FundamentalsComponentsSignals & Interfaces

RC and RL time constants: charging, discharging, and transients

Calculate the time constant τ of RC and RL circuits, interpret charging, discharging, and transient response, then apply the 5τ rule correctly.

Level

Intermediate

Reading time

17 min

Primary standard or source

First-order linear circuit theory and IEC 80000-6

Last verified

13 September 2026

Associated tool

RC and RL filter calculator

Written and technically reviewed byElectroDesignForge Engineering Team

View the editorial process

Key point: the time constant τ measures how quickly a first-order circuit responds. For RC, τ = R × C; for RL, τ = L / R. After 1τ, the response has travelled 63.2% of the way to its final value; after 5τ, only about 0.67% of that difference remains. The circuit is usually treated as settled, although it is never mathematically instantaneous.

Quick reference

Use SI units: R in ohms (Ω), C in farads (F), L in henrys (H), and τ in seconds (s). Thus, 10 kΩ × 100 nF = 1 ms and 10 mH / 10 Ω = 1 ms.

Transients: the difference from the final state decays

A transient is the interval between two steady states, such as immediately after a switch closes. A capacitor opposes an instantaneous change in voltage; an inductor opposes an instantaneous change in current:

V_C(0+) = V_C(0−)       I_L(0+) = I_L(0−)

This does not mean capacitor current or inductor voltage stays constant: both can be large at the start. The general expression for a quantity x moving to a final value is:

x(t) = x_final + [x(0+) − x_final] × e^(−t/τ)

It covers charging and discharging. Every τ, the difference from the final value is divided by e. Source resistance, ESR, winding resistance, and supply limitation set real peaks.

RC: capacitor charging

With a source V_S, resistor R, and initially discharged capacitor C:

+V_S ──[ R ]───┬──
               │
              [ C ]
               │
0 V ───────────┴──

τ = R × C
V_C(t) = V_S × (1 − e^(−t/τ))
I(t) = (V_S / R) × e^(−t/τ)

Initially, V_C = 0 and ideal current is V_S / R; at steady state, V_C approaches V_S and current approaches zero. Final stored energy is E = ½ × C × V_S².

RC example

For V_S = 12 V, R = 10 kΩ, and C = 100 nF:

τ = 10,000 Ω × 100 × 10⁻9 F = 1 ms
V_C(1 ms) = 12 V × (1 − e⁻¹) = 7.59 V
I(0+) = 12 V / 10 kΩ = 1.2 mA

After roughly 5 ms, the capacitor reaches 11.92 V. This sizes a delay, but a logic circuit can become usable before 5τ as soon as it crosses its real threshold.

RC discharge and bleed resistor

If a capacitor initially at V_0 discharges through R toward 0 V:

V_C(t) = V_0 × e^(−t/τ)
I(t) = −(V_0 / R) × e^(−t/τ)

Use the general formula for a non-zero final voltage. For a target voltage during a zero-volt discharge:

t = −τ × ln(V_target / V_0)

Example: 470 µF at 24 V must fall below 5 V through 10 kΩ. τ = 4.7 s and t = −4.7 × ln(5/24) = 7.37 s. Initial resistor dissipation is 24² / 10 kΩ = 57.6 mW; check power, energy, and voltage rating.

RL: current build-up

In a series R-L circuit driven by V_S, the inductor slows current build-up. Its voltage is V_L = L × dI/dt:

+V_S ──[ R ]──( L )── 0 V

τ = L / R
I_L(t) = (V_S / R) × (1 − e^(−t/τ))
V_L(t) = V_S × e^(−t/τ)

At t = 0+, the inductor carries nearly the whole source voltage and current is zero. At DC steady state, an ideal inductor is a short circuit and current tends to V_S/R. For τ, use the resistance seen by the inductor with independent sources zeroed, including source, winding, and series-load resistance.

RL example

With L = 10 mH, total resistance R = 10 Ω, and V_S = 5 V:

τ = 10 mH / 10 Ω = 1 ms
I_final = 5 V / 10 Ω = 0.5 A
I_L(1 ms) = 0.5 A × 0.632 = 0.316 A

Include winding resistance. Omitting it overestimates τ and can underestimate final current or dissipation.

RL current decay and turn-off overvoltage

When a switch opens, inductor current must remain continuous. The coil generates the voltage needed to find a path; without a freewheel path, it may damage a transistor, switch, or insulation. A flyback diode reduces overvoltage but extends turn-off because it limits coil voltage. A diode plus Zener/TVS or a snubber can balance protection, speed, and EMC.

Initial stored energy is E = ½ × L × I_0²; it must be dissipated in a resistor, diode, TVS, arc, or another path.

The 5τ rule, precisely

The completed fraction of a rise after nτ is 1 − e^(−n); the remaining fraction of a decay is e^(−n).

5τ is a design convention, not a physical boundary. A precise timer can need more margin when R and C have tolerance, capacitance shifts with bias/temperature, or a threshold is near the final value. Conversely, a digital function can be available before 5τ.

Reliable calculation workflow

1. Set conditions before and immediately after switching (V_C(0+) or I_L(0+)).
2. Determine the final DC value.
3. Find equivalent resistance seen by the reactive element; zero independent sources only to calculate τ.
4. Use τ = R_eq × C for RC or τ = L / R_eq for RL.
5. Check voltage, current, energy, and initial power with the general expression.

Common pitfalls

• Forgetting source resistance, load, ESR, or winding resistance.
• Treating a circuit as exactly finished at 5τ: its response remains asymptotic.
• Confusing an ideal capacitor’s steady DC current (zero) with initial charging current.
• Forgetting the current path when an inductor turns off.
• Applying first-order equations to several independent reactive elements without reduction or simulation.

Associated calculators

• Open the RC and RL filter calculator to connect R, C, or L with cutoff frequency, load, and filter behaviour.
• Open the capacitor charge calculator to size RC precharge, visualise voltage and current, and check safe discharge.

Sources

• IEC 80000-6 — Quantities and units: electromagnetism.
• C. K. Alexander and M. N. O. Sadiku, Fundamentals of Electric Circuits.
• Application notes on RC/RL transients, relay drive, and flyback diodes.