Valkenburg Problems PDF

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ok Equations / Ch. 3

uic method for form. The so(u-solution is to be rmulation offers he only require-ement or several I .the state-space on by computer is to be accorn-inarily simpler to I Book Company, cusses a graphical ms. in Linear Circuits

,

Theory, McGraw-1 and McGraw-12. Viley & Sons, Inc.,

ring Circuit Analy-'ork,1971.

Theory: An Intro-nnpany, Reading,

and Bacon, Inc.,

vetworks for Elect-Vinston, Inc., New

the State Variable ark, 1970. Analysis, Prentice-is a programmed of state equations.

~s

e digital computer . .described in nethod from

refer-87 IProblems

in Appendix E-4.1. Consider also the analysis of resistive ladder rks as described in references in Appendix E-4.2. For specific

sugges-see Huelsman, reference 7 of Appendix E-I0, for the resistive network ated to the solution of simultaneous equations in Chapter 7 and the ion of equations for the RLC networkS of Chapter 6. More advanced bilities include the solution of state equations by methods described ferences given in Appendix E-4.3 and the use of canned programs for ork analysis as given in Appendix E-8.4.

PROBLEMS

What must be the relationship between

C.

and Cl and C2 in (a) of the figure of the networks if(a) and (c)are equivalent? Repeat for the network shown in(b). 0-

1 ..

"

(c) (b) (a) Fig. P3-1.

What must be the relationship between Le. and Lt. L2and M for the networkS of (a) and of(b) to be equivalent to that of (c)?

]

M

•

(c) (b) (a) Fig. P3-2.

Repeat Prob. 3-2 for the three networks shown in the accompanying figure.

(b)

(e)

la)

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88 Network Equations/ Ch. 3 Ch.3/ Problems

3-4. The network of inductors shown in the figure is composed of a J-H inductor on each edge of a cube with the inductors connected to the vertices of the cube as shown. Show that, with respect to vertices a and b, the network is equivalent to that in (b) of the figure when

Leq

=

i

H. Make use of symmetry in working this problem, rather than writing Kirchhoff laws.

The series ( tain to the netwo specified in the ta connection of ele connection of elet to zero. For thest mine 'VI in the for on a cathode ray0 and so on. (a) Fig. P3-4.

10--1

?L,q

1'~ 2 (a) V2 2 2 -3 (c) V2 volts +1 -1 (e)

"

,

[

(b)

3-5. In the rietworks of Prob. 3-4, each I-H inductor is replaced by a J-H capacitor, and L,q is replaced by C,q' What must be the value of Ceq for the two networks to be equivalent?

3-6. This problem may be solved using the two Kirchhoff laws and voltage-current relationships for the elements. At time to after the switch K was closed, it is found that t'2

=

+5 V. You are required to deter-mine the value of i2(lo) and di2(tO)/dl.

'K 111 + 10v -=-Fig. P3-6. 211 + 111 ~h

3-7. This problem is similar to Prob. 3·6. In the network given in the figure, it is given that 1'2(10) ~ 2 V, and (dl:2/dt)(to) = -10 V/sec, where la is the time after the switch K wasclosed. Determine the value of C.

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rations / Ch. 3 ;ed of a I-H ected to the :0vertices a figure when ilern,rather Ch. 3/Problems 89

The series of problems described in the following table all per-tain to the network of (g) of the figure with the network In A and B specified in the table. In A, two entries in the column implies a series connection of elements, while in B, two entries implies a parallel connection of elements. In each case, all initial conditions are equal to zero. For the specified waveform for V2, you are required to deter-mine VI in the form of a sketch of the waveform as it might be seen on a cathode ray oscilloscope. Evaluate significant amplitudes, slopes, and so on.

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C (b) 3-19. Demonstrate the so establish a inductor intoan C2 3-20. Demonstrate tha 3-21. Write a setof appropriate 1 R3 3-17. (d) 90 Network Equatiolls/

c»

.

J Network ofA Network of B Waveforms ofV2 3-8. R=2 L =:l- a, b,c, d, e,f

•

3-9. C=! L= 1 a,b,e,d,e.f 3-10. C=

f

,

R= 1 L=2 (I,b,c,d,e.f 3-11. C=J,R=t L=~, R= J (I,b, c, d,e,f 3-12. R=2 C= 1 b,d,f 3-13. R= 1 R=2,C= 1 b,d,f 3-14. R= 2 R= I,C= 1

s.a.]

3-15. L=1: R=l,C=! b,d,f 3-16. L= 1,R= 1 R=l,C=! b,d,f

3-17. For each of the four networks shown in the figure, determine the number ofindependent loop currents, and the number of independent node-to-node voltages that may be used in writing equilibrium equa -tions using the Kirchhoff laws. R2 2

f

i

0

3 RI L C v(t) +

~

v(t} R3 4 (a) 2 v(t} +

3

_Rr_;J C v(t} 3 (c) Fig. P3-17. 2

3-18. Repeat Prob. 3-17 for each of the four networks shown in the figure on page 91. Ch.

3/

Problems v(t) (b) 3-23. Write aset0 network in one controll equations

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Ch.3/Problems v{t) (b) 91 (a) v{t) (cl Fig. P3-18.

3-19. Demonstrate the equivalence of the networks shown in Fig. 3-17 and so establish a rule for converting a voltage source in series with an inductor into an equivalent network containing a current source. 3-20. Demonstrate that the two networks shown in Fig.3-18 are equivalent. 3-21.Write a set of equations using the Kirchhoff voltage law in terms of

appropriate loop-current variables for the four networks of Prob. 3-17.

3-22. Make use of the Kirchhoff voltage law to write equations on the loop basis for the four networks of Prob. 3-18.

3-23. Write a set of equilibrium equations on the loop basis to describe the network in the accompanying figure. Note that the network contains one controlled source. Collect terms in your formulation so that your equations have the general form of Eqs, (3-47).

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92 Network EquationsI Ch.3

Fig. P-3-23.

3-24. For the coupled network of the figure, write loop equations using the Kirchhoffvoltage law. In your formulation, use the three loop currents which are identified.

3-25. The network of the figure is that of Fig. 3-30 but with different loop-current variables chosen. Using the specified currents, write the Kirch-hoff voltage law equations for this network.

vlt)

Fig. P3-2S.

3-26. A network with magnetic coupling is shown in the figure. For the network, M\2 =O. Formulate the loop equations for this network using the Kirchhoff voltage law.

· Y

M

23 L3

f:

\

i2)R3 R2

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he rk

c

i

.

3IProblems

3.27. Write the loop-basis voltage equations for the magnetically coupled network of Fig. P5- 22 with K closed.

3.28. Write equations using the Kirchhoff current law in terms of no de-to-datum voltage variables for the four networks of Prob. 3-17. 3.29. Making use ofthe Kirchhoff current law, write equations on the node

basis for the four networks of Prob. 3-18.

3.30. For the given network, write the node-basis equations using the node-to-datum voltages as variables. Collect terms in your formula -tion sothat the equations have the general form of Eqs. (3-59).

2

AIIR~~ohm All C~ ~farad

4

Fig. P3·30.

