87 ok Equations / Ch. 3
I Problems in Appendix E-4.1. Consider also the analysis of resistive ladder es rks as described in references in Appendix E-4.2. For specific sugg see Huels , reference 7 of Appendix E-I0, for the resistive network man 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
uic method for form. The so(usolution is to be rmulation offers he only requireement or several I . the state-space on by computer is to be accorninarily simpler to
ork analysis as given in Appendix
E-8.4.
PROBLEMS 2
What must be the relationship between C. and Cl and C in (a) of the figure of the networks if (a) and (c) are equivalent? Repeat for the network
shown in (b).
I Book Company, cusses a graphical ms. in Linear Circuits
1
.."
0-
,
(c) (b)
Theory, McGraw1 and 12.
(a)
Fig. P3-1.
Viley & Sons, Inc., ring Circuit Analy'ork,1971. Theory: An Intronnpany, Reading,
What must be the relationship between Le. and Lt. L2 and M for the networkS of (a) and of (b) to be equivalent to that of (c)?
•
]
M
and Bacon, Inc., vetworks for ElectVinston, Inc., New
(c) (b) (a)
the State Variable ark, 1970. Analysis, Prenticeis a programmed of state equations.
Fig. P3-2. Repeat Prob. 3-2 for the three networks
shown in the accompanying
figure.
~s
e digital computer . . described in refernethod from refer-
(e) (b)
la)
Fig. P3-3.
88 3-4.
Network Equations / Ch. 3 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.
Ch. 3/ Problems The series ( tain to the netwo specified in the ta connection of ele connection of elet to zero. For the st mine 'VI in the for on a cathode ray 0 and so on.
10--1 ?L,q 2
1'~
(a) (b)
(a)
V2
2
Fig. P3-4.
3-5.
3-6.
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 C eq for the two networks to be equivalent?
2
This problem may be solved using the two Kirchhoff laws and voltagecurrent 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 determine the value of i2(lo) and di2(tO)/dl.
-3 (c)
V2
'K
+
111
volts
111 + 10 v-=-
+1
211 ~h
Fig. P3-6.
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 was closed. Determine the value of C.
-1 (e)
",[
rations / Ch. 3 ;ed of a I-H ected to the :0 vertices a figure when ilern, rather
Ch. 3 / Problems
89
The series of problems described in the following table all pertain 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 determine 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.
Network Equatiolls /
90
c». J Ch.
Waveforms of
Network of A
Network of B
3-8.
R=2
L =:l-
a, b, c, d, e,f
3-9.
C=!
L= 1
a,
b,
3-10.
C=
(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
s.a.]
f, R
L=2
= 1
3-14.
R = 2
R = I, C = 1
3-15.
L=1:
R=l,C=!
b,d,f
R=l,C=!
b,d,f
L= 1,R=
3-16.
1
e,
3/
Problems
V2
•
d, e.f
3-17. For each of the four networks shown in the figure, determine the number of independent loop currents, and the number of independent node-to-node voltages that may be used in writing equilibrium equations using the Kirchhoff laws.
v(t)
(b)
R2 2
3
fi0
2
RI L C v(t)
+
v(t}
R3
~
C
4 (a)
(b)
2
v(t}
+
3
Rr;J
3-19. Demonstrate the so establish a inductor into an C
C2
v(t}
3
R3 (c)
3-20. Demonstrate tha 3-21. Write a set of appropriate 1 3-17.
(d)
Fig. P3-17. 3-18. Repeat Prob. 3-17 for each of the four networks on page 91.
shown in the figure
3-23. Write a set 0 network in one controll equations
91
Ch. 3 / Problems
v{t)
(a)
v{t)
(cl
(b)
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).
Network Equations I Ch. 3
92
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 loopcurrent variables chosen. Using the specified currents, write the Kirchhoff 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.
·YM23
f:\
L3
i2)R3
R2
Fig. P3-26.
ci. 3 I Problems
93
3.27. Write the loop-basis voltage equations network of Fig. P5- 22 with K closed.
for the magnetically
coupled
3.28. Write equations using the Kirchhoff current law in terms of nodeto-datum voltage variables for the four networks of Prob. 3-17. 3.29. Making use of the Kirchhoff current law, write equations basis for the four networks of Prob. 3-18.
on the node
3.30. For the given network, write the node-basis equations using the node-to-datum voltages as variables. Collect terms in your formulation 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 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,
"'t_Cl
~i2
-1--R---L,2
__
f....--=-.l.....-.~--.J
R6
Fig. P3-31.
he
).32. The network of the figure is a model suitable for "rnidband" operation of the "cascode-connected" MOS transistor amplifier. Analyze the
rk
+
Fig. P3-32.
Network Equations / Ch. 3
94
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
Fig. P3-33.
2h
Fig. P3-34
2h
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 equations on (a) the loop basis, and (b) the node basis, and simplify the equations to the form of the chart used in Examples 11 and 13. R
R
R
R
R
R
R
Fig. P3-3S.
R
3-36. For the network shown in the figure, determine the numerical value of the bi ~11chcurrent i I. All sources in the network are time invariant. 2fl
H2
Fig. P3·36.
2v
3
ci. 3/
e
3-37. In the network of the figure, all sources are time invariant. the numerical value of i2•
95
Problems
Determine
d n
a 2v
Fig. P3-37. 3-38. In the given network, all sources are time invariant. branch current in the 2-0 resistor.
Determine
In ns
ahe
the
2
Fig. P3-38. 3-39. In the network of the figure, all voltage sources and current source are time invariant, and all resistors have the value R = O. Solve for the four node-to-datum voltages.
t
All R=~ 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-todatum voltages.
Network
96
Equations
/ Ch. 3
b
Fig. P3-40.
3-41. Evaluate the determinant: -1
2 -1
O. -2
0 -1
3 -1
0
0
0
0 -2
3
2
3-42. Evaluate the determinant: -2
1
0
-1
4
-1
-2
4
3
1
0
2
4 1
3
4
1
1
0
2
3
3 -1
-2
1
3-43. Solve the following system of equations for i 1> iz, and i3, Cramer's rule. 3i 1
-
2i2
+ Oi3 = 5
+ 9i2
- 4i3
=
Oil - 4i2
+ 9i3
= 10
-2il
0
3-44. Solve the following system of equations for the three unknowns, i 1> iz, and i3 by Cramer's rule. -
3i2 -
5i3 =
5
-3il
8i1
+
7i2 -
Oi3
=
-10
-5il
+ Oiz +
1113 = -10
97
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 Si ,
Si2
10iJ
-
4i2 + 5iJ + 20i2 + 14iJ :'-il + 7i2 + 2iJ
2il -Sil
+ +
system of equations.
12i.
=
S
6i4
=
33
-- 16i.
=
10
- 10i4 = -15
3-48. Consider the equations 3x -
4x
3z = 1
y -
x - 3y
+ Oy
+
z = I
-- 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 conclude regarding the three lines represented by these equations? 3-49. Find duals for the four networks 3-S0. Find the dual networks
of Prob. 3-17.
for the four networks
3-S1. Find the dual of the network
of Prob. 3-31.
3-S2. If one exists, find a dual of the network 3-S3. Analyze the network
given in Prob. 3-IS.
of Prob. 3-40.
of Prob. 3-17(c) using the state variable formu-
lation. 3-S4. Consider the network shown in Prob. using appropriate state variables. 3-SS. Analyze the network formulation.
3-23. Analyze
this network
shown in Fig. P3-IS(b) using the state variable
3-56. Analyze the network of Prob. 3-30 using state variables. 3-S7. Apply the method in Fig. P3-31.
of state variables
3-S8. The element represented by the equations
shown
in the network is a gyrator which is described
sing Find the two-element
to analyze the network
'VI
=
V2
=
equivalent
Roi2 ---Roil network
rns,
(a)
Fig. P3-SS.
shown in (b) of the figure.
98
Network Equations
t Ch. 3
3-59. For the gyrator-RL network of the figure, write the differential equation relating VI to il• Find a two-element equivalent network, as in 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. Find L.q, the equivalent inductance, 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 is intended 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 values for Ct. Cz, and C3? ~I
1$?t?L
2
II
I
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 a linear co solution of the vet) is someti Assume sources which' and currents. system is alte or closing of obtain equati
112
Cox,
First-Order Differential Equations I Ch. 4 CYRUS W., AND WILLIAM
The Macmillan
Company,
M. E. VAN Miffiin Company,
CRUZ, JOSEB., JR., AND
Houghton
Circuits, Signals, and Networks, New York, 1969. Chapter 4.
L. REUTER,
4-3.
Signals in Linear Circuits, Mass., 1974. Chapter 5.
VALKENBURG,
Boston,
P., Basic Circuit Theory with Digital Computations, Inc., Englewood Cliffs, N.J., 1972. Chapter 5.
HUELSMAN, LAWRENCE
Prentice-Hall,
J., AND PAUL A. WINTZ, Basic Linear Networks Jar Elettrical and Electronics Engineers, Holt, Rinehart & Winston, New York, 1970. Chapter 2.
LEaN, BENJAMIN
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 7 in Appendix E-IO.
4-4.
PROBLEMS 4-1.
In the network of the figure, the switch K is moved from position I 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(/).
4-5.
Fig. P4-t. 4-2.
The switch K is moved from position a to b at I ~ U, having been in position a for a long time before I ~--O. Capacitor C2 is unchargedat 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.
