Modern Filter Design: Active RC and switched capacitor
2: Department of Electrical Engineering, University of Pennsylvania, Philadelphia, PA, USA
Originally published in 1981, Modern Filter Design remains a classic statement of the principles underlying the analysis and design of active RC and switched capacitor filters. Among other topics, the authors discuss the design of continuoustime, second order active sections (biquads), various measures of sensitivity, and the basic properties and classification of continuoustime and sampled data systems, together with filter transfer functions and approximations.
Inspec keywords: RC circuits; statistical analysis; switched capacitor filters; active filters; operational amplifiers; MOS integrated circuits; approximation theory
Other keywords: switched capacitor filters; higherorder filter design; active filter design analysis; active RC; operational amplifiers; MOS integrators; filter transmission; modern filter design; bipolar integrators; filter approximations; statistical sensitivity measurement; basic systems classifications; continuoustime secondorder active sections
Subjects: Other MOS integrated circuits; Amplifiers; Interpolation and function approximation (numerical analysis); Active filters and other active networks; General electrical engineering topics
 Book DOI: 10.1049/SBCS005E
 Chapter DOI: 10.1049/SBCS005E
 ISBN: 9781884932380
 eISBN: 9781613530528
 Format: PDF

Front Matter
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1 Filter Transmission and Related Topics
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In this chapter, the fundamental concepts relevant to continuous and sampled data systems have been reviewed. Although this chapter cannot possibly provide a complete treatment of these subjects, this material has been found to be a sufficient background for understanding and applying the tools to the design of practical active filters.

2 Operational Amplifiers
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In this chapter, the operational amplifier is used to realize simple functions used in active filters, such as summing amplifiers, integrators, inductor simulations, and FDNRs. Although in our pencilandpaper analyses, for simplicity, assume ideal Op Amp with infinite gain, the omission of their finite gainbandwidth products results in some degree of error. Depending on the frequency of operation and the form of compensation used, this error may or may not be negligible. When the error is nonignorable, designs can frequently be nominally predistorted to minimize the error.

3 Sensitivity
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One of the most important criteria for comparing equivalent circuit configurations and for establishing their practical utility in meeting desired requirements is sensitivity. In practice, real circuit components will deviate from their nominal design values due to manufacturing tolerances, environmental changes such as in temperature and humidity, and chemical changes which occur as the circuit ages. The causeandeffect relationship between the circuit element variations and the resulting changes in the response or some other network function is referred to as sensitivity.

4 ContinuousTime Active Filters  Biquadratic Realizations
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In this chapter, the design and performance of singleamplifier and multipleamplifier activeRC and activeR(C) biquad circuits has been considered. Negativefeedback singleamplifier circuits have been shown to have lowQ sensitivities to passive components but large ω_{o} and Q sensitivities to the Op Amp GB. The latter property restricted negative feedback circuits to low Q. On the other hand, positivefeedback circuits have been shown to offer the ability of tradeoff between active and passive sensitivities and can be designed for higher Q than the negativefeedback circuits.

5 HighOrder Filter Realization
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In this chapter we have presented numerous alternative means for realizing highorder filters. These techniques are: 1. Direct synthesis. 2. Cascade synthesis. 3. GIC immittance simulation. 4. Multipleloop feedback (and feedforward) synthesis. The technique that will serve a given application best, is the one that meets the frequencydomain and statistical requirements with minimum hardware and engineering time. For all but the most tolerant applications, the direct approach is impractical. To minimize Op Amps, designs based on the use of singleamplifier biquads (SABs) are encouraged. A simple design method requiring minimum engineering time is the cascade approach. In most applications we find that cascade designs, with a 0.2 % resistor tuning capability, provide sufficient precision. To meet more challenging requirements, the SAB MF topologies in Fig. 549 are recommended. When the sections are assigned identical Q's, the FLF (PRB) design will provide lower sensitivities than will a corresponding cascade design. In fact, all MF designs can provide lower sensitivities than the cascade within the passband. However, lower inband sensitivities will be achieved with either an LF or MLF design. Since LF designs can be derived directly from a properly terminated LC ladder network, the very low LF sensitivities can often be obtained with modest engineering effort. FLF filter is not only easy to design, it is the only topology whose output noise can remain independent of the overall filter gain and the type of secondorder section used [P27]. Finally, when the LF, MLF, and FLF designs do not quite satisfy requirements, an optimized MSF design should be tried.

6 Active Switched Capacitor SampledData Networks
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In this chapter we describe and demonstrate techniques for the analysis and design of active switched capacitor networks. Because of their sampleddata character, switched capacitor networks are most conveniently analyzed and designed, in the ztransform domain, like digital filters. However, SC filters are analog networks; thus, the analog concepts of impedance and loading, which are absent in digital filters, are retained. In this chapter a library of zdomain equivalent circuits will be developed to facilitate the analysis, design, and, more importantly, to aid the reader's understanding of switched capacitor networks. With these equivalent circuits, the familiar circuit theory tools, which we use routinely in the analysis of continuous active networks, are extended for use in switchedcapacitor networks. Like activeRC networks, there are many SC topologies that can be used to realize a given zdomain transfer function. We will examine some of the more interesting and useful ones.

Appendix A: Selected Topics in PassiveNetwork Properties
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Some selected topics in passivenetwork properties have been briefly discussed. In the design of activeRC networks, active devices and passive RC networks are employed. It is helpful to the designer to be familiar with some of the basic properties of RC networks. Before considering passiveRC networks, it is useful to know some basic properties of passive networks in general.

Appendix B: Tables of Classical Filter Functions
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This book chapter presents the table for filter function with different filter types discussed. Ideal lowpass filter characteristics is approximated by an allpole function. Butterworth filters table B1 gives the polynomials for n up to 10 and a<sub>o</sub> normalized to unity. For n < 4, see Table 11. Chebyshev filters when the Chebyshev filter requirements call for 1/2dB ripple, the polynomials for n up to 10 and a<sub>n</sub> normalized to unity are listed in Table B2. For n < 4, see Table 13. The corresponding polynomials for 1dB ripple are in table B3 and for 2dB ripple, in Table B4. Thomson filters the Bessel polynomials for Thomson filters for n up to 10 are given in table B5 for τ = a<sub>1</sub>/a<sub>o</sub> = 1. For n < 4, see Table 14. The coefficients of the elliptic filters for one zero pair and two poles, one zero pair and three poles, and two pairs of zeros and four poles are given in Tables B6, B7, and B8, respectively [P1]. The parameters A_{1}, A_{2}, and ω_{s} are defined in Fig. B1. The reader is referred to many excellent tabulations for elliptic filter data [B1B4].

Appendix C: Op Amp Terminologies and Selected Data Sheets
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Back Matter
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