ACTIVE FILTER SYNTHESIS BASED ON MESH CURRENT EMULATION OF LC LADDER STRUCTURES

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1 ACTIVE FILTER SYNTHESIS BASED ON MESH CURRENT EMULATION OF LC LADDER STRUCTURES Andrei Câmpeanu Universitatea "Plitehnia" Timişara, Departamentul de Cmuniaţii, Bv. V. Parvan Nr., Rmania, ABSTRACT In a series f papers, []-[3], A. Câmpeanu and M. Nafrniţă prpsed a new ative RC filter synthesis methd based n mesh urrent emulatin f LC ladder filter netwrs. Initially, the methd was suessfully applied t highpass and lwpass plynmial filters. This paper extends the methd t the very imprtant lasses f lwpass and highpass finite-zers (ellipti) filters. Based n the mesh urrent methd, simple and systemati design rules are prpsed, whih permits a diret ative RC filter implementatin f the passive netwr. It is als utlined a methd f reduing the number f amplifiers required. Sme results btained using iruit simulatin shw the effiieny f the methd and the imprtane f the results.. INTRODUCTION Beause f their lw-sensitivity prperties, the duble terminated lssless ladder filters are very suitable t be implemented as RC-ative filters. The mst used apprah t the design f ative filters based n LC ladder simulatins is based n simulating the urrent vltage relatinships existing in the LC-ladder prttype, [4]-[6]. This paper is speifially nerned with this ind f apprah. A. Campeanu and M. Nafrniţă []-[3], have given a predure t derive an RC-ative filter frm the passive ladder netwr using the mesh urrents desriptin f the passive iruit. The prpsed methd have used as building bls, mdified biquad resnatr filter ells with multiple inputs t implement lwpass and highpass RC plynmial ative filters. Here we prpse an extensin f the mesh urrents simulatin methd t lwpass and highpass finite-zers filters. Fr this very imprtant ase, mst prpsed RC-ative filter synthesis methds [6]-[7] fr emulating high-rder LC ladder filters are subjet t either mpliated design predures r extra numbers f ative elements. Mesh urrents emulatin ative filters have the advantages that we an realize every lp in the riginal ladder prttype by a speifi multiple-inputs multiple-utputs RC-ative ell. Furthermre, systemati design rules and equivalenies that simplify the design predures are established. The multiple-inputs multiple-utputs RC-ative ells are implemented using nventinal summatin and integratin devies, that nsist nly f p-amps with grunded inputs. These iruits are insensitive t parasiti input apaitanes f p-amps and are nvenient fr eletrni ntrl f parameters. CIRCUIT TOPOLOGIES The predure fr the mesh urrents emulatin methd starts by taing an LC-ladder prttype netwr and sliing it int individual lps. We desribe the wr f the passive netwr presented in Fig. using mesh urrents that flws in the lps f the netwr: I, I,, In. Kirhff s therem applied in any f the netwr lps gives: V Z Z I I l l I l () Z Z Z Fig. Using mesh urrents t desribe the wr f a passive ladder filter

2 l where Z is the mesh impedane f the -th lp. V represents the value f vltage sures in the lp. Only fr 0, V E 0 and I 0. Fr the last mesh f the netwr, I 0. Arding t (), the ladder filter netwr an be implemented using RC ative iruits ells with multiple inputs. Mrever, as the denminatrs f all the terms n the right side f () are idential, l Z, a unique ative iruit struture with multiple inputs an be used t implement the desired funtins. As will be shwn in the next Setin, lined t the psitin f the lp in the ladder, the implementatins are first r send rder RC ative iruits having multiple inputs and utputs. Fig. π-shaped passive filter struture: a. lwpass filter, b. highpass filter. Previus papers, []-[3], used T-shaped LC ladder strutures t synthesize RC-ative filters. This apprah gives the advantage f using a unique ative ell repeatedly fr eah lp f the passive struture. Extending the methd t the ase f finite-zers filters, the previus apprah bemes unsatisfatry beause in a single iruit lp there are as far as tw arms with LC resnant series iruits. A better slutin is t use π-shaped filter strutures (Fig. ), beause in this ase eah iruit lp has n mre than a single LC resnant parallel iruit. This send apprah is used in this paper unless it gives different transfer funtins and different ative filter ell struture fr almst every lp f the riginal ladder prttype. Furthermre, in the next setin, systemati design rules are established t simplify design predures. 3. MESH CURRENTS SIMULATION OF LC LADDER LOOPS The examinatin f π-shaped duble terminated passive filter struture has identified distint basi types f iruit lps. Based n the mesh urrents desriptin f a ertain lp, the strutures f equivalent RC-ative ell with p amps were established. Instead f the riginal urrents I, the RCative ell wr is desribed by vltages V having the same subsripts and supersripts as the riginal urrents. In the ase f netwr lps having n series arms LC parallel iruits, the need f eeping the number f p amps in the synthesized ell t a minimum, impses the separatin f the lp urrent, I in tw terms, the first with a highpass behaviur, filter: I, the send ne, I behaving lie a lwpass I α I I () where α is a nstant fr a given lp. Arding t (), the utputs f the rrespnding RC-ative ell will satisfy a similar equatin: V α V V (3) Atually, the transpsitin f (3) in the ative iruit ell means the use f tw separate utputs fr the same ell: a highpass utput, V, and a lwpass utput, V. Finally, the multipliatin with α and the additin f the tw terms in (3) is made in the input stage f the adjaent ells whih use as input signal the result f the equatin. Cnsequently, the p amp needed t implement (3) is spared. Next, the first and send rder RC-ative ells used t emulate the passive iruit unterparts are presented. A. First rder transfer funtin ladder netwr lps synthesis The input and utput terminatins f π-shaped passive ladder netwr lwpass and highpass filters present a first rder transfer funtin between the mesh urrent f the lp and the vltage sure frm the lp and the mesh urrent f the adjaent ell. Fr the utput terminatins, this is true nly in the

