VALKENBURG-PROBLEMS.pdf

ok Equations / Ch. 3

uic method forform. The so(u-solution is to bermulation offershe only require-ement or several

I. the state-spaceon by computeris to be accorn-

inarily simpler to

I Book Company,cusses a graphicalms.in Linear Circuits ,

Theory, McGraw-1 and 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 Variableark, 1970.

Analysis, Prentice-is a programmedof state equations.

~s

e digital computer .. described in refer-nethod from refer-

87

I Problems

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

es-

see Huelsman

, reference 7 of Appendix E-I0, for the resistive networkated to the solution of simultaneous equations in Chapter 7 and theion of equations for the RLC networkS of Chapter 6. More advancedbilities include the solution of state equations by methods describedferences 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 C 2 in (a) ofthe 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. L2 and M for thenetworkS 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)

Fig. P3-3.

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

3-4. The network of inductors shown in the figure is composed of a J-Hinductor on each edge of a cube with the inductors connected to thevertices of the cube as shown. Show that, with respect to vertices aand b, the network is equivalent to that in (b) of the figure whenLeq = i H. Make use of symmetry in working this problem, ratherthan writing Kirchhoff laws.

The series (tain to the netwospecified in the taconnection of eleconnection of eletto zero. For the stmine 'VI in the foron a cathode ray 0

and so on.

(a)

Fig. P3-4.

10--1?L,q

1'~ 2

(a)

V22

2

-3(c)

V2volts

+1

-1

(e)

",[

(b)

3-5. In the rietworks of Prob. 3-4, each I-H inductor is replaced by aJ-H capacitor, and L,q is replaced by C,q' What must be the value ofC eq 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 Kwas 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

+10 v-=-

Fig. P3-6.

211

+

111

~ h

3-7. This problem is similar to Prob. 3·6. In the network given in thefigure, 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 was closed. Determine the valueof C.

rations / Ch. 3

;ed of a I-Hected to the:0 vertices afigure whenilern, 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 Bspecified in the table. In A, two entries in the column implies a seriesconnection of elements, while in B, two entries implies a parallelconnection of elements. In each case, all initial conditions are equalto 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 seenon a cathode ray oscilloscope. Evaluate significant amplitudes, slopes,and so on.

C

(b)

3-19. Demonstrate theso establish ainductor into an

C2 3-20. Demonstrate tha

3-21. Write a set ofappropriate 1

R3 3-17.(d)

90 Network Equatiolls / c». J

Network of A Network of B Waveforms of V2

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 thenumber of independent loop currents, and the number of independentnode-to-node voltages that may be used in writing equilibrium equa-tions using the Kirchhoff laws.

R2

2 fi0 3

RIL

Cv(t) +

~

v(t}

R3

4(a)

2

v(t} + 3 Rr;JC v(t}

3

(c)

Fig. P3-17.

2

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

Ch. 3/ Problems

v(t)

(b)

3-23. Write a set0network inone controllequations

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 andso establish a rule for converting a voltage source in series with aninductor 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 loopbasis for the four networks of Prob. 3-18.

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

92Network Equations I Ch. 3

Fig. P-3-23.

3-24. For the coupled network of the figure, write loop equations using theKirchhoffvoltage 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 thenetwork, M \2 = O. Formulate the loop equations for this networkusing the Kirchhoff voltage law.

·YM23

L3f:\i2)R3

R2Fig. P3-26.

herk

ci. 3 I Problems

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

3.28. Write equations using the Kirchhoff current law in terms of node-to-datum voltage variables for the four networks of Prob. 3-17.

3.29. Making use of the Kirchhoff current law, write equations on the nodebasis for the four networks of Prob. 3-18.

3.30. For the given network, write the node-basis equations using thenode-to-datum voltages as variables. Collect terms in your formula-tion so that 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 sourceand two controlled sources. Using the Kirchhoff current law, writenode-basis equations. Collect terms in the formulation so that theequations 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 a model suitable for "rnidband" operationof the "cascode-connected" MOS transistor amplifier. Analyze the

+

Fig. P3-32.

93

94 Network Equations / Ch. 3

network on (a) the loop basis, and (b) the node basis. Write theresulting equations in matrix form, but do not solve them.

3-33. In the network of the figure, each branch contains a 1-n resistor, andfour branches contain a I-V voltage source, Analyze the network onthe loop basis, and organize the resulting equations in the form of achart 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. Inaddition, write equations on the node basis, and arrange the equationsin 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 theequations to the form of the chart used in Examples 11 and 13.

R R

R

RR

R

R R Fig. P3-3S.

3-36. For the network shown in the figure, determine the numerical valueof the bi ~11chcurrent iI. All sources in the network are time invariant.

H2 2v

2fl

Fig. P3·36.

3 ci. 3/ Problems

e 3-37. In the network of the figure, all sources are time invariant. Determinethe numerical value of i2•

dna

2v

Fig. P3-37.

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

Fig. P3-38.

2Inns

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. Solvefor the four node-to-datum voltages.

All R=~ ohm

Fig. P3-39.

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

95

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 i 1> iz, and i3,Cramer's rule.

3i 1 - 2i2 + Oi3 = 5

-2il + 9i2 - 4i3 = 0

Oil - 4i2 + 9i3 = 10

3-44. Solve the following system of equations for the three unknowns,i 1> iz, and i3 by Cramer's rule.

8i1 - 3i2 - 5i3 = 5

-3il + 7i2 - Oi3 = -10

-5il + Oiz + 1113 = -10

ci. 3 / Problems

3-4S. Solve the equations of Prob. 3-43 using 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 theseequations 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 networkusing appropriate state variables.

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

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

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

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

'VI = Roi2

V2 = ---Roil

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

sing

rns,

(a)

Fig. P3-SS.

97

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, as inProb. 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. Find L.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 withrespect to the pair of terminals which are identified. What must bethe values for Cl, L2' and L3 ?

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

3-62. It is intended that the two networks of the figure be equivalent withrespect to two pairs of terminals, terminal pair I-I' and terminal pair2-2'. For this equivalence to exist, what must be the values for Ct. Cz,and C3?

~I

1$?t?L2

III

l'().o----.L------;o 2'

Fig. P3-62.

In this chapterof the simplestcoefficients whiwritten

In these equativariable, is usindependent ;'a"ing a linear cosolution of thevet) is someti

Assumesources which'and currents.system is alteor closing ofobtain equati

112 First-Order Differential Equations I Ch. 4

Cox, CYRUS W., AND WILLIAM L. REUTER, Circuits, Signals, and Networks, 4-3.The Macmillan Company, New York, 1969. Chapter 4.

CRUZ, JOSEB., JR., AND M. E. VAN VALKENBURG, Signals in Linear Circuits,Houghton Miffiin Company, Boston, Mass., 1974. Chapter 5.

HUELSMAN, LAWRENCE P., Basic Circuit Theory with Digital Computations,Prentice-Hall, Inc., Englewood Cliffs, N.J., 1972. Chapter 5.

LEaN, BENJAMIN J., AND PAUL A. WINTZ, Basic Linear Networks Jar Elet-trical and Electronics Engineers, Holt, Rinehart & Winston, NewYork, 1970. Chapter 2.

DIGITAL COMPUTER EXERCISES

Exercises relating to the topics of this chapter are concerned with thenumerical 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 7 in Appendix E-IO.

PROBLEMS

4-1. In the network of the figure, the switch K is moved from position Ito position 2 at I = 0, a steady-state current having previously beenestablished in the RL circuit. Find the particular solution for thecurrent i(/).

Fig. P4-t.

4-2. The switch K is moved from position a to b at I ~ U, having been inposition a for a long time before I ~--O. Capacitor C2 is unchargedatt --- O. (a) Find the particular solution for i(t) for t > O. (b) Find ti'tparticular solution for 1'2(t) for t > O.

Fig. P4·2.

4-4.

4-5.

4-6.

I Ch. 4

elworks,

iuuits,5.

t-een inged at

md the

ci. 4 I Problems

4·3. In the network given, the initial voltage on C. is V, and on C2 is 1'2such that 1",(0) ~c V, and 1"2(0) = If~. At 1= 0, the switch K is closed.(a) Find i(1) for all t imc. (b) Find 1',(/) for I " O. (c) Find I'~(/) forI> 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 -- 2 V, I': ~c I V, sketch i(1) and I'"" amI idcntify thelime com,lam of each,

Fig. P4·3.

4·4. In the network of the figure, the switch K is in position a for a longperiod of time. At I = 0, the switch is moved from a to b (by a "make-before-break" mechanism), Find 1'2(1) using the numerical valuesgiven in the nctw ark. Assume that the initial current in the 2-1 iinductor is zero.

KIQ

~

-1 Iv I_h-L ~ _L__ ~FiJ,:. P4·4.

4·5. The network of the figure reaches a steady state with the switch Kopen, At I = 0, switch K is closed. Find i(/) for the numerical valuesgiven, sketch the current waveform, and indicate the value of the limeconstant,

30 C! 20n

.J:20V1-~' '7)10 v-=-

Fig. P4·5. -L4-6. 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 numericalvalues given for the element, sketch the waveform, and show the valt.eof the time constant.

4·7. In the given network, t', ~ e : for 12:0 and is zero for all I < 0If 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) identifyingthe value of the ti.nc constant on the sketch.

113

+

114

Fig. P4-7.

4-8. In the network shown in the figure, switch K is closed at I =necting a Source e-t to the RC network. At t = 0, it is observthe capacitor voltage has the value re(O) = 0.5 V. For the evalues 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 switchclosed. 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 switis opened. For the element values given on the figure, determinefor 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 the equon equals the derivative d(Ri)/df. (a) Show that the required i

'nsl Ch. 4

= 0 con-rved thatelement

andL ='tch K is

c« 4 / Problems

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

4-12. In the network shown in the accompanying figure, the switch K isclosed at I 0, a steady-state having previously been attained. Solvefor 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 is closed at I '= O. (a) Solvefor the current assuming that (I, oF R/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) = tfor I ~ O. Show that i(/) "', I .- I .,- e-t for 12: 0, and sketch thiswaveform,

4-15. In the network shown, the switch is closed at I = 0 connecting avoltage Source r(t) - V sin WI to a series RL circuit. For this system,solve for the response i(t).

