**Torque in DC Motor with Formula**

Turning or twisting moment of a force about the axis is called torque in DC Motor. The unit of torque is Newton-metre. It is measured by the product of the force and the radius at which this force acts. Consider a pulley of radius R metres acted upon by a force on the circumference with F Newtons to cause the pulley to rotate at a speed of N rev. per second as shown in Figure.

Then as per the definition torque = F x r Newton-metre

Work done by this force in one revolution (Force x distance) = F x 2πr Joules

If N is the speed in r.p.s. then Work done/second = F x 2πr x N = (F.r) x 2πN

2πN is called the angular velocity and is represented by ‘ω'. It is measured in radians/second.

Power developed (P

_{m}) = Two Joules/second or watts
The torque developed at the shaft in DC Motor is always lesser than the torque developed at the armature because of losses.

The formula of torque developed by the armature = T

_{a}.2πN watts
The mechanical power developed in the armature =, E

_{b}l_{a}watts
But the mechanical power developed in the armature is equal to the armature torque i.e.,

E

_{b}l_{a}= T_{a}.2πN/60 where N is in r.p.m. and
N

_{rpm}/60 = N_{rps}
But E

_{b}= ZΦNP/60A
Substituting for E

_{b}
ZΦNP/60A x l

_{a}= T_{a}.2πN/60 or
T

_{a}= ZΦ l_{a}/2π x P/A
= 1/2π Φ Zl

_{a}x P/A = 0.159 Φ Zl_{a}[P/A] N-m
= 0.159/9.81 x Φ Zl

_{a}x P/A kg-m
= 0.0162 Φ Zl

_{a}x [P/A] kg-m
This is also known as the gross torque. The torque developed at the shaft is the useful torque and is known as shaft torque (T

_{sh}).
BHP (metric) = Tshaft x 2πN/735.5 where N is in r.p.s.

Or T

_{sh}= (BHP)_{m}x 735.5/2πN
The motor output is given by T

_{sh}x 2πN watt
T

_{sh}= output in watts/2πN , where N is in r.p.s.
= 60/2π x Output/N = 9.55 x Output/N N-m

Where N is in r.p.m

The difference between armature torque and shaft torque is the torque lost.

Torque lost = 0.159 x iron and frictional loss/N N-m

**Case 1 Shunt Motor:**

The flux is constant in shunt motor and is dependent on the terminal voltage and shunt field resistance. Therefore,

T

_{a}∝ I_{a}

**Case 2 Series Motor:**

In the case of a series motor the armature and series field currents are same i.e., I

_{a}= I_{se}= I_{L}, therefore
Φ ∝ I

_{a}and T_{a}∝ I_{a}^{2}
and also the torque in dc motor formula,

T

_{a}= E_{b}I_{a}/2πN = 0.159 E_{b}I_{a}/N N-m
= 0.0162 E

_{b}I_{a}/N kg-m
= 0.0162/N x Power developed in armature Kg-m

**Difference between Developed Torque, Useful Torque and Lost Torque**

The whole of the armature torque is not available for doing any useful work because a certain percentage of it is required for supplying iron and frictional losses in the motor. Therefore, the net torque available is the shaft torque. The difference between the armature torque and shaft torque is the torque lost. The torque developed in the armature is called gross torque and the torque which is available at the shaft which is much less than the torque developed in the armature is called shaft torque or useful torque.

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