The correct answer is: A. Proportional to the armature current
A DC series motor is a type of electric motor that uses a series connection between the armature and field windings. This means that the current flowing through the armature also flows through the field windings. The strength of the magnetic field produced by the field windings is proportional to the current flowing through them. The torque produced by the motor is proportional to the strength of the magnetic field. Therefore, the speed of a DC series motor is proportional to the armature current.
Option B is incorrect because the speed of a DC series motor is not proportional to the square of the armature current. The square of the armature current only affects the torque produced by the motor, not the speed.
Option C is incorrect because the speed of a DC series motor is not proportional to the field current. The field current only affects the strength of the magnetic field, not the speed.
Option D is incorrect because the speed of a DC series motor is not inversely proportional to the armature current. The armature current only affects the torque produced by the motor, not the speed.