Torque and Speed Equations of DC Motor

TORQUE AND SPEED EQUATIONS

Torque is the turning or twisting moment of a force about an axis. It is measured by the product of the force and the radius at which this force acts.

Consider a pulley of radius r metre acted upon by a circumferential force of F newton which causes it to rotate at speed N rpm.

Torque T= F×r Newton ‒ metre (Nm)

Work done by this force in one revolution.

= Force × distance

= F × 2πr Joules

and P = Power developed =  Work done / time

= F×2πг  / time for one revolution

= F×2πг / (60/N)

= (F×r) × (2πΝ/60)

∴ P=T×W watts

where,

T= Torque in Nm

ω = Angular speed in rad/sec = 2πΝ / 60

 ∴ P = 2πΝΤ / 60  W

1. Armature Torque of a motor

Let T_a be the gross torque developed by the armature of motor running at N r.p.m. It is also called armature torque.

Power developed in armature = T_a (2πΝ / 60) W

We also know that electrical power converted into mechanical power in the Armature = E_bI_a W

Equating (i) and (ii) we get

 T_a × [2πΝ / 60] = E_bI_a

but E_b in a motor is given by

This is the armature torque equation of DC motor.

From the above equation it is clear that

T_a ∝ I_a

(i) In case of series motor  ϕ ∝ I_a

∴ T_a ∝ I²_a

(ii) In case of shunt motor flux is practically constant,

∴ T_a ∝ I_a

2. Shaft Torque (T_sh)

The torque which is available for doing useful work is known as Shaft Torque (T_sh).

The gross torque T_a developed in the armature of a motor is not available at the shaft because a part of it is lost, in overcoming the iron and frictional losses in the motor. Therefore, shaft torque T_sh is some what less than the armature torque, T_a. The difference (T_a‒T_sh) is known as Lost torque,

The motor output is given by

Output = T_sh×2πN / 60

T_sh = (output in Watts) / (2πN/60) Nm.

T_sh = 9.55 × (output in Watts / N) Nm

1. Speed of a DC Motor

From the voltage equation of a motor, we get,

 E_b = V ‒ l_aR_a

But,

 N ∝ k E_b/ϕ

or

N ∝ E_b/ϕ

Therefore, in a DC motor, speed is directly proportional to back emf, E_b and inversely proportional to flux per pole.

2. Speed Relations

If a DC motor has initial values of speed, flux per pole and back emf as N₁, ϕ₂ and E_b1 respectively and the corresponding final values as N₂, ϕ₂ and E_b2, then

For a shunt motor, flux practically remains constant so that,

 ϕ₁ = ϕ₂

N₂ / N₁ = E_b2 / E_b1

For a shunt motor, ϕ ∝ I prior to saturation.

N₂ / N₁ = E_b2/E_b1 × I_a1/I_a2

where,

I_a1 = Initial armature current.

I_a2 = Final armature current.

Speed Regulation

The speed Regulation is defined as the change in speed when the load on the motor is reduced from rated value to zero, expressed as percent of the rated load speed.

 % Speed regulation = [ (No load speed ‒ Full load speed) /  Full load speed ] × 100