Universal Torque Equation of Relays - its Derivation

Most of the protection relays consist of some arrangement of electromagnets with armature or induction disc, which carry contacts. The relays also carry the closing or opening of contacts control devices like trip coils of the circuit breaker. The electromagnets are wound with either current coil, voltage coil, or both current and voltage coils.


The coils produce magnetic flux which develops torque in the rotor when interacts with the eddy currents that are induced in the rotor by these fluxes. The universal torque equation of relay gives the torque exerted on the rotor by all the quantities. Let us see the expression for the universal torque equation of relay.


Let us assume the following,
  • V = Voltage applied
  • I = Current through winding
  • θ = Angle between voltage and current
  • 𝜏 = Maximum torque angle
  • T = Universal torque equation
  • T1 = Torque set up by the current coil
  • T2 = Toque set up by the voltage coil
  • T3 = Torque set up by both voltage and current coil
  • T4 = Torque produced by control spring
  • K1, K2, K3, K4 = Constants.

The protective relay consists of electromagnets with armature or induction disc. Some of the electromagnets will have windings for current or voltage, or for both current and voltage. The torque set by the current coil is directly proportional to the square of the current i.e.,
T1 = K1 I2
The torque set by the voltage coil is directly proportional to the square of the voltage i.e.,
T2 = K2 V2
The torque produced by both the voltage and current coils is directly proportional to the product of voltage and current i.e.,
T3 = K3 V I cos(θ - 𝜏)
The torque set up by the control spring is given by,
T4 = K4
The universal torque equation is obtained when all the elements are considered in the relay i.e.,
T = T1 + T2 + T3 + T4
Substituting equations of T1, T2, T3, and T4 in the above equation, we get,
T = K1 I2 + K2 V2 + K3 V I cos(θ - 𝜏) + K4
The above equation represents the universal torque equation of relay.


Previous Post Next Post