switched capacitors

Volume 3 No.3, Fall 1999 ISSN# 1523-9926

Applications of Switched-Capacitor Circuits inActive Filters and Instrumentation Amplifiers

Dr. William R. Grisé[email protected]

Department of IETMorehead State University

ABSTRACT

This paper will explain the basic concepts underlying the operation of the switched capacitor, as well asthe use of switched-capacitors to realize compact and versatile circuits already familiar to theundergraduate student of electronics. One set of example circuits include easily tunable active filters;specific examples of filter designs that incorporate switched-capacitors will be developed, and the use of acommercially available switched-capacitor integrated circuit, the MF10, to implement the designs will beshown. Another example circuit is an instrumentation amplifier that is more compact and has a higherCMRR than the conventional realization. Linear Technology's LTC1043 serves as the vehicle for thiscircuit. By demonstrating the utility of the modern switched-capacitor IC in these two importantelectronic functions, it is hoped that instructors and students in engineering technology will include thestudy of the switched-capacitor in advanced electronics courses.

I. INTRODUCTION

This paper aims to show how the switched-capacitor concept can be used to realize a wide variety ofactive filters that have the advantages of compactness and tunability. In particular, the explanations anddesign examples presented here will use mathematical tools familiar to the electronics technology andengineering undergraduate student. We will not use the Z-transform, which is the rigorously correct toolfor analyzing sampled-data waveforms.

The paper will present the following topics. First, the basic ideas behind the use of the switched-capacitorto replace resistors in active filter circuits will be explained. Second, the use of the switched-capacitor toimplement lossless, lossy, and differential integrators, which are the backbone of many switched-capacitorfilter circuits, especially those based on National Semiconductor's MF10 IC [1]. Third, example designsof active filters using the MF10 will be presented.

Before detailing the operation of switched-capacitor circuits, it will be useful to understand the motivationbehind, and applications of, these circuits. Basically, switched-capacitor techniques have been developedin order to allow for the integration on a single silicon chip of both digital and analog functions. Becausevery large scale integrated (VLSI) circuits rely on MOS transistors and pico-farad range MOS capacitors,

Switched Capacitors file:///G:/Windswept1/WINDSWEPT/S5EC/sc_fil.htm

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any realization of analog circuits on a chip will have to use these elements. By comparison, conventionalanalog circuits use the ratio of resistances to set the transfer functions of amplifiers, and the magnitudesof resistances to determine the operation of current-to-voltage and voltage-to-current converters. Finally,the values of RC products are used in active filters and signal generators to determine the frequencyresponses of those circuits. When one moves to the silicon chip and strives to achieve the samefunctionality in a much reduced area and using the tools of MOS technology, this is what one discovers.First, switches, small-value capacitors, and decent op-amps are easy enough to realize in MOStechnology. Second, using that same technology, it is very difficult and wasteful of silicon die area tomake resistors and capacitors with the values and accuracy encountered in audio and instrumentationapplications [2,3]. As we will see in the subsequent sections, designers have overcome these difficultiesby realizing that (1) resistors can be replaced by MOS switches that are rapidly turned on and off, andMOS capacitors, and that (2) the time constants arising from these simulated resistances and the MOScapacitors are given in the form of capacitance ratios. The fact that capacitor ratios control the timeconstants means that these constants now can take advantage of the superior matching of capacitancesfabricated on silicon, as well as their ability to track each other with temperature.

If these are undoubted advantages for the VLSI designer, what can the board-level circuit design expect toachieve with the use of switched-capacitors? For one thing, as we will see shortly, not only are the timeconstants of the switched-capacitor circuit superior in their control, but these time constants are tunablethrough the simple expedient of changing the frequency of the clock pulses that drive the circuit.Furthermore, the integrated circuit packages that are now available support a number of filtering functionsin one package, thus reducing footprints needed on circuit boards to realize a given set of analogfunctions.