4-6.
I Ch. 4 elworks,
ci. 4 I Problems 4·3.
113
In the network
given,
such that 1",(0) (a) Find
iuuits, 5.
~c
the initial
on C. is V, and on C2 is 1'2
voltage
V, and 1"2(0) = If~. At 1=
i(1) for all t imc.
(b) Find
0, the switch
for I "
1',(/)
I> O. (d) From your results on (b) and (c). show that (e) For the following values of the ctcrncnts, R C2 = ~F, 1'1 -- 2 V, I': I V, sketch i(1) and lime com,lam of each,
l.
"',(cy
n,
\
for
I'~(/)
I',('Y.)
0,
Cl
~c
F,
\
amI idcntify
I'""
~c
K is closed.
O. (c) Find
the
Fig. P4·3.
4·4.
In the network period
of the figure,
of time.
At I
before-break"
=
mechanism),
given in the nctw ark. inductor is zero.
from a to b (by a "make-
is moved
Find
Assume
using
1'2(1)
that
a for a long
K is in position
the switch
0, the switch
the
numerical
the
initial
current
values 2-1 i
in the
K
IQ
+
~
-1
FiJ,:. P4·4.
4·5. The network open,
of the figure
At I = 0, switch
given, sketch constant,
Iv
I_h-L
reaches
a steady
K is closed.
the current
~
waveform,
state
i(/)
Find
_L __ ~
with
and indicate
K
the switch
for the numerical the value
values
of the lime
20n
30 C!
.J:
20V1-~' '7)
t-een in ged at md the
10 v-=-
-L
Fig. P4·5.
4-6.
The network I =
4·7.
°
of Prob,
the switch
4-5 reaches
is moved
a steady
to position
state
in position i(/)
1, Find
values given for the element, of the time constant.
sketch
In the given
e : for 12:0 and is zero
If
the
network,
capacitor
is
t', ~
initially
the waveform,
unchargcd,
find
R2 -, 20, and C = -to' F, and for these values the value of the ti.nc constant on the sketch.
2 and
'I'
for the numerical and show
t'2(1), sketch
the valt.e
for
all
Let
R ,~-'
"2(t)
I
<0 10,
identifying
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 observ the capacitor voltage has the value re(O) = 0.5 V. For the e 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 di -;- P(I)i dt
equation
== Q(I)
by an "integrating factor" R such that the left-hand side of the eq uon equals the derivative d(Ri)/df. (a) Show that the required i
'nsl Ch. 4
c« 4 / Problems 115
grating factor is R eS "", (b) Using this integrating solution to the differential equation that corresponds
= 0 conrved that element
factor, find the to Eq. (4-30).
4-12. In the network shown in the accompanying figure, the switch K is closed at I 0, a steady-state having previously been attained. Solve for the current in the circuit as a function of time.
+ V-=-
Fig. P4-I2.
andL = 'tch K is
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) Solve for the current assuming that (I, oF R/L. (b) Solve for the current when R/L.
(J,'
K vlt)
L -lH
Fig. P4-13.
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) - 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 -;-: at . dl
= Jr (t ) k
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 (e)!s = sin- i (b)J~' te=> (f) !6 cc cos- I (c) Ji sin Wol (fJ,)f~ " I sin '21 (d) f~ cos Wot (h) J8= e- sin 2t c~
t
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 is closed. Find the current in the inductor for I :> 0, sketch this current, and identify the time constant.
10 10
n
n
+ -=5v
2H
Fig. P4-17. 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 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 is 0.008 amp. The switch K is closed at 1 -~ 0, and it is observed that the relay is actuated when I = 0.1 sec. Find: (a) the indu.:tance L of the coil, (b) the equation of i(1) with all terms evaluated.
117
Ch..4 / Problems ~
100V~
10,0000
~
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
of a resistor
so that it is . Theswitch uatedwhen equation of
132
Initial Conditions in Networks / Ch. 5
ci. 5/ Proble 5-7.
In the solve r, and C
5-8.
The ru Solve and L
5-9.
In the switch given,
PROBLEMS 5-1.
In the network of the figure, the switch K is closed 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 L = 1 H, and V = 100 Y.
n,
Fig. PS-2. 5-3.
In the network of the figure, K is changed from position a to b at = 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.
t
Fig. PS-3. 5-4.
For the network and the conditions stated in Prob. 4-3, determine the values 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 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.
5-10. In tH state
h.5
cs. 5 / Problems 5-7.
133
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
the It,
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 and L = 1 H.
=
0+ if 1=
1 amp, R
=
100
n,
v
the
Fig. P5-S. 5-9.
In the network shown in the figure, a steady state is reached with the switch K open. At t = 0, the switch is closed. For the element values given, determine the value of v.(O-) and v.(O+). 10 ~!
at 10 I!
20 ~!
+ 5 V-=-
Fig. P5-9. the
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
ate ine
20Q
lOH
vb
+
"-1
Ton Fig. PS-lOo
J"
134 Initial Conditions
For the element v.(O+).
values given, determine
5-11. In the network of Fig. P5-9, determine ditions stated in Prob. 5-9.
in Networks
/ Ch.. 5
the values of v.(O-)
iL(O +) and i ( L
5-12. In the network given in Fig. P5-1O, determine the conditions stated in Prob. 5-10.
(0)
I
Problems
an
for the cor
Vb(O+) and Vb(oo) fo
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 a t = 0+, (d) d2V2/dt2 at t = 0+.
!,~ the given m rh; switch K
R2'
I Mr!,
t . , 0·; .
S-19. In the circui connecting a (a) dil/cll and 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. Fig. PS-l3.
5-15. The switch K in the network of the figure is closed at t = 0 connecting the 2 2battery to an unenergized network. (a) Determine i, dildt, and d i/dt at t = 0+. (b) Determine 1'1, do-Jdt, and d2Vl/d/2 at t = 0+.
+
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. 5-15 at 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 , V
5-20. In the net open with and C I integrodifli closed. (b)
ks / Ch. 5
13S
a: 5 / Problems
0-) and
: the con-
Fig. PS-17. =0 with 'or (a) t'l dvz/dt at
5-18. "~ the given network, the capacitor Cl is charged to voltage Vo and rh, switch K is c'osed at T ,,0. When RI ·2 Mn, Vo 1000 y, 2 2 Rz I Mn, c, 10 J1F, and c, - 20 J1F, solve for d iz/dT 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, Vo sin WT, to the parallel RL-RC circuit. Find (a) dil/df
and (b) diz/df
at T
0 i .
switch K 'led. Solve onnecting difdT, and t T = O-l .
Fig. PS-i9. 5-20. In the network shown, a steady state is reached with the open with V . lOO Y, RI" 10 n, Rz ·20 n, RJ --= 20 n, and C I J1F. At time f 0, the switch is closed. (a) integrodifTerential equations for the network after the closed. (b) What is the voltage Vu across C before the
switch K L I H, Write the switch is switch is
:onditions the switch III =
0+.
witch K i, stablished
Fi~. PS-20.
T
_-L----
c
Initial Conditions
136
ill Networks
i
Ch. 5
closed? What is its polarity? (c) Solve for the initial value of i, ami i2Ct ~= 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 at t = 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 the instant t = 0, connecting an unenergized system to a voltage source. Let M 12 = O. Show that if v(O) = V, then:
Ch. 5 / Problems
5-24. The given netv I' 0, the swin V sin (I/./MC /,.(0+)
= 0,
5-25. In the network network has at an expression f parameters are what is the val dVK/dt (O+)?
di dt1 (0 -t-. ) (L1
+ L3 + 2M13)(L2 + L3 + 2M23)
-
(L3
+ M13 + M23)2
di2(0+ ) dt
5-26. In the network connecting the age Va at t = O·
•
Fig. PS-22. 5-23. For the network 2
d i1(0+) dt2
of the figure, show that if K is closed at t
= _1- (-1 R1lR1C
2
[v(O) _ dV(O)-J_ d v(0)} R1C dt dt2
=
0,
F
5-27. In the network I = 0-, all cap node-to-datum and dc d dt at t V3 and dV3/dl at fig. PS-D.
· Ch. 5
it and ~ 0+. pairs. ties at
ci. 5 !Problems
137
5-24. The given network consists of two coupled coils and a capacitor. At t : 0, the switch K is closed connecting a generator of voltage, r(f) V sin (If"'; MC). Show that ~o
/,.(0+) = 0,
(;;/(0+)
=
(V/L)"'; M/C,
and
~ K +
~
Viii
L
(/2V·(O+) df2
=
0
a
1-~L C
+ v.
Fig. P5-24.
at the source.
5-25. In the network of the figure, the switch K is opened at t = 0 after the network has attained a steady state with the switch closed. (a) Find an expression 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 voltage v. at t = 0+. (b) Find the voltage across capacitor Cl at t = CD.
r -=-V
,0, Fig. PS-26.
5-27. In the network of the figure, the switch K is closed at t ,-c O. At t 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·. e-c,
Initial Conditions ill Networks I Ch. 5
138
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 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+). O~
+
c
The differential eq uations of the we will continue restrictions as to 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
L
6-1. SECOND·O EXCITATIO Fig. PS-29.
