Ac motor control and ekectrical vehicle applications

Tai ngay!!! Ban co the xoa dong chu nay!!! i i “K32224˙FM” — 2018/9/21 — 14:48 — page — #2 i i AC Motor Control and Electrical Vehicle Applications Second Edition i i i i i i “K32224˙FM” — 2018/9/21 — 14:48 — page — #3 i i i i i i i i “K32224˙FM” — 2018/9/21 — 14:48 — page — #4 i i AC Motor Control and Electrical Vehicle Applications Second Edition Kwang Hee Nam i i i i i i “K32224˙FM” — 2018/9/21 — 14:48 — page — #6 i i CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2019 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S Government works Printed on acid-free paper Version Date: 20180921 International Standard Book Number-13: 978-1-138-71249-2 (Hardback) This book contains information obtained from authentic and highly regarded sources Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint Except as permitted under U.S Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400 CCC is a not-for-profit organization that provides licenses and registration for a variety of users For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com i i i i ✐ ✐ “book18-March-fin” — 2018/10/3 — 15:59 — page v — #2 ✐ ✐ Contents Preface xiii About the Author xvii Preliminaries for Motor Control 1.1 Basics of Electromagnetics 1.1.1 Tensors 1.1.2 Riemann Integral and Fundamental Theorem of Calculus 1.1.3 Ampere’s Law 1.1.4 Faraday’s Law 1.1.5 Inductance 1.1.6 Analogy of Ohm’s Law 1.1.7 Transformer 1.1.8 Three Phase System 1.2 Basics of DC Machines 1.2.1 DC Machine Dynamics 1.2.2 Field Weakening Control 1.2.3 Four Quadrant Operation 1.2.4 DC Motor Dynamics and Control 1.3 Dynamical System Control 1.3.1 Gain and Phase Margins 1.3.2 PD Controller 1.3.3 PI Controller 1.3.4 IP Controller 1.3.5 PI Controller with Reference Model 1.3.6 2-DOF Controller 1.3.7 Variations of 2-DOF Structures 1.3.8 Load Torque Observer 1.3.9 Feedback Linearization References Problems 1 12 13 14 18 19 21 23 25 25 28 30 31 34 37 39 44 45 46 47 49 50 v ✐ ✐ ✐ ✐ ✐ ✐ “book18-March-fin” — 2018/10/3 — 15:59 — page vi — #3 ✐ ✐ vi CONTENTS Rotating Magnetic Field 2.1 Magneto Motive Force and Inductance 2.1.1 Single Phase Inductance 2.1.2 Inductance of Three Phase Uniform Gap Machine 2.2 Rotating Field 2.2.1 Rotating Field Generation by Inverter 2.2.2 High Order Space Harmonics 2.3 Change of Coordinates 2.3.1 Mapping into Stationary DQ Coordinate 2.3.2 Mapping into Synchronous Frame 2.3.3 Formulation via Matrices 2.3.4 Power Relations 2.3.5 Transformation of Impedance Matrices 2.4 PI Controller in Synchronous Frame References Problems 57 58 59 61 62 64 65 68 69 71 73 75 77 80 83 84 Induction Motor Basics 3.1 IM Construction 3.2 IM Operation Principle 3.2.1 IM Equivalent Circuit 3.2.2 Torque-Speed Curve 3.2.3 Breakdown Torque 3.2.4 Stable and Unstable Regions 3.2.5 Parasitic Torques 3.3 Leakage Inductances 3.3.1 Inverse Gamma Equivalent Circuit 3.4 Circle Diagram 3.4.1 Torque and Losses 3.5 Current Displacement 3.5.1 Double Cage Rotor 3.5.2 Line Starting 3.6 IM Speed Control 3.6.1 Variable Voltage Control 3.6.2 VVVF Control References Problems 87 87 89 90 93 96 99 100 101 103 105 107 110 112 115 116 116 116 119 119 123 123 123 129 133 138 138 Dynamic Modeling of Induction Motors 4.1 Voltage Equation 4.1.1 Flux Linkage 4.1.2 Voltage Equations 4.1.3 Transformation via Matrix Multiplications 4.2 IM Dynamic Models 4.2.1 ODE Model with Current Variables ✐ ✐ ✐ ✐ ✐ ✐ “book18-March-fin” — 2018/10/3 — 15:59 — page vii — #4 ✐ ✐ CONTENTS 4.2.2 IM ODE Model with Current-Flux 4.2.3 Alternative Derivations 4.2.4 Steady State Models 4.3 Power and Torque Equations References Problems vii Variables Induction Motor Control 5.1 Rotor Field Oriented Scheme 