34 L. DOLÍVKA, J. HOSPODKA, SWITCHED-CAPACITOR FILTER OPTIMIZATION Swtched-Capactor Flter Optmzaton wth Respect to Swtch On-State Resstance and Features of Real Operatonal Amplfers Lukáš DOLÍVKA, Jří HOSPODKA Dept. of Crcut Theory, Czech Techncal Unversty, Techncká, 66 7 Prague, Czech Republc dolvl@fel.cvut.cz, hospodka@fel.cvut.cz Abstract. The optmzaton of a swtched-capactor flter, whch mplements a bquadratc secton, s descrbed n ths paper. The am of the optmzaton s to obtan a requred magntude frequency response of the flter. The optmzaton takes nto account both one of the features of real swtches ther on-state resstance, and the features of real operatonal amplfers fnte voltage gan and fnte unty-gan bandwdth. An optmal dynamc range s to be acheved as well. The dfferental evoluton a knd of evolutonary algorthms s employed for the optmzaton. The flter s desgned by the usual way wth deal swtches and deal operatonal amplfers at frst. The analyss of ths flter wth real swtches and real operatonal amplfers proves that there s a sgnfcant dfference between ts magntude frequency response and the one wth deal components. Hence, the optmzaton s appled for fndng component values so that the magntude frequency response s as smlar to the one wth deal components as possble. As for other man real features of operatonal amplfers nput and output resstance t s shown that ther effect s small. Keywords Swtched-capactor crcut, band-pass flter, optmzaton, evolutonary algorthm, dfferental evoluton.. Introducton One of common methods for crcut mplementaton, n partcular ntegrated crcuts, s the swtched-capactor (SC) technque. Ths technque s wdespread because t has a few advantages n comparson wth other technques [], for nstance: The transfer of SC crcuts depends not on capactor values, but on the ratos of them. These ratos can be substantally more accurate than the capactor values. A clock frequency sgnal, whch s needed for SC crcut operaton, can be used for ther tunng. SC crcuts do not requre resstors, whose mplementaton s dffcult n ntegrated form. As the swtches n SC crcuts, feld effect transstors are commonly used []. However, ths swtch mplementaton has several nondealtes: nonzero off-state conductance, nonzero on-state resstance, and parastc capactances. The transfer of SC crcuts s affected negatvely by these nondealtes. From the mentoned swtch nondealtes, one can say that nonzero on-state resstance R ON shows tself mostly. Therefore, just ths nondealty (lnear resstance) was taken nto account n ths paper. On-state resstance causes that a capactor C n a SC crcut s charged not n zero but n nonzero tme. It s apparent that the hgher on-state resstance s, the longer the tme s, and the stronger effect of on-state resstance on the SC crcut behavor s. Another nondealty that can occur n SC crcuts s the effect of the features of real operatonal amplfers. Ther man nondealtes are these: fnte nput resstance, nonzero output resstance, fnte slew rate, fnte unty-gan bandwdth, and fnte voltage gan. The effect of nput resstance s usually nsgnfcant, especally when feld-effect transstors on the operatonal amplfer nputs are used. The output resstance effect s also less mportant. The slew rate s not consdered n ths paper. However, the remanng two features unty-gan bandwdth and voltage gan affect the transfer functon of an SC crcut substantally. In ths paper, the effect of the three chosen nondealtes on the magntude frequency response of an SC crcut was elmnated usng optmzaton method based on one of evolutonary algorthms. Evolutonary algorthms are a group of optmzaton technques, see, e.g., [], [3], and [4], whch are applcable for fndng the global extreme of a mathematcal functon, called the objectve functon. Hence, they try to ether maxmze or mnmze ts value. Ths s accomplshed by fndng sutable values for ts varables. A few methods belong n evolutonary algorthms, e.g., genetc algorthms [5] and the dfferental evoluton (DE) [6], [7]. For the purpose of optmzaton n ths paper, the DE was chosen. The applcatons of the DE are, e.g., [7], [8], and [9].
