CHAP. 7]

129

FIRST-ORDER CIRCUITS

The equivalent resistance of the two parallel resistors is R ¼ 2 k. 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

7.3

and

i ¼ v=6000 ¼ 0:67e100t ðmAÞ

ESTABLISHING A DC VOLTAGE ACROSS A CAPACITOR

Connect an initially uncharged capacitor to a battery with voltage V0 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 V0 which, after substituting i ¼ Cðdv=dtÞ, becomes dv 1 1 þ v¼ V dt RC RC 0

t>0

ð5aÞ

with the initial condition vð0þ Þ ¼ vð0 Þ ¼ 0

ð5bÞ

The solution should satisfy both (5a) and (5b). The particular solution (or forced response) vp ðtÞ ¼ V0 satisﬁes (5a) but not (5b). The homogeneous solution (or natural response) vh ðtÞ ¼ Aet=RC can be added and its magnitude A can be adjusted so that the total solution (6a) satisﬁes both (5a) and (5b). vðtÞ ¼ vp ðtÞ þ vh ðtÞ ¼ V0 þ Aet=RC From the initial condition, vð0þ Þ ¼ V0 þ A ¼ 0 or A ¼ V0 .

Thus the total solution is

ð6aÞ

Mahmood_Nahvi_eBook_Schaum_s_Outlines_Theory_An
Mahmood_Nahvi_eBook_Schaum_s_Outlines_Theory_An