ELECTRIC MOTOR DRIVES Modeling, Analysis, and Control ELECTRIC MOTOR DRIVES Modeling, Analysis, and Control R. Krishn
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ELECTRIC MOTOR DRIVES Modeling, Analysis, and Control
ELECTRIC MOTOR DRIVES Modeling, Analysis, and Control
R. Krishnan Virginia Tech. Blacksburg. VA
Pn'nlin'

11,,11
Upper Saddle River, New Jersey 0745S
Ubrary of Congress Cl.taJog_in·PubliQllion DaH' Krishnan. R. (Ramu) Electric motor drivt:5: modeling. analysis. and mntroll R. Knshnal!.
p.em. Includea bibliographical rckrcnces and irKkx_ ISBN 0.130910147 \. E1cctric driving. I,Tllle. TK40S8.K73 2001 62146'dc21
00050216
Vrec l>resldenl and Editorial Dircctor. ECS. MtI'C"Ul J. HOfll/tl Acquisitions Edilor. Eric: F,... Editorial Assistanl:Jr.uit:" H.(;o VICC President of Prodoction and Manufacturing. ESM: Oovid IV HIl'co"l; Executlvc Managing Edilor: V"mu O'Brien Managing Editor: Dllvid A ~ Plodut:tion Editor. Lmuhmi 8010Speciallhank!> and grato itude. They are Mr. Pnwccn ViFIY r\\'M 1I01/ogn
7.4.10.4 Flux Weukl!triug Opera/ioll 377 7.4.11.1 Flul( Wcakemng 377 7.4.11.2 Calculation of Slip 379 7.4.11.3 Muirnum St:llor Frequt:ncy
7.5./ 7.5.1
7.6
361
General 362 Phase·Shifting Conlrol 362 Pulse·Width Modulation (PWM)
7.4.10.1 7.4.10.2 7.4.10.3
7.4. / I
38/
3~1
Commutatltln 382 PhaseSequence RCI'crsal 383 Regeneration J84 CompanSCln of Con,erters for AC and DC MOlor Drives
3H5
S(t'o(/I"SIU/(~ fJ;'rforw(l/Ict'
385 Dlrl'Cl Sltad.,,Swle Em/llllliOIl of Si.r·SIt'p C"rrt'tI/SOllrce Im·trwr·Fer! 1",IIICllml MOlOr reSIAt) Dr/I'I' SIS/I'm JH9 ClosedLoop CSIAt Om/e SyStem 396 Dyllamic SimI/la/forI of/ht ClosedLao" CStM Om'e Sysfem
398
Applications 405 Rercrcnces 406 Discussion Questions to? Exercise Problems 409
.11
VectorControlled Induction Motor Drives
8.1 K2 8.3
Introduction 411 Principle of Vector Conlrol 412 Direcl Vector Control .\ 15 8..1.1 8.3.2
DescriplIO/l 4/5 FI/Lf /lIJd Tonl/lc
""I( l'H"r no
10.1.1 8.J.2.2
8.3..1 83.1 8.35
$.4 8..5 Kfl
xvii
Ca~c
(1):Tcrmmal \'nltagcs 4t7 Case (u). Induecu emf from nu.~ "Cn'lIle ((JIh nr Hall ImplellltlllUIIOIl" lilt .\IX·.\lt'lI (//rrell1 SOl/rcc' 422 Implt'tII('IlI/l/IOII" 1/11 l/ollIIJ:t' SOl/rl" 71872K.NuI IDce. PHI. h. J. P. Succna·l'all..t. R I lcrnundcz. and L.L. Frcn.... !>. "Slahlht~ st ud) uf contHJll~d reclltl":I' u)ing a new di,crele mudd:' Ijroe. fllSlilllle IJ{ Ef",'me,,1 E/!!:Im!'rr!. Lnndnn. '01 11~ pp.128s..I29J.SePI 1'J72. 7 C. E. Robinson. "Re(lc~lgn 01" de mOlors fm ,tppllciltlons "llh thynstnr rO'l~1 ~up· plic tof4ue. mduced emf. ptl...·cr, and field nux vs. speed In nmlllah,...J Ull1" fot nlled art1lillur~' curren I and for ,HI Inll.'rnllllenllll:k:ral,on :II 1,2 I' u. arm:l1llre curr.:nt r.'r lhL' 'l:k:ed r,1I1ge ,,(0 to 2 p.ll.
122
Chapter 3
PhaseControlled DC Motor Drives
]. The dc motor given in Problem 2 is operaTed from a fullycontrolled threephase converter fed from a 460 V. 6OHz. ]·phase ac main. Calculate the triggering anglc whcn The machine is delivcring raTed IOrque at rated speed. The armature currenT is assumed to be continuous. 4. Assuming The current is continuous. draw 0 vs. speed. maintaining the load torquc aT rated value. for Problem]. for a speed range of 0 to I p.lI.
5. A separatelyexciTed dc mOlar is controlled from a threephase fullwave converter fed from a 460V. ]phase_ 6O·Hz ac supply. The dc mOTOr details are as follows: 250 hp. 500 V. 1250 rpm, R. = 0.052 O. L. = 2 mHo (i) Find the rated current and K b when the field is maintained at rated value_ (ii) Draw the torquespeed characteristics as function of triggering angle_ o. 6. The moTor drivc given in Examplc ].1 is driving a load proportional to the square of The speed. Draw The tOrquespeed charactcrlsTics of the drive s~,qem ami 0) vs. speed characteristics_ The combined mOTOr and loaJ mechanical constantS lire a, follows: 8, = 0_06 N·m/rad/sec.J = 5 kgm:
,I
7. DeTcrmine the electrical and mechanical Time cnnstants ,If Ihe dc motor in Problems 5 and 6. respecTively I~ nOI nldified. nlC conlrol voltage;s::': III V lit ax 111 b.... th cases. DeSign lhe eurrent controller ~ain, (PI) for the mOTOr drive gwen in Prohlt:m 5. COIlSIJ 100 hp) motor drives, but it is possible to increase the armature inductance of the machine during the design of the motor or to include an external inductor to increase the effective inductance in the armature path. The latter solution is the only practical approach in retrofit applications. Example 4.4 A separatelyexcited de motor is controlled by a chopper whose input dc voltage is 180 V. This motor is considered for lowspeed applications requiring less than 20/0 pulsating torque at 300 rpm. (i) Evaluate its suitability for that application. (ii) If it is found unsuit
Section 4.11
Pulsating Torques
149
able, what is the chopping frequency that will bring the pulsating torque to the specification? (iii) Alternatively, a series inductor in the armature can be introduced to meet the specification. Determine the value of that inductor. The motor and chopper data are as follows: 3 hp, 120 V, 1500 rpm, Ra = 0.8 n, La
=
0.003 H, Kb
=
0.764 V/rad/sec, fc = 500 Hz
Rated torque.
Solution
T er
=
3 x 745.6 21T x 1500/60
Maximum pulsating torque permitted
=
14.25 N· m
=
0.02 x T er
=
0.02 x 14.25
=
0.285 N'm
To find the harmonic currents. it is necessary to know the duty cycle. That is approximately deternlined by the averaginganalysis technique. by assuming that the motor delivers rated torque at 300 rpm.
Ter
[11 = 
K I1
14.25 0.764
=    = 18.65 A
Va = E + IavR a
d
Va
38.91
Vs
180
= Kbw m
+
[~lVRa =
0.764 x
21T
X
60
300
+ 18.65 x 0.8
=
38.91 V
.
=  =  = 0.216
(i) The fundamental pulsating torque is assumed to be predominant for this analysis. (al
2Vs
= n
vi R~
2 x 180
sin(7id) =
+ w;L;'
2
VO.8 + (21t X 500 x 0.(03)2 Kl1 i al = 0.764 x 7.6 = 5.8 N· m n
Tel
=
sin(0.2161T)
=
7.6 A.
TIlis pulsating torque exceeds the specification. and, hence. in the present condition. the drive is unsuitable for use. (ii) The fundamental current to produce 20/0 pulsating torque is
0.285 = 0.373 A 0.764
Td(spec)
lalr"pt'c\
=  = 
lalisp~c' =
Kh ,
2V s
nVR;
+ w~L~
sin(1Td)
from which the angular switching frequency to meet the fundamental current specification is obtained as
Wei
f et = 
2n
=
10.23 kHz
150
Chapter 4
ChopperControlled DC Motor Drives
Note that fel is the chopping frequency in Hz, which decreases the pulsating torque to the specification.
(iii)
Let Lex be the inductor introduced in the armature circuit. Then its value is
. "( .., .,,, (4 x 1802 Sln0.2161T))/(1T(21T x 5(0)(0.373)) 
(
0.8 ) 21T x 500
2
 0.003
= 71.5 mH
Example 4.5 Calculate (i) the maximum harmonic resistive loss and (ii) the derating of the motor drive given in Example 4.4. The motor is operated with a base current of 18.65 A, which is inclusive of the fundamentalharmonic current. Consider only the dominantharmonic component. to simplify the calculation.
Solution By dividing by the square of the base current. the equation is expressed in terms of the normalized currents as
I~vn + lin == 1 p.u. where
where
Vs
Za
YR; + w~L;
Vb
Zb
Zh
Vb
=   = 6.43 n
Vsn = ~ ~ ~an
Zh
=
Ib
120
18.65
180
V sn = 120 = 1.5 p.u.
Za
=
YO.8 2 + (21T
Z3lI
=
9.42 6.43 = 1.464 p.u.
* 500 * 0.(03)2
=
9.42 fl
For a duty cycle of 0.5, the dominantharmonic current is maximum and is given as
V2
1.5 I 1n =   6  = O.46p.u. 1T 1.4 4 (i) The dominantharmonic armature resistive loss is ?
PIn
= IlnR an
.,
0.8 6.43
= 0.46   = 0.02628
p.u.
Section 4. 12
ClosedLoop Operation
151
(ii) For equality of losses in the machine with pure and choppedcurrent operation, the aver
age current in the machine with choppedcurrent operation is derived as la\'n
=~=
V1=A62 = 0.887 p.u.
which translates into an average electromagnetic torque of 0.887 p.u.. resulting in 11.3% derating of the torque and hence of output power.
4.12 CLOSEDLOOP OPERATION 4.12.1 SpeedControlled Drive System The speedcontrolled dcmotor chopper drive is very similar to the phasecontrolled dcmotor drive in its outer speedcontrol loop. The inner current loop and its control are distinctly different from those of the phasecontrolled dc motor drive. This difference is due to the particular characteristics of the chopper power stage. The current loop and speed loop are examined. and their characteristics are explained. in this section. The closedloop speedcontrolled separatelyexcited dc motor drive is shown in Figure 4.20 for analysis. but the drive system control strategy is equally applicable to a series motor drive. 4.12.2 Current Control Loop With inner current loop alone, the motor drive system is a torque amplifier. The commanded value of current is compared to the actual armature current and its error is processed through a current controller. The output of the current controller. in conjunction with other constraints, determines the base drive signals of the chopper switches. The current controller can be either of the following types: (i) PulseWidthModulation (PWM) controller (ii) Hysteresis controller
DC Supply
I Current Controller PI Speed Controller & Limiter
L.i
Figure 4.20
IChopper Po\ver
ICircuit
He ...1
Speedcontrolled dcmotor chopper dri\'e
152
ChopperControlled DC Motor Drives
Chapter 4
The selection of the current controller affects its transient response (and hence the overall speed loop bandwidth indirectly). These two controllers are described in the following sections. 4.12.3 PulseWidthModulated Current Controller
The current error is fed into a controller, which could be proportional (P), proportional plus integral (PI), or proportional, integral, and differential (PID). The most commonly used controller among them is the PI controller. The current error is amplified through this controller and emerges as a control voltage, ve . It is required to generate a proportional armature voltage from the fixed source through a chopper operation. Therefore, the control voltage is equivalent to the duty cycle of the chopper. Its realization is as follows. The control voltage is compared with a ramp signal to generate the 00 and offtimes, as shown in Figure 4.21. On signal is produced if the control voltage is greater than the ramp (carrier) signal; olf signal is generated when the control signal is less than the ramp signal.
o TP
I
0
I I I
I
1
I

I
I
I
I
I
ontime signal
T]
t
t_ _~   _ i
I I I f
I I I I
o T~
I
I
I
I I I
I I I
I I I
0 Figure 4.21
Generation of basedrive signals from current error for forward motoring when One is using the chopper shown in Figure 4.2
Section 4.12
ClosedLoop Operation
153
This logic amounts to the fact that the duration for which the control signal exceeds the ramp signal determines the duty cycle of the chopper. The on and offtime signals are combined with other control features, such as interlock, minimum on and offtimes, and quadrant selection. The interlock feature prevents the turning on of the transistor (top/bottom) in the same leg before the other transistor (bottom/top) is turned off completely. This is ensured by giving a time delay between the turnoff instant of one device and the turnon instant of the other device in the same phase leg. Simultaneous conduction of the top and bottom devices in the same leg results in a short circuit of the de source~ it is known as shootthrough failure in the literature. Figure 4.21 corresponds to the forward motoring operation in the fourquadrant chopper shown in Figure 4.2. When the motor drive is to operate in the third and fourth quadrants, the armature current reverses. This calls for a change in the currentcontrol circuitry. A block diagram of the current controller is shown in Figure 4.22, including all the constraints pertaining to the operational quadrant. The speed and current polarities, along with that of the control voltage, determine the quadrant and hence the appropriate gating signals. The ontime is determined by comparing the ramp signal with the absolute value of the control voltage. The currenterror signal, which determines the control voltage vc' is rectified to find the intersection point between the carrier ramp and V c A unidirectional carrierramp signal can be used when the control voltage is also unidirectional, but the control voltage will be negative when the current error becomes negative. It happens for various cases, such as reducing the reference during transient operation and changing the polarity of the reference to go from quadrant one to three or four. Taking the polarity of the control voltage and
ia
S of the mOlordrive systems. eliminating inadvertent design mistakes and the rc~ulting errors in the prototype construction and lesting and hence the need for dyllllmic modds of the induction machine. The dynamic model of lhe induction machine in direcd 10 obtain transient responses. smallsignal equations. and a multitude of transfer functions. all ot which arc usdul in the study of converterfed induction· motor drives. Spacephasor approes
Fil:"rc 5.17
8
f1""·ch,,n for dynam,c "mula'h'" "I Ihe ,nducl,,,,, I1u""r
m
226
Chapter 5
Polyphase Induction Machines
Solution The plots are shown in llgurcs 5.IX(u). (b) ,tnd (cl. The llI11chinc outputs. such as air gap torque, speed. actual stator ,tnd rotor phase currents. and magnitude of st,1I0r nux linkages are all the same rcgardlcss of reference franles Also note that the flux linkagcs are in real units. white other variables arc in normalized units. Line start produces higher currents. flux linkilgCs. and dc offsets in them. The torque pulsations arc vcry severe. and repealed line starting could cndanga thc I11cch,tnical intc~rily of thc motor. Higher stator and rotor curn:nts also producc resistiv..:: losses thm ,Ire multiple till1e~ the design linlll.
5.9 SMAllSIGNAL EQUATIONS OF THE INDUCTION MACHINE S.9.1 Derivation 'nle electrical equations of the induction machine and the ('leclromcchanical sub· systems given in (5.IR7). (5.194). and (5.195) combine to give Ihc dynamic equations of the motorload system. Thesl' dynamic equ
'.
'" UIJ5
'" WJ
!Ill)
UI~
I ;.
II.?!I
II
!~
fj.:'~
uu~
n.w
UI_~
II :.'11
(ll~
•
Uti u~
Ifl
'.
'"
IIH~
Hili
UlH
1I~
II
.~
IIfU
, ,"
'"
'"
u~
,•
l/ll U~
'"
uu~
1/111
II
(I
!II
II !~
n(l~
!lUI
:
' II
I~
II lU
Ul~
, ," ,"
'II WI
IlIJ~
1110 U l~ n .). lime. ~ l,'J I "e~""u'ekr_"'OI1 ,h,..... re.,.,,,... HI ....h .. refncnc, 'ram.:. !Pan II lilTk' ,
I',~urt· f;. III
I~
A
~v
,"
II.'
"" ... '"
u!t)
,
• ,• , ," ,
II~
(11"1
I ;.
,
IfI"
IIU!
,
"" '"
U"I
(J 1$
,
• ,"
""
fl.'
'"
1110
,"
I
II!
nu~
,•"
H!
1
22'
,
II:~
228
Chapter 5
Polyphase Induction Machines
, rrr::r,,,
, 1
~
1
,
0.7
0_6
03 0.1 0.1
6
• L..I._'_L'' 0.00 0.05 0.10 0.15 0.20 0_25
. . ~'_
'r,.,,,
06 ,     .       ,   ,  
0.00
_'___.JL___'__=I
n LU
0.15
0.20
liZ.."
03
1
,
U
,. deg
Vas' V
20.6 31.4
153.2 70.7
The triggering angles of (XI and (X2 with respective conduction angles ~l and ~2 yield the two stator voltages for chosen operating conditions. There are four unknowns (aI' a l , ~I' ~2) and two known values of vas' so it is necessary to resort to Table 6.1 to find various choices of (X and ~ for the given phase angles. They then are substituted in the voltage equation to verify that they yield the desired voltages. By that procedure,
al (Xl
== ==
102 0 with 135° with
PI == 98° P2 == 68°
are obtained. thus giving the range of triggering angle variation as 102° to 135°.
Example 6.2 Determine the currentvs.slip characteristics for the phasecontrolled induction motor drive whose details are given in Example 6.1. Assume fan and frictional loads, and evaluate the load constant based on the fact that 0.2 p.u. torque is developed at 0.7 p.u. speed.
Solution VII 460 Base voltage, Vb = .. ~ = .. ;: = 265.58 V
v3
v3
. Pb 150 x 745.6 Base current, I b = 3 = = 140.37 A Vb 3 x 265.58 Base speed,
Wb =
Base torque, T b
=
21T x 60 21T X 60 P/2 = ~ = 188.49 rad/sec Pb Wb
=
150 x 745.6 188.49
= 593.33 N ·m.
Case (i): Friction load 0.2Tb 0.2 Load constant, B, =   = 0.7Wb 0.7
Kr =
2 Ws
x 593.33 X
88 9 = 0.8994 N'm/(rad/sec) 1 .4
B (P)2  220.03 3 ·R _
I
2
1m =
r
Kr ~
In
. =
I r     .
1.5675 \! s(l  s) p.u.
278
Chapter 6
PhaseControlled InductionMotor Drives
The normalized stator current is
_ J(R )2 2[1
lasn  1m
r

+ X 1r R
~ +'X s J ir
s
1]
+ .
(p.u.)
JX m
Case (ii): Fan load Load constant, 8,
O.2Tb
""I
= " =
Kr =
(O. 7W b)" ~
W
0.0068 Nm/(rad/sec).c
B, (P)3 3 ·R
s
2
=
262.99
r
The stator current is computed as in the frictionload case. The normalized currentvs.slip characteristics are shown in Figure 6.11. Note that, at synchronous speed. the current drawn is the magnetizing current: the rotor current is zero. It is to be noted that the speedcontrol range is very limited in phasecontrolled drives for currents lower than the rated value.
o.g rrr,r=====F==r,r~____,
Friction Load
0.7
0.0
0.5
0.3
0.2
0.]
OL..~...l.I
o Figure 6.11
0.1
0.2
_ __"'__ _
0.3
...L.._
0.4
0.5 slip
___I_ __ " ' __ _...L..__
0.6
0.7
0.8
____L_ _....Jl
0.9
Normalized fundamental stator current vs. slip for friction and fan loads
Section 6.2
Stator Voltage Control
279
6.2.7 ClosedLoop Operation
Feedback control of speed and current, shown in Figure 6.12, is employed to regulate the speed and to maintain the current within safe limits. The inner currentfeedback loop is for the purpose of current limiting. The outer speed loop enforces the desired speed in the motor drive. The speed command is processed through a soft start/stop controller to limit the acceleration and deceleration of the drive system. The speed error is processed. usually through a PItype controller. and the resulting torque command is limited and transformed into a statorcurrent command. Note that for zero torque command. the current command is not zero but equals the magnetizing current. To reduce the noload running losses, it is advisable to run at low speed and at reduced flux level. This results in lower running costs compared to rated speed and flux operation at no load. The current command is compared with the actual current, and its error is processed through a limiter. This limiter ensures that the control signal veto the phase controller is constrained to a safe level. The rise and fall of the controlsignal voltage are made gradual, so as to protect the motor and the phase controller from transients. In case there is no feedback control of speed. the control voltage is increased at a preset rate. The current limit might or might not be incorporated in such openloop drives. 6.2.8 Efficiency
Regardless of the openloop or closedloop operation of the phasecontrolled induction motor drives, the efficiency of the motor drive is proportional to speed. The efficiency is derived from the steadystate equivalent circuit of the induction
abc
L T lit.? e*
PI speed controller
+
rT*
.e Function generator
.
Limiter
14
I
..Ja
Tachogenerator
Absolute value circuit
W mr
t=
Filter
Figure 6.12
Closedloop schematic of the phasecontrolled induction n10lor drive
280
Chapter 6
PhaseControlled InductionMotor Drives
motor. Neglecting stator copper losses, stray losses, friction, and windage losses, the maximum efficiency is given as 2
. . Ps Pm  Pfw Efficiency, 1") = = P
p
a
a
1m
(1  s) .  •
S
Rm
2
1m ·R S m
= (1 
S)
=
Wm
(6.33)
Here, Pa and Pm denote the air gap and the mechanical output power of the induction motor, respectively, and Pfw denotes the friction and windage losses. Slip variation means speed variation, and it is seen from (6.33) that the efficiency decreases as the speed decreases. The rotor copper losses are equal to slip times the input air gap power. This imposes severe strain on the thermal capability of the motor at low operating speeds. Operation over a wide speed range will be restricted by this consideration alone more than by any other factor in this type of motor drive. The efficiency can be calculated more precisely than is given by equation (6.33) in the following manner: (6.34) where 3I;R (1  s)
Pm
= Power output =  r  
Prc
= Rotor copper losses = 31;R r
S
(6.35)
Psc = Stator copper losses = 3I;sR s
Pco
= Core losses = 31~Rc
Pst is the stray losses, and the shaft output power is the difference between the mechanical power and the friction and windage losses. This calculation yields a realistic estimate of efficiency. Energy savings are realized by reducing the motor input compared to the conventional rotorresistance or statorresistance control. Consider the case where speed is controlled by adding an adjustable external resistor in the stator phases. That, in turn, reduces the applied voltage to the motor windings. Considerable copper losses occur in the external resistor, Rex. In the phasecontrolled induction motor, the input voltage is varied without the accompanying losses, as in the case of the external resistor based speed control system. The equivalent circuit of the resistancecontrolled induction motor is shown in Figure 6.13. The equivalent normalized resistance and reactance of the induction motor are obtained from equations (6.2) and (6.3). The applied voltage is then written, from the equivalent circuit shown in Figure 6.13, as (6.36)
Section 6.2
Stator Voltage Control
281
Vas
o4_J
Figure 6.13
E4uivalent circuit of the statorresislancccontrnlkd induction motor
The external resistor value for each slip is found by substituting for rated applied voltage and the stator current required to meet the load characteristics. The latter is obtained fronl the rotor current, calculated fronl load constants and slip given hy (6.28) and (6.29). Then the stator current is computed fronl equation (6.32) for each value of slip. Sunlmarizing these steps gives the following: (h.J7)
I( ;;R )2 + X;
\j Ias( s) =
X
. IJ ~ )
(6.3~)
III
"[he external resistor for each operating slip is
Rex =
JC~a~J2 
X;m  R,,"
(6.39 )
lne energy savings hy using phase control of the stator voltages are approxirnateJy (6.40)
where las is obtained from equation (6.38) and R~,\ is obtained fronl equation (6.40). For Exanlple 6.1, consider a fan load requiring 0.5 p.Ll. torque at rated speed. "nle speed variation possihle with external statorresistor control is () to 0.35 p.ll. For this range of speed variation, the external resistor required and normalized po\ver input Pin' and power savings, Pexn ' are shown in Figure 6.14(i). "[he efficiency of the drive with statorphase and resistor control is shown in Figure 6.14(ii). TIle efficiency of the statorphase control is computed by subtracting the external resistor losses fro'TI the input and treating that as the input in the statorphase control case. ~Ille fact that statorphase control is very much superior to th~ statorresistor control is clearly demonstrated from Figure 6.14(ii).
284
Chapter 6
PhaseControlled InductionMotor Drives
~phase. O+i~~
AOHz ac supply
"'___,
T 1 :T~ Stator
Rotor
Figure 6.16
PhaseRectifier de Link controlled Transformer converter
Slippower recovery scheme
+
II
a
8

From }r_o_l.o_f   l     _ _ _ e b termll1als
."
T 1 I~
T
~,
' 7
~,
T) "} c
3PE ase
h
Ot
...... ....,
ac su pply
t
I
0....,+1++
!, I
I
I
~~ D~ I
!
C
1 
T.t
I JL, + +_ _.._ _+_ _~__+
L  _ _
...... ....,
a ." I~
~,
T(1 7
T2
;'
~..__J
1
Rectifier
figure 6.17
de link
Phasecontrolled converter
Rectifierinverter power stage for slippower recovery scheme
ternlinals. The rotor voltages are rectified and. through an inductor, fed to the phasecontrolled converter. The detailed diagranl of the bridgediode rectifier and phasecontrolled converter are shown in Figure 6.17. The operation of the phasecontrolled converter has been described in Chapter 3. The phasecontrolled converter is operated in the inversion mode by having a triggering angle greater:than 90°. Then the inverter voltage Vi is in opposition to the rectified rotor voltage V dc in the dc link. A current IJc is established in the dc link by adjusting the triggering angle of the phasecontrolled converter. The diodebridge rectifier on the rotor side of the induction motor forces power to flow only away fronl the rotor windings. A transfornler is generally introduced between the phasecontrolled converter and the ac supply to conlpensate for the low turns ratio in the induction nlotor and to improve the overall po\ver factor of the system. The latter assertion will be proved in the subsequent section.
SlipEnergy Recovery Scheme
Section 6.3
285
6.3.3 SteadyState Analysis The equivalent circuit is used to predict the steadystate motor drive performance. The steadystate performance is computed by proceeding logically from the rotor part of the induction motor to the stator part through the converter subsystem. The dc link current is assumed to be ripplefree. The rotor phase voltage, Yap and the dc link current, Ide, are shown in Figure 6.18(a). The current is a rectangular pulse of 120degree duration. The magnitude of the rotor current is equal to the dc link current, Idc . The phase and line voltages and phase current in the phasecontrolled converter are shown in Figure 6.18(b). There is a phase displacement of 0: degrees between the phase voltage and current, which is the triggering angle in the converter. The magnitude of the converter line current is equal to the dc link current but has 120degree duration for each halfcycle. The rms rotor line voltage in terms of stator line voltage is Vrt ==
(kkw,T, )sv" W2
T
2
(6.42)
where k W1 are k W2 are the stator and rotor winding factors and T, and T 2 are the stator and rotor turns per phase, respectively. Denoting the effective turns ratio by a, we write (6.43) The rotor line voltage is then derived from equations (6.42) and (6.43) as (6.44 ) The average rectified rotor voltage is 1.35sV" Vdc == 1.35Vrt ==   a
(6.45)
This has to be equal to the sum of the inverter voltage and the resistive drop in the dc link inductor. Neglecting the resistive voltage drop, we get (6.46) where the phasecontrolledconverter output voltage is given by Vi
=:
1.35Vt == cos
0:
(6.47)
where (6.48)
282
Chapter 6
PhaseControlled InductionMotor Drives 2.5
r~_or_,.~___r"_r____r____,
2.0 o
~l
 110 ) ...
~~in(
7w,I >
(7.11 )
1211 l 
}
l~O}
} r7.IJ)
(7.12)
The phase voltages arc shifted from the lillc \'ollage' by JH degree, and lheir magnitudes are
2
11" V",.
Only the fumJamenlal
produce~
useful wrquc. and h",ncc
needs to be considered for the sth and in vOltlUllf'. which is the
"'
"' u~
"'
ILl
112 Il.!
(1.11
no
tillurr 7.1('
.~.
'"
tl2
," '"
OJ f.... p.u
'" '"
(Iii
'"
'0
""n",allrnpkrn,·III••I,,'n HI Ihe ,,,Ilag"·lo·fr,,qu,,nn !',,,(,It III 1I1''''l
'HI
nil
w,
"'" "'" 0.65
"
10
171)
1.)(1
OJ >
,.,
~,
9,. .:1«"
0.0 0.5
10 0
'" ''''
e•. elccik&
tlgu~
1.40
I'WM "oIlaFcs
L~
l
In
IJ~ lind
~. L.
0
L.
L,
0
0
L,
0
Lm
0
G~p L.
0 0
"
L.
0 0
0
L,
dq ham..'
~. ]
In
p_U
i7 1&1)
L,
n
(7, IXI)
whIch is caSI III Sllu(:spacc form as
X
AX I Btl
(7,lX2l
372
Chapter 7
Frequency"Controlled Induction Motor Drives
where
A
=
L 1[R + w,G]
(7.183)
L 1
(7.184)
B=
X = i
(7.185)
u = V
(7.186)
next step is to use this set of statespace equations to obtain tne stcady·state current vector directly,
111I)]
J~O
(7. I Y7).
k
where [is the 4 x 4 identity matrix. For the example under consideralioll.lhe sampling time corresponds to 2 electrical degrees. given as T,
2
:=
f' s
(7.ltJH)
360 >
The exponential of ATs can be evaluated from the series with!:! 10 IS value of k is given by k
+
I
:=
360/2
:=
180
lcrlll~_lh:
(7.199)
which gives k := 179. For wave forms with halfwave symmetry.lhe final and initial values are related by other than idenlity, as is shown in the section on the sixstep inverterfed induction motor drive. 7.4.10.4 Computation of steadystate performance. nle currel1l vcctor is evaluated for the entire cycle from the discretized statespace equation discussed and derived above. The electromagnetic torque is compuled as (7.2ll0}
where lhe currents are the components of the state vector X(k). Bv inverse lransformation.the phase currents are computed as
it>o(k)
:=
iAk} = i".(k) O.5i",(k)  O.866iurcc feed at base \'ohage. Find the slip and electromagnetic IOrque althc deMred opcr.tlln~ point. The machine details arc as follows: ~o
hI'. 460 V.~ pllk. ] ph,l....:.HI III. Y cunnected
R," 112090:L,. "" 0041i.L,'" 00·*25 H:L, =0.043 H
Ignore the stator resistance and c()n~idcr lhallh.: ma~ne\1l:lIlg branch is placed ah.:ad or the stator impedance 11\ lhe .:qulvalcnI CirCUit in lln.1cr III ~Imphfy the analytical exprCS~I(ms in this example. Solution
w.. .. 2'1l"t X..
=
~
2'Il" ·6U • 376.99 rad sec
" IS.1+.. d axis
~
axis
Stationary reference Figure 9.39
Phasor diagram corresponding to an error between the actual and assumed rotor position
frames. Therefore, the machine equations in the assumed rotorspeed reference frames are
Rs
W:
~:
[ P~exm] Pl~m
sLq
wrm L d
[wrm~af]
Ld [:::]
~
+
~:
+
Lq
(9.146)
Ld
In the model a and ~ reference frames, the actual machine equations are written from the d and q axes as
Rs
W
rm
Lq Lq Ld
 W
rm
Ld Lq
Rs Lq
[
l ~ a~]
Wr~af
+

L: cos 08
wr~
f
__ a
Ld
sin 88
Vex +
Lq v~
(9.147)
Ld
The variables without the second subscript indicate that they are actual machine variables~ the machine model (or estimated) variables end with subscript tn. 'The actual machine equations are derived on the understanding that a~ axes are the
Section 9.8
Sensorless Control
565
considered reference axes and hence the rotor flux linkages have components on them from the d axis given as a function of the error in rotor position, Be. It is assumed that the entire rotor field is aligned on the d axis. Discretize the two sets of model and actual current equations, respectively, as
~am (kT)] = [~am (k  IT)] + [P~am (k  1T)]T
[ 113m (kT)
113m
(k  IT)
(k  IT)
pl13m
(9.148)
(9.149) where T is the sampling time and k is the present sampling instant. Substituting for the derivative terms in terms of the currents, induced emfs, and input voltages from equations (9.146) and (9.147) into (9.148) and (9.149) and finding their respective current errors yields the following:
[
B~a (kT)] = [~a (kT) BI13
(kT)
 ~am (kT)] 113 (kT)  113m (kT)
Aaf
= T  L q (w r cos 09  wrm ) WrAaf
L:
. SIn
(9.150)
Be
A number of assumptions have been made to derive (9.150): (i) The difference between the model and actual currents, when multiplied by T, becomes negligible: (ii) The sampling time is very small compared to the electrical and mechanical time constants of the drive system. If Be is smalL then the following approximations are valid for use in interpreting the above results: sin BO == Be
(9.151 )
cos Be == 1
(9.152)
Hence, substituting these into error currents results in (9.153) (9.154 ) from which the actual rotor speed is obtained as Wr
Lq 1 . T Bl a (kT) +
= 
W rm
(9.155 )
Aaf
and the error in estimated rotor position is
L d 1 Bi 13 (kT)
BO=Aaf T Wr
(9.156)
566
Chapter 9
PermanentMagnet Synchronous and Brushless DC Motor Drives
rI I I
Initial W
rm
, 8rmo+~~
I I
Gain
1 I
8 rm :
i as O4~ i bs Transformatio ~~M+)(~~ to a and f3 axes
I I I I I
+_+'_ . 
1....._ _
Vas
v~
f\
o++~~ PMSM Model
+
I I
I
ransformation W rm
I
t
~~ Figure 9.40 Blockdiagram realization of electrical rotor position and speed estimation
Substitution for the rotor speed in the error rotorposition equation gives 88 as
Ld ( TA
af
)&i (kT) J3
88=Lq . ] [ W rm  TAaf SI,,(kT)
(9.157)
Note that 8e, W rm , and W r are for the sampling instant of kT also. They have to be evaluated for each sampling instant to follow the rotor position closely. The rotor position then is (9.158) which becomes the estimate for 8rm in the next sampling interval for feeding into the controller to compute statorcurrent commands. Filters are required to sn100th ripples in the current errors due to PWM voltages fed to the machine. The realization of the position and speed estimator is shown in blockdiagram form in Figure 9.40. Starting from standstill is achieved by bringing the rotor to a particular position by energizing the stator phases accordingly. Alternatively, a sequence of pulse patterns is issued to the inverter, which would enable the motor to achieve a significant but small rotor ~peed to start the estimation. During this time, the estirnator is kept inactive: it is brought into the control process by switching off the starting process. The input of initial values of speed and rotor position serves to sn100th the transition process from initial starting to estimation for continued operation and control. Lowspeed operation continues to be a challenge with this technique. Several other methods exist to estimate the rotor position. For one. the induced emf of the stator phases could be estimated from the measured stator currents and voltages. This method has the problem of finding the position at zero speed; there are no induced emfs at that point, and at low speeds they would be dif
Section 9.9
Parameter Sensitivity
567
ficult to estimate accurately because of small magnitude. Hence, this method has to incorporate an initialstarting process, as discussed above. Ideally, rotor position can be obtained from stator selfinductances. The reluctance variation between, say, q and d axes in the machine occurs when the magnets have a pole arc less than 180 electrical degrees, regardless of the methods of magnet placement in the rotor. This variation in the reluctance can be measured by noninvasive measurement techniques, and then they can be used to extract the rotor position information. This reluctance variation is prominent in machines with high saliency. Even the surfacemagnet machines exhibit a 100/0 variation in their reluctance between the d and q axes. References contain research papers discussing alternative methods of estimating rotor position.
9.9 PARAMETER SENSITIVITY
Temperature variation changes the stator resistance and flux remanence in the permanent magnets. The loss of magnetism for a IOOoe rise in temperature in ceramic~ neodymium, and samariumcobalt magnets is 19%, 11 % , and 4 ok, respectively. from their nominal values. The effect due to the loss of magnetism with temperature variations is predominant compared to the effect of statorresistance variations on the performance of the drive system. Further, the statorresistance sensitivity is overcome in currentregulated drives by the nature of closedloop control. That the currentcontrol has no impact on the drive system is due to temperature sensitivity of the magnets. A closedloop speedcontrolled drive system will minimize. the effects due to temperature sensitivity of the magnets. To have an understandin·g· of· this operation, consider that the drive system is in steady state and that the magnet flux density has decreased in a step fashion (not usually the case, but taken up here as an extreme case for illustration). The torque will decrease instantaneously, because the currentmagnitude command is a constant in steady state. With the decrease in the torque, the rotor will slow down, resulting in higher speed error, higher torque command, and hence higher current command. The torque then will rise and hence the rotor speed. and this cycle of events will go on. depending on the dynamics of the system, until the system reaches steady state again. In the wake of such a disturbance in the magnet flux linkages, the drive system will encounter electromagnetic torque oscillations as explained, and that may not be very desirable in highperformance drives. Saturation usually will affect the quadratureaxis inductance rather than the directaxis inductance. It is due to the fact that the direct axis, with its magnets. presents a very highreluctance path~ the magnets have a relative permeability comparable to that of air. In the quadrature axis, the reluctance is lower, because most of the flux path is through the iron. By denoting the temperaturevariation effects by ex and the saturation effects by ~, the relationship of the torque to its reference and of the mutual flux linkage to its reference are derived with the speed loop open for the PMSM drive. It is further assumed that the drive has inner current loops and that the stator currents equal their references.
568
Chapter 9
PermanentMagnet Synchronous and Brushless DC Motor Drives
0.9
f~I fv 0.8
0.7
0.6 L     . . . . . _   '   _ ' O     ' 0.7 0.8
'O..._......_'O
'
0.9
1.0
0.
Figure 9.41
Ratio of electromagnetic torque to its command for various saturation levels, with speed loop open
9.9.1 Ratio of Torque to Its Reference Ratio of torque to its reference is derived as
Te
T;
aA:fi~s * + (L d A:fi~s*
+ (L d

f3 L q )i ds *i~s *

Lq)ids*i~s*
aA:f + Ld(l  ~p )i ds * A:f + Ld(l  p )i ds *
(9.159)
Note that the ratio of torque to its reference is independent of the q axis stator current. By dividing the numerator and denominator by the base flux linkages, Lbl b, the torquetoreference is derived as
Te
aAafn + (1  f3p) Ldnidsn
T;
Aafn + (1  P )Ldnidsn
(9.160)
where the normalized rotor flux linkage is given as
Aafn
=
Aaf Lbl
b
=
Aaf Ab' p. u.
(9.161 )
Note that i~sn == (i~sn) *
(9.162)
For the same machin~,detailsused in the previous illustrations, the ratio of torque to its reference, plotted against a, is shown in Figure 9.41 for various values of ~ ranging from 80% to 100% of the nominal value of Lq . Note that i dn = 1 p.u., p = 1.607, and Ldn = 0.435 p.u.. a is varied from 0.7 to 1. Lower values of a indicate increased rotor temperature: the value of 1 corresponds to the operation at ambient temperature. Higher rotor temperature reduces the output torque for the same stator current, and saturation further decreases it from the nUllliodl \ alul:. 111c variation is linear. unlike in the case of the indirect vector controlled induction motor drive described in Chapter 8.
Section 9.9
Parameter Sensitivity
569
1.0
0.9
.11 * 1T 1T g
(9.182)
Note that the product of flux and number of conductors in series has the dimension of flux linkages and is denoted by "p. Since this is proportional to phase a flux linkages by a factor of l. it is hereafter referred to as modified flux linkages. 1T
PM Brushless DC Motor (PMBDCM)
Section 9.10
579
If there is no change in the rotor reluctance with angle because of a nonsalient rotor, and assuming three symmetric phases, the following are obtained:
L aa ==L bb ==
Lee
== L ~ and Lan == L ha = Lac = Lea = L he = L eh == M (H) (9.183)
Substituting equations (9.182) and (9.183) in equation (9.180) gives the Pi\1BDCM model as
vas] [
Vhs
V es
[1 01 O][i o ~hS as
== R s 0 0
0
]
+
[LM MM] ria] [easl L M P ~h + ebSJ M
lIes
M
L
Ie
(9.184)
e cs
The stator phase currents are constrained to be halanced, i.e., ias + ihs + i c" == O. which leads to the simplification of the inductance matrix in the model as
[
:::] = V es
[~s 0
o
o ] [ ~ as.. ] o Ills + Rs
i cs
[( L
 M) 0 0
o
~
(L  ·M)
o
(L  M)
] p[ ::] + [:;:: (l). 185 ) ic e c,
The electromagnetic torque is given by (9.186)
The instantaneous induced en1fs can be written fron1 Figure 9.8 and equation (9.181) as
e as == fas(8r)Apwm
(9.187)
== fhs( 8r ) ApW m
(9.188)
e c, == fcs(8r)Arwm
(l).l Xl))
ehs
where the functions (,Jar)' fns(flr)' and fcsC8J have the same shape as eas ' en,,' and elY with a maximum magnitude of 2: 1. The induced emfs do not have sharp corners. as is shown in trapezoidal functions, but rounded edges.lllc emfs are the result of the fluxlinkages derivatives, and the flux linkages are continuous functions. Fringing also makes the flux density functions sn100th with no abrupt edges. The electron1agnetic torque then is (9.1l)())
It is significant to observe that the phasevoltage equation is identical to the annaturevoltage equation of a dc machine. TIlat is one of the reasons for nan1ing this machine the PM brushless dc machine. The equation of Illotion for a simple SYStCI11 with inertia 1. friction coefficient B. and load torque 'f, is
dW m
J   + BW m == (Tc dt
~

1 1)
(9.1 1
machine. It is easier to cool fractionalhorsepower machines without significant additional resources: the thermal mass and surface area per unit output watt are higher in fup machines compared to the integralhp mac~ines. Therefore, halfwave converter drives might be ideally suitable in fup sizes. Further, the increase in cost to handle the thermal effects of higher copper losses in the machine has to be viewed from the overall perspective of the total cost of the motor drive system. The option (ii) is very preferable, to inherently exploit the full dclink voltage in the halfwaveconverterbased PMBDCM drive as compared to its counterpart and therefore to achieve a higher speed. That such a characteristic might be possible to exploit in many pump and fan drive applications is to be noted., thus making the halfwavebased PMBDCM drive an attractive technical solution. If the route through option (ii) is taken for comparison., a number of choices come into play, but they are not looked into for lack of space. Option (iii) might he the least desirable prima facie hut has to be viewed in terms of the overall cost of the PMBDCM drive system to assess its suitability for a given application.
E. Impact of the Motor Inductance on the Dynamic Performance From the machine equations., it is seen that the selfinductance of the phase winding plays a crucial role in the dynanlics of the current loop and hence in the torque generation. In the case of the fullwaveoperated PMBDC machine, the electrical time constant is given by (9.241 )
where M is the mutual inductance. Consider a PMBDCM with twice the number of turns per phase as compared to the full\vaveoperated machine, for operation with the halfwave converter and its selfinductance in ternlS of Ls is equal to 4L, and its resistance for equal copper losses is 2R". thus giving its electrical time constant as 4L, 'hw
2L,
(9.242 )
== 2R., == ~
To find the ratio between these two time constants, it is necessary to express M in terms of L,. frolll \vhich we get
2L s
2
(9.243 )
596
Chapter 9 TABLE 9.5
PermanentMagnet Synchronous and BrushlessDC Motor Drives Ratio of electrical time constants for various designs of the PMBDCM
Stator Slots per Pole per
Ratio of Time Constants Thw/'rtw for Turns per Phase
Ratio of Mutual to SelfInductance,
Phase
2 3
km
2N b
V2N b
Nh
0.333 0.400 0.415
1.500 1.428 1.413
1.050 1.010 0.999
0.750 0.714 0.706
a
A
N
AI
B
•
C
B'
b Figure 9.57
C'
c
New machine winding connections for the halfwave converter topology
where (9.244 ) and k m is used to evaluate the ratio of the time constants. given in Table 9.5 for a PMBDCM with turns per phase twice, v2, and equal to those of the fullwaveoperated PMBDCM denoted by N b . From the table. it is seen that the equivalent PMBDCM for operation with halfwave converter has the same electrical time constant compared to the fullwave PMBDCM drive, thus nlaking the proposed drive suitable for very highperformance applications.
F. Winding Connections The motor windings have to be connected as sho\vn in Figure 9.57 for the splitsupply converter as opposed to the halfwave converter connection, with all the phase wires forming the neutral having the same polarity for each winding. These connections will not create any additional manufacturing or design burden. thus having no impac~,on the cost.
G. DriveSystem Description The speedcontrolled PMBDCM drive system schematic is shown in Figure 9.58. The feedback signals available for control are the phase currents, discrete rotor position signals from the Hall sensors to generate the gating instances for the phase switches. and rotor speed signal from a tachogenerator or fronl the position signal itself. The inner current loops enforce current commands, and the outer speed loop enforces the speed command. The speed signal is passed through a filter. and the resulting modified speed "Iignal is compared with the speed reference to produce the speederror signal. The
Section 9.10
PM Brushless DC Motor (PMBDCM)
597
Hall Sensors
Figure 9.58
Schematic of the speedcontrolled PMBDC motor drive system
torquecommand signal is obtained from the speederror signal through a speed controller that is a proportionalplusintegral (PI) type. The current magnitude reference is derived from the torque reference by a divider circuit, and the phase current comnlands are generated in combination with respective· Hall position sensor signals through a steering circuit. The gating signals are generated for each phase switch by obtaining the phase current errors and processing these errors with PI current controllers and then combining them with carrier frequency to generate the pulsewidth modulated signals.
H. Modeling, Simulation, and Analysis of the PMBDC Drive System The various subsystems shown in Figure 9.58 are taken up for modeling. simulation, and analysis in this section. Modeling of the PMBDCM with Converter Modes: PMBDCM .model in ahc phase variables is used in this simulation. Further, an ideal model with zero conduction voltage drop and zero switching times is utilized in this simulation for the switches and diodes. The operational modes determine whether one phase or two phases conduct at a given time~ accordingly. the system equations emerge. To model when only phase A is conducting, the system equation is given by (9.245) where R s is stator phase resistance. Ls is selfinductance of a phase. e as is a phaseinduced emf. p is the derivative operator, and V s is the dclink voltage of the top half. The induced emf is given by
(9.246) where fas is a unit function generator to correspond to the trapezoidal induced emf of the PMBDCM as a function of 8r (which is the rotor electrical position). K b is the emf constant. and W r is the rotor electrical speed. fas is given by 6 TI f as (8 r) == (Or);. 0 < 8 r < (;
==
6
(n  6r )  . n
1.
1T 5TI er < 66 51T 7n 
1.0
~ 0.5 in the case of the Cdump. and the diode losses are half those of the fullwave converter. This reduction in losses translates into heat sink and thermal management reduction by the same measure, resulting in a sizable reduction in packaging size. Further~ it is helped by the requirement of smaller numbers of logic power supplies. snubbers, and gate drivers.
D. System Performance The simulation takes the same drivesystem parameters as given for the splitsupply converteroperated drive system. The target value for the Cdump capacitor voltage is set to 175 V. The Cdump capacitor is initially charged to 100 V, the magnitude of the supply voltage. Figure 9.64 shows the commanded speed (w;), actual speed (w r ), induced phase emf of phase ll, phasea current air gap torque, and the voltage of the Cdf~mp capacitor for the simulated speed loop with a speed command of ~ 1000 r/min. The dump capacitor is charged to its target value of 175 V in less than 0.05 s. This voltage is then maintained in the proximity of 175 V. The transition from 1000 to + 1000 r/min takes 0.9 seconds. As long as the commanded and actual speed are significantly different. the current command is at its maximum value of 20 A. Once the desired speed is achieved, the current drops to 8 A in order to support the motor load of 0.48 N·nl. While the direction of rotation and the sign of the air gap torque are different the load torque acts in unison with the air gap
Section 9.10
PM Brushless DC Motor (PMBDCM)
607
100 50
o 50 1 00
C_l...:::::==:1_ _L_ _L_ _L_~::::t:::==::i:==:::::::::1
5
>
0
,j"
5
E
Z
~
0 1
25 20
~
~
250 200 150 100
50 0
0
0.05
0.1
n.l:'
0.2
0.25
0.3
0.35
0,4
Time.s Figure 9.64
Dynamic simulation results of Cdump based PMBDCM drive systems
torque to decelerate the rotor. This results in faster deceleration of the rotor. The situation is different during acceleration, when the load opposes the air gap torque. resulting in slower acceleration than deceleration. 9.10.7.3 Variabledelink converter topology. One of the converter topologies \vith the advantage of varying the dc input voltage to the Inachine but with lower switch voltage not exceeding that of the dc source has other significant advantages also. Such a circuit is shown in Figure 9.65. In addition, this power converter topology has the advantages of the Cdump and splitsupply converter topologies.
