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Handbook of Asynchronous Machine with Variable Speed


Hubert Razik

First published 2011 in Great Britain and the United States by ISTE Ltd and John Wiley amp Sons Inc Adapted and updated from La machine asynchrone agrave vitesse variable 1amp2 capteurs modegraveles controcircle et diagnostic published 2006 in France by Hermes ScienceLavoisier copy LAVOISIER 2006

Apart from any fair dealing for the purposes of research or private study or criticism or review as permitted under the Copyright Designs and Patents Act 1988 this publication may only be reproduced stored or transmitted in any form or by any means with the prior permission in writing of the publishers or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address

ISTE Ltd John Wiley amp Sons Inc 27-37 St Georgersquos Road 111 River Street London SW19 4EU Hoboken NJ 07030 UK USA

wwwistecouk wwwwileycom

copy ISTE Ltd 2011 The rights of Hubert Razik to be identified as the author of this work have been asserted by him in accordance with the Copyright Designs and Patents Act 1988 ____________________________________________________________________________________

Library of Congress Cataloging-in-Publication Data Razik Hubert Handbook of asynchronous machines with variable speed Hubert Razik p cm Includes bibliographical references and index ISBN 978-1-84821-225-1 1 Electric machinery Induction--Automatic control 2 Electric motors Induction--Automatic control 3 Electric driving Variable speed I Title TK2731R37 2011 6213136--dc22

2010048625

British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN 978-1-84821-225-1 Printed and bound in Great Britain by CPI Antony Rowe Chippenham and Eastbourne

Table of Contents

Foreword xiii

Introduction xvii

Chapter 1 Sensors and Electrical Measurements 1

11 Optical encoder 2111 Technical aspect 2112 Absolute encoder 3113 Incremental encoder 6

12 The velocity measurement 7121 Method of the frequency counter 7122 Method of the period measurement 8

13 The resolver 914 The isolated measurement 14

141 The isolated ammeter 14142 The isolated voltmeter 15

15 The numerical aspect 1516 The analog to digital converter 16

161 Principle of the flash converter 17162 Principle of the successive approximation converter 18163 The zero-order hold 18164 The multiplexer 19165 Principle of converter using slope(s) 20

17 The digital-to-analog converter 2118 The digital output 2219 The arithmetic logic unit 22110 Real time or abuse language 23111 Programming 24

v

vi Asynchronous Machine with Variable Speed

Chapter 2 Analog Numerical Control 25

21 Structure of a regulator 2522 Stability of a system 26

221 Introduction 26222 A formal criterion 27223 A graphical criterion 28224 The stability criterion 29

23 Precision of systems 30231 The initial and final value 30232 The precision of systems 31

24 Correction of systems 31241 The lag and lead corrector 32242 Other correctors 33

25 Nonlinear control 34251 First harmonic method 34252 The oscillation stability 34

26 Practical method of identification and control 35261 Broiumldarsquos method 35262 Zieglerrsquos and Nicholsrsquos method 36

27 The digital correctors 36271 Digital controller 36272 The Z-transform 37273 The Z-transform of a function 38274 Advanced Z-transform 39275 The Z-transform of a loop 40276 Some theorems 41

2761 The initial and final value 412762 The recurrence relation 412763 The fraction expansion 42

277 The Jury stability criterion 42278 Stability graphical criterion 43

2781 The bilinear transform 442782 The formal criterion 442783 The graphical criterion 45

28 Classical controllers 45281 The PID structure 46282 The PI anti-windup structure 46283 Conversion of an analog controller to a digital controller 48

2831 Approximation of the integrator 482832 Use of the bilinear transform 49

29 Disadvantages of digital controller 52291 Choice of the sampling period 52292 Noise 53

Table of Contents vii

2921 Reminder of some concepts 532922 Quantization by truncation 532923 Quantization by rounding 542924 Quantization of a product using tworsquos complement 552925 Quantization of a product by truncation 562926 The signal-to-quantization noise ratio 57

293 Cycles limits and limitations 58

Chapter 3 Models of Asynchronous Machines 59

31 The induction motor 59311 The electromagnetic torque 62312 The equivalent scheme 63

32 The squirrel cage induction motor 66321 The stator inductances 67322 The stator mutual inductances 69323 The rotor inductances 70324 The rotor mutual inductances 72325 The stator-rotor mutual inductances 73326 The rotor voltage equations 75327 The voltage and mechanical equations 75328 Reduction of the model 77

33 The static and dynamic behavior 82331 The steady state of the induction machine 82

3311 Assessment of the power 823312 Characteristics of the electromagnetic torque 84

332 Some practical characteristics 88333 The dynamics of the induction motor 92

3331 No choice of reference frame 943332 Choice of rotor reference frame 943333 Choice of stator reference frame 953334 Choice of synchronous reference frame 953335 Arrangement of variables 96

334 Some electromagnetic torque expressions 9834 Winding and induced harmonics 99

341 Principle of the rotating field 100342 The effect of currents 103

3421 Effect of unbalanced currents 1043422 Effect of non-sinusoidal currents 1063423 Effect of non-sinusoidal winding 1073424 Effect of harmonic components and winding 108

343 Choices of winding 1083431 Single-layer winding 1093432 Concentric and distributed winding 112

viii Asynchronous Machine with Variable Speed

3433 Double-layer winding 11335 Squirrel cage 115

351 The fundamental component of MMF 115352 Effect of harmonics due to slots 116353 Effect of harmonic components on the torque 117

36 Variation in air-gap permeance 118361 Effect of the rotor and stator slots 119362 Effect of magnetic saturation 120363 Effect of eccentricity 120

37 Noise and vibrations 121371 The first harmonics approach 122372 Choice of the number of rotor bars in squirrel-cage induction 124

38 Influence of rotor frequency 125381 One ideal rotor bar at null frequency 126

3811 Aspects of the rotor bar 1263812 The aspect of the isthmus 1273813 Synthesis 128

382 One ideal rotor bar at non-null frequency 1283821 The aspect of inductance 1293822 The aspect of resistance 1293823 Synthesis 129

39 Thermal behavior 130391 Insulation classes 131392 Static thermal model 132393 A dynamic hybrid thermal model 134

Chapter 4 Speed Variation 137

41 Cases of multiphase machines 137411 Motors with a high number of phases 138

4111 Type-I motors 1384112 Type-II motors 140

412 Interactions between harmonics 141413 Three-phase induction machine 144

4131 Three-phase model 1444132 Application in another frame 146

414 Five-phase induction machine 151415 Double-star induction motor 155

4151 Six-phase induction motor version 1 1564152 Six-phase induction motor version 2 161

42 Control of asynchronous motors 164421 The basic environment 166422 Scalar control Vf 167423 Vector control Vf 169

Table of Contents ix

4231 A classical approach 1714232 Variant without a speed sensor 172

424 Direct torque control (DTC) 1754241 The concept 1784242 Strategy of vector choice 1814243 Torque ripple 1824244 Three-level inverter 1844245 Influence of voltage limitation 1894246 The DTC-SVM approach 1894247 Prediction of the torque ripple 1924248 Application to a five-phase induction motor 193

