Q:

How does current vary with time through a motor?

Say an electric motor is connected via a switch to a battery. What would a graph to show how current varies with time look like (from when the switch is first closed).
Apparently the graph would look like this:

I understand that when the switch is closed, initially the current increases rapidly because the motor just acts as a resistor. However, as the motor spins the flux linkage through the motor's coil changes.

When parallel to field lines the rate of change of flux linkage is at a max + when perp. the rate of change is zero. Hence, shouldn't the back emf (and therefore current) vary as the motor spins causing the current to vary continuously? - not tend towards a single value as suggested by the first graph
Any explanation would be greatly appreciated but I'm only doing A level physics so please keep as simple as possible!

A:

The current through the motor will vary continuously and you should be able to see a current ripple on top of whatever slower changes of the current may be due to speeding up or load changes.
The picture though will be more complicated than suggested by the EMF action alone because most motors have multiple windings and the currents in the windings overlap and switch on and off. 
Still, you should be able to see that the frequency of the ripples is a multiple of RPM. 

