Question

When load current of a dc series motor is increased, then

• A. \( \text{speed decreases non-linearly and torque increases non-linearly} \)
• B. \( \text{flux increases non-linearly and torque increases linearly} \)
• C. \( \text{speed decreases linearly and torque increases non-linearly} \)
• D. \( \text{speed decreases non-linearly and torque increases linearly} \)

Hint:

For a DC series motor prior to saturation: - Torque vs Current is parabolic ($T \propto I_a^2$) $\rightarrow$ Non-linear! - Speed vs Current is hyperbolic ($N \propto 1/I_a$) $\rightarrow$ Non-linear! Both curves are distinctly non-linear lines.

Answer 1

The Correct Option is A

Concept: In a DC series motor, the field winding is connected in series with the armature winding. Therefore, the armature current ($I_a$), line current ($I_L$), and series field current ($I_{se}$) are all identical: $$I_a = I_{se} = I_L$$ The operating characteristics of a DC motor are governed by two fundamental physical relationships:
• Torque equation: $T \propto \phi \cdot I_a$
• Speed equation: $N \propto \frac{E_b}{\phi}$ Where $\phi$ is the magnetic flux and $E_b$ is the back EMF ($E_b = V - I_a(R_a + R_{se})$).

Step 1: Analyze the torque behavior below magnetic saturation.
Before the ferromagnetic core reaches magnetic saturation, the flux $\phi$ in the machine increases linearly with the current flowing through the series field: $$\phi \propto I_a$$ Substituting this proportional relationship into the core torque equation: $$T \propto (I_a) \cdot I_a \quad \Rightarrow \quad T \propto I_a^2$$ This quadratic relationship means that the torque graph versus load current is a parabola, which represents a non-linear increase.

Step 2: Analyze the speed behavior below magnetic saturation.
Now let us substitute the linear flux relationship ($\phi \propto I_a$) into the speed equation: $$N \propto \frac{V - I_a(R_a + R_{se})}{I_a}$$ Since the voltage drop $I_a(R_a + R_{se})$ is relatively small under standard operating limits compared to the applied line voltage $V$, the numerator remains roughly constant. Therefore, the expression simplifies to an inverse relationship with current: $$N \propto \frac{1}{I_a}$$ The mathematical graph of $N$ versus $I_a$ takes the shape of a rectangular hyperbola, which represents a non-linear decrease.

Step 3: Combine the speed and torque characteristics.
As the load current increases, the speed drops non-linearly (hyperbolically) and the torque rises non-linearly (parabolically). Therefore, option (1) accurately describes both conditions.