5.2 Stator Field Oriented Scheme 5.3 Field Weakening Control 5.3.1 Current and Voltage Limits 5.3.2 Torque–Speed Curve 5.3.3 Torque and Power Maximizing Solutions 5.4 IM Sensorless Control 5.4.1 Voltage Model Estimator 5.4.2 Current Model Estimator 5.4.3 Closed-Loop MRAS Observer 5.4.4 Dual Reference Frame Observer 5.4.5 Full Order Observer 5.4.6 Reduced Order Observer 5.4.7 Sliding Mode Observer 5.4.8 Reduced Order Observer by Harnefors 5.4.9 Robust Sensorless Algorithm 5.4.10 Relation between Flux and Current Errors References Problems 139 141 144 144 150 151 155 156 165 167 167 168 170 173 174 175 175 176 179 181 182 185 187 190 194 196 201 202 202 204 209 209 209 213 218 225 226 229 230 232 234 234 Permanent Magnet AC Motors 6.1 PMSM and BLDCM 6.1.1 PMSM Torque Generation 6.1.2 BLDCM Torque Generation 6.1.3 Comparison between PMSM and BLDCM 6.2 PMSM Dynamic Modeling 6.2.1 Types of PMSMs 6.2.2 SPMSM Voltage Equations 6.2.3 IPMSM Dynamic Model 6.2.4 Multiple Saliency Effect 6.2.5 Multi-pole PMSM Dynamics and Vector Diagram 6.3 PMSM Torque Equations 6.4 PMSM Block Diagram and Control 6.4.1 MATLAB Simulation References Problems ✐ ✐ ✐ ✐ ✐ ✐ “book18-March-fin” — 2018/10/3 — 15:59 — page viii — #5 ✐ ✐ viii CONTENTS PMSM Control Methods 7.1 Machine Sizing 7.1.1 Machine Sizes under Same Power Rating 7.2 Current Voltage and Speed Limits 7.2.1 Torque versus Current Angle 7.3 Extending Constant Power Speed Range 7.3.1 Torque Speed Profile 7.4 Current Control Methods 7.4.1 Maximum Torque per Ampere Control 7.4.2 Transversal Intersection with Current Limit 7.4.3 Maximum Power Control 7.4.4 Maximum Torque per Voltage Control 7.4.5 Combination of Maximum Power Control Methods 7.4.6 Unity Power Factor Control 7.4.7 Current Control Contour for SPMSM 7.4.8 Properties when ψm = Ld Is 7.4.9 Per Unit Model of PMSM 7.4.10 Power-Speed Curve 7.4.11 Wide CPSR References Problems 243 243 245 246 248 250 251 253 253 255 257 259 262 264 267 267 271 273 274 276 277 Magnetism and Motor Losses 8.1 Soft and Hard Ferromagnetism 8.1.1 Permanent Magnet 8.1.2 Air Gap Field Determination 8.1.3 Temperature Dependence and PM Demagnetization 8.1.4 Hysteresis Loss 8.1.5 Skin Depth and Eddy Current Loss 8.1.6 Electrical Steel 8.2 Motor Losses 8.3 Loss Minimizing Control for IPMSMs 8.3.1 PMSM Loss Equation and Flux Saturation 8.3.2 Solution Search by Lagrange Equation 8.3.3 LMC Based Controller and Experimental Setup 8.3.4 Experimental Results References Problems 281 281 283 283 285 287 288 292 293 296 297 299 301 303 305 307 PMSM Sensorless Control 9.1 IPMSM Dynamics in a Misaligned Frame 9.1.1 Different Derivation Using Matrices 9.1.2 Dynamic Model for Sensorless Algorithm 9.2 Back-EMF Based Angle Estimation 9.2.1 Morimoto’s Extended EMF Observer 309 310 311 312 313 313 ✐ ✐ ✐ ✐ ✐ ✐ “book18-March-fin” — 2018/10/3 — 15:59 — page ix — #6 ✐ ✐ CONTENTS ix 9.2.2 Ortega’s Nonlinear Observer for Sensorless Control 9.2.3 Bobtsov’s Initial Parameter Estimator 9.2.4 Comparison between Back EMF and Rotor Flux Estimate 9.2.5 Starting Algorithm by Signal Injection 9.3 Sensorless Control by Signal Injection 9.3.1 Rotating Signal Injection in Stationary Frame 9.3.2 Signal Injection in a Synchronous Frame 9.3.3 PWM Level Square-Voltage Injection in Estimated Frame References Problems 10 Pulse Width Modulation and Inverter 10.1 Switching Function and Six Step Operation 10.2 PWM Methods 10.2.1 Sinusoidal PWM 10.2.2 Injection of Third Order Harmonics 10.2.3 Space Vector Modulation 10.2.4 Sector Finding Algorithm 10.2.5 Space Vector Waveform 10.2.6 Discrete PWM 10.2.7 Overmodulation Methods 10.3 Common Mode Current and Countermeasures 10.4 Dead Time and Compensation 10.5 Position and Current