Although switched-capacitors were developed in order to meet the need to incorporate analog, activefilters on silicon along with digital functions, they have since found many other uses [2]. These include,besides filters, instrumentation amplifiers, voltage-to-frequency converters, data converters,programmable capacitor arrays, balanced modulators, peak detectors, and oscillators.

II. BASIC SWITCHED-CAPACITOR OPERATION

The essence of the switched-capacitor is the use of capacitors and analog switches to perform the samefunction as a resistor. This replacement resistor, along with op-amp based integrators, then forms anactive filter. Before delving too far into actual filter designs, however, it makes sense to ask why onewould want to replace the resistor with such an apparently complex assembly of parts as switches andcapacitors. It would seem from the multiplication of parts that the switched-capacitor would be areaintensive. As a matter of fact, for the resistor values that one seeks in certain filter designs, this is not thecase. Furthermore, the use of the switched-capacitor will be seen to give frequency tunability to activefilters. Figure 1[2, 3] shows the basic setup for a switched-capacitor, including two N-channelMetal-Oxide Semiconductor Field-Effect Transistors (NMOS) and a capacitor. There are two clock

phases, , which are non-overlapping. The MOSFET's, either M1 or M2, will be turned ON whenthe gate voltage is high, and the equivalent resistance of the channel in that case will be low,

. Conversely, when the gate voltage goes LOW, the channel resistance will look

like . With such a high ratio of OFF to ON resistances, each MOSFET can be taken fora switch. Furthermore, when the two MOSFET's are driven by non-overlapping clock signals, then M1and M2 will conduct during alternate half-cycles.


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Figure 1. Two NMOSFET's, driven by alternating, non-overlapping clock signals, comprise the basicswitched capacitor network.

This makes the two-MOSFET arrangement equivalent to a single-pole, double-throw switch (SPDT). One can now use a symbolic switch picture, as in Figure 2a below, to represent the circuit. The operation

of this circuit is as follows. When the switch is thrown to the left, the capacitor will charge up to .

When the switch is thrown to the right, the capacitor will discharge down to/charge up to . As a resultof these consecutive switching events, there will be a net charge transfer of

. Now, if one flips the switch back and forth at a rate of

cycles/sec, then the charge transferred in one second is , which of

course has the units of current. One can claim that an average current, .

If is much higher than the frequency of the voltage waveforms, then the switching process can betaken to be essentially continuous, and the switched-capacitor can then be modeled as an equivalentresistance, as shown below in Figure 2b. The value of the equivalent resistance is given by:

(Eq. 1)

Therefore, this equivalent resistance, in conjunction with other capacitors, and Op-amp integrators, can beused to synthesize active filters. It is now clear from Equation (1) how the use of the switched-capacitorleads to tunability in the active filters, by varying the clock frequency.

(a) (b)

Figure 2. Equivalent resistor model for switched capacitor circuit in Fig. 1.

This equivalent resistance has features which make it advantageous when realized in integrated-circuitform:


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(a) High-value resistors can be implemented in very little silicon area. For example, a 1-MWresistor can be realized with a 10-pF capacitor switched at a clock rate of 100 kHz.

(b) Very accurate time constants can be realized, because the time constant is proportional to theratio of capacitances, and inversely proportional to the clock frequency:

. Capacitor ratios, especially in monolithic form, are very robustagainst changes in temperature, and clock frequencies can also be strictly controlled, so thataccurate time constants are now available in the switched-capacitor technology.

The principal constraint in using the switched-capacitor is that inherent in all sampled-data systems: theclock frequency must be much higher than the critical frequency set by the RC products in the circuit.Furthermore, on either side of the analog switches, i.e., the MOSFET's, there must be essentiallyzero-impedance nodes (voltage sources). There are a number of other constraints which the unsuspectingdesigner/user might overlook [3, p. 725]:

(a) The equivalent resistance formed by the action of the switched-capacitor cannot be used toclose the negative-feedback path in an op-amp all by itself. One must recall that to ensurestability, the op-amp's feedback path must be closed continuously, while the switched-capacitor is a sampled-data construction of a resistor, and thus not continuous.

