Problems
lt
It
247
Problems
Section 7.1
c) Find i 1(t) fort ;:::: O.
d) Find i2(t) for t ;:::: 0+.
7.1 In the circuit in Fig. P7.1, the voltage and current
expressions are
v
=
160e- lOr V,
t;::::
i = 6.4e- lOr A,
e) Explain why i2 (0-)
0+;
Figure P7.4
t ;:::: O.
soon
6ki1
Find
(""
a) R.
b)
T
"* i2(0+).
()
40V
400mH
(in milliseconds).
c) L.
d) the initial energy stored in the inductor.
e) the time (in milliseconds) it takes to dissipate
60% of the initial stored energy.
7.5 The switch shown in Fig. P7.5 has been open a long
time before closing at t =
a) Find io(O-).
Figure P7.1
o.
b) Find iL(O-).
c) Find io(O+).
d) Find i [(0+).
e) Find io(oo).
f) Find iL(oo).
7.2 a) Use component values from Appendix H to create
a first-order RL circuit (see Fig. 7.4) with a time
constant of 1 ms. Use a single inductor and a net­
work of resistors, if necessary. Draw your circuit.
g) Write the expression for iL(t) for t ;:::: O.
h) Find VL(O-).
i) Find v L(O+).
j) FindvL(oo).
k) Write the expression for VL(t) for t ;:::: 0+.
b) Suppose the inductor you chose in part (a) has
an initial current of 10 rnA. Write an expression
for the current through the inductor for t ;:::: o.
I) Write the expression for VI) for I ;:::: 0+.
c) Using your result from part (b), calculate the
time at which half of the initial energy stored in
the inductor has been dissipated by the resistor.
Figure P7.5
401}
10,0,
20,0,
7.3 The switch in the circuit in Fig. P7.3 has been open
PSP!CE
MUlTISlf·l
for a long time. At t = 0 the switch is closed.
120 n
12V
a) Determine io(O+) and io(oo).
100 mH r l ,
b) Determine ioCt) for t ;:::: 0+.
c) How many milliseconds after the switch has been
closed will the current in the switch equal 3 A?
7.6 The switch in the circuit in Fig. P7.6 has been closed a
long time.At t = 0 it is opened. Find ioCt) for I ;:::: O.
Figure P7.3
5,0,
PSPICE
10,0,
]25 V
HUlTlSI/.1
50mH
7.4 The switch in the circuit in Fig. P7.4 has been closed
PSPICE
/.lUlTISII·l
for a long time before opening at / = O.
Figure P7.6
.. ---'{.= I)
4,0,
1.5,0,
12.45,0,
48V
18,0,
0.5 H
a) Find iJ(O-) and i 2 (0-).
b) Find i l (0+) and i 2 (0+).
2,0,
JOn
i" ..
54,0,
26
n
248
Response of First-Order RL and RC Circuits
7.7 In the circuit shown in Fig. P7.7, the switch makes
contact with position b just before breaking contact
with position a. As already mentioned, this is
known as a make-before-break switch and is
designed so that the switch does not interrupt the
current in an inductive circuit. The interval of time
between "making" and "breaking" is assumed to be
negligible. The switch has been in the a position for
a long time. At t = the switch is thrown from posi­
tion a to position b.
7.10 In the circuit in Fig. P7.10, the switch has been
closed for a long time before opening at t = 0.
a) Find the value of L so that vo(t) equals 0.5 vo(O+)
when I = 1 ms.
b) Find the percentage of the stored energy that
has been dissipated in the 10.n resistor when
t = 1 ms.
°
Figure P7.10
9kD
a) Determine the initial current in the inductor.
1= U
b) Determine the time constant of the circuit
for I > 0.
c) Find i,
VJ,
and
V2
for
I 2:
0.
d) What percentage of the initial energy stored in
the inductor is dissipated in the 72.n resistor
15 ms after the switch is thrown from position a
to position b?
7.11 In the circuit shown in Fig. P7.11, the switch has
been in position a for a long time. At t = 0, it moves
NULTISII·I
instantaneously from a to b.
PSPICE
a) Find Lo(t) for t
O.
b) What is the total energy delivered to the 8 n
resistor?
c) How many time constants does it take to deliver
95% of the energy found in (b)?
Figure P7.7
4D
laD /'" L
1 kD
30mA
8D
1.6 B
1'1
2:
Figure P7.11
30D a
,
7.8 The switch in the circuit seen in Fig. P7.8 has been
in position 1 for a long time. At t = 0, the switch
moves instantaneously to position 2. Find the value
of R so that 10% of the initial energy stored in the
10 mR inductor is dissipated in R in 10 f.-Ls.
12A
8D
lS0D
2mH
7.12 The switch in the circuit in Fig. P7.12 has been in
position 1 for a long time. At t = 0, the switch moves
/·\ULTISU·\ instantaneously to position 2. Find vo(t) for t 2: 0+.
PSPICE
Figure P7.8
1
Figure P7.12
100D
SA
R
12D
1
240 V
7.9 In the circuit in Fig. P7.8, let I g represent the dc cur­
rent source, 0' represent the fraction of initial
energy stored in the inductor that is dissipated in [0
seconds, and L represent the inductance.
a) Show that
R=
Lin [1/(1 - 0')]
2t o
.
b) Test the expression derived in (a) by using it to
find tbe value of R in Problem 7.8.
4D
72mB
I'"
400,
Ion
7.13 For the circuit of Fig. P7.12, what percentage of the
initial energy stored in the inductor is eventually
dissipated in the 40 .n resistor?
7.14 The switch in Fig. P7.14 has been closed for a long
time before opening at t = O. Find
a) i L(t), t 2: O.
b) VL(t), t 2: 0+.
c) il;.(t), t 2: 0+.
Problems
figure P7.14
Figure P7.18
1= ()
40D
249
6D
l20V <GOD
~rA
[ = ()
20i"
~
~
T: ~ /',
'.
•
IOOD<
7.15 What percentage of the initial energy stored in the
inductor in the circuit in Fig. P7.14 is dissipated by
the 60 D resistor?
