Post History

Unless you know the *exact* formula for the I-V curve, you will never find out anything analytically. Fair warning: very unlikely you will get such formula, since they are strongly non-linear in nature. The analysis of such oscillators is done based on approximations (see the [Esaki diode](https://www.hindawi.com/journals/apec/2011/830182/), for example), and it involves a lot of estimations.

But if your *purpose* is a [relaxation oscillator](https://en.wikipedia.org/wiki/Relaxation_oscillator) then you'll be much better off using some of the already known circuits: reverse BE junction, multivibrator, comparator, etc. All of them rely on a much more analytically-inclined RC time constant, combined with simple thresholds. The tunnel diode has a continuously variable threshold. Plus, they will likely consume much less current.

If you insist in pursuing the tunnel diode analysis then you should know that the way you started is not the way to go. The small signal does not apply here, since the whole behaviour of the oscillator relies on the point on the I-V curve (as depicted on your I-V curve picture) moving all the time. I can't find the right words to describe now, but a picture should be worth a thousand words: take whatever simulator you want, that lets you zoom in and analyze the waveforms.