The filter cutoff calculator above answers the question every passive filter design starts with: at what frequency does this circuit begin to throw signal away? For a resistor and capacitor, or a resistor and inductor, the answer is a single division. For a resistor, inductor and capacitor together the answer involves a square root and a damping term. All three cases are covered here, in both directions, so you can start from the parts you have or from the frequency you want.
Arb Digital builds free tools that pick one job and finish it properly. This page reports more than the cutoff, because the cutoff alone tells you little. What matters is how much unwanted signal survives at a specific frequency, and how much phase the wanted signal picks up. Both are in the result grid.
What This Filter Cutoff Calculator Does
A passive filter is a frequency-dependent voltage divider. In an RC low-pass filter the resistor is fixed but the capacitor's impedance falls as frequency rises, so the fraction of the input that reaches the output shrinks with frequency. Swap the two components and the same circuit becomes a high-pass filter. The frequency at which the resistance and the reactance are numerically equal is the cutoff, and at that point the output is exactly one over the square root of two of the input.
The tool handles three topologies. RC is the workhorse: cheap, stable, used everywhere from anti-aliasing to supply decoupling. RL does the same job with an inductor, rare in signal work and common in power work. LC is second order and rolls off twice as fast, at the price of a resonance that has to be damped.
Running it backwards is often more useful. Capacitors come in far fewer values than resistors, so the normal workflow is to fix the capacitor, enter the cutoff you want, and let the calculator return the resistance, snapped to the nearest standard E24 value because no supplier stocks 15,915 ohms.
How to Use It
- Pick the topology and the response first. Low pass and high pass share the same cutoff formula, so the frequency will not change, but the gain and phase at your test frequency will change completely.
- Decide which unknown you want. Leave the mode on cutoff frequency if you have the parts in front of you. Switch to a component mode if you have a target frequency instead.
- Fix the capacitor or inductor before the resistor. Reactive components have coarse value steps and poor tolerance; resistors are cheap, precise and available in ninety-six values per decade.
- Set the test frequency to something you actually care about. The frequency of the interference you want gone, or the highest frequency in the signal you want kept.
- Read the gain figure, not just the cutoff. A first-order filter one decade past cutoff has only removed 20 dB. If you needed 60 dB, this topology cannot do it and the answer is more stages or a different order.
The Formula: How the Cutoff Frequency Is Calculated
For a first-order RC filter the cutoff is fc = 1 ÷ (2πRC), with resistance in ohms, capacitance in farads and the result in hertz. The same relationship written as a time constant is τ = RC, and the two are linked by fc = 1 ÷ (2πτ). OpenStax University Physics Volume 2, section 10.5 on RC circuits, gives the time constant directly and confirms that the product of resistance and capacitance has units of seconds.
For a first-order RL filter the cutoff is fc = R ÷ (2πL), and the time constant is τ = L ÷ R. Note that the roles invert: raising the resistance raises the cutoff in an RL filter and lowers it in an RC filter, which trips people up when they swap topologies mid-design.
For the second-order LC case the corner sits at fc = 1 ÷ (2π√(LC)), the same expression as a resonant tank. The resistor no longer sets the frequency; it sets the damping ratio ζ = (R ÷ 2) × √(C ÷ L) for the series arrangement used here, and the quality factor is Q = 1 ÷ (2ζ). OpenStax University Physics Volume 2, section 15.3 on RLC series circuits with AC, sets out the impedance behaviour that produces this.
Gain at any frequency comes from the ratio x = f ÷ fc. For a first-order low pass the magnitude is 1 ÷ √(1 + x²) and the phase is −arctan(x). For a first-order high pass the magnitude is x ÷ √(1 + x²) and the phase is +arctan(1 ÷ x). Decibels are 20 × log10 of the magnitude, because these are voltage ratios rather than power ratios.
Work the defaults. With R = 10,000 ohms and C = 100 nanofarads, the product RC is 0.001 seconds, so fc = 1 ÷ (2π × 0.001) = 159.15 hertz. At a test frequency of 1,000 hertz, x = 1000 ÷ 159.15 = 6.283. The low-pass magnitude is 1 ÷ √(1 + 39.48) = 0.1572, which is −16.07 dB, and the phase is −arctan(6.283) = −80.96 degrees.
Why the Cutoff Is Defined at Minus Three Decibels
The choice looks arbitrary until you look at power rather than voltage. At the cutoff frequency the resistance and the reactance are equal in magnitude, and because they combine as the square root of the sum of squares, the total impedance is √2 times the resistance alone. The output voltage is therefore 1 ÷ √2 of the input, which is 0.7071, or 70.71 per cent.
Square that voltage ratio and you get 0.5: the output power is exactly half the input power. Half power in decibels is 10 × log10(0.5) = −3.01 dB, rounded to −3. The −3 dB point is therefore not a tidy convention but the frequency at which the filter has given away half the power, which is why the same number defines bandwidth for amplifiers and antennas too.
The practical consequence is that a filter is already working well before its cutoff. At half the cutoff a low-pass filter passes 89 per cent, a loss of 1 dB, which is audible in a loudspeaker crossover. If you need a genuinely flat passband up to some frequency, put the cutoff two or three times higher than that frequency.
What Twenty Decibels Per Decade Really Buys You
Far above the cutoff of a first-order low-pass filter, the magnitude approaches 1 ÷ x, so every tenfold increase in frequency divides the output by ten. That is 20 dB per decade, or equivalently 6 dB per octave. It is a fixed property of the topology and cannot be improved by choosing better components.
