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I i~ I ·~ : 10-< iQI - - -t ! ctlU i! . . . . . . . I; ':: U ! <U I . . . . . . ! ~ !~ 1 0 I . . . . . . 18 , . . . . . . i S:: 1 0 lU 1 " ti) I " 'O S i ~ v l . . . .c: +-> , s:: tI) ! O ~ ! .~ i ~ II ~ ~ I e O ,~ ~I;> SI lrillgcr /l a/ iII l/rldt"!{Jag Nnl' Yo rJe Hilrct:!ollll [[oJlg KOllg I./JJJd(}}/ Mi/tl IIO f'lI! i ·o ~ (n.l:(llllll t ' 1: ,/111[ ' . ' . 1ONIINI LIIIIII\RY li.ll",II11'l'r ing I III I) 1.//WWW."1 II II 1/ II' I,dl l/" 1 \11 I! II ,/ 6'< 1" n " 1,-) L581 v -< o~( Werner Leonhard Contrai of Electrical Drives Third Edition F~J= ~~D1,~ U.F.M.G.· BIBLIOTECA UNIVERSITÁRIA rll rlllfl lrlllrl~lllllllllllr III 26780505 NAo DANIFIQUE ESTA ETlQUETA With 299 Figures ' , ., ~ t> _._._-- - _ -~.. ~ . ~ Spri ngcr BIBLIU 1 1' C " i ~t~~ •.'" I q/" l lI .iI. n.. I , N/II III1 IAIJIA ill~. I Professor WERNER LEONHARD Tl'chnische Universitãt Braunschweig IlIsti tut für Regelungstechnik IIa ns Sommer Str. 66 ,\1) 106 Braunschweig I' II/(./il: W.Leonhard@tu-bs.de 1/llcrnet: http://www.ifr.ing.tu-bs.de 'JÇLtb 14 r ~: : F' ' :~ (\ ,~'~ 1 1 \::1 ,'~{~t. ' :: ...6',l,: '.p.,~' ,:'; :! ':.' ~~~ c'::; . !.~~ . IS I\N \ :,~()-41820-2 Springer-Verlag Bedin Heidelberg NewYork I 01" ,II Y"I ( :0111'.1'<:" Cataloging-in-Publication Data 1 "01111111.1. W('rncr: 1 ",,\, 01 01 l' lc. (rical Drives. - Third edition.1 Werner Leonhard. - Berlin; Heidelberg; New York; 11 ,", ,,111,0:1; Iln"I', Kong; London; Milan; Paris; Singapore; Tokyo: Springer 2001 (IIIIV'I,I\'[' SYSlt'IIIS) I:;III~ I ','lO 111\20 .. 2 TI ... : worl, i, slIbjcct to copyright. All rights are reserved, whether the whole Dr part ofthe material is , "'I< .. roo('d, ,pccifically the rights Df translation, reprintíng, reuse Df illustrations, recitation, I" " " .I. "s\ illg , reproduction on microfilm or in otherways, and storage in data banks. Duplication of I 100',p""li ..,,\ ion or parts thereofis permitted only under the provisions ofthe German Copyríght Law "I :;I'\,\('m"er 9, 1965, in its current version, and permission for use must always be obtained from :'1" illl',"r Vcrlag. Violatíons are liable for prosecutíon act under German Copyright Law. :qll '001',<'1' Verlag is a company in the BertelsmannSpringer publishing group ,,\ I I"Iwww.springer.de ~, ' S" r;III',('r , V crlag Berlin Heidelberg New York 2001 l' rI "I",1 iII (;cnnany '1'\". 00 SI' "fl',ellcral dcscriptive names, registered names, trademarks,etc. in this publication does not ioo'ply, I'vcn iII lhe "bsencc or a spccific statement, that ,uch names are excmpl from the relevant "r"I,.• 'liv(' law, a"d regulat;o"s llnd tht:rdore freI: for I',cllnal US(' . 'L'yp rtJdltnH: ( :111111'1 ~t 1~';lIly by Illlllltlr ( ' ''V I' ' Ih'NII",, : .i<- \olil'" 11",1;" l'I " ~1c'dtlll ll . h Lfl"' ~ I"'111'1 .'~I'IN : IOH\\II\)C\ (1 _~ /\O .loJl.II '1.1 \.' 10 Preface Electrical drives play an important role as electromechanical energy convert ers in transportation, material handling and most production processes. The ease of controlling electrical drives is an important aspect for meeting the in creasing demands by the user with respect to flexibility and precision, caused by technological progress in industry as well as the need for energy conser vation. At the sarne time, the control of electrical drives has provided strong incentives to control engineering in general, leading to the development of new control structures and their introduction to other areas of controI. This is due to the stringent operating conditions and widely varying specifications - a drive may alternately require control of torque, acceleration, speed or position - and the fact that most electric drives have - in contrast to chem ical or thermal processes - well defined structures and consistent dynamic characteristics. During the last years the field of controlled electrical drives has undergone rapid expansion due mainly to the advances of semiconductors in the form of power electronics as well as analogue and digital signal electronics, eventu ally culminating in microelectronics and microprocessors. The introduction of electronically switched solid-state power converters has renewed the search for adjustable speed AC motor drives, not subject to the limitations of the mechanical commutator of DC drives which dominated the field for a century. This has created new and difficult control problems since the mechanically simpler AC machine is a much more involved control plant than a DC motor; on the other hand, the fast response of electronic power switching devices and their limited overload capacity have made the inclusion of protective control functions essentiaI. The present phase of evolution is likely to continue for some years, a new steady-state is not yet in sight. This book, originally published 1974 in German as "Regelung in der elek trischen Antriebstechnik", was an outcome of lectures the author held for a number of years at the Technical University Braunschweig. ln its updated En glish version it characterises the present state of the art without laying claim to complete coverage of the field. Many interesting details had to be omitted, which is not necessarily a disadvantage since details are often bound for early obs()les(:(~Ilc(~ . ln selecting and presenting the material, didactic view points ha.vc also hC(~1I clllIsidcrpr]. A prerequisite for the reader is a basic knowledge VI Preface "r !l0wer electronics, electrical machines and control engineering, as taught in 1111d, I1Ilcler-graduate electrical engineering courses; for additional facts, re (11IllSC is made to specialliterature. However, the text should be sufficiently ::,,11' coutained to be useful also for non-experts wishing to extend or refresh Ilwir kuowledge of controlled electrical drives. Tbcy consist of several parts, the electrical machine, the power converter, !.II(' ('oIltrol equipment and the mechanicalload, all of which are dealt with in v:uyiJlg depths. A brief resume of mechanics and of thermal effects in electri ()II IIlachines is presented in Chaps, 1-4 which would be skipped by the more 1 ':: I'I ~ ri(~llCed reáâer. Chaps. 5-9 deal with DC drives which have for ove r a "'III,lIry been the standard solution when adjustable speed was required. This p;\.rl. or the text also contains an introduction to line-commutated converters ,I:: IISI:<I for the supply ofDC machines. AC drives are introduced in Chapo lO, I>l'1';iII Jlillg with a general dynamic model of the symmetrical AC motor, valid iII IIpC" the steady-state and transient condition. This is followed in Chapo 11 II\' ali overview of static converters to be employed for AC drives. The con 1.1'01 ;1.spects are discussed in Chaps. 12-14 with emphasis on high dynarnic l'I'rI'Ol'lllanCe drives, where microprocessors prove invaluable in disentangling LIli' 1I1111Civariate interactions present in AC machines. Chapter 15 finally de ::nil)('s some of the problems connected with the industrial application of .IriVl's , 'l'ltis cannot by any means cover the wide field of special situations Wil.lt wllich the designer is confronted in practice but some frequently en '''IIIII.I'II'd katures of drive system applications are explained there. It will 111''''"111' ~"Ifliciently clear that the design of a controlled drive, in particular "I. Iilrl','" pow(~r ratings, cannot stop at the motor shaft but often entails an ;(11:0.11':" :: ,.I' L"(~ whole electro-mechanical system. III v inv 01' Lhe fact that this book is an adaptation and extension of an "111 dical.i, 'll-oricIltated text in another language, there are inevitably problems 1.' 1111 1" 'I ';: \.rd 1.0 symbols, the drawing of circuit diagrams etc. After thorough ","'::lIll.a.t.iolls with competent advisors and the publisher, a compromise so I"Li!)1I was adopted, using symbols recommended by IEE wherever possible, 1.111. rdailliug the authors usage where confusion could otherwise arise with I,i:: r(';ukrs at home. A list of the symbols is compiled following the table of ,'IIIII,I'III.S, The underlying principIe employed is that time varying quantities ;),1'" IISll1\.lly dcnoted by lower case latin letters, while capitalletters are ap pli,'" 1.11 parameters, average quantities, phasors etc; greek letters are used pt'l'dolllÍllalltly for angles, angular frequencies etc. A certain amount of over 1:'1 1 iH Illl<lvoidable, since the number of available symbols is limited. AIso the l.tI dillgl'aplty still exhibits a strong continental bias, eventhough an attcmpt III~:; IH:t!lt Illadc to halance it with titlcs in english la\lf(llagl~ . The list is cer I.niJlJy III' uo lIWilllS ('.()Inpl<!te hut it contaiJlH t.b(~ illfp\'lllnl.ioll r\,adiJy availahle to t.!,, \ anl.ho!', Dired rdl : reIl('.(:~i i1l 1.111: t.I'''1. hllVI' 11"1'11 II:II'c! :Iparillj~''y, wlwlI 1.1" , Ilril':ill WIl,H Lo 1)(' al'!llIowlc'dl (nd, l!opl'l'IIII 'y 1,11 11 1"'11""1 111 1.1'" w \ "il,, ~ (,II IU'I'I'P\. 1.II ":il' idl,"'L""llIit" I,i l wil." 1.111' IIc'c" 'HH lI ry 1'01,lnlll 'l' 111101 1111.