A second-o stant
coefficients
The solution of the solution itsel
Ch. 6 / Problems
163
Continued / Ch. 6
otherwise this . the derivative
(6-137)
(6-138)
current will be sustained indefinitely. However, if there is resistance present, the current through the resistor will cause energy to be dissipated, and the total energy will decrease with each cycle. Eventually all 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 each cycle, the oscillations can be sustained. This is accomplished in the electronic oscillator to produce audio frequency or radio frequency sinusoidal signals.
ise is FURTHER
(6-139)
READING
ne appearance
BALABANIAN, NORMAN,Fundamentals of Circuit Theory, Allyn and Bacon, Inc., Boston, 1961. Chapter 3.
: Rcn and the or the current :ten
CHIRLlAN, PAUL~t, Basic Network Theory, McGraw-Hill New York, 1969. Chapter 4.
(6-140) idition of the
Book Company,
CLEMENT, PRESTONR., AND WALTER C. JOHNSON, Electrical Engineering Science, McGraw-Hill Book Company, New York, 1960. Chapter 7. CLOSE,CHARLESM., The Analysis of Linear Circuits, Harcourt, World, Inc., New York, 1966. Chapter 4.
Brace &
HUELSMAN, LAWRENCEP., Basic Circuit Theory with Digital Computations, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1972. Chapter 6. r)]
(6-141 )
SKILLlNG, HVGH H., Electrical Engineering Circuits, 2nd ed., John Wiley & Sons, Inc., New York, 1965. Chapter 2.
(6-142)
WYLlE,CLARENCER., JR., Advanced Engineering Mathematics, 3rd ed., McGraw-Hill Book Company, New York, 1966. Chapters 2,3, and 5.
DIGITAL
(6-143) e is shown in ng factor and I envelope or ermines how es zero, the IS result. .ult may be 1 the electric rage element tored in the rergy, When ) the electric s as long as e oscillatory
COMPUTER
EXERCISES
References that are useful in designing exercises to go with the topics of this chapter are cited in Appendix £-6.3 and are concerned with the numerical solution of higher-order differential equations. In particular, the suggestions contained in Chapters 5, 6, and 7 of Huelsman, reference 7, Appendix E-10, are recommended.
PROBLEMS 6-1.
Show that equation
i
=
ke=> and i 2
di
dt»
= ke= are solutions of the differential
+ 3 dtdi + 2'I =
0
164
Differential Equations, Continued / Ch, Ch. 6 / Problems
6-2.
Show that i = ke= and i equation di dt? Find the general solution dZ' (a) -.-! dt2
+
(b) d2i dtZ
+ 5 dtdi + 6'I
(c)
;t21 +
(d) d2i df2 6-4.
7 ::
+ 2i =
+
+ 5 dtdi + 4'I =
0
dv
+ 4 v = 0.
d-» ; do (c) dtZ -r- 4dt
+ 2v =
-L
4 di -"4'
di dt (0+)
=
di (0 +-) dt
co
i
6-10. Solve the differential
dO(o'_r) clt
=
J
I- 8
--=:
d" dt ?
where (J., is real and positive. i maximum value. 6-12. In a certain network, sion
it is found
Show that i(t) reaches a maxi t =--
1-
1
1X1-
homogeneo
0
6-13. The graph shows a damped si form Ke-at si From
the graph, determine n
- 0. of Prob. 6-)
0. equations
.;
of Prob, 6.)
I equations
of Prob. 6-4 Fig.
I equations
given in Prm
Prob. 6-13 for the wa
6-14. Repeat
I
'" '"
a. E
equation
J'
r
equations
~~ (0 +-) ,--
Find particular solutions to the differential 6-4, given the initial conditions:
3~ dt
dt
+ 3 do + 5v dt
Find particular solutions to the differential subject to the initial conditions:
Solve the differential
of a network is fr i= Kite:'
-- 0.
d,; , 16. + 8 dt -;- t--
Find particular solutions for the differential subject to the initial conditions: 2,
0
(h) d2i
Find particular solutions for the differential subject to the initial conditions:
=
=
2 di .: '= 0 dt r I
(f) d 0 dt 2
I,
0
-L
2
0
=
2i
=
( ) dl; g d{2'
d 21) (d) 2 dt Z
0.
(b) d2V +: ? dv dt2 - dt
+-
of each of the following
('(0.+) ~, 2, 6-9.
di
d{2'
r(O+) 0= I, 6-8.
6-11. The response
d2'
0
+ 2 dt + 20 =
i(o'+) 6-7.
0
I
(0 df~ -T' d:
-
i(O+) 6-6.
0
=
=
subject to the initial conditions d+ildt? = --I at t = 0.+.
Z' d' -~ 6i (e) d --.! -i- _l.. df2 dt'
- 0
12;
-I- .
of the differenti
of each of the following equations:
Find the general solution differential equations: d2L' (a) dt2
6-5.
d' 3 -.: dt
kte= an: solutions
+ 2 dt' di
2
6-3.
=
-j-
di 10 -.: ., 3i dt
=
0.
equation
J
d ; -;- 9 d~; -:- 13 dr d; __ 6,' == 0. 2c1tJ dt2 Fig.
Continued I Ch. 6
Ch.6 I Problems
165
the differential subject to the initial conditions d2i/dt2 = ·-1 at t = 0+. 6-11.The response of a network
i(O+) = 0, dildt
=
0+, and
f::::: 0 where (I, is real and positive. maximum value.
=0
6-12. In a certain network, sion
=0 , =0
Find the time at which i(t) attains
t
li =0
=
0
In (1,2
(l,lK1 (l,2Kz
6-13. The graph shows a damped sinusoidal waveform form Ke:= sin(eui -;- ifJ) From the graph, determine
Iv = 0
value at time
1 (1,1 -
g homogeneous
-,
having the general
values for K,
numerical
(1,
co,
and ifJ.
'.
=0
5v lOS
of Prob.
6-3
lOS
of Prob.
6-3
ms of Prob.
6-4 Fig. P6-13.
IS
given in Prob.
6-14. Repeat Prob. 6-13 for the waveform +1
'" '"
a. E
/"
V
o -1
o
a
it is found that the current is given by the expres-
Show that i(t) reaches a maximum
=
1 vt t
is found to be
uations:
16v
=
of the accompanying
figure.
,
r-.
-,
-H--
I"", <; /
1/
V
2
3 t, msec
Fig. P6-14.
4
5
166 Differential Equations,
6-15. In the network of the figure, the switch K is closed and is reached in the network. At f = 0, the switch is 0 expression for the current in the inductor, i (t). 2
~
-=- 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. At switch K is closed. Determine an expression for the vel
Fig. P6-16. 6-17. The voltage SOurce in the network of the figure is descri equation, VI = 2 cos 2t fer t ~ 0 and is a short circuit p' time. Determine V2(t). Repeat if '1.·1 = KIt for t ~ 0 and s t
< O.
Fig. P6-17. 6-18. Solve the following nonhomogeneous differential equationsI ( ) d2i a dt2 (b)
+ 2 dtdi + i =
1
g:.! + 3 dtdi + 2i = St
dt? 2
(c) ;t ;
+ 3:: + 2i
=
(d) d2q dt2
+
Sdq dt
+
6q
= te=
(e) ;t2~
+
5;~+
6v
= e=» + Se-3r
10 sin lOt
6-19. Solve the differential equations given in Prob. following initial conditions: x(O+) = 1
and
dx
dr(O+)
where x is the general dependent variable.
= -1
167 is closed and a steady sta switch is opened. Find u '2(t).
Find the particular solutions
to the differential
equations
of Prob,
6-18for the following initial conditions: x(O+)
=
2
and
dx dt(O+)
=
-1
where x is the dependent variable in each case.
~ll. Solvethe differential equation dJ' 2dt~ P6-15.
+ 9 ddt~' + 2
di 13 d;
+ 6i =
r
Kote- sin t
which is valid for t ~ 0, if i(O+) = 1, di/dl(O +) = -1, and d+il
'oIt~ge vC and tor IS zero. At t = 0, the n for the voltage V2(t).
dl'(O"t') = O.
~ll. A special generator has a voltage variation
given by the equation
1 V, where t is the time in seconds and 1 ~ O. This generator
is connected to an RL series circuit, where R = 2 nand L = I H, at urne 1 = 0 by the closing of a switch, Find the equation for the current
t,l)
as a function of time i(t).
6-13. A bolt of lightning having a waveform
which is approximated as 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 will be in amperes per unit volt of the lightning; likewise 1'(1)
'6-16. igure is described by the ~hort circuit prior to that :or t ~ 0 and VI = 0 for
=
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). i-17. itial equations
for t ~ O. Fig. P6-24. 6-25. In the network 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::\,5IlF ilt))
ob. 6-18 subject
-1
to the
Fig. P6-2S. 6-26. In the network shown in Fig. P6-2S, a steady state is reached with the switch K 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
1
2 O.
vlt)
~
Fig. P6-23.
Differential Equations, Continued I Ch. 6
Ch.,6 I Problem!
6-27. Consider the network shown in Fig. P6-24. The capacitor has an initial voltage, Vc = 10 V. At I = O. the switch K is closed. Determine
6-33. A switch series RI of time i:
168
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 for
w (b) Find tion of t steady-si as 1-" in the st
the Voltage, v(l) for t :2: O. +
c
t
10 sin wt
u( t)
K
6-34.
