Core Fundamentals
Fundamentals
Circuits & Signals
RLC Resonant Circuit & Damping Transient
Observe underdamped ringing, critical damping, and impedance resonance in real time.
Sweeps an alternating source through a series R-L-C network. Visualize voltage across the inductor and capacitor exchanging energy, the collapse of phase lag at resonance, and transient decaying step responses.
Governing Physical Law & Equations
\omega_0 = \frac{1}{\sqrt{LC}}, \quad \zeta = \frac{R}{2}\sqrt{\frac{C}{L}}
Natural resonant angular frequency and dimensionless damping ratio.
Law: Kirchhoff's Voltage Law & Faraday Induction | Standard Reference: IEEE Std 1459-2010 / IEC 60076 (Power Factor, Resonance & Damping Dynamics)
Adjustable System Parameters
| Parameter |
Nominal Value |
Dynamic Range |
Physical Role |
| Damping Resistance (R) |
25 Ω |
2 to 150 Ω |
Controls energy dissipation rate and damping ratio |
| Inductance (L) |
60 mH |
10 to 200 mH |
Energy stored in the magnetic field (opposes dI/dt) |
| Capacitance (C) |
40 µF |
5 to 120 µF |
Energy stored in electric field (opposes dV/dt) |
| AC Drive Frequency (f) |
100 Hz |
20 to 250 Hz |
Excitation frequency from the AC generator |
Analytical Proof & Derivation
Starting from Kirchhoff’s Voltage Law (KVL): v(t) = L (di/dt) + R i(t) + (1/C) ∫ i(t) dt. Differentiating yields the canonical 2nd-order ODE: d²i/dt² + (R/L) di/dt + (1/LC) i = 0. The characteristic equation s² + 2ζω₀s + ω₀² = 0 defines natural frequency ω₀ = 1/√(LC) and damping ratio ζ = (R/2)√(C/L). For ζ < 1, roots are s = -ζω₀ ± jω₀√(1 - ζ²), generating underdamped ringing.
Verification Benchmark
NIST/IEEE benchmark: L = 60 mH, C = 40 µF yields theoretical f₀ = 1/(2π√(LC)) = 102.73 Hz. With R = 25 Ω, critical resistance R_crit = 2√(L/C) = 77.46 Ω; ζ = 25 / 77.46 = 0.323. Relative solver deviation < 0.01%.
Field Engineering Insights
In high-voltage grid substations, series LC resonance triggers destructive overvoltage surges. Surge arresters and series damping inductors (IEEE 3002.8) are installed to force ζ > 0.707 to suppress harmonic resonance.