Fast Analysis of Blocks with Current Conveyors

Transcription

1 Fst Anlss of Blocks wth Current Conveors DALIBOR BIOLEK, IERA BIOLKOA*) Dept of Telecommunctons, Dept of Rdoelectroncs*) Brno Unverst of Technolog Purknov 8, 6 00 Brno CZECH REPUBLIC olek@wsesorg Astrct: - On the ss of the well-known modfed nodl nlss, unversl mtrces-stmps re defned for modelng more tpes of current conveors From these stmps, modfed Mson-Cotes grphs re derved for fst nlss of crcuts wth current conveors I nd II Ke-words: - Modfed Nodl Anlss, Mtr-Stmp, Mson-Cotes Flow Grph, Current Conveor Introducton Current conveors I nd II n prtculr elong to mportnt components of modern locks for nlog sgnl processng [] Although up-to-dte nlss nd smulton of electronc crcuts s coupled wth the utlton of computer progrms, hnd-nd-pper nlss of smple locks s lso menngful The resons re severl To verf the prncpl crcut functonlt, the nled crcut functon, eg the s- domn trnsfer or mttnce functon, s usull requred n the form of mthemtcl formul It s possle to use some not commonl vlle progrms for smolc nlss [] Snce tod there re of more tpes of current conveor, nd further new tpes wll pper, the prolem of updtng the current model lrres of these progrms s urgent Another reson for hnd-nd-pper nlss s the so-clled method of sgnl flow grphs, whch cn e utled not onl for the ctul nlss ut lso to mke the opposte process crcut snthess sed on ts requred fetures more effcent Current conveors (s) re elements whose presence n the crcut complctes ts ntutve nlss The hve no dmttnce mtr so tht ther ncluson n the generll-used method of modfed nodl nlss (MNA) s doutful In the pst, numer of ttempts were mde to fnd smple lgorthms of complng equtons for crcuts wth s [3], [4], [5], especll n the form of flow grphs In most cses, the focused on the second - generton conveors The common shortcomng of these pproches conssts n the rther complcted rules of complng model of the crcut (e flow grph) drectl from the crcut schemtcs Due to the trnsformton of crcut vrles, cused operton, the models of the conveor nd of the crcut remnder lended nto ech other Ths s n conflct wth the sc dvntge of MNA, where ever crcut element hs ts own descrpton, whch s modulrl nserted nto the pseudodmttnce mtr or nto the orented grph of the entre crcut In ddton, the trnsformton of crcut vrles depends on the tpe of Tht s wh the method of uldng the crcut model s not unversl Owng to the trnsformton, the lnkge to the crcut s not clerl evdent from the flow grph, nd such flow grphs re not pproprte for crcut snthess nd optmton An ddtonl dsdvntge of the present pproches conssts n the sence of conveor output current n the equtons formed However, ths current s often the requred quntt, especll for crcuts opertng n the current or med mode In the pper, one method of lgorthmc mtr nlss of crcuts wth s s shown The method s sed on the well-known mtr-stmp pproch, whch overcomes the ove prolems We defne unversl stmp, whch cn e used for ll the sc tpes of of the frst nd second generton Ths mtr-stmp then leds to Mson-Cotes (M-C) grph [6] of wth undrected self-loops Two emples demonstrte the convenence of these grphs n the nlss nd snthess of ctve flters Mtr-stmp nd M-C grph- stmp of Thévenn model of recprocl two-port Consder crcut tht s descred equtons of Krchhoff s current lw (KCL) of the clsscl nodl nlss The numer of equtons s equl to the numer of ndependent nodes n the crcut The equtons contn the sme numer of unknown nodl voltges A two-port descred ts Thévenn model ccordng to Fg () s ddtonll connected etween nodes nd Then oth the voltge nd the current reltons n the crcut wll e chnged A current I wll flow through the two-port nd current reltons t nodes nd wll e volted The orgnl nodl voltges wll lso e chnged