(b) Circuit nodes cannot be left floating. That is, there must always be a resistive path to groundso that charge does not build up on the capacitor plates.

(c) The bottom plates of the MOS capacitors must be connected to ground or to a voltage source.There is an intrinsic, parasitic capacitance associated with the MOS capacitor's bottom plate[4]. This parasitic capacitance can be between 5% and 20% of the desired value; furthermore,it behaves nonlinearly with voltage [4]. Therefore, it must be connected to AC ground or avoltage source so that this nonlinear portion of the capacitance will not affect the overallresponse of the switched-capacitor filter. In practical terms, this means that capacitive voltagedividers with three or more capacitors, and circuits that switch both ends of a capacitor insequence to the inputs of an op-amp, are used.

(d) The noninverting pin of the op-amp should be kept at a constant voltage. If this pin isconnected to the signal in some way, then the virtual short circuit between op-amp inputsmeans that the inverting input is no longer a virtual ground, and so an undesirable alteration offilter response due to the MOS capacitor's parasitic capacitance will occur (see item (c)above).

III. SWITCHED CAPACITOR INTEGRATORS

The op-amp integrator is the most frequently chosen building block for switched-capacitor filters. Thestandard RC integrator is shown in Figure 3a, and its analysis and description can be found in anyelectronics text [5, 6].


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Figure 3a. Standard op-amp integrator, with transfer curve

To review, the voltage transfer function of this circuit is given by

(Eq. 2)

where . Now, if one replaces the resistor by its switched-capacitor equivalent, as shown inFigure 3b, and use Eq. 1 as the resistor's value, one finds

(Eq. 3)

Figure 3b. Switched-capacitor integrator.

Again, one notes the fact that this new integrator has no resistors, which take up excessive silicon die

area. Also, the -3 dB frequency, , depends on a ratio of capacitances, not on an RC product. Thetolerances for ratios are much easier to control than the tolerances for products. Finally, this characteristicfrequency of the integrator is inherently settable with a simple change in the clock frequency.

The typical values of capacitances used in switched-capacitor technology range from 0.1 pF to 100 pF.These are low enough values that the stray capacitances of the MOS switches, of the interconnects, and ofthe "plates" of the switched-capacitors themselves can all have a significant effect on the desiredfrequency response of the filters designed with switched-capacitors. The effects of stray capacitance havebeen reduced greatly by dual-switch configurations [2, 7]. Figure 4 shows explicitly the clock phasing ofthe MOS switches which acts to eliminate the transient charge transfer through the stray capacitances,Cs1 and Cs2, also indicated in the figure. In essence, charge transfer only takes place through the

capacitor . Figures 5a and 5b show both the inverting and noninverting stray-insensitive integrator.The noninverting stray-insensitive integrator is obtained simply by switching the clock phasing ontransistors M2 and M4.


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Figure 4. Arrangement of extra MOSFET's and clock signals to make switched capacitor circuitinsensitive to stray capacitances.

Figure 5a. Switch setting used to realize stray-insensitive inverting integrator.

Figure 5b. Switch setting for stray-insensitive non-inverting integrator.

Because of the importance of the integrator to switched-capacitor filters, it is necessary to be familiarwith the variants of the integrator. These include the summing integrator, the differential integrator, theintegrator/summer, and the lossy integrator. All of these play a role in the synthesis of switched-capacitorfilters. The summing integrator, shown below in Figure 6, has a response given by:


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(Eq. 4)

Figure 6. Summing integrator.

Figure 7 shows the differential integrator. The easiest way to understand this circuit is to look at what

happens to the charge accumulation on the capacitor when the switches are thrown to the left. In this

case, the capacitor charges up to a value of . When the switches are thrown to the right, thecharge on the capacitor is poured into the op-amp's summing node. The average current, assuming theswitching rate (= clock frequency) is high enough, is given by

(Eq. 5)

This results in a stray-insensitive output voltage of

(Eq. 6)

where .


