Sallen KeyAnalysis

7/29/2019 Sallen KeyAnalysis

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TECHNICAL REPORT #07-1106

Second Order Frequency Domain

Effects of the SallenKey Filter

Onder OzDoctoral Student, USC Electrical Engineering

&

Dr. John ChomaProfessor of Electrical & Systems Architecture Engineering

Fellow, Scintera Networks (San Jose, CA)

University of Southern California

Ming Hsieh Department of Electrical EngineeringUniversity Park: Mail Code: 0271

Los Angeles, California 900890271

2137404692 [USC Office]

6269150944 [Home Fax][email protected]

ABSTRACT:

This report provides a detailed analysis of the popular Sallen-Key filter, which is

ubiquitously used in a variety of system applications requiring lowpass filtering at

baseband frequencies. In the course of developing this analysis, engineeringproblems deriving from a finite and nonzero output impedance in the amplifier

utilized as the active element of the filter are revealed. Unless suitable design

care is exercised to mitigate these problems, the performance of the Sallen-Key

filter can deviate substantively from the somewhat idealized performancepostulated in the technical literature.

November 2006


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Sallen-Key Filter University of Southern California Oz/Choma

USCintera - 2 - November 2006

1.0. LOWPASS FILTER APPLICATIONS

Lowpass filters are required in amplitude demodulators, sampled data signalprocessors, and a host of other subcircuits encountered in state of the art communication systems.An example that highlights the utility of a lowpass filter is the synchronous detector abstracted inFigure (1). The applied radio frequency (RF) signal, vrf(t), is the amplitude modulated sinusoid,

rf m cv (t) r (t) ( t) ,cos (1)

LowpassFilter:H(s)

Mixer

v (t)rfv (t)m

v (t)lo

v (t)o

Fig. (1). Simplified block diagram of a synchronous amplitude detector.where c is the radial carrier frequency, which is the frequency of the sinusoid whose amplitudeis modulated by the time varying information signal, rm(t). In conventional amplitude modula-tion, the spectrum of frequencies embodied by rm(t) span from a stipulated low frequency to a ra-dial frequency ofm, where generally, m


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loo m

m

Vv (t) r (t) h(t) ,

2V

(5)

where h(t) is the impulse response corresponding to the filter transfer function, H(s), and theasterisk(*) signifies the mathematical operation of convolution. To the extent that the filter fre-quency response, |H(j)|, is constant from zero frequency through the highest frequencycomponent, m, associated with the modulation signal, vo(t) is rendered directly proportional torm(t). In short, the detector captures the information signal to within a presumably predictablescale factor determined by the amplitude, Vlo, of the local oscillator signal and the conversiongain, (1/Vm), of the mixer.

In principle, either a passive or an active circuit can be deployed for the requisite low-pass filter in Figure (1). An active filter is generally preferred for at least two reasons. First, anactive filter affords more independent control of the filter quality factor and bandwidth than dopassive counterparts, thereby allowing for enhanced flexibility in the choice of desired frequencyresponse. This is to say that while a particular application may call for a maximally flat magni-tude filter response, others may function more optimally as a Tchebyschev, Bessel, or other type

of architecture[1]

. A second reason underlying a preferred active filter embodiment, particularlyfor detector applications stems from the fact that the filter must be capable of sufficiently largeattenuation for frequencies lying outside the modulation passband. This large attenuation inhib-its aliasing of information frequencies lying in the neighborhood of frequency (2c) withinformation in the neighborhood of frequency m. In the system of Figure (1), for example,modulation information is contained at the highest frequency, m, of the modulating passbandand, because of the second term on the right hand side of (4), at the frequency, (2c m). Inorder to mitigate the potential interference between these two spectral components, the gain ob-served at (2c m) must be substantially smaller (usually by 60 dB or more) than the filter gainat frequency m. Such extreme attenuation generally presupposes a multi-order filter. Whilemulti-order filters can be realized as passive structures, they are simpler to design and implement

reliably as cascades of active biquadratic structures because of the inherent interstage bufferingafforded by appropriately designed active cells.

