Fundamental loop-current method using virtual voltage sources technique for special cases

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1 Fundamental loop-current method usng vrtual voltage sources technque for specal cases George E. Chatzaraks, 1 Marna D. Tortorel 1 and Anastasos D. Tzolas 1 Electrcal and Electroncs Engneerng Departments, School of Pedagogcal and Technologcal Educaton (ASPETE), Athens, Greece Computng Department, Arstotelan Unversty of Thessalonk, Greece E-mal: Abstract A new technque based on the use of vrtual voltage sources makes any electrc crcut solvable by the fundamental loop-current method n an easy and formulated way for students. Thus, the fundamental loop-current method s systematsed (especally for nonplanar crcuts) and the dffcultes presented up to now for specal crcut categores cease to exst. Keywords source fundamental loop; lnk; non-convertble current source; specal cases; tree; vrtual voltage Solvng an electrc crcut n a systematc way demands topologcal concepts lke those referred to n many textbooks on electrc crcuts. 1,,3 The most mportant concepts are the graph, the tree and the fundamental loop. To each electrc crcut corresponds a graph, whch s the geometrc fgure resultng from the crcut, when each branch of a non-actve crcut element s substtuted by a lne part or a curved part; the crcut voltage sources are short-crcuted and the current sources are open-crcuted. If a graph can be drawn on a plane n such a way that any two of ts branches are not to be crossed at any pont that s not a crcut node, then ths graph s named a planar graph and the crcut, a planar crcut (Fg. 1(a, b)). Otherwse, they are called a nonplanar graph and a nonplanar crcut, respectvely (Fg. 1(c)). A graph for whch the transton from a node to another node s made by followng contnuous branches s called a connected graph (Fgure (a)). In the opposte case, t s called a non-connected graph (Fgure (b)). Gven a connected graph for a crcut, a tree (T) s a connected part of the graph whch contans all the nodes and smultaneously does not contan any loops. The branches of a tree are named tree branches and the rest of the branches (consttutng the graph) are called lnks (LT) or chords. A basc property resultng drectly from the defnton of the tree s that f a graph has n nodes, then the tree has n nodes as well, but t has n - 1 branches. Fgure 3 shows two trees T 1 and T of a connected graph and ther lnks LT 1 and LT, respectvely. Obvously, the possble trees of a connected graph result from all the combnatons of the graph branches that satsfy the tree propertes. For a gven tree n a connected graph, a fundamental loop (FL) s each loop whch

2 Vrtual voltage sources technque 189 Fg. 1 (a), (b) Planar graphs; (c) non-planar graph. Fg. (a) Connected graph; (b) non-connected graphs. Fg. 3 Trees of a connected graph.

3 190 G. E. Chatzaraks, M. D. Tortorel and A. D. Tzolas ncludes only one lnk, whle all the other branches of ths tree are tree branches of t. Fgure 4 shows the fundamental loops of a tree selected from a connected graph. The analyss of d.c. crcuts by the fundamental loop-current method s based on the above-mentoned topologcal concepts and apples n general to connected planar or nonplanar crcuts, n whch most of ther sources are voltage sources. 1,,5 Ths method s based on two theorems: The number m of the lnks (and therefore of the fundamental loops) of a tree of a connected graph havng b branches and n nodes s m = b - n + 1. The equatons resultng from applyng Krchhoff s voltage law to each fundamental loop of a tree of a connected graph are ndependent of each other. Also, ths method s relatvely easy and systematc for crcuts that contan only ndependent voltage sources. The dffculty of ths method from a systematc standardsaton and a pedagogcal effectveness pont of vew starts when the crcut also contans dependent sources. It s most dffcult when there are current sources (ndependent, dependent) whch are not transformable nto voltage sources (ndependent and dependent, respectvely), because there are no resstances (at least n an obvous way) parallel to them. The problem of the non-convertble current sources s usually tackled by the use of supermeshes and only for planar crcuts. For nonplanar crcuts, the problem s not confronted at all by usng loops; the only method that can be used s the nodevoltage method (where supernodes are used when there are non-convertble voltage Fg. 4 Fundamental loops for a selected tree of a connected graph.

