Rl Time Constant⚠ unverified
Electrical / Basic · Compute the time constant of a resistor-inductor (RL) circuit
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| resistance | resistance | — | 1.0 | Resistance of the circuit |
| inductance | inductance | H | 1.0 | Inductance of the circuit |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | τ | s | Time constant, in seconds (s) |
The science & history
Understanding the Parameters
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Inductance $L$ — opposes changes in current ($v = L\,di/dt$); larger $L$ slows the current's rise and decay.
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Resistance $R$ — sets the final steady current ($V/R$) and how quickly stored magnetic energy is dissipated; larger $R$ makes the RL transient faster (note $\tau = L/R$, the inverse dependence compared with RC).
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$\tau$ — their ratio $L/R$, in seconds.
Derivation (Approaching a Proof)
For an inductor energised from a source $V$ through series resistance $R$, Kirchhoff's voltage law gives
$$V = L\,\frac{di}{dt} + iR.$$
This first‑order linear ODE has the solution (with $i(0)=0$)
$$i(t) = \frac{V}{R}\left(1 - e^{-t/(L/R)}\right),$$
so the natural time scale is $\tau = L/R$. At $t=\tau$ the current reaches $0.632\,V/R$; when the source is removed the current decays as $i(t) = I_0\,e^{-t/(L/R)}$.
Units. $\dfrac{\text{H}}{\Omega} = \dfrac{\text{V}\cdot\text{s/A}}{\text{V/A}} = \text{s}$ — genuinely seconds. (Note: the live calculator currently marks the resistance input "dimensionless"; it should be ohms — flagged for calculator verification.)
History
The RL transient is the magnetic dual of the RC transient, following from Faraday's law and Kirchhoff's laws. Its abrupt‑interruption behaviour — a rapidly collapsing field producing a large $L\,di/dt$ voltage — is why switches arc and why flyback diodes and snubbers are standard around relays and motors.
Related Concepts: RC Time Constant, Inductive Reactance, Inductor Energy, Ohm's Law solve for current
Notes: Registry calculator rl-time-constant (unverified). First‑order single‑R single‑L model.