2.0. SALLEN-KEY FILTER

A variety of active filters capable of responding to the foregoing and related otherengineering requirements is available in the literature[2]-[4]. Among the most popular of these isthe venerable Sallen-Key filter, whose basic schematic diagram is submitted in Figure (2a)[5].The Sallen-Key structure uses two resistors, R1 andR2, two capacitors, C1 andC2, and a nonphase-inverting amplifier of voltage gain, K. Because K = Vout/Vi > 0,capacitance C1 forms apositive feedback loop, which generates the complex closed loop poles that are required of

Butterworth, Tchebyschev, and Bessel filters. An advantage that the Sallen-Key circuit enjoysover many of its competitive active RC topologies is its ability to operate with unity amplifiergain (K = 1). Aside from ensuring unconditionally stable closed loop dynamics,K = 1 affordshigh amplifier linearity even when the applied input signal amplitude, Vin, is relatively large.Additionally,K = 1 does not inhibit the broadband amplifier frequency responses that may berequired of a particular system application. In the specific case of the detector in Figure (1), thisfrequency response attribute accommodates a potentially large signal passband for the modula-tion signal, rm(t).


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K

R2

R1

C2

Co

C1

Vout

Vin

Vi

(a)

R2

R1

C2

C1

Vout

Vin

Vi

(b)

Ro

KVi

Fig. (2). (a). Basic schematic diagram of the Sallen-Key

filter. (b). Linear model of the filter in (a).Capacitive output port loading in the amount ofCo ispresumed, as is a nonzero amplifier output portresistance ofRo.

The linearized equivalent circuit of the filter in Figure (2a) is offered in Figure (2b).The amplifier is modeled at its output port as a Thvenin equivalent circuit comprised of the se-ries interconnection of the voltage controlled voltage source,KVi, and the amplifier output resis-

tance, Ro. The inclusion of a nonzero Thvenin output resistance is a necessary undertaking ifthe amplifier is realized in deep submicron MOSFET technology. Achieving very low outputresistances in this device technology is a daunting challenge, particularly if the amplifier is real-ized as a low power, broadband buffer. Note that a capacitive load, Co, is appended to thelinearized model. The inclusion ofCo is practical when system specifications call for a multi-order filter that is realized as a cascade of individual Sallen-Key structures. Finally, the ampli-fier is modeled with infinitely large input impedance, which is reasonable for MOSFETrealizations operating at baseband frequencies.

2.1. CIRCUIT ANALYSIS

In the model of Figure (2b), assumeK = 1 and take

2 1R NR NR , (6)

2 1C MC MC , (7)

o cC k C , (8)

and

o rR k R . (9)


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Additionally, introduce the normalized complex frequency variable,p, such that

p sRC . (10)

Then, the transfer function,H(p), of the model in Figure (2b) can be shown to be

2r r

2 3r c r r c r c

out

in

1 pk p k MN .

1 p M N 1 k k 1 p MN k M N 1 k k 1 M MN p k k MN

VH(p)

V

(11)

This cumbersome expression can be put into engineering perspective by considering the ideal-ized Sallen-Key transfer relationship, sayHI(p), that results when the utilized unity gain ampli-fier is ideal in the sense of delivering zero output resistance; that is, Ro = 0. From (9), kr = 0whenRo = 0, which results in the considerably simpler transfer relationship,

o

outI 2

in R 0

V 1H (p) .

V 1 pM N 1 p MN

(12)

2.2. PERFORMANCE ASSESSMENT

A comparison of (11) with (12) reveals interesting effects exerted on Sallen-Key filterperformance by the amplifier output resistance, Ro. Aside from altering the coefficients of thefirst and second order frequency terms in the network characteristic polynomial, nonzero Roestablishes a third order response, as opposed to the second order frequency response indigenousto the idealized model. This fact is hardly surprising in light of the capacitive load, Co, imposedat the output port of the filter. But an additional effect of nonzero output resistance is the genera-tion of two, invariably complex, zeros. These two zeros lie at the normalized frequencies,z1 and

z2, (normalized to the frequency, 1/RC) which are given by

r1,2

4MN1 j 1

kz .

