**1. Electrical Characteristics :**

In a d.c. series motor, the armature current and series field current are same and therefore Φ ∝ I

_{a}. The armature torque is equal to
T

_{a}∝ Φ I_{a}or
T

_{a}∝ I_{a}^{2}
This is true till the point of magnetic saturation. When I

_{a}is zero, torque is also zero and when I is small torque is also small. Since is proportional to square of the armature current the curve is parabolic. After saturation Φ is almost constant and for any increase in armature current torque increases linearly i.e., T_{a}∝ I_{a}. The shaft torque is less than armature torque by rotational losses, The curve is shown in Fig.**2. Speed and Armature Current Characteristics :**

For a d.c. series motor

E

_{b}= V - I_{a}(R_{a}+R_{se}) and E_{b}= ΦN x ZP/60A
or E

_{b}= K_{a}ΦN
Where K

_{a}= ZP/60A
Substituting the values,

K

_{a}ΦN = V - I_{a}(R_{a}+R_{se})
or N = V - I

_{a}(R_{a}+R_{se})/ K_{a}Φ
or N = V/ K

_{a}Φ - I_{a}(R_{a}+R_{se})/ K_{a}Φ
But for a dc series motor, Φ ∝ I

_{a}, therefore
N = V/ K

_{b}I_{a}– (R_{a}+R_{se})/ K_{b}; where K_{b}= I_{a}/K_{a}
From the above equation, it is seen that neglecting armature reaction and with saturation the speed-current characteristics of d.c. series motor is hyperbolic as shown in Fig. When l

_{a}increases Φ also increase but due to demagnetisation effect of armature reaction and saturation, the air gap flux tends to remain constant and for constant flux Φ the term V/K_{a}Φ remains constant and the term l_{a}[R_{a}+ R_{se}]/ K_{a}Φ increases with armature current linearly.
Thus for larger values of armature current, the curve takes a straight line path. At no load the armature current is small and so the armature drop I

_{a}(R_{a}+ R_{se}) and can be considered negligible as compared to terminal voltage.
Then N = V/K

_{a}Φ, But Φ ∝ I_{a},therefore N = V/K_{b}l_{a}
On no load when the armature current tends to zero, the speed tends to infinity. Therefore, the no load speed of d.c. series motor is dangerously high and due to this reason the d.c. series motor must not be started without a load. The curve is shown in Fig.

**3. Speed-Torque Characteristics :**

The torque of d.c. series motor is proportional to the square of armature current i.e.

I

_{a}^{2}∝ T_{a}or
I

_{a}∝ √T_{a}
Neglecting the armature reaction and saturation

N = V/K

_{b}I_{a}or N = V/K_{b}√T_{a}
Squaring both sides

N

^{2}= V^{2}/K_{c}T_{a}
or T

_{a}= V^{2}/N^{2}K_{c}
or T

_{a}∝ 1/N^{2}
The speed torque characteristics is also hyperbolic in nature. But with saturation and armature reaction large torque requires larger currents and these large currents tends to make the air gap flux constant and the effect is T

_{a}∝ I_{a}instead of T_{a}∝ I_{a}^{2}. The curves approaches straight line as shown in Fig.
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