To add some perspective, Danyuk's triode emulator is intended for music reproduction, i.e. Beethoven's 5th Symphony. In this context, triodes in a hi-fi amplifier do not get close to clipping limits, and so this circuit doesn't address clipping characteristics. The idea is to add a subtle to moderate amount of harmonic distortion, let's say between 1% to 5% depending on the listener's tastes. There is a mix knob that blends the pure signal with the same signal processed by a JFET circuit that approximates to the 1.5 exponent. If you think about this, a mix of these two signals will no longer follow the 1.5 exponent rule anyway!

The relevant part here is that adding a controlled amount of low-order harmonics (mostly 2nd a bit of 3rd) is pleasant to the ear. Whether these harmonics follow a 1.3, 1.5 or 1.7 exponent rule is not as relevant as having the *right* amount in the mix, which by the way will be different from person to person.

In addition, a practical 12AX7 triode stage with bypassed cathode won't follow a pure 1.5 exponent law either, even if popular knowledge says the triode plate current follows a 1.5 exponent law with regards to the grid voltage. Why?

Let's take Child's law for the plate current: Ip = (Vg + Vp/mu)^1.5

There you have the gate voltage (Vg) and the 1.5 exponent, but you also have another term (Vp/mu) which to make things worse depends on the instantaneous plate voltage as well.

The plate voltage will be something like Vp = Vcc - Rp*Ip

where Vcc is the supply voltage and Rp is the plate resistance (here we assume there is no load on the stage)

Replacing into the first equation you get:

Ip = [ Vg + (Vcc - Rp * Ip) / mu ] ^ 1.5

Notice that Ip is on both sides and cannot be isolated on one side (as far as I know) to obtain a closed equation in the form: Ip = function( Vg, Vcc, Rp, mu )

This shows that in practice you don't get a PURE 1.5 exponent law. In addition we know that Child's law is just an approximation, so the "real world" may differ even more. *** The moral here is that the 1.5 exponent law is not as important as it appears at first sight. ***

If you want to know, I've run many simulations with different triode models comparing with a JFET circuit with different source resistors, and have come to the conclusion that a JFET with a source resistor chosen as Rs = K * Vp / Idss, where K lies between 1.5 to 2.0 is a much closer approximation in terms of the harmonic content of the resulting signal (as seen running an FFT analysis) than the 0.83 or 0.84 "magic value". Of course this is only valid in the central third of the dynamic range, as the JFET won't follow the triode saturation characteristics.

So, do we need to change the paradigm on the 0.83 magic source value? Absolutely not! Based on listening tests I really like this 0.83 value on electric guitar, and I'm sure many other people do. I've also done listening tests bypassing the source resistor in the JFET circuit to approximate to the JFET theoretical 2.0 exponent (and also adding some attenuation to compensate for the extra gain obtained when the capacitor is added, so as to compare things at a similar level). In summary, I don't like much the result with the 2.0 exponent since it gets on the harsh side to my liking. Moving in the opposite direction, I also liked the case where Rs = 1.0 * Vp/Idss, as it sounds fuller than the original guitar signal.

CONCLUSION: The above is proof enough for ME that it is a waste of energy to pursue an EXACT triode emulation. Instead, it pays-off to see which the basic distortion mechanisms are and try to get something similar that's pleasant to the ear. As the saying goes, "there are many ways to skin a cat."