How this instrument works
Nothing actually cuts off at a cutoff frequency. Pick any series resistor feeding one capacitor to ground and you have built a voltage divider whose lower leg changes with frequency: capacitive reactance 1 ⁄ (2πfC) starts enormous and shrinks decade by decade, while R holds still. Somewhere those two meet as equals; solving 1 ⁄ (2πfC) = R for f hands you 1 ⁄ (2πRC) directly. That crossing is your corner. Because reactance and resistance add in quadrature rather than head-on, amplitude there falls to 1 ⁄ √2, or 0.7071 of input — half power, arriving 45 degrees late.
Why three decibels and not some rounder loss? That threshold is an artefact of one unit invented for telephone circuits. Bell System engineers graded lines in miles of standard cable until W. H. Martin proposed a logarithmic transmission unit in 1924; four years later it was renamed decibel, honouring Alexander Graham Bell. Half power in that arithmetic works out at 10 log₁₀(0.5) = −3.0103 dB, near enough whole that −3 dB became shorthand for it. Hendrik Bode, also at Bell Laboratories, supplied a second reason to care in 1940: sketch gain against frequency on log paper, draw one flat asymptote plus one sloping at 20 dB per decade, and they intersect exactly at 1 ⁄ (2πRC). Measured response bends smoothly around that intersection, passing 3 dB beneath it — which is how a curve with no corner acquired its corner frequency.
Three assumptions hold this expression up, and hardware tests each one. R must include whatever source impedance drives that node, so one filter drawn around 1 kΩ but fed from a 600 Ω generator really corners at 1 ⁄ (2π × 1600 × C). Whatever comes next must draw negligible current, since any load resistance across your capacitor parallels itself into R and drags your corner upward. And C must genuinely be C — Class II dielectrics surrender capacitance as bias voltage and temperature climb, so an X7R part marked 1 µF can behave like 600 nF once installed, throwing your corner two thirds higher than intended. One pole is all this describes. Cascade sections, add inductance, or build a Sallen-Key stage and 1 ⁄ (2πRC) stops naming your −3 dB point, though it still names each individual pole.
- Enter Resistance in ohm, kohm or Mohm — total series resistance, including output impedance of whatever drives this network.
- Set Capacitance in F, mF, uF, nF or pF, straight off part markings, and derate Class II ceramics if bias voltage sits across them.
- Read Cutoff (−3 dB) frequency in Hz, kHz or MHz. That figure holds whether your capacitor sits to ground (low-pass) or in series (high-pass).
- Sanity-check by decade: ten times either component gives one tenth of that frequency, since both sit in one denominator.
Worked example — 1 µF coupling into a 1 kΩ pedal input
A stompbox input stage: one 1 µF film capacitor blocking DC, feeding 1 kΩ of bias resistance to ground. Put 1000 into Resistance and 1 µF into Capacitance, and Cutoff (−3 dB) frequency returns 159.154943 Hz — that being 1 ⁄ (2π × 1000 × 0.000001), or 1 ⁄ 0.0062831853. Above 159 Hz your signal passes essentially untouched; below it, capacitor reactance climbs past 1 kΩ and begins winning that divider.
Which is a problem, because a guitar's low E string rings at 82.4 Hz. Sitting an octave under that corner, it arrives 6.8 dB down; a bass guitar's 41.2 Hz open E loses 12 dB, a quarter of its amplitude, and both collect phase shift that blunts pick attack. Swap 1 µF for 10 µF and Cutoff (−3 dB) frequency drops to 15.9 Hz, safely below anything a player can produce. Coupling capacitors get chosen exactly this way in every audio stage: name your lowest wanted note, then size C so 1 ⁄ (2πRC) sits well beneath it.
Questions
Why is cutoff defined at −3 dB and not where output stops?
Because first-order filters never stop anything — they only roll off, forever approaching zero without arriving. Engineers needed one agreed landmark, and half power was an obvious pick: it is where resistor and capacitor contribute equally, where phase has swung 45°, and where amplitude sits at 0.7071 of passband. Expressed logarithmically that loss is 10 log₁₀(0.5) = −3.0103 dB, which everybody rounds to 3. Other fields chose other landmarks — −1 dB for passband ripple specifications, −6 dB in ultrasound and imaging — so check which convention a datasheet means before comparing numbers.
Does this formula work for high-pass filters as well?
Yes, identically. Swap positions — capacitor in series, resistor to ground — and you get high-pass behaviour whose corner still lands at 1 ⁄ (2πRC). Only which side survives changes. Same parts, same 159 Hz corner for 1 kΩ with 1 µF; low-pass keeps everything beneath that figure, high-pass everything above. Phase differs in sign: low-pass output lags 45° at cutoff, high-pass leads 45°. Chain one of each with well-separated corners and you have crude band-pass response.
Why is my measured corner higher than calculated?
Almost always because R or C is not what you typed. Load resistance from a following stage sits in parallel with your shunt element, dropping effective R and pushing that corner upward — one 10 kΩ filter feeding a 10 kΩ input really corners at double its design figure. Class II ceramics, X5R and X7R especially, shed much of their marked capacitance under DC bias, where 40 percent loss near rated voltage is unremarkable. Source impedance pushes matters back down instead, adding to R. Measure C on an LCR bridge at operating bias before blaming arithmetic.
How does cutoff frequency relate to a time constant?
Through 2π, and dropping that factor is a genuinely common slip. Since f_c = 1 ⁄ (2πRC) while τ = RC, one is simply 1 ⁄ (2πτ): a network settling on a scale of 1 ms corners near 159 Hz, not at 1000 Hz as people tend to guess. That factor exists because capacitor behaviour is written against angular frequency, counted in radians per second, whereas filters get specified in cycles per second. Any time your answer lands about 6.28 times adrift, suspect this.
Can I cascade two RC sections for a sharper cutoff?
You can steepen roll-off, though not at an unchanged corner. Two isolated identical sections give 40 dB per decade instead of 20, yet each contributes its own 3 dB of loss, so combined response is 6 dB down at that frequency. Your true −3 dB point therefore slides to 0.6436 of single-section value — 159 Hz becomes 102 Hz. Skip buffering between them and it is worse, since section two loads section one and moves both poles. Passive cascades also round off gently near their knee; anything needing sharp transitions wants active topology.
What is happening to phase at the corner?
Output lags input by exactly 45° for low-pass RC, and leads by 45° for high-pass. That shift is not confined to one frequency: it runs from near 0° a decade below to near 90° a decade above, crossing 45° precisely at cutoff. Gradual swing of that kind matters inside feedback loops, where lag accumulated across several poles eats phase margin and can turn a stable amplifier into an oscillator. It matters in audio measurement too — filters can look flat in amplitude across a band while still smearing transient edges through phase alone.