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Comments: Note that the worst-case percent error in the ampli er gain is double the
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resistor tolerance
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circuit of Figure 125 can be found in the accompanying CD-ROM
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The Summing Ampli er
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A useful op-amp circuit that is based on the inverting ampli er is the op-amp summer, or summing ampli er This circuit, shown in Figure 127, is used to
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Operational Ampli ers
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RF RS1 iF vS1
+ vout
+ vS2 _
add signal sources The primary advantage of using the op-amp as a summer is that the summation occurs independently of load and source impedances, so that sources with different internal impedances will not interact with each other The operation of the summing ampli er is best understood by application of KCL at the inverting node: the sum of the N source currents and the feedback current must equal zero, so that i1 + i2 + + iN = iF But each of the source currents is given by the following expression: (1224)
in =
vSn RS n
n = 1, 2, , N
(1225)
Figure 127 Summing ampli er
while the feedback current is vout iF = RF
(1226)
Combining equations 1225 and 1226, and using equation 1215, we obtain the following result:
N n=1
vS n vout = RS n RF
(1227)
vout =
RF vS RS n n
(1228)
That is, the output consists of the weighted sum of N input signal sources, with the weighting factor for each source equal to the ratio of the feedback resistance to the source resistance
The Noninverting Ampli er
RF RS iS R
iin v v+ iin +
To avoid the negative gain (ie, phase inversion) introduced by the inverting ampli er, a noninverting ampli er con guration is often employed A typical noninverting ampli er is shown in Figure 128; note that the input signal is applied to the noninverting terminal this time The noninverting ampli er can be analyzed in much the same way as the inverting ampli er Writing KCL at the inverting node yields
+ vout
iF = iS + iin iS where iF = vout v RF v RS
(1229)
(1230) (1231)
Figure 128 Noninverting ampli er
iS =
Now, since iin = 0, the voltage drop across the source resistance, R, is equal to zero Thus, v + = vs (1232)
Part II
Electronics
and, using equation 1222, v = v + = vS (1233)
Substituting this result in equations 1229 and 1230, we can easily show that iF = iS or vS vout vS = RF RS It is easy to manipulate equation 1235 to obtain the result vout RF =1+ vS RS (1235) (1234)
Noninverting ampli er closed-loop gain
(1236)
which is the closed-loop gain expression for a noninverting ampli er Note that the gain of this type of ampli er is always positive and greater than (or equal to) 1 The same result could have been obtained without making the assumption v + = v , at the expense of some additional work The procedure one would follow in this latter case is analogous to the derivation carried out earlier for the inverting ampli er, and is left as an exercise In summary, in the preceding pages it has been shown that by constructing a nonideal ampli er with very large gain and near-in nite input resistance, it is possible to design ampli ers that have near-ideal performance and provide a variable range of gains, easily controlled by the selection of external resistors The mechanism that allows this is negative feedback From here on, unless otherwise noted, it will be reasonable and suf cient in analyzing new op-amp con gurations to utilize the two assumptions 1 iin = 0 2 v = v +
Approximations used for ideal op-amps with negative feedback
(1237)
EXAMPLE 122 Voltage Follower
Problem
Determine the closed-loop voltage gain and input resistance of the voltage follower circuit of Figure 129
Solution
Known Quantities: Feedback and source resistances, source voltage
12
Operational Ampli ers
Find:
i in vin
AV =
+ vout
vout vin
ri =
vin iin
Assumptions: The ampli er behaves ideally; that is, the input current into the op-amp is zero and negative feedback forces v + = v Analysis: From the ideal op-amp assumptions, v + = v But v + = vin and v = vout ,
thus:
Figure 129 Voltage follower
vin = vout
The voltage follower s name derives from the ability of the output voltage to follow exactly the input voltage To compute the input resistance of this ampli er, we observe that since the input current is zero, vin ri = iin
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