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For a di erential gain of 106 we must have
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AMPLIFIERS AND OPERATIONAL AMPLIFIER CIRCUITS
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The circuit of Fig. 5-49 has an in nite input resistance, employs resistors within ordinary range, and uses three op amps.
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Show that in the circuit of Fig. 5-50 i1 i2 , regardless of the circuits of N1 and N2 .
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Fig. 5-50 Nodes A and B are at the same voltage vA vB . Since the op amp draws no current, i1 and i2 ow through the two resistors and KVL around the op amp loop ABC gives Ri1 Ri2 0. Therefore, i1 i2 .
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Let N1 be the voltage source v1 and N2 be the resistor R2 in the circuit of Fig. 5-50. input resistance Rin v1 =i1 .
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From the op amp we obtain vA vB and i1 i2 . From connections to N1 and N2 we obtain v1 vB v2 vA and v2 i2 R2 , respectively. The input resistance is v1 =i1 i2 R2 =i2 R2 which is the negative of the load. The op amp circuit is a negative impedance converter.
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A voltage follower is constructed using an op amp with a nite open-loop gain A and Rin 1 (see Fig. 5-51). Find the gain G v2 =v1 . De ning sensitivity s as the ratio of percentage change produced in G to the percentage change in A, nd s.
Fig. 5-51 From Fig. 5-51 we have v2 Avd . Applying KVL around the ampli er, obtain
v1 vd v2 vd Avd vd 1 A v2 1 A =A G v2 A v1 1 A
AMPLIFIERS AND OPERATIONAL AMPLIFIER CIRCUITS
[CHAP. 5
The rate of change of G with respect to A is dG 1 dA 1 A 2 from which dG dA 1 A 2
The percentage change produced in G is 100 dG=G . dG dA 1 A 1 dA G A 1 A A 1 A 2 and the sensitivity is s The percentage change in G depends on A. dG=G 1 dA=A 1 A
Samples of dG=dA and s are shown in Table 5-8. Table 5-8
A 10 11 100 1000
G v2 =v1 0.909 0.917 0.990 0.999
dG=dA 0.008 0.007 0.0001 0
s 0.091 0.083 0.01 0
For high values of A, the gain G is not sensitive to changes in A.
Supplementary Problems
5.30 Repeat Problem 5.3 by replacing the circuit to the left of node B (including vs , R1 , and Ri ) by its Thevenin equivalent (see Fig. 5-33) Solve the problem by applying the results of Example 5.4. Find the Thevenin equivalent of the circuit to the left of nodes A-B in Fig. 5-52 with k 10 for (a) R2 1 and (b) R2 50 k
. Ans: a vTh 100 V; RTh 100
; (b vTh 31:22 V; RTh 37:48
Fig. 5-52
Repeat Problem 5.31 for R2 50 k
and k 100.
Ans:
vTh 47:16 V; RTh 5:66
Determine the relationship between R, R1 , and R2 in Fig. 5-41 such that the circuit has a gain of v2 =i1 106 V/A. Ans: RR2 =R1 106
CHAP. 5]
AMPLIFIERS AND OPERATIONAL AMPLIFIER CIRCUITS
In the circuit of Fig. 5-13, Vcc 10 V, R1 2 k
and v1 1 V. Find the maximum value of R2 before the op amp is saturated. Ans: R2 20 k
Let the summing circuit of Fig. 5-14 have two inputs with v1 1 and v2 sin t (V). Let R1 3 k
, R2 5 k
, and Rf 8 k
. Apply superposition to nd vo . Ans: vo 8 8 sin t 3 5 In Fig. 5-17 let R1 4 k
and R2 8 k
. Ans: vo v1 v2 v3 Apply superposition to nd vo in terms of the input voltages.
5.37 5.38
Find the input resistance seen by vf in Fig. 5-19.
Ans:
Rin 2R1
Use superposition to nd vo in Fig. 5-20 for R1 2, R2 7, R3 10, R4 5, all values in k
. Ans: vo 1:5v2 3:5v1 In the circuit of Fig. 5-20 nd (a) v0 for R1 1, R2 3, R3 2, and R4 2, all values in k
; (b) the input resistance R2 in seen by v2 ; (c) i1 as a function of v1 and v2 and show that v1 sees a variable load which depends on v2 . Ans: a vo 2v2 3v1 ; b R2 in 4 k
; c i1 v1 v2 =2 Using a single op amp, design an ampli er with a gain of v2 =v1 3=4, input resistance of 8 k
, and zero output resistance. Ans: See Fig. 5-53.
Fig. 5-53
Show that, given R1 1 and R2 0, the noninverting op amp circuit of Fig. 5-15 and (12) is reduced to a voltage follower. In the circuit of Fig. 5-22 let Rs 10 k
. Discuss. Ans: a 40 k
; b yes (a) Find Rf such that is 0. (b) Is Rf independent of Rs
The input to the circuit of Fig. 5-23 with RC 1 is v1 sin !t. Ans: v2 1=! cos !t C
Write KCL at node B and solve for v2 .
Show that the output v2 in Fig. 5.54 is the same as the output of the integrator in Fig. 5-23.
Fig. 5-54
AMPLIFIERS AND OPERATIONAL AMPLIFIER CIRCUITS  1V 0
[CHAP. 5 t>0 . t<0
Find v2 in the leaky integrator of Fig. 5-24 with R1 Rf 1 k
, C 1 mF, and v1 ( Ans: v2 t 1 e 1000t V 0  1V 0 t>0 t<0 t<0 . t>0  Ans: v2 t e 1000t V 1 V t>0 t<0
Repeat Problem 5.45 for v1
In the di erential equation 10 2 dv2 =dt v2 vs , vs is the forcing function and v2 is the response. Design an op amp circuit to obtain v2 from vs . Ans: See Fig. 5-24, with R1 Rf ; RC 10 2 , and v1 vs .
Design a circuit containing op amps to solve the following set of equations: y 0 x vs1 2y x 0 3x vs2 Ans. See Fig. 5-55, with R1 C R4 C 1 s, R2 C 1 s, R3 C 1 s. 3 2
Fig. 5-55
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