2MN

(13)

The subject zeros are complex for the pragmatic operating circumstance of a small output resis-tance conducive to kr< 4MN. For very small, but nonzero, kr, the subject zeros are very near theimaginary axis in the complex frequency plane, which suggests a convergence toward a fre-quency notch in the magnitude response of the filter. With krsmall, (13) projects an approximatenormalized notch frequency,yn = nRC, of

n n r RC 1 k MN . (14)

Precise notching in the sense of zero signal transmission at = n cannot occur because thecomplex zeros in (11) have finite quality factor. In particular, the coefficient of the first orderfrequency term of the numerator polynomial on the right hand side of (11) is not zero forkr 0.Nonetheless, the assurance of a smooth, monotonically decreasing frequency response in theSallen-Key filter requires that steps be taken to ensure that the notch frequency approximated by(14) lies well outside the passband of the information signal.


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The foregoing contention is best placed into design-oriented perspective by returning tothe idealized transfer function of (12). Note therein that the normalized self-resonant frequency,sayyo, is

ro o n RC y k1 MN . (15)

Since

o

M N 11

,Q y

(16)

the quality factor, Q, of the idealized filter is

Q1 N

.N 1 M

(17)

It follows that (12) is expressible as

o

out

I 2in R 0

o o

V 1

H (p) ,V p p1

Q y y

(18)

which can be shown to imply a normalized3-dB bandwidth,yb, deriving from

2b

b Q2 2o

y 1 1 RC MN 1 1 1 f .

y 2Q 2Q

(19)

In order for the notch region to lie substantially outside the passband of the baseband signal, thenotch frequency, n, must be significantly larger than the radial 3-dB bandwidth, b. To this

end, (14) and (19) combine to yield the design requirement,

r Q 1k f . (20)

Figure (3) plots the bandwidth function,fQ, in (19) as a function of the quality factor, Q,of the Sallen-Key filter. Superimposed on this graph is the minimum resistance ratio, 1/kr =R/Ro, required to ensure the satisfaction of (20) by a factor of ten. The figure limits considera-tion to quality factors no smaller than one-half since (18) establishes complex poles, which arerequired of Butterworth, Tchebyschev, and other commonly adopted filter architectures, only forQ > . An assiduous consideration of (19) and the figure at hand shows that fQ = 1 for Q

=1 2 , whereupon the corresponding 3-dB bandwidth, using (19) and (17), is seen to be

b Q 1 2

N 1 1 .

N 2 RC

(21)

A design reflecting fQ = 1 is significant in that it corresponds to a two-pole, Butterworth filterrealization.


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0

50

100

150

200

250

0.50 1.00 1.50 2.00 2.50 3.00

Filter Quality Factor, Q

0.00

0.32

0.64

0.96

1.28

1.60

Minimum R/R o Bandwidth Function, fQ

R/Ro

fQ

Fig. (3). The dependence of the bandwidth metric,fQ, on the filter quality factor, Q, and the

minimum resistance ratio, R/Ro, commensurate with a notch frequency lying adecade above the 3-dB bandwidth of the Sallen-Key filter.

0.00

0.20

0.40

0.60

0.80

1.00

0.01 0.10 1.00 10.00 100.00

Normalized Frequency

Ideal:

k r = k c = 0

Nonideal:

k r = 0.12

k c = 0.10

Gain Magnitud e (Volts/Volt)

Fig. (4). Frequency responses for the Sallen-Key filter for the ideal case of kr = kc = 0 and

the more practical, non-ideal case ofkr = 0.12 andkc = 0.10. The frequency scaleis normalized to the inverse of time constantRC.


