Problems
Problems
------- -
Sections 5.1-5.2
MUmSI!~
--
-
-
-- ---
-- - - - -
--
-----
5.4 Find i L (in microamperes) in the circuit in Fig. P5.4.
folUlTISIH
Figure P5.4
a) Label the five op amp terminals with their names.
5kD
b) What ideal op amp constraint determines the
value of i'l? What is this value?
10 k.fl
c) What ideal op amp constraint determines the
value of (v p - VII)? What is this value?
d) Calculate
---
PSPICE
5.1 The op amp in the circuit in Fig. P5.1 is ideal.
ISPICE
-
165
5V
-5 V
va'
i .I
figure P5.1
,
4 kf}
15 kD
5.5 A voltmeter with a full-scale reading of to V is used
to measure the output voltage in the circuit in
/.!UlTlSII·\
Fig. P5.5. What is the reading of the voltmeter?
Assume the op amp is ideal.
PSPICE
+
3V
1""
8kD
Figure P5.5
2.2MD
5.2 Find i a in the circuit in Fig. P5.2 if the op amp is ideal.
ISPICE
MumSIH
Figure P5.2
lOkD
3.5 fJ..A
-i"
a.5mA
~
2.5 k.fl
5 k.fl
5.6 The op amp in the circuit in Fig. P5.6 is ideal.
Calculate the following:
PSPICE
/·1 UlTISII·!
MUlTlSIH
a) Calculate
va
if Va
b) Calculate va if Va
=
=
4 V and
Vb
2 V and
Vb
c) Calculate va if Va = 2 V and
d) Calculate
va
if Va
=
1 V and
e) Calculate
va
if Va
=
1.5 V and
Vb
d) ia
= 0 V.
= 0 V.
Vb =
=
Vb
Vj
b) va
c) i 2
5.3 The op amp in the circuit in Fig. P5.3 is ideal.
FSPICE
a)
Figure P5.6
40 k.fl
1 V.
2 V.
50 kO,
20ko'
..
= 4 V.
f) If Vb = 1.6 V, specify the range of va such that
the amplifier does not saturate.
i"
/',
25ko'
Figure P5.3
100 kD
5.7 A circuit designer claims the circuit in Fig. P5.7 win
PRACTICAL
PERSPECTIVE
PSPICE
+
/·\ULTISU.\
I'"
50 kD
produce an output voltage that will vary between
±5 as v g varies between 0 and 5 V. Assume the op
amp is ideal.
a) Draw a graph of the output voltage Va as a func­
tion of the input voltage v g for 0 :::; v g :::; 5 V.
b) Do you agree with the designer's claim?
166
The Operational Amplifier
Figure P5.7
Figure P5.10
IOkfl
50 kfl
aR", { R::,
2kfl
~
+1
-7V
10 kfl
Section 5.3
5.8 a) Design an inverting amplifier using an ideal op
amp that has a gain of 3. Use a set of identical
resistors from Appendix H.
DESlGN
PROBLEM
b) If you wish to amplify a 5 Y input signal using
the circuit you designed in part (a), what are the
smallest power supply signals you can use?
5.9 The op amp in the circuit in Fig. P5.9 is ideal.
PSPlCE
/.\UlTISlM
a) Find the range of values for
8mp does not saturate.
(T
b) Find i o (in microamperes) when
in which the op
if
= 0.272.
Section 5.4
5.11 Refer to the circuit in Fig. 5.12, where the op amp
is assumed to be ideal. Given that Ra = 4 k.o.,
Rb
5 k.o., Rc = 20 k.o., va
200 mY.
Vb = 150 mY, V c = 400 mY, and Vee = ±6 V, spec­
ify the range of R f for which the op amp operates
within its linear region.
5.12 The op amp in Fig. P5.12 is ideal.
PSPICE
1·IULTlSI~1
PSPICE
/·IULTlsm
Figure P5.9
a) What circuit configuration is shown in this figure?
b) Find
cr 50
Vu
if
Va
=
1 Y,
Vb
1.5 Y,
=
and
-4 V.
Vc =
12 kfl
c) The voltages va and V c remain at 1 V and -4 V,
respectively. What are the limits on Vb if the op
amp operates within its linear region?
kfl
--------.