A. Principle of Operation The converter circuit for a threephase output has four switches and diodes. with the additional capacitor and inductor for its operation. The converter has two stages.
608
Cha pter 9
PermanentMagnet "Synchronous and Brushless DC Motor Drives
L
TT +
iL
ic
+
+ 0
V dc
v·
I
C
Cd
Figure 9.65
Variabledelink converter topology for PMBDC drives
The first stage is the chopper, which allows the variation of the input voltage to the machine. Switch T, diode 0, inductor L, and capacitor C form a stepdownchopper power stage. The input voltage applied to the phases, Vi' is regulated by the operation of the chopper switch T. The inductor L and the capacitor C reduce the ripple content of the voltage Vi. The second stage of the converter is the n1achine side, for handling the energy from the dc link to the machine and from the machine to the source. The chopper switch can be coordinated with the phase switches to regulate the current without having to switch the phase windings at carrier frequency. Because of the coordination option between the chopper switch and the phase switches. many modes of operation are possible in this drive system. The motoring (I quadrant) and regenerative (IV quadrant) control of the PMBDCM are briefly described with this converter. Motoring Assume the direction of the motor is clockwise, which may be considered as positive with a phase sequence of abc of motor phase windings. The motoring operation is initiated when the phase voltage is constant positive for a fixed speed and with the duration of 120 0 electrical. Phase a is energized when the switch T 1 is turned on. To regulate current, T 1 is turned oft which initiates routing of the current through the freewheeling diode D I' source voltage V dc' and capacitor C~ applying a voltage of (Vi  V de) across the machine phases. The motoring operation is similar in the reverse direction~ except that the phase energization sequence will be acb in the motor phase windings. This corresponds to quadrant III operation. Regenerative Operation To transfer energy from the load to the source~ the PMBOeM has to be . operated as a generator, i.e., by providing negative torque to the machine. Negative torque is achieved by injecting a positive current during the negative constantemf period. On the basis of the rotorposition information and the polarity of i*, the appropriate machine phase is turned on. The phase switch is turned on and modulated only if the current in the phase increases beyond a current window over the reference current by hysteresis control. Normally, there is precise control of the machine input voltage through the chopper switch T, so the phase switch is rarely modulated to regulate the phase current. The phase switch is turned off only during comolutation of the phase.
Section 9.10
W mn
er
~ [
:
t
; :
PM Brushless DC Motor (PMBDCM)
609
1
5 0
T;n' Ten
Vi
j E\: :1!\1I1~"C1 o.~ b;:   1 5
Ln
_~ ~dl~'
Cn
_~ ~ ~:
i
i
: OJ.
J
0.4
0.6
O.X
0.2
mIIillIIillIIil ITill1llIill1Il
0 ~ 0.2 0.2
0_ 0.2
ebn
I
...
Pan_:.~ r'If"~'D'"IlIlIllD 0.2
e an
1
Pin_
o.~ O.S
[fIiY"'D",m:"D 0.2
0.4
1;inle. s Figure 9.66
0.6
0.8
1
Time.s Torquedrive perfornlance
B. System Performance Consider the drivesystem configuration as shown in Figure 9.58. For the torquedrive system, the speedfeedback loop is open. Both the torque and speedcontrolled drive systems are sinlulated. and results are given below.
TorqueDrive Performance
Figure 9.66 shows the simulation results of torquedrive system performance. i.e.. a system operating with only inner current loops. The machine is operating at 50% of the rated speed. The reference torque T: is set at 1 p.u. by setting i* to 1 p.u.. The phase currents i as ' ins' and ics are seen to reach the same value as the reference current i* in the machine. It can be observed that the actual torque developed in the machine T e has dips when there are transitions from one phase to another. Coordinating the currents in the incoming and outgoing phases can solve this problem. After 0.03 seconds. the commanded torque is s\vitched from 1 p.u. to 1 p.u. while the speed remains the same, i.e., the system is operating in the IV quadrant. It can be observed that the voltage applied to the machine phases is about 0.5 p.u .. unlike in other converter topologies, where a voltage switching between 0 and 1 p.u. is nornlally applied to the machine phases.
SpeedControlled Drive Performance Figure 9.67 shows the sinlulation of a speedcontrolled system operating in both forward and reverse directions. The current and torque are limited to 1.4 p.u. 111e current in the machine phases is unidirectional. hut
610
PermanentMagnet Synchronous and Brushless DC Motor Drives
Chapter 9
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the torque developed by the machine is bidirectional. The input voltage to the phases, Vi' is varied with the speed. The current through the chopper inductor iL has a high rip
ple content, but the currents in the machine phases have very little ripple. The variation in the ripple is due to machine inductances: additionally, it occurs because no subsequent switching occurs in the machine phases.
C. Merits and Demerits The merits of the proposed topology are briefly listed next. (i) Only four switches and diodes are required for fourquadrant operation with a threephase PMBDCM.
(ii) Reducing gating driver circuits and logic power supplies saves money. (iii) Full fourquadrant operational capability is available.
(iv) There is reduced possibility of shootthrough fault; switch is in series \vith the machine phase winding. (v) Capability of operating with one switch failure or one phasewinding failure raises reliability. (vi) Lower phaseswitch losses raise efficiency. (vii) Highfrequency ripple in steadystate operation can be considerably lo\ver than that of the fixeddclinkbased converters; this is a distinct advantage for highperformance drive systenls. lJnlike the Cdump converter. this con
Section 9.10
PM Brushless DC Motor (PMBDCM)
611
verter has no circulating energy, resulting in better efficiency and very low torque ripple. (viii) The power switches have a voltage rating equal to that of the source voltage, which is much lower than that of theedump and singleswitchperphase converter. The topology has the demerits associated with any halfwave converter topology, such as poorer utilization of the machine and a larger electrical time constant. In addition, this converter has twostage power conversion, resulting in a slightly lower efficiency compared to singlestage powerconverter topologies. Unlike the Cdump converter, this converter has no circulating energy, resulting in better efficiency and very low torque ripple. 9.10.8 Sensorless Control of PMBDCM Drive
The drive system is dependent on the position and current sensors for control. Elimination of both types of sensors is desirable in many applications, particularly in lowcost but highvolume applications, for cost and packaging considerations. Between the two sensors, the current sensor is easier to accommodate in the electronic part of the system; the position sensor requires a considerable labor and volume in the motor for its mounting. That makes it all the more important to do without the position sensor for the control of the PMBDCM drive system. Current Sensing At least two phase currents are required for the current control of a threephase machine. The phase currents can be sensed from the dc link current: hence, one sensor is sufficient for current control of the machine. The current sensors are relatively expensive if galvanic isolation is required. If isolation is not necessary. then the currents can be sensed inexpensively with precision resistors by Illeasuring the voltage drops across them. The latter solution is used widely in lowcost motor drives. Another approach is to use MOSFET devices, with inbuilt currentsensing capability, to measure the currents. Alternatively, the MOSFET device itself serves as a sensing resistor during its conduction. The use of the drainsource voltage drop to estimate currents is fraught with inaccuracies due to temperature effect, and, for precise current control, the feedback from this voltage drop is not a viable method. Halleffect current sensors are ideal for sensing the currents with galvanic isolation. At this stage, it is very nearly impossible to do away with current feedbacks for the control of the PMBDC machine to deliver high performance. If precise torque and speed controls are not required, current feedback control and hence current sensing can be dispensed with. Then, a simple duty cycle using an openloop PWM voltage controller is sufficient. However, the steering of the current to the appropriate machine phases requires the rotorposition information. A number of methods have caine into practice to estimate rotor position without an externally mounted sensor. Position Estimation
Position can be sensed by Hall sensors overlooking a olagnet
\\~heel mounted on the shaft of the rotor extension with the magnets. This will pro
vide just sufficient commutation signals, i.e., six per electrical cycle for a threephase machine. Such a low discrete pulse count is not suitable for highperfornlcHlce
612
Chapter 9
PermanentMagnet Synchronous and Brushless DC Motor Drives
applications. Optical encoders and resolvers provide the rotor position with high resolution, but they are expensive. Further, the position sensors require extensive mounting arrangements. Highvolume applications demand that they be dispensed with, on account of the cost and manufacturing burdens. Many methods are possible to estimate the commutation signals~ they are briefly described here. (i) Estimation by using machine model: The inducedemf can be sensed from the
