# Charging current of capacitor

In one of my books there is a figure

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.

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.