and k = 1 in Eq. (8.38). However, these control parameters do offer extra design flexibility. With the control law in Eq. (8.38), the global stability is reached since 1 V˙ = − eTx Qex < 0 2 Nevertheless, the integral-like law in Eq. (8.38) could be extended to a PI5 -like function as below without affecting the overall system stability      eiβ Ψˆ r α − eiα Ψˆ rβ dt ωˆ r e = k p eiβ Ψˆ r α − eiα Ψˆ rβ + ki

(8.39)

When comparing to Eq. (3.76) in Chap. 3, it becomes clear that the rotor speed is ∗ actually estimated based on the virtual torque sensor. Take ΔTem as ∗ ΔTem = eiβ Ψˆ r α − eiα Ψˆ rβ

5

Proportional and integral.

(8.40)

278

8 Rotor Field Oriented Control and Senseless Control

Then the rotor speed estimation can be further simplified as ∗ + ki ωˆ r e = k p ΔTem



∗ dt ΔTem

Based on the above equation, the incorrect torque (delivered by the the machine) will lead to the rotor speed being continuously updated, till the torque error diminishes. The use of the torque error to estimate the rotor speed is somehow understandable since the stator current is indeed an accurate measure of the torque. It is noteworthy that the speed estimation law derived as above is not well applicable at extremely low speed. By investigating Eq. (8.36), it is clear that at zero or low speed, the error ex is hardly affected by the rotor speed variation. In other words, the error ex contains almost no information of the rotor speed within such low speed range. One root cause of the current error comes from the incorrect observation of the back EMF of the rotor, which will be either too weakened (at low speed) or completely diminished (if the machine is blocked). A more robust approach is still required.

8.5 Model Reference-Based Sensorless Control The sensorless control is referred to as the no-need of a rotor speed sensor, which can be either an encoder or a resolver. Basically, this means that the estimation of rotor speed is required to have sufficient accuracy. The rotor speed sensing is so critical, not only for torque control, but for many other aspects such as functional diagnosis and safety monitoring. Therefore, in automotive applications, it is not recommended to adopt sensorless control for traction motors and other safety critical applications. Nevertheless, in addition to the adaptive parameter identification theorem presented in last section, the model reference type of parameter identification is introduced.

8.5.1 Rotor Flux-Based Model Reference The rotor flux-based model reference is to estimate the rotor speed by [5, 7] • Computing the rotor flux through different channels, one being from the stator side, the other form the rotor side; • Using the difference in the rotor flux calculation to estimate the rotor speed. Regardless of whether with the dq or αβ-frame model (or any other rotating frame), as shown in Eqs. (3.39) and (3.56), the relation between the stator and rotor flux persists. This relation is represented below.

8.5 Model Reference-Based Sensorless Control

279

 s = L s Is + L m Ir  r = L m Is + L r Ir It is noteworthy that the above expressions are valid in either way of matrix vector and complex vector space. However, the complex vector space is adopted in this section by default. The current vector Is and Ir can be equally expressed in terms of the stator and rotor flux  s and  r as Is =

Ls L s L r −L 2m

Ir =

Lr L s L r −L 2m

 s −  r −

Lm r Lr Lm s Ls

 

From the first equation,  r can be derived from the stator flux  s and the stator current Is as r =

1  s − L δ Is ) ( κr

According to the the stator model in the αβ frame as given in Eq. (3.39),  s is driven by the stator voltage Vs and the stator current Is as s d = Vs − Rs Is dt

(8.41)

Combining all above, there is a way to determine the rotor flux  r by the stator voltage Vs and the stator current Is without the need of the rotor speed ωr e . This will be taken as the reference model for  r . According to Eq. (8.24), which is the same as Eq. (3.39) except that Ir is eliminated, the rotor flux ˆ r can also be derived by the stator current alone, but requires the knowledge of the rotor speed ωr e . To see this more clearly, the following is given ˙ ˆ r =



Lm 1 Is − ˆ r τr τr



− jωr eˆ r

Here ˆ r is used to indicate the rotor flux that is determined by a channel different from the reference model. The difference eΨ r eΨ r =  r − ˆ r

(8.42)

is applied to estimate the rotor speed. Since the referenced rotor flux  r needs no knowledge from ωr e , this method also works well in the low rotor speed region. The estimation law is defined as  ωˆ r e = k p |eΨ r | + ki |eΨ r | dt (8.43)

280

8 Rotor Field Oriented Control and Senseless Control Reference model

ψs

Vs +

+

-

-

1/ κ r

ψr +

eψ r

-

Integral

PI

Lδ

Rs

ψr

τr

-

Lm

+

Is

ω re

|.|

j

First order dynamics

X

Observer model

Adaptive model

Fig. 8.12 Overall architecture for ωr e identification

Since ωˆ r e is a scalar signal while eΨ r a vector signal in complex space, |.| is introduced here as a norm representing the length of the vector. eΨ r is actually a cross product of vectors as eΨ r = Ψr α Ψˆ rβ − Ψrβ Ψˆ r α The overall calculation architecture for ωr e identification is presented in Fig. 8.12. Some comments are made as below: • Ignoring the proportional term in Eq. (8.43) results in  r − ˆ r | ω˙ˆ r e = ki | Thus, at steady state

 r = ˆ r

Given that the overall system is stable6 and all the parameters (e.g. stator and rotor resistance) are known, then ωˆ r e = ωr e This is the power of the integral term; • The integral term has intrinsic adaptation function. Take the rotor and stator resistance identification as an example. The model given in Eq. (8.32) is not complete and the disturbances do exist. All these could be modelled as   e˙ x = Aex + ΔReq Aeq + ΔRr Ar xˆ +  where  represents these uncertainties. However, with the integral function in Eqs. (8.34) and (8.35) eid = 0 and eiq = 0 6

The system stability requires a limit on the integral gain.

