RF Filter Design — Butterworth, Chebyshev & LC Ladders

RF Filters

RF filters are two-port networks that pass signals in certain frequency bands while attenuating others. They are fundamental in every RF system for channel selection, harmonic suppression, and image rejection.

Filter Types

Low-pass (LPF) — passes below cutoff fc; used for harmonic suppression, anti-aliasing
High-pass (HPF) — passes above fc; used to block DC or low-frequency interference
Band-pass (BPF) — passes a band centred on f₀; used for channel selection
Band-stop / Notch — rejects a specific band; used to suppress interferers

Butterworth Response

n=3 low-pass filter responses: Butterworth (flat passband, purple) vs Chebyshev (steeper roll-off, orange).

Maximally flat magnitude in the passband — no ripple. The magnitude response is:

Butterworth LP pole-zero plot. n=4 poles equally spaced on the left half of the unit circle in the s-plane.

\(|H(j\omega)| = \dfrac{1}{\sqrt{1+(\omega/\omega_c)^{2n}}}\)

Roll-off: 20n dB/decade beyond \(\omega_c\). Gentle transition band.

Chebyshev Response

Allows equiripple in the passband in exchange for a steeper transition band. For the same order, Chebyshev gives more attenuation in the stopband than Butterworth. The ripple (dB) is a design parameter.

LC Ladder Synthesis

3rd-order LC ladder low-pass filter between source and load impedances.

Prototype low-pass filter element values \(g_k\) are tabulated for Butterworth and Chebyshev responses. These normalised values are frequency- and impedance-scaled to the target cutoff frequency and port impedance using:

\(L_k = \dfrac{g_k R_0}{\omega_c}\quad C_k = \dfrac{g_k}{R_0 \omega_c}\)

🔧 Related Calculators

LC Filter Designer → Bandpass Filter Calculator → RF Circuit Simulator → Filter Prototype g-Values → Coupled-Line Filter → Stub Filter (Richards) →