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Figure 7. Differential integrator.

The lossy integrator provides a simple, first-order lowpass response with gain. This circuit is realized byplacing a switched-capacitor (i.e., a simulated resistor) in parallel with a feedback capacitor, Figure 8. Ingeneral, the easiest way to analyze the response of more complex switched-capacitor circuits such as thisone is to replace all switched-capacitors with their resistor equivalents. Once the transfer function is foundfor a circuit with resistors (and discrete capacitors), then the switched-capacitor equivalents of theresistors (Eq. 1) can be placed back in the transfer function to obtain the final result. For the lossyintegrator, the analysis proceeds as follows:

where the "0" in refers to the virtual ground at the op-amp's inverting input.The transfer function is obtained using these resistor equivalents:

(Eq. 7)

Now substituting the switched-capacitor equivalents for the resistors from Eq. 1, one obtains

(Eq. 8)


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where is the critical, or -3 dB, frequency of the lowpass filter. Eq. 8 has the form ofa lowpass filter multiplied by a gain proportional to the ratio of the two switched-capacitors.

Figure 8. Lossy integrator, or first-order lowpass filter with gain.

IV. SWITCHED CAPACITOR BIQUADRATIC FILTERS

The biquad configuration [8] normally features a lossy inverting integrator, a lossless inverting integrator,and a unity-gain-inverting amplifier. In the standard active RC configuration, this requires threeop-amps. However, the switched-capacitor realization of the biquad needs only two op-amps to performthe same function. One op-amp performs the lossy inverting integration function, while the secondop-amp performs lossless, noninverting integration. Although one can design an adequate switched-capacitor version of the biquad by making a resistor-by-resistor replacement in the standard RC biquadfilter, such an implementation has been found to have unacceptably wide capacitance spreads, especiallywhen higher filter Q's are sought [2]. Instead, Figure 9 shows the biquad filter with improved capacitanceratios. This circuit provides the highpass and bandpass responses. Just as with the analysis of the lossyintegrator, a fairly complete analysis of this circuit will be made here.


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Figure 9. Biquadratic bandpass and highpass filter provided by switched capacitor network, withimprovements needed to reduce spread in capacitor values.

The form of the bandpass filter function is at least clear:

(Eq. 9)

The use of a stray-insensitive switched-capacitor ( ) with alternating clock phases makes possible thenoninverting form above. In order to complete the analysis, one has to calculate the highpass filterresponse. The output node of the first op-amp, which gives the highpass filter response, can be seen to bethe superposition of two signals at the summing node of the op-amp:

(Eq. 10a)

With the expression in curly brackets simplifies, giving

(Eq. 10b)


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In order to obtain finally the bandpass transfer function, Eq. 10b is inserted into Eq. 9. The result of this

manipulation is

(Eq. 11)

Gathering like terms together, and multiplying both sides by , one obtains

(Eq. 12)

With , and isolating , the transfer function for the bandpass filter is finally found:

(Eq. 13)

The corresponding expression for the highpass filter is simply obtained by inverting Eq. 9. Recalling that

, one finds readily

(Eq. 14)

Although Eq. 13 does in fact display the standard form for the frequency response of a bandpass filter,there is nothing in the expression that gives the gain of the filter circuit. In fact, from the form of thetransfer function, and from simulation, it can be seen that the circuit in Figure 9 has no resonant gain. Thesame applies to the highpass filter expression in Eq. 14. It is clear that a useful active filter circuit formedfrom switched-capacitors must possess some voltage gain in the passband. Of course, it is possible tosimply "tack on" an amplifier to the output of the circuit. However, an elegant solution is to turn thesecond op-amp in Figure 9 into a summing integrator, in which the input signal is injected into the secondop-amp via an equivalent resistance derived from another switched-capacitor. Figure 10 shows theimplementation of this idea. In Figure 10, the SPDT switches, labeled S1, are shown closed in the first

half of the clock cycle .

