## 11/01/2018

### Full load torque The torque in the DC Motor is related by the product of field flux and armature current whereas the torque in the induction motor is product of rotor flux, rotor current and rotor power factor.

Tf a FI2Cos F2
= KF [ SE2 / Z2 ] [ R2 / Z2 ]
= KFSE2 R2 / Z22

= KFSE2 R2 / [ R22 + ( SX2 )2 ]……………… ( 1 )

Condition for Maximum Full load torque
dTf / dS = 0
= d { KFSE2 R2 / [ R22 + ( SX2 )2 ] } / dS  = 0
= [ R22 + ( SX2 )2 ] KFE2 R2 – KFSE2R2 [2( SX2 )( X2 ) ] / [ R22 + X22 ]2 = 0
= KFE2 R2 { R22 + ( SX2 )2 – 2( SX2 )2 } = 0
The rotor induced emf should not be zero therefore
{ R22 + ( SX2 )2 – 2( SX2 )2 } = 0
{ R22 – ( SX2 )2 } = 0
{ R2  – ( SX2 ) } = 0 OR { R2  + ( SX2 ) }= 0
Therefore R2  = ( SX2 ) or R2  = – ( SX2 )
R2  = – ( SX2 ) is not possible therefore R2  = ( SX2
• When the rotor resistance is slip times the rotor reactance, the maximum torque occurs in the three phase induction motor at full load condition.

Putting R2  = ( SX2 )  in the equation ( 1 )
Tf( MAX ) = KFSE2 R2 / [ R22 + ( SX2 )2 ]
= KFS2X2E2 / [( SX2 )2 + ( SX2 )2 ]     { As R2  = ( SX2 ) }
Tf( MAX ) = KFE2 / 2X2
• The maximum full load torque does not depend upon rotor resistance.
• As the rotor resistance increases, the maximum full load torque does not change but speed or slip at which maximum torque occur change.

S = R2 / X2
• The maximum full load torque is inversely proportional to rotor reactance.
• Higher the rotor reactance, lesser the maximum starting torque and vice versa.

Tf( MAX ) a ( 1 / X2 )
Ratio of full load torque to maximum torque
Full load Tf = KFSE2 R2 / [ R22 + ( SX2 )2 ] and full load maximum torque
Tf( MAX ) = KFE2 / 2X2
The ratio of full load torque to maximum torque
Tf / Tf( MAX ) = { KFSE2 R2 / [ R22 + ( SX2 )2 ]  / KFE2 / 2X2
Tf / Tf( MAX ) = { 2SR2X2 / [ R22 + ( SX2 )2 ]…….. ( 2 )
Multiply and dividing equation ( 2 ) by ( X2 )2
Tf / Tf( MAX ) = { 2SR2 / X2 } / [ R22 / ( X2 )2 + ( SX2 )2 / ( X2 )2 ]
Putting R2 / X2 = a
Tf / Tf( MAX ) = 2aS / [ ( a )2 + ( s )2  ]
If S = 1 or standstill condition

Tf / Tf ( MAX ) = 2a / [ ( a )2 + 1 ]………….( 3 )

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