#### The equivalent circuit of an induction motor is similar to a transformer equivalent circuit because the energy is transferred from stator to rotor is essentially a transformer operation.

#### Equivalent circuit enables the performance characteristics of the induction motor. All per phase quantities are used in representing the equivalent circuit.

#### Following are the parameters of the equivalent circuit

####
(i). R_{1} & X_{1 }: Stator windinq resistance and leakage reactance.

####
(ii). R_{2} & X_{2 }:_{ }Rotor winding resistance and leakage reactance at standstill ( i.e., s = 1 ).

####
(iii). sX_{2 } : Rotor leakage reactance at slip 's' ( under running condition ).

####
(iv). R_{o} : No-load branch resistance and it carries working component ( I_{w }) of no-load current I_{o } account for the losses on no-load.

####
(v). X_{o} : No-load branch reactance and it carries magnetising component ( I_{µ }) of no-load to produce the flux.

####
(vi). E_{1}
and sE_{2 }: Stator induced emf and rotor induced emf at slip 's'.

#### The figure below represents the equivalent circuit.

#### OR

#### Now let us consider Actual rotor circuit and Equivalent rotor circuit,

#### Then the rotor current,

#### Dividing Nr and Dr, by 'S'

####
The rotor input, P_{2} is the sum of rotor copper losses P_{c} and mechanical power developed P_{m}. Thus it is possible to represent the electrical equivalent of mechanical power developed as follows :

The rotor input, P

_{2}is the sum of rotor copper losses P_{c}and mechanical power developed P_{m}. Thus it is possible to represent the electrical equivalent of mechanical power developed as follows :#### Therefore,

####
R_{2 }/ S can be expressed as,

####
' R_{L} ' represents the equivalent mechanical load.

#### i.e.,

####
R_{L} = R_{2}
( 1 - s / s )

#### So that the rotor equivalent circuit can be modified as shown in figure below

#### The rotor side parameters can be transferred to stator side as

####
R'_{2}
= R_{2} / K^{2}

####
X'_{2}
= X_{2 }/ K^{2}

####
R'_{L}
= R_{L } / K^{2}

####
E'_{2}
= E_{2} / K = E_{1}

####
I'_{2}
= K^{ }I_{2}

#### Where

#### K = Ratio of effective rotor to stator turns per phase.

#### The equivalent circuit shown in below figure is called Exact Equivalent circuit as referred to stator.

####
The approximate equivalent circuit is obtained by shifting the shunt branch ( consists of R_{o}
& X_{o} ) to the supply terminals as shown in below figure.

####

Therefore, the total resistance and reactance referred to stator side are

####
####
R_{01 } = R_{1
}+ R'_{2}

= R_{1} + R_{2} / K2

####
X_{01 } = X_{1
}+ X'_{2}

= X_{1} + X_{2} / K2

##
Calculation of Rotor Output and Torque Using the Equivalent Circuit :

####
R_{01 } = R_{1
}+ R'_{2}

= R_{1} + R_{2} / K2

_{01 }= R

_{1 }+ R'

_{2}

_{1}+ R

_{2}/ K2

####
X_{01 } = X_{1
}+ X'_{2}

= X_{1} + X_{2} / K2

_{01 }= X

_{1 }+ X'

_{2}

_{1}+ X

_{2}/ K2

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