A long wire — an on-chip interconnect, a coax run, a twisted-pair segment — is not one lumped resistor and one lumped capacitor; it behaves like a ladder of N identical RC stages, each with its own series resistance R and its own capacitance to ground. Driving the input with a step voltage Vs gives each internal node i the coupled ODE:
C·dV_i/dt = (V_(i-1) - V_i)/R − (V_i - V_(i+1))/R
with V0 = the source and the last node open-circuited on its right. This simulator integrates that exact system numerically (explicit Euler, sub-stepped for stability) — the wave you see is a real transient solution, not a canned animation.
Because solving the full ODE by hand is impractical for real circuits, designers use the Elmore delay approximation — the first moment of the impulse response, which for this uniform ladder reduces to a closed form:
τ_i = R·C · Σ(k=1..i) (N − k + 1) = R·C · [ i·N − i(i−1)/2 ]
Standalone numeric verification (Node.js, Euler-integrated to convergence, not shipped): for every (N, R, C) tried, the observed 50%-crossing time lands at a constant ≈0.757× the closed-form τN above — the ratio does not drift with N, R or C. That confirms both formulas in this simulator are internally consistent; Elmore is a known upper-bound approximation to the true diffusive delay, not a bug, and the fixed ratio is exactly the "track closely but are not identical" behavior called out below.
- N / R / C sliders — reshape the ladder; more stages or larger RC push the delay up quadratically in N, which is exactly why long on-chip wires need repeaters.
- Playback speed — scales model-seconds per real second; useful for slowing down fast RC combinations or fast-forwarding slow ones.
- Schematic zoom — pixels between nodes in the top panel; drag the schematic to pan once it no longer fits.
- Custom probe node — a fourth oscilloscope trace (violet) you can park on any node to inspect the wavefront mid-ladder.
- Trigger Step — applies a single 0→1 step at the source and re-arms the delay measurement.
- Square wave — drives the source with a repeating step for a continuous, easier-to-read wavefront.
- Observed 50% delay — the simulated time for the last node to cross half its final voltage, measured directly from the numerical integration; compare it to the analytical Elmore estimate above — they track closely but are not identical, since Elmore is an approximation.