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Unformatted text preview: 111 Chapter 11
Induction Motors:
Balanced, Sinusoidal Steady
State Operation Exit 2001 by N. Mohan Print TOC " ! 112 Induction Motors
! Adjustable speed drives
! Servo drives
•
•
• Exit 2001 by N. Mohan Induction motors under balanced sinusoidal steady
state (Rated voltage at rated frequency)
Speed control using V/f
Field oriented control TOC " ! 113 Linefed Induction Motors
! Structure
! Principle of operation
! Equivalent circuits
! Performance characteristics Exit 2001 by N. Mohan TOC " ! 114 Structure
b − axis
ib 2π / 3 a − axis 2π / 3
2π / 3 ia ic c − axis Simple representation of three phase stator
windings Exit 2001 by N. Mohan Squirrelcage rotor TOC " ! 115 Stator Representation
Vc
−
−
− va
vb
vc + +
vb ib − +
+ ia − n − va + Va vc
+ ic Vb ! Assumptions : Rs, Ls,leakage = 0
va (t ) = 2 E cos(2π ft ) 2π
)
3
4π
vc (t ) = 2 E cos(2π ft )
3
(for a 2pole machine)
ω syn = ω = 2π f
vb (t ) = ω syn =
Exit 2001 by N. Mohan 2 E cos(2π ft  2
2
ω =
(2π f )
p
p (for a ppole machine)
TOC " ! Electrically Opencircuited Rotor
116 Vc − va − vb − vc + +
vb ib
+ − ia I mc I mb Va − n − va + vc + + ic Vb I ma ! Only magnetizing currents are present because rotor is inert
! Magnetizing currents set up rotating flux
ˆ
Im = ˆ
V
ω Lm ˆ
ima (t ) = I m cos(ω t  π / 2), etc. Exit 3 ˆ
!
vs (t ) = V ∠ω t
2 2001 by N. Mohan !
π
3ˆ
ims (t ) = I m ∠(ω t  )
2
2
3ˆ
ˆ
I ms = I m
2
3 ˆ
ˆ
Vs = V
2
TOC " ! 117 Electrically Opencircuited Rotor
Fields
at t = 0 I ma ω syn !
vs """" """
! !
Bms , ims ! Exit ω syn """"
!
Bms phase
a − axis 90 o Va + Va
I ma − is a constant magnitude, rotating flux !
3ˆ
ˆ
ims (t ) = I m ∠(ω t  π / 2) = I ms ∠(ω t  π / 2)
2
!
!
N s ims
H ms (t ) =
2# g
!
!
!
µ0 N s ˆ
∴
I ms ∠(ω t  π / 2)
Bms (t ) = µ0 H m (t ) Bms (t ) =
2# g 2001 by N. Mohan jω Lm TOC " ! Shortcircuiting the Rotor
(Rs, Ls,leakage = 0)
" Transformer Analogy 118 φm ′
im + i2 i2 +
v1 − N1 N2 Load ! Assuming no resistances or leakage inductance in the
stator windings, the stator voltages completely determine
the motor flux regardless of any rotor currents
! Flux Φm is unaffected by the load
Exit 2001 by N. Mohan TOC " ! Induced Voltages on Rotor 119 at t = 0 − − − −
ωm − + + + + + ω syn "" a − axis
!
vs ω syn """"
!
Bms θ Flux rotating at speed ωsyn
Rotor rotating at speed ωm
Rotor conductors cutting flux at speed:
ωsyn – ωm = ωslip (slip speed)
Cutting flux generates voltage across rotor conductors:
ebar(θ) = Bms(θ) # r ωslip
Exit 2001 by N. Mohan TOC " ! Induced Currents in Rotor 1110 at t = 0
− −
− ω
m −
− +
+ + + −
− ω syn
""
!
vs ω syn
θ +
+ Rbar −
−
− a − axis θ """"
!
Bms ebar front +
+
+ + − + − ibar (θ ) +
+
+ −
−
− back endring endring ! Rotor conductors (bars) shorted together by end rings
! Because of symmetry of induced bar voltages, end rings are at
same potential, therefore bar voltage is dropped across bar
resistance (assuming Lr,l = 0) generating currents by Ohms Law
ibar
Exit 2001 by N. Mohan Bm #rω slip
ebar
=
=
Rbar
Rbar
TOC " ! Rotor MMF – Reflected Rotor
MMF – Reflected Rotor Current
1111 φm,ir ' φm,ir at t = 0
at t = 0 net flux = 0 ωm ""
!
vs
ima + ira ' ω syn ""
!
Fr ""
!
Fr′ ""
!
vs ""
!
ir ' a − axis """
!
ims
""""
!
