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Analog Filters

Cutoff, resonance and component-selection calculators for RC, RLC and Sallen-Key filters, plus reference tables for common filter families.

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Cutoff, Q and roll-off are idealized. Real op-amps add open-loop limits, offset and GBP effects — verify against the op-amp datasheet and simulate with your tool of choice.

RC Low-Pass Cutoff & Attenuation

fc = 1 ÷ (2π·R·C); attenuation A = −20·log10(√(1 + (f/fc)²)) dB at frequency f.

Resistance R

Capacitance C

Probe frequency f

Cutoff fc (−3 dB)

—

Attenuation at f

—

Below fc the signal passes with little loss; well above fc it rolls off at 20 dB/decade. Leave f empty to read only the cutoff.

RC High-Pass Cutoff & Attenuation

fc = 1 ÷ (2π·R·C); attenuation A = −20·log10(√(1 + (fc/f)²)) dB at frequency f.

Resistance R

Capacitance C

Probe frequency f

Cutoff fc (−3 dB)

—

Attenuation at f

—

Useful for AC coupling and DC removal. Frequencies well below fc are blocked; above fc it passes with 20 dB/decade rise toward 0 dB.

Series RLC Resonance

f0 = 1 ÷ (2π√(L·C)); Q = √(L/C) ÷ R; bandwidth BW = f0 ÷ Q.

Inductance L

Capacitance C

Series R (loss)

Resonance f0

—

Quality factor Q

—

Bandwidth (−3 dB)

—

At resonance the series impedance is minimum (= R). Lower R gives a higher Q and a sharper, narrower passband.

2nd-Order LC Crossover (Load Z)

fc = 1 ÷ (2π√(L·C)); Q = R·√(C/L); damping ζ = 1 ÷ (2Q).

Inductance L

Capacitance C

Load impedance Z

Crossover fc

—

Quality factor Q

—

Damping ζ

—

Common for speaker crossovers and LC networks. Q = 0.707 (Butterworth) gives a maximally flat response; higher Q adds a peak near fc.

Sallen-Key Unity-Gain Low-Pass (Component Pick)

For R1 = R2 = R: Q = ½·√(C1/C2); R = 1 ÷ (2π·fc·√(C1·C2)).

Target cutoff fc

Pick C1

Pick C2

Realized Q

—

Required R1 = R2

—

Response

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Butterworth needs Q ≈ 0.707, i.e. C1 ≈ 2·C2. Larger C2 (lower Q) overdamps and rounds the corner; smaller C2 (higher Q) peaks near fc.

High-Order Low-Pass: Required Order

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n = ceil( log10(10^(As/10) − 1) ÷ (2·log10(fs/fc)) ) for Butterworth. fs must be well above fc.

Passband edge fc

Stopband freq fs

Min stopband reject

Required order n

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Reject at fs (realized)

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2nd-order sections

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Round up to the nearest integer order. Each pole adds 20 dB/decade of asymptotic roll-off; cascade the sections shown in the Butterworth Q table below.

Twin-T / Notch Center Frequency

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f0 = 1 ÷ (2π·R·C) with a T of R, C plus a second T of R/2 (series) and 2C (shunt). Notches one frequency sharply.

Arm resistance R

Arm capacitance C

Notch center f0

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Period T = 1/f0

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Notch depth and Q depend on how well the 2R and C/2 arms are matched (0.1% or better keeps the null deep). Add a buffer around the network to avoid loading it.

🎚️ Low-Pass Filter Families

Design families trade passband flatness, phase linearity and roll-off slope.

Family	Passband ripple	Phase / delay	Roll-off past fc
Butterworth	None — maximally flat	Slightly nonlinear, flat near fc	20n dB/dec
Bessel	None — smooth	Linear group delay (best transient)	< 20n dB/dec
Chebyshev 0.5 dB	0.5 dB	Ripples in passband	~20n–30n dB/dec
Chebyshev 1 dB	1 dB	Ripples in passband	20n–40n dB/dec
Linkwitz-Riley	18 dB dip at fc	Cascaded Butterworth, flat sum	2·20n dB/dec

⤵️ Roll-Off vs Filter Order

Far from the corner, low-pass attenuation rises by 20 dB per decade per pole.

Order (poles)	Typical Sallen-Key stages	Asymptotic roll-off (low-pass)
1	—	20 dB/dec
2	1	40 dB/dec
3	1 + passive	60 dB/dec
4	2	80 dB/dec

📐 Butterworth Section Q by Order

For a maximally-flat all-pole cascade, each 2nd-order Sallen-Key section runs at its own Q (odd orders add one 1st-order RC pole).

Order	Section Q values	Structure
2	0.707	1 × 2nd-order
3	1.000 + 1st	1 × 2nd + RC pole
4	0.541, 1.307	2 × 2nd-order
5	0.618, 1.618 + 1st	2 × 2nd + RC pole
6	0.518, 0.707, 1.932	3 × 2nd-order

🔧 Active Filter Topologies

Topology	Good for	Notes
Sallen-Key	Low-Q, low-noise, non-inverting	Few parts; hard to realize Q above ~10
Multiple-Feedback	High-Q, inverting	Positive-Q stable; load affects response
State-Variable	LP, HP and BP outputs at once	3 op-amps; independent fc and Q
Biquad	Synthesizable 2nd-order filters	DSP standard; low sensitivity to parts

📊 Schematic Diagrams

Simplified passive and active filter stages showing where each component sits in the signal path.

RC low-pass — R feeds the node while C shunts high frequencies to ground. At fc the output is −3 dB; below fc it passes, above fc it rolls off at 20 dB/decade.

RC high-pass — C blocks DC and low frequencies in series; R defines the corner with C. At fc the output is −3 dB; above it passes, below it rolls off at 20 dB/decade.

Series RLC resonator — L, R and C sit in series. At resonance f0 the reactances cancel, current is maximum and the impedance is just R. Q = √(L/C)·1/R sets the bandwidth.

Sallen-Key unity-gain low-pass — R1, R2 and C2 form the phase network at the non-inverting input; C1 feeds back from the output to tune the pole pair. The op-amp is wired as a follower (unity gain), and the component ratio sets Q.

Butterworth vs Chebyshev — Butterworth stays flat in the passband but rolls off more gently; Chebyshev trades a ripple in the passband for a steeper transition into the stopband for the same order. Bessel keeps the best phase linearity but the softest roll-off.

Cascaded high-order Sallen-Key — Higher order is built by cascading 2nd-order sections. Put the low-Q (gentler) section first and the high-Q (peaky) section second to keep each stage's peak below clip. All sections share one fc.