8.5 Model Reference-Based Sensorless Control Fig. 8.13 Integral function improvement

281 Amp

Amp

First order system

Integral

Integral

Freq

Freq

fL Reject low frequency noise Amp

Amp

Integral

Integral Freq

fH

Freq

Reject high frequency noise

regardless. This is the adaptive function of an integral term. Keep in mind that the presence of these uncertainties does affect the accuracy of the parameter estimation. Nevertheless, appropriate tuning of the gains will greatly attenuate (or reduce) this effect; • The integral function also integrates the noises in the system. This is the fundamental issue with the integral function. If the signal contains none-white-noise or there is an offset in the signal, the integral of such signals can lead to anywhere. This is true for the measured voltages and current signals in the sensorless control, and also in the Rs and Rr estimation. This must be improved. To reject the constant and low frequency noise is one of the keys for any integral function design. This is particularly true for the stator flux linkage  s calculation, which is a pure integral over time. In addition, the inputs to this integral are of relatively high frequency in the αβ frame normally. In other words, the inputs with zero (or constant input) or low frequency signals are noises typically. One way to improve the integral performance over the low frequency is to convert the pure integral function into a first-order filter as illustrated in Fig. 8.13 (The rejection of high frequency noise is also shown in the figure). To covert the integral calculation of  s as in Eq. (8.41) into a first-order system (or low pass filter), the following is given s 1 d +  s = Vs − Rs Is (8.44) dt τL where τ L is the time constant. the frequency f L over which the integral function becomes prominent is determined by τ L as f L = 1/τ L . As shown in Fig. 8.13, the first-order system manifests the integral characteristics at high frequency, but shows significant difference at low frequency. Nevertheless, this improvement has an obvious drawback that it is not working well for rotor speed

282

8 Rotor Field Oriented Control and Senseless Control

estimation at low speed, which is the issue to be handled in this section particularly. Thus, caution must be taken when applying this sensorless control technique.

8.5.2 Induced Voltage-Based Model Reference Similar to the rotor flux-based approach, the induced voltage-based method estimates the rotor speed by • Computing the induced voltage of the magnetic field through different channels: one being from the stator side, the other from the rotor side; • Using the difference in the induced voltage calculation to estimate the rotor speed; • One channel of calculation, particularly the one that requires no speed information, is taken as reference. According to Eq. (3.39), the induced voltage Vir based on the stator model is calculated as   dIs 1 ˙ s − L δ I˙s Virs = = Vs − Rs Is − L δ (8.45) κr dt κr  s /dt in (3.39) contains both the voltage across Bear in mind that the voltage term of d the mutual inductance of the stator and rotor, the voltage drop at the respective stator and rotor self leakage inductance. The term 1/κr is to convert the voltage at the stator side to the rotor side. This is taken as the reference model for the induced voltage. Please note that L δ I˙s is the voltage loss (seen at the stator side) at both the stator and rotor leakage inductances. On the other hand, the induced voltage Vir can also be derived from the rotor side.  r /dt in Eq. (8.24) is the induced voltage viewed at the rotor side. This The term d voltage is consumed by the rotor resistance loss, rotor self-inductance loss and the back-EMF at the rotor (which forms the useful work by an ACIM). Vrir

= ˙ r =



Lm 1 Is −  r τr τr

 − jωr e r

(8.46)

Regardless of whether the induced voltage Vir is calculated from the stator or rotor side, they must be the same in reality. This forms the basis for the speed estimation. Similar to the rotor flux-based approach, the estimation law is designed as  ωˆ r e = k p |eir | + ki

|eir | dt

where eir represents the voltage difference between Virs and Vrir ir ir eir = Virs − Vrir = Vsα Vrβ − Vsβir Vrirα

(8.47)

8.5 Model Reference-Based Sensorless Control Lsl

Rs

283

ψs

Rr*

ψ *r

IM

VM

Vs

* Lrl

Ir*

Is

+

ω reψ *r j LM

Rs

Lsl

Is Lm

ψs

S

VM

Vs

Lrl

Ir

Vsir Vrir VRM

Rr

ψr

+

ω reψ r j -

Fig. 8.14 Equivalent electric circuit Reference model

Vs +

+

-

-

1/ κ r

V sir +

eir

-

Differential

PI

Lδ

Rs

ω re

|.|

Is

V rir

+

-

Lm / τ r

ψr

Integral +

1/ τ r

+

j

Observer model

X

Adaptive model

Fig. 8.15 Over-all architecture for ωr e identification

Figure 8.15 illustrates the overall calculation architecture for the induced voltagebased model reference method. This is quite analogous to what is presented in Fig. 8.12. The model for deriving the speed estimation algorithm is better explained by the equivalent electric circuit as in Fig. 8.14, in which V M = Virs κr = Vrir κr .

284

8 Rotor Field Oriented Control and Senseless Control

8.5.3 Phase Alignment Approach Having a close look at the equivalent circuit model in Fig. 8.14, the induced voltage at the rotor side consists of the voltage drop at Rr , L rl , and the back-EMF VRM = Rr Ir + L rl

dIr − jωr e r dt

Here VRM = Vrir is the rotor mutual reluctance voltage. At steady state, the leakage voltage drop is diminished. At high speed, voltage drop at the rotor resistance becomes insignificant, which leaves the back-EMF as the only dominating term. Thus VR VR ωr e = M = M r L r ir It is worth noting that the vector signals deteriorate to scalar signals as only their magnitudes are of interest. In order to make this expression more accurate in the low speed range, a factor ε which accounts for the voltage drop at the resistance is introduced. 1 VR (8.48) ωr e = M  r 1 + ε(ωr e ) In the meanwhile, the rotor current is derived from the stator current according to Eq. (3.71) as ir = κr i sq Bear in mind that the induced voltage estimation is conducted in the αβ frame, while the rotor speed estimation in the dq frame. Figure 8.16 presents the signal flow diagram for the speed estimation. The above estimation of the induced voltage is mainly viewed from the stator side rather than the rotor side. Nevertheless, the above analysis clearly indicates that the back-EMF is a good source for the estimation of the flux, speed, and phase angle. In particular, the phase angle estimation is very useful for the rotor field-oriented control, as it works well in the low speed range. This method is called the Phase Loop Lock (PLL) [2]. PLL is a feedback approach. The phase error calculation is illustrated as in Fig. 8.17. From the simplification point of view, the magnetic field voltage is mapped onto the stator side. If the system orients to the rotor flux correctly, then from the dq frame pint of view v RMd = 0, v RMq = ωr e  r Nevertheless, if there exists a phase error Δθ , then the dq frame is not aligned with the rotor flux. The induced voltage at the field direction will not be diminished, and is determined by

8.5 Model Reference-Based Sensorless Control Vsα i sα

+

-

1/ κ r

285 R

VMR α

VM d

R

|.|

αβ

VM

ω re

÷

Low Pass Filter

δLs

d/ dt

+

+

Rs

Vsβ i sβ

+

-

1/ κ r

R

VMR β

VM q

dq

ψr

Low Pass Filter δLs

d/ dt

+

+

Lr

Rs

αβ

Ird

Low Pass Filter

dq

Fig. 8.16 Over-all architecture for ωr e calculation Fig. 8.17 Phase error calculation

q

Δθ

VM

Δθ

d

v Md = |VM | sin Δθ v Mq = |VM | cos Δθ Particularly, when Δθ is small v Md = |VM | sin Δθ

(8.49)

Thus, v RMd could be utilized to estimate the rotor speed and in turn the phase angle as below  (8.50) ωˆ s = −k p v Md − ki v Md dt This is a negative feedback since v Md ↑ → Δθ ↓, and v Md ↓ → Δθ ↑ The architecture of PLL is given as in Fig. 8.18.

286

8 Rotor Field Oriented Control and Senseless Control

Finally, Table 8.1 summarizes the rotor speed estimation technologies being employed in this book for the sensorless control, and Fig. 8.19 gives an overall control architecture as a reference.

αβ

VM α

VMd

VM αβ

ωs

-+ 0

PI

dq θ

Fig. 8.18 Phase-loop lock algorithm architecture Table 8.1 Rotor speed estimation summary Method Sections Frame αβ

Adaptive law 8.4.2

αβ

Model reference 8.5.1 Model reference

Mixed 8.5.2

Phase loop lock

dq 8.5.3

Ttemp

Low speed estimation deteriorated Use rotor flux error Low speed estimation improved Use induced voltage Low speed estimation error improved Use field voltage error Better low speed estimation

|vs| Pv Current Determination

raw

dmd isd

dmd

Tem

idmd sq

Tem

ωr

-

PId

-

| fb vsd

|

vsd

+

dq

Modulation

vsα

PWMa

Inveter

PWMb

ψ lim

Torque Capability cap Tem

Attributes

Use torque error

Vlim

Ilim

Tem

Brief

dmd isq

PIq

-

fb vsq

+

vsq

vsβ

PWMc

αβ

idmd sd

ωr

ω rm

Np

dmd isd

ff vsd

dmd isq

ff vsq

ω re

ACIM

ψ r Observer ω s Calculator

ωs

θ

act

isd

¹rm Observer

dq

act

isq ψr

Fig. 8.19 Sensorless architecture

isα

αβ

αβ

isa isb

isβ abc

References

287

References 1. Clark RN (1996) Control system dynamics. Cambridge University Press 2. Comanescu M, Xu L (2006) An improved flux observer based on pll frequency estimator for sensorless vector control of induction motors. IEEE Trans Ind Electron 53(1):50–56 3. Finch JW, Giaouris D (2008) Controlled ac electrical drives. IEEE Trans Ind Electron 55(2):481–491 4. Franklin GF, Powell JD, Emami-Naeini A, Powell JD (2002) Feedback control of dynamic systems, vol 4. Prentice Hall, Upper Saddle River 5. Holtz J (2002) Sensorless control of induction motor drives. Proc IEEE 90(8):1359–1394 6. Holtz J (2006) Sensorless control of induction machines–with or without signal injection? IEEE Trans Ind Electron 53(1):7–30 7. Holtz J, Quan J (2002) Sensorless vector control of induction motors at very low speed using a nonlinear inverter model and parameter identification. IEEE Trans Ind Appl 38(4):1087–1095 8. Nam KH (2018) AC motor control and electrical vehicle applications. CRC Press 9. Novotny DW, Lipo TA (1996) Vector control and dynamics of AC drives, vol 41. Oxford University Press 10. Trzynadlowski AM (2000) Control of induction motors. Elsevier 11. Xu D, Wang B, Zhang G, Wang G, Yu Y (2018) A review of sensorless control methods for ac motor drives. CES Trans Electr Mach Syst 2(1):104–115

Chapter 9

Inverter PWM Control

9.1 Introduction Induction motors are not typically used as variable speed drives in industrial applications, where the motor speed is relatively fixed in accordance with the supply electrical frequency. Slip only changes within a minor range with respect to the load torque, which is predictably constant as well. Sufficiently, in those cases, the motor is directly connected to the 3-phase AC power supply and an inverter is not mandatory. Automotive traction motors are however required to work dynamically in response to the vehicle or driver’s torque demand. Motor drives will therefore work as variable speed drives, and usually operate in the torque control mode.1 For the AC motors, this variable speed operation has to be fulfilled with the assistance of a pulse width modulation (PWM) inverter. In automotive applications, the motor drive system is to discharge/charge power from/to the battery pack or other energy source if applicable. In motoring, electrical power is converted into mechanical power, and vice versa when in generating. The motor exchanges its AC power with the DC power from the battery via a PWM voltage source inverter (VSI). Control outputs of voltage signals, in magnitudes, frequencies or even phase shift, from either scalar control or vector control, will ultimately emerge as duty ratio switching signals to control the power switches in the inverter. An usual way of regulating the voltage is via the PWM control, which outputs highfrequency switching signals to the inverter and generates the AC voltage waveform from the DC voltage supply. There are numerous PWM methods, where the most commonly used are the continuous PWM techniques include the sinusoidal pulse width modulation (SPWM), the third harmonic injection sinusoidal PWM (THIPWM or THISPWM) and the space vector PWM (SVPWM). Here, continuous refers to

1

Speed control is usually enabled for the suppression of the powertrain oscillation.

© Springer Nature Switzerland AG 2024 S. Shen and Q.-z. Chen, Practical Control of Electric Machines for EV/HEVs, Lecture Notes in Electrical Engineering 1064, https://doi.org/10.1007/978-3-031-38161-4_9

289

290

9 Inverter PWM Control

continuous resultant common mode voltage. On the contrary, discontinuous PWM leads to discontinuous common mode voltage. The sinusoidal PWM method, as its name suggests, is to use a sinusoidal waveform as the reference modulating signal. The PWM duty cycle is calculated via comparing the modulating sinusoidal waveform and the carrier wave—a triangular waveform in general. SPWM can only modulates the phase voltage amplitude up to 1/2 of dc bus votage in the linear region. In order to extend the linear modulation regions, third harmonic injection method was introduced. Soon after, the space vector PWM method was brought up for the vector control. Both methods have higher percentage of linear utilization of the DC bus voltage than the SPWM and extend the voltage linearity by around 15%. In the meanwhile, they result in less harmonic distortion than the SPWM scheme. The THIPWM scheme was not initially proposed for the vector control and therefore does not necessarily require Park transform in implementation. The dynamic performance is considered to be inferior compared to the SVPWM. However, when adopted in conjunction with the Park transform, its performance is substantially improved to a degree that is comparable to the SVPWM, which as well has intrinsic third harmonic injection due to the space vector decomposition. Over the decades, the SVPWM technique has been widely used in high-performance traction motors. In this section, the three continuous PWM methods and their implementation are introduced and the comparison is highlighted as well. Presumably, the PWM control is applied to a 2-level half-bridge inverter. The schematic diagram of the ACIM motor and the inverter can be seen in Fig. 9.1. Here in the schematic diagram, point ‘O’ denotes the the mid-point voltage reference of the input DC bus; ‘N’ denotes the neutral point of the motor phases. [K a , K a ] refer to the boolean signals for the upper and bottom switches for phase A respectively; Likewise, [K b , K b ] are [K c , K c ] are the switch signals for phase B and C correspondingly.

Udc/2

Kc

+

Kb

Ka

O

A

DC

– Kc'

Kb'

Ka'

ua uCA

u AB

ub

- Udc/2

N

uc B

Fig. 9.1 Induction motor drive schematic diagram

uBC

C

9.1 Introduction

291

It is obvious that two switches on the same phase leg cannot be both simultaneously on as this will present as a shoot-through hazard to the DC supply. From the control perspective, ignoring the lockout between two switches to avoid shoot through, they cannot be both simultaneously off either, as this will make the terminal voltage undetermined depending on the polarity of that instant phase current. Therefore, switching states of the bottom and upper switches are logical NOT to each other by the following relations: K a = K a 

Kb = Kb

(9.1)



Kc = Kc The phase terminal voltage with reference to the middle point of ‘O’ is then given by ⎤ ⎡ ⎤ ⎡ ⎡ ⎤ Ka U AO 1 U ⎣ U B O ⎦ = Udc ⎣ K b ⎦ − dc ⎣ 1 ⎦ . (9.2) 2 UC O Kc 1 Likewise, the line-line voltage can be written in terms of the boolean switching signals as ⎤ ⎡ ⎤⎡ ⎤ ⎡ 1 −1 0 Ka U AB ⎣ U BC ⎦ = Udc ⎣ 0 1 −1 ⎦ ⎣ K b ⎦ . (9.3) −1 0 1 UC A Kc In a non-fault situation, the three phases of a motor is assumed to be ideally balanced and there exists a neutral point ‘N’ so that the phase-to-neutral voltage follows the relationship below at any time: U AN + U B N + UC N = 0

(9.4)

Here, potential at ‘N’ signifies the common mode voltage sequence U N O (t), which is typically time dependent. Consequently, the phase voltage with reference to the middle point can also be given by U AO = U AN + U N O (t) U B O = U B N + U N O (t)

(9.5)

UC O = UC N + U N O (t) Equation (9.2) shows that the transient U AO , U B O and UC O pose a modulated rectangular pulse waveform reciprocating within the limits of [− U2dc , U2dc ] due to the switching behaviour. In the PWM techniques, a modulating signal shall fit well within the peak envelopes of the carrier wave in order to maintain a linear amplification effect. Because of this, the SPWM method can only represent linearly a fundamental amplitude up to U2dc . A higher amplitude will however lead to nonlinear

292

9 Inverter PWM Control

saturation. With harmonic injection of U N O (t), the fundamental modulating signal is superimposed by a harmonic component. If the injected harmonic is appropriately chosen, the resultant modulating signal resulting from the superimposed components will be sustained within the peak envelopes of the carrier wave, while being capable of incorporating a higher-amplitude fundamental component. Consequently, it can represent linearly a higher fundamental phase voltage. This is the principle of the harmonic injection PWM methods. It is straightforward that line-line voltage can be written as U AB = U AO − U B O U BC = U B O − UC O

(9.6)

UC A = UC O − U AO and equally,

U AB = U AN − U B N U BC = U B N − UC N

(9.7)

UC A = UC N − U AN Clearly, common mode voltage harmonic U N O (t) does not appear at the line-line voltage. By combining Eqs. (9.3)–(9.4) and (9.7), the phase-to-neutral voltage can be given in expression of the switching states as ⎡

⎤ ⎤ ⎡ ⎤⎡ U AN 2 −1 −1 Ka ⎣ U B N ⎦ = Udc ⎣ −1 2 −1 ⎦ ⎣ K b ⎦ . 3 UC N Kc −1 −1 2

(9.8)

9.2 Sinusoidal PWM The inverter voltage control is to apply desirable, balanced sinusoidal voltages to the motor phases so that the phase-to-neutral voltages are purely sinusoidal with 120◦ phase shift to each other as U AN = u cos (ωe t + ϕ)   2π U B N = u cos ωe t + ϕ − 3   2π UC N = u cos ωe t + ϕ + 3

(9.9) (9.10) (9.11)

where u is the amplitude, ωe the stator frequency and ϕ the initial phase angle for phase A.