\" '1'111,111 1111111 '" Preface VII The author wishes to express his sincere gratitude to two British col leagues, R. M. Davis, formerly of Nottingham University and S. R. Bowes of the University of Bristol who have given help and encouragement to start the work of updating and translating the original German text and who have spent considerable time and effort in reviewing and improving the ini tial rough translation; without their assistance the work could not have been completed. Anyone who has undertaken the task of smoothing the transla tion of a foreign text can appreciate how tedious and time-consuming this can be. Thanks are also due to the editors of this Springer Series, Prof. J. G. Kassakian and Prof. D. H. Naunin, and the publisher for their cooperation and continued encouragement. Braunschweig, October 1984 Werner Leonhard Preface to the 2nd edition During the past 10 years the book on Control of Electrical Drives has found its way onto many desks in industry and universities all over the world, as the author has noticed on numerous occasions. After a reprinting in 1990 and 1992, where errors had been corrected and a few technical updates made, the book is now appearing in a second revised edition, again with the aim of offer ing to the readers perhaps not the latest little details but an updated general view at the field of controlled electrical drives, which are maintaining and extending their position as the most ftexible source of controlled mechanical energy. The bibliography has been considerably extended but in view of the con tinuous strearn of high quality publications, particularly in the field of con trolled AC drives, the list is still far from complete. As those familiar with word processing will recognise, the text and figures are now produced as a data set on the computer. This would not have been possible without the ex pert help by Dipl.-Ing. Hendrik Klaassen, Dipl.-Math. Petra Heinrich, as well as Dr.-Ing. Rüdiger Reichow, Dipl.-Ing. Marcus Heller, Mrs. Jutta Stich and Mr. Stefan Brix, to whom the author wishes to express his sincere gratitude. Peter Wilson has been reading some of the chapters, offering valuable sugges tions, The final layout remained the task of the publishers, whose patience and helpful cooperation is gratefully appreciated. Braunschweig, May 1996 Werner Leonhard Prefacc to the 3rd Edition '1'1111. (,;\.\'Iy c\"l'ldio" of LIli: pllblislllll's :.;Locks oll('I'< ,d Hll opportullit,y of again II 1''':!.I,i "I': I.ltc' IIClClII , IIlakil'l', IIliJlOI' 1'01'l'I'cl.i'III:; :utel I'lllal'l':illJf, SOIlIt' SlIh.i,'i'I ,H I V III Preface whilc t.hc main part of the book was left unchanged. Tackling this work 1(''1uircd the encouragement by good friends but the author would still have 1)( '(:11 lIuable to realise it alone. He expresses his sincere gratitude to the helpful \(': ;('archers at the Institut für Regelungstechnik, particular1y Dipl.-Ing. Jan I \\ ...kcr and Dipl.-Ing. Frithjof Tobaben who were always ready to resolve ::Illl.wilrc-related crises on the computer, as well as Dipl.-Ing. Klaus Jaschke ;1.lId C;\llél. Wirtsch.-Inform. Danny Wallschlager, our definitive ~TEX-experts wliosc participation was essential for undertaking the task. Dr.-Ing. Sõnke 1\ ()ck suggested a clearer definition of the magnetic leakage which had gone II II lIot.iccd beforej finally, I want to thank Prof. Walter Schumacher, now head (Ir Ui(' laboratory, for his continued interest. Werner Leonhard 1 \ r:tllllscltweig, Spring 2001 Contents Introduction ................................................. . 1 1. Elementary PrincipIes of Mechanics ........ . ............ . 7 1.1 Newtons Law ......................................... . 7 1.2 Moment of Inertia ............. .. ...................... . 9 1.3 Effect of Gearing . ..... .......... . .............. ... .... . 11 1.4 Power and Energy .. .. .......................... ...... . . 12 1.5 Experimental Determination of Inertia ................... . 14 2. Dynamics of a Mechanical Drive ......................... . 17 2.1 Equations Describing the Motion of a Drive with Lumped Inertia .......................................... . .... . 17 2.2 Two Axes Drive in Polar Coordinates . .... .......... . .... . 20 2.3 Steady State Characteristics of Motors and Loads ......... . 22 2.4 Stable and Unstable Operating Points .................... . 26 3. Integration of the Simplified Equation of Motion ........ . 29 3.1 Solution of the Linearised Equation ............. . ........ . 29 3.1.1 Start of a Motor with Shunt-type Characteristic at No-Ioad ........................................ . 30 3.1.2 Starting the Motor with a Load Torque Proportional to Speed ........................ . .............. . 32 3.1.3 Loading Transient of the Motor Initially Running at No-Ioad Speed .. . .. . .......... . 33 3.1.4 Starting of a DC Motor by Sequentially Short-circuiting Starting Resistors ... . 35 3.2 Analytical Solution of Nonlinear Differential Equation ...... . 38 3.3 Numerical and Graphical Integration ..................... . 39 4. Thermal Effects in Electrical Machines .................. . 43 4.1 Power Losses and Temperature Restrictions ............... . 43 4.2 Hcating of a Homogeneous Body ....... .. ....... . ..... . . . 44 1.:~ Difrer('lll: Morles of Operation ...... . . .............. ... .. . 48 '1.:\.1 (;()ilt.illIlOIlS Dllty ..... ' ..... . . .. . . ...... . . . ...... . 1K x Contents 4.3.2 Short Time Intermittent Duty ..................... 48 4.3.3 Periodic intermittent duty . . . . . . . . . . . . . . . . . . . . . . . .. 49 ~ ',. SeparateIy Excited DC Machine . . . . . . . . . .. . . .. . . . . . . . . . .. 51 5.1 Introduction .................... ·.····················· 51 G.2 Mathematical Model of the DC Machine ..... ..... ........ 54 G.3 Steady State Characteristics with Armature and Field Control 56 5.3.1 Armature Control . . . . . . . . . . . . . . . . . . . .. . . . . .. . . . .. 57 5.3.2 Field Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 58 5.3.3 Combined Armature and Field Controlo . . . . . . . . . . . .. 61 !:iA Dynamic Behaviour of DC Motor with Constant Flux .. . . . .. 64 DC Motor with Series Field Winding . . . . . . . . . . . . . . . . . . . .. 69 0.1 Block Diagram of a Series-wound Motor. . . . . . . . . . . . . . . . . .. 70 O. G.2 Steady State Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 "( . ControI of a Separately Excited DC Machine .. . . . . . . . . . .. 77 7.1 Introduction .................... ······················· 77 7.2 Cascade Control of DC Motor in the Armature Control Region 79 7.:3 Cascade Control of DC Motor in the Field-weakening Region 90 'IA Supplying aDC Motor from a Rotating Generator.. . . . . . . .. 93 Static Converter as a Power Actuator for DC Drives ..... 97 ~u Electronic Switching Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 97 K. H.2 Line-commutated Converter in Single-phase Bridge Connection102 ~u Line-commutated Converter in Three-phase Bridge Connection119 H.tJ Line-commutated Converters with Reduced Reactive Power .. 130 H.:) Control Loop Containing an Electronic Power Converter .... 133 n. (;oulrol of Converter-supplied DC Drives ................. 139 ~u DC Drive with Line-commutated Converter ........ . ....... 139 ~).2 DC Drives with Force-commutated Converters ............. 148 Syrnrnetrical Three-Phase AC Machines .. , ............... 163 10.1 Mathematical Model of a General AC Machine .......... ·.· 164 10. 10.2 Induction Motor with Sinusoidal Symmetrical Voltages in Steady State ........ .. ............ . . . .................. 176 10.2.1 Stator Current, Current Locus .................. ... 176 10.2.2 Steady State Torque, Efficiency .................. . . 182 10.2.3 Comparison with Practical Motor Designs .......... , 186 10.204 St.art.ing of lhe Induction Molor ......... .. .... .... . 187 .Ill. :.\ Indlld.ioll Mot.()r with lmpresscd Volt.ar;c·s or t\ I'hit.r,\ry Wave- rOI'lIl :l . . ' .... . . . . ........ , ••• . . • , .. . . , •• .... ... 190 10/1 111.11\1'1.1011 M,J\,,!' wit.h lJn:iyllllllC't.1 i,';" 1,11 ... VIIII,il.l',C·: i iII S t.c ·;\.dy :J.m ~; 111.1." " . . •• , • , ••• Contents XI 1004.1 Symmetrical Components ......................... 202 10.4.2 Single-phase Induction Motor ........ ..... .... .. ... 206 10.4.3 Single-phase Electric Brake for AC Crane-Drives ..... 209 10.4.4 Unsymmetrical Starting Circuit for Induction Motor .. 211 11. Power Supplies for Adjustable Speed AC Drives . ... ...... 215 11.1 Pulse width modulated (PWM) Voltage Source Transistor Converter (IGBT) .................... . ................. 218 11.2 Volt age Source PWM Thyristor Converter ................. 225 11.3 Current Source Thyristor Converters ....... ........ ..... .. 232 11.4 Converter Without DC Link (Cycloconverter) .............. 236 12. Control of Induction Motor Drives ....................... 241 12.1 Control of Induction Motor Based on Steady State Machine Model ................................................. 242 12.2 Rotor Flux Orientated Control of Current-fed Induction Motor252 12.2.1 PrincipIe of Field Orientation .......... .. .......... 252 12.2.2 Acquisition of Flux Signals .. ..... ......... . .. . .... 260 12.2.3 Effects of Residual Lag of the Current Control Loops . 262 12.2.4 Digital Signal Processing ... .. ..................... 265 12.2.5 Experimental Results ............................. 268 12.2.6 Effects of a Detuned Flux Model ...... . . . .......... 269 12.3 Control of Voltage-fed Induction Motor ................... 275 12.4 Field Orientated Control of Induction Motor with a Current Source Converter ......... . .......... ...... ............. 281 12.5 Control of an Induction Motor Without a Mechanical Sensor. 289 12.5.1 Machine Model in Stator Flux Coordinates .......... 