It is shown that in slip ring induction motor, externally resistance can be added in the rotor. Let us see the effect of change in rotor resistance on the torque produced.

Let R

_{2 }= Rotor resistance per phase Corresponding torque, T α (s E

_{2}^{2}R_{2}_{})/√(R_{2}^{2}+(s X_{2})^{2}) Now externally resistance is added in each phase of rotor through slip rings.

Let R

_{2}**'**= New rotor resistance per phase Corresponding torque T

**'**α**(s E**_{2}^{2}R_{2}**'**_{})/√(R_{2}**'**^{2}+(s X_{2})^{2}) Similarly the starting torque at s = 1 for R

T

_{2 }and R_{2}**'**can be written asT

_{st }**α (E**_{2}^{2}R_{2})/√(R_{2}^{2}+(X_{2})^{2}) and T

**'**st α (E_{2}^{2}R**'**_{2})/√(R**'**_{2}^{2}+(X_{2})^{2})Maximum torque T

_{m}**α (E**_{2}^{2})/(2X_{2})**Key Point**: It can be observed that T

_{m}

**is independent of R**

_{2}hence whatever may be the rotor resistance, maximum torque produced never change but the slip and speed at which it occurs depends on R

_{2}.

For R

_{2}, s_{m}= R_{2}/X_{2}where T_{m}**occurs** For R

_{2}**'**, s_{m}**'**= R_{2}**'**/X_{2}**'**where same T_{m}**occurs** As R

_{2}**'**> R_{2}, the slip s_{m}**'**> s_{m}. Due to this, we get a new torque-slip characteristics for rotor resistance . This new characteristics is parallel to the characteristics for with same but T_{m}**occurring at s**_{m}**'**. The effect of change in rotor resistance on torque-slip characteristics shown in the Fig. 1. It can be seen that the starting torque T

**'**st for R_{2}**'**is more than Tst for R_{2}. Thus by changing rotor resistance the starting torque can be controlled. If now resistance is further added to rotor to get resistance as R

_{2}**'**and so on, it can be seen that T_{m}**remains same but slip at which it occurs increases to s**_{m}**'**and so on. Similarly starting torque also increases to T**'**st**and so on.**Fig. 1 Effect of rotor resistance on torque-slip curves |

_{m}= 1 as at start slip is always unity, so

s

_{m }= R

_{2}/X

_{2}= 1

R

_{2}= X

_{2}Condition for getting Tst = T

_{m}

**Key Point**: Thus by adding external resistance to rotor till it becomes equal to X

_{2}, the maximum torque can be achieved at start.

If such high resistance is kept permanently in the circuit, there will be large copper losses (I

Thus good performance at start and in the running condition is ensured.^{2}R) and hence efficiency of the motor will be very poor. Hence such added resistance is cut-off gradually and finally removed from the rotor circuit, in the normal running condition of the motor. So this method is used in practice to achieve higher starting torque hence resistance in rotor is added only at start.**Key Point**: This is possible only in case of slip type of induction motor as in squirrel cage due to short circuited rotor, extra rotor resistance can not be added.

**Example**: Rotor resistance and standstill reactance per phase of a 3 phase induction motor are 0.04 Ω and 0.2 Ω respectively. What should be the external resistance required at start in rotor circuit to obtain.

**Solution**:

R

_{2}= 0.04 Ω, X

_{2}= 0.2 Ω

i) For T

_{m}

**= T**

_{st}

**, s**

_{m}

**= R**

_{2}

**'**/X

_{2}= 1

**.**R

^{.}._{2}

**'**= X

_{2}= 0.2

Let R

_{ex}= external resistance required in rotor.

R

_{2}

**'**= R

_{2}+ R

_{ex}

**.**R

^{.}._{ex}= R

_{2}

**'**- R

_{2}= 0.2 - 0.04 = 0.16 Ω per phase

ii) For T

_{st}

**= 0.5 T**

_{m},

Now T

_{m}= (k E

_{2}

^{2})/(2 X

_{2}) and

T

_{st}

**= (k E**

_{2}

^{2}R

_{2})/(R

_{2}

^{2}+ X

_{2}

^{2})

But at start, external resistance R

_{ex}is added. So new value of rotor resistance is say R

_{2}

**'**.

R

_{2}

**'**= R

_{2}+ R

_{ex}

**.**T

^{.}._{st}

**= (k E**

_{2}

^{2}R

_{2}

**'**

_{})/(R

_{2}

**'**

_{}

^{2}+ X

_{2}

^{2}) with added resistance

but T

_{st}

**= 0.5T**

_{m}required.

Substituting expressions of T

_{st}

**and T**

_{m}, we get

(k E

_{2}

^{2}R

_{2}

**'**)/(R

_{2}

**'**

^{2}+ X

_{2}

^{2}) = 0.5 (k E

_{2}

^{2})/ (2X

_{2})

**.**4 R

^{.}._{2}

**'**X

_{2}= (R

_{2}

**'**

_{}

^{2}+ X

_{2}

^{2})

**.**(R

^{.}._{2}

**'**

^{2}) - 4 x 0.2 x R

_{2}

**'**+ 0.2

^{2}= 0

**.**(R

^{.}._{2}

**'**

^{2}) - 0.8 R

_{2}

**'**+ 0.04 = 0

**.**R

^{.}._{2}

**'**=

**{**0.8 + √(0.8

^{2}- 4 x 0.04)

**}**/2

**.**R

^{.}._{2}

**'**= 0.0535 , 0.7464 Ω

But R

_{2}

**'**can not greater than X

_{2}hence,

R

_{2}

**'**= 0.0535 = R

_{2}

**+ R**

_{ex}

**.**0.0535 = 0.04 + R

^{.}._{ex}

**.**R

^{.}._{ex }= 0.0135 Ω per phase

This is much resistance is required in the rotor externally to obtain T

_{st}

**= 0.5 T**

_{m}.

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