9.2 Sinusoidal PWM

293

Given the voltage source inverter with input DC voltage rails, the analog phase voltage signals need to be translated into switching signals to control the power switches at each corresponding phase leg. PWM is the process that the reference signal modulates a carrier wave and yields the switching pulses. A carrier wave can be a sawtooth, backward sawtooth or isosceles triangular wave, among which the isosceles triangular carrier wave is most widely used. Isosceles triangular carrier wave offers better harmonics behavior than the other as it allows for center alignment of the reference signal, which is beneficial for reducing the harmonics of the modulated signals. A comparison of the harmonics spectrum using the bipolar sawtooth and isosceles triangular carrier waves can be found in [7], where it shows that the modulation with isosceles triangular carrier results in much fewer harmonics. Ideally, the fundamental voltage signals are to be applied to the three-phase terminals. The SPWM technique is to obtain the desired phase voltage by comparing the respective reference sinusoidal modulating wave, denoted by λr , with an isosceles triangular carrier wave, denoted by λcw . The instant switching state of the upper switch in each phase is decided by:  Ki =

1, 0,

if λr,i ≥ λcw Otherwise

(9.12)

where the subscript i stands for a, b or c, corresponding to the three phases respectively as shown in Fig. 9.1. Figure 9.2 shows the SPWM in a natural sampling manner, meaning that the analog modulating signals are not sampled as per the carrier wave frequency before being used to compare with the carrier wave. This figure shows phase A as an example for the modulated output phase voltage with respect to the middle point and neutral point respectively. If the sinusoidal modulating wave fits within the boundary limits of the carrier wave, the PWM works similarly to a linear amplifier, i.e., the output voltage waveform will be linearly proportional to the input modulating wave with identical frequency and phase angle. In this linear region, the modulating wave has two intersections (ignoring the boundary situation) with the carrier wave per period, signifying the occurrence of two switching events of ‘on’ and ‘off’. Therefore, period of the carrier wave is also said to be the switching cycle, or equally, the fundamental frequency of the carrier wave is referred to as the switching frequency of the inverter. The carrier wave needs to have a higher frequency than the modulating wave in order to attain satisfactory resolution of the modulated output signal. Denote the frequency of the carrier wave by f cw and that of the reference signal by f s —the synchronous speed of the motor. The frequency modulation index is defined as mf =

f cw fs

(9.13)

The modulated signals can be analyzed using double Fourier analysis method. The PWM modulated phase-to-middle point voltage will contain the DC component

294

9 Inverter PWM Control

VP VT

-VT -VP

Fig. 9.2 Natural sampling sinusoidal pulse width modulation with bipolar carrier wave

9.2 Sinusoidal PWM

295

(if any) and fundamental sinusoidal component, and on top of that, it also contains the harmonic components in the following groups [1, 7]: • Fundamental and harmonic components of the carrier wave; • Sideband harmonics of the above fundamental and harmonic components of the carrier wave. Frequencies of these harmonics can be expressed as: f = M f cw ± N f s

s.t.

M + N = odd integer

(9.14)

where M = 1, 2, 3, ..., +∞ and N = 0, 1, 2, ..., +∞. Figure 9.3 demonstrates the spectral analysis of the PWM voltage with m f = 30 as seen in the example in Fig. 9.2. It is straightforward that the phase-to-middle voltage V AO (given the example of phase A) spectrum comprises the following components: • Fundamental component from the reference modulating signal; • The harmonic orders of 30, 90,..., which correspond to the frequencies of the carrier wave and its harmonics; • The harmonic orders of such as 28, 32, 57, 59, 61, 63,..., which correspond to the sideband harmonic frequencies of the carrier wave and its harmonics. Harmonics with orders at multiples of 3 go with the common mode voltage, while the phase-to-neutral voltage inherits the fundamental component and other harmonics. In the motor behaviour, these harmonics components are reflected in the phase current waveforms as switching harmonics. Selection of higher frequency for carrier wave, i.e. higher switching frequency, will be able to shift the modulated phase voltage harmonics to a frequency band, that can be better damped out by the inductive load of the motor. Consequently, it can reduce harmonic loss from the motor. The cost is however the increased switching loss on the inverter side. Hence, the switching frequency will be constrained and mainly determined by the thermal behaviour of the power devices. An optimization analysis on the motor drive level can therefore be carried out in this regard. In the linear region, the PWM inverter has a linear amplification effect, and therefore the amplitude of the modulating signal is linearly proportional to the fundamental amplitude of the PWM phase voltage signal. This linearity however does not hold if the amplitude of the modulating signal goes beyond the peak value of the carrier wave, and literally, it then enters the nonlinear region. In order to reflect this amplification ratio, modulation index2 is then defined as the ratio of the amplitude of output fundamental phase voltage versus the peak of the carrier wave. In order to keep consistency with the after-mentioned SVPWM scheme, which is most widely used in EV applications for the time being, modulation index reference is selected for the sake of convenience for the implementation of SVPWM. Therefore, we define the unity of modulation index at the linear boundary of the SVPWM such that

2

The expression of modulation index and the rationale are given in the SVPWM Sect. 9.4.

296

9 Inverter PWM Control

Fig. 9.3 Harmonic spectrum of the modulated phase-to-middle voltage, phase-to-neutral voltage and common mode voltage respectively with respect to Fig. 9.2

√ Definition 9.1 m = 1 when the fundamental output phase voltage reaches Udc / 3, where m denotes the modulation index. With the modulating signal being within the boundary of the carrier wave, the maximum amplitude of linear fundamental phase voltage that can be achieved by SPWM becomes Udc Umax = (9.15) 2

9.2 Sinusoidal PWM

297

and therefore the corresponding maximum linear modulation index for SPWM is √ m=

3 2

(9.16)

Using this definition of modulation index, the amplitude of the modulating signal, as seen in Fig. 9.2, can therefore be expressed as a ratio to the peak of the carrier wave as 2m VT = √ V P 3

(9.17)

Carrier wave is usually normalized and scaled in a unipolar manner between [0, 1] so that the modulating signal serves as the duty cycle input to control the corresponding switch. That is to say, 1 of the instant value of the modulating signal represents 100% duty cycle, while 0 means 0% duty cycle. Correspondingly, the normalized three-phase modulating signals are as below m 1 + √ cos (ωe t + ϕ) 2 3 m 1 2π ) = + √ cos (ωe t + ϕ − 2 3 3 m 1 2π ) = + √ cos (ωe t + ϕ + 2 3 3

λr,a =

(9.18)

λr,b

(9.19)

λr,c

(9.20)

