Full graph method of switched-capacitor circuit analysis

Transcription

1 Bohumil BRNÍK ollege Polytechnics Jihlava Full graph method switched-capacitor circuit analysis bstract. ircuits with switched capacitors are described by a capacitance matrix and seeking voltage transfers then means calculating the ratio algebraic supplements this matrix. s there are also graph methods circuit analysis in addition to algebraic methods, it is clearly possible in theory to carry out an analysis the whole switched circuit in two-phase switching exclusively by the full graph method as well. In this case the summary graph can be constructed by the transformation graph or two graphs and can be simply evaluated by the Mason s relation. Strescenie. Predstawiono metodę wykorystania grafów do analiy obwodów prełącanym kondensatorem. Sumarycny graf był konstruowany wykorystanie grafów transformacji i metody Masona. (Wykorystanie metody grafów do analiy układów prełącanym kondensatorem Keywords: M- graph, summary graph, transformation graph, two graphs. Słowa klucowe: obwody prełącanym kondensatorem, teoria grafów. Introduction he analysis the electric circuits is necessary not only for computing circuit properties but also for understanding their principles. he computer methods are a powerful tool for symbolic analysis circuit parameters. But they can be considered only tools. hanks to its clarity, the graphic method is extremely suitable even for understanding these networks. clearly arranged set transformation graphs derived for different types switching circuits can be used for analying capacitor switched networks and, course, for understanding them, too. he M- signal flow graphs are used to design and analye continuous time circuits as well as periodically switched linear circuits. ransformation graphs [] and two-graphs [, ] are commonly used for assembling the final matrix considering all phases solving circuits. he matrix is calculated by algebraic minors. It means that this method is a combination both graph and numerical methods. However, only selected circuits can be solved by graphs as described below, and also the circuits which contain switched capacitors. he summa graph can be constructed by transformation graphs (-graphs or by two-graphs as the resulting graph and is evaluated by Mason s formula. Solving ircuits onsidering Operational mplifier with Diffraction its Frequency haracteristics mplification by ransformation Graphs his calculation will be illustrated by an example solving a circuit with a switched capacitor whose schematic wiring diagram is shown in Fig... E O.. 5 Fig.. ircuit diagram for the solving he phases are indexed as E for even and O for odd, because the nodes are numbered. he circuit in Fig. has five nodes; therefore the starting graph the circuit in Fig. E 4. O has five nodes, too. he operational amplifier is connected to the third node by its inverting input and into the fifth node by its output. onsequently, the branch with the charge transfer the transformation graph goes from node, the branch with the voltage transfer the transformation graph enters node 5 and the branch with the voltage transfer expressing the final amplification the operational amplifier by the value / enters node [, 4]. Following this transformation graph, the capacity connected between nodes and 5 then transforms into the resulting capacity the amount (/ -. he capacitor is now connected to node by one its ends, therefore the inherent loop at this node has transfer and is ~ Q transformed according to the equation a.. a., to the inherent loop (/, where a v = /. he branch between nodes and 5 with transfer is transformed to the inherent loop with transfer -, because ~ Q in the relation a.. a. [5, 6] is now, as the branch the original graph converts to the inherent loop in the resulting transformed graph. In the odd phase OO, by closing the switch, nodes and will be connected, which will be demonstrated in the graph by their transformation uniting into a single node O.=O. t the same time, this resulting node is the input node the operational amplifier. herefore the branch with charge transfer a Q the operational amplifier s transformation graph issues from this node. In the remaining phases EO and OE, we start, according ~ Q to the equation a.. a., along the branch with voltage transfer a from the resulting node to the original node. We enter back to the resulting node along the branch Q with charge transfer a. he transformation graphs for all the four phases are in Fig.. he summary graph obtained from the partial transformed graphs from Fig. by the above mentioned procedure is then shown in Fig.. First the results the transformed graphs for EE and OO phases are plotted (in case this example only as three nodes: E.=E., E.=5E., O.=. with the transfers,,. PRZEGLĄD ELEKROEHNIZNY (Electrical Review, ISSN , R. 88 NR 6/0

