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

Background.

An RC circuit is the simplest useful network in electronics: one resistor, one capacitor, and behaviour that turns up in almost every circuit anyone builds. It sets how fast a signal can change, how long a timer runs, which frequencies get through and which are thrown away. This calculator answers the whole family of questions at once — the time constant, the corner frequency, the voltage on the capacitor at any instant during a charge or discharge, and the gain and phase the network applies to a signal at any frequency you name.

The two answers people usually want are the same number seen from two directions. In the time domain the network is described by its time constant, tau, equal to resistance multiplied by capacitance. In the frequency domain it is described by its corner frequency, one divided by two pi times that same product. A network with a long time constant has a low corner frequency, and the two are linked by nothing more than a factor of 2π. Once you see that, a great deal of otherwise mysterious circuit behaviour becomes obvious: the reason a decoupling capacitor slows an edge is the same reason it removes high-frequency noise, and it is not a coincidence.

The transient side follows an exponential. Charging from zero, the capacitor voltage rises as the supply times one minus e to the minus t over tau; discharging, it falls as the supply times e to the minus t over tau. The landmarks are worth memorising because they come up constantly: 63.2 % after one time constant, 86.5 % after two, 95.0 % after three, and 99.3 % after five. Five time constants is the conventional engineering definition of settled, and it is the figure to use when sizing a power-up delay, an anti-bounce filter or a sample-and-hold acquisition window.

One number on this page is misread more often than any other, so it gets its own output: the 10 % to 90 % rise time. That is not the time constant. It is the time constant multiplied by the natural log of nine, about 2.197 times longer. Oscilloscope bandwidth specifications, logic-family datasheets and signal-integrity budgets are all quoted in 10–90 rise time, so treating tau and rise time as interchangeable understates a settling budget by more than a factor of two. Both appear here so the conversion never has to be done in your head.

The frequency side is a voltage divider between a resistor and a capacitor whose reactance falls as frequency rises. Take the output across the capacitor and you have a low-pass filter — Fiore's lag network — which passes low frequencies and rolls off above the corner. Take it across the resistor instead and you have a high-pass, the lead network, with the same corner frequency and the mirror-image response. The two are complementary: at every frequency their squared magnitudes sum to exactly one, and their phase shifts are exactly 90° apart. Both arrangements are available from the same dropdown here because they are the same two components.

A first-order RC filter rolls off at 20 dB per decade, which is gentle. At the corner frequency the output is 3.0103 dB down and shifted 45°; a full decade past the corner it is only about 20 dB down. If you need a sharper transition you need a higher-order filter, and this page will not design one for you — but its corner frequency is still the right starting point for each section.

A scope limit worth reading before you trust the last digits: this is the ideal, first-order, unloaded model. It assumes ideal components, a source with zero output impedance and nothing drawing current from the output node. Real capacitors have equivalent series resistance and inductance, class-2 ceramics such as X7R and Y5V lose a substantial fraction of their marked capacitance under DC bias, and a ±5 % resistor with a ±10 % capacitor already moves the corner frequency by roughly ±15 %. That tolerance is far larger than any digit this calculator prints, which is why the result is shown with the caveat attached rather than buried in an FAQ.

What is rc circuit calculator?

An RC circuit is a resistor and a capacitor connected so that current through the resistor charges or discharges the capacitor. It is a first-order system: one energy-storage element, one exponential, one corner frequency. Its defining constant is the time constant τ = RC, which has units of seconds because an ohm times a farad is a second. In the time domain τ is the time taken to cover 63.2 % of the remaining distance to the final value — a proportion that is the same at every point on the curve, which is what makes the response exponential rather than linear. In the frequency domain the same network has a corner frequency f_c = 1/(2πRC), the frequency at which the output power is half the input power and the output voltage is 1/√2 of the input. Which of the two roles the network plays depends only on where you measure. Output taken across the capacitor gives a low-pass filter, because a capacitor's reactance falls with frequency and increasingly shorts the output to ground. Output taken across the resistor gives a high-pass filter, because at low frequencies the capacitor blocks and almost all the input appears across it rather than the resistor. The same R and C, the same τ, the same f_c — only the node changes. RC networks appear as decoupling, as debounce, as anti-aliasing filters ahead of an ADC, as the timing element of a 555, as the feedback compensation around an op-amp, and as the RC snubber across a switching device. All of them are this one calculation.

How to use this calculator.

