Graph Method for Solving Switched Capacitors Circuits

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1 Recent Advances n rcuts, ystems, gnal and Telecommuncatons Graph Method for olvng wtched apactors rcuts BHUMIL BRTNÍ Department of lectroncs and Informatcs ollege of Polytechncs Jhlava Tolstého 6, 586 Jhlava ZH RPUBLI brtnk@vspjc Abstract: - rcuts wth swtched capactors are descrbed by a capactance matrx and seekng voltage transfers then means calculatng the rato of algebrac supplements of ths matrx As there are also graph methods of crcut analyss n addton to algebrac methods, t s clearly possble n theory to carry out an analyss of the whole swtched crcut n two-phase swtchng exclusvely by the graph method as well For ths purpose t s possble to plot a Mason graph of a crcut, use transformaton graphs to reduce Mason graphs for all the four phases of swtchng, and then plot a summary graph from the transformed graphs obtaned ths way Frst we draw nodes and possble branches, obtaned by transformaton graphs for transfers of even-even and odd-odd phases In the next step, branches obtaned by transformaton graphs for and phase are drawn between these nodes, whle ther resultng transfer s multpled by Ths summary graph can then be nterpreted by the Mason s relaton to provde transparent voltage transfers Therefore t s not necessary to compose a sum capactance matrx and to express ths consequently n numbers, and so t s possble to reach the fnal result n a graphcal ey-words: - wtched capactors, two phases, transformaton graph, Mason s formula, voltage transfer, summary graph ummary Graph onstructon olvng crcuts wth swtched capactors by means of nodal charge equaton method system [], [], [5], leads generally to an equaton system Q Q whch can have the followng from, for nstance Ths system can also be llustrated by a graph, the constructon of whch can proceed ths way: for example the last fourth equaton can be consdered n the followng form whle for the sake of clarty we lad, so the product s thus ero and wll fall out of the equaton Ths fourth equaton wll be rewrtten so that the fourth varable wll be expressed from t The equaton can be nterpreted so that the addton to the varable, multpled by the coeffcent from the varable has the value of multpled by the coeffcent, and the addton from the varable has the value of multpled by the coeffcent Therefore the very loop at the node has the transfer, the branch from the node to the node has L the transfer and fnally, the branch from the node to the node has the transfer mentoned constructon s shown by the graph n Fg : : : The above Fg Graph of equaton -/ -/ IBN:

2 Recent Advances n rcuts, ystems, gnal and Telecommuncatons onstructon Generally The above mentoned constructon can be used for plottng a summary graph of the crcut, whch wll thus be plotted by frst drawng the nodes and possble branches, obtaned by transformaton graphs for transfers of even-even and odd-odd phases, as shown n Fg Fg The frst step: and transfers In the next step, branches obtaned by transformaton graphs for and phase are drawn between these nodes, whle ther resultng transfer s multpled by For ths reason eg the resultng graph wth an phase, ncludng ts own loop wth the transfer, wll be represented by a branch gong from the node to the node n the summary graph and havng the transfer In the same way the resultng graph n the phase ncludng ts own loop wth the transfer wll be represented by a branch gong from the node to the node n the summary graph and havng the transfer, as shown n Fg Fg The second step: and transfers Thereby obtaned summary graph s then evaluated by means of the Mason s rule for the T transfer of the graph p T [] Therefore t s not necessary to compose a sum capactance matrx and to express ths consequently n numbers, and so t s possble to reach the fnal result n a graphcal way valuaton of the T-graph valuaton of the transformaton graph n and phases can proceded ths way: In an equaton system, we lad, so the product s 5 Q [ ] 5 Q quaton Q can be nterpreted so that the addton to the varable Q from the varable The ~ Q capacty s gven by the relaton a a α therefore for ths stuaton s, and Q Q, and the transfer of the voltage branch s and transfer of the charge s Therefore the loop from the node to the node ~ has the transfer of the voltage, and return to the node has the transfer of the charge Descrbed constructon s shown n Fg for phase and for phase, too ~ Fg The constructon of the T-graph for and xample of olvng The above descrbed way of a graph evaluaton wll be llustrated by the followng example A crcut wth a swtched capactor has got the schematc wrng dagram shown n Fg5 Fg5 chematc dagram from the example The crcut has four nodes; therefore the startng graph of the crcut n Fg6 has also four nodes The capactor s connected to the second node, the capactor then between the thrd and fourth nodes, whch n the smplfed startng graph n Fg 5 s marked by notng above the second node and between the thrd and fourth nodes In the even-numbered phase nodes and wll be connected by closng the swtch, whch s demonstrated n the graph by ther transformaton _ ~ IBN:

3 Recent Advances n rcuts, ystems, gnal and Telecommuncatons untng nto a sngle node The capacty n ths resultng node s gven generally by the relaton ~ Q a a α, where ~ s the capacty of the orgnal Q node, a, a are then the branches of the transformaton graph wth the transfers of voltage and of charge Thus the resultng capacty here wll be to the nherent loop n the resultng transformed graph In the odd phase by closng the swtch the nodes and wll be connected, whch wll demonstrate n the graph by ther transformaton untng nto a sngle node, and ths resultng node s at the same tme the nput node of the operatonal amplfer In the remanng phases and, we start, accordng ~ Q to the equaton a a α, along the branch wth the voltage transfer a from the resultng node to the orgnal node and we enter back to the resultng node along the branch wth the charge transfer transformaton graphs for all the four cases are n Fg6 The summary graph obtaned from the partal transformed graphs from the Fg6 by the above mentoned procedure s then shown n Fg7 Frst the results of the transformed graphs for and phases are plotted n case of ths example only as nodes Q a The -/ - - -/ - -/ - - Fg7 ummary graph of the crcut from Fg5 - - Fg6 T-graphs for,, and phases The operatonal amplfer s connected to the thrd node by ts nvertng nput and nto the fourth node by ts output, and consequently the branch wth the charge transfer of the transformaton graph goes from the node, the branch wth the voltage transfer of the transformaton graph enters the node Followng ths transformaton graph, the capacty connected between nodes and then transforms nto the resultng capacty of the amount, as the capactor s connected to the node by one of ts ends, therefore the nherent look at ths node has the transfer and s ~ Q transformed accordng to the equaton a a α The branch between the nodes and wth the transfer s transformed to the nherent loop wth the transfer ~ Q, because n the relaton a a α s nowα, as the branch of the orgnal graph converts - In the next step, the results of the transformed graph for the and phases multpled by are then drawn between these nodes as branches, e the branch wth the transfer between the nodes, and, and the branches wth the transfers between the nodes, and By evaluatng ths summary graph, whch s done by substtuton nto the Mason s formula p T, we get the followng fnal results ths way From the graph t s obvous that the entry node s: or the frst node n the even phase, therefore there wll only be transfers from the even phase of the frst node It s further evdent from the graph that the ext e fourth node exsts here both n the even phase as: and n the odd phase as: It s thus possble to express n numbers the two followng transfers: and:, for whch t holds that: IBN:

4 p 6 and for the second one p 7 olvng by a Matrx Method To compare the soluton of a crcut wth swtched capactors by the above mentoned purely graph method, we wll present a calculaton of the same crcut by the usually used method of nodal charge equatons usng matrx calculus n the concluson In dong so, ths soluton wll be done n just as detaled steps as the graph method so that we can compare both methods The crcut n Fg5 has four nodes, so the partal capactance matrx wll be of the fourth 8 An operatonal amplfer, connected by ts nput nto the node and by ts output nto the node, when applyng a modfcaton of the node voltage method so called reduced nodal method [], [], modfes ths matrx, so the matrx wll get the shape 9 9 The capactance matrx of the crcut wll be composed of four of those sub-matrces 9, whle the sub-matrx lyng n the adjacent dagonal of the man matrx wll be multpled by In the next step ths matrx wll be reduced due to closng the swtches n the followng way: losng the swtch n the even phase wll be manfested by connectng nodes and n the even phase, and thus by untng the voltages and, whch means that the matrx columns and can be summared, and by summarng charges, whch means that the matrx rows and can be summared The swtch n the odd phase connects the node to the node, whch s, though, the entry node of an deal operatonal amplfer, whch has however ero voltage due to the nfnte voltage gan of the deal operatonal amplfer, whch s manfested by the possblty to leave out the column from the matrx The capactance matrx wll thus be of the three grade, and wll have the followng form As the crcut s strred only n the even phase, the only non-ero nput charge s the charge Q, so only two transfers can be calculated, namely and, whch can be expressed by applyng the algebrac complements theory lke ths : : and for the second one Recent Advances n rcuts, ystems, gnal and Telecommuncatons IBN:

5 Recent Advances n rcuts, ystems, gnal and Telecommuncatons : : oncluson Whle n case of usng the graph method a graph was ndcated, a transformaton graph was plotted and from ts results a summary graph was drawn and evaluated by the Mason s rule, after whch the result was obtaned by an easy smplfcaton, n case of solvng by the matrx calculus the procedure was much more complcated As we can see n comparson chapter and chapter, the graph method presents the numercal method In case of solvng by the matrx calculus by the modfed nodal method [6] the procedure was much more complcated Frst a partal capactance matrx had to be composed, n the next step t was modfed by an deal operatonal amplfer From four matrces obtaned by ths a capactance matrx was constructed and was reduced by the actvty of swtches; from the reduced matrx three algebrac complements were made up and they had to be expressed by means of an expanson because they were of a hgher grade than It means, the solvng by modfed nodal method s rather dffcult References: [] Bolek, D, olvng lectronc rcuts, BN Publsher Praque, [] Čajka, J, vasl, J, Theory of Lnear Networks NTL/ALFA Praque, 979 [] Dostál, T, The Analyss of the Actve omponents ontanng wtched apactors by Nodal oltage Method lectroncs horont, ol 5, NoI, 98, pp -6 [] Martnek, P, Boreš, P, Hospodka, J,lectrcs Flters UT Publsher Praque, [5] Toumaou, h, rcuts nad ystems Tutorals I Press Inc, New York, 996 [6] lach, J 99 omputer Methods for rcut Analyss and Desgn New York, NY: an Nostrand Renhold, 99 IBN: