RC Circuit Calculator — Time Constant, Cutoff Frequency and Filter Response

Background.

How long does a capacitor take to charge through a resistor? The answer turns on a single product. Multiply the resistance in ohms by the capacitance in farads and you get the time constant τ, in seconds — an ohm times a farad is exactly a second. A 10 kΩ resistor feeding a 100 nF capacitor gives 1 ms; swap the capacitor for 1 µF and it becomes 10 ms. After one τ the capacitor has covered 63.2 % of the way to the supply, after five it is within 0.7 % of it, and in theory it never quite arrives.

The same pair of parts also sorts signals by frequency. A capacitor's opposition to alternating current, its reactance, falls as frequency rises, so a resistor and capacitor in series form a voltage divider whose ratio depends on frequency. Take the output across the capacitor and slow signals pass while fast ones are shunted away: a low-pass filter. Take it across the resistor and the reverse happens: a high-pass filter. Either way the dividing line sits at 1/(2πRC), which for the 1 ms pair is 159.15 Hz. Time constant and corner frequency are one fact written in two languages, which is why the capacitor that slows a logic edge is also the one that strips high-frequency noise from it.

One set of entries gives both views: τ and the corner, the capacitor voltage at any moment after a step, the 10–90 % rise time, the five-τ settling time, and the gain and phase at a frequency you choose.

All of it assumes ideal parts — a source with no internal resistance, no load on the output, and a capacitor exactly at its marked value. Capacitors are where real boards stray most, since a common ceramic can lose much of its printed capacitance once a DC voltage sits across it; the tolerance notes below show how far a built circuit may land from the figures.

What is rc circuit calculator?

An RC circuit is any arrangement in which current through a resistor charges or discharges a capacitor. Because it stores energy in only one place, the capacitor's electric field, it is a first-order system: its response to a sudden change is a single exponential, and its response to a sine wave has a single corner. The time constant τ = RC is the time needed to close 63.2 % of whatever gap remains between the capacitor's voltage and its final value. That proportion holds at every point on the curve, so the second τ closes 63.2 % of the remaining 36.8 %, bringing the total to 86.5 %, and so on. The corner frequency f_c = 1/(2πRC) is where the output voltage has fallen to 1/√2, about 0.7071, of the input, so the output power is half the input power. You meet RC networks as switch debouncers, power-on reset delays, supply decoupling, anti-aliasing filters ahead of an analogue-to-digital converter, coupling capacitors between amplifier stages, the timing element of a 555 timer, and snubbers across relay contacts and switching transistors.

How to use this calculator.

1. 01Enter the resistance and pick Ω, kΩ or MΩ. If the source driving the network has its own output resistance, add it to the resistor: a 600 Ω signal generator feeding a 1 kΩ resistor behaves as a 1.6 kΩ network.
2. 02Enter the capacitance and its unit as marked on the part. Small ceramics are often coded in picofarads with three digits — 104 means 10 followed by four zeros, 100,000 pF, which is 100 nF or 0.1 µF — and European schematics write the same part as 100n.
3. 03For a timing question, set the step voltage, the time since the step with its unit, and whether the capacitor is charging from 0 V or discharging from the supply. A negative or zero step is accepted, and the percentage output ignores the voltage entirely.
4. 04For a filter question, enter the signal frequency in hertz and choose low-pass (output across the capacitor) or high-pass (output across the resistor). The frequency has no effect on τ or on any of the transient results.
5. 05Compare the 10–90 % rise time, not τ, against a scope reading or a datasheet edge figure; it is 2.1972 times longer. Use the five-τ figure when the capacitor must have effectively finished moving, such as the wait before reading a sensor after power-up.

The formula.

Charging. Around the loop, the step voltage V₀ splits between the resistor and the capacitor: V₀ = iR + v, where v is the capacitor voltage. The capacitor's current is i = C·dv/dt, so V₀ = RC·dv/dt + v. That first-order differential equation, starting from 0 V, has the solution v(t) = V₀(1 − e^(−t/τ)) with τ = RC. Discharging from V₀ through the same resistor gives v(t) = V₀·e^(−t/τ). These are equations 10.8, 10.9 and 10.11 in §10.5 of OpenStax University Physics Volume 2. The percentage output is simply (1 − e^(−t/τ)) or e^(−t/τ) multiplied by 100, which is why it does not depend on the supply.