289 12.5.2 Example of an "Encoderless Control" ... ... ......... 291 12.5.3 Simulation and Experimental Results ............... 296 12.6 Control of an Induction Motor Using a Combined Flux Model 298 13. Induction Motor Drive with Reduced Speed Range ....... 303 13.1 Doubly-fed Induction Machine with Constant Stator Fre quency and Field-orientated Rotor Current ................ 303 13.2 Control of a Line-side Voltage Source Converter as a Reactive Power Compensator ...................... .. 317 13.3 Wound-Rotor Induction with Slip-Power Recovery .......... 323 14. Variable Frequency Synchronous Motor Drives ........... 329 14.1 Control of Synchronous Motors with PM Excitation .. . . .... 331 14.2 Synchronous Motor with Field- and Damper-Windings ..... . 342 14.3 Synchronous Motor with Load-commutated Inverter (LCI- Driv(') ........ . .......... .... .......... ..... ......... . 349 X II Contents I !:i . Some Applications of Controlled Electrical Drives ........ 363 15.1 Speed Controlled Drives ................................. 364 15.2 Linear Position Control ................................. 373 15.3 Linear Position Control with Moving Reference Point ....... 383 15.4 Time-optimal Position Control with Fixed Reference Point .. 389 15.5 Time-optimal Position Control with Moving Reference Point . 396 Bibliography .................................................. 402 IIHlcx .... , ........... , ........................................ 455 Abbreviations and Symbols 1 Equations ln all equations comprising physical variables, they are described by the prod uct of a unit and a dimensionless number, which depends on the choice of the unit. Some variables are nondimensional due to their nature or because of nor malisation (p.u.). 2 Characterisation by Style of Writing i(t), u(t), etc. instantaneous values 1:, Id , u, Ud , etc. average values I, U, etc. RMS-values L U, etc. complex phasors for sinusoidal variables i.(t), :!!(t), etc. complex time-variable vectors, used with multiphase systems f (t), :!!* (t), I*, U', etc. conjugate complex vectors or phasors li(t), l:!!(t), etc. vectors in special coordinates 1(8) = L(i(t)) etc. Laplace transforms 3 Symbols Abbreviation Variable Unit a(t) current distribution A/m linear acceleration m/s2 nondimensional factor A area m2 b nondimensional field factor D magnetic flux density T = Vs/m2 C " clectrical capacity F = As/V theonal storaw' capnci t.y J;oC = Ws;oC f) dlLl II pill r~ r aLio '\IV Abbreviations and Symbols ,(/), I~, E induced voltage, e,m.f, r , frequency force ,,' (.'i ) transfer function 'I gravitational constant 'I (I) unit impulse response weight ( " gain ii airgap ,(1) , I , I current ,/ inertia I,; nondimensional factor /I torsional stiffness , length inductance 1/1",( /) torque II/ mass mutual inductance speed, rev /min N number of turns I' ( /) , I' power () reactive power / ' radius I,' resistance Laplace variable '/I " I J W distance , " ' slip / time :1 \ " " ' time constant /I (I), II, II voltage velo city " (I) unit ramp response \I volume /1/(1 ) unit step response energy ,/' control variable actuating variable :'1 disturbance variable ('.'."" discrete Laplace variable \ ' admitt.a.nce /. illlj)('dallce /I' codlicicut, or II(';\'/. t,rnll :';I'('I ' lil'ill/', ;\.11/,:1(' 1111['," LII ' ,1.1 '''' '1''I'a/,IIII' v Hz = l/s N m/s2 N m A kg m2 Nm/rad m H = Vs/A Nm kg H l/min W VA m n rad/s m s S V m/s m3 J =Ws 1/f2 = S .rl W /1I1;~ "(; I I i 1\ , I !' I I 0:, {3, 15, (, ç, >., /l , {2 etc. "1= 21f/3 15 L1 ê ." {) e /lo v (J' T = Jw dt, wt <p cos<p P 'ljJ w 41ndices ia ie UF ZS iR iSd, ZSq iRd, iRq Zm i mR ZmS mL mM mp Sp Abbreviations and Symbols xv angular coordinates rad load angle rad difference operator angle of rotation rad efficiency temperature oe absolute temperature K magnetomotive force, m.m.r. A coefficient of permeability H/m integer number leakage factor normalised time, angle rad phase shift rad power factor magnetic flux Wb = Vs flux linkage Wb = Vs angular frequency rad/s armature current exciting current field voltage stator current rotor current direct and quadrature components of stator current direct and quadrature components of rotor current magnetising current magnetising current representing rotor flux magnetising current representing stator flux load torque motor torque pull-out torque pull-out slip Abbreviations and Symbols XVIIxV I Abbreviations and Symbols Y1 X., (;('aphical Symbols ----171 • x =Yt!Y2 division integrator4kí~ T~~=y x = f(y) nonlinearity v _~ T dx = G (T !EL + y) PI controller ~8t!j~~ - ~ ~ first order lag x = y for Xmin < y < Xmax4l~::í~ T~~+x=Gy ~~I~) X = Xmin for y ~ Xmin limiter x = Xmax for y ~ XmaxY ~ ............... xmin G r;; r, first order lead/lag~k~ Tl~~+X=G(T2~+Y) -------..~ . A/D- or D/A converter x(t) = G y (t - T) delay ---+I-~r---41lit--:x YI X current sensor ~O • summing point x = Yl - Y2 ~i YI X lu I~ --_.~ .. volt age sensor multiplicationx = Yl Y2 I I +U xV III R L (/)1~ m R · L di +Ri [!fi ~! = dt L% U 2 + e = 'fi! 'fi2u dt e 1 Ir? o T '1'1 t( ~ volt age arrowS indicating volt age sources (u, e) or volt age drops (R i, L ~) 1('pl( ~Sent the differences of electrical potential, pointing from the higher to I . h( ~ lower assumed potential. Hence the volt ages in any closed mesh have zero :: 11111 , L: u = O. Introduction Energy is the basis of any technical and industrial development. As long as only human and animal labour is available, a main prerequisite for social progress and general welfare is lacking. The energy consumption per capita in a country is thus an indicator of its state of technical development, ex hibiting differences of more than two orders of magnitude between highly industrialised and not yet developed countries. ln its primary form, energy is widely distributed (fossil and nuclear fuels, hydro and tidal energy, solar and wind energy, geothermal energy etc.), but it must be developed and made available at the point of consumption in suitable form, for instance chemical, mechanical or thermal, and at an acceptable cost. This creates problems of transporting the energy from the place of origin to the point of demand and of converting it into its final physical formo ln many cases, these problems are best solved with an electrical intermediate stage, Fig. 0.1, where the bold numbers refer to the European grid, because electricity can be pri7;{J _. ;ower ------LT;ansmiSSiOntPOWer electronics+ Final I ene~tat,onT,stobuflon consume~ Fuelcel/s Electrical drives Solar (PV) i 46% Fossil--..... 84% 100% "~ Control/ad electrical árives 38% Nuclear_ Thermal _ Mechanical _Elektrical f- Elsctrical .......Mechênical >50% Solar --- -~ 'o,Uo=con '1' UI = variable ;.>, .Biomass' / Water, Wind, Waves _ Mechanical < 16%,« 1% - Electrical Electrical energy _ThermalUCTE- European Grid per capita and year _Chemical > 350 GW, 1 800 TWh/a world wide: 0.02 - 28 MWh Fig. 0.1. From primary energy to final use, a chain of conversion processes 3 ~ Introduction • ~enerated from primary energy (chemical energy in fossil fuel, potential hydro energy, nuclear energy) in relatively efficient generating stations, • transported with low los ses over long distances and distributed simply and at acceptable cost, • converted into any final form at the point of destination. 'I'his ftexibility is unmatched by any other form of energy. Of particular importance is the mechanical form of energy which is needed i Il widely varying power ratings wherever physical activities take place, involv ill~ the transportation of goods and people or industrial production processes. I,'or this final conversion at the point of utilisation, electromechanical devices i II the form of electrical drives are well suitedj it is estimated, that about half I.he clectricity generated in an industrial country is eventually converted to Ill( ~chanical energy. Most electrical motors are used in constant- speed drives I.hat do not need to be controlled except for starting, stopping or protection, 1)11 t there is a smaller portion, where torque and speed must be matched to I.he need of the mechanical loadj this is the topic of this book. Due to the progress of automation and with a view to energy conservation, the need for col1trol is likely to become more important in future. As an example, Fig. 0.2 ~h()ws the mechanical power needed by a centrifugal pump, when the flow is couLrolled by a variable speed drive or, still in frequent use, by throttle and !).\' pass valves. mecham"cal input power Pm Po l Bypass 1,0 ~t_c~,,!~ary_se,,-e~_ _ _ __ _ Throtlle rt<í '-~---, Thro",e "I l ' at constant speedCentrifugai 3'( ) pumpI Bypass Load variabfe speed , O l-.. Flow 1,0 00 Fig. 0.2. Mechanical input power to a centrifugai pump IIKillg difrcrent methods of fiow controi 'J'hc predomil1ance of electrical drives is caused by several aspects: • I':kdric drlv( ~s aT<' available for any power, frOlIl 10° W in electronic wnt.