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Figure (4) is submitted as a means of assessing the impact of nonzero amplifier outputresistance and nonzero load capacitance. It effectively compares the idealized Sallen-Key fre-quency response implied by (12) with the frequency response deriving from the non-idealtransfer function given by (11). The quality factor of the idealized two-pole system is set to Q

=1 2 , thereby prescribing an unambiguous relationship between the resistive and capacitive

parameter ratios,N

andM

. These same values ofN

andM

are invoked in the three-pole transferfunction of (11). In concert with filter designs processing frequencies up to the mid hundreds ofmegahertz, kr is set to 0.12 andkc is chosen to be 0.10. The graph at hand shows flat, monotonedecreasing frequency responses for both the ideal and non-ideal cases at low signal frequencies.On the other hand, the 3-dB bandwidth is deleteriously impacted by kr= 0.12 and, to a lesser ex-tent, by kc = 0.10 in the amount of better than 17.5%. As expected, partial notching is evidencedin the non-ideal response. The extremum of this notching is evidenced in the specific exampleconsidered herewith at a signal frequency that is roughly 3.8-times larger than the non-idealbandwidth. For particularly demanding applications, this notching frequency may not be suffi-ciently removed from the information passband. The problem can be mitigated by an amplifierpresenting smaller output resistance and thus, a smaller value of the parameter, kr.

3.0. OPTIMAL ELEMENT RATIOS

In much of the textbook literature, the resistance and capacitance ratios, N andM,respectively, are routinely equated and are often set to unity. While the simplicity of this designapproach cannot be argued, particularly from an analytical perspective, it proves inappropriatewhenever the quality factor of a stage exceeds unity. To wit, ifN = Min (17),Nmust be nega-tive, which is a daunting challenge in light of (6) and (7), if Q > 1. Moreover, N = M = 1 isappropriate if and only if the desired quality factor is 0.5.

A more rational design tack exploits the fact that in an integrated circuit, only reason-able ratios of resistance or capacitance values can be realized predictably and accurately. Thus,design strategies forging overtly large or small element ratios in a Sallen-Key topology areimprudent. In order to address this issue, return to (17) to solve for parameterM; specifically,

22N

,Q N 1

(22)

which indicates that a maximum value ofMcan be found with respect toNfor a given qualityfactor, Q. In particular, the slope, dM/dN, is zero at N = 1, for which the corresponding maxi-mum value ofMis 1/4Q2. Since Q must be at least as large as to produce the complex filterpoles required for maximally flat magnitude responses, the largest pertinent value ofM is one.Observe in (22) that Q > 1 precipitates a small value ofM, which means that the capacitance ra-

tio in a Sallen-Key structure may be distressingly large whenever significant underdamping isrequired of a frequency response.

Figure (5) plots the capacitance ratio,M, as a function of the resistance ratio,N, in (22)for various values of the circuit quality factor, Q. Aside from confirming the extrema inMatN= 1, the curves also indicate that parameterMis considerably more sensitive toNforN < 1 thanforN > 1, particularly if Q < 1.5. In light of the uncertainties routinely encountered duringmonolithic circuit processing, designs invoking values ofN that are nominally of the order of15% larger than the optimizedN-value of one may be warranted. Note thatN = 1.15, which al-


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lows for a generous 15% uncertainty in circuit element ratio, achieves anM-value that remainsnear its maximum value, thereby complementing the engineering goal of avoiding excessivelylarge ratios between filter capacitances in the Sallen-Key structure.

0

0.2

0.4

0.6

0.8

1

0.1 1 1.9 2.8 3.7 4.6 5.5

Resistance Ratio, N

Capacitance Ratio, M

Q = 0.5

Q = 0.7

Q = 1.0

Q = 1.5

Fig. (5). Capacitance ratio Mas a function of resistance ratio N for various values of thequality factor, Q, of a Sallen-Key filter.

