Analysis of circuits containing active elements by using modified T  graphs


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1 Anlsis of circuits contining ctive elements using modified T  grphs DALBO BOLEK *) nd EA BOLKOA**) Deprtment of Telecommunictions *) dioelectronics **) Brno Universit of Technolog Purknov 8, 6 Brno CECH EPUBLC Astrct:  The soclled modified trnsform (T) grphs re descried in the pper, which re grph nlog to wellknown voltge nd current incidence mtrices. The procedure is shown how to pss from Tgrphs to the clssicl MsonCotes (MC) flow grph. The explntion is illustrted with exmples of circuits contining opertionl mplifiers nd current conveors. KeWords:  Signl flow grph, flow grph, Tgrph, incidence mtrix, liner trnsformtion, opertionl mplifier, current conveor. ntroduction Mson s (M) nd MsonCotes' (MC) flow grphs s tools of hndndpper nlsis of reltivel simple liner circuits hve een in existence for mn ers [], []. However, their sic form is not suitle for direct solution of circuits contining some ctive elements, such s tod's much promising current conveors. Severl tpes of signl flow grph hve een developed for circuits contining ctive elements [], [4], [5]. Their nlsis shows tht common utilition of given grph is conditionl on the following fctors:. Simple rules of grph construction directl from the circuit schemtics.. Economicl grph structure, i.e. the requirement of reltivel simple grph nd its simple evlution.. Simple rules of grph evlution. 4. Possiilit of evlution oth voltge & current gins nd immittnces. ules No. nd No. re often in contrdiction: The w to n economicl grph leds mostl through hrdtorememer rules of thum [4]. ule No. is fulfilled utomticll if the resulting grph is the M or the MCgrph. Then Mson s wellknown generl gin formul cn e used. ule No. 4 is not fulfilled for some grphs due to the reduction of some vriles s consequence of ppling rule No.. However, these vriles re necessr, for exmple, for nling immittnce reltions. From the point of view of rules No.,, nd 4, the soclled trnsform (T) grphs introduced in [5] re interesting. The strt from the grph interprettion of voltge nd current incidence mtrices. For the purpose of fulfilling rule No., we developed the mtrix method of soclled twin vriles. The soclled modified Tgrphs re introduced in the pper, which represent snthesis of Tgrphs nd the method of twin vriles. These grphs were developed with the im to rech welllnced compromise while fulfilling rules No. to 4. As finl structure, we otin the clssicl MCgrph with its common evlution. This pper is orgnied s follows: n Section we summrie sic knowledge from the theor of T grphs. The T to MCgrph conversion is demonstrted on n exmple of pssive liner circuit. n Section we then explin the sic ide of twin vriles nd its utilition in the construction of economicl Tgrphs. We use the circuits with OpAmps nd current conveors for explntion. The ppliction Section 4 shows mong others the prolem of serching for permitted twin vriles while nling circuits comprising severl ctive elements. ncidence mtrices, T nd MCgrphs Consider pssive liner circuit consisting of m oneports with the dmittnces k, k =,,.., m. Let the circuit contin n independent nodes to which we ssign n node voltges i, i =,,,n, directed from node i towrd the reference node, nd n node currents i, i =,,,n, which flow into the circuit from outer sources. We introduce loding orienttion of voltges k nd currents k of rnch dmittnces k. Then the equtions hold B B B = () N B = A () B N = A () where B (m,), B (m,), N (n,), N (n,) re the vectors of rnch currents, rnch voltges, node currents nd node voltges. Here B (m,m) is digonl mtrix of rnch dmittnces, A (n,m) nd A (m,n) re the current nd voltge incidence mtrices. For the pssive circuits, the incidence mtrices re connected through trnsposition.