7.16 The switch in the circuit in Fig. P7.16 has been
closed for a long time before opening at t = O. Find
~ULnsll·l
vo(t) for t 2: 0+.
F5PlCE
Figure P7.16
(
,
11(1<; rn.6.
<4 kD
:> 1 kD
<20 kD <80 kfl
7.19 The two switches shown in the circuit in Fig. P7.19
operate simultaneously. Prior to t = 0 each switch
'·\ULTISIM
has been in its indicated position for a long time. At
t = 0 the two switches move instantaneously to
their new positions. Find
PSPICE
a) vo(t), t
2:
b) io(t), t
1= ()
4A
"
i;,.
60D~
f = ()
6H
2:
0+.
O.
Figure P7.19
2D
1',,~5mH
7.17 The 240 V,2 D source in the circuit in Fig. P7.17 is
inadvertently short-circuited at its terminals a,b. At
MUlTISH·l
the time the fault occurs, the circuit has been in
operation for a long time.
FSPICE
a) What is the initial value of the current i ab in the
short-circuit connection between terminals a,b?
..
10D
2A
1"
7.5 kD
)
..,6H
7.20 For the circuit seen in Fig. P7.19, find
b) What is the final value of the current i ab ?
a) the total energy dissipated in the 7.5 kD resistor.
c) How many microseconds after the short circuit
has occurred is the current in the short equal
to 114 A?
b) the energy trapped in the ideal inductors.
Figure P7.17
2,0,
Section 7.2
7.21 In the circuit in Fig. P7.21 the voltage and current
expressions are
a
v = 72e- 5001 V, t
IOD
15 D
i
= ge- 5001 rnA,
2:
t
0;
2:
0+.
240V' +
2mH
~6
Find
mH
b
7.18 The two switches in the circuit seen in Fig. P7.18 are
synchronized. The switches have been closed for a
long time before opening at t = O.
a) How many microseconds after the switches are
open is the energy dissipated in the 4 kD resis­
tor 10% of the initial energy stored in the 6 H
inductor?
b) At the time calculated in (a), what percentage of
the total energy stored in the inductor has been
dissipated?
a) R.
b) C.
c) T (in milliseconds).
d) the initial energy stored in the capacitor.
e) how many microseconds it takes to dissipate
68% of the initial energy stored in the capacitor.
Figure P7.21
i
..
cO
250
Response of First-Order RL and RC Circuits
7.22 a) Use component values from Appendix H to cre­
ate a first-order RC circuit (see Fig. 7.11) with a
time constant of 50 ms. Use a single capacitor
and a network of resistors, if necessary. Draw
your circuit.
a) How many microjoules of energy have been
dissipated in the 12 kfl resistor 12 ms after the
switches open?
b) How long does it take to dissipate 75% of the
initially stored energy?
b) Suppose the capacitor you chose in part (a) has an
initial voltage drop of 50 V. Write an expression for
the voltage drop across the capacitor for t 2': O.
I -= (J
c) Using you resull from part (b), calculate the
time at which the voltage drop across the capac­
itor has reached 10 V.
7.23 The switch in the circuit in Fig. P7.23 has been in
position a for a long time and VI = 0 V. At t = 0,
the switch is thrown to position b. Calculate
a) i, VI, and
VI
for t
68 kfl
7.26 Both switches in the circuit in Fig. P7.26 have been
closed for a long time. At t = 0, both switches open
/·\ULTIS!I·\
simultaneously.
PSPICE
0+.
2':
12 kfl
b) the energy stored in the capacitor at t = O.
a) Find io(t) for t
c) the energy trapped in the circuit and the total
energy dissipated in the 25 kfl resistor if the
switch remains in position b indefinitely.
b) Find vo(t) for t
2':
0+.
O.
2':
c) Calculate the energy (in microjoules) trapped in
the circuit.
Figure P7.23
3.3 kfl
~
Figure P7.26
25 kfl
b
a
~"-
40V
1. J.LF
/.
I
4 "F
r-
40mA
t·~UlT!Sl;·i
a) Find i1(0-) and iI(O-).
b) Find i 1(0+) and iI(O+).
c) Explain why i1(0-) = i[(O+).
2':
2':
=j.
lkfl
­
7.24 The switch in the circuit in Fig. P7.24 is closed at
t = 0 after being open for a long time.
e) Find i] (t) for t
f) Find i2 (t) for t
= ()
r,
PSPICE
d) Explain why i2CO-)
I
300 of r"
7.27 After the circuit in Fig. P7.27 has been in operation
for a long time, a screwdriver is inadvertently con­
r·\ULTISlf.!
nected across the terminals a,b. Assume the resist­
ance of the screwdriver is negligible.
PSPICE
a) Find the current in the screwdriver at t
t =
i2 (0+).
b) Derive the expression for the current in the
screwdriver for t 2': 0+.
O.
0+.
Figure P7.27
Figure P7.24
a
100mA
0.2fl
20fl
IV
= 0+ and
00.
Sfl
2fl
.­
/,
ij
SA
3fl
30fl
100 J.LF
O.Sfl
b
1= ()
7.25 In the circuit shown in Fig. P7.25, both switches
operate together; that is, they either open or close at
the same time. TIle switches are closed a long time
before opening at t = O.
7.28 The switch in the circuit seen in Fig. P7.28 has been
in position x for a long time. At t = 0, the switch
moves instantaneously to position y.
a) Find (l' so that the time constant for t > 0 is
40ms.
b) For the ex found in (a), find
vb,'
Problems
been
er the
. Figure P7.28
20kD
c) Find Vl(t) fort
2:
O.
d) Find V2(t) for t
2:
O.
251
e) Find the energy (in millijoules) trapped in the
ideal capacitors.
)f the
Figure P7.3 2
7.29 a) In Problem 7.28, how many microjoules of
energy are generated by the dependent current
source during the time the capacitor discharges
to OV?
+
7.30 The switch in the circuit in Fig. P7.30 has been in
position 1 for a long time before moving to posi­
HUlTISlI-I
tion 2 at t = O. Find io(t) for t 2: 0+.
