Single Amplifier, Active-RC, Butterworth, and Chebyshev Filters Using Impedance Tapering

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1 Sigle Amplifier, Active-RC, Butterworth, ad Chebyshev Filters Usig Impedace Taperig Draže Jurišić Faculty of Electrical Egieerig ad Computig, Uska 3, HR-0000, Zagreb, Croatia George S. Moschytz Swiss Federal Istitute of Techology (ETHZ-ISI) Sterwartstr. 7, CH-809, Zürich, Switzerlad Neve Miat Faculty of Electrical Egieerig ad Computig, Uska 3, HR-0000, Zagreb, Croatia ABSTRACT I this paper the sigle-amplifier active-rc filter desig procedure for some commo filter types, usig tables with ormalized filter compoet values, is preseted. The cosidered filters cosist of a RC ladder etwork i a positive feedback loop of a operatioal amplifier. Tables for the filter structures havig equal capacitors ad equal resistors were already preseted [],[]. I this commuicatio, we preset tables for desigig filters havig low sesitivities to variatios of passive compoets achieved by applyig the cocept of impedace taperig. The Schoeffler sesitivity measure is used as a basis for a sesitivity compariso of the filters desiged with equal capacitors, equal resistors ad with tapered capacitors. By usig a impedace taperig desig techique a cosiderable improvemet i sesitivity is achieved. Low-pass filters of up to 6 th -order are preseted. KEY WORDS: active-rc filters, sigle-amplifier filters, low-power, low-sesitivity, ormalized compoets.. INTRODUCTION Evaluatio of active-rc filter quality ivolves various parameters such as: simple realizability, repeatability, a possibility of straightforward procedure of parameter calculatio, small umber of compoets, low power cosumptio, low oise performace, ad most ofte, a low filter amplitude sesitivity to passive ad/or active compoet toleraces [],[3]. I this work passive sesitivities of sigle amplifier low-pass (LP) active filters are cosidered ad improved usig a desig techique, called impedace taperig [4]. It is well kow that sesitivities of active filters deped o the trasfer fuctio pole Q-factors. This is the reaso why the applicatios of this kid of filters are limited to the realizatio of trasfer fuctios with low pole Q-factors. It also meas that the order of the trasfer fuctio should be as low as possible, because low-order filters have lower pole-q factors tha high-order filters. This is particularly true for a Butterworth filter, which has maximally flat amplitude respose ad correspods to the limit case of o ripple i the filter pass-bad. Compared to a Chebyshev filter of equal order, it has lower pole Qs. As was show i [4], i order to realize a filter with low sesitivities to its compoet toleraces, the desiger should choose a filter with the lowest possible pole Q-factors. For example, a Butterworth filter is always preferable to a Chebyshev filter ad a lowripple Chebyshev filter is preferable to a Chebyshev filter with higher ripple, whe the sesitivity to compoet toleraces is to be held small. Thus, for low- ad high-pass filters of reasoably low order (as for example 6), the use of sigle-amplifier filters ca be advatageous i compariso to the cascade realizatio with d - ad 3 rd -order sectios, which are preseted i [5],[6]. Although the latter already have smaller sesitivities to compoet toleraces, they ca be very improved by applyig the impedace taperig techique proposed i [4]. Beside the low-power cosumptio, the advatages of the sigle-amplifier filters preseted here, are i havig less passive compoets. The impedace taperig techique ca be used for the desesitizatio of the sectios i a cascade structure as.transfer FUNCTION COEFFICIENTS Cosider the th order allpole low-pass filter circuits with positive feedback preseted i Fig.. V V C C - C - R R - R - R C C 3 R C R 3 C 3 C - (a) R C C - R C C A R=R F G ( β-) R G A R=R F G ( β-) R G (b) Fig.. Geeral th-order sigle-amplifier low-pass filter. (a) With reverse otatio. (b) With ormal otatio. The descedig otatio of R, C to R, C from the drivig source to the amplifier iput i Fig. (a) is R - R V V

