3. Slip and Rotor frequency of IMThe revolving magnetic field of the stator of 3 phase IM rotates in a given direction (clockwise/counter based on the phase sequence of supplied voltage) with synchronous speed Ns.

The rotor of 3 phase induction motor starts running in the direction of rotating magnetic field.

At standstill, rotor conductors are being cut by rotating flux at synchronous speed. At this instant, frequency f2

of rotor emf and current is equal to the supply frequency f1.

When the rotor rotates at a speed nr in the direction of rotating field, the relative speed between synchronously rotating stator flux and rotor conductors becomes:-

In practice the rotor never succeeds in catching up with the stator field.

If it really did so, then, there would be no relative speed between the two; hence no rotor emf, no rotor current, and so no torque to maintain rotation.

That is why, in the rotor windings emf and current are induced and the rotor develops torque .

The difference in speeds depends upon the load on the motor .

rss nNn Where, ns - Slip speed

Ns - synch.speed

nr – rotor speed

Therefore, frequency of the rotor emf is:-

,rps

Slip is defined as

22

rs nNpf

rss

rs nSNN

nNS

1;

We know that,

12

2

1

1

2

,2

,2

Sff

pSNf

rpspN

f

rpsp

fN

s

s

s

T

hu

s, t

he

freq

uen

cy o

f th

e ro

tor

emf

and

cu

rren

t in

an

in

du

ctio

n

mo

tor

f2 =

Sf 1

; an

d f

2 i

s ca

lled

th

e sl

ip f

req

uen

cy.

Stator field speed

Rotor

speed

Relative

speed slip Rotor emf

Ns 0 Ns 1 E2

Ns 0.4Ns 0.6Ns 0.6 0.6E2

Ns 0.8Ns 0.2Ns 0.2 0.2E2

Rotor emf, current and power of IMAt standstill, relative speed between magnetic field of the stator and rotor conductors is synchronous speed of the field Ns and slip S = 1. Let the per phase induced emf in rotor circuit at this moment be E2.

In general, for any value of slip S, the per phase generated emf in the rotor conductors is SE2

2212

2212

44.4

44.4

NKSfE

NKfSSE

w

w

212 2 Lf

212222 22 LSfLfSSXX

• Rotor leakage reactance at standstill is:-

• Rotor leakage reactance at any slip:-

• Rotor leakage impedance at standstill:-

222

22 XrZ

• Rotor leakage impedance at any slip is:-

22222 SXrZ

• Per phase rotor current at standstill:-

• Per phase rotor current at any slip:-

• Per phase power input to rotor is:-

• Where,

pg – the power transferred from stator to rotor across the air gap.

θ - power factor angle of the rotor which I2 lugs E2.

θ - arc tan

22

22

2

2

22

Xr

E

Z

EI

22

2

2

2

22

22

22

XSr

E

SXr

SEI

222 cosIEPg

2

2

r

SX

222 cosIEPg

22

2

2

2

22

2

2

22 ..

Xsr

sr

Xsr

EEPg

X2

r2/SSX2 = X2`

r2

Z2

E2 2

22

2

2

2

2

22cos

XSr

Sr

Z

r

Rotor equivalent circuit

22

2

2

2

22

2

2

22 ..

Xsr

sr

Xsr

EEPg

s

rIPg

22

2 .

Pg can be expressed in an other way

s

sIrIsrIP

s

srIrI

s

rIP

g

g

222

222

22

2222

22

222

1

s

g

s

g

r

Me

P

s

sPPT

1

1

s

rIs

rIP

Tsss

ge

222

222

.1

sPSPP

ss

rI

s

rSIP

ggg

g

1

1222

222

gSPWhere, - rotor ohmic loss

- internal mechanical power

developed in the rotor Mg PsP 1

s

srIsPP gM

11 2

22

ss

PSP M

g .1

• Thus from rotor Equivalent circuit of IM, ohmic loss in the rotor circuit is equal to slip times power input to the rotor. i.e.

gSPrI 222

Internal Torque (gross) developed per phase is:_

s

g

s

g

r

Me

P

s

sPPT

1

1

sec/,2 radN ss

sec/,2 1 revP

fN s

sec/,4

, 1 radP

fThus s

Where,

• OR,

Power output at the shaft can be obtained from Pg

Pshaft = PM – Pmech loss (friction & windage)

