Voltage across the capacitor

voltage across the capacitor

The charge "Q" of the capacitor is related to the potential difference by the following relation:

\[dQ=CdU_{c}\]

and the current i to the quantity of electricity (or charge) is given by the relation

\[i=\frac{dQ}{dt}\]

The first relationship can be written as a function of the output voltage “Uc" as follows

\[E=RC\frac{U_{c}}{dt}+U_{c} \Rightarrow \frac{R}{RC} = \frac{d U_{c}}{dt} +\frac{U_{c}}{RC}\]

Under the following initial conditions, t = 0; Uc = 0, the voltage across the capacitor is the solution to the above differential equation, and is written in the form:

\[U_{c}(t)=E(1-e^{-t/RC})\]

and the current of the capacitor is

\[i(t)=\frac{E}{R}e^{-t/RC}\]