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Werner Leonhard
Contrai of Electrical Drives Third Edition F~J= ~~D1,~
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Preface
Professor WERNER LEONHARD
Tl'chnische Universitãt Braunschweig IlIsti tut für Regelungstechnik II a ns Sommer Str. 66 ,\1) 106 Braunschweig I' II/(./il:
[email protected] 1/llcrnet: http://www.ifr.ing.tu-bs.de
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:,~()-41820-2 Springer-Verlag Bedin Heidelberg NewYork
I 01" ,II Y "I ( :0111'.1'
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 '(: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
1)(
Introduction ................................................. .
1
1. Elementary PrincipIes of Mechanics ........ . ............ .
1.1 1.2 1.3 1.4 1.5
7
Newtons Law ......................................... . 7
Moment of Inertia ............. .. ...................... . 9
Effect of Gearing . ..... .......... . .............. ... .... . 11
Power and Energy .. .. .......................... ...... . . 12
Experimental Determination of Inertia ................... . 14
2. Dynamics of a Mechanical Drive ......................... . 2.1 Equations Describing the Motion of a Drive with Lumped
Inertia .......................................... . .... . 2.2 Two Axes Drive in Polar Coordinates . .... .......... . .... . 2.3 Steady State Characteristics of Motors and Loads ......... . 2.4 Stable and Unstable Operating Points .................... .
17
3. Integration of the Simplified Equation of Motion ........ . 3.1 Solution of the Linearised Equation ............. . ........ . 3.1.1 Start of a Motor with Shunt-type Characteristic at
No-Ioad ........................................ . 3.1.2 Starting the Motor with a Load Torque Proportional
to Speed ........................ . .............. . 3.1.3 Loading Transient of the Motor
Initially Running at No-Ioad Speed .. . .. . .......... . 3.1.4 Starting of a DC Motor
by Sequentially Short-circuiting Starting Resistors ... . 3.2 Analytical Solution of Nonlinear Differential Equation ...... . 3.3 Numerical and Graphical Integration ..................... .
29
29
4. Thermal Effects in Electrical Machines .................. . 4.1 Power Losses and Temperature Restrictions ............... . 4.2 Hcating of a Homogeneous Body ....... .. ....... . ..... . . . 1.:~ Difrer('lll: Morles of Operation ...... . . .............. ... .. . '1.:\.1 (;()ilt.illIlOIlS Dllty ..... ' ..... . . .. . . ...... . . . ...... .
17
20
22
26
30
32
33
35
38
39
43
43
44
48
1K
x
Contents
Contents
4.3.2 4.3.3 ~ ',.
O.
"(
Short Time Intermittent Duty ..................... 48
Periodic intermittent duty . . . . . . . . . . . . . . . . . . . . . . . .. 49
SeparateIy Excited DC Machine . . . . . . . . . .. . . .. . . . . . . . . . .. 5.1 Introduction .................... ·.····················· G.2 Mathematical Model of the DC Machine ..... ..... ........ G.3 Steady State Characteristics with Armature and Field Control 5.3.1 Armature Control . . . . . . . . . . . . . . . . . . . .. . . . . .. . . . .. 5.3.2 Field Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 5.3.3 Combined Armature and Field Controlo . . . . . . . . . . . .. !:iA Dynamic Behaviour of DC Motor with Constant Flux .. . . . ..
51
51
54
56
57
58
61
64
DC Motor with Series Field Winding . . . . . . . . . . . . . . . . . . . .. 69 0.1 Block Diagram of a Series-wound Motor. . . . . . . . . . . . . . . . . .. 70
G.2 Steady State Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
. ControI of a Separately Excited DC Machine .. . . . . . . . . . .. 77
7.1 Introduction .................... ·······················
7.2 7.:3 'IA K.
77
Cascade Control of DC Motor in the Armature Control Region 79
Cascade Control of DC Motor in the Field-weakening Region 90
Supplying aDC Motor from a Rotating Generator.. . . . . . . .. 93
Static Converter as a Power Actuator for DC Drives ..... 97 ~u Electronic Switching Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 97
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
10. Syrnrnetrical Three-Phase AC Machines .. , ............... 163
10.1 Mathematical Model of a General AC Machine .......... ·.· 164
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 . .
' .... . . . . ........ ,
••• . . • , .. . . , •• . . . . . . .
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 ~; 111.1."
"
. . •• , • , •••
190
:J.m
1004.1 10.4.2 10.4.3 10.4.4
Symmetrical Components ......................... Single-phase Induction Motor ........ ..... .... .. ... Single-phase Electric Brake for AC Crane-Drives ..... Unsymmetrical Starting Circuit for Induction Motor ..
XI
202
206
209
211
11. Power Supplies for Adjustable Speed AC Drives . ... ...... 215
11.1 Pulse width modulated (PWM) Voltage Source Transistor
Converter (IGBT) .................... . ................. 11.2 Volt age Source PWM Thyristor Converter ................. 11.3 Current Source Thyristor Converters ....... ........ ..... .. 11.4 Converter Without DC Link (Cycloconverter) ..............
218
225
232
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 ........ 15.1 Speed Controlled Drives ................................. 15.2 Linear Position Control ................................. 15.3 Linear Position Control with Moving Reference Point ....... 15.4 Time-optimal Position Control with Fixed Reference Point .. 15.5 Time-optimal Position Control with Moving Reference Point .
363
364
373
383
389 396
Abbreviations and Symbols
Bibliography .................................................. 402
IIHlcx .... , ........... , ........................................ 455
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.
1:, I d , u, Ud , etc. I, U, etc. U, etc. i.(t), :!!(t), etc.
L
f (t), :!!* (t), I*, U', etc. li(t), l:!!(t), etc. 1(8) = L(i(t)) etc.
instantaneous values average values RMS-values complex phasors for sinusoidal variables complex time-variable vectors, used with multiphase systems conjugate complex vectors or phasors vectors in special coordinates Laplace transforms
3 Symbols
Abbreviation
Variable
Unit
a(t)
current distribution linear acceleration nondimensional factor area nondimensional field factor magnetic flux density clectrical capacity theonal storaw' capnci t.y
A/m m/s2
A b D C "
f)
dlLl II pill r~ r a Lio
m2 T = Vs/m 2 F = As/V J;oC = Ws;oC
,(/), I ~, E
r ,,' (.'i ) 'I 'I (I)
"
(
ii ,(1) , I , ,/
I
I,;
, " /I
1/1,( /) II/
'/I
N
I' ( /) , I' () /'
I,'
" :1 \
I J
W
"
," '
/ "
'
/I
(I), II,
" (I) \I
/1/(1 ) ,/'
:'1 ('.'.""
\' /. /I'
Abbreviations and Symbols
A bbreviations and Symbols
'\IV
II
induced voltage, e,m.f, , frequency force transfer function gravitational constant unit impulse response weight gain airgap current inertia nondimensional factor torsional stiffness length inductance torque mass mutual inductance speed, rev /min
number of turns
power
reactive power
radius
resistance
Laplace variable distance
slip
time
time constant
voltage
velo city
unit ramp response
volume
unit step response
energy
control variable
actuating variable
disturbance variable discrete Laplace variable admitt.a.nce illlj)('dallce 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 m 2 Nm/rad m H = Vs/A Nm kg H l/min
I
I
i
1\ ,I
!' I
W VA m
n
I
0:, {3, 15, (, ç, >., /l, {2 etc. angular coordinates "1= 21f/3 15 load angle L1 difference operator angle of rotation ê efficiency ." {) temperature absolute temperature magnetomotive force, m.m.r. e coefficient of permeability /lo v integer number (J' leakage factor normalised time, angle T = J w dt, wt phase shift
(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 dt .
Ml2 3
(1.19)
h is the force driving wheel 2. Since the forces at the point of contact are in balance and the two wheels move synchronously,
Ml2 12
h=h, ( ~ Iimination
O b
[I
1,'1,,;. I.r., M"I","I,
"r \ '"' 1' 1. 1"
"I' (\
,."eI, piv",-,'d
_J~
2 0111. ,,(" , " '111.,-('
of h,
1 ~mMl
rI Wl
=
= r2 W2
(1.20)
,
h, W2 results in
dW l dt
.lI -
tlWI
.II,' dI.
ri
dW2
+ r2 - 1 2 - dt =
[
II
rI
1dWI
2 lz + ( -) 1'2 dt
(1.21 )
1. Elementary PrincipIes of Mechanics
12
1.4 Power and Energy
dw wmM=wmL+Jwdj'
v
13
(1.26)
where PM = w mM 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.
~IIJII :
2r3
M3
F ig. 1.6. Effect of gearing on inertia
h + (::) 2 J 2
= J 1 + (::) 2 h + (::) 2
[J3
+ M3 r~l
'
mMl
= J 1e -d
t
W3
+ -Wl
r3g M 3
.
(1.24)
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
w. /, I .I ti,
Mlll !. q,l ill d ,IIIII I,y
( 01
,vi.. ldn UIos il.iollf:(} vcrtically dC(l':lIdillf!: OH th,: angle of rotation of 1.11
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 J 1 , J 2 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,
Power
supp/y
YM
o
Motor
co
mM
mL
me Fig. 2.1. Simplified block diagram of lumped inertia drive
me
~1fl>, Motor
))K)(1 me
Cü2,E2
J2 Load
=K
(2.3)
(C1 - c2) ,
and neglecting internal friction effects, the following state equations result
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 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,
J
dw
di = mM (w,
= ma (w,
t) - mL (w, t)
(3.1)
t) ,
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), d(Llw) Tm~
+ Llw
= 0,
Tm
J
is Llw a small deviation from the steady state speed
k = ii.
(3.2)
=k Wl
and
~ (mL - mM)1 âw
(3.3) w]
II 1dweell lhe load a.nel lhe ek('(,ri(:al sllJ>ply fccding the 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 fiy Ht.(1II1 r""11I h ip,h !, 1I11 1 11111'1( " ;1 Hllrll n~i ti IO/i(> ('1I 1l 1.~ , d 1,1' 111>1'1'." 10 11 1111 ', IlIill d dvl 'rlj """VI1l'lll'l y, II /11' 11 /111. 11', , 1,,1101 , 1'''1 "',11.1111'' ' ii
I HO l.or .
From Fig. 3.10 the following relation is found rn,2 1/1,2
mI
w
(v + 1) - w (v) Wo - W(v)
Wl,idl 1" ':-:1111.::, wil.lt Ul(: ;-\.IJI,rnvia(.joll '11/.1
(3.16)
jm./,
U·
-I, iII a. recllrsiv(' forlllula
:~ (;
3.1 Solution of the Linearised Equation
3. Integration of the Simplified Equation of Motion
37
mM
w (l)(N)
m1
(()(v+ 1)
(ú(v)
(ú(1)
..... ..... ..... ,>,
I
w(O)=O jo'i~. 3.10.
m
m:z=mo{O)
1
•
mdO)
m
: - Tm(I)--: I I I I
Torque/speed-curves for different values of armature resistor
cu W
CUo
(1/ + 1) = (1 - a) Wo + aw (v) = w(l) + aw (v) .
i\SSlIlllÍug w(O)
+ a) w(l) (1 + a + a2 ) w(l)
w(:\) =
(,!( II)-CC (1
I
I
I
I
-------i------I I I I
= O, this leads to a geornetric progression
w(2) = (1
I
I
I
a
CU 1
b
+ a + ... + aV-I) w(l)
Fig. 3.11. Starting a motor by successively reducing the starting resistor I.h,~ ~;1I11l
wit.ll
1,1(11)
I
Il
w(t)
v
w(l) = (1 -
I -- a
a
V )
(3.17)
Wo .
'1'11
II 'I I L I,
(5.14)
", I,i,·11 1"'IH'('SCllts ali envelope for the family of torque/speed-curves with pa I ; t.lII'·1,1"I' I, . lVIaximum power is obtained at the point, where the straight line i:: 1. :1.1'1';"111. 1.0 the hyperbola. To avoid the situation mentioned before, the
sl!oltld lIever be operated at the right hand si de of the tangent point.
11101,01
~
IIlachillcs without compensating windings show a pronounced arma in the field weakening range because the field distribution in 1.1 1
("( ;a("(.i O Il
61
ia iao
(1-~)
(5.15)
iao'
which only depends on the normalised armature current. Rence, with lim ited armature current, the motor can be operated at variable speed with a given maximum power , indicating that field weakening has a similar effect as a variable gear for increasing the speed of the load. For the reasons al ready mentioned and because of the mechanical stresses, the maximum speed achieved by field weakening is sei dom higher than two or three times base speed depending on the design and the size of the machine. Rowever, there are exceptions, such as on vehicle or servo drives, where a much higher speed ratio may be necessary. The similarity of a speed increase by field control and by a variable gear is corroborated by their effects on the mechanical time constant. At nominal field, Pe = Peo, the nominal time constant is Tm = Jwo/mo, where Wo is the base (no load) speed and mo the extrapolated stalled torque at nominal armature voltage and field . With reduced field, the no load speed is raised to b-lwo , while the stalled torque is lowered to b mo. Rence the mechanical time constant is a quadratic function of the field factor,
Tm (b ~ 1) = J Wo b- 2 == Tm b- 2
•
mo
A similar effect is observed if the load is coupled to the motor through a step-up gear having the speed ratio b- I ; as explained in Sect. 1.3, this would increase the effective inertia of the load, as seen from the motor side, by b- 2 • 5.3.3 Combined Armature and Field Control
From the preceding paragraph it is seen that armature and field-control of De motors each have their merits but that their normal applications exclude each other. Below base speed Wo the main :fl.ux is best kept at nominal value PeO while the speed is varied through the armature voltage U a . This is called the base speed- or armature control range. When the nominal armature voltage bl a O is reached, a further speed increase is only possible by lowering the l\laill flll X, thus creating separate field-control regions which extend the base ;) I w l ~ d r a ll g (~ iII cadl (lirectioll of rotation . Thi s is depicted in Fig. 5.7. li. i ~: fI(' I'1I I.ltal. e= 4>eO
-2'71:
07 -1
Field contrai Ua == -UaO 4>e< 4>eO
Commutation limit
I
I
I
Ua = UaO
§I
I
0- I
II_I ....:u· • .'t1J' 2 ~
._ ~
,
\ili \
\
\
.
\
1-2
'
'
/
I
,
mM
ua- O mn Ua = -uaO
I
,I
Fig. 5.8. Operating regions of separately excited DC motor in torquejspeed plane
The figure clearly shows that the DC motor represents a linear control plant only in the armature control regionj beyond base speed, there are con siderable non-linearities even in steady state condition. Simultaneous field- and armature-control is sometimes proposed as a means of reducing motor losses through diminished copper losses in the field windings and iron losses in the armature; of course, this is only applicable at very light load when the increased conduction los ses in the armature are not masking this effect. AIso, this may cause an objectionable delay in building up motor torque, should an unexpected load surge occur, because the flux would first have to be raised to full value. Hence, this practice can only be a lIIurr;inal option in few applications, such as battery supplied electric car dri v(~s, w II\~ rI ' (~I\( ~J',.;y conservation is of overriding importance and where the dyll;Ullil' drawll1l.('h :: ;I.r
Acceleration
at no load
(6.18)
-~ b
Wc
f4\ ia -.::..;/_ia1
Wl
- !b- ! - !r -R a
UUal
- RI
-
ie ia
(J)
®
® 8)
Load applied
mL mI
=0,25 =const
Nominal operation
Fig. 6.5. Transient of self-excitation of a series-wound motor 0,5
1 1
I
4 1
I
0,5
p
- p; =const 1111
"" t< ' liA .
75
:-1 1,' ·II .~ y .'I l.Id." .. l. ar l"·C," ill l,i" r! I,r 11 11' \1' 1" 11
\V""""'
1)1 :
,,,,,1.,,,'
It is assumed that the field winding has been reversed and the circuit is elosed through a braking resistor. The motor is running with W > O, so that braking calls for operation in the second quadrant of the torque/speed plane; it is controlled by the effective armature circuit resistance, Eq. 6.15. A detailed analysis of the braking transient requires the inclusion of iron saturation, but the principIe may be deduced qualitatively from the curves in Fig. G.a h, WIICl'C e(ia) is the induced armature voltage at a given overcriticaI :·qH'd nlld U. '':" i:; t.IlC voltage across the total braking resistor including the 1'(·:1i: ;1.1\1I("(· 11'." 1 ,"I'!.lll' arlllat\l),(~ itsdf. III Lhe initinl pha..se, the current is elriven II,Y 1.1", 1" '11 ' fi 111'11' ,. 1'1111.11.1',(' ('II, wil,ltolll. ,'x·I.('J'lIal V()It. nl ~(' lwillf~ appli(~d . Tlw
7(;
6. DC Motor with Sedes Field Winding
e, Ria
e(ia , (O) (O
R
-mn
mn
2m n
m
liamaxll
~
c::
C> ()
- -Ú)2 - , - - - - - - - - - - 11 1
"'i~.
c,rol
lia
lUa
Rotating generator ar converter
I lU a
7.1. Steady-state torquejspeed curves of a controlled drive with torque limit
angular position must be controlIed according to prescribed references. AIso l11achine tools or robots, the feed drives in the different axes must accu ral.dy follow prescribed angular commands as functions of time, in order to 1I10V(' the tool or workpiece along a desired spatial trajectory. Ali these requirements can be fulfilIed with the general scheme shown in Fig. 7.2 where the armature and field of the motor are fed from separate con I,mllaillc power supplies. Rotating generators in the form of the welI known Wal'd Leonard scheme have been the common choice in the past but, after l, ri,;I" 111(.('rllldes by magnetic amplifiers and mercury-arc converters, power ,'I,~,·(.r,,"ic COllvcrters employing solid state semiconductor switching elements (\';lli"II:: I.ypes o[ thyristors or power transistors) are now the standard solu
()Il
I. i" II,
W iI, h a "r()1If quadrant" armature supply allowing both polarities of volt
1\.,"1 Cllrl'(~lIt, the power flow to the motor can be reversed, with the III III 'I,ill" "perating in alI four quadrants of the torquejspeed plane. This is 1.111' (':1."', ir a rotating DC generator driven by a line-fed AC motor serves as n, P"WI'I' :-iIlPply; the sarne can be achieved with switched converters. With 1I11I/',II('l,i.: alllplifiers, containing only diodes instead of switches, regeneration WIL; : 11,,1. possible. Of course, there is always the option of electrical braking I',Y "laóllg resistors in the armature circuito 'I'he :-;tcady state power required for the field winding is only a few per ,','11(. of I.ltat for the armature; a "single quadrant" rectifier, controlIed or 1II1'·oll'l.rolkd is adeqllate for feeding the field winding. li"i/',. '1.2 contains a block "control equipment" which is controlIed by ref "I"~I((," ~;i/';Ilal:::; én,'!, wne!, CXRe!, 'ia mal" as welI as feedback signals é, w, a, I.,,, a~ i 11,:,''](' V,,
I, I" i ~( •
[k ~,.1, 1''llli 'l! d l"nl l1 J1l f' ct t 1)(:
1111
ai.II I'
rl l,1.
"..
1I 111 1lI 11 11 1I 1'" \'\11' 1
II l'pl Y, 'I',
'/' , II
('(lIll.lolln 11:1:: 1.(, 4 Ta) the lead time constants T l , T 2 of the PID-controller could be tuned to Ta and one of the emerging lag time constants of the drive, while the rest can be dealt with by approximation on the basis of the principie of "equivalent lag" , e.g. [38]. The closed current controlloop is now inserted into the speed loop form ing the next higher levei of control, as is shown in Fig. 7.8. Provided the current loop is well damped, it is acceptable to replace its closed loop trans fer function by a proportionallag element having the gain Gc L,i and the time constant Tc L,i' Since this equivalent lag, being a first order, low frequency approximation of a much more complicated transfer function, cannot be sim plified by lead time constants, a PI-controller for speed is the appropriate choice. Hence the transfer function of the open speed loop is approximated by
FOL,w( S) ~ G ow
1 Tc,w s + 1 GCL,i T c ,w S TCL , iS + 1 T m S
(7.4)
'-v-'~
PI· Controller
l.i()lJ,
Current control
The pa rameters G c , w , Tc, w may be chosen in accordance with the "symmetri cal Opt.illIHIlI" IX 2:i,L33,38], which is a standard design procedure for transfer fllJldioll s cU Ilt.a i II i II g a double integratioll (indllding the controller). '1'1", ll lll,i ll i el" II, ln I.() c11()W, (~ t.h(~ cross-()vc'r frc'CjIl('IWY at t\w gC'ornetri c IIlca n III' 1.11 ~~
-
.o~.~~,--------------
---t
Fi,l. 7.111. (~I)III.I' 1)1
e max
lil'ill'llIl: ()[
WMel L(~OIlIl.rd · drivc:
!II'
7. ContraI of a SeparateIy Excited DC Machine
drive assumes well defined operating characteristics and may be inserted into n,lIl.olllatic production processes. By exciting the generator and the drive mo 1,,1 I.ltrough static converters with high ceiling volt age (2 to 3 times steady ::Iill.(' voltage) the speed of response of the drive may be sufficiently improved 1m Illeding most dynamic specifications. 'l'lw control structure shown in Fig. 7.16 is essentially in line with that in I"il': · 7.13. The only difference is the possible introduction of an inner control loop ror the armature voltage U a of the generator [876]. 8ince the pertinent 1'()Jlioll of the control plant comprises only the field time constant of the 1',('llr
blocking
reverse blocking c)
. I
o
o •
-u
~
!ig
f>(
o
-u
Control/ed rectifier Thyristor
Gate tum·of( Thyristor (GTO)
conducting
/ ~. JI , VO / !Jt l
" " '/Iludiu!}
"J
, conducting
~"
electronic \ control
forWârdU blocking
u
u
Insulaled gale bipolar Transistor (IGBT)
... l,~ .
H.
\
e)
electronic contraI
:. . ... .'. . .. .
blocking u
electronic \ control"-.., conducting
-1 \f.
~
,
blocking
1
i
LfJt
Ug2
--.J \.
/
L
~ u Symmetrical switch (IGBT)
I. I)II COII trolleel anel controlJed electroni c switdlC::;
III I.he pax(. Lhe fllllct.ion of controlled d(~cl.J'()lIi (' x w;Ldws lias been realised with vm;"lIs dl'VÚ'('S Ill,tkiug \l*~ o[ diJkre eurly Ilrtllll,ioll: ; nll' 1I1l/l11kf,(' (.odny. Siul't: l'I(\('.(.I'OIl Llrl1oc 'lI wít.1t Itl.'i t!.l:d· cal.hodl~s cOllld 1I"t. IllIpply "II"II I \ IJ l ' IIIT"IIt. 1,0 drivo IIIOt.OHI II JltI \\11 ' 1< ' pl nl~ll 7r, the thyristors 1, l' cannot be made conducting since they are reversely biased. This is also seen from the curve of the volt age across the thyristor, UTI (T), drawn in Fig. 8.6 c for different firing angles a. Later the range of the control angle will have to be further narrowed. The trace of the thyristor current iTl (T), also shown in Fig. 8.6 c, elearly indicates the switching action of the thyristors; at any time at least one of the quantities UTl (T), iTl (T) is zero. ln steady state condition with a = consto the volt ages and currents in thyristors 2, 2' pertaining to the opposite diagonal pair are identical, being shifted by a half period. Finally, in Fig. 8.6 d the input current iA(T) for the three different values of the firing angle a is drawn. As a consequence of the periodic commuta tion between the two pairs of switches the impressed continuous current ID appears at the line side of the converter as a square wave alternating current iA(T), whose phase lag ep with respect to the line volt age UA(T) is prescribed by the firing delay, ep = a. The active power (mean of the instantaneous power) at the line side is - assuming sinusoidalline volt age determined by the fundamental component iAl (T) of the line current
PAC
= UA IAI
cosep
2V2 =- UA ID 7r
cosa,
(8.2)
whe['(~ (J II, I II' are H.MS-values of the line voltage and the fundamental cur n:lll. ('.OIIlj>OIlC·I\I ,. For LI < a < ~ , the power flows from the AC - to the DC :;id", wilil., ("r :: . 11 ' n""",,, (./1I~ (lO WN f10w is ITv( ~ rs( : d.
104
8. Static Converter as a Power Actuator for DC Drives
8.2 Line-commutated Converter in Single-phase Bridge Connection
105
rot
a
Inverter I Rectifier
IX
1
=
lX.2= 90°
30°
a=7C
(X3=150°
2 Fig. 8.7. Phasor diagram of a single phase converter
rot Uo
blocking voltage V,T > O is created for a < O, during which the thyristor can be fired. Forced commutation calls for additional components and thyristors with short recovery time which involves increased cost and complexity; this wiU be discussed later.
2.
rot
Üo
-u 11: Reclifier
c lX.2
«( I
'II I
I-----
.
~ ~ /II
_'AI \
/
I
- , 1\ Ir
,
I
I
rot
I
~ ~
I
7C
I
\ I
\ ~
rol
~Tr
I
'_I
,. ~Al . ----T-J IA
I
I
\
I
I
iA \
I
I
a
a3
,~
Inverter
\
t7' \
/
d 1"'1(' H.G. V()ltages and currents of an ideal single phase converter
'I'II(~
first mode corresponds to controlled rectifier-, the other to inverter
"llCral.iof! . 111 Fig. 8.7 a phasor diagram is showing the phasors of the line volt !l~" I J A
aud of the fundamental component of the line current,
I Al .
Clearly,
a cllul.roUcd converter appears as an inductive load at the line side, drawing Ja.j':I':ill p; J'( ~ a("t.iv(! cunent; this is due to the control principIe allowing only ,kla.y ( ~ d lirill,~.
Fig. 8.8. Control curve of phase controlled converter
The possibility of reversing the power flow by electronic control is of importance if 'the converter is to suppIy a DC motor since it permits regen eration. It is pointed out, however, that firing controI affects not onIy the active but also the reactive power flow at the line side. At a = ~ the reactive power assumes a maximum value (for I D = const.) while the active power is zero; this is the case when the motor produces torque at standstill. Identical results are obtained when examining the DC side of the con verter, where the mean voltage is
Firiup; adv;mce is uot possible with line-commutatcd convcrt
1H".ctUl s(' Lhe jiriug plllses would bc appli(~d (.0 tltyristors having rcver s (~ Il ta!-l volL i'lt'í< ' whirll i'f('v('uLH tlH:lll [('IIJH (·OJldlld.illg; advaúC(~ firillg i R ollly !,ow d ld(' wil.h· r"I" '" ,·o)[)l ll u(.n.I.('d '~OIlV .. [ I.'· lii , wI"," : illI iul. e, because otherwise the thyristors would be reversely biased; the current ceases to flow as soon as this condition is violatedj Fig. 8.13 shows the effect. A furtlu:r cornplication arises, if the DC circuit contains an inductance J~ II in ,t E, 1,11,
O ai1
c
a
I
I I I I T2111
Ro
iT2(0:
wll(')"(:
;',/"1
Uo
I I
I I
I
.
iA(T) ~ iA sin(T - arctanwTA) -[iD(O:) + iA sin(o: - arctanwTA)] e-(T-a)/wTA
'tA
I
These results apply only as long as the four thyristors are conducting. The commutation terminates at time o: + Tc, when
O,
It.. , 01"':(' of" Lhe llIuch smaller inductance LA, the line current changes rapidly wlll'f"(':ls Llw load current may be assumed to be approximately constant dur 1111', III(' shorL commutation interval, iD(T) ~ iD(O:). Hence, with the abbre
"I:d.ioJl
\
/1
ue
(8.20)
= UA ,
. + RD ZD =-E .
diD . dT
w Lo - -
io'/'I
t1Uo
---,-,,-
+ iT2
'iTi
115
a
~
T
~
o:
+ Tc
i
';'A(T)I ,
iI.
ill( r }l·
c;m;;(';; ii drop
01"
i.Il(~
JIlcan voltage by
(8.24)
8.2 Line-conunutated Converter in Single-phase Bridge Connection
8. Static Converter as a Power Actuator for DC Drives
II(i
_
"J+
1
L1uD ~
1r
TC
-L1uD (8.25)
UA dr ,
" Wllich may be estimated from the current change in the AC circuito With the mentioned before,
~;illl[llification
RA «WLA, \VI'
lilld from Eq. (8.20)
---
.d"lll)
~
1
-
1r
"J+
TC UA dr
~
1 -wLAiA(r)
I"+TC
1r
"
"
1
= -WLA [iA (a + rc) - iA(a)].
L1uD ~ wLAIDn iD = k iD . UDO .,fiUA IDn I Dn
1 kllce
n
Uoo
(8.28)
The normalised impedance factor k is mainly determined by the leakage reactance of the line transformerj a common value is k ~ 0.05 to 0.10. The commutation interval rc, which usually lasts only a few degrees, de pends also on the firing angle aj this is seen from a rough estimate of the integral in Eq. (8.25)
discontinuous current
"2
(8.27)
which indicates that the line-commutated converter acts like a controlled voltage source having an apparent internal resistance proportional to the reactance of the AC linej of course, this drop of the terminal volt age does not involve real power losses, as it is caused by commutation at the AC side of the converter. As a consequence, the load curves of the converter uD(i D) for 0'= const. are drooping in the continuous current region (Fig. 8.16), even though all ohmic resistances apart from that in the load branch have been neglected. ln contrast to the intermittent current range where the curves are strongly nonlinear, the characteristics in the continuous current region are linear, exhibiting only a slight droop. By normalising Eq. (8.27) with the no-Ioad volt age UDO and nominal current I Dn , we find
(8.26)
the voltage-time-area lost during commutation is related to the flux ('liallgc in the line inductance.
2 -WLA iD, 1r
1r
Uo
~
117
~ 4.J.UD ~ -1
a= 0°
1r
J' .
"+T
c
UA sm r dr
~
M2
. a -v rc UA sm
(8.29)
1r
"
continuous current
which shows that for constant load current the overlap is minimal at a ~, when the AC volt age has its peak amplitude. A more accurate result is obtained by evaluating the integral, Eq. (8.25),
60° 0,5
1\\ ':
'
i
r/ "'it~.
900
a
":n
120°
150°
8.10. Clmractcristics of single phase converter with inductive load
rc
~
arccos cosa - .,fiwLA-] UA iD - a.
[
The knowledge of the overlap is of particular importance when the con verter operates as inverter, i.e. with large firing delay a, because then it must be assured that the commutation is completed and in addition the outgoing thyristors have recovered and are ready to block forward volt age when the line voltage changes signj otherwise an unintentional refiring may occur. Since the back voltage is negative during inverter operation, E < O, this would con nect in series two voltages (E, UA) having (for a half period) the sarne signj the result wOllld be a short circuit condition with a fast current rise which is only limit (1)O 540 V. 'I'IU' olltput voltal~e 'uJ)(r) in Figs. 8.19 and 8.20 contains harmoncis, thc rn' 1r/3) so that they overlap or by generating for each thyristor two short firing pulses, 1r /3 apart. Normally the second pulse would be applied to a thyristor that is already conducting and be of no consequence. If the first solution is preferred, the firing pulse may be chopped into narrow pulses to redu ce the voltage-time-area and the size of the pulse transformers. The iupllt current ofthe converter, drawn in Figs. 8.19 d and 8.19 d with oul t;ü.illg COllllllllt.a.tioH into account, is the sarne as the current that would fl ow iII I.hl' prilll1\.ry willdiug of a Y /Y- O1\ ia = O] --+ CI On, [iaRef < O1\ ia = O] --+ C I Off,
I. I
;0 úJ
Fig. 9.3. Four quadrant operation of a DC drive by armature reversal
one of the converters can be allowed to conduct at any one time; hence only one thyristor of each pair produces conduction- and switching- losses so that the pair can be mounted on the sarne heat sink. However, having opposite polarity, they must be electrically insulated. At lower power ratings, com plete thyristor modules are available, having the necessary interconnections built-in. Supp/y oncommon heat s;nk
(ú
ia
úJ
Fig. 9.4. DC drive employing a three-phase dual converter without circulating current
Tlw cllrrent controller supplies 1. a:: I " ' ! I('/,I V"
I'H\V I "
(9.3)
1.,.
[' "IIII':!
1111,,111 a~; ()II
145
9.1 DC Drive with Line-commutated Converter
1'":i::iI,j',, 1." 1',',
ú>1
Lo ia
\.:J
-
e
L(7RC
e
C I Rectifier
C I Rectifier
C2 1nverter
C2 1nverter
C I Inverter C2 Rectifier
C I Inverter C2 Rectifier
ia" mM
i2 -------'
a
b
Fig. 9.5. Reversible converter with circulating current supplying DC machine
continuous so that there are no unnecessary waiting intervals delaying possi ble control action, The operating states of the two converters are indicated in the four quad rants of the torque/speed- plane in Fig. 9.5 b; for mM > O, the armature and circulating currents flow through C 1 whereas the auxiliary converter C 2 carries only the circulating current; the inverse is true for mM < O. To keep Lhe circulating current at a sufficiently low leveI, such as 10% of nominal cur rent, the mean voltage of the auxiliary converter must c10sely track the mean voltage of the main converter which is determined by the armature of the Illotor. If this condition is maintained all the time, both converters are active ;wd ready to accept the motor current in a continuous transient, without any waiting interval. This two-variable control is effected with the help of the two lirillg circuits. The auxiliary converter must be controlled to satisfy the condition UDl
+ UD2
= Rc(i1
+ i 2)
~ O,
(9.4)
I . (~ . the sum of the mean volt ages should be approximately zero; however, Lili,; is not true for the instantaneous voltages UD1(T), UD2(T), because one III' Lhe converters operates as rectifier, the other as inverter. The two volt ages 1\.1111 th( ~ ir sum are plotted in Fig. 9.6 for three different conditions, neglecting ,·I)III1I1I1t.af:ion and internal voltage drops. Clearly at 01 = 02 = ~ the sum 01' !.lu' iJlsLanl.all<XHlS voltagcs assumes very large values eventhough the sum "I' I.he 1IH':t1l va.lues is :r.( ~ ro. To prevent I.hcsc large alternating voltages from I'II.IIJi illl ( I' )( ', 'HH iv,' (' II ITI'II L", Lhe c.Íl'clllal.il\l!; I:IIIT('IIL rcad.ors Dc' seen in Fig. 9.5 1 111(' II 'qu i n'.! , Si ""I> 1.111'1'1' i i; nlw ll,YH "III' whirh (,;I.rrit's 1.111'. slllall ('ollt.iIIIlOIl S
1,1(;
!lo Control of Converter-supplied DC Drives
II/II
...
a1 !: a2 !:
= 45 (X2 = 135 ~
;1 II I \ \ J
90 0
0
(XI
II II I I
9.1 DC Drive with Line-commutated Converter
I' I \ \.J
,i2Ref
,,
h I' I' j' 1 \ 1 \ 1 \ 1\ \ , ,I , 1 , 1 1 'I 1 1 ,1 ,1
: 'I '/ '~ 1
a2 =45
0
a1
,
~
,,
0
= 135
0
rot!:
147
T
,,
,,
,,
..
~
iaRef
v
I, 1 \ 1
b
ii
~
UDt W Ref
$~
--1M Lo
W
rot = T
I"ig. 9.6. Voltages in reversible converter with circulating current
-LC
~I
~
U02
Lc
a Fig. 9.7. Control of reversible converter with controIled circulating current
cir O, until the next switching cycle begins. ln Fig. 9.10 the commutation interval has been enlarged for better visibility. Clearly it is criticaI for the functioning of the circuit that the alternating volt age of the commutating capacitor is of sufficient magnitude to ensure an adcquate hold-off interval t 3 - t2 when the main thyristor is reversely biased. I[ this time becomes too short because the charge on the capacitor was in ~i il{fic ient or there is excessive load current, TI will refire and a short circuit colldition develops which can only be cleared by a fuse link or a breaker. ln I,"is r< :sp ect a commutation failure of this circuit is more serious than with a l i ll('('d ('()lIl.roller. The next higher levei of control could be a position control !\lO p :IS showlI in Fig. 15.9, where the response may be further improved by 1" "/, is the load-side link current; furthermore 11./1
.: (,('l.ioIlS oe 'U'N; hence it is necessary, as shown in Fig. 9.17 b, to create ', t'rll i Ill.crv;ds of the voltage UB for storing in the line-side inductance energy I.Il il.1. i:, slli>sequ(~ ntly transferred in the form of a current pulse to the link I'II,p;u 'iI.OI, "'hus it is necessary to employ a unipolar or cyclical modulation, w I w J'\ ~ I.hc COllverter bridge is temporarily short circuitcd at a DC bus. '1 '11(> switchillg frcquency of the converter is best chosen as an integer 1IIIdl.ipl(~ II of the line frequency to avoid in st ead y st.at(~ sllhharmonics of the l i lll ~ 1'11/'1' wou ld fit illto one lillC voll.iI,l':(> Jlcriod, rcsultillg in a r.d rl.v :n llool.lt w;l.vd<mll ()f Lhe ('.111'1'I'" \, () II tlIe other hand, the equivalent circuits, derived for steady state "1)('r:ll.ioll with sinusoidal volt ages and currents, are inadequate when dealing wiUl I.rallsients or when the motor is supplied from a switching converter. ln I.h(' ['ollowing, the steady state condition will be treated as a special case of Uw dyuamic case. The mathematical model to be used is tailored to the needs of controlled d ri ves. It incorporates most of the qualitative features of an actual motor lilll. would not, of course, be accurate enough for the purpose of designing !.II(' Illaehine.
1.-:1 -
---,.p.
b)
S"ri e:~-wound
Lnll kW, 7730 Nm
Conto rating
,d, 1~):lIlJllin - l H~dll NIIl
IlilUI
III;"
:I ~ , ~ ,II h 1" 1~~ II 1'1 '. ' >I ;:
920mm
Three-phase AC induction motor
tator motor
1428 kW, 9155 Nm at 1490 win- 1
Torque (5 min) Max, speed Mass Inertia
11600 Nm 4200 min- 1
a
2660 kg
0 s (ex,t)
22 kg m 2
,
F I,( . 10 . 1. ( ;lIlllparison of traction motors (Siemens). (II ) S ' "I·.I,· I'I"\.~,, o;eries-wound AC commutator motor, ( I, ) 'l'1>I ' I"'-pll;c~e AC induction motor
a.
I
ex
-TI
b ~s(ex-4TI/3)
S -TI
zs:
~s(ex)
'S:6~SV
~s(ex-2TI/3)
zs:
3
TI7KZ
c
10. 1 Mathematical Model of a General AC Machine I L i:: ...sslIllwd that the stator S of the machine is a hollow iron cylinder with ,.; rl'lda, (TOSS sectioIl, containing a concentric rotor R $0 that a llarrow airgap "I' I·{)W;(.alll. radial length h exists between the SIlI()oth cyJillctrical surfaces to wlll O, is 2, iss < O. Magnitude and angle each vary with time according to
The current vector determines the instantaneous magnitude and angular po sition of the peak of the sinusoidally distributed ampereturns wave produced by the three spatially displaced stator windings. By combining Eqs. (1004, 10.7) the ampereturns rnay be expressed as a travelling wave, the peak of which follows the angle a = ((t) of the current vector, 8s(a , t) I
r t1w
. I)
,I,
N :.'
1':1 (1 ),'
I"
I
/7,(/ ) " I"
1
I
I
f
~"
I
1
( 10.,1)
cos(((t) - a) .
(10.8) three phase and velocity
d ( / di, i II I,) I( ~ ai q~ap of the mo(;or . Siw(' 1.111' CC>lllp)(' X v('d()r~; ddillt'd iu )';(p :. (10.;, In.S) rJ('scribe t,he spat.i:tl dl ll l.r'il'lll,i ,," (Ir II 1I ,1I 11 ~ III'I , iI' lidd ill II, 1'""1.l2:
where I is the effective axiallength and r the radius of the rotor. The integra tion over is a consequence of the non-uniform field in the airgap while the integration ove r À is required by the nonuniform distribution of the winding. Inserting Eqs. (10.4, 10.14, 10.16) results in
,';illC" I.IL
may be defined as follows, using complex notation,
ej
(10.61)
we find
10.2 l.lHluction Motor with Sinusoidal Symmetrical V olta/,!;es in Steady State
I'r(:<jIH:JlC:y Wl
= -/2 [I e jw1t + -s r e- jw1t ] , etc. 2-S
. . _ 3 -/2 I ej(WI-W)t. L/I(t) 2 -R
(10.64)
TI\(: rotor Cllrrcnt in stator coordinates, i.e. as seen by a stationary observer, III with ~ - wl. lWC;lUSC of constant sp(:ed, i /{(I)
, ,, I tI {l}
., ..
J/. '0
I/{
i (",I
t ..
(IO.(;[i)
I '/ H
lO
''-;'y"11Ic s!lol'l,,'d illt.l'I'l lII. lly wit.h ii ~ (I('('i ll. l 1111 '1' 111 11 11 11 111 wl li ll' t.l1(,'IH'II :d ll': i 11.1'11 lil'l,('" 1.(, !-tv" i ,1 LI,, ·, 1I1111< 1I'(' WII II' y f l'lt- Ll oll (lI ,,1 W(' 1'.I .
fi. 1""1., ,,· wl l. "
L2
(10.94)
/I
o,() r"
II I.I t l' l. I "I "
a
b
R I "R2
LI "L 2 Fig. 10.14. Cross section of eddy current rotors and equivalent circuit
There are various designs which tend to emphasise the effect of eddy currents in the rotor bars. Qne solution employing two cage windings is shown in Fig. 10.14 a. The starting cage (1), being of higher resistivity, is placed in open slots immediately below the rotor surface and exhibits low magnetic leakage; as a consequence this winding produces a high pull-out slip BpI. Because of the low leakage this winding dominates at high rotor frequency, i.e. at low speed. The operational rotor winding (2), in contrast, is characterised by a larger cross section of the conductors, possibly employing also material of higher conductivitYj the leakage reactance is increased because the bars are now embedded in the rotor iron. Clearly, this winding becomes mainly effective at low slip fre(]IlPIlcy, Figure 10.14 b shows the equivalent circuit with the two rotor Willdilli\S COllIH',c[,cd in parallcl. 1\';'
corresponds to the angular frequency of the stator voltages. Should the mo tor be supplied with sinusoidal symmetrical three-phase voltages of constant frequency and amplitude, Eq. (10.60) again becomes yalid
'Y,S (t) = 3 y'2 US ej
SlIpplied by Impressed Voltages of Arbitrary Waveforms Arl.,'!' I.his discussion of the steady state with sinusoidal symmetrical three voltages let us now return to the more general model of the symmetrical i11..) di s _ jÀ .d>'. -jÀ d is --=-zse =-e -J-zse dt dt dt dt -
193
'1/' "
0.,
(10.112)
voH'lgcs lmve been imposed so far. WiI.h 1.11('1:(' ddillitiollS thr V()Jl.al~(' (~q1\atiolls (10.111, 10.112) and thc e.' ·
:i/,;II,()I'
pr":ir:i"ll
rOl'
LIli'
I,")'qll('
1-;'1' (IO.llli)
: 1.':: :11111
1
dxs
Wl
2 + 2 T s dt + [ ( Wo ) 2] di2 1 + S;s
xS
=O,
(10.133)
with the eigenvalues I+-t---ó+ - - - - I
YSXR
81,2
.!:!!.... OJo
OJ2
Wõ "' i ,~. I O. LO . lllock diagram of symmetrical induction motor
r.."
h.v illq .ri (l)
(10.5 5) follows
= 1Re [is(t)]
J :~ (/ '~'II (Ai l ) ri
/, ::
f(
:/:,'; -
1 - Xli ) CO:-) W I/' -1 I {r ll
eIS
1 (T
li
lI n
) Sill Ü)lt'l, c
(10 , 111)
This is also plotted in Fig. 10.20; it shows the characteristic large inrush current which, as the motor approaches synchronism, settles down to the small sinusoidal no-load current until the load torque is applied, The currents in the other stator phases are similar. There are strong alternating components of torque which also affect the speed, even though with reduced amplitude because of the mechanical inertia. If the inertia is predominantly on the load side, the shaft and the mechanical coupling may be severely stressed during the start. The speed, as a function of time, follows a similar curve as obtained in Sect. 3,2 when the electrical transients were totally ignored, Fig. 3.12; this explains also the absence of the overshoot in speed which is present in Fig. 10.20, As an intermediate step between the quasi-steady-state model and the more complete model of the induction motor, the heuristic block diagram in Fig. 10.21 a may be helpful. It differs from the purely mechanical model by the inclusion of a lag T~, following the steady-state torque-slip-function. For small values of slip, where the torque function is linear, this clearly re sembles the block diagram of a De machine. A comparison of the starting transient computed with this model shows good qualitative agreement with Lhe transiellts obtained by integrating the Eqs. (10.127-10.131). Of course, Lh e os(:illa(,iom; Hl1perimposed on speed and torque cannot be represented but I.I)( ~ lJl( :all viI,I'WH ! ~ i'( : wdl luo
202
10. Symmetrical Three-Phase AC Machines
10.4 Induction Motor with Unsymmetrical Line Voltages
10.4 Induction Motor with Unsymmetrical Line
Voltages in Steady State
U5 + --
3"1 [U -51
+ eh -52 U + e_53 -h U ] ,
10.4.1 Symmetrical Components
U5 - --
3"1 [u -51
+ e -h -52 U + e_53 h U ] ,
For intermittent duty drives, such as cranes or hoists, where simplicity of hardware is more important than high efficiency, induction motors with un symmetrical voltage supply may be used. By way of example, a phasor di agram of sinusoidal but unsymmetricaJ line-to-neutral volt ages is shown in Fig. 10.22 a. Due to the isolated neutral of the symmetrical stator winding, the sum of the instantaneous phase voltages is still zero, Eq. (10.37); hence the phasor diagram forms a elosed triangle,
fl.o
e -fYl}se -jy l}s+
b
fl5-
u5 ++
U 52 = e-h U 5 +
U 5-
!)S
c
+ fl.o ,
211" + eh U5 _ + fl.o , 1=3'
=t
U 12
[U 12
(10.14G)
eh U 23 ]
-
+ U 23 + U 31 = O
u 23 = (10.143)
ehfl5++e-hU5_+fl.o,
where U 5+ is the positive-, U5- the negative- and fl.o the zero-sequence volt age component per phase. The symmetrical components fl5+' U 5 .- form two symmetrical three-phase systems oE oppositc phase-sequencc. Solvilll( Eq. (10.14.1) yi(dds
(10.147)
is valid by definition. General1y the use of line-to-line volt ages does not ex elude the appearance of an arbitrary zero sequence component fl.o but, as seen before, it vanishes with a symmetrical load. If the motor is fed from a line possessing noticeable internal impedance, the symmetrical components of the terminal volt ages depend also on the load, i.e. the operating state of the motor. Should the voltages U 12' U 23' ~1 form a symmetrical three-phase system, e-j,,!
U 12
(10.148)
,
the result is
-
U53 =
(10.145)
Two of the line-to-line volt ages are sufficient to determine the positive ,Uld negative sequence components, because
It is known, that an unsymmetrical three-phase system may be de com posed into "symmetrical components" defined by
=
+ e-h)]
and correspondingly
Fig. 10.22. Decomposition of unsymmetrical balanced three-phase system
into symmetrical components
U 51
v
=0
-- 3"1 [U -12 - e -hu] -23
~s+À
~S2
!L23
!L12
The sarne holds naturally if the stator windings are Ll-connected.
!)S1
O.
1[U' -51- v- -U- 52 -e-h (fl52 - u 53) + u 52(1 + eh -' '--v---'
(10.142)
ejY!)s+
(10.144)
The expression for the positive sequence component is similar to that for the time dependent volt age vector, Eq. (10.34). Since the volt ages between the motor terminals and the isolated neutral are not directly available (the motor may be Ll-connected), it is appropriate to introduce the line-to-line voltages instead. By expanding Eq. (10.144) a we find, analogous to Eq. (10,38),
U 5+ =
fl 51 + U52 + U53 = O .
a
1 [U 51 + U 52 + u53] =
=
203
U -5+ -
1 e- j7r / 6 -12 U J3
-U - -51'
U-O - 5- -
.
(10.H!))
'l'hc case of a symmetricaJ induction motor fed by unsymmetrical snpply vol (.agci-i is fully containecl in the thcory developed in Scct. 10.1 bllt til
§~o
.... "" Q).3 ...c:: > o
Q)
-§,o~-5-§,gbO§S
~~$~~2-~9
. ~..oo"",o. ~::g(j o..~ VJ
..
o r.l,+'
.~ 's-s ~ w.;,l
p'i. 0',
,. , Ir;
I~
-
~
~ .... ro
te
~ § t)
...,+-> u o
E S
~
f1
. 1-
S~ .... '"
'-'
Uo '
8.~ al ~·C
2: o. o."d 8 "'dO) 3.-!:r t.lt al. tll(: COll verter circnit of Fig_ 11.3, which in
·.! 22
;/
11. Power Supplies for Adjustable Speed AC Drives
.m.~; iReI (t), i(t)
/)
223
11.1 Pulse width modulated Voltage Source Transistor Converter
protective interval which lasts only a few rnicroseconds when transistors are used, can be assigned to a finite switching time in the converter model. When writing the volt age vector 1f.s(t) in terms of line- to-line voltages (10,38) which can assume the values UD, O and -UD , 1f.s(t)
" 2 = USI + US2 eJ"Y + US3 eJ "'( = U12
-
U23
-'
e
(11.1)
J"'(
u(t)
OOllPl~i
six distinct voltage vectors and two zero vectors result, as seen in Fig. 11.7 b, Clearly, when going from one comer of the hexagon to the next, only one leg of the converter needs to change its state,
1l.2
"Zero vectors' 2x
S1
Uo
S2
,,
S3
U
~ -SRef
'.
-1
(
\ .' U l -I
()ff
I I I I
I (.'
i(t)
u(t)
U23
"51 [
a
"'2"52 [
uS3
[
b
Available voltage vectors
Fig. 11.1. Switching model of converter in Fig. 11.3 and generated output voltage
, .,
"
vectors
,.,' iRe/t)
S;LI.llration effects of single-phase switched transistor converter with ( .,, / ( Hr cml't:1I1. control supplying a passive R-L load impedance 10' 1/\.
I I. H.
I",!" . I I. 7 a is represented by a switching model, must be so operated that 11t0l11' 0(' I.h(~ Icgs is short- circuiting the link voltage UD and that each motor 1.('J'llIill;d
lI"w .
11.2 Voltage Source PWM Thyristor Converter
227
It is realised that the initial transient, when iS l is commutated from T to AT, is of course also continuous because smalI inductances, not shown in Fig. 11.9, limit the rate of change of the thyristor currents. Likewise there are paralIel RC snubber circuits connected to each thyristor to limit the rate of change of the voltages. A detailed analysis of converter circuits, taking alI these effects into account, is best performed by digital simulatioll'j this is particularly true for transient operation, when the initial conditions, for instance the charge on the capacitor when AT is fired, are changing. As with alI converter circuits, the function of the converter is eventualIy endangered when the required recovery time of the thyristors cannot be maintained, for example due to excessive load current or insuilicient initial charge on the commutating capacitor. Rence the accurate analysis of the commutating transients is of major concem to the circuit designer. Special types of thyris tors with short recovery time « 20f,Ls) are usualIy specified for pulse-width modulated converters, which limits the power rating to some MW. AIso, the occurrence of unacceptable load currents must be prevented by fast control action. The firing instants for the main- and auxiliary thyristors are determined by volt age and/or current controlIers. ln view of the limited switching fre quency, which is usualIy below 1 kRz for thyristor converters of higher power, the load currents can no longer be considered close to sinusoidal as was pos sible with transistor converters operating at a higher switching frequency. Rence PWM- thyristor converters of the type shown in Fig. 11.9 are nat uralIy suited to act as controlIable voltage sources, presenting to the load pulse-width modulated rectangular voltages, the fundamental components of which are prescribed by volt age command signals. Because of the large superimposed harmonics, caused by the chosen switching strategy, closed loop control of the fundamental currents at variable frequency is normalIy not practicalj it was different with transistor converters, where the switch ing frequency is higher and the currents are suiliciently filtered by the load impedances. Therefore, open loop voltage control through a pulse-width mod ulator is the standard solution, while the current loops are normally closed on the next higher levei of drive control, as will be shown in Chapo 12, The pulse-width modulators may again be of a variety of designs. One frequently used scheme with converters of higher power (Fig. 11.10) employs a triangular sampling signal [J8, M42, S29], the frequency to of which is a multiple n of the stator fundamental frequency ft to avoid subharmonics; n is reduced in stages as the stator frequency and volt age rise, to limit the switching frequency and to fulIy utilise the voltage capability ofthe converter. A drawback of this otherwise simple scheme is the appearance of large har monics ai the frequencies to ± 2ft . The load currents are smoothed by the leakage illductance of the motor which should not be chosen too smalI with this tyP(' 01' COllv(·rl.(\r. Ali tripkll harlllollic; a.re climinated from the cur l'('uLH iII :; 1., ':' 01.1' :: l.:t.I.,. dup 1.0 I.lw i :;ola.t., :d IlPIlLral 1:rfonll:lIlCI; drivr:s. 'l'lte curve in Fig. 12.3 b is calculated for Rs/wo Ls = 0.01, S
I.inll s
[rom
()\, I' II
~il L iv , \ 1. ..
Innp ('nllll W II S ll.t.inl.~ ("ollt.rol s("It' : I1I" ~ Iii\('
LlI
rrOll1
i'i LaLor
CIIIT'·III.:i .
III'(,W( '('II
Lilc roLor · CllITCIILH alld
SI IH '" 1.II" · rol.or
.
= ~ ~ 1m [is (imR 3 1 + O'R
is)*],
(12.23)
~ (1 -
(12.24)
0') Ls 1m [is CnR] .
1nserting the magnetising current vector according to Eq. (12.22) yields
'I'11(! heuristic control scheme described in the preceding section is, in spite of L!te acceptable results obtained, not best suited for high dynamic performance cI ri ves . Because of the open loop flux control it would be difficult or impos ~: ii>I( ! Lo operate the motor with ful! torque at low speed or even standstill w lli (' il (!xcludes, for example, its application as a servo-drive. o remove these I (·::l. ricLions it is necessary to return to the dynamic model equations of the I 1Ic1Ilcl.iOIl motor, based on instantaneous currents and voltages, as derived in ~ ;( '(1.. lO .l. Of particular interest must be the interaction of stator and rotor "lIr)(!II L~ resulting in fiux and torque. When initially assuming that the stator ('11 rJ'(!n Ls are impressed by fast current control loops considerable simplifica l.iollS result since the stator volt age equation (10.50) becomes a concern of 1.11(' cnrrent controllers and can be omitted from the model equations of the cI ri ve. The restrictions of steady state are now avoided which is a precondi I.ioll for achieving rapid transients. The power supply could be a transistor collverter with high switching frequency and adequate ceiling voltages or a cyc!oconverter, resulting in very fast response of the stator currents. The expression for the electric torque (10.48) =
+ (1 + O'R)iR ej E]
which is simplified to
12.2.1 Principle of Field Orientation
'/Il M
= Lo [is(t)
Eliminating the rotor current from Eq. (12.21) results in
mM(t) =
.
Ii(t)
(12.22)
mM(t) =
2
253
12.2 Control of Current-fed Induction Motor
12. Control of Induction Motor Drives
CIII"I"I'1I1.:i (';1.1111111.
2
3 (1 -
.
0') Ls imR 1m [is e- J li] ,
(12.25)
where
is e-j li = is e j
Ó
= iSd
(12.26)
+ j iS q
rcpresents the stator current vector in a moving frame of reference defined \)y the rotor flux or the magnetising current vector imR · The angular relationships of the current vectors are depicted in Fig. 12.11; Lhe stator current vector in field coordinates or the "field orientated" stator wrrent vector is seen to consist of two orthogonal components,
iSd = Re [is e-j li] = is cos Ó iS q = 1m [is e-j li] = is sin ó in the direction of the magnetising current vector and the other perpen dicular to it. ln steady state condition iSd and iS q are constant apart from ('(IilVerter induced ripple, i.e. the stator current vector is and the magnetising (,IIIT(·m t vector imR rotate in synchronism. The steady state current trajecto til!S iII Fig. 12.11 were taken from actual measurements with the drive shown
III1C
iII I"ig. 12.28. IIlt,rod\l cing ;t
the quadrature current component into Eq. (12.25) leads to si III pie (!xprcss ioll ')
'III'A/(/.)