(·II('~i 1.0 I ()H W for drivill~ plllllpS iII hydro hl.orHI~t' líln llt.f!, • 'I'1L< ~ 'y e nV I ' .' tL wid,' rI,W I',C' or I.orqlll' 11.11(1 " p"( 'd, . IO'f NIII , ror il,\I ore IHill lIud,"I' , • 10)' ' /I l1itr , Cm l i (·(·l1l.riflll ( · driv(· . Introduction • Electric drives are adaptable to almost any operating conditions such as forced air ventilation or totally enclosed, submerged in liquids, exposed to explosive or radioactive environments. Since electric motors do not require hazardous fuels and do not emit exhaust fumes, electrical drives have no detrimental effect on their immediate environment. The noise leveI is low compared, for instance, with combustion engines. • Electric drives are operable at a moment's notice and can be fully loaded immediately. There is no need to refuel, nor warm-up the motor. The ser vice requirements are very modest, as compared with other drives. • Electrical motors have low no-Ioad losses and exhibit high efficiencYj they normally have a considerable short-time overload capacity. • Electrical drives are easily controllable. The steady state characteristics can be reshaped almost at will, so that railway traction motors do not require speed-changing gears. High dynamic performance is achieved by electronic controI. • Electrical drives can be designed to operate indefinitely in alI four quad rants of the torque-speed-plane without requiring a special reversing gear, Fig. 0.3. During braking, i.e. when operating in quadrants 2 or 4, the drive is normally regenerating, feeding power back to the line. A comparison with combustion engines or turbines makes this feature look particularly attractive. W, mM (~J Motor) )) () Load) mL mM =Motor torque mL =Load torque V-úJ - mMtM Fil';. ().:\. 0l" ~ ral.illg ma des of nll d('ctric drive: iII 11 11 'f"il,c1 rr,"f.,; (lI' 1.11<' l.o rqllc:j" pc't· c1 plallCi 1 5 Introduction • The rotational symmetry of electrical machines and (with most motors) the smooth torque results in quiet operation with little vibrations. Since there are no elevated temperatures causing material fatigue, long operating life can be expected. • Electrical motors are built in a variety of designs to make them compatible with the load; they may be foot- or fiange-mounted, or the motor may have an outer rotor etc. Machine-tools which formerly had a single drive shaft and complicated mechanical internal gearing can now be driven by a multitude of individually controlled motors producing the mechanical power exactly where, when and in what form it is needed. This has removed constraints from machine tool designers . III special cases, such as machine-tools or the propulsion of tracked vehicles, linear electric drives are also available. I Signallevel I Power levei -+ I Voltages, currents Torque, acceleration, speed, angular position etc ~ Signal acquisition: Reconstruction of unmeasurable quantities, A Elimination of sensors Contrai: Oecoupling, feed - forward, limiting Coordination: Coordinate transformation .c .-=: Identification: Estimation of variable parameters Self tuning: Computer aided commissioning ~ A daptation: Adjustment of contrai parameters Optimisation: Minimisation of objective functions Fig. 0.4. Digital control structure of an electrical drive 1\ s wou lei Iw (~xpccted , this long list of remarkalJle characteristics is to IH ' :-; lIppkllwllt,(~d by Jisa.c1va.ntages of electric drives which lirnit or preclude U1I·ir 11 SC: • 'I'1!t'. dqH'lIdHII("( ' 011 it COJlLiIlIIOll:-l POW('I' sllpply C'ill lllPS [)I· ..hl n lll !-J witll vdti.. (' II' !II'"p"I: :i<l11. Iii II" I'''WI 'I' r 1l.il UI' (,id,I'llilry lli lI,valln l lk , /l ll I ·II'r(,rir 1:1U :I'J-(Y Ilull r ... • 11.111 11 1. I... ' ·III ' r l , ·.\. "" 1''' .'\1'<1, whi<-h ln ll il il rdl,Y IlItlllY. I ...." vy n llel " XIH'Il ntv" ( 111.0' " 1"'" II,nU.' " y, I <lUd,I III'. 1',I'III l h,I,, " wil.l, 1111 " , ii I.! 1',,"d'lInl.i'J! 1 I·,III',h ... ' Introduction or turbine, fuel- or solar cells). The lack of a suitable storage battery has so far prevented the wide-spread use of electric vehicles. The weight of a present day lead-acid battery is about 50 times that of a liquid fuel tank storing equal energy, even when taking the low efficiency of the combustion engine into account. • Due to the magnetic saturation of iron and cooling problems, electric mo tors are likely to have a lower power-to--weight ratio than, for instance, high pressure hydraulic drives that utilise normal instead of tangential forces. This is of importance with servo drives on-board vehicles , e.g. for position ing the control surfaces of aircraft. The electromechanical energy conversion in controlled drives is subject to the terminal quantities of the electrical machine which can be changed with low losses by controllable power electronic converters consisting of semicon ductor switches; they can produce voltages and currents of almost any wave form as prescribed by control which today is executed by microelectronic components. Thus semiconductor technology, combining power conversion and high speed signal processing at reasonable cost, has been the essential force behind the development of todays high performance drives; this is part of the general transition from analogue to digital control systems using mi crocomputers and signalprocessors. Fig. 0.4 gives an impression of how the mechanical, power electronic and control functions, combining hardware and software, are interleaved in a modern drive system. 1. Some Elementary Principies of Mechanics Since electrical drives are linking mechanical and electrical engineering, let us recall some basic laws of mechanics. 1.1 Newtons Law A mass M is assumed, moving on a straight horizontal track in the direction of the s-axis, Fig. 1.1 a. Let IM(t) be the driving force of the motor in the direction of the velocity v and h(t) the load force opposing the motion, then Newtons law holds d dv dM1M - h = - (Mv) = M - + v- (1.1)dt dt dt' where M v is the mechanical momentum. Usually the forces are dependent on velocity v and position s, such as gravitational or frictional forces. li the mass is constant, M = Mo = const., Eq. (1.1) is simplified, dv 1M - h = Mo dt ; (1.2) with the definition of velo city v = ds Idt, this results in a second order differ ential equation for the displacement, (f( &J m,mM tM M s V /:1 b I"il-';. J .1 . ' I 'n.. , ..hd.;oll .d Jl llcl ""ULI.;oll a l 11101.;011, "r IaIlIlP"cI llm..;S('~ 9 ~l 1. Elementary PrincipIes of Mechanics d2 s (1.3)f M - !L = Mo dt2 ' wh( ~ re dv d2 s a = -=- (1.4) dt dt2 is I.he acceleration. If the motion is rotational, which is usualIy the case with ('h:cLrical drives, there are analogous equations, Fig. 1.1 b, d dw dJ 1I/.M - mL = dt (Jw) = J dt + w di ' (1.5) wil.h mM being the driving- and mL the load torque. w = 21l" n is the angular vdocity, in the folIowing called speed. J is the moment of inertia of the 1'( d.:d:iIlg mass about the axis of rotation, J w is the angular momentum. The 1,'1'111 w (dJ/ dt) is of significance with variable inertia drives such as centrifuges 01' n :eling drives, where the geometry of the load depends on speed or time, or illdllst rial robots with changing geometry. ln most cases however, the inertia C; lJI ue assumed to be constant, J = Jo = const., hence dw mM - mL = Jo di . (1.6) Wit.h E the angle of rotation and w = dE / dt the angular velocity, we have d2E 'II/.M - mL = Jo dt2 ' (1.7) wh('n~ (lw d2E ( r (1.8) (ft dt2 ' i:: Ih( ' ;\.I1i ~\Ilar acceleration. II. ~; II()\Ild be noted that mM is the internal or electrical motor torque, not id( 'llli cal with the torque available at the motor shaft. The difference between i111"rllal torque and shaft torque is the torque required for accelerating the il ... rlia of Lhe motor itself and overcoming the internal friction torque of the III< 1101'. ' I'ranslational and rotational motions are often combined, for example in v(:hich~ propulsion, elevator- or rolling mill-drives. Fig. 1.2 shows a mechanical II1 0dd , where a constant mass M is moved with a rope and pulIey; when 1I<'I',kct.iIlg the mass of the pulley and with '/lI' M = '/' fM , mL = r fL und v = rw \V(' Jiud wilh M = consto a z dw 'II/.M 1/I. J. '/' (M'II) = M l' - . (1.9) di, di. ./,. 1\'/ .,.:' 1'1'111'1': :"111.: : til,' ('qllivilklll. 1110111('111. "r illl'l'lin, 01' I.h,' liI)( :arly UIOV illl.~ rtllI )I::, l',·t'<-n:,'d 1." 1,1 ... I l. x i ll (Ir U", plIlI, 'y. "I'P'""III.I V 1.1 ... 111:1 :::: f\;1 ( 'a,1I II<: 1.1I<1I1J ·,.hl. "I /i') 1,,,1.1( ' ,[ j,iI ,! 11 1111., '" " I"" r~ 1.11" , ll'(' lIlId"" '1I1 (I "I"!.II<' \\,11<'1,1 :, 1.2 Moment of Inertia GtM/ t L / "'I .(( \) .. L M - ---=--. ( vrM ' úJ mL '~r -"'r ---. v Fig. 1.2. Linking linear and rotational motion 1.2 Moment or Inertia The moment of inertia, introduced in the preceding section, may be derived as follows: A rigid body of arbitrary shape, having the mass M, rotates freely about a vertical axis orientated in the direction of gravity, Fig. 1.3. An element of the mass dM is accelerated in tangential direction by the force element dfa, which corresponds to an element dma of the accelerating torque dv dwdm = l' d+ = l ' dM - = '1'2 dM - . a ~ a dt dt The total accelerating torque follows by integration ma M ma = j dma = j '1'2 ~ dM . (1.10) O O Due to the assumed rigidity of the body, alI its mass elements move with the sarne angular velocity; hence M = dWj 2dM=Jdw ma dt r dt . (LU) o The moment of inertia, referred to the axis of rotation, M J = j r 2 dM (1.12) o ~ dm . a I '2 p '1 Fig. 1.3. M II IIII·/I1. "r ill....l.ia Fig. 1.4. MCl\wlni of inertia of cOIlc<·"Ll'i(' (·. 'ylilHlcr 11 lO 1. Elementary PrincipIes of Mechanics i~ a three-dimensional integral. III many cases the rotating body possesses rotational symmetry; as an (')(;tlllple, consider the hollow homogeneous cylinder with mass density º,Fig. I ,ti, As volume increment dV we define a thin concentric cylinder having the r;tdius r and the thickness dr; its mass is dM = ºdV = º21f r l dr . 