If we plot and compare v-t or I-t curves with actual exponential curve we can see that they are same, but why? Is there any proof?
From the equation we can see that e is creating bridge between electrical property and time. Why e?
I have a good text book that covers the curve’s characteristics and how it can be used but doesn’t show any mathematical proof.
Best Answer
Capacitance is defined as:
\$C = \frac{Q}{V}\$
and equally:
\$C = \frac{\delta Q}{\delta V}\$
Since current is defined as the rate of change of charge:
\$I(t) = \frac{\delta Q(t)}{\delta t}\$
Thus:
\$I(t) = C \frac{\delta V(t)}{\delta t}\$
If we set up a circuit with a voltage source, Resistor and a capacitor
\$V = V_r + V_c\$ and this creates a differential equation
\$v = i(t)R + \frac{1}{C}\int i(t) \delta t \$
Solving such differential equations produces an equation:
\$I(t) = \frac{V}{R}e^\frac{-t}{RC}\$
Which equally can be re-arranged to be
\$V(t) = V(1-e^\frac{-t}{RC}) \$ taking into account initial conditions