How this instrument works
Wire one resistor to one capacitor and you have built a clock running on nothing but component values. Charge cannot arrive all at once — resistance meters it — so voltage across those plates climbs along a curve that never quite lands. Solve RC·dv/dt + v = V₀ and an exponential drops out, carrying τ = RC as its only timescale. One τ into charging, your capacitor has covered 63.2 percent of its journey; one τ into discharging, 36.8 percent still remains. Units agree without any fudge factor, since ohms multiplied by farads reduce to coulombs per ampere — which is precisely what one second means.
That timescale first cost real money under an ocean. William Thomson, analysing submarine telegraph cables for a Royal Society paper in 1855, treated a cable as resistance and capacitance smeared along its whole length, then concluded that signal retardation grows with distance squared — lay twice as much cable and delay quadruples. His law of squares was unwelcome news to Atlantic Telegraph backers and to Wildman Whitehouse, who preferred brute voltage; Thomson's mirror galvanometer, reading feeble currents instead of forcing large ones, settled that argument in 1858. Silicon designers relearn his result constantly, since RC delay along on-chip interconnect is precisely why long wires get chopped up by repeaters.
Two assumptions sit underneath τ = RC. First, R must be whatever resistance a capacitor genuinely looks back into — Thévenin equivalent at its own two terminals, including a driving source's output impedance and equivalent series resistance inside that capacitor, not merely whichever part you can point at on your schematic. Second, C must hold still: one ceramic part may shed a third of its marked value once bias voltage sits across it, and a 555 timer fills through one path while emptying through another, giving two constants in a single circuit. Add another capacitor or stray inductance and one exponential stops describing anything — expect two timescales, or ringing.
- Put your figure into Resistance — ohms, kilohms or megohms. Use resistance looking outward from capacitor terminals, not simply whichever resistor sits nearest.
- Enter Capacitance in farads, millifarads, microfarads, nanofarads or picofarads, straight off part markings or meter readings.
- Read Time constant, switching that display to milliseconds for most electronics work and to minutes for supercapacitor backup rails.
- Weigh Time to ~99% charged (5τ) against your timing budget. Any square wave driving this network wants half-periods longer than that figure, or your capacitor never arrives.
Worked example — 1 kΩ and 1 µF on a scope
A function generator drives square waves through one 1 kΩ resistor into one 1 µF capacitor, with your probe clipped across that capacitor. Resistance is 1000 ohms, capacitance is 0.000001 farads, so τ = 1000 × 0.000001 = 0.001 s — one millisecond, no rounding involved. Five of those, 0.005 s or 5 ms, carries voltage to 99.33 percent of supply, which any eight-bit reading happily calls done.
Mismatch that pace and your trace changes character entirely. At 50 Hz each half-cycle lasts 10 ms, ten constants, so flat tops and textbook exponentials appear just as diagrams promise. Wind up to 1 kHz, each half-cycle becomes 0.5 ms — half a constant — voltage reaches barely 39 percent before polarity flips, and those curves degenerate into a triangle. Identical parts, identical 1 ms figure, unrecognisable waveform. That same millisecond also fixes a corner frequency at 1 ⁄ (2π × 0.001) = 159 Hz and a 10-to-90-percent rise time of 2.2 ms.
Questions
Why does an ohm times a farad give seconds?
Because those units cancel down to time. An ohm is one volt per ampere and a farad is one coulomb per volt, so multiplying them kills volts and leaves coulombs per ampere. An ampere is one coulomb per second, meaning what survives is simply seconds. That is why 1 kΩ paired with 1 µF lands on exactly 1 ms with no conversion factor hiding anywhere. Every first-order network puts t/RC into an exponent, where it must be dimensionless; if your answer refuses to come out in seconds, a unit prefix went astray.
Is a capacitor fully charged after one time constant?
No, and assuming otherwise is a genuinely common slip. One τ reaches 63.2 percent of final voltage, two reaches 86.5, three 95.0, four 98.2, five 99.33. An exponential never truly arrives; it only runs out of headroom. Discharge mirrors that shape — after one τ, 36.8 percent of starting voltage still sits on those plates. Treat τ as characteristic pace, never as some finish line, then quote whichever percentage your own application honestly needs.
Where does the five-time-constant rule come from?
Convention rather than physics. Five constants leave 0.67 percent of error behind, which usually vanishes inside measurement noise, so bench work adopted it. Precision timing wants something sharper. Settling within one least-significant bit of an n-bit converter takes roughly (n+1) × ln 2 constants — about 6.2τ at 8 bits, 9.0τ at 12 bits, 11.8τ at 16 bits. Sample-and-hold designers size acquisition windows from that expression, never from a rule of thumb.
Which resistance should I enter if my circuit has several?
Whatever a capacitor sees looking outward with sources zeroed, which is Thévenin equivalent resistance at its own terminals. Two 2 kΩ resistors forming a divider present 1 kΩ, not 2 kΩ and not 4 kΩ. Add output impedance of whatever drives that node, plus equivalent series resistance within your capacitor; both start mattering once R falls below a few hundred ohms. Where filling and emptying travel separate paths, as inside a 555 astable, run this sheet twice, once per path.
How does the time constant relate to a filter's cutoff frequency?
They sit a factor of 2π apart: cutoff frequency equals 1 ⁄ (2πRC), which is simply 1 ⁄ (2πτ). A 1 ms constant puts a corner at 159 Hz, where output has dropped 3 dB and lags input by 45 degrees. Rise time follows from that same figure — 10 to 90 percent takes 2.197τ, which is exactly where an old bench shortcut originates: rise time multiplied by bandwidth is about 0.35. A single resistor and capacitor form both low-pass filter and step delay at once; those are two views of one behaviour.
Does supply voltage change the time constant?
Not at all. τ depends purely on resistance and capacitance, so a 3.3 V rail and a 400 V rail fill on identical schedules through identical parts — vertical scale differs, timing does not. That independence is what makes RC networks dependable timers. Voltage matters for one practical reason though: leakage. Aluminium electrolytics pass a small current through their own dielectric, and beyond roughly ten seconds of constant that path effectively parallels your resistor, capping how high voltage ever climbs. Long timers use film parts for exactly that reason.