Question

A LCR series circuit driven with $E_{\text{rms}} = 90 \text{ V}$ at frequency $f_d = 30 \text{ Hz}$ has resistance $R = 80 \text{ } \Omega$, an inductance with inductive reactance $X_L = 20.0 \text{ } \Omega$ and capacitance with capacitive reactance $X_C = 80.0 \text{ } \Omega$. The power factor of the circuit is ________.

• A. 0.8
• B. 0.64
• C. 0.9
• D. 0.5

Hint:

Power factor is defined as R/Z. Calculate Z using the formula for series LCR circuits: Z = sqrt(R^2 + (X_L - X_C)^2).

Answer 1

The Correct Option is A

The power factor represents how effectively real power is delivered in an AC circuit and is given by $\cos \phi$. In a series combination of Resistance ($R$), Inductance ($L$), and Capacitance ($C$), the impedance $Z$ is the vector sum of these components' resistances.

Step 1: Identify given values.
$R = 80 \text{ } \Omega$, $X_L = 20 \text{ } \Omega$, $X_C = 80 \text{ } \Omega$. Note that the driving frequency and RMS voltage are provided but are not necessary for calculating the power factor once the reactances are known.

Step 2: Use the impedance triangle relationship.
The impedance $Z$ is calculated as:
$$ Z = \sqrt{R^2 + (X_C - X_L)^2} $$
$$ Z = \sqrt{80^2 + (80 - 20)^2} $$
$$ Z = \sqrt{80^2 + 60^2} $$
$$ Z = \sqrt{6400 + 3600} = \sqrt{10000} = 100 \text{ } \Omega $$

Step 3: Compute the power factor $\cos \phi$.
$$ \cos \phi = \frac{R}{Z} $$
$$ \cos \phi = \frac{80}{100} = 0.8 $$
The result is $0.8$, which corresponds to option 1.