In this article we will study about the Relation between torque and rotor power factor and also read different types of torque like starting torque, starting torque of a squirrel-cage motor, Condition for maximum starting torque in induction motor, Torque under running condition, Full load torque and maximum torque, Relation between torque and slip. You will get to study all this in this article.
The relation between torque and rotor power factor
In case of induction motor the torque is proportional to the product of flux per stator pole and the rotor current. and power factor of the rotor. Therefore, T is proportional to ΦI2 cosΦ2 or T = kΦI2 cosΦ2.
Where, I2 = rotor current at standstill, Φ2 = is the angle between rotor E.M.F. and rotor current, k = is a constant. E.M.F. at standstill is denoting by E2.
At standstill condition E2 is proportional to Φ, therefore T ∝ E2I2 cosΦ2 or T = k1E2I2 cosΦ2. The effect of rotor power factor on rotor torque is given for various value of Φ2.
From the above relation it is conclude that if I2 increase cos Φ2 decrease and also the torque decrease and vice versa.
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( 1 ) If the Rotor is Assumed to the Non-inductive or Φ2 = 0
In this case, the rotor current I2 is in phase with E.M.F. induce in the rotor E2. The power factor cos Φ = become 1 hence, the torque is always +ve and that is unidirectional. and this +ve torque is know as forward torque.
( 2 ) Rotor Assumed to the Inductive
Φ2 = tan-1 X2/R2 , In this case rotor current I2 lags behind E2 by an angle Φ2 = tan-1 X2/R2. In this case a portion of torque of -ve that is it is revers. The -ve torque is also know as backward torque.
Hence, the total torque is the difference of forward torque and backward torque. In this case the torque is bidirectional.
Φ2 = 90° in this case the total torque is zero. Because the backward torque and forward torque becomes equal and opposite.
Starting torque of induction motor
The starting torque is that torque produce in the motor or develop by the motor at the instant of starting. It is denoted by the letter Tst .
Let E2 = rotor e.m.f. at standstill per phase
I2 = Rotor current at standstill
R2 = rotor resistance at standstill
X2 = rotor reactance at standstill
Z2 = rotor impedance at standstill
Φ2 = Phase angle between E2 and I2
So the Standstill or starting torque is Tst = K1 E2 I2 Cos Φ2
We know , K1 = 3 / 2𝛑Ns
Therefore,
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Some numerical of starting torque – Relation between torque and rotor power factor
Q-1. A 3-phase induction motor having a star-connected rotor has an induced E.M.F. of 80 volts between slip-ring at standstill on open-circuit. and The rotor has a resistance and reactance per phase of 1 Ω and 4 Ω respectively. Calculate current per phase and power factor when (i) slip rings are short-circuited (ii) slip-rings are connected to a star-connected rheostat of 3 Ω per phase.
Solution :
( a ) Standstill e.m.f. / rotor phase ( E2 ) = 80/√3 = 46.18 volt
Impedance phase ( Z2 ) =
=
= 4.12 ohm
Current per phase ( I2 ) = e.m.f. per phase / Impedance per phase
= 46.18 / 4.12
= 11.2 A
Power factor ( Cos Φ2 )
= 0.242
( b ) Rotor resistance / phase = 1+3 = 4 ohm
Rotor impedance / phase =
=
= 5.65 ohm
Current per phase ( I2 ) = e.m.f. per phase / Impedance per phase
= 46.18 / 5.65
= 8.17 A
Power factor ( Cos Φ2 )
= 0.70
Starting torque of a squirrel cage induction motor
The resistance of squirrel cage motor is fixed and small as compared to its reactance which is very large especially at the start because at standstill, the frequency of the rotor current is equal to the supply frequency.
Hence, the starting current I2 of the rotor though very large in magnitude, lags by very large angle behind E2, with the result the starting torque per ampere is very poor.
It is roughly 1.5 times the full load torque although the starting current is 5 to 7 times the full load current. Hence, such motor are not useful where the motor has to start against heavy load.
Starting torque of slip ring induction motor.(What is the Relation between torque and rotor power factor ?)
