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Crystal ladder filter calculator

Why these filters usually fail

Not because of the formulas. The frequency spread across crystals from one batch runs to hundreds of hertz, and the filter bandwidth is also hundreds of hertz. Without selecting the crystals on an instrument the response comes out random, however carefully you compute the coupling capacitors. The calculator shows straight away what spread you have to stay within.

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A wider band comes from less coupling capacitance — counter-intuitive, but that is how it works
Response
Number of crystals
Crystal frequency (MHz)
Required bandwidth (Hz)
Motional capacitance Cm (fF)
Holder capacitance C0 (pF)
The step from a low-pass prototype to coupled resonators: q₁ = g₀·g₁, k(i,i+1) = 1/√(gᵢ·gᵢ₊₁). This is the standard relation, which handbooks give as ready-made tables of k and q.

📐 The formulas​

From the prototype's normalised g values come the coupling coefficients and the external Q — the standard step towards coupled resonators:

q₁ = g₀ · g₁              q_n = g_n · g(n+1)
k(i,i+1) = 1 / √( g_i · g(i+1) )

Scaling to the real bandwidth:

Qe = q₁ · f₀ / BW
k_actual = k · BW / f₀

Coupling capacitance:

C(i,i+1) = Cm / k_actual = Cm · f₀ / (BW · k)

Filter termination impedance:

x = 1 / (2π·f₀·Cm)        ← the reactance slope parameter
R = x / Qe

Crystal matching tolerance: the spread must be no greater than BW/10.

Where this calculator stops:
  • The crystals have to be selected. Without that the filter will not work, and no amount of calculation will save it
  • The effect of C0 is not accounted for: with six crystals or more it spoils the attenuation on the upper skirt and needs compensating
  • Cm and C0 come from measuring your particular batch, not from a datasheet: the spread between specimens is larger than the difference between types
  • The termination impedance is almost never 50 Ω — matching networks are needed at both ends

🎯 In practice​

  • An SSB filter for a homebrew transceiver — 4 to 8 crystals, 2.4 kHz wide.
  • A CW filter — the same ladder, 400–500 Hz wide.
  • Reworking a published design for the crystals you actually have.
Practical advice:
  1. Buy the crystals as one batch and select with room to spare: out of twenty you will usually find eight that match.
  2. A wider band needs smaller coupling capacitors — the most counter-intuitive thing in this subject.
  3. Match the termination impedance with a capacitive divider or a binocular transformer.
  4. Test the finished filter with a high-impedance analyser input or through the calculated matching networks — otherwise the picture will mislead you.

© 2026 UR3PKI · CyberDev.Space · Content licensed under CC BY-NC-SA 4.0.

How to cite this calculator
UR3PKI. «Crystal ladder filter calculator». CyberDev.Space. https://cyberdev.space/en/radio/calculators/others/crystal_ladder (licence CC BY-NC-SA 4.0).

The licence lets you use this material freely, including in teaching materials, but only with attribution to the author and a link to the source.

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