27103242 Inductionc Machine

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Induction Machine

Chapter No 7



2

Home Work Number 4

Home Work Problems:

7.2, 7.3, 7.4, 7.5, 7.7, 7.8, 7.14.



3

Induced Torque in an AC Machine

In an AC machine under normal operating

condition, there are two magnetic fieldspresent- a magnetic field from the rotor circuit and another magnetic field from thestator circuit. The interaction of these two

magnetic fields produces the torque in themachine, just as two permanent magnetsnear each other will experience a torque

which causes them to line up. (4.5 details)

S Rind B BK ×= τ



4

Introduction to Induction Machine

! The name induction machine because the

rotor voltage (which produces the rotor current and rotor magnetic field) is induced

in the rotor winding rather than being

physically connected by wires.

! A distinguishing feature is that no DC field

current is required to run this machine.



5

Motor Construction:

Two types of rotor.

1.

Squirrel Cage Rotor or simply cage rotor 2. Wound rotor

Squirrel Cage Rotor:In squirrel cage rotor series of conducting

bars are laid in the slots carved in to the face

of rotor and shorted at either end by largeshortening rings.



6

Wound Rotor:" It has a complete set of three phase

windings that are mirror image of the

windings on the stator. Three phases are

usually y-connected. The ends of rotor

wires are tied through slip rings on therotor shaft. The rotor windings are shorted

through brushes riding on the slip rings.

" Rotor current can be examined and extraresistance can be added to rotor circuit



7

When three phase set of voltages are

applied to stator, three phase current flowsin the stator winding.

!

A magnetic field BS is produced, which isrotating counter clock wise.

! The speed of magnetic field

# is frequency in hertz

#P is no of poles

p

f n esynch

120==

e f

Basic Induction Motor Concepts



8

! when rotating magnetic field BS passesover rotor bars, it induces voltage in them

V: velocity of bars relative to magnetic field

B: magnetic flux densityI: length of conductor in the magnetic field

Then

l BV e Sind ).( ×=

S Rind B BK ×=



9

! The motor will start rotating, there is a limitto the speed of rotor.

! If rotor speed becomes equal to

synchronous speed then rotor bars will bestationary relative to the magnetic field,

and there would be no induced voltage.! If then current in rotor bars is zero

No rotor current

No torque

⇒

⇒

0=ind e



10

Normal Operation:Both BR and BS rotate together.

Rotor Slip

nslip : Slip Speed

nsync: Synchronous Speed

nm : Mechanical Shaft Speed

%)100(×=

synch

slip

n

nSOr Slip =

msyncslip nnn −=



11

…..(1)

….. (2)

Or in terms of angular velocity

If => S = 0

=> S = 1

Normal operation S is between 0 & 1

%100×

−

=

synch

msynch

n

nnS

%100×

−

=

synch

msynchS

ω

ω ω

syncm

=

0=m



12

synchm

synchm

SnSnω ω )1()2(

)1()1(−=⇒−=⇒



13

Electrical Frequency on Rotor Rotating Transformer

- Induction motor works by induced voltage

and current in rotor, therefore some timescalled as rotating transformer.

- Primary (Stator) induces voltage inSecondary (Rotor)

- Secondary frequency?

#If rotor is locked then secondaryfrequency is same as primary.

#If rotor turns at synchronous speed the

Rotor frequency will be zero.



14

synch

msynch

n

nnS

−=

e

synch

msynch

r f n

nn f ×−

=

er

r syncm

er m

sf f

sand f nn

sand f f n

=⇒

==⇒=

==⇒=

0 0

1 0



15

p

f n e

synch

120=

e

e

msynchr f

f

Pnn f ××−=

120

)(

)(120 msynchr nn

P

f −=



16

Equivalent Circuit of an Induction

Machine

! An induction motor is called a singly

excited machine (as opposed to a doublyexcited synchronous machine), sincepower is supplied only to the stator circuit.

Because induction motor does not have anindependent field circuit its model will notcontain an internal voltage source such as

internal generated voltage EA in asynchronous machine.

! We will begin with the transformer model.



17

The Transformer Model of an InductionMachine



18

Transformer

Induction Motor

φφφφ

FFFF



19

! Due to the air gap in an inductionmachine, the reluctance of the flux path is

increased greatly, which reduces coupling

between primary (stator) and secondarywinding (rotor). The higher reluctance

caused by the air gap means that a higher

magnetizing current is required to obtain a

given flux. Therefore, magnetizing

reactance will be much smaller.! E1 coupled to ER by an ideal transformer

with an effective turn ratio aeff .



