Frequency Response and Filters

Nearly every circuit on this site is a filter of some kind — a tone control, a pickup's resonant peak, a coupling capacitor's bass rolloff, an output transformer's bandwidth — and all of them reduce to the same handful of RC, RL, and LC building blocks. This chapter teaches the cutoff-frequency formula, why a decibel is logarithmic, and why doubling a filter's reactive elements doubles its rolloff steepness, so every later chapter's filter math has a home to point back to.

Nearly every circuit covered anywhere on this site is a filter, whether or not it’s labeled that way. A guitar’s tone control is a low-pass filter. A pickup’s resonant peak is an LC circuit. A coupling capacitor sets an amp’s bass rolloff. An output transformer and a speaker cabinet are both band-limited on purpose. Every one of these reduces to the same small set of building blocks — one resistor and one capacitor, or an inductor and a capacitor together — and once those blocks are understood once, they don’t need re-deriving every time a new chapter mentions a “cutoff frequency.”

The decibel: a logarithmic unit for a logarithmic ear

Audio measurements are expressed in decibels because human hearing itself responds to loudness logarithmically, not linearly — doubling a signal’s voltage reads to the ear as a noticeable but not overwhelming jump (+6dB), not literally “twice as loud” in a linear sense. The number that matters most for filters specifically is -3dB, the point where power has dropped to half its original value — this is the standard, universal definition of a filter’s cutoff frequency, and it’s why every cutoff-frequency claim in this book means specifically “the point where the signal is down 3dB,” not some vaguer notion of “starting to roll off.”

Capacitors and inductors are frequency-dependent resistors

In a DC circuit, only resistance opposes current. In an AC circuit, capacitors and inductors oppose current too, and — critically — how much they oppose it depends on frequency. A capacitor’s reactance (X_C = 1/(2πfC)) falls as frequency rises, so a capacitor increasingly behaves like a short circuit to high frequencies and an open circuit toward DC — which is exactly why a coupling capacitor blocks DC while passing an audio signal through. An inductor’s reactance (X_L = 2πfL) does the opposite, rising with frequency — a dead short to DC, a high impedance to treble. Every filter in this book is built from these two frequency-dependent behaviors combined with an ordinary resistor.

The RC low-pass and high-pass: one arrangement, two behaviors

A resistor followed by a capacitor to ground forms a low-pass filter: at low frequencies the capacitor’s reactance is high and most signal reaches the output; at high frequencies the capacitor’s reactance drops and increasingly shunts signal to ground instead. Swap the resistor and capacitor’s positions — capacitor first, resistor to ground — and the same two parts form a high-pass filter, blocking DC and low frequencies while passing highs. Both share the identical cutoff-frequency formula, f_c = 1/(2πRC): a 100kΩ resistor and a 10nF capacitor put the cutoff around 159Hz, the exact math behind a guitar tone control’s rolloff (see Pots, Caps, and Tone Controls) and an amp’s coupling-capacitor bass limit alike — the same formula, two different jobs depending only on which side of the circuit the output is taken from.

LC resonance: the mechanism behind every “voiced” frequency response

Put an inductor and capacitor together instead of a resistor and capacitor, and the circuit stops being a simple rolloff and becomes resonant — at the specific frequency where the inductor’s and capacitor’s reactances become equal and cancel (f_res = 1/(2π√(LC))), the circuit’s impedance hits an extreme and the frequency response shows a peak (or notch, depending on the topology) instead of a smooth slope. This is the exact mechanism behind a magnetic pickup’s characteristic voice, covered in full in Pickup Theory and Types: the coil’s inductance and the combined self- and cable-capacitance form an LC pair, and the frequency where they resonate — not the pickup’s DC resistance — is what actually defines whether a pickup reads as bright or dark.

Filter order: why some rolloffs are gentle and others are steep

A filter built from one reactive element (one capacitor or one inductor) is first-order and rolls off at a fixed, gentle rate — 20dB per decade, roughly 6dB per octave — no matter what the specific component values are. Add a second reactive element and the filter becomes second-order, doubling that rolloff steepness to 40dB per decade; each additional order adds another 20dB per decade of steepness, at the cost of more parts and more phase shift near the cutoff. A simple guitar tone control is first-order. A wah pedal’s swept filter is second-order, which is part of why it sounds like a distinct, narrower “voice” sweeping through the spectrum rather than a gentle tilt. Recognizing a filter’s order from its schematic — count the capacitors and inductors actually in the signal path — predicts how sharply it’ll cut, before ever calculating a specific cutoff frequency.

Common mistake: assuming a filter’s nominal cutoff is the whole story

Calculating f_c = 1/(2πRC) from a schematic’s labeled resistor and capacitor values gives the textbook cutoff — but any circuit’s real-world behavior also depends on the source impedance feeding it and the load impedance it’s driving into, both of which effectively add to or interact with the filter’s own resistance. This is precisely why a guitar’s tone control at its “off” position doesn’t audibly do nothing even though the bare pot-and-cap formula suggests a cutoff below the audible range — the pickup’s own output impedance is part of the real circuit, not just the two parts drawn in the tone control’s own little box on the schematic. Whenever a calculated cutoff frequency doesn’t match what’s actually heard, the source and load impedance around the filter — not the filter’s own two components — are usually where the discrepancy lives.