# The back emf in a d.c motor

As soon as the coil in the motor starts rotating, a back e.m.f. will be induced in it due to the flux that it cuts, and this will tend to reduce the current through it.
Let the supply e.m.f. be E, the back e.m.f. be e, the resistance of the coil R and the current through the coil I. Then

I = [E – e]/R since e is proportional to the angular speed (w) the greater w the smaller I.

For practical motors with E = 100 V, the back e.m.f. may be great as 95 V!

The resistance of the coil R is usually small (less than 1W) and therefore when it is at rest a large current may flow through it. When the coil speeds up this is reduced, since the back e.m.f. is proportional to the rate of rotation of the coil. The starting current can be as large as 1000 A, and a protective resistor must be incorporated in series with the coil during starting. This can be removed when the motor is running. This is why a d.c. motor that is running should never be stopped with the supply connected. If this is done the back e.m.f. will fall to zero, the current will become very large and the coil may burn out.

The diagram shows an electric car run by a 60 V battery going over a hill. It should help to explain what happens when the motor runs at different speeds. As the car climbs the hill AB on the left the motor is running slowly, the back e.m.f. is therefore low (say 5 V) and this means that a large current flows through the motor, giving a large torque. Chemical energy from the battery is converted to potential energy of the car.

The car now goes up section BC. The slope is much shallower, the motor speeds up and so the back e.m.f. rises to say 59 V. The current through the motor is therefore low.

The car now descends the section CD. The speed increases so that the back e.m.f. rises to 60 V, and energy is supplied to just overcome friction. Further down the hill, however, the car is moving faster and the back e.m.f. is greater than 60 V and so the motor acts as a dynamo, storing up energy in the battery. The current flowing produces a torque which tends to oppose the motion and so acts as a brake.

As long as electromagnets are used for the field, a d.c. motor will run on a.c., although very inefficiently owing to the large self-inductance of its coils.