RC Time Constant

Where τ comes from

Stand a discharged capacitor C through a resistor R against a step source V₀. Kirchhoff's voltage law gives:

V₀ = i·R + V_C
   = R·C·(dV_C/dt) + V_C

This is a first-order linear ODE. Its solution is

V_C(t) = V₀ · (1 − e^(-t/τ))     where τ = R·C

τ is the time at which the exponent equals 1, i.e. where 1 − e⁻¹ = 0.632 of the final value has been reached. It is not "how long until the system finishes" — it is the natural time scale of the response. After 5τ the system is within 0.7% of its asymptote; after 7τ within 0.1%; after 10τ within 5 parts per million.

The exponential is universal because the differential equation is universal. Anything obeying dx/dt = (x_∞ − x)/τ — where the rate of approach to a target is proportional to remaining distance — converges exponentially. Heat soaking into a metal block, water draining through an orifice, charge flowing through a resistor: same equation, same shape, same 63.2% at one τ.

Worked examples

1 µF × 1 kΩ:

τ = 1×10⁻⁶ × 1000 = 1×10⁻³ s = 1 ms

So at t = 1 ms, V_C = 0.632·V₀. At t = 5 ms, V_C = 0.993·V₀. The associated low-pass cutoff is

f_c = 1 / (2π·τ) = 1 / (2π · 0.001) ≈ 159 Hz

Anything below 159 Hz passes; anything above is attenuated 6 dB per octave.

Audio anti-alias filter at 20 kHz:

τ = 1 / (2π · 20000) = 7.96 µs
   pick R = 1 kΩ → C = τ/R = 7.96 nF (use 8.2 nF E12 standard value)

Touchscreen pad RC scan, 50 pF + 5 MΩ pull-up:

τ = 5×10⁶ × 50×10⁻¹² = 250 µs
   With finger touch raising C to 100 pF, τ rises to 500 µs.
   Compare counts: 250 µs vs 500 µs → finger detected.

Capacitive sensing is exactly an RC time constant measurement, repeated thousands of times per second across every pad on a screen.

The 63 / 86 / 95 / 99 ladder

The shape repeats forever — knowing the percentages by heart is one of the few mnemonics every electrical engineer keeps for life:

t = 0:    0.00 %   ← starting line
t = 1τ:  63.21 %   ← "one tau done"
t = 2τ:  86.47 %
t = 3τ:  95.02 %
t = 4τ:  98.17 %
t = 5τ:  99.33 %   ← "fully settled" by convention
t = 7τ:  99.91 %
t = 10τ: 99.995%

For discharge, swap "1−" for the bare exponential and you get the dual ladder — 36.79 % at 1τ, 13.53 % at 2τ, etc.

RC vs RL vs RLC time-domain responses

Topology	Order	Time constant	Step response shape	Cutoff	Typical role
RC low-pass	1st	τ = RC	Pure exponential rise	f_c = 1/(2πRC)	Audio, ADC anti-alias, debounce
RC high-pass	1st	τ = RC	Exponential decay from peak	f_c = 1/(2πRC)	DC blocking, AC coupling
RL low-pass	1st	τ = L/R	Same shape (current ramp)	f_c = R/(2πL)	Inrush limit, DC chokes
RL high-pass	1st	τ = L/R	Same shape (voltage decay)	f_c = R/(2πL)	Power-supply pre-filters
Series RLC (underdamped)	2nd	1/(ζω₀)	Damped oscillation, ζ<1	f₀ = 1/(2π√LC)	Bandpass tuner, ringing
Series RLC (critically damped)	2nd	1/ω₀	Fastest non-oscillating	f₀	Servo and motor controls
Series RLC (overdamped)	2nd	two roots	Slow droop, no ring	f₀	Snubbers, surge absorbers

First-order RC and RL circuits share the universal 63.2 % shape. RLC is fundamentally different — two energy stores can swap energy back and forth, producing oscillation when damping is light enough.