250 kD
/',
2/-LF
b) Show that for t 2: 0 the total energy stored and
generated in the capacitive circuit equals the
total energy dissipated.
een
pen
I'"
P511CE
lin
Figure P7.30
4.7 kD
Section 7.3
7.33 The current and voltage at the terminals of the
inductor in the circuit in Fig. 7.16 are
1
i(t) = (4 + 4e-40') A,
2
2:
0;
v(t) = -80e- 401 V,
15V
15 D
n
t
a) Specify the numerical values of v" R, 10 , and L.
7.31 At the time the switch is closed in the circuit in
Fig. P7.31, the voltage across the paralleled capaci­
tors is 50 V and the voltage on the 250 nF capacitor
is 40 V.
a) What percentage of the initial energy stored in
the three capacitors is dissipated in the 24 kD,
resistor?
b) Repeat (a) for the 400 D, and 16 kD, resistors.
c) What percentage of the initial energy is trapped
in the capacitors?
b) How many milliseconds after the switch has
been closed does the energy stored in the induc­
tor reach 9 J?
F5PJ([
HUlTISl~1
Figure P7.31
250 of
r+l
40 V _
200 of
~\
+
50 V
400D
I
= +()-----'INv_---,
24kSl
16kSl
800 of
7.32 At the time the switch is closed in the circuit shown
in Fig. P7.32, the capacitors are charged as shown.
a) Find vo(t) for t 2: 0+.
b) What percentage of the total energy initially
stored in the three capacitors is dissipated in the
250 kD, resistor?
rtrtt.~
7.34 a) Use component values from Appendix H to
create a first-order RL circuit (see Fig. 7.16)
with a time constant of 8 fLS. Use a single induc­
tor and a network of resistors, if necessary.
Draw your circuit.
b) Suppose the inductor you chose in part (a) has
no initial stored energy. At t = 0, a switch con­
nects a voltage source with a value of 25 V in
series with the inductor and equivalent resist­
ance. Write an expression for the current
through the inductor for l 2: O.
c) Using your result from part (b), calculate the
time at which the current through the inductor
reaches 75% of its final value.
7.35 The switch in the circuit shown in Fig. P7.35 has
been closed for a long time before opening at t = O.
PSP!CE
:.IULTlSll-l
a) Find the numerical expressions for iL(t) and
vO(t) for t 2: O.
b) Find the numerical values of vL(O+) and vo(O+).
_
252
Response of First-Order RL and RC Circuits
c) Find Vsw as a function of JfI' RI> Rz, and L.
d) Explain what happens to vsw as Rz gets larger
and larger.
Figure P7.35
4mB
4n
.f-
1'1.
i,
--
40V
..
+
1'"
16n
Figure P7.3 9
5A
1= (J
7.36 After the switch in the circuit of Fig. P7.36 has been
open for a long time, it is closed at t = O. Calculate
(a) the initial value of i; (b) the final value of i;
(c) the time constant for t :=:: 0; and (d) the numeri­
cal expression for i(t) when t :=:: O.
Figure P7.36
5 k.(1
4k.(1
20 k.(1
150V
75rnH
20 k.(1
20 k.(1 0.5 rnA
[ = ()
7.37 The switch in the circuit shown in Fig. P7.37 has
been in position a for a long time. At t = 0, the
/·IULTISHoI
switch moves instantaneously to position b.
a) Find the numerical expression for io(t) when
t :=:: O.
PSPICE
-I-
t',,, ­
L
1',.(1)
7.40 The switch in the circuit in Fig. P7.40 has been
closed for a long time. A student abruptly opens the
switch and reports to her instructor that when the
switch opened, an electric arc with noticeable per­
sistence was established across the switch, and at
the same time the voltmeter placed across the coil
was damaged. On the basis of your analysis of the
circuit in Problem 7.39, can you explain to the stu­
dent why this happened?
Figure P7.40
R
b) Find the numerical expression for vo(t) for
t :=:: 0+.
L
Figure P7.3 7
IOn
a
+
50A
sn
'-'"
7.41 The switch in the circuit in Fig. P7.41 has been
open a long time before closing at t = O. Find voCt)
l·lULTISI/·\
for t :=:: 0+.
PSPICE
40n 40mB
Figure P7.41
10.(1
5.(1
+
7.38 a) Derive Eq. 7.47 by first converting the Thevenin
equivalent in Fig. 7.16 to a Norton equivalent
and then sununing the currents away from the
upper node, using the inductor voltage v as the
variable of interest.
b) Use the separation of variables technique to find
the solution to Eq. 7.47. Verify that your solution
agrees with the solution given in Eq. 7.42.
7.39 The switch in the circuit shown in Fig. P7.39 has
been closed for a long time. The switch opens at
t = O. For t :=:: 0+:
a) Find vo(t) as a function of I g , R 1, R z, and L.
b) Explain what happens to vo(t) as Rz gets larger
and larger.
20 rnA
15.(1
I'"
7.42 The switch in the circuit in Fig. P7.42 has been open a
long time before closing at t = O. Find ioCt) for t :=:: O.
PS?ICE
/·1 ULTISII·\
Figure P7.42
80mH
i,,( I)
15 n
20n
In
f­
t
1',1,
.­
+ 480 V
0.8 vI/>
2S0V
Problems
7.43 The switch in the circuit in Fig. P7.43 has been
open a long time before closing at t = O. Find vo(t)
"umsl/~ for ( ;::: 0+.
253
Figure P7.46
PiPICE
----+--,2
+
Figure P7.43
SOY
1.5 H /'"
40 D
1= ()
20 mH
lOD
1'"
ISA
40n
+
SOmH
7.47 For the circuit in Fig. P7.46, find (in joules):
SOY
a) the total energy dissipated in the 40
n
resistor;
b) the energy trapped in the inductors;
c) the initial energy stored in the inductors.
7.44 There is no energy stored in the inductors L j and L 2
at the time the switch is opened in the circuit shown
in Fig. P7.44.