Pshaft = Pg – (rotor ohmic loss + fric. & windage)

s

rIs

rIP

Tsss

ge

222

222

.1

s

rI

NT

se

222.2

1

• Output or shaft torque is –

• If the stator input is known, then, air gap power is –

statorstatorlossing corelossRIPP 2

s

shaft

r

shaftshaft s

PPT

1

Losses and efficiency of 3-phase IM• There are two major losses in 3-phase IM. 1) Fixed losses (Rotational losses) 2) Variable losses Fixed losses – are usually taken to be constant at a constant speed over the normal working range of the IM; even though friction losses may vary slightly with load and speed. Fixed losses are composed of:-

a) Core losses - are a function of speed and flux - Hyteresis loss –-

- Eddy current loss –

fVBKP xhh

t

VfBtKPe222

1

fVBKP xhh

VfBtKPe222

1

where, – is constant for a given iron used. - flux density raised to Steinmetz exponent, and the value of x is 1.6 – 2.00

– frequency of flux

- volume of iron used.

– eddy current constant for the conductive material used.

– thickness of the conductive material used.

b) Mechanical loss

- Friction losses ( bearing & brush friction in WRIMs) - Windage losses.

hKxB

f

V

1K

t

• Under usual working conditions, the rotor current of 3-phase IMs is of very low frequency. As a result, rotor core losses which are almost proportional to frequency squared are negligible.

• Fixed losses (rotational losses) of IMs can be obtained directly by performing no-load test on the IM.

Fixed losses = (Pinput– I2R stator loss) at no-load

Variable losses – The value of these losses vary with load variation and are composed of:-

a) Stator ohmic losses

b) Rotor ohmic losses

c) Brush contact losses for WRIMs only

d) Stray load losses – are losses due to step harmonics, skin effect losses in stator conductors and iron losses in structural parts of machines.

• The stator and rotor ohmic losses (total ohmic loss) can be obtained from block rotor test.

• The brush contact loss for WRIMs and stray losses are very small & can be taken as 0.5 – 1% of the total losses.

• Power flow diagram of IM.

Power input

Pin = √3VlIlcosθ

Stator I 2R loss

Stator core lossR

otor I 2R loss

Rotor core lossFriction and w

indage loss

Pg

Rotor input power

PM

Shaft powerMech. power developed

eout TP

• Efficiency of IM.• Actually, the IM experiences a change in rotor speed with load as well

as a change in the rotor frequency resulting from the speed change. i.e. neither the rotational losses (which are a functions of speed and frequency ) nor the stator and rotor electrical losses ( which are functions of load) are constant.

• However, the efficiency of a 3-phase induction motor is:-

%100var

PPP

P

PP

P

fixedoutput

output

lossesoutput

output

%100input

output

P

P %1001

losses

out

P

P

Phasor diagram of IM• An IM is similar to transformers in many respects.

• IM at stand still

• Application of 3-phase balanced voltage at frequency f1to the stator winding causes the development of rotating magnetic field; which induces rotor and stator emfs.

• Line frequency f1 appears in E2 also, because rotor is at stand still.

• The emf ratio for the IM is:-

2212

1111

44.4

44.4

NKfE

NKfE

w

w

2

11

2

2

1

w

wEE

KN

KN

• Where, Kw1 and Kw2 are stator and rotor winding distribution factors.

• Stator and rotor windings which obviously possess resistances and reactance are treated as transformer primary and secondary windings.

• Thus, the phasor diagram and equivalent circuit of a three-phase IM are almost similar to those in a trans-r. The main differences are;

- Appearance of winding factors Kw1 and Kw2 in the

voltage ratio in IMs.

- The no-load current in IM varies from 30-50% of full load current, where as in transformers it varies from 2-6% of the full load current. This is because, the resultant flux in IM completes its path through high reluctance air-gap.

• Phasor diagram of IM at standstill.

E1

E2=I2Z2

I2

I2r2

jI2 x

2

фIm

Ic

-I2

I1

V1= -E1

I1r1

jI 1x 1

Io

θ1

θ2

• Pasor diagram of IM at full load slip

E1

фIm

Ic

I1

V1= -E1

I1r1

jI 1x 1

V1

I0

θ0

E2

I2

jI2 x

2

I2r2

θ1

θ2

Observe that, • the power factor angle at stand still condition is large. i.e. the

stator power factor at starting is very poor.

• At normal operating slip, θ1 is small. The rotor voltage equation now becomes SE2 = I2(r2 + jX2) and it is illustrated in the phasor diagram.

θ1