How this instrument works
An RC pair is a single first-order system, and a first-order system has exactly one number describing how urgently it responds — everything else is that number wearing different units. Solve the charging equation and the natural answer is τ = RC, a duration: after one τ a capacitor has covered 63.2 percent of a step change, after five it is close enough to done that most instruments call it settled. Ask the same circuit how it treats a continuous sine wave instead of a single step, and the identical R and C reappear as f_c = 1 ⁄ (2πRC), the frequency at which the capacitor's reactance equals the resistor's value and the output has fallen to 70.7 percent of the input. Nothing new was measured; the same physical pair was simply asked a different question.
The factor of 2π connecting the two is not decoration. τ falls out of an equation written in plain seconds, with no angle anywhere in it. f_c, by contrast, describes a rotating sinusoid, and a sinusoid completes one full turn — 2π radians — for every cycle. Converting the pair's natural time constant into a frequency in hertz means first finding the angular rate ω = 1 ⁄ τ, then dividing by 2π to count whole revolutions per second rather than radians per second. Skip that division and you get a number in the right units but off by a factor of roughly 6.28 — a common slip for anyone porting a control-systems time constant into an audio or RF cutoff spec.
Both readouts assume one clean pole: an ideal voltage source driving R into an ideal capacitor with nothing downstream drawing current. A real op-amp input, ADC sample-and-hold, or second RC stage placed right after this one adds its own impedance and its own pole, and the moment that happens neither τ nor f_c alone still describes the full circuit — the two numbers are only exact for this one RC section in isolation, which is precisely the case this instrument computes.
- Enter the resistor's value into Resistance, in ohms or kilohms — whichever part actually sits between the source and the capacitor.
- Enter the capacitor's value into Capacitance, switching the unit menu to nF or µF to match the marking on the part.
- Read RC time constant for how the pair answers a step: multiply it by five for a practical settling time to near-final value.
- Read Cutoff frequency for how the same pair answers a continuous sine wave: the point where output power has already halved.
- If a design brief specifies one of the two readouts, adjust Resistance or Capacitance until it lands there, then check the other readout still fits your timing or bandwidth budget.
Worked example — a 10 kΩ, 100 nF anti-alias stage on a scanned channel
An instrumentation engineer is building an eight-channel thermocouple scanner: one analog multiplexer feeds a single ADC, stepping through channels in turn, and each channel gets its own small RC stage ahead of the mux to knock down mains hum and switching noise before it aliases into the reading. Sizing that stage with a 10 kΩ resistor and a 100 nF capacitor, she enters 10 into Resistance (kΩ) and 100 into Capacitance (nF). RC time constant returns τ = 10,000 Ω × 0.0000001 F = 0.001 s, shown as 1 ms. Cutoff frequency returns f_c = 1 ⁄ (2π × 10,000 × 0.0000001) = 159.154943092 Hz, shown as 159.155 Hz — comfortably below the mains-hum band she wanted attenuated.
The frequency spec alone looks finished, but the same 1 ms time constant carries a settling requirement the cutoff number never mentions: after the multiplexer steps to a new channel, this stage needs roughly five time constants, about 5 ms, before its output has stopped carrying a trace of the previous channel's voltage. If the scanner's firmware only dwells 2 ms per channel before sampling, every reading still includes a sliver of whatever came before it — a crosstalk error a frequency-domain datasheet spec would never flag, because it only shows up when the same RC pair is read as a settling-time problem instead of a filtering one.
Questions
Why does one resistor-capacitor pair produce two different numbers?
Because a time constant and a cutoff frequency are the same physical behavior read in two domains. τ = RC describes how the pair answers a single step, in seconds; f_c = 1 ⁄ (2πRC) describes how it answers a continuous sine wave, in hertz. A 10 kΩ, 100 nF pair is simultaneously a 1 ms step response and a 159 Hz corner — nothing about the circuit changed between those two descriptions, only the question asked of it.
What happens to both readouts if I double the resistance?
Time constant doubles and cutoff frequency is cut in half, since R sits directly in τ = RC and in the denominator of f_c = 1 ⁄ (2πRC). Taking the worked example's 10 kΩ and 100 nF to 20 kΩ moves τ from 1 ms to 2 ms and f_c from 159.155 Hz down to 79.577 Hz — a slower-responding, lower-cutoff filter from the same capacitor alone.
Does doubling the capacitance instead do the same thing?
Yes — R and C enter both formulas identically, so doubling either one has the identical effect. Holding resistance at 10 kΩ and raising capacitance from 100 nF to 200 nF also carries τ from 1 ms to 2 ms and f_c from 159.155 Hz to 79.577 Hz, the exact same pair of results doubling R alone produces.
Why does 2π appear in the cutoff-frequency formula but not in the time constant?
Because τ is a plain duration with no angle in it, while f_c counts complete sinusoidal cycles per second. The natural rate that falls out of the circuit equation is an angular one, ω = 1 ⁄ τ, measured in radians per second; dividing by 2π radians per cycle converts that into hertz. Forgetting that division is a common source of being off by roughly 6.28 when converting a control-systems time constant into a frequency spec.
Can a filter that looks correct at its cutoff frequency still fail on timing?
Yes, and it is a genuine trap in multiplexed or sampled systems. A stage sized purely to hit a target f_c also carries a fixed settling time of about 5τ; if a multiplexer or sample-and-hold circuit switches faster than that, the output still carries a remnant of the previous signal even though the frequency-domain spec was met exactly.