3 ase f dd-rder filters. Fig. 3 presents the fur different ases that lead t first rder transfer funtins. Fig. 3 First rder transfer funtin passive filter ladder lps: a. and b. input and utput terminatins f a lwpass filter,. and d. input and utput terminatins f a highpass filter As example, the mesh urrent I n the input terminatin f a π-shaped passive highpass filter (Fig. 3..) has the expressin E r sl r I I (4) sl r sl r There are tw distint behaviurs in the same expressin: a lwpass transfer funtin given the vltage E and a highpass transfer funtin relative t the mesh urrent f the adjaent ell, I. A similar ase urs fr the lwpass iruit in Fig. 3. a. The iruits in Fig. 3. b. and d. have simpler first rder behaviurs: nly lwpass fr the first f them, respetively nly highpass fr the send ne. Fig. 4 a. The ative ell used t emulate first rder transfer funtin passive filter ladder lps, b. Symbl f the ative filter ell The ative filter ell that implements equivalents first rder transfer funtins is presented in Fig. 4. a. It uses nly simple p-amp building bls with grunded inputs: tw adders and an integratr. T enable the additin f the tw utput vltages given by a neighburing ative filter ell, the iruit has tw lwpass and tw highpass inputs. The symbl used t represent the ative ell is shwn in Fig. 4.b. As in the next bl symbls presented, the ntatins αlp and α HP, used n the inputs V LP and V HP, indiate that the utput V f the adjaent ell will be multiplied at ne f these inputs by the suitable effiient. Returning t the input terminatin f a π-shaped passive highpass filter in Fig. 3.., the design relatinships fr the parts in the equivalent ative iruit are very easy t establish. Chsing the value f the resistr R, the ther parts in the shemati are: l C, R LP R, R HP R α, R HP R (5) r R The input remains unused and the effiient α is established when the adjaent ative ell is mputed. V LP

4 B. Send rder transfer funtin ladder netwr lps synthesis Fr the π-shaped passive lssless filters, there are fur different tplgies fr the interir netwr lps that leads t a send rder dependeny between the mesh urrent f the lp and the mesh urrents f the adjaent lps (Fig. 5). Fig. 5 Send rder transfer funtin passive filter ladder lps: a. and b. interir lps f lwpass and highpass plynmial filters,. and d. interir lps f lwpass and highpass finite-zers filters The iruits represented in Fig. 5. a. and. are used in plynmial lwpass and highpass filter netwrs. They have a single reative element n eah lp f the mesh. The mesh urrent I is expressed in these ases by a lwpass, respetively a highpass undamped send rder transfer funtin f adjaent lps urrents, I and I. The interir lps f finite zers lw-pass and highpass filters inlude n the series arms LC parallel iruits as it shw Fig. 5. b. and d. Their send rder transfer funtins are undamped and exhibit tw imaginary finite zers in additin t a pair f imaginary ples. In the ase f lwpass finite-zers filter lp (Fig. 5. b), the mesh urrent I has an expressin whih allw a very nvenient dempsitin in tw terms lie in (): the first, I, with a highpass behaviur, the send ne, I, with a lwpass behaviur. where I I I I I s l α s l s l α s l I I I,, I I I s l It is very simple t prve that the utput vltages V and V f the RC-ative iruit in Fig. 6. a. perfrm idential behaviurs with regard t the input vltages as the passive iruit: s R C R R R R V 0 ( V i) ( V i) ( V j) ( V j), s R C R i R i R j R j (8) R R R R V 0 ( V i) ( V i) ( V j) ( V j) s R C R i R i R j R j Sine the mplete synthesis f the funtin in (6) needs a supplementary amplifier t multiply V by α and then t add the result t V, the slutin, whih eeps at a minimum the number f p-amps, nsists t transfer this peratin t the input stages f the adjaent filter ells. (6) (7)