Bms "!
is a − axis !
!
!
Fs (t ) = Fms (t ) + Fr′(t )
!
!
!
is (t ) = ims (t ) + ir′ (t )
ˆ
ˆ
I′ = k B ω
r i ms slip """"
!
Bms !
F
! ! r produced by rotor currents
! Fr′ produced by additional stator currents to keep total flux
unchanged (transformer analogy)
!
! These currents are viewed as a current space vector ir′
! Total stator current is magnetizing current plus this reflected
rotor current
Exit
TOC
"
2001 by N. Mohan ! 1112 Slip frequency (fslip) in the rotor
circuit
ω slip
ω syn  ω m
=
slip : s =
ω syn
ω syn
f slip ω slip
=
f = sf
ω syn ω slip + ω m = ω syn ! Slip is rotor speed normalized to synchronous speed
! Slip generally small (< 3%), therefore rotor current
frequency is very low
Exit 2001 by N. Mohan TOC " ! Electromagnetic Torque
Production 1113 # at t = 0 ""
!
Fr """
!
Fr ' ""
!
vs a − axis
""
!
ir ' """"
!
Bms !
!
! Current ir′ , in field Bms , produces torque Tem Exit 2001 by N. Mohan N
ˆ ˆ
ˆ2
Tem = π s r # Bms I r′ = keω Bmsω slip
$&'
%2 %
kt TOC " ! Torque – Speed Characteristics
(slip small ; ωm near ωsyn) 1114 ωm
Tem,rated ω syn,rated ω slip, rated ω m,rated 0 Tem,rated Tem 0 ( ω
ω syn,ratedm ω m,rated ω slip, rated ! Linear relationship
! These curves are valid up to rated torque
Exit 2001 by N. Mohan TOC " ! Generator (Regenerative Braking)
Mode
1115 ωm
"""
!
Fr ' ""
!
Fr a − axis ω syn
""""
!
Bms ! For generation or for braking – in either case rotor speed exceeds
synchronous speed, ωm > ωsyn
! ωslip < 0
" Bar voltage polarities reversed
!
" Rotor currents and mmf ( Fr ) reversed
!
" Reflected rotor currents and mmf ( Fr′) reversed
" Torque reversed
Exit
TOC
" !
2001 by N. Mohan Rotor Leakage Inductance
φm,ir′ net=0 φm,ir at t = 0 ""
!
Fr
θr θr ""
!
vs (t ) a − axis θr ""
!
ir ' ""
!
vs """
!
Fr ' a − axis """!
ims """"
!
Bms """"
!
Bms ""
!
Fr ω syn at t = 0 θr ""!
Br 2001 by N. Mohan φ# r ""
!
Fr θ r ω syn Exit at t = 0 θ ω syn
"""
!
F '
"""!r
"
ims 1116 θr 90 o ""
!
Fr′ ""
!
vs a − axis "!
′
ir """"
!
90 o Bms
"""
!
Blr TOC " ! Rotor Leakage Inductance
(cont…) 1117 ! Effect of rotor leakage inductance is to reduce
Tem at high slip
! Rotor leakage inductance is often neglected
when motor is operating near synchronous
speed (below the rated torque) Exit 2001 by N. Mohan TOC " ! 1118 PerPhase Equivalent Circuit
at t = 0 ′
I ra Ia
θr ""
!
Fr ωm """!
"
ims """
!
Fr ' ""
!
ir ' ""
!
vs a − axis + ω at f =
V
2π a jω Lm I ma Req I ra '
I ma − ′
Req = Rr
"
!
is """"
!