2 Associting equtions (), () nd () ields the nodl dmittnce equtions: N N N = (4) where N B = A A. (5) The Tgrphs represent equtions ()(). Then equtions (4) nd (5) give directions on how to pss from the T to the MCgrphs. As n exmple, consider the pssive Tcheshev ldder filter in Fig.. C L Q S C Fig.. Pssive filter nled. The independent nodes, node currents nd specified directions of rnch voltges nd currents re denoted in the schemtic. For the circuit we get C C L = /sl L () C C B B B =  C  L () C N A B  C L =  () C B A N  N =A B A =  /sl /sl (5) /sl /sl Equtions () () cn e represented the Tgrph in Fig.. The following structures cn e recognied in the trnsform grph: /sl Fig.. Trnsform grph of the circuit in Fig..  The twins node voltge/node current.  The rnch dmittnces.  The voltge nd current rnches of the Tgrph, mrked the open nd contoured rrows; the trnsform coefficients corresponding to the elements of incidence mtrices re plced ner the rrows. As consequence of the circuit pssivit nd the smmetr of voltge nd current incidence mtrices, the voltge nd current gins of the corresponding rnches of the Tgrph re equl. Pssing from the T to MCgrphs cn e done ccording to the following lgorithm:  The selfloop gin of node voltge i is otined s sum of products where nd ki m k= k ki, (6) re the voltge nd the current gin of the rnch tht directl connects twins i i with dmittnce k.  The gin of pth directed from node i to node j is otined s negtive sum of products m k =. (7)  The pth with gin of is directed from node i to node i. The ppliction of the procedure given is in Fig.. The MCgrph cn e constructed directl ove the T grph. The resulting MCgrph full conforms to eqution (4), where N is given eqution (5 ). /sl /sl /sl /sl  /sl Fig.. T to MCgrph construction (circuit from Fig. ). k kj   
3 t should e noted tht the corresponding unitgin pths from the node currents to the node voltges re constructed onl if it is dvisle to consider the given node current. The Tgrph cn e constructed directl from the schemtic, following this procedure:  n the schemtic, we numer the nodes, we drw the rnches of outer sources nd direct them inside the circuit, nd we direct (in n ritrr w) ech internl rnch comprising the oneport dmittnce.  We drw smols of twin vriles, ordering them s unknown vrile known vrile (for pssive circuits we hve node voltge node current ). Below them we drw grph nodes, lelled the corresponding rnch dmittnces.  Utiliing Kirchhoff s voltge lw, we drw rnches of the voltge Tgrph. We dd gins ( or ) to individul rnches.  We complete the Tgrph rnches smmetricll (for pssive circuits) the current gins. After tht, the MCgrph is constructed nd evluted s descried ove.. Method of twin vriles nd modified Tgrph. Circuits with idel OpAmps Consider n idel opertionl mplifier (OPA) connected to the circuit s shown in Fig. 4. c Fig. 4. Exmple of OPA connection. c c The OPA mintins ero voltge etween its inputs, such tht the equlit = is true. This will e utilied for the reduction of unknown vriles. f we re not interested in the OPA output current (it cn e clculted post fcto from the voltges), we need not compile the eqution of Kirchhoff s current lw for node c. Be wre tht the numer of node voltges nd currents must e the sme in the set of circuit equtions. After the reduction descried, these vriles re [ =, c ] nd [, ]. The circuit in Fig. 4 gives the equtions...,.... = = The dots indicte other possile elements which re generted the floting dmittnces nd. n the mtrix form, = = c. (8) There re two corresponding MCgrphs in Fig = () c = () c Fig. 5. Two versions of MCgrphs corresponding to eqution (8). Two possile vrints of twin vriles result form the grphs: ) =, c, ) =, c. Conclusion: n the cse of idel OpAmp, we onl consider unknown vriles = nd c. Either current or current is llocted to the first unknown voltge vrile (OpAmp input). The second current will e llocted to the OpAmp output voltge. Both vrints of the T nd the MCgrphs of the circuit in Fig. 4 re shown in Fig. 6. Note tht the grphs re equl to those in Fig = c = c c ()  c Fig. 6. Construction of grphs of circuit in Fig. 4.. Circuits with current conveors CC± Shown in Fig. 7 is current conveor with the wellknown internl structure, connected in generl network. The circuit cn e descried the following equtions: =... =... =... At the sme time the equlit = holds, which will e used to reduce the vrile. ()
