What an RC filter does and where the corner comes from
An RC filter is a resistor and a capacitor in series across a signal, with the output taken from the junction between them. It works because the capacitor's reactance depends on frequency while the resistor's does not. Capacitive reactance is X_C = 1/(2πfC), so at low frequencies the capacitor looks like a large resistance and at high frequencies it looks like a short. The pair is a frequency-dependent voltage divider, and which side you take the output from decides which half of the spectrum survives.
Take the output across the capacitor and you have a low-pass. At DC the capacitor blocks everything, no current flows, and the output equals the input. Climb in frequency and the capacitor's reactance falls until it is small compared with R, at which point almost all of the input is dropped across the resistor and the output falls away. Take the output across the resistor instead and every statement reverses: you have a high-pass that blocks DC entirely and passes high frequencies with essentially no loss.
The corner frequency is the point where the two elements contribute equally — where the capacitive reactance equals the resistance. Set 1/(2πfC) = R and solve for f and you get fc = 1/(2πRC). At that frequency the divider is not half, because the two contributions are 90° apart: they add as the hypotenuse of a right triangle, giving √2 rather than 2, so the output is 1/√2 = 0.7071 of the input. Power is voltage squared over resistance, so half the voltage-squared is half the power, and half the power is −3.0103 dB. That is the whole reason the “−3 dB point” is the standard definition of a cutoff.
If you want just the reactance term at a single frequency, the capacitive reactance calculator gives X_C directly. If your network also contains an inductor, the response is second-order and you want the resonant frequency calculator instead.
Gain, phase and the 20 dB per decade slope
Far from the corner, a first-order filter has a simple, memorable behaviour: every factor of ten in frequency changes the gain by a factor of ten, which is 20 dB. That is the definition of 20 dB per decade, equivalently 6.02 dB per octave. One decade past the corner a low-pass is 20.04 dB down; two decades past it is 40.00 dB down. The tiny excess over the round number at one decade comes from the “1 +” in the denominator and vanishes as you go further out.
Close to the corner the response is a smooth curve, not two straight lines. At half the corner frequency a low-pass is only 0.97 dB down; at twice the corner it is 6.99 dB down. Sketching the asymptotes and marking the true −3.01 dB at the corner reproduces the exact curve to better than a decibel everywhere, which is the classic Bode approximation.
Phase is the part designers underestimate. A first-order low-pass shifts phase by −45° exactly at the corner, tends to 0° well below it and approaches −90° well above. The shift is already −26.6° at half the corner and −11.3° a full octave and a bit below. If the filter sits inside a feedback loop, that phase eats directly into your phase margin at frequencies where the amplitude loss still looks negligible. A high-pass has the mirror behaviour: +90° at DC, +45° at the corner, tending to 0° far above.
One consequence worth internalising: for a first-order filter the gain and phase are locked together. You cannot get more attenuation at a given frequency without also getting more phase shift, because both come from the same single pole. Buying a steeper skirt means adding poles — cascading sections, or moving to an active Sallen-Key or multiple-feedback topology — and each pole brings up to another 90° with it.
Worked example: 10 kΩ with 10 nF, low-pass
You are anti-aliasing an audio-band signal with a 10 kΩ resistor and a 10 nF capacitor and want to know what it does to a 15.915 kHz tone.
- Time constant. τ = R × C = 10,000 × 1.0000 × 10⁻⁸ = 1.0000 × 10⁻⁴ s, or 100 µs.
- Corner frequency. fc = 1/(2πτ) = 1/(6.283185 × 1.0000 × 10⁻⁴) = 1/(6.283185 × 10⁻⁴) = 1,591.549 Hz.
- Frequency ratio. f/fc = 15,915.494 / 1,591.549 = 10.0000. Exactly one decade above the corner.
- Gain. |H| = 1/√(1 + 10²) = 1/√101 = 1/10.049876 = 0.0995037.
- Decibels. 20 × log₁₀(0.0995037) = 20 × (−1.0021607) = −20.0432 dB.
- Phase. φ = −arctan(10.0000) = −84.2894°.
Sanity check the “20 dB per decade” claim against the corner itself: at f/fc = 1 the gain is 1/√2 = 0.707107, which is −3.0103 dB. From −3.0103 dB to −20.0432 dB across one decade is 17.03 dB, not 20 dB, because the corner sits on the curved part of the response rather than on the asymptote. Go out another decade to f/fc = 100 and the gain is 1/√10001 = 0.00999950, which is −40.0004 dB — and −40.0004 minus −20.0432 is 19.96 dB, essentially the full 20 dB slope now that both points are on the asymptote.
Now the design direction. You want the corner at 1.000 kHz instead. Keeping the 10 nF capacitor, R = 1/(2π × 1,000 × 1.0000 × 10⁻⁸) = 1/(6.283185 × 10⁻⁵) = 15,915.49 Ω, so you fit the nearest standard value, 16 kΩ, which lands the corner at 1/(2π × 16,000 × 10⁻⁸) = 994.72 Hz. Keeping the 10 kΩ resistor instead, C = 1/(2π × 1,000 × 10,000) = 1/(6.283185 × 10⁷) = 1.59155 × 10⁻⁸ F, or 15.92 nF.
How to choose the corner and read the result
Place the corner by asking what has to survive and what has to die, then check both ends. For an anti-aliasing filter ahead of an ADC, the requirement is that anything above half the sample rate is attenuated below the converter's resolution. A single RC gives 20 dB per decade, so a 12-bit converter needing about 74 dB of rejection needs nearly four decades of separation between the corner and the offending frequency — which is usually impossible, and is the honest reason most anti-aliasing filters are multi-pole active designs rather than one R and one C.
