Direct Current and Transient Analysis 129
following fi rst-order linear differential equation is obtained in terms of the
unknown loop current i(t) illustrated as follows:
1
C
∫itdt Rit() () V
then
it
V
R
() et/
where τ = RC.
vt VeR t
()t/ for 0
and
vt V vt V VeCR t
()() t/ for 0
R.2.99 When a sudden DC voltage V is applied to a simple series RC circuit (with vc(0) =
0 V), the capacitor voltage charges up to
vC(t) = V (1 − e−t/)
and the current through C is then given by
iCt t
V
R
() e/
where τ = RC (s) is the time constant of the circuit.
R.2.100 A charged capacitor in an RC circuit with an initial voltage V(0) and no sources,
as indicated in Figure 2.29, discharges with the following current and voltage
relations:
vt VoeC
() t/
FIGURE 2.28
RC series circuit.
V
R
C
Switch closes at t = 0
i(t)