2 nd World Conference on Technology and Engneerng Educaton 2 WIETE Lublana Slovena 5-8 September 2 Nodal analyss of fnte square resstve grds and the teachng effectveness of students proects P. Zegarmstrz S.A. Mtkowsk A. Porębska & A.M. Dąbrowsk AH Unversty of Scence and Technology Krakow Poland ABSTRACT: In ths paper the ssue of teachng the node voltage method by a student s proect s presented. In the frst part basc nformaton about the subect of the proect s presented and dscussed. Ths concerns the computng of the resultant resstance between two chosen nodes of a resstve grd. In the next part the authors descrbe assumptons of the proect gven to undergraduate students. Then students results and mprovements to the proect as well as students nvolvement are dscussed. In concluson the authors elaborate on how the students proects mproved the effectveness of the teachng of the method. INTRODUCTION In the years 28 2 students at the AH Unversty of Scence and Technology Department of Electrcal Engneerng specalsng n Computer Engneerng n Electrcal Systems durng the 7 th semester of ther studes partcpated n lectures on the subect Computer-aded Analyss of Electronc Systems. Both lectures and laboratory/proect actvtes of that subect were carred out n the Englsh language. Semester credt was bult usng the marks of fve proects whch were prepared durng laboratory actvtes. The obect of the frst proect was the nodal analyss of a fnte square resstve grd on a plane. Students mplemented an algorthm for computng the resstance between two nodes of a grd based on the node voltage method. In the next step of the proect students had to compare dfferent methods for solvng the resultng set of equatons. The authors could compare the effectveness of teachng the node voltage method wth the same method as taught to students of ths Department earler durng the 2 nd and rd semesters of ther studes n the subect Electrcal Crcut Theory. In ths case students partcpate only n lectures and classes where they solve some problems theoretcally and only learn how to wrte down the set of equatons for a gven crcut. FINITE SQUARE RESISTIE RIDS ON A PLANE Consder fnte square resstve grds on a plane. Resstve grds are electrcal crcuts n whch the resstve elements connect neghbourng ponts of a square lattce. The case wll be consdered where the nodes fll a square area of the plane. It s assumed that each nteror node s connected wth all of ts four neghbours by means of a conductance whch s fnte and postve whle each boundary node s connected only to one nteror node (Fgure ). The assumpton that boundary nodes are not connected to each other s not a lmtaton. If a grd contans connectons at the edges auxlary nodes can be added around the network to connect them wth the network by means of a fxed conductance to obtan a grd wth the shape shown n Fgure. The sze of a grd s defned as the number of rows (or columns). The square grd shown n Fgure s of sze 7. It can be seen that ths grd has 25 nteror nodes and groups of 5 boundary nodes. In general a grd wth sze n has (n-2) 2 nteror nodes and (n-2) boundary nodes whch gves a total number of (n 2 -) nodes [2]. 8
Fgure : Example of fnte square resstve grd [2]. CALCULATION OF RESISTANCE - FORMULATION OF THE PROBLEM Consder the set of m n nodes arranged n n columns and m rows (see Fg. 2.) The potental of the node lyng n the -th row and the -th column s. If () and (kl) are neghbourng nodes the equaton -k -l = s true. Call the resstance connectng nodes () and (kl) R k. The current nected nto the node () s I. Assume that the sum of l all currents for all nodes of the grd s equal to zero whch follows the Krchhoff s Current Law. Consder the problem of calculatng the potental at the nodes for gven values of resstance R k and currents I. l Fgure 2: Fnte square resstve grd of sze m x n []. 85
86 Calculatng the resstance between two chosen ponts of the grd s a specal case of the problem above. Choosng the nodal analyss method to fnd the potentals at the nodes of the grd Krchhoff s Current Law for a node () s: I ) ( ) ( ) ( ) ( = () Ths set of equatons does not have a unque soluton changng the potental of one node by a fxed value does not change the currents n all branches of the crcut. Fxng the potental of a chosen node.e. m n = and deletng the approprate equaton gves a lnear set of equatons wth a unque soluton. In matrx form the set of equatons can be wrtten as: I = (2) where s the conductance matrx resultng from the topology of the grd )...... ( 2 2 n n m = s a vector of potentals at the nodes of the grd and I s a vector of the currents nected at the nodes. The problem of the calculaton of the resultant resstance seen from the nodes () and (kl) s reduced to solvng the set of equatons above for the case where two elements of the vector I are non-zero: I =-I kl =. Resultant resstance s a quotent of a dfference of the potentals - k l and the value of the current. CONDUCTANCE MATRIX CHARACTERISTIC PROPERTIES When constructng algorthms for solvng a gven problem the specfc propertes of the conductance matrx for the resstve grd are very mportant. It s a square matrx wth sze n m- whch means that even for small grds we have a large matrx. Furthermore ths s a band matrx wth the bandwdth 2n. In each row and each column there are no more than fve non zero elements.e. when all resstances n a grd of sze n=m= has a value of the set of equatons s: = 2 2 2 2 2 2 2 22 2 2 () The observatons above mean that for solvng ths specfc problem sparse matrx methods can be used. Furthermore the elements on the man dagonal are larger than or equal to the sum of the magntudes of the rest of the elements n the same row (or column). Therefore the matrx s dagonally domnant for the rows and the columns of the matrx. STUDENTS PROJECTS REALISATION As mentoned n the ntroducton students n the 7 th semester of Electrcal Engneerng at the AH Unversty of Scence and Technology partcpated n lectures on the subect Computer-aded Analyss of Electronc Systems. The lectures were led by Prof. Zbgnew alas and the laboratory/proect actvtes were led by one of the authors Potr Zegarmstrz both employed n the Department of Electrcal and Power Engneerng. In the years 28-2 there were more than 5 partcpants n the lectures. The frst task of the laboratory actvtes was to mplement an algorthm for computng the resstance between two chosen nodes of the grd usng nodal analyss. Frstly the teacher presented a theoretcal ntroducton to the proect. Informaton gven to the students was smlar to that above n ths paper. Then students mplemented ther own algorthm by wrtng down the set of equatons (n partcular the conductance matrx ) descrbng the gven problem.
Choosng the software soluton for that was an ndvdual decson of the student. Most of them have chosen MATLAB software but some decded to mplement ther own applcaton usng a well known programmng language (mostly C or Java). The smplest verson of the algorthm assumed the user only chooses the sze of the grd wth all resstances havng a fxed value of. The mplementaton of the algorthm led to the creaton of an applcaton allowng the user to change resstances whch then creates the conductance matrx for a gven grd (wth both grd sze and element values set by the user). In the next step the user chooses nodes between whch the resultant resstance wll be calculated. In practce t s realsed by connectng the chosen nodes wth current sources of a fxed value. In the applcaton the problem s restrcted by havng the current vector I wth only two non-zero elements (the same value dfferent sgn) n postons correspondng to the chosen nodes. When the conductance matrx and current vector I are created there s enough nformaton to solve the set of equatons and fnd the vector of the potentals at all of the nodes. The method of solvng a lnear set of equatons was freely chosen by the students. Some who had decded to mplement ther own applcaton were forced to create functons whch could then be used to fnd the soluton. Most had chosen methods such as auss-jordan elmnaton or LU factorsaton. The rest who had chosen the MATLAB software tred to compare the effectveness of the MATLAB functons wth ther own algorthms. The measure of the effectveness was the tme requred to compute the soluton. Furthermore all of the students tred to show the relatonshp between the sze of the grd and the tme to compute. Some also tred to prove that there exsts a lmt of resstance between two chosen nodes for an nfnte sze grd wth elements wth resstance value of. Ths was undertaken by the calculaton of the resultant resstance for two nodes wth the grd growng n sze. STUDENTS PROJECTS RESULTS The most nterestng results found n the reports from students' proects are presented below. The authors have chosen one example for each obect of research mentoned n the prevous part of ths paper. It should be emphassed that almost 5 reports from students proects were analysed. Student Janusz Duc (Electrcal Engneerng 26-2) showed the relaton between the number of nodes n the network and the operaton tme for the MATLAB functon (see Reference [5]). Results of the students research are shown n Fgure below. Fgure : Dependence between the number of nodes n the network and the operaton tme - Janusz Duc 29..29 [5]. In the same paper there s ncluded a very nterestng comparson of the MATLAB functon for solvng the set of equatons wth the student s own algorthm based on the auss-jordan elmnaton method. Table presents the results of that research. Table : Comparson of computng tme for the chosen methods - Janusz Duc 29..29 [5]. 87