4 Vrtual voltage sources technque 191 sources).,4,5 However, the use of supermeshes or supernodes s somethng that students are not able easly to understand and apply; more specfcally, the generalsaton and standardsaton nvolved n dealng wth specal cases n electrc crcuts s not easy for them. Ths paper, beyond the effort of facng the fundamental loop-current method n a systematc way (especally for nonplanar crcuts) also faces the problem of the nonconvertble current sources by ntroducng the concept of vrtual voltage sources. The term vrtual voltage source means that a non-convertble current source (ndependent or dependent) s substtuted by a voltage source (ndependent or dependent respectvely), whch has a value equal to the voltage at ts termnals and whch s obvously unknown. Fundamental loop-current method Facng connected planar and nonplanar crcuts n a systematc and standard way usng the fundamental loop-current method depends on the knd of sources that exst n the crcut and also on whether the exstng current sources are convertble or non-convertble. Based on all these, the fundamental loop-current method can be examned for four dfferent cases. Case A. Connected planar or nonplanar crcut wth ndependent (voltage or current) sources but wth all possbly exstng current sources convertble In such a case, the current sources are ntally transformed nto voltage sources and then n the resultng equvalent crcut the followng steps are executed: Step 1. From the graph of the crcut, a tree s selected whch, as s known, has m lnks. Arrangng each tme a lnk, a fundamental loop results. Thus, m fundamental loops are found that correspond to the specfc tree whch s selected. Step. In all the m fundamental loops, the fundamental loop currents 1,, 3,..., m are defned, havng any drecton. Step 3. The fundamental loop equatons are wrtten n matrx form as follows: È R11 R1 R13... R1 m È 1 È Sv1 R1 R R3... R m Sv R31 R3 R33... R3 m 3 Sv = R m1 Rm Rm3... Rmm m Sv m where: R, " = 1,, 3,..., m denotes the self-resstance of the (FL) and s equal to the sum of all resstances of ths fundamental loop.

5 19 G. E. Chatzaraks, M. D. Tortorel and A. D. Tzolas R j = R j, " π j,, j = 1,, 3,..., m denotes the mutual resstance of the (FL) and (FL) j and s equal to the sum of all resstances n the common branches of these fundamental loops. Its sgn s (+) f the loop current drectons on the common branches are the same, otherwse s (-). Sv, " = 1,, 3,..., m s the algebrac sum of the values of all voltage sources of the (FL). The values of those sources whose loop current goes from the negatve to the postve pole are taken to be postve; otherwse, they are taken to be negatve. Step 4. The resultng m m lnear system s solved usng the Cramer method or by the matrx nverson method and the currents 1,, 3,..., m are thus known. Step 5. The currents of all branches are calculated combnng the fundamental loop currents and as a consequence the voltages of all crcut elements are known. In other words, the soluton of the electrc crcut s completed. Notes The resstance matrx s symmetrcal snce R j = R j, " π j. There s no partcular reason to take the fundamental loop currents n the same drecton, snce the sgn of the R j, " π j s not known from the begnnng as n the mesh current method, but attenton must be pad to t. Example For the crcut of Fg. 5, usng the fundamental loop method show that the power developed s equal to the power dsspated. Soluton. Selectng a tree from the nonplanar graph of the crcut, the fundamental loops are found correspondng to t and ther currents are defned (Fg. 6). The fundamental loop equatons n matrx form are: È R11 R1 R13 R14 R15 È1 ÈSv1 R R R R R Sv R31 R3 R33 R34 R35 3 = Sv3 R41 R4 R43 R44 R45 4 Sv4 R51 R5 R53 R54 R55 5 S v5 È È1 È f = 0 f =-831. A = A = A = A = A

