Control of Electrical Drives, 3ª ed Werner Leonhard

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Werner Leonhard 
Contrai 
of Electrical Drives 
Third Edition 
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U.F.M.G.· BIBLIOTECA UNIVERSITÁRIA 
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26780505 
NAo DANIFIQUE ESTA ETlQUETA 
With 299 Figures 
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_._._-- - _ -~­.. ~ . ~ 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 
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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) 
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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