Electric Circuits, 9th Edition P54 docx

• If a circuit is driven by a unit impulse, xt = 8t, then the response of the circuit equals the inverse Laplace transform of the transfer function, yt = !£~ l {Hs} = ht.. b Give the n

Summary

• We can represent each of the circuit elements as an

6-domain equivalent circuit by Laplace-transforming

the voltage-current equation for each element:

• Resistor: V = RI

• Inductor: V - sLl - LI ()

• Capacitor: V = (l/sC)I + VJs

In these equations, V = Z£{v}, I = ?£{i}, /0 is the

ini-tial current through the inductor, and V0 is the initial

voltage across the capacitor (See pages 468-469.)

• We can perform circuit analysis in the s domain by

replacing each circuit element with its s-domain

equiva-lent circuit The resulting equivaequiva-lent circuit is solved by

writing algebraic equations using the circuit analysis

techniques from resistive circuits Table 13.1

summa-rizes the equivalent circuits for resistors, inductors, and

capacitors in the s domain (See page 470.)

• Circuit analysis in the s domain is particularly

advanta-geous for solving transient response problems in linear

lumped parameter circuits when initial conditions are

known It is also useful for problems involving multiple

simultaneous mesh-current or node-voltage equations,

because it reduces problems to algebraic rather than

differential equations (See pages 476-478.)

• The transfer function is the s-domain ratio of a circuit's

output to its input It is represented as

where Y(s) is the Laplace transform of the output

nal, and X(s) is the Laplace transform of the input

sig-nal (See page 484.)

• The partial fraction expansion of the product H(s)X(s)

yields a term for each pole of H(s) and X(s) The

H(s) terms correspond to the transient component of

the total response; the X(s) terms correspond to the

steady-state component (See page 486.)

• If a circuit is driven by a unit impulse, x(t) = 8(t), then

the response of the circuit equals the inverse Laplace

transform of the transfer function, y(t) = !£~ l {H(s)}

= h(t) (See pages 488-489.)

• A time-invariant circuit is one for which, if the input is

delayed by a seconds, the response function is also delayed by a seconds (See page 488.)

• The output of a circuit, y(t), can be computed by con-volving the input, x(t), with the impulse response of the circuit, h(t):

y{t) = h{t) * x{t) = / h{k)x{t - \)dk

Jo

= x{t) * h{t) = j x(\)h(t - A)d\

JO

A graphical interpretation of the convolution integral often provides an easier computational method to

gen-erate y(t) (See page 489.)

• We can use the transfer function of a circuit to compute its steady-state response to a sinusoidal source To do so,

make the substitution s = j<o in H(s) and represent the

resulting complex number as a magnitude and phase angle If

x(t) = A cos((ot + ¢),

Hijco) = \H(jco)\e m "K

then

yjjt) = A\H(ja>)\ cos[e*t + ¢ + $(<*>)),

(See page 496.)

• Laplace transform analysis correctly predicts impulsive currents and voltages arising from switching and impul-sive sources You must ensure that the ^-domain

equiva-lent circuits are based on initial conditions at t = (T,

that is, prior to the switching (See page 498.)

Problems

Section 13.1

13.1 Derive the s-domain equivalent circuit shown in

Fig 13.4 by expressing the inductor current i as a

function of the terminal voltage v and then

find-ing the Laplace transform of this time-domain

integral equation

13.2 Find the Thevenin equivalent of the circuit shown

in Fig 13.7

13.3 Find the Norton equivalent of the circuit shown

in Fig 13.3

Section 13.2

13.4 A 2 k i l resistor, a 312.5 mH inductor, and a 12.5 nF

capacitor are in parallel

a) Express the s-domain impedance of this parallel

combination as a rational function

b) Give the numerical values of the poles and zeros

of the impedance

13.5 A 1 kd resistor is in series with a 625 nF

capaci-tor This series combination is in parallel with a

100 mH inductor

a) Express the equivalent s-domain impedance of

these parallel branches as a rational function

b) Determine the numerical values of the poles

and zeros

13.6 A 8 kO resistor, a 1 H inductor, and a 40 nF

capaci-tor are in series

a) Express the s-domain impedance of this series

combination as a rational function

b) Give the numerical value of the poles and zeros

of the impedance

13.7 Find the poles and zeros of the impedance seen

looking into the terminals a,b of the circuit shown

in Fig PI3.7

Figure PI3.7

13.8 Find the poles and zeros of the impedance seen

looking into the terminals a,b of the circuit shown

in Fig P13.8

Figure P13.8

h

1 F ^

<

I F ; :