50 kfl
Figure P5.12
5V
1.6 kfl
r---AIIIV------.----<lI-----/-e - I
+
250 mV
6.4 kD
220 kD
44 kfl
-5V
+
+
+
lOkD
80 kfl
+
+
1
5.10 a) The op amp in the circuit shown in Fig. P5.10 is
ideal. The adjustable resistor R;; has a maxi­
mum value of 100 k.o., and ex is restricted to the
range of 0.2 :s; ex :s; 1. Calculate the range of
V u if v g = 40 mY.
27.5 kfl
I'"
l'b
1'1'
l" c
3.3 kD
PSPlCE
.... .... ....
r·IUlTISl~1
b) If ex is not restricted, at what value of ex will the
op amp saturate?
5.13 Design an inverting-summing amplifier so that
DESIGN
PROBLH\
PSPICE
'·\ULTISl~1
V"
=
-(3va
+
5Vb
+
4vc
+ 2Vd)'
Start by choosing a feedback resistor (Rf ) from
Appendix H. TI1en choose single resistors from
Appendix H or construct resistor neworks from resis­
tors in Appendix H to satisfy the design values for Ra,
R b , Reo and Rd' Draw your final circuit diagram.
Problems
5.14 a) The op amp in Fig. P5.14 is ideal. Find Va if
Va = 4 V, Vb = 9 V, V C = 13 V, and Vd = 8 V.
Figure P5.17
28 k11
15PIC[
HUlTISII·\
b) Assume va v c , and
167
retain their values as given
in (a). Specify the range of Va such that the op
amp operates within its linear region.
Vd
8V
4kD
,..----/\Iv\r------.-----te+
12V
Figure P5.14
v,
220 kD
+
7 kD
10kD
/'"
5.18 The op amp in the circuit shown in Fig. P5.18 is ideal,
a) Calculate V o when v g equals 4 V.
1·IULTlSm
rSPICE
I'"~
10 k11
b) Specify the range of values of v g so that the op
amp operates in a linear mode.
)
c) Assume that v~ equals 2 V and that the 63 kfl
resistor is replaced with a variable resistor. What
value of l.he variable resistor will cause the op
amp to saturate?
5.15 The 220 kD feedback resistor in the circuit in
Fig. P5.14 is replaced by a variable resistor R r . The
HUlTlSJ:~
voltages Va - Vd have the same values as given in
Problem 5,14(a).
?5PIC[
Figure P5.18
63 k11
a) What value of R r will cause the op amp to satu­
rate? Note that 0 ::s; R f ::s; 00.
30 k11
b) When RJ has the value found in (a), what is the
current (in microamperes) into the output ter­
minal of the op amp?
12 kD.
-l2V
/'"
27 kD
68 kD
Section 5.5
5.16 The op amp in the circuit of Fig. P5.16 is ideal.
PSfICE
HUlTlSll1
a) What op amp circuit configuration is this?
b) Calculate va'
Figure P5.16
80kD
40kD
T
5.19 a) Design a non-inverting amplifier with a gain of
4. Use resistors from Appendix H. You might
need to combine resistors in series and in par­
allel to get the desired resistance. Draw your
final circuit.
b) If you use ± 12 V power supplies for the op amp,
what range of input values will allow the op amp
to stay in its linear operating region?
5.20 The op amp in the circuit of Fig. P5.20 is ideal.
PSPICE
/·\ULTISm
3V
..,
a) What op amp circuit configuration is this?
b) Find va in terms of v, .
c) Find the range of values for V s such that V u does
not saturate and the op amp remains in its linear
region of operation.
Figure P5.20
5.17 The op amp in the circuit of Fig. P5.17 is ideal.
60 kD.
a) What op amp circuit configuration is this?
b) Find
Vo
in terms of VI'
c) Find the range of values for V s such that Va does
not saturate and the op amp remains in its linear
region of operation.
2 kD
1'"
168
The Operational Amplifier
5.21 The op amp in the circuit shown in Fig. P5.21 is
ideal. The signal voltages va and Vb are 800 mY and
/·lULTISH·l
400 mY, respectively.
b) Using the values found in part (a) for R f , R b , and
R c, find (in microamperes) i a , ib , ie, i g , and is
when va = 0.5 Y, Vb = 2.5 V, and V e = 1. v.