Sensors 10.5.1 Encoder 10.5.2 Resolver and R/D Converter 10.5.3 Hall Current Sensor and Current Sampling References Problems 11 Basics of Motor Design 11.1 Winding Methods 11.1.1 Full and Short Pitch Windings 11.2 MMF with Slot Openings 11.2.1 MMF with Current Harmonics 11.3 Fractional Slot Machines 11.3.1 Concentrated Winding on Segmented Core 11.3.2 Feasible Slot-Pole Number Combination 11.3.3 Torque-Producing Harmonic and Subharmonics 11.4 Demagnetization Analysis 11.4.1 PM Loss Influential Factors 11.4.2 PM Loss and Demagnetization Analysis 11.4.3 Armature Reaction 11.5 Torque Analysis 11.5.1 Torque Ripple 318 324 327 328 331 332 333 336 340 343 347 349 352 353 354 354 356 358 361 362 366 368 371 372 373 376 378 380 385 385 387 393 396 400 400 401 406 409 409 410 411 412 417 ✐ ✐ ✐ ✐ ✐ ✐ “book18-March-fin” — 2018/10/3 — 15:59 — page 368 — #383 ✐ ✐ 368 CHAPTER 10 PULSE WIDTH MODULATION AND INVERTER mode noise levels are strongly regulated by the rules of Federal Communications Commission (FCC) in the U.S housing coil stator rotor load bearing ball shaft inverter D air gap vcm inverter power cable rotor bearing E Figure 10.26: (a) Common mode current path and (b) an equivalent circuit The common mode current can be suppressed by the common mode noise filter as shown in Fig 10.27 (a) [16] Note that the coils of common mode inductor are wound with the same orientation for the three power lines As a result, no field is induced by the differential mode, whereas a big inductance is developed to the common mode current So, it does not inhibit the flow of differential currents, while causing an impedance to the common mode current It is a method of impeding the common mode current Another way of avoiding the possible damage is to provide a low impedance path as shown in Fig 10.27 (b) It consists of coaxial shield, RF connectors, metal cases, and the ground system Then, the common mode current will be bypassed through coaxial shield and metal conduit, and to the ground Impedance matching is made via RF connectors between the coaxial shield and inverter case/motor housing 10.4 Dead Time and Compensation It is shown in Fig 10.2 (a) that the switch current does not fall down to zero immediately after the gating signal becomes low This is because IGBTs have a relatively long tail current (about µs.) and the switching circuit has stray inductances Therefore, the complementary switch should be turned on after the current tail disappears Otherwise, a shoot-through phenomenon takes place To prevent it, a dead interval, in which both upper and lower switches are low, should be provided for each signal transition The dead time can be set by putting off all rising edges of the gating signals The dead time period depends on current tail, ✐ ✐ ✐ ✐ ✐ ✐ “book18-March-fin” — 2018/10/3 — 15:59 — page 369 — #384 ✐ ✐ 10.4 DEAD TIME AND COMPENSATION 369 Common mode filter Motor Ground (a) Shielded power cable Motor RF socket RF socket Common mode current Ground (b) Figure 10.27: Methods of attenuating common mode noise: (a) common mode filter and (b) coaxial cable dV /dt capability of IGBT, stray inductance of the switching loop, and the DC link voltage In the case of IGBTs, the typical value is ∼ µs Fig 10.28 shows a method of generating the dead time and the voltage error caused by the dead-time When the load is inductive, the load current continues to flow through an anti-parallel (free wheeling) diode during the dead time: Therefore, the pole voltage (inverter terminal voltage) is determined to be either V2dc or − V2dc depending on the current direction: If the current flows into the load (ias > 0), the load current flows through the anti-parallel diode of lower