425 Direct self-control approach (DSC) 194426 Vector control FOC 197

4261 Application to three-phase induction motors 2004262 Application to five-phase induction motors 2034263 Application to six-phase induction motors 207

427 Control without a position sensor 208428 Exploitation of natural asymmetries 209

4281 The static and dynamic eccentricity 2094282 The rotor slots effect 2104283 The magnetic saturation effect 2114284 The estimation of the velocity 2114285 Spectrum estimation 213

429 Estimation by high-frequency injection 21343 Identification of parameter aspects 216

431 Classical methods 2164311 The step method 2174312 Empirical method 219

432 Generic methods 2214321 Principle of the method based on the model 2214322 The gradient method 2224323 The Newton-Raphson method 2224324 The Marquardt-Levenberg method 2224325 The genetic algorithm 2234326 Identification of electrical and mechanical parameters 225

433 Conclusion 22644 Voltage inverter converters 227

441 Inverters using the pulse width modulation technique 2274411 Two-level inverter 2284412 Over-modulation 2324413 Three levels inverter 2334414 Three-level inverter using clamped capacitor 2354415 Four-level inverter 236

x Asynchronous Machine with Variable Speed

4416 Multi-levels inverter 239442 The inverters using the space vector modulation 243

4421 Application to the three-phase induction motor 2454422 Application to the five-phase induction motor 2494423 Application to the six-phase induction motor 2534424 Multilevel aspect 257

443 The matrix converter 2614431 Direct matrix converter 2634432 Indirect matrix converter 266

45 Rectifiers based on the PWM 268451 Two-level rectifier 268452 Three-level rectifier 270

Chapter 5 Tools of Fuzzy Logic 273

51 Preamble 27352 Introduction 27453 Fuzzy logic 275

531 Definitions and norms 275532 Some variants 276533 T -norm and T -conorm 276534 Membership functions 277535 Inference engine 278536 Defuzzification 280

54 Fuzzy logic controller 28055 Fuzzy and adaptive PI 284

551 Examples of programs to calculate a fuzzy surface 2865511 The layout of a fuzzy surface 2865512 Routine of a PI-fuzzy controller 287

552 Examples of application 288553 Examples of simulation results 289

5531 Controller based on a fuzzy PI 2895532 A controller based on a fuzzy PID 291

554 Examples of tables of rules 29156 Conclusion 295

Chapter 6 Diagnostics and Signals Pointing to a Change 297

61 Signals and measurements 29862 Defects 299

621 Problems with broken bars 300622 Problems in the stator 302623 Problems due to eccentricities 304624 Problems due to speed ripples 307625 Problems with ball bearings 307

Table of Contents xi

63 Analysis of signals 309631 Fast Fourier transform analysis of the stator current 309632 Fast Fourier transform 309633 Discrete fast Fourier transform 311634 Windows functions 312

6341 The Hamming function 3136342 The Hanning function 3136343 The Blackmann function 3136344 The Bartlett function 3136345 The Kaiser function 313

635 Sliding discrete fast Fourier transform 3146351 Zoom effect 316

64 Some considerations regarding broken bar defects 317641 Model of the induction motor 317642 Inherent frequencies in the broken bar defect 318643 Evaluation of the magnitude of the left line 320644 Equivalent model in the steady state 320

65 Evaluation of the severity of broken bars 322651 Some spectra results 322652 Evaluation of the severity of broken bars 326

6521 Analytical approach 3266522 Artificial intelligence approach 3286523 Self-extraction of signatures an application of PSO 330

653 Wireless communication 335

Exercise No 1 Fuzzy Logic 337

11 Adaptive k and ki coefficients in function of the error 33712 Adaptive k and ki coefficients in function of the error and

its derivative 33813 Answers 339

Exercise No 2 The Stator Defect 345

21 Equations of the induction motor under stator defect 34722 Torque ripple due to a stator defect 34823 Fault current estimation 34924 Schematic model of three-phase induction motor under a stator

defect 35025 Answers 351

Exercise No 3 The Control of Five-Phase Induction Motors 357

31 The five-phase system 35832 Distribution of active currents 359

xii Asynchronous Machine with Variable Speed

33 A model for control 36234 Answers 364

Exercise No 4 The Control of Serial Connected Induction Motors 373

41 Study about the serial connection of two five-phase inductionmotors 374

42 Study on the serial connection of several seven-phaseinduction motors 375

43 Study on the serial connection of multi-phase induction motors 37744 Answers 378

Exercise No 5 Fault Detection of a Three-Phase Voltage InverterConverter 385

51 A conducting fault 38652 Fault detector 38753 Monitoring of the DC component 38954 Answers 390

Appendix Some Mathematical Expressions 393A1 Laplace transforms 393A2 Z transforms 394A3 W transforms 395A4 Common expressions 395A5 Trigonometric identities 395

A51 Addition 396A52 Sum identities 396A53 Product identities 397A54 The product 397A55 Sum of sinus and cosinus 397

A6 Mathematical series 398A7 Greek numbers 398

Bibliography 399

Index 407

Foreword

The asynchronous machine also known as the induction machine has been anindustry workhorse for more than 120 years The first rotating machine to work usingalternating current its advent started a new chapter in the evolution of our societywith the propagation of long-distance electrical lines ensuring a large distributionof electrical energy and a systematic replacement of steam-powered machines withelectrical ones This expansion still continues today as much in industrial applicationsand transport as home automation The induction machine is a part of our everydaylives in the broadest sense it is present in all Western households and emerging anddeveloping countries strongly depend on its use

Naturally since its first appearance the induction motor has undergonerevolutions and changes through the development of techniques and progress inthe field of magnetic or dielectric materials This can easily be seen from the sizeof the machines with a power-to-mass ratio that has considerably evolved and canoperate at variable speeds which makes it a broad-spectrum actuator The intrinsicperformances of this machine have considerably progressed first through a moreoptimized construction of its structure and by the use of better quality materials butmostly from the set-up of an adapted electronic environment allowing the optimizedmanagement of energy processing

To this end this book by Professor Hubert Razik provides an eloquent and originaltestimonial by discussing in an educational and rational way all the constituentelements of a variable speed drive at the base of the asynchronous machine In thisway we can easily navigate between methods and tools principles and rules in orderto cover all the angles of modern control of this machine by making the link betweensignal processing control and diagnostics

This book is split into six chapters and accompanied by a bibliography consistingof the greatest contributions to the field

xiv Asynchronous Machine with Variable Speed

Chapter 1 covers sensors and electrical measurements and brings to light in aconcise way the notions of coding and restitution of information that can be read by acontroller

Chapter 2 clarifies the fundamental elements for the control of the systems througha review of the representation and synthesis of corrector tools in the context of electricmotor control (identification nonlinear control stability data quantization etc)

