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The gain of an ampli er may be controlled by feeding back a portion of its output to its input as done for the ideal ampli er in Fig. 5-4 through the feedback resistor R2 . The feedback ratio R1 = R1 R2 a ects the overall gain and makes the ampli er less sensitive to variations in k.
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AMPLIFIERS AND OPERATIONAL AMPLIFIER CIRCUITS
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Fig. 5-4
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EXAMPLE 5.3 Find v2 =vs in Fig. 5-4 and express it as a function of the ratio b R1 = R1 R2 . From the ampli er we know that v2 kv1 Applying KCL at node A, v1 vs v1 v2 0 R1 R2 Substitute v1 in (3) into (4) to obtain v2 R2 k k 1 b 1 bk vs R2 R1 R1 k where b R1 R1 R2 5 4 or v1 v2 =k 3
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EXAMPLE 5.4 In Fig. 5-5, R1 1 k
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and R2 5 k
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. (a) Find v2 =vs as a function of the open-loop gain k. (b) Compute v2 =vs for k 100 and 1000 and discuss the results.
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Fig. 5-5 (a) Figures 5-4 and 5-5 di er only in the polarity of the dependent voltage source. Example 5.3 and change k to k in (5). v2 k 1 b 1 bk vs where b R1 1 R1 R2 6 To nd v2 =vs , use the results of
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v2 5k vs 6 k (b) At k 100, v2 =vs 4:72; at k 1000, v2 =vs 4:97. Thus, a tenfold increase in k produces only a 5.3 percent change in v2 =vs ; i.e., 4:97 4:72 =4:72 5:3 percent. Note that for very large values of k, v2 =vs approaches R2 =R1 which is independent of k.
OPERATIONAL AMPLIFIERS
The operational ampli er (op amp) is a device with two input terminals, labeled and or noninverting and inverting, respectively. The device is also connected to dc power supplies ( Vcc and
CHAP. 5]
AMPLIFIERS AND OPERATIONAL AMPLIFIER CIRCUITS
Vcc ). The common reference for inputs, output, and power supplies resides outside the op amp and is called the ground (Fig. 5-6).
Fig. 5-6
The output voltage vo depends on vd v v . Neglecting the capacitive e ects, the transfer function is that shown in Fig. 5-7. In the linear range, vo Avd . The open-loop gain A is generally very high. vo saturates at the extremes of Vcc and Vcc when input vd exceeds the linear range jvd j > Vcc =A.
Fig. 5-7
Figure 5-8 shows the model of an op amp in the linear range with power supply connections omitted for simplicity. In practice, Ri is large, Ro is small, and A ranges from 105 to several millions. The model of Fig. 5-8 is valid as long as the output remains between Vcc and Vcc . Vcc is generally from 5 to 18 V.
EXAMPLE 5.5 In the op amp of Fig. 5-8, Vcc 15 V, A 105 , and v 0. tude of v for linear operation. jvo j j105 v j < 15 V Find the upper limit on the magni-
jv j < 15 10 5 V 150 mV
EXAMPLE 5.6 In the op amp of Fig. 5-8, Vcc 5 V, A 105 , v 0 and v 100 sin 2 t mV . Find and sketch the open-loop output vo . The input to the op amp is vd v v 100 sin 2 t 10 6 (V). When the op amp operates in the linear range, vo 105 vd 10 sin 2 t (V). The output should remain between 5 and 5 V (Fig. 5-9). Saturation starts when vo 10 sin 2 t reaches the 5-V level. This occurs at t 1=12 s. The op amp comes out of 5-V saturation at
AMPLIFIERS AND OPERATIONAL AMPLIFIER CIRCUITS
[CHAP. 5
Fig. 5-8 t 5=12. Similarly, the op amp is in 5-V saturation from t 7=12 to 11/12 s. in volts, from t 0 to 1 s is 8 5 1=12 < t < 5=12 < 5 7=12 < t < 11=12 vo : 10 sin 2 t otherwise One full cycle of the output, given
Fig. 5-9 EXAMPLE 5.7 Repeat Example 5.6 for v 25 mV and v 50 sin 2 t mV). vd v v 50 sin 2 t 10 6 25 10 6 50 10 6 sin 2 t 1=2 V When the op amp is within linear range, its output is vo 105 vd 5 sin 2 t 1=2 V vo saturates at the 5-V level when 5 sin 2t 1=2 < 5, 7=12 < t < 11=12 (see Fig. 5-10). One cycle of vo , in volts, from t 0 to 1 s is
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