Fig. P4-15.

4-16. Consider the differential equation

.u : . r ( )dl -;- at = J k t

where a is real and positive. Find the general solution of this equatio..if all J~ ~ 0 for I < 0 and for I 2. 0 have the following values:(a)!1 kIt

(b)J~' te=>(c) Ji sin Wol

(d) f~ c~ cos Wot

(e)!s = sin- i

(f) !6 cc cos- I

(fJ,)f~ " I sin '21

(h) J8= e-t sin 2t

115

116 First-Order Differential Equations I Ch. 4

4-17. In the network (If the figure, the switch K is open and the networkreaches a steady state. At I = 0, switch K is closed. Find the currentin the inductor for I :> 0, sketch this current, and identify the time

constant.

10 n10 n

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 K open.At I = 0, the switch is closed. Find the current in the capacitor forI > 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 currentwaveform is observed with a cathode ray oscilloscope. The initial valueof the current is measured to be 0.01 amp. The transient appears todisappear 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 resistorand a relay with inductance L. The relay is adjusted so that it isactuated when the current through the coil is 0.008 amp. The switchK is closed at 1 -~ 0, and it is observed that the relay is actuated whenI = 0.1 sec. Find: (a) the indu.:tance L of the coil, (b) the equation of

i(1) with all terms evaluated.

Ch..4 / Problems 117

~ 10,0000

100V~ ~

Fig. P4-21.

4-22. A switch is closed at ( = 0, connecting a battery of voltage V witha series RC circuit. (a) Determine the ratio of energy delivered to thecapacitor to the total energy supplied by the source as a function oftime. (b) Show that this ratio approaches 0.50 as 1 -, 00.

4-23. Consider the exponentially decreasing function i ~~Ke=u? where Tis 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'tialvaluepears to

ofC, and

of a resistorso that it is.Theswitchuatedwhenequation of

132 Initial Conditions in Networks / Ch. 5

PROBLEMS

5-1. In the network of the figure, the switch K is closed at t = 0 with thecapacitor 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 theinductor. 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 att = O. Solve for i, di/dt, and d+ildt? at t = 0+ if R = 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 thevalues of dvJ!dt and dVz/df at f = 0+.

5-5. For the network described in Prob. 4-7, determine values of dZvz/dtZ

and d3vz/dt3 at t = 0+.

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

Fig. P5-6.

ci. 5/ Proble

5-7. In thesolve r,and C

5-8. The ruSolveand L

5-9. In theswitchgiven,

5-10. In tHstate

h.5

theIt,

the

at

the

ateine

cs. 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+rl dt? if I ~" I 0 amp, R == lOOOn,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 is reached with theswitch K open. At t = 0, the switch is closed. For the element valuesgiven, determine the value of v.(O-) and v.(O+).

10 ~!

20 ~!10 I!

+5 V-=-

Fig. P5-9.

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

lOQ

lOH 20Qvb

+

Ton"-1 J"Fig. PS-lOo

133

1345 I ProblemsInitial Conditions in Networks / Ch..

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

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

( (0) for the corditions stated in Prob. 5-9.

5-12. In the network given in Fig. P5-1O, determine Vb(O+) and Vb(oo) fothe 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 V2 at t = 0+, (b) VI and V2 at t = 00, (c) dVI/dt and dV2/dt at = 0+, (d) d2V2/dt2 at t = 0+.

Fig. PS-l3.

!,~ the given mrh; switch KR2' I Mr!,t . , 0·; .

5-14. The network of Prob. 5-13 reaches a steady state with the switch Kclosed. At a new reference time, t = 0, the switch K is opened. Solvefor the quantities specified in the four parts of Prob. 5-13.

5-15. The switch K in the network of the figure is closed at t = 0 connectingthe battery to an unenergized network. (a) Determine i, dildt, andd2i/dt

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

S-19. In the circuiconnecting a(a) dil/cll and

+

5-20. In the netopen withand C Iintegrodifliclosed. (b)Fig. PS-IS.

5-16. The network of Prob. 5-15 reaches a steady state under the conditionsspecified in that problem. At a new reference time, t = 0, the switchK is Opencd. Solve for the quantities specified in Prob. 5-15 at t = 0+.

5-17. In the network shown in the accompanying figure, the switch K ischanged from a to b at I = 0 (a steady state having been establishedat position a). Show that at f = 0-1 ,

V

ks / Ch. 5

0-) and

: the con-

=0 with'or (a) t'l

dvz/dt at

switch K'led. Solve

onnectingdifdT, andt T = O-l .

:onditionsthe switchIII = 0+.witchK i,stablished

a: 5 / Problems

Fig. PS-17.

5-18. "~ the given network, the capacitor Cl is charged to voltage Vo andrh, switch K is c'osed at T ,,0. When RI ·2 Mn, Vo 1000y,Rz I Mn, c, 10J1F, and c, - 20 J1F, 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 ~. 0connecting a voltage, Vo sin WT, 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 Kopen with V . lOOY, RI" 10n, Rz ·20 n, RJ --= 20 n, L I H,and C I J1F. At time f 0, the switch is closed. (a) Write theintegrodifTerential equations for the network after the switch isclosed. (b) What is the voltage Vu across C before the switch is

Fi~. PS-20.

c_-L----T

13S

136 Initial Conditions ill Networks i Ch. 5

closed? What is its polarity? (c) Solve for the initial value of i, amii2Ct ~= 0+). (J) Solve for the values of di.ldt and di-f dt 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 att = 0+: (a) VI, (b) V2, (c) do-f dt, (d) dV2/dt.

Fig. PS-2I.

5-22. In the network shown in the figure, the switch K is closed at theinstant t = 0, connecting an unenergized system to a voltage source.Let M 12 = 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 if K is closed 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 netvI' 0, the swinV sin (I/./MC

/,.(0+) = 0,

5-25. In the networknetwork has atan expression fparameters arewhat is the valdVK/dt (O+)?

5-26. In the networkconnecting theage Va at t = O·

F

5-27. In the networkI = 0-, all capnode-to-datumand dc d dt at tV3 and dV3/dl at

· Ch. 5

it and~ 0+.

pairs.ties at

at thesource.

,0,

ci. 5 !Problems

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

V sin (If"'; MC). Show that

/,.(0+) = 0, (;;/(0+) = (V/L)"'; M/C, and (/2V·(O+) = 0df2

~ a

~K L 1-~L ++ Viii C v.

Fig. P5-24.

5-25. In the network of the figure, the switch K is opened at t = 0 after thenetwork has attained a steady state with the switch closed. (a) Findan expression for the voltage across the switch at f = 0+. (b) If theparameters 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 = 0connecting 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. Att e-c, 0 -, all capacitor voltages and inductor currents are zero. Threenode-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) FindV3 and dVl/df at t = 0-1·.

137

138 Initial Conditions ill Networks 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 att ~=0+. (b) Find dVK/dt at t = 0+.

Fig. PS-28.

5-29. In the network of the accompanying figure, a steady state is reachedwith 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 differentialeq uations of thewe will continuerestrictions as toThe mathematicunder the head inthe classical metdifferential equatconceptual advatransformation iswhich are ordinmore easily devebe reserved for t

6-1. SECOND·OEXCITATIO

A second-ostant coefficients

The solution ofthe solution itsel

Continued / Ch. 6

otherwise this. the derivative

(6-137)

(6-138)

ise is

(6-139)

ne appearance

: Rcn and theor the current:ten

(6-140)

idition of the

r)] (6-141 )

(6-142)

(6-143)

e is shown inng factor andI envelope orermines howes zero, theIS result..ult may be1the electricrage elementtored in therergy, When) the electrics as long ase oscillatory

Ch. 6 / Problems163

current will be sustained indefinitely. However, if there is resistancepresent, the current through the resistor will cause energy to be dissi-pated, and the total energy will decrease with each cycle. Eventuallyall the energy will be dissipated and the current will be reduced to zero.If a scheme can be devised to supply the energy that is lost in eachcycle, the oscillations can be sustained. This is accomplished in theelectronic oscillator to produce audio frequency or radio frequencysinusoidal signals.

FURTHER READING

BALABANIAN,NORMAN,Fundamentals of Circuit Theory, Allyn and Bacon,Inc., Boston, 1961. Chapter 3.

CHIRLlAN,PAUL~t, Basic Network Theory, McGraw-Hill Book Company,New York, 1969. Chapter 4.

CLEMENT,PRESTONR., AND WALTERC. JOHNSON,Electrical EngineeringScience, McGraw-Hill Book Company, New York, 1960. Chapter 7.

CLOSE,CHARLESM., The Analysis of Linear Circuits, Harcourt, Brace &World, Inc., New York, 1966. Chapter 4.

HUELSMAN,LAWRENCEP., Basic Circuit Theory with Digital Computations,Prentice-Hall, Inc., Englewood Cliffs, N.J., 1972. Chapter 6.

SKILLlNG,HVGH H., Electrical Engineering Circuits, 2nd ed., John Wiley& Sons, Inc., New York, 1965. Chapter 2.

WYLlE,CLARENCER., JR., Advanced Engineering Mathematics, 3rd ed.,McGraw-Hill Book Company, New York, 1966. Chapters 2,3, and 5.


References that are useful in designing exercises to go with the topicsof this chapter are cited in Appendix £-6.3 and are concerned with thenumerical solution of higher-order differential equations. In particular, thesuggestions contained in Chapters 5, 6, and 7 of Huelsman, reference 7,Appendix E-10, are recommended.

