Are there any commonly accepted algebraic models for vacuum tube operation (triode, tetrode, and pentode)? In the same way that BJTs have a Gummel-Poon or Ebers-Moll model, and (macroscale) MOSFETs have a cutoff/linear/saturation algebraic model, is there a similar model for vacuum tubes? A DC-accurate model plus some dynamic components (dominant capacitances) would be excellent, but I'm having trouble finding any references. Higher order effects (equivalent to something like the Early effect in BJTs) would be good to know about as well, especially if they tend to impact practical designs.

I am one of the developers of CircuitLab and vacuum tubes are one of our most requested features, and I'm investigating whether it'd be practical to implement in our simulator. Thanks!

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    \$\begingroup\$ I cannot help, but congratulation for the nice tool! \$\endgroup\$
    – clabacchio
    Mar 2, 2012 at 21:31
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    \$\begingroup\$ An AC model of a triode tube - is this a good start? \$\endgroup\$ Mar 2, 2012 at 21:45
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    \$\begingroup\$ Did they even have math back then? \$\endgroup\$
    – user3624
    Mar 2, 2012 at 21:49
  • \$\begingroup\$ Thanks! The AC-only (small-signal) model is a start, and is helpful for showing dominant capacitances for dynamic effects, but they've already taken linearized derivatives (such as gm, rp). What I really need next is the full non-linear equations from which you could calculate those derivatives. \$\endgroup\$
    – compumike
    Mar 2, 2012 at 21:50
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    \$\begingroup\$ could you simulate the vacuum part using 'void'? =P \$\endgroup\$
    – JustJeff
    Mar 2, 2012 at 23:35

1 Answer 1


The general small-signal model of a tube is pretty much outlined at the link Madmanguruman posted: a current source in parallel with a plate resistance and the associated inter-electrode capacitances.

For non-linear time domain analysis, the situation is more complicated as outlined here. As the article states, it's possible to derive a mathematical model of a triode or pentode based on the Langmuir-Child law, but that model doesn't accurately represent the real behavior of a tube in certain areas of its operation. The best models are "phenomenological" i.e. designed to fit a tube's actual performance curves as closely as possible without regard to underlying physics.

There is a program I've used available that will let you take published tube curves, fit the model to the curves, and then spit out a SPICE subcircuit. It works well for triodes, I don't know if it can be used for pentodes though. There are also many, many ready-to-run SPICE models of varying quality for a variety of different tubes on the web.

  • \$\begingroup\$ That first link, and the finding parameters page linked from it, are great. It'll take me some time to dig in to all of it, but this seems to be just what I was looking for. Many thanks! \$\endgroup\$
    – compumike
    Mar 3, 2012 at 5:46

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