[Al (astral highway)]

I have discovered (for myself, by thought-experiment!) a magical property hidden in an equation, which may appeal to those interested in numbers -- just as much as in the maths describing the behaviour of resonant circuits.

I'm pleased to share it, as well as my conclusions.

This is a thematic development from a previous thread where Argus25 (Hugo, thank you again) discussed the equation derived by Otto Schade. It describes the conditions when for a parallel resonant circuit with L and C, with some lumped R, current builds with time to some peak value; Ipk.

Then, when the switch opens, the peak value of voltage across the resonant circuit, given by Schade's 1947 equation is:

Vpk=Ipk * Sqrt (L/C) *e^-Pi/4Q


This is of interest to me in modelling stress conditions in a current project.

For those not familiar, e is the base of natural logarithms, after Napier. It is transcendent but can approximate to 2.71828...

I started by asking: what's the difference between the above circuit, with the exponential multiplicand, and the 'stripped-down' version, which stops with the product of peak current and the square root of the ratio of inductance to capacitance?

I became fascinated by the expression e^-Pi/4Q. I plugged in various numbers for Q, (using the approximation for Pi of 3.1415 and e as above) and was instantly struck by how these converged on values close to unity.

So, what if the whole expression was a remarkably intuitive way of expressing a coefficient?, I asked.

Since, when:

Q=1,000, the expression resolves to 9.9921 e-1, or, as a coefficient, 0.9921

When Q=100, the expression resolves to 9.9218 e-1, or as a coefficient,0.99218...

A factor of ten smaller, and it's not making a huge difference! And

when Q=10, the expression resolves to 9.2447 e-1, or as coefficient, 0.92447

All of which so far shows that Shade's equation has a very subtle way of showing the impact of Q under these conditions.

But here's the exciting thing.

As we go up the powers of 10, the coefficient gets closer and closer to unity. We might think it would never reach unity, given the wonders of the two transcendent numbers, pi and e.

But...

When Q=1 e4, the coefficient becomes 0.99999

And when Q=1 e5, voila! It reaches exactly 1.

And not only that, for all powers of ten above the fifth power, it remains at unity. That is the maximum value of the coefficient.

Why 10,000 should be the theoretical limit of Q is an interesting question, way beyond my knowledge, experience, or powers of conjecture. But according to Schade's equation, this appears to be true.

That's my insight, and I feel quite excited to share it!

And the take away, I suppose, is that in the instance described in the opening paragraph, Q is far less significant determinant than the relationship between inductance and capacitance. Simply, the smaller the capacitance for a given inductance, the greater by far the peak voltage.