Torque slip characteristics ,effect of change of supply voltage on torque and speed.

As we know the torque equation of induction motor is

T = K1 * φ * sE2 * R2/( R2² + (sX2)²)

So , if s=0 then T=0

Let start the curve from point 0.

Let's say, motor is running near to synchronous speed. The term  sX2 = 0 negligible

Therefore T =  K1 * φ * sE2 /R2

T > s [ other is let say constant]

So for low value of slip, torque slip curve is straight line.

as a load increases sleep increases. As the load further increases it the touches Tmax point when s= R2/ X2 called pull out torque.

Any further increases in load i.e slip increases R2 becomes negligible

 Therefore T > (1/s)

Hence after pull out aur breakdown torque the curve will be rectangular hyperbola. Hance we can see that any further increases in load motor slow down and eventually stop. If we provide some electrical protection for overload it will trip circuit. From figure it is also clear that by increasing the value of rotor resistance we can get maximum torque at the s = 1 or zero speed

Effect of change in supply voltage is on torque and speed.

T = K1 * φ * sE2 * R2/( R2² + (sX2)²)

Since ,φ >   E2

And     E2   > V

Therefore    T > V²

if there is any change in supply voltage then it is not only changes the starting torque but also running torque. So if there is any decrease in voltage to maintain same torque increases slip.

Induction motor torque slip characteristics.

Since we know the motor running torque is

T = K1 * φ * sE2 * R2/( R2² + X2²)

Where φ  - flux per status pole

I₂ - rotor current

E2 rotor emf per phase.

So if S = 0 then T = 0

Let's start from right hand side of the curve from point 0.

Let's say, motor is running near to synchronous speed. The term sX2 approximately equals to zero negligible.

Therefore, T = K1 * s * φ * E2 /R2

Assuming stator flux per pole, rotor EMF per phase, rotor resistance per phase is constant.

T ⇒ s 

So for low value of sleep torque slip service straight line.

As the load increases sleep increases. As the load further increases it touches the maximum torque point when S = R2/X2 called pull- out torque or breakdown torque. Any further increase is in load that is slip increases R2 becomes negligible.

 So,  T ⇒ 1/s

Hence after pull out torque or breakdown torque the curve will be rectangular hyperbola. Hence we can see that any further increases in load motor slow down and eventually stopped. If we provide some electrical protection for overload it will trip the circuit.

the circuit.

From figure it is also clear that by increasing the value of rotor resistance we can get maximum torque at s= 1 or zero speed.

To start a motor successfully, the load torque always be less than motor torque. If motor start successfully and when due to some reason load torque becomes more than motor torque than motor will stop. So to run motor successfully motor torque must be greater than load torque all the time.

What is a slip and frequency of rotor current in stand still and running condition

When we connect three phase applied to stator winding then rotating flux will set up due to this flux rotor start rotating but it will never catch up the synchronous speed of flux Ns=120 f/p. if it did so then no relative speed between flux and rotor then no EMF will induces, no current and so no torque to maintain rotation. So motor will always rotate less than synchronous speed of magnetic flux Ns = 120 f/p.

The difference between synchronous speed and actual speed Nr is known as slip.

     Slip = Ns - Nr. 

Unit revolution per second.

Express percentage of synchronous speed 

% slip S = ( (Ns -  Nr) /NS) * 100 %

One more term we call it as slip speed

Slip speed = sNs = Ns- Nr 

when rotor is stationary then frequency of rotor current is same as supply frequency. But when rotor start rotating then frequency depend upon relative speed between the two rotating magnetic flux and rotor. Let's say at any sleep Speed the frequency of rotor current is f' .

Ns - Nr = 120f'/ p   -------  (1)

Ns = 120f / p -------------------(2)

Divide equation 1 by question 2 we get

Ns-Nr/ Ns = f'/ f

 Sf = f'

Comment your doubt.

So frequency of rotor current under running condition is

 f' = sf.

Torque in induction motor under running condition, condition for maximum torque at running and maximum torque equation.

The torque in induction motor is proportional to the product of flux per stator pole and the rotor current one more term we will take in account is power factor of the rotor.

  T →  φ  *  I₂  * cos∅₂ 

Where φ  - flux per stator pole

I₂ - rotor current

cos∅₂ - rotor power factor

∅₂ phase angle  rotor current and rotor EMF.

The induced rotor EMF E2 is proportional to flux per status φ.

Therefore T = K1 * E₂ *  I₂ * cos φ2 

Where K1 is another constant.

This is torque at standstill condition.

Under running condition induced EMF is

T = K1 * Er *  Ir * cos ∅2 ---------------------- (1)

Where Er = rotor induced EMF per phase under running condition.

Ir = rotor current per phase under running condition.

φ = flux per stator pole.

rotor current,

Ir = Er / Zr = sE2/  √( R2² + (sX2)²)  ----------- (2)

power factor,

cos∅2 = R2 / √( R2² + (sX2)²)   ----------------(3)

So the rotor torque is, 

After putting Ir from equation (2) and cos∅2 from equation (3) in equation no. (1), we we get the torque equation

T = K1 * φ * sE2 * R2/( R2² + X2²)  ---------------(4)

The condition of maximum torque may be obtained if it differentiated equation number (4 )with respect to s.

