Live Curves' Math Model
What we like best about a triode vacuum tube is its amazing linearity, the consistent incremental change in plate current relative to incremental changes in the grid-to-cathode voltage. But not even the 300B or 845 is perfectly linear: as voltage increases and/or current decreases, nonlinearity increases. To compound problems, this movement into nonlinearity is in itself very nonlinear. Fingerprinting this nonlinearity has proven elusive. Here the irony of the situation is made plain:  what we need to know about a triode in order to make the best use of its linearity is knowing where it is nonlinear. Tolstoy might have said "All linear devices are linear in the same way; but each nonlinear device is nonlinear in its own way." Remember it is this unique bending that makes a 300B and a triode-connected KT88 sound different.

Most old electronic textbooks give formulas based on Child's Law; these formulas are, unfortunately, next to worthless. Here is the problem with any of the textbook variations of the Child's Three-Halves Law formulas:  each assumes that all the plate current vs. plate voltage lines are identical in shape and differ only in x-axis spacing, which would be defined by the mu (amplification factor) of the triode. But any cursory examination of the plate lines of a triode reveals that the mu is not constant, for example, that it decreases at cutoff. Consequently, the Three-Halves power formula must be superseded by a less optimistic mathematical model.

Recently in the professional audio press and in the electronic hobby magazines there have been attempts to define such a formula, mostly for use in a Spice program. Some have been little more than mathematical tricks to curve the lines. 

If we wish to stick to the actual tube curves, a better attempt would begin with two lines describing the plate current vs. plate voltage, with the grid-to-cathode voltage held at some constant value at each (for example one at 0v and another at -80v). The first thing we note is that the lines look very different from each other; the slopes and the basic amount of curvature appear unrelated.

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