PROBLEMS

6-1. Show that i = ke=> and i = ke= are solutions of the differentialequation

d2i + 3 di + 2' = 0

dt» dt I

164Differential Equations, Continued / Ch,

6-2. Show that i = ke= and i = kte= an: solutions of the differentiequation

Ch. 6 / Problems

d2i + 2 di -I- . = 0

dt? dt' I

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

dZ' d' dZ' d'(a) -.-! + 3 -.: + 2i = 0 (e) --.! -i- _l.. -~ 6i = 0dt2 dt df2 dt'

(b) d2i + 5 di + 6' - 0dtZ dt I -

(c) ;t21 + 7:: + 12; = 0

(d) d2i + 5 di + 4' = 0df2 dt I

d2' di(0 df~ -T' d: +- 2i = 0

( ) dl; -L 2 di .: '= 0g d{2' dt r I

(h) d2i -L 4 di -"4' -- 0.d{2' dt r 1-

6-4. Find the general solution of each of the following homogeneodifferential 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 d 21) 8 d,; , 16. - 0dt Z + 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.dt 2 dt v -

i(O+) = I, didt (0+) = 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 .; Idt

Find particular solutions to the differential equations of Prob. 6-4subject to the initial conditions:

r(O+) 0= I,

6-8.

~~ (0 +-) ,-- I

Find particular solutions to the differential equations given in Prm6-4, given the initial conditions:

('(0.+) ~, 2, dO(o'_r) = Iclt6-9. Solve the differential equation

iJ' d" di3 ~ I- 8 --=: -j- 10 -.: ., 3i = 0.dt J dt ? dt

6-10. Solve the differential equation

dJ; d~; 13 d; __ 6,' == 0.2c1tJ -;- 9dt2 -:- dr

subject to the initial conditionsd+ildt? = --I at t = 0.+.

6-11. The response of a network is fr

i= Kite:'

6-12.

where (J., is real and positive. imaximum value.

In a certain network, it is foundsion

Show that i(t) reaches a maxi

1t =--1X1-

6-13. The graph shows a damped siform

Ke-at si

From the graph, determine n

Fig.

6-14. Repeat Prob. 6-13 for the wa

'"a.E'"

Fig.

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

IS given in Prob.

Ch.6 I Problems 165

subject to the initial conditions i(O+) = 0, dildt = 1 vt t = 0+, andd2i/dt2 = ·-1at t = 0+.

6-11.The response of a network is found to be

f::::: 0

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

6-12.In a certain network, it is found 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 generalform

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

-1o 2 3 4 5t, msec

Fig. P6-14.

166Differential Equations,

6-15. In the network of the figure, the switch K is closed andis reached in the network. At f = 0, the switch is 0

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

~-=- 100 v

Fig. P6-15.

6·16. The capacitor of the figure has an initial voltage vc(o-)at the same time the current in the induct or is zero.Atswitch K is closed. Determine an expression for the vel

Fig. P6-16.

6-17. The voltage SOurce in the network of the figure is descriequation, VI = 2 cos 2t fer t ~ 0 and is a short circuitp'time. Determine V2(t). Repeat if '1.·1 = KIt for t ~ 0 and st < O.

Fig. P6-17.

6-18. Solve the following nonhomogeneous differential equationsI( ) d2i + 2 di + i = 1a dt2 dt

(b) g:.! + 3 di + 2i = Stdt? dt

(c) ;t2

; + 3:: + 2i = 10 sin lOt

(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 dxand dr(O+) = -1

where x is the general dependent variable.

is closed and a steady staswitch is opened. Find u'2(t).

P6-15.

'oIt~ge vC<O-) = VI> andtor IS zero. At t = 0, then 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:

dxx(O+) = 2 and dt(O+) = -1

wherex is the dependent variable in each case.

~ll.Solvethe differential equationdJ' 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. A special generator has a voltage variation given by the equation

t,l) 1V, where t is the time in seconds and 1~ O. This generator isconnected to an RL series circuit, where R = 2 nand L = I H, aturne1 = 0 by the closing of a switch, Find the equation for the current

as a function of time i(t).

6-13. A bolt of lightning having a waveform which is approximated as1'(1) = te-r strikes a transmission line having resistance R = 0.1 nand inductance L = 0,1 H (the line-to-line capacitance is assumednegligible). An equivalent network is shown in the accompanyingdiagram. What is the form of the current as a function of time?(Thiscurrent will be in amperes per unit volt of the lightning; likewise

the time base is normalized.)6-24.In the network 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. In the network shown in the accompanying figure, a steady state isreached with the switch K open. At r = 0, the switch is closed. Forthe element values given, determine the current, i(t) for 1 ~ 0,

R-103 (l

r::\,5IlFilt))

Fig. P6-2S.

6-26. In the network shown in Fig. P6-2S, a steady state is reached with theswitch K open. At t = 0, the value of the x resistor R is changed tothe critical value, Ra defined by Eq, (6-88). For the element valuesgiven, determine the current i(t) for 1 2 O.

168 Differential Equations, Continued I Ch. 6

6-27. Consider the network shown in Fig. P6-24. The capacitor has aninitial voltage, Vc = 10 V. At I = O. the switch K is 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 is closed. Find an expression forthe Voltage, v(l) for t :2: O.

c u( t)

+

10 sin wt tK

Fig. P6-28.

6-29. Consider a series RLC network which is excited by a voltage source.(a) Determine the characteristic equation corresponding to the differ-ential equation for i(t). (b) Suppose that Land C are fixed in valuebut that R varies from 0 to 00. What will be the locus of the roots ofthe characteristic equation? (c) Plot the roots of the characteristicequation in the s plane if L = 1 H, C = 1 J.l.F, and R has the followingvalues: 500 n. 1000 n, 3000 n, 5000 n.

6-30. Consider the RLC network of Prob. 6-16. Repeat Prob. 6-29, exceptthat in this case the study will concern the characteristic equationcorresponding to the differential equation for V2(t). Compare resultswith 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 asa function of Kt. Find the value(s) of Kt for which the roots of thecharacteristic equation are on the imaginary axis of the s plane. Findthe range of values of Kt for which the roots of the characteristicequation 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 I Problem!

6-33. A switchseries RIof time i:

w

(b) Findtion of tsteady-sias 1-"

in the st

6-34. In the sfrequent

(1) CO=

(2) CO =These fexperimwhen thsteady-sthe rna:is, whicgreater'

· Continued / Ch. 6

:apacitor has anosed. Determine

y state with then expression for

-0

+

)(t)

voltage SOurce.ig to the differ-fixed in value

of the roots ofcharacteristic

) the following

). 6-29, exceptistic equationrnpare results

.is, and deter-e network as: roots of thes plane. Findcharactensnc

~q. (6-121).

o. 6/ Problems 169

6-33.A switch is closed at t = 0 connecting a battery of voltage V with aseriesRL circuit. (a) Show that the energy in the resistor as a functionof time is

V2( 2L R'L L 2R'L 3L). IWR = R t + R c: t - 2Re- t, - 2R JOU es

(b) Find an 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 thesteady-state asymptotes, that is, the values that WR and WL approachas I -4 eo. (d) Find the total energy supplied by the voltage sourcein the steady state.

6-34.In the series RLC circuit shown in the accompanying diagram, thefrequency of 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 eachexperiment we measure (a) the peak value of the transient currentwhen the switch is closed at I = 0, and (b) the maximum value of thesteady-state current. (a) In which case (that is, which frequency) isthe maximum value of the transient greater? (b) In which case (thatis, which frequency) is the maximum value of the steady-state currentgreater?

Fig. P6-34.

~100 sin wt ~ lJ1F!It,)

i Network Theorems / CIr.9

(b)

: 6 for which the

rolled source which

(9-94)

find the impedanceng a voltage sourcerrent I(s) under thezero, meaning that

'k(S) (9-95)

equired impedance

(9-96)

(9-97)

(9-98)

rk is constructed

seful artifice thathe operation ofIg the amount ofrnplish this.

". 9/ Problems 271

FURTHER READING

CHoo.IAN,PAULM., Basic Network Theory, McGraw-Hill Book Company,New York, 1969. Chapter 5.

DfsoER, CHARLESA., AND ERNESTS. KUH, Basic Circuit Theory, McGraw-Hill Book Company, New York, 1969. Chapters 16 and 17.

Kuo, FRANKLlNF., Network Analysis and Synthesis, 2nd ed., John Wiley& Sons, Inc., New York, 1966. Chapter 7.


The topics of this chapter are not directly related to the use of thedigitalcomputer, since new concepts and theorems are stressed. Use thetimeavailable for computer exercises in completing more of those suggestedat the end of Chapter 3.

PROBLEMS

9-1. In the network of (a) of the accompanying figure, '1:1 = Voe-Zt

cos t u(t), and for the network of (b), i, = 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 + 1Z(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.

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 L for (a) to hold. (c) Discussthe 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 steadystate having previously been established. With the switch open, drawthe transform network for analysis on the loop basis, representing allelements 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 area consequence of active elements removed at t = O.

Fig. P9-S.

9-6. In the network of the figure, the switch K is closed at t = 0 and att = 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 inthe accompanying figure.

z~g'1~\ I

Fig. P9-7.

Ch. 9 / Problems

9-8. For the RCance, Z(s), ip(s) andq(sof Prob. 9-1

9-9. Repeat Pro

9-10. Repeat Prfigure.

9-11. Repeat Pthis case

9-12. Two blacknown thcontainsthe input(b) Invesnetwork.conditio!

or- r :

IIII

1I

IIII1

0>---+-1-L.

9-13. Repeatpanying

5Slepian,6Macklel

September, 191

seorems I CIr. 9

all t> Q. (b)d. (c) Discusscomplex fre-

= 0, a steady~ open, drawiresenting all

nsform net-loop basis,

/oltages are

~ 0 and atpeat Prob.

shown in

a. 9/ Problems

f.I. For the RC network shown in the figure, find the transform imped-ance, Z(s), in the form of a quotient of polynomials, p(s)/q(s). Factorpes) and q(s) so that Z(s) may be written in the form of the impedanceofProb.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 accompanyingfigure,

Fig. P9-10.