X = 1/T = ( R2² + X2²) / (K1 * φ * sE2 * R2)

Since E2 is proportional to φ

X = 1/T = ( R2² + X2²) / (K1 * s E2² * R2)

After solving the above equation we get,

R2 = sX2                         ------------------ (5) 

Hence under running condition torque is maximum at that value of slip which makes rotor resistance per phase equal to rotor reactance per phase.

Hence slip at T maximum is

Smax = R2 / X2 

After putting equation (5) in equation (4)

T max= K1 E2 / (2 X2)      ---------------------       ( 6)

Or 

Tmax = T = K1 sE2²/( 2R2) ------------------          (7)

From equation number (6) it is clear that maximum torque is independent of rotor resistance.

But at the same time maximum torque is inversely proportional to rotar reactance per phase so it's should be a small as possible.

Although maximum torque is not dependent upon total resistance per phase but speed or slip at which maximum torque is dependent upon rotor resistance.

thanks by wearing the rotary distance we can get maximum target any value of slip.

See the diagram below.

How will change rotar EMF and reactance under running condition.

 When we supply three phase input to the rotor of the motor then voltage is induced in the rotar e2 and current I to start flowing. The frequency of rotar and voltage of rotor at stand still is equal to supply frequency. Because relative velocity is maximum at a stand still when s= 1 between status flux and rotar.

Let a stand still S =1 ,

E2= stand still rotar induced EMF per phase

f2 = rotar current frequency at stand still

X2 = standd still router induced reactance per phase.

when total starts rotating then relative velocity between rotar and stator flux decreases then induced voltage E2  start decrease.

When  rotor speed catches rotating flux speed than relative velocity becomes zero that is S= 0 then induced voltage them become zero.

let's say for a slip s the rotor induced EMF will be s times the induced emf at a stand still.

Therefore under running condition

Rotar induced EMF running condition

Er = s E2

Frequency of induced EMF at running condition

fr = sf2

Since frequency of induced EMF is reduced so the reactance will also reduce.

Xr = sX2

Where Xr, Er, fr are induced reactance,  rotor EMF and frequency under running condition respectively.

Torque in induction motor and relationship with rotor power factor.

The torque in induction motor is proportional to the product of flux per stator pole and the rotor current one more term we will take in account is power factor of the rotor.

  T →  φ  *  I₂  * cos∅₂ 

Where φ  - flux per status pole

I₂ - rotor current

cos∅₂ - rotor power factor

∅₂ phase angle  rotor current and rotorEMF.

The induced rotor EMF E2 is proportional to flux per status φ.

Therefore T = K1 * E₂ *  I₂ * cos φ2 

Where K1 is another constant.

From the above equation it is clear that when φ2 increases cos φ2 decreases and vice versa.

Due to the revolving stator flux EMF is induced in rotor conductor, and this EMI is also sinusoidal.

When rotor is non inductive.

φ₂=0

in this case rotor current is in phase with rotor EMF. So the instantaneous value of torque is product of instantaneous value of flux and current. It is seen that torque is always positive.

When rotor is inductive load.

in inductive load I₂ lags behind E₂ bY an angle φ2.

φ2 = tan-¹(X₂/R₂ )

R2 = rotor  resistance per phase

X2 = rota reactance per phase at standstill.

Its clear from diagram that portion of torque is reverse direction and hence the total torque is difference of forward torque and reverse torque. When φ2 = 90 degree then reverse torque equals to forward torque and total torque is equals to zero so the motor will not run.

​Starting torque of squirrel cage induction motor and slip ring induction motor.

The resistance of squirrel cage rotor is fixed and small as compared to reactance. Reactance of rotor is high as frequency of a current is equals to supply frequency. Hance at starting power factor is poor. starting current is although very high but lags by very large angle behind E2. So the torque is only 1.5 times the full load torque but starting current is 5 to 7 times the full load current. Hence squirrel cage induction motor is not suitable for load which required high starting torque.

where as in slip ring induction motor the starting torque of motor increases by improving the power factor of motor. In this we connect the start connected rheostat to the rotor circuit. So the current decreases but torque increases. When motor catches speed this external resistance gradually cut off and all the slip rings are short circuit by copper bar and the brushes are lifted up and work as squirrel cage induction motor.

Impedance traingle of rotor circuit of induction motor.

Let E2 = rotar EMF per phase at stand still

 R2 = rotar resistance per phase

X2 = rota reactance per per phase at stand still

Z2 = √( R2² + X2²) rotar impedance per phase at stand Still

Rotor current,

I2 = E2 / √( R2² + X2²) 

power factor,

cos∅2 = R2 / √( R2² + X2²) 

T = K1 E₂ R2/( R2² + X2²)  

Rotor torque 

= K1 * E2 / √( R2² + X2²) * E2 * R2 / √( R2² + X2²)

= K1 E₂² R2 /(R2² + X2²)

if we increase the resistance of rotor circuit then we can improve the power factor.

E2 rotor induced EMF

I2 rotor current

R2 rotor resistance per phase

X2 rotor reactance per phase at stand still

from the above equation it is clear that if we increases external resistance R2 then torque will increase but are two total current will decreases and power factor also improves.