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Unformatted text preview: 11-1 Chapter 11
Balanced, Sinusoidal Steady
State Operation Exit 2001 by N. Mohan Print TOC " ! 11-2 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 " ! 11-3 Line-fed Induction Motors
! Principle of operation
! Equivalent circuits
! Performance characteristics Exit 2001 by N. Mohan TOC " ! 11-4 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 Squirrel-cage rotor TOC " ! 11-5 Stator Representation
vc + +
vb ib − +
+ ia − n − va + Va vc
+ ic Vb ! Assumptions : Rs, Ls,leakage = 0
va (t ) = 2 E cos(2π ft ) 2π
vc (t ) = 2 E cos(2π ft )
(for a 2-pole machine)
ω syn = ω = 2π f
vb (t ) = ω syn =
Exit 2001 by N. Mohan 2 E cos(2π ft - 2
(2π f )
p (for a p-pole machine)
TOC " ! Electrically Open-circuited Rotor
11-6 Vc − va − vb − vc + +
+ − 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
Im = ˆ
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 !
ims (t ) = I m ∠(ω t - )
I ms = I m
Vs = V
TOC " ! 11-7 Electrically Open-circuited Rotor
at t = 0 I ma ω syn !
vs """" """
Bms , ims ! Exit ω syn """"
a − axis 90 o Va + Va
I ma − is a constant magnitude, rotating flux !
ims (t ) = I m ∠(ω t - π / 2) = I ms ∠(ω t - π / 2)
N s ims
H ms (t ) =
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 " ! Short-circuiting the Rotor
(Rs, Ls,leakage = 0)
" Transformer Analogy 11-8 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 11-9 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 11-10 at t = 0
+ + + −
− ω syn
vs ω syn
+ Rbar −
− a − axis θ """"
Bms ebar front +
+ + − + − ibar (θ ) +
− back end-ring end-ring ! 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
Exit 2001 by N. Mohan Bm #rω slip
TOC " ! Rotor MMF – Reflected Rotor
MMF – Reflected Rotor Current
11-11 φm,ir ' φm,ir at t = 0
at t = 0 net flux = 0 ωm ""
ima + ira ' ω syn ""
ir ' a − axis """
is a − axis !
Fs (t ) = Fms (t ) + Fr′(t )
is (t ) = ims (t ) + ir′ (t )
I′ = k B ω
r i ms slip """"
! ! 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
2001 by N. Mohan ! 11-12 Slip frequency (fslip) in the rotor
ω syn - ω m
slip : s =
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 11-13 # at t = 0 ""
Fr ' ""
vs a − axis
ir ' """"
! Current ir′ , in field Bms , produces torque Tem Exit 2001 by N. Mohan N
Tem = π s r # Bms I r′ = keω Bmsω slip
kt TOC " ! Torque – Speed Characteristics
(slip small ; ωm near ωsyn) 11-14 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)
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
2001 by N. Mohan Rotor Leakage Inductance
φm,ir′ net=0 φm,ir at t = 0 ""
θr θr ""
vs (t ) a − axis θr ""
ir ' ""
Fr ' a − axis """!
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
ims 11-16 θr 90 o ""
vs a − axis "!
90 o Bms
Blr TOC " ! Rotor Leakage Inductance
(cont…) 11-17 ! 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 " ! 11-18 Per-Phase Equivalent Circuit
at t = 0 ′
I ra Ia
Fr ωm """!
Fr ' ""
ir ' ""
vs a − axis + ω at f =
2π a jω Lm I ma Req I ra '
I ma − ′
Req = Rr
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
2001 by N. Mohan ! Power Into Rotor – Power Lost
In Rotor – Power Out Of Rotor 11-19 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
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
2001 by N. Mohan ! Stator Winding Resistance and
Leakage Inductance 11-20 Rs
(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 " ! 11-21 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)
(at ω ) I ra ' Ia jω Lls + Ema
− 11-22 Rs jω Llr '
I ma jω Lls I ra ' ≈ 0 + Rr ' ω syn Va ω slip − Ia jω Lm − Equivalent Circuit under no
(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 )
(at ω ) I ra ' Ia
jω Lls + Ema
− Rs jω Llr '
I ma − Equivalent Circuit + Rr ' ω syn
ω slip Va
− Ia jω Lls 11-23 I ra ' ′
jω Llr Rr '
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
! To find Lls and Llr we can often assume that Lls = Llr
2001 by N. Mohan Characteristics at Rated Voltage
and Rated Frequency
I ra Tem
I ra ,rated pull-out 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 (start-up condition)
Exit 2001 by N. Mohan TOC " ! Motor Currents, Efficiency, Power
Factor As a Function of Load
90 Efficiency 80
Power Factor 60
Power Factor (%) 50
30 20 10
0 25 75 100
Load (%) 125 Typical for design B 10 kW, 4 pole, three-phase induction motor
Exit 2001 by N. Mohan TOC " ! 11-26 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
I ra ′
I ra ,rated
6.0 Tem 5.0 Tacc = Tem − TL
Tem 4.0 Tem 3.0 I ma 2.0
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
11-27 ia ia
ac input 0 n t ea
0 t Tem
motor 0 t ! Circuit applies reduced voltage to motor during start-up to avoid
large currents and over heating
! Circuit also used to reduce voltage to motor under light load
steady state conditions. This improves efficiency
2001 by N. Mohan 11-28 Summary/Review ! Describe the construction of squirrel-cage 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 open-circuited.
! Under a balanced, three-phase, sinusoidal steady state
excitation, what is the speed of the rotating flux-density
distribution called? How is this speed related to the angular
frequency of the electrical excitation in a p-pole machine.
Exit 2001 by N. Mohan TOC " ! 11-29 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 flux-density 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
represented by Bms (t ) .
Exit 2001 by N. Mohan TOC " ! 11-30 Summary/Review ! The rotor bars located around the periphery of the rotor are of
uniform cross-section. 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 per-phase equivalent circuit of a loaded induction motor.
! In the equivalent circuit of Problem 9, what quantities does the
rotor-bar 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 " ! 11-31 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
Tem produced? How and why does it depend on ωslip and Bms ?
Draw the torque-speed characteristic.
! Assuming L′r to be zero, explain how induction motors meet
Exit 2001 by N. Mohan TOC " ! 11-32 Summary/Review ! What makes an induction machine go into the regenerativebraking mode? Draw the space vectors and the corresponding
phasors under the regenerative-braking condition.
! Can an induction machine be operated as a generator that feeds
into a passive load, for example a bank of three-phase
! How is it possible to reverse the direction of rotation of an
! 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 " ! 11-33 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 brush-type and brushless dc motors discussed in Chapters 7 and 10?
Describe how to obtain the per-phase equivalent circuit, including
the effect of the rotor leakage flux.
What is the difference between I ra in Fig. 11-18c and in Fig. 1119c, in terms of its frequency, magnitude, and phase angle? 2001 by N. Mohan TOC " ! 11-34 Summary/Review ! Is the torque expression in Eq. 11-41 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 per-phase equivalent circuit, including the stator
! Describe the tests and the procedure to obtain the parameters of
the per-phase 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 " ! 11-35 Summary/Review ! Do induction machines have voltage and torque constants similar
to other machines that we have studied so far? If so, write their
! Plot the torque-speed characteristic of an induction motor for
applied rated voltages. Describe various portions of this
! What are the various classes of induction machines? Briefly
describe their differences.
! What are the problems associated with the line-starting of
induction motors? Why is the starting currents so high?
! Why is reduced-voltage starting used? Show the circuit
implementation and discuss the pros and cons of using it to save
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