Step Response

Step Response of RC Circuits

What happens if we excite an RC circuit with Vs at t=0?

when t=0

\(i_c = V_s/R\)

for t>0

\(i_c = (V_s-V_c)/R\)

\(i_c = C \dfrac{dV_c}{dt}\)

Then;

\(C \dfrac{dV_c}{dt} = (V_s-V_c)/R\)

which can be represented as:

\(\dfrac{dV_c}{dt} = \dfrac{V_s-V_c}{RC}\)

\(\dfrac{dV_c}{V_c-V_s} = -\dfrac{dt}{RC}\)

integrating both sides and taking exponentials:

\(V_c(t)=V_s + (V_o - V_s)e^{-t/RC}\)

where \(V_0\) is the initial voltage of the capacitor.

If the initial voltage of the capacitor is zero then:

\(V_c(t)=V_s (1-e^{-t/RC})\)