In the s frequent (1) CO = (2) CO =
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 differential equation for i(t). (b) Suppose that Land C are fixed in value but that R varies 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 s plane 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 determine 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 i
=
Ke-'W"'cos(con~i
in the form
+ 1/»
Give the values for K and I/> in terms of K, and K6 of Eq. (6-121).
These f experim when th steady-s the rna: is, whic greater'
· Continued / Ch. 6
o. 6/
:apacitor has an osed. Determine
6-33.A switch is closed at t = 0 connecting a battery of voltage V with a series RL circuit. (a) Show that the energy in the resistor as a function of time is
Problems
y state with the n expression for
-0
+
169
WR
=
V2( R
t
2L c: R'L +R t
-
L 2Re-
3L). 2R
2R'L t,
-
JOU
Ies
(b) Find an expression for the energy in the magnetic field as a function 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 as I eo. (d) Find the total energy supplied by the voltage source in the steady state. -4
)(t)
6-34.In the series RLC circuit shown in the accompanying frequency of the driving force voltage is (I) W =
eo,
(2) W = Wn~
voltage SOurce. ig to the differfixed in value of the roots of characteristic ) the following
charactensnc
~q. (6-121).
the
(the undarnped natural frequency) (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 the maximum value of the transient greater? (b) In which case (that is, which frequency) is the maximum value of the steady-state current greater?
). 6-29, except istic equation rnpare results .is, and detere network as : roots of the s plane. Find
diagram,
~
100 sin wt
Fig. P6-34.
~
!It,)
lJ1F
i Network Theorems / CIr.9
". 9/ Problems
271
FURTHER
READING
CHoo.IAN, PAULM., Basic Network Theory, McGraw-Hill New York, 1969. Chapter 5.
Book Company,
DfsoER, CHARLES A., AND ERNESTS. KUH, Basic Circuit Theory, McGrawHill Book Company, New York, 1969. Chapters 16 and 17. Kuo, FRANKLlNF., Network Analysis and Synthesis, & Sons, Inc., New York, 1966. Chapter 7.
2nd ed., John Wiley
(b)
: 6 for which the
rolled source
DIGITAL
which
COMPUTER
The topics of this chapter are not directly related to the use of the digitalcomputer, since new concepts and theorems are stressed. Use the timeavailable for computer exercises in completing more of those suggested at the end of Chapter 3.
(9-94) find the impedance ng a voltage source rrent I(s) under the zero, meaning that
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 impedance of the passive network N is found to be Z(s) = (s
'k(S)
(s
(9-95)
equired
impedance
(9-96)
EXERCISES
(a) With N connected to what will be the complex (b) With N connected to what will be the complex
+ 2Xs + 3) + IXs + 4)
the voltage source as in (a) of the figure, frequencies in the current i, (t)? the current source as in (b) of the figure, frequencies in the voltage VI(t)?
2s'
rk
(S2
+ 3s + 5s + Ss + + IX2sz + 2s + 4) 3
2
1
N
VI
Solve part (b) only. 9·3. Consider the two series circuits shown in the accompanying Given that VI(t) = sin 103t, vz(t) = e-IOOOtfor t > 0, and C
is constructed
L'
R
C
C
seful
artifice that he operation of Ig the amount of rnplish this.
(a)
+
Z(s) =
(9-98)
N
VI
9·2. Repeat Prob. 9-1 if
(9-97)
-"
~
~
(b)
(a)
Fig. 1'9-3.
=
figure. I j.l.F.
(b)
Fig. P9-1.
Impedance Functions and Network Theorems / Ch. 9
9-4.
Ch. 9 / Problems
(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) Discuss the physical meaning of this problem in terms of the complex frequencies of the two series circuits.
9-8.
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.
9-9.
rr-
For the RC ance, Z(s), i p(s) andq(s
of Prob. 9-1 Repeat Pro
9-10. Repeat Pr figure.
0::-
V -
Fig. P9-4. 9-5.
This problem is similar to Prob. 9-4, except that the transform network 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. 9-11. Repeat P this case
Fig. P9-S.
9-6.
9-12. Two blac known th contains the input
In the network of the figure, the switch K is closed at t = 0 and at t = 0 - the indicated voItages are on the two capacitors. Repeat Prob.
(b) Inves
network. conditio!
9-4 for this network.
r-
o
r
:
I I I I
1 I
Fig. P9-6.
I I I I
1 0>---+-1-
9-7.
Determine the transform impedances for the two networks shown in the accompanying figure.
z~g'1~\ I Fig. P9-7.
L.
9-13. Repeat panying
5Slepian, 6Macklel September, 191
seorems I CIr. 9
a. 9/ Problems
all t> Q. (b) d. (c) Discuss complex fre-
f.I. For the RC network shown in the figure, find the transform impedance, Z(s), in the form of a quotient of polynomials, p(s)/q(s). Factor pes) and q(s) so that Z(s) may be written in the form of the impedance ofProb.9-1.
= 0, a steady
,.9. Repeat Prob. 9-8 for the LC network of the accompanying
273
figure.
2F
2F
z~
~ open, draw iresenting all
Fig. P9-S.
Fig. P9-9.
,.10. Repeat Prob. 9-8 for the RC network figure,
shown in the accompanying
nsform netloop basis, /oltages are 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).
~ 0 and at peat Prob.
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
R
:
:R- VCfT I
I
CRI I
I L
J
I
L
shown in
J
(b)
(a)
Fig. P9-12.
9-13.Repeat Prob. 9-12 by comparing the network shown in the accompanying 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.
O-------r------,
zIH
Fig. P9-11.
Ch. 9 / Problems 274
Impedance Functions and Network Theorems / Ch. 9
r----------, I
R R=
,fF.c
c
R
Fig. P9·13. 9·18. The accom 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 switch K
is opened. Starting with the transform network found in Prob. 9-4, determine the voltage across the switch, Vk(t), for t :2: O.
sources in network, fi expression
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
=
ll(s)
(S2
IO(s2 + s + 1) lXs2 + 2s + 2)
+
9·19. Th1netwo
IH IF
current so determine
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 _ l( ) - (S2
s(s
+ 4s +
+ 2X5s + 6) 13XlOs2 + 18s + 4) 9·20. The ne
this netw RL•
1n
Fig P9·16. 9·17. For the network of the figure, show that the equivalent Thevenin
network is represented by Vs
=
--tV (1 + a + b -
and 3-b z, =-2-
ab)
Theorems /
275
cs. 9 1n
Fig. P9-17.
switch K closed - 0 the switch K d in Prob. 9-4, ~ O.
9-18.The accompanying network consists of resistors and controlled sources in 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.
current zero at orm of the gen-
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',
It the generator
Fig. P9-19. 9·20. The network shown is a simple representation this network, determine the Thevenin equivalent RL•
rlent Thevenin
Fig. P9·20.
of a transistor. For network for the load
276
Impedance Functions and Network Theorems I Ch. 9
9-21. The network in the figure contains a resistor and a capacitor in addition to various sources. With respect to the load consisting of RL in 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.
9-30. Using alent ditions.
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.
9-31. The values dete equiva
eorems /
cs. 9
lcitor in addisting of RL in ~rk.
Ch.9 I Problems
277
J.28.Thenetwork shown in the figure is a low-pass filter. The input voltage VI(t) is a unit step function, and the input and load resistors have the value R = ...;LIe. By using Thevenin's theorem, show that the transform of the output voltage is
+
R
~Itl
n equivalent Fig. P9-28. n equivalent
>rk given in described in
9·29.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 of I-V amplitude and T-sec duration, show that vz(t) is also a pulse, and find its amplitude and time duration.
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 equivalent network for the terminals a-b in the figure for zero initial conditions. 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 at a-a', (b) Determine the equivalent 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 network to compute the transform of the current in RL•
Fig. P9-31.
9-33. Assuming zero initial voltage on the capacitor, determine 1 he equivalent Norton network for the resistor Rx. +
-
In this char admittance extended. F different par mathematic, functions arl
10·1. TERMI]
Fig. P9-33.
Consid elements. T( represented I fastened to a access, the en are required necting some ments. The IT the terminal! another pair name terminc
I Terminal This results in: this chapter.
tnd Zeros / Ch. 10
o, /0 I Problems
i). The stability
whichis a quad, indicating that pes) has two zeros in the right halfplanefrom the quad. Dividing Eq. (10-123) into Eq. (10-121) gives the factor 2S2 s 1 which may be analyzed by the quadratic formula.
317
+ +
aial or an odd e even polynoS" ja)(s - ja)
+
other possibiIlay be reached D-31 which are
;b) (10-120) ) if b > a. In on applies for
FURTHER
READING
CHARLESA., AND ERNESTS. KUH, Basic Circuit Theory, McGrawHill 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., AND Joss B. CRUZ, JR., Engineering of Dynamic Systems, John Wiley & Sons, Inc., New York, 1969. Chapter 8. DESOER,
DIGITAL
COMPUTER
EXERCISES
Two topics of this chapter which lend themselves to computer solutionare the determination of the roots of a polynomial and the determination of the locus of roots. The sections of Appendix E devoted to these topicsare E-l and E-9.5. In particular, see Huelsman, reference 7, Appendix E-IO, and his discussions of root-locus plots in Section 10.3, and MeCracken, reference 12, Case Studies 21 and 23. zeros, symvith respect to irm a quad of
.rr
PROBLEMS 10-1. For the network Z12 = V2(s)jII(s).
