A DC motor works in the same principle as an AC motor, with a mechanism to allow it to rotate even if the supply is not alternating. After all, simple motors work because the the direction of current reverses (alternates) every half cycle, rendering the armature to rotate. Now why does the current have to alternate to allow for rotation? The left hand motor rule will show you their relation. I'll elaborate on that later.
Of course, as mentioned earlier, DC motors will also rotate only when the current direction changes, although the supply is unidirectional. The commutator is the mechanical device that allows the current in a conductor to reverse its direction.
To understand intuitively, read on:
AC means the direction of current is reversing every half cycle, about 100 times a second if it is a 50 Hz supply. DC just means the current is maintaining the same direction. As a principle, the current direction must change, in order for the motor to actually rotate. Why? Because the direction of current and the direction of motion of conductor are related - Take left hand rule for motors.
Looking at the hand picture above, let’s work out our motion of conductor. Look at the diagram above. There are two conductors, each connected to its one slip ring. Take one conductor for reference and let's say the red conductor, which is horizontal on the diagram, has reached the top(vertical). This is indicated by the thumb in picture on the top left.
The field provided by permanent magnets is not going to change its direction(north to south as indicated by your index finger)
So this is the reason that a conductor must have currents that change direction at the instant its upward motion is complete. If the current direction didn't change, your red conductor would have just stayed on top! At that instant, in AC, the current cycle reverses, so you have your conductor moving down. This repeats and you have a rotation. Motor works!
But now in a DC, you have current that is in the same direction even after your conductor has moved up by 180 degrees. So that's why they place a commutator. A split ring that touches the opposite polarity of the supply at the exact moment when its direction needs to change. So in actuality, the red conductor will receive current whose direction is different because the ring will switch from a BRUSH of one polarity to a brush of OPPOSITE polarity.
Look at the diagram below and look how one conductor gets both positive and negative currents in a 360 degree cycle:
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