Back EMF in a Simple DC Motor

Last Update: 3 December 2025

This is part of the HSC Physics syllabus under the topic Applications of the Motor Effect.

HSC Physics Syllabus

investigate the operation of a simple DC motor to analyse:

– the functions of its components

– production of a torque `\tau = nIAB_(_|_) sin \theta`

– effects of back emf (ACSPH108)

What is Back EMF?

This video investigates the operation of a simple DC motor to analyse the effects of back emf.

How Does a DC Motor Produce Back EMF?

As the armature rotates within the external magnetic field, the amount of magnetic flux passing through the coil changes. This is because whilst the field strength is constant, the angle between the surface’s normal (area vector) and the flux lines changes. This causes the coil to experience an induced emf according to Faraday's Law. The magnitude of this emf is directly proportional to the rate of flux change which is determined by the rotational speed of the armature.

$$\varepsilon = -\frac{\Delta\Phi}{\Delta t}$$

Lenz's Law states that the current associated with this induced emf will produce a magnetic field that opposes the external magnetic field. Thus, the induced emf’s direction is always opposite to the battery's supply voltage This is why this emf is called back emf.

As such, the net voltage of a DC motor can be described by:

$$V_{net} = V_{supply} - \varepsilon_{back}$$

As the coil's angular velocity increases, the rate of change in flux increases, which increases the back emf as `\varepsilon = -N (\Delta \Phi)/(\Delta t)`. This means that the net voltage will decrease, causing the current to also decrease, according to Ohm's Law (`V = IR`). Since `\tau = nIAB \sin \theta`, a decreasing current causes torque to decrease.

Graph shows the changes in a motor's rotational speed and current with time from the beginning when it's switched on to when it reaches steady state.

Back EMF, Torque and Angular Speed

When a motor is operating without a load (no counter torque), the back emf will continue to decrease until it equals the supply voltage, at which point the net voltage and current will both equal zero. At this instance, the coil would have reached its maximum rotational velocity since torque will be zero (no further rotational acceleration).

In majority of cases when a motor is required to rotate against a load (e.g. turning against weight), some amount of torque is required by the armature to turn against the load (and its counter torque), so the back emf will only decrease until a non-zero value. This allows there to be a non-zero value of net voltage and current in the armature to produce the aforementioned torque. The rotational speed at this steady state in the presence of a load is slower than without one. From the perspective of energy, some amount of armature's kinetic energy is required to do work against the weight of the load.

Stalling Torque

The stalling torque of a motor is the torque it produces when its armature is fixed or made to stall (imagine physically holding the coils to prevent its rotation). When there's no rotation, the coils do not experience any changes in magnetic flux, and thus there would be no induced emf (back emf). In the absence of back emf, the net voltage in the coils would equal to the supply voltage and therefore be at its maximum.

By Ohm's law, when the current would also be at its maximum magnitude, resulting in the greatest torque. Therefore, the stalling torque of a motor is also the greatest torque the motor can generate.

Operational Speed

When the armature is again allowed to rotate against a load or weight, its angular speed will increase until its torque decreases (due to increasing back emf) to equal the counter torque of the load. When this occurs, the net torque of the armature is zero, and as a result, the armature will continue to rotate at constant angular speed - this is the operational speed of the motor for a given load.

Feature	No Mechanical Load	Operating Against a Load
Required Torque	Approaching Zero. Only needs to overcome minor friction and air resistance.	Non-zero and Constant. Must be equal to the load's counter-torque
Rotational Speed	Maximum possible speed	Lower, constant working speed
Back EM	Almost Equal to Supply Voltage	Less than Supply Voltage
Net Voltage	Approaching Zero	Non-zero
Armature Current	Approaching Zero. Limited only to maintaining angular speed against friction	Non-zero, steady working current. Proportional to the required torque.

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