2 coveiet for developig recursive formulas which determie the trasfer fuctio coefficiets of the circuits i Fig. as fuctios of resistors R i, capacitors C i ad amplifier gai β. The filter i Fig. has a ladder etwork i the positive feedback loop of a amplifier with gai β = + R F / RG represetig the gai i the class-4 filter circuit, i.e. filter with positive feedback loop. The trasfer fuctio of the th-order filter preseted i Fig. (a) is a allpole trasfer fuctio havig the form: β T ( s) =. () D' ( s) As show i [] ad [] the coeficiets of th -order deomiator polyomial i trasfer fuctio (), i.e. D ' ( s) = d s + () = ca be calculated usig the coeficiets of polyomials of order - ad -: ' ( ) = D s c s + (3) = ad ' ( ) = D s b s + (4) = usig the recursio formula R (5) ( ) + d = ( c b ) + c + c RC δ RCβ, R where δ =0, for ; ad δ =, for = where. Note that b 0 =c 0 =d 0 =. Note also that for the start of the recursive process polyomials D 0 = ad D =R C s+ are eeded. At the ed of recursive process the ascedig otatio is chaged, i.e. we substitute R R, C C, R - R, C - C,, R R, C C, resultig i the otatio show i Fig. (b). Cosequetly we perform multiplicatio of umerator ad deomiator with the same factor: N(s)=N (s)/d =β a 0, D(s)=D (s)/d (6a) a =d /d, 0. (6b) We obtai the form of the trasfer fuctio of the thorder filter give by N( s) βa0 T ( s) = =.(7) D( s) s + a s + K+ a s + K+ a s + a 0 I this paper we preset tables with ormalized compoets, usig expoetialy tapered capacitor values of various orders ad types (up to 6 th -orders), which are show i Table ad Table. The filters are optimised for miimum sesitivity to compoet toleraces of the circuit for a chose taperig factor. As show i [4], for the 3 rd -order low-pass filter case, the optimizatio of the filter s sesitivities ca be performed by choosig the appropriate desig frequecy ω 0 =(R C ) -, which will produce the filter with tapered capacitor values, havig miimal sesitivities. Note that i Table ad Table we obtai various optimal values for R (i.e. for ω 0, where ω 0 =(R C ) - ; C =), ad gai β. They are calculated usig the procedure show i Fig., which is implemeted usig the symbolic ad umeric calculatio program MATHEMATICA [7]. This method is exteded to produce low-sesitivity high-pass filters, as 3. CALCULATION AND OPTIMIZATION The set of oliear equatios, developed from the use of recursive formulae, after equatig each of the coefficiets i the polyomial to the appropriate Butterworth or Chebyshev polyomials, are solved umerically. Tables with elemet values for the circuits havig equal capacitors ad equal resistors are give i [] ad []. They are calculated for some typical amplifier gais (β=.0 ad.), ad are ot optimized for miimum sesitivity. Fig.. Block-diagram for solvig capacitive tapered 4 th -, 5 th - ad 6 th -order filter ad optimisig desig frequecy ω 0 for mi. sesitivity (choose C = ad optimal R ). It ca be see that the capacitive-taperig factors ρ C for higher-order filters are lower, tha those for filters of lower-order. They are limited by a possibility of achievig the optimal solutio. If we try to fid the solutio of 6 th -order Butterworth filter with higher ρ C (for example ρ C =3), it is ot possible, because the Newto s