3.31. The network in the figure contains one independent voltage source and two controlled sources. Using the Kirchhoff current law, write node-basis equations. Collect terms in the formulation so that the equations have the general form of Eqs. (3·59).

n,

~i2

"'t

_

Cl

-1--R---L,2 __ f....--=-.l...-.~--.J R6 Fig. P3-31.

).32. The network of the figure is amodel suitable for "rnidband" operation of the "cascode-connected" MOS transistor amplifier. Analyze the

+

Fig. P3-32.

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94 Network Equations / Ch. 3

network on (a) the loop basis, and (b) the node basis. Write the

resulting equations in matrix form, but do not solve them.

3-33. In the network of the figure, each branch contains a 1-n resistor, and four branches contain a I-V voltage source, Analyze the network on the loop basis, and organize the resulting equations in the form of a chart as in Example 11. Do not solve the equations.

2h 2h Iv 2h Fig. P3-34 2h Fig. P3-33.

3-34. Repeat Prob. 3-33 for the network of the accompanying figure. In

addition, write equations on the node basis,and arrange the equations in the form of the chart of Example 13.

3-35. In the network of the figure, R

=

2

n

and RI'

=

1

n.

Write

equa-tions on (a) the loop basis, and (b) the node basis, and simplify the

equations to the form of the chart used in Examples 11and 13.

R R R R R R R R Fig. P3-3S.

3-36. For the network shown in the figure, determine the numerical value

of thebi~11chcurrent iI.All sources in the network are time invariant.

H2 2v

2fl

(9)

3

ci

.

3/Problems

e 3-37. In the network of the figure, allsources are time invariant. Determine the numerical value ofi2•

d n a

2v

Fig.P3-37.

3-38. In the given network, all sources are time invariant. Determine the branch current in the 2-0 resistor.

Fig. P3-38.

2 In

ns

a

-he 3-39. In the network of the figure, all voltage sources and current source are time invariant, and all resistors have the value R =

t

O. Solve for the four node-to-datum voltages.

AllR=~ ohm

Fig.P3-39.

3-40. In the given network, node d is selected as the datum. For the specified element and source values, determine values for the four node-to-datum voltages.

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96

b

Fig. P3-40.

3-41. Evaluate the determinant:

2 -1 0 0

-1 3 -2 0

O. -2 3 -1

0 0 -1 2

3-42. Evaluate the determinant:

1 -2 0 3 4 -1 4 -1 1 0 2 0 1 1 3 4 -2 4 2 -1 3 1 3 -2 1 Network Equations /Ch. 3

3-43. Solve the following system of equations for i1> iz, and i3,

Cramer's rule.

3i1 - 2i2

+

Oi3=5

-2il

+

9i2 - 4i3

=

0 Oil - 4i2

+

9i3 =10

3-44. Solve the following system of equations for the three unknowns, i1> iz, and i3by Cramer's rule.

8i1 - 3i2 - 5i3 = 5 -3il

+

7i2 - Oi3

=

-10

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ci.3/Problems

3-4S. Solvethe equations ofProb. 3-43using the Gauss elimination method. 3-46. Solve the equations of Prob. 3-44 using the Gauss elimination method.

3-47. Determine il, i2, iJ, and i, from the following system of equations.

Si , Si2 - 10iJ

+

12i. =S 2il 4i2

+

5iJ

+

6i4 =33 -Sil

+

20i2

+

14iJ -- 16i. = 10 :'-il

+

7i2

+

2iJ - 10i4 =-15 3-48.Consider the equations 3x - y - 3z =1 x - 3y

+

z =I 4x

+

Oy -- 5z =1

(a) Is (4, 2, 3) a solution? Is (- I,-1, -I) a solution? (b) Can these equations be solved by determinants? Why? (c) What can you

con-clude regarding the three lines represented by these equations? 3-49.Find duals for the four networks of Prob. 3-17.

3-S0.Find the dual networks for the four networks given in Prob. 3-IS.

3-S1.Find the dual of the network of Prob. 3-31.

3-S2. If one exists, find a dual of the network of Prob. 3-40.

3-S3. Analyze the network of Prob. 3-17(c) using the state variable formu -lation.

3-S4.Consider the network shown in Prob. 3-23. Analyze this network using appropriate state variables.

3-SS. Analyze the network shown in Fig. P3-IS(b) using the state variable formulation.

3-56. Analyze the network of Prob. 3-30 using state variables.

3-S7. Apply the method of state variables to analyze the network shown in Fig. P3-31.

3-S8. The element represented in the network is a gyrator which is described bythe equations

'VI = Roi2

V2 = ---Roil

Find the two-element equivalent network shown in (b) of the figure. sing

rns,

(a)

Fig.P3-SS.

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98 _{Network Equations}

t

_Ch_.₃

3-59. For the gyrator-RL network of the figure, write the differential

equa-tion relating VI to il• Find a two-element equivalent network, asin

Prob. 3-49, in which neither of the elements is a gyrator.

Fig. P3-59.

3-60. In the network of(a)of the figure, all self inductance values are 1 H,

and mutual inductance values are

i

H. FindL.q,the equivalent

induc-tance, shown in(b)of the figure.

l~Leq

l'~

(a) (b)

Fig. P3-60.

3-61. It is intended that the two networks of the figure be equivalent with

respect to the pair of terminals which are identified. What must be

the values for Cl, L2' and L3 ?

(a) (b) Fig.P3-61.

3-62. It isintended that the two networks of the figure be equivalent with

respect to two pairs of terminals, terminal pair I-I'and terminal pair 2-2'.For this equivalence to exist, what must be the valuesforCt. Cz, and C3? ~I

1$?t?L

2

I

II l'()----.o .L---;o 2' Fig. P3-62. In this chapter of the simplest coefficients whi written In these equati variable, is us independent ;'a" ing alinear co solution of the vet) is someti Assume sources which' and currents. system is alte or closing of obtain equati

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112 _F_i_r_s_t-Or_d_{er Di}_f_f_e_r_e_{ntial Equat}_io_n_s_ICh.4

Cox, CYRUSW., AND WILLIAM L. REUTER,Circuits, Signals, and Networks, 4-3.

The Macmillan Company, New York, 1969. Chapter 4.

CRUZ, JOSEB., JR., ANDM. E. VANVALKENBURG, Signals in Linear Circuits,

Houghton Miffiin Company, Boston, Mass., 1974. Chapter 5.

HUELSMAN, LAWRENCEP., Basic Circuit Theory with Digital Computations,

Prentice-Hall, Inc., Englewood Cliffs, N.J., 1972. Chapter 5. LEaN, BENJAMINJ., AND PAUL A. WINTZ, Basic Linear Networks Jar Elet

-trical and Electronics Engineers, Holt, Rinehart & Winston, New York, 1970. Chapter 2.

DIGITAL COMPUTER EXERCISES

Exercises relating to the topics of this chapter are concerned with the numerical solution of first-order differential equations in Appendix £-6.1, and the solution of the RLC series circuit in Appendix E-6.2. In particular,

see Section 5.2 of Huelsman, reference 7in Appendix E-IO.

PROBLEMS

4-1. In the network of the figure, the switch K ismoved from positionI

to position 2 at I =0, a steady-state current having previously been

established in the RL circuit. Find the particular solution for the

current i(/).

Fig. P4-t.