7.48 The current and voltage at the terminals of the
capacitor in the circuit in Fig. 7.21 are
a) Derive the expressions for the currents ij(t) and
tzCt) for t ;::: O.
i(t)
= 3e-2500t
rnA,
v(f) = (4D - 24e-'l':RR") V,
b) Use the expressions derived in (a) to find ij(OO)
and i 2(00).
( ;::: 0+;
( ;::: D.
a) Specify the numerical values of Is. Yc" R, C,
and T.
Figure P7.44
b) How many microseconds after the switch has
been closed does the energy stored in the capac­
itor reach 81 % of its final value?
7.45 The make-before-break switch in the circuit of
PiPICE
MUmSI/·\
Fig. P7.45 has been in position a for a long time. At
( = 0, the switch moves instantaneously to posi­ tion b. Find
7.49 a) Use component values from Appendix H to cre­
ate a first-order RC circuit (see Fig. 7.21) with a
time constant of 250 ms. Use a single capacitor
and a network of resistors, if necessary. Draw
your circuit.
b) Suppose the capacitor you chose in part (a) has an
initial voltage drop of 100 V. At t = 0, a switch con­
nects a current source with a value of 1 rnA in par­
allel with the capacitor and equivalent resistance.
Write an expression for the voltage drop across
the capacitor for ( ;::: O.
a) vo(t), t ;::: 0+.
b) tJ(t),
t ;:::
O.
c) i2(t), t :=:: O.
Figure P7.45
c) Using your result from part (b), calculate the
time at which the voltage drop across the capici­
tor reaches 50 V.
a
~----'c--
/
r-..,1=()
-'­
40 mH
I',.
...-----,
120 D
SOmA
7.50 The switch in the circuit shown in Fig. P7.50 has
been closed a long time before opening at t = O.
PSP[CE
'·\ULnSII·\
a) What is the initial value of ioCt)?
b) What is the final value of ioCt)?
c) What is the time constant of the circuit for ( ;::: O?
7.46 The switch in the circuit in Fig. P7.46 has been in
position 1 for a long time. At t = 0 it moves instan­
~!UlTISI/·\
taneously to position 2. How many milliseconds
after the switch operates does Va equal 100 Y?
FSPICE
4
d) What is the numerical expression for to(t) when
( ;::: O+?
e) What is the numerical expression for vo(t) when
t ;::: O+?
_
254
Response of First-Order RL and RC Circuits
Figure P7.50
2kl1
7.54 The switch in the circuit seen in Fig. P7.54 has been
in position a for a long time. At t = O. the switch
/·\UlTlSIi·!
moves i'nstantaneously to position b. Find vo(t) and
io(l) for I ~ 0+.
PSPICE
3.2 kD,
-~
i,,(1 )
40Y
0.8/LF
18kl1
Figure P7.54
7.51 The switch in the circuit shown in Fig. P7.51 has
been closed a long time before opening at t = O.
NULTlSII·t
For t ~ 0+, find
PSPICE
20 kD,
lOrnA
a) VoCE).
50 kD,
15 mA
16 nF
b) io(t).
c) ij(t).
d) i 2 (t).
a
e) il(O+).
7.55 Assume that the switch in the circuit of Fig, P7.55
has been in position a for a long time and that at
t = 0 it is moved to position b. Find (a) vcCO+);
(b) vc(oo); (c) T for t > 0; (d) i(O+); (e) vc, t ~ 0;
and (f) i, t ~ 0+.
Figure P7.51
Skn
+
SmA
Hi)
-
, 15 kO,i,,(1) , 60 kfl ('"
500 nF
Figure P7.55
400 kD,a
SD,
b
r = ()
7.52 The switch in the circuit seen in Fig. P7.52 has been in
PSPICE
position a for a long time. At t = 0, the switch moves
instantaneously to position b. For I ~ 0+, find
a) V,,(l).
20n
SOY
30Y
r·lULTlSJ~l
b) io(t).
c) vit).
d) vn(O+).
"
Figure P7.52
i,,(1)
150 kfl
7.56 The switch in the circuit of Fig. P7.56 has been in
position a for a long time. At t = 0 the switch is
moved to position b. Calculate (a) the initial voltage
on the capacitor; (b) the final voltage on the capaci­
tor; (c) the time constant (in microseconds) for
t > 0; and (d) the length of time (in microseconds)
required for the capacitor voltage to reach zero
after the switch is moved to position b.
+
50 kO,
('Jr)
Figure P7.56
4mA
lOkD
3kD
7.53 The circuit in Fig. P7.53 has been in operation for a
PSPICE
long time. At t = 0, the voltage source reverses
MUlTlSII,\
polarity and the current source drops from 3 rnA to
2 mA. Find vo(t) for I ~ O.
+
40kD
120Y
9kD
L.5 mA
I'·(
Figure P7.53
lOkn
SOY
40 kl1
4kl1
3 rnA
24 kl1
0.05/LF
7.57 The switch in the circuit in Fig. P7.57 has been in
position a for a long time. At t = 0, the switch
/·lULTlSIt·!
moves instantaneously to position b. At the instant
the switch makes contact with terminal b, switch 2
opens. Find vo(t) for t ~ O.
PSPICE
Problems
Figure P7.5 7
40kfl
255
Figure P7.62
a
1
Ib
r-.llM~_~"1 = ()/~-'---_----~
+
50V
I'"~
20 kfl
7.58 The switch in the circuit shown in Fig. P7.58 has
been in the OFF position for a long time. At t = 0,
~UlTISIM
h
.
t e swItch moves instantaneously to the ON posi­
tion. Find vo(t) for t ;::= O.
PSPICE
Figure P7.58
7.63 The switch in the circuit in Fig. P7.63 has been in
position x for a long time. The initial charge on the
10 nF capacitor is zero. At t = 0, the switch moves
instantaneously to posi tion y.
a) Find vo(t) for t ;::= 0+.
20k[1
b) Find vJ(t) for t ;::= O.
i .:;
lOk[1
80 k[1
100Y
Figure P7.63
10k[1
x
/
10nF
,••- - /
7.59 Assume that the switch in the circuit of Fig. P7.58
PSPICE
has been in the ON position for a long time before
MUlTISII·\
switching instantaneously to the OFF position at
t = O. Find vo(t) for f ;::= O.
