Question

An ac voltage $V = 280 \sin(100\pi t)$ volt is connected across a series LCR circuit in which $R = 400 \Omega$, $L = 5/\pi \text{ H}$ and $C = 50/\pi \mu\text{F}$. Taking $\sqrt{2} = 1.4$, calculate power factor of the circuit.

Hint:

The absolute value of any physically valid power factor inherently must always rigidly fall strictly between 0 and 1 inclusive; getting a number larger than 1 means a fraction was flipped upside down.
While merely providing the numerical value usually earns full marks, additionally specifying whether the factor is 'leading' or 'lagging' vividly demonstrates a supreme mastery of the core AC concepts.

Answer 1

Concept:
• In any complex alternating current setup, power is not merely the simple product of total voltage and total current; it is heavily dictated by relative phase.

• The power factor fundamentally acts as a strict efficiency multiplier, geometrically quantifying exactly what fraction of the total apparent power is actively converted into highly useful true real power.

• Mathematically, it is elegantly defined precisely as the exact cosine of the phase angle separating the total voltage waveform and the operational current waveform.

Step 1: Understand the Power Factor Formula
From the standard foundational impedance triangle specifically constructed for series AC circuits, the base strictly represents the true Resistance $R$, while the long hypotenuse rigidly represents the total Impedance $Z$.
The angle securely nestled exactly between them is the fundamental phase angle $\phi$.
Therefore, the power factor, denoted officially as $\cos\phi$, can be extracted geometrically straight from the lengths of the triangle's rigid sides:
\[ \text{Power Factor } (\cos\phi) = \frac{\text{Adjacent Side}}{\text{Hypotenuse}} = \frac{R}{Z} \]
This elegant fractional formula effectively serves as the absolute fastest and most reliable method to precisely determine the required factor.

Step 2: Extract Required Values
From the original problem statement text, the pure ohmic resistance heavily installed in the circuit is rigidly given as:
\[ R = 400 \Omega \]
From our own meticulous, detailed calculation securely performed back in sub-question (I), we robustly established the total operational circuit impedance:
\[ Z = 500 \Omega \]
We also previously calculated $X_L = 500 \Omega$ and $X_C = 200 \Omega$, establishing firmly that $X_L > X_C$, which fundamentally means the circuit is definitively inductive overall.

Step 3: Calculate the Power Factor
We systematically substitute our definitively verified resistance and impedance values directly into the established geometric power factor ratio:
\[ \cos\phi = \frac{R}{Z} \]
\[ \cos\phi = \frac{400}{500} \]
Simplify the numerical fraction completely by canceling the clearly obvious trailing hundreds:
\[ \cos\phi = \frac{4}{5} \]
Convert the remaining simple fraction smoothly into a final readable decimal format:
\[ \cos\phi = 0.8 \]
Because the overall inductive reactance heavily outweighs the overall capacitive reactance in this specific setup, this circuit operates definitively with a "lagging" power factor of precisely 0.8, meaning the current waveform trails behind the voltage waveform.