4-2. The switch K is moved from position a to bat I ~ U,having beenin position a for a long time before I ~--O.Capacitor C2 isunchargedat

t ---O.(a) Find the particular solution for i(t) for t

>

O.(b) Find ti't particular solution for 1'2(t) for t> O.

Fig.P4·2.

4-4.

4-5.

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ICh.4 elworks, iuuits, 5. t-een in ged at mdthe

ci.

4IProblems

4·3. In the network given, the initial voltage on C. is V, and on C2 is 1'2

such that 1",(0) ~c V, and 1"2(0) = If~. At 1= 0, the switch K is closed.

(a) Find i(1) for all timc. (b) Find 1',(/) for I " O. (c) Find I'~(/) for

I> O.(d) From your results on (b) and (c). show that I',('Y.) "',(cy l.

(e) For the following values of the ctcrncnts, R 0, \

n

,

Cl ~c \ F, C2 = ~F, 1'1 -- 2V, I': ~c I V, sketch i(1) and I'"" amI idcntify the

limecom,lam of each,

Fig. P4·3.

4·4. In the network of the figure, the switch K is in position a for a long

period of time. At I

=

0,the switch is moved from ato b(by a" make-before-break" mechanism), Find 1'2(1) using the numerical values given in the nctw ark. Assume that the initial current in the 2-1 i inductor is zero. K IQ

~

-1

Iv I_h-L ~ _L__~ FiJ,:. P4·4.

4·5. The network of the figure reaches a steady state with the switch K open, At I =0, switch K is closed. Find i(/) for the numerical values

given, sketch the current waveform, and indicate the value of the lime constant, 30C! 20n

.

J

:

2

0 V

1 -

~'

'7

)

10 v-= -Fig. P4·5.

-L

4-6. The network of Prob, 4-5 reaches a steady state in position 2 and 'I' I =

°

the switch is moved to position 1, Find i(/) for the numerical values given for the element, sketch the waveform, and show the valt.e of the time constant.

4·7. In the given network, t', ~ e : for 12:0 and is zero for all I

<

0 If the capacitor is initially unchargcd, find t'2(1), Let R ,~-' 10,

R2 -, 20, and C

=

-

to

'

F, and for these values sketch "2(t) identifying the value of the ti.nc constant on the sketch.

113

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114

Fig. P4-7.

4-8. In the network shown in the figure, switch K is closed atI

=

necting a Source e-t to the RC network. At t =0, it is observ

the capacitor voltage has the value re(O)

=

0.5 V. For thee values given, determine t'2(t).

+

Fig. P4-8.

4-9. In the network shown, Vo

=

3 V, RI == 10

n,

Rz =c 5

n,

and

±

H. The network attains a steady state, and at t =0 switch

closed. Find V.(I) for t~ O.

K

Fig. P4-9.

4-10. The network of the figure consists of a current source of val

(a constant), two resistors, and a capacitor. At I.' 0, the swit

is opened. For the element values given on the figure, determine

for t~ O.

+ 1!!

Fig. P4-10. 4-11. We wish to multiply the differential equation

di -;-P(I)i == Q(I)

dt

by an "integrating factor" R such that the left-hand side of theeq

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'nsl Ch. 4 =0 con-rved that element andL

=

'tch K is

c

«

4/Problems

grating factor is R eS"", (b) Using this integrating factor, find the solution to the differential equation that corresponds to Eq. (4-30).

4-12. In the network shown in the accompanying figure, the switch K is closed at I 0, asteady-state having previously been attained. Solve for the current in the circuit as a function of time.

+ V-=

-Fig. P4-I2.

4-13. In the network shown, the voltage source follows the law L-(/) o.~

Ve'at, where (I,is a constant. The switch isclosed at I '=O. (a) Solve for the current assuming that (I,oFR/L. (b) Solve for the current when (J,' R/L.

K

L -lH Fig. P4-13.

vlt)

4-14. In the network: shown in Fig. P4-13, V(/) =0 for I

<

0, and vet) =t for I~ O. Show that i(/) "', I .- I .,- e-t for 12:0, and sketch this waveform,

4-15. In the network shown, the switch is closed at I

=

0 connecting a

voltage Source r(t) - Vsin WI to aseries RL circuit. For this system,

solve for the response i(t).

Fig. P4-15. 4-16. Consider the differential equation

.u

:

.

r ( )

dl -;-at

=

Jk t

where a is real and positive. Find the general solution of this equatio.. if allJ~~ 0 for I <0and for I2.0 have the following values:

(a)!1 kIt (b)J~' te=> (c)

Ji

sinWol (d)f~c~ cos Wot (e)!s

=

sin- i (f)

!

6

cc cos- I (fJ,)f~ " Isin '21 (h)

J8

=

e-t sin 2t 115

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116 First-Order Differential Equations

I

Ch.4 4-17. In the network (If the figure, the switch K is open and the network

reaches a steady state. At I=0,switch K isclosed. Find the current in the inductor for I:>0, sketch this current, and identify the time

constant. 10

n

10n Fig. P4-17. +

-

=

-

5v 2H

4-18. Repeat Prob. 4-13, determining the voltage at node a,v.(I) for I

>

O. 4-19. The network of the figure is in a steady state with the switch Kopen. At I=0, the switch is closed. Find the current in the capacitor for

I

>

0, sketch this waveform, and determine the time constant.

Fig. P4-19.

4-20. In the network shown, the switch K is closed at 1

=

O. The current

waveform is observed with a cathode ray oscilloscope. The initial value

of the current is measured to be 0.01 amp. The transient appears to

disappear in 0.1 sec. Find (a) the value of R, (b) the value of C, and

(c) the equation of i(t).

Fig. P4-20.

4-21. The circuit shown in the accompanying figure consists of a resistor

and a relay with inductance L. The relay is adjusted so that it is actuated when the current through the coil is0.008 amp. The switch

Kisclosed at 1-~ 0,and it isobserved that the relay isactuated when

I=0.1 sec. Find: (a) the indu.:tance Lof the coil, (b) the equation of i(1) with all terms evaluated.

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Ch..4/Problems 117

~ 10,0000

100V~ ~

Fig. P4-21.

4-22. A switch is closed at (=0, connecting a battery of voltage V with

a series RC circuit. (a) Determine the ratio of energy delivered to the capacitor to the total energy supplied by the source as a function of time. (b) Show that this ratio approaches 0.50 as 1-, 00.

4-23. Consider the exponentially decreasing function i~~Ke=u? where T

is the time constant. Let the tangent drawn from the curve at t=(1 intersect the line i=0 at 12' Show that for any such point, i(lI), (2 - 11 =T. current 'tialvalue pears to ofC, and ofa resistor sothat it is .Theswitch uatedwhen equation of

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132 Initial Conditions in Networks / Ch.5

PROBLEMS

5-1. In the network of the figure, the switch K isclosed at t =0 with the capacitor uncharged. Find values for i, di/dt and d+iidt? at t =0+, for element values as follows: V

=

100 V, R

=

1000

n,

and C

=

l.uF.

Fig. PS-I.

5-2. In the given network, K is closed at t =0 with zero current in the inductor. Find the values of i, di/dt, and d+iidt? at t =0+ if R

=

10

n,

L

=

1 H, and V

=

100 Y.

Fig. PS-2.