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

9-12.Two black boxes with two terminals each are externally identical. It isknown that one box contains the network shown as (a) and the othercontains the network shown as (b) with R = ..;L/e. (a) Show thatthe input impedance, Zin(S) = Vin(s)/Iin(s) = R for both networks.'(b) Investigate the possibility of distinguishing the purely resistivenetwork. Any external measurements may be made, initial and finalconditions may be examined, etc.

r-----------, .------------,II

L :

:R- fTI VCI

CRIIII

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.

274 Impedance Functions and Network Theorems / Ch. 9

r----------,I

Fig. P9·13.

R

R= fF., c

R c

9·14. The network shown in Fig. P9-4 is operated with switch K closeduntil a steady-state condition is reached. Then at t = 0 the switch Kis opened. Starting with the transform network found in Prob. 9-4,determine the voltage across the switch, Vk(t), for t :2: O.

9·15. If the capacitors are uncharged and the inductor current zero att = 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)

IHIF 10

Fig. P9·1S.

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

I s _ s(s + 2X5s + 6)l( ) - (S2 + 4s + 13XlOs2 + 18s + 4)

1 n

Fig P9·16.

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

VVs = --t (1 + a + b - ab)

and3-bz, =-2-

Ch. 9 / Problems

9·18. The accomsources innetwork, fiexpression

9·19. Th1netwocurrent sodetermine

9·20. The nethis netwRL•

Theorems / cs. 9

switch K closed- 0 the switch Kd in Prob. 9-4,

~ O.

current zero atorm of the gen-

It the generator

rlent Thevenin

275

1 n

Fig. P9-17.

9-18.The accompanying network consists of resistors and controlledsources in addition to the independent voltage source v,. For thisnetwork, find the Thevenin equivalent network by determining anexpression for the voltage V8 and the Thevenin equivalent resistance.

fig. P9-1S.

9-19.ThJnetwork of the figure contains three resistors and one controlledcurfent 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. Forthis network, determine the Thevenin equivalent network for the loadRL•

Fig. P9·20.

276 Impedance Functions and Network Theorems I Ch. 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 RL inseries 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 equivalentnetwork.

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

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

9-25. Determine the Norton equivalent network for the system described inProb.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 thatcondition, determine the transform of the voltage across the 0.5-Fcapacitor 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 steadystate having previously existed. Find the current in the resistor R3using (a) Thevenin's theorem, and (b) Norton's theorem.

10 n

Fig.P9-17.

9-30. Usingalentditions.

9-31. Thevaluesdeteequiva

eorems / cs. 9 Ch.9 I Problems 277

lcitor in addi-sting of RL in~rk.

J.28.Thenetwork shown in the figure is a low-pass filter. The input voltageVI(t) is a unit step function, and the input and load resistors have thevalue R = ...;LIe. By using Thevenin's theorem, show that the trans-form of the output voltage is

n equivalent

+

R ~Itl

Fig. P9-28.n equivalent

>rk given in9·29.In the network shown in the accompanying sketch, the elements are

chosen such that L = eRr and RI = Rz. If v\(t) is a voltage pulse ofI-V amplitude and T-sec duration, show that vz(t) is also a pulse, andfind its amplitude and time duration.

described in

JaOY state is,Under that

the O.S-Fs 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 elementvalues given, with v\(t) = u(t), and for zero initial conditions: (a)determine the equivalent Thevenin network at a-a', (b) Determine theequivalent Thevenin network at bob'.

Impedanc« Functions and Network Theorems I Ch. 9

Fig. P9-3I.

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

Fig. P9-31.

9-33. Assuming zero initial voltage on the capacitor, determine 1he equiv-alent Norton network for the resistor Rx.

+ -

Fig. P9-33.

In this charadmittanceextended. Fdifferent parmathematic,functions arl

10·1. TERMI]

Considelements. T(represented Ifastened to aaccess, the enare requirednecting somements. The IT

the terminal!another pairname terminc

ITerminalThis results in:this chapter.

tnd Zeros / Ch. 10

i). The stability

aial or an odde even polyno-S" + ja)(s - ja)other possibiI-lay be reachedD-31which are

;b)(10-120)

) if b > a. Inon applies for

.rr zeros, sym-vith respect toirm 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) givesthe factor 2S2 + s + 1 which may be analyzed by the quadraticformula.

FURTHER READING

DESOER, CHARLESA., AND ERNESTS. KUH, Basic Circuit Theory, McGraw-Hill Book Company, New York, 1969. Chapter 15.

KARNI,SHLOMO,Intermediate Network Analysis, AlIyn and Bacon, Inc.,Boston, 1971. Chapter 6.

LATHI,B.P., Signals, Systems, and Communication, John Wiley & Sons,Inc., New York, 1965. Chapter 7.

MELSA, JAMESL., AND DONALD G. SCHULTZ, Linear Control Systems,McGraw-Hill Book Company, New York, 1969. Chapter 6.

PERKINS,WILLIAMR., ANDJoss B. CRUZ, JR., Engineering of Dynamic Sys-tems, John Wiley & Sons, Inc., New York, 1969. Chapter 8.


Two topics of this chapter which lend themselves to computer solu-tionare the determination of the roots of a polynomial and the determina-tion of the locus of roots. The sections of Appendix E devoted to thesetopicsare E-l and E-9.5. In particular, see Huelsman, reference 7, AppendixE-IO, and his discussions of root-locus plots in Section 10.3, and Me-Cracken, reference 12, Case Studies 21 and 23.

PROBLEMS

10-1. For the network shown in the accompanying figure, determineZ12 = V2(s)jII(s).

Fig_ PlO-I.

10-2. Consider the RC two-port network shown in the accompanyingfigure. For this network show that

G - r S2 + (R1C1 + R2C2)SjRIR2CIC2 + 1jR1R2C1C2 ]12 - l..$2+ (R1C1 + R1C2 + R2C2)SjRtR2CIC2 + 1jR1R2C1C2

318 Network Functions; Poles and Zeros I Ch. 10

Fig. PlO-I.

10-3. (a) For the given network, show that with port 2 open, the inputimpedance at port 1 is 1 n. (b) Find the voltage-ratio transfer func-tion, G12 for the two-port network.

1+ ~------r---------~2+10

2F

10

10

~------------------------~~--------~2Fig. PI0-3.

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

U!

Fig. PI0-4.1 n

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

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

Fig. PIO-5.

I~ 1n 2V;v,u::Jln

Fig. PI0-6.

a.. 10 I Problems

10-7. For the net'specified, dr

10-8. Fur the RI

10-'). For the g

and dete

10-10. For thetransfer

10-11. Foreala voltaV. at I

~eros/ cs. 10

\' the inputlnsfer func-

02~

2

ine the

re is to1%12.

s. For

u: 10 I Problems

10-7. For the network of the accompanying figure and the element valuesspecified, determine IX 12 = 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-ratiotransfer function is

(S2 + 1)2G 12 = 5s4 + 5s2 +

I H I H

+ +

~T_l _ ______lll~2Fig. r-io-io.

10-11. For each of the networks shown in the accompanying figure, connecta voltage source VI to port I and designate polarity references forV2 at port 2. For each network, determine G 12 = V2/ VI'

319

320 Network Functions; Poles and Zeros / Ch. 10

1 n 2

if 2

(g)

Fig. PlO-H.

10-12. For the network given in Fig. PlO-ll(a), terminate port 2 in a I-Qresistor and connect a voltage source at port I. Let 11 be the currentin the voltage source and 12 be the current in the I-n load. Assignreference 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 atport I and a I-n resistor at port 2. Assign reference directions forall voltages and currents. For this network, compute Z12 = V21I1and 0(12 = 121/1,

10-16. The network shown in (a) of the figure is known as a shunt peakingnetwork. Show that the impedance has the form

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

and determine ZI, p i, and P2 in terms of R, L, and C. If the poles andzeros 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 awhich may be asystem to a stepof K, as a funciidone by the

.l0-18.

10-19. A system hass = -3, andOne term ofK3e-r sin (t +of a between

id Zeros / Ch. 10

rrt 2 in a I-Qre the currentload. Assign

npute G12 =

source I( atlirections forZ12 = V2/1(

hunt peaking

he poles and'e with ZUO)

Ch. JOI Problems 321

I

IIIII*-- --

(J

s plane jw

JTITrr--2 I 1.5

cZlsI -3vrrr

2L

(c) Ibl

Fig. PIO-16.

10·17.A system has a transfer function with a pole at s = - 3 and a zerowhich may be adjusted in position at s = -a_The response of thissystem to a step input has a term of the form K,e:», Plot the valueof K( as a function of a for values of a between 0 and 5. This may bedone by the graphical procedure of Section 10-7.

10-18.A system has a transfer function with poles at s = -1 ± j 1 and azero which may be adjusted in position at s = -a. The response ofthis 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 a for values of a between 0 and5. This may be done graphically.

10·19.A system has a transfer function with poles at s = -1 ± j 1and ats = - 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 formK3e-r sin Ct + rjJ). Plot the value of K3 as a function of a for valuesof 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 anddetermine: (a) the number of roots with positive real parts, (b) thenumber of roots with zero real parts, and (c) the number of rootswith negative real parts.(a) 4s3 + 7s2 + 7s + 2 = 0(b) S3 + 3s2 + 4s + 1 = 0Cc) 5s3 + S2 + 6s + 2 = 0(d) SS + 2S4 + 2s3 + 4S2 + l l s + 10 = 0


10-21. Given the equationS3 + 5s2 + Ks + 1 = 0

(a) For what range of values of K will the roots of the equation havenegative real parts? (b) Determine the value of K such that the realpart vanishes.