machine model by using the applied currents and voltages and machine parameters of resistance, selfinductance, and mutual inductance. The advantage of this method is that an isolated signal can be extracted, because the input currents and voltages are themselves isolated signals. The voltages can be extracted from the base or gate drive signals and the dclink voltage. The variations in the dclink voltage can be estimated from the dclink filter parameters and the dclink current. Parameter sensitivity~ particularly that of the stator resistance, will introduce an error in the induced emf estimation, resulting in inaccurate commutation signals to the inverter. (ii) Induced emf from sensing coils: Sensing coils in the machine can be installed inexpensively to obtain inducedemf signals. The advantages of this method are that the signals are fairly clean, parameterinsensitive, and galvanically isolated. The disadvantages are in the additional manufacturing process and additional wire harness from the machine. The latter is not acceptable in refrigerator compressor motor drives, because of hermetic sealing requirements. (iii) Sensing emfs from inactive phases: One of the most commonly used methods for acquiring position information is to monitor the induced emf of the machine phases when they are not being energized. Note that a machine phase is inactive for 33.33% of the time and that only two phases conduct at any given time. During the inactive time, an induced emf appears across the machine winding, which can be sensed. The induced emf of the phase yields the information on zero crossing and on when the emf reaches the constant region, indicating when that phase has to be energized. The polarity of the induced emf determines the appropriate polarity of the current to be injected into that machine phase. Instead of waiting for the constant region of the induced emf for energizing a machine phase, the induced emf on integration from its zero crossing will attain a particular value corresponding to thirty degrees from the zero crossing instant. The integrator output corresponding to thirty degrees from the positive zero crossing could be termed the threshold value used in energizing a phase. This threshold is independent of the rotor speed, as is shown belo\\'. Assuming a .trapezoidal induced emf whose peak is E p at the rotor electrical speed of W b , the slope of the rising portion of the induced emf for any speed Ws is given by dividing the peak value of the voltage at that speed by the time interval corresponding to thirty electrical degrees. Then, the instantaneous value of the induced emf during its rising instant is given by
(9.266)
Section 9.10
PM Brushless DC Motor (PMBDCM)
613
which, upon integration from 0 to 'IT/6ws ' yields the sensor output voltage, V vs: Vvs
iT/6w~
'IT E p
o
1
= f e as (t)dt = 2 
(9.267)
Wb
It can accordingly be proven that this algorithm also works for machines with sinusoidal induced emf, even though the sensor output voltage will be different. Note that the sensor output voltage is a constant, independent of the motor stator parameters, and that its magnitude is always the same for any operating speed of the machine. The only thing that could adversely affect the sensor output voltage is the inducedemf peak's decreasing with partial loss of rotor flux due to temperature sensitivity of the rotor magnets. This will clearly introduce errors, in that energization may not be exactly at thirty electrical degrees from the zero crossing as desired. Optimal utilization of the machine might not be possible in this condition, unless other corrective measures are taken. (iv) Thirdharmonic induced emf: An alternative method is to detect the thirdhar
monic induced emf in the machine windings and use them to generate the control signals. A threephase, starconnected, fourwire system will allow the collection of the thirdharmonic induced emf, and this can be inexpensively instrumented with four resistors. All the methods that rely on the induced emf have the disadvantage that, at standstilL the position information is not available, as there is no induced emf at zero speed. Even at very low speeds, the induced emf might not be easily detectable. Therefore, a method to generate the control signals at and around zero speed has to be incorporated for successful starting of the machine and up to a speed at which the inducedemf methods can come in to generate the position information reliably. Therefore, a starting procedure at standstill is required. This procedure can consist of two steps: Step (i): Exciting one or two phases, the rotor can be aligned to a predetermined rotor position. This way, the starting position is known~ hence. correct starting control signals are generated. When the rotor starts moving at slow speed, the induced emf is so small that it cannot be used for generating the commutation pulses until the rotor speed reaches a certain level. This fact necessitates a second step to complete the starting process. Step (ii): Once the rotor starts moving, the stator phases are energized at a slowly varying frequency, keeping the stator currents constant. The rate of frequency variation is kept low so that synchronism is maintained and can be controlled modestly if the load is known a priori. If not, the stator frequency is altered by trial and error until it reaches the minimum speed at which the induced emfs are of sufficient magnitude to render them useful for control. This constitutes the second step in the starting process. The problem with this approach is that this is not precise; some jitter and vibrations can be felt during the starting, which may not be significantly adverse in many applications. In many cases, step (i) is skipped, and only step (ii) is used for starting of the machine.
614
Chapter 9
PermanentMagnet Synchronous and Brushless DC Motor Drives
A method based on the saliency of the rotor is another alternative, but caution must be used here: the saliency in the PMBDCMs is not very significant. This requires a detection of the machine inductance and its profile, from which the rotor position can be extracted. 9.10.9 TorqueSmoothing
It is not possible to generate ideal rectangular currents, because of the time delay introduced by the machine inductance. Therefore, the currents become more or less trapezoidal and produce a large commutationtorque ripple, as much as 10 to 15% of the rated torque. Further, the induced emfs are not exact trapezoids, because of significant slot harmonics. They, in turn, will generate harmonic ripple torques, resulting in poorer torque performance. The quality of the induced emfs is further affected by the type of armature winding. Windings are chosen for lowcost manufacturing for highvolume applications, and they invariably cause a greater deviation from the ideal waveforms. The cumulative effect of all these imperfections leads to a drive with uneven torque over an electrical cycle of its operation. That makes the drive highly unsuitable for highperformance applications. To overcome these disadvantages, methods based on currentshaping to counter the ill effects of the flux distribution are successful. To overcome the unevenness in the fluxdensity distribution, it is measured or computed, and the current is continuously adjusted accordingly. to generate a constant torque. To counter the commutationtorque pulsation, the incomingphase and outgoingphase currents are coordinated in such a manner that the sum of the torque produced by the two phases is kept constant. All of the algorithms require a set of fastacting currentcontrol loops to shape the current, with no deviation either in magnitude or phase from their references. 9.10.10 Design of Current and Speed Controllers
Since the PMBDCM is similar to the separatelyexcited armaturecontrolled dc machine, as can be seen from its model, the methods that are appropriate to the design of the current and speed controllers for that can be gainfully employed here. The design of current and speed controllers directly relevant to this motor drive can be obtained from Chapters 3 and 4. 9.10.11 Parameter Sensitivity of the PMBDCM Drive
The motor parameters that are sensitive to variations in temperature are stator resistances and rotor magnets. The use of inner current loops overcomes the effect of statorresistance variations. The use of speedcontrol loop counters the rotor t1uxlinkages variation. In that process, the linearity of the torque with its reference might be lost. In order to preserve the torque linearity in the drive system, methods similar to the air gappower feedback control have to be resorted to. The inductance variation is a function of saturation and hence of the exciting current. Therefore, it is easy to counter the inductance variations if the excitation current is measured and made available.
Section 9.11
References
615
9.11 REFERENCES 1. DC Motors, Speed Controls and Servo Systems, An Engineering Handbook by ElectroCraft Corporation, Fifth Edition, August, 1980, pp. 612. 2. R. Krishnan and P. N. Materu, "Analysis and design of a low cost converter for switched reluctance motor drives," Conf Record, IEEElAS Annual Meeting, San Diego, CA. October 1989, pp. 561567. 3. 1. Bass, M. Ehsani, T. 1. E. Miller, and R. L. Steigerwald, "Development of a unipolar converter for variable reluctance motor drives," IEEElAS Annual Meeting, Toronto. Canada, Oct. 1985, pp. 10621068. 4. R. Krishnan and P. N. Materu, "Design of a singleswitchperphase converter for switched reluctance motor drives," Conf Record of IEEE IECON 1988. pp. 773779. 5. T.1. E. Miller. Brf1shless PermanentMagnet and Reluctance Motor Drives, Clarendon Press, Oxford, 1989, pp. 7880. 6. R. Krishnan and S. Lee. "PM brushless dc motor drive with a new power converter topology," Conf Record, IEEElAS Annual Meeting, Orlando. FL, pp. 380387, Oct. 1995. 7. P.1. Lawrenson, 1. M. Stephenson, P. T. 81enkinsop, 1. Corda, and N. N. Fulton, "Variablespeed switched reluctance motors," lEE Proc. B, Power Applications, vol. 127, no. 4. pp. 253265, 1980. 8. C. Pollock and 8. W. Williams, "Power converter circuits for wwitched reluctance motors with minimum number of switches," Proc. lEE, London, Electric Power Applications, vol. 137, pt. B, no. 6, pp. 373384, Nov. 1990. 9. R. Krishnan and P. N. Materu. "Analysis and design of a low cost converter for switched reluctance motor drives:' Conf Record, IEEElAS Annual Meeting, San Diego, CA. October 1989, pp. 561567. 10. R. Krishnan, "Analysis and design of switchedreluctance motor drives," Course Notes for EE 6444, MCSRG, Virginia Tech., pp. 150, Aug. 1995. 11. R. Krishnan, "Modeling. simulation, and analysis of permanentmagnet motor drives. Part II: The Brushless DC Motor Drive," ~ Pillay and R. Krishnan, IEEE Trans. on Industry Applications, vol. 25, no. 2, pp. 274279, March/April 1989. 12. R. Krishnan, "Control and operation of PM synchronous motor drives in the field weakening region," Con! Record, IEEE Ind. Electronics Conf. Invited Paper, pp. 745750. Nov. 1993. 13. S. Morimoto et ai, "'Servo drive system and control characteristics of salient pole permanent magnet synchronous motor," IEEE Trans. on Industry Applications, vol. 29, no. 2. pp. 338343, March/April 1993. 14. Marco Bilewski, A. Frena. L. Giordano, A. Vagati, and F. Villata, "Control of high performance interior PM synchronous drives," IEEE Trans. on Industry Applications, vol. 29. no. 2, pp. 328337, Marchi April 1993. 15. P. Pillay and R. Krishnan. "Modeling, Simulation, and Analysis of Permanent Magnet Motor Drives, Part I: The Permanent Magnet Synchronous Drives," IEEE Trans. on Industry Applications, vol. 25, no. 2, pp. 265273, March/April 1989. 16. ~ Pillay and R. Krishnan. "Application characteristics of PM synchronous and BLOC motor servo drives," (·onf. Record, IEEE lAS Annual Meeting. Atlanta, pp. 380390. Oct. 1987. 17. T. M. Jahns, "Fluxweakening regime operation of an interior PMSM drive," Cont." Record, IEEElAS Annua/l\1eeting, pp. 814823, Oct. 1986.
