Analyze Resonant Frequency, Damping, and Transient Response
Resistance R (Ω)
Inductance L (H)
Capacitance C (F)
Result
Resonant Freq f₀
Damping Factor ζ
Damping Type
Quality Q
Step-by-Step Calculation
RLC Circuit Formulas
ω₀ = 1/√(LC), f₀ = ω₀/(2π)
ζ = R/2 × √(C/L) (damping ratio)
Q = 1/(2ζ) = (1/R) × √(L/C)
ωd = ω₀ × √(1-ζ²) (damped freq if ζ<1)
The RLC circuit is a second-order system with both an inductor and capacitor. It can resonate at frequency f₀ and exhibit damped oscillations depending on the damping factor ζ. Understanding RLC behavior is essential for filter design, tuned circuits, and transient analysis.
⚠At resonance, inductive reactance equals capacitive reactance (XL = Xc). Impedance is minimum (= R). Current is maximum. Voltage across L and C can be much higher than input (resonant step-up in series RLC).
RLC Circuit Damping Types
The damping factor ζ determines the circuit response. Underdamped circuits oscillate at the damped frequency before settling. Critically damped circuits return to equilibrium fastest without overshoot. Overdamped circuits are sluggish but stable. The transition from underdamped to overdamped occurs at R = 2√(L/C).
Underdamped (ζ<1)
Oscillatory response. Rings at ωd = ω₀√(1-ζ²). Overshoot then settles. ζ=0 = no damping (continuous oscillation).
Critically Damped (ζ=1)
Fastest return to zero without oscillation. R_critical = 2√(L/C). Optimal for control systems, meters, and servo mechanisms.
Overdamped (ζ>1)
Slow decay, no oscillation. Two real time constants. Used where overshoot cannot be tolerated (e.g., sensitive equipment).
Resonance (ζ=0)
Ideal LC circuit (no R). Infinite Q. Continuous oscillation at f₀. Real circuits always have some resistance (loss).
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