5
shown
in the accompanying
figure,
determine
(l0-121)
Fig_ PlO-I.
(10-122) 10-2.
Consider the RC two-port network figure. For this network show that
:) is G
(10-123)
12 -
r l..$2
shown
in the accompanying
S2 + (R1C1 + R2C2)SjRIR2CIC2 + 1jR R2C C2 + (R1C1 + R C2 + R2C2)SjRtR2CIC2 + 1jR R2C C2 1
1
1
1
1
]
Network Functions; Poles and Zeros I Ch. 10
318
a..
10 I Problems
10-7.
For the net' specified, dr
Fig. PlO-I. 10-3.
(a) For the given network, show that with port 2 open, the input impedance at port 1 is 1 (b) Find the voltage-ratio transfer function, G12 for the two-port network.
n.
1
10
+
10-8.
Fur the RI
~------r---------~2 +
10 10
2F
10-').
For the g
~------------------------~~--------~2 Fig. PI0-3. 10-4.
and dete
For the resistive two-port network of the figure, determine numerical value for (a) G12, (b) Z12, (c) Y12, and (d) tX12•
the
U!
10-10. For the transfer
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 this network, compute G12 = Vz/V!.
Fig. PIO-5.
and controlled
v,u::Jln I ~
1n
Fig. PI0-6.
sources.
For
2V;
10-11. Foreal a volta V. at I
~eros/
cs. 10
u: 10 I Problems
319
10-7. For the network of the accompanying specified, determine IX 12 = 12//1,
figure and the element values
In
\' the input lnsfer func-
Fig. PIO-7. 10·8. Fur the RC two-port G
-12 -
i-
network
shown in the figure, show that 1/R1R2CIC2
LS2
+ (RICI + RIC2 + R2C2)S/RIR2CIC2
+
] 1/R1R2CIC2
02
~
+
Fig. PlO-So 10·). For the given network, 2
o
show that YI2
and determine
=
K(s
(s
+
+
2)(s
1)
+
4)
the value and sign of K.
ine the
Fig. PIO·9. 10·10. For the network shown in the figure, show that the voltage-ratio transfer function is (S2 + 1)2
re is to G 12
1%12.
s. For
=
5s4
+ 5s2 +
I H
I H
+
~T_l
+
_ Fig.
______lll~2 r-io-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 G 12 = V2/ VI'
Network Functions; Poles and Zeros / Ch. 10
320
Ch. 10/ Problems
R Z(s)
L
(a)
1n
2
if
2
10-17. A system has a which may be a system to a step of K, as a funcii done by the .l0-18.
(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
10-14. Repeat Prob. 10-12 for the network
of Fig. PIO-ll(b). 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 = 121/1, 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
-
(s - Pl)(S
ZI)
- P2)
and determine ZI, p i, 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.
10-19. A system has s = -3, and One term of K3e-r sin (t + of a between
id Zeros / Ch. 10
Ch. JOI Problems
321
s plane
jw
JTITrr-2
I
1.5
(J
c
ZlsI
I
-3
I I I I
vrrr
L
2
I
*-- -Ibl
(c)
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 of K( 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.A system has a transfer function with poles at s = -1 ± j 1 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 a for values of a between 0 and 5. This may be done graphically. rrt 2 in a I-Q
re the current load. Assign npute G12 =
10·19.A system has a transfer function with poles at s = -1 ± j 1and 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 a for values of a between 0 and 5. jw x
source I( at lirections for Z12 = V2/1(
-4
-3
-2
he poles and 'e with ZUO)
(J
-1 x
hunt peaking
j1
-a
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 (d) SS + 2S4 + 2s3 + 4S2 + l l s + 10 = 0
Network Functions; Poles and Zeros / Ch. 10
322 10-21. Given the equation S3
+
5s2
+
+
Ks
1 =0
(a) For what range of values of K will the roots of the equation have negative real parts? (b) Determine the value of K such that the real part 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 the following polynomials, (I) determine the number of zeros in right half of the s plane, (2) determine the number of zeros on imaginary axis of the s plane. Show method. 2s6 + 2s5 + 3s4 + 2s3 + 4S2 + 3s 2 = PI(S) S6 + 2s5 + 6s4 + 1Os3 + l1s2 + 12s + 6 = P2(S) 2s6 2s5 + 4S4 + 3s3 + 5s2 4s + 1 = P3(S)
+
+
+
10-25. For the following polynomial, determine the number of zeros in the right half of the s plane, the left half of the s plane, and on the imaginary 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
10-29. The am analyzed. for the istic eq without and K? amplifier
+
+
10-24. For the the (a) (b) (c)
10-28. For the Dete . the syst equation
the equation aos4 + als3
+
a2s2
+
a3s
+ a4
=
0
Use the Routh-Hurwitz criterion to determine a set of conditions necessary in order that all roots of the equation have negative real parts. Assume that all coefficients in the equation are positive.
n,
10-27. For the network of the figure, let RI = R2 = 1 Cl = 1 F and C2 = 2 F. For what values of k will the network be stable? In other words, for what values of k will the roots of the characteristic equation have real parts in the left half of the s plane?
Fig. PI0-27.
10-30. The ne oscil/at gmRL~ = 29 is
tnd Zeros /
cs. 10 Cl. 10 I Problems
e equation have eh that the real
itions-
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 for the system to oscillate, that is, for the roots of the characteristic equation to be conjugate and have zero real parts. 10-29. The amplifier-network shown in the accompanying figure is to be analyzed. (a) What must be the relationship between RI, Rz, and K for the system to be stable (real parts of the roots of the characteristic equation are zero or negative)? (b) For the system to oscillate without damping, what must be the relationship between RI> Rz, and K? What will be the frequency of oscillation? Assume that the amplifier has infinite input impedance and zero output impedance.
Amplifier gain -K
er of zeros in . of zeros on
+
cJ
V2
Fig. PIO-29.
zeros in the and on the 24s
10-30. The network of the accompanying figure represents a phase-shift oscillator. (a) Show that the condition necessary for oscillation is gmRL ~ 29. (b) Show that the frequency of oscillation when gmRL = 29 is Wo = 1/../6 RC.
+9 +
+
18 Vg
Fig. PIO-30. :onditions tative real tive, ·1 F and 'In other :tic equa-
10·31. Show that with Z.Zb panying figure,
=
Rij in the bridged-T Vz
of the accom-
1
VI = 1
and the input impedance
network
+ Z./Ro
at port 1 is
Zin
=
Ro.
Fig. PIO-31. 10-32. An active network
is described S2
by the characteristic
+ (3 + 6K )s +- 6K I
2
=0
equation
324
Network Functions; Poles and Zeros / Ch. 10 It is required that the network be stable and that no component of its response decay more rapidly than Kte-31• Show that these conditions are satisfied if K2 > 0, I Kt I < i, and K2 > 3Kt. Crosshatch the area of permitted values of Kt and K2 in the Kt-K2 plane.
10-33. Values for the elements of the Routh array can also be expressed in terms of second-order determinants multiplied by - 1. Thus the formulas shown in Fig. 10-30 become
a5 \
b5
Using the indexing scheme suggested on page 312, give a general formula for the elements of the Routh array.
We next turn 0 is useful in de to describe an tions are like I restrictions im open or be sho
In the I identified+tw and currents box enclosing voltages and important ina transformq and V2 and 11 the four vari of them dele specified, then four variabl depending on variables. In in Table 11-1.
Two-Port
342
DIGITAL
COMPUTER
Parameters
( Ch. I
EXERCISES
In connection with the matrix multiplication of the ABeD parameter matrices for networks connected in cascade, see the exercises in references cited in Appendix E-3.1. The determination of the other parameters involves ordinary network analysis with the special condition that the one pair 01 network terminals be either open or shorted. These topics are considered in references cited in Appendix E-8.
PROBLEMS In the problems
to follow, all element values are in ohms, farads, or
henrys. 11-1.
Find the y and z parameters the figure if they exist.
11-2.
For the two networks eter's if they exist.
11-3.
Find the y and z parameters panying figure.
1~2
1'0>__------02'
for the two simple networks
shown in
shown in the figure, find the z and y paramfor the resistive network
of the accom-
la)
11-8.
The and det
11-9.
Find
1~2 ~ I1
1"0
1'~2'
--'
-'-_~
2' Fig. Pll·3.
(bl
11-4.
Fig. PIl-l.
The network of the figure contains a current-controlled source. For this network, find the y and z parameters.
current
+
Fig. Pll·4.
(a)
11-5. l:n
[ (b)
Fig. Pll-2.
Find the y and z parameters for the resistive network containing a controlled source as shown in the accompanying figure.
+
Ideal
Fig. Pll·S.
11-10. The
343
SES
The accompanying figure shows a resistive network containing a singlecontrolled source. For this network, find the y and z parameters.
e ABeD parameter ercises in referel1Cel parameters involveI that the one pair 01 cs are considered iD
:,
2Q
IQ
~
~ Fig. Pl1-6.
ohms, farads, or
11·7. Thenetwork of the figure contains both a dependent current source and a dependent voltage source. For the element values given, determinethe y and z parameters.
etworks shown in e z and y paramk of the accom-
Fig. PH-7.
11·8. The accompanying network contains a voltage-controlled source and a current-controlled source. For the element values specified, determine the y and z parameters.
rolled current ).
f--,-----jf---,-----o2
+
+ + 2\l
1 \1
\.j 1'0-----'-------'---02'
Fig. PH-9.