Chapter 3 covers the modeling of induction motors with a gradual evolutionfrom simple models valid in nominal cases up to taking into account very unusualphenomena such as magnetic saturation rotor eccentricity or thermal incidence Itcontains a very explicit and relevant contribution

Chapter 4 the longest of all splits speed variation into its different aspects Itdiscusses scalar then vector control then passes through direct torque control andcontrols without mechanical sensors This part is subject to several declinations beit over the phases of the motor (five- or six-phase motor) or even over the numberof inverter output levels (three level four or multi-level) This chapter is even morevaluable since it is based on the real know-how of the author It is the result of longand meticulous practice searching for better solutions

Chapter 5 covers fuzzy logic by simply setting down the meets and bounds ofthis approach and its applications for control of the asynchronous motor The basis iscarefully shown allowing quick and relevant applications for the given potentials

The final chapter ndash Chapter 6 ndash covers the delicate problem of diagnostics ofelectric motors which is currently the subject of ever more numerous and openresearch Professor Razikrsquos contribution to the field is very well developed He hascompiled an inventory of different problems and proposed methods of detectingand identifying defects Ultimately machine monitoring is proposed via the use ofwireless communication

The authorrsquos educational background is present throughout this book Forcompletion the book proposes a series of corrected exercises that allow theassimilation of the concepts needed for this discipline

The operation and diagnostics of electric motors such as induction motorsremains a very complex discipline since it brings into play conversion and processingphenomena of electrical energy with a distinct nonlinear character This is an exampleof an application that joins significant know-how and we thank Professor Razikfor this accessible work demystifying the laws of control and diagnostics of thesedevices

Foreword xv

This book will be useful for postgraduate students engineering students theirtutors and other young researchers starting a career in this field where much progressis awaited The interconnection of the notions covered make this a complete book forthe field of electric motors

Professor Maurice FADELENSEEIHT

Deputy Director of LAPLACE

xvi

Introduction

This book is aimed as much at students and postgraduates whether working inresearch or not as it is for tutors and engineers

It offers students knowledge that goes from the basics to a much more advancedlevel the broad spectrum offered to readers allows easy comprehension of morespecialized books Exercises are included at the end of the book

Engineers will be able to find elements corresponding to their needs whetheracademic or providing the in-depth knowledge that will give them the tools for broaderthinking

This book covers the asynchronous machine in its immediate environment Itstarted out as a reflection on the electromagnetic converter whose integration inindustrial environments is taking on an increasingly important role Historically thismotor was used in a chain of a fixed speed process Now it is increasingly beingintegrated in variable speed processes It is for this reason it seemed useful if notnecessary to write a book covering the different aspects from the motor itself passingthrough control and finishing with the diagnostic aspect In fact asynchronousmotors are today in an industry where variation of speed and reliability are requiredFor sensitive systems we should always know the state of the conversion chain andinform the operator of the appearance of any anomaly and its severity

All these approaches are described in a rigorous manner and numerous referenceswill allow readers to further develop their comprehension and knowledge

I would like to thank my wife and my daughters for their loving support throughoutthis endeavour

xviii

Chapter 1

Sensors and Electrical Measurements

Measurement is an obligatory phase that we cannot avoid In fact in allelectromagnetic processes ndash whether at a variable or constant speed ndash instructedmeasurements are carried out These measurements are compared to those taken tocorrect the process so that they conform to the desired values This phase is used as acontrol and in some cases is incorporated into regulations Such a system can generallybe split into four parts as we see in Figure 11 which are

(1) instruction(s) and protection these different parts allow us to take into accountthe needs and constraints

(2) control and regulation(s) this is dedicated to control loop that can consist ofseveral interleaved loops

(3) electric supply this is an interface allowing application of the desired voltagesto the motor in alternating current

(4) electric motor this converts electrical energy into mechanical energy

Between these parts connections are necessary either to transmit information(instructions between parts) or to transmit energy from a source of power towards theelectric motor For this analog lines such as coaxial or fiber optics are used for powerFor instruction(s) and regulation(s) digital connections are preferred It will benecessary to represent the numbers in digital form which will lead to a quantizationthat will be represented in binary form The choice is not without consequencesince this noise of quantization will alter the performance of the system (closed loopprocess) Nevertheless we can reasonably say that a representation between 12 and16 bits is acceptable for the majority of microcontrollers for regulation

1

2 Asynchronous Machine with Variable Speed

Figure 11 Flow diagram representing the process with regulation

Let us approach the measurement aspect of the angular position of a mechanicalshaft then its rotation speed For this two categories of coders exist one is opticaland the other is electromagnetic

11 Optical encoder

Different types of optical coders exist Certain ones have many advantages andothers fewer along with the inherent constraints of the choice of technology used Forthis reason we will first cover the measurement of the absolute position and secondthe measurement of the relative position of the mechanical shaft

111 Technical aspect

What is an optical encoder

An optical encoder is a system that allows us to provide logic level transitions 1and 0 depending on the mechanical rotation It consists of a disc with cuts similar to amultitude of lines that allow light to pass though In this way a beam of light will excitea receiver which will provide either level 1 or level 0 transitions after transformationof the collected signals Figure 12 is a synoptic diagram of an optical encoder wherewe can discern the disc itself with it axis of revolution light emitter receiver and spaceletting the beam of light pass through

The optical encoder is free of restraints since mechanical disturbances interferewith its lifespan and its performance For this reason overspeeds are not allowed sincethis will destroy the cuttings as well as causing mechanical vibrations that will vibratethe disc bringing it into involuntary contact with the receiver or the emitter In bothcases the defect encountered is irreversible since the cutting is affected by an anomalyBefore mentioning some of the defects of encoders let us now look at the differenttypes of optical encoders starting with the absolute encoder

Sensors and Electrical Measurements 3

Figure 12 Representative diagram of an optical encoder

112 Absolute encoder

The advantage of this encoder is to provide precise information of the angularposition since during the operation of the sensor it is the real position of the shaftthat is provided Nevertheless for problems of cutting quality only the Gray codeis recommended In fact there is one and only one transition of logic state 0 to 1or 1 to 0 This is not the case for classic binary code For the purposes of structuralcomparison Figure 13(a) shows the said relative absolute encoder and Figure 13(b)the Gray encoder The angular representation is given in Table 11

(a) The natural code (b) The Gray code

Figure 13 Different types of absolute encoder (a) the natural or binary code and (b) theGray encoder whose angular resolution is Δθ = 360˚23


These two coders have a resolution of eight positions This means that we will onlyhave the position of the measurable shaft in the sectors that have an angular definitionof 2π8 radians or 45˚ For this reason Table 12 lists all the values for this type ofcoder as well as the codes and angular sectors For illustrative purposes code No 2corresponds to an angular position of the shaft of 112˚ plusmn 225˚ The correspondingGray code is 010 and the natural binary code is 011 It is obvious that the Gray codesuch as it is does not give the direct correspondence of the angular position Also weare brought to convert the Gray code to a natural binary code This can easily be doneby using logic gates whose function is ldquoor exclusiverdquo (see Figure 14) represented bythe symbol oplus