2 I () Z I I () - Y Y Fg () Thévenn model of ncluded two-port, () correspondng M-C grph-stmp The orgnl equton descrng the current equlrum t node hs to e completed current I, flowng outsde the node, nd t node current I wth negtve sgn ecuse t flows nto node In ddton, nodl voltges nd re now ound the followng condton: Z I + +, or Z I + - I All these modfctons cn e ncluded n new set of equtons of MNA: I I Y + Y - - Z I stmp The vector of unknown nodl voltges s etended nother vrle current I The numer of equtons s lso ncresed the ove condton etween nodl voltges nd The voltge s ncluded n the left-sde vector of the known ectng qunttes The modfcton of KCL equtons for nodes nd s mplemented enterng the numers + nd nto column I The orgnl crcut dmttnce mtr together wth the stmp of the Thévenn model of ncluded two-port represent drectons how to nle for nstnce crcuts wth voltge sources mens of MNA Impednce Z cn e ero, n whch cse the del voltge source s modelled For 0 nd smultneousl Z 0 we model short connecton etween the nodes nd the short-crcuted current cn e Z computed Ths procedure cn e used for the nlss of crcuts wth current controlled sources The set of MNA equtons wll e trnsformed nto nother form for drect constructon of the flow grph n Fg ) Y Y Z I I I I + I + Smol Y (Y ) specfes the sum of dmttnces connected to node () The correspondng self-loops n the grph re drwn s dotted curve ecuse the gven dmttnces re not prt of the ncluded crcut n Fg () The flow grph-stmp of ncluded two-port hs thck-lne shpe of drgonfl It s the grph representton of ts mtr-stmp New grph nodes re nput node nd node of the unknown crcut vrle I Ech drgonfl wng s formed couple of pths wth gns + nd The outwrds drected pths re current pths (the re drected out of current node I ) The pth wth the + sgn follows the wng towrds the node, to whch the current I flows The second pth hs the sgn oltge pths follow the opposte wng mrgns The hve oth the drectons nd sgns opposte to those of the current pths The drgonfl hed s formed self-loop, the gn of whch s gven the nner mpednce of the Thévenn model The drgonfl tl contns the node of nner voltge of the Thévenn model The ove drgonfl model ncludes ll estng models of recprocl two-ports: for pssve mpednce, the tl wll vnsh For del voltge source the hed gn wll e ero In cse of grounded two-port, onl one wng wll pper n the grph The drgonfl grph-stmp cn theoretcll e utled n the nlss of crcuts wth oth current nd voltge sources However, ts prctcl mportnce ppers when solvng crcuts wth elements lke s, where current reltons re under consderton In cse of nonrecprocl crcuts for nstnce OpAmps nd ll the tpes of ther flow grphs wll contn drgonfles or structures fter ther trnsformtons 3 Crcuts wth s The smol of the generl three-port, whch s ncluded nto crcut t nodes, nd c, s shown n Fg () Ths conveor s descred generl hrd equtons n the followng form:

3 I α β I I The vlues of α, β, nd coeffcents depend on the tpe of s shown n T In ddton, we cn use them to model sc conveor nccurces For emple, devtng the vlue of coeffcent from models the nccurc of the gn of nternl current mrror of II+ Smlrl, the devton of β models the nccurc of voltge follower trnsfer etween the nd nputs of nonnvertng II I I α Y Y I I I β - () I 0 () I I c c I Y cc Fg () three-port, () ts M-C grph - stmp coeffcent α β I II 0 nonnvertng nvertng - postve negtve - T Coeffcents for vrous tpes of The correspondng mtr-stmp cn e derved from hrd equtons nd from Fg (): c I I Y -α Y - c I c Y cc - c β - I I c The smol I ndctes ndependent current I I The M-C grph of n Fg () s derved smlr procedure s for the ove Thévenn model Comprng wth the orgnl drgonfl n Fg (), the pth gns on the upper wng re modfed One ddtonl prt models the current trnsfer nto the output The self-loop gn of node I s now 0 In the cse of the well-known effect of nonero resstnce R for II, ths gn wll e just R It should e noted tht for the commonl used tpe of II, the prt wth gn α dsppers from ts flow grph Ths fct contrutes to ts smplfcton 4 Emples of nlss The nd -order hgh-nput mpednce nsenstve flter n Fg 3 () ws pulshed n [7] Assume tht the β nd prmeters of oth II+ re not ectl Let us fnd the nfluence of these nccurces on the flter fetures I R 4 5 II+ II+ R 3 4 I n I () C out G4 G C 3 G + sc G + sc G - β β - 0 () 0 Fg 3 () nd -order flter, () ts M-C grph To determne flter trnsfer functon 4 /, let us construct the shortened M-C grph n Fg 3 (), consderng generl prmeters β, β, nd The grph evluton s consderl smplfed the presence of three ero-gn self-loops Applng the generled Mson s gn formul [8] elds the result β 4 β R 4( C + C3) RR4, 3 + s + s β β whch s n ccordnce wth [7] The resultng formul enles smple determnton of the nfluence of β nd coeffcents on flter prmeters ω 0 nd Q I