RADIOENGINEERING, VOL. 6, NO., JUNE 7 35. Desgn of Swtched-Capactor Flter wth Ideal Swtches The SC crcut chosen for the optmzaton was an SC bquad (bquadratc secton) wth schematc dagram n Fg. []. Four knds of flters can be mplemented by ths bquad: low-pass, hgh-pass, band-pass, and notch flter. From these types, the band-pass flter was chosen. VI Sample & Hold Crcut phase phase C7 S C4 A C6 S C3 S7 VO C5 S8 S3 C S6 C S4 Fg.. SC bquad. The Sample & Hold Crcut n Fg. converts the nput sgnal V I, whch s contnuous-tme, to a dscrete-tme one. In phase, the crcut samples the nput sgnal and n phase, the crcut holds t. The flter was requred to have these parameters: center frequency: f = 4 khz, clock frequency: f C = 6 MHz, gan at f : G = db, qualty factor: Q =, transfer functon mplemented from the nput V I to the output V O of the bquad. The transfer functon from the nput V I to the output V O s denoted P and the transfer functon from the nput V I to the output V O s denoted P. Ths transfer functon labelng s for the bquad wth deal components. Hence, the followng equatons are vald for P and P VO VO P =, P =. (), () V V I I In ths paper, the transfer functons of the bquad wth both deal and nondeal components are consdered from phase on the nput to phase on the output. The way of obtanng component values of the flter was accordng to the common method [], whch s descrbed brefly below.. Prewarpng the frequency f owng to the blnear transformaton that s used afterwards for creatng a dscrete-tme transfer functon. Ths prewarpng yelds a prewarped frequency f P.. Substtutng the prewarped frequency f P, the gan G at frequency f, and qualty factor Q nto the general S5 A VO form of the transfer functon of an analogue bandpass flter. 3. Usng the blnear transformaton, whch gves the requred transfer functon of the equvalent SC bandpass flter (bquad). 4. Calculatng capactor values by comparng the coeffcents of the requred and symbolc transfer functon of the SC bquad. 5. Scalng capactor values so that an optmal dynamc range s acheved (.e., the magntude maxma of the transfer functons P and P are at the same level). The resultng capactor values gven by the prevous desgn are lsted n Tab.. These values were adjusted so that the lowest capactor value s equal to. Ths could be done thanks to the fact that the transfer of SC crcuts depends on capacty ratos (mentoned n Secton ). Ths fact s also the reason why the unt of capacty n Tab. s not specfed. 3 4 5 6 7 C.35 5.86 4.5.35 5 Tab.. The resultng capactor values for the transfer functons P and P wth deal components. 3. Analyss of Swtched-Capactor Bquad Mathematcal program Maple TM was used for both the analyss and the optmzaton of the bquad. The analyss was carred out by PraSCAn [], whch s a package for analyzng both deal and real swtched-capactor and swtched-current crcuts. All the frequency responses presented n ths paper were verfed by WnSpce a general-purpose crcut smulaton program []. Ths was done accordng to [] and t confrmed the correctness of PraSCAn s results. 3. Bquad wth Ideal Components At frst, the bquad wth deal swtches and deal operatonal amplfers was analyzed and ts magntude frequency responses were obtaned. The magntude of a transfer P(z) s symbolzed by M(f). In case of the transfers P (z) and P (z), the magntudes are calculated as follows πf πf j j C C ( ) f, ( ) = f M f = P e M f P e. (3), (4) In Fg., there are the magntude frequency responses M and M of the bquad wth the capactors lsted n Tab..
36 L. DOLÍVKA, J. HOSPODKA, SWITCHED-CAPACITOR FILTER OPTIMIZATION log M(f) and log M(f) [db].5.5 3 3 Fg.. The magntude frequency responses of the SC bquad, dotted lne: the magntude M, sold lne: the magntude M. 3. Bquad wth Real Components In ths case, swtches and operatonal amplfers wth real features were used. The features that were consdered for real swtches and operatonal amplfers have been already presented n Secton. For fndng out the effect of swtch on-state resstance R ON, knowng the ratos of capactors s not suffcent. It s necessary to known also concrete capactor values. The reason s that the effect of the resstance corresponds to the tme constant τ of chargng the capactors. The value of the constant s dependent not on the capacty ratos but on the capacty tself. Thus, the unt pcofarad was chosen for the calculated capactor values n Tab., whch were multpled by two. Therefore, the capactor values used for analyss wth real components were, e.g., C 5 = pf. The value of on-state resstance has to be determned as well. A sutable value for the SC crcut swtches s kω, whch was used. The model n Fg. 3 was used for the operatonal amplfers n the bquad. R IN means the nput resstance and R OUT means the output resstance. The value of the transadmttance g s S and the value of the voltage gan a s. nonnvertng nput R OUT output nvertng nput.5 R IN V IN gv IN R OA C OA V RC av RC Fg. 3. Model of operatonal amplfer. The value of the resstor R OA and the capactor C OA depends on the unty-gan bandwdth BB and the voltage gan A of the appled operatonal amplfer and t can be calculated accordng to the followng formulae R OA A = A, COA =. (5), (6) πb A For the transfer functons of the bquad wth nondeal components, the labelng P N and P N are used nstead of P and P, respectvely. The magntudes of the transfer functons P N (z) and P N (z) are denoted M N (f) and M N (f), respectvely. They can