5 Fig. 6 a. The ative ell used t emulate send rder transfer funtin f passive filter ladder lps, b. Symbl f the ative filter ell Finally, the emulatin f the passive filter lp in Fig. 5. b. by the ative filter ell in Fig. 6. a. uses the fllwing relatinships: l R R, R R, C, i i α R R R, R R j j α In (9), we suppse the value f R being hsen. The send rder ative filter ell in Fig. 6. a. is used suessfully t implement all the passive filter lps in Fig. 5. If the passive filter is plynmial, nly ne f tw ative filter ell utputs is used: the lwpass utput, V, fr the iruit in Fig. 5. a. and the highpass utput, V, fr the iruit in Fig. 5.. Finally, the finite-zers highpass ladder netwr filter in Fig. 5. d. reveals the same transfer funtin as its lwpass unterpart in Fig. 5. b. 4. DESIGN PROCEDURE AND SIMULATION RESULTS The design predure f an RC-ative filter based n the mesh urrents emulatin methd is straightfrward. First, divide the riginal ladder prttype in distint lps and hse apprpriate filter ells amng the ells intrdued in the previus Setin. Next, nnet neighburing ells with apprpiate internnetins. Finally, determine the values f resistrs and apaitanes via the derived design relatinships. T demnstrate the flexibility f the prpsed ative filters synthesis, a fifth-rder LC ellipti lwpass filter is realised firstly. Its prttype is shwn in Fig. 7.a., having 0.dB ripple in the passband and minimum 43dB attenuatin in the stpband. The bandwidth f the ative filter was 50 Hz. Fig. 7. b. shws the implementatin f the LC ladder netwr filter at the ative filter ell level. The synthesis emplys a ttal number f p amps. The simulatin f the ative iruit emplyed LT5 (LINEAR TECHNOLOGY) p amps and gave the result shwn in Fig. 7.. (9) Fig. 7 a. Fifth-rder LC lwpass filter prttype, b. The rrespnding RC-ative filter derived frm the prttype n the basis f mesh urrents emulatin methd,. Amplitude respnse f the ative lwpass filter

6 The send design ase refers t a third-rder ellipti highpass filter. The LC π-shaped passive prttype is shwn in Fig. 8. a. It has 0.5 db ripple in the passband and at least 35 db attenuatin in the stpband. The bandwidth f the ative filter is 0 Hz. The ative filter implementatin, shwn in Fig. 8. b. emplys als LT5 p-amps. Fig. 8 a. Third-rder LC highpass filter prttype, b. The rrespnding RC-ative filter derived frm the prttype n the basis f mesh urrents emulatin methd,. Amplitude respnse f the ative highpass filter 5. CONCLUSIONS A new passive filter implementatin based n mesh urrents emulatin f LC ladder netwrs was intrdued fr finite-zers lwpass and highpass ative filters. The ative filter is btained by the adequate internnetin f different types ative ells. Design priniples and predures jintly with the results f iruit simulatins are als presented. In nlusin, the mesh urrents emulatin f LC ladder netwrs represents a very pwerfull methd fr realising ative filters. 6. REFERENCES [] A. Câmpeanu and M. Nafrniţă, Implementatin f HP filters by a leapfrg realizatin, Pr. Intern. Cnf. On Mireletrn. and Cmp. Siene, Kishinev, Mldavia, pp , Ot [] A. Câmpeanu and M. Nafrniţă, A new biquad ell imprves the perfrmane f a high-pass filter, Bul. Tehni Univ. Tehnie Timişara, tm 38 (5), pp. 3-38, 993. [3] A. Câmpeanu and M. Nafrniţă, Multiple-inputs ative filter ells fr a new leap-frg filter struture, Pr. ECCTD 95 Eurpean Cnf. n Cir. Thery & Design, pp , Istanbul, Turey, August 995. [4] F.E. Girling and E. F. Gd, Ative filters : the leap-frg r ative ladder synthesis, Wireless Wrld, vl. 76, pp , July 970. [5] K. Martin and A. S. Sedra, Design f signal-flw graph (SFG) ative filters, IEEE Trans. Ciruits Syst., vl. CAS-5, pp , April, 978. [6] H. G. Dimpuls and A. G. Cnstantinides, Linear transfrmatin ative filters, IEEE Trans. Ciruits Syst., vl. CAS-5, pp , Otber, 978. [7] L. P. Calba and A. C. M. De Queirz, OTA-C simulatin f passive filters via embedding, Pr. IEEE ISCAS 989, pp