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

In an AC series LCR circuit, the power factor is defined as the cosine of the phase angle $\phi$ between the voltage and the current. It is mathematically expressed as the ratio of the resistance $R$ to the total impedance $Z$ of the circuit:
$$ \text{Power Factor} = \cos \phi = \frac{R}{Z} $$

The total impedance $Z$ of a series LCR circuit is given by the formula:
$$ Z = \sqrt{R^2 + (X_L - X_C)^2} $$
where $X_L$ is the inductive reactance and $X_C$ is the capacitive reactance.

Given values:
$R = 80 \text{ } \Omega$
$X_L = 20.0 \text{ } \Omega$
$X_C = 80.0 \text{ } \Omega$

First, calculate the net reactance:
$$ X = X_L - X_C = 20.0 - 80.0 = -60.0 \text{ } \Omega $$
(The negative sign indicates the circuit is capacitive, but for $Z$ we only need the magnitude squared).

Now, calculate the impedance $Z$:
$$ Z = \sqrt{80^2 + (-60)^2} $$
$$ Z = \sqrt{6400 + 3600} $$
$$ Z = \sqrt{10000} = 100 \text{ } \Omega $$

Finally, calculate the power factor:
$$ \cos \phi = \frac{R}{Z} = \frac{80}{100} = 0.8 $$
Thus, the power factor of the circuit is $0.8$.