Number of bits Number of sectors Angle in radian Angle in degrees

1 2 3141592 180

4 16 0392699 225

8 256 0024543 140625

10 1024 0006135 0351562

12 4096 0001534 0008789

Table 11 Table illustrating the angular resolution of an optical encoder

Decimal code Binary code Gray code Angular sector

0 000 000 0ndash45˚

1 001 001 45ndash90˚

2 010 011 90ndash135˚

3 011 010 135ndash180˚

4 100 110 180ndash225˚

5 101 111 225ndash270˚

6 110 101 270ndash315˚

7 111 100 315ndash360˚

d2 d1 d0 g2 g1 g0

Table 12 Table illustrating the conversion of the Gray code to a natural one

For a number n of bits

ndash dn = gn

ndash di = gn oplus gnminus1 oplus gnminus2 middot middot middot gi so that i lt n


Figure 14 shows the use of these logic gates

Figure 14 Conversion of the Gray code to a natural code

In the case of a cutting fault the absolute encoder using the natural code will beinterrupted Figure 15 shows two examples of interruptions

Case 1 (the code changes from 111 to010 then 000) The angular sectors are

therefore 315ndash360˚90ndash135˚ then 0ndash45˚



Figure 15 Absolute natural encoder with two cutting faults

In any event we note a failure in the transitions For this the Gray code will befavored and only one transition at a time can occur Therefore we can overcome thesector imperfections due to imprecise cutting through an adapted digital system It is


to be noted that with an increasing number of tracks the angular resolution increasesin the same way leading to a significant number of light emitters and receivers nearthe disc In order for an encoder to be of a smaller size the collection of componentsmust coexist in a confined space

113 Incremental encoder

A relative encoder is an encoder that cannot provide the exact position of the shaftduring the operation of the device In fact it provides a position relative to that of itsstarting point As a result its simplicity comes down to the requirement for one cuttrack instead of a huge number as in the case of an absolute encoder However we adda second track shifted by 90˚ electric in such a way as to discern the rotation of theshaft As we have shown in Figure 16 these two tracks are shifted in relation to eachother but with a surprising allowance (manufacturerrsquos data) Track B does not requireparticular attention during cutting since it is only dedicated to obtaining the directionof rotation A third track commonly called zero generates one pulse per revolutionThis allows us to lock a position

Figure 16 The relative or incremental encoder

The angular resolution is therefore inversely proportional to the number of pointsper revolution In order to obtain the angular movement it is necessary to use acountercount-down counter that track A will excite Obtaining the direction ofrotation will be achieved using an RS edge latch Track B connected to the input ofthe clock will change output Q of the latch to the value of track A connected to thedata input at each rising edge of B

In order to measure the position we can opt for several methods that all haveadvantages and disadvantages We can proceed as follows

(1) s = A simple coding on level A


(2) s =uarr A simple coding on rising edges of A

(3) s =uarr A and uarr B double coding on rising edges

(4) s =uarr A and darr A uarr B and darr B quadruple encoding on rising and falling edges

The HCTL 2000 is one of the circuits allowing the measurement of positionIt is very versatile in its use while requiring the minimum number of componentsThis integrated circuit (Hewlett Packard) is a decodercounter interface for a relativeencoder We can also refer to Texas Instrumentsrsquo THCT 12316 circuit which presentsa high quality interface allowing us to decode three simultaneous tracks Each trackis compatible with the basic TCHT 2000 circuit It detects the direction of rotationmeasures the position in simple double or quadruple coding either length of thepulses or frequency and provides the result on a number of 16 bits This circuit iscommercialized with a PLCC casing with 68 pins The monotrack version is calledTHCT 12016 (dual-in-line 28 PIN casing)

12 The velocity measurement

The measurement of rotation speed can be done in three ways The first consistsof calculating the difference in position between two precise instants and dividingthe result by the measurement time interval The second calls upon the method of afrequency indicator the third the method of a period meter These two last methodshave the advantage of directly providing information relative to the speed of rotationon a while number of 16 bits

121 Method of the frequency counter

The method of the frequency counter is based on counting the number pulses ina given time frame From a precise time base we count the number of pulses Nappearing from the encoder over an interval Tm In this way the frequency of thesignal received has for a relationship

F = NTm (11)

If the number of pulses per revolution of the encoder is Ni then the rotationfrequency Fr in Hz is

Fr = NNiTm (12)

We would not be able to finish this approach to measurement without concerningourselves with the precision of the measurement In fact if we commit a counting


error of a pulse then the measurement error of the frequency of rotation is written

Error(N) = Fr(N + 1)minus Fr(N) =1

NiTm(13)

or in percent

Error(N) =100Fr(N + 1)minus Fr(N)

Fr(N)=

100N

(14)

As a result the higher the number of pulses counted the lower the error on thefrequency of rotation It is convenient to change the measurement method when thisreaches excessive values Now we will look at the method of the measuring the period

122 Method of the period measurement

As with the method of frequency counting period measurement is based oncounting the number of pulses in a given time frame The difference resides in thesignal to be analyzed It concerns taking the signal of track A as a counting frameThe precise time base is a very high frequency signal that stable in the time Fhf We therefore count the number of these high frequency pulses N appearing duringa measurement time interval Tm linked to the speed of rotation of the shaft In thisway the frequency observed is

F = 1Tm (15)

If the number of pulses per rotation of the encoder is Ni the rotation frequency Fr

in Hz is reduced to

Fr = FhfNiN (16)

As a result the lower the rotation speed the higher the number of pulses Thismethod is therefore adapted for the measurement of low rotation speeds Once againit is necessary to concern ourselves with measurement error In the same way as beforewe will proceed to a variation of a unit over the number of pulses We therefore get

Error(N) = Fr(N + 1)minus Fr(N) =Fhf

Ni

minus1N(N + 1)

(17)

or in percentage


Fr(N)=minus100N + 1

(18)


0

10

20

30

40

50

60

70

0 10 20 30 40 50 60

N

Rotation speed

Frequency counter

Speed limitation

Transitionzone

Period measurement

Limitation of N

Figure 17 The number of pulses counted as a function of rotation frequency

The result is the same as with the frequency counter the higher the number ofpulses counted the lower the relative error

Figure 17 contains numerous data relative to precision and limits The countingof N is limited by the number of bits of the counter (Nmax) The frequency ofthe reproduced speed is also limited (Fmax) This figure illustrates a remarkablephenomenon created by the transition zone This zone defines the operating modegiving the best precision We therefore find that at low speed the method of periodcounting is advised At high speeds it is the method of frequency counting that isrecommended There is therefore a critical frequency that if the electronics permitwill change the measurement mode of the rotation speed

