Let us consider a three-phase synchronous alternator that is driven by a prime mover. The equation of motion of the machine rotor is given by

1 |

where

J |
is the total moment of inertia of the rotor mass in kgm^{2} |

T_{m
} |
is the mechanical torque supplied by the prime mover in N-m |

T_{e} |
is the electrical torque output of the alternator in N-m |

θ |
is the angular position of the rotor in rad |

Neglecting the losses, the difference between the mechanical and electrical torque gives the net accelerating torque *T _{a} *. In the steady state, the electrical torque is equal to the mechanical torque, and hence the accelerating power will be zero. During this period the rotor will move at

**synchronous speed**

*ω*in rad/s.The angular position

_{s}*θ*is measured with a stationary reference frame. To represent it with respect to the synchronously rotating frame, we define

2 |

where *δ *is the angular position in rad with respect to the synchronously rotating reference frame.

Defining the angular speed of the rotor as we can write as

3 |

Differentiating the equation 2 w.r.t. time & using the equation 3, we can write : |

We can therefore conclude that the rotor angular speed is equal to the synchronous speed only when *d δ / dt *is equal to zero. We can therefore term

*d*as the error in speed. Taking derivative of above equation , we can then rewrite as

*/ dt**δ*Multiplying both side by *ω _{m} *we get

4 |

where *P _{m} *,

*P*and

_{e}*P*respectively are the mechanical, electrical and accelerating power in MW.

_{a}We now define a normalized inertia constant as

Substituting the value of H in above equation 4, we get

In steady state, the machine angular speed is equal to the synchronous speed and hence we can replace *ω _{r}* in the above equation by

*ω*. Note that

_{s}*P*,

_{m}*P*and

_{e}*P*are given in MW. Therefore dividing them by the generator MVA rating

_{a}*S*we can get these quantities in per unit. Hence dividing both sides by

_{rated}*S*in above equation we get

_{rated}
per unit The above equation is known as Swing Equation in P.U. Email id: er.sohail.ansaari@gmail.com 2007.ansari@gmail.com Facebook id:https://www.facebook.com/sohail.ansari.itm linkedin id:http://in.linkedin.com/pub/sohail-ansari/73/17a/547/ |