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[CHAP. 9
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Fig. 9-14 The principle of superposition is applicable to this linear circuit. With vS2 0 (shorted), the voltage appearing at the noninverting terminal is found by voltage division to be v2 Let vo1 be the value of vo with vS2 0. vo1 Similarly, with vS1 0, R v v S1 R R S1 2 1
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By the result of Example 9.3 and (1),     R R vS1 1 2 v2 1 2 R1 R1 2   R vS2 vo2 1 2 R1 2   1 R 1 2 vS1 vS2 2 R1
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By superposition, the total output is then vo vo1 vo2 The circuit is a noninverting adder.
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The circuit of Fig. 9-15(a) (represented in the s domain) is a more practical di erentiator than that of Fig. 9-5, because it will attenuate high-frequency noise. (a) Find the s-domain transfer
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0.1 RC
1 RC
10 RC
IF (s) IS (s) VS (s) 1/sC + Vo (s)
Slope = _ 20 db/decade Slope = 20 db/decade
Fig. 9-15
CHAP. 9]
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function relating Vo and VS . (b) Sketch the Bode plot (Mdb only), and how high-frequency noise e ects are reduced. Assume an ideal op amp.
(a) In an ideal op amp the inverting terminal is a virtual ground, so IS s IF s . ZF s Then But whence R sRC 1 V s sRC 1 Vo s IF s o ZF s R sRC 1 sRC 1 Vo s R sC As in Example 9.9,
VS s IS s Zin s IF s Zin s A s  Vo s sRC VS s sRC 1 2
(b) From the result of part a,  Mdb  20 log jA j! j 20 log !RC 40 log j j!RC 1j % 20 log !RC 20 log !RC for !RC 1 for !RC ! 1
Figure 9-15(b) is a plot of this approximate (asymptotic) expression for Mdb . For a true di erentiator, we would have vo K dvS dt or Vo sKVS
which would lead to Mdb 20 log !K. Thus the practical circuit di erentiates only components of the signal whose frequency is less than the break frequency f1  1=2RC Hz. Spectral components above the break frequency including (and especially) noise will be attenuated; the higher the frequency, the greater the attenuation.
In analog signal processing, the need often arises to introduce a level clamp (linear ampli cation to a desired output level or value and then no further increase in output level as the input continues to increase). One level-clamp circuit, shown in Fig. 9-16(a), uses series Zener diodes in a negative feedback path. Assuming ideal Zeners and op amp, nd the relationship between vo and vS . Sketch the results on a transfer characteristic.
_ VZ2 + Z2
VZ1 _ Z1
R2 R1
LS Lo
_ _ R1 V R2 Z2 _V (a)
R1 V R2 Z1
Fig. 9-16
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[CHAP. 9
Since the op amp is ideal, the inverting terminal is a virtual ground, and vo appears across the parallelconnected feedback paths. There are two distinct possibilities: Case I: vS > 0. For vo < 0, Z2 is forward-biased and Z1 reverse-biased. The Zener feedback path is an open circuit until vo VZ1 ; then Z1 will limit vo at VZ1 so that no further negative excursion is possible. Case II: vS < 0. For vo > 0, Z1 is forward-biased and Z2 reverse-biased. The Zener feedback path acts as an open circuit until vo reaches VZ2 , at which point Z2 limits vo to that value. In summary, for both cases, 8 R >V > Z2 for vS < 1 VZ2 > > > R2 > > < R R1 R1 2 v for V vS V vo > R1 S R2 Z2 R2 Z1 > > > > R > > for vS > 1 VZ1 : VZ1 R2 Figure 9-16(b) gives the transfer characteristic.
The circuit of Fig. 9-17 is an adjustable-output voltage regulator. Assume that the basic op amp is ideal. Regulation of the Zener is preserved if iZ ! 0:1IZ (Section 2.10). (a) Find the regulated output vo in terms of VZ . (b) Given a speci c Zener diode and the values of RS and R1 , over what range of VS would there be no loss of regulation
+ VS iS R2 RS i1 a + VZ iZ _ + R1 _
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