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
i t dt Ri t
V
( )
( )
ϩ
ϭ
∫
then
i t
V
R
e
t
( )
/
ϭ
Ϫ
where τ = RC.
v t
Ve
t
R
t
( )
/
ϭ
Ͼ
Ϫ
for
0
and
v t
V v t
V Ve
t
C
R
t
( )
( )
/
ϭ
Ͼ
Ϫ
ϭ Ϫ
Ϫ
for
0
R.2.99 When a sudden DC voltage V is applied to a simple series RC circuit (with v c (0) =
0 V), the capacitor voltage charges up to
v C (t) = V (1 − e
−t/
)
and the current through C is then given by
i C
t
t
V
R
e
( )
/
ϭ
Ϫ
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:
v t
Voe
C
t
( )
/
ϭ
Ϫ
FIGURE 2.28
RC series circuit.
V
R
C
Switch closes at t = 0
i(t )
CRC_47760_CH002.indd 129
CRC_47760_CH002.indd 129
7/23/2008 1:38:42 PM
7/23/2008 1:38:42 PM