be calculated n the same way as n case of P and P accordng to (3) and (4). Fg. 4 shows the magntude frequency response of the bquad wth both deal and real components. The used value of the swtch on-state resstance has been already mentoned ( kω). The parameter values of the operatonal amplfers were the followng: R IN = TΩ, R OUT = 5 Ω, BB 5 = MHz, A =. From ths fgure, one can see that the magntude frequency responses are dfferent. log M(f) and log MN(f) [db] 4 3.5.5 3 3 Fg. 4. The magntude frequency responses of the SC bquad before the optmzaton, dotted lne: the magntude M, sold lne: the magntude M N. 4. Optmzaton of Swtched-Capactor Bquad Transfer Functon The optmzaton had the followng ams, whch shall have been satsfed by fndng sutable capactor values: The magntude frequency response M N should fulfll a defned magntude flter specfcaton, whch was derved from the magntude frequency response M. The magntude frequency responses M N and M N should have ther maxmum values as smlar as possble (because of obtanng an optmal dynamc range). Achevng the stablty of the optmzed bquad was the thrd am. However, ths am s evdent. (The condton of fulfllng t s well known all the poles of the transfer functon have to have the absolute value lower than.) The spread of capactor values was not consdered n the optmzaton. Nevertheless, the obtaned capactor values (see Tab. ) have a spread, whch s acceptable. The only publcatons dealng wth the optmzaton of an SC crcut whch the authors know are [8], [3], and [4]. In [8], swtch on-state resstance s consdered; parastc capactances are respected n [3]. Ref. [4] descrbes fndng capactor values of an SC flter so that a requred transfer functon s obtaned and smultaneously the spread of the capactor values and the dynamc range are optmzed, but for all the components beng deal..5
RADIOENGINEERING, VOL. 6, NO., JUNE 7 37 The magntude flter specfcaton mentoned n the frst requrement conssts of ranges BBL(f ), B UB (f ) for several frequences f [7], [8]. After the optmzaton, the magntude M N at a frequency f should be n the range that s assgned to ths frequency,.e., BBL(f ), B UB (f ). The followng formulae express ths fact M N ( f) BL( f), BU( f) f. (7) If the unt of the magntude M N s decbel, formulae (8), (9), and () are used nstead of (7) logm N( f) BLdB( f), BUdB( f) f, (8) B f ) = logb ( f ), (9) LdB( L B f ) = log B ( f ). () UdB( U However, n general, both of the bounds need not be determned for all frequences. The upper bound BBU(f ) for some frequences f can be equal to. Alternatvely, for other frequences f, the lower bound B LB (f ) can be equal to (generally,, but magntude cannot be negatve). For these frequences f, the formula (7) can be modfed to one of nequaltes M N( L N U f ) B ( f ) or M ( f ) B ( f ). (), () It s also possble that both of the bounds are the same at some frequences f. Then (7) s changed nto ths equaton M f ) = B ( f ) = B ( f ) (3) N( L U and the magntude M N at these frequences should be equal to one value. The values of the lower and upper bounds were derved from the magntude frequency response M due to the frst am of the optmzaton. Addton of a small number to M (f ) was carred out to obtan the value of the upper bound BBU(f ). The value of the lower bound B LB (f ) was obtaned by a smlar way by subtracton of a small number from M (f ). Chosen frequences f for the magntude flter specfcaton are apparent from Fg. 5. In Fg. 6, there s a detal of Fg. 5 around the frequency f. The number of the frequences f s 3. Only the upper bounds are specfed at fourteen of them. Both the upper and the lower bounds are defned at the others. In order to determne the peak of the magntude frequency response M N exactly, the upper and lower bound at the frequency 4 khz (f ) are the same ( db). Ths means that the value of the optmzed magntude M N at ths frequency should be just ths one. As mentoned n Secton, the optmzaton employed the DE. For the descrpton of t, refer to the presented references. In ths paper, only the appled objectve functon s descrbed. Its value was mnmzed n case of ths optmzaton. Usually, several forms of the objectve functon can be used for a partcular optmzaton task. The followng form of the objectve functon F was found to be the most convenent for ths optmzaton [7], [8]: max MN( C, M Nmax + L U + FL ( C, + FU( C, F( C, K, = (4) = = f the bquad s stable, f the bquad s unstable, where BL( f) M N( f, C, BL( f) FL( C, K, = (5) f M N( f, C, < BL( f), else, wth ths symbol meanng: C, C,, C 7 capactor values, L the number of the lower bounds, L = 9, U the number of the upper bounds, U = 3, max M N the maxmal value of the magntude M N, M Nmax the requred maxmal value of the magntude M N, M Nmax = (= db). BLdB(f), BUdB(f), and log M(f) [db] BLdB(f), BUdB(f), and log M(f) [db] 3 4.5.5.5 3 Fg. 5. The magntude flter specfcaton for the optmzaton, crcles: the lower bounds BBLdB of the magntude ranges, crosses: the upper bounds B UdB B of the magntude ranges, dotted lne: the magntude frequency response M. 9.7 9.4 f [khz] 9. 396 398 4 4 Fg. 6. Detal of Fg. 5 for a vcnty of the frequency f. 44 If the requrements for the optmzaton result are satsfed, the value of the objectve functon s. Capactor values for the optmzaton could be wthn a range of pf, pf. One generaton was composed of 7 members and the values of optmzaton parameters F and CR were.5 and.9, respectvely.
