In this case, the armature is connected in series with the field coils. Thus
I_a  = I_f = I

NB. At starting, the back emf is zero (because the motor is initially stationary) and hence the current is very large (limited only by the field and armature resistances). Hence the starting torque will be extremely large and if no load is applied (so that the torque requirement is small) the motor will accelerate rapidly to dangerously high speeds. A series connected motor must be started with a mechanical load attached to prevent the speed at start up rising to damaging speeds.

Note that saturation of the magnetic material in the motor can occur at high currents and then the flux generated by the field coils becomes independent of current. This causes the characteristics to become similar to that of a shunt wound motor that also has an approximately constant field flux.

The series motor gives a high starting torque (proportional to 1/I2 ), 2.5 to 3 times the full load (rated) torque) but it falls rapidly as the speed increases. Applications: heavy duty, with intermittent starting under conditions where a mechanical load is always present - e.g. traction, cranes, hoists. etc

The speed of a series wound motor may be controlled by the addition of a resistor in parallel with the field coil. This diverts
some of the current away from the field coil, reducing the field strength and hence the back emf at a given speed. The additional resistor is called a 'diverter'. The speed must increase to compensate this reduction in back emf. . Hence a reduction in the diverter resistance reduces the field strength and increases the speed of the motor.