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In one of my books there is a figure

enter image description here

where G is a neon lamp. Basically the capacitor gets charged once the switch is closed up to a certain spark-current $U_Z$ where the neon lamp gets switched on so the capacitor can discharge to a certain charge-current $U_L$. Further it says that from the charging current

$U(t)=U_0(1-\exp(-t/RC))$

of the capacitor it follows that the periodicity is

$\displaystyle T=RC\cdot\log\frac{U_0-U_L}{U_0-U_Z}$.

How exactly does this equation follow? I am not familiar with the proper english terms in electrical engineering so I might have mixed up voltage, current, etc. I hope, it's still clear what I mean.

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The time to get from zero to $U_L$ is obtained by solving (putting $RC=\tau$)

$$U_L=U_0\left(1-e^{-t/\tau}\right)\\ U_0-U_L = U_0 e^{-t/\tau}\\ \log(U_0-U_L) = \log(U_0)-t/\tau\\ t = \tau \log\frac{U_0}{U_0-U_L}$$

The time to get from $U_0$ to $U_H$ is similarly obtained. When you take the difference between these numbers and rearrange, you get the expression from your book.

Do you think you can do the rest of the derivation yourself now?

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