How to use the RL / RC / RLC Transient Response Calculator
- 1. Pick RC, RL or RLCSeries RLC handles all three damping cases.
- 2. Enter the source and partsAdd initial vC(0) and iL(0) if they are not 0.
- 3. See the responseFormula, graphs, a τ table and values at your time t.
First-order and second-order transients
In an RC or RL circuit every voltage and current follows x(t) = x(∞) + [x(0) − x(∞)]e^(−t/τ), with τ = RC or τ = L/R. The capacitor voltage and the inductor current cannot change instantly, so they give x(0). After 5τ the circuit is within 0.7% of steady state. A series RLC circuit has the characteristic equation s² + (R/L)s + 1/(LC) = 0, with α = R/2L and ω₀ = 1/√(LC). If ζ = α/ω₀ is greater than 1 the response is overdamped, if it equals 1 it is critically damped, and if it is less than 1 it is underdamped (it rings at ωd = ω₀√(1 − ζ²)).
Worked example: R = 1 kΩ and C = 100 µF charged from 10 V give τ = 0.1 s, so vC(t) = 10(1 − e^(−10t)) V. At t = τ, vC = 6.32 V and i = 3.68 mA. For a series RLC with R = 20 Ω, L = 100 mH and C = 100 µF: α = 100, ω₀ = 316.2 and ζ = 0.316, so the response is underdamped with 35% overshoot.
Exam tips
- Find vC(0⁻) or iL(0⁻) from the circuit before switching. They stay the same at 0⁺.
- The final value comes from the DC steady state: a capacitor is open and an inductor is a short.
- Critical damping needs R = 2√(L/C).
- For an RLC circuit, the second initial condition is dvC/dt(0) = iL(0)/C.
Frequently asked questions
What does 'use 0 for a discharge' mean?
If the source is removed after switching, the final voltage is 0. Set the source to 0 and give the starting capacitor voltage or inductor current.
How is the RLC damping type decided?
Compare α = R/2L with ω₀ = 1/√(LC). If α > ω₀ it is overdamped, if they are equal it is critically damped, and if α < ω₀ it is underdamped. The tool also shows ζ = α/ω₀.
Why is overshoot shown only sometimes?
The standard overshoot formula e^(−ζπ/√(1−ζ²)) applies to an underdamped step response starting from rest (vC(0) = 0, iL(0) = 0).
Is this tool free?
Yes. You fill in a short form once to see your result, and the other Uncodemy tools then open without it. All the maths runs in your browser.