'I'bis redu ces the volume integral to a simple integration along the radius, 2 3 .l c fM r dM = º21f i fr2 r dr = %ºi (r~ - rt) . (1.13) o TI 1l"!lce the moment of inertia increases with the 4th power of the outer radius. IlIl.roclucing the weight of the cylinder G = ºgl1f(r~ - ri) (1.14) l"('slIlts in G r~ +d G 2 J = 9 --2- = 9 ri , (1.15) wl!('rc.ll is the gravitational acceleration. The quadratic mean of the radii 'ri 21 (ri + r~) (1.16) I ;; c:t1I, ~ d Lhe radius of gyration; it defines the radius of a thin concentric "ylill<!(,r with length l and mass M, that has the sarne moment of inertia as Lh,: urigillal cylinder. J Ml2 3 Ml2 12 O _J~ 1 ~- [I b 2 1,'1,,; . I.r., M"I","I, "r \ '"' 1' 1. 1" "I' (\ ,."eI, piv",-,'d 0111. ,,(" , " '111.,-(' 1.3 Effect Df Gearing Another example is seen in Fig. 1.5 a, where a homogeneous thin rod of length i and mass M is pivoted around a point P, the distance of which from one end of the rod is a. With the mass element dM = (M/i) dr we find for the moment of inertia a 2 2 2 1 = 1r dM = ~ [J r dr + 7r dr] o o o M 12 [ ( a) 2] = 12 1+3 1-2y . (1.17) The minimum inertia is obtained, when the rod is pivoted at the centre, Fig. 1.5 b. 1.3 Effect of Gearing Many applications of electrical drives call for relatively slow motion and high torque, for instance in traction or when positioning robots. Since the tangen tial force per rotor surface, i.e. the specific torque of the motor, is limited to some N/c:m2 by iron saturation and heat los ses in the conductors, the direct coupling of a low speed motor with the load may result in an unnecessarily large motor. It is then often preferable to employ gears, operating the motor at a higher speed and thus increasing its power density. This also affects the inertia of the coupled rotating masses. ln Fig. 1.6 an ideal gear is shown, where two wheels are engaged at the point P without friction, backlash or slip. From Newtons law it follows for the left hand wheel, assumed to be the driving wheel, dWl mM1 - rI fI = li dt ' (1.18) where iI is the circumferential contact force exerted by wheel 2. lf there is no load torque applied we have, correspondingly, for wheel 2 r2 fz = lz dw2 (1.19)dt . h is the force driving wheel 2. Since the forces at the point of contact are in balance and the two wheels move synchronously, h=h, rI Wl = r2 W2 , (1.20) ( ~ Iimination of h, h, W2 results in dWl ri dW2 rI dWI mMl = .lI - + - 12 - = II + ( -) 2 lz1dt r2 dt [ 1'2 dt tlWI (1.21 ) .II,' dI. 13 ~IIJII 12 1. Elementary PrincipIes of Mechanics v : 2r3 M3 F ig. 1.6. Effect of gearing on inertia Fig. 1.7. Hoist drive with gear J le is the moment of inertia effective at the axis of wheel 1; it contains a component reflected from wheel 2. ln most cases it is easier to determine the speed ratio rather than the radii, J le = h + (::) 2 J2 , (1.22) which indicates that a rotating part, moving at higher speed, contributes 1I1Ore strongly to the total moment of inertia. ln Fig. 1.7 an example of a multiple gear for a hoist drive is seen. J l , .J2 , Js are the moments of inertia of the different shafts. The total effective illertia referred to shaft 1 is J1e = J1 + (::) 2 h + (::) 2 [J3 + M3 r~l ' (1.23) illduding the equivalent inertia of the mass M3 being moved in vertical direc lion. Applying Newtons law, taking the load of the hoist into account, results iu dWl W3 mMl = J1e -d + - r3g M 3 . (1.24) t Wl 1.4 Power and Energy The rotational motion of the mechanical arrangement shown in Fig. 1.8 is dC!icrihcd by a first. order dilferential equation for speed Ii.w 'III'M w. /, I .I ti, (1.25) Mlll !. q,l ill d ,IIIII I,y ( 01 ,vi .. ldn UI<' PClW(" 1.,11 " 11(' (' 1.4 Power and Energy dw wmM=wmL+Jwdj' (1.26) where PM = wmM is the driving power, PL = w mL the load power and J w (dw / dt) the change of kinetic energy stored in the rotating masses. M m 'W5EJ J )-) mL - ...... .... PM PL Fig. 1.8. Power flow of drive By integrating Eq. (1.26) with the initial condition w (t = O) = Owe find the energy input WM(t) = tf PMdT = t tf PLdT +f Jw: dT o o o t w = f PL dT + J f n dn o o 1 2 =wL(t)+2"Jw. (1.27) The last term represents the stored kinetic energy; it is analogous to 1 1 1 -Mv2 - Li2 -Cu2 2 ' 2 2 . of other energy storage devices. Since the energy content of a physical body cannot be changed instantaneously - this would require infinite power the linear 01' rotational velo city of a body possessing mass must always be a continuous function of time. This is an important condition of continuity which will frequently be used. Because of the definitions ds de v = - and w = dt dt the positional quantities s and € are also continuous functions of time, due to fiuite speed. This is also understandable from an energy point of view, since position lIlay be associated with potential energy, as seen in Fig. 1.7, where I.h, : Ill as ~; 1I{, i:-l ]>os il.iollf:(} vcrtically dC(l':lIdillf!: OH th, : angle of rotation of 1.11<' d ri Vl' sJlid'i. . 14 15 1. Elementary PrincipIes of Mechanics 1.5 Experimental Determination of Inertia The moment of inertia of a complex inhomogeneous body, such as the rotor of an electrical machine, containing iron, copper and insulating material with complicated shapes can in practice only be determined by approximation. The problem is even more difficult with mechanicalloads, the constructional details of which are normally unknown to the user. Sometimes the moment of inertia is not constant but changes periodically about a mean value, as in the case of a piston compressor with crankshaft and connecting rods. Therefore experimental tests are preferable; a very sim pIe one, called the run-out or coasting test, is described in the following. Its main advantage is that it can be conducted with the complete drive in place and operable, requiring no knowledge about details of the planto The accuracy obtainable is adequate for most applications. First the input power PM (w) of the drive under steady state conditions is measured at different angular velocities w and with the load, not contributing to the inertia, being disconnected. From Eq. (1.26), dw (1.28)PM = PL + J w dt ' the last term is omitted due to constant speed, so that the input power PM corresponds to the los ses including the remaining load, PM = PL. This power is corrected by subtracting loss components which are only present during power input, such as armature copper los ses in the motor. From this corrected power loss P~ the steady-state effective load torque m~ = p~jw is computed for different speedsj with graphical interpolation, this yields a curve m~ (w) as shown in Fig. 1.9. For the run-out test, the drive is now accelerated to some initial speed wo, where the drive power is switched off, so that the plant is decelerated by the loss torque with the speed measured as a function of time, w(t). Solving the equation of motion (1.25) for J results in '" -m~(w)J mM=O. (1.29) '" dw ' âf(w) Rence the inertia can be determined from the slope of the coasting curve, as shown in Fig. 1.9. Graphical constructions, particularly when a differentiation is involved, are only of moderate accuracy. Therefore the inertia should be computed at different speeds in order to form an average. The accuracy requirements regarding inertia are modest; when designing a drive control system, an error of ± 10% iil IlsualJy Itcccptable without al1y ~criom; effect. Two spcc.j;d CiLS(': ; }PaI! to [>itrticlllarly sill1pl(~ iJlt.(:rprüt,ations: ii,) Ã n:i11 III i "I'. 1.11 (1 1" 01"1,,,( ' 1,'.1 IOI'!Il t.OI·qlle' '111,'1. to !lo Il,pproxilJlat.dy (:OIlK(:i\.Ut. ill n. Jilldl. \,d Iii>!"'" ,"I,,·, vld , 1.5 Experimental Determination of Inertia ú) 00, coasting curve oo(t)Corrected steady state load curve m'L (00) m' ,L m'L (00 1) Fig. 1.9. Run - out test m~ ~ const, for Wj < w < w2 , then w(t) resembles a straight line; the inertia is determined from the slope of this line. b) li a section of the loss torque may be approximated by a straight line, m~ ~ a+bw, for Wj < w < W2 . a linear differential equation results, dw J dt +bw= -a. The solution is, with W(t2) = W2, w(t) = -~ +(w2 +~) e-b (t-t2)/J t 2: t2 •b b ' Plotting this curve on semi-Iogarithmic paper yields a straight line with the slope -bjJ, from which an approximation of J is obtained. 