7.60 The switch in the circuit shown in Fig. P7.60 opens at
f = 0 after being closed for a long time. How many
MUlTlSI/\
milliseconds after the switch opens is the energy
stored in the capacitor 36% of its final value?
PSPICE
75 Y 20k[1
f.--~
-:-­
+
I"
250 k[1
7.64 The switch in the circuit of Fig. P7.64 has been in
position a for a long time. At ( = 0, it moves instan­
~\ULTISH·l
taneously to position b. For t ;::= 0+, find
PSPICE
a) vo(t).
7.61 a) Derive Eq. 7.52 by first converting the Norton
equivalent circuit shown in Fig. 7.21 to a Thevenin
equivalent and then summing the voltages around
the closed loop, using the capacitor current i as the
relevant variable.
b) Use the separation of variables technique to find
the solution to Eq. 7.52. Verify that your solution
agrees with that of Eg. 7.53.
7.62 There is no energy stored in the capacitors C j and
C2 at the time the switch is closed in the circuit seen
in Fig. P7.62.
a) Derive the expressions for VI(t) and V2(t) for
t ;::= O.
b) Use the expressions derived in (a) to find Vl(oo)
and V2(00).
Figure P7.60
120MA
33 k[1
47k[1
!=()
~===;=:=;=:======._---
16 k[1
0.25 MF
b) ioCt).
c) VI(t).
d) V2(t).
e) the energy trapped in the capacitors as f ~ 00.
Figure P7.64
2.2 k[1
a
Ib
6.25 k[1
I = ()
40 Y
+
+
/'"
i"
+ SOY
256
Response of First-Order RL and RC Circuits
Section 7.4
Figure P7.69
250 n
7.65 Repeat (a) and (b) in Example 7.10 if the mutual
inductance is reduced to zero.
,
lOY
0.25 H
i,
•
7.66 There is no energy stored in the circuit in Fig. P7.66
at the time the switch is closed.
PSPICE
f1ULTISlfI
a) Find i(t) for t
b) Find
VI (t)
2:
O.
for t
2:
0+.
Section 7.5
c) Find V2(t) for t
2:
O.
7.70 In the circuit in Fig. P7.70, switch A has been open
and switch B has been closed for a long time. At
;·IULTlSIM
t = 0, switch A closes. Five seconds after switch A
closes, switch B opens. Find iL(t) for t 2: O.
d) Do your answers make sense 1n terms of known
circuit behavior?
PSPICE
Figure P7.66
Figure P7. 70
40
n
.. i (I)
SOV
In
lOV
5H
•
7.67 Repeat Problem 7.66 if the dot on the 10 H coil is at
PSPICE
the top of the coil.
1·1ULTISIM
7.68 There is no energy stored in the circuit of Fig. P7.68
at the time the switch is closed.
a) Find io(t) for t
b) Find vo(t) for t
2:
2:
c) Find iJ(t) for t
2:
cl) Find i 2(t) for t
2:
O.
0+.
O.
O.
7.71 The action of the two switches in the circuit seen in
Fig. P7.71 is as follows. For t < 0, switch 1 is in posi­
'·Iumml
tion a and switch 2 is open. This state has existed for
a long time. At t = 0, switch 1 moves instanta­
neously from position a to position b, while switch 2
remains open. Ten milliseconds after switch 1 oper­
ates, switch 2 closes, remains closed for 10 ms and
then opens. Find v o(£) 25 ms after switch 1 moves to
position b.
PSPICE
Figure P7.71
5 Sl () + lO ms
e) Do your answers make sense in terms of known
circuit behavior?
2
+
/'"
15 A
50 mH 20 n
Figure P7.68
20n
5H '"
SOV
5H
10H
•
7.72 For the circuit in Fig. P7.71, how many milliseconds
after switch 1 moves to position b is the energy
stored in the inductor 4% of its initial value?
7.73 The switch in the circuit shown in Fig. P7.73 has
been in position a for a long time. At t = 0, the
MumSlfl
switch is moved to position b, where it remains for
1 ms. The switch is then moved to position c, where
it remains indefinitely. Find
PSPICE
7.69 There is no energy stored in the circuit in Fig. P7.69
at the time the switch is closed.
PSPICE
~IULTlSIM
a) Find io(£) for t
b) Find vo(t) for I
c) Find il(t) for t
d) Find i 2 (t) for (
2:
2:
O.
0+.
2:
O.
2:
O.
e) Do your answers make sense in terms of known
circuit behavior?
Fi
a) i(O+).
b) 1(200 fLS).
c) i(6 ms).
d) v(l- ms).
e) v(l+ ms).
7.
Problems
7.77 For the circuit in Fig. P7.76, what percentage of the
initial energy stored in the 500 nF capacitor is dissi­
r·w lTISH·\
pated in the 3 kfl resistor?
Figure Pl.l3
PSPICE
40D,
a
.... +
c
b
!
60D,
40D
257
120D
I'
80mH
7.78 The switch in the circuit in Fig. P7.78 has been in
position a for a long time. At t = 0, it moves instan­
1·\UlTISH,1
taneously to position b, where it remains for five
seconds before moving instantaneously to position
c. Find va for t 2': O.
PSPICE
7.74 There is no energy stored in the capacitor in the cir­
cuit in Fig. P7.74 when switch 1 closes at t = O. Ten
HUlTISIH
microseconds later, switch 2 closes. Find vo(t) for
t 2': O.
PSPiCE
Figure P7.78
b
3.3kn!=O~s
Figure P7.74
.e---~
a
5mA
/
C
•
1 kD
100 kD
100 f.LF
30V
4kfl
7.79 The voltage waveform shown in Fig. P7.79(a) is
applied to the circuit of Fig. P7.79(b). The initial
1·\ULTISII,\
current in the inductor is zero.