3 method does ot coverge. Furthermore, larger capacitive 5 taperig factor ρ C is ot permitted sice C 6 =C /ρ C becomes too small ad comparable to the parasitic capacitace of the circuit (i the case of itegrated filters o a chip). However, i higher-order filters, eve small taperig factor satisfies our eeds i degree of desesitisatio (for example: ρ C =.0 is good eough for =6). The block diagram show i Fig. is primarily iteded for solvig high-order (i.e. 4 th -, 5 th - ad 6 th -order) capacitively tapered allpole low-pass filters. Optimizatio of d - ad 3 rd -order filters follows the steps i the block diagram show i Fig. as well, but it is simpler because the calculatio of filter compoets ca be performed aalytically. The procedure show i Fig. is briefly explaied i what follows. At the begiig the iput parameters are etered, i.e., the values of coefficiets a i (i=0,, -), the chose taperig factor ρ C, ad C =. Because of capacitive taperig the rest of the capacitors have the values C k =C /ρ k- C ; k=,,. I oe step of a solutio fidig procedure the resistor R has to be defied first. The resistor R is a desig parameter to be adusted, sice we have chose C =. By varyig the value of R we vary the value of the desig frequecy ω 0. For this value R we solve the system of o-liear equatios for the vector R,, R ad β. To achieve a solutio, we start with vector of radom values for R 0,, R 0, ad β 0. Radom 0 iitial resistors values R k are iside the iterval <0., 0> ad the gai β has values from iside <, 5>. If the proper value of the startig vector is chose, Newto s method will, with prescribed accuracy, coverge i several steps to the solutio, i.e. to the vector R,, R ad β. If the method fails to coverge, we try aother radom startig vector. If the covergece is achieved but we have a solutio with egative resistor values or gai β less tha uity, the we, agai, choose aother radom startig vector. We perform radom startig vectors for maximum 000 times. This process of fidig a solutio is kow as Radom Search. Choosig startig vectors for Newto s iterative solvig method ca be performed by applyig a rule, which tries to fid all possible solutios, i which case we have a Exhaustive Search. If we do ot fid all real ad positive compoet values R,, R ad gai β, we proceed with aother value of resistor R. If we fid real ad positive filter elemets ad gai, we calculate the multiparametrical statistical measure M, which is defied i [8],[9] as: ω M = S ( ω) dω. (8) ω M represets the area uder the fuctio S (ω), with borders of itegratio from ω to ω. S (ω) represets the Schoeffler sesitivity fuctio of frequecy, while M is a umber, ad ca be used as a goal for the optimizig process. A disadvatage is the depedece of the umber M o the selected boudaries of itegratio (i.e. ω ad ω ). Durig the whole optimizig process we, therefore, choose the same pair of frequecies ω ad ω. The above procedure is repeated with a ew value for R, util the miimum value of M is foud. Note that i Table ad Table, we obtaied a low-q realizatio of secod-order filters with uity-gai (β=). The capacitive taperig factor ρ C follows from ( + r) ρ ρmax = q p, (9) r ad for equal resistors, (i.e. r=); we have the miimum ρ ( ρmax ) = ρ mi max = 4q r = p, (0) ad with ρ=(ρ max ) mi =4q p we obtai β=. Furthermore, ote that for the 3 rd -order filter the values of R ad R 3 are very close. This correspods to the coclusios for 3 rd -order low-pass circuits with miimum sesitivity derived i [4]. ρ C C C C 3 C 4 C 5 C 6 R R R 3 R 4 R 5 R 6 β Table. Normalized compoets of capacitive tapered LP filters: Butterworth trasfer fuctios. ρ C C C C 3 C 4 C 5 C 6 R R R 3 R 4 R 5 R 6 β Table. Normalized compoets of capacitive tapered LP filters: Chebyshev trasfer fuctios, with 0.5 db pass-bad ripple.