/; i'"II ·i.,';,/,
f..:
~ (I
:\
0)/' ::,
(12.27)
:!51
12. ContraI of Induction Motor Drives 12.2 Control of Current-fed Induction Motor
di T R -mR dt
" Measured steady state
Current traj~onés
I
(12.30)
dt = WmR(t) ,
d( dt
d8
= Wl (t) = WmR + dt
Multiplying Eq. (12.29) by sults in dimR TR ~
UI! iII
+ Lo dt [P + O'R) iR + is e- J ~J v
i --mR
wiL/) '1'11
e-
J
1' 11/ /lu , tlli:-l J(~ads to
c
e-jl!
and expanding the left hand expression re
' + J. WmR TR '2mR + ( 1 - J' WT) R 2mR = '!.S. e - j I! ,
(12.32)
which may be split into real and imaginary parts,
which gives a hint, why the transformation of the stator current vector into fídd coordinates, also called field orientation, is the key to rapid control 01' AC machines. This principIe has been proposed by Blaschke [B33-B38J, ( ~ xtcnding and generalising earlier work by Hasse [H26J. Clearly Eq. (12.27) reminds one of the expression for the electric torque of d DC machine, Eq. (5.3), where i mR corresponds to the main flux c!lllv,'rl.l'l' j 1.\1111 i d pl ~) whjl(' h Iii "fl.I'1. 'J/"""'.\.J
10" 11-e written in the form (5,5 () t
w hcn ~
= (1 -
(J"
di.mR () R ' )LS ~ = l!s t - s :?os -
(J'
L
S
di.s dt '
(12.51)
RR TR
s 0.200
0.150
0.100
o
s j
5
10
Fig. 12.26. Rw self- tuning employing terminal voltages. (a) Self-tuning scheme j (b) experimental results with 22 kW wound-rotor motor
This scheme is shown in Fig. 12.26 a; to obtain convergence, the signs of WmR and i S q have to be correctly chosen. AIso, there are lower practical limits of speed and load, for which the adaptation yields reliable results. When the rotor current is dose to zero at low load, a correct setting of T.k is not important, provided the parameter is quickly adjusted when load is applied. A test result is seen in Fig. 12.26 b taken on a 22 kW wound rotor motor, with external rotor resistors being switched in and out. The lag of a few seconds would normally be acceptable for correcting a temperature induced variation of the rotor resistance.
c.dt) is the "volt age behind the transient stator impedance". Assuming
/l .,; to i> c kllown, for instance from a temperature senSor in the stator winding, "".! (/ /' S to be relatively unaffected by saturation, ~s (t) may be derived
ri (" II
275
a 111( ~ aSllrement
of terminal volt ages and currentS. When converting this field coordinates defined by the rotor flux mo dei in Fig. 12.20 or nnd for the direct component of the volt age
12.3 Rotor FIux Orientated ControI of Voltage-fed Induction Motor
(·qll itl.ioll lo 1 'L.
'2:1 , wc Co d
.
= (1 -
(J"
dimR . ) L s - - = USd - Rs tSd -
&
(J'
disd Ls - -
. & + WmR (J' L s ts q ,
(12.52) wlaiell reduces in steady state to CSrl
~ 'USd - Rs iSd
+ WmR
(J"
L s iS q
:::::J
O.
(12.53)
tu; I.be angle of transformation p', being derived from an initially detuned JllOdd,
rnay be incorrect this condition is not automatically satisfied but may
1)(' Ils(~d for sdf- tuning the model according to
,, /",
+j
USq)Ref
ej
(2'
;
(12.58)
I' I: : ; 1I ~aill the angle of the fundamental flux wave as determined by a flux iI,('([llisitiOll scheme, for instance the flux model shown in Fig. 12.16 b. The Jllaill dilrcrence between the control schemes in Fig. 12.14 and Fig. 12.27 is l.h;tI, I.he current control is now performed in field coordinates, where the con I.roll(!rs are processing DC signals in steady state, thus avoiding the problem or pltase shift of AC current loops. AIso, the digital part of the control may 1I0W illclude the pulse-width-modulator and can be extended down to thc I~(!ll(!ration of the switching signals for the converter. Th(! rCllIaillder of the control system is similar to the one shown in Fig. I :~. 1,1. TI.e ~tructure in Fig. 12.27 is somewhat redundant since the task of is d - 1Llld isC/-mutrol could also be assumed by the flux and torque controlknl, (lrllil.i.illg th(! CIIIT('1I1; COl1t,fo\l('rs. Field weakening is adricvctl with a fllllCt.ioll r:llIlt'J'ill.or
286
12. Control of Induction Motor Drives
287
12.4 Induction Motor with a Current Source Converter
~~/'-------- Currentlimit
es () t = (1 -
-
(J
dimR )Ls - - = (1 -
dt
(J
mR ) L s [di --
dt
. ] + J. WmR ZmR
e j{! .
(12.62) This is now supplemented in Fig. 11.16 by the stator current equation
~---~
~
.
1~'
\;/-\
d'gs . . Zs, C -dt- = -zp - -
+~R(t;
-
(12.63)
where i p is the current vector supplied by the GTO-converter,
ip = V3iD(t) ejW ) ,
No load current
a
b
Fig. 12.37. Stator and magnetising current vectors of current source converter drive during a speed reversal at 200 1/min
possible with either volt age source or current source converters discussed so far. As a consequence, the additional copper- and iron- losses in the motor are reducedj there are also lower volt age stresses on the insulation of the willdings. Sillce the converter is now feeding a low impedance load, the commutation al',;~iu l>(~comes independent of the motor, as with a voltage source converter, i tlld ('ali operate at a comparable switching frequency, producing a smooth 1II"l.or Lo[que. Instead of zero volt age intervals there is now the possibility of "lllpl"yillg zero current intervals where the link current bypasses the motor, I.J 111:-: (' n ~ al.i llg additional degrees of freedom for designing PWM strategies. At 1,11
'1/) (t)
= Lo lmS,
(12.65)
where ~.m8 = ims( t) ej l'I ~ [> r ese llts
p.(t)
= (1
+ (J's) 15 + IR e j
, '"
1 __ I'" + : +
:~
-~:, ~~
,
(1--o--.t. 1 ~
I
~
I ~ I L r - - __I.J
,, 'ti ,, o'" , ~~ :s. ::J
o:: o:: :-...5'
0lV)
CC C\j
>
C\,i
'+-.
-..;::
CC C\j
>
"
vi
&"
8 1 81
..... (l)
h~ 1
h
ê: (l)
ço cc ~ L0
cc
2a
C\j
c:
oS
.~
"Q
.g
(l)
«l
~Q ~
Q)
Q iJ '-
~.§
~
r.nVí (l)
~a
t t t
E: (l)
>
"
vi
&"
,
Q) "Q
a E:
......
Ol
c:
~
(l)
~
t::
::J
SJ
~
'
-o...
ti
::J
uls( ~ width-lIlodnlat.cd converter voltagcs y(t) serve a!i Lhe aetnaLiIlf( variabJes. Fig. 13. Ui slIowH Lhe v{:cLor diag-ram of tlw lill(·~ · sid( ~ C(>lIv( ~ rt.(~ r. 'l'll(~ t.rallsforllwd voll.;I./',"" ; lIId '·III.,.c:"I.~l .,,(1) ,.
Fig. 13.15. Vector diagram of the line-side converter
( 1:1.:\:1)
LdiNq -dt
' + R 2Nq
= -vq -WoL'2Nd,
(13.36)
The instantaneous direct and quadrature components ofthe currents, iNd and i Nc11 are a measure for the active and reactive currents flowing from the line to the converter, as expressed by thc instantaneous complex power, 11."rv
'lN
(:IJ:~/2) lI N " .i>-' i N - :IJ:~/'2 (/NI'iNdU)
I " !i 1.( ·ll.dy nU d,
Reactive companent af fine current
40
350 >
Ir (j e·/
INq/A
o)
uD
/ 1/
300
I
•t 20ms _ _ _o
I~ I~IT
()
.
.
1
.
50
I.
~-a
The doubly-fed machine described in the Sect. 13.1 can operate as a motor or generator in a speed band above or below synchronism, drawing leading or lagging stator current from the line. Depending on the operating condi tion power :fI.ows from the converter to the rotor or in the reverse direction. This high degree of :fI.exibility calls for a relatively costly converter but sim plifications are possible when the specifications are relaxed. For example, when rectifying the three-phase rotor currents with diodes and feeding a DC link, the power flow becomes unidirectional. This is so because both volt age and current at the DC side of the rectifier are restricted to positive valuesj seen from the AC side the rectifier appears like a nonlinear, purely resistive network. The purpose of this scheme may therefore be to recover the slip power that would otherwise be lost in external rotor resistorsj it can either be converted to mechanical power in a DC motor coupled to the shaft of the induction motor (Kramer-drive) or fed back to the AC line with the help of a line-commutated or other type of inverter (static Scherbius-drive) as is explained in Fig. 13.19 a.
20ms
-I....l-i W-\ ff-
i,
JActl e il A
o
ilA
o
i
..,.,
(j)
1\ iNd
Reac ive iNq
\ '\,:;
I
ÜDmax
~
.
IS' - /' )
om, one ts af fine urre t
(j)
o - Ll
UR=O
---.L
~--
UDmax
·50
·30
Loadcurve
b
Fi~ .
13.18. Measured step responses on a line-side IGBT converter
wil.h v,~ct.orial PWM at 4 kHz clock frequency
(a) Itc·.versa.1 o[ Lhe rea.ctive current, (b) Change ofDe link voltage
TIl
1.1
1.2
1.3
1.4
1.5
~I ~ ~ ~ fl ~ ~
1.1
1.2
'p I_
:
$:
i,!
1.6
S
I 1.6
S
I
~
~ • ~ , ~ ~~~~~~~ A
1.3
1.4
1.5
I;'ig. 13.26. Simulation results with a Scherbius drive
currents is c1early visible. A recorded step response of the inner current con trol loop is depicted in Fig. 13.24 showing good dynamic performance of the dectronic torque controI. The effect on the stator and rotor currents when changing of the link current between O and 50 A at constant speed (1725 l/min) is shown in Fig. 13.25. As an example of the results obtainable with digital simulation, Fig. 13.26 di spl ays some computed curves of voltages and currents as weII as electrical I.orque. Clearly, the deviations from sinusoidal waveforms are substantial; of particular interest are the long commutation intervals of the uncontrolled n~ctifi(!r, caused by thc small slip-proportiona! driving vo!tagc and the ro(.(11' lpaka/'.(! n'iLct rmce. On the other hand, rcclncing thc higher freqllcncy hanuOll" ic: ; (,(~lIds
(,0
:aJlool.h tll( ~
CUUC1ÜH alI(!
I.he !:orqll(!.
