Torque Equation of a DC Motor

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 a 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 is same for the generator and motor.

When the current carrying current is placed in the magnetic field, a force is exerted or it which exerts turning moment or torque F x r. This torque is produced due to the electromagnetic effect, hence is called Electromagnetic torque.

Since,

torque-equation-of-dc-motor-eq1

Multiplying the equation (1) by Ia we get

torque-equation-of-dc-motor-eq2

Where,

VIa is the electrical power input to the armature.

I2aRa is the copper loss in the armature.

We know that,

Total electrical power supplied to the armature = Mechanical power developed by the armature  + losses due to armature resistance

Now, the mechanical power developed by the armature is Pm.

torque-equation-of-dc-motor-eq3

Also, the mechanical power rotating armature can be given regarding torque T and speed n.

torque-equation-of-dc-motor-eq4

Where n is in revolution per seconds (rps) and T is in Newton-Meter.

Hence,

torque-equation-of-dc-motor-eq5

But,

torque-equation-of-dc-motor-eq6

Where N is the speed in revolution per minute (rpm) and

torque-equation-of-dc-motor-eq7

Where, n is the speed in (rps).

Therefore,

torque-equation-of-dc-motor-eq8

So, the torque equation is given as

torque-equation-of-dc-motor-eq9

For a particular DC Motor, the number of poles (P) and the number of conductors per parallel path (Z/A) are constant.

torque-equation-of-dc-motor-eq10

Where,

torque-equation-of-dc-motor-eq11

Thus, from the above equation (5) it is clear that the torque produced in the armature is directly proportional to the flux per pole and the armature current. Moreover, the direction of electromagnetic torque developed in the armature depends upon the current in armature conductors. If either of the two is reversed the direction of torque produced is reversed and hence the direction of rotation. But when both are reversed, and direction of torque does not change.

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