Section 5-2: Capacitors
Problem 5.8 After plotting the voltage waveform, obtain expressions and generate
plots for i(t), p(t), and w(t) for a 0.2-mF capacitor. The voltage waveforms are given
by
(a) υ1 (t) = 5r(t) − 5r(t − 2) V
(b) υ2 (t) = 10u(−t) + 10u(t) − 5r(t − 2) + 5r(t − 4) V
(c) υ3 (t) = 15u(−t) + 15e−0.5t u(t) V
(d) υ4 (t) = 15[1 − e−0.5t ] u(t) V
Solution: (a)
!
5t (V) for 0 ≤ t ≤ 2 s
υ1 (t) = 5r(t) − 5r(t − 2) = 5t u(t) − 5(t − 2) u(t − 2) =
10 (V) for t > 2 s
!
d υ1
1 (mA) for 0 ≤ t ≤ 2 s
i1 (t) = C
=
dt
0
for t > 2 s
!
5t (mW) for 0 ≤ t ≤ 2 s
p1 (t) = i1 (t) υ1 (t) =
0
for t > 2 s
!
1
2.5t 2 (mJ) for 0 ≤ t ≤ 2 s
w1 (t) = Cυ12 =
2
10 (mJ)
for t > 2 s
(b)
(c)
υ2 (t) = 10u(−t) + 10u(t) − 5r(t − 2) + 5r(t − 4)
(V)
⎧
⎪
for t < 2 s
⎨10 (V)
= 20 − 5t (V) for 2 ≤ t ≤ 4 s
⎪
⎩0
for t > 4 s
⎧
⎪0
for t < 2 s
d υ2 ⎨
i2 (t) = C
= −1 (mA) for 2 ≤ t ≤ 4 s
⎪
dt
⎩0
for t > 4 s
⎧
⎪
for t < 2 s
⎨0
p2 (t) = i2 (t) υ2 (t) = (5t − 20) (mW) for 2 ≤ t ≤ 4 s
⎪
⎩0
for t > 4 s
⎧
⎪
for t < 2 s
⎨10 (mJ)
1
2
2
w2 (t) = Cυ2 = 0.1(20 − 5t) (mJ) for 2 ≤ t ≤ 4 s
⎪
2
⎩0
for t > 4 s
υ3 (t) = 15u(−t) + 15e−0.5t
d υ3
i3 (t) = C
=
dt
!
15 (V)
for t ≤ 0
u(t) (V) =
−0.5t
15e
(V) for t ≥ 0
!
for t < 0
0
−0.5t
−1.5e
(mA) for t > 0
All rights reserved. Do not reproduce or distribute. © 2016 National Technology and Science Press
!
0
for t < 0
p3 (t) = i3 (t) υ3 (t) =
−22.5e−t (mW) for t > 0
!
1
22.5 (mJ)
for t < 0
w3 (t) = Cυ32 =
−t
2
22.5e (mJ) for t > 0
(d)
υ4 (t) = 15[1 − e−0.5t ] u(t) (V)
d υ4
= 1.5e−0.5t u(t) (mA)
dt
p4 (t) = i4 (t) υ4 (t) = 22.5e−0.5t [1 − e−0.5t ] u(t) (mW)
i4 (t) = C
w4 (t) =
1
Cυ42 = 22.5(1 − e−0.5t )2 u(t) (mJ)
2
All rights reserved. Do not reproduce or distribute. © 2016 National Technology and Science Press