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PermanentMagnet Synchronous and Brushless DC Motor Drives
18. R. Monajemy and R. Krishnan, "Implementation strategies for concurrent flux weakening and torque control of the PM synchronous motor," IEEElAS Conference, pp. 238245, vol. 1, Oct. 1995. 19. R. Krishnan, N. Tripathi, and R. Monajemy, "Neural control of high performance drives: An application to the PM synchronous motor drive," Invited Paper. IEEE Ind. Electronics Conf, pp. 3843, vol. 1, Nov. 1995. 20. R. Krishnan.. S. Lee~ and R. Monajemy, "Modeling, dynamic simulation and analysis of a Cdump brushless dc motor drive," Proceedings of Applied Power Electronics Conference,pp. 745750, vol. 2, March 1996. 21. M. 1. Corley and R.D. Lorenz, "Rotor position and velocity estimation for a salient pole PMSM," IEEE Trans. In lA, vol. 34, no. 4, pp. 784789, July/Aug. 1998. 22. S. Ogasawara and H. Akagi, "Implementation and position control performance of a positionsensorless rPM motor drive system based on magnetic saliency:' IEEE Trans. In fA .. vol. 34~ no. 4, pp. 806812, July/Aug. 1998. 23. R. Mizutani, T. Takashita, and N. Matsui. "Current modelbased sensorless drives of salientpole PMSM at low speed and standstill," IEEE Trans. In IA. vol. 34~ no. 4, pp. 841846.. July/Aug. 1998. 24. R. Krishnan and P. Vijayraghavan, "A new power converter topology for P~1 brushless dc motor drives.·· IEEE Ind. Electronics Conf. Sept. 1998.
9.12 DISCUSSION QUESTIONS 1. A PM synchronous motor can also be realized with permanent magnets on the stator and armature windings on the rotor. Discuss its merits and demerits compared to a PMSM with magnets on the rotor. 2. A cage rotor with PMs (called a linestart PM synchronous motor) can be used to start the motor as an induction motor and run it as a synchronous machine at utility frequency. Such an arrangement will improve the efficiency of the motor in con1parison to induction motors. Discuss its construction, detailed operation, and possible applications. 3. The linestart PMSMs can also be operated from an inverter. In that case. the cage rotor will act as damper windings to damp out oscillations and. in case of loss of synchronization. the machine can run as an induction motor without disrupting the motion. Discuss an application scenario for such a motor drive. (Hint: Highreliability propulsion applications.) 4. Synchronous machines with surfacemount magnets have very little difference between directaxis and quadratureaxis inductances. Explain why. 5. Interior PMSMs are preferred for high Lq/Ld ratios. Where would such a feature find application? 6. If the rotor is made salient with neither windings nor PMs. then the machine is a synchronousreluctance motor. Its stator is that of the conventional PM synchronous n10tor. Due to its difference in quadrature and directaxis path reluctances. a torque is produced for an armature excitation. Since its field excitation is not from PMs. it must come from stator excitation. Since the inverter gets only active power from the dc link. where will the reactive power be generated for the excitation of the machine? 7. For equal power rating of the synchronous reluctance and Pf\1 synchronous motor, will the ratings of their inverters be equal? Explain.
Section 9.12
Discussion Questions
617
8. Is L q greater than L d in the woundrotor salientpole synchronous machine?
9. What is the consequence of Lq > Ld in control of the PMSM? (Hint: Consider maximum torque generated per unit input current.) 10. Highspeed PMSMs have the rotor enclosed in a stainless sleeve to restrain the magnets .against centrifugal forces. Ideally, will there be losses in the sleeve?
11. If the stator currents are sinusoids at fundamental frequency superposed with harmonics, will there be losses in the magnet sleeves? If there are losses, will they increase with rotor speed? 12. Injecting ac rectangular currents into PMSM is used in lowperformance applications. Give reasons for such a control strategy and discuss the disadvantages of this control. 13. Torque pulsation is one of the measures for evaluating the suitability of a motor drive for an application. For a critical application requiring minimum torque pulsation, which is a suitable candidate between PMSM and PMBDCM drive?
14. A large number of control strategies have been developed and discussed for PMSM drives. Consider a fourquadrant application requiring speed variation to a maximum of base speed only. Which control strategy will be ideal, based on each one of the following considerations? (i) Simplicity in implementation. (ii) Maximum utilization of the inverter or minimum rating of the inverter. (iii) Optimal output of the machine. (iv) Optimal voltage utilization of the inverter and machine. 15. For implementation of any control strategy. a mapping from the torque and flux commands to the stator current and torque angle has to be performed. Is it a good choice to recommend online computation for this mapping? If so, justify it in the context of present computational capability available with processors. 16. A lookuptable implementation is considered for the control of a PMSTv1 and PMBOeM drive system. Discuss the merits and demerits of the performance ohtained with this implementation. Relate the bit resolution and accuracy in stator current and torque with the memory requirement for the implementation for each motor drive.
17. The inner currentcontrol loops can use stator currents in stator or rotor reference frames. Discuss the merits and demerits of using each of the reference frame currents for currentfeedback control. 18. An interior PMSM is completely demagnetized by injecting a negative d axis current. What is the magnitude of stator current to achieve demagnetization? 19. An interior PMSM is completely demagnetized by injecting a negative d axis current. Assume also a q axis stator current in the machine during this operation. Detennine the electromagnetic torque generated in the machine.
20. 21. 22. 23.
('an magnet flux linkages be reduced by stator currents? Can magnet flux linkages be increased by stator currents? Air gap flux linkage is varied for fluxweakening. Will this affect the stator tl ux linkages? Discuss the effects of losing control over d axis current in the highspeed operational region in a prvrSM.
24. What are the salient differences between the maximumtorqueperampere strategy and
the constantmutualtluxlinkages strategy?
25. Including the effects of core losses in the formulation of control strategy is desirable. How can it be achieved?
618
Chapter 9
PermanentMagnet Synchronous and Brushless DC Motor Drives
26. Core losses are constant in one of the following schemes for constantspeed but variable
torque operation in a PMSM drive: (i) maximumtorqueperampere control; (ii) constantfluxlinkages control; (iii) constanttorqueangle control. Identify the control strategy that gives constant core losses. 27. Propose a method of measurement for Ld and Lq of a PMSM, using the model developed in the text. {Hint: (i) Connect two phases together in a starconnected machine. (ii) Lock the rotor for each measurement.) 28. Almost all fluxweakening control schemes are dependent on machine parameters. Is there a control method to weaken the flux independent of machine parameters? If so, how can it be achieved? 29. Js nuxweakening possible in surfaccmountmagnet machines? 30. Discuss a discreteICchipbascd implementation of a PMSf\1drive constanttorqueangle control strategy. 31. Is it possible to implement other control strategies with discrete Ie chips? 32. Why are processorbased implenlentations popular. and what are their advantages over discrcte Iechipbased inlplcnlcntations? 33. Paranleter sensitivity of PMSrv1 has hecn discussed in this chapter. Can the state of the rotor nlagnet flux linkages bc used to approxinlately predict stator tenlperature? If so, justify it with reasoning. 34. How critical is it to have paranlcter adaptation for a torquecontrolled PMSM drive? 35. In the flux\veakening region. eventually the currcnt loops saturate and sixstep voltage operation will result. Discuss the cffects of such an operation. 36... Resorting to sixstep operation in the nuxweakening region \vill givc an enhanced torquevs.speed characteristic coolpared to the currentcontrolled operating region." Is this true'! Develop a justification. 37. Sensorlcss operation is desirahk. Enunlerate the reasons and c:xplain thcrn. 38. Starting WIth precision froln any rotor position without position transducers is difficult \vith Inany of the sensorlcss control algorithnls. Explain the underlying reason for this statcrncnt. 39. TIle directaxis selfinductance of the Prv1S~1 in stator reference franles can be modeled as L:1 = L; + L 2 cos 28... Could this infof1nation he used to identify the rotor position. 8.. ? {Hint: Ref.[21].} 40. A voltagcsource P\VM inverter applies a fundamental and a nun1her of higherorder harmonics into a PMSM. The dOlninantharnlonic current can he detected. from which the inductance can be calculate(i. TIl is is achievahle. because the harnHJnic inductive voltage drops are doolinant compared to the resistive voltage drops. Fronl inductances. the instantaneous rotor position can hc extracted for control. The following steps are involved in the detection algorithol: \h =
Ji h LLit
r l ~ L L"(lS '1H r  L= I l L j sin 28, (I'
:
L;sin2H, L'i  L cos 2H 1
1
_
Section 9.12
41.
42. 43,
44. 45. 46. 47,
4S. 49.
50. 51.
52. 53. 54.
Discussion Questions
619
where Vh = [Vqsh Vdsh]T and ih = [i qsh idsh]T. vh is the stator qd axis harmonicvoltages vector, and ih is the stator qd axis harmoniccurrent vector. By measuring v h and ih and by using the inductances of the machine, Or can be estimated. What is the difference between this method and the one given in discussion question 39? {Hint: Ref. [22].} A sensorlesscontrol algorithm injects a voltage signal at high frequency into the d axis of the PMSM. The current response is correlated with the rotor position by using machinemodelbased current estimation. Fronl the rotor position, rotor speed is derived. Compare this method with the Inethod described in discussion questions 39 and 40, {Hint: Ref. [23].} What is the effect of parameter sensitivity on the sensorless methods described in discussion questions 39,40 and 41? The nlutual tlux linkage increases with stator current. Will this saturate the stator core? In PMBDCM. the induced eOlfs might not be exactly trapezoidal. \Vhat can cause their distonion'! "PMBDCMs have surfacemounted magnets'"Is this true'? Considering fundaolentals of voltages and currents only, the oper