Rc Time Constant⚠ unverified
Electrical / Basic · Compute the time constant of a resistor-capacitor (RC) circuit
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| resistance | resistance | — | 1.0 | Resistance of the circuit |
| capacitance | capacitance | F | 1.0 | Capacitance of the circuit |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | τ | s | Time constant, in seconds (s) |
The science & history
Understanding the Parameters
- Resistance $R$ — limits the current available to move charge; larger $R$ slows charging.
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Capacitance $C$ — the amount of charge to be moved per volt; larger $C$ stores more charge and takes longer to fill.
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$\tau$ — their product, in seconds; it sets the speed of every RC transient and the corner frequency $f_c = 1/(2\pi RC)$ of the corresponding first‑order filter.
Derivation (Approaching a Proof)
Consider a capacitor charging from a source $V$ through a series resistor $R$. Kirchhoff's voltage law gives $V = iR + v_C$, and the capacitor relation is $i = C\,\dfrac{dv_C}{dt}$. Substituting:
$$RC\,\frac{dv_C}{dt} + v_C = V.$$
This first‑order linear ODE has the solution (with $v_C(0)=0$)
$$v_C(t) = V\left(1 - e^{-t/RC}\right),$$
so the natural time scale of the exponential is $\tau = RC$. At $t = \tau$, $v_C = V(1 - e^{-1}) = 0.632\,V$; discharging follows the dual $v_C(t) = V_0\,e^{-t/RC}$.
Units. $\Omega \cdot \text{F} = \dfrac{\text{V}}{\text{A}} \cdot \dfrac{\text{C}}{\text{V}} = \dfrac{\text{C}}{\text{A}} = \text{s}$ — the product is genuinely seconds. (Note: the live calculator currently marks the resistance input "dimensionless"; it should be ohms — flagged for calculator verification.)
History
The exponential charging/discharging of an RC circuit is the archetypal first‑order transient, arising directly from Ohm's and Kirchhoff's laws combined with the capacitor's definition. It became a workhorse of timing, filtering, and pulse circuits through the 20th century (e.g. the ubiquitous 555 timer sets delays from $R$ and $C$).
Related Concepts: Rl Time Constant, Capacitive Reactance, Capacitor Energy, Ohm's Law solve for current
Notes: Registry calculator rc-time-constant (unverified). First‑order, single‑R single‑C model;
networks with multiple energy‑storage elements have several time constants.