arm Thus, − V2dc appears at the terminal during the dead time, i.e., a negative error voltage is made, as shown in Fig 10.28 (a) On the other hand, if current flows out from the load (ias < 0), the load current flows through the anti-parallel diode of upper arm Thus, Vdc appears at the terminal during the dead time, i.e., a positive error voltage is made as shown in Fig 10.28 (b) When the current level is low, the pole voltage during the dead time is not determined definitely Distorted voltage and current wave forms are depicted in Fig 10.29 It is obvious from Fig 10.29 that the percentage of distortion is large when the current level is low The distortion caused by dead time can be corrected by adding or subtracting the dead time interval, Td , depending on the current polarities: To compensate the dead time error, on-duty of high side switch is increased by Td , if ia < Conversely, if ia < 0, then on-duty is decreased by Td :  Ta + Td , if ia > 0, Ta − Td , if ia < 0, ✐ ✐ ✐ ✐ ✐ ✐ “book18-March-fin” — 2018/10/3 — 15:59 — page 370 — #385 ✐ ✐ 370 CHAPTER 10 PULSE WIDTH MODULATION AND INVERTER Sa Sa Sa’ Sa’ Original PWM Original PWM Gate signal with dead time Gate signal with dead time Sa Sa’ Sa’ Vdc/2 Vdc/2 Terminal voltage Terminal voltage -Vdc/2 Dead time voltage error Sa -Vdc/2 Vdc Dead time voltage error -Vdc (a) (b) Figure 10.28: Dead-time effects depending on current direction: (a) ias > and (b) ias < Distorted voltage by dead time Distorted voltage by dead time current current (a) (b) Figure 10.29: Voltage distortion due to the dead time when the current level is (a) high and (b) low ✐ ✐ ✐ ✐ ✐ ✐ “book18-March-fin” — 2018/10/3 — 15:59 — page 371 — #386 ✐ ✐ 10.5 POSITION AND CURRENT SENSORS 371 where Ta is on-duty of the upper switch before compensation Fig 10.30 shows the dead time compensation method Sa Sa Sa’ Sa’ Original PWM Original PWM With dead time compensation Gate signal after providing dead time With dead time compensation Sa Gate signal after providing dead time Sa’ Sa Sa’ (a) (b) Figure 10.30: Dead time compensation depending on the current polarities: (a) ia < and (b) ia > Fig 10.31 shows experimental current wave forms before and after the dead time correction It is observed that the dead time also causes a reduction in the fundamental component of the output voltage, apart from distortion In the extreme case, the distorted output voltage generates subharmonics, which result in torque pulsation and instability at low-speed and light-load operation [17], [18] In the sensorless vector control, the dead time voltage error causes a negative impact on the low-speed performance [19] Some dead time compensation schemes are found in [17],[18],[20],[21] 10.5 Position and Current Sensors In order to construct a current controller in the reference frame, the measured current vector should be mapped into the reference frame Therefore, the rotor angular position and phase currents need to be measured For position sensing, the absolute encoder or resolver is utilized Since absolute encoders are normally expensive, resolvers are popularly utilized for PMSMs However, an incremental encoder can be used with some starting methods For current sensing, Hall effect sensors are widely used ✐ ✐ ✐ ✐ ✐ ✐ “book18-March-fin” — 2018/10/3 — 15:59 — page 372 — #387 ✐ ✐ 372 CHAPTER 10 PULSE WIDTH MODULATION AND INVERTER 10A 10A -10A -10A 25ms 25ms (b) (a) Figure 10.31: Current distortion due to the dead time: (a) before compensation and (b) after compensation 10.5.1 Encoder An incremental rotary encoder generates electrical pulses when the shaft rotates By counting the pulses, angular position or speed is measured Fig 10.32 shows the structure of