13 The resolver

A relevant question here is what is a resolver

A resolver is an analog sensor of angular position Its make-up is similar to thatof a rotary transformer Traditionally the measurement of speed and position is doneby the use of an optical encoder Nevertheless this technique is not without its limitsThe precision of the measurement is inversely proportional to the number of lines perrotation The most frequently used have between 500 and 4096 lines per revolutionOne of the disadvantages is related to the cost which is exponential depending on thenumber of lines and therefore on the angular resolution Its use requires mechanicalprecautions It is necessary to avoid mechanical disturbances (vibrations) that lead to


premature wearing of the encoder and implicates the use of specific coupling (and istherefore expensive) that allows mechanical defaults It is also necessary to ensure thatthe maximum rotation speed is not exceeded due to the risk of irreversible alterationsto the lines leading to irreparable distortion of the measurements We must not forgetthat that an encoder whichever it is requires an electronic conversion interface

Before going into more detail on the operation and the interface necessary for thereconstruction of the angular position we can set out some properties of this device

A resolver is a rotary transformer whose rotor and stator are both coiled (seeFigure 18)

Figure 18 Principle diagram of a resolver

There is an advantage in the external aspect of the resolver In effect this isjoined to the electric motor If this is explosion-proof then the resolver will alsobe so The axis of the said rotor will be coupled with the machine whose speed wewant to measure This will be excited by sinusoidal voltage of constant amplitude andfrequency (for example Vexc = V cos(ωt) with V = 7 v effective and f = 5 kHz)

As shown in Figure 19 the analog-digital converter has three basic components

ndash a phase comparator

ndash a ldquolooprdquo filter

ndash a voltage controlled oscillator

The phase comparator consists of a ldquocosinusoidalrdquo multiplier a sinusoidalmultiplier a subtractor and a demodulator The basis function of the voltagecontrolled oscillator (VCO) is a digital integrator

The voltage v(t) from

v(t) = V sin θ cos(ωt) cosφminus V cos θ cos(ωt) sinφ (19)



Hubert Razik







2010048625


Table of Contents

Foreword xiii

Introduction xvii









v






































































































Bibliography 399

Index 407

Foreword














Foreword xv




xvi

Introduction







xviii

Chapter 1








1




11 Optical encoder















1 2 3141592 180

4 16 0392699 225

8 256 0024543 140625

10 1024 0006135 0351562

12 4096 0001534 0008789



0 000 000 0ndash45˚

1 001 001 45ndash90˚

2 010 011 90ndash135˚

3 011 010 135ndash180˚

4 100 110 180ndash225˚

5 101 111 225ndash270˚

6 110 101 270ndash315˚

7 111 100 315ndash360˚

d2 d1 d0 g2 g1 g0



ndash dn = gn





























F = NTm (11)


Fr = NNiTm (12)





NiTm(13)

or in percent


Fr(N)=

100N

(14)




F = 1Tm (15)


in Hz is reduced to

Fr = FhfNiN (16)



Ni

minus1N(N + 1)

(17)

or in percentage


Fr(N)=minus100N + 1

(18)


0

10

20

30

40

50

60

70

0 10 20 30 40 50 60

N

Rotation speed

Frequency counter

Speed limitation

Transitionzone

Period measurement

Limitation of N




13 The resolver

















Hubert Razik







2010048625


Table of Contents

Foreword xiii

Introduction xvii









v






































































































Bibliography 399

Index 407

Foreword














Foreword xv




xvi

Introduction







xviii

Chapter 1








1




11 Optical encoder















1 2 3141592 180

4 16 0392699 225

8 256 0024543 140625

10 1024 0006135 0351562

12 4096 0001534 0008789



0 000 000 0ndash45˚

1 001 001 45ndash90˚

2 010 011 90ndash135˚

3 011 010 135ndash180˚

4 100 110 180ndash225˚

5 101 111 225ndash270˚

6 110 101 270ndash315˚

7 111 100 315ndash360˚

d2 d1 d0 g2 g1 g0



ndash dn = gn





























F = NTm (11)


Fr = NNiTm (12)





NiTm(13)

or in percent


Fr(N)=

100N

(14)




F = 1Tm (15)


in Hz is reduced to

Fr = FhfNiN (16)



Ni

minus1N(N + 1)

(17)

or in percentage


Fr(N)=minus100N + 1

(18)


0

10

20

30

40

50

60

70

0 10 20 30 40 50 60

N

Rotation speed

Frequency counter

Speed limitation

Transitionzone

Period measurement

Limitation of N




13 The resolver






















2010048625


Table of Contents

Foreword xiii

Introduction xvii









v






































































































Bibliography 399

Index 407

Foreword














Foreword xv




xvi

Introduction







xviii

Chapter 1








1




11 Optical encoder















1 2 3141592 180

4 16 0392699 225

8 256 0024543 140625

10 1024 0006135 0351562

12 4096 0001534 0008789



0 000 000 0ndash45˚

1 001 001 45ndash90˚

2 010 011 90ndash135˚

3 011 010 135ndash180˚

4 100 110 180ndash225˚

5 101 111 225ndash270˚

6 110 101 270ndash315˚

7 111 100 315ndash360˚

d2 d1 d0 g2 g1 g0



ndash dn = gn





























F = NTm (11)


Fr = NNiTm (12)





NiTm(13)

or in percent


Fr(N)=

100N

(14)




F = 1Tm (15)


in Hz is reduced to

Fr = FhfNiN (16)



Ni

minus1N(N + 1)

(17)

or in percentage


Fr(N)=minus100N + 1

(18)


0

10

20

30

40

50

60

70

0 10 20 30 40 50 60

N

Rotation speed

Frequency counter

Speed limitation

Transitionzone

Period measurement

Limitation of N




13 The resolver
















Table of Contents

Foreword xiii

Introduction xvii









v






































































































Bibliography 399

Index 407

Foreword














Foreword xv




xvi

Introduction







xviii

Chapter 1








1




11 Optical encoder















1 2 3141592 180

4 16 0392699 225

8 256 0024543 140625

10 1024 0006135 0351562

12 4096 0001534 0008789



0 000 000 0ndash45˚

1 001 001 45ndash90˚

2 010 011 90ndash135˚

3 011 010 135ndash180˚

4 100 110 180ndash225˚

5 101 111 225ndash270˚

6 110 101 270ndash315˚

7 111 100 315ndash360˚

d2 d1 d0 g2 g1 g0



ndash dn = gn





























F = NTm (11)


Fr = NNiTm (12)





NiTm(13)

or in percent


Fr(N)=

100N

(14)




F = 1Tm (15)


in Hz is reduced to

Fr = FhfNiN (16)



Ni

minus1N(N + 1)

(17)

or in percentage


Fr(N)=minus100N + 1

(18)


0

10

20

30

40

50

60

70

0 10 20 30 40 50 60

N

Rotation speed

Frequency counter

Speed limitation

Transitionzone

Period measurement

Limitation of N




13 The resolver





















































































































Bibliography 399

Index 407

Foreword














Foreword xv




xvi

Introduction







xviii

Chapter 1








1




11 Optical encoder















1 2 3141592 180

4 16 0392699 225

8 256 0024543 140625

10 1024 0006135 0351562

12 4096 0001534 0008789



0 000 000 0ndash45˚

1 001 001 45ndash90˚

2 010 011 90ndash135˚

3 011 010 135ndash180˚

4 100 110 180ndash225˚

5 101 111 225ndash270˚

6 110 101 270ndash315˚

7 111 100 315ndash360˚

d2 d1 d0 g2 g1 g0



ndash dn = gn





























F = NTm (11)