10-22. Repeat Prob. 10-20 for the equations:(a) 5s4 + 6s3 + 4S2 + 2s + 3 = 0(b) S4 + 3s3 + 2S2 + s + 1 = 0(c) 2S4 + 3s3 + 6s2 + 7s + 2 = 0(d) 3s6 + S5 + 19s4 + 6s3 + 81s2 + 25s + 25 = 0

10-23. Repeat the tests of Prob. 10-20 for the following equations:(a) 720s5 + 144s4 + 214s3 + 38s2 + 10s + 1 = 0(b) 25s5 + 105s4 + 120s3 + 120s2 + 20s + 1 = 0(c) S5 + 5.5s4 + 14.5s3 + 8s2 - 19s - 10 = 0(d) S5 - S4 - 2s3 + 2S2 - 8s + 8 = 0(e) S6 + 1 = 0

10-24. For the following polynomials, (I) determine the number of zeros inthe right half of the s plane, (2) determine the number of zeros onthe imaginary axis of the s plane. Show method.(a) 2s6 + 2s5 + 3s4 + 2s3 + 4S2 + 3s + 2 = PI(S)(b) S6 + 2s5 + 6s4 + 1Os3 + l1s2 + 12s + 6 = P2(S)(c) 2s6 + 2s5 + 4S4 + 3s3 + 5s2 + 4s + 1 = P3(S)

10-25. For the following polynomial, determine the number of zeros in theright half of the s plane, the left half of the s plane, and on theimaginary axis (the boundary) of the s plane:(a) PI(S) = 2s7 + 2s6 + 15s5 + 17s4 + 44s3 + 36s2 + 24s + 9(b) S6 + 3s5 + 4S4 + 6s3 + 13s2 + 27s + 18 = P2(S)(c) S8 + 3s7 + 5s6 + 9s5 + 17s4 + 33s3 + 31s2 + 27s + 18

=P3(S).

10-26. Consider the equation

aos4 + als3 + a2s2 + a3s + a4 = 0Use the Routh-Hurwitz criterion to determine a set of conditionsnecessary in order that all roots of the equation have negative realparts. Assume that all coefficients in the equation are positive.

10-27. For the network of the figure, let RI = R2 = 1 n, Cl = 1 F andC2 = 2 F. For what values of k will the network be stable? In otherwords, for what values of k will the roots of the characteristic equa-tion have real parts in the left half of the s plane?

Fig. PI0-27.

10-28. For theDete .the systequation

10-29. The amanalyzed.for theistic eqwithoutand K?amplifier

10-30. The neoscil/atgmRL~= 29 is

tnd Zeros / cs. 10

e equation haveeh that the real

itions-

er of zeros in. of zeros on

zeros in theand on the

24s + 9

+ 18

:onditionstative realtive,

·1 F and'In other:tic equa-

Cl. 10I Problems 323

11-28. For the network of Prob. 10-27, let k = 2, Cl = 1 F and Rz = 1 n.Determine the relationship that must exist between RI and Cz forthe system to oscillate, that is, for the roots of the characteristicequation to be conjugate and have zero real parts.

10-29. The amplifier-network shown in the accompanying figure is to beanalyzed. (a) What must be the relationship between RI, Rz, and Kfor the system to be stable (real parts of the roots of the character-istic equation are zero or negative)? (b) For the system to oscillatewithout damping, what must be the relationship between RI> Rz,and K? What will be the frequency of oscillation? Assume that theamplifier has infinite input impedance and zero output impedance.

cJ+

Amplifiergain -K V2

Fig. PIO-29.

10-30. The network of the accompanying figure represents a phase-shiftoscillator. (a) Show that the condition necessary for oscillation isgmRL ~ 29. (b) Show that the frequency of oscillation when gmRL= 29 is Wo = 1/../6 RC.

Fig. PIO-30.

Vg

+

10·31. Show that with Z.Zb = Rij in the bridged-T network of the accom-panying figure,

Vz 1VI = 1 + Z./Ro

and the input impedance at port 1 is Zin = Ro.

Fig. PIO-31.

10-32. An active network is described by the characteristic equation

S2 + (3 + 6KI)s +- 6K2 = 0

We next turn 0

is useful in deto describe antions are likeIrestrictions imopen or be sho


It is required that the network be stable and that no component ofits response decay more rapidly than Kte-31• Show that these condi-tions are satisfied if K2 > 0, I Kt I < i, and K2 > 3Kt. Crosshatchthe area of permitted values of Kt and K2 in the Kt-K2 plane.

a5 \b5

10-33. Values for the elements of the Routh array can also be expressed interms of second-order determinants multiplied by - 1. Thus the for-mulas shown in Fig. 10-30 become

Using the indexing scheme suggested on page 312, give a generalformula for the elements of the Routh array.

In the Iidentified+twand currentsbox enclosingvoltages andimportant inatransformqand V2 and 11the four variof them delespecified, thenfour variabldepending onvariables. Inin Table 11-1.

342 Two-Port Parameters ( Ch. I


In connection with the matrix multiplication of the ABeD parametermatrices for networks connected in cascade, see the exercises in referencescited in Appendix E-3.1. The determination of the other parameters involvesordinary network analysis with the special condition that the one pair 01network terminals be either open or shorted. These topics are considered inreferences cited in Appendix E-8.

PROBLEMS

1~2

In the problems to follow, all element values are in ohms, farads, orhenrys.

11-1. Find the y and z parameters for the two simple networks shown inthe figure if they exist.

11-2. For the two networks shown in the figure, find the z and y param-eter's if they exist.

11-3. Find the y and z parameters for the resistive network of the accom-panying figure.1'0>__------02'

la)

1~2

1'~2'(bl

Fig. PIl-l.

~ I1

1"0 --' -'-_~ 2' Fig. Pll·3.

11-4. The network of the figure contains a current-controlled currentsource. For this network, find the y and z parameters.

(a)Fig. Pll·4.

+

(b)

Fig. Pll-2.

l:n

[Ideal

11-5. Find the y and z parameters for the resistive network containing acontrolled source as shown in the accompanying figure.

+

Fig. Pll·S.

11-8. Theanddet

11-9. Find

11-10. The

343

SESThe accompanying figure shows a resistive network containing asinglecontrolled source. For this network, find the y and z param-eters.

e ABeD parameterercises in referel1Celparameters involveIthat the one pair 01cs are considered iD

containing a

:,~ ~

2Q IQ

Fig. Pl1-6.

ohms, farads, or

11·7. Thenetwork of the figure contains both a dependent current sourceanda dependent voltage source. For the element values given, deter-minethe y and z parameters.

k of the accom-

etworks shown in

e z and y param-

Fig. PH-7.

11·8. The accompanying network contains a voltage-controlled sourceand a current-controlled source. For the element values specified,determine the y and z parameters.

rolled currentf--,-----jf---,-----o2

).

+ +

1 \12\l \.j

1'0-----'-------'---02'

+

Fig. PH·S. Fig. PH-9.

11·9. Find the y and z parameters for the RC ladder network of the figure.

11·10. The network of the figure is a bridged- T RC network. For the valuesgiven, find the y and z parameters.

Il-Il, Determine the ABCD (transmission) parameters for the network ofProb. 11-10.

~F

~~1~02

1 F

1'0 T 2 02'

Fig. PH-lOo

n·12. The accompanying figure shows a network with passive elementsand two ideal transformers having I:I turns-ratios. For the elementvalues specified, determine the z parameters.

344 Two-Port Parameters / Ch. 11

1'0-------'-------" '------02'

r-----<> 2

f\fv--+~\.f\fv~-~2Fig. PH-12.

1'0-------'-------02'11-13. The network of the figure represents a certain transistor over a given

range of frequencies, For this network, determine (a) the h param-eters, and (b) the g parameters. Check your results using Table 11-2.

11-14. The network of the figure represents the transistor of Prob. 11-13over a different range of frequencies. For this network, determine (a)the h parameters, and (b) the g parameters.

-11

o--'----AJV''v--+--il-----'---<>211-15. Show that the standard T section representation of a two-port net-work may be expressed in terms of the h parameters by the equationsshown in the accompanying figure.

Fig. PH-13.

1'0-----'-------02'

Fig. Pll·14.

1'0------'----------<>2'

Fig. PH-IS.

11·16. The network of the figure may be considered as a two-port networkembedded in another resistive network. The resistive network is

1/2f!

2+ +

la 2f! V; N ~ If!

J' 2'

Fig. Pll-16.

Parameters / c« 11

02'

stor over a given(a) the h param-ising Table 11-2.

. of Prob. 11-13'k, determine (a)

a two-port net-'y the equations

-port networke network is

1n

Ch.lJ / Problems 345

described by the following short-circuit admittances: YII = Y22 =2 U, Y21 = 2 U, and Y12 = 1 U. If la is a constant equal to 1 amp,find the voltages and the two ports of the network N, VI and V2•

11·17.The network shown in the figure consists of a resistive T-and a resis-tive It-network connected in parallel. For the element values given,determine the Y parameters.

1/2 I! 1 n

22n

10--

2

Fig. PH-17.

11·18.The resistive network shown in the figure is to be analyzed to deter-mine the Y parameters.

In 2n In

I'o------~L...---'------'-.--'-- .<J2'

Fig. PH-IS.

11-19.The accompanying figure shows two two-port networks connectedin parallel. One two-port contains only a gyrator, and the other is aresistive network containing a single controlled source. For this net-work, determine the Y parameters.

K

+

Fig. PH-19.

346 Two-Port Parameters / Ch. 11

11-20. In the network of Fig. 11-16, let Z. = s/2, Zb = 2/s, and Ro = I.For these specific element values, determine the y parameters.

11-21. The network of the figure is of the type used for the so-called "notchfilter." For the element values that are given, determine the y param-eters.

2F 2F

IIIIF

1'0---------'.--...L.-------o2'

Fig. Pll-21.

11-22. Let the element values for the network shown in Fig. 11-15 be asfollows: Cl = C2 = 1 F, RI = 1 Q, R. = Rb = 2 Q, C. = t F.Using these values, determine the y parameters.

11-23. The figure shows two networks as (a) and (b). It is asserted that one isthe equivalent of the other. Is this assertion correct? Show reasoning.If it is, might one network have an advantage over the other as far asthe calculation of network parameters is concerned?