Real-world τ values

• 0.1 µF × 100 Ω bypass loop: τ ≈ 10 ns. Fast enough to source a CMOS gate's switching transient.
• 10 µF × 1 Ω power-supply input filter: τ = 10 µs. Sets soft-start ramp visible on a scope at power-up.
• 100 nF × 100 kΩ debounce on a momentary switch: τ = 10 ms. Filters out 1–5 ms contact bounce while allowing 100 Hz button presses to register.
• 1 µF × 1 MΩ analog integrator: τ = 1 s. Op-amp integrators with these values measure charge over seconds — common in piezoelectric and pyroelectric front-ends.
• Thermal resistance 5 °C/W × thermal mass 10 J/°C (a TO-220 transistor on a small heatsink): τ_thermal = 50 s. Heat soaks for a minute before steady state. Same 1 − e^(-t/τ) curve, plotted as temperature rather than voltage.
• Earth ionosphere D-layer recombination: τ ≈ 30 minutes. Electrons drop out after sunset on a 30-minute exponential, opening up MF/HF radio propagation.

Variants and equivalents

• Charging vs discharging. Mirror images on the same τ. Charge: V₀(1 − e^(-t/τ)). Discharge: V₀·e^(-t/τ). The 63.2 % charge number and the 36.8 % discharge number are 1 − e⁻¹ and e⁻¹ respectively.
• Net vs gross capacitance. When series resistors load both sides of a cap, the effective τ uses the Thévenin equivalent resistance seen by the cap. A 10 kΩ source feeding 1 µF feeding a 10 kΩ load has τ = (10k‖10k)·1µ = 5 ms, not 10 ms.
• Multiple poles. Cascading two RC sections gives a second-order Bessel- or Butterworth-shaped roll-off if buffered, or a stretched-out single-time-constant approximation if direct-coupled. Never assume two τ's just multiply.
• Open-circuit time-constant method. A useful trick for analyzing complex multi-cap analog stages: short each independent source, look at the resistance seen by each cap one at a time, sum the τs to estimate dominant pole frequency.
• Thermal RC. Same equations describe heat flow: thermal resistance R_θ (°C/W) takes the role of electrical R, thermal capacitance C_θ (J/°C) takes the role of C. Junction-to-ambient temperature transients are read off the same 63.2 % ladder.

Failure modes

• RC droop on long pulse trains. AC-coupling capacitors block DC, but a long string of same-polarity pulses lets the cap charge up, biasing the input. Video signals (long blanking intervals) and serial data with long runs of 1s droop visibly. Mitigated with DC restore clamps or with high-pass corner well below the lowest signal frequency.
• Settling slower than expected at low temperature. Electrolytic ESR rises 5–10× from 25 °C to −40 °C, stretching τ on bulk filter caps. Power supplies that pass at room temperature fail to start in cold storage units.
• Anti-alias not matched to sample rate. If f_c sits above f_sample/2, aliased signals fold into the band. The classic 1980s digital audio bug: 22 kHz hiss because the anti-alias was at 30 kHz, not 20 kHz, sampling at 44.1 kHz.
• Tolerance stack-up. Class-2 ceramics have ±20% tolerance, then lose 50% to DC bias, then drift another 15% over temperature. A nominal 10 ms τ can land anywhere from 4 ms to 14 ms in production. Designs that need precision pick C0G ceramics or film caps.
• Hidden parallel paths. Surface contamination (flux residue, fingerprints, humidity) puts megohms in parallel with high-impedance nodes. A 1 GΩ leakage on a 10 pF hold-cap drops τ from "infinite" to 10 ms — enough to wreck a scope's sample-and-hold accuracy.
• Reset circuits that don't reset. Power-on RC reset networks need τ much longer than supply rise time. If τ = 10 ms but the supply takes 50 ms to ramp, the reset pin never sees the active-low edge and the MCU boots from indeterminate state.

Where τ is the answer

• Pick filter component values from a target cutoff frequency.
• Estimate settling time of any first-order analog stage.
• Size debounce networks for mechanical switches and relays.
• Set ramp rate of soft-start, brown-out detect, and reset circuits.
• Measure unknown capacitance or resistance by timing a step response.
• Predict thermal lag of any heatsink-and-mass system using R_θ × C_θ.