Torque Equation of DC Motor

Torque equation of DC machine

The term torque is a quantitative measure of the tendency of a force to cause a rotational motion. It is a turning and twisting moment of a force about an axis. In this article, we will learn to find the torque equation of DC Motor.

The torque of the DC motor calculated by the product of force and the radius at which this force acts.

The armature conductor of DC motor experiences a circumferential force at a distance r, which tends to rotate the armature.

Torque Equation of DC Motor

Total torque developed by an armature conductor known as armature torque or gross torque and it denoted by Ta.

    \[ T_a = F \times r \quad \tagaddtext {Nm}\]

Work done (W) by this force in one revolution of the armature expressed as,

    \[ W = F \times 2 \pi r \quad Joule \]

Mechanical power developed by the armature,

    \[ P_m = F \times 2 \pi r \times n \quad J/sec (watt) \]

    \[P_m = F \times r \times  2 \pi n \]

where; n = speed of motor (RPS)

    \[ T_a = F \times r \quad and \quad \omega = 2 \pi n \]

Therefore, mechanical power developed by the armature,

    \[ P_m = T_a \times \omega \quad watt \]

Now, the mechanical power developed in armature converted into the electrical power. Hence, the electrical power expressed as (Pe),

    \[ P_e = E_b \time I_a \quad watt \]

Let’s assume that the mechanical power is same the electrical power,

    \[ P_m = P_e \]

    \[ T_a \times \omega = E_b \times I_a \]

    \[ T_a = \frac{E_b I_a}{\omega} \]

    \[ T_a = \frac{E_b I_a}{2 \pi n} \]

In this equation, Eb is back EMF. And it is expressed as,

    \[ E_b = \frac{P \phi n Z}{A} \]

Where, ф = flux per pole, Wb
Z = number of armature conductors
n = speed of armature, (RPS)
P = number of poles
A = number of parallel paths

Now, place the value of back EMF in the torque equation,

    \[ T_a = \frac{P \phi n Z}{A} \times \frac{I_a}{2 \pi n} \]

    \[ T_a = \frac{1}{2 \pi} \phi Z I_a \frac{P}{A} \quad Nm \]

    \[ T_a = 0.159 \phi Z I_a \frac{P}{A} \quad Nm \]

Now, once the motor designed, number of armature conductors (Z), number of poles (P), and number of parallel paths (A) are constant.

    \[ T_a = k_a \phi I_a \]

    \[ k_a = \frac{PZ}{2 \pi A} \]

    \[ T_a \propto \phi I_a \]

For, DC shunt motor, the flux per pole is constant,

    \[ T_a \propto I_a \]

For, DC series motor, the flux per pole is directly proportional to armature current.

    \[ T_a \propto \phi I_a \]

    \[ T_a \propto I_a I_a \]

    \[ T_a \propto I_a^2 \]

Shaft Torque

The torque which is available at the motor shaft for doing useful work is known as shaft torque Tsh.

    \[ Net \, motor \, output = T_s_h \time \omega \]

    \[ Net \, motor \, output = T_s_h \time \frac{2 \pi N}{60} \]

    \[ T_s_h = 9.55 \frac{Net \, motor \, output}{ N} \quad Nm \]

The whole of the torque developed by the armature is not available at the shaft. Because some part of this torque is lost to overcome the iron and mechanical losses. Hence, the shaft torque is less than the armature torque or gross torque.

    \[ T_s_h = T_a - (Iron + Friction Loss) \]

    \[ T_a - T_s_h = Iron + Friction Loss \]

    \[ Total loss of Torque = T_a - T_s_h \]

The difference between the gross torque and shaft torque is loss of torque due to iron and mechanical losses occurs in the motor.

Break Horse Power (BHP)

The horse power (HP) developed by the shaft torque is known as Break Horse Power (BHP).

    \[ Useful \, Power \, Output = T_s_h \times \omega \]

    \[ BHP = \frac{T_s_h}{746} \times \frac{2 \pi N}{60} \]

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