4 Fig. 3. Schoeffler s sesitivity of ormalized Butterworth low-pass filter circuits (up to 6 th -order), with compoets give i Table ad i tables preseted i [] ad [] pp. 5. Fig. 4. Schoeffler s sesitivity of ormalized Butterworth low-pass filter circuits i Fig. 3, sorted by type of impedace taperig. 4. COMPARISON OF SENSITIVITY A sesitivity aalysis was performed assumig the relative chages of the resistors ad capacitors to be ucorrelated radom variables, with a zero-mea Gaussia distributio ad % stadard deviatio. The stadard deviatio (which is related to the Shoeffler sesitivities) of the variatio of the logarithmic gai α= T BP (ω) / T BP (ω), with respect to the passive elemets, is calculated for the ormalized values of filter compoets, for Butterworth filter approximatios, for the capacitively tapered filters give i Table, ad equal capacitors ad equal resistors case from tables i [] ad [] pp. 5, respectively. We have preseted the obtaied results i Fig. 3. Observig the stadard deviatio σ α (ω)[db] of the variatio of the logarithmic gai α i Fig. 3 we coclude that the capacitively impedace tapered filters have miimum sesitivity to compoet toleraces of the circuits for all filter orders. The secod best results, which are very ear, show filter circuits, with equall resistors. This is because, they also have slightly tapered capacitor values, which ca be cocluded observig Table (Butterworth filter example). The worst sesitivity performaces is obtaied for filters with equal capacitors. I fact it is usually ot practical to mass produce discrete compoet active-rc filters havig uequal capacitors. The same ivestigatio was performed for Chebyshev filter approximatios ad the same results were obtaied. The sesitivity curves i Fig. 3, were repeated i Fig. 4, but they are sorted by impedace taperig type. Observig sesitivity curves i Fig. 4 we coclude that with icreasig filter order, sesitivities icrease, as 5. CONCLUSIONS A procedure for the desig of allpole low-sesitivity, lowpower active-rc filters usig tables with ormalized filter compoet values has bee preseted. The filters use oly oe operatioal amplifier, ad a miimum umber of passive compoets. The amplifier itself esures realizatio of cougate-complex filter poles, ad a low output impedace. The desig procedure usig impedace

5 taperig adds othig to the cost of covetioal circuits; compoet cout ad topology remai the same. For reasos related to the filter topology, applyig the capacitive impedace taperig, we ca improve the sesitivity of the low-pass filters magitude to compoet toleraces [4]. The desig is uiversal, ad ca be exteded to the desig of sigle-amplifier, lowsesitivity high-pass filters. Because, the high-pass filters are dual to the low-pass filters, resistive taperig should be applied to reduce sesitivity of the high-pass filter. Furthermore, the reductio i power ad compoet cout achieved with the sigle-amplifier LP filters is obtaied at a price: a cascade of impedace-tapered biquads or bitriplets has a lower sesitivity tha capacitively tapered sigle-amplifier filters. Thus the decisio o which way to go is typically oe of tradeoffs: low power ad compoet cout versus low sesitivity. REFERENCES [] R. S. Aikes ad W. J. Kerwi, Sigle amplifier, miimal RC, Butterworth, Thompso ad Chebyshev filters to 6 th order, Proc. of Iter. Filt. Symposium, Sata Moica, Califoria, April 97, 8-8. [] G. S. Moschytz, Liear itegrated etworks: desig (Bell Laboratories Series, New York: Va Nostrad Reihold Co., 975). [3] G. S. Moschytz, Liear itegrated etworks: fudametals. (Bell Laboratories Series, New York: Va Nostrad Reihold Co., 974). [4] G. S. Moschytz, Low-Sesitivity, Low-Power, Active-RC Allpole Filters Usig Impedace Taperig, IEEE Tras. o Circuits ad Systems, vol. CAS-46(8), Aug. 999, [5] G. S. Moschytz ad P. Hor, Active filter desig hadbook. (Chichester, U.K.: Wiley 98). [6] R. P. Salle ad E. L. Key, A practical Method of Desigig RC Active Filters, IRE Trasactios o Circuit Theory, vol. CT-, 955, [7] S. Wolfram, The MATHEMATICA Book, (Wolfram Media, IL-USA ad Cambridge Uiversity Press, Cambridge, UK, 999). [8] K. R. Laker ad M. S. Ghausi, Statistical Multiparameter Sesitivity A Valuable Measure for CAD, Proc. of the IEEE It. Symp. o Circuits ad Systems, ISCAS 975, April 975, [9] C. Acar, M. S. Ghausi ad K. R. Laker, Statistical multiparameter sesitivity measures i high Q etworks. J. Frakli Ist., vol. 300, October 975, 8-97.