Question

If a circuit has unity power factor, then reactive power is:

• A. Maximum
• B. Zero
• C. Minimum
• D. Inductive

Hint:

Using the trigonometric identity \(\sin^2\theta + \cos^2\theta = 1\): If \(\cos\theta = 1\), then \(\sin\theta = \sqrt{1 - 1^2} = 0\). Consequently, Reactive Power \(Q \propto \sin\theta = 0\).

Answer 1

The Correct Option is B

Concept: In alternating current (AC) circuit engineering, total power is modeled via the complex power triangle framework. The relationship between Apparent Power (\(S\)), Active Power (\(P\)), and Reactive Power (\(Q\)) is governed by: \[ S = P + jQ \] The components are evaluated using the phase displacement angle \(\theta\) between the voltage waveform and current waveform:
• Active Power (Real Power): \(P = V I \cos\theta\) (measured in Watts, W)
• Reactive Power (Imaginary Power): \(Q = V I \sin\theta\) (measured in Volt-Amperes Reactive, VAR) The expression \(\cos\theta\) is known as the Power Factor (\(PF\)) of the circuit.

Step 1: Evaluating the phase angle from the given unity power factor constraint.
The question states that the system operates at a "unity power factor". Mathematically, this means: \[ PF = \cos\theta = 1 \] To find the phase angle \(\theta\), we take the inverse cosine: \[ \theta = \cos^{-1}(1) = 0^\circ \] A phase angle of \(0^\circ\) implies that the voltage and current waveforms cross zero and reach their peak amplitudes in perfect synchronization (in-phase behavior).

Step 2: Calculating the corresponding reactive power \(Q\).
Now, substitute our phase angle \(\theta = 0^\circ\) into the formula defining reactive power: \[ Q = V I \sin(0^\circ) \] Since the trigonometric value of \(\sin(0^\circ)\) is identically zero: \[ Q = V I \times 0 = 0\text{ VAR} \] Hence, when the power factor becomes unity, the circuit behaves purely resistively and stores no net steady-state reactive energy, reducing the reactive power exactly to zero.