RC Time Constant Calculator — τ, Charging & Discharging Curve

Enter resistance \(R\), capacitance \(C\), and initial voltage \(V_0\). We’ll compute τ = RC and plot charging (to \(V_s\)) and discharging (to 0 V) curves with τ markers. Everything runs locally in your browser.

Inputs

Charging Discharging

Results

τ (time constant): — (also —)
Key times: 1τ ≈ 63.2% step, 3τ ≈ 95%, 5τ ≈ 99.3%
Inputs used: R = —, C = —, V0 = —, Vs = —

Equations: \( \tau = RC \), \( v_{charge}(t) = V_s - (V_s - V_0)e^{-t/\tau} \), \( v_{discharge}(t) = V_0 e^{-t/\tau} \).

RC Circuits — Quick Primer

A first-order RC network (one resistor, one capacitor) responds to a step input with an exponential characterized by the time constant \( \tau = RC \). After one time constant the output has moved about 63.2% toward its final value; after three it is ~95%; after five it is ~99.3%. For a charge from \(V_0\) toward a supply \(V_s\): \( v(t) = V_s - \bigl(V_s - V_0\bigr)\,e^{-t/\tau} \). For a discharge toward 0 V: \( v(t) = V_0\,e^{-t/\tau} \).

These curves come from the differential equation \( \dfrac{dv}{dt} + \dfrac{1}{\tau}v = \dfrac{V_s}{\tau} \) (charging) or \( \dfrac{dv}{dt} + \dfrac{1}{\tau}v = 0 \) (discharging). You can invert the exponential to find time to reach a target fraction \( \alpha \) of the step: \( t = -\tau \ln(1-\alpha) \). Common checkpoints are 50% (\(0.693\,\tau\)), 90% (\(2.303\,\tau\)), and the 10–90% rise time \( t_r \approx 2.2\,\tau \).

RC networks also act as simple filters. A series-R, shunt-C “low-pass” has a –3 dB cutoff at \( f_c = \dfrac{1}{2\pi RC} \) (angular cutoff \( \omega_c = 1/RC \)). Below \( f_c \) the output tracks the input; above \( f_c \) amplitude rolls off at 20 dB/decade and the phase lags by up to 90°. Swapping the element order yields a “high-pass” with the same cutoff but complementary behavior. Time and frequency views are two sides of the same first-order system.

Design from timing. If you need a specific delay to a threshold \(V_T\), pick \( \tau = \dfrac{t}{-\ln\!\bigl(1 - (V_T - V_0)/(V_s - V_0)\bigr)} \), then choose convenient R and C whose product equals that \( \tau \). For example, to hit 90% of \(V_s\) in 10 ms, use \( \tau = 10~\text{ms}/2.303 \approx 4.34~\text{ms} \); \( R=10~\text{k}\Omega \) with \( C=0.434~\mu\text{F} \) (or any pair with the same product) will work.

Real-world notes. Electrolytic capacitors have wide tolerances (±20% is common) and effective series resistance (ESR) plus leakage, all of which perturb the ideal \( \tau \). Resistors change value with temperature. The capacitor’s voltage rating and ripple current must not be exceeded. If you interface to logic, design for the actual threshold of the input (e.g., 0.7·\(V_s\) for some CMOS) rather than an abstract “100%”. For power-up delays, add a discharge path so the node reliably resets.

Units & ranges. Use Ω, kΩ, MΩ for R and F, mF, µF, nF, pF for C. Typical hobby values might be \( R = 1~\text{k}\Omega \) with \( C = 100~\mu\text{F} \Rightarrow \tau = 0.1~\text{s} \), or \( R = 10~\text{k}\Omega \) with \( C = 10~\text{nF} \Rightarrow \tau = 100~\mu\text{s} \). The energy stored at voltage \(v\) is \( E = \tfrac{1}{2} C v^2 \), useful when estimating inrush and discharge safety.

Tip: If your plot looks “flat,” increase the range to \(5\tau\) or \(7\tau\) so the exponential visibly settles. If it clips at the top, reduce \(V_s\) or scale the vertical axis.

Explore more tools