5-3. In the network of the figure, K is changed from position a to b at t

=

O. Solve for i,di/dt, and d+ildt? at t

=

0+ ifR

=

1000

n

,

L =

1 H, C

=

0.1.uF, and V=100 Y.

Fig. PS-3.

5-4. For the network and the conditions stated in Prob. 4-3, determine the values ofdvJ!dt and dVz/df at f =0+.

5-5. For the network described in Prob. 4-7, determine values ofdZvz/dtZ and d3vz/dt3 at t =0+.

5-6. The network shown in the accompanying figure is in the steady state with the switch K closed. At t =0, the switch is opened. Determine the voltage across the switch, VK, and dVK/dt at t =0+.

Fig. P5-6. ci.5/ Proble 5-7. In the solve

r,

and C 5-8. The ru Solve and L 5-9. In the switch given, 5-10. In tH state

(20)

h.5 the It, the at the ate ine

c

s.

5/Problems

5-7. In the given network, the switch K is opened at t

=

O.At t

=

0+, solve for the values of v,dcldt, and d+rldt? if I~"I 0amp, R == lOOO

n

,

and C ~= IILF.

v

Fig. PS-7.

5-8. The network shown in the figure has the switch K opened at t

=

O. Solve for 1',doldt, and d+oldt» at t

=

0+ if 1= 1 amp, R

=

100

n,

and L

=

1 H.

v

Fig.P5-S.

5-9. In the network shown in the figure, a steady state isreached with the switch K open. At t

=

0, the switch isclosed. For theelement values given, determine the value of v.(O-) and v.(O+).

10~! 20~! 10I! + 5 V-= -Fig. P5-9.

5-10. In the accompanying figure is shown a network in which a steady state is reached with switch K open. At t =0, the switch is closed.

lOQ lOH 20Q vb +

Ton

"-1

_J"

Fig. PS-lOo 133

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134

5 IProblems

Initial Conditions in Networks / Ch..

For the element values given, determine the values of v.(O-) an v.(O+).

5-11. In the network of Fig. P5-9, determine iL(O+)and iL( (0) for the cor

ditions stated in Prob. 5-9.

5-12. In the network given in Fig. P5-1O, determine Vb(O+) and Vb(oo) fo

the conditions stated in Prob. 5-10.

5-13. In the accompanying network, the switch K is closed at t

=

0 wit! zero capacitor voltage and zero inductor Current. Solve for (a) t'_IS.

and V2at t

=

0+, (b) VIand V2at t

=

00, (c) dVI/dt and dV2/dt a t

=

0+, (d)d2V2/dt2 at t

=

0+. Fig. PS-l3. !,~ the given m rh; switch K R2' IMr!, t .,0·; .

5-14. The network of Prob. 5-13 reaches a steady state with the switch K

closed. At a new reference time, t

=

0, the switch K is opened. Solve

for the quantities specified in the four parts of Prob. 5-13.

5-15. The switch K inthe network of the figure isclosed at t

=

0connecting

the battery to an unenergized network. (a) Determine i, dildt, and

d2i/dt2 at t

=

0+. (b) Determine 1'1,do-Jdt, and d2Vl/d/2 at t

=

0+. S-19. In the circui connecting a (a)dil/cll and + 5-20. In the net open with and C I integrodifli closed. (b) Fig. PS-IS.

5-16. The network of Prob. 5-15 reaches a steady state under the conditions

specified in that problem. At a new reference time, t

=

0, the switch

K is Opencd. Solve for the quantities specified in Prob. 5-15at t =0+. 5-17. In the network shown in the accompanying figure, the switch K is

changed from a to b at I=0 (a steady state having been established

at position a). Show that at f

=

0-1,

(22)

ks / Ch.5 0-) and :the con -=0 with 'or (a) t'l dvz/dt at switch K 'led. Solve onnecting difdT, and tT =O-l . :onditions the switch III=

0 +

.

witch K i, stablished

a:

5/Problems Fig. PS-17.

5-18. "~the given network, the capacitor Cl ischarged to voltage Voand

rh, switch K is c'osed at T ,,0. When RI ·2 Mn, Vo 1000y,

Rz I Mn, c, 10 J1F, and

c

,

-

20J1F, solve for d2iz/dT2 at

t .·0; .

Fig.PS-IS.

~-19. in the circuit shown in the figure, the switch K IS closed at t ~. 0

connecting a voltage, VosinWT, to the parallel RL-RC circuit. Find

(a)dil/df and (b) diz/df at T 0 i .

Fig. PS-i9.

5-20. In the network shown, a steady state is reached with the switch K

open with V . lOOY,RI" 10n, Rz ·20 n, RJ --= 20n, L I H,

and C IJ1F. At time f 0, the switch is closed. (a) Write the

integrodifTerential equations for the network after the switch is closed. (b) What is the voltage Vu across C before the switch is

Fi~. PS-20.

c

_-L----

T

(23)

136 Initial Conditions illNetworks

i

Ch.5

closed? What is its polarity? (c) Solve for the initial value of i,ami

i2

Ct

~=0+). (J)Solve for the values of di.ldt and di-fdt at I'"0+. (c) What is the value of di-fdt at t ~=co?

5-21. The network shown in the figure has two independent node pairs.

If the switch K is opened at t =0, find the following quam ities at

t

=

0+: (a) VI, (b) V2, (c) do-fdt,(d) dV2/dt.

Fig. PS-2I.

5-22. In the network shown in the figure, the switch K is closed at the instant t =0, connecting an unenergized system to a voltage source.

Let M12 =O.Show that if v(O) = V, then: di1 (0 .) dt -t -(L1

+

L3

+

2M13)(L2

+

L3

+

2M23) - (L3

+

M13

+

M23)2 di2(0+ ) dt

•

Fig. PS-22. 5-23. For the network of the figure, show that ifK isclosed at t =0, d2i1(0+)

=

_

1 -

(-1 [v(O) _ dV(O)-J_ d2v(0)} dt2 R1lR1C R1C dt dt2 fig. PS-D. Ch. 5/Problems 5-24. The given netv I' 0, the swin Vsin (I/./MC /,.(0+) =0, 5-25. In the network network has at an expression f parameters are what is the val dVK/dt (O+)? 5-26. In the network connecting the age Vaat t =O· F 5-27. In the network I=0-, all cap node-to-datum and dcddt at t V3 and dV3/dl at

(24)

·Ch.5 it and ~0+. pairs. ties at at the source. ,0,

ci.

5!Problems 5-24. The given network consists of two coupled coils and a capacitor. At t : 0,the switch K is closed connecting a generator ofvoltage, r(f) ~o

V sin (If"'; MC). Show that /,.(0+)=0, (;;/(0+)

=

(V/L)"'; M/C, and (/2V_df₂·(O+)

=

0 ~ a

~

K L

1 -

~L

+ + Viii C v. Fig. P5-24.

5-25. In the network of the figure, the switch K isopened at t

=

0 after the

network has attained a steady state with the switch closed. (a) Find

anexpression for the voltage across the switch at f =0+. (b) If the

parameters are adjusted such that i(O+)

=

I and dildt (0+) ~, - I, what is the value of the derivative of the voltage across the switch.

dVK/dt (O+)?

Fig. P5-2S.

