Torque of a DC Machine

When a dc machine is loaded either as a motor or as a generator, the rotor conductor carry current.These conductor lie in the magnetic field of air gap. thus each conductor experiences a force. the conductor lie near the surface of the rotor at a common radius from its centre. Hence a torque is produced around the circumference of the rotor and the rotor starting rotating.

When the machine operates as a generator at constant speed, this torque is equal and opposite to that provided by the prime-mover. when the machine is operating as a motor the torque is transferred to the shaft of the rotor and drives the mechanical load.

The expression for torque is the same for generator or motor. It can be derive as

The voltage equation of d.c. motor
                                          V = E + I_aR_a  
Multiplying both sides of equation by  I_a  we, obtain
                                        VI_a = EI_a + I_a²R_a  
But 
           VI_a  = Electrical power input to the armature
          I_a²R_a = Copper loss in  the armature

                        We also known that  input = output + losses

EI_a=Electrical equivalent of gross mechanical power developed by the armature         ( electromechanical power )    
𝜏_av = Average  electromechanical torque developed by the armature in newton            meters (Nm) 

At this value of torque the electromechanical power conversion takes place.

Mechanical power developed by the armature,

This equation is called the torque equation of d.c. motor.

For a given d.c. machine P Z snd A are constant, therefore

Hence the torque developed by the a d.c. motor is directly proportional to the flux per pole and armature current.