In implementation, the modulating signal needs to be discretized. Instead of having the natural sampling with an analog signal input, the modulating signal needs to be sampled, very often at the peak, or at both the peak and valley in a carrier wave cycle if computational load is not a concern. In contrast to the natural sampling in Fig. 9.2, this sampling method is called regular sampling [7]. Figure 9.4 shows an example of the regular sampling SPWM method with normalized unipolar carrier wave for one phase. The reference signal is typically updated at the switching frequency, i.e. the frequency of the carrier wave ωc , starting at the trailing edge of the carrier wave. In this figure, the output of the modulated signal shows pulses and notches resulting from the switching events. The width of the output pulses represents the conduction time of the switch, while the notches denote the switching state being ‘off’. The modulating signal in this example has reached its maximum amplitude of 0.5 p.u.—the boundary that can be linearly represented by the carrier wave. Otherwise, further increase of the amplitude results in the saturation of duty cycle as it cannot possibly go higher than 100%. As a consequence, modulating wave with amplitude higher than 0.5p.u. will fall in the nonlinear modulation region and will need to be mapped to the actual modulation index using a nonlinear relationship. This will be further analyzed in Sect. 9.5. More often in EV traction control, half-cycle update, or sometimes referred to as half-cycle reload, are adopted as it exhibits clear benefits compared to the full

298

9 Inverter PWM Control

Fig. 9.4 Full-cycle update of SPWM for one phase with normalized unipolar carrier wave

Fig. 9.5 Half-cycle update of SPWM for one phase

cycle update as in Fig. 9.4. Half-cycle update is to update the modulating signal at a rate twice as fast as the carrier wave, typically with once at the peak and once at the trough in each carrier wave cycle. As such, the phase voltage harmonics is reduced and therefore leads to reduction of current harmonics as well. Particularly, in the vector torque control scheme, where there is an inner current control loop, dynamics of the phase current control can be significantly improved. Figure 9.5 shows the half-cycle update of the SPWM method.

9.3 Third Harmonic Injection PWM The SPWM technique is easy to be implemented; however, its major drawback is the limited usage of the DC bus voltage in the linear region. Third harmmonic injection PWM technique (THIPWM), or sometimes referred to as third harmmonic injection

9.3 Third Harmonic Injection PWM

299

sinusoidal PWM (THISPWM), is to inject a small amount of third order harmonics into the fundamental modulating signal. Typically, the third harmonic is with 1/6 the amplitude of the fundamental, and has a difference of π in phase shift compared to the fundamental [4, 5]. As such, taking phase A as an example (likewise applicable to phase B or C), the modulating signal for the THIPWM can be written as: λr,a =

1 m m + √ cos (ωe t + ϕ) + √ cos [3(ωe t + ϕ + π )] 2 3 6 3

(9.21)

Let θ = ωe t + ϕ for short, the above equation can be rewritten as λr,a = Local extrema occur when

m 1 m + √ cos (θ ) − √ cos (3θ ) 2 3 6 3

∂λr,a ∂θ

(9.22)

= 0, from which we get sin (θ ) −

1 sin (3θ ) = 0 2

1 sin (θ ) − [3 sin (θ ) − 4 sin3 (θ )] = 0 2 1 |sin (θ )| = 2

(9.23)

π. It is clear that the extrema are located in the positions θ ∗ = 16 π, 56 π, 76 π and 11 6 As stated in Sect. 9.2 of SPWM, a modulation is said to be linear if the peak of the modulating signal is no greater than the peak of the carrier wave—or within [0,1] after normalization. That is, λr,a (m, θ ∗ ) =

m 1 + √ cos θ ∗ ≤ 1 2 3

(9.24)

Substituting θ ∗ into Eq. (9.24), we have the linear modulation index of THIPWM ranging within [0, 1]. The maximum linear modulation index that can be achieved using THIPWM is therefore m=1 (9.25) As per the Definition 9.1 of the modulation index reference in Sect. 9.2, the maximum amplitude of the fundamental phase voltage from THIPWM evolves as Udc Umax = √ 3

(9.26)

Figure 9.6 shows the natural sampling of the THIPWM compared to the carrier wave for one phase. The modulating signal has the third harmonic superimposed on the fundamental component. The modulating signal has reached its boundary limit

300

9 Inverter PWM Control

VT VP

-VP -VT

Fig. 9.6 Natural sampling of THIPWM for one phase

Fig. 9.7 Half-cycle update of THIPWM for one phase with normalized unipolar carrier wave

of the linear modulation region, and therefore the modulation index is m = 1 in this example. Compared to the SPWM technique, the THIPWM extends the linear modulation region by 15.47%. The extension of the linearly proportional modulation allows better extraction of the DC bus voltage and leads to less harmonic distortion in current if a higher voltage is required. Consequently, it also extends the operating range of the optimization method such as maximum torque per Ampere (MTPA). In EV applications, this is particularly advantageous in high power demand or high speed operation, where the motor control comes often with field weakening to meet the DC voltage constraints.

9.4 Space Vector PWM

301

Figure 9.7 shows the half-cycle update of the regular sampling THIPWM method with the peak linear modulation index for one phase. The fundamental amplitude from the PWM phase voltage (phase-to-middle) reaches √13 the DC bus voltage.

9.4 Space Vector PWM The SVPWM technique is to map the desirable resultant voltage vector of the three phases into the voltage vector space, which is composed of the base voltage vectors in accordance with respective permissible switching states [4, 5, 8]. The SVPWM method depicted in this section is based on the 2-level 6-step inverter as shown in Fig. 9.1. Ignoring the deadtime effects, the upper and bottom switches from the same phase leg cannot be in the same switching states, see in Eq. (9.1). The inverter has therefore altogether 23 = 8 permissible switching states represented by the array [K a K b K c ] . We denote these eight switching states, namely [000], [100], [110], [010], [011], [001], [101] and [111] by S0 , S1 , S2 ,..., S7 respectively. From Eq. (9.8), each switching state corresponds to a deterministic combination of the three phase voltages, named as a base voltage vector in accordance with that switching state. Table 9.1 summarizes the eight permissible switching states and the corresponding voltage space vectors, denoted by base voltage vectors hereinafter. Among these switching states, S0 and S7 are called the zeros states, as they do not apply voltages over the three phases. The corresponding base voltage vectors − → − → U0 (000) and U7 (111) are called the zero voltage vectors. By contrast, the other switching states are referred to as active switching states, where the applied voltage is not zero. Given that the three phases are spatially distributed 120◦ to each other, the non-zero base voltage vectors have an amplitude of Udc given in the spatial reference frame as π (9.27) Ui∗ = Udc ej(i−1) 3 , i = 1, 2, ..., 6

