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ampli er of Figure 124 is a voltage ampli er; another type of operational ampli er, called a current or transconductance ampli er, is described in the homework problems
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Negative power supply Simplified circuit symbol
Offset null 1 Inverting input 2 7 8
No connection VS
Inverting input in Noninverting input in + Output
Noninverting input 3 VS 4 6
Output Offset null 5
Integrated circuit operational amplifier (IC op-amp)
VS IC op-amp diagram
Figure 124 Operational ampli er model symbols, and circuit diagram
The Operational Ampli er in the Closed-Loop Mode
The Inverting Ampli er
One of the more popular circuit con gurations of the op-amp, because of its simplicity, is the so-called inverting ampli er, shown in Figure 125 The input signal to be ampli ed is connected to the inverting terminal, while the noninverting terminal is grounded It will now be shown how it is possible to choose an (almost) arbitrary gain for this ampli er by selecting the ratio of two resistors The analysis is begun by noting that at the inverting input node, KCL requires that iS + iF = iin (1211)
+ _
RF iF + + vout
RS iS vS
v iin v+
The current iF , which ows back to the inverting terminal from the output, is appropriately termed feedback current, because it represents an input to the ampli er that is fed back from the output Applying Ohm s law, we may determine
Figure 125 Inverting ampli er
12
Operational Ampli ers
each of the three currents shown in Figure 125: iS = vS v RS iF = vout v RF iin = 0 (1212)
(the last by assumption, as stated earlier) The voltage at the noninverting input, v + , is easily identi ed as zero, since it is directly connected to ground: v + = 0 Now, the open-loop model for the op-amp requires that vout = AV (OL) (v + v ) = AV (OL) v or v = vout AV (OL) (1214) (1213)
Having solved for the voltage present at the inverting input, v , in terms of vout , we may now compute an expression for the ampli er gain, vout /vS This quantity is called the closed-loop gain, because the presence of a feedback connection between the output and the input constitutes a closed loop2 Combining equations 1211 and 1212, we can write iS = iF and vout vout vout vS + = RS AV (OL) RS RF AV (OL) RF leading to the expression vout vout vout vS = RS RF AV (OL) RF AV (OL) RS or vS = vout 1 1 1 + + RF /RS AV (OL) RF /RS AV (OL) (1218) (1217) (1216) (1215)
If the open-loop gain of the ampli er, AV (OL) , is suf ciently large, the terms 1/(AV (OL) RF /RS ) and 1/AV (OL) are essentially negligible, compared with 1/(RF /RS ) As stated earlier, typical values of AV (OL) range from 105 to 107 , and thus it is reasonable to conclude that, to a close approximation, the following expression describes the closed-loop gain of the inverting ampli er: RF vS RS
vout =
Inverting ampli er closed-loop gain
(1219)
That is, the closed-loop gain of an inverting ampli er may be selected simply by the appropriate choice of two externally connected resistors The price for this extremely simple result is an inversion of the output with respect to the input that is, a negative sign
2 This
terminology is borrowed from the eld of automatic controls, for which the theory of closed-loop feedback systems forms the foundation
Part II
Electronics
Next, we show that by making an additional assumption it is possible to simplify the analysis considerably Consider that, as was shown for the inverting ampli er, the inverting terminal voltage is given by vout v = (1220) AV (OL) Clearly, as AV (OL) approaches in nity, the inverting-terminal voltage is going to be very small (practically, of the order of microvolts) It may then be assumed that in the inverting ampli er, v is virtually zero: v 0 (1221)
This assumption prompts an interesting observation (which may not yet appear obvious at this point):
The effect of the feedback connection from output to inverting input is to force the voltage at the inverting input to be equal to that at the noninverting input
This is equivalent to stating that for an op-amp with negative feedback, v v+ (1222)
The analysis of the operational ampli er can now be greatly simpli ed if the following two assumptions are made: 1 iin = 0 2 v = v + (1223)
This technique will be tested in the next subsection by analyzing a noninverting ampli er con guration Example 121 illustrates some simple design considerations
Why Feedback Why is such emphasis placed on the notion of an ampli er with a very large open-loop gain and with negative feedback Why not just design an ampli er with a reasonable gain, say, 10, or 100, and just use it as such, without using feedback connections In these paragraphs, we hope to answer these and other questions, introducing the concept of negative feedback in an intuitive fashion The fundamental reason for designing an ampli er with a very large open-loop gain is the exibility it provides in the design of ampli ers with an (almost) arbitrary gain; it has already been shown that the gain of the inverting ampli er is determined by the choice of two external resistors undoubtedly a convenient feature! Negative feedback is the mechanism that enables us to enjoy such exibility in the design of linear ampli ers To understand the role of feedback in the operational ampli er, consider the internal structure of the op-amp shown in Figure 124 The large openloop gain causes any difference in voltage at the input terminals to appear greatly ampli ed at the output When a negative feedback connection is provided, as shown, for example, in the inverting ampli er of
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