A series LCR circuit is connected to an AC source of 220 V, 50 Hz. The circuit contains a resistance $$R = 80$$ $$\Omega$$, an inductor of inductive reactance $$X_L = 70$$ $$\Omega$$, and a capacitor of capacitive reactance $$X_C = 130$$ $$\Omega$$. The power factor of circuit is $$\frac{x}{10}$$. The value of $$x$$ is:

For a series LCR circuit, power factor is:

$$\cos\phi=\frac{R}{Z}$$

where impedance,

$$Z=\sqrt{R^2+(X_L-X_C)^2}$$

Given:

• R=80 Ω
• XL=70 Ω
• XC=130 Ω

Net reactance:

$$X=X_L-X_C=70-130=-60$$

Impedance:

$$Z=\sqrt{80^2+60^2}$$

$$Z=\sqrt{6400+3600}$$

$$Z=\sqrt{10000}=100$$

Therefore,

$$\cos\phi=\frac{80}{100}=0.8$$

Given power factor is written as:

$$\frac{x}{10}$$

So,

$$\frac{x}{10}=0.8$$

x=8