6 Vrtual voltage sources technque 193 Fg. 5 Crcut for case A. Hence, p( 16V) = 16 4 = W ( delvered) p( 10V) = 10 1 = W ( delvered) p( 15V) = 15 5 = W ( delvered) p( 1V) = 1 = W ( delvered) 4 presist. = 5( ) + 4( ) + 3( ) + ( + - ) = W Thus, pdeliv. = p( 16V) + p( 10V) + p( 15V) + p( 1V) = W pdeliv. p pdissip. = presist. = W f = DISSIP. Case B. Connected planar or nonplanar crcut wth ndependent (voltage or current) sources but, wth at least one current source not transformable to voltage source or for whch the transformaton s dffcult (specal case) In such a case, the followng steps are executed:

7 194 G. E. Chatzaraks, M. D. Tortorel and A. D. Tzolas Fg. 6 Currents of fundamental loops for crcut shown n Fg. 5.

8 Vrtual voltage sources technque 195 Step 1. In the locatons of the current sources that are non-convertble, vrtual voltage sources are consdered wth values equal to the correspondng voltage values present at the termnals of the non-convertble current sources of the gven crcut. Step. From the vrtual graph of the crcut, a tree s selected whch has, as t s well known m lnks. Placng each tme one lnk, a fundamental loop results. Thus, m fundamental loops are found correspondng to the specfc selected tree. Step 3. In all the m fundamental loops, the fundamental loop currents 1,, 3,..., m are defned havng any drecton. Step 4. The fundamental loop equatons are wrtten n matrx form as n case A. However, n ths case the vrtual voltage sources are smlarly taken as the exstng (non-vrtual) ones and they are ncluded n the terms Sv. Step 5. For each vrtual voltage source, an equaton s ntroduced n the matrx that descrbes the correspondng non-convertble current source wth a lnear combnaton of the unknown fundamental loop currents of the problem, elmnatng each tme an equaton that contans a vrtual voltage. The remanng equatons needed for the soluton are taken from the ntal form of the matrx, as they are (those that do not contan other unknown currents than the fundamental loop currents) or as they result after the approprate addtons or subtractons n order to elmnate the vrtual voltages appearng ntally. By dong so, the new matrx form of the equatons does not any longer represent Ohm s Law, but t s smply an algebracally m m equvalent system, whch can lead to the fndng of the fundamental loop currents. Step 6. The resultng m m lnear system s solved as n the prevous case and so the fundamental loop currents are readly avalable. Step 7. The currents of all branches are calculated by combnng the fundamental loop currents, and as a consequence, the voltages of all crcut elements are known, except those at the termnals of the non-convertble current sources (that s the vrtual voltages). The calculaton of these voltages s done usng the equatons elmnated from the ntal form of the system, snce the fundamental loop currents are already known. In other words, the soluton of the electrc crcut s completed. Example For the crcut of Fg. 7, usng the fundamental loop-current method show that the power developed s equal to the power dsspated. Soluton. Snce the ndependent current sources of the A and 3 A are nonconvertble (there are no resstances parallel to them), t s consdered that, nstead

9 of them, vrtual voltage sources exst at ther locaton wth values v x and v y equal to the voltages at ther termnals respectvely. By selectng a tree from the vrtual nonplanar graph of the crcut, the fundamental loops are found correspondng to ths tree and ther currents are defned (Fg. 8). The fundamental loop equatons n matrx form are: (1) Substtutng the frst lne of relatonshp (1) by the equaton 3 =- A whch apples to the current source of A, the second lne by the equaton = 3 A whch apples to the current source of 3 A, the thrd lne by the frst and the fourth lne as s, the followng algebracally equvalent system results: È È = È f =- = =- =-.. A A A A R R R R R R R R R R R R R R R R v v v v È È = È f È S S S S È = È v v y x G. E. Chatzaraks, M. D. Tortorel and A. D. Tzolas Fg. 7 Crcut for case B.