5-26. In the network shown in the figure, the switch K is closed at t

=

0

connecting the battery with an unenergized system. (a) Find the volt -age v. at t

=

0+. (b) Find the voltage across capacitor Cl at t

=

CD.

Fig. PS-26.

r

-=-V

5-27. In the network of the figure, the switch K is closed at t ,-c O.At

t e-c, 0-, all capacitor voltages and inductor currents are zero. Three

node-to-datum voltages are identified as '1.'1,1'2, and 1'3. (a) Find VI

and dvr/df at t =0+. (b) Find 1'2 and de2/df at t

=

0+. (c) Find V3 and dVl/df at t =0-1·.

(25)

138 _Initial _Co_n_di_ti_o_n_s_ill_Ne_tw_o_rk_s _I_Ch_{. 5}

K

vitl +

Fig. PS-27.

5-28. In the network of the figure, a steady state is reached, and at t =0, the switch K is opened. (a) Find the voltage across the switch, 1"K at

t ~=0+. (b) Find dVK/dt at t =0+.

Fig. PS-28.

5-29. In the network of the accompanying figure, a steady state is reached with the switch K closed and with i O~ 10' a constant. At t

=

0, switch K is opened. Find: (a) t'2(0-), (b) t'2(0+), and (c) (dt"2/dl) (0+). +

c

L Fig. PS-29. The differential eq uations ofthe we will continue restrictions asto

The mathematic

under the head in the classical met

differential equat

conceptual adva

transformation is

which are ordin

more easily deve

be reserved for t

6-1. SECOND·O

EXCITATIO

A second-o

stant coefficients

The solution of

(26)

Continued / Ch. 6 otherwise this .the derivative (6-137) (6-138) ise is (6-139) ne appearance :Rcn and the

or

the current :ten (6-140) idition of the r)] ₍_{6-141 )} (6-142) (6-143) e is shown in ng factor and Ienvelope or ermines how es zero, the ISresult. .ult may be 1the electric rage element tored in the rergy, When ) the electric sas long as e oscillatory Ch.6/Problems 163

current will be sustained indefinitely. However, if there is resistance

present, the current through the resistor will cause energy to be

dissi-pated, and the total energy will decrease with each cycle. Eventually allthe energy will be dissipated and the current will be reduced to zero.

Ifa scheme can be devised to supply the energy that is lost in each cycle, the oscillations can be sustained. This is accomplished in the

electronic oscillator to produce audio frequency or radio frequency

sinusoidal signals.

=

ke=> and i

=

ke= are solutions of the differential equation

d2i

+

3 di

+

2'

=

0

(27)

164

Differential Equations, Continued/ Ch,

6-2. Show that i=ke= and i

=

kte= an: solutions of the differenti

equation

Ch. 6 / Problems

d2i

+

2 di -I- .

=

0

dt? dt' I

6-3. Find the general solution ofeach of the following equations:

dZ' d' dZ' d' (a) -.-!

+

3-.:

+

2i

=

0 (e) --.!-i- _l..-~ 6i

=

0 dt2 dt df2 dt' (b) d2i

+

5di

+

6' - 0 dtZ dt I -(c)

;t

2

1 +

7 ::

+

12;

=

0 (d) d2i

+

5 di

+

4'

=

0 df2 dt I d2' di (0 df~-T'd:

+

-

2i

=

0 ( ) dl; -L 2 di .: '= 0 g d{2' dt r I (h) d2i -L 4di -"4' -- 0. d{2' dt r 1

-6-4. Find the general solution of each of the following homogeneo differential equations:

d2L' dv (a) dt2

+

2 dt

+

20

=

0. (b) d2V +:?dv

+

4

=

0. dt2 - dt v d-» ; do (c) dtZ -r-4dt

+

2v

=

0 (d) 2 d21) 8d,; , 16. - 0 dtZ

+

dt -;- t-

-6-5. Find particular solutions for the differential equations of Prob. 6-) subject to the initial conditions:

(f) d20

+

3 do

+

5 - 0.

dt2 dt v

-i(O+)

=

I, _{dt (0}di ₊₎

₌

_0.

6-6. Find particular solutions for the differential equations of Prob, 6.) subject to the initial conditions:

i(o'+)

=

2,

6-7.

di (0+-) co .; I

dt

Find particular solutions to the differential equations of Prob. 6-4

subject to the initial conditions:

r(O+) 0= I,

6-8.

~~(0+-) ,-- I

Find particular solutions to the differential equations given in Prm 6-4, given the initial conditions:

('(0.+) ~,2, dO(o'_r)

=

I

clt

6-9. Solve the differential equation

iJ' d" di

3 ~ I- 8

-

=:

-j- 10-.: ., 3i

=

0.

dtJ dt? dt

6-10. Solve the differential equation

dJ; d~; 13d; __6,' ==0.

2c1tJ -;-9dt2 -:- dr

subject to the initial conditions

d+ildt? = --I at t

=

0.+.

6-11. The response of a network is fr

i= Kite:'

6-12.

where (J., is real and positive. i

maximum value.

In acertain network, it is found

sion

Show that i(t) reaches a maxi

1 t = --1X1 -6-13. The graph shows a damped si form Ke-at si From the graph, determine n Fig. 6-14. Repeat Prob. 6-13 for the wa

'

"

a. E

'

"

Fig.

(28)

Continued I Ch. 6 the differential uations: =0 =0 ,=0 li=0 g homogeneous 16v

=

0 Iv =0 5v

=

0 lOS of Prob. 6-3 lOS of Prob. 6-3 ms of Prob. 6-4 ISgiven in Prob. Ch.6IProblems 165

subject to the initial conditions i(O+) =0, dildt

=

1 vt t

=

0+, and

d2i/dt2

=

·-1 at t=0+.

6-11.The response of anetwork is found to be f:::::0

where (I, is real and positive. Find the time at which i(t) attains a maximum value.

6-12.Ina certain network, it isfound that the current is given by the expres-sion

Show that i(t) reaches a maximum value at time

t

=

1 In(l,lK1

(1,1 - (1,2 (l,2Kz

6-13. The graph shows a damped sinusoidal waveform having the general form

Ke:= sin(eui -;-ifJ)

From the graph, determine numerical values for K,(1,

co,

and ifJ.

-,

'.

Fig. P6-13.

6-14. Repeat Prob. 6-13 for the waveform of the accompanying figure.

+1

/"

r

-

.

,

V

-,

1 /

-H

-

I"",

/

<;V

'

"

a. E

'

"

o

-1

o

2 3 4 5 t,msec Fig. P6-14.

(29)

166

Differential Equations,

6-15. In the network of the figure, the switchKis closedand is reached in the network. At f

=

0, the switch is0

expression for the current in the inductor, i2(t).

~

-

=

-

100v

Fig. P6-15. 6·16. The capacitor of the figure has an initial voltagevc(o-)

at the same time the current in the induct or is zero.At switch K is closed. Determine an expression for thevel

Fig. P6-16. 6-17. The voltage SOurce in the network of the figure is descri

equation, VI =2 cos2t fer t~ 0 and is a short circuitp' time. Determine V2(t). Repeat if'1.·1 =KIt for t~ 0ands t

<

O.

Fig. P6-17.

6-18. Solve the following nonhomogeneous differential equationsI

( ) d2i

+

2 di

+

i

=

1 a dt2 dt (b)

g:

.