Fig. PH·S. 11·9. Find the y and z parameters for the RC ladder network of the figure.
containing a
~F
11·10. The network of the figure is a bridged- T RC network. For the values given, find the y and z parameters. Il-Il, Determine the ABCD (transmission) parameters for the network of Prob. 11-10. n·12. The accompanying figure shows a network with passive elements and two ideal transformers having I: I turns-ratios. For the element
values specified, determine the z parameters.
~~ 1~02
1'0
T
1F 2
Fig. PH-lOo
02'
344
Two-Port Parameters
/ Ch. 11
r-----<> 2
1'0-------'-------"
'------02'
f\fv--+~\.f\fv~-~2
Fig. PH-12.
1'0-------'-------02'
Fig. PH-13.
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. 11-13 over a different range of frequencies. For this network, determine (a) the h parameters, and (b) the g parameters.
11 o--'----AJV''v--+--il-----'---<>2 11-15. Show -
that the standard T section representation of a two-port network may be expressed in terms of the h parameters by the equations shown in the accompanying figure.
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 network
embedded in another resistive network. The resistive network is 1/2 f!
2
+
+ la
V;
2f!
N
If!
~ 2'
J'
Fig. Pll-16.
Parameters /
c« 11
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. 1n
1/2 I! 02'
2n 10--
2
stor over a given (a) the h paramising Table 11-2.
2
. of Prob. 11-13 'k, determine (a) a two-port net'y the equations
Fig. PH-17.
11·18.The resistive network shown in the figure is to be analyzed to determine the Y parameters. In
2n
I'o------~L...---'------'-.--'--
In
.
Fig. PH-IS.
-port network e network is
11-19.The accompanying figure shows two two-port networks connected in parallel. One two-port contains only a gyrator, and the other is a resistive network containing a single controlled source. For this network, determine the Y parameters. K
+
1n
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 For these specific element values, determine the y parameters.
=
I.
11-21. The network of the figure is of the type used for the so-called "notch filter." For the element values that are given, determine the y parameters. 2F
2F
IF
III
1'0---------'.--...L.-------o2'
Fig. Pll-21. 11-22. Let the element values for the network shown in Fig. 11-15 be as follows: 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 is the equivalent of the other. Is this assertion correct? Show reasoning. If it is, might one network have an advantage over the other as far as the calculation of network parameters is concerned?
--
-12
II
+
~
+
V;
C3
C2
Cl
lal
C3
II
:
E fk~ IR' It
~R'
Ib,
12
+~
Fig. Pll-23. 11-24. Two two-port networks are said to be equivalent if they have identical y or z parameters (or other of the characterizing parameters). In this problem, we wish to study the conditions under
eters /
cs. J J 347
and Ra = 1. meters. called "notch they param-
which the z-network of (c), is equivalent to the T-network Show that the two networks are equivalent if Ya
=
Z2 D'
Yb
Z3
= 75'
an
d
of Cb).
Y =Zt D C
where
1:!L:2
f---,.----o2
1"C>---'-----'---o2'
r
2'
(a)
(b)
Fig. Pll-24.
1-15 be a5
c, = i-F.
that one is easoning. er as far as
11·25. Derive equations similar to those given in Prob. 11-24 expressing ZI> Z2, and Z3 in terms of Yr, Yb, and Ye' This result and that given in 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 network of the figure to obtain an equivalent (a) T-network, (b) zr-network.
1 !!
211
1F
l~y~2
r----o
+ Fig. Pll-26.
1'0>------L-----_02'
2F
11·27. Apply the T-n transformation to obtain an equivalent (a) T-network and (b) zr-network for the capacitive network given in the figure.
l'o~----
I
Fig, Pll-27.
11·28. Apply the T-7t transformation as many times as is necessary to the inductive ladder network shown in the accompanying figure in order to determine the numerical values for the equivalent (a) T-network, (b) z-network.
13JI2 1H
--0
Fig. Pll·28.
, have X1ramunder
1'0
1H
~
2 H
2'
11·29. The network given in the figure is known as a lattice network; this lattice is symmetrical in the sense that two arms of the lattice have impedance Z, and two have impedance Zb' For this network, (a) determine the z parameters, and (b) express Z; and Z; in terms of z parameters.
Fig. Pll-29.
02'
348
Two-Port Parameters / Ch. If
ll-30. In this problem, we consider two-port networks having a symmetry property illustrated in (a) of the figure: If the network is divided at the dashed line, the two half networks have mirror symmetry with respect to the dashed line. The two half networks are connected by any number of wires as shown, and we will consider only the cases in which these wires do not cross. If a network meeting these specifications is bisected at the dashed line, then with the connecting wires open, the input impedance at either port is Z 1/20c as shown in (b). Similarly, with the connecting wires shorted, the impedance at either port is ZI/2,c as shown in (c). A theorem due to Bartlett states that these impedances are related to those given for the arms of the lattice in Prob. 11-29 by the equations
This is known as Bartlett's bisection theorem, and permits an equivalent lattice network to be found for any symmetrical network. Prove the theorem.
11
~,
---12
I
~N
I
r---L--l
!N
I
I
~, 11-35. Th z
(a)
:,0,-1
~N
~----~
:,,,-1
~N
r----~
(c)
(b)
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 obtain a lattice equivalent network.
11-36. Th
ha
11-32. Apply the theorem of Prob. 11-30 to the network of Prob. 10-31 with the terminating resistor Ro removed to find the lattice equivalent of the given network. 11-33. (a) Show that the network of the accompanying figure satisfies the requirements described in Prob. 11-30. (b) Find the lattice equivalent of the network.
11-37. T ZL
tters / Ch. J I 349 'ng a sym, network is nirror syrn:tworks are ill consider Irk meeting h the con~ ZI/20c as heimpedhe to Barten for the
Cl
0--L-fV\/v--,--JV\/V __ ~2
1'0------------'---
Fig. Pll-33.
--<:
2'
U·34, Find the lattice equivalent
of the network of the accompanying figure making use of the results of Prob. 11-30.
n equivark. Prove
12
~
+ ~~~/\/\~-+-J\AJ\'~~2
V2
r--o
Fig. Pll-34.
_____T_
3 F _
1'0--
, 02
JI·JS, The network N in the accompanying figure may be described z parameters. Show that with port 2 open,
by the
---Fig. Pll-3S. Prob. obtain
JI·36. The network N in the figure is terminated having impedance ZL = Jf YL. Show that
at port 2 with a network
10-31
uivaes the valent
Fig. PII-36. 1l·37. The network N of the figure is terminated at port 2 in impedance ZL = J/ r L· Show that the transfer impedance for the combination is Z
12
-
ZL +ZL
Z21 Z22
350
Two-Port Parameters / Ch. //
Fig. Pll-37. 11-38. The figure shows two two-port networks connected in cascade. The two networks are distinguished by the subscripts a and b. Show that the combined network may be described by the equations
and _ Y 12 -
-Y12aY12b Y11b
for the transfer
+
Y22a
functions.
Fig. Pll-3S.
Stated in function when s= In generated swinging these dev' voltage is
Sinusoidal Steady-State Analysis I Ch. 12
366
FURTHER
READING
BALABANIAN,NORMAN, Fundamentals of Circuit Theory, Allyn and Bacon, Inc., Boston, 1961. Chapter 4. CHIRLIAN, PAUL M., Basic Network Theory, McGraw-Hill New York, 1969. Chapter 6.
Ch. 12 I Problems 12-3.
Starting \\ similar to
12-4.
Given the
Book Company,
CLOSE, CHARLES M., The Analysis nf Linear Circuits, Harcourt ovich, Inc., New York, 1966. Chapter 5.
sin determin
Brace Jovan12-5.
HUANG, THOMASS., AND RONALD R. PARKER, Network Theory: An Introductory Course, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1971. Chapter 10.
Show tha
In other arbitrar a sinuso
LEO"'",.BENJAMINJ., AND PAUL A. WINTZ, Basic Linear Networks for Electrical and Electronics Engineers, Halt, Rinehart & Winston, Inc., New York, 1970. Chapter 4. •
12-6.
MANNING, LAURENCEA., Electrical Circuits, McGraw-Hill Book Company, New York, 1965. Chapter 6.
Using t in term
12-7.
Using equati in Cha di (a) lit
WING, OMAR, Circuit Theory with Computer Methods, Holt, Rinehart Winston, Inc., New York, 1972. See Chapter 7.
DIGITAL
COMPUTER
&
(b) di
EXERCISES
This chapter is devoted to a discussion of networks operating in the sinusoidal steady state. Analysis of large systems in this condition is straightforward but tedious if done with pencil and paper, and the computer can be used to advantage. See the references cited in Appendix E-8.3 for suggested exercises. In particular, see Chapters 9 and 10 of Huelsman, reference 7 in Appendix £-10, and Chapters 3 and 11 of Ley, reference I1 in Appendix
E-I0.
dt
Cc) ;' 1
(d) d
12-8.
Repe onl (a)
PROBLEMS
(b)
12·1.
Let v(t) = VI cos Wit for Eq, (12-1) and carry out the derivation leading 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. I I
f-+lO
,v(t) I
J 0
0.1
f--lO
t',~f-
I-
•j
1\ f-
Thl o~ del
12-11. F N
02
I
12-9.