iia

Section 13.3 13.9 The switch in the circuit shown in Fig PI 3.9 has

* switch moves instantaneously to position y

a) Construct an v-domain circuit for t > 0

b) Find V ot

c) Find v 0

Figure P13.9

13.10 The switch in the circuit in Fig P13.10 has been in

position a for a long time At t - 0, it moves

instan-taneously from a to b

a) Construct the s-domain circuit for t > 0

b) Find V a (s)

c) Find v„(t) for t > 0

Figure P13.10

12511

10 mH

13.11 The switch in the circuit in Fig. P13.ll has been

MULTISIM

a) Construct the y-domain equivalent circuit for

t > 0

b) Find V ,

c) Find v() for t > 0

Figure P13.ll

10 mF

8(2

2 H

PSPKE

MULTISIM

13.12 T h e switch in t h e circuit in Fig P13.12 h a s b e e n in

position a for a long time A t if = 0, t h e switch

moves instantaneously t o position b

a) Construct the ^-domain circuit for t > 0

c) Find / L

d) Find v() for t > 0

e) Find iL for t > 0

Figure P13.12

6.25 JX¥

/ = 0

20 ft

son

+

6.4 rnH v„

13.13 T h e switch in t h e circuit in Fig P13.13 h a s b e e n

PSPICE closed for a long time A t t = 0, t h e switch

MULTISIM • j

is o p e n e d

a) Find va for t ^ 0

b ) Find ia for t > 0

Figure P13.13

t = 0

PYn

125 nF

4kfl

A W

H-13.14 T h e make-before-break switch in the circuit in

PSPICE Fig PI3.14 has b e e n in position a for a long time A t MULTisiM ? = o, it moves instantaneously t o position b Find i0 for t > 0

Figure P13.14

13.15 Find V 0 a n d va in t h e circuit s h o w n in Fig P13.15 if

PSPICE t h e initial energy is z e r o a n d t h e switch is closed at

MULTISIM f = 0

Figure P13.15

2.8 kll 200 mH

t = 0

13.16 R e p e a t P r o b l e m 13.15 if t h e initial voltage o n t h e

PSPICE capacitor is 30 V positive at t h e u p p e r terminal

MULTISIM

13.17 T h e switch in the circuit seen in Fig PI3.17 has been

instanta-MULTISIM neously to position b at t = 0

a) Construct the s-domain equivalent circuit for

t > 0

b) Find V\ a n d v^

c) Find V2 a n d v2

Figure P13.17

450 V

1.25 mH

13.18 T h e switch in the circuit seen in Fig P13.18 has b e e n

MULTISIM i n s t a n t a n e o u s l y to p o s i t i o n b

a) F i n d V f r

b) F i n d e r

Figure P13.18

20 a

13.19 The switch in the circuit in Fig P13.19 has been

MULTISIM c + - ^ n

v n for t > 0

Figure P13.19

13.20 Find v () in the circuit shown in Fig PI3.20 if

PSP1CE L = 5u(t) mA There is no energy stored in the

cuit at t = 0

Figure P13.20

13.21 There is no energy stored in the circuit in Fig PI3.21

PSPICE at the time the switch is closed

MULTISIM

a) Find v 0 for J s O

b) Does your solution make sense in terms of

known circuit behavior? Explain

Figure P13.21

2H t = 0 + "A

+ 1H

_ 4mF

13.22 There is no energy stored in the circuit in Fig PI3.22

PSPICE

MULTISIM at t = 0~

a) Use the mesh current method to find i a

b) Find the time domain expression for v 0

c) Do your answers in (a) and (b) make sense in

terms of known circuit behavior? Explain

Figure P13.22

10//(0 V

i n

13.23 a) Find the s-domain expression for V 0 in the circuit _™_ in Fig PI3.23

MULTISIM

b) Use the ^-domain expression derived in (a) to

predict the initial- and final-values of v a c) Find the time-domain expression for v ()

Figure P13.23

|15w(0mA j l H

7n

^vw-+

13.24 Find the time-domain expression for the current in

PSPICE the inductor in Fig P13.23 Assume the reference

MULTISIM ¢ ^ ¢ ^ 0 ] } f or / £ j s d o w n

13.25 There is no energy stored in the capacitors in the

PSPICE circuit in Fig P13.25 at the time the switch is closed

a) Construct the s-domain circuit tor t > 0

b) Find I h V h and V 2 c) Find z'i, Vi, and i>2

d) Do your answers for i h V\, and v 2 make sense in terms of known circuit behavior? Explain