Fr = NNiTm (12)





NiTm(13)

or in percent


Fr(N)=

100N

(14)




F = 1Tm (15)


in Hz is reduced to

Fr = FhfNiN (16)



Ni

minus1N(N + 1)

(17)

or in percentage


Fr(N)=minus100N + 1

(18)


0

10

20

30

40

50

60

70

0 10 20 30 40 50 60

N

Rotation speed

Frequency counter

Speed limitation

Transitionzone

Period measurement

Limitation of N




13 The resolver






































































































Bibliography 399

Index 407

Foreword














Foreword xv




xvi

Introduction







xviii

Chapter 1








1




11 Optical encoder















1 2 3141592 180

4 16 0392699 225

8 256 0024543 140625

10 1024 0006135 0351562

12 4096 0001534 0008789



0 000 000 0ndash45˚

1 001 001 45ndash90˚

2 010 011 90ndash135˚

3 011 010 135ndash180˚

4 100 110 180ndash225˚

5 101 111 225ndash270˚

6 110 101 270ndash315˚

7 111 100 315ndash360˚

d2 d1 d0 g2 g1 g0



ndash dn = gn





























F = NTm (11)


Fr = NNiTm (12)





NiTm(13)

or in percent


Fr(N)=

100N

(14)




F = 1Tm (15)


in Hz is reduced to

Fr = FhfNiN (16)



Ni

minus1N(N + 1)

(17)

or in percentage


Fr(N)=minus100N + 1

(18)


0

10

20

30

40

50

60

70

0 10 20 30 40 50 60

N

Rotation speed

Frequency counter

Speed limitation

Transitionzone

Period measurement

Limitation of N




13 The resolver
























































































Bibliography 399

Index 407

Foreword














Foreword xv




xvi

Introduction







xviii

Chapter 1








1




11 Optical encoder















1 2 3141592 180

4 16 0392699 225

8 256 0024543 140625

10 1024 0006135 0351562

12 4096 0001534 0008789



0 000 000 0ndash45˚

1 001 001 45ndash90˚

2 010 011 90ndash135˚

3 011 010 135ndash180˚

4 100 110 180ndash225˚

5 101 111 225ndash270˚

6 110 101 270ndash315˚

7 111 100 315ndash360˚

d2 d1 d0 g2 g1 g0



ndash dn = gn





























F = NTm (11)


Fr = NNiTm (12)





NiTm(13)

or in percent


Fr(N)=

100N

(14)




F = 1Tm (15)


in Hz is reduced to

Fr = FhfNiN (16)



Ni

minus1N(N + 1)

(17)

or in percentage


Fr(N)=minus100N + 1

(18)


0

10

20

30

40

50

60

70

0 10 20 30 40 50 60

N

Rotation speed

Frequency counter

Speed limitation

Transitionzone

Period measurement

Limitation of N




13 The resolver







































































Bibliography 399

Index 407

Foreword














Foreword xv




xvi

Introduction







xviii

Chapter 1








1




11 Optical encoder















1 2 3141592 180

4 16 0392699 225

8 256 0024543 140625

10 1024 0006135 0351562

12 4096 0001534 0008789



0 000 000 0ndash45˚

1 001 001 45ndash90˚

2 010 011 90ndash135˚

3 011 010 135ndash180˚

4 100 110 180ndash225˚

5 101 111 225ndash270˚

6 110 101 270ndash315˚

7 111 100 315ndash360˚

d2 d1 d0 g2 g1 g0



ndash dn = gn





























F = NTm (11)


Fr = NNiTm (12)





NiTm(13)

or in percent


Fr(N)=

100N

(14)




F = 1Tm (15)


in Hz is reduced to

Fr = FhfNiN (16)



Ni

minus1N(N + 1)

(17)

or in percentage


Fr(N)=minus100N + 1

(18)


0

10

20

30

40

50

60

70

0 10 20 30 40 50 60

N

Rotation speed

Frequency counter

Speed limitation

Transitionzone

Period measurement

Limitation of N




13 The resolver



























































Bibliography 399

Index 407

Foreword














Foreword xv




xvi

Introduction







xviii

Chapter 1








1




11 Optical encoder















1 2 3141592 180

4 16 0392699 225

8 256 0024543 140625

10 1024 0006135 0351562

12 4096 0001534 0008789



0 000 000 0ndash45˚

1 001 001 45ndash90˚

2 010 011 90ndash135˚

3 011 010 135ndash180˚

4 100 110 180ndash225˚

5 101 111 225ndash270˚

6 110 101 270ndash315˚

7 111 100 315ndash360˚

d2 d1 d0 g2 g1 g0



ndash dn = gn





























F = NTm (11)


Fr = NNiTm (12)





NiTm(13)

or in percent


Fr(N)=

100N

(14)




F = 1Tm (15)


in Hz is reduced to

Fr = FhfNiN (16)



Ni

minus1N(N + 1)

(17)

or in percentage


Fr(N)=minus100N + 1

(18)


0

10

20

30

40

50

60

70

0 10 20 30 40 50 60

N

Rotation speed

Frequency counter

Speed limitation

Transitionzone

Period measurement

Limitation of N




13 The resolver











































Bibliography 399

Index 407

Foreword














Foreword xv




xvi

Introduction







xviii

Chapter 1








1




11 Optical encoder















1 2 3141592 180

4 16 0392699 225

8 256 0024543 140625

10 1024 0006135 0351562

12 4096 0001534 0008789



0 000 000 0ndash45˚

1 001 001 45ndash90˚

2 010 011 90ndash135˚

3 011 010 135ndash180˚

4 100 110 180ndash225˚

5 101 111 225ndash270˚

6 110 101 270ndash315˚

7 111 100 315ndash360˚

d2 d1 d0 g2 g1 g0



ndash dn = gn





























F = NTm (11)


Fr = NNiTm (12)





NiTm(13)

or in percent


Fr(N)=

100N

(14)




F = 1Tm (15)


in Hz is reduced to

Fr = FhfNiN (16)



Ni

minus1N(N + 1)

(17)

or in percentage


Fr(N)=minus100N + 1

(18)


0

10

20

30

40

50

60

70

0 10 20 30 40 50 60

N

Rotation speed

Frequency counter

Speed limitation

Transitionzone

Period measurement

Limitation of N




13 The resolver



























Bibliography 399

Index 407

Foreword














Foreword xv




xvi

Introduction







xviii

Chapter 1








1




11 Optical encoder















1 2 3141592 180

4 16 0392699 225

8 256 0024543 140625

10 1024 0006135 0351562

12 4096 0001534 0008789



0 000 000 0ndash45˚

1 001 001 45ndash90˚

2 010 011 90ndash135˚

3 011 010 135ndash180˚

4 100 110 180ndash225˚

5 101 111 225ndash270˚

6 110 101 270ndash315˚

7 111 100 315ndash360˚

d2 d1 d0 g2 g1 g0



ndash dn = gn





























F = NTm (11)


