The real equation for quenching frequency

I found this super regen circuit from http://www.vk2zay.net/article/129 The author stated:

Best over-all performance was achieved with 10 K (for R1) and 6.8 nF (for C1) in the source circuit.

and in his notes:

Typically the quench is set around 30 kHz.

Applying the basic RC equation doesn't work and someone who did the same also posted his findings in the circuit comments as follows:

You mentioned that the optimal quench frequency should not be below 15Khz. According to your design, your quench frequency is defined by R1 and C1. In your circuit, R1 is 10K and C1 is 6.8nF. According to my calculations, your quench frequency is 2.3Khz. Shouldn't C1 be more like 1nF or am I doing my calculations wrong?

The formula I'm using is 1/(2 * pi * C1 * R1)

Then the author responded as follows:

The 1/(2.Pi.R.C) formula is for the cut-off frequency of a simple 1st order RC filter. The RC time constant is how long it takes to charge the capacitor through R to (1 - 1/e) times the supply voltage (about 63.2%), or discharge it down to 36.8% of its initial voltage.

In most RC timing circuits (and this particular circuit) the quench frequency is not determined by the cut-off frequency. There is generally a fixed reference voltage at which point the charging (or discharging) will be stopped and the cycle repeated. How long it takes to reach that point and how quickly it is re-estabilished determines the frequency. Generally the full solution involves two differential equations (one for the charge, the other for the discharge part of the cycle) with boundary conditions determined by other circuit parameters which are often constant voltage/resistance or constant current approximations. A completely analytical solution, even for simple models can be truly nasty!

I haven't modelled this circuit extensively so as to give a nice formula for the quench frequency as a function of the emitter RC values. It would not be a simple thing to calculate, and would be an approximation at best as the quench frequency changes with the transistor operating point (emitter current changes, temperature, noise/signal forcing of the oscillator start-up, etc).

So where do I get such differential equations from and how do I apply them to the circuit? I want to be able to properly calculate the quench frequency. And if it means factoring in source voltage and/or any transistor properties such as junction capacitance, I'd like to know. If I can get answers then I won't have to constantly replace capacitors in my circuit with different valued ones. I want the equation to be compatible with BJT transistors since my superregen uses an NPN transistor.

My superregen is more similar to this circuit: • Why don't you ask the author - he seems willing to answer questions raised and he's in the best position to give you the facts. – Andy aka Apr 26 '17 at 17:35
• Are you familiar with the operation of superregenerative receivers in general? – duskwuff Apr 26 '17 at 17:38
• Why do I suddenly have a hankering for a Guinness... – Trevor_G Apr 26 '17 at 17:43
• This isn't my field, but I was under the impression that regenerative receivers went obsolete as soon as transistors became cheaply available. Why do you want to make one? – Hearth Apr 26 '17 at 17:46
• Eddy Insam goes through the math: eix.co.uk/Articles/Radio/Welcome.htm but you'll find that parasitic reactances are important too, so that models are difficult to develop...especially for these self-quenching circuits. – glen_geek Apr 26 '17 at 18:18