II 12-- --+ +

~C3 V;

Cl C2

lal

II C3 12

: ~R' E It fk~ IR' +~Ib,

Fig. Pll-23.

11-24. Two two-port networks are said to be equivalent if they haveidentical y or z parameters (or other of the characterizing param-eters). In this problem, we wish to study the conditions under

eters / cs. J J

and Ra = 1.meters.

called "notchthey param-

1-15 be a5c, = i-F.

that one iseasoning.

er as far as

r----o+

--0

, haveX1ram-under

347

which the z-network of (c), is equivalent to the T-network of Cb).Show that the two networks are equivalent if

Z2 Z3 d Y =ZtYa = D' Yb = 75' an C Dwhere

1:!L:2r 2'

(b)

f---,.----o2

1"C>---'-----'---o2'

(a)

Fig. Pll-24.

11·25. Derive equations similar to those given in Prob. 11-24 expressingZI> Z2, and Z3 in terms of Yr, Yb, and Ye' This result and that givenin Prob. 11-24 are used in obtaining a T-7t transformation.

11·26. Apply the T-7t transformation of Prob. 11-24 or 11-25 to the networkof the figure to obtain an equivalent (a) T-network, (b) zr-network.

1 !! 211 1 F

l~y~2I2F

l'o~----11·27. Apply the T-n transformation to obtain an equivalent (a) T-networkand (b) zr-network for the capacitive network given in the figure.

11·28. Apply the T-7t transformation as many times as is necessary to theinductive ladder network shown in the accompanying figure in orderto determine the numerical values for the equivalent (a) T-network,(b) z-network.

Fig. Pll-26. 1'0>------L-----_02'

02'

Fig, Pll-27.

Fig. Pll·28.

1 H 1 H 2 H13JI2~1'0 2'

11·29. The network given in the figure is known as a lattice network; thislattice is symmetrical in the sense that two arms of the lattice haveimpedance Z, and two have impedance Zb' For this network, (a)determine the z parameters, and (b) express Z; and Z; in terms of zparameters. Fig. Pll-29.

348 Two-Port Parameters / Ch. If

ll-30. In this problem, we consider two-port networks having a sym-metry property illustrated in (a) of the figure: If the network isdivided at the dashed line, the two half networks have mirror sym-metry with respect to the dashed line. The two half networks areconnected by any number of wires as shown, and we will consideronly the cases in which these wires do not cross. If a network meetingthese specifications is bisected at the dashed line, then with the con-necting wires open, the input impedance at either port is Z 1/20c asshown in (b). Similarly, with the connecting wires shorted, the imped-ance at either port is ZI/2,c as shown in (c). A theorem due to Bart-lett states that these impedances are related to those given for thearms of the lattice in Prob. 11-29 by the equations

This is known as Bartlett's bisection theorem, and permits an equiva-lent lattice network to be found for any symmetrical network. Provethe theorem.

11 I12-

I r---L--l I----

~, ~,~N !N

I

(a) 11-35. Thz

:,0,-1 ~----~:,,,-1 r----~~N ~N

(b) (c)

Fig. Pll-30.

11-31. Apply the theorem of Prob. 11-30 to the network given in Prob.10-2 with the terminating resistor at port 2 removed, and so obtaina lattice equivalent network.

11-32. Apply the theorem of Prob. 11-30 to the network of Prob. 10-31with the terminating resistor Ro removed to find the lattice equiva-lent of the given network.

11-33. (a) Show that the network of the accompanying figure satisfies therequirements described in Prob. 11-30. (b) Find the lattice equivalentof the network.

11-36. Thha

11-37. TZL

tters / Ch. J I349

'ng a sym, Clnetwork isnirror syrn-:tworks areill considerIrk meetingh the con-

~ ZI/20c asheimped-

he to Bart-en for the

n equiva-rk. Prove

12

~+V2

r--o

----

Prob.obtain

10-31uiva-

es thevalent

0--L-fV\/v--,--JV\/V __~2

Fig.Pll-33. 1'0------------'--- --<: 2'

