Torque Equation of DC Machines

TORQUE EQUATION

Torque is nothing but turning or twisting force about an axis. Torque is measured by the product of force and radius which the force acts. Consider a wheel of radius 'r' meters acted on by a circumferential force 'F' Newton as Figure 2.15.

Let the force 'F' cause the wheel to rotate at 'N' rpm. The angular velocity of the

wheel is:

Torque, T = F x r N-m

Work done per revolution = F x distance moved = F x 2πr joules.

Power developed, P = Work done / Time = (F x 2πr) / Time for one revolution = (F x 2πr) / (60/N)

rps = rpm/60 ; rps = N/60 ; time for one revolution = 60/N

P = (F x r) 2πN/60

P=T ω Watts.

where, T = torque in N - m

ω = angular speed in rad/sec.

The torque developed by a DC Motor is obtained by looking at the electrical power supplied to it and mechanical power produced by it. It is also called armature torque. The gross mechanical power developed in the armature is E_b I_a.

Then, power in armature = Armature torque x ω.

The above equation is torque equation of a DC motors.

'T' is proportional to φ I_a. Hence the torque of a given DC motor is proportional to the product of the armature current and the flux.

Shaft Torque (T_sh)

The torque developed by the armature is called armature torque. It is denoted as T_a. The full armature Torque is not available for doing useful work. Some amount of torque is used for supplying iron and friction losses in the motor.

This torque is called torque lost. It is denoted as T_f The remaining torque is available in the shaft. It is used for doing useful work. This torque is known as shaft torque. It is denoted as T_sh. The armature torque is the sum of the torque lost and shaft torque.

T_a = T_f + T_sh

The output power of the motor is:

Speed and Torque Equations

For a DC motor, the speed equation is obtained as follows:

We know,

Hence the speed of the motor is directly proportional to back emf E_b and inversely proportional to flux φ. By varying the flux and voltage, the motor speed can be changed.

The emf change from instant to instant and becomes alternately positive and negative. Such an emf is called an alternating emf. If the coil sides are connected to two slip rings 'a' and 'b' and an external resistance R connected across them, a current flows through the resistor, which is again alternating.

The induced emf in the coil can be increased by:

Increasing the flux density (B) and

Increasing the angular velocity (ω).

In commercial generators a large number of coils are used and they are housed in the armature, which rotates on a shaft at high speed.

The current flowing the external resistance to a DC generator is made unidirectional by replacing the slip rings by replacing the slip rings by a split rings. The ring is split into two equal segments are insulated from each other and also from the shaft. The coil side AB is always attached to the segment P and likewise CD to Q. It is indicated in Figure 3.9. The brushes B₁ and B₂ touch these segments and are meant to collect the current.

After time "t" secs, the coil would have rotated through an angle of radians in the anti-clockwise direction. The flux then linking with the coil is B lb cos θ.

When θ = 90°, the coil sides are moving at right angles to the flux lines. The flux lines are cut at the maximum rate and the emf induced is maximum. When θ = 180°, the coil sides are again moving parallel to the flux lines (AB and CD have exchanged) and the emf induced is zero once again. When θ = 270°, the coil sides again move at right angles to the flux lines but their position reversed when compared with θ = 90°. Hence the induced emf is maximum in the opposite direction. When θ = 360°, the coil sides once again move parallel to the magnetic field making the induced emf equal to zero. The coil has now come back to the starting point.

If the rotating of the coil is continued, the changes in the emf are again repeated. For the two pole generator shown one complete cycle of changes occurs in one revolution of the coil. The changes in voltage, 'e' with respect to the angle or even time can be plotted.

The torque equation of a DC motor is given by:

T α φ I_a

Here, the flux is directly proportional to the current flowing through the field winding.

φ α I_f

For DC shunt motor, the shunt field current I_sh is constant as long as input voltage is constant. Therefore flux is also constant.

T α I_a

So, for DC shunt motor torque is directly proportional to the armature current. For DC series motor, the series field current is equal to the armature current I_a. Here, the flux φ is proportional to the armature current I_a.

Here T α φ I_a becomes:

T α I_a²

So, for DC series motor, the torque is directly proportional to the square of the armature current. The speed and torque equations are mainly used for analyzing the various characteristics of DC motors.