PSPICE
7.75 The capacitor in the circuit seen in Fig. P7.75 has
been charged to 300 V. At t = 0, switch 1 closes,
HUlTlSIH
causing the capacitor to discharge into the resistive
network. Switch 2 closes 200 J-LS after switch 1
closes. Find the magnitude and direction of the cur­
rent in the second switch 300 J-Ls after switch 1
closes.
PlPICE
a) Calculate vo(t).
b) Make a sketch of vo(t) versus I.
c) Find io at t = 5 ms.
Figure P7. 79
Vs
Figure P7. 75
1
(V)
20D,
80/--­
+
I =()
+
300V
2
10 nF
3
I
120 kD
i", 40 mH
l'.\
/'"
60kD,
30kD,
= () +
o
2.5
t (ms)
(b)
(a)
2()() fJ.'
40 kD,
7.80 The current source in the circuit in Fig. P7.80(a)
generates the current pulse shown in Fig. P7.80(b).
I-\ULTISlfI
There is no energy stored at t = O.
P5PIC<
7.76 In the circuit in Fig. P7.76, switch 1 has been in posi­
tion a and switch 2 has been closed for a long time.
At t = 0, switch 1 moves instantaneously to posi­
tion b. Eight hundred microseconds later, switch 2
opens, remains open for 300 J-LS, and then recloses.
Find V o 1.5 ms after switch 1 makes contact with
terminal b.
a) Derive the numerical expressions for va(t) for
the time intervals t < 0, 0 :0; t :0; 75 J-LS, and
75 J-LS :0; I < co.
b) Calculate V o (75- J-Ls) and
Va
(75+ J-Ls).
c) Calculate i o (75- J-Ls) and i o (75+ J-Ls).
Figure P7.80
is (rnA)
Figure P7. 76
251------.
7.5 rnA
10kfl
3kD
2kD,
250mH
o
(a)
75
(b)
I
(f.Ls)
258
Response of First-Order RL and RC Circuits
7.81 The voltage waveform shown in Fig. P7.81(a) is
applied to the circuit of Fig. P7.81(b). The initial
nUlTISH·1
voltage on the capacitor is zero.
d) Sketch io(t) versus
-1 ms < t < 4 ms.
PSPICE
e) Sketch vJt) versus
-1 ms < t < 4 ms.
a) Calculate v()(t).
t
for
the
interval
for
the
interval
b) Make a sketch of vo(t) versus t.
Figure P7.83
Figure P7.81
VS
(V)
..
~f--..----e
50/-----,
1­
'T'
1',
o
1
ig (rnA)
4kD.
10 nF
ig
400 kf2
16 kD.
I'"~
20
i(,
0.2p.,F
o
( (ms)
(a)
(b)
7.82 The voltage signal source in the circuit in Fig. P7.82(a)
is generating the signal shown in Fig. P7.82(b). There is
r·1ULTISlll
no stored energy at t = O.
P5?ICE
a) Derive the expressions for vo(t) that apply in the
intervals t < 0; 0 ::5 t ::5 4 ms; 4 ms ::5 t ::5 8 ms;
and 8 ms ::5 t < 00.
b) Sketch
Vo
and
Vs
on the same coordinate axes.
2 {(ms)
(b)
(a)
Section 7.6
7.84 The capacitor in the circuit shown in Fig. P7.84 is
charged to 20 V at the time the switch is closed. If
t,lULTISH·l
the capacitor ruptures when its terminal voltage
equals or exceeds 20 kV, how long does it take to
rupture the capacitor?
P5PlCE
c) Repeat (a) and (b) with R reduced to 50 kD..
Figure P7.84
Figure P7.82
12 X 104 iil
R = 200 kf2
,-----./ \----<+
v'8----------.;
80 kf2
-'>----..--~'\N\~----,
+
i.l
20Y
20 kf2
(a)
V,(Y)
100
k-----,
o
-100
7.85 The switch in the circuit in Fig. P7.85 has been
closed for a long time. The maximum voltage rating
r.1ULTIsrn
of the 1.6 p.,F capacitor is 14.4 kV. How long after
the switch is opened does the voltage across the
capacitor reach the maximum voltage rating?
PSPICE
4
I
(ms)
I­
Figure P7.85
(b)
lkD.
7.83 The current source in the circuit in Fig. P7.83(a)
generates the current pulse shown in Fig. P7.83(b).
There is no energy stored at t = O.
2kD.
FSP!CE
SmA
t·~ULnSPI
a) Derive the expressions for io(t) and vo(t) for the
time intervals t < 0; 0 < t < 2 ms; and
2 ms < t < 00.
b) Calculate i()(O-);
i o(O.002+).
c) Calculate vo(O-);
v o (O.002+).
io(O+);
io(O.OOT);
and
v()(O+);
v()(O.OOT);
and
7.86 The inductor current in the circuit in Fig. P7.86 is
25 rnA at the instant the switch is opened. The
nULTlSfI.\
inductor will malfunction whenever the magnitude
of the inductor current equals or exceeds 5 A. How
long after the switch is opened does the inductor
malfunction?
PSPICE
Problems
Figure P7.86
259
Figure P7.88
2kD
+
1'-" (J
2
1',"
Push button
X 10- 3 Vrj>
..
a ------=--:=~=.:~tof==---·t
­
b
4kD
4kD
7.87 The gap in the circuit seen in Fig. P7.87 will arc over
whenever the voltage across the gap reaches 45 kY.
The initial current in the inductor is zero. The value
of (3 is adjusted so the Thevenin resistance with
respect to the terminals of the inductor is -5 kfl.
a) What is the value of (3?
Electric
relay
25 kD
PSPICE
MULTI5I~I
b) How many microseconds after the switch has
been closed will the gap arc over?
+
--=-SOY
Section 7.7
7.89 The voltage pulse shown in Fig. P7.89(a) is applied
to the ideal integrating amplifier shown in
!·IULTISm
Fig. P7.89(b). Derive the numerical expressions for
vo(t) when vo(O) = 0 for the time intervals
a) t < O.
b) 0 s t :S 250ms.