4 x x x CC ± ± Fig. 7. Sucircuit with current conveor CC±. Two grph vrints in Fig. 8 correspond to the ove equtions: x = x x x x =  x x ±   ()  x ()  Fig. 8. rph representtion of circuit in Fig. 7. Conclusion: n the cse of CC± elements, we onl consider the unknown vriles =,, nd. Either current x or current is llocted to the first unknown voltge vrile ( = ). Either current or current will e llocted to the second unknown voltge vrile ( ). Either current or current x will e llocted to vrile. Considering the sequence of unknown vriles =,,, the corresponding sequence of twin currents is either,, or the sequence,,, which is its cclicshifted equivlent.  4 llustrtive exmples 4. Circuits with opertionl mplifiers Let us consider the nd order ndpss filter in Fig. 9 [6] nd the corresponding MCgrph constructed from the Tgrph in Fig.. C Q C S T OPA C Fig. 9. nd order ndpss filter = OPA Fig.. rph representtion of filter in Fig. 9. Evluting the MCgrph, we otin the voltge trnsfer function 5 = C ( ) s C C s C s = s C 4. Circuits with current conveors Shown in Fig. is simple circuit contining CC. The voltge gin from node to node is s follows: =. Q CC Fig.. Circuit contining CC. S U
5 The corresponding grphs re in Fig.. Evluting the MCgrph ields the ove trnsfer function. = Fig.. rph representtion of circuit in Fig.. An impednce converter with oth positive nd negtive current conveors is shown in Fig. [7]. The unknown vriles re s follows: =, = 4, (CC current), ( CC current). Q CC CC Fig.. mpednce converter with CC nd CC [7].  = 4  =  S Fig. 4. rph representtion of circuit in Fig.. From the viewpoint of CC, the following twin vriles could e possile:  ) = 4 4 or = 4 ) 4 c)  Now we complete them from the viewpoint of CC: = conflict with ) or = see ) see ) The onl possile piring is given in Fig. 4. Evluting the MCgrph, we get the input impednce:..( ).( ) = = =. inp 5 Conclusions The descried method of grph solution of liner sstems comines the soclled trnsform grphs nd the method of twin vriles. As result, the lgorithm of grph construction is reltivel simple, nd the sie of the resulting MCgrph is reduced enough. As shown in the finl exmple, the selection of permitted twin vriles must e performed crefull if there is numer of trnsform elements in the circuit nled. This method cn e used to solve generl liner prolems, not onl in the nlsis of electricl circuits. eferences: [] Mson, S.J., Feedck Theor: Further Properties of Signl Flow rphs. Proc. E, ol. 44, No. 7, pp. 996, 956. [] Cotes, C.L., Flowgrph Solution of Liner Algeric Equtions. E Trns. Circuit Theor, CT 6, pp. 787, 959. [] Mul, J., SignlFlowrphgin with espect to the enerl Node of rph. Electronics Letters, August 969, No. 6, pp [4] Biolek,D., Novel Signl Flow rphs of Current Conveors. 8th MWSCAS, io de Jneiro, Bril August 6, 995, pp [5] Mul, J., Pospisil,J., Anlsis of circuits contining ctive trnsform elements. Electronic Horion, ol. 5, No. 4, pp. 667, 974 (in Cech). [6] BOLEK,D., BOLKOA,., Optimition of Liner Sstems using Smolic Modelling. MS, Ls Plms de rn Cnri (Spin), pp [7] Tomou,C., Lidge,F.J, High,D.., Anlogue C Design: the CurrentMode Approch. EE Circuits nd Sstems Series, Peter Peregrinus Ltd, London 99,UK. Acknowledgement: This work is supported the rnt Agenc of the Cech epulic under grnts No. //4 nd //97, nd the reserch progrmme of BUT eserch of electronic communiction sstems nd technologies.
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