A SIMPL MOD OF GOAL DIRCD LOSSY SYNSIS AND NWORK OPIMIZAION Karel ájek a), ratislav Michal, Jiří Sedláček a) Uiversity of Defece, Kouicova 65,63 00 Bro,Czech Republic, Bro Uiversity of echology, Kolejí 4, 6 00 Bro,Czech Republic e-mail: sedlacj@feec.vutbr.cz, www.postreh.com/vmichal Summary A simple method for sythesis of goal directed lossy filters is described i the preset paper. his method of sythesis is based o the applicatio of optimizatio procedures used i usually accessible professioal software foetwork aalysis. he preseted method of etwork optimizatio eable to improve etwork performace without requiremet of speciaumerical programs what brigs ew possibility for may desigers i area of etwork desig ad optimizatio.. INRODUCION By classical filter desig there are usually sigle or double sided termiatio RLC filter prototypes used. I may catalogues are these stadard LC ladder prototypes which are usig ideal loss les reactive L ad C elemets wide tabled. ere the termial resistaces are trasformed to the iteral structure ad dumped the LC circuit to realize the required trasfer fuctio. I [] was described a ew method how to optimize resultig active filter structures based o classical ladder prototypes usig goal lossy filter prototype desig, where the losses are dispersed to the whole ladder structure. It was show that the performaces of these filters desiged with the lossy structures - like sesitivities, compoets ratio etc, may be better tha the sythesis based o the classical loss les RLC ladder prototypes. heoretically ca be used goal lossy ladder prototypes with losses dispersed to the all braches of etwork. I the practice there are most ofte used structures, where losses are dispersed to all parallel (see Fig. or to all serial braches of filter structure (Fig.c) what correspods with resultig parameters of active elemets simulatig passive ladder elemets (iductors, FDNR). I [],[3], the desig algorithm of these goal lossy ladder a) c) c r c r c r prototypes with parallel lossy resistors (Fig. was described ad special software to goal lossy filter sythesis was preseted, mai possible etwork goal lossy filter value elemets was preseted for stadard types of approximatios i form of tables too. owever for may cases the width of approximatio types or parameter rages is ot sufficiet, requiremet of special software is also some disadvatage for may desigers. herefore a ew method how to desig the goal lossy filter structures has bee developed.. A PRINCIPL OF SYNSIS MOD I the preset time there are at disposal may aalysis software products [4],[5] which ca be used geerally by a process of etwork optimizatio. hese products ca be used also directly to sythesis of goal directed lossy structures without requiremet of the speciaumerical algorithm implemetatio or software. he software blocks called usually etwork optimizers ad allow to I I Ideal Real Out Out Step : Ideal leader -> Loosy leader prototype Step : Loosy leader prototype -> ideal active realizatio with ideal active elemet Step 3: Compesatio of effects itroduced by active elemets real properties Fig.. RLC ladder filter prototype a) ideal, with parallel loses c) with serial loses Fig.. he process of active filter desig ad process of fial filter optimizatio
compute (i may cases by selected umerical method) all values of etwork elemets with aims to reach the prescribed (ideal) trasfeetwork characteristic. A priciple ad process of a ew sythesis method is briefly give as rough draft i Fig.. he mai procedures of filter compoet computig ca be divided to these mai steps: Lossy etwork aalysis aalysis of lossy structure behavior ad ivestigatio of maximum lossy values or determiatio of limitig lossy parameters Geeratio of required trasfer fuctio table (trasfer ratio value versus frequecy), Fial optimizatio of passive ladder etwork Fial optimizatio of active etwork with regard to real active elemet performace. approximatio filter types - Butterworth ad schebyshev. he values (ormalized to frequecy ω) of parallel goal lossy ladder prototypes for 5 th to 9 th filter order were ordered i table. he preseted values ca be directly to filter desig of low pass filters with FDNR active elemets (as filter from Fig.3) or high pass filters with active lossy (o ideal) iductors successfully used. As a example of low pass filter desig which has bee based o derived lossy ladder prototype elemets (From able ) here is preseted resultig etwork of filter i Fig.3. I the preseted low pass filter are used as active elemets simple parallel lossy FDNR etworks with miimized umbers of operatioal amplifiers [], []. 