EMF equation and Torque equation of a DC machine : Explained with Derivations

DC machines play a vital role in electromechanical energy conversion. Whether operating as a generator or motor, two key performance parameters define their operation — EMF (Electromotive Force) and Torque. This article breaks down the EMF equation of a DC generator and the torque equation of a DC motor, with clear derivations and practical insights.

emf equation of dc generator and torque equation of dc motor

EMF equation of a DC generator

To derive the EMF equation, consider a DC generator with the following parameters:

P = number of field poles
Ø = flux produced per pole in Wb (weber)
Z = total no. of armature conductors
A = no. of parallel paths in armature
N = rotational speed of armature in revolutions per min. (rpm)

Step by step derivation

Average EMF generated per conductor:
EMF per conductor = dΦ/dt (volts) ....... eq.1
Flux cut by one conductor in one revolution:
dΦ = P × Φ (weber)
Time for one revolution:
dt = 60 / N
Substituting value of dΦ and dt in eq.1, EMF generated per conductor:
e = PΦN / 60
This equation gives the EMF induced in a single armature conductor. Since the armature conductors are arranged in series within each parallel path, the total generated EMF across the generator’s terminals is:
Total generated EMF:
Eg = PΦNZ / 60A

For simplex lap winding, number of parallel paths is equal to the number of poles (i.e. A=P),
Therefore, for simplex lap wound dc generator, Eg = PΦNZ / 60P

For simplex wave winding, number of parallel paths is equal to 2 (i.e P=2),
Therefore, for simplex wave wound dc generator, Eg = PΦNZ / 120

Example Problem

Q: A 4-pole DC generator has 500 conductors, flux per pole of 0.02 Wb, lap winding, and runs at 1500 RPM. Calculate the generated EMF.

Solution:

Given:
P = 4, Z = 500, Φ = 0.02, N = 1500, A = P = 4

Eg = (P × Φ × N × Z) / (60 × A)
   = (4 × 0.02 × 1500 × 500) / (60 × 4)
   = 250 Volts

Torque equation of a DC motor

When the armature conductors of a DC motor carry current and interact with the magnetic field produced by the stator, a mechanical torque is generated. This torque results from the electromagnetic force acting on the conductors, and is given by the product of the force and the radius at which it acts:
T = F × r.

T = Torque (N·m)
F = Force (N)
r = Radius of armature (m)
N = Speed in RPM
ω = Angular velocity = 2πN / 60 (rad/s)

Torque T = F × r (N·m)
Work done by this force in once revolution = Force × distance = F × 2πr (where, 2πr = circumference of the armature)
Net power developed in the armature = Work done / time
= (force × circumference × no. of revolutions) / time
= (F × 2πr × N) / 60 (Joules per second) .... eq. 2.1
But, F × r = T and 2πN/60 = angular velocity ω in radians per second. Putting these in the above equation 2.1
Net power developed in the armature = P = T × ω (Joules per second)

Armature torque (Ta)

The power developed in the armature can be expressed as:
Pa = Ta × ω = Ta × (2πN/60)
This mechanical power is derived from the electrical input power.
Therefore, mechanical power = electrical power:
Ta × (2πN/60) = Eb × Ia
Here, Eb is the back EMF (electromotive force) developed in the armature, which opposes the applied voltage and regulates the armature current.
We know: Eb = (PΦNZ) / (60A)
Substituting this in, we get:
Ta × (2πN/60) = [(PΦNZ) / (60A)] × Ia
Rearranging the equation, the armature torque becomes:
Ta = (PZ / 2πA) × Φ × Ia (in N·m)

The term (PZ / 2πA) is constant for a given DC machine. Thus, the armature torque is directly proportional to the product of the flux and the armature current:
Ta ∝ Φ × Ia

Shaft Torque (Tsh)

Due to iron and friction losses in a DC machine, not all of the developed armature torque is available at the shaft. A portion of torque is lost internally, so the shaft torque is always less than the armature torque.

Shaft torque of a DC motor is given by:
Tsh = Output Power (W) / (2πN / 60)
(where N is the motor speed in RPM)

What is Back EMF and Why Is It Important?

Back EMF, or back electromotive force (denoted as Eb), is the voltage induced in the armature windings of a DC motor when it rotates in the presence of a magnetic field. According to Faraday’s Law of Electromagnetic Induction, any conductor moving through a magnetic field will experience an induced EMF.

In a DC motor, this induced EMF acts in the opposite direction to the applied supply voltage. That’s why it’s called "back" EMF. As the motor picks up speed, the back EMF increases, which in turn reduces the net voltage across the armature and limits the current flowing through it.

Back EMF plays a crucial role in regulating the motor’s operation. It:

Prevents the motor from drawing excessive current during steady-state operation.
Provides a natural feedback mechanism for motor speed control.
Ensures efficient energy conversion from electrical to mechanical form.

The magnitude of back EMF is given by:
Eb = (PΦNZ) / (60A)

Is Back EMF the Same as Generated EMF?

Yes, back EMF in motors and generated EMF in generators arise from the same principle of electromagnetic induction. The difference lies in the direction of energy conversion. In a generator, the induced EMF is the output. In a motor, it's an internal feedback mechanism that helps regulate current flow.