Student Darusz Kowalk (Electrcal Engneerng 26-2) presented the dependence between the resultant resstance for two chosen nodes wth fxed placement and grd sze and the grd where all resstances has a value equal to. He also showed how the resstance depends on the poston n the grd and that ths dependence decreases wth the grd sze. Ths proves that there exsts a lmt of resstance for nfnte grds. Fgure presents that part of the student s report. Fgure : Lmt of the resstance for nfnte grd - Darusz Kowalk 8..29 [6]. As a comment on Fgure the student wrote: The chart represents values of resstance measured at ponts: n the centre of the grd away from the centre n the dagonal drecton by nodes and by 8 nodes. The value of the resstance s ncreased away from the grd s centre. In small grds the resstance strongly depends on the dstance from the centre. The greater the grd s sze the smaller the resstance dependence on dstance [6]. The same student presented a very nterestng comparson of the mathematcal methods mplemented by hmself. As mentoned earler as the measure of effectveness was the tme requred to compute the soluton. It s clear that the tme requred to compute depended on the machne used. Therefore the exact tme of computaton tself does not matter but the relaton between results for dfferent methods on a gven machne s mportant. The student tested auss-jordan elmnaton LU factorsaton wth partal pvotng matrx left dvson usng an nverted matrx and a sparse matrx. A sparse matrx s a partcular way of keepng a matrx n memory. Unlke the common way where all elements of a matrx are kept n memory wth a sparse matrx only coordnates and values of non-zero elements of matrx are kept n memory whch s more memory effcent [6]. Table 2: Results of smulaton for x resstve grd comparson of methods - Darusz Kowalk 8..29 [6]. In concluson the student noted: As t can be seen above the auss-jordan elmnaton s a hundred tmes slower and naccurate by comparson wth other methods. Therefore takng nto consderaton the speed of computaton and memory effcency the sparse matrx left dvson wll be used [6]. Student rzegorz ancarczyk (Electrcal Engneerng 25-2) decded to check the correctness of calculatons by comparng hs results from MATLAB wth a smulaton of the crcut usng PSPICE software. He bult a small grd (x) n the PSPICE envronment and compared the resultng potentals of the nodes. Results of ths experment are shown on Table. Table : Comparson of results MATLAB and PSPICE - rzegorz ancarczyk 7..28 []. 88
CONCLUSIONS Nodal analyss of electrcal crcuts s not very easy for students to learn. Academc teachers experence shows that students choose that method only when forced to when solvng problems n Electrcal Crcut Theory. Probably they try to avod methods whch they do not understand. A much more popular method s to use the superposton theorem the mesh current method or smply to wrte down the set of equatons resultng from Ohm s and Krchhoff s Laws. Students say those methods are more ntutve. In addton t s much easer to check the correctness of the wrtten equatons. It s not so easy to realse that when checkng the correctness of node voltage equatons t s necessary to sum the currents n the node and then multply the potental dfferences and conductance. The observatons of the authors are that a student s proect wth an nterestng problem to solve s more effectve n teachng the method than a theoretcal soluton consstng of many lttle tasks for smple electrcal crcuts solved on paper or a board. When students worked on the mplementaton of the algorthm and wrote down the conductance matrx for a gven problem they easly understood how the method worked and how t s mplfed dervng the soluton. Another concluson s that ths type of proect can change a student s decson when choosng the method to solve the problem. Up untl then they tred to solve all crcut theory tasks begnnng from the mesh current method. Now they check to see f the node voltage method gves an easer soluton. It s also very mportant that f the proect problem s nterestngly presented the teacher observes hgh student nvolvement; that students propose ther own mprovements to the algorthm and try to fnd some nterestng results and conclusons from ther computatons. REFERENCES. Zegarmstrz P. and alas Z. Comparson of methods for the computaton of resstance for resstve grds. Proc. XXIX IC-SPETO 26 Conf. 2 2-26 (26). 2. Zegarmstrz P. and alas Z. Study of the algorthm for reconstructon of conductances n square resstve grds. Proc. Inter. Conf. on Sgnals and Electronc Systems (ICSES'6) 9-96 (26).. Curts E.B. and Morrow J.A. Determnng the resstors n a network. SIAM J. Appled Math. 5 98-9 (99).. ancarczyk. Analyss and Smulaton of a Square Resstant Lattce - Appontment of Substtute Resstance Usng Nodal Analyss. Computer-Aded Analyss of Electronc Systems Student s Proect Report (28). 5. Duc J. Computng Resultant Resstance for Resstve rds. Computer-Aded Analyss of Electronc Systems Student s Proect Report (29). 6. Kowalk D. Resstve rd. Computer-Aded Analyss of Electronc Systems Student s Proect Report (29). 89