Fundamentals-of-Microelectronics-Behzad-Razavi.pdf

Cls v 2006 at 1342 388 1 388 chap 8 operational

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[Razavi.cls v. 2006] June 30, 2007 at 13:42 388 (1) 388 Chap. 8 Operational Amplifier As A Black Box If the decay time constant, , is sufficiently small, the passive circuit can be viewed as an approximation of the ideal differentiator. out V C 1 R 1 X in V I in 0 V 1 C 1 R in V 0 V 1 1 out V V 1 (a) (b) X Figure 8.18 Comparison of differentiator and RC circuit. Let us now study the differentiator with a finite op amp gain. Equating the capacitor and resistor currents in Fig. 8.15 gives (8.53) Substituting for , we have (8.54) In contrast to the ideal differentiator, the circuit contains a pole at (8.55) Example 8.8 Determine the transfer function of the high-pass filter shown in Fig. 8.19 and choose and such that the pole of this circuit coincides with (8.55). C R in V out V X X Figure 8.19 Simple high-pass filter. Solution The capacitor and resistor operate as a voltage divider: (8.56) (8.57)
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BR Wiley/Razavi/ Fundamentals of Microelectronics [Razavi.cls v. 2006] June 30, 2007 at 13:42 389 (1) Sec. 8.2 Op-Amp-Based Circuits 389 The circuit therefore exhibits a zero at the origin ( ) and a pole at . For this pole to be equal to (8.55), we require (8.58) One choice of and is (8.59) (8.60) Exercise What is the necessary value of if ? An important drawback of differentiators stems from the amplification of high-frequency noise . As suggested by (8.42) and Fig. 8.16, the increasingly larger gain of the circuit at high frequencies tends to boost noise in the circuit. The general topology of Fig. 8.9 and its integrator and differentiator descendants operate as inverting circuits. The reader may wonder if it is possible to employ a configuration similar to the noninverting amplifier of Fig. 8.5 to avoid the sign reversal. Shown in Fig. 8.20, such a circuit provides the following transfer function Z 1 V in out V Z 2 Figure 8.20 Op amp with general network. (8.61) if the op amp is ideal. Unfortunately, this function does not translate to ideal integration or dif- ferentiation. For example, and yield a nonideal differentiator (why?). 8.2.4 Voltage Adder The need for adding voltages arises in many applications. In audio recording, for example, a number of microphones may convert the sounds of various musical instruments to voltages, and these voltages must then be added to create the overall musical piece. This operation is called “mixing” in the audio industry. For example, in“noise cancelling” headphones, the environmen- The term “mixing” bears a completely different meaning in the RF and wireless industry.
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BR Wiley/Razavi/ Fundamentals of Microelectronics [Razavi.cls v. 2006] June 30, 2007 at 13:42 390 (1) 390 Chap. 8 Operational Amplifier As A Black Box tal noise is applied to an inverting amplifier and subsequently added to the signal so as to cancel itself. Figure 8.21 depicts a voltage adder (“summer”) incorporating an op amp. With an ideal op amp, , and and carry currents proportional to and , respectively. The two currents add at the virtual ground node and flow through : R R X out V F R V 1 V 2 1 2 Figure 8.21 Voltage adder.
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