2. Dynamics of a Mechanical Drive 2.1 Equations Describing the Motion of a Drive with Lumped Inertia The equations derived in Chapo 1 J dw dt = mM (w, é, YM,t) - mL (w, é, YL,t) , (2.1) dE: dt =w, (2.2) describe the dynamic behaviour of a mechanical drive with constant inertia in steady state condition and during transients. Stiff coupling between the different parts of the drive is assumed so that all partial masses may be lumped into one common inertia. The equations are written as state equations for the continuous state variables w, é involving energy storage, e. g. [38,88]). Only mechanical transients are consideredj a more detailed description would have to take into account the electrical transients defined by additional state variables and differential equations. The sarne is true for the load torque mL which depends on dynamic effects in the load, such as a machine tool or an devator. AIso the control inputs YM, YL to the actuators on the motor and load side have to be included. Fig. 2.1 shows a block diagram, representing the iuteractions of the mechanical system in graphical formo The output variables of the two integrators are the continuous state variables, characterising the (~l1ergy state of the system at any instant. Linear transfer elements, such ;\.S integrators with fixed time constants, are depicted by blocks with single !"rames containing a figure of the step response. A block with double frame ,lCllominates a nonlinear function; if it represents an instantaneous, i. e. static, Ilonlinearity, its characteristic function is indicated. The nonlinear blocks i Il Fig. 2.1 may contain additional dynamic states described by differential ('qllations. Dcpenc\cIlce of the driving torque mM on the angle of rotation is a char ad(~ristic fCiI.tUfe of synchronous motors. However, the important quantity is II' d. Lhl' allgl!' of roCaLiol\ é itself but the difrc r('uc( ~ itllgle 8 against the no-load 11.111,:1" 1(( w ii ich is dd(~rIlJiu(~d hy 1.111' l.o('(pl(· . () ud('r ~lt.('ady statl' condi I.io/l s 1.1«' loa.! 11111',1(, S iH (,oll:d.alll.. 19 li> 2. Dynamics of a Mechanical Drive Power supp/y YM co mM mL Motor o Fig. 2.1. Simplified block diagram of lumped inertia drive ln order to gain a better insight, let us first assume that the electrical transients within the motor and the internalload transients decay consider ably faster than the mechanical transients of w and ci as a consequence, it r()l1ows that the motor and load torques mM, mL are algebraic, i.e. instan (,allcouS functions of w, c and 8. Hence, by neglecting the dynarnics of motor ;Lmlload, we arrive at a second order system that is completely described by lhe two state equations (2.1), (2.2). So [ar we have assumed that alI moving parts of the drive can be combined lo forrn a single effective inertia. However, for a more detailed analysis of dy n<tlIlic effects it may be necessary to consider the distribution of the masses aud the linkages between them. This leads to multi-mass-systems and in the liwit to systems with continuously distributed masses, where transients of lJigher frequency and sometimes insuflicient damping may be superimposed O!l Lhe common motion. The frequency of these free oscillations, describing l.h(~ relative displacement of the separate masses against each other, increases wit.h Lhe stiffness of the connecting shaftsi they are usually outside the fre qll\!llCy range of interest for control transients but must be considered for the llH:dw,llical design of the drive. /TIM , <Út,f,1 me Cü2,E2 J2 ~1fl>, ))K)(1 ))) Motor me Load mL Jo'ilf . 2,2. I )riv!' l'1I1I:. i/<(, I.'I: 01' mot,or nlld 101\.1 '-,0111'1".1 1,.1 ""xij,k J;ltart. 2.1 Equations Describing the Motion of a Drive with Lumped Inertia However, when the partial masses are coupled by flexible linkages, such as with mine hoists, where drive and cage are connected by the long winding rope or in the case of paper mill drives with many gears, drive shafts and large rotating masses particularly in the drying section, a more detailed description becomes necessary. The free oscillations may then have frequencies of a few Hertz which are well within the range of a fast controlloop. ln Fig. 2.2 an example is sketched, where the drive motor and the load having the moments of inertia J1 , J2 are coupled by a flexible shaft with the torsional stiffness K. The ends of the shaft, the mass of which is ignored, have the angles of rotation C1, C2 and the angular velocities W1, W2 . Assuming a linear torsionallaw for the coupling torque me, me = K (C1 - c2) , (2.3) and neglecting internal friction effects, the following state equations result dw1J1 dt = mM - me = mM (Wt, C1, YM) - K (c1 - c2) , (2.4) dw2J2 dt = me - mL = K (cl - c2) - mL (W2, c2, YL) , (2.5) de1 dt =W1, (2.6) de2 -=W2· (2.7)dt A graphical representation is seen in Fig. 2.3 a. Here too, the torques III IV!, mL are in reality defined by additional differential equations and state vdriables. If only the speeds are of interest, the block diagram in Fig. 2.3 b "':1..)' be useful, which, containing three integrators, is described by a third lIJ'dc[' differential equation. With increasing stiffness of the shaft, the quantities C1, C2 and W1, W2 I"'collle tighter coupledi in the limit the case of lumped inertia emerges, w lU'r<: C1 = C2, W2 = W1. Ir Lhe transmission is affected by mechanical backlash, where the two "",rLias separate when the shaft torque changes sign, the linear torsional III " Il('h f{ is replaced by a nonlinear function, as shown in Fig. 2.3 c. This is III' i IlIportance with reversible drives, for instance servo drives. (lltviollSly the subdivision of the inertia may be continued indefinitely; o' Vo -J'y Lime a new partial inertia is separated, two more state variables have to 100 ' ,I, -Ii 11(:(1, t.ransforming Fig. 2.3 into a chainlike structure. A typical example, w l",I'" 1I1,tlly partial masses must be taken into account for calculating stress 111141 I'n.t.igw!, is a turbine rotor. 20 21 2. Dynamics of a Mechanical Drive PowerPower supplysupp/y fO ba c~JP F ig. 2.3. Block diagram of twin-inertia drive with flexible shaft (a) Model of mechanical plant, (b) Reduced order model, c) Mechanical transmission with backlash 2.2 Two Axes Drive in Polar Coordinates On machine tools or robots there are normally several axes of motion, that rnust be independently driven or positioned. An example is seen in Fig. 2.4 a, where an arm, carrying a tool or workpiece, is rotated by an angle e:(t) around a horizontal axis. The radial distance r(t) from the axis to the cen ter of the mass M2 represents a second degree of freedom, so that M2 can bc positioned in polar coordinates in a plane perpendicular to the axis. The rotary and radial motions are assumed to be driven by servo motors, produc ing a controlled driving torque mM and a driving force 1M through a rotary gear and a rotary to translational mechanical converter, for instance a lead screw. With fast current control the motors are generating nearly instanta \ICOIlS impressed torques, serving as control inputs to the mechanical planto For simplicity, the masses are assumed to be concentrated in th.e joints, re sltlt.ing in the inertias h, J2 . The coupling terms of the motion can be derived by (~xpressillg t,he acceh~ration of the mass M 2 in complex formo dr !l . ) (2.8) (1' ( . .1 ' ('II I :i r w) (.3 ~ , dI (li ,, ~\ ,(J. )' (/L'II I_,.} .~ ;I ,. ) t' I (,..\1 " (, '~~ (LI 'fi I 'I' IllU ) (' j ' ' I (/.. !J ) . (1< i' ) dI ,11 ',\ ,II ~ , d' 2.2 Two Axes Drive in Polar Coordinates where w = dE; / dt and v = dr / dt are the rotational and radial veloci ties . After separating the terms of acceleration in tangential and radial direction and superimposing frictional and gravitational components, Newtons law is ap plied in both directions, resulting in the equations for the mechanical motion of the centre point of M 2 J r,,-_~A , dw (ll + h +M2r2) dt = Coriolis Gravitation ,...-"'-... ,.-"'-----.... mM - 2M2 rwv-M2 gr cose:-mF -mL, (2 .10) de: dt = w , (2.11) C entrifugai Gravitation dv M 2 dt = 1M + ,-"--.. M2 r w 2 ,...-"'-... - M2 g sine: - lF h, (2.12) dr dt = v . (2.13) The equations (2.10)-(2.13) are depicted in Fig. 2.4 b in the graphical form of a block diagram, containing four integrators for the state variables. Despite the simple mechanics, there are complicated interactions, which become more prominent with increasing rotary and radial velocities. The control of this mechanical structure is dealt with in a later chapter. Clearly, the two motions are nonlinearly coupled though gravitational, Coriolis- and centrifugaI effects; they are described by four nonlinear state equations. mF, lF and mL, h are due to friction and external load forces with may exhibit their own complicated geometric or dynamic dependencies, If it is important for the application to express the position of mass M 2 in cartesian coordinates, this is achieved by a polar-cartesian conversion x (t) = r cos e: , (2 .14) y(t) = r sine: . (2.15) Moving the arm also in the direction of the axis of rotation, so that the mass M 2 can be positioned in cylindrical coordinates, would introduce a third dccoupled degree of freedom . The dynamic interactions for a general motion, involving six degrees of r)'(~edom (three for the position, three for the orientation of the tool) are ex ('cedilJgly complicated, they must be dealt with when controlling the motions 01' Jllult.i ax( ~s rouots with high dynamic performance [M53, 013, 849]. 23 '22 2. Dynamics of a Mechanical Drive mLosd Friction MV J ,+J2+M2r2 rw2 grcos E gsinE I y/, xfLoad b Fig. 2.4. Two axes drive in polar coordinates (a) Mechanical plant (b) Block diagram 2.3 Steady State Characteristics of Different Types of Motors and Loads Consider first the steady state condition, when the torque and speed of a single axis lumped inertia drive are constant and the angle changes linearly with timej this condition is reached when mM - mL = O. With some motors, such as single phase induction motors, or loads, for example piston compres sors or punches, the torque is a periodic function of the angle of rotationj in this case, the steady state condition is reached, when the mean values of both torques are equal, mM - mL = O. The speed then contains periodic oscillations, which must be kept within limits by a sufficiently large inertia. The stcady state characteristics of a motor or 1030<1 are often functions given in graphical [o fi II , <:onnN:ting main varíilhl(~K, sllch as speed and torque; lh(' provisíou i:i 1.11;t\. :lllxili<tJ"y 01' COlltrol íllPIlII.H, ror ('X;UllPlc Sllpply voltage, li(.ld (·II!"l"('IIt., rll ' illJ ~ flllJ ~ It', hrw;h pm;ll.ioll 01" r,·(·" ru \.