an encoder consisting of LEDs, photo transistors, and a disk with slits Normally, the disk is made of glass and the slits are created by etching When the disk rotates, flickering light is converted into electrical pulses by the photo transistors The voltage pulses are transmitted outside after being conditioned Incremental encoders output three pairs of signals: A, A, B, B, and Z, Z The A and B phase signals are 90◦ out of phase, as shown in Fig 10.32 The signal states are summarized in Table 10.1 The direction is determined based on which signal comes first after resetting, (A, B) = (0, 0) If B changes from to while A = 0, then the device rotates in the clockwise (CW) direction Alternatively if B changes from to while A = 0, then it L rotates in the counterclockwise (CCW) direction The ‘exclusive OR’ operation, A B is used to increase the resolution by a factor of The disk has one slot for Z-phase, so that the Z-phase pulse arises once Table 10.1: State diagram for encoder pulses Direction Phase CW A B 0 1 1 L A B 1 Phase CCW A B 0 1 1 L A B 1 per revolution The Z-phase signal is used for resetting the data with an absolute position MT Methods There are two speed measuring methods With the M -method, the speed is measured by the number of encoder pulses, mm for a given fixed time interval, Ts ✐ ✐ ✐ ✐ ✐ ✐ “book18-March-fin” — 2018/10/3 — 15:59 — page 373 — #388 ✐ ✐ 10.5 POSITION AND CURRENT SENSORS 373 CCW Disk Z B A Slit 4 4 CW Z B A LED 3 Photo TR Figure 10.32: Incremental encoder and its signal patterns Suppose that an encoder resolution is Ppr pulses per revolution (PPR) Then, the number of pulses per second is mTm Therefore, the shaft speed, Nf , in rpm is s calculated as 60 mm Nf = (rpm) Ts Ppr Note however that the encoder pulse width becomes wider and wider as the speed decreases, while Ts is fixed Finally, mm will be a fractional number at a low speed The truncation error of mm /Ts becomes serious at a low speed In short, the M -method error increases as the speed decreases On the other hand, the T -method utilizes high frequency reference pulses to measure the encoder pulse width It counts the number of pulses captured between the rising edges of encoder pulses Let ft be the frequency of reference pulses, and  mt be the number of reference pulses captured in an encoder period Then, m ft t  is the encoder period in seconds, and ft mt is the number of encoder pulses per second Therefore, the shaft speed, Nf in rpm is calculated as Nf = 60ft (rpm) Ppr mt The T -method is more accurate in a low speed region, but becomes less accurate with increasing speed, because mt is decreasing Thus, it is not appropriate for high speed measurements Fig 10.34 compares the measurement errors of the M -method and T -method An integrated method that combines both is called the M/T -method A synchronous speed measurement method was proposed in [22] 10.5.2 Resolver and R/D Converter Resolver is a type of angular position sensor that provides an absolute position [23] Thus, it is suitable for PMSM control Most resolvers consist of two parts, stator ✐ ✐ ✐ ✐ ✐ ✐ “book18-March-fin” — 2018/10/3 — 15:59 — page 374 — #389 ✐ ✐ 374 CHAPTER 10 PULSE WIDTH MODULATION AND INVERTER Encoder signal Reference pulses Frequency (b) (a) Measurement error Figure 10.33: Speed measuring methods: (a) M -method and (b) T -method M-method T-method Speed Figure 10.34: Measurement errors versus speed and rotor without a housing Furthermore, the rotor of such resolver is mounted directly on the motor shaft, thereby it requires neither bearing nor shaft coupler Since the resolver is mechanically robust, it is used in the harsh environments such as in electric vehicles Resolver is basically a transformer with a moving object, rotor It has three sets of windings: one is for field excitation and the other two are for signal detection The variable reluctance type has the exciting coil on the stator, as well as the detection coils The rotor of reluctance resolver has a curved surface without any winding On the other hand, the synchro type has an exciting coil on the rotor Fig 10.35 shows a eight pole reluctance type resolver This type of resolver is favored for its simplicity in structure and use Note that two sets of sensing coils are placed at different positions, and they are coupled to the exciting coils via a rotor as shown in Fig 10.35 (b) Then the amount of coupling between the sensing coil and the exciting coil varies when the curved rotor surface rotates In the following, we assume for convenience that the pole number is Suppose that a high frequency carrier signal, E0 sin ωrs t is injected to the exciting coil and the rotor axis is offset by θr from the magnet axis of a sensing coil, namely coil X Then the induced voltage is X = E sin ωrs t cos θr The other sensing coil, namely coil Y is shifted by π from coil X The induced voltage of coil Y is Y = E sin ωrs t sin θr ✐ ✐ ✐ ✐ ✐ ✐ “book18-March-fin” — 2018/10/3 — 15:59 — page 375 — #390 ✐ ✐ 10.5 POSITION AND CURRENT SENSORS 375 Cosine coils Sine coils Exciting coils (a) (b) Figure 10.35: Reluctance type resolver: (a) stator and (b) rotor Fig 10.36 illustrates the principle of the synchro-type resolver, and Fig 10.37 shows a sample photo It has a ring-type rotary transformer through which carrier signal is delivered to the rotor winding without a contact Normally an AC source (2∼8 V, 10∼20 kHz) is applied as a carrier to the stationary primary coil Rotary transformer Flux linkage Rotating coil Output Stationary coil + Output Input + Figure 10.36: Schematic diagram and signals of a synchro-type resolver A block diagram of a resolver-to-digital converter (RDC) is shown in Fig 10.38 The induced voltages of the two sensing coils have the same form as in the case of reluctance type Fig 10.38 shows a signal processing block diagram of a resolver to digital converter (RDC) Let ϕ be an internal variable of RDC which is set to track θr , i.e., ϕ is a digital value which is transmitted to a digital signal processor (DSP) as an angle estimate As a first step, the above two signals are multiplied by cos ϕ and sin ϕ, respectively Then, we have E0 sin θr cos ϕ sin ωrs t − E0 cos θr sin ϕ sin ωrs t = E0 sin(θr − ϕ) sin ωrs t Second, the carrier signal is eliminated by the synchronous rectification: The signal ✐ ✐ ✐ ✐ ✐ ✐ “book18-March-fin” — 2018/10/3 — 15:59 — page 376 — #391 ✐ ✐ 376 CHAPTER 10 PULSE WIDTH MODULATION AND INVERTER Rotary transformer core Sine coil Coine coil Secondary coil Field winding Figure 10.37: Synchro-type resolver demodulator + LPF PI - V/F converter + PI Pulse counter to processor (a) (b) Figure 10.38: RDC block diagram: (a) functional blocks and (b) simplified equivalent block diagram is multiplied by sin ωrs t and passed through a low pass filter: E0 sin(θr − ϕ) sin2 ωrs t = E0 sin(θr − ϕ) − cos 2ωrs t LPF ⇒ E0 sin(θr − ϕ) The voltage-to-frequency (V/F) converter generates pulses in proportion to the input voltage level Thus, a pulse counter yields digital values Note that the V/F converter together with the pulse counter functions as an integrator, and these two are symbolized by 1/s in the block diagram Then, a closed loop is completed with the resolver, as shown in Fig 10.38 (b) The PI controller is needed for loop tracking performance Note that the input to the PI controller will be regulated ultimately due to the integral action In other