Which DC motor has the least percentage increase in Input current for the same percentage increase in torque?

Concept: Torque current relation in DC machines:

T = K_a ϕ Ia

i.e., it is proportional to the product of flux and armature current.

In DC series matrix, field current and armature current are equal. Therefore,

T ∝ Ia² 

\(\frac{\delta(T)}{\delta T} = 2 \frac{\delta (Ia)}{\delta Ia}\)

∴ \(\frac{\delta T}{\delta Ia} = 2\)

or \(\frac{\delta Ia}{\delta T} = \frac{1}{2}\)

In DC shunt motor, field current is constant.

∴ T ∝ Ia

∴ \(\frac{\delta T}{\delta Ia}=1\)

In compound motor, it will initially behave as series motor and when saturation sets in, it will behave like shut motor. Hence, current to torque ratio will lie b/w 0.5 to 1.

Hence, the corresponding ratio will be minimum in DC series motor.

Therefore, correct option is (1).