10 Vrtual voltage sources technque 197 Fg. 8 Currents of fundamental loops for crcut shown n Fg. 7. In order to fnd the voltages at the non-convertble current source termnals, the followng are consdered: From the thrd lne of relatonshp (1) results: 14 - vx = f vx = V

11 198 G. E. Chatzaraks, M. D. Tortorel and A. D. Tzolas From the second lne of relatonshp (1) results: vy + 14 = f vy = V Hence, p( 3 A) =-3 v y = W ( delvered) p( A) =- v x = W ( delvered) p( 15V) = 15 ( 1 + 4) = W ( delvered) p( 10V) =-10 4 = W ( dsspated) p( 14V) =-14 ( )= W ( delvered) presist = 5( 1 + 4) + 4( ) + 7( ) = W Thus, 1 4 pdeliv. = p( 3A) + p( A) + p( 15V) + p( 14V) = W pdeliv. p pdissip. = presist. + p V = W f = ( 10 ) DISSIP. Case C. Connected planar or nonplanar crcut wth ndependent and dependent (voltage or current) sources, but wth all current sources (ndependent and dependent) that possbly exst n the crcut convertble In ths case, the current sources (ndependent and dependent) are ntally transformed nto voltage sources (ndependent and dependent respectvely), and then n the resultng equvalent crcut the followng steps are executed: Step 1. From the graph of the crcut, a tree s selected whch has, as s well known, m lnks. Placng each tme one lnk, a fundamental loop results. Thus, m fundamental loops are found correspondng to the selected tree. Step. In all the m fundamental loops, the fundamental loop currents 1,, 3,..., m are defned havng any drecton. Step 3. The fundamental loop equatons are wrtten n matrx form as n case A. Step 4. The dependent quanttes appearng n the matrx form are expressed wth respect to the unknown fundamental loop currents. However, ths mples that the unknown fundamental loop currents appear n the second part of the matrx form of the equatons as well. Step 5. The elements of the equaton lnes are rearranged (when needed) so that the unknown fundamental loop currents appear only n the left-hand part of the equatons. Step 6. The resultng m m lnear system s solved as n the prevous cases and so the fundamental loop currents are readly avalable.

12 Vrtual voltage sources technque 199 Step 7. The currents of all branches are calculated by combnng the fundamental loop currents and as a consequence the voltages of all crcut elements are known. In other words, the soluton of the electrc crcut s completed. Notes A dependent current source s consdered convertble when there s a resstance parallel to t and smultaneously the dependent quantty of ths or other dependent source s not located at ths parallel resstance. If somethng lke ths were to happen, the source transformaton would result n the elmnaton of the dependent quantty and therefore further steps for the problem soluton would be dffcult or even mpossble. An ndependent current source s consdered convertble when there s a resstance parallel to t and smultaneously the dependent quantty of a dependent current or voltage source does not appear at ths resstance or at the source (for the same reason as prevously referred). Example For the crcut of Fg. 9, usng the fundamental loop-current method show that the power developed s equal to the power dsspated. Fg. 9 Crcut for case C.

13 00 G. E. Chatzaraks, M. D. Tortorel and A. D. Tzolas Soluton. By selectng a tree from the nonplanar graph of the crcut, the fundamental loops correspondng to t and ther currents are defned (Fg. 10). The fundamental loop equatons n matrx form are: È R11 R1 R13 R14 R15 È1 ÈSv1 È È1 R1 R R3 R4 R 5 Sv R31 R3 R33 R34 R35 3 = Sv3 f R41 R4 R43 R44 R45 4 Sv R51 R5 R53 R54 R55 5 S v È -15 È = 0 = 0-0 -x vx -15( ) È È1 È = A = A f = 0 f 3 = A = A = A Hence, p( 0V) = 0 ( )= W ( delvered) p( 15V) = 15 1 = W ( delvered) p( x ) = x 4 = 3 4 = W ( delvered) p( 3v 3vx x ) = 5 = 15( ) 5 = W ( delvered) presist. = ( ) + 4( ) + 5( ) = W Thus, pdeliv. = p( 0V) + p( 15V) + p( p x) + ( 3v = W x) pdeliv. p pdissip. = presist. = W f = DISSIP. Case D. Connected planar or nonplanar crcut wth ndependent and dependent (voltage or current) sources but, wth at least one current source (ndependent or dependent) not transformable to voltage source (ndependent or dependent respectvely) or for whch the transformaton s dffcult (specal case) In such a case, the followng steps are executed:

14 Vrtual voltage sources technque 01 Fg. 10 Currents of fundamental loops for crcut shown n Fg. 9.

15 0 G. E. Chatzaraks, M. D. Tortorel and A. D. Tzolas Step 1. In the locatons of the current sources that are non-convertble, vrtual voltage sources are consdered wth values equal to the correspondng voltage values present at the termnals of the non-convertble current sources of the gven crcut. Step. From the vrtual graph of the crcut, a tree s selected whch has, as s well know m lnks. Placng each tme one lnk, a fundamental loop results. Thus, m fundamental loops are found correspondng to the selected tree. Step 3. In all the m fundamental loops, the fundamental loop currents 1,, 3,..., m are defned havng any drecton. Step 4. The fundamental loop equatons are wrtten n matrx form as n case A. However, n ths case the vrtual voltage sources are smlarly taken as the exstng (non-vrtual) ones and they are ncluded n the terms Sv. Step 5. For each vrtual voltage source, an equaton s ntroduced n the matrx that descrbes the correspondng non-convertble current source wth a lnear combnaton of the unknown fundamental loop currents, elmnatng each tme an equaton that contans a vrtual voltage. The remanng equatons needed for the soluton are taken from the ntal form of the matrx, as they are (those that do not contan other unknown currents than the fundamental loop currents) or as they result after the approprate addtons or subtractons n order to elmnate the vrtual voltages appearng ntally. By dong so, the new matrx form of the equatons does not any longer represent Ohm s Law, but t s smply an algebracally m m equvalent system, whch can lead to the fndng of the fundamental loop currents. Step 6. The dependent quanttes appearng n the matrx form are expressed wth respect to the unknown fundamental loop currents. However, ths mples that the unknown fundamental loop currents appear n the second part of the matrx form of the equatons as well. Step 7. The elements of the equaton lnes are rearranged so as the unknown fundamental loop currents appear only n the left-hand part of the equatons. Step 8. The resultng m m lnear system s solved as n case A and so the fundamental loop currents are readly avalable. Step 9. The currents of all branches are calculated combnng the fundamental loop currents and as a consequence the voltages of all crcut elements are known, except those at the termnals of the non-convertble current sources (that s the vrtual voltages). The calculaton of these voltages s done usng the equatons that were elmnated from the ntal form of the system, snce the fundamental loop currents are already known.

16 Vrtual voltage sources technque 03 In other words, the soluton of the electrc crcut s completed. Example For the crcut of Fg. 11, usng the fundamental loop-current method show that the power developed s equal to the power dsspated. Soluton. Snce the ndependent current source of 5 A s non-convertble (there s no resstance parallel to t), t s consdered that, nstead of t, a vrtual voltage source exsts at ts locaton wth a value v x equal to the voltage at ts termnals. Also, for the same reason, the dependent current source of v j s non-convertble and therefore a vrtual voltage source s consdered to exst at ts locaton wth a value v y equal to the voltage at ts termnals. By selectng a tree from the nonplanar graph of the crcut, the fundamental loops are found correspondng to ths tree and ther currents are defned (Fg. 1). The fundamental loop equatons n matrx form are: Fg. 11 Crcut for case D.

17 04 G. E. Chatzaraks, M. D. Tortorel and A. D. Tzolas Fg. 1 Currents of fundamental loops for crcut shown n Fg. 11.