This matters when a single RC section is asked to remove interference just outside the passband. If the wanted signal ends at 1 kHz and the interference sits at 3 kHz, a 1.5 kHz cutoff leaves the interference only a factor of two out, where a first-order filter attenuates by about 7 dB. That is a halving, not the elimination people expect.
Cascading helps, but not as simply as multiplying. Two identical RC sections each drop 3 dB at the original cutoff, so the pair is 6 dB down there and its effective corner has moved lower, while the second section loads the first unless a buffer separates them. An LC section reaches 40 dB per decade in one go without loading, which is why it dominates power supply filtering.
Phase Shift Is the Cost Nobody Budgets For
A first-order low-pass filter lags its output behind the input by 45 degrees at the cutoff frequency, approaching 90 degrees far above it. A high pass leads by 45 degrees at cutoff. That phase shift is not a defect; it is inseparable from the amplitude response of any causal filter.
In audio it goes unnoticed alone but becomes obvious when a filtered path is summed with an unfiltered one, as in a crossover, where the phase difference turns a flat sum into a notch or a peak. In a feedback loop it is far more serious: every filter eats phase margin, and enough accumulated lag turns negative feedback into oscillation. Inside a control loop, the phase figure in the result grid is the number to watch, not the gain.
Choosing Component Values That Behave
An infinite number of resistor and capacitor pairs give the same cutoff, and they are not equivalent in practice. Very high resistances pick up noise and let the next stage's input bias current create an offset. Very low resistances load whatever drives the filter. For small-signal work, 1 kilohm to 100 kilohms is the comfortable band.
Capacitor dielectric matters more than tolerance. A class II ceramic such as X7R can lose a large fraction of its nominal capacitance when a DC bias is applied across it, so a filter designed for 1 kHz can end up at 2 kHz in the finished board with no component fault to find. Film capacitors and C0G ceramics hold their value and are what you want anywhere the cutoff frequency is specified rather than approximate. The capacitance converter and the resistance converter handle the unit changes between farads, microfarads, nanofarads, ohms and megohms.
Source and load impedance are part of the filter whether you intended them to be or not: the driving stage's output resistance adds to the filter resistance and the load shunts it, and both shift the cutoff. The voltage divider calculator shows how much signal a loaded divider really delivers, and the Ohm's law calculator covers current and dissipation.
Arb Digital builds free tools like this one because useful pages earn attention. If you want tools, calculators or content built for your own audience, we can help.
Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Treating the cutoff as a wall — the output is already down 3 dB there and has been falling for an octave before it. Place the cutoff well outside the band you want untouched.
- Mixing up RC and RL behaviour — raising the resistor lowers an RC cutoff and raises an RL cutoff. The formulas invert, so a habit built on one topology misleads on the other.
- Ignoring source and load impedance — the driving stage's output resistance adds to the filter resistance and the following stage's input resistance shunts it. Both move the cutoff away from the calculated value.
- Using an X7R or Y5V ceramic where the frequency matters — DC bias and temperature can move the capacitance far enough to shift the cutoff by an octave.
- Expecting one RC stage to remove nearby interference — at twice the cutoff a first-order filter attenuates by only about 7 dB, which is a halving, not a removal.
Related Free Tools From Arb Digital
The same RC product read as a settling time is what the RC time constant calculator reports, the companion page for step-response work. The LC resonant frequency calculator covers tuned circuits, and the reactance calculator gives the impedance of a single capacitor or inductor, the quantity the cutoff condition sets equal to the resistance. Decode parts with the resistor colour code calculator, change units with the frequency converter, and browse the free online tools hub.
Frequently Asked Questions
The cutoff is one divided by the product of two pi, the resistance in ohms and the capacitance in farads. With 10 kilohms and 100 nanofarads that gives 159.15 hertz. The same circuit has a time constant of RC, which is one millisecond in this example.
Because at that frequency the resistance and the reactance are equal, so the output voltage is one over the square root of two of the input, which is 70.71 per cent. Squaring that gives exactly half the power, and half power in decibels is minus 3.01.
No. The same resistor and capacitor give the same cutoff frequency in either arrangement, because the cutoff is the point where the two impedances are equal. What changes is which side of that frequency is kept and the sign of the phase shift.
About 20 decibels, which is a factor of ten in voltage. That rate, 20 dB per decade or 6 dB per octave, is fixed by the topology and cannot be improved by using better components.
Because inductive reactance rises with frequency while capacitive reactance falls. The RL cutoff is the resistance divided by two pi times the inductance, so increasing the resistor raises the cutoff, where in an RC filter increasing the resistor lowers it.
A first-order low pass lags by 45 degrees at cutoff and approaches 90 degrees far above it. A first-order high pass leads by 45 degrees at cutoff. This shift is inseparable from the amplitude response and matters most inside feedback loops.
Usually the capacitor or the surrounding impedances. Class II ceramics such as X7R lose capacitance under DC bias, and the driving stage's output resistance adds to the filter resistance while the load shunts it. Both move the corner frequency.
The far-out rolloff does double to 40 dB per decade, but the combined response is 6 dB down at the original cutoff rather than 3, so the effective corner moves lower. The second stage also loads the first unless a buffer separates them.
This tool is provided for educational and design-estimating use. Component tolerances and surrounding circuit impedances mean a built filter will differ from the calculated figures, and nothing on this page is electrical safety guidance.