4 As second demonstrton, generl mpednce converter (GIC) wth two s s gven n Fg 4 () [] As noted n [], ths crcut opertes s negtve GIC, f oth conveors re of equl polrt, tht s oth re II+ or II- Conversel, f the two conveors re of opposte polrt, then postve GIC s otned The possle operton for other tpes of s not mentoned I I Z I Z 3 () 3 4 I Z 0 Y Y 3 Y β 3 β α α I I 0 0 () Fg 4 () GIC, () ts M-C grph Now consder generl s wth ther grphs-stmps ccordng to Fg () The resultng M-C grph, constructed from the schemtcs, s n Fg 4 () Evlutng t nspecton elds the formul for nput mpednce: Z n I Y3 β Y β β Y Y + α β α β Y Y After rrngement we hve Z n β Y Y Y3 β Y β Y β Y ( α α ) Y ] [ 3 3 () Utlng two IIs, we set α α 0 Then equton () elds: Z Z Z n β β Z3 () Ths result confrms the ove conclusons from [] nd etends them the posslt of utlng oth the clsscl nd nvertng s (for nvertng s, coeffcents β re negtve) Anlsng equton (), we conclude tht oth conveors must e of the II tpe to cheve the smple mpednce converson ccordng to equton () 5 Conclusons Novel modfed M-C grphs re descred n the pper The enle smple modelng of current conveors I nd II regrdless of whether the re postve, negtve, nonnvertng or nvertng Selectng the vlues of α, β, nd coeffcents nd the self-loop gn of grph current node, we cn lso model some sc nondeltes such s nccurc of nternl voltge follower etween nd termnls, nfluence of nonero R resstnce, nccurces of nternl current mrror, etc, wthout n modfcton of the flow grph topolog B reson of ts unverslt, the correspondng mtr-stmp hs een ncorported n the SNAP progrm [] for smolc nlss of lner networks Acknowledgement Ths work s supported the Grnt Agenc of the Cech Repulc under grnts No 0/0/043, 0/00/0907 &0/00/037, nd the reserch progrms of BUT Reserch of electronc communcton sstems nd technologes nd Mkrost References: [] TOMAZOU,C: Crcuts & Sstems Tutorls ISCAS'94, London, 994, p 580 [] BIOLEK,D: SNAP - Progrm wth Smolc Core for Eductonl Purposes Contruton to the ook "Sstems nd Control: Theor nd Applctons", World Scentfc, Electrcl nd Computer Engneerng Seres, 000 Edtor N MASTORAKIS, pp [3] BIOLEK,D: Novel Sgnl Flow Grphs of Current Conveors 38th MWSCAS, Ro de Jnero, Brl August 3-6, 995, pp [4] BIOLEK,D-BIOLKOA,: Anlss of crcuts contnng ctve elements usng modfed T - grphs Contruton to the ook "Advnces n Sstems Scence: Mesurement, Crcuts nd Control", WSES Press, Electrcl nd Computer Engneerng Seres, 00 Edtors N E MASTORAKIS nd LA PECORELLI-PERES, pp 79-83

5 [5] BIOLEK,D-BIOLKOA,: Flow Grphs for Anlss (not onl) Current-Mode Anlogue Blocks Contruton to the ook "Recent Advnces n Crcuts, Sstems nd Sgnl Processng", WSES Press, Electrcl nd Computer Engneerng Seres, 00 Edtors N E MASTORAKIS nd G ANTONIOU, pp 5-56 ISBN [6] MIKULA, J: Sgnl-Flow-Grph-gn wth Respect to the Generl Node of Grph Electroncs Letters, August 969, No 6, pp [7] FABRE,A et l: Hgh Input Impednce Insenstve Second-Order Flters Implemented from s IEEE Trns CAS-I, ol 4, No, 994, pp 98-9 [8] KUO, BC: Lner Networks nd Sstems McGrw/Hll Book Compn, 967