U·34, Find the lattice equivalent of the network of the accompanyingfiguremaking use of the results of Prob. 11-30.

~~~/\/\~-+-J\AJ\'~~2

Fig. Pll-34. 1'0-- _____T_3_

F

02,

JI·JS, The network N in the accompanying figure may be described by thez parameters. Show that with port 2 open,

Fig. Pll-3S.

JI·36. The network N in the figure is terminated at port 2 with a networkhaving impedance ZL = Jf YL. Show that

Fig. PII-36.

1l·37. The network N of the figure is terminated at port 2 in impedanceZL = J/ r L· Show that the transfer impedance for the combination is

Z - Z21ZL12 - Z22 +ZL

350 Two-Port Parameters / Ch. //

Fig. Pll-37.

11-38. The figure shows two two-port networks connected in cascade. Thetwo networks are distinguished by the subscripts a and b. Show thatthe combined network may be described by the equations

and

Y_ -Y12aY12b

12 -Y11b + Y22a

for the transfer functions.

Fig. Pll-3S.

Stated infunctionwhen s=

Ingeneratedswingingthese dev'voltage is

366 Sinusoidal Steady-State Analysis I Ch. 12

FURTHER READING

BALABANIAN,NORMAN, Fundamentals of Circuit Theory, Allyn and Bacon,Inc., Boston, 1961. Chapter 4.

CHIRLIAN, PAUL M., Basic Network Theory, McGraw-Hill Book Company,New York, 1969. Chapter 6.

CLOSE, CHARLESM., The Analysis nf Linear Circuits, Harcourt Brace Jovan-ovich, Inc., New York, 1966. Chapter 5.

HUANG, THOMASS., AND RONALD R. PARKER, Network Theory: An Intro-ductory Course, Addison-Wesley Publishing Co., Inc., Reading, Mass.,1971. Chapter 10.

LEO"'",.BENJAMINJ., AND PAUL A. WINTZ, Basic Linear Networks for Elec-trical and Electronics Engineers, Halt, Rinehart & Winston, Inc., NewYork, 1970. Chapter 4. •

MANNING, LAURENCEA., Electrical Circuits, McGraw-Hill Book Company,New York, 1965. Chapter 6.

WING, OMAR, Circuit Theory with Computer Methods, Holt, Rinehart &Winston, Inc., New York, 1972. See Chapter 7.


This chapter is devoted to a discussion of networks operating in thesinusoidal steady state. Analysis of large systems in this condition is straight-forward but tedious if done with pencil and paper, and the computer canbe used to advantage. See the references cited in Appendix E-8.3 for sug-gested exercises. In particular, see Chapters 9 and 10 of Huelsman, reference7 in Appendix £-10, and Chapters 3 and 11 of Ley, reference I1 in AppendixE-I0.

PROBLEMS

12·1. Let v(t) = VI cos Wit for Eq, (12-1) and carry out the derivationleading to a result similar to Eq, (12-9).

12-2. For the sinusoidal waveform of the figure, write an equation for vet)using numerical values for the magnitude, phase, and frequency.

f-+lO I Il-

v,v(t)

J I

0 0.1 02I t',~f-I-1\ •j

f--lOf- I I

I I Fig. Pl1-2.

Ch. 12 I Problems

12-3. Starting \\similar to

12-4. Given the

sin

determin

12-5. Show tha

In otherarbitrara sinuso

12-6. Using tin term

12-7. Usingequatiin Cha

di(a) lit(b) di

dt

Cc) ;'

(d) d1

12-8. Repeonl

(a)

(b)

(c)

12-9. Thlo~del

12-10. Inis12

12-11. FN

12-12. 11f

lysis / cs. 11

and Bacon,

Company,

race Jovan-

An Intro-ing, Mass.,

for Elec-Inc., New

~ompany,

nehart &

Ig in thestraight-ter can

Cor sug-:ference!)pendix

Ivation

or vet)ncy,

Starting with the rotating phasors, e->', show by a constructionsimilar to that illustrated in Figs. 12-4 and 12-5 that

sin? WI + cos- WI = I

Given the equation

sin 377t + 3ft sin (377t + ~) = A cos (377t + e)determine A and e.

12·5. Show (ha t

i: Aksin (WIt + CPk) = Csin(wlt + e)k"1

In other words, show that the sum of any number of sinusoids ofarbitrary amplitude and phase angle but all of the same frequency isa sinusoid of the same frequency.

12.6. Using the equation of Prob. 12-5 with 1/ = 2, determine C and ein terms of AI, A2, CPI,and CP2'

12·7. Using the method of Section 12-3, solve the following differentialequations for the steady-state solution (called the particular integralin Chapter 6):

C ) di 2' . 2a dt + I = Sin t

Cb) :;. + i = cos 3t

(c) :: + 3i = cos (21 + 45°)

(d) ~:; + 2 :~ -I- i = 5 sin (2t -I- 30°)

d2i. .(e) dt2 -I- 1 = 2 Sin t

12·8. Repeat Prob. 12-7 for the following differential equations, solvingonly for the steady-state solution:

d+! di(a) dt~ + 2 d; + 2i = 3 cos (I + 30°)

dt i(b) dl~ + 4i = 3 cos (21 + 45°)

(c) :: -I- 2i = sin 2t + cos t

12.9. The network of the figure has a sinusoidal voltage source and isoperating in the steady state. Use the method of Section 12-3 todetermine the steady-state current i(t) if VI = 2 cos 2t.

12.10. In the network of the figure, i, = 3 cos (t + 45°) and the networkis operating in the steady state, Make use of the method of Section12-3 to determine the node-to-datum voltage VI(t). VI

12.11. For the given network, find v.(t) in the steady state if VI = 2 sin 2t.Make use of the method of Section 12-3.

12.12. In the resistive network shown in the figure, VI = 2 sin (2t -I- 45°)for all t. (a) Determine i.Ct). (b) Determine ib(t).

Fig. P12-9.

Fig. P12-10.

Fig. PI2-lI.

+

L'd

368 Sinusoidal Steady-State Analysis I Ch. 12

Fig. P12-12.

12-13. The network shown in the accompanying figure is operating in thesteady state with sinusoidal voltage sources, If t'l ,= 2 cos 21 andV2 = 2 sin 21, determine the voltage v.Ct).

~F

'---_~_.L-13 Fig. P12-13.

12-14. The inductively coupled network of the figure is operating in thesinusoidal steady state with 1'1(1) = 2 cos I, Jf LI ~-,L: - I H,M = * H, and C = 1 F, determine the voltage dO,

M

+

Va cFig. P12-I4.

12-15. The network of the figure is operating in the sinusoidal steady state,In the network, it is determined that 1'. ,= 10 sin (10001 -- 60) andt'b = 5 sin (10001 - 45°), The magnitude of the impedance of thecapacitor is 10 Q. Determine the impedance. at the input terminals

of the network N.

+

+

Fig. Pl2-IS.

12-16. In the network shown, 1'1 c- 10 sin 106t and i, 10 Cl)S JO"t. andthe network is operating in the steady state, For the ,'kment valuesgiven, determine the node-ta-datum voltage /'/t),

ting in theLz - 1 H,

Ch.12 I Problems 369talvsis / ci. Il

ating in the2 cos 21 and

Fig. P12-16.

12·17.For the hridged-Tnetwork of the accompanying figure, t'l =·2 cos tand the system is in the steady state, For this network, (a) determinei.Ct), and (b) determine ib(t),

t'l

+

Fig. P12-17.

ady state,60') and

cc of thererlllinals

12·111.The network of the figure is operating in the steady state with1'1 . 2 sin 21 and K I ,= -~, Under these conditions, determine i2Ct).

The following series of problems are intended to give practiceIn constructing phasor diagrams. The network shown in the figurefor Prob. 12-19 is assumed to be operating in the sinusoidal steadystate, In the element values given in the table, a double entry incolumn 1 implies a series connection, in column 2 a parallel connec-tion, For each problem, (a) determine VICt), (b) Draw a completephasor diagram showing all voltagcs and all currents, as well as allrelationships between the voltages and the currents,

Fig. P12-18.

If!

Fig. P\2-19.

+

16/. andI values

cz= V", sin (wt+CP)

SinusoidalSteady-State Analysis / Ch. 12

Network 1 Network 2 Vm eo ifJ

R = 1 C=2 2 1 -30°R =2 C=1 10 2 45°R = 20 C=! 1 0.1 0°R =2 L=2 100 1 30°L=1 R=1 10 ! 0°C=2 R =2 3 1 45°L=3 C=1 10 ! -45°C=1 L=! 1 2 0°

R = I, C = 1 L=2 2 1 30°R = I, L = 2 C=! 2 1 45°L=I,C=2 R = 1 10 ! 0°R = I, C = 1 R = I, L = 2 10 1 90°R = 3, L = 2 R=I,C=! 1 ! 0°L = I, C = 2 R = I, C = 1 100 1 -90°L = I, C = 2 R = I, L = 2 1 1 0°

370

12-19.12-20.12-21.12-22.12-23.12-24.12-25.12-26.12-27.12-28.12-29.12-30.12-31.12-32.12-33.

12-34. The network of the figure is operating in the sinusoidal steadystate and it is known that V3 = 2 sin 21. For the element values given,determine Vz/V 1 = AeJ~.

+

Fig. P12-34.

+

12-35. The network of the figure is adjusted so that RL = Re = ..;LIe.(a) Draw a complete phasor diagram showing all voltages and cur-rents (and their relationships to each \other) for the condition\ lL \ = \le \. (b) Let the frequency for the condition of part (a) be Wt.Draw a phasor diagram for a frequency W2 > Wt. (c) Repeat part(b) for a frequency W3 < Wt.

Fig. PI2-3S.

Analysis / cs. 12

co

!2

0.1!!1

!211

!1

!11

-300

450

00

300

00

450

-450

00

300

450

00

900

00

-900

00

soidal steadyvalues given,

c = -.!L/C.es and Cur-: conditiont (a) be CDJ•

.epeat part

Ch.12 / Problems 371

12·36.The network of the figure is adjusted so that RI Cl = R2C2 = T.Let the phase angle of '1:2 with respect to VI be cp. (a) Show how cpvaries with T. (b) For a fixed T, show how cp varies with CD. (c) Fora fixed T, show how the maximum amplitude of V2 is related to themaximum amplitude of VI as a function of CD.

+"2

Fig. P12-36.

-:408 Frequency Response Plots I Ch. 13

PROBLEMS

13-1. Sketch the (a) magnitude, (b) phase, (c) real part, and (d) imaginarypart variation of the following network functions with ro for bothro> 0 and ro < 0;

(a) 1 +j2ro1

(b) 1 _ j2ro

( ) (1 - 2roZ) +jroc 1 +j2ro

13-2. Consider the RLC one-port network shown in the figure. For thisnetwork, determine the driving-point functions Z(jro) and Y(jro).For each of these functions, plot the magnitude, phase, real part, andimaginary part as a function of frequency for ro > 0 and ro < O.

13-3. For the two-port network of the figure, determine the voltage-ratiotransfer function, Gll(jro) = VZ(jro)/V1(jro). Plot the variation ofthis function with ro for the two methods employed in Fig. 13-7.

2H111

Fig. Pl3-1.

13-4. The two-port network of the figure shows an RL network. Repeatthe plots specified in Prob. 13-3.

Fig. P13-4.

13-5. Repeat Prob. 13-3 for the RC two-port network shown in theaccompanying figure.

0 'VV'v

lR'0

+ RI+

VI V2

- Te -Fig. Pl3-5. 0 0

13-6. Show that the locus plot of Eq. (13-15) shown in Fig. 13-7 is asemicircle centered at Gll(jro) = 0.5 + jO for the frequency range0< ro < co.

["11. 13/

13-7.

13-8.

13-5

13·

1

s / Ch./J 409

aginaryfor both

Consider the locus plot required in Prob. 13-5. Show that this locusis a circle for the frequency range, - 00 < Cl) < 00. Determine thecenter of the circle and its radius.

Jl.8.Consider the RLC series circuit shown in the fipure. (a) Suppose thatthis network is connected to a sinusoidal voltage source. Plot thevariation of the current magnitude and phase with frequency. (b)Suppose that the same network is connected to a current source of asinusoidal waveform. Plot the variation of the voltage across thethree elements using the same coordinates as in part (a). Elementvalues are in ohms, farads, and henrys.

+or this f7)Y(jw).

,and + il t<0. VI V2

-ratioon of ~ F ~F-7.

(a) (b)

Fig. P13-S.

13·9. The figure shows a network which functions as a low-pass filter. Forthis network, determine the transfer function V2/11 and plot themagnitude and phase as a function of frequency for this ratio.

Fig. P13-9.

e

13-10. The network shown in the accompanying figure serves a similarfunction to that considered in Prob. 13-9, namely, it is a low-passfilter. For this network, determine the transfer function V2/11 andplot the magnitude and phase as a function of frequency.

Fig. PI3-10.

13-11. A network is analyzed and it is found that the transfer function is

V2 Iv;- = S3 + 2S2 + 2s + 1

)J1 F

2\1 ~H

Fig. Pl3-12.

IF

Fig. Pl3-l3.

Fig. Pl3-14.

410 Frequency Response Plots I c« 13

For this function, plot the magnitude and phase as a function offrequency for the range 0 < Cl) < 4.

13-12. For the RLC network shown in the figure, plot (a) the locus of theimpedance function, and (b) the locus of the admittance function.

13-13. Plot (a) the admittance locus, and (b) the impedance locus for theRLC network shown in the figure.

