Read the Characteristics of Series wound DC Generator

1. Torque and armature current (T_a/I_a characteristic). It is known as an electrical characteristic. 
2. Speed and armature current (N/I_a characteristic)
3. Speed and torque (N/T_a characteristic) It is also known as mechanical characteristic. It can be found from (1) and (2) above.

While discussing dc motor characteristics, the following two relations should always be kept in mind :

T_a ∝ ΦI_a  and 

N ∝ E_b/Φ

T_a/I_a Characteristic (Electrical)

We have seen that in series motor T_a ∝ ΦI_a. In a series motor, as field windings also carry the armature current, Φ ∝ I_a up to the point of magnetic saturation. Hence, before saturation,

T_a ∝ ΦI_a and ∴ T_a ∝ I_a²

At light loads, I_a and hence Φ is small. But as I_a increases, T_a increases as the square of

the current.  Hence, T_a/I_a curve is a parabola as shown in the figure. 

After saturation, Φ is almost independent of Ia hence T_a ∝ I_a only.  So the characteristic becomes a straight line. The shaft torque T_sh is less than armature torque due to stray losses. It is shown dotted in the figure. 

 So we conclude that (prior to magnetic saturation) on heavy loads, a series motor exerts a torque proportional to the square of armature current. 

Hence, in cases where huge starting torque is required for accelerating heavy masses quickly as in hoists and electric trains, etc., series motors are used.

N/I_a Characteristic

Variations of speed can be deduced from the formula :

                                                                        N ∝ E_b/Φ

Change in E_b, for various load currents is small and hence may be neglected for the time being. With increased I_a, Φ also increases. Hence, speed varies inversely as armature current as shown in the figure below.

When the load is heavy, I_a is large. Hence, speed is low (this decreases E_b and allows more armature current to flow). 

But when load current and hence I_a falls to a small value, speed becomes dangerously high. Hence, a series motor should never be started without some mechanical (not belt-driven) load on it otherwise it may develop excessive speed and get damaged due to heavy centrifugal forces so produced. 

It should be noted that the series motor is a variable speed motor.

N/T_a Characteristic (Mechanical)

It is found from above that when speed is high, torque is low and vice-versa.

It is clear that series motor develops high torque at low speed and vice-versa. 

It is because an increase in torque requires an increase in armature current, which is also the field current. The result is that flux is strengthened and hence the speed drops