Consider the approximate equivalent circuit as shown in the Fig.7

In this circuit, the exciting current I

_{o }is neglected hence the exciting no load branch is not shown.**.**I

^{.}._{1 }= I

_{2r}

**'**

The total impedance is given by,

Z

_{T}= (R_{1e}+ R_{L}**'**)+ where R_{L}**'**= R_{2}**'**(1-s)/s I

_{1 }= V_{1 }/√**(**(R_{1e}+ R_{L}**'**)^{2}+(X_{1e})^{2}**)** The power supplied to the load i.e. P

_{out }per phase is, Per phase P

_{out }= I_{1}^{2}_{ }R_{L}**'**watts per phase**.**Total = 3 I

^{.}._{1}

^{2}

_{ }R

_{L}

**'**

To obtain maximum output power, differentiate the equation of total P

_{out }with respect to variable_{ }R_{L}**'**and equal to zero.But Z

_{1e}= √**(**R_{1e}^{2}+X_{1e}^{2}**) =**Leakage impedance referred to stator**.**Z

^{.}._{1e}

^{2}

**=**R

_{L}

**'**

_{}

^{2}

Thus the mechanical load on the induction motor should be such that the equivalent load resistance referred to stator is equal to the total leakage impedance of motor referred to stator.

**Slip at maximum P**

_{out }: This can be obtained as,

R

_{L}**'**= Z_{1e}= R_{2}**'**(1-s)/s where R_{L}**'**= R_{2}/K^{2}**.**s Z

^{.}._{1e}= R

_{2}

**'**- sR

_{2}

**'**

**.**s(Z

^{.}._{1e}+ R

_{2}

**'**) = R

_{2}

**'**

This is slip at maximum output.

**Expression for maximum**

**P**

_{out }: Using the condition obtained in expression of total P

_{out }, we can get maximum P

_{out}.

**.**(P

^{.}._{out})

_{max }= 3 I

_{1}

^{2}Z

_{1e}

_{ }as R

_{L}

**'**= Z

_{1e}

_{ }

But R

_{1e}^{2 }+ X_{1e}^{2}= Z_{1e}__1.4 Maximum Torque__

In case of induction motor, the speed of the motor decreases with increase in load. Thus the maximum power output is not obtained at a slip which corresponds to maximum torque. In the previous section we have seen the condition for maximum power output. In this section we will find the condition which gives maximum torque.

The expression for torque is given by,

The condition for maximum torque can be obtained from maximum power transfer theorem. When I

_{2r}**'**^{2}R_{2}**'**/s is maximum consider the approximate equivalent circuit of induction motor as shown in The Fig. 8.Fig. 8 |

The value of R

_{o}is assumed to be negligible. Hence the circuit will be reduced as shown below.Fig. 9 |

The thevenin's equivalent circuit for the above network is shown in the Fig.10 across the terminals x and y.

Fig. 10 |

The mechanical torque developed by rotor is maximum if there is maximum power transfer to the resistor R

_{2}**'**/s. This takes place when R_{2}**'**/s equals to impedance looking back into the supply source. This is the slip corresponding to the maximum torque. The maximum torque is given by,

Substituting,

From the above expression, it can be seen that the maximum torque is independent of rotor resistance.

__1.5 Synchronous Watt__

The torque produced in the induction motor is given by,

Thus torque is directly proportional to the rotor input. By defining new unit of torque which is synchronous watt we can write,

T = P

_{2 }synchronous-watts If torque is given in synchronous-watts then it can be obtained in N-m as,

**Key Point**: Unit synchronous watt can be defined as the torque developed by the motor such that the power input to the rotor across the air gap is 1 W while running at synchronous speed.

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