18 Vrtual voltage sources technque 05 È R11 R1 R13 R14 R15 È1 ÈSv1 R R R R R Sv R31 R3 R33 R34 R35 3 = Sv3 R41 R4 R43 R44 R45 4 Sv4 R51 R5 R53 R54 R55 5 S v5 È È1 È f = vx vy Substtutng the frst lne of relatonshp () by the equaton 4 = 5 A whch apples to the ndependent current source of 5 A, the second lne by the equaton 5 = v j whch apples to the dependent current source of v j, the thrd lne by the frst, the fourth lne by the second and the ffth lne by the thrd, the followng algebracally equvalent system results: È È1 È 5 È v 1 j = 5 + 3j = 5 + 3( ) È È1 È 5 1 = A = A f = 5 f 3 = A = 5A = A In order to fnd the voltages at the non-convertble current source termnals the followng are consdered: The fourth lne of relatonshp () gves: vx = f vx = V The ffth lne of relatonshp () gves: vy = f vy = V Hence, p( 5 A) =-5 v x = W ( delvered) p( v v v v ) =- j y =-1 y = W ( delvered ) j p( 30V) =-30 ( )= W ( delvered) j ( )

19 06 G. E. Chatzaraks, M. D. Tortorel and A. D. Tzolas p( 0V) = 0 ( )= W ( delvered) p( 15V) = 15 j = 15 ( )= W ( delvered) p( j ) =- j =- ( ) = -. W ( delvered) presist. = 5( ) + ( ) ( ) = W Thus, pdeliv. = p( 5A) + p( + p p p p v V + V + V + = W ) ( 30 ) ( 0 ) ( 15 ) ( ) j 3 j pdissip. = presist. = W f p = p DELIV. 3. DISSIP. Conclusons The classfcaton of connected planar or nonplanar electrc crcuts nto four categores, as have been examned n ths paper, enables the student to solve easly, followng smlar procedures, any crcut by the fundamental loop-current method. However, ths method bascally apples to nonplanar crcuts for whch the meshcurrent method cannot be used. For planar crcuts, the mesh-current method can be appled and ts use s recommended, snce t s an easer method than the fundamental loop-current method. Therefore, all the examples presented n ths artcle deal wth nonplanar crcuts. Wth respect to the specal cases, the second and fourth (B, D) dealt wth vrtual voltage sources; ths results n the non-dfferentaton of these cases regardng the overall methodologcal steps to be followed. Another equally mportant advantage of usng vrtual voltage sources s the mmedate fndng of the voltages present at the termnals of the non-convertble current sources, gven that the fundamental loop currents are known, snce ther voltages are already expressed by the way the equatons are wrtten n matrx form. So, the power developed by these sources s easy to calculate and therefore the proof of the power balance does not present any dffcultes. Specal attenton must be pad to the condtons under whch a current source s transformed to a voltage source; ths s because whereas a current source s convertble when there s a resstance parallel to t, for methodologcal purposes t should be consdered non-convertble, when the condtons mentoned n the fourth case (D) are not met. Fnally, the fundamental loop-current method, as has been analysed already, can obvously be used for crcuts n the snusodal steady state (a.c. crcuts). However, the necessary condton s that all crcut sources are of the same frequency (otherwse the prncple of superposton s used). So, f all sources are of the same frequency, the fundamental loop-current method starts after the crcut transformaton to the frequency doman.

20 Vrtual voltage sources technque 07 References 1 G. E. Chatzaraks, Electrc Crcuts, vol. II (Tzolas, Thessalonk, 000). J. W. Nlsson and A. Susan Redel, Electrc Crcuts (Addson Wesley, New York, 1996). 3 C. A. Desoer and S. Ernest Kuh, Basc Crcut Theory (McGraw Hll, New York, 1969). 4 C. K. Alexander and N. O. Matthew Sadku, Fundamentals of Electrc Crcuts (McGraw Hll, New York, 000). 5 W. H. Hayt and E. Jack Kemmerly, Engneerng Crcut Analyss (McGraw Hll, New York, 1993).

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