About the RLC Circuit Resonance Simulator
This simulation models a series RLC circuit, the electrical twin of a damped mass-spring oscillator. It integrates the governing equation L d²Q/dt² + R dQ/dt + Q/C = Vs using a fourth-order Runge-Kutta scheme at a 1 ms time step, plotting charge Q(t) and current I(t) on a live canvas. The circuit oscillates at its natural angular frequency ω₀ = 1/√(LC), where impedance falls to a purely resistive minimum.
Switch between Free Oscillations and Driven (AC) modes, then drag the sliders for resistance R (0.5–80 Ω), inductance L (0.01–1.0 H) and capacitance C (10–1000 µF). Driven mode adds a sinusoidal source and a frequency control to sweep through resonance. Derived readouts show ω₀, f₀, Q-factor, damping regime and decay time τ. RLC resonators underpin radio tuning, filters and oscillators across all of modern electronics.
Frequently Asked Questions
What is a series RLC circuit?
A series RLC circuit connects a resistor, an inductor and a capacitor in a single loop. Energy sloshes back and forth between the magnetic field of the inductor and the electric field of the capacitor, while the resistor dissipates it as heat. The result is a damped electrical oscillation analogous to a mass on a spring with friction.
What is resonance and how is it shown here?
Resonance occurs at the natural angular frequency ω₀ = 1/√(LC), where the inductive and capacitive reactances cancel and the impedance drops to just the resistance R. The simulator plots the impedance curve Z(ω) and marks ω₀ with a red dashed line, where the curve reaches its minimum and current peaks.
What does the Q-factor tell me?
The quality factor here is computed as Q = ω₀L/R. A high Q means low resistance relative to reactance, giving a sharp, narrow resonance peak and many oscillation cycles before the signal decays. A low Q gives broad, heavily damped behaviour. The derived-values panel shows Q updating live as you move the sliders.
What is the difference between Free and Driven modes?
In Free Oscillations mode the capacitor starts with a small charge and the circuit rings down naturally with no source, so you see decaying Q(t) and I(t) traces. In Driven (AC) mode a sinusoidal voltage source is applied; you can sweep its frequency to see the steady-state amplitude swell as you approach ω₀.
What do underdamped, critical and overdamped mean?
The damping regime depends on the sign of R² − 4L/C. When it is negative the circuit is underdamped and oscillates; near zero it is critically damped and returns to rest as fast as possible without overshoot; positive means overdamped, settling slowly with no oscillation. The damping readout colours these cases differently.
What numerical method does the simulation use?
It uses the classic fourth-order Runge-Kutta (RK4) method to integrate the coupled equations dQ/dt = I and dI/dt = (Vs − Q/C − RI)/L. RK4 is far more accurate than simple Euler stepping, so the energy and oscillation frequency stay faithful even over many cycles at the 1 ms time step.
Is the model physically accurate?
The differential equation is the exact ideal-component model for a series RLC loop, and RK4 integration introduces only tiny numerical error. It assumes linear, lossless inductors and capacitors and a perfectly resistive R, ignoring real effects like core saturation, dielectric loss and parasitic capacitance. For teaching the core physics it is highly accurate.
How does decreasing R change the behaviour?
Lowering the resistance raises the Q-factor and lengthens the decay time τ = 2L/R. The oscillations persist for many more cycles and the resonance peak in the impedance plot becomes sharper and deeper. Increasing R does the opposite, broadening the peak and quickly damping the ring-down.
What is the impedance curve Z(ω)?
Impedance is the total opposition to alternating current, given by Z(ω) = √(R² + (ωL − 1/ωC)²). The plot shows it across roughly two decades of frequency on a logarithmic axis. It is large at very low and very high frequencies and dips to its minimum value of R exactly at resonance.
Why does charge lead or lag the current?
Current is the time derivative of charge, I = dQ/dt, so the two traces are ninety degrees out of phase: current peaks when charge crosses zero and vice versa. This quadrature relationship is the electrical signature of energy shuttling between the capacitor's electric field and the inductor's magnetic field.
Where are RLC resonators used in the real world?
They are everywhere in electronics: radio and television tuners select a station by matching an RLC resonance to the carrier frequency, while band-pass and band-stop filters shape audio and signal frequencies. High-Q crystal and LC oscillators provide the precise clocks inside computers, GPS receivers and atomic timekeeping.