Fr = NNiTm (12)





NiTm(13)

or in percent


Fr(N)=

100N

(14)




F = 1Tm (15)


in Hz is reduced to

Fr = FhfNiN (16)



Ni

minus1N(N + 1)

(17)

or in percentage


Fr(N)=minus100N + 1

(18)


0

10

20

30

40

50

60

70

0 10 20 30 40 50 60

N

Rotation speed

Frequency counter

Speed limitation

Transitionzone

Period measurement

Limitation of N




13 The resolver
















Foreword














Foreword xv




xvi

Introduction







xviii

Chapter 1








1




11 Optical encoder















1 2 3141592 180

4 16 0392699 225

8 256 0024543 140625

10 1024 0006135 0351562

12 4096 0001534 0008789



0 000 000 0ndash45˚

1 001 001 45ndash90˚

2 010 011 90ndash135˚

3 011 010 135ndash180˚

4 100 110 180ndash225˚

5 101 111 225ndash270˚

6 110 101 270ndash315˚

7 111 100 315ndash360˚

d2 d1 d0 g2 g1 g0



ndash dn = gn





























F = NTm (11)


Fr = NNiTm (12)





NiTm(13)

or in percent


Fr(N)=

100N

(14)




F = 1Tm (15)


in Hz is reduced to

Fr = FhfNiN (16)



Ni

minus1N(N + 1)

(17)

or in percentage


Fr(N)=minus100N + 1

(18)


0

10

20

30

40

50

60

70

0 10 20 30 40 50 60

N

Rotation speed

Frequency counter

Speed limitation

Transitionzone

Period measurement

Limitation of N




13 The resolver

























Foreword xv




xvi

Introduction







xviii

Chapter 1








1




11 Optical encoder















1 2 3141592 180

4 16 0392699 225

8 256 0024543 140625

10 1024 0006135 0351562

12 4096 0001534 0008789



0 000 000 0ndash45˚

1 001 001 45ndash90˚

2 010 011 90ndash135˚

3 011 010 135ndash180˚

4 100 110 180ndash225˚

5 101 111 225ndash270˚

6 110 101 270ndash315˚

7 111 100 315ndash360˚

d2 d1 d0 g2 g1 g0



ndash dn = gn





























F = NTm (11)


Fr = NNiTm (12)





NiTm(13)

or in percent


Fr(N)=

100N

(14)




F = 1Tm (15)


in Hz is reduced to

Fr = FhfNiN (16)



Ni

minus1N(N + 1)

(17)

or in percentage


Fr(N)=minus100N + 1

(18)


0

10

20

30

40

50

60

70

0 10 20 30 40 50 60

N

Rotation speed

Frequency counter

Speed limitation

Transitionzone

Period measurement

Limitation of N




13 The resolver
















Foreword xv




xvi

Introduction







xviii

Chapter 1








1




11 Optical encoder















1 2 3141592 180

4 16 0392699 225

8 256 0024543 140625

10 1024 0006135 0351562

12 4096 0001534 0008789



0 000 000 0ndash45˚

1 001 001 45ndash90˚

2 010 011 90ndash135˚

3 011 010 135ndash180˚

4 100 110 180ndash225˚

5 101 111 225ndash270˚

6 110 101 270ndash315˚

7 111 100 315ndash360˚

d2 d1 d0 g2 g1 g0



ndash dn = gn





























F = NTm (11)


Fr = NNiTm (12)





NiTm(13)

or in percent


Fr(N)=

100N

(14)




F = 1Tm (15)


in Hz is reduced to

Fr = FhfNiN (16)



Ni

minus1N(N + 1)

(17)

or in percentage


Fr(N)=minus100N + 1

(18)


0

10

20

30

40

50

60

70

0 10 20 30 40 50 60

N

Rotation speed

Frequency counter

Speed limitation

Transitionzone

Period measurement

Limitation of N




13 The resolver
















xvi

Introduction







xviii

Chapter 1








1




11 Optical encoder















1 2 3141592 180

4 16 0392699 225

8 256 0024543 140625

10 1024 0006135 0351562

12 4096 0001534 0008789



0 000 000 0ndash45˚

1 001 001 45ndash90˚

2 010 011 90ndash135˚

3 011 010 135ndash180˚

4 100 110 180ndash225˚

5 101 111 225ndash270˚

6 110 101 270ndash315˚

7 111 100 315ndash360˚

d2 d1 d0 g2 g1 g0



ndash dn = gn





























F = NTm (11)


Fr = NNiTm (12)





NiTm(13)

or in percent


Fr(N)=

100N

(14)




F = 1Tm (15)


in Hz is reduced to

Fr = FhfNiN (16)



Ni

minus1N(N + 1)

(17)

or in percentage


Fr(N)=minus100N + 1

(18)


0

10

20

30

40

50

60

70

0 10 20 30 40 50 60

N

Rotation speed

Frequency counter

Speed limitation

Transitionzone

Period measurement

Limitation of N




13 The resolver
















Introduction







xviii

Chapter 1








1




11 Optical encoder















1 2 3141592 180

4 16 0392699 225

8 256 0024543 140625

10 1024 0006135 0351562

12 4096 0001534 0008789



0 000 000 0ndash45˚

1 001 001 45ndash90˚

2 010 011 90ndash135˚

3 011 010 135ndash180˚

4 100 110 180ndash225˚

5 101 111 225ndash270˚

6 110 101 270ndash315˚

7 111 100 315ndash360˚

d2 d1 d0 g2 g1 g0



ndash dn = gn





























F = NTm (11)


Fr = NNiTm (12)





NiTm(13)

or in percent


Fr(N)=

100N

(14)




F = 1Tm (15)


in Hz is reduced to

Fr = FhfNiN (16)



Ni

minus1N(N + 1)

(17)

or in percentage


Fr(N)=minus100N + 1

(18)


0

10

20

30

40

50

60

70

0 10 20 30 40 50 60

N

Rotation speed

Frequency counter

Speed limitation

Transitionzone

Period measurement

Limitation of N




13 The resolver
















xviii

Chapter 1








1




11 Optical encoder















1 2 3141592 180

4 16 0392699 225

8 256 0024543 140625

10 1024 0006135 0351562

12 4096 0001534 0008789



0 000 000 0ndash45˚

1 001 001 45ndash90˚

2 010 011 90ndash135˚

3 011 010 135ndash180˚

4 100 110 180ndash225˚

5 101 111 225ndash270˚

6 110 101 270ndash315˚

7 111 100 315ndash360˚

d2 d1 d0 g2 g1 g0



ndash dn = gn





























F = NTm (11)


Fr = NNiTm (12)





NiTm(13)

or in percent


Fr(N)=

100N

(14)