For noise filtering and supply decoupling the demands are gentler and a single RC is often exactly right. A 100 µs time constant on a sensor line removes megahertz-scale pickup while leaving a 100 Hz signal untouched at 0.02 dB. Read the phase figure at your signal frequency before you accept that, though: 0.02 dB of amplitude loss comes with 3.6° of phase lag, which matters in a servo loop and does not matter in a temperature reading.
Watch the impedances at both ends. The R in the formula is the total series resistance, so a 600 Ω source driving a 1 kΩ filter resistor makes the effective R 1,600 Ω and moves the corner by 60%. At the output, any load resistance appears in parallel with the capacitor for a low-pass and changes both the corner and the passband gain. The classic fix is to make the filter impedance low compared with the load and high compared with the source, or to buffer both ends.
Finally, respect the capacitor. Class II ceramics such as X7R lose a large fraction of their capacitance under DC bias and drift with temperature, so a corner set by an X7R part is not a corner you can hold to a few percent. C0G/NP0 ceramics, polypropylene and polystyrene films are the parts to use when the frequency has to be repeatable. If you need to express the resulting attenuation alongside other gains and losses, the decibel calculator handles the arithmetic of combining them.
First-order RC response by frequency ratio
| f / fc | Low-pass gain | Low-pass (dB) | Low-pass phase | High-pass (dB) | High-pass phase |
|---|---|---|---|---|---|
| 0.1 | 0.99504 | −0.043 | −5.71° | −20.043 | +84.29° |
| 0.2 | 0.98058 | −0.170 | −11.31° | −14.150 | +78.69° |
| 0.5 | 0.89443 | −0.969 | −26.57° | −6.990 | +63.43° |
| 1 | 0.70711 | −3.010 | −45.00° | −3.010 | +45.00° |
| 2 | 0.44721 | −6.990 | −63.43° | −0.969 | +26.57° |
| 5 | 0.19612 | −14.150 | −78.69° | −0.170 | +11.31° |
| 10 | 0.09950 | −20.043 | −84.29° | −0.043 | +5.71° |
| 100 | 0.01000 | −40.000 | −89.43° | −0.0004 | +0.57° |
Read the table as ratios and it applies to every RC filter ever built. Multiply the ratio column by your own corner frequency to get real frequencies.
What a single RC does not do
- It does not give a flat passband and a sharp skirt. The response starts bending an octave either side of the corner. If you need flat to 20 kHz and 60 dB down at 24 kHz, no single RC will do it and no combination of R and C will change that.
- It does not isolate the source from the load. Source impedance adds to R and load impedance shunts the output, and both move the corner. Buffer the filter or design its impedance to sit well between the two.
- Cascading two RC sections does not simply double the slope near the corner. Each section loads the other, so two identical 1 kHz sections in series do not give a 1 kHz two-pole response — the combined −3 dB point moves down and the transition is soft. Buffer between sections or use a proper two-pole topology.
- It cannot amplify. The passband gain of a passive RC is at best unity, and it is exactly unity only well away from the corner. Anything needing gain and filtering together is an active filter.
- The capacitor is rarely the value printed on it. X7R and Y5V ceramics lose capacitance with applied DC voltage and with temperature. Tolerance on the corner frequency is dominated by the capacitor in nearly every design.
- Phase shift arrives long before attenuation does. At a tenth of the corner a low-pass has lost 0.04 dB but has already shifted the signal by 5.7°. In control loops the phase, not the amplitude, is usually what bites.
Key terms
- Corner or cutoff frequency
- The frequency at which output power falls to half the passband value, so voltage falls to 1/√2 = 0.7071 and the gain is −3.0103 dB. Also called the break or −3 dB frequency.
- Time constant (τ)
- The product RC, in seconds. A step applied to the network reaches 63.2% of its final value in one time constant, and the corner frequency is 1/(2πτ).
- Decade and octave
- A decade is a factor of ten in frequency, an octave a factor of two. A first-order filter rolls off 20 dB per decade, which is the same slope as 6.02 dB per octave.
- Pole
- Each independent energy-storage element that shapes the response contributes one pole. One pole gives 20 dB per decade and up to 90° of phase shift; n poles give 20n dB per decade.
When to reach for something other than one R and one C
A single RC is the right answer far more often than its reputation suggests. Debouncing a switch, slowing an edge to cut radiated emissions, removing supply noise from a reference, blocking DC ahead of an AC-coupled input, taming a sensor line — all of these want gentle, unconditionally stable filtering with no active parts, and one resistor and one capacitor deliver it for pennies with no power supply and no stability question.
Move to an active filter when you need a steeper transition, a gain other than unity, or immunity to source and load impedance. A Sallen-Key or multiple-feedback section gives you two poles per op-amp and a designable Q, so you can choose a Butterworth response for flatness, a Bessel for phase linearity, or a Chebyshev for the steepest skirt you can tolerate ripple for. When the signal has already been digitised, a digital filter gives arbitrary responses with no component tolerance at all — which is why so many modern signal chains use one modest RC in front of the converter and do the rest in software.
An LC filter is the choice at radio frequencies and in power circuits, where the resistor's loss is unacceptable and an inductor is practical. Where an RC dissipates the energy it rejects, an LC reflects it, and an LC section gives two poles with no active device. The trade is that the LC pair resonates, so the response near the corner depends on damping — see the RLC impedance calculator for how the magnitude and phase behave through that region, and the LC resonance calculator for the peak itself.
Finally, remember the time-domain view. The same RC that gives a 1,591 Hz corner gives a 100 µs time constant, so a step at the input reaches 63.2% in 100 µs and 99.3% in five time constants, 500 µs. Rise time from 10% to 90% is 2.2τ = 220 µs, and the product of rise time and bandwidth is the familiar 0.35. Frequency response and step response are the same fact stated twice.