3. SYNSIS OF GOAL - DIRCD LOSSY PROOYPS WI PARALLL RSISORS he procedure of filter sythesis ca be leade to compute lossy ladder structure: a) to obtai the required quality factor of passive elemets, to reach the trasfer fuctio with required resistor value. While i the above metioed literature the first process was chose, i preseted ew method the secod possibility was selected. he accuracy of goal lossy ladder prototype trasfer respose with compariso to trasfer respose of ideal loss-less ladder ca be i a case of 3 rd order ladder RLC filter aalysis demostrated. rasfer fuctio of LP filters from Fig.a) ca be writte as:, () a0 + as + as +... + as Fig. 3. he low pass filteetwork desiged usig goal directed lossy prototype I the Fig.4 we ca see a magitude trasfer respose of desiged (o optimized) active etwork with sigificat deviatio from ideal trasfer respose due to real active elemet properties. he resultig trasfer respose (labeled as optimized) has bee obtaied after fial filter optimizatio process (Step 3 of filter desig ad optimizatio process i Fig.), where the real active elemets performace was elimiated. where a 0 -a are coefficiets of deomiator ad s the complex frequecy jω. he voltage trasfer of goal - lossy structure from Fig. is give: a + 0 ( a + k ) s + ( a + k ) s +... + a s, () where k, are the parts which are iserted due to losses. By compariso of this two equatio, we ca observe, that the exact solutio of problem (trasfer fuctio of goal lossy prototypes with parallel dumped resistors which is equivalet with ideal trasfer respose) evidetly ca be (i limited rage of parameters ) foud. Applyig above metioed method by usage of software INA optimizer [5] were calculated parameter values of two most ofte used stadard Fig. 4. he resultig trasfer respose of desiged filter of 5 th order low pass Butterworth
4. SYNSIS OF LOSSY GOAL DIRCD PROOYPS WI SRIAL RSISORS he sythesis of goal lossy etwork with serial resistors (Fig. brigs more problems. rasfer fuctio of 3 rd order metioed filter etwork with serial lossy resistors ca be expressed as: r + 3 ( r r c + l + l ) s + ( r l + r l + r l ) c s + l c l s r + r r c s. (3) owever by compariso with eq.(), it is clear, that exact solutio (equivalet trasfer fuctio by lossy structure) caot be foud. he omiator of trasfer fuctio (3) of goal - lossy RLC ladder prototype with serial dumped resistors exhibit iserted zeros of trasfer with time costat c. It meas that from frequecy ω is the r c slope of trasfer fuctio reduced to -0dB. herefore for this etwork structures the rage of possible filter solutios is limited. Fig. 5. he resultig trasfer respose i a case of etwork with serial lossy resistor prototype By proper optimized etwork value parameters ca be successfully foud solutio with allowable differece of trasfer fuctio from ideal required trasfer respose, how declare a typical example of trasfer fuctio of schebyshev goal lossy prototype of 5 th filter order from Fig.5. From compariso of trasfer resposes is evidetly see ifluece of parasitic pole for RLC lossy structure ype o R p c l 3 c 3 l 4 c 4 l 5 -.5400.6900.3800 0.8940 0.3089 - - - - B U R W O R S C B Y S 5 7 9 5 7 9 6.040.440.400.090 0.3336 - - - - 4 0.9630.3700.590.0980 0.3450 - - - - 0.698 3.0570 0.9805.80 0.376 - - - - 0.4370 4.4430 0.750.6500 0.4 - - - - -.5900.800.6600.4000.0500 0.6550 0.0 - - 6 0.975.5700.700.800 0.938 0.768 0.4 - - 4 0.867.9500.400.000 0.8876 0.83 0.507 - - 0.5438 4.00 0.8605.600 0.7593 0.9585 0.75 - -.5 0.448 4.9000 0.7406 3,0000 0.6905.0500 0.588 - - -.5600.8400.7800.600.400.400 0.840 0.550 0.730 6 0.868.900.000.3500.0800.4800 0.7567 0.6009 0.869 -.500.3000.600.500.7400 - - - - 6 0.85.0080.3880.548.639 - - - - 4 0.5053.86.035.6660 3.0700 - - - - -.800.3300.700.3600.6700.700.7600 - - 6 0.3768 3.500.0600.0600.5300.4700 3.0600 - - 4 0.07 5.7500.5700.3800.00.5900 3.800 - - 0.084 4.980 0.758 3.9600.4500.00 5.4800 - - -.000.3400.7400.3800.7600.3700.6900.700.7700 6 0.887 6.4000.6500.4900.8900.9000.6800.4300 3.7300 4 0.098.790.0600 3.00.4600.3500.3900.6000 4.9300 0.0550.540 0.733 4.0800.700.8700.0500.800 5.9300 ab.. he elemets of schebyshev ad Butterworth sigle termiated RLC filter prototype (Fig., ormalized to ω, termial resistor r Ω.