PSPICE
Figure P7.87
5kD
c) 250 ms s t s 500 ms.
d) 500 ms s t < (x).
20 kD
Figure P7.89
vg (mY)
200 I­
7.88 The circuit shown in Fig. P7.88 is used to close the
switch between a and b for a predetermined length
of time. The electric relay holds its contact arms
down as long as the voltage across the relay coil
exceeds 5 Y. When the coil voltage equals 5 V, the
relay contacts return to their initial position by a
mechanical spring action. The switch between a and
b is initially closed by momentarily pressing the
push button. Assume that the capacitor is fully
charged when the push button is first pushed down.
o
I
250
500
I
(ms)
-2001---1
(a)
400nF
The resistance of the relay coil is 25 kD, and the
inductance of the coil is negligible.
a) How long will the switch between a and b
remain closed?
b) Write the numerical expression for i from the
time the relay contacts first open to the time the
capacitor is completely charged.
c) How many milliseconds (after the circuit
between a and b is interrupted) does it take the
capacitor to reach 85% of its final value?
(b)
7.90 Repeat Problem 7.89 with a 5 Mfl resistor placed
across the 400 nF feedback capacitor.
PSPICE
r·\ULTlSI1·\
~--------------
----_._._------------------------.­
260
Response of First-Order RL and RC Circuits
7.91 The energy stored in the capacitor in the circuit
shown in Fig. P7.91 is zero at the instant the switch
/\UlTISII·\
is closed. The ideal operational amplifier reaches
saturation in 15 ms. What is the numerical value of
R in kilo-ohms?
PSPICE
7.94 There is no energy stored in the capacitors in the
circuit shown in Fig. P7.94 at the instant the t\Vo
r·1ULTISIM
switches close. Assume the op amp is, ideal.
PSPICE
a) Find V o as a function of Va'
Vb,
R, and C.
b) On the basis of the result obtained in (a),
describe the operation of the circuit.
Figure P7.91
c) How long will it take to saturate the amplifier
if Va = 40mV; Vb = 15mV; R = 50kD;
C = 10 nF; and Vee = 6 V?
500nF
Figure P7.94
1'"
R
5.1 kf)
[~()
7.92 At the instant the switch is closed in the circuit of
Fig. P7.91, the capacitor is charged to 6 V, positive at
r1UlTlSIM
the right-hand terminal. If the ideal operational
amplifier saturates in 40 ms, what is the value of R?
PSPICE
7.93 The voltage source in the circuit in Fig. P7.93(a) is
generating the triangular waveform shown in
r1UlTlSIr-I
Fig. P7.93(b). Assume the energy stored in the
capacitor is zero at t = 0 and the op amp is ideal.
PSPICE
a) Derive the numerical expressions for vo(t) [or
the following time intervals: 0 s t s 1 f.Ls;
1 f.LS S t s 3 fJ.-s; and 3 f.Ls s t s 4 f.Ls.
b) Sketch the output waveform between 0 and 4 f.Ls.
c) If the triangular input voltage continues to repeat
itself for t > 4 f.Ls, what would you expect the
output voltage to be? Explain.
7.95 At the time the double-pole switch in the circuit
shown in Fig. P7.95 is closed, the initial voltages on
r·\UlTlSlI·\
the capacitors are 12 V and 4 V, as shown. Find the
numerical expressions for vo(t), V2(t), and VI (t) that
are applicable as long as the ideal op amp operates
in its linear range.
PSPICE
Figure P7.95
-
+
12V
50 nF
Figure P7.93
l=lJ
800 pF
~
lOOkfl
-
20V
15V
lkfl
r=
1',-(11
-;­
+
(J
1';.(1)
T
-15V
+
1'(1
V,(Vt
_+----'---~--L----}--[-(fL-S)
(b)
T
..,
7.96 At the instant the switch of Fig. P7.96 is closed, the
voltage on the capacitor is 56 V. Assume an ideal
r·!ULTISm
operational amplifier. How many milliseconds.
after the switch is closed will the output voltage Vo
equal zero?
PSPICE
(a)
4V
Problems
Figure P7.96
- 56V +
+
Sections 7.1-7.7
7.97 The circuit shown in Fig. P7.97 is known as a
MULT15IH
monostable l11ultivibrator. The adjective l11onostable
is used to describe the fact that the circuit has one
stable state. That is, if left alone, the electronic
switch T? will be ON, and T j will be OFF. (The opera­
tion of the ideal transistor switch is described in
detail in Problem 7.99.) T 2 can be turned OFF by
momentarily closing the switch S. After S returns to
its open position, T 2 will return to its ON state.
a) Show that ifT2 is ON, T j is OFF and will stay
b) Explain why T 2 is turned
tarily closed.
OFF
7.99 The circuit shown in Fig. P7.99 is known as an
astable multivibrator and finds wide application in
MULTISIM
pulse circuits. The purpose of this problem is to
relate the charging and discharging of the capaci­
tors to the operation of the circuit. The key to ana­
lyzing the circuit is to understand the behavior of
the ideal transistor switches T 1 and T z. The circuit is
designed so that the switches automatically alter­
nate between ON and OFF. When T] is OFF, T z is ON
and vice versa. Thus in the analysis of this circuit, we
assume a switch is either ON or OFF. We also assume
that the ideal transistor switch can change its state
instantaneously. In other words, it can snap from
OFF to ON and vice versa. When a transistor switch is
ON, (1) the base current i b is greater than zero,
(2) the terminal voltage vbe is zero, and (3) the ter­
minal voltage vee is zero. Thus, when a transistor
switch is ON, it presents a short circuit between the
terminals b,e and c,e. When a transistor switch is
OFF, (1) the terminal voltage vbe is negative, (2) the
base current is zero, and (3) there is an open circuit
between the terminals c,e. Thus when a transistor
switch is OFF, it presents an open circuit between
the terminals b,e and c,e. Assume that T z has been
ON and has just snapped OFF, while T j has been OFF
and has just snapped ON. You may assume that at
this instance, C2 is charged to the supply voltage
Vee, and the charge on C 1 is zero. Also assume
C j = C 2 and R J = R2 = 10R L .