The analysis of this circuit in order to derive the bandpass filter transfer function follows a path similar tothe one followed in Eq. 13, although more tedious. We will simply present the main results here andproceed to the simulation of the circuit. Superposition of input signals to each of the op-amp's input pinsgives


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(Eq. 15)

After much manipulation, one finds that the transfer function for the bandpass filter is

(Eq. 16)

In order to simulate the performance of switched-capacitor networks, it is necessary to realize that thesecircuits are a hybrid of both continuous and sampled-data signals. In fact, switched-capacitor circuits arecontinuous in amplitude and discrete in time. Because of this combination, simulation with conventionalcircuit simulators such as PSPICEÔ presents a problem. The presence of switched voltages means that atransient analysis must be performed. At the same time, the desire to determine the frequency responseacross a wide range of frequencies means that a transient analysis must be performed for each desiredfrequency. This is a very time-consuming process because one has to wait until the steady state is reached.One way to overcome this problem within a SPICE-type simulator is to implement all the designequations in z-transforms. These z-domain models will allow one to perform frequency-domainsimulation of complex switched-capacitor circuits. Although the z-transform is the rigorously correctmathematical tool for the analysis of sampled-data systems, it is not really available to the majority ofengineering technology students. The model element used as the simulation kernel in SPICE for z-domainanalysis is called the storistor [10]. It consists of conductances, a lossless transmission line for delayeffects, and controlled sources. In order to simulate the frequency response of something as simple as anintegrator, one is required to model as many as seven storistors, four capacitors, and an op-amp subcircuit.Given the long experience of many students and instructors with the SPICE program, this might be anacceptable alternative. However, the size of the input files for even simple switched-capacitor circuits(beyond the integrator), together with the mathematics required to understand the z-transform, willprobably deter many from this approach.

This paper will use a perhaps less well-known simulation package called APLACÔ [11-12] (originallyAnalysis Program for L inear Active Circuits). The APLAC program has been under continuousdevelopment since 1972. Since 1988, the Nokia Corporation, developers of wireless communicationproducts, has sponsored continued improvement in the system design and electromagnetic capabilities ofthe APLAC program. The particular strengths of the APLAC program are its use of object orientedprogramming techniques, which permit easy adaptation of models to the circuit environment in which acomponent finds itself. Also, the program has a very extensive library of system level blocks, and theability to model electromagnetic behavior of components.


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For our present purposes, one of the attractions of the APLAC program is its use of the convolutionintegral to model circuits with both frequency-dependent and transient behaviors. This feature isimportant for switched-capacitor circuits. The frequency-dependent portions of the circuit can be analyzedstraightforwardly. However, the frequency response of the time-dependent portions of the circuit (such asswitches and sources) is calculated by creating a frequency-domain equivalent circuit by means of theconvolution integral.

In the circuit of Figure 10, the following values of components and parameters are used:

These values give a resonant (center frequency) voltage gain of A0 = 10, a Q = 50, and a center frequency

of f0 = 20 kHz. The results of the simulation are shown below, in Figure 11.

Figure 10. Biquadratic switched-capacitor filter with gain setting from summing integrator. The switchesS1 are closed on the first half-cycle of the clock waveform.

Figure 11. APLAC simulation of bandpass filter with gain, from Figure 10. From Probe tool of simulator,peak gain is 3.32 dB, and center frequency is at f0 = 17.8 kHz.


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The results show a simulated gain of only 3.3 dB, and the center frequency is off by ~2 kHz. Thisinaccuracy is due to the use of ideal MOS switches in the hand calculations leading up to Eq. 16. TheAPLAC model for the switches assumes an Ron = 100 W, and an Roff = 100 kW. APLAC has the

capability to optimize the circuit's component values to achieve desired circuit behaviors.

