P7.70, switch A has been open PSPICE and switch B has been closed for a long time.. P7.71, how many milliseconds after switch 1 moves to position b is the energy stored in the inductor 4
Trang 1Section 7.4
7.65 Repeat (a) and (b) in Example 7.10 if the mutual
inductance is reduced to zero
7.66 There is no energy stored in the circuit in Fig P7.66
PSPICE at the time the switch is closed
MULTISIM
a) Find /(/) for t > 0
b) Find v^t) for t > 0+
c) Find v 2 (t) for t > 0
d) Do your answers make sense in terms of known
circuit behavior?
Figure P7.66
80 V
Figure P7.69
250 O
10 V
Section 7.5 7.70 In the circuit in Fig P7.70, switch A has been open
PSPICE and switch B has been closed for a long time At
t = 0, switch A closes Five seconds after switch A closes, switch B opens Find i L {t) for t a 0
Figure P7.70
5
• A W
t = 5s
*—T—v\—r
i L (t)
7.67 Repeat Problem 7.66 if the dot on the 10 H coil is at
PSPICE the top of the coil
MULTISIM
7.68 There is no energy stored in the circuit of Fig P7.68
at the time the switch is closed
a) Find i 0 {t) for t > 0
b) Find v 0 (t) for t > 0+
c) Find /, (r) for/ a 0
d) Find i 2 {t) for t > 0
e) Do your answers make sense in terms of known
circuit behavior?
Figure P7.68
7.69 There is no energy stored in the circuit in Fig P7.69
PSPICE at the time the switch is closed
WUTSIM
a) Find i a (t) for t > 0
b) Find v 0 (t) for t > 0+
c) Find i^t) for t > 0
d) Find /2(f) for t > 0
e) Do your answers make sense in terms of known
circuit behavior?
7.71 The action of the two switches in the circuit seen in
PSPICE Fig P7.71 is as follows For t < 0, switch 1 is in
posi-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 vjt) 25 ms after switch 1 moves to
position b
Figure P7.71
0+ 10
ms-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
PSPICE MULTISIM
The switch in the circuit shown in Fig P7.73 has
been in position a for a long time At t = 0, the
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 a) /(0+)
b) /(200/AS)
c) /(6 ms)
d) -y(l"ms)
e) -y(l+ms)
Trang 2Figure P7.73
20 A ( f ) 4 0 a H 8 0 m H
7.74 T h e r e is n o energy stored in t h e capacitor in t h e
cir-PSPICE c ui t in Fig P7.74 when switch 1 closes at t = 0 Ten
microseconds later, switch 2 closes Find v a {t) for
t > 0
Figure P7.74
30 V
7.75 T h e capacitor in t h e circuit seen in Fig P7.75 has
PSPICE been charged to 300 V A t t = 0, switch 1 closes,
causing t h e capacitor to discharge into t h e resistive
network Switch 2 closes 2 0 0 / t s after switch 1
closes Find the m a g n i t u d e a n d direction of the
cur-r e n t in the second switch 300 /AS aftecur-r switch 1
closes
Figure P7.75
60 kfl
300 V
40 kfl
7.76 In t h e circuit in Fig P7.76, switch 1 has b e e n in
posi-tion a a n d switch 2 has b e e n closed for a long time
A t t = 0, switch 1 moves instantaneously to
posi-tion b Eight h u n d r e d microseconds later, switch 2
opens, remains o p e n for 300 tts, a n d then recloses
Find v a 1.5 ms after switch 1 m a k e s contact with
terminal b
Figure P7.76
a 1
7.5mA( M l O k a J v„
0 + 800 /.is
2 Ml ^ 2
-^vw—
r = 0
500 n F :
Ml
^
0 + 1.1 ms
3 kfl
7.77 For t h e circuit in Fig P7.76, what p e r c e n t a g e of t h e PSPICE initial energy stored in t h e 500 n F capacitor is
dissi-MumsiM pated in the 3 k f l resistor?