Bms Va jω Leq ω syn
ω slip ′
Leq = Llr Ia ω syn Space Vectors Equivalent Circuit Phasor Diagram ! Includes rotor leakage inductance
! Does not include stator leakage inductance or resistance
! Req depends on slip
Exit
TOC
"
2001 by N. Mohan ! Power Into Rotor – Power Lost
In Rotor – Power Out Of Rotor 1119 jω Llr '
+ Va jω Llr ' ′
I ra jω Lm ′
I ra +
Rr ' ω syn
ω slip − Power in resistor is power into the rotor circuit Va
− jω Lm ω syn Rr '
ω slip Rr '
Rr ' ⇒ Pr , loss ωm
⇒ Pem
ω slip Resistor split to indicate rotor loss and
mechanical power ! Power in equivalent resistance represents power entering
rotor across air gap
! Depending on slip, some or all of this power becomes
losses in the rotor
Exit
TOC
"
2001 by N. Mohan ! Stator Winding Resistance and
Leakage Inductance 1120 Rs
+ Va
(at ω ) I ra ' Ia
jω Lls Va Ema
− jω Llr + − I ma
jω Lm Equivalent Circuit Exit 2001 by N. Mohan ' Rr ' ω syn
ω slip I ma Ia ′
I ra Ema jω Lls I a
Rs I a Phasor Diagram TOC " ! 1121 Motor Tests
! DC – Resistance Test ( Rs )
! No Load Test ( Lm )
′
! Blocked Rotor Test ( Rr′ , Lls , Llr ) Exit 2001 by N. Mohan TOC " ! No Load Test (Lm)
Rs
+ Va
(at ω ) I ra ' Ia jω Lls + Ema
− 1122 Rs jω Llr '
I ma jω Lls I ra ' ≈ 0 + Rr ' ω syn Va ω slip − Ia jω Lm − Equivalent Circuit under no
load conditions
(ω slip = 0 ∴ Req = ∞ ) Equivalent Circuit jω Lm Approximate Circuit
( Lm >> Ll ) (Rs negligible) ! Under no load conditions the equivalent circuit is
dominated by the magnetizing inductance
Exit 2001 by N. Mohan TOC " ! Blocked Rotor Test (Ll )
Rs
+ Va
(at ω ) I ra ' Ia
jω Lls + Ema
− Rs jω Llr '
I ma − Equivalent Circuit + Rr ' ω syn
ω slip Va
− Ia jω Lls 1123 I ra ' ′
jω Llr Rr '
ω syn
I ma ≈ 0 ω slip =1 Approximate Circuit With Rotor Blocked
(R eq << ωL m ) and (ωL#r << ωL m ) ! With the rotor blocked, the magnetizing inductance is nearly
shorted out and can be neglected
! Measurements give real power (into Rs and Rr′ ) and reactive
′
power (into Lls and Llr )
! Rr′ can be found since Rs was previously determined through the
DC test
2
′
′
! To find Lls and Llr we can often assume that Lls = Llr
3 TOC
Exit
" !
2001 by N. Mohan Characteristics at Rated Voltage
and Rated Frequency
1124 ′
I ra Tem
Tem,rated
2.0 ′
I ra ,rated pullout torque ˆ
Bms decreases 6.0
5.0 1.5 ˆ
Bms,rated 1.0 3.0 (rated) I ma 2.0 0.5
0 4.0 0.2 0.4 0.6 ω slip 0.8 1.0 ωm
ω syn ω slip ,rated 1.0
0 0.2 0.4 0.6 ω slip 0.8 ωm
1.0 ω syn
ω slip,rated ! Nearly linear near ωsyn
! At higher slip (ωm smaller) leakage inductances and
stator resistance reduce torque
! High currents at low speeds (startup condition)
Exit 2001 by N. Mohan TOC " ! Motor Currents, Efficiency, Power
Factor As a Function of Load
1125 100
90 Efficiency 80
70
Power Factor 60
Current (A)
Efficiency (%)
Power Factor (%) 50
40 Current
30 20 10
0 25 75 100
50
Load (%) 125 Typical for design B 10 kW, 4 pole, threephase induction motor
Exit 2001 by N. Mohan TOC " ! 1126 Line Start ! When started directly off the line, induction motor draws a
very large current (approx. 8 x rated)
! At the same time the torque available to accelerate the
motor/load is limited
! Motor can quickly overheat – Solution: Reduced voltage
soft start
′
I ra ′
I ra ,rated
6.0 Tem 5.0 Tacc = Tem − TL
Tem 4.0 Tem 3.0 I ma 2.0
1.0
0 0.2 0.4 0.6 0.8 ω slip Current vs. Speed
Exit 2001 by N. Mohan ωm
1.0 ω syn
ω slip, rated 0 TL ωm steady state speed Accelerating Torque
TOC " ! Reduced Voltage Starting (Soft Start)
Energy Savings in Lightly – Loaded
Machines
1127 ia ia
a 3phase
ac input 0 n t ea
0 t Tem
Threephase
induction
motor 0 t ! Circuit applies reduced voltage to motor during startup to avoid
large currents and over heating
! Circuit also used to reduce voltage to motor under light load
steady state conditions. This improves efficiency
Exit
TOC
" !
2001 by N. Mohan 1128 Summary/Review ! Describe the construction of squirrelcage induction machines.
! With the rated voltages applied, what does the magnetizing
current depend on? Does this current, to a significant extent,
depend on the mechanical load on the motor? How large is it in
relation to the rated motor current?
! Draw the space vector diagram at t = 0, and the corresponding
phasor diagram, assuming the rotor to be opencircuited.
! Under a balanced, threephase, sinusoidal steady state
excitation, what is the speed of the rotating fluxdensity
distribution called? How is this speed related to the angular
frequency of the electrical excitation in a ppole machine.