2 EE OO O.=O. 4O.=. E.=E. E.=5E. O.=. EO OE Fig. ransformation graphs for EE, OO, EO and OE phases Fig. he summary graph the S circuit from Fig.. In the next step, the results the transformed graph for the EO and OE phases multiplied by or are then drawn between these nodes as branches, i.e. the branch with the transfer. between the nodes E.=E. and O.=. and the branches with the transfers. E.=E. E.=5E. O.=.. (. between the nodes E.=5E. and O.=. By evaluating this summary graph which is done by substitution into the Mason s formula p( i. ( K ( K S.. ( [5, 7] we get the following final results this way: From the graph it is obvious that the entry node is E or the first node in the even phase, therefore there will only be transfers from the even phase the first node. It is further evident from the graph that the exit (i.e. fourth node exists here both in the even phase as 5E (5E.=E. and in the odd phase as (.=O.. It is thus possible to express in numbers the following transfer 4 O by Mason s formula, for which it holds that: E p ( ( K ( K S E where is the voltage and or IJ ( K is the transfer the loop, by cancelling out and removing the complex fractions. PRZEGLĄD ELEKROEHNIZNY (Electrical Review, ISSN , R. 88 NR 6/0

3 By substituting 0. or. we s s s can calculate the frequency dependence. Solving ircuits onsidering Operational mplifier with Diffraction its Frequency haracteristics mplification by wo-graphs solution a circuit by the described method will be shown by solving a particular circuit with two switched capacitors, whose wiring diagram is in Fig.. First we draw a partial diagram for the even phase and for the odd phase separately; these diagrams are shown in Fig.4. he node numbers in the squares are the numbers the nodes the charge graph and the node numbers in triangles are the numbers the nodes voltage graph after renumbering the nodes. For orientation there are the original numbers nodes from the diagram in Fig...=..=4..=. 4.= Fig.4. Diagrams circuits for even and odd phases For both even and odd phases it is necessary to draw a special voltage (-graph and charge (Q-graph graphs. hese graphs are in Fig.5 and include capacitors only. summary M-graph is now constructed by first finding the EO : incomplete common skeletons the -graph and the Q- graph in the EE, OO, EO and OE phases by formula (. (E, O (E, O - (E, O (E, O.=..=4..=..=4..=..=. 4.= Q-gr.: -gr.: Q-gr.: -gr.: EE : OO : (E, E OE : (O, O (E, E (O,O - (E, E (E, O (E, E (E, O - (O,O - (E, O Fig. wo-graphs for even and odd phases s described in [8], [9] determinant the matrix Y i.e. the matrix in this case switched capacitors circuits is given by ( ( ( product capacitors where set trees spanning set graph trees spanning Q graph. In other words, there is a term in the expression for corresponding to each spanning tree that is common to the charge (Q-gr. and voltage graphs (-gr.. his principle is used for constructing summary Mgraph by two-graphs. 4 PRZEGLĄD ELEKROEHNIZNY (Electrical Review, ISSN , R. 88 NR 6/0

4 First we draw the nodes in both phases. Because graph in Fig.5 has three nodes (.=.,.=4., in the even phase and two nodes (.=., 4.= in the odd one, the summary M-graph in Fig.6 has three nodes in the even phase and two nodes in the odd one, too. In the second step, between thus obtained nodes E, E, O, O we will consequently draw branches and inherent loops the S, according to the rules stated in []. his step is illustrated in Fig.6. EE: OO: E. E. E. O. O Fig.6 Graph with S after second step Between thus obtained nodes and branches, we will consequently draw branches and inherent loops as the results finding the incomplete common skeletons the -graph and the Q-graph in the event phase and in the odd one. EE: fter completing the summary M-graph we will get the form shown in Fig.7. OO: E. E. E. O. O Fig.7 Resulting summary M-graph he voltage transfer for example E ( will now be obtained from a shortened graph, i.e. a graph in which there will not be the entry node s own loop and branches going into entry node, by means the Mason rule [5, 7]: ( E p ( K ( K S...(. (.(..(.(. (.(...(. (. (.(. ( (... (.(.. (...(.. PRZEGLĄD ELEKROEHNIZNY (Electrical Review, ISSN , R. 88 NR 6/0 5