PSPICE
a) What circuit configuration is shown in the figure?
b) Calculate
va
in volts.
Figure P5.23
c) Find i a and i b in microamperes.
Rc
d) What are the weighting factors associated with
Va and Vb?
R s = 15 k.ll
Figure P5.21
110k.ll
-
+
13 k.ll
+
27 k.ll
i';J
..
ih
1'"
1;\
•
Co
47 k.ll
L'(I
Rc
+
I'b
iL" .
'g,
/' c
..
ih
...
+
3.3 kD
•
+
+
+
R g = 3 k.ll
...
...
I'h
...
...
Section 5.6
5.22 The circuit in Fig. P5.22 is a noninverting summing
p~~si~i;'l amplifier. Assume the op amp is ideal. Design the
PSPICE
circuit so that
5.24 a) Use the principle of superposition to derive
Eq.5.22.
b) Derive Eqs. 5.23 and 5.24.
/·1 ULTISm
+
Va = Va
+
2vb
3v e ·
a) Specify the numerical values of R a and R e .
b) Calculate i a , i b , and i e (in microamperes) when
Va = 0.7 Y, Vb = 0.4 Y, and Vc = 1.1 Y.
5.25 The resistors in the difference amplifier shown
in Fig. 5.15 are R a = 24 kD, R b = 75 k,(1,
/·IULTlSII·'
R e = 130 kD and R d = 120 kD. The signal volt­
ages Va and Vb are 8 and 5 Y, respectively," and
Vee = ±20 V.
PSPICE
a) Find
Figure P5.22
b) What is the resistance seen by the signal
source va?
100 k.ll
c) What is the resistance seen by the signal
source Vb?
20 kD
+
R"
I'"
i
4.7 k.ll
5.26 The op amp in the circuit of Fig. P5.26 is ideal. What
value of Rf will give the equation
Vo
R h = 15 k.ll "
+
c
+
i'e
...
...
..
..
=5
-
4v a,
for this circuit?
R ii,
1'"
I'h
va'
Figure P5.26
ic
...
5k.ll
5.23 The op amp in the noninverting summing amplifier
of Fig. P5.23 is ideal.
Va
PSPICE
/·lULTISllol
a) Specify the values of R f , R b , and R c so that
Va = 6v a
+
3Vb
+
4v e ·
5V
8k.ll
2k.ll
"'"
Problems
5:27 Design the difference-amplifier circuit in Fig. P5.27
so that va = W( Vb - Va), and the voltage source Vb
sees an input resistance of 220 kn. Specify the val­
PSPICE
MUlTIS!!·1
ues of R 8 ,Rb , and R f using single resistors or com­
binations of resistors from Appendix H. Use the
ideal model for the op amp.
DESIGN
PROBlE/·j
Figure PS.27
Rr
169
5.30 Design a difference amplifier (Fig. 5.15) to meet
the following criteria: Va = 3vb - 4v a. The resist­
ance seen by the signal source Vb is 470 H2, and
PSPICE
/·1ULTISIN the resistance seen by the signal source Va is
22 kn when the output voltage Va is zero. Specify
the values of R a , R b , Rc ' and Rd using single
resistors or combinations of resistors from
Appendix H.
DESIGN
PROBLEI·1
5.31 The resistor Rr in the circuit in Fig. P5.31 is
adjusted until the ideal op amp saturates. Specify
R r in kilohms.
4.7 kn
Figure PS.31
-9V
1'"
I~: •
22 kD
1.6kft
5.28 The op amp in the adder-subtracter circuit shown in
PSPICE
Fig. PS.28 is ideal.
MUlTlSlH
a) Find
Vd
Va
when
Va
= 1 V, Vb = 2 V, V C = 3 V, and
7.5 kD
e-3
+
18V
1.5kD
-9V
5.6kft
= 4 V.
b) If Va' Vb, and Vd are held constant, what values of
V c will not saturate the op amp?
Figure PS.28
180 kD
20kft
I'"
b) Write an equation for the straight line you plot­
ted in (a). How are the slope and inter­
cept of the line related to vg and the ratio Rd R1?
c) Using the results from (b), choose values for vg
and the ratio Rd R 1 such that va = -6a + 4.
