0195136047.pdf

184 TIME-DEPENDENT CIRCUIT ANALYSIS

1 mH 100 Ω

10 cos 10^6 t vC(t)

i(t)

S

t = 0

0.1 μF

99 i(t)

+

−

+

−

Figure P3.2.14

2 Ω

2 Ω 2 A

1

10 sin (3t + 20 °) V L^ = 3 H

vC

iL

S

t = 0

3

C = 2 F

+

−

+

−

Figure P3.2.15

3.2.16DetermineiL(t) andvC(t) in the circuit of Figure
P3.2.16.

3.2.17Express the waveform of the staircase type shown
in Figure P3.2.17 as a sum of step functions.

3.2.18The voltage waveform of Figure P3.2.18 is applied
to anRLCseries circuit withR= 2 , L=2H,
andC=1 F. Obtaini(t) in the series circuit. (Note
that the capacitor and inductor values are chosen
here for calculational ease, although they are too
big and not typical.)

3.2.19(a) Let a unit impulse of currenti(t)=δ(t)be
applied to a parallel combination ofR= 3 
andC =^12 F. Determine the voltagevC(t)
across the capacitor.
(b) Repeat (a) fori(t)=δ(t− 3 ).
(Note that the capacitance value is chosen here
for calculational ease, even though it is too big
and not typical.)

3.2.20(a) Let a unit impulse of voltagev(t)=δ(t)be
applied to a series combination ofR= 20 
andL=10 mH. Determine the currenti(t)in
the series circuit.
(b) Repeat (a) forv(t)=δ(t)+δ(t− 3 ).
3.3.1Determine the Laplace transform for each of the
following functions from the basic definition of
Equation (3.3.1).
(a)f 1 (t)=u(t)
(b)f 2 (t)=e−at

(c)f 3 (t)=

df (t)

dt

, assumingf(t) is transform-

able.

(d)f 4 (t)=t

(e)f 5 (t)=sinωt

(f) f 6 (t)=cos(ωt+θ)

(g)f 7 (t)=te−at

(h)f 8 (t)=sinht

(i) f 9 (t)=cosht

(j) f 10 (t)= 5 f(t)+ 2

df (t)

dt

3.3.2Using the properties listed in Table 3.3.2, deter-

mine the Laplace transform of each of the follow-

ing functions.

(a)te−t

(b)t^2 e−t

(c)te−^2 tsin 2t

(d)

1 −cost

t

(e)

e−^2 tsin 2t

t

*3.3.3Given the frequency-domain response of anRL

circuit to be

I(s)=

10

2 s+ 5

determine the initial value and the final value of the

current by using the initial-value and final-value

theorems given in Table 3.3.2.