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# Starting and Maximum Torque Equation of 3 Phase Induction Motor

By May | Published on Sep 16,2015
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The torque generated by three phase induction motor depends upon the following three factors:

1.The magnitude of rotor current.

2.The flux which interact with the rotor of 3 phase induction motor and is responsible for generating emf in the rotor part of induction motor.

3.The power factor of rotor of the 3 phase induction motor.

Combining all these above-mentioned factors together we get the equation of torque as the below:

Where, T represents the torque generated by induction motor,φ represents flux responsible of producing induced emf, I2 represents rotor current, and cosθ2 represents the power factor of rotor circuit.

The flux φ generated by the stator is proportional to stator emf E1.

i.e φ ∝ E1

We know that transformation ratio K is defined as the ratio of secondary voltage (rotor voltage) to that of primary voltage (stator voltage).

Rotor current I2 is defined as the ratio of rotor induced emf under running condition , sE2 to total impedance, Z2 of rotor side,

and total impedance Z2 on rotor side is given by ,

Putting this value in above equation we get,

We know that power factor is defined as ratio of resistance to that of impedance. The power factor of the rotor circuit is

Place the value of flux φ, rotor current I2, power factor cosθ2 in the equation of torque we can get,

Combine similar term we get,

Remove proportionality constant we get,

Where ns is synchronous speed in r. p. s, ns = Ns / 60. So, finally the equation of torque is,

Derivation of K in torque equation.

In case of three phase induction motor, there occur copper losses in rotor. These rotor copper losses are expressed as Pc = 3I22R2

We know that rotor current,

Substitute this value of I2 in the equation of rotor copper losses, Pc. So, we get

The ratio of P2 : Pc : Pm = 1 : s : (1 - s)

Where P2 is the rotor input,

Pc is the rotor copper losses,

Pm is the mechanical power developed.

Substitute the value of Pc in above equation we can get,

after simplifying we get,

The mechanical power developed Pm = Tω,

Substituting the value of Pm

We know that the rotor speed N = Ns(1 - s)

Substituting this value of rotor speed in the above equations we get,

Ns is speed in revolution per minute (rpm) and ns is the speed in revolution per sec (rps) , and the relation between the two is

Substitute this value of Ns in above equation and simplifying it we get

After comparing both the equations, we get, constant K = 3 / 2πns

## Equation of Starting Torque of 3 Phase Induction Motor

Starting torque is the torque generated by induction motor when it is started. We know that at start the rotor speed, N is zero.

So, the equation of starting torque is easily got by simply putting the value of s = 1 in the equation of torque of the three phase induction motor,

## Maximum Torque Condition for 3 Phase Induction Motor

In the equation of torque,

The rotor resistance, rotor inductive reactance and synchronous speed of induction motor keeps constant. The supply voltage to the three phase induction motor is usually rated and keeps constant so the stator emf also keeps the constant. The transformation ratio is defined as the ratio of rotor emf to that of stator emf. So if stator emf keeps constant , So does the rotor emf.

If we need to find the maximum value of some quantity, then we have to figure out the difference among quantity with respect to some variable parameter and then put it equal to zero. In this experiment, we have to find out the condition for maximum torque, so we have to differentiate torque with respect to some variable quantity which is slip, in this case as all other parameters in the equation of torque remains constant.

So, for torque to be maximum

Now differentiate the above equation by using division rule of differentiation. On differentiating and after putting the terms equal to zero we get,

Neglecting the negative value of slip we get

So, when slip s = R2 / X2, the torque will be maximum and this slip is called maximum slip Sm and it is defined as the ratio of rotor resistance to that of rotor reactance.

## Equation of Maximum Torque

The equation of torque is

The torque will reach its maximum point when slip s = R2 / X2

Substitute the value of this slip in above equation we obtain the maximum value of torque as,