The speed of a synchronous motor with DC excitation in the rotor is de termined by the stator frequency and the number of poles. As long as effi cient, variable frequency power supplies were unavailable this meant constant speed operation at fixed frequency. There are drive applications , where con stant speed is desired or where the reactive power that can be generated with line- connected synchronous motors is an important feature; these are, besides electromechanical docks, mainly high power drives, such as for pis ton compressors in the chemical industry. Another field of application exists in pumped storage power plants, where the synchronous generators serve as constant-speed-drives for pumps in periods of low demand for electrical power, feeding water into e!evated reservoirs for later use during hours of peak demando ln this instance the type of motor is, of course, not a matter of choice but the synchronous machine is very welI suited for this duty; it is, in fact, the on!y one that could be used at a power leveI of, possibly, several hundred MW. Problems with !arge synchronous motors operating on a constant fre quency supply may be caused by their inherent oscillatory response since the torque now depends on the load ang!e, and the required start-up procedure. Asynchronous starting at fuII or reduced line voltage with the help of the damper winding and the short circuited field winding as welI as special start ing motors are common practice; more recent!y, large synchronous machines are also started with variable frequency supplied from static converters. This can be extended to variable speed drives with many interesting features. Historically, the synchronous motor fed with variable frequency from a DC link by thyratrons represented the first attempt at assigning the task of commutation to static equipment [A15, W29j. The thyratrons, gas filIed discharge tubes with heated cathodes, that were available in those days, were of course hardly suitable for converter duty. Thereafter, the converter-fed synchronous motor did not receive much attention until this was completely changed by the advent of high power solid state switching devices available today. As :;( ' (' 11 til t.!t(! s'yllopsis in Table 11.1 there are three main fields of appli ca l.ioll
rOI
II djll:lbhl,· ~peed RyJlchroJlOWi lllotor drivcs ,
330
14. Variable Frequency Synchronous Motor Drives
• Large, low speed reversing drives, such as needed for gear-Iess rolling mills or mine hoists, with stringent requirements for high dynamic performance. ln the past, these were mainly DC motors but they can now be built as AC synchronous motors supplied by cycloconverters, thereby eliminating all the design- and operating- restrictions inherent in DC machines [J1,L6, N4, R23, 84, 85, 867, 879]. • Large, high speed compressor drives up to 100 MW for pipe-lines or wind tunnels, where the induced source voltages and, as a consequence, the ca pability of the synchronous machine of supplying reactive current, permits the use of simple DC link converters with natural or load commutation. DC motors could not be built at this power rating and speed because of the commutator [G25,H5,K63,N15,Vll]. 8ince a synchronous motor does not have to be magnetised through the con verter, as is the case with induction motors, a larger airgap is no disadvantage and can even be desirable for mechanical reasons. The slip-rings for supplying the rotating field winding with direct current can be avoided by employing an AC exciter with rotating rectifiers as is common on large turbo-generators. • A third important and growing field of application are low power, high dy namic performance servo drives (usually < 10 kW) with permanent mag net excitation and transistor converter supply (MOSFET or lGBT). Their main advantage, when compared with induction motors, is the almost com plete elimination of rotor losses; on the other hand, field weakening is more difficult [B29,G35,J2,L48,L51,P28]. After the initial problems of achieving a wider field weakening range have been solved with internal magnets built into the rotor, permanently excited synchronous motors are also an option for vehicle drives . Large synchronous motors with permanent magnets are of interest for the propulsion of military ships, because of the improved cfficiency, Table 11.1; the stator phases may there be fed from separate four-quadrant converters to attain redundancy in case of faults. The analysis and design of a control system for an electric drive cal Is for a dynamic model of the motor; with a synchronous machine this may be of considerable complexity if details such as saliency and unsymmetrical rotor windings are to be taken into account; as this is not the aim of the discus sion here , substantial simplification may be achieved without much loss of validity by restricting ourselves to machines with cylindrical rotors, when : the mathematical model of a symmetrical three-phase machine described in Sect. 10.1 remains applicable. With large machines and high speed a solid st.ecl nOllsalient ro tor is the normal choice for mechanical reasons but eV(~ll a.t low spved cylindrical rotor designs with symmetrical damper windings 1\.1"F
(14.10)
US 1 = Us cos(19 1 -19)
RS/Sd
li =-~ 2
wLsIs q E+RsIsq
By now adding a direct component, ISd < O, while maintaining ISq and hence torque, the volt age phasor U S moves along the dashed straight line in the indicated direction. The minimum value of the volt age is reached when the phasor U S is orthogonal to this line, hence
'Sdf :l ppli(";d,i()ll~;, as !Ollg as thc [requency mllg(~ of t.hc cycloconverter slIllire:j OH' In r \',"(' 11I11IIhcr oF t.hyriK!.or brandlcs i~ !lOt. ()IJj uow(~r (·(llIV(:rl.(:1's, the schenw in Pir~ . ] 4 . .1. 7 a wOllld call for a second DIA ('OIJVmi,l lI ' Um!. ('1 \.11 IJarate
14.3 Synchronous Motor with Load-commutated Inverter (LeI-Drive) The variabIe frequency synchronous motor drive with cycloconverter sup pIy discussed in the preceding paragraph is suitabIe for Iarge drives with high dynamic performance. 8ince the cycloconverter is line-commutated, onIy converter-grade thyristors are needed, although in a Iarge quantity. On the other hand there is the frequency limitation of cycloconverters; with a 50 Hz supply and 6-puIse circuits the range extends to about h = 20 H z. This is sufficient for a rolling mill motor having a base speed of 60 l/min and a maximum speed of 120 l/min; it could be realised with up to 20 poles at practically any power rating [84,879]. However, when Iarge high speed drives are needed, for exampIe 4500 l/min on a compressor for a pipe-line or a bIast fumace, a maximum stator fre quency of 75 Hz wouId be needed even with a two poIe motor, so a cyclo converter can offer no solution; neither could a De drive have been built for this duty. The rotor of the synchronous motor is made of solid steeI, similar to that of a turbogenerator. The simplest way to eliminate the restrictions imposed by the line fre quency is a two-stage conversion, inserting a De link for decoupling of the line- and the machine-side converters. Fig. 11.14 showed such a circuit con taining only 12 thyristors; the machine-side converter can be further simpli fied with a synchronous motor because the stator winding contains a volt age source fs in series with a transient impedance, Eq. (14.31). Hence the machine-side converter can be commutated by the Ioad without resorting to forced commutation as was necessary with the induction motor; in other words the synchronous machine is operated in the overexcited region so that it can supply the reactive current neeeded by the phase-controlled machine-side converter. The basic circuit of a Ioad-commutated inverter drive (LeI-drive) is seen in Fig. 14.18. This leaves the problem of commutation at start-up, where the induced volt age is insufficient, but this can be circumvented by intermittently bIank ing the link current with the heIp of the line-side converter. This produces Lorqnc pulsations, but there are many applications where smooth operation ill tlw very low speed range, say beIow 10% of nominal speed, is not neces sary; examples are the turbo-compressors mentioned before, also gas-turbine 01' PlllllP(~d storage sets, which are started with the synchronous generator a.diJlI'. a.s
ii. 11101.01'.
()III.Ni d('
a.
low s p ~'t:d l'l1,1I J«: , wli!'!'(· UI IIIllsl. ),(' IIlad(' »11"\\1 (·"IIIllIlll.lll. l. 'l1 01' 1.11" "' /ir llÍl ... lIi d,· (·""V(',·I. O "
I
max
• __ ~_
Uo> O , \
max
Pulsafing forque
'
(ÚS
"
iomax' uo
~ -' \
m
, .,. Uo < O
,,
,;--'Omax
• permit safe commutation, • result in minimum link current for a given torquej the line-side converter would otherwise operate at an unnecessarily low power factor.
.)
Fig. 14.19. Operating range of synchronous motor drive with direct current link
can operate smoothly in all four quadrants of the torquejspeed plane, as is illust.rated in Fig. 14.19. During regeneration the voltage tiD is reversed by ddayed firing of the line-side converter whereas the direction of rotation is dctcnuined by the sequence of the firing pulses for the machine-side converter. 'l'lw voltage UD rises with the speed up to the li.mit given by the converter, II furt.hnr incrcase of speed is then achieved by reducing the field currcnt, i.e. I.y lidd weakeuing as with a De drive. The line interactiow; of this tyP(~ (II' drive ar ['( >111"' 1" '111, 1.111' illplll. i1i l':naINj t.h i H wOllldl d\'lir d Ld,Y f"f>qllin> ,t Itli('l >()('or ~ ll)lII > , >q 1,,1 " ~ "(,III i f!f', I,II!' \llI l i"lI :i lIuld i l\('rtr>rlll,l(' l.If 'll lI 'I'ft", [iI II I. "r UI 1I11;lIl.t>IILillllalloa,d dqwlld.>1I1. :1.1'111;1
:154
14. Variable Frequency Synchronous Motor Drives
14.3 Synchronous Motor with Load-commutated Inverter
Lme reaction. This can be offset by a corresponding change of field current. Olle possibility is to control the magnitude of the induced motor voltages es 1.0 a reference value rising with speedj when limiting the voltage reference above base speed, field weakening is initiated.
Microcomputer 8085 Minimum current
_1_
Supp/y
Converter and motor Link current control/er
~
Firing circuit Line-side converter
Machine-side converter
355
link current reference begins to respond to increased torque demand, The in verter limit angle Qmax must be continuously adjusted to achieve minimal but safe extinction timej it can be determined either by open loop computation based on current, voltage and speed or by closed loop control, possibly with feed-forward compensation from the current. The control loop for the field voltage Up is included in Fig. 14.21 which corresponds to the low impedance rotor model in Fig. 14.13. AIso, control of the field voltage responds more favourably when transient currents are induced in the field winding when load is applied. A control loop for the field current ip would tend to tem porarily reduce the field voltage at a time, when it should be increased. On the other hand, control of the field current would eliminate the effect of changing field winding resistance due to temperature variations and improve the dynamic response in the field weakening region. The sequence of the firing pulses for the machine-side converter can be based on the estimated angular position of the flux model, Fig. 14.16, or on the rotor position ê, if a voltage model according to Eq. (14.33) is used, whose accuracy is questionable at low speed, An important feature is the starting control algorithm in the very low speed range by temporarily blanking the link current via the line-side con verter and firing the shunt thyristor. This is a convenient way of quickly altering the mode of operation, such as torque reversal at low speed which could take too long when waiting for the next natural commutation.
n 1/min
Jo'i~.
14.21. Microcomputer based speed control of LeI synchronous motor drive
wil.h I)C link converter
A control scheme incorporating these features is shown in Fig. 14.21. The cil'cuit contains four converters of different types and sizes.
p()w(~r
• • • •
Intermittent Link current
1 IIII,II~" 0;5
t:he refercncc values for the De link current and field voltage as wdJ II:·; :;i,~lIals for lhe digital firing circuit of the machine-side cOllvertcr a.nel tIl(' lidllllll.illg I,hyrislol', 'l'he function gmleralor 1)t'(~scl'Íhillg lhe Jillk cnlTcnt rd'e r
()r a:;snl'ing cont.iulIOtlH ClllT(~llt;
w!.ic 'h c.ptl ,' hl.v ll !l~H II" ( 'I :
i..Q
20
;1.1'('
1IlOdc' ('(,"!.r()11Ül d lh'l.('d t.llrOIlI~h 1./1 0 Jirillf~ I lll! ';IC~
,/
~
40
f'lI' which the microcomputer is producing coordinated control comman(h;.
('II«T Iaa:; (.llI' pllrpO:j(;
o
-200
- 400
- 600
- 800
A
Linc-sid e converter, Machinc-side converter, ShUllt thyristor, J,'idd power supply,
'l'IH'y
800 600 400 200
iu
t\w lllillÍlllal
c·.tlrrell!.
(,r II.ilc' HlII.dlÍll C:~H iel(' C( 'IlV (', ['i,(' 1
il tl IIppC' 1' III' l"wC'r 111111 ", "
(I CW
(\1
n"'ilHI I,IJ
II H
1.1.,.
1;0
1;5
2;0
11111
,"
2;5
3;0
3;5
4;0
1 5
Fig. 14.22. Reversing transient of a 20 kW synchronous motor drive at half nominal speed
/I. :W kW syllchrouous Illotor drive wit.h microcomputer control has heen c\c'si ).{lIrt! iii UI(' lahonüol'Y aliei t.(~sl.(~d l1un,lu7,IUH1· Fjr;lln ~ 11.22 ckpic:t.s a n·nll,I,·,llc ' vc ·I'nÍtl g 1,1'1I.1I: 1'iC'llt.nt iraI!' IICIIIJillll.l :iJltTCI wil.lr 1.1iC' IIlC'J'tiacl!' LliC' clriv\'
14. Variable Frequency Synchronous Motor Drives
356
increased by a dynamometer without load. The intermittent link current at low speed as well as the upper and lower current limits are clearly visible. The oscillogram demonstrates that the dynamic response is slower than could be expected with a field orientated type of controI. However, there are many applications where fast response is not a primary concern.
