Chebyshev Filters — Equiripple LC Filter Design

The Chebyshev filter makes a deliberate trade: it accepts a controlled amount of ripple in the passband in exchange for a dramatically sharper rolloff. Named after the Russian mathematician Pafnuty Chebyshev, who studied equiripple approximation polynomials in the 1850s, this filter type allows the designer to specify exactly how much the passband can deviate from flat — and uses that allowance to push the transition band as steep as possible. The result is that a 5th-order Chebyshev with 0.5 dB ripple can attenuate harmonics 6–10 dB more than a 5th-order Butterworth with the same component count.

This makes the Chebyshev the working ham's choice for transmitter harmonic filters. A 0.5 dB ripple introduces only minimal signal variation across a 300 kHz ham band, yet it buys enough extra stopband rejection to meet FCC requirements with fewer components than the Butterworth. The 40m filter that needs eight or nine Butterworth poles might need only five Chebyshev poles — halving the component count, reducing insertion loss, and shrinking the enclosure.

The Chebyshev Equiripple Response

The Chebyshev low-pass filter is defined by its magnitude-squared response:

The Chebyshev polynomials T_n(x) are defined recursively:

• T₀(x) = 1
• T₁(x) = x
• T_n(x) = 2x·T_n−1(x) − T_n−2(x)

For |x| ≤ 1 (inside the passband), T_n(x) oscillates between −1 and +1. This is the mathematical source of the equiripple behavior: within the passband, the response ripples up and down with constant amplitude, reaching exactly 0 dB and −R_dB alternately (where R_dB is the ripple). Every Chebyshev filter of order n has exactly n peaks and valleys in the passband. At the passband edge (x = 1), T_n(1) = 1 always, so the response always equals −R_dB at f = f_c.

For |x| > 1 (above the passband), T_n(x) grows rapidly — much faster than xⁿ. This rapid growth is why Chebyshev filters roll off so much more sharply than Butterworth filters of the same order.

The ripple factor ε is related to the passband ripple in decibels:

Chebyshev low-pass filter response (dashed, 0.5 dB ripple) compared to Butterworth (solid) of the same 5th order. The Chebyshev exhibits equiripple oscillations in the passband but achieves significantly greater attenuation above the passband edge frequency f_c.

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Choosing Passband Ripple

The ripple specification is the key design choice for a Chebyshev filter. More ripple means sharper rolloff but more amplitude variation in the passband. The table below summarizes the practical implications:

For transmitter harmonic filters, 0.5 dB ripple is the most common choice. The 0.5 dB amplitude variation is negligible in practice — a transmitter running 100 W will vary by ±0.5 dB across the entire ham band, from about 97 W to 103 W. No operator or recipient would notice. Yet this small ripple buys enough extra stopband attenuation to significantly reduce the required filter order.

Normalized Component Value Tables

As with the Butterworth, Chebyshev component values are presented as normalized prototypes referenced to 1 ohm and ω_c = 1 rad/s. The values depend on both the filter order n and the chosen passband ripple. For equal source and load impedances (50 Ω), odd-order Chebyshev filters have equal terminations automatically; even-order filters have unequal terminations and require a transformer or resistive pad unless modified.

This is a critical practical difference from Butterworth: even-order Chebyshev filters (n = 2, 4, 6) have different source and load impedances in the normalized prototype. For this reason, odd-order designs (n = 3, 5, 7) are universally preferred for transmitter harmonic filters with matched 50 Ω source and load.

Chebyshev prototype values — 0.1 dB ripple (ε = 0.153):

Chebyshev prototype values — 0.5 dB ripple (ε = 0.349):

Chebyshev prototype values — 1.0 dB ripple (ε = 0.509):

Notice how the component values change with ripple. Higher ripple gives higher g values overall — which translates to larger component values, or alternatively, more reactive impedance variation across the passband (consistent with the ripple). The odd-order symmetry (g₁ = g_n) holds for all Chebyshev odd orders with equal terminations, just as with Butterworth.

Stopband Attenuation Formula

For a Chebyshev low-pass filter with passband ripple ε, the attenuation at any frequency f above the passband edge f_c is:

For frequencies well above the passband (f ≫ f_c), this simplifies to an asymptotic approximation — but for engineering calculations at specific harmonic frequencies, the full formula is needed. Many scientific calculators and filter design programs can evaluate cosh and arccosh directly. For hand calculation, use the identity: arccosh(x) = ln(x + √(x² − 1)) for x ≥ 1.

Quick comparison at f = 2·f_c (frequency ratio = 2.0):

The Chebyshev advantage grows with filter order. A 5th-order Chebyshev at 0.5 dB ripple provides 45 dB at twice the cutoff — more than the FCC minimum 43 dB requirement, in a 5-pole filter. The equivalent Butterworth needs 7 poles for 43 dB at the same frequency ratio. Two fewer components, smaller enclosure, lower insertion loss.

Worked Design Example: 40m Transmitter Low-Pass Filter

Let's design a 5th-order Chebyshev LPF for a 40m transmitter at 100 W. This time the tighter rolloff works in our favor — we can place the cutoff closer to the operating band while still achieving adequate harmonic suppression.

Recommended physical layout for a 7-pole Chebyshev 40m transmitter LPF. The series inductors (toroids) alternate in orientation by 90° to minimize coupling. Shunt capacitors mount directly to ground plane. Input/output via BNC or PL-259 connectors.

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Chebyshev vs. Butterworth: When to Use Each

Now that we have seen both filters in detail, the decision is straightforward:

For ham radio transmitter harmonic filters — by far the most common homebrew filter project — the Chebyshev at 0.5 dB ripple is the standard choice. For any application involving digital signals, pulse waveforms, or precise group delay requirements (such as SSB speech processing), the Butterworth or Bessel is preferred.

Group Delay and Phase Distortion

The equiripple passband of the Chebyshev filter comes with a cost: its phase response is not linear, and its group delay varies across the passband. Group delay is defined as the negative derivative of phase with respect to frequency. When group delay is constant (flat), all frequency components of a signal experience the same time delay — the signal's shape is preserved. When group delay varies, different frequencies are delayed by different amounts, distorting the signal's envelope.

For SSB voice, the group delay variation of a 7th-order Chebyshev transmitter LPF at audio frequencies is negligible — the filter is far from cutoff during voice operation and the group delay is essentially flat. For CW keying waveform shape, the filter may slightly round the leading edge of key-down pulses, but this is equally negligible. The Chebyshev is entirely suitable for all normal ham radio voice and CW applications.

Where group delay becomes critical is in digital mode bandpass filtering and audio processing chains where the filter is operating at or near its cutoff with complex waveforms. In those cases, a Bessel filter (covered in Module 16A) is appropriate.

Practical Construction Considerations

The Chebyshev filter has one notable difference from the Butterworth in construction: the outer inductors (L₁ and Lₙ) have relatively high g values (around 1.7 for 0.5 dB ripple), while the inner inductors have even higher values. This means the Chebyshev filter uses more inductance overall than the equivalent Butterworth, which translates to more turns on the toroid cores — but also means higher Q requirements to keep insertion loss low.

At HF, a well-wound T68-2 toroid in a well-designed ladder filter should have insertion loss under 0.3 dB across the passband for filter orders up to 7. The main insertion loss culprit is low-Q inductors with high wire resistance. For transmitter filters above 10 W, always use Litz wire or 22 AWG or heavier enameled copper to keep winding resistance low. Measure the Q of your inductors with an RF bridge or antenna analyzer if possible; a Q above 150 at the operating frequency is sufficient for a 0.1 dB additional insertion loss contribution.