18 kft
J'j,
+
30 kD
-I­
.-20V
1'"
I'L'
5.32 The op amp in the circuit of Fig. P5.32 is ideal.
a) Plot va versus a when Rf = 4R[ and vg = 2 V.
Use increments of 0.1 and note by hypothesis
tha t 0 ::5 a ::5 1.0.
47 left
20kft
I'd
Figure PS.32
20kft
5.29 Select the values of Ra and Rf m the circuit
Fig. P5.29 so that
1ll
OESI@
PROBlEfl
mICE
va
MUlTlSlfl
= SOOO(i b
-
ia)·
Use single resistors or combinations of resistors
from Appendix H. The op amp is ideal.
Figure PS.29
170
J
The Operational Amplifier
5.33 In the difference amplifier shown in Fig. P5.33, what
range of values of R c yields a CMRR 2 lOOO?
Figure P5.36
vg
O.SV
cle.
Figure P5.33
50kD
20 kD
-0.5 V
R,
50 kD
(a)
120 kD
5.34 In the difference amplifier shown in Fig. PS.34,
compute (a) the differential mode gain, (b) the
common mode gain, and (c) the CMRR.
Figure P5.34
/'"
6.8 kD
25 kD
lkD
(b)
+
1 kD
/'"
24 kD
5.37 The signal voltage v g in the circuit shown in Fig. PS.37
is described by the following equations:
PSPICE
j·IULTlSlI·1
Vg =
vg
=
0,
t :5
0,
o :5 t :5
10 sin(1T/3)t Y,
00.
Sections 5.1-5.6
Sketch V o versus t, assuming the op amp is ideal.
5.35 Assume that the ideal op amp in the circuit seen in
Fig. PS.3S is operating in its linear region.
a) Show that V o = [(R l + R2 )/ Rdv s .
b) What happens if R J --700 and R 2 --7 a?
Figure P5.37
15 kH
75 kD
c) Explain why this circuit is referred to as a volt­
age follower when R] = 00 and R2 = O.
Figure P5.35
Rs
l\
1'"
5.6 kD
+
6.8 kD
5.38 a) Show that when the ideal op amp in Fig. PS.38 is
operating in its linear region,
r .'
b) Show that the ideal op amp will saturate when
5.36 The voltage v g shown in Fig. PS.36(a) is applied to
the inverting amplifier shown in Fig. PS.36(b).
1·lumml
. 'd
I
Sketch V o versus t, assuming the op amp lS 1 ea.
PSPICE
R"
R(±Vcc - 2v g )
=
3v"
"
.
Problems
Figure P5.38
171
5.40 The circuit inside the shaded area in Fig. P5.40 is a con­
stant current source for a limited range of values of R L .
R
PSPICE
(·\ULTlSIM
a) Find the value of i L for R L = 4 kn.
b) Find the maximum value for RL for which
have the value in (a).
R
IL
will
c) Assume that R L = 16 kD. Explain the operation
of the circuit. You can assume that 1/1 = I p ~ 0
under all operating conditions.
R
d) Sketch
IL
versus R L for 0
~
RL
~
16 kD.
Figure P5.40
50 kfl
5.39 Assume that the ideal op amp in the circuit in
Fig. P5.39 is operating in its linear region.
MUlTlSII·1
a) Calculate the power delivered to the 16 kD
resistor.
b) Repeat (a) with the op amp removed from the
circuit, that is, with the 16 kD resistor connected
in the series with the voltage source and the
48 kD resistor.
c) Find the ratio of the power found in (a) to that
found in (b).
d) Does the insertion of the op amp between the
source and the load serve a useful purpose?
Explain.
20V
PSPICE
8V
5.41 The two op amps in the circuit
ideal. Calculate Vol and V o2.
III
Fig. PS.41 are
PSPICE
iWLTiSIt·l
Figure P5.41
15 V
15V-----~+
Figure P5.39
48 kfl
16 kfl
320mV
Load
5.42 TIle op amps in the circuit in Fig. PS.42 are ideal.
PSPICE
/·1 UlTISlI-\
a) Find la.
b) Find the value of the left source voltage for
which la = O.
Figure P5.42
47 kfl
220kfl
lkfl
IV
de
150mV
172
The Operational Amplifier
Section 5.7
5.46 a) Find the Thevenin equivalent circuit with
respect to the output terminals a,b for the
/·IULTlSIM
inverting amplifier of Fig. P5.46. The dc signal
source has a value of 880 mY. The op amp has
an input resistance of 500 kD, an output
resistance of 2 kD and an open-loop gain
of 100,000.
