Page 140 - Schaum's Outline of Theory and Problems of Electric Circuits

129
FIRST-ORDER CIRCUITS
CHAP. 7]
The equivalent resistance of the two parallel resistors is R ¼ 2k
. The time constant of the circuit is
RC ¼ 10 2 s. The voltage and current in the 6-k
resistor are, respectively,
v ¼ 4e 100t ðVÞ and i ¼ v=6000 ¼ 0:67e 100t ðmAÞ

7.3 ESTABLISHING A DC VOLTAGE ACROSS A CAPACITOR
Connect an initially uncharged capacitor to a battery with voltage V 0 through a resistor at t ¼ 0.
The circuit is shown in Fig. 7-3(a).

Fig. 7-3

For t > 0, KVL around the loop gives Ri þ v ¼ V 0 which, after substituting i ¼ Cðdv=dtÞ, becomes
dv 1 1
þ v ¼ V 0 t > 0 ð5aÞ
dt RC RC
with the initial condition

þ
vð0 Þ¼ vð0 Þ¼ 0 ð5bÞ
The solution should satisfy both (5a) and (5b). The particular solution (or forced response) v p ðtÞ¼ V 0
satisfies (5a) but not (5b). The homogeneous solution (or natural response) v ðtÞ¼ Ae t=RC can be added
h
and its magnitude A can be adjusted so that the total solution (6a) satisfies both (5a) and (5b).
t=RC
vðtÞ¼ v p ðtÞþ v h ðtÞ¼ V 0 þ Ae ð6aÞ
þ
From the initial condition, vð0 Þ¼ V 0 þ A ¼ 0or A ¼ V 0 . Thus the total solution is