TORQUE OF A DC MACHINE

When a dc machine is loaded either as a motor or as a generator,the rotor conductors carry current .These conductors lie in the magnetic field of the air gap.Thus each conductor experiences a force.The conductors lie near the surface of the rotor at a common radius from its center. Hence a torque is produced around the circumference of the rotor and the rotor starts rotating.
When the machine operates as a generator at constant speed,this torque is equal and opposite to that provided by the prime-motor.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 the torque is the same for the generator and the motor.It can be deduced as follows:
The voltage equation of a d.c motor is
V=E+IaRa------(1)

Multiplying both the sides of Eq. (1) we obtain
VIa=EIa + (Ia*Ia)*Ra-------(2)
But
VIa = electrical power input to the armature
 (Ia*Ia)*Ra=copper loss in the armature
we also known that input=output+losses-----(3)
comparison of Eq(2) and  Eq(3) show that
EIa=electrical equivalent of gross mechanical power developed by the armature (electromagnetic power)
Let T= average electromagnetic 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,
Pm=wT=2πnT
Therefore 
Pm=EIa=wT=2πnT---------(4)
but E=(nPϕZ)/A
Therefore
(nPϕZ)*Ia/A=2πnT
T=(PZϕIa)/2πA---------(5)
Eq(5)is called the torque equation of d.c motor.
For a given d.c.machine,P,Z and A are constant,therefore (PZ/2πA) is also constant.
Let 
(PZ/2πA)=k------(6)
T=kϕIa------(7)
T∝ϕIa------(8)
Hence the torque developed by a d.c. motor is directly proportional to the flux per pole and armature current..