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121 Consider an op-amp connected in the inverting con guration with a nominal closed-loop gain of RF /RS = 1,000 (this would be the gain if the op-amp had an in nite open-loop gain) Determine the value of the closed-loop gain that includes the open-loop gain as a parameter, and compute the closed-loop gain for the following values of AV (OL) : 107 , 106 , 105 , and 104 How large do you think the open-loop gain should be for this op-amp, to achieve the desired closed-loop gain [Hint: Do not assume that AV (OL) is negligible in Equation 1218] 122 Repeat Check Your Understanding Exercise 121 for RF /RS = 100 What is the smallest value of AV (OL) you would recommend in this case 123 Derive the result given for the differential ampli er by utilizing the principle of superposition (Think of the differential ampli er as the combination of an inverting ampli er with input = v2 , plus a noninverting ampli er with input = v1 ) 124 For Example 124, nd R if the supply voltages are symmetrical at 15 V and
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a 10-k potentiometer is tied to two 10-k resistors
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125 For the circuit of Example 124, nd the range of values of Vref if the supply voltages are symmetrical at 15 V and a 1-k potentiometer is tied to two 10-k resistors 126 Find the numerical values of RF /RS and Vref if the temperature sensor of in Focus on Measurements: Sensor Calibration Circuit has = 0235 and V0 = 07 V and the desired relationship is vout = 10T
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The range of useful applications of an operational ampli er is greatly expanded if energy-storage elements are introduced into the design; the frequency-dependent properties of these elements, studied in s 4 and 6, will prove useful in the design of various types of op-amp circuits In particular, it will be shown that it is possible to shape the frequency response of an operational ampli er by appropriate use of complex impedances in the input and feedback circuits The class of lters one can obtain by means of op-amp designs is called active lters, because op-amps can provide ampli cation (gain) in addition to the ltering effects already studied in 6 for passive circuits (ie, circuits comprising exclusively resistors, capacitors, and inductors) The easiest way to see how the frequency response of an op-amp can be shaped (almost) arbitrarily is to replace the resistors RF and RS in Figures 125 and 128 with impedances ZF and ZS , as shown in Figure 1220 It is a straightforward matter to show that in the case of the inverting ampli er, the expression for the closed loop gain is given by Vout ZF (j ) = VS ZS whereas for the noninverting case, the gain is Vout ZF (j ) = 1 + VS ZS (1246) (1245)
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ZF IF ZS + IS ~ VS + + Vout Inverting ZF IF ZS IS + VS Noninverting ~ + + Vout
where ZF and ZS can be arbitrarily complex impedance functions and where VS , Vout , IF , and IS are all phasors Thus, it is possible to shape the frequency response of an ideal op-amp lter simply by selecting suitable ratios of feedback impedance to source impedance By connecting a circuit similar to the low-pass lters studied
Figure 1220 Op-amp circuits employing complex impedances
12
Operational Ampli ers
ZF CF RF
in 6 in the feedback loop of an op-amp, the same ltering effect can be achieved and, in addition, the signal can be ampli ed The simplest op-amp low-pass lter is shown in Figure 1221 Its analysis is quite simple if we take advantage of the fact that the closed-loop gain, as a function of frequency, is given by ALP (j ) = ZF ZS (1247)
RS + VS
where ZF = RF and ZS = RS (1249) 1 RF = j CF 1 + j CF RF (1248)
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