1. 01Enter the resistance and choose its unit. If your source has significant output impedance — a 50 Ω generator, a 600 Ω audio output, an op-amp's open-loop output resistance — add it to the resistor value, because the network sees the sum.
2. 02Enter the capacitance and choose its unit. Use the marked nominal value; the derating caveat beside the result covers what a real class-2 ceramic actually delivers under bias.
3. 03Read the time constant and corner frequency. These two are properties of the network alone and do not depend on any of the fields below them.
4. 04For a timing problem, set the step voltage, the elapsed time and whether the capacitor is charging or discharging. The calculator returns the voltage at that instant and the same figure as a percentage of the step.
5. 05Use the 10 %–90 % rise time, not the time constant, whenever you are comparing against an oscilloscope, a logic-family datasheet or a signal-integrity budget. It is 2.1972 times longer than τ.
6. 06For a filter problem, set the signal frequency and choose the arrangement: low-pass takes the output across the capacitor, high-pass across the resistor.
7. 07Read the gain in dB and the phase shift. At the corner frequency these are always −3.0103 dB and ∓45°, whatever the component values.
8. 08Treat the printed precision as arithmetic, not as a prediction. Component tolerance moves the real corner frequency by 10–20 % in a typical build, and further still with a class-2 ceramic under DC bias.

The formula.

Everything on this page comes from two equations and one conversion.

The transient. A capacitor's current is proportional to the rate of change of its voltage, and the resistor's current is set by the voltage across it, which gives a first-order differential equation whose solution is an exponential. OpenStax University Physics Volume 2 §10.5 prints it as Eq. 10.8 — q(t) = Cε(1 − e^(−t/RC)) = Q(1 − e^(−t/τ)), with V_C(t) = ε(1 − e^(−t/τ)) — and defines τ = RC as Eq. 10.9. Discharging is Eq. 10.11, q(t) = Qe^(−t/τ). The same text records that at t = τ the charge has reached 1 − e⁻¹ = 1 − 0.368 = 0.632 of maximum, which is where the familiar 63.2 % comes from. Note that the percentage output on this page is derived from the exponential alone, so it is identical whether your supply is 5 V, 1.8 V or −12 V.

Rise time. Solving V(t)/V₀ = 0.9 and 0.1 and subtracting gives t = τ·ln(0.9/0.1) = τ·ln 9 = 2.1972245773 τ. This is a different quantity from τ, and confusing the two is the single most common arithmetic error in RC timing work.

The frequency response. Fiore's AC Electrical Circuit Analysis §10.3 gives the corner frequency for both arrangements as the same expression, Eq. 10.15, f_c = 1/(2πRC), and records that break frequencies are also known as the half-power points or corner frequencies. The lag network's gain is Eq. 10.14, A'_v = −10·log₁₀(1 + f²/f_c²), and the lead network's is Eq. 10.11, 1/√(1 + f_c²/f²). Both fall out of the divider between R and the capacitor's reactance X_C = 1/(ωC), OpenStax Eq. 15.3. Writing x for f/f_c in the low-pass case and f_c/f in the high-pass case collapses both to one expression, |H| = 1/√(1 + x²), which is what this calculator evaluates.

Phase. The low-pass output lags by arctan(f/f_c) and the high-pass leads by arctan(f_c/f). At f = f_c both arctangents are exactly 45°, which is Fiore's printed −45° lag and +45° lead at the critical frequency. This is why the corner frequency can be found on a bench without a spectrum analyser: sweep until the scope shows 45° of phase shift.

Rounding stage, stated explicitly. Every quantity above is carried at full decimal precision and rounded once, at the return boundary, to ten decimal places. The gain in decibels is computed directly as −10·log₁₀(1 + x²) rather than by taking 20·log₁₀ of an already-rounded linear ratio, because rounding first and taking the logarithm second moves the fourth decimal of the dB figure. The band description beside the result classifies the unrounded gain, not the displayed one.

What the model leaves out. Equivalent series resistance and inductance in the capacitor; dielectric absorption, which is why a discharged capacitor can recover a fraction of its voltage; DC-bias and ageing derating in class-2 ceramics, which can remove more than half of a marked X7R value; source output impedance, unless you add it to the resistance; loading on the output node, which forms a second divider; and self-heating. Above roughly 100 kHz the ESR of an electrolytic dominates its impedance, and at that point this model is describing a different component from the one on your bench.

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.