7.78 T h e switch in t h e circuit in Fig P7.78 has been in
PSPICE position a for a long time Alt = 0, it moves
instan-taneously to position b, w h e r e it r e m a i n s for five seconds before moving instantaneously t o position
c Find v a for t ^ 0
Figure P7.78
100 kfl
7.79 T h e voltage waveform shown in Fig P7.79(a) is PSPICE applied to t h e circuit of Fig P7.79(b) T h e initial
mTISIM current in t h e inductor is zero
a) Calculate v ( ,(t)
b) M a k e a sketch of v 0 (t) versus t
c) Find i () at t = 5 ms
Figure P7.79
%(V)
!40mH v,
2.5 t (ms)
7.80 T h e current source in t h e circuit in Fig P7.80(a) PSPICE g e n e r a t e s t h e current pulse shown in Fig P7.80(b) HULTISIH T h e r e j s n o e n e rg y stored at t = 0
a) Derive t h e numerical expressions for v (> (t) for
the time intervals / < 0, 0 < t < 75 /AS, a n d
75 /ts < t < oo
b) Calculate v a ( 7 5 " /AS) a n d v 0 ( 7 5 + /AS)
c) Calculate i a (75~ tis) a n d i 0 (75 + /AS)
Figure P7.80
is (mA)
25
if \ ) 2 kfl J 9,, j250mH
75 t(fjs)
(b)
Trang 37.81 The voltage waveform shown in Fig P7.81(a) is
PSPICE applied to the circuit of Fig P7.81 (b) The initial
voltage on the capacitor is zero
a) Calculate v 0 {t)
b) Make a sketch of v () (t) versus t
d) Sketch i a {t) versus t for the interval
- 1 ms < t < 4 ms
e) Sketch v a (t) versus t for the interval
- 1 ms < t < 4 ms
Figure P7.81
v s (V)
50
10 nF
i\ 400 kft
1 t (ms)
Figure P7.83
L (inA)
20
0.2 /xF
(a)
0 2 /(ms) (b)
7.82 The voltage signal source in the circuit in Fig P7.82(a)
PSPICE is generating the signal shown in Fig P7.82(b).There is
mnm no stored energy at f = 0
a) Derive the expressions for v 0 {t) that apply in the
intervals t < 0; 0 < t < 4 ms; 4 ms < t < 8 ms;
and 8 ms < t < oo
b) Sketch v a and v s on the same coordinate axes
c) Repeat (a) and (b) with R reduced to 50 kfi
Figure P7.82
R = 200 kO
AM,
25 nF:
(a)
»,00
100
0
tooh
t (ms)
(b)
7.83 The current source in the circuit in Fig P7.83(a)
PSPICE generates the current pulse shown in Fig P7.83(b)
mnsiM T h e r e i s Q O e n e r g y s t o r e d a t t = Q
a) Derive the expressions for i 0 (t) and v 0 (t) for the
time intervals / < 0 ; 0 < r < 2 ms; and
2 ms < t < oo
b) Calculate i o (0~); i o (0 + ); /o(0.002"); and
/;/0.002+)
c) Calculate v Q (0~); v o (0 + ); t?o(0.002~); and
^(0.002+)
Section 7.6 7.84 The capacitor in the circuit shown in Fig P7.84 is
PSPICE charged to 20 V at the time the switch is closed If the capacitor ruptures when its terminal voltage equals or exceeds 20 kV, how long does it take to rupture the capacitor?
Figure P7.84
7.85 The switch in the circuit in Fig P7.85 has been
PSPICE closed for a long time The maximum voltage rating
m n s , M of the 1.6 ^ F capacitor is 14.4 kV How long after the switch is opened does the voltage across the capacitor reach the maximum voltage rating?
Figure P7.85
PSPICE MULTISIM
7.86 The inductor current in the circuit in Fig P7.86 is
25 mA at the instant the switch is opened The 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?
Trang 4Figure P7.86
10 H
Figure P7.88
4kO
7.87 The gap in the circuit seen in Fig P7.87 will arc over
PSPICE whenever the voltage across the gap reaches 45 kV
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 kO
a) What is the value of /3?
b) How many microseconds after the switch has
been closed will the gap arc over?
