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VXO calculator: pulling a crystal

Upwards only — and not far

Load capacitance pulls a crystal only upwards from series resonance: it is between fs and fp that it behaves as an inductance, and that is where the oscillator works. The whole available span is fp minus fs, usually a few tens of kilohertz, and in practice you can use barely half of it.

fsfpworking rangerange
A capacitor pulls a crystal upwards from fs only; going lower needs a series inductor
Crystal frequency fs (MHz)
Motional capacitance Cm (fF)
Holder capacitance C0 (pF)
Load capacitance from (pF)
to (pF)
Series coil (µH, 0 — none)
Tuning by capacitance
Offset above fs
Tuning range
Theoretical maximum (fp − fs)
Used, of the maximum
Parallel resonance fp

📐 The formulas​

Frequency shift from load capacitance:

Δf = fs · Cm / ( 2 · (C0 + CL) )

Tuning range between two capacitances:

Δf = fs · (Cm/2) · [ 1/(C0 + CL1) − 1/(C0 + CL2) ]

The theoretical limit is the whole span between the resonances:

fp = fs · √( 1 + Cm/C0 )
maximum = fp − fs

A series coil adds negative reactance and extends the span downwards; the limit is where oscillation stops, when the reactance leaves the crystal's inductive region.

A check point (fs = 10 MHz, Cm = 20 fF, C0 = 4 pF):​

CL = 10 pF → +7143 Hz, CL = 50 pF → +1852 Hz, a range of 5.29 kHz. The theoretical maximum (fp − fs) is 24.97 kHz, so 21 % of it is used.

Where this calculator stops:
  • Capacitance cannot pull the frequency below fs — that needs a series coil
  • Near the edges of the range the amplitude falls and stability degrades sharply: in practice take no more than half the theoretical maximum
  • Cm and C0 have to be measured for your particular crystal, not taken from a datasheet
  • Stray wiring capacitance, which adds to CL, is not accounted for

🎯 In practice​

  • Tuning across part of a band with crystal stability.
  • Judging whether a VXO will be enough for what you have in mind — it usually turns out not to be.
  • Choosing a crystal: the larger Cm and the smaller C0, the wider the range.
Practical advice:
  1. Crystals with a larger Cm pull better: fundamental-mode ones beat overtone ones.
  2. A series coil widens the range but lowers the stability — a trade-off.
  3. Several crystals of slightly different frequencies in parallel give a wider tuning range.
  4. If you need a whole band, use a DDS or a PLL rather than a VXO.

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

How to cite this calculator
UR3PKI. «VXO calculator: pulling a crystal». CyberDev.Space. https://cyberdev.space/en/radio/calculators/others/vxo (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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