INDECTION

MOTOR
MECHANICAL ENGINEER 3A

Present by : Shashikant
Gurjit
Arshad
 Construction

a Stator windings of practical machines are

120o 120o
distributed
c’
b’ Coil sides span can be less than
180o – short-pitch or fractional-pitch
or chorded winding

c If rotor is wound, its winding the same

b
as stator
a’

120o

Stator – 3-phase winding

Rotor – squirrel cage / wound
 Construction

The harmonics in the mmf can be further reduced by

increasing the number of slots: e.g. winding of a phase are
placed in 12 slots:
 Construction

In order to obtain a truly sinusoidal mmf in the airgap:

• the number of slots has to infinitely large

• conductors in slots are sinusoidally distributed

In practice, the number of slots are limited & it is a lot easier

to place the same number of conductors in a slot
 Phase a – sinusoidal distributed winding

Air–gap mmf

F()

 
2
• Sinusoidal winding for each phase produces space sinusoidal MMF
and flux

• Sinusoidal current excitation (with frequency s) in a phase

produces space sinusoidal standing wave MMF
i(t)
This is the excitation
t current which is sinusoidal
with time
F()


• Sinusoidal winding for each phase produces space sinusoidal MMF
and flux

• Sinusoidal current excitation (with frequency s) in a phase

produces space sinusoidal standing wave MMF
i(t)

0
t

F()

t=0


• Sinusoidal winding for each phase produces space sinusoidal MMF
and flux

• Sinusoidal current excitation (with frequency s) in a phase

produces space sinusoidal standing wave MMF
i(t)

t1
t

F()

t = t1
 2

• Sinusoidal winding for each phase produces space sinusoidal MMF
and flux

• Sinusoidal current excitation (with frequency s) in a phase

produces space sinusoidal standing wave MMF
i(t)

t2
t

F()

t = t2
 2

• Sinusoidal winding for each phase produces space sinusoidal MMF
and flux

• Sinusoidal current excitation (with frequency s) in a phase

produces space sinusoidal standing wave MMF
i(t)

t3
t

F()

t = t3
 2

• Sinusoidal winding for each phase produces space sinusoidal MMF
and flux

• Sinusoidal current excitation (with frequency s) in a phase

produces space sinusoidal standing wave MMF
i(t)

t4
t

F()

t = t4
 2

• Sinusoidal winding for each phase produces space sinusoidal MMF
and flux

• Sinusoidal current excitation (with frequency s) in a phase

produces space sinusoidal standing wave MMF
i(t)

t5
t

F()

t = t5
 2

• Sinusoidal winding for each phase produces space sinusoidal MMF
and flux

• Sinusoidal current excitation (with frequency s) in a phase

produces space sinusoidal standing wave MMF
i(t)

t6
t

F()

t = t6
 2

• Sinusoidal winding for each phase produces space sinusoidal MMF
and flux

• Sinusoidal current excitation (with frequency s) in a phase

produces space sinusoidal standing wave MMF
i(t)

t7
t

F()

t = t7
 2

• Sinusoidal winding for each phase produces space sinusoidal MMF
and flux

• Sinusoidal current excitation (with frequency s) in a phase

produces space sinusoidal standing wave MMF
i(t)

t8
t

F()

t = t8
 2

Combination of 3 standing waves resulted in ROTATING MMF wave
Frequency of rotation is given by:

2 p – number of poles
s  2f
p f – supply frequency

known as synchronous frequency

 • Rotating flux induced:
Emf in stator winding (known as back emf)
Emf in rotor winding
Rotor flux rotating at synchronous frequency

Rotor current interact with flux to produce torque

Rotor ALWAYS rotate at frequency less than synchronous, i.e. at

slip speed:
sl = s – r

Ratio between slip speed and synchronous speed known as slip

s  r
s
s
Induced voltage
Flux density distribution in airgap: Bmaxcos 
/2
Flux per pole: p  
 / 2
 Bmax cos  l r d = 2 Bmaxl r