14.3 Synchronous Motor with Load-commutated Inverter
357
UTh
-V 200 Thyristor voltage
O -200
n min
-1
- 400
2000
Speed
aO
1500
1600 1000
/
1400
io
Load applíed 1/
A 50
40
30
20
10
O
:
delayed firing causes commutation failure
1200
De Iínk current
ia
1 1 1
A 60
I
I
De Iink current 40
I,
A
20
Field current
5
O
4
b
3
20
30
40
50
t
ms
2
Fig. 14.24. Intentional commutation failure Df machine-side converter
16;~
1400
Firing angle of machine-side ._ __ converteI
1200
~
ms
2
Extinction time
~Ii!.or d l'i v(' 011 ro\.ary 11I' ill\.il q( prt'; IN":'t , wlwJ'(' ('ip:h\. or JllOn' i II . (1, 1'1
I,wo illll"r
I'IIU ('l.[,,1I
1... I.W''I'fl Lor,!,/(' lLIHI
IOO(l!i
1110(.01'
!i lll·,'d i:l,
mo
T 2 = hwo
'
mo '
il01
=
K
(~+~). J1 J2
il02 =
I!f. (15.2)
2
F2 (
c
,·olll .l l,ill !i
(15.1)
OIH'!I
-
T1 T 2
»
1
i.e. if poles and zeros are dose together and are nearly cancelling. This is seen in Fig. 15.5, where measured step responses of the speed control loop are shown for different cases following changes of the speed reference W1 Ref and of the load torque mL. The transients were recorded on a DC drive, with the resonant load being simulated by another controlled DC drive. The parameters of the Pl-controller were in each case so chosen as to obtain optimal results [B72,B73]. It is realised that, even when the motor speed follows the reference speed in a swift and well damped transient, the load speed, being coupled to the motor speed through the transfer function
me
IFI
T1
J
E) - E2
: :
, - - - , O) 1
1 (8Iil o2 )2+1 (T1 +T2 )s (8Iilo1 )2+1'
with the following abbreviations
_ J1 _
:
/,,_,
I 1_ .------------.
L (wt/wo)
= L (mM Imo)
It contains imaginary pairs of poles and zeros which make this a diflicult plant to control; the Bode-diagram is sketched in Fig. 15.4 c. When attempting to control the motor speed W1, which is the usual prac tice because the speed sensor is normally attached to the free end of the motor shaft, it is realised that a reasonably damped speed loop with a Pl-controller can only be achieved under the condition
úl2,E2
~~)mL ) me me:
F 1(8)
369
Joop
I.rall! if'·1
8
)
L (w2Iwo) _ _ _1 - (8 I il02 )2
= L (wt/wo)
+1
(15.3)
remains outside the control loop and can exhibit practically undamped os cillations. Obviously, this calIs for a different approach to the control of the planto When assuming for a moment that in an ideal case both speeds W1, W2 as well as the torsional displacement ê1 - ê2 of the shaft are measurable, then a very effective scheme for decoupling the mechanical plant could be devised, which i~ indicated by the dashed signal paths in Fig. 15.4 b: By feeding back to thc torqnc controller a. signal whirh is proportional to differential ::ipccd, dalllpiJlg is achicved h(!CiWS(' tJl!~ two 11l;\S:';(!S are kept aligncd, thu::i r(!llJovil1l~ t.Il!! f:allN(' ror :mhSC
15.1 Speed Control\ed Drives
15. Some Applications of Controlled Electrical Drives
:170
J1
mm-'~ ~
II
200
= 10
J2
L:=
ro1
-20~P
mln-'
200
~
measures is that the torque loop is responding sufficiently fast, Te « Tt, T 2 . The effectiveness of L1w-feedback for the speed control ofthe two mass system is apparent from the recorded transients in Fig. 15.6. UnfortunateIy, feedback signaIs for speed and position of the load, W2 and C2, are usually not available. There may be practical reasons, for example that the load is driven though a gear and rotates at a very Iow speed so that mounting a speed sensor would be difficult and expensive. Similar problems exist if the motion of the load inertia is translationaI, for instance on an elevator or mine hoist.
J l = 0.2
m,
Jtvv\
J
2
fvv:=
t -20~ ~
Step change of speed reference
m;~t~
~ m~~ ~ t_20~J P ·
P
_20~J
j
---J'
~ rol
ro1 --..
m"0:::1))) 00,." m'f) T)J m
f\-...
t
L
me K
Microcomputer --i-Drive I
v"-...
t
371
ContraI plant
Step change of load torque
1~t
a
b
~m2
v
me
,,_
\.--õ+--mL
Fig. 15.5. Transients of speed control loop with two-mass system, recorded with :. 1\0 kW DC drive (a) J 1 = 10 J2 ; (b) J1 = 0.2 h
II
~.L
Drive
__________ _ ----------ControT:--------- .. ------,
"Load feed-forward ~ú)-
~ú)-
Feedback
P
of(
of(
on
. -1 /1)/1.' 200 .
.o~p Jl J 2
_
10
2s
m;nl~ 200
' 20~
DamPing~I
I
t=
11
on
~
t
min-~ I 200 o
-200
1
,-
roi-roi 1
r----- 4
-'+
-'+ I I
II
Feedback
L
~
~
t
J 'f = 0.2 2
t:=== min-'~ 200 ~
\;--J t -20~1P ·
Fig. 15.6. Effect of auxiliary Llw-feedback on speed control of a two-mass system, recorded with a 40 kW DC drive
'-... Input to model
®
II
Output ------ ofmodel
I~ I
[email protected]~_~i-i
rol ___
-
- ro
II
k
2
I~
...---+-I
k3
I
1 1
~e
i
i I
1
11
II
II
l_ rol_- ro?________________ ....!
~~
(Observer)
Pig. 15.7. Speed control of a two-mass drive system with auxiliary feedback signals derived from an observer
ln such a situation one could attempt to derive the necessary information from a dynamic model, called an observer [W17, W18], which is shown in Fig. 15.7 in simplified formo Thc observ(~r is a mathematical model of the plant tllat is (~valllated in. real time iII pamlld witb the actual transients of the pJ a.1I 1.. 1':Kl.ablishiJlg thiR modd 1'('qui"" 1"! I.hf~ s t.l"IIdlll'( ~ Rlld thc parnlllc krs or 1.11(' plalll, t.o Iw knowlI fnirly ILI" 'III'il,I,,·ly, wllich iK IIsllal1 y uo I.~n ' at. prohl"11I wll.h 1I1""hHlli(",d Ky fi t."III IL Til, · Iii " !'I V"II hy t,lu' 1)1( ~ I I.H llrll.hl('
III""'"
i IlPIlI,(I "I' 1.1", ,,111111.,
'III
I.h ln
C / Iii"
1,lI u 1."1 ''1''''
1" '1"1' 1'.1 11' "
wh1rh iII ,dl'Pfu l.v IWII,II
15.2 Linear Position ControI
15. Some Applications of Controlled Electrical Drives
372
able in the microcomputer; the remaining inputs, particularly the load and frictional torques, are unknown and have to be estimated. Keeping the model in line with the actual plant is achieved by comparing some accessible output variable from the plant, for example the motor speed Wl, with its estimated counterpart Wl and correcting the model by additional inputs so that the er ror vanishes. If the model parameters are correctly representing the plant, it can be assumed that the remaining estimated variables in the model are also tracking the real variables, i.e. that the model serves as an accurate observer. Clearly, designing an observer calls for a detailed analysis of the system to be modelled [L80]. The structure of the observer seen in Fig. 15.7 gives an idea of how the correction of the model can be achieved; each of the integrators in the model is influenced by the error Wl - Wl in order to obtain maximum flexibility in selecting the eigenvalues of the observer by suitable choice of the gain factors k i . This is so because the observer represents itself a feedback system, the clynamic properties of which are the subject of a separate analysis. Realisation of observers for complex applications has only become prac tical since distributed digital signal processing with microelectronic compo lIents is possible; neither analogue techniques nor a main-franle process com puter could provi de an economic solution.
Ú)1
Ú)1
ú)o
ú)o
0.2
0.2
0.1
0.1
373
rather unsatisfactory transient considerably by applying differential speed feedback and damping signals from the observer; the oscillations are now all but eliminated. The algorithms for the speed controller and the observer have been implemented in a single board 8086-microcomputer; the sampling period for the complete algorithm was 2 ms, the storage requirement < 1 k byte of ROM. The characteristics of the plant and its parameters must be known for the computed variables produced by the observer to be trustworthy. While the dynamic structure of mechanical systems is usually known from the ge ometry, the distribution of masses with their linkages etc. may not be known accurately or may change under varying operating conditions. EXanlples are mine hoists, where the winding rope at different length acts like a variable spring, or robots with changing geometry when positioning a mass by a ro tary movement at a varying radius. li the parameter changes are known or can be derived from other sources of information, it is of course possible to include them in the model, forming a time varying observer; this is feasible if the observer is implemented in a microcomputer. However, if the variations of plant parameters are unknown or even the structure of the plant is uncertain, the problem of estimating internal vari ables becomes much more complex because it involves the identification of the plant as well as the adaptation of the observer and the controllers. This task has also come now within reach of a microcomputer but it would require a more detailed presentation than is possible in the present con text [B73, M21, WI3].
15.2 Linear Position ControI
O
a
2
t /s
0 1
b
2
t /s
Fig. 15.8. Step response of speed control of a two-mass system, recorded with a I kW DC drive. (11) PI-speed controller; (b) P-speed controller and observer
Tlte cstimate of the load torque, mL, is generated by integrating the Sl)(-'(~d (~rror WI - Wl; this permits the use of a proportional instead of a proportiollal~ pllls-illl/~g;ntl (PI) speed controller thus reducing the speed overshoot followiu ii, cllallg(~ o[ lhe rderence without sacrificing steady statc accuracy. Ileslll t.H ror a I kW De drive wit.h microcompntcr cOlllrol af(~ shown in Fig. lG.H \ W I ~ , W 1:1\. '1'1'1(' :iI,cp rcaring of the main construction. The two or more servo motors could I II : arranged at symmetrical points arollnd the circumference and exert torqlle I.hmlll·;1t pillioll drives to the central gear. A simplified schematic is drawlI ill I"i/':. I ri. 1:\ ass1lmiu!; two motors, with the main strueture of thc étutellUi1 (tlld 1.11(' I.wo drivI~ lIIoLors I>eillg IUlI1ped illto thr
whcr'
/" 1
lnitial position
of shear
çt)V2
I c:
x 10
~I
I"ig. 15.31. Principie of a fiying shear
'l'he basic layout of a rotating shear is sketched in Fig. 15.31. The strip- or rod-shapccl material coming from the last stand of a rolling mill moves with vdoci l,V /I j through the shear, which consists of two mechanicalIy coupled drlllw; lH!aring opposing cutting blades. InitialIy the shear may be at rest; :d. a pl'(~ddcnllined time tI is is accelerated, 50 that the tools separate the lII : d.(~rial al a. prescribcd point. The blades and the material should move in :l.ppr(\ X illlal.( ~ s,VI\ChrOllislll during the scparation, lmt a slightly higher vdoc.. iLy 01' 1.\1(' I.ools cunl.! also bc sp ecified for separatillf, t.he ma.terial. F()lI()willl~ UI!' ~ ';)h q.r "'( ,d I II I.('IUI I,h n ll I.w.. 1.' i l.y 11.11" f'/l,Il l.t'fll. IH Iil:ü hlc r :~H pOIlH~~ , coul.rolkd. A
I.collhard, W. (Ed.), Control in power electronics and electrical drives. 2''''
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