4.0. FOUR-POLE BUTTERWORTH FILTER

The Sallen-Key architecture shown in Figure (2a) delivers a lowpass, two-pole filterwhose transfer function, assuming negligibly small amplifier output resistance, is the normalizedexpression in (12). The normalized frequency variable,p, in this relationship can be de-normal-ized with the help of (17) and (19) to arrive at the equally convenient form,

out

2in Q Q

b b

V 1H(s) ,

V f s f s11

Q

(23)

where b is recalled as the radial 3-dB bandwidth of the filter, and the approximation reflects the

tacit presumption of satisfying (20). IfQ =1 2 ,fQ in (19) is one, and (23) collapses to


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Q 1 2 2

b b

1H(s) ,

s s1 2

(24)

which is precisely the transfer characteristic of a second order, lowpass, Butterworth filter.

A lowpass Sallen-Key filter obviously delivers a magnitude attenuation of 20dB/decade at high signal frequencies. If greater response selectivity at the passband edge is re-quired of a particular application, the number of circuit pole pairs can be increased by cascadingseveral Sallen-Key stages. For example, the two-stage structure of Figure (6) delivers 40dB/decade of attenuation at the high frequency edge of the passband. This contention reflectsthat fact that, assuming non-interactivity between the two stages, the overall transfer function is

R2a

R1a

C2a

C1a

Vout

Via

K=1

K=1

R2b

R1b

C2b

C1b

Vib

Vin

Fig. (6). Schematic diagram of a four-pole lowpass filter realized as a cascade of two

Sallen-Key stages.

out

2 2in

Qa Qa Qb Qb

a ba ba b bb bb

V 1H(s) ,

V f s f s f s f s1 11 1

Q Q

(25)

where the appended subscripts, a andb, respectively refer to the first and second stages of thesubject topology. For overall Butterworth response characteristics, the desired3-dB bandwidth,b, must satisfy

ba bbb

Qa Qb

,f f

(25)

where ba is the 3-dB bandwidth of the first filter stage, andbb symbolizes the 3-dB bandwidthof the second stage. Additionally, the quality factors, Qa andQb, associated with the poles of thefirst and second stages are disclosed in the literature to be Qa = 0.5412 andQb = 1.307

[6].

5.0. DESIGN EXAMPLE

Consider the design of a four-pole, lowpass Butterworth filter realized as two Sallen-Key stages in cascade. The filter is to deliver an overall 3-dB bandwidth ofb = 2(300 MHz).Assume that the amplifiers utilized in the cascade provide an output resistance ofRo = 50 andan output port capacitance (inclusive of parasitic loading phenomena) ofCo = 30 fF. Assumethat partial notching in the frequency response can be tolerated at a frequency that is at least 1GHz. A viable design tack is itemized below.


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(1). ForQa = 0.5412, (19) givesfQa = 0.7195, while Qb = 1.307corresponds tofQb = 1.390.

(2). It is prudent to satisfy (20) for the largest anticipated value offQ, which isfQ = fQb = 1.390. Forpartial notching at a frequency of at least a gigahertz, which is a factor of3.3 above the targetedfilter bandwidth goal,

r 22

Qb

k 0.04751

.

3.3 f

Choosing kr = 0.04 and recallingRo = 50 , (9) and (6) stipulate baseline resistance values forboth filter stages ofR R1a R1b = 1.25 K.

(3). Recall thatNa = Nb = 1.15 is a reasonable design tack It follows from (6) that withR1a = R1b =1,250 ,R2a = NaR1a = 1,438 , andR2b = NbR1b = 1,438 .

(4). Appealing to (22), the capacitance ratio,Ma, for the first stage is, with Qa = 0.5412 andNa =1.15,Ma = 0.8494. Similarly, for the second stage, for which Qb = 1.307andNb remains equalto 1.15,Mb = 0.1457.