F = 1Tm (15)


in Hz is reduced to

Fr = FhfNiN (16)



Ni

minus1N(N + 1)

(17)

or in percentage


Fr(N)=minus100N + 1

(18)


0

10

20

30

40

50

60

70

0 10 20 30 40 50 60

N

Rotation speed

Frequency counter

Speed limitation

Transitionzone

Period measurement

Limitation of N




13 The resolver
















Chapter 1








1




11 Optical encoder















1 2 3141592 180

4 16 0392699 225

8 256 0024543 140625

10 1024 0006135 0351562

12 4096 0001534 0008789



0 000 000 0ndash45˚

1 001 001 45ndash90˚

2 010 011 90ndash135˚

3 011 010 135ndash180˚

4 100 110 180ndash225˚

5 101 111 225ndash270˚

6 110 101 270ndash315˚

7 111 100 315ndash360˚

d2 d1 d0 g2 g1 g0



ndash dn = gn





























F = NTm (11)


Fr = NNiTm (12)





NiTm(13)

or in percent


Fr(N)=

100N

(14)




F = 1Tm (15)


in Hz is reduced to

Fr = FhfNiN (16)



Ni

minus1N(N + 1)

(17)

or in percentage


Fr(N)=minus100N + 1

(18)


0

10

20

30

40

50

60

70

0 10 20 30 40 50 60

N

Rotation speed

Frequency counter

Speed limitation

Transitionzone

Period measurement

Limitation of N




13 The resolver



















11 Optical encoder















1 2 3141592 180

4 16 0392699 225

8 256 0024543 140625

10 1024 0006135 0351562

12 4096 0001534 0008789



0 000 000 0ndash45˚

1 001 001 45ndash90˚

2 010 011 90ndash135˚

3 011 010 135ndash180˚

4 100 110 180ndash225˚

5 101 111 225ndash270˚

6 110 101 270ndash315˚

7 111 100 315ndash360˚

d2 d1 d0 g2 g1 g0



ndash dn = gn





























F = NTm (11)


Fr = NNiTm (12)





NiTm(13)

or in percent


Fr(N)=

100N

(14)




F = 1Tm (15)


in Hz is reduced to

Fr = FhfNiN (16)



Ni

minus1N(N + 1)

(17)

or in percentage


Fr(N)=minus100N + 1

(18)


0

10

20

30

40

50

60

70

0 10 20 30 40 50 60

N

Rotation speed

Frequency counter

Speed limitation

Transitionzone

Period measurement

Limitation of N




13 The resolver

























1 2 3141592 180

4 16 0392699 225

8 256 0024543 140625

10 1024 0006135 0351562

12 4096 0001534 0008789



0 000 000 0ndash45˚

1 001 001 45ndash90˚

2 010 011 90ndash135˚

3 011 010 135ndash180˚

4 100 110 180ndash225˚

5 101 111 225ndash270˚

6 110 101 270ndash315˚

7 111 100 315ndash360˚

d2 d1 d0 g2 g1 g0



ndash dn = gn





























F = NTm (11)


Fr = NNiTm (12)





NiTm(13)

or in percent


Fr(N)=

100N

(14)




F = 1Tm (15)


in Hz is reduced to

Fr = FhfNiN (16)



Ni

minus1N(N + 1)

(17)

or in percentage


Fr(N)=minus100N + 1

(18)


0

10

20

30

40

50

60

70

0 10 20 30 40 50 60

N

Rotation speed

Frequency counter

Speed limitation

Transitionzone

Period measurement

Limitation of N




13 The resolver



















1 2 3141592 180

4 16 0392699 225

8 256 0024543 140625

10 1024 0006135 0351562

12 4096 0001534 0008789



0 000 000 0ndash45˚

1 001 001 45ndash90˚

2 010 011 90ndash135˚

3 011 010 135ndash180˚

4 100 110 180ndash225˚

5 101 111 225ndash270˚

6 110 101 270ndash315˚

7 111 100 315ndash360˚

d2 d1 d0 g2 g1 g0



ndash dn = gn





























F = NTm (11)


Fr = NNiTm (12)





NiTm(13)

or in percent


Fr(N)=

100N

(14)




F = 1Tm (15)


in Hz is reduced to

Fr = FhfNiN (16)



Ni

minus1N(N + 1)

(17)

or in percentage


Fr(N)=minus100N + 1

(18)


0

10

20

30

40

50

60

70

0 10 20 30 40 50 60

N

Rotation speed

Frequency counter

Speed limitation

Transitionzone

Period measurement

Limitation of N




13 The resolver











































F = NTm (11)


Fr = NNiTm (12)





NiTm(13)

or in percent


Fr(N)=

100N

(14)




F = 1Tm (15)


in Hz is reduced to

Fr = FhfNiN (16)



Ni

minus1N(N + 1)

(17)

or in percentage


Fr(N)=minus100N + 1

(18)


0

10

20

30

40

50

60

70

0 10 20 30 40 50 60

N

Rotation speed

Frequency counter

Speed limitation

Transitionzone

Period measurement

Limitation of N




13 The resolver

































F = NTm (11)


Fr = NNiTm (12)





NiTm(13)

or in percent


Fr(N)=

100N

(14)




F = 1Tm (15)


in Hz is reduced to

Fr = FhfNiN (16)



Ni

minus1N(N + 1)

(17)

or in percentage


Fr(N)=minus100N + 1

(18)


0

10

20

30

40

50

60

70

0 10 20 30 40 50 60

N

Rotation speed

Frequency counter

Speed limitation

Transitionzone

Period measurement

Limitation of N




13 The resolver

























F = NTm (11)


Fr = NNiTm (12)





NiTm(13)

or in percent


Fr(N)=

100N

(14)




F = 1Tm (15)


in Hz is reduced to

Fr = FhfNiN (16)



Ni

minus1N(N + 1)

(17)

or in percentage


Fr(N)=minus100N + 1

(18)


0

10

20

30

40

50

60

70

0 10 20 30 40 50 60

N

Rotation speed

Frequency counter

Speed limitation

Transitionzone

Period measurement

Limitation of N




13 The resolver



















NiTm(13)

or in percent


Fr(N)=

100N

(14)




F = 1Tm (15)


in Hz is reduced to

Fr = FhfNiN (16)



Ni

minus1N(N + 1)

(17)

or in percentage


Fr(N)=minus100N + 1

(18)


0

10

20

30

40

50

60

70

0 10 20 30 40 50 60

N

Rotation speed

Frequency counter

Speed limitation

Transitionzone

Period measurement

Limitation of N




13 The resolver

















0

10

20

30

40

50

60

70

0 10 20 30 40 50 60

N

Rotation speed

Frequency counter

Speed limitation

Transitionzone

Period measurement

Limitation of N




13 The resolver





























Documents

Handbook of Asynchronous Machine with Variable Speeddownload.e-bookshelf.de/download/0000/7504/64/L-G-0000750464... · 3.1. The induction motor . . . ... 4.1.4. Five-phase induction