38 L. DOLÍVKA, J. HOSPODKA, SWITCHED-CAPACITOR FILTER OPTIMIZATION The optmzaton was performed whle usng the parameter values of the real components lsted n Secton 3.. However, the parameters R IN and R OUT n the model of the operatonal amplfers were not used (so R IN = Ω and R OUT = Ω). The nput and output resstances of the operatonal amplfers were neglected because of smplfyng the analyss of the bquad durng the optmzaton and thereby speedng t up. Ths smplfcaton was done snce ther effect was supposed to be not sgnfcant. After the optmzaton, the analyses of the bquad wth the nput and output resstances and wthout them were carred out and these analyses confrmed ths assumpton. The dfference between these analyses can be seen n Fg. and. 5. Results of Optmzaton The optmzaton reached the value of the objectve functon of.34 durng 53 generatons. Usng more generatons dd not mprove the objectve functon value (the total number of generatons was 4). Tab. shows the capactor values arsen from the optmzaton. 3 Fg. 7. The magntude frequency responses of the SC bquad after the optmzaton, dotted lne: the magntude M, dashed lne: the magntude M N, sold lne: the magntude M N. Fg. 9 depcts the magntude frequency response M N computed by WnSpce (see Secton 3). It s obvous that t s very smlar to the magntude M N n Fg. 7. The magntudes M N and M N whch are plotted n Fg. 7 and 8 are wth R IN = Ω and R OUT = Ω these parameters were not consdered snce the optmzaton was carred out wthout them (see Secton 4). However, ther effect on the magntude frequency responses of the bquad s not sgnfcant. The magntude frequency responses M N and M N wth usng the parameters R IN and R OUT n the model of the operatonal amplfers are denoted M NR and M NR, respectvely. log M(f), log MN(f), and log MN(f) [db] 9.5 9 8.5 39 395 4 45 Fg. 8. Detal of Fg. 7 for a vcnty of the frequency f. f [khz] 4 3 4 5 6 7 C [pf] 65.39 43.877 69.6 58.68 3.45 4.664 5.68 Tab.. The resultng capactor values for the transfer functons P N and P N wth real components. The magntude frequency responses M N and M N wth usng the resultng capactor values are shown n Fg. 7. The magntude frequency response M s also shown n ths fgure for comparng. In Fg. 8, there s a detal of Fg. 7 around the frequency f. The magntudes M N and M N have ther maxma on almost the same level; the dfference between them s only about.5 db. The dfference between the frequences of the maxma occurs even n case of the magntudes M and M. log M(f), log MN(f), and log MN(f) [db].5.5 3.5 Fg. 9. The optmzed magntude frequency response M N of the SC bquad computed by WnSpce. In Fg., there s the dfference between the magntude frequency responses M NR and M N. Fg. shows the dfference between M NR and M N. It s apparent from these two fgures that the dfferences are not hgh. Hgher values (but not too hgh) are only n the stop-band of the dfference between M NR and M N. log MNR(f) log MN(f) [db].3.....5.5.5 3 Fg.. The dfference between the magntude frequency responses of the SC bquad M NR and M N.