words, the PI controller will force sin(θr − ϕ) ≈ (θr − ϕ) → Therefore, the convergence, ϕ → θr is achieved 10.5.3 Hall Current Sensor and Current Sampling When the current flows through a semiconductor piece that is laid in a magnetic field, the carriers (holes and electrons) experience a force in an orthogonal direc- ✐ ✐ ✐ ✐ ✐ ✐ “book18-March-fin” — 2018/10/3 — 15:59 — page 377 — #392 ✐ ✐ 10.5 POSITION AND CURRENT SENSORS 377 tion That force makes a carrier concentration gradient, so that a voltage appears across the device width The voltage is proportional to the magnet field, and this effect is called the Hall effect Hall effect sensors are most widely used for current measurements, since they provide natural isolation between the measured line and the sensing circuit + + PI - - (a) (b) Figure 10.39: Hall effect current sensors: (a) voltage type and (b) current type Fig 10.39 shows two types of current sensors The voltage type directly utilizes the Hall sensor voltage, vh as a measured current But, the core material shows nonlinear characteristics as the B field increases Further, this method contains an offset error due to the hysteresis loop Thus, the voltage type current sensors have a relatively large error (typically ±1%) The current type Hall current sensor employs an extra winding and a current servo amplifier The Hall sensor is used for detecting the air gap flux If the gap field is not equal to zero, then the servo amplifier forces current, isv , to flow in the opposite direction until the air gap field is nullified Since the closed loop is formed with a PI controller, the air gap field remains zero In this case, isv is proportional to I Here, resistor, rsv , is inserted to measure isv That is, I is detected by measuring vsv = rsv isv Since the current type sensor maintains the core field equal to zero, it is less affected by the material properties of the core Therefore, it yields more accurate measurements (0.1% offset error) To implement the vector control, it is necessary to sample the current for feedback However, the current changes widely even in a single PWM interval Therefore, the sampled value is desired to be the average value of the PWM interval Fig 10.40 shows currents of symmetric and asymmetric PWMs with a current sampling point in the middle In case of the symmetric PWM, the center value represents the average value However, the middle point sampling does not give an average value in the asymmetric PWM Therefore, it is better to use the symmetric PWM with the center point sampling ✐ ✐ ✐ ✐ ✐ ✐ “book18-March-fin” — 2018/10/3 — 15:59 — page 378 — #393 ✐ ✐ 378 REFERENCES current sampling current current average current average current current sampling 1 1 (a) 1 0 1 1 1 (b) Figure 10.40: Current sampling points and the average value: (a) symmetric PWM and (b) asymmetric PWM References [1] N Mohan, T M Undeland, and W.P Robbins, Power Electronics, Converters, Applications, and Design, John Wiley & Sons Inc., 1995 [2] J Holtz, “Pulsewidth modulation a survey,” IEEE Trans Ind Electron., vol 39, no 5, pp 410−420, Oct 1992 [3] J Holtz, “Pulsewidth modulation for electronic power conversion,” Proc IEEE, vol 82, no 8, pp 1194−1214, Aug 1994 [4] D G Holmes and T A Lipo, Pulse Width Modulation for Power Converters: Principles and Practice, Wiley, 2003 [5] D Grahame Holmes and Thomas A Lipo, Pulse Width Modulation for power converters: Principles and Practice, Wiley-IEEE Press, Sep 2003 [6] A M Hava, R J.Kerkman, and T A Lipo, “A high performance generalized discontinuous PWM algorithm,” IEEE Trans on Ind Appl., vol 34, no 5, pp 1059−1071, Sep 1998 [7] J Holtz, W Lotzkat, and A Khambadkone, “On continuous control of PWM inverters