12-10. In is 12
l-
v
(c)
I I I I
Fig. Pl1-2.
12-12. 11 f
lysis /
cs.
11
Starting with the rotating phasors, e->', show by a construction similar to that illustrated in Figs. 12-4 and 12-5 that
and Bacon,
sin?
WI
+ cos-
WI =
I
Given the equation
Company,
+ 3ft
sin 377t
determine A and
race Jovan-
sin (377t
+ ~)
=
A cos (377t
+ e)
e.
12·5. Show (ha t
i:
An Introing, Mass.,
+ e)
12·7.
Using the method of Section 12-3, solve the following differential equations for the steady-state solution (called the particular integral in Chapter 6): Ca )
di dt
+ 2'I
+ i = cos
(c) ::
+ 3i
(d) ~:;
Cor sug:ference
d2i. (e) dt2 -I-
=
+ 2 :~ 1
=
=
3t
cos (21
+ 45°)
-I- i
5 sin (2t -I- 30°)
=
.
2 Sin t
12·8. Repeat Prob. 12-7 for the following differential only for the steady-state solution: d+! di (a) dt~ + 2 d; + 2i = 3 cos (I + 30°) dt i (b) dl~
Ivation
(c) :: 12.9.
+ 4i -I- 2i
C and
. 2 = Sin t
Cb) :;.
in the straightter can
Ig
1/
2, determine
e
Using the equation of Prob. 12-5 with in terms of AI, A2, CPI,and CP2'
nehart &
or vet)
= Csin(wlt
12.6.
~ompany,
ncy,
+ CPk)
(WIt
In other words, show that the sum of any number of sinusoids of arbitrary amplitude and phase angle but all of the same frequency is a sinusoid of the same frequency.
for ElecInc., New
!)pendix
Aksin
k"1
= 3 cos (21 =
sin 2t
equations,
solving Fig. P12-9.
+ 45°)
+ cos
t
The network of the figure has a sinusoidal voltage source and is operating in the steady state. Use the method of Section 12-3 to determine 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 network is operating in the steady state, Make use of the method of Section 12-3 to determine the node-to-datum voltage VI(t). 12.11. For the given network, find v.(t) in the steady state if Make use of the method of Section 12-3. 12.12. In the resistive network shown in the figure, VI for all t. (a) Determine i.Ct). (b) Determine ib(t).
=
VI =
Fig. P12-10.
+ L'd
VI
2 sin 2t.
2 sin (2t -I- 45°)
Fig. PI2-lI.
Sinusoidal
368
Steady-State
Analysis
I Ch. 12
Fig. P12-12.
12-13. The network shown in the accompanying figure is operating in the steady state with sinusoidal voltage sources, If t'l ,= 2 cos 21 and V2 = 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 the sinusoidal 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
c Fig. 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) and t'b = 5 sin (10001 - 45°), The magnitude of the impedance of the capacitor is 10 Q. Determine the impedance. at the input terminals of the network
N.
+
+
Fig. Pl2-IS.
12-16. In the network shown, the network is operating given, determine
10 sin 106t and i, 10 Cl)S JO"t. and in the steady state, For the ,'kment values
1'1
c-
the node-ta-datum
voltage
/'/t),
talvsis /
ci. Il
Ch.12 I Problems
369
ating in the 2 cos 21 and
Fig. P12-16.
12·17. For the hridged-Tnetwork of the accompanying figure, t'l =·2 cos t and the system is in the steady state, For this network, (a) determine i.Ct), and (b) determine ib(t),
+ t'l
ting in the Lz - 1 H,
Fig. P12-17.
12·111.The network of the figure is operating in the steady state with 1'1 . 2 sin 21 and K I ,= -~, Under these conditions, determine i2Ct).
ady state, 60') and cc of the rerlllinals
The following series of problems are intended to give practice In constructing phasor diagrams. The network shown in the figure for Prob. 12-19 is assumed to be operating in the sinusoidal steady state, In the element values given in the table, a double entry in column 1 implies a series connection, in column 2 a parallel connection, For each problem, (a) determine VICt), (b) Draw a complete phasor diagram showing all voltagcs and all currents, as well as all relationships between the voltages and the currents,
If!
Fig. P12-18.
+
cz= V", sin (wt+CP)
16/.
and I values Fig. P\2-19.
SinusoidalSteady-State Analysis / Ch. 12
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.
Network 1
Network 2
Vm
R = 1 R =2 R = 20 R =2 L=1 C=2 L=3 C=1 R = I, C = 1 R = I, L = 2 L=I,C=2 R = I, C = 1 R = 3, L = 2 L = I, C = 2 L = I, C = 2
C=2 C=1 C=! L=2 R=1 R =2 C=1 L=! L=2 C=! R = 1 R = I, L = 2 R=I,C=! R = I, C = 1 R = I, L = 2
2 10 1 100 10 3 10 1 2 2 10 10 1 100 1
eo
1 2 0.1
1
! 1
! 2 1 1
! 1
! 1 1
ifJ
-30° 45° 0° 30° 0° 45° -45° 0° 30° 45° 0° 90° 0° -90° 0°
12-34. The network of the figure is operating in the sinusoidal steady state 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 currents (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
! 2 1 1
-30 45 00 30 00 450 -45 00 30 45 0
0
+ "2
0
0
00
1 1
00
0
soidal steady values given,
c
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 cp varies with T. (b) For a fixed T, show how cp varies with CD. (c) For a fixed T, show how the maximum amplitude of V2 is related to the maximum amplitude of VI as a function of CD.
0
900 00 -90
!
371
0
! 1
Ch.12 / Problems
= -.!L/C.
es and Cur: condition t (a) be CDJ• .epeat part
Fig. P12-36.
-: Frequency Response Plots I Ch. 13
408
["11.
13/
13-7.
PROBLEMS 13-8. 13-1.
Sketch the (a) magnitude, (b) phase, (c) real part, and (d) imaginary part variation of the following network functions with ro for both ro> 0 and ro < 0; (a) 1
+ j2ro
1 (b) 1 _ j2ro ( ) c 2H
+
13-2.
Consider the RLC one-port network shown in the figure. For this network, determine the driving-point functions Z(jro) and Y(jro). For each of these functions, plot the magnitude, phase, real part, and imaginary 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-ratio transfer function, Gll(jro) = VZ(jro)/V1(jro). Plot the variation of this function with ro for the two methods employed in Fig. 13-7.
111
Fig. Pl3-1.
+
(1 - 2roZ) jro 1 j2ro
13-5
13-4.
The two-port network of the figure shows an RL network. the plots specified in Prob. 13-3.
Repeat
Fig. P13-4. 13-5.
Repeat Prob. accompanying
13-3 for the RC two-port figure. 0
+
VI
Fig. Pl3-5. 13-6.
-0
13·
network
'VV'v RI
shown
in the
0
+
lR' T
e
V2
-0
Show that the locus plot of Eq. (13-15) shown in Fig. 13-7 is a semicircle centered at Gll(jro) = 0.5 + jO for the frequency range
0<
ro
<
co.
1
s / Ch./J
409 Consider the locus plot required in Prob. 13-5. Show that this locus is a circle for the frequency range, - 00 < Cl) < 00. Determine the center of the circle and its radius.
aginary for both
Jl.8. Consider the RLC series circuit shown in the fipure. (a) Suppose that this network is connected to a sinusoidal voltage source. Plot the variation of the current magnitude and phase with frequency. (b) Suppose that the same network is connected to a current source of a sinusoidal waveform. Plot the variation of the voltage across the three elements using the same coordinates as in part (a). Element values are in ohms, farads, and henrys. +
f7)
or this Y(jw).
,and <0.
VI
+
il
-ratio on of -7.
t
V2
~F
~F
(a)
(b)
Fig. P13-S. 13·9. The figure shows a network which functions as a low-pass filter. For this network, determine the transfer function V2/11 and plot the magnitude 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 similar function to that considered in Prob. 13-9, namely, it is a low-pass filter. For this network, determine the transfer function V2/11 and plot 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 V2
I
v;- = S3 + 2S2 + 2s + 1
function
is
Frequency Response Plots I
410
)J
For this function, plot the magnitude frequency for the range 0 < Cl) < 4.
1F
2\1
c«
and phase as a function
13 of
13-12. For the RLC network shown in the figure, plot (a) the locus of the impedance function, and (b) the locus of the admittance function.
~H
13-13. Plot (a) the admittance locus, and (b) the impedance RLC network shown in the figure.
Fig. Pl3-12.
cs.
13-1
locus for the
13-14. The four-element network shown in the figure is to be analyzed to determine (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 impedance function, and (b) the locus of the admittance function.
IF
Fig. Pl3-l3.
1\1
I~
2F
Fig. Pl3-15.
~F
13-16. The RL network shown in (a) of the figure has element values such that the phase of the voltage measured with respect to the current is
Fig. Pl3-14.
(a)
+0'
+30'
r-.r-.
13
f
Phase of voltage with respect. to current in series RL circuit
------ I---- l--V
-
I--
+60'
+90'
o
10
20
30 Frequency
40 in cycles/sec (b)
Fig. Pl3-16.