V. APPLICATION OF MF-10

The MF-10 is a universal switched-capacitor filter supplied by National Semiconductor [1]. The MF-10uses the two-integrator loop structure to realize lowpass, highpass, bandpass, notch, and allpass functionsthrough externally chosen, discrete, resistors. The actual switched-capacitor integrators are internal to thechip, while the external resistors give the user flexibility in configuring his/her own response. However, totake advantage of component tracking with temperature, etc., all responses are designed to be functions ofresistor ratios only. Figure 12 shows the summing amplifier and two-integrator cascade internal to each

section of an MF-10. The tunability of a particular filter's critical frequency, , is determined by a logiclevel applied to a 50/100/CL frequency ratio programming pin. In other words, the critical frequency willbe

; if the programming pin is tied to ground, the divisor is 100, otherwise, iftied to a HIGH (positive power supply), the divisor will be 50. Figure 12 shows how the notch, bandpass,and lowpass filter functions are realized by the MF-10. Because the summing amplifier is outside thetwo-integrator loop, this configuration will be faster and allow a greater range of operating frequencies.

Figure 12. Two-integrator loop, with external resistors, used in MF-10 to realizenotch, lowpass, and bandpass filters.

The analysis of the transfer functions for the three transfer functions mentioned above follows the patternin Eq. 13-14. By inspection, one sees that

(Eq. 17a)

(Eq. 17b)

(Eq. 17c)


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Eliminating to obtain , the transfer function for the notch filter is obtained:

(Eq. 18)

Similar manipulations give the bandpass and lowpass responses. The bandpass response is:

(Eq. 19)

The lowpass response is:

(Eq. 20)

VI. SWITCHED-CAPACITOR IMPLEMENTATION APPLIFIER ¾ LTC1043 Although the initial impetus for the development of the switched-capacitor was the opportunity and needto synthesize active filters that would be compatible with MOSFET technology, the early 1980's foundmany other uses for the switched-capacitor. Linear Technology has developed the LTC1043 [9], whichcontains dual switched capacitor networks, along with an on-chip non-overlapping clock generator,oscillator, and charge balancing circuitry. The clock generator controls both of the switch networks, whilethe charge balancing circuitry is designed to cancel any effects due to stray capacitance. The on-chiposcillator has a fixed frequency of 185 kHz. An external capacitor can be connected across pins 16 and 17(for the instrumentation amplifier) to yield any desired clock rate. The desired clock rate can be foundfrom

; the 24-picofarad capacitance is the internal capacitance responsible for theoscillator’s fixed frequency.

Among the circuits developed from the LTC1043 are instrumentation amplifiers, lock-in amplifiers fordetecting extremely small parameter shifts in sensor applications, and signal conditioners for platinumresistance temperature detectors (RTD), relative humidity sensors, and LVDT’s. The instrumentationamplifier is a standard op-amp circuit presented in many electronics texts [5-6], and is designed toamplify small difference signals such as might be found in measurement or transducer applications. At thesame time, common-mode or noise signals picked up by the lines feeding the amplifier must besuppressed, especially as these signal levels are often larger in amplitude than the sought-for differencesignals. Figure 13 shows the LTC1043 combined with a standard non-inverting op-amp to give aninstrumentation amplifier with a common-mode rejection ratio (CMRR) of >120 dB. Figure 14 shows the


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same circuit with the ½ LTC1043 as a black box.

Figure 13. Instrumentation amplifier using ½ of LTC 1043 switched-capacitor, along with LF356/353op-amp in non-inverting configuration.

Figure 14. Switched-capacitor-based instrumentation amplifier, with ½ LTC1043 shown as a black box.

The pin numbers in Figure 13 are the pins in the black box in this figure.

