Question

In an a.c. circuit containing L, C and R in series, the ratio of apparent power to the true power is
(Z= impedance of the circuit and R= resistance )

• A. \(RZ\)
• B. \(cosφ\)
• C. \(cotφ\)
• D. \(\frac{Z}{R}\)

Hint:

True power = V I cos phi, apparent power = V I, and cos phi = R/Z.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
Apparent power is \(P_{app}=V_{rms}I_{rms}\). True power is \(P_{true}=V_{rms}I_{rms}\cos\phi\).

Step 2: Take the ratio:
\(\dfrac{P_{app}}{P_{true}}=\dfrac{1}{\cos\phi}\). For a series LCR circuit \(\cos\phi=\dfrac RZ\).

Step 3: Result:
\(\dfrac{P_{app}}{P_{true}}=\dfrac ZR\). Option D.

Step 4: Why the other options are wrong.
\(RZ\) is not even dimensionless. \(\cos\phi\) is the inverse of the ratio. \(\cot\phi\) equals \(X/R\), which is a different quantity.

Final Answer:
The ratio is Z / R. \[ \boxed{\text{(D) }\dfrac ZR} \]