<', ;I],(~ lIlailll ,;ú,\(~d COIl Ht.iI.lIl.. I" 1'11\ '). ,1, /1 ,1,11,1"" 1.,Ypi(,iI.I ri II ll' Jl l'\.l' I'l ll I. Ic- II "I' (,1(,( ' 1.1'1(' JlJoI,ol"X 11.1'(' :; \iOWII. '1'1" . "11,\1 ",111,,,,,,,11 11" (h ll ' I,( · t."r1 rll. iI' li l ( IId ,y vtd l" 1,"' ( ' t lll::L ;UIl :;p(,,'cI, ::11\('(' 1.111' Friction r a 2.3 Steady State Characteristics of Motors and Loads variable is the load angle 0, i. e, the displacement of the shaft from its no load angular position. When the maximum torque is exceeded, the motor falls out of step; asynchronous operation of larger motors is not allowed for extended periods of time because of the high currents and pulsating torque, The electrical transients usually cannot be neglected with synchronous motor drives. The rigid speed of synchronous motors when supplied by a constant fre quency source makes them suitable for only a few applications, for exarnple large slow-speed drives for reciprocating compressors or synchronous gener ators operating as motors in pumped storage hydro power stationsj at the low end of the power scale are electromechanical clocks, The situation is dif ferent, when the synchronous motor is fed from a variable frequency inverter because then the speed of the drive can be varied freely (Chap. 14). With the progress of power electronics, these drives are now more widely used. mM Fig. 2.5. Steady state torque-speed characteristics of (li) electrical and (b) mechanical drives The "asynchronous" or "shunt-typ" characteristic in Fig, 2.5 a is slightly clrooping; often there is also a pronounced maximum torque. The lower por Lioll of the curve is forbidden in steady state in view of the high losses. With I.hree- phase asynchronous motors the rotor angle has no effect on the torque i Il steady state. Motors with "series-type" characteristic show considerably larger speed drop under loadj with DC or AC commutator motors, this is achieved by ;1. suitable connection of the field winding. The main area of applications at larger ratings are traction drives because the curved characteristic resembling ;1. hyperbola facilitates load sharing on multi pie drives and permits nearly collst.ant power operation over a wide speed range without gear changej this is par\.icularly suited to a Diesel-electric or turbo- electric drive, where the 1',,11 !)()Wl'r o[ lhe thermal engine must be used. For ('olllparisoll, SOIlJC t:ypical charactcrist.ics of a Lurbine anel a Diesel ('I\/'; i ll(' ai. ('ClIlHl.n,lll. flld illjc('.t.ioJl pcr Ht.I'III(' lU'" i'iC'('1\ iII l"il':. :~.S h. 2. Dynamics of a Mechanical Drive21. The curves in Fig. 2.5 a are "n~tural" characteristics which can be modi [ied at will by different control inputs, e. g. through the power supply. With dosed loop control a shunt motor could assume the behaviour of asynchronous O[ of a series type motor. As an example, typical steady state curves of a con l.rolled DC drive are shown in Fig. 2.6; they consist of a constant speed branch (normal operation) which is joined at both ends by constant torque sections aetivated under overload condition through current limito Figure 2.7 depicts Lhe steady state curves of the motor for driving a coiler. li the electrical power rcference is determined by the feed velo city v of the web or strip to be wound and includes the friction torque, the coiler operates with constant web force f independent of the radius r of the coil, PL = vi + PF· The steady state characteristics of mechanicalloads are of great variety; however, they are often composed of simple elements. This is seen in Fig. L.8 with the example of a hoist and a vehicle drive. The gravitational lift Lorque mL is independent of speed (Fig. 2.8 c); in the first quadrant the load is lifted, increasing its potential energy, hence the drive must operate in the Illotoring region. ln the fourth quadrant, the power flow is reversed with the load releasing some of its potential energy. Part of that power is flowing back Lo the line, the remainder is converted to heat losses. The lower half of the winding rope, seen in Fig. 2.8 a, serves to balance the torque caused by the ú) m = const 1 ú)tl r" = r:onst m =const Ú)2 L __ _ Torque limit m Fi~. 2.6. Torque/speed curves of controlled motor wiLlt constant speed branch and torque limit ú) ~ ---, r/'lM[)n lt\i,-,. ~ . 7 . ('~Iil~ll' dll\'(~ ('1) 1\~1· (" IL·HI(('~ 1 lIllel (II) I d, f,t,k IllIHI ('IIl"V(':1 2.3 Steady State Characteristics of Motors and Loads 25 mL = r 9 (MI - M2 ) (J) Gravitational torque mL =mL r fL = r9 M sin a a b c Fig. 2.8. Drives involving gravitational forces weight of the rope; this effect can be substantial on a winder for a deep mine, tending to destabilise the drive. All mechanical motion is accompanied by frictional forces between the surfaces where relative motion exists. There are several types of friction, some of which are described in Fig. 2.9. ln bearings, gears, couplings and brakes one observes dry or Coulomb friction (a), which is nearly independent of speed; however one has to distinguish between sliding and sticking friction, the difference ofwhich may be considerable, depending at the roughness ofthe surfaces. The forces when cutting or milling material also contain Coulomb type friction. ln welllubricated bearings there is a component offrictional torque which rises proportionally with speed; it is due to laminar flow of the lubricant and is called viscous friction (b). At very low speed and without pressurised lubrication, Coulomb type friction again appears. aú) /// /// /..,./ __--:/' / / /./ /i /// /b /~......,c ~ mL ú) a mG I I G .. II ravltatlona r---- torque I mL Fi~. 2.U. llirrnnnL typcs of frictional Fig. 2.10. Torque/speed curves of a l.o!''I''I: hniHI. d ri v(' 26 2. Dynamics of a Mechanical Drive With pumps and ventilators, where turbulent flow occurs, the torque rises with the square of speed; the drag force due to air flow around vehicles or l,!Je torque required by a centrifugaI blower pushing cooling air through an clectric motor also have this characteristic (c). ln practical drives, with the motor as well as load, all these types of friction cxist simultaneously, with one or the other component dominating. When driving a paper mill, printing presses or machine tools, Coulomb friction is usually the main constituent but with centrifugaI pumps and compressors the lorque following the square law is most important, representing the useful lllechanical output. Note that frictional torques are always opposed to the direction of relative motion. ln Fig. 2.10 various torques, acting on a crane under load, are drawn; me is the constant gravitational torque caused by the load of the crane, mF is the Cri ctional torque, resulting in the totalload torque curve (a). The speed Wo corresponds to the run- away speed with no external braking torque. When for safety reasons a self-Iocking transmission, such as a worm gear (b), is eruployed the crane must be powered even when lowering the load; this is due to the large sticking friction. 2 .4 Stable and Unstable Operating Points I~'y ignoring the dependence of driving and load torques on the angle of rota t.iOll, the corresponding interaction seen in Fig. 2.1 vanishes and so does the df'('ct of Eq. (2.2) on the static and transient behaviour of the drive. Equa I.joll (2 .2) then becomes an indefinite integral having no effect on the drive. Ir wc also neglect the electrical transients and the dynamics of the load, the n:lllaillillg mechanical system is described by a first order, usually nonlinear, d iífcrential equation but, in view of the simplifications introduced, its validity is rcstricted to relatively slow changes of speed, when the internal transients alld interactions in the electrical machine and the load can be neglected. dwJ di = mM (w, t) - mL (w, t) . (2.16) Apparently, a steady state condition with a constant rotational speed Wl is possible, if the characteristics are intersecting at that point, i. e. for mM (Wl) - mL (WI) = o. III order to test wltether this condition is stable, Eq. (2.16) can be linearised ai I.h(: o])(:ratitt /-Ç poiltt W I, assumiug a smn.ll displa(:ernent Llw. With G:J = (ti l I ,1 (cl , wc littd 1.1\(: lill('aris(~d ('quat.iotl d !\(,) lI'111 nl l ,\,,) Ihll.J, I !\/,; , .I ,/I IIt .J ' u I }(4 ) 1.11 2.4 Stable and Unstable Operating Points 27 or in normalised form, J dLlw + Llw = O â k dt where k = - (mL - mM) IWl (2.17)âw This is illustrated in Fig. 2.11 with some examples. The assumed steady state is stable for k > O; in this case a small displacement Llw, that may have been caused by a temporary disturbance, is decaying along an exponential function with the time constant Tm = J / k. k can be interpreted as the slope of the retarding torque at the operating point, as seen in Fig. 2.11. For k < O the operating point at Wl is unstable, i. e. an assumed deviation of speed incre ases with time; a new stable operating point may 01' may not be attained. The case k = O corresponds to indifferent stability; there is no definite operating point, with the speed fiuctuating due to random torque variations. 