The term torque is meant turning or twisting movement of force about an axis. When dc machine is loaded either as dc motor or as a generator. the rotor conductors carry current, these conductors are in the magnetic field of the air gap. thus each rotor conductor experiences a force and torque developed in the rotor. the conductor which lies near the surface of the rotor at a common radius from its center. hence torque developed around the circumference of the rotor and the rotor starts rotating. Here we will see derivation for the torque equation of dc motor.

When the Dc machine operates as a generator at constant speed this torque is equal and opposite to that torque provided by the prime mover.

On the other hand, when the dc machine operates as the motor it transfers torque to the shaft and drives the mechanical load.

The torque equation is the same for the motor and generator.

The voltage equation of a Dc motor is,

**V=E+IaRa**

Multiplying both sides by Ia

But,

VIa is the power input to the armature and,

I2aRa= copper loss in armature

**Total Electrical Input to Armature=Mechanical Output Developed by Armature+losses**

Hence,

EIa= electrical equivalent of gross mechanical power developed by the armature.

𝜏av= average torque developed by armature in Nm (electromagnetic torque)

Mechanical torque developed by the armature,

Pm=EIa=ω𝜏av =2πn𝜏av

Hence, rearranging the above formula the torque equation of dc motor become

This is called a **Torque equation of DC motor**.

Here P, Z, and A are constant

𝜏av=KΦIa

From the above equation, we can see that torque developed in the Dc motor is directly proportional to flux per pole **Φ** and the armature current **Ia**.

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