!

+

3 di

+

2i

=

St dt? dt (c) ;t2;

+

3::

+

2i=10sinlOt (d) d2q

+

Sdq

+

6q

=

te= dt2 dt

(e) ;t2~

+

5;~

+

6v

=

e=»

+

Se-3r

6-19. Solve the differential equations given in Prob. following initial conditions:

x(O+) =1 dx

and dr(O+) =-1 wherex is the general dependent variable.

(30)

isclosed and asteady sta

switch is opened. Find

u

'2(t).

P6-15.

'oIt~ge vC<O-) = VI> and

tor IS zero. At t =0, the

n for the voltage V2(t).

'6-16.

igure is described by the

~hort circuit prior to that

:or t~ 0 and VI =0 for i-17. itial equations for t~ O. ob. 6-18 subject to the -1 167

Find the particular solutions to the differential equations of Prob,

6-18for the following initial conditions:

dx

x(O+)

=

2 and dt(O+)

=

-1

wherexis the dependent variable ineach case.

~ll.Solvethe differential equation

dJ' d2' di

2dt~

+

9 dt~

+

13 d;

+

6i

=

Kote-r sin t

which is valid for t~ 0, if i(O+) = 1,di/dl(O +) = -1, and d+il dl'(O"t') =O.

~ll.Aspecialgenerator has a voltage variation given by the equation

t,l) 1V, where t isthe time in seconds and 1~ O.This generator is

connected to an RL series circuit, where R

=

2 nand L =I H, at

urne1=0bytheclosing of aswitch, Find the equation for the current as a function of timei(t).

6-13.A bolt of lightning having a waveform which is approximated as

1'(1)

=

te-r strikes a transmission line having resistance R =0.1 n and inductance L =0,1 H (the line-to-line capacitance is assumed

negligible). An equivalent network is shown in the accompanying

diagram.What is the form of the current as a function of time?

(Thiscurrent willbe in amperes per unit volt of the lightning; likewise

the timebase isnormalized.)

6-24.In thenetwork of the figure, the switch K is closed at 1=0 with the capacitor initially unenergized. For the numerical values given, find

i(I).

Fig. P6-23.

vlt) ~

Fig.P6-24.

6-25. Inthenetwork shown in the accompanying figure, a steady state is

reached with the switch K open. At r=0, the switch is closed. For the element values given, determine the current, i(t) for 1~ 0,

R-103 (l

r

:

\

,

5I

lF

ilt))

Fig.P6-2S.

6-26. In the network shown in Fig. P6-2S, a steady state isreached with the switchK open. At t =0, the value of the x resistor R is changed to the critical value, Ra defined by Eq, (6-88). For the element values given,determine the current i(t) for 12O.

(31)

168 Differential Equations, Continued I Ch.6

6-27. Consider the network shown in Fig. P6-24. The capacitor has an initial voltage, Vc =10 V. At I=O. the switch Kis closed. Determine i(t) for I:2:O.

6-28. The network of the figure is operating in the steady state with the

switch K open. At t

=

0, the switch isclosed. Find an expression for the Voltage, v(l) for t:2:O.

c

u(t)

+

10sinwt

t

K

Fig. P6-28.

6-29. Consider a series RLC network which is excited by avoltage source.

(a) Determine the characteristic equation corresponding to the differ -ential equation for i(t). (b) Suppose that Land C are fixed in value but that Rvaries from 0 to 00. What will be the locus of the roots of the characteristic equation? (c) Plot the roots of the characteristic equation in the splane if L

=

1 H, C

=

1 J.l.F, and R has the following values: 500

n

.

1000

n

,

3000

n

,

5000

n

.

6-30. Consider the RLC network of Prob. 6-16. Repeat Prob. 6-29, except that in this case the study will concern the characteristic equation corresponding to the differential equation for V2(t). Compare results with those obtained in Prob. 6-29.

6-31. Analyze the network given in the figure on the loop basis, and deter -mine the characteristic equation for the currents in the network as a function of Kt. Find the value(s) of Kt for which the roots of the characteristic equation are on the imaginary axis of the s plane. Find the range of values of Kt for which the roots of the characteristic equation have positive real parts.

Fig. P6-31. 6-32. Show that Eq. (6-121) can be written in the form

i

=

Ke-'W"'cos(con~i

+

1/»

Give the values for K and I/> in terms of K, and K6 of Eq. (6-121).

Ch.,6 IProblem! 6-33. Aswitch series RI of time i: w (b) Find tion of t steady-si as 1-" in the st 6-34. In the s frequent (1) CO= (2) CO

=

These f experim when th steady-s the rna: is,whic greater'

(32)

· Continued / Ch.6

:apacitor has an

osed. Determine

y state with the nexpression for -0 + )(t) voltage SOurce. igto the diffe r-fixed in value of the roots of characteristic

)the following

). 6-29, except istic equation rnpare results .is,and deter-e ndeter-etwork as :roots of the s plane. Find charactensnc ~q.(6-121).

o

.

6/Problems 169

6-33.A switch is closed at t

=

0 connecting a battery of voltage Vwith a

seriesRLcircuit. (a) Show that the energy in the resistor as afunction of time is

V

2(

2L R'L L 2R'L 3L). I

WR =

R

t

+

R

c: t - 2Re- t, - 2R JOU es

(b) Findan expression for the energy in the magnetic field as a func-tion of time. (c) Sketch WR and WL as a function of time. Show the

steady-state asymptotes, that is, the values that WR and WL approach

asI-4 eo.(d) Find the total energy supplied by the voltage source inthe steady state.

6-34.In the series RLC circuit shown in the accompanying diagram, the frequencyof the driving force voltage is

(I) W =

eo

,

(the undarnped natural frequency)

(2) W =Wn~ (the natural frequency)

These frequencies are applied in two separate experiments. In each experiment we measure (a) the peak value of the transient current

when the switch is closed at I =0,and (b) the maximum value of the steady-state current. (a) In which case (that is, which frequency) is

themaximum value of the transient greater? (b) In which case (that is, whichfrequency) is the maximum value of the steady-state current

greater?

Fig.P6-34.

~

100 sinwt ~ lJ1F

!

I

t,)

(33)

iNetwork Theorems / CIr.9

(b)

: 6 for which the

rolled source which

(9-94)

find the impedance ng a voltage source

rrent I(s) under the

zero, meaning that

'k(S) (9-95)

equired

impedance

(9-96)

(9-97)

(9-98)

rk

is constructed

seful

artifice that he operation of Ig the amount of

rnplish

this.

"

.

9/

Problems 271

=

loe-t sin 31 u(t). The imped-ance of the passive network N is found to be

Z(s) =(s

+

2Xs

+

3) (s

+

IXs

+

4)

(a) With N connected to the voltage source as in (a) of the figure, what will be the complex frequencies in the current i,(t)?

(b) With N connected to the current source as in (b) of the figure, what will be the complex frequencies in the voltage VI(t)?

9·2. Repeat Prob. 9-1 if

2s'

+

3s3

+

5s2

+

Ss

+

1

Z(s) = (S2

+

IX2sz

+

2s

+

4) Solve part (b) only.

9·3. Consider the two series circuits shown in the accompanying figure.