5 By substituting 0. or. s s s we can calculate the frequency dependence. his formula ( is much more complicated than (. However, the results must be the same. herefore, quite a complicated adjustment is needed. fter this relatively laborious adjustment ( the result will be the same as (. onclusion While in case using the graph method, first a graph was indicated, then a transformation graph was plotted, and from its results a summary graph was drawn and evaluated by the Mason s rule, after which the result was obtained by an easy simplification, in case solving by the matrix calculus the procedure was much more complicated. First a partial capacitance matrix had to be composed, in the next step it was modified by an operational amplifier. From four matrices obtained by this a capacitance matrix was constructed and was reduced by the activity switches; from the reduced matrix three algebraic complements were made up and they had to be expressed by means an expansion because they were a higher grade than. fter an elaborate simplification in four steps, the same result was reached. In case using the graph method a graph was solved, but modified nodal method is rather difficult. While in case using the graph method, first a graph was indicated, then voltage and charge graphs were plotted and from these two-graphs a summary graph was drawn and evaluated by the Mason s rule, after which the result was obtained, in case solving by the matrix calculus the procedure was much more complicated. First a partial capacitance matrix had to be composed, in the next step it was modified by an operational amplifier. From four matrices obtained by this, a capacitance matrix was constructed and was reduced by the activity switches. In literature [],, 8] etc. we can find a description a solution procedure by a method two graphs, which leads to the construction a matrix, from which the desired voltage transfers for corresponding phases are calculated by the method algebraic complements. However, the above described method enables us to carry out the whole solution procedure in the graphic form. With regard to the evaluation the resulting summary M graph it is suitable for manual solution rather simple circuits. Its certain disadvantage is the renumbering node in different ways for the charge graph and for the voltage graph, which may seem complicated. While in case using a transformation graph for plotting the summary graph, the result is gained from Mason s formula without simplification, in case the two graphs evaluation is more complicated, because simplification the result from Mason s formula is necessary. he t-graph is much more advantageous for the assembling the summa graph. his work was supported by the ollege Polytechnics Jihlava, Department Electronics and Informatics, ech Republic. REFERENES [] Dostál,., he nalysis the ctive omponents ontaining Switched apacitors by Nodal oltage Method. Electronics horiont, 45, (984, nr, -6. [] lach, J., Basic Network heory with omputer pplication, an Nostrand Reynhold, New York, 99. [] lach, J., Singhal, K., omputer Methods for ircuit nalysis and Design, nd ed. an Nostrand Reynhold, New York, 0 [4] Martinek, P., Bores, P., Hospodka, J., he Electric Filters. U Publisher Prague, 00 (in ech. [5] Čajka, J., he ircuits heory. Linear ircuits. SNL/LF Prague, 979 (in ech. [6] Biolek, D., Biolkova,., Flow Graphs Suitable for eaching ircuit nalysis, in Proceedings the 4 th WSES International onference on pplications Electrical Engineering E 05, Praque, ech Republic, March -5, 00 [7] Biolek, D., Solving Electronic ircuits, BEN Publisher Praque, 004. [8] Grimbleby, J., B., lgorithm for Finding the ommon Spanning rees wo Graphs. Electronics Letters, 7 (98, nr, [9] Grimbleby, J. B., Symbolic nalysis ircuits ontaining ctive Elements. Electronics Letters, 7 (98, nr. 0, uthor: dr. ing. Bohumil Brtník, ollege Polytechnics Jihlava, Department Electronics and Informatics, olstého 6, Jihlava, brtnik@vspj.c. 6 PRZEGLĄD ELEKROEHNIZNY (Electrical Review, ISSN , R. 88 NR 6/0