PSPIEE
5.43 Repeat Assessment Problem 5.6, given that the
inverting amplifier is loaded with a 500 D resistor.
PSPICE
/·\ULTlSli·!
5.44 Assume the input resistance of the op amp in
Fig. P5.44 is infinite and its output resistance is zero.
PSPICE
r·1UlTISIi·l
b) What is the output resistance of the inverting
amplifier?
a) Find va as a function of v g and the open-loop
gainA.
c) What is the resistance (in ohms) seen by the sig­
nal source V" when the load at the terminals a,b
is 330 D?
= 1 V and A = 150?
What is the value of V o if vg = 1 V and A = oo?
b) What is the value of V o if vg
c)
d) How large does A have to be so that V o is 99% of
its value in (c)?
Figure P5.46
24 kfl
Figure P5.44
1.6kfl
10 kn
~~--+----e
2kn
v,
e-3------e
a
1',.
5.47 Repeat Problem 5.46 assuming an ideal op amp.
5.45 The op amp in the noninverting amplifier circuit of
PSPICE
Fig. P5.45 has an input resistance of 560 kD, an out­
/·IlJLTlSIM
put resistance of 8 kD, and an open-loop gain of
50,000. Assume that the op amp is operating in its
linear region.
a) Calculate the voltage gain (va/v g ).
PSP[CE
j·1UlTISH-l
5.48 Derive Eg. 5.60.
Sections 5.1-5.7
b) Find the inverting and noninverting input volt­
ages v" and v p (in millivolts) if v g = 1 V.
c) Calculate the difference (v p - v n ) in microvolts
when vI? = 1 V.
d) Find the current drain in picoamperes on the
signal source v g when v g = 1 V.
e) Repeat (a)-(d) assuming an ideal op amp.
Figure P5.45
5.49 Suppose the strain gages in the bridge in Fig. 5.21
have the value 120 D ± 1 %. The power supplies
to the op amp are ± 15 V, and the refer­
ence voltage, Vrer, is taken from the positive
power supply.
PR,\CTIEAL
PERSPECTIVE
a) Calculate the value of RJ so that when the strain
gage that is lengthening reaches its maximum
length, the output voltage is 5 Y.
b) Suppose that we can accurately measure
50 mV changes in the output voltage. What
change in strain gage resistance can be
detected in milliohms?
200 kn
16 kn
+
-15V
I'"~
20kfl
Problems
5.50 a) For the circuit shown in Fig. P5.50, show that if
6.R « R, the output voltage of the op amp is
approximately
PSPICE
173
show that the percent error in the approxima­
tion of va in Problem 5.50 is
PRAmCAL
PERSPEmVE
MUmSI/·1
% error
b) Find va if R r = 470 kn, R
and vin = 15 V.
= 10 kG, 6.R = 95 G,
b) Calculate the percent error in V o for Problem 5.50.
PRACTICAL
PERSPECTIVE
PSPICE
Figure P5.50
+ 6.R
r
5.52 Assume the percent error in the approximation of
Va in the circuit in Fig. P5.50 is not to exceed 1%.
What is the largest percent change in R that can be
MULTISI/·!
tolerated?
c) Find the actual value of va in (b).
R
6.R (R + R r)
(R + 2R ) x 100.
=R
5.53 Assume the resistor in the variable branch of the
bridge circuit in Fig. P5.50 is R - 6.R instead of
PRACTICAL
PERSPECTIVE
PSPICE
R
r·IULTlSlfl
R + 6.R.
a) What is the expression for va if 6.R
« R?
b) What is the expression for the percent error in
va as a function of R, R r, and 6.R?
c) Assume the resistance in the variable arm of
the bridge circuit in Fig. P5.50 is 9810 n and the
values of R, R r, and vin are the same as in
Problem 5.50(b). What is the approximate value
of va?
5.51 a) If percent error is defined as
d) What is the percent error in the approximation
of va when the variable arm resistance is
9810 n?
PRACTICAL
PfRSPEmVE
PSPICE
MumSIM
d,
% error =
[apprOXimate value
true value
-
1
]
x 100,
_