0195136047.pdf

138 TIME-DEPENDENT CIRCUIT ANALYSIS

The complete response isvC(t)= 1. 0 +(A 1 +A 2 t)e−t. Evaluating bothvCand dvC/dt

att= 0 +, we have

vC( 0 +)= 0 = 1 +A 1

dvC

dt

( 0 +)= 0 =A 2 −A 1

These equations result inA 1 =A 2 =−1, from which it follows that

vC(t)= 1 −e−t−te−t= 1 −(t+ 1 )e−tV

(c) ForC= 0 .5 F, the values ofs 1 ands 2 are obtained ass 1 =(− 1 +j 1 )ands 2 =(− 1 −j 1 )

(see Case 3 with complex conjugate roots). The transient response is of the form of

Equation (3.2.38),

vC, n(t)=Ae−αtsin(βt+φ)

whereα=1 andβ=1 in our case. The complete solution is thenvC(t)= 1. 0 +

Ae−t sin(βt+φ)V. From the initial conditionsvC( 0 +)=0 anddvC/dt

(

0 +

)

=0,

evaluating both att= 0 +yields

vC( 0 +)= 0 = 1 +Asinφ

dvC

dt

(

0 +

)

= 0 =A(cosφ−sinφ)

Simultaneous solution yieldsA=−

√

2 andφ=π/4. Thus the complete response is

given by

vC(t)= 1. 0 −

√

2 e−tsin

(

t+

π

4

)

V

The total responses obtained for the three cases are plotted in Figure E3.2.5. These cases are

said to be overdamped for case (a), critically damped for case (b), and underdamped for case (c).

(c) 1 −√ 2 e−t sin (t + (^) )
(b)
critically
damped
(a) overdamped
1 −1.125e−0.2t + 0.125e−1.8t
012345
Time, s
Capacitor voltage, V
0
0.2
0.4
0.6
0.8
1.0
1.2
678910
Under damped
1 −e−t (t + 1)
π
4
Figure E3.2.5Total responses
obtained.
A system that is overdamped [Figure E3.2.5(a)] responds slowly to any change in excitation.
The critically damped system [Figure E3.2.5(b)] responds smoothly in the speediest fashion
to approach the steady-state value. The underdamped system [Figure E3.2.5(c)] responds most
quickly accompanied by overshoot, which makes the response exceed and oscillate about the