13-14. The four-element network shown in the figure is to be analyzed todetermine (a) the locus of the impedance of the network, and (t)the locus of the admittance function for the network.

13-15. For the network of the figure, plot (a) the locus of the impedancefunction, and (b) the locus of the admittance function.

1\1

2FFig. Pl3-15.

~F

13-16. The RL network shown in (a) of the figure has element values suchthat the phase of the voltage measured with respect to the current is

(a)

r-.r-. f Phase of voltage with respect.to current in series RL circuit------VI---- l--- I-- -

+0'

+30'

+60'

+90' o 40 50 6010 20 30Frequency in cycles/sec

(b)

Fig. Pl3-16.

cs.

13-1

I~

70

13

/ ci. 1J

et ion of

s of thenction.

for the

zed tod (t)

dance

cht is

(1.13/ Problem s 411

that shown in (b) of the figure. From this information, determine thepole and zero locations for Y(s).

1),17.The figure shows the variation of the magnitude of the current withw for an RLC series network with an applied sinusoidal voltage ofconstant magnitude. From the figure, determine the locations of thepoles and zeros of the admittance of the network.

IFrequency response of _

/ \ / RLC series circuit

/ ~.

I \I <,

~

/ <,/

~/

1.00

0.80

III 0.60

0.40

0.20

o 10 20 30 40 50 60Frequency in cycles/sec.

Fig. P13-17.

13-18. The pole-zero configuration shown in the figure represents theadmittance function for the series RLC circuit. From the pole-zeroconfiguration, determine: (a) the undamped natural frequency Wn,

(b) the damping ratio (, (c) the circuit Q, (d) the bandwidth (to thehalf-power points), (e) the actual frequency of oscillation of thetransient response, (f) the damping factor of the transient response,(g) the frequency of resonance, (h) the parameter values (in terms ofL if the values cannot otherwise be uniquely determined). (i) Sketchthe magnitude of the admittance I Y(jw) I as a function offrequency.(j) If the frequency scale is magnified by a factor of 1000, how do thevalues of the parameters, R, L, and C change?

13-19. The figure shows two configurations of poles and zeros for a certaintransfer function. Use a graphical procedure to determine the varia-lion of the magnitude of the network function for the two configura-t ions. Superimpose the two plots on the same system of coordinates.

r----,.----, +j5

1 poleI----t-"*----j +j4

1----+-----1 +)3

1----t----1 +j2

----1---- -- +jl

1 zero1----:

2:---+---:-

1----jOj)0

'-------- -jl

------ =j?

t------ ------ -j3

1 pole-- ------ -- ...•--- - j4

'-- __ -1- __ --' - j5

Fig. P13-JS.

jw

Fig. P13·22.

412 Freouency Response Plots / Ch. J3

-0.5+j2.0x jca -0.5+j2x jw

-0.5+j 1.5 x

-O.5-j2x(b)

<T

(Scale factor - 1)

(Scale factor -1)

-0.5+j1x

3 zeros3 zeros<T

-0.5-jl x

-0.5 - j 1.5 x

-O.5-j2 x(a)

Fig. P13-19.

13-20. Show that the bandwidth B varies inversely with the circuit Q fora series RLC circuit.

13-21. Show that for an RLC series network the product of I Y Imax and thebandwidth B equals I/L, where L is the inductance ..

13-22. The two poles and zero shown in the s plane of the accompanyingsketch are for the transfer function of a two-terminal-pair network,G(s) = V2(s)/ VI (s). The zero is on the real axis at a position to cor-respond with the same real part of the poles. The poles have positionscorresponding to ( = O.707«() ~ 45°); eo, is the distance from theorigin to the pole as shown. In this problem, we will investigate theeffect of the finite zero by computations with and without the zero.(a) The bandwidth of the system is modified from the definitiongiven in the chapter as the range of frequencies from ()) = 0 to thehalf-power point. Compute the bandwidth of the system with thepole-zero configuration shown above; compute the bandwidth withthe zero removed. In which case is the bandwidth greater and bywhat factor? A graphical construction is suggested.

13-23. The Q of a series RLC network at resonance is 10 The maximumamplitude of the current at resonance is I amp when the maximumamplitude of the applied voltage is 10 V. If L = 0.1 H, find the valueof C in microfarads.

13-24. A coil under test may be represented by the model of L in series withR. The coil is connected in series with a calibrated capacitor. A sinewave generator of 10 V maximum amplitude and frequency (l) =-1000 radians/sec is connected to the coil. The capacitor is varied andit is found that the current is a maximum when C = lOO J.l.F. Also,when C = 12.5 J.l.F, the current is 0.707 of the maximum value.Find the Q of the coil at ()) = 1000 radians/sec,

13-

Ch. I

13-

13-

13-

· / c». 11 413

1),25. The network of the figure is found to have the driving-point imped-ance

106(s + I)Z(s) = (s -+- I + jlOOXs + I - jlOO)

From this information, determine the values of R J, Rz, L, and C.

Fig. P13-25.

1J·26.For the following network function, plot the straight-line asymptoticmagnitude response and the phase response. Use 4- or 5-cycle semilogpaper.

100

for G(s) = s(l + O.OlsXI + O.OOIs)

]3·27.Given the network function,

G(s) = (1 + O.IsXI + O.Ols)(l + sXI + O.OOls)

Plot the straight-line asymptotic magnitude response and the phaseresponse. Use 4- or 5-cycle semilog paper.

13-28.Plot the straight-line asymptotic magnitude response and phase anglefor the network function

the

eehy

S2

G(s) = 100(1 + 0.17sXI + 0.53s)

Use 3- or 4-cycle semilog paper.

13-29.(a) Plot the straight-line asymptotic magnitude response, and (b)determine the actual (or true) response for the network function

G( ) = 1000(1 + 0.25sXI + O.ls)s (1 + s)( I + 0.025s)

(c) On the same coordinate system, plot the phase response. Use 4-or 5-cycle semilog paper for the plotting.

13-30. Repeat Prob. 13-29 for the following network functions:

(a) G(s) = 50~~1-~°o~~~~~1000s

(b) G(s) = (I + O.OlsXI + 0.0025s)

~(l + O.Ols)(c) G(s) = 180(1 + 0.05sXl + O.ools)

13-31. (a) Plot the straight-line asymptotic magnitude response, and(b) determine the actual (or true) response curve for the network

Fig. PI3-36.

~:1 J2F IFJ~2Fig. PI3-37.

Fig. PI3-38.

Fig. PI3-39.

414 Frequency Response Plots I Ch. JJ G.

function(l + 0.2s)

G(s) = 120s(S2 + 2s + 10)

(c) On the same coordinate system, plot the phase response. Use 3-or 4-cyc1e semilog paper.

13-32. Repeat Prob. 13-31 for the fo\1owing network functions:s

(a) G(s) = 1000(1 + O.OOlsXI+ 4 x to-5s + 10 8S2)

lOOs(b) G(s) = (I + s + 0.Ss2XI + O.4s + 0.2s2)

13-33. We are required to construct a network function G(s) satisfying thefo\1owing specifications: The asymptotic curve should have a low-frequency response of 0 db/octave slope, and the high-frequencyresponse has a slope of - 24 db/octave. The break frequency betweenthese two slopes is at (J) = 1 radian/sec. At no frequency should thedifference between the asymptotic and the true response exceed ± 1db.

13-34. The figure shows two straight-line segments having slopes of ±n6db/octave. The low-and high-frequency asymptotes extend indefi-nitely, and the network function the response represents has first-orderfactors only. Find G(s) and evaluate the constant multiplier of thefunction.

Mdb

..,(Iog scale)

+

Fig. Pl3-34.

13-35. Repeat Prob. 13-34 if the response is changed only by the high-frequency asymptote having a slope of -18 db/octave.

13-36. For the two-port network shown in the figure, determine V1/VI andplot the magnitude response (Bode plot) showing both asymptoticand true curves.

13-37. Prepare a Bode plot for the network function V1/VI for the networkshown in the accompanying figure.

13-38. Prepare a Bode plot for the voltage-ratio transfer function GIl =V1/VI for the two-port network shown in the figure.

13-39. The figure shows an RLC network. For this two-port network, plotthe transfer function GIl = V2/V\ showing both the asymptotic andtrue curves.

· Consider the following transfer functions:

s-1(a) G(s)H(s) = K s + I

s + 1(b) G(s)H(s) = K s _ 1

K(c) G(s)H(s) = s( I + 0.05s)

For each of these functions: (a) plot GUw)H(jw) in the complexCH plane from w = 0 to w = 00 with K = 1. (b) Determine therange of values of K that will result in a stable system by means ofthe Nyquist criterion.

13-41. For the locus plot shown in Fig. 13-45, sketch the correspondingBode plots for the magnitude and phase, making some assumptionas to the frequency scale. Estimate the gain and phase margins andindicate these on the Bode plots.

13042. Repeat Prob. 13-41 for the locus plot shown in Fig. 13-48.

13·43. Starting with the locus plot shown in the figure for Prob. 13-4;.sketch the corresponding magnitude and phase plots using Bodecoordinates. Make an assumption about the frequency scale alongthe locus. Indicate on the figure the gain and phase margins.

13·44. The Nyquist plot of the figure is made for a system for which P == O.Analyze the system by applying the Nyquist criterion, indicatingwhether the system is stable, conditional\y stable, or unstable.

j ImCH

Re CH

0+Fig. P13-44.

13-45. The locus plot is made for a system for which P = O. It is given thatA = -0.75, B = -1.3, and C = -2. Assuming that the plot is

jlmCH

c

Fig. Pl3-45.

415

416 Frequency Response Plots / Ch. 13 Ch.

made for a gain K, what is the range of values of gain for which thesystem will be (a) stable, and (b) unstable.

13-46. Repeat Prob. 13-45 if P= 1.13-47. The figure shows a locus plot made for a system for which P = O.

Is the system stable? Determine your answer to this question byapplying the Nyquist criterion. Repeat if P = I, P = 2.

13-~

jlmGH

13Re GH

Fig. Pl3-47.

13-48.' The locus plot shown in the figure is made for a system with P = 2,two poles with positive real parts. Apply the Nyquist criterion tothis system to determine the stability of the system.

j Im G(jwl H(jw)

Fig. PI3-48.

13-49. The locus plot of G(jw)H(jw) shown in the figure is made for a sys-tem with P = O.For this system, apply the Nyquist criterion to studythe stability of the system.

j lm Gfjw) H(jw)

GH plane

Re GfjuJ I H(juJ)

Fig. PI3-49.

ponse Plots / Ch. J3

13·50. The accompanying figure shows a plot of the locus of G(jw)H(jw)from w = 0 to o: = DJ. From this plot determine everything you canabout G(s)H(s) as a quotient of polynomials in s.

Ch. 13 / Problems 417

gain for which the

for which P = o.this question by

t>=2.j Im GH

w=o"Re GH

Fig. P13-S0.

13-51. The figure shows the feedback system for which the Nyquist criterionhas been developed. For this problem, let H = 1, and

G(s) _ K- (s - aXs + 2Xs + 3)

Make use of the Nyquist criterion to study this system for stabilityfor the case a = 1.

ern withP = 2ist criterion t;

Fig. P13-SI.

13-52. Repeat Prob. 13-S1 if a = 2.

13-53. Repeat Prob. 13-S1 if a = 4.

13-54. A system is described by the transfer functions which relate to thesystem of Fig. P13-S1.

iade for a sys-:rion to study

IOSG(s) = (s + 2Xs + IOXs + 20)

and H = 1. Make use of the Nyquist criterion TO determine if thissystem is stable.

13-55. Repeat Prob. 13-S4 for the given G(s), but for a new feedback transferfunction

H(s) = s t020

This causes cancellation in the product H(s)G(s) and is a form ofcompensation of a system to improve stability. Comment on theeffectiveness of this compensation function.

4111 Frequency Response Plots I Ch. J3

~3-S6. The figure shows a model of a feedback amplifier. For this system,identify G(s) and H(s) as in Fig. P13-51 and express each as a quo-tient of polynomials in s. Is this system capable of oscillation? Makeuse of the Nyquist criterion in answering this question and in a gen-eral study of the system stability.

+

Fig. Pl3-56.

VALKENBURG-PROBLEMS.pdf

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