Rise time. Setting v/V₀ to 0.1 and to 0.9 and subtracting the two times gives t₉₀ − t₁₀ = τ·ln(0.9/0.1) = τ·ln 9 ≈ 2.1972τ. The same algebra answers any threshold question: the time to reach a fraction p of the step is τ·ln(1/(1 − p)).

Frequency response. A capacitor's reactance is X_C = 1/(2πfC). With the output across the capacitor, the divider ratio has magnitude 1/√(1 + (f/f_c)²), where f_c = 1/(2πRC) is the frequency at which X_C equals R. With the output across the resistor, f and f_c swap places. Writing x for f/f_c (low-pass) or f_c/f (high-pass), both cases become |H| = 1/√(1 + x²), and the gain in decibels is −10·log₁₀(1 + x²), Fiore's equation 10.14. The low-pass output lags the input by arctan(f/f_c); the high-pass output leads it by arctan(f_c/f). At f = f_c, x = 1, the gain is 1/√2 (−3.0103 dB) and the phase is 45° in either direction.

Two consequences fall out of the algebra. The squared low-pass and high-pass gains always add to exactly 1, and their phases are always 90° apart, so the two outputs split the input's power between them. And because f_c = 1/(2πτ), rise time and bandwidth are locked together: t_r = ln 9/(2πf_c) ≈ 0.35/f_c, the rule used to read an oscilloscope's bandwidth as an edge speed.

The model leaves out the capacitor's equivalent series resistance and inductance, which take over at high frequency; dielectric absorption, which lets a discharged capacitor creep back up a little; leakage; the source's output resistance unless you add it to R; and any load on the output node, which forms a second divider.

A worked example.

Frequently asked questions.

References& sources.

1. [1]OpenStax (Rice University), University Physics Volume 2, §10.5 'RC Circuits'. Source of the transient equations used here: Eq. 10.8 q(t) = Cε(1 − e^(−t/RC)) = Q(1 − e^(−t/τ)) with V_C(t) = ε(1 − e^(−t/τ)); Eq. 10.9 τ = RC; Eq. 10.11 q(t) = Qe^(−t/τ) for discharge; and the printed statement that at t = τ the charge has reached 1 − e^(−1) = 1 − 0.368 = 0.632 of maximum. Open access; retrieved 2026-07-29.
2. [2]James M. Fiore, AC Electrical Circuit Analysis: A Practical Approach, §10.3 'Bode Plots' (Engineering LibreTexts). Source of the frequency-domain equations: Eq. 10.15 f_c = 1/(2πRC) for both arrangements; Eq. 10.14 lag-network gain A'_v = −10·log₁₀(1 + f²/f_c²); Eq. 10.11 lead-network gain 1/√(1 + f_c²/f²); the printed −45° lag and +45° lead at the critical frequency; and the statement that break frequencies are 'also known as the half-power points, or corner frequencies'. Open access; retrieved 2026-07-29.
3. [3]OpenStax, University Physics Volume 2, §15.3 'RLC Series Circuits with AC'. Source of the capacitive reactance the divider rests on, Eq. 15.3 X_C = 1/(ωC), together with Eq. 15.9 φ = tan⁻¹((X_L − X_C)/R) and Eq. 15.11 Z = √(R² + (X_L − X_C)²) for the phase and magnitude algebra. Open access; retrieved 2026-07-29.
4. [4]James M. Fiore, AC Electrical Circuit Analysis: A Practical Approach, §8.2 'Series Resonance' (Engineering LibreTexts). Cited as an independent restatement of the half-power definition this page's −3.0103 dB band edge rests on: Eq. 8.2 f₀ = 1/(2π√LC) and Eq. 8.3/8.4 BW = f₂ − f₁ with Q = f₀/BW, where f₁ and f₂ are defined as the half-power frequencies. Open access; retrieved 2026-07-29.
5. [5]IEC 60050, International Electrotechnical Vocabulary (IEV Online / Electropedia), entry 131-12-46 'reactance', X = Im(Z) — the normative source for the vocabulary used on this page. ACCESS NOTE: electropedia.org returns HTTP 403 to automated retrieval, so the entry number and term were confirmed from the IEC's indexed entry title but the definition text was not opened. No figure on this page is taken from it.