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Starting torque of slip ring induction motor
The starting torque of slip ring induction motor is increase by improving power factor by addition resistance in the rotor circuit from the star-connected the rheostat, the rheostat resistance being progressively cut out as the motor gather speed.
Addition of external resistance increase the rotor impedance and so reduce the rotor current. The effect of improve power factor predominates the current decreasing effect of impedance .Hence, starting torque is increased.
Condition for maximum starting torque in induction motor
Tst = k2R2 / (R2)2 + (X2)2
So for maximum condition dTst / dR2 = 0
( R2 )2 + ( X2 )2 = 2( R2 )2
( X2 )2 = ( R2 )2
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Rotor E.M.F. and reactance under running condition
Let E2 = Standstill rotor induced E.M.F. per phase
X2 = Standstill rotor reactance per phase
f2 = Rotor current frequency at standstill
When rotor is stationary that is s = 1, the frequency of rotor E.M.F. is the same as that of the stator supply frequency the value of E.M.F. induced in the rotor at standstill is maximum.
Because the relative speed between the rotor and the revolving stator flux is maximum. In fact, the motor is equivalent to a 3-Φ transformer with short circuit rotating secondary.
When rotor start running the relative speed between it and the rotating stator flux is decrease. Hence, the rotor induce E.M.F. which is directly proportional to this relative speed is also decrease.
Hence, for a slip ‘s’ the rotor induced E.M.F. will be ‘s’ time the induced E.M.F. at standstill. Therefore, under running condition Er = sE2. The frequency of the induced E.M.F. will be fr = sf2.
Due to decrease in frequency of the rotor E.M.F. the rotor reactor also decrease Xr = sX2 . Where Er and Xr are rotor E.M.F. and the reactance under running condition.
Torque under running condition – Relation between torque and rotor power factor
Tr ∝ Er Ir CosΦ2 or T ∝ Φ Ir Cos Φ2
Where, Er = is the rotor e.m.f. per phase under running condition
Ir = is the rotor current per phase under running condition
Er = sE2
Tr ∝ Er Ir Cos Φ2
So we know that k1 is constant and its value can be prove = 3 / 2𝛑Ns . Hence in that case the expression for torque is
At standstill when s = 1
Relation between torque and slip in induction motor
So , T = kΦsE2R2 / (R2)2 + (sX2)2
When s = 0 and T = 0
Normal speed is directly proportional to slip and the curve start from the origin 0. At normal speed, close to synchronism the term ‘ sX2 ‘ is small and hence, negligible with respect to R2. Therefore T ∝ s/R2 or T = s [ If R2 is constant].
Hence, for low value of slip the torque /slip curve is approximately is straight line. As slip increases the torque also increases and becomes maximum when s = R2/X2.
This torque is also know as pull out torque or maximum torque or breakdown torque. denoted by Tb or stalling torque. As the slip further increases with further increase in motor load then R2 becomes negligible as compared to sX2.
High speed condition
At high speed or at last speed R2 is small and hence negligible with respect to sX2. Therefore T ∝ 1/sX2 if X2 is constant. therefore T ∝ 1/s. ( Torque inversely proportional to slip.)
Hence, the torque slip curve is a rectangular hyperbola. So, we see that beyond the point of maximum torque any further increase in motor load results in decrease of torque developed by the motor.
The result is that motor slow down and eventually stop the circuit breaker will be tripped open if the circuit has been protected.
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Full load torque and maximum torque
Tf ∝ Sf R2 / ( R2 )2 + ( sf X2 )2 ———- (1)
Tmax ∝ 1 / 2×X2 ———- (2)
Now dividing both numerator and denominator by ( X2 )2 then we get
Where, R2 / X2 = a |
and ‘a’ is the resistance per reactance per standstill
So , Tf / Tmax = 2sf a / a2 + s2f |
Starting torque and maximum torque – Relation between torque and rotor power factor
Tst ∝ R2 / ( R2 )2 +( X2 )2
Tmax ∝ 1 / 2X2
So , Tst / Tmax = 2R2 X2 / ( R2 )2 + ( X2 )2
Now dividing both numerator and denominator by ( X2 )2
Where , a = R2 / X2 |
‘a’ is the rotor resistance per standstill per phase
So , Tst / Tmax = 2a / a2 + 1 |