Question

If the locus of minimum armature currents of V-curves of synchronous motor is a straight line, then the slope of the line is

• A. \( \text{positive} \)
• B. \( \text{negative} \)
• C. \( \text{zero} \)
• D. \( \text{infinite} \)

Hint:

The locus of minimum armature currents is also called the Unity Power Factor Locus. As load increases, both $I_a$ and $I_f$ must increase to maintain $\cos\theta = 1$. Since both variables grow together, the slope is always positive.

Answer 1

The Correct Option is A

Concept: The V-curves of a synchronous motor plot the armature current ($I_a$) as a function of the field excitation current ($I_f$) for various constant mechanical load conditions.
• For a given load, as $I_f$ is varied, $I_a$ reaches a distinct minimum point.
• This minimum armature current point corresponds exactly to operation at unity power factor ($\cos\theta = 1$).
• Connecting the minimum points of the V-curves for different loads forms a curve known as the compounding curve or the locus of minimum armature current.

Step 1: Analyze minimum armature current across different mechanical loads.
When the mechanical shaft load on a synchronous motor is increased: 1. The real active power ($P$) demanded by the motor increases. 2. The real power expression at unity power factor is: $$P = \sqrt{3} \cdot V_L \cdot I_a \cdot \cos\theta \quad \Rightarrow \quad P = \sqrt{3} \cdot V_L \cdot I_a \quad (\text{since } \cos\theta = 1)$$ Thus, as load power ($P$) increases, the minimum armature current ($I_a$) required must also increase. This means the points on the locus move upward on the y-axis ($I_a$).

Step 2: Determine the shift in the required field current \( I_f \).
To maintain a unity power factor under an increased mechanical load, the motor requires additional excitation to overcome the demagnetising effect of the increased armature reaction. Therefore, the field current ($I_f$) must be increased to achieve the minimum armature current condition at a higher load. This causes the minimum points to shift to the right on the x-axis ($I_f$).

Step 3: Evaluate the slope of the resulting locus line.
Since an increase in the vertical coordinate ($I_a$) corresponds to an increase in the horizontal coordinate ($I_f$), both $\Delta I_a > 0$ and $\Delta I_f > 0$. The slope ($m$) of this straight line locus is: $$m = \frac{\Delta I_a}{\Delta I_f} = \frac{\text{Positive}}{\text{Positive}} = \text{Positive}$$ Thus, the locus line slants upward to the right, which indicates a positive slope. This matches option (1).