Figure P7.87
5kft
^VW-i = 0
140V 20 kO i /3/,, ( f ) i 200 mH *Gap
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 V 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
The resistance of the relay coil is 25 kO, 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?
Push button
2/JLF
Section 7.7 7.89 The voltage pulse shown in Fig P7.89(a) is applied
PSPICE to the ideal integrating amplifier shown in Fig P7.89(b) Derive the numerical expressions for
v (> (t) when v o (0) = 0 for the time intervals a) t < 0
b) 0 < t < 250 ms
c) 250 ms < t < 500 ms
d) 500 ms < t < oo
Figure P7.89
v g (mV)
200
0 -200
250 500 t(ms)
(a)
400 nF
(b)
7.90 Repeat Problem 7.89 with a 5 Mft resistor placed
PSPICE across the 400 nF feedback capacitor
MULTIS1M
Trang 57.91 The energy stored in the capacitor in the circuit
PSPICE shown in Fig P7.91 is zero at the instant the switch
is closed The ideal operational amplifier reaches
saturation in 15 ms What is the numerical value of
R in kilo-ohms?
Figure P7.91
7.92 A t t h e instant t h e switch is closed in t h e circuit of
PSPICE Fig P7.91, the capacitor is charged t o 6 V, positive at
HULTISIM t h e right-hand terminal If the ideal operational
amplifier saturates in 40 ms, what is the value of /??
7.93 The voltage source in the circuit in Fig P7.93(a) is
PSPICE generating the triangular waveform shown in
MULTISIM F i g P 7 9 3(b) Assume the energy stored in the
capacitor is zero at t = 0 and the op amp is ideal
a) Derive the numerical expressions for v a {t) for
the following time intervals: 0 < t < 1 /xs;
1 /xs < / < 3 /xs; and 3 /xs ^ t ^ 4 /xs
b) Sketch the output waveform between 0 and 4 /xs
c) If the triangular input voltage continues to repeat
itself for t > 4 /xs, what would you expect the
output voltage to be? Explain
Figure P7.93
800 pF
7.94 There is no energy stored in the capacitors in the
PSPICE cir c ui t shown in Fig P7.94 at the instant the two
MULTISIM , , » i • • ,
switches close Assume the op amp is ideal
a) Find v () as a function of v & , v b , R, and C
b) On the basis of the result obtained in (a), describe the operation of the circuit
c) How long will it take to saturate the amplifier
if v a = 40 mV; v h = 15mV; R = 50 kO;
C = 10 nF; and V cc = 6 V?
Figure P7.94
7.95 At the time the double-pole switch in the circuit
PSPICE shown in Fig P7.95 is closed, the initial voltages on
MULTISIM - r t i r AVT I T - I t
the capacitors are 12 V and 4 V, as shown Find the
numerical expressions for v t> (t), v 2 (t), and vAt) that
are applicable as long as the ideal op amp operates
in its linear range
Figure P7.95
7.96 At the instant the switch of Fig P7.96 is closed, the PSPKE voltage on the capacitor is 56 V Assume an ideal operational amplifier How many milliseconds
after the switch is closed will the output voltage v„
equal zero?
