Resonance in an RLC Circuit

In an RLC circuit, which consists of a resistor (R), an inductor (L), and a capacitor (C) connected in series, resonance occurs when the frequency of the applied voltage matches the natural frequency of the circuit. At resonance, the reactance of the inductor and the capacitor cancel each other out, resulting in a maximum current flow through the circuit.

Given Parameters:
- Inductance (L) = 1 mH
- Capacitance (C) = 0.1 uF
- Applied voltage (V) = 9 sin(wt)

Finding Resonance Frequency:
The resonance frequency (fr) of an RLC circuit can be calculated using the formula:

fr = 1 / (2π√(LC))

Substituting the given values, we have:
L = 1 mH = 1 × 10^(-3) H
C = 0.1 uF = 0.1 × 10^(-6) F

fr = 1 / (2π√(1 × 10^(-3) × 0.1 × 10^(-6)))
= 1 / (2π√(10^(-7)))
≈ 1591.55 Hz

Resonance Peak Voltage across the Capacitor:
At resonance, the impedance of the circuit is purely resistive, and the voltage across the capacitor is at its maximum value. The impedance (Z) of an RLC circuit is given by the formula:

Z = √(R^2 + (XL - XC)^2)

Where:
R is the resistance,
XL is the inductive reactance (2πfL),
XC is the capacitive reactance (1/(2πfC)), and
f is the frequency of the applied voltage.

At resonance, XL = XC, so the impedance simplifies to:

Z = R

Since the voltage across the capacitor (Vc) is given by:

Vc = V × (XC / Z)

At resonance, XC = XL, so the voltage across the capacitor simplifies to:

Vc = V × (XC / R)

Substituting the given values:
V = 9 sin(wt)
XC = 1 / (2πfC) = 1 / (2π × 1591.55 × 0.1 × 10^(-6))
≈ 99.89 Ω
R (Resistance) = 102 Ω

Vc = 9 sin(wt) × (99.89 / 102)

10^5