(5). Figure (4) and its associated analytical disclosures confirm that nonzero amplifier output portresistance and capacitance deleteriously impact the achievable filter bandwidth by as much as15% -to- 20%. Although the bandwidth specification of the filter is 300 MHz, engineering

pragmatics call for a design strategy that increases the bandwidth goal by nominally 15% toperhaps 340 MHz. In short, the desired frequency domain performance of the filter might bemore satisfactorily achieved by designing for a bandwidth ofb = 2(340 MHz). From (25),the radial 3-dB bandwidth of the first stage follows as ba = fQab = 2(244.6 MHz) and for thesecond stage, bb = fQbb = 2(472.6 MHz). It is interesting to observe that one of the twoSallen-Key stages operates at a bandwidth that is better than 50% larger than the desired overallfilter bandwidth. Thus, some of the amplifier buffers utilized in respective filter stages must becapable of operating through frequencies that are appreciably larger than the requisite overallbandwidth.

(6). Recalling (19) and (17), the baseline capacitance, C1a, for the first stage of the filter is

Qa aa1a

a ba 1a

378.9 fFf QN 1

C .N R

(29)

In this expression, the parameter values invoked areNa = 1.15,fQa = 0.7195, Qa = 0.5412, ba= 2(244.6 MHz), andR1a = 1,250 . For the second stage,

Qb bb1b

b bb 1b

914.7 fFf QN 1

C ,N R

(30)

whereNb = 1.15,fQb = 1.390, Qb = 1.307, bb = 2(472.6 MHz), andR1b = 1,250 .

(7). SinceMa = 0.8494,Mb = 0.1457, C1a = 378.9 fF, andC1b = 914.7 fF, C2a = MaC1a = 321.8 fF,andC2b = MbC1b = 133.3 fF.

Figure (7) depicts the schematic diagram of the completed design, where all resistancesare in units of ohms and all capacitances are dimensioned in femtofarads. Figure (8) shows theresults of HSPICE frequency response simulations. In the latter diagram, the curve labeledTwo-Stage Filter, pertains to the overall magnitude response, H(s) = Vout/Vin. The two othercurves reflect frequency responses of the individual filter stages. In particular, the First-StageFilter curve pertains to the transfer function, Vab/Vin in Figure (7), while the curve labeled Sec-ond-Stage Filter corresponds to the gain function, Vout/Vab in Figure (7). The overall magnituderesponse delivers a 3-dB bandwidth of301.9 MHz, which is 0.63% larger than the design goal of300 MHz. The first stage response delivers a 3-dB bandwidth of 233.8 MHz, which is 4.4%smaller than the calculated first stage bandwidth ofba/2= 244.6 MHz. On the other hand, thebandwidth delivered by the second stage is 433.0 MHz, which is 8.4% smaller than the calculated


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second stage bandwidth ofbb/2 = 472.6 MHz. It is intriguing that the overall filter bandwidthis very near its design target, despite the fact that the 3-dB bandwidths of both the first and sec-ond stages are smaller than anticipated. Evidently, the quality factors, Qa andQb, which impactthe respective bandwidth functionsfQa andfQb, must decrease from their idealized design targetsbecause of nonzero output resistance and capacitance in each of the two utilized amplifiers.

14381250

321.8

378.9

Vout

Via

K=1

K=1

1438

1250

133.3

914.7

Vib

Vin

Vab

Fig. (7). Schematic diagram of a four-pole, lowpass, Butterworth filter designed for a 300

MHz 3-dB bandwidth and realized as a cascade of two Sallen-Key stages. Allresistance values are in units of ohms, and all capacitances are in units offemtofarads. Each unity gain amplifier is presumed to extol a Thvenin output

port resistance of50 and a shunt output port capacitance of30 fF.

-80

-70

-60

-50

-40

-30

-20

-10

0

10

0.01 0.1 1 10

I/OGa

in(decibels)

Signal Frequency (GHz)

Two-StageFilter

First-StageFilter

Second-StageFilter

Fig. (8). Simulated frequency response of the two-stage, four-pole Butterworth filter shown

schematically in Figure (6). Also shown are the frequency responses of the firstand second stages of the filter.