(b)
Trang 6Figure P7.96
33 kii > 47 kn
-^Wv *
- 56V +
— 1 ( —
/ = 0
© 14 V
Sections 7.1-7.7
7.97
PSPICE
MULTISIM
The circuit shown in Fig P7.97 is known as a
monostable multivibrator.The adjective monostable
is used to describe the fact that the circuit has one
stable state That is, if left alone, the electronic
switch T2 will be ON, and Tj will be OFF (The
opera-tion of the ideal transistor switch is described in
detail in Problem 7.99.) T2 can be turned OFF by
momentarily closing the switch S After S returns to
its open position, T2 will return to its ON state
a) Show that if T2 is ON, T { is OFF and will stay OFF
b) Explain why T2 is turned OFF when S is
momen-tarily closed
c) Show that T2 will stay OFF for RC In 2 s
Figure P7.97
7.98 The parameter values in the circuit in Fig P7.97
are V cc = 6 V; R x = 5.0 kft; R L = 20 kH;
C = 250 pF; and R = 23,083 H
a) Sketch v ce2 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 = 0 Make your sketch for the
inter-val - 5 < t < lOjus
b) Repeat (a) for /b2 versus t
7.99 PSPICE MULTISIM
The circuit shown in Fig P7.99 is known as an
astable multivibrator and finds wide application in
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 Ti and T2 The circuit is designed so that the switches automatically
alter-nate between ON and OFF When T { is OFF, T2 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 v bc is zero, and (3) the
ter-minal voltage v ce 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 v he 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 T2 has been
ON and has just snapped OFF, while Tj has been OFF and has just snapped ON You may assume that at this instance, C2 is charged to the supply voltage Vcc, a nd t n e charge on C\ is zero Also assume
C x = C2 and R x = R 2 = 10R L a) Derive the expression for v bc2 during the inter-val that T2 is OFF
b) Derive the expression for v cc2 during the inter-val that T2 is OFF
c) Find the length of time T2 is OFF
d) Find the value of v ce2 at the end of the interval that T2 is OFF
e) Derive the expression for /bl during the interval that T2 is OFF
f) Find the value of i bx at the end of the interval that T2 is OFF
g) Sketch v cc2 versus t during the interval that T2
is OFF
h) Sketch /M versus t during the interval that T2
is OFF
Trang 7Figure P7.99
PSPICE
MULTISIM
7.100 The component values in the circuit of Fig P7.99
are V cc = 9 V; R L = 3 kH; C, = C2 = 2 nF; and
i?i = i?2 = 18kfl
a) How long is T2 in the OFF state during one cycle
of operation?
b) How long is T2 in the ON state during one cycle
of operation?
c) Repeat (a) for Tj
d) Repeat (b) for Tj
e) At the first instant after T] turns ON, what is the
value of//,1 ?
f) At the instant just before Ti turns OFF, what is
the value of//,]?
g) What is the value of v ce2 a t the instant just
before T2 turns ON?
7.101 Repeat Problem 7.100 with C { = 3 nF and
C2 = 2.8 nF All other component values are
unchanged
7.102 The astable multivibrator circuit in Fig P7.99 is to
satisfy the following criteria: (1) One transistor
switch is to be ON for 48 /AS and OFF for 36 (xs for
each cycle; (2) R L = 2 kH; (3) V cc = 5 V;
(4) R\ = R 2 \ and (5) 6R L < R^ ^ 50R L What are
the limiting values for the capacitors C\ and C2?
7.103 Suppose the circuit in Fig 7.45 models a portable
PRACTICAL flashing light circuit Assume that four 1.5 V
batter-ies power the circuit, and that the capacitor value is
10 /JLF 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 kO when it is conducting and has an infinite
resistance when it is not conducting
a) Suppose we don't want to wait more than 10 s in
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?
PSPICE
MULTISIM
7.104 In the circuit of Fig 7.45, the lamp starts to conduct
PRACTICAL whenever the lamp voltage reaches 15 V During
PERSPECTIVE r O &
the time when the lamp conducts, it can be modeled
as a 10 kfl 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 s = 40 V; R - 800 kO; and
C = 25 fiF
a) How many times per minute will the lamp turn on?
b) The 800 kfl 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 /??
7.105 In the flashing light circuit shown in Fig 7.45, the
PRACTICAL lamp can be modeled as a 1.3 kO resistor when it is
PERSPECTIVE r
PSPICE conducting The lamp triggers at 900 V and cuts off MULTISIM a t 3 0 0 V
a) If V s = 1000 V, R = 3.7 k O , and C = 250 fiF,
how many times per minute will the light flash?
b) What is the average current in milliamps deliv-ered by the source?