Exit 2001 by N. Mohan TOC " ! 1129 Summary/Review ! In our analysis, why did we initially assume the stator leakage
impedance to be zero? How does the analogy to a transformer,
with the primary winding leakage impedance assumed to be
zero, help? Under the assumption that the stator leakage
!
impedance is zero, is the fluxdensity space vector Bms (t)
completely independent of the motor loading?
! What is the definition of the slip speed ωslip ? Does ωslip depend
on the number of poles? How large is the rated slip speed,
compared to the rated synchronous speed?
! Write the expressions for the voltage and the current (assuming
the rotor leakage inductance to be zero) in a rotor bar located at
an angle θ from ! peak of the flux density distribution
the
represented by Bms (t ) .
Exit 2001 by N. Mohan TOC " ! 1130 Summary/Review ! The rotor bars located around the periphery of the rotor are of
uniform crosssection. In spite of this, what allows us to
represent the mmf produced by the rotor bar currents by a space
!
vector Fr (t ) at any time t?
! Assuming the stator leakage impedance and the rotor inductance
to be zero, draw the space vector diagram, the phasor diagram,
and the perphase equivalent circuit of a loaded induction motor.
! In the equivalent circuit of Problem 9, what quantities does the
ˆ′
rotorbar current peak, represented by I ra , depend on?
! What is the frequency of voltages and currents in the rotor circuit
called? How is it related to the slip speed? Does it depend on the
number of poles?
Exit 2001 by N. Mohan TOC " ! 1131 Summary/Review ! What is the definition of slip s, and how does it relate the
frequency of voltages and currents in the stator circuit to that in
the rotor circuit?
! What is the speed of rotation of the mmf distribution produced
by the rotor bar currents: (a) with respect to the rotor? (b) in the
air gap with respect to a stationary observer?
! Assuming L′ to be zero, what is the expression for the torque
lr
ˆ
Tem produced? How and why does it depend on ωslip and Bms ?
Draw the torquespeed characteristic.
! Assuming L′r to be zero, explain how induction motors meet
l
loadtorque demand.
Exit 2001 by N. Mohan TOC " ! 1132 Summary/Review ! What makes an induction machine go into the regenerativebraking mode? Draw the space vectors and the corresponding
phasors under the regenerativebraking condition.
! Can an induction machine be operated as a generator that feeds
into a passive load, for example a bank of threephase
resistors?
! How is it possible to reverse the direction of rotation of an
induction machine?
! Explain the effect of including the rotor leakage flux by means
of a space vector diagram.
! How do we derive the torque expression, including the effect
′
of Llr ?
Exit 2001 by N. Mohan TOC " ! 1133 Summary/Review
!
!
!
!
! Exit !
!
!
What is Br (t ) and how does it differ from Bms (t ) ? Is Br (t )
!
perpendicular to the Fr (t ) space vector?
Including the rotor leakage flux, which rotor bars have the highest
currents at any instant of time?
What clue do we have for the vector control of induction
machines, to emulate the performance of brushtype and brushless dc motors discussed in Chapters 7 and 10?
Describe how to obtain the perphase equivalent circuit, including
the effect of the rotor leakage flux.
ˆ′
What is the difference between I ra in Fig. 1118c and in Fig. 1119c, in terms of its frequency, magnitude, and phase angle? 2001 by N. Mohan TOC " ! 1134 Summary/Review ! Is the torque expression in Eq. 1141 valid in the presence of the
rotor leakage inductance and the stator leakage impedance?
! When producing a desired torque Tem, what is the power loss in
the rotor circuit proportional to?
! Draw the perphase equivalent circuit, including the stator
leakage impedance.
! Describe the tests and the procedure to obtain the parameters of
the perphase equivalent circuit.
! In steady state, how is the mechanical torque at the shaft
different than the electromechanical torque Tem developed by the
machine? Exit 2001 by N. Mohan TOC " ! 1135 Summary/Review ! Do induction machines have voltage and torque constants similar
to other machines that we have studied so far? If so, write their
expressions.
! Plot the torquespeed characteristic of an induction motor for
applied rated voltages. Describe various portions of this
characteristic.
! What are the various classes of induction machines? Briefly
describe their differences.
! What are the problems associated with the linestarting of
induction motors? Why is the starting currents so high?
! Why is reducedvoltage starting used? Show the circuit
implementation and discuss the pros and cons of using it to save
energy.
Exit 2001 by N. Mohan TOC " ! ...
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This note was uploaded on 02/06/2012 for the course EE 4002 taught by Professor Scalzo during the Fall '06 term at LSU.
 Fall '06
 Scalzo

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