RADIOENGINEERING, VOL. 6, NO., JUNE 7 39 log MNR(f) log MN(f) [db].8.6.4...5.5.5 3 Fg.. The dfference between the magntude frequency responses of the SC bquad M NR and M N. Of course, the optmzaton could be accomplshed ncludng the parameters R IN and R OUT but t would take a longer tme than wthout them (about three tmes). 6. Concluson A possble method for elmnaton of nondealtes that affect SC bquad features was presented n ths paper. These nondealtes were represented by one of the real features of swtches on-state resstance and the real features of operatonal amplfers. The elmnaton was acheved by utlzaton of the dfferental evoluton, whch proved to be suffcently powerful. The method was successful n meetng the determned requrements. The appled way of optmzaton can be consdered as sutable for ths ntenton. Acknowledgment The research descrbed n ths paper was fnancally supported by the research program Research n the Area of the Prospectve Informaton and Navgaton Technologes No. MSM684774 of the Czech Techncal Unversty n Prague. Maple s a trademark of Waterloo Maple Inc. References [] ANANDA MOHAN P. V., RAMACHANDRAN V., SWAMY M. N. S. Swtched Capactor Flters Theory, Analyss and Desgn. Prentce Hall Internatonal, 995, ISBN -3-87988-4. [] CORNE D., DORIGO M., GLOVER F. New Ideas n Optmzaton. London: McGraw-Hll, 999, ISBN -7-7956-5. [3] MAŘÍK V., ŠTĚPÁNKOVÁ O., LAŽANSKÝ J. et al. Artfcal Intellgence (n Czech), vol.,, 3, and 4. Prague: Academa, 993, 997,, and 3, ISBN 8--496-3, 8--54-8, 8- -47-6, and 8--44-. [4] ZELINKA I. Artfcal Intellgence n Global Optmzaton Problems (n Czech). Prague: BEN,, ISBN 8-73-69-5. [5] GOLDBERG D. E. Genetc Algorthms n Search, Optmzaton, and Machne Learnng. Readng, Massachusetts: Addson-Wesley Publshng Co., 989, ISBN --5767-5. [6] STORN R., PRICE K. Dfferental evoluton a smple and effcent heurstc for global optmzaton over contnuous spaces. Journal of Global Optmzaton, 997, vol., no. 4, pp. 34 359. Kluwer Academc Publshers, ISSN 95-5. [7] STORN R. Dfferental Evoluton Desgn of an IIR-flter wth Requrements for Magntude and Group Delay. Techncal report TR- 95-6. ICSI, 995. [8] DOLÍVKA L., HOSPODKA J. Elmnaton of swtch on-state resstance effect on a swtched-capactor flter characterstc. In Proceedngs of Conference PRIME 6, pp. 77 8. IEEE, 6, ISBN -444-56-9. [9] TICHÁ D., MARTINEK P. OTA-C lowpass desgn usng evolutonary algorthms. In Proceedngs of ECCTD 5, vol., pp. 97. Cork: Unversty College Cork, 5, ISBN -783-966-. [] BIČÁK J., HOSPODKA J. PraSCAn Maple package for analyss of real perodcally swtched crcuts. In Proceedngs of Maple Conference 5, pp. 8 8. Waterloo Ontaro: Maplesoft, a dvson of Waterloo Maple Inc., ISBN -8945-85-9. [] SMITH M. WnSpce User s Manual. http://www.wnspce.com. [] BIČÁK J., HOSPODKA J. Frequency response of swtched crcuts n SPICE. In Proceedngs of ECCTD 3, pp. I-333 I-336. Krakow (Poland): IEEE, 3, ISBN 83-8839-95-. [3] STORN R. System Desgn by Constrant Adaptaton and Dfferental Evoluton. Techncal report TR-96-39. ICSI, 996. [4] VESTENICKÝ M., VESTENICKÝ P. Evolutonary algorthms n desgn of swtched capactors crcuts. In Proceedngs of Dgtal Technologes 4, pp. 34 37. Žlna (Slovaka): EDIS Žlna Unversty publsher, 4, ISBN 8-87-334-5. About Authors... Lukáš DOLÍVKA was born n 98. He graduated from the Czech Techncal Unversty n Prague n 5. Snce then, he has been a student of a doctoral study program at the Department of Crcut Theory at the same unversty. Hs research nterests are concerned wth dscrete-workng crcuts and the optmzaton of them. Jří HOSPODKA was born n 967. He receved the M.Sc. and Ph.D. degree 99 and 995 from the Czech Techncal Unversty n Prague. Research nterests: crcut theory, analog electroncs, flter desgn, swtched-capactor and swtched-current crcuts.