We now examine the performance of the integrator for

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We now examine the performance of the integrator for . Denoting the potential of the virtual ground node in Fig. 8.10 with , we have (8.34)

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BR Wiley/Razavi/ Fundamentals of Microelectronics [Razavi.cls v. 2006] June 30, 2007 at 13:42 385 (1) Sec. 8.2 Op-Amp-Based Circuits 385 out V C 1 R 1 X in V 0 V 1 R V 1 1 C R 1 out V in V 1 R V 1 1 0 V 1 V out Figure 8.13 Comparison of integrator with and RC circuit. and (8.35) Thus, (8.36) revealing that the gain at is limited to (rather than infinity) and the pole frequency has moved from zero to (8.37) Such a circuit is sometimes called a “lossy” integrator to emphasize the nonideal gain and pole position. Example 8.6 Recall from basic circuit theory that the RC filter shown in Fig. 8.14 contains a pole at . Determine and such that this circuit exhibits the same pole as that of the above integrator. C R out V in V X X Figure 8.14 Simple low-pass filter. Solution From (8.37), (8.38) The choice of and is arbitrary so long as their product satisfies (8.38). An interesting choice is (8.39) (8.40) It is as if the op amp “boosts” the value of by a factor of .
BR Wiley/Razavi/ Fundamentals of Microelectronics [Razavi.cls v. 2006] June 30, 2007 at 13:42 386 (1) 386 Chap. 8 Operational Amplifier As A Black Box Exercise What value of is necessary if ? Differentiator If in the general topology of Fig. 8.9, is a resistor and a capacitor (Fig. 8.15), we have in V out V C 1 R 1 X Figure 8.15 Differentiator. (8.41) (8.42) Exhibiting a zero at the origin, the circuit acts as a differentiator (and a high-pass filter). Figure 8.16 plots the magnitude of as a function of frequency. From a time-domain perspec- tive, we can equate the currents flowing through and : f out V V in R C 1 1 Figure 8.16 Frequency response of differentiator. (8.43) arriving at (8.44) Example 8.7 Plot the output waveform of the circuit shown in Fig. 8.17(a) assuming an ideal op amp. Solution At , and (why?). When jumps to , an impulse of current flows through because the op amp maintains constant: (8.45)

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BR Wiley/Razavi/ Fundamentals of Microelectronics [Razavi.cls v. 2006] June 30, 2007 at 13:42 387 (1) Sec. 8.2 Op-Amp-Based Circuits 387 out V C 1 R 1 X in V 0 V 1 T b 0 V 1 T b 0 t out V in V I C1 (a) (b) I in 0 0 Figure 8.17 (a) Differentiator with pulse input, (b) input and output waveforms. (8.46) The current flows through , generating an output given by (8.47) (8.48) Figure 8.17(b) depicts the result. At , returns to zero, again creating an impulse of current in : (8.49) (8.50) It follows that (8.51) (8.52) We can therefore say that the circuit generates an impulse of current and “amplifies” it by to produce . In reality, of course, the output exhibits neither an infinite height (limited by the supply voltage) nor a zero width (limited by the op amp nonidealities). Exercise Plot the output if is negative. It is instructive to compare the operation of the differentiator with that of its “passive” coun- terpart (Fig. 8.18). In the ideal differentiator, the virtual ground node permits the input to change the voltage across instantaneously. In the RC filter, on the other hand, node is not “pinned,” thereby following the input change at and limiting the initial current in the circuit to .
BR Wiley/Razavi/ Fundamentals of Microelectronics [Razavi.cls v. 2006]

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