What is the back EMF in a DC motor and derive the EMF equation of a DC machine?

Back EMF of a DC Motor

In a DC motor, as the armature rotates underneath the influence of the provided voltage, it passes through the magnetic field created by the field magnets. According to Faraday’s Law of Electromagnetic Induction, cutting the magnetic field with armature conductors produces an electromotive force (EMF) in those conductors. This produced EMF opposes the applied voltage in one direction, hence the name back EMF (Eb) or counter EMF.

Consider the motor push back against the applied voltage through the generation of its own voltage. The higher the motor speed, the more powerful the magnetic field and back EMF.

Back EMF Equation

The equation for back EMF in a DC machine is equivalent to the equation for generated EMF in a DC generator. This is because the same basic mechanism is at work in both conditions. Here is the derivation:

Flux per pole (Φ): The flux per pole (Φ) indicates the strength of the magnetic field created by each motor/generator pole. It is measured in Weber units (Wb).

Number of poles (P): The total number of magnetic poles (north & south pairs) in the equipment.

Number of conductors (Z): The number of conductors (Z) refers to the total number of individual conductors in armature winding.

Speed (N): The armature’s rotational speed, measured in RPM.

Number of parallel paths (A): The number of parallel routes (A) indicates how the armature winding is divided.

EMF Equation

Eb = (PΦNZ) / (60A)


Eb is back EMF in volts (V)

P is number of poles.

Φ refers to flux per pole in Weber (Wb)

N is speed in RPM.

Z represents the total number of conductors

A represents the number of parallel paths

Understanding the Equation

  • PΦ is the total magnetic flux associated with a single conductor. As the conductor spins, it cuts this flux, causing an EMF.
  • NZ represents the frequency at which each conductor cuts the flux.
  • 1/60 converts RPM to revolutions/second.
  • A represents the number of parallel paths, each providing an EMF in series.

As a result, the equation calculates the total back EMF generated by the armature winding.

Back EMF is a key element in DC motor performance, affecting current flow and speed control. Understanding the back EMF equation assists in predicting motor behavior and designing effective control systems.