PSPICE
-25V
PSPICE
261
OFF.
when S is momen­
c) Show that T 2 will stay OFF for RC In 2 s.
Figure P7. 97
a) Derive the expression for
val that T z is OFF.
vbe2
during the inter­
b) Derive the expression for
val that T z is OFF.
V ce 2
during the inter­
c) Find the length of time T z is OFF.
d) Find the value of vee 2 at the end of the interval
that T z is OFF.
e) Derive the expression for i bj during the interval
that T 2 is OFF.
f) Finel the value of i b1 at the end of the interval
that T z is OFF.
7.98 The parameter values in the circuit in Fig. P7.97
are
Vee = 6 V;
R 1 = 5.0 kn;
C = 250 pF; anel R = 23,083 n.
RL
= 20 kn;
g) Sketch
is OFF.
V ee 2
versus t during the interval that T z
h) Sketch i bJ versus t during the interval that T 2
is OFF.
a) Sketch vee 2 versus t, assuming that after S is
momentarily closed, it remains open until the
circuit has reached its stable state. Assume S is
closed at t = O. Make your sketch for the inter­
val -5 =5 t =5 10 J.LS.
b) Repeat (a) for i bZ versus t.
~"-----------------
262
Response of First-Order RL and RC Circuits
Figure P7.99
PSPICE
'·lUlTISft.l
R2
RI­
RL
C2
+ Vee
fbI
~
-l­
1"""1
7.104 In the circuit of Fig. 7.45, the lamp starts to conduct
whenever the lamp voltage reaches 15 V. During
the time when the lamp conducts, it can be modeled
"ULTlSJI.\ as a 10 leD resistor. Once the lamp conducts, it will
continue to conduct until the lamp voltage drops to
5 V. When the lamp is not conducting, it appears as
an open circuit. V, = 40 V; R = 800 kn; and
C = 25 pF
PRilCT1CAL
PERSPECTIVE
PSPICE
T[
cl
..
bj
b 2
-l-
e[
ih~
i'bc-I
+
t'h..:2
a) How many times per minute will the lamp
turn on?
+
C2
T2
l'e>.::':'
b) The 800 kD resistor is replaced with a variable
resistor R. The resistance is adjusted until the
lamp flashes 12 times per minute. What is the
value of R?
e2
7.100 The component values in the circuit of Fig. P7.99
are Vee = 9 V; R L = 3 kD; C 1 = C 2 = 2 nF; and
R[ = R2 = 18 kn.
a) How long is T 2 in the OFF state during one cycle
of operation?
7.105 In the flashing light circuit shown in Fig. 7.45, the
lamp can be modeled as a 1.3 ld 1 resistor when it is
conducting. The lamp triggers at 900 V and cuts off
/·\ULTlSlt.l at 300 V.
PRACTICAL
PERSPECTIVE
PSPICE
b) How long is T 2 in the ON state during one cycle
of operation?
a) If V, = 1000 V, R = 3.7 leD, and C = 250 p.F,
how many times per minute will the light flash?
c) Repeat (a) for T I .
b) What is the average current in milliamps deliv­
ered by the source?
d) Repeat (b) for T i .
e) At the first instant after T turns ON, what is the
value of i b1 ?
1
f) At the instant just before T I turns OFF, what is
the value of i b1 ?
g) What is the value of
before T 2 turns ON?
V ce 2
at the instant just
7.101 Repeat Problem 7.100 with C I = 3 nF and
C2 = 2.8 nP. All other component values are
unchanged.
7.102 TIle astable multivibrator circuit in Fig. P7.99 is to
satisfy the following criteria: (1) One transistor
switch is to be ON for 48/.LS and OFF for 36 f.LS for
each cycle; (2) R L = 2 kD; (3) Vee = 5 V;
(4) R] = R2 ; and (5) 6R L s R 1 S 50R L . What are
the limiting values for the capacitors C 1 and C2 ?
7.103 Suppose the circuit in Fig. 7.45 models a portable
flashing light circuit. Assume that four 1.5 V batter.
les power tI
Je"
CIrCUIt, an d th at t h e capacItor
vaI
ue'IS
10 f.LF. Assume that the lamp conducts when its
voltage reaches 4 V and stops conducting when its
voltage drops below 1 V. The lamp has a resistance
of 20 kD when it is conducting and has an infinite
resistance when it is not conducting.
c) Assume the flashing light is operated 24 hours
per day. If the cost of power is 5 cents per kilowatt­
hour, how much does it cost to operate the light
per year?
7.106 a) Show that the expression for the voltage drop
PRACT1CAL
across the capacitor while the lamp is conduct­
PERSPECTIVE
ing in the flashing light circuit in Fig. 7.48 is
given by
- VTh + (V max - V Til ) e-(1-111)/7
V L (t) -
where
VTI1 =
RL
V
R + R L S
PRACTICAL
PERSPECTIVE •
a) Suppose we don't want to wait more than 10 sin
between flashes. What value of resistance R is
required to meet this time constraint?
b) For the value of resistance from (a), how long
does the flash of light last?
b) Show that the expression for the time the lamp
conducts in the flashing light circuit in Fig. 7.48
is given by
Problems
7.107 The relay shown in Fig. P7.107 connects the 30 V dc
generator to the dc bus as long as the relay current
~
is greater than 0.4 A. If the relay current drops to
0.4 A or less, the spring-loaded relay immediately
connects the dc bus to the 30 V standby battery. The
resistance of the relay winding is 60 D. The induc­
tance of the relay winding is to be determined.
Figure P7.107
PRACTICAL
/t\5PErnVE
a) Assume the prime motor driving the 30 V dc
generator abruptly slows down, causing the gen­
erated voltage to drop suddenly to 21 V. What
value of L will assure that the standby battery
will be connected to the dc bus in 0.5 seconds?
b) Using the value of L determined in (a), state
how long it will take the relay to operate if the
generated voltage suddenly drops to zero.
.
+
....:::::....--30Y
30Y
de
gen
+
-
relay
coil
(R,L)
DC loads
263