A series LCR circuit contains an inductor $L = 0.1 ext{ H}$, a capacitor $C = 10\,\mu ext{F}$, and a resistor $R = 20\,\Omega$ connected across an alternating voltage source of $V = 220 ext{ V}$ and frequency $f = 50 ext{ Hz}$.
Calculate: Inductive Reactance ($XL$) Capacitive Reactance ($XC$) Total Impedance ($Z$) of the circuit RMS current ($I{rms}$) flowing through the circuit Power factor ($\cos\phi$) of the circuit Resonant frequency ($fr$) of the given circuit.

Q: A series LCR circuit contains an inductor $L = 0.1\text{ H}$, a capacitor $C = 10\,\mu\text{F}$, and a resistor $R = 20\,\Omega$ connected across an alternating voltage source of $V = 220\text{ V}$ and frequency $f = 50\text{ Hz}$. \nCalculate: Inductive Reactance ($XL$) Capacitive Reactance ($XC$) Total Impedance ($Z$) of the circuit RMS current ($I{rms}$) flowing through the circuit Power factor ($\cos\phi$) of the circuit Resonant frequency ($fr$) of the given circuit.

Complete Step-by-Step Numerical Solution Given Values: Inductance ($L$) = $0.1\text{ H}$ Capacitance ($C$) = $10\,\mu\text{F} = 10 \times 10^{-6}\text{ F} = 10^{-5}\text{ F}$ Resistance ($R$) = $20\,\Omega$ RMS Voltage ($V{rms}$) = $220\text{ V}$ Frequency ($f$) = $50\text{ Hz}$ Angular frequency ($\omega$) = $2\pi f = 2 \times 3.1416 \times 50 = 314.16\text{ rad/s}$ --- Inductive Reactance ($XL$): $$XL = 2\pi f L = \omega L$$ $$XL = 314.16 \times 0.1 = 31.42\,\Omega$$ --- Capacitive Reactance ($XC$): $$XC = \frac{1}{2\pi f C} = \frac{1}{\omega C}$$ $$XC = \frac{1}{314.16 \times 10^{-5}} = \frac{10^5}{314.16} = \frac{100000}{314.16} \approx 318.31\,\Omega$$ --- Total Impedance ($Z$) of the Circuit: $$Z = \sqrt{R^2 + (XC - XL)^2}$$ $$XC - XL = 318.31 - 31.42 = 286.89\,\Omega$$ $$Z = \sqrt{20^2 + (286.89)^2}$$ $$Z = \sqrt{400 + 82305.87} = \sqrt{82705.87} \approx 287.59\,\Omega$$ --- RMS Current ($I{rms}$): $$I{rms} = \frac{V{rms}}{Z}$$ $$I{rms} = \frac{220}{287.59} \approx 0.765\text{ A}$$ --- Power Factor ($\cos\phi$): $$\cos\phi = \frac{R}{Z}$$ $$\cos\phi = \frac{20}{287.59} \approx 0.0695 \text{ (Leading, since } XC > XL)$$ --- Resonant Frequency ($fr$): $$fr = \frac{1}{2\pi \sqrt{LC}}$$ $$\sqrt{LC} = \sqrt{0.1 \times 10^{-5}} = \sqrt{10^{-6}} = 10^{-3}\text{ s}$$ $$fr = \frac{1}{2 \times 3.1416 \times 10^{-3}} = \frac{1000}{6.2832} \approx 159.15\text{ Hz}$$ --- Summary of Results: $XL$ = $31.42\,\Omega$ $XC$ = $318.31\,\Omega$ Impedance ($Z$) = $287.59\,\Omega$ RMS Current ($I{rms}$) = $0.765\text{ A}$ Power Factor ($\cos\phi$) = $0.0695$ Resonant Frequency ($fr$) = $159.15\text{ Hz}$

Question

A series LCR circuit contains an inductor $L = 0.1	ext{ H}$, a capacitor $C = 10\,\mu	ext{F}$, and a resistor $R = 20\,\Omega$ connected across an alternating voltage source of $V = 220	ext{ V}$ and frequency $f = 50	ext{ Hz}$. 
Calculate: Inductive Reactance ($XL$) Capacitive Reactance ($XC$) Total Impedance ($Z$) of the circuit RMS current ($I{rms}$) flowing through the circuit Power factor ($\cos\phi$) of the circuit Resonant frequency ($fr$) of the given circuit.

Accepted Answer

Complete Step-by-Step Numerical Solution

Given Values:

Inductance ($L$) = $0.1\text{ H}$
Capacitance ($C$) = $10,\mu\text{F} = 10 \times 10^{-6}\text{ F} = 10^{-5}\text{ F}$
Resistance ($R$) = $20,\Omega$
RMS Voltage ($V_{rms}$) = $220\text{ V}$
Frequency ($f$) = $50\text{ Hz}$
Angular frequency ($\omega$) = $2\pi f = 2 \times 3.1416 \times 50 = 314.16\text{ rad/s}$

1. Inductive Reactance ($X_L$):

$$X_L = 2\pi f L = \omega L$$ $$X_L = 314.16 \times 0.1 = 31.42,\Omega$$

2. Capacitive Reactance ($X_C$):

$$X_C = \frac{1}{2\pi f C} = \frac{1}{\omega C}$$ $$X_C = \frac{1}{314.16 \times 10^{-5}} = \frac{10^5}{314.16} = \frac{100000}{314.16} \approx 318.31,\Omega$$

3. Total Impedance ($Z$) of the Circuit:

$$Z = \sqrt{R^2 + (X_C - X_L)^2}$$ $$X_C - X_L = 318.31 - 31.42 = 286.89,\Omega$$ $$Z = \sqrt{20^2 + (286.89)^2}$$ $$Z = \sqrt{400 + 82305.87} = \sqrt{82705.87} \approx 287.59,\Omega$$

4. RMS Current ($I_{rms}$):

$$I_{rms} = \frac{V_{rms}}{Z}$$ $$I_{rms} = \frac{220}{287.59} \approx 0.765\text{ A}$$

5. Power Factor ($\cos\phi$):

$$\cos\phi = \frac{R}{Z}$$ $$\cos\phi = \frac{20}{287.59} \approx 0.0695 \text{ (Leading, since } X_C > X_L)$$

6. Resonant Frequency ($f_r$):

$$f_r = \frac{1}{2\pi \sqrt{LC}}$$ $$\sqrt{LC} = \sqrt{0.1 \times 10^{-5}} = \sqrt{10^{-6}} = 10^{-3}\text{ s}$$ $$f_r = \frac{1}{2 \times 3.1416 \times 10^{-3}} = \frac{1000}{6.2832} \approx 159.15\text{ Hz}$$

Summary of Results:

$X_L$ = $31.42,\Omega$
$X_C$ = $318.31,\Omega$
Impedance ($Z$) = $287.59,\Omega$
RMS Current ($I_{rms}$) = $0.765\text{ A}$
Power Factor ($\cos\phi$) = $0.0695$
Resonant Frequency ($f_r$) = $159.15\text{ Hz}$