Sinusoidally distributed flux rotates at s and induced voltage in the phase coils

Maximum flux links phase a when t = 0. No flux links phase a when t = 90o
Induced voltage

a  flux linkage of phase a

a = N p cos(t)

By Faraday’s law, induced voltage in a phase coil aa’ is

d
ea    N p sin t
dt

N p
Erms   4.44f N p
2

Maximum flux links phase a when t = 0. No flux links phase a when t = 90o
Induced voltage

In actual machine with distributed and short-pitch

windinds induced voltage is LESS than this by a
winding factor Kw

N p
Erms   4.44f N pK w
2
N p
Erms   4.44f N p
2
Stator phase voltage equation:

Vs = Rs Is + j(2f)LlsIs + Eag

Eag – airgap voltage or back emf (Erms derive previously)

Eag = k f ag

Rotor phase voltage equation:

Er = Rr Ir + js(2f)Llr

Er – induced emf in rotor circuit

Er /s = (Rr / s) Ir + j(2f)Llr
 Per–phase equivalent circuit

Rs Lls Llr Ir
+ + +
Is
Vs Lm Eag Er/s Rr/s
Im
– – –

Rs – stator winding resistance

Rr – rotor winding resistance
Lls – stator leakage inductance
Llr – rotor leakage inductance
Lm – mutual inductance
s– slip
 We know Eg and Er related by

Er s
 Where a is the winding turn ratio = N1/N2
Eag a

The rotor parameters referred to stator are:

Ir
Ir '  , R r '  a R r , L lr '  a L lr
2 2

a
 rotor voltage equation becomes

Eag = (Rr’ / s) Ir’ + j(2f)Llr’ Ir’

 Per–phase equivalent circuit

Rs Is Lls Llr’ Ir’

+ +
Lm Rr’/s
Vs Eag
Im
– –

Rs – stator winding resistance

Rr ’ – rotor winding resistance referred to stator
Lls – stator leakage inductance
Llr’ – rotor leakage inductance referred to stator
Lm – mutual inductance
Ir ’ – rotor current referred to stator
 Power and Torque

Power is transferred from stator to rotor via air–gap,

known as airgap power

Rr ' Rr '
Pag  3Ir' 2  3Ir' 2R r '  3Ir' 2 1  s
s s

Lost in rotor Converted to mechanical

winding power = (1–s)Pag= Pm
 Power and Torque

Mechanical power, Pm = Tem r

But, ss = s - r  r = (1-s)s

 Pag = Tem s

Pag 3Ir'2R r '

Tem  
s ss

Therefore torque is given by:

3R r ' Vs2
Tem 
ss  Rr ' 
2

   X ls  X lr '
2
R s 
 s 
 Power and Torque

This torque expression is derived based on approximate equivalent circuit

A more accurate method is to use Thevenin equivalent circuit:

3R r ' VTh2
Tem 
ss  Rr ' 
2

R
3R r 'Th  
 Vs Th X
2  X lr '  2

Tem   s  2
ss  Rr ' 
   X ls  X lr '
2
R s 
 s 
 Power and Torque
Tem
Rr
Pull out s Tm  
R s   X ls  X lr 
2 2
Torque
(Tmax)
 
3  Vs2 
Tmax 
ss  R  R 2   X  X  2 
 s s ls lr 
Trated

r
0 sTm rated syn
s
1 0
 Steady state performance

The steady state performance can be calculated from

equivalent circuit, e.g. using Matlab

Rs Is Lls Llr’ Ir’

+ +
Lm Rr’/s
Vs Eag
Im
– –
 Steady state performance

Rs Is Lls Llr’ Ir’

+ +
Lm Rr’/s
Vs Eag
Im
– –

e.g. 3–phase squirrel cage IM

V = 460 V Rs= 0.25  Rr=0.2 
Lr = Ls = 0.5/(2*pi*50) Lm=30/(2*pi*50)
f = 50Hz p=4