About the 2D RLC Circuit Oscilloscope
This simulation drives a series RLC circuit with a sinusoidal source V(t) = V₀sin(ωt) and integrates the exact second-order circuit equation L dI/dt + RI + Q/C = V(t) sample-by-sample with fourth-order Runge–Kutta, exactly like a real oscilloscope would capture a physical bench circuit. The top panel is a live scope trace of the source voltage and the resulting current; the bottom panel is a frequency-response curve built the honest way — by actually re-running the simulation at each of many drive frequencies, letting the transient die out, and measuring the true steady-state current amplitude, rather than plotting the algebraic impedance formula.
Drag R, L and C to reshape the resonance, move the Drive frequency slider to hear…er, see the scope trace change phase and amplitude in real time, then press "Run Frequency Sweep" to measure the full response curve at the current component values. The red marker shows the theoretical ω₀ = 1/√(LC); the cyan marker tracks wherever the sweep actually measured its peak, and normally the two coincide, which is itself a nice numerical-accuracy check.
Frequently Asked Questions
How is this different from a plain impedance formula?
Nothing here is precomputed analytically. Every point of the sweep curve comes from actually numerically integrating the circuit's ODE at that frequency for many drive cycles until the transient decays, then measuring the amplitude of the resulting steady-state current. It happens to match the algebraic impedance prediction because the model is exact, but the number on screen was measured, not looked up.
What is the oscilloscope panel showing?
The top panel plots two live traces against time: the driving voltage V(t) (yellow) and the resulting circuit current I(t) (orange), both integrated in real time at the current slider settings. Watch the current trace's amplitude grow as you tune the drive frequency toward resonance, and its phase shift from leading to lagging the voltage as it crosses resonance.
What does the sweep button actually do?
Pressing "Run Frequency Sweep" runs a separate, independent batch of simulations: for roughly 160 frequencies spanning a decade below to a decade above the natural frequency, it resets the circuit to rest, integrates for 40 drive periods, discards the first 30 as transient settling time, and records the peak current magnitude over the last 10 periods. That measured curve is drawn in the bottom panel.
Why does the measured peak land exactly at ω₀?
Because a series RLC circuit's steady-state current amplitude is I₀(ω) = V₀/√(R² + (ωL − 1/ωC)²), which is maximised exactly when the reactive term ωL − 1/ωC vanishes, i.e. at ω = ω₀ = 1/√(LC). The simulation reproduces that maximum empirically because RK4 integration of the true ODE converges to the same steady state.
Why discard the first cycles of each sweep run?
Right after the source switches on, the circuit carries a transient response (a decaying free oscillation at its natural frequency) superimposed on the eventual steady-state response at the drive frequency. Measuring amplitude too early would mix the two. Waiting 30 periods lets e−γt decay far enough that only the steady-state sinusoid remains, for all but pathologically high-Q settings.
What is Q-factor and bandwidth here?
Q = ω₀L/R measures how sharply peaked the resonance is. The half-power bandwidth Δω ≈ ω₀/Q is the width (in rad/s) over which the measured current stays above 1/√2 of its peak. A high-Q circuit (low R) gives a narrow, tall peak in the sweep curve; a low-Q circuit gives a broad, shallow one.
Is the numerical method accurate enough to trust the sweep?
Yes: each sweep point uses fourth-order Runge–Kutta with roughly 200 steps per drive period, which keeps phase and amplitude error many orders of magnitude below plotting resolution over the 40 periods simulated. The same integrator drives both the live scope and the sweep, so the two views are always numerically consistent with each other.
How is this a genuinely different simulation from the 3D RLC page?
The 3D companion shows free ring-down and an algebraic impedance curve computed directly from the formula. This page only ever drives the circuit with a continuous AC source and never plots a formula — its frequency-response curve is entirely the output of repeated, independent time-domain simulations, alongside a true real-time oscilloscope view that the 3D page does not have.