# For a solution to exist we require that 1451 or 1452

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For a solution to exist, we require that (14.51) or (14.52) Comparison with (14.41) reveals the poles are complex here. The reader is encouraged to plot the resulting frequency response for different values of and prove that the peak value increases as decreases. Exercise Compare the gain at with that at . The peaking phenomenon studied in the above example proves undesirable in many appli- cations as it disproportionately amplifies some frequency components of the signal. Viewed as ripple in the passband, peaking must remain below approximately 1 dB ( ) in such cases. Example 14.15 Consider the low-pass circuit shown in Fig. 14.30 and explain why it is less useful than that of C out V in V L 1 1 R 1 Figure 14.30 Fig. 14.29. Solution This circuit satisfies the conceptual illustration in Fig. 14.28(a) and hence operates as a low-pass

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BR Wiley/Razavi/ Fundamentals of Microelectronics [Razavi.cls v. 2006] June 30, 2007 at 13:42 742 (1) 742 Chap. 14 Analog Filters filter. However, at high frequencies, the parallel combination of and is dominated by because , thereby reducing the circuit to and . The filter thus exhibits a roll-off less sharp than the second-order response of the previous design. Exercise What type of frequency response is obtained if and are swapped? High-Pass Filter To obtain a high-pass response, we swap and in Fig. 14.29, arriving at the arrangement depicted in Fig. 14.31. Satisfying the principle illustrated in Fig. 14.28(b), the out V in V R 1 L 1 C 1 Figure 14.31 High-pass filter obtained from Fig. 14.27. circuit acts as a second-order filter because as , approaches an open circuit and a short circuit. The transfer function is given by (14.53) (14.54) The filter therefore contains two zeros at the origin. As with the low-pass counterpart, this circuit can exhibit peaking in its frequency response. Band-Pass Filter From our observation in Fig. 14.28(c), we postulate that, must contain both a capacitor and an inductor so that it approaches zero as or . Depicted in Fig. 14.32 is a candidate. Note that at , the parallel combination of and acts as out V in V L 1 C 1 R 1 Figure 14.32 Band-pass filter obtained from Fig. 14.27. an open circuit, yielding . The transfer function is given by (14.55) (14.56)
BR Wiley/Razavi/ Fundamentals of Microelectronics [Razavi.cls v. 2006] June 30, 2007 at 13:42 743 (1) Sec. 14.4 Active Filters 743 14.4 Active Filters Our study of second-order systems in the previous section has concentrated on passive RLC realizations. However, passive filters suffer from a number of drawbacks; e.g., they constrain the type of transfer function that can be implemented, and they may require bulky inductors. In this section, we introduce active implementations that provide second- or higher-order responses. Most active filters employ op amps to allow simplifying idealizations and hence a systematic procedure for the design of the circuit. For example, the op-amp-based integrator studied in Chapter 8 and repeated in Fig. 14.10(a) serves as an ideal integrator only when incorporating an ideal op amp, but it still provides a reasonable approximation with a practical op amp. (Thus, the term “integrator” is a simplifying idealization.)

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