**Universal Relay Torque Equation Derivation:**

Universal Relay Torque Equation – Most of the protection relays consist of some arrangement of electromagnets. These electromagnets have either current windings or voltage windings and in some cases both the windings. Currents through the windings produce magnetic fluxes and the torque is produced by interaction between the fluxes of the same winding or between the fluxes of both the windings.

If both the current and voltage windings are used, the torque developed by the interaction between the fluxes

where

- θ is the angle between V and I and τ is the relay maximum torque angle.

If a relay consists of all the three elements, the torque will be developed due to all the three causes and therefore, its torque in general will be given as

where

- K
_{1}, K_{2}, K_{3}are tap settings or constants of I and V and - K
_{4}is the mechanical restraint due to spring or gravity.

By assigning plus or minus signs to certain of the constants and letting others be zero, and sometimes by adding other similar terms, the operating characteristics of all types of protective relays can be obtained.

For example, in case of an overcurrent relay K_{2} = K_{3} = 0 because of absence of voltage windings and therefore universal relay torque equation becomes

-ve sign is assigned to K_{4} as the torque produced by spring is restraining one.

Similarly, for directional relay K_{1} = K_{2} = 0 and the torque developed will be given as