Given that VI(t)

=

sin 103t, vz(t) =e-IOOOtfor t

>

0,and C

=

I j.l.F.

"

-VI N (a) + VI N (b) Fig. P9-1. R L'

~

C

~

C (a) (b) Fig. 1'9-3.

(34)

Impedance Functions and Network Theorems / Ch. 9

(a) Show that it is possible to have ;1(t) =;z(t) for all t

>

Q. (b) Determine the required values of Rand Lfor (a) to hold. (c) Discuss the physical meaning of this problem in terms of the complex fre-quencies of the two series circuits.

9-4. In the network of the figure, the switch is opened at t

=

0, a steady state having previously been established. With the switch open, draw the transform network for analysis on the loop basis, representing all elements and all initial conditions.

rr-V

₀

_:

-

:-Fig. P9-4.

9-5. This problem is similar to Prob. 9-4,except that the transform net-work required should be prepared for analysis on the (a) loop basis, and (b) node basis. In this network, initial currents and voltages are a consequence of active elements removed at t

=

O.

Fig. P9-S.

9-6. In the network of the figure, the switch Kisclosed at t=0 and at

t=0 - the indicated voItages are on the two capacitors. Repeat Prob. 9-4 for this network.

Fig. P9-6.

9-7. Determine the transform impedances for the two networks shown in the accompanying figure.

z~g'1~\

I

Fig. P9-7. Ch. 9 / Problems 9-8. For the RC ance, Z(s),i p(s) andq(s of Prob. 9-1 9-9. Repeat Pro 9-10. Repeat Pr figure. 9-11. Repeat P this case 9-12. Two blac known th contains the input (b) Inves network. conditio! o r-r : I I I I 1 I I I I I 1 0>---+- 1-L. 9-13. Repeat panying 5Slepian, 6Macklel September, 191

(35)

seorems ICIr. 9 all

t>

Q. (b) d.(c) Discuss complex fr e-=0, a steady ~open, draw iresenting all nsform ne t-loop basis, /oltages are ~0 and at peat Prob. shown in

a.

9/Problems

f.I. For the RC network shown in the figure, find the transform i mped-ance,Z(s), in the form of a quotient of polynomials, p(s)/q(s). Factor

pes) andq(s) so that Z(s) may be written in the form of the impedance ofProb.9-1.

,.9. Repeat Prob. 9-8 for the LC network of the accompanying figure.

Fig. P9-9.

,.10. Repeat Prob. 9-8 for the RC network shown in the accompanying

figure,

Fig. P9-10.

9-11.Repeat Prob. ~-8 for the RLC network of the figure, except that in this case determine yes) rather than Z(s).

9-12.Two black boxes with two terminals each are externally identical. It is

known that one box contains the network shown as (a) and the other contains the network shown as (b) with R

=

..;

L/e. (a) Show that the input impedance, Zin(S)

=

Vin(s)/Iin(s)

=

R for both networks.' (b) Investigate the possibility of distinguishing the purely resistive network. Any external measurements may be made, initial and final conditions may be examined, etc.

r-----, .------------, I I L :

:R-

f

T

I VC I CRI I I I L J R R L J (a) (b) Fig. P9-12.

9-13.Repeat Prob. 9-12 by comparing the network shown in the accom-panying figures to that given in (a) of the figure for Prob. 9-12.

~Slepian,J., letter in Elec. Engrg., 68,377; April, 1949.

6Macklem, F. S., "Or. Slepian's black box problem," Proc. IEEE, 51,1269; September,1963. 273 2F 2F z~ Fig. P9-S. O---r---,

z

-IH Fig. P9-11.

(36)

274 Impedance Functions and Network Theorems / Ch. 9 r---, I Fig. P9·13. R R=

f

F.

,c R

c

9·14. The network shown in Fig. P9-4 is operated with switch K closed until a steady-state condition is reached. Then at t

=

0 the switchK

is opened. Starting with the transform network found in Prob. 9-4, determine the voltage across the switch,Vk(t), fort:2:O.

9·15. If the capacitors are uncharged and the inductor current zero at

t =0-, in the given network, show that the transform of the gen -erator current is IO(s2

+

s

+

1) ll(s)

=

(S2

+

lXs2

+

2s

+

2) IH IF 10 Fig. P9·1S.

9·16. Repeat Prob. 9-15 for the network given to show that the generator current is given by the transform

I s _ s(s

+

2X5s

+

6)

l( ) - (S2

+

4s

+

13XlOs2

+

18s

+

4)

1n

Fig P9·16.

9·17. For the network of the figure, show that the equivalent Thevenin network is represented by V Vs

=

-

-t

(1

+

a

+

b - ab) and 3-b

z,

= -2-Ch. 9 / Problems 9·18. The accom sources in network, fi expression 9·19. Th1netwo current so determine 9·20. The ne this netw RL•

(37)

Theorems /

cs

.

9 switch K closed - 0 the switch K d in Prob. 9-4, ~O. current zero at orm of the ge n-It the generator rlent Thevenin 275 1n Fig. P9-17.

9-18.The accompanying network consists of resistors and controlled

sourcesin addition to the independent voltage source

v,.

For this

network, find the Thevenin equivalent network by determining an

expression for the voltage V8 and the Thevenin equivalent resistance.

fig. P9-1S.

9-19.ThJnetwork of the figure contains three resistors and one controlled

curfent source in addition to independent sources. For this network,

determine the Thevenin equivalent network at terminals I-I',

Fig. P9-19.

9·20. The network shown is a simple representation of a transistor. For

thisnetwork, determine the Thevenin equivalent network for the load

RL•

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276 Impedance Functions and Network Theorems ICh. 9 9-21. The network in the figure contains a resistor and a capacitor in addi-tion to various sources. With respect to the load consisting of RLin series with L,determine the Thevenin equivalent network.

+

111 IJ••

Fig.1'9-21.

9-22. Using the network of Prob. 9-18, determine the Norton equivalent network.

9-23. For the network used in Prob. 9-19, determine the Norton equivalent network.

9-24. Determine the Norton equivalent network for the network given in Prob.9-20.

9-25. Determine the Norton equivalent network for the system described in Prob.9-21.

9-26. In the given network, the switch is in position a until a steady state is.

reached. At t

=

0, the switch is moved to position b. Under that condition, determine the transform of the voltage across the 0.5-F capacitor using (a) Thevenin's theorem, and (o) Norton's theorem.

Fig. 1'9-26.

9-27. In the network of the figure, the switch K is closed at 1

=

0, a steady state having previously existed. Find the current in the resistor R3 using (a) Thevenin's theorem, and (b) Norton's theorem.

10

n

Fig.P9-17. 9-30. Using alent ditions. 9-31. The values dete equiva

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eorems /

cs.

9 Ch.9IProblems 277

lcitor in a ddi-sting of RL in ~rk.

J.28.The network shown in thefigure isa low-passfilter.The input voltage

VI(t) is a unit step function, and the input and load resistors have the value

R

=...

L

;

I

e

.

ByusingThevenin's theorem, show that the trans-form of the output voltage is

nequivalent + R ~Itl Fig. P9-28. nequivalent >rk given in

9·29.In the network shown in the accompanying sketch, the elements are chosen such thatL

=

eRr and RI

=

Rz. Ifv\(t) is a voltage pulse of

I-V amplitude and T-sec duration, show that vz(t) is also a pulse, and find its amplitude and time duration.

described in JaOY state is, Under that the O.S-F s theorem. + Fig. P9-29.