Table 9.1 Voltage space vectors Switching Switching U AN state state value [K a K b K c ]

UB N

UC N

S0

[000]

0

0

0

S1

[100]

2Udc /3

−Udc /3

−Udc /3

S2

[110]

Udc /3

Udc /3

−2Udc /3

S3

[010]

−Udc /3

2Udc /3

−Udc /3

S4

[011]

−2Udc /3

Udc /3

Udc /3

S5

[001]

−Udc /3

−Udc /3

2Udc /3

S6

[101]

Udc /3

−2Udc /3

Udc /3

S7

[111]

0

0

0

Base voltage vector − → U0 (000) − → U1 (100) − → U2 (110) − → U3 (010) − → U4 (011) − → U5 (001) − → U6 (101) − → U7 (111)

302

9 Inverter PWM Control

Assuming the three phases are spatially balanced, the resultant voltage vector can be written as: 2 2 (9.28) U∗ = U AN ej0π + U B N ej 3 π + UC N e−j 3 π Substitute the desirable sinusoidal phase voltages as seen in Eqs. (9.9)–(9.11) into the Eq. (9.28), the resultant voltage vector then becomes U∗ =

3 u [cos(θ ) + j sin(θ )] 2

(9.29)

where θ = ωe t + ϕ, denoting the rotor position. For orthogonal vector decomposition, the three-phase system is converted into the two-axis Cartesian coordinate system using the Clarke Transform. Equation (9.29) indeed shows this static transformation. Here, we use the amplitude-invariant Clarke Transform as the convention for the α-β coordinate system [2, 6]. Equation (9.29) can then be rewritten in the α-β reference frame by applying a scaling factor of 2/3 as U=

2 ∗ U = Uα + jUβ 3 = u cos(θ ) + ju sin(θ )

(9.30)

This scaling factor brings the representation of voltage vector space from threephase spatial superimposition to a per-phase amplitude scale. The scaled base voltage vectors become 2 π U ej(i−1) 3 , i = 1, 2, ..., 6 (9.31) Ui = 3 dc 0, i = 0 or 7 These eight base voltage vectors form a hexagonal voltage vector space composed of six sectors, where any voltage vector that falls within the space boundary can be represented by an appropriate combination of these base voltage vectors. The benefit from applying the scaling factor 2/3 is that, in implementation, the ratio of projection of the targeted voltage vector U over the base voltage vectors literally equals to the duty cycles for the corresponding switching states. For instance, a targeted voltage vector U = 2Udc /3 + j0 requires full duty cycle on the switching state S1 . Figure 9.83 shows the hexagonal voltage vector space in the per-phase amplitude scale. The space is partitioned into six sectors—each of them is shaped by two adjacent base voltage vectors, as shown in the figure. As will be elaborated in Sect. 9.4.1, the inscribed circle of the hexagon is the boundary of the linear modulation region. A desirable voltage vector within this boundary can be linearly decomposed using the base voltage vectors. As such, the length of this voltage vector is linearly proportional to the amplitude of the fundamental output phase voltage. Using u to 3 Over arrows highlight the vectors, which are symbolized by bold fonts in the text expressions throughout this book.

9.4 Space Vector PWM

303

Fig. 9.8 Hexagonal voltage vector space

denote the amplitude of the fundamental phase voltage as shown in Eqs. (9.9)–(9.11), we define the modulation index (MI) as m=

u

√ Udc / 3

(9.32)

so that m = 1 corresponds to the inscribed circle of the voltage trajectory. The phase amplitude at this trajectory is given by Udc u = Uamp,0 = √ 3

(9.33)

This is the rationale behind the definition of unity reference modulation index as aforementioned in Sect. 9.2.

9.4.1 Implementation of SVPWM To convert a voltage vector to the form of switching states, we can express it using the base voltage vectors as 7 U= ηi Ui (9.34) i=0

304

9 Inverter PWM Control

where ηi is the mapping coefficient along that base voltage vector Ui , and thus it can be regarded as the duty cycle of the corresponding switching state. In one switching cycle, ηi can be written as Ti ηi = (9.35) Ts with

7

ηi = 1

(9.36)

i=0

where Ts is the switching period and Ti is the duration with respect to switching state Si . Assuming a voltage vector resides within any sector of the space, it can be vector-decomposed using the two adjacent base voltage vectors, therefore the − → corresponding switching states, that shape the sector. Zero voltage vectors U0 (000) − → and U7 (111) are typically needed in order to complete a switching cycle to satisfy Eq. (9.36). We normalize the quantities in a p.u. system, so that 1 p.u. corresponds to those with modulation index m = 1. Thus, Eq. (9.31) can be rewritten as Ui [p.u.] Uamp,0  2 j(i−1) π √ e 3 , i = 1, 2, ..., 6 3 = 0, i = 0 or 7

Ui :=

(9.37)

Figure 9.9 shows an example of the voltage vector decomposition in sector I. We use the generalized expression of any voltage vector U = Uα + jUβ , see in Eq. (9.30). Decomposition of the voltage vector is given by U = η1 U1 + η2 U2 + η0 U0 + η7 U7

(9.38)

Here, the zero voltage vectors are to complete a switching cycle. These components do not contribute to the fundamental of the output voltage. From Eq. (9.36), we have the coefficients given as below η0 + η7 = 1 − (η1 + η2 )

(9.39)

The trade between η0 and η7 yields a bunch of variants of the PWM strategies, including the SVPWM, which is a continuous PWM scheme and will be elaborated in this section, and a group of discontinuous PWM schemes. Being continuous or discontinuous is referenced to (dis)continuity of the zero sequence voltage, equally the common mode voltage if not taking into account the switching harmonics. Based on this, the discontinuous PWM schemes include • If η0 = 0 and η7 = 1 − (ηm + ηn ), the PWM scheme turns into DPWMMAX, which takes the maximum value (positive peak) of the common mode voltage;