with serial resistors brigig effect of slope degradatio i the stop-bad. he computed characteristic are agree with ideal up to -80dB what is i practice fully acceptable i more cases. hus the active realizatio of serial lossy structures ca be successfully used by sythesis with simple oe op-amp sythetic elemets with serial losses. his realizatios ca exhibit i some cases a advatage as better properties i the stop-bad (smaller effect of trasfer respose degradatio due to absece of parasitic trasfer zeros). realize the active equivalet of the passive RLC laddeetwork with simple ( o ideal) sythetic fuctio block, like FDNR, GIC or sythetic r s l c l 3 -,5440,6936,383 0,8940 0,3089 B 80m,04,7040,4300 0,900 0,3593 00m,43,7870,46 0,9450 0,358 -,500,3000,600,500,7400 50m,5300,360,7780,3640,0740 60m,80,40,9380,450,850 ab.. he elemets of schebyshev ad Butterworth RLC ladder prototypes with serial lossy resistors Usig above prescribed optimizatio method have bee calculated parameters of RLC ladder prototypes with serial losses i the case of Butterworth (B) ad schebyshev () low pass prototype of 5 th order, which are summarized i table. Startig from these parameters as illustrative example we have desiged resultig filter topology with simple serial lossy FDNR active elemets (Fig.6). A iitial desiged active filter structure has bee optimized i fial step of optimizatio procedure to compesate real parameters of active elemets. ow it is see from resultig trasfer respose of desiged low pass filter (Fig.7), the resultig optimized magitude respose is idetical with required (ideal) trasfer respose up value Fig. 7. he resultig trasfer respose i a case of etwork based o serial lossy resistor RLC prototype iductors with miimized umbers of active elemets []. he AC trasfer respose of filter desiged from the RLC ladder prototype should exactly correspod with the required AC trasfer respose oly if the ideal Op-amp or other active elemets are used. he real frequecy respose of active elemets caused shiftig of the cut-of frequecy ad quality factor of filter. his fact is domiat by all active realizatios (e.g. cascade realizatio etc.), if domiat real parameters of active elemets are ot take i to accout durig the desig procedure. he above described usage of etwork optimizers ca be used also to fial correctio of the Step Step a) Z x Z x Z Z A A Fig.6. he resultig active low pass Butterworth filter of 5 th order with serial lossy FDNR elemets about - 80dB, what is i practice fully acceptable result. 5. LIMINAION OF RAL ACI LMN PROPRIS Most importat advatage of goal - directed lossy frequecy filter desig is the possibility to Fig.8. he priciple of optimizatio method to compesatio of real active elemets properties by -a) o cascade realizatio, cascade realizatio active filter performace with real models of active elemets. he process of the optimizatio must correspod with the priciple of the used filter sythesis method. By the case of o-cascade method of filter realizatio as the mai optimized parameter durig
sythesis optimizatio process was foud crossimpedace of each sub-blocks (by the coditio where the source geerator is switched off). I the case of cascade filter realizatio was successfully used as optimized parameters AC voltage trasfer fuctio. he priciple of active elemet optimizatio method ad process of optimizatio procedure is roughly sketched i Fig. 8. Applyig above metioed optimizatio method the o cascade sythesized active filter from Fig.9, tab. 3 has bee optimized to elimiate real properties of operatioal amplifiers. he low - pass RCD By the above preseted compesatio method (usig ia optimizer) was used to optimize the fial etwork parameters as best goal fuctio the cross impedace of each block was foud. ach active block was optimized separately, what icreased the speed of optimizatio. he fial optimized characteristic is show i Fig.0. Durig optimizatio process as most importat part was idetified procedure to determiatio of goal ideal AC trasfer fuctio (i tabled form) used by optimizatio procedure. his goal fuctio must be specified i the trasfer area, where are ot preseted the further parasitic effects of active elemets. herefore i the case of preseted example of optimizatio process, the stop frequecy of tabled AC trasfer fuctio has bee selected the cut - off frequecy of filter, it meas 0kz. CONCLUSION Fig. 9. he resultig active RCD structure of 5 th order schebyshev low pass filter structure of the 5 th filter order has bee desiged from the passive RLC lossy ladder prototype usig Bruto s trasformatio []. he equivalet compoet values (R, C,...R 5 ) create iitial vector of optimized parameters. he required AC trasfer fuctio was determied from ideal trasfer respose of etwork. From the Fig.0 we ca observe the ideal characteristic (with ideal operatioal amplifier) ad the characteristic obtaied with real two-pole operatioal amplifier model LM74. From curves is very good see the cut-of frequecy shift due to the real properties of the operatioal amplifiers. I the paper there is described the simple etwork sythesis method which brigs further possibility to improve filter desig ad optimizatio i area passive as well active filters. ere described method allows wider expasio of the filter sythesis based o goal directed lossy filter prototypes. he preseted method of compesatio of active elemet iflueces ca be widely used for all active elemets like FA, CFA, OA, CCII etc. he great advatage of method is that eable to optimize resultig etworks (icludig passive ad active filters) by usage of usually accessible software for etwork aalysis without requiremet of special umerical programs what brigs ew possibility for may desigers i area of etwork desig ad optimizatio. RFRNCS [] ájek K., Sedláček J. Kmitočtové filtry. ydavatelství BN Praha 00 [] ÁJK, K. - SDLÁČK, J.: Lossy LC Ladder Prototypes ad their use for ARC Filter Optimizatio. WSAS RANSACIONS o LCRONICS,Issue3, olume, July 005, pp. 94-99, ISSN 09-9445. [3] MARINK,P.-DAŠA,. volutioary algorithms by ARC filter sythesis. CCD 05, Cork, 005,pp.55-59. Fig.0. he trasfer resposes obtaied by optimizatio process i case of active LP filter from Fig.7. [4] PSPIC desig software (http://www.orcad.com/) [5] PC Simulator ia 6 pro (www.tia.com) R C R C R 3 C 3 R 4 C 4 R 5 C 5 Ideal 80,0,35 4k,35 4,4k 769p 440k 769p 4,56k 0 Comp 833,54 490p 400,5k,79 4,49k 458,p 470,3k,09 4,56k 9 ab 3 ideal ad optimized compoets value for 0 kz low pass filter (fig 9,0)