The operation of this circuit is as follows. First, the dual switch, when flipped to the left, charges thecapacitor C1 up to the difference V1 – V2. Second, on the next clock pulse, the switches will then dump

the charge represented by that voltage difference onto C2. Third, the continuous clocking from the

oscillator will force C2 to eventually develop a voltage equal to the difference voltage. Finally, the

difference voltage, with the common-mode signal stripped off by the LTC1043 is amplified by theop-amp. It is interesting to observe several features of this circuit and compare them to the standardinstrumentation amplifier. By using the capacitor C1 (the so-called “flying capacitor”), the common-mode

voltage present at the inputs is looking into a capacitive voltage divider, between the C1 and the

LTC1043’s parasitic capacitance. This parasitic capacitance is typically less than 1 picofarad, so the ACvalue of the CMRR is > 120 dB. By comparison, Analog Device’s AD624 instrumentation amplifier cango as high as 130 dB for high gains, up to 60 Hz. Because of the capacitive voltage divider from theLTC1043, this instrumentation amplifier shows higher CMRR, over a wider range of voltage gains, and toa higher frequency.


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VII. CONCLUSIONS

This paper has presented the essentials of operation of switched-capacitor networks, with a specialemphasis on its use in designing active filters. Unlike active filters based on the conventional op-amp,switched-capacitor filters have critical frequencies that are easily pin-settable. Furthermore, they requireless power than the conventional op-amp based network because of their reliance on CMOS technology.Finally, for the functionality provided on a single chip, they take up less room on circuit boards. Alternateuse of the switched-capacitor network in an instrumentation amplifier has also been presented. Theoperation of this device is a little easier to digest for some students than discussion of active filters; it ishoped that instructors and students can use the information herein to extend their acquaintance withmodern integrated circuits.

The presentation of results here is in a form which instructors and upper-level students in electronicstechnology can adapt to the curriculum in technology programs.

REFERENCES

1. Introducing the MF-10: A Versatile Monolithic Active Filter Building Block, by Tim Regan.National Semiconductor Application Note 307, August 1982.

2. Design with Operational Amplifiers and Analog Integrated Circuits, by Sergio Franco.McGraw-Hill Book Company, New York, 1988. Chap. 13.

3. Bipolar and MOS Analog Integrated Circuit Design, by A.B. Grebene. Wiley-Interscience,John Wiley & Sons, New York, 1984. Chap. 13, pp. 703-752.

4. Fundamentals of MOS Digital Integrated Circuits, by J.P. Uyemura. Addison-Wesley,Reading, MA, 1988. Chap. 8.

5. Electronic Devices, 5th Edition, by T. L. Floyd. Prentice-Hall, Englewood Cliffs, NJ, 1997.Chap. 14.

6. Microelectronic circuits, 3rd Edition, by A. Sedra and K.C. Smith. Saunders CollegePublishing/HRW, Philadelphia, PA, 1991. Chap. 2.

7. "Stray Capacitance Insensitive Switched Capacitor Filters," M. Hasler. Proceedings of theIEEE International Symposium on Circuits and Systems, 1981.

8. Analog Filter Design, by M.E. Van Valkenburg. Holt, Rinehart, and Winston, New York,1982. Chap. 5.

9. Applications for a Switched-Capacitor Instrumentation Building Block, by Jim Williams.Linear Technology Application Note 3, July 1985.

10. CMOS Analog Circuit Design, 2nd Edition, by P. Allen and D. Holberg. Saunders CollegePublishing/HRW, Philadelphia, PA, 1998. Chap. 9.

11. APLACÔ 7.0 User's Manual, Helsinki University of Technology, Circuit Theory Laboratory& Nokia Corporation Research Center, 1998. Available at http://www.aplac.hut.fi/

12. "Fast Analysis of Nonideal Switched-Capacitor Circuits using Convolution," H. Jokinen,

M. Valtonen, and T. Veijola. 11th European Conference on Circuit Theory and Design, Davos,Switzerland, 1993, pp.941-946.


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