in the overmodulation range including the six-step mode,” IEEE Trans Power Electron., vol 8, no 4, pp 546−553, Oct 1993 [8] A M Hava, R J Kerkman, and T A Lipo, “Carrier-based PWM-VSI overmodulation strategies: Analysis, comparison, and design,” IEEE Trans Power Electron., vol 13, no pp 674−689, July 1998 [9] G Narayanan and V T Ranganathan, “Extension of operation of space vector PWM strategies with low switching frequencies using different overmodulation algorithms,” IEEE Trans Power Electron., vol 17, no 5, pp 788−798, Sep 2002 ✐ ✐ ✐ ✐ ✐ ✐ “book18-March-fin” — 2018/10/3 — 15:59 — page 379 — #394 ✐ ✐ REFERENCES 379 [10] T G Habetler, F Profumo, M Pastorelli, and L M Tolbert, “Direct torque control of induction machines using space vector modulation,” IEEE Trans Ind Appl., vol 28, pp 1045−1053, Sep./Oct 1992 [11] H Mochikawa, T Hirose, and T Umemoto, “Overmodulation of voltage source PWM inverter,” Conf Rec JIEE-IAS Conf., pp 466−471, 1991 [12] J.-K Seok and S Sul, “A new overmodulation strategy for induction motor drive using space vector PWM,” Conf Rec IEEE APEC95, pp 211−216, Mar 1995 [13] S Bolognani and M Zigliotto, “Novel digital continuous control of SVM inverters in the overmodulation range,” IEEE Trans Ind Appl., vol 33, no 2, pp 525−530, Mar./Apr 1997 [14] Y Kwon, et al., “Six-Step Operation of PMSM With Instantaneous Current Control,” IEEE Trans on Ind Appl., vol 50, no 4, pp 2614−2625, Jul./Aug 2014 [15] W Oh, “Preventing VFD/AC Drive Induced Electrical Damage to AC Motor Bearings,” Electro Static Technology, ITW, 2006 [16] H Akagi et al., “Design and Performance of a Passive EMI Filter for Use With a Voltage-Source PWM Inverter Having Sinusoidal Output Voltage and Zero Common-Mode Voltage,” IEEE Trans Power Electron., vol 19, no 4, pp 1069−1076, July 2004 [17] R C Dodson, P D Evans, H T Yazdi, and S C Harley, “Compensating for dead time degradation of PWM inverter waveforms,” IEE Proc B, Electr Power Appl., vol 137, no 2, pp 73−81, Mar 1990 [18] D Leggate and R J Kerkman, “Pulse-based dead-time compensator for PWM voltage inverters,” IEEE Trans Ind Electron., vol 44, no 2, pp 191−197, Apr 1997 [19] J Lee, T Takeshita, and N Matsui, “Optimized stator-flux-oriented sensorless drives of IM in low-speed performance,” in Conf Rec IEEE IAS Annu Meeting, pp 250−256, 1996 [20] N Hur, K Nam, and S Won, “A two degrees of freedom current control scheme for dead-time compensation,” IEEE Trans Ind Electron., vol 47, no 3, pp 557−564, June 2000 [21] G Liu, D Wang, Y Jin, M Wang, and P Zhang, “Current-detectionindependent dead-time compensation method based on terminal voltage A/D conversion for PWM VSI,” IEEE Trans on Ind Electron., vol 64, no 10, pp 7689−7699, 2017 ✐ ✐ ✐ ✐ ✐ ✐ “book18-March-fin” — 2018/10/3 — 15:59 — page 380 — #395 ✐ ✐ 380 REFERENCES [22] T Tsuji, T Hashimoto, H Kobayashi, M Mizuochi, and K Ohnishi, “A widerange velocity measurement method for motion control,” IEEE Trans on Ind Electron., vol 56, no 2, pp 510−519, 2009 [23] C.W de Silva, Sensors and Actuators: Control Systems Intrumentation, CRC Press, 2007 Problems 10.1 Calculate vas[5] /vas[1] and vas[7] /vas[1] for six step inverter operation 10.2 Suppose that the DC link voltage is Vdc = 300 V and that the switching frequency is kHz Calculate T1 and T2 in the symmetric PWM to synthesis a voltage vector, 100∠30◦ V 10.3 Construct a DPWM1 generator using MATLAB Simulink a) Construct a block module for the third order harmonic wave of DPWM1 referring to the block code, T hirdS V M in the following figure: b) Construct a PWM generation block and a low pass filter, τ0 s+1 , where τ = 7.96 × 10−5 Obtain the filtered PWM output and its spectrum as shown below: ✐ ✐ ✐ ✐