It should be noted in Figure (8) that the second stage response displays observablepeaking. Such peaking is to be expected in view of the relatively large quality factor (Qb =1.307) of the second stage. On the other hand, no such peaking is evidenced in the frequency re-sponse of the first stage, which is designed for a comparatively small quality factor, Qa, of0.5412. The individual stage frequency responses serve to dramatize the effects of the individualButterworth characteristic polynomials. In particular, the low-Q stage has relatively anemic


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bandwidth, while the second stage has observable peaking in the vicinity of the 3-dB bandwidthof the first stage. It appears that the peaking of the second stage offsets the inferior first stagebandwidth, and the response attenuation of the first stage mitigates the peaking of the secondstage. The result is that both stages interact in an ostensibly optimal fashion to produce a maxi-mally flat overall two stage frequency response.

-0.2

0

0.2

0.4

0.6

0.8

1.0

1.2

0 20 40 60 80 100 120

Time (nSEC)

I/O

Responses(volts)

Fig. (9). Simulated transient response of the two-stage, four-pole Butterworth filter

shown schematically in Figure (6). The input signal, shown as the dashedcurve, is a 1-voltpulse train having 10 pSEC rise and fall times, an initial

delay of1 nSEC, a pulse width of20 nSEC, and a period of40 nSEC.

The transient pulse response shown in Figure (9) conveys no drama. The input signalapplied for this test is a pulse train having an initial delay of 1 nSEC, pulse widths of20 nSEC, aperiod of40 nSEC, and rise and fall times of10 pSEC. The 10% overshoot evidenced during therise and fall portions of the input pulse are stereotypical of the transient response of aButterworth filter. A detailed investigation of the simulated transient response data indicates thatin the steady state, response settling to with 0.5 % occurs in nominally 6.4 nSEC.

6.0. CONCLUSIONS

The popular Sallen-Key filter is definitively analyzed in this report, and an examplefour-pole design is presented and assessed. An advantage posited by the Sallen-Key structureover many competitive active RC architectures is its ability to operate with unity gain amplifiersin each of its stages. Unity gain amplifiers beget wide frequency response capabilities and adynamic range with respect to the input signal excitation that is larger than that of high gaincells. But the Sallen-Key structure is not without its shortfalls. In particular, the unavoidablepresence of nonzero amplifier output resistance produces complex left half plane zeros that lieuncomfortably close to the imaginary axis of the complex frequency plane. As a result, partialnotching of the frequency response is unavoidable at high signal frequencies, thereby precluding


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a strictly monotone decreasing frequency response. Moreover, capacitive loads imposed on theoutput port of the amplifier interact with the amplifier output resistance to degrade the observ-able bandwidth by as much as 15% -to- 20%, depending on the quality factor to which a stage isdesigned. Finally, in a multiple pole Sallen-Key filter designed for maximally flat magnitude re-sponse, it is impossible to maintain the small passive element ratios that are desired for accurateand reliable monolithic circuit processing. In the four-pole example addressed in this report, the

capacitance ratio required of the second stage is almost seven. Unfortunately, even wider ele-ment ratios are required in maximally flat Sallen-Key filters designed for a greater than four-poleclosed loop responses.

7.0. REFERENCES

[1]. W-K. Chen,Passive and Active Filters: Theory and Applications. New York: John Wiley andSons, 1986, chap. 2.

[2]. Ibid., chap. 7.[3]. P. E. Allen and D. R. Holberg, CMOS Analog Integrated Circuit Design. New York: Oxford

University Press, 2002, pp. 595-599.

[4]. L. P. Huelsman (ed.),Active Filters: Lumped, Distributed, Integrated, Digital, and Parametric.New York: McGraw-Hill Book Company, 1970, chaps. 2, 4.

[5]. R. P. Sallen and E. L. Key, A Practical Method of Designing RC Active Filters,IRE Trans.Circuit Theory, vol. CT-2, pp. 74-85, March 1955.

[6]. R. Schaumann and M. E. Van Valkenburg, Design of Analog Filters. New York: OxfordUniversity Press, 2001, pp. 256-261.

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