Induction Motors Equivalent Circuit

8/18/2019 Induction Motors Equivalent Circuit

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Equivalent Circuit

The induction motor is similar to the transformer with theexception that its secondary windings are free to rotate

As we noticed in the transformer, it is easier if we can combinethese two circuits in one circuit but there are some difficulties


2/27

When the rotor is locked or blocked!, i"e" s #$, thelargest voltage and rotor frequency are induced in the

rotor, Why%

&n the other side, if the rotor rotates at synchronous

speed, i"e" s # ', the induced voltage and frequency in therotor will be equal to (ero, Why%

Where E R0 is the largest value of the rotor)s induced voltage

obtained at s # $loacked rotor!

Equivalent Circuit

' R R E sE =


3/27

The same is true for the frequency, i"e"

*t is known that

+o, as the frequency of the induced voltage in the rotor

changes, the reactance of the rotor circuit also changes

Where X r0 is the rotor reactance

at the supply frequencyat blocked rotor!

Equivalent Circuit

r e f s f =

, X L f Lω π = =

'

,,

r r r r r

e r

r

X L f L sf L

sX

ω π π = ==

=


4/27

Then, we can draw the rotor equivalent circuitas follows

Where ER is the induced voltage in the rotor and RR

is the rotor resistance

Equivalent Circuit


5/27

Equivalent Circuit

-ow we can calculate the rotor current as

.ividing both the numerator and denominator by s so

nothing changes we get

Where E R0 is the induced voltage and X R0 is the rotor reactance

at blocked rotor condition s # $!

'

'

!

!

R R

R R

R

R R

E I

R jX

sE

R jsX

=+

= +

'

' !

R

R R

R

E

I R jX s

= +


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-ow we can have the rotor equivalent circuit

Equivalent Circuit


7/27

We can rearrange the equivalent circuit asfollows

Actualrotorresistance

Resistanceequivalent tomechanical

load

Equivalent Circuit


8/27

-ow as we managed to solve the induced voltage and differentfrequency problems, we can combine the stator and rotor

circuits in one equivalent circuit

Where

Equivalent Circuit

'

$ '

eff R

eff R

R

eff

eff R

S eff

R

X a X

R a R

I I

a

E a E

N a

N

=

=

=

=

=


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Active Power in a Induction Motor


10/27

/ower losses in *nduction machines

Copper losses

Copper loss in the stator P SCL! # I 12 R1

Copper loss in the rotor P RCL! # I 22

R2

Core loss P core!

0echanical power loss due to friction and windage

1ow this power flow in the motor%


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/ower flow in induction motor


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/ower relations

2 cos 2 cosin L L ph ph P V I V I θ θ = =

,

$ $2SCL P I R=

! AG in SCL core P P P P = − +,

, ,2 RCL P I R=

conv AG RCL P P P = −

!out conv f stra! P P P P += − + conv

in"

#

P τ

ω =


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/ower relations

2 cos 2 cosin L L ph ph P V I V I θ θ = =

,

$ $2SCL P I R=

! AG in SCL core P P P P = − +,

, ,2 RCL P I R=

conv AG RCL P P P = −

!out conv f stra! P P P P += − +

conv RCL P P = + , ,,2 R I s=

, ,

,

$ !2

R s I

s

−=

RCL P s

=

$ ! RCL P s

s

−=

$ !conv AG P s P = −conv

in"

#

P τ

ω =

$ !

$ !

AG

s

s P

s ω

−=

−


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/ower relations

1

s

1-s

AG P

RCL P

conv P

3 3

$ 3 3 $4

AG RCL conv P P P

s s


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Torque, power and Thevenin)s Theorem

Thevenin)s theorem can be used to transform the network

to the left of points 5a) and 5b) into an equivalent voltage

source V $% in series with equivalent impedance R$% 6 jX $%


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$ $ !

& $%

&

jX V V R j X X

φ = + +

$ $ ! 77$% $% & R jX R jX jX + = +

, ,

$ $

8 8 8 8 !

& $%

&

X V V R X X

φ =

+ +


17/27

+ince X & ''X 1 and X & ''R1

9ecause X & ''X 1 and X & (X 1''R1


$

& $%

&

X V V

X X φ ≈ +

,

$

$

$

& $%

&

$%

X R R X X

X X

≈ ÷+

≈


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Then the power converted to mechanical P conv!

And the internal mechanical torque $ conv

!


!

$% $%

$

$% $%

V V I

) R R X X

s

= = + + + ÷

, ,,

$ !2conv

R s P I

s

−=

convin"

#

P τ

ω =

$ !

conv

s

P

s ω =

−

, ,,2

AG

s s

R I

P s

ω ω = =


19/27


2

!

$% in"

s

$% $%

V R

s R R X X

s

τ ω

÷ ÷ = ÷ ÷ ÷+ + +

÷ ÷

2$

!

$%

in"

s

$% $%

RV

s R

R X X s

τ ω

÷

= + + + ÷


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Torque4speed characteristics

Typical torque-speed characteristics of inductionmotor


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Comments

$" The induced torque is (ero at synchronous speed"

.iscussed earlier"

" The curve is nearly linear between no4load and full load"

*n this range, the rotor resistance is much greater than thereactance, so the rotor current, torque increase linearly

with the slip"

2" There is a maximum possible torque that can)t beexceeded" This torque is called pu**out tor+ue and is to 2

times the rated full4load torque"


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:" The starting torque of the motor is slightly higher than itsfull4load torque, so the motor will start carrying any load it

can supply at full load"

;" The torque of the motor for a given slip varies as the squareof the applied voltage"


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Complete +peed4torque c7c


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0aximum torque

0aximum torque occurs when the power transferred to R27 s is maximum"

This condition occurs when R27 s equals the magnitude of the

impedance R$% 6 j X $% 6 X 2!

max

, ,,, !$% $%

$

R R X X

s= + +

max

,

, ,

, !$

$% $%

R s R X X

=+ +


25/27

The corresponding maximum torque of an induction motorequals

The slip at maximum torque is directly proportional to the

rotor resistance R2

The maximum torque is independent of R2

0aximum torque

max

2$

!$%

s $% $% $%

V

R R X X τ ω

÷= ÷+ + +


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=otor resistance can be increased by inserting externalresistance in the rotor of a wound4rotor induction motor"

The

value of the maximum torque remains unaffected

but

the speed at which it occurs can be controlled"

0aximum torque


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Effect of rotor resistance on torque4speed characteristic

0aximum torque

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Induction Motors Equivalent Circuit

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