20

! For wound rotor it is ratio of theconductors per phase on stator to the

conductors per phase on rotor.

! Difference:

Primary difference is effect of varying

rotor frequency on the rotor voltage ERand rotor impedance RR and j XR.

If varies, varies.

120)(

P

nn f msynchr ×−=

r f R X



21

Rotor Circuit Model! Voltage applied to stator induces voltage

in the rotor.

! The greater the relative motion (Slip

Speed) between rotor and stator magnetic

fields, the greater the resulting voltage andfrequency.

! The largest relative motion occurs when

the rotor is stationary, called Locked Rotor or Blocked Rotor condition.



22

=>Largest voltage and frequency are induced.

⇒Magnitude and voltage

induced in rotor α Slip of Rotor

Let ,voltage induced at

locked rotor condition

Then, the voltage

induced at any Slip

And, frequency er sf f ==

0 R R E S E ==

0 R E =



23

-Rotor Resistance is fixed RR

-Rotor Reactance = Rr R L X =

)( 22 er Re Rr sf f L f s L f === ∵π π

0

)2(

R R

Re

X S X

L f S

=

=

is blocked rotor reactance0 R X



24

Rotor Equivalent Circuit



25

0

0

0

0

R R

R R

R R

R R

R R

R R

jX S

R

E I

jSX R

SE I

jX R

E I

+

=

+

=

+

=



26



27

=>It is possible to treat all of the rotor effects

due to varying rotor speed as beingcaused by varying impedance suppliedwith power from a constant voltage source

ER0.

-very low Slip RR/S >> XR0, so rotor resistance dominates at high Slip XR0 >>

RR/S-Current reaches steady state value as slip

becomes very large.

0, R R

eq R jX S

R Z +=



28

Final Equivalent Circuit

! Need to refer rotor part to the stator side.

! Speed effects are concentrated in impedance.

! In transformer secondary can be referred by turnratio.



29

SS

SSP

SSP

Z a Z

a I

I I

aV V V

2'

'

'

=

==

==

Let, a eff = turn ratio

-Prime means

referred quantities



30

Then,

)(

&

&

0

2

2

2

0

'

1

R R

eff

eff

R

Reff R

jX S

Ra Z impedance Rotor

a

I I

E a E E

+==

=

==



31

Let,

Then the final per phase equivalent circuit is

0

2

2

2

2

Reff

Reff

X a X Ra R

=

=

φ

V



32

!

The rotor resistance and locked rotor reactance XR0 are very difficult or

impossible to determine on a cage rotor

also the effective turn ratio aeff is difficult toobtain for squirrel-cage rotor .

! Fortunately it is possible to make

measurements that will directly give the

referred resistance and reactance R2 and

X2, even though RR1 ,XR0 and aeff are notknown separately.



33

Power and Torque in Induction Machine

! Induction motor same like transformer.

! Input is 3-φ voltage & currents.! Output of transformer is electric power

from secondary winding.

! The secondary (rotor) of an inductionmachine is shorted, therefore no electricaloutput form induction machine insteadmechanical output.

out in PP → Mechanical Electrical→



34

Rotational Loss! Core losses

! Friction & winding losses

! Stray Losses

Pin=√√√√3VTIL

cosθθθθ

Pconv=ΓΓΓΓ

indωωωω

m

Pout =τωτωτωτω

load

PSCL

stator

copper

Pcore(core

losses)

(rotor+sta

tor

PRCL

(Rotor

copper

losses)

Pfriction&

windage

PAG =

Air gap power

P STRAY

(Pmisc)



35

eq Z

V I

ϕ =1

φ V

22

11

/

11

jX S R jBG

jX R Z

M c

eq

+

+−

++=,1

eq Z

V I

ϕ =



36

Stator Copper Losses:

Core Losses:

Air gap power:

In the equivalent circuit it can be seen that this

power can be consumed in

…(1)

C CORE G E P2

1

3=

coreSCLin AG PPPP −−=

S R 2

S

R I P AG

22

23=∵

1

2

13 R I P SCL =



37

Actual resistive losses in rotor

When referred to stator side P remains the same

….(2)

R R RCL R I P2

3=→

2

2

23 R I P RCL =

−=

−=

−=

11

3

33

2

2

2

2

2

222

2

S

R I

R I

S

R I

PPP RCL AGCONV

Now,



38

(1) and (2) implies that

Rotor copper losses = Air gap power * Slip

! The lower the slip the lower the rotor copper losses.