00"t ,"", O w, +4:t __ ~~4W<:/", .": =='1( 00, W, ---/ /--;/' ......mM / mL / mM mmL mM L ~ m m mk> O k=O k<O stable indifferent unstable a b c Fig. 2.11. Stable and unstable operating points cu L3 cu 3/ m --J.___...... unstableM mM) ~ ~ stable2/ ~ ~ L2 mL m m Fig. 2.12. Induction motor with differ- Fig. 2.13. Rising torquejspeed curve as ent types ar load cause of instability Figure 2.12 depicts the steady state characteristic of an induction motor (7I/.M) /.ogd.l\(:r with some load curves (Sect. 10.2). LI could be the character i ~ l.i(' Or ; l vClll.ila tiug fali ; thc intersedioll ] is s t.ahlf:, rOltghly corrcsponding to IICIIll i ll HI I"ad . Wil.h IJ~ 1.I11~J'(' is N.l SO ;t fi l.nl)I, · "1'<')'ni..iJlI '; pnill(.) bltt. tlw 11101.0)' 11'1 2. Dynamics of a Mechanical Drive would be heavily overloaded. With the ideallift characteristic L3, there is an IIllstable (3) and a stable (3') operating point, where the motor would also h(' overloaded. In addition, the drive would fail to start since the load would pllll the motor in the lowering direction when the brakes are released and I.!lere is insufficient frictional torque. A particularly criticaI case is seen in Fig. 2.13. A slightly rising motor characteristic, which on a De motor could be caused by armature reaction dll( ~ to incorrect brush setting, leads to instability with most load curves n ccpt those intersecting in the shaded sector. The stability test based on a linearised differential equation does not fully (:xdude instability if electrical transients or angle-dependent torques should II:tvc been included. The condition k > °is only to be understood as a Iwccssary condition, even though it is often a sufficient condition as well. 3. Integration of the Simplified Equation of Motion With the assumptions introduced in the preceding section the motion of a single axis lumped inertia drive is described by a first order differential equation, Fig. 3.1, dw J di = mM (w, t) - mL (w, t) = ma (w, t) , (3.1) which upon integration yields the mechanical transients. Several options are available for performing the integration. mM' úJ, ê =E) ~ ~- ~} mL Fig. 3.1. Drive with concentrated inertia 3.1 Solution of the Linearised Equation In the linearised homogeneous equation, Eq.(2.17), Jd(Llw) + Llw = 0,Tm~ Tm = k (3.2) is Llw a small deviation from the steady state speed Wl and k = ~ (mL - mM)1 (3.3) âw w] ii. II 1<:<l.SII rc ror UL(~ I·;]ope of th~ rr.tardill j.z; I,()rqll\~ aI. Lhe operating point. Tm =: .I/ J..: 1111 1' UI(" IIW:tllilll', of a IlIcchalliraJ tilli(' rolii i l.nlll.. 'I'll!" 1~( ~ II( ~ral KO!lll.ioll iK 31 :30 3. Integration of the Simplified Equation of Motion t / TmL1w(t) = L1w(O) e- (3.4) where L1w(O) is the initial deviation, that may be caused by switching from one motor characteristic to another resulting in a new steady state speed. Because of the stored kinetic energy, the speed is a continuous state variable (the only one because of the simplifications introduced). cu ,ócu mL ---,mM1;\w(O) m a b Tm F ig. 3.2. Mechanical transient of linearised drive system Til Fig. 3.2 a it is assumed that the motor, initially in a steady state ('(Jlldition at point 1, is switched to another supply voltage so that a new ('iJaraderistic for the driving torque mM2 is valido This causes an initial devi ;i.I.io ll L1w(O) with respect to the new operating point . The deviation vanishes IIlolI/,. ; LII exponential function (Fig. 3.2 b), the time constant of which is de- 1."l'llIillpd hy the inertia and the slope of the load- and the driving-torques. 'l'hiN is di scu::ised with the help of some simple examples. :1.1. 1 St.art of a Motor with Shunt-type Characteristic at No-Ioad fi. IJlolor which initially is at rest is started at t = O, Fig. 3.3. The 1 .() I'Ill\"jsp( ~cd-curve of the motor is assumed to be linear between no-load liI)('( 'd WI) (lIeglecting friction load) and stalled torque mo'. With normal mo 1.111:: or lIwdium size, the stalled torque may be 8 or 10 times nominal torque; i I. is dd.cnnined by extrapolating the drooping shunt-type characteristic to Hla lldsLill anel hence represents only a reference quantity that cannot be mea :: lIl'cd iII pmctice. However, in the present example we assume that a starting 1'(·::i::(.OI" lia.s been inserted, reducing the stalled torque to perhaps twice nomi lI a l Lorqlw alld thus extending the validity of the linear characteristic down to HL;\.lIdsLill. This, by the way, makes the simplifying assumptions more realistic :l llIn : Lhe dectric;J.l tmnsicllts becornc faster while the mechanical transients alT <lday("1. 1111.1,'1" id,':d 110 · 10;),.1 ('olldilioIlS, w(~ hnv(' '11/./ , (l; nlso, (rll e Lo tlw a:;sllllled lill( '1 1I iLy 01' i.lw I'IlrV( 'I: '//1. n/ (Ui), "II/.d lt! ) 1.11(' d( ·vill l.i()1I / \ ,,/ i:; lJoL ", ·:: I,l'i('I.,,<I 1.0 :l lIl ldl v.dll'· : 3.1 Solution of the Linearised Equation The slope of the retarding torque is â mok = - (mL - mM) = - > O;âw Wo hence the differential equation is stable, it has the form d(L1w) + L1w = O, Tm~ (3,5) where JwoTm mo is the mechanical time constant.The steady state operating point at the in tersection of the two torque curves is Wl =wo; due to the initial condition at standstill, we find L1w(O) = -Wo, which leads to the particular solution t / TL1w(t) = -Wo e- = or wi th w = Wo + L1w w(t) = Wo (1 - e- t / T=) . (3.6) From the torquejspeed curve of the motor ~ =1- mM (3.7)Wo mo the motor torque during the start- up is obtained (Fig. 3.4), t / TmmM(t) = mo e- . (3.8) The discontinuity of the motor torque is caused by the omission of the elec trical transients. ln reality the torque is also associated with energy states and, hence, is continuous. cu mM cu cuo' mo 1~k--------cuo'" t~ 00 Uort0 mM - mL=O t =O mà m Tm I"il;. :1 . :1. SI.fIl' l.illl'. "r 1l1ll1.0!' ai. 110 10ft" Fil-'( . :1.'1 . St.art,iIl14 t.rall fi it~ lIl, 33 :1:2 3. Integration of the Simplified Equation of Motion :1.1.2 Starting the Motor with a Load Torque Proportional to Speed 111 Fig. 3.5 the curves of a drive with speed-proportional load torque are displayed. The steady state operating point is now at WI < Wo, hence L1w(O) = ~ WI' úJ {Oo.... mM (1)1 m1 m I"ig. 3.5. Starting of motor with linearly rising load torque The slope of the retarding torque is ô mo kl = - (mL - mM) = - > O, (3.9)ÔW WI wlüch leads to a reduced mechanical time constant, JWI = WI T . Tml m (3.10)= mo Wo TIl\' speed transient is W(t) = WI (1 - e- t / Tm ,). (3.11) 1';lilIlination of speed from the motor characteristic Eq. (3.7) results in the. d ri vi11g torque t mM(t) = mo (1- :J = mo [1- :~ (1- e- / Tm ,)] =mI + (mo - mI) e- t / T"" . (3.12) I lol.h tnl.nsi(~J1l.s af( ~ plol.ted in Fig. 3.6, together with the no-Ioad starting 1.1';t.J1Hi(~J1(. (111./, - O) . AI. I. = 0, t.he Rlopc of tltc specd curves is the sal1lC, 1)('("a.lI :ic- tlI(" ;,1<'("(,1('1 n l.illl', l.orqll(,i' ;1.1"(' idcul.ieal n.1. (v · O. 3.1 Solution of the Linearised Equation w mM Wo mo 1 1 W1 m1Wo mo " " , ' ....... mL =O ....... .... tTm1 Tm Tm1 Tm t a b Fig. 3.6. Starting transient with load 3.1.3 Loading Transient or the Motor Initially Running at N o-load Speed The motor is now supposed to be initially in a no-Ioad condition at the speed Wo with the starting resistor short circuited, so that the speed droop is reduced and the extrapolated stalled torque assumes its large nominal value mOn· The torque/speed-curve is then linear only in a narrow speed range around no-Ioad speed. úJ t = O ........................................ mL ................ mM ................ ........ m1 mOn m Fig. 3.7. Applying constant load torque to motor running at no-load At t = O a constant torque load without inertia is applied to the motor Iii I iloft, possibly by a mechanical brake, Fig. 3.7, causing the motor to slow dowlI [.0 Lhe steady state speed wI; hence the initial deviation is L1w(O) ( ' ) 0 W I . III vil'W <Ir I.Iw ("ow;L;mt load torqll(', WI: lilld 35 :11 3. Integration of the Simplified Equation of Motion JwO = T · (3.13)k = ~ (mL - mM) I mOn Tm mn = mOâw Wo nWl Sillce mO corresponds to the extrapolated stalled torque with short circuitedn sl.arting resistor, Tmn is also called short circuit mechanical time constant. The solution of Eq. (3.5) is t / TmnLlw(t) = (wo - wd e 01' t Tmnw(t) = Wl + (wo - wd e- / • (3.14) The driving torque is again obtained from Eq. (3.7), t Tmn mM(t) =mOn (1- ~J = mOn (1- ~:) (1- e- / ) e- t / Tmn= mI (1 - ) • (3.15) III Fig. 3.8 the transients are drawn for loading and unloading the motor. mM ..ill.. - - womOn 11 ___ 1_ IL-1 ----1- ..ill.. 1 (O Wo~~-~ (1)0 1 1 mM 1 1_____ _mi mOn =-=- I'm() n 1 Tmn t 1"11';. ~L8. Loading transient 'l'l\( ~ drivillg torque apparently follows the applied load torque with a lag d!'l.(~rlllillcd by Tmn . The reason for this is that during the deceleration phase hOl1W of til(' kinctic energy stored in the inertia is released and makes up for pMI. 0[' I.he loat! torque; when the drive is accelerated after disconnecting the !O,H!, I.he 1l1ÜiSill~ kinetic encrgy is restored. Thc mcchanical energy storage t.ltll S act.s HS a hl\fr(~r !>dweell lhe load a.nel lhe ek('(,ri(:al sllJ>ply fccding the I HO l.or . Tltin dr,'d Ill ay b(: n.cn~Ilt.\1at.(:d hy iI.dtlilll~ It f1yw'llt'd lo Lhe: drive:, (II IIl'Ikl' l.tI 1'1'1",,'( ' " tllt' :lllpp! y fiyHt.(1II1 r""11I h ip,h
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