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
across the capacitor while the lamp is conduct-ing in the flashconduct-ing light circuit in Fig 7.48 is given by
v L (0 = Vm + (Vmax - VTh)t'-<'-"^
PRACTICAL
PERSPECTIVE
where
Vi R>
R + RL
RR L C
7 R + R L '
b) Show that the expression for the time the lamp conducts in the flashing light circuit in Fig 7.48
is given by
(t c ~ Q RR L c , V U - vTh
R + R, In v„ K, ih
Trang 8PRACTICAL 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 ft The
induc-tance of the relay winding is to be determined
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
30 V • r v ,
DC
Trang 9\, _Y
Natural and Step
Responses of RLC Circuits
C H A P T E R C O N T E
8.1 Introduction to the Natural Response of a
Parallel RLC Circuit p 266
8.2 The Forms of the Natural Response of a
Parallel RLC Circuit p 270
8.3 The Step Response of a Parallel
RLC Circuit p 280
8.4 The Natural and Step Response of a Series
RLC Circuit p 285
8.5 A Circuit with Two Integrating
Amplifiers p 289
1 Be able to determine the natural response and
the step response of parallel RLC circuits
2 Be able to determine the natural response and
the step response of series RLC circuits
In this chapter, discussion of the natural response and step
response of circuits containing both inductors and capacitors is
limited to two simple structures: the parallel RLC circuit and the series RLC circuit Finding the natural response of a parallel RLC
circuit consists of finding the voltage created across the parallel branches by the release of energy stored in the inductor or capac-itor or both The task is defined in terms of the circuit shown in
Fig 8.1 on page 266 The initial voltage on the capacitor, V (h repre-sents the initial energy stored in the capacitor The initial current
through the inductor, I {h represents the initial energy stored in the inductor If the individual branch currents are of interest, you can find them after determining the terminal voltage
We derive the step response of a parallel RLC circuit by using
Fig 8.2 on page 266 We are interested in the voltage that appears across the parallel branches as a result of the sudden application
of a dc current source Energy may or may not be stored in the circuit when the current source is applied
Finding the natural response of a series RLC circuit consists
of finding the current generated in the seriesconnected elements
by the release of initially stored energy in the inductor, capacitor,
or both The task is defined by the circuit shown in Fig 8.3 on
page 266 As before, the initial inductor current, I {h and the initial
capacitor voltage, V {h represent the initially stored energy If any
of the individual element voltages are of interest, you can find them after determining the current
We describe the step response of a series RLC circuit in terms
of the circuit shown in Fig 8.4 on page 266 We are interested in the current resulting from the sudden application of the dc volt-age source Energy may or may not be stored in the circuit when the switch is closed
If you have not studied ordinary differential equations, deri-vation of the natural and step responses of parallel and series
RLC circuits may be a bit difficult to follow However, the results
are important enough to warrant presentation at this time We
begin with the natural response of a parallel RLC circuit and
cover this material over two sections: one to discuss the solution
of the differential equation that describes the circuit and one to present the three distinct forms that the solution can take After
264
Trang 10Practical Perspective
An Ignition Circuit
In this chapter we introduce the step response of an RLC
cir-cuit An automobile ignition circuit is based on the transient
response of an RLC circuit In such a circuit, a switching
oper-ation causes a rapid change in the current in an inductive
winding known as an ignition coil The ignition coil consists
of two magnetically coupled coils connected in series This
series connection is also known as an autotransformer The
coil connected to the battery is referred to as the primary
winding and the coil connected to the spark plug is referred
to as the secondary winding The rapidly changing current in
the primary winding induces via magnetic coupling (mutual
inductance) a very high voltage in the secondary winding
This voltage, which peaks at from 20 to 40 kV, is used to
ignite a spark across the gap of the spark plug The spark
ignites the fuel-air mixture in the cylinder
Ignition coil (autotransformer;'
Secondary
| # Primary
Battery i
Switch^ | •
(distributor point) * \ ^ ^ J
Spark plug
Capacitor (condenser)
A schematic diagram showing the basic components of an ignition system is shown in the accompanying figure In today's automobile, electronic (as opposed to mechanical) switching is used to cause the rapid change in the primary winding current An understanding of the electronic switching circuit requires a knowledge of electronic components that is beyond the scope of this text However, an analysis of the older, conventional ignition circuit will serve as an introduc-tion to the types of problems encountered in the design of a useful circuit
265