! When the rotor is stand still all the air gap

power is consumed in rotor and

−

=

SS R I PCONV 13 2

22

(Also known as developed mechanical power)

AG RCL PSP =



39

0== mload OUT T P ω

AGConv

AG AG

RCL AGConv

PSP

SPP

PPP

)1(

−=

−=

−=

Lastly if friction, windage and stray losses

are known, the output power is

miscw f convOUT PPPP −−= &



40

Tind: Torque generated by internal electric tomechanical power conversion

synch

AG

ind S

PS

ω τ

)1(

)1(

−

−=

Developed torque of the

machine

synch

AG

ind

P

ω τ =

Important because it provides Tind in the from of

PAG & ωsynch, where ωsynch is constant.

m

conv

ind

P

T ω =



41

Separating Rotor Copper Losses and Power

Converted in induction Machine

−=

−=

−=

−=

S

S R R

S R I P

R I S

R I P

PPP

conv

conv

conv

RCL AGconv

1

11

3

33

2

2

2

2

2

2

222

2⇒)2(&)1(



42

Per phase equivalent circuit with Rotor

copper losses & Pconv separated.



43

Derivation of Induction Motor Induced

Torque

! We will use the equivalent circuit of an induction

machine and power flow diagram to drive ageneral expression for induced torque as a

function of speed.

synch

AG

ind

M

convind

P

P

ω τ

ω τ

=

=

Quite useful because WSYNC is constant so

knowledge of air gap tells us about Tind



44

Air Gap Power:! Power crossing the gap from stator circuit

to rotor circuit.

! This is equal to power absorbed inresistance R2/S.

S

R I P

S R I P

AG

AG

222

22

21,

3=

=φ

If I2 is known, PAG can be calculated.





46

( )2

1

2

1

.

M

M TH

X X R

X V V ++

= ϕ



47

XM

>>X1

and XM

>>R1

=>

M

M

TH X X

X V V

+

≈

1

.ϕ



48

( )

( ) M

M

TH

TH TH TH

M

M

TH

X X j R

jX R jX

Z

jX R Z Z Z

Z Z Z

++

+

=

+=

+

=

11

11

1

1.



49

XM >> X1 and X1 + XM >> R1

=>2

1

1

+≈

M

M TH X X

X R R

1

X X TH

≈



50



51

( )22

2

2

2

22

2

2

2

X X S

R R

V I

jX jX

S

R R

V I

Z Z V I

TH TH

TH

TH TH

TH

TH

TH

++

+

=

+++

=

+

=



52

SYNC

AGind

P

ω τ =

S

R I P AG

22

23=∵

( )

++

+

=

2

2

2

2

22 3

X X S

R R

S

RV

TH TH SYNC

TH

ind

ω

τ



53

Starting

torque

N sync

Pull out

torque

Full load

torque

nm

Motor region

0 1 -1N sync



54

1. T at synchronous speed is zero.

2. T vs ω is linear between no load and full

load.

3. Maximum torque is 2 to 3 times rated fullload torque.

4. Starting torque is slightly larger than itsfull load torque.

5. For a given slip

6. If rotor runs at a speed faster than

synchronous speed then Tind is negative,

machine acts as a generator.

2V ∝τ



55

Maximum Pull out Torque in Induction

Machine

! Air gap power is power consumed in the

resistor R2/S, the maximum inducedtorque will occur when power consumed

by resistor is maximum.

! Max power transfer theorem

2 jX jX R Z

Z Z

TH TH source

load source

++=

=



56

For Pmax

S (at pull out torque) =

( ) ( )2

22 X X RS

RTH TH ++=

( ) ( )2

2

2

2

max X X R

R

STH TH ++

=

( )

+++

=2

2

2

2

max

2

3

X X R R

V T

TH TH TH SYNC

TH

ω

…(1)

…(2)

T (Supply Voltage)2



57

Tmax (Supply Voltage)2

Tmax 1/ Size of stator impedance androtor reactance

(1) =>Slip at which

Tmax occurs α Rotor Resistance

(2) => The value of Tmax is independent of rotor resistance.

For wound rotor machine insert the

resistance in the beginning, to get Tmax atlow speed then take it out to move Tmax toa higher speed.

∝

∝

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27103242 Inductionc Machine

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