Figure PI3.25

50 kn

300 nF

20 V

100 nF

13.26 There is no energy stored in the circuit in Fig PI3.26

MULTISIM ^ = 7 5 u { ( ) y

a) Find V 0 and I 0 b) Find v 0 and i a c) Do the solutions for v a and i a make sense in terms of known circuit behavior? Explain

Figure P13.26

4 m F

if

:20 II

13.27 T h e r e is no energy stored in the circuit in Fig PI 3.27

MULTISIM

a) Find / a and Ib

b) Find / a and / b

c) Find V&i V bi and Vc

d) Find t* a, vh , a n d v c

e) A s s u m e a capacitor will break down w h e n e v e r

its terminal voltage is 1000 V H o w long after the

current source turns on will o n e of the capacitors

b r e a k d o w n ?

Figure P13.27

lOOmF

+ v«

-ion

9«(0A(t

100 mF

Z 100 mF ion

PSPICE

MULTISIM

13.28 T h e r e is no energy stored in the circuit in Fig PI3.28

at t = 0"

c) D o e s your solution for v0 m a k e sense in terms of

known circuit behavior? Explain

Figure P13.28

30 n

w v

13.29 T h e r e is no energy stored in the circuit in Fig PI 3.29

PSPICE at the time the sources are energized

MULTISIM

a) Find I^s) a n d Ijis)

b) Use the initial- and final-value theorems to check

the initial- and final-values of i\{t) and /'2(f)

c) Find i{ (t) a n d i (t) for t > 0

Figure P13.29

2.5 H

6«(/)A( f

200 mF

13,30 T h e r e is no energy stored in the circuit in Fig P13.30

PSPICE at t h e time the current source turns on Given that

MULTISIM ig = 5 Q H W A

a) Find V„(s)

b) U s e the initial- a n d final-value t h e o r e m s to find

v o (0 + ) and yf) (oo)

c) D e t e r m i n e if the results obtained in (b) agree with known circuit behavior

Figure P13.30

13.31 T h e initial energy in the circuit in Fig P13.31 is zero

PSPICE T h e ideal voltage source is 120«(7) V

MULTISIM

b) U s e the initial- and final-value t h e o r e m s to find

i a (Q + ) and f0 (oo)

c) D o the values o b t a i n e d in (b) agree with k n o w n circuit behavior? Explain

d) Find / „ ( 0 Figure P13.31

20

<e>

_4 CYV-V>_

+

13.32 There is n o energy stored in the circuit in Fig P13.32

PSPICE at the time the voltage source is energized

MULTISIM

a) Find V() , I () , and I L

Figure P13.32 Figure P13.35

-25f

13.33 Beginning with Eq 13,65, show that the capacitor

current in the circuit in Fig 13.19 is positive for

0 < t < 200 (is and negative for t > 200 {is Also

show that at 200 (is, the current is zero and that this

corresponds to when dv c /dt is zero

PSPICE

MULTfSIM

13.34 The two switches in the circuit shown in Fig P13.34

operate simultaneously There is no energy stored

in the circuit at the instant the switches close Find

/(f) for t & 0+ by first finding the s-domain

Thevenin equivalent of the circuit to the left of the

terminals a,b

Figure P13.34

13.35 The switch in the circuit shown in Fig P13.35

has been open for a long time The voltage of

the sinusoidal source is v g = V m sin {cot + cj>)

The switch closes at / = 0 Note that the angle

cf) in the voltage expression determines the value

of the voltage at the moment when the switch

closes, that is, v g (0) = V m sin

4>-a) Use the Laplace transform method to find

/" for t > 0

b) Using the expression derived in (a), write the

expression for the current after the switch has

been closed for a long time

c) Using the expression derived in (a), write the

expression for the transient component of /'

d) Find the steady-state expression for i using the

phasor method Verify that your expression is

equivalent to that obtained in (b)

e) Specify the value of c/> so that the circuit passes

directly into steady-state operation when the

switch is closed

13.36 The magnetically coupled coils in the circuit seen

PSPICE m pjg_ pi 3.36 carry initial currents of 15 and 10 A,

MULTISIM ,

as shown

a) Find the initial energy stored in the circuit

b) Find I { and /2

c) Find i] and i 2

d) Find the total energy dissipated in the 120 and

270 H resistors

e) Repeat (a)-(d), with the dot on the 18 H induc-tor at the lower terminal

Figure P13.36

6 H

120ft:

/

8 H

T

15 A r

18 H ' hj

i

t

10 A

:270 0

13.37 The switch in the circuit seen in Fig PI3.37 has

1 Use the Laplace transform method of analysis to

find v„

Figure P13.37

X = 0

13.38 The make-before-break switch in the circuit seen in

PSPICE pig P13.38 has been in position a for a long time At

t = 0, it moves instantaneously to position b Find

L for t > 0

Figure P13.38 Figure P13.42

90 V

10 a

13.39 There is no energy stored in the circuit in Fig PI3.39

PSPICE at the time the switch is closed

MULTISIM

a) Find / ,

b) Use the initial- and final-value theorems to find

/t(0+) and/j(oo)

c) Find /,

Figure P13.39

13.40 a) Find the current in the 40 XI resistor in the

cir-pspicE cuit in Fig PI3.39 The reference direction for

the current is down through the resistor

b) Repeat part (a) if the dot on the 1.25 H coil

is reversed

13.41 In the circuit in Fig P13.41, switch 1 closes at t = 0,

instanta-MULTISIM n e o u si y from position a to position b

a) Construct the A-domain equivalent circuit for

t > 0

b) F i n d / ,

c) Use the initial- and final-value theorems to

check the initial and final values of /,

d) Find /, for t > 0+

Figure P13.41

120 a

1 0 a

20 V

13.42 There is no energy stored in the circuit seen in

a) Use the principle ot superposition to find V 0

b) Find v for t > 0

10 a

A A A r

-6 0 K ( / ) V V 0

10H

12.5 mF f \ J1.5w(r)A | 2 0 a

13.43 Verify that the solution of Eqs 13.91 and 13.92 for V 2

yields the same expression as that given by Eq 13.90

13.44 The op amp in the circuit shown in Fig P13.44 is

fR™ ideal There is no energy stored in the circuit at the

MULTISIM time it is energized If v g = 16,000ta(/) V, find (a) V ( „ (b) v 0 , (c) how long it takes to saturate the

operational amplifier, and (d) how small the rate of

increase in v g must be to prevent saturation

Figure P13.44

12.5 nF

13.45 The op amp in the circuit seen in Fig P13.45 is ideal

MULTISIM ^ m& ^ g c r r c mt js energized Determine (a) V () , (b)

v m and (c) how long it takes to saturate the opera-tional amplifier

Figure P13.45

200 k a 200 k a

• — w v +

250 nF

-1(-250 nF

:100 Ml

13.46

PSPICE

MULTISIM

0.5K(J) V ^ - 5 0 0 n F

Find v () (t) in the circuit shown in Fig P13.46 if the

ideal op amp operates within its linear range and

v s = \6u(t) mV

Figure P13.46

13.47 The op amp in the circuit shown in Fig PI3.47 is

MUITISIM jj1 £ jn s t a n t the cir c ui t is energized

a) Find v a if v gi = 40i/(f) V and V H2 = 16//(/) V

b) How many milliseconds after the two voltage

sources are turned on does the op amp saturate?

Sections 13.4-13.5 13.49 a) Find the numerical expression for the

trans-fer function H(s) = V„/Vi for the circuit in

Fig PI3.49

b) Give the numerical value of each pole and zero

of H{s)

Figure P13.49

16 kO

100 kO 13.50 Find the numerical expression for the transfer

func-tion (VJV,) of each circuit in Fig P13.50 and give

the numerical value of the poles and zeros of each transfer function

Figure P13.47

w«fion

Figure P13.50

100 kO

40 nF

40 n F

r-K->\ 100 kO

(a) 2kH

v, 250 mPH v <> v < 2kfi

13.48 The op amps in the circuit shown in Fig P13.48 are

MULTISIM t = ( )- T f ^ = 1 6 K ^ m V i h o w m a n y m i l l i s e c o r i d s

elapse before an op amp saturates?

Figure P13.48

25 kf!

