Armature Torque of D.C. Motor

Torque is the turning moment of a force about an axis and is measured by the product of force (F) and radius (r) at right angle to which the force acts.
T = F x r
In a d.c. motor, each conductor is acted upon by a circumferential force F at a distance r, the radius of the armature. Therefore, each conductor exerts a torque, tending to rotate the armature. The sum of the torques due to all armature conductors is known as gross or armature torque (Ta).
Let in a d.c. motor
r = average radius of armature in m

l = effective length of each conductor in m
Z = total number of armature conductors
A = number of parallel paths
i = current in each conductor = Ia/A
B = average flux density inWb/m2
f = flux per pole inWb
P = number of poles
Force on each conductor, F = B i l newtons
Torque due to one conductor = F x r (newton- metre)
Total armature torque, Ta = Z F r (newton-metre)
= Z B i l r
Now i = Ia/A, B = /a where a is the x-sectional area of flux path per pole at
radius r. Clearly,
   a = 2  r l /P.

Since Z, P and A are fixed for a given machine,

 

Hence torque in a d.c. motor is directly proportional to flux per pole and armature
current.
(i) For a shunt motor, flux f is practically constant.

(ii) For a series motor, flux f is directly proportional to armature current Ia
provided magnetic saturation does not take place.