Analog Filter & Bode
Design analog filters up to second order: set the cutoff frequency (and Q) and get the R/L/C component values rounded to the E-series, the realized f₀ and Q, and the Bode plot. Demonstrative tool.
Topology
Requirements
You set the frequency (and Q for 2nd order). C is fixed — R for the RL — and R, L and the rest are the result.
Computed components
Magnitude |H| in dB (left axis) and phase in degrees (right axis) on a log frequency scale. The vertical line marks the cutoff f. Demonstrative tool.
What is a filter for?
A filter lets the useful frequencies through and attenuates the rest. It is everywhere: anti-aliasing ahead of an ADC, removing 50/60 Hz mains hum, an audio crossover, cleaning a switching-supply output, isolating a carrier. Choosing the cutoff frequency, slope and shape around the cutoff is exactly what this tool sizes.
A real signal is a sum of frequencies: the one you want, plus noise, hum and out-of-band interference. It is everywhere: band-limiting a signal before sampling it (anti-aliasing ahead of an ADC), removing 50/60 Hz mains hum, splitting an audio crossover, cleaning a switching-supply output, isolating or extracting a carrier. Choosing the right cutoff frequency, slope and shape around the cutoff is the craft.
Transfer function
A filter is described by its transfer function H(jω): how much it attenuates and how much it phase-shifts each frequency. The Bode plot draws it in two traces — magnitude in dB and phase in degrees — on a logarithmic frequency scale, where slopes become straight lines. In normalized form, with x = f/f₀:
Natural frequency and −3 dB
The cutoff frequency is where the magnitude drops 3 dB below the passband, i.e. where the power halves. For an RC it is f = 1/(2πRC); for an RLC it is f = 1/(2π√(LC)). It is the reference point around which everything else is read.
What is the Q factor?
In 2nd-order filters the Q factor governs the shape around the cutoff: Q = 0.707 (Butterworth) gives the flattest response with no peak; a higher Q produces a resonance peak and a sharper transition; a lower Q softens it. In a band-pass, Q is the selectivity — f₀ divided by the bandwidth.
Which family: Butterworth, Bessel, Chebyshev, elliptic
At a given order, the "family" (or alignment) is the trade-off between three things you cannot maximise together: passband flatness, cutoff steepness and phase / transient quality.
- Butterworth — maximally flat passband, no ripple, moderate cutoff and phase. The neutral default when you have no particular constraint.
- Bessel — nearly linear phase (constant group delay): clean step response, no overshoot or ringing. In return, the gentlest cutoff. Used when the waveform shape matters (audio, pulses, data).
- Chebyshev — steeper cutoff than Butterworth at the same order, at the cost of passband ripple (more ripple = steeper). When you need selectivity and ripple is acceptable.
- Elliptic (Cauer) — the steepest possible cutoff for a given order, but with ripple in both passband and stopband, and the worst phase. Chosen when the transition band must be minimal.
For an all-pole 2nd-order cell (RLC, Sallen-Key) the family reduces to a Q value: Bessel ≈ 0.58, Butterworth = 0.707, Chebyshev > 0.707 (with ripple). Elliptic and inverse-Chebyshev also need ZEROS (a notch) and cannot be realised with a plain Sallen-Key low-pass — they need topologies with zeros. That is why, at 2nd order, the tool offers Bessel/Butterworth/Chebyshev.
Roll-off: 20 and 40 dB/decade
Past the cutoff the magnitude falls at a constant slope: −20 dB/decade for 1st order, −40 dB/decade for 2nd. Each extra order adds 20 dB/decade of attenuation — and 90° of phase shift. That is why stages are cascaded.
Sallen-Key: why active
A 2nd-order RLC needs an inductor — bulky and non-ideal at low frequencies. The Sallen-Key cell achieves the same 2nd order with an op-amp and only Rs and Cs: no inductors, low output impedance (stages cascade without loading each other) and a Q set by component ratios. It is the most common active topology.
What are the design formulas?
You start from the requirements — cutoff frequency f₀ and, at 2nd order, Q — fix a reference component (the capacitor C, the resistor R for the RL) and derive the others, rounded to the E-series. With ω₀ = 2π·f₀: for the RC R = 1/(ω₀C); for the RLC series L = 1/(ω₀²C). The tool shows the ideal and nearest E-series value and recomputes the realized f₀ and Q.
You start from the requirements (cutoff frequency f₀ and, for 2nd order, Q), fix a reference component (the capacitor C — the resistor R for the RL) and derive the others, then rounded to the E-series. With ω₀ = 2π·f₀:
The tool shows the ideal value and the nearest E-series value, and recomputes the realized f₀ and Q with the real values (rounding introduces a small error).
What are the tool’s limitations?
The tool assumes ideal components: no tolerances, no real ESR/ESL, no downstream load. The Sallen-Key uses an ideal op-amp (infinite bandwidth, gain and slew-rate), so real op-amp limits move the high-frequency response. It covers up to 2nd order — higher orders come from cascading stages. It is demonstrative, not a SPICE simulator.
- Ideal components: no tolerances, no real ESR/ESL, no downstream load (an unloaded output is assumed).
- Sallen-Key uses an ideal op-amp: infinite bandwidth, gain and slew-rate. Real op-amp limits move the high-frequency response.
- Up to 2nd order; higher orders come from cascading stages (not covered here).
- Demonstrative tool, not a SPICE simulator.