»*

(c)

40 kO

(d)

•—

+

•—

10kO<

(

i 1

t 250 n F ^

» (

•

*

o

+

o

(e)

13,51 a) Find the transfer function H(s) = VJVj for the

circuit shown in Fig PI3.51 (a)

b) Find the transfer function H(s) = V 0 /V t for the circuit shown in Fig PI 3.51(b)

c) Create two different circuits that have the transfer

function H(s) = V () /Vi = 1000/(5+1000) Use

components selected from Appendix H and Figs.P13.51(a)and(b)

Figure PI3.51

+

• —

R

<>—

!-—•

(a)

+

(b)

13.54 The operational amplifier in the circuit in Fig PI3.54

is ideal

a) Find the numerical expression for the transfer

function H(s) = VJV S

b) Give the numerical value of each zero and pole

of H(s)

13.52 a) Find the transfer function H(s) = VJV, for the

circuit shown in Fig PI3.52(a)

b) Find the transfer function H(s) = V 0 /V t for the

circuit shown in Fig PI3.52(b)

c) Create two different circuits that have the

trans-fer function H(s) = VJV-, = s/(s + 10,000)

Use components selected from Appendix H and

Figs P13.52(a) and (b)

Figure P13.52

• — 1 (

-Figure P13.54

13.53

+ +

a) Find the transfer function H(s) = V () /V, for the

circuit shown in Fig P13.53 Identify the poles

and zeros for this transfer function

b) Find three components from Appendix H which

when used in the circuit of Fig P13.53 will result in

a transfer function with two poles that are distinct

real numbers Calculate the values of the poles

c) Find three components from Appendix H which

when used in the circuit of Fig PI3.53 will result

in a transfer function with two poles, both with

the same value Calculate the value of the poles

d) Find three components from Appendix H which

when used in the circuit of Fig P13.53 will result

in a transfer function with two poles that are

complex conjugate complex numbers Calculate

the values of the poles

C? = 25 nF

+

l k Q 200nF

^vw } | —

ft c

13.55 The operational amplifier in the circuit in Fig PI3.55

is ideal

a) Find the numerical expression for the transfer

function//(5) = VJV r

b) Give the numerical value of each zero and pole

of H(s)

Figure P13.55

400 pF

Figure P13.53

R

13.56 The operational amplifier in the circuit in

Fig PI3.56 is ideal

a) Derive the numerical expression of the

trans-fer function H(s) = VJV g for the circuit in Fig P13.56

b) Give the numerical value of each pole and zero

of H(s)

Figure P13.56 13.59 a) Find the transfer function I( ,/I s as a function of

MULnSIM

b) Find the largest value of i± that will produce a

b o u n d e d output signal for a b o u n d e d input signal

c) Find it) for /x = - 3 , 0 , 4 , 5 , and 6 if L = 5u(t) A

Figure P13.59

8 k O

O

13.57 T h e r e is no energy stored in the circuit in Fig P13.57

PSPICE a t the time the switch is opened.The sinusoidal current

MULTISIM s o u r c e i s g e n e r a t j n g the signal 100 cos 10,000/ m A

The response signal is the current iir

c) Describe the n a t u r e of the transient c o m p o n e n t

of 4 ( 0 without solving for in (t)

d) Describe the n a t u r e of the steadystate c o m p o

-n e -n t of i0 (t) without solving for i 0 {t)

e) Verify the observations m a d e in (c) and (d) by

Figure P13.57

Section 13.6

13.60 a) Find h{t) * x{t) w h e n h(t) a n d x(t) are the

rec-tangular pulses shown in Fig P13.60(a)

b) R e p e a t (a) w h e n x(t) changes to the rectangular

pulse shown in Fig P13.60(b)

c) R e p e a t (a) when h(t) changes to t h e rectangular

pulse shown in Fig P13.60(c)

Figure P13.60

HO

25

100 nF

x{t)

25

10

(a)

13.58 In the circuit of Fig P13.58 i (> is the o u t p u t signal

and vg is the input signal Find the poles and zeros

of the transfer function, assuming t h e r e is n o initial

energy stored in the linear transformer or in the

capacitor

Figure P13.58

x(t)

12.5

0

/7(0

25

10

5 H

v P \

o

25 H

10 H

10 kO

62.5 nF

13.61 a) Given y{t) = h(t) * x(t), find y(t) when h(t) and

x(t) are the rectangular pulses shown in

Fig PI3.61 (a)

b) R e p e a t (a) w h e n h{t) changes to the rectangular

pulse shown in Fig PI3.61(b)

c) R e p e a t (a) when h(t) changes t o the rectangular

pulse shown in Fig PI 3.61(c)

d) Sketch y(t) versus t for ( a ) - ( c ) on a single graph

e) D o the sketches in (d) m a k e sense? Explain