9.4 Space Vector PWM

305

• If η7 = 0 and η0 = 1 − (ηm + ηn ), the PWM scheme turns into DPWMMIN, which takes the minimum value (negative peak) of the common mode voltage; • If the above two conditions are applied in sequence depending on the voltage space sectors and/or the voltage phase angle, there produces the generalized DPWM (GDPWM) schemes including DPWM0, DPWM1 and DPWM2. Here, m and n denote respectively the subscripts of two adjacent nonzero base voltage vectors that shape any sector in the voltage vector space. We can see that discontinuous PWM schemes only have one of the zero voltage vectors clamped at a time. Advantages of discontinuous PWM schemes are reduction of the number of switching events and thus reducing the switching loss. The cost however is the increase of current harmonics, in most cases higher common mode voltage and in some cases uneven thermal stresses on the power modules. We refer to [4] for more information regarding the discontinuous PWM strategies. The continuous PWM scheme in this space vector description comes from the following condition: • η0 = η7 = 21 (1 − ηm − ηn ). This then turns into the conventional SVPWM scheme. Study has shown that high order harmonics can be minimized when zero voltage components are equally distributed on switching states S0 and S7 . Continuing with the example in Fig. 9.9, we have η0 = η7 =

U3(010)

β

Uβ

U4(011)

U0(000)

U7(111)

U5(001)

1 (1 − η1 − η2 ) 2

(9.40)

U2(110)

η2U2

U

θ

1p.u

η1U1

Uα

U6(101)

Fig. 9.9 Voltage vector decomposition in Sector I (in p.u. system)

U1(100) 2 p.u. 3

α

306

9 Inverter PWM Control

The vector components can be written using the α-β coordinates as

π  η1 |U1 | = Uα − Uβ tan

π  6 η2 |U2 | = Uβ / cos 6 The normalized amplitude of any active base voltage vector is (9.37). Thus √ 3 π η1 = Uα − Uβ tan( ) 6 √2

3 π η2 = Uβ / cos( ) 2 6

(9.41) (9.42) √2 3

as shown in Eq.

(9.43) (9.44)

where both Uα and Uβ are in per unit expression. It is straightforward that the duty cycles corresponding to the three phases, i.e., the switching signals K a , K b and K c to the upper switches of each phase leg respectively, then become ⎧ ⎪ ⎨η A = η1 + η2 + η7 (9.45) η B = η2 + η7 ⎪ ⎩ ηC = η7 Figure 9.10 shows an example of the switching sequence and the corresponding duty ratios in Sector I for one switching period. Much the same as the voltage vector decomposition in the space sector I, Fig. 9.11 shows the decomposition of a voltage vector in Sector II. This generalization is applicable to any other sector by changing the active states accordingly. Similarly, any voltage vector in Sector II can be decomposed using the following expression U = η2 U2 + η3 U3 + η0 U0 + η7 U7

(9.46)

Assuming this voltage vector now resides in a new coordinate system α-β, which is transformed from rotating α-β anticlockwise through an angle of π3 . U and U denote this voltage vector in the original reference frame and the new reference frame respectively, and the corresponding coordinates are denoted by (Uα , Uβ ) and (U α , U β ) respectively. This voltage vector in the new α-β reference frame emerges as π

U = Ue− 3 j  π  = Uα + jUβ e− 3 j The corresponding coordinates become

(9.47)

9.4 Space Vector PWM

307

1

Switch Ka

0

Switch Kb Switch Kc Switching states

S0

S1

η0/2 η1/2

S2

S7

S2

η2/2

η7

η2/2

S1 η1/2

S0 η0/2

ηC ηB ηA

Phase duty ra o

1 p.u. ( = 1 switching period) Fig. 9.10 Switching sequence and duty cycles in Sector I in one switching cycle

U3(010)

α

β

U2(110) β

Uβ

U

Uα η2U2 η3U3

U4(011)

Uβ U7(111)

U5(001)

θ

1p.u

Uα

U1(100) 2 p.u. 3

U0(000)

U6(101)

Fig. 9.11 Voltage vector decomposition in Sector II (in p.u. system)

α

308

9 Inverter PWM Control

√ 3 1 U α = Uα + Uβ 2√ 2 3 1 Uα + Uβ Uβ = − 2 2

(9.48) (9.49)

Using the newly defined U α and U β , the decomposition can be done in an analogous way as in Sector I, see Eqs. (9.41) and (9.42), as

π  η2 |U2 | = U α − U β tan

π  6 η3 |U3 | = U β / cos 6

(9.50) (9.51)

Combine Eqs. (9.48)–(9.51) and take into account the normalized amplitude of the base voltage vectors, then correspondingly, the duty ratio of each base voltage vector becomes √ 3 1 η2 = Uα + Uβ (9.52) 2√ 2 3 1 η3 = − Uα + Uβ (9.53) 2 2 Likewise, the zero voltage components are expressed as η0 = η 7 =

1 (1 − η2 − η3 ) 2

(9.54)

Duty cycles to the phase leg switches for a voltage vector in Sector II can be decomposed as ⎧ ⎪η A = η2 + η7 ⎨ (9.55) η B = η2 + η3 + η7 ⎪ ⎩ ηC = η7 Generalization of the voltage vector decomposition for other space sectors can be done by rotating the reference frame α-β accordingly. Indeed, Eq. (9.47) can be generalized as (i−1)π Ui = Ue− 3 j (9.56) where i = 1, 2, ..., 6, denotes the number of space sector. The rest of calculation is much of the same and is not reiterated. Table 9.2 summarizes the duty ratio calculation for the six sectors. Figure 9.12 gives an example of the SVPWM in a natural sampling manner. The space vector harmonics, which is the modulating signal less the fundamental, pose as a triangular wave and has an intrinsic third order harmonic to the fundamental. With this natural inclusion of harmonics from the space vector decomposition, linear

9.4 Space Vector PWM

309

Table 9.2 Summary of duty cycles in the six sectors Switching states duty ratio Sector I 0 ≤ θ