50
60
70
/ ci. 1J et ion of s of the nction. for the
(1.13/ Problem s
411
that shown in (b) of the figure. From this information, pole and zero locations for Y(s).
determine
the
1),17. The figure shows the variation
of the magnitude of the current with for an RLC series network with an applied sinusoidal voltage of constant magnitude. From the figure, determine the locations of the poles and zeros of the admittance of the network.
w
zed to d (t)
I
dance 1.00
/
/
0.80
III 0.60
ch t is
+j5
r----,.----,
0.40
/
I I
/
_
1 pole
I----t-"*----j
+j4
1----+-----1
+)3
1----t----1
+j2
~.
\
<, ~
<,
----1----
/
0.20
o
\
Frequency response of RLC series circuit
~
+jl
1 zero
1----: :---+---:- ----jOj)0 2 1
/ 10
--
20
30 Frequency
40
50
60
'--------
-jl
------
=j?
in cycles/sec.
Fig. P13-17. 13-18. The pole-zero configuration shown in the figure represents the admittance function for the series RLC circuit. From the pole-zero configuration, determine: (a) the undamped natural frequency Wn, (b) the damping ratio (, (c) the circuit Q, (d) the bandwidth (to the half-power points), (e) the actual frequency of oscillation of the transient response, (f) the damping factor of the transient response, (g) the frequency of resonance, (h) the parameter values (in terms of L if the values cannot otherwise be uniquely determined). (i) Sketch the 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 the values of the parameters, R, L, and C change? 13-19. The figure shows two configurations of poles and zeros for a certain transfer function. Use a graphical procedure to determine the varialion of the magnitude of the network function for the two configurat ions. Superimpose the two plots on the same system of coordinates.
t------
- j3
------
1 pole
-- ------
'-- __
-- ...•---
-1- __
- j4
--' -
Fig. P13-JS.
j5
412
Response Plots / Ch. J3
Freouency
Ch. I
13-0.5+j2.0x -0.5+j
-0.5+j2x
jca
1.5 x
jw
(Scale factor -1)
-0.5+j1x
(Scale factor - 1)
3 zeros
3 zeros
-0.5-jl
x
-0.5 - j 1.5 x -O.5-j2
x
-O.5-j2x
13-
(b)
(a)
Fig. P13-19. 13-20. Show that the bandwidth a series RLC circuit.
B varies inversely with the circuit Q for
13-
13-21. Show that for an RLC series network the product of I Y Imax and the bandwidth B equals I/L, where L is the inductance .. jw
Fig. P13·22.
13-22. The two poles and zero shown in the s plane of the accompanying sketch 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 correspond with the same real part of the poles. The poles have positions corresponding to ( = O.707«() ~ 45°); eo, is the distance from the origin to the pole as shown. In this problem, we will investigate the effect of the finite zero by computations with and without the zero. (a) The bandwidth of the system is modified from the definition given in the chapter as the range of frequencies from ()) = 0 to the half-power point. Compute the bandwidth of the system with the pole-zero configuration shown above; compute the bandwidth with the zero removed. In which case is the bandwidth greater and by what factor? A graphical construction is suggested. 13-23. The Q of a series RLC network at resonance is 10 The maximum amplitude of the current at resonance is I amp when the maximum amplitude of the applied voltage is 10 V. If L = 0.1 H, find the value of C in microfarads. 13-24. A coil under test may be represented by the model of L in series with R. The coil is connected in series with a calibrated capacitor. A sine wave generator of 10 V maximum amplitude and frequency (l) =1000 radians/sec is connected to the coil. The capacitor is varied and it 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-
·/ c».
11
413
1),25. The network of the figure is found to have the driving-point ance
-+-
Z(s) = (s From this information,
I
imped-
106(s + I) + jlOOXs + I - jlOO)
determine
the values of R J, Rz, L, and C.
Fig. P13-25.
1J·26.For the following network function, plot the straight-line asymptotic magnitude response and the phase response. Use 4- or 5-cycle semilog paper. G(s)
for ]3·27.Given the network
100
s(l
+ O.OlsXI + O.OOIs)
function, G(s)
the
=
=
(1 (l
+ O.IsXI + O.Ols) + sXI + O.OOls)
Plot the straight-line asymptotic magnitude response. Use 4- or 5-cycle semilog paper. 13-28.Plot the straight-line asymptotic for the network function
magnitude
response
and the phase
response and phase angle
S2
G(s) = 100(1
+ 0.17sXI + 0.53s)
Use 3- or 4-cycle semilog paper. e
e
13-29.(a) Plot the straight-line asymptotic magnitude response, and (b) determine the actual (or true) response for the network function
h y
G( ) = 1000(1 s (1
+ 0.25sXI + O.ls) + s)( I + 0.025s)
On the same coordinate system, plot the phase response. or 5-cycle semilog paper for the plotting. (c)
13-30. Repeat (a) G(s)
Prob.
=
13-29 for the following
50~~1-~
network
Use 4-
functions:
°o~~~~~ 1000s
+ O.OlsXI + 0.0025s) ~(l + O.Ols) 180(1 + 0.05sXl + O.ools)
(b) G(s) = (I (c) G(s) =
13-31. (a) Plot the straight-line asymptotic magnitude response, and (b) determine the actual (or true) response curve for the network
G.
Frequency Response Plots I Ch. JJ
414
function (l
G(s)
+ 0.2s)
= 120s(S2 + 2s + 10)
(c) On the same coordinate system, plot the phase response. Use 3or 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) (b) G(s)
= (I +
lOOs
s
+ 0.Ss2XI +
O.4s
+ 0.2s2)
13-33. We are required to construct a network function G(s) satisfying the fo\1owing specifications: The asymptotic curve should have a lowfrequency response of 0 db/octave slope, and the high-frequency response has a slope of - 24 db/octave. The break frequency between these two slopes is at (J) = 1 radian/sec. At no frequency should the difference between the asymptotic and the true response exceed ± 1 db. 13-34. The figure shows two straight-line segments having slopes of ±n6 db/octave. The low-and high-frequency asymptotes extend indefinitely, and the network function the response represents has first-order factors only. Find G(s) and evaluate the constant multiplier of the function. M
Fig. PI3-36.
:1~
db
J2F IFJ~2
..,(Iog scale)
Fig. PI3-37. Fig. Pl3-34. +
Fig. PI3-38.
13-35. Repeat Prob. 13-34 if the response is changed only by the highfrequency asymptote having a slope of -18 db/octave. 13-36. For the two-port network shown in the figure, determine V1/VI and plot the magnitude response (Bode plot) showing both asymptotic and true curves. 13-37. Prepare a Bode plot for the network function V1/VI for the network shown 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.
Fig. PI3-39.
=
13-39. The figure shows an RLC network. For this two-port network, plot the transfer function GIl = V2/V\ showing both the asymptotic and true curves.
415 · Consider the following
transfer functions:
s-1
(a) G(s)H(s)
= K s
+
I
(b) G(s)H(s)
= Ks _
s
+
1 1
(c) G(s)H(s)
= s(
I
+ 0.05s)
K
For each of these functions: (a) plot GUw)H(jw) in the complex w = 0 to w = 00 with K = 1. (b) Determine the range of values of K that will result in a stable system by means of the Nyquist criterion. CH plane from
13-41. For the locus plot shown in Fig. 13-45, sketch the corresponding Bode plots for the magnitude and phase, making some assumption as to the frequency scale. Estimate the gain and phase margins and indicate 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 Bode coordinates. Make an assumption about the frequency scale along the 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, indicating whether 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 that A = -0.75, B = -1.3, and C = -2. Assuming that the plot is jlmCH
c
Fig. Pl3-45.
Frequency
416
Response
Plots / Ch. 13
made for a gain K, what is the range of values of gain for which the system will be (a) stable, and (b) unstable.
Ch.
13-~
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 by applying the Nyquist criterion. Repeat if P = I, P = 2. jlmGH
Re 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 to this 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 system with P = O.For this system, apply the Nyquist criterion to study the stability of the system. j lm Gfjw)
H(jw)
GH plane
Re GfjuJ I H(juJ)
Fig. PI3-49.
13
ponse Plots / Ch. J3 gain for which the
417
Ch. 13 / Problems
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 can about G(s)H(s) as a quotient of polynomials in s.
o.
for which P = this question by
j Im GH
t>=2.
w=o" Re GH
Fig. P13-S0. 13-51. The figure shows the feedback system for which the Nyquist criterion has been developed. For this problem, let H = 1, and G(s) _ K - (s - aXs 2Xs
+
Make use of the Nyquist for the case a = 1.
ern withP
criterion
+ 3)
to study this system for stability
= 2
ist criterion
t;
Fig. P13-SI. 13-52. Repeat
Prob. 13-S1 if a
13-53. Repeat
Prob.
=
2.
13-S1 if a = 4.
13-54. A system is described by the transfer system of Fig. P13-S1. G(s) iade for a sys:rion to study
=
functions
which relate to the
IOS
(s
+ 2Xs + IOXs + 20)
and H = 1. Make use of the Nyquist system is stable.
criterion
TO
determine
if this
13-55. Repeat Prob. 13-S4 for the given G(s), but for a new feedback transfer function H(s)
=
s t020
This causes cancellation in the product H(s)G(s) and is a form of compensation of a system to improve stability. Comment on the effectiveness of this compensation function.
Frequency Response Plots I Ch. J 3
4111
~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 quotient of polynomials in s. Is this system capable of oscillation? Make use of the Nyquist criterion in answering this question and in a general study of the system stability.
+
Fig. Pl3-56.