9-30.Using either Thevenin's or Norton's theorem, determine an equiv-alent network for the terminals a-b in the figure for zero initial con-ditions.

I,a steady :sistor RJ

Fig. P9-JO.

9-31.The network given contains a controlled source. For the element values given, with v\(t) =u(t), and for zero initial conditions: (a) determine the equivalent Thevenin network ata-a', (b) Determine the equivalent Thevenin network at bob'.

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Impedanc« Functionsand Network Theorems

I

Ch. 9

Fig. P9-3I.

9-32. For the given network, determine the equivalent Thevenin network to compute the transform of the current in RL•

Fig. P9-31.

9-33. Assuming zero initial voltage on the capacitor, determine 1he equ

iv-alent Norton network for the resistor Rx.

+ -Fig. P9-33. In this char admittance extended. F different par mathematic, functions arl 10·1. TERMI] Consid elements. T( represented I fastened to a access, the en are required necting some ments. The IT the terminal! another pair name terminc ITerminal

This results in:

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tnd Zeros / Ch. 10

i).The stability

aial or an odd eeven polyno-S"

+

ja)(s - ja) other pos sibiI-lay be reached D-31which are ;b) (10-120) ) if b

>

a.

In

on

applies

for

.rrzeros, sy m-vith respect to irm a quad of 5 (l0-121) (10-122) :) is (10-123)

o,

/0

I

Problems 317 whichis a quad, indicating that pes) has two zeros in the right half-planefrom the quad. Dividing Eq. (10-123) into Eq. (10-121) gives the factor 2S2

+

s

+

1 which may be analyzed by the quadratic

formula.

r

S2

+

(R1C1

+

R2C2)SjRIR2CIC2

+

1jR1R2C1C2 ]

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318 Network Functions;Poles and Zeros ICh. 10

Fig. PlO-I.

10-3. (a) For the given network, show that with port 2 open, the input impedance at port 1 is 1

n.

(b) Find the voltage-ratio transfer func-tion, G12for the two-port network.

1 + 10

~---r---~2

+ 2F 10 10

~---~~---~2

Fig. PI0-3.

10-4. For the resistive two-port network of the figure, determine the numerical value for (a) G12, (b) Z12, (c) Y12, and (d)tX12•

U!

Fig. PI0-4.

1n

10-5. The resistive bridged-T, two-port network shown in the figure is to be analyzed to determine (a) G\2, (b) Z12, (c) Y12, and (d) tX12•

10-6. The given network contains resistors and controlled sources. For this network, compute G12 =Vz/V!.

Fig. PIO-5. I ~ 1

n

2V;

v,u::Jln

Fig. PI0-6.

a

.

10IProblems 10-7. For the net' specified, dr 10-8. Fur the RI 10-'). For the g and dete 10-10. For the transfer 10-11. Foreal avolta V.at I

(43)

~eros/

cs

.

10 \' the input lnsfer func-02

~

2 ine the re isto 1%12. s. For

u

:

10IProblems

10-7.For the network of the accompanying figure and the element values specified, determine IX12

=

12//1,

In

Fig. PIO-7.

10·8. Fur the RC two-port network shown in the figure, show that

G --

i

-

1/R1R2CIC2 ]

12 - LS2

+

(RICI

+

RIC2

+

R2C2)S/RIR2CIC2

+

1/R1R2CIC2

+

Fig. PlO-So _o

10·). For the given network, show that

K(s

+

1)

YI2

=

(s

+

2)(s

+

4)

and determine the value and sign of K.

Fig. PIO·9.

10·10. For the network shown in the figure, show that the voltage-ratio transfer function is (S2

+

1)2 G12

=

5s4

+

5s2

+

I H I H + +

~T_l

_

______ lll~

2

Fig.

r-i

o-

io.

10-11. For each of the networks shown in the accompanying figure, connect a voltage source VI to port I and designate polarity references for

V2 at port 2.For each network, determine G12

=

V2/ VI'

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320 Network Functions; Poles and Zeros/ Ch. 10

1n 2

if 2

(g)

Fig. PlO-H.

10-12. For the network given in Fig. PlO-ll(a), terminate port 2 in a I-Q

resistor and connect a voltage source at port I. Let 11 be the current in the voltage source and 12 be the current in the I-n load. Assign reference directions for each. For this network, compute G12 =

V21V1 and 0(12 =12112,

10-13. Repeat Prob. 10-12 for the network of Fig. PIO-ll(b).

10-14. Repeat Prob. 10-12 for the network of Fig. PlO-Il(g).

10-15. For the network of Fig. PlO-II(g), connect a current source 11 at port I and a I-n resistor at port 2. Assign reference directions for all voltages and currents. For this network, compute Z12 = V21I1

and 0(12 =12

1/

1,

10-16. The network shown in (a) of the figure is known as a shunt peaking

network. Show that the impedance has the form

Z(s)

=

K(s - ZI) (s - Pl)(S - P2)

and determine ZI, pi, and P2 in terms of R, L, and C.If the poles and

zeros of Z(s) have the locations shown in(b) of the figure with Z(jO) =I, find the values for R, L, and C.

Ch. 10/ Problems R Z(s) L (a) 10-17. A system has a which maybe a system toastep of K, asafuncii done by the .l0-18. 10-19. A systemhas s = -3, and One term of K3e-r sin(t

+

ofabetween

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id Zeros / Ch. 10 rrt 2 in a I-Q re the current load. Assign npute G12 = source I( at lirections for Z12 = V2/1( hunt peaking he poles and 'e with ZUO) Ch.JOIProblems 321 I I I I I I

*-

-

--(J splane jw

JTITrr-

₂

-I 1.5

c

ZlsI _-3

v

r

2 L (c) Ibl Fig. PIO-16.

10·17.A system has a transfer function with a pole at s

=

-

3 and a zero which may be adjusted in position at s = -a_The response of this system to a step input has a term of the form K,e:», Plot the value ofK( as a function of a for values of a between 0 and 5.This may be done by the graphical procedure of Section 10-7.

10-18.Asystem has a transfer function with poles at s = -1

±

j1 and a zero which may be adjusted in position at s

=

-a. The response of this system to a step input has a term of the form K2e-r sin (t

+

rjJ). Plot the value of K2 as a function of afor values of abetween 0 and

5.This may be done graphically.

10·19.A system has a transfer function with poles at s

=

-1

±

j1and at s= - 3, and a zero which may be adjusted in position at s = -a. One term of the response of this system to a step input is of the form K3e-r sin

Ct

+

rjJ).Plot the value of K3 as a function of afor values of a between 0 and 5. jw x j1 -a -4 -3 -2 -1 (J x j1 Fig. PIO-19.

10·20. Apply the Routh-Hurwitz criterion to the following equations and determine: (a) the number of roots with positive real parts, (b) the number of roots with zero real parts, and (c) the number of roots with negative real parts.

(a) 4s3

+

7s2

+

7s

+

2

=

0 (b) S3

+

3s2

+

4s

+

1=0 Cc) 5s3

+

S2

+

6s

+

2=0