Formulating structural matrices of equations of electric circuits containing dependent sources, with the use of the Maple program

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1 1 st World Conerence on echnology and Engineering Education 2010 WIEE Kraków, Poland, Septemer 2010 Formulating structural matrices o equations o electric circuits containing dependent sources, with the use o the Maple program A.M. Dąrowski, S.A. Mitkowski, A. Poręska & E. Kurgan AGH University o Science and echnology Kraków, Poland ABSRAC: he method o deriving loop matrix impedances BZB and the loop voltage vector B(E-ZJ) in electric linear circuits containing dependent sources, is descried in this paper. Primary equations o the electric circuit were ormulated, using the classical method, or the loop current method o network analysis. he transormation o them was derived within the mathematics sotware program, Maple. he method o analysis o the electric circuit presented in this paper was illustrated with an example. IRODUCIO Circuit theory is a undamental engineering discipline that pervades all electrical engineering, thereore learning electric circuit theory in electrical engineering education, is important. he goal o circuit theory is to make quantitative and qualitative calculations o the electrical ehaviour o circuits [3]. For electrical engineers not only is it important to know elementary circuit theory, it is also important to understand the methods or handling more complex circuit theory. Students who specialise in electrical engineering or electronics typically need to study not only DC and AC circuits with techniques such as Kirchho s voltage law, Kirchho s current law and Ohm s law, ut also topology methods with matrix or handling more complex networks. hey are usually required to learn how to apply various transorm methods (phasor, Fourier and Laplace) and Fourier-series, in circuit analysis. Understanding concepts rom circuit theory, especially DC and AC electricity, periodic signals and transients, is essential to understanding sujects such as electronics, telecommunication, power engineering, system theory and automated technology. Studies into the teaching o electricity and circuit theory have shown that students, even at university or college level, have diiculties in acquiring a unctional understanding o, and distinguishing etween, undamental concepts such as current, voltage, energy and power. Students lack o qualitative understanding is the reason why diiculties arise in correctly solving quantitative prolems [1]. Insuicient mathematical knowledge is oten a arrier or students in the understanding o the theory o circuits. he way one thinks aout mathematics in circuit theory diers rom what is taught in the suject o mathematics. his process is not transparent or students. Mathematical computer programs can e helpul as a solution to this prolem [4][5]. EWORK AALYSIS he goal o network analysis is to determine the voltages and currents associated with the elements o an electrical network so as to predict the electrical ehaviour o real physical circuits. he purpose o these predictions is to improve their design; particularly to decrease their cost and improve their perormance under all conditions o operation. Kirchho laws are undamental postulates o circuit theory. hey are valid regardless o the nature o circuit elements. hereore, the separation etween the equations o Kirchho's laws and that o elements characteristics is natural. Kirchho s equations are prescried y the topology o the circuit, that is, the way circuit elements are interconnected. he element characteristic is prescried y the voltage-current (v-i) relations given y the laws o physics. However, these equations involve a large quantity o variales. Using graph theory diminishes the numer o unknowns. he theory o graphs plays a undamental role in exploring the structural properties o electrical circuits [7].

2 he graphs are good pictorial representations o circuits and capture all their structural characteristics. In circuit analysis using graph theory, there is compliance with three transormations: loop transormations, cutest transormations and node transormations. Only the method o analysis o electric circuits ased on loop transormation is applied in this paper. Given solutions can e applied in the manner o methods o analysis ased on extant transormations. LOOP RASFORMAIOS AD LOOP SYSEM OF EQUAIOS An electrical circuit is an interconnection o electrical network elements, such as resistances, capacitances, inductances, independent and dependent voltages and current sources. Each network element is associated with two variales: the voltage variale v(t ) and the current variale i(t ). he electrical network can e represented y corresponding with its directed graph G. Let e a spanning tree o an electrical network. Let I c e the column vectors o chord currents and I e the column vectors o ranch currents with respect to. he sumatrix o circuit matrix B c corresponding to the undamental circuits deined y the chords o a spanning tree is called undamental circuit matrix B o G with respect to the spanning tree. he equation elow presents the loop transormation. I = B I (1) c As can e seen rom the loop transormations, not all these variales are independent. Furthermore, in place o Kirchho s voltage law equations, the loop transormation, which involves only chord currents as variales, can e used. Kirchho s voltage law is given elow: B V 0 (2) = he voltage-current relations or every ranch o the electric network are: V = Z I (3) where Z is the ranch impedance matrix. he advantage o these transormations is that it enales dierent systems o network equations known as loop systems to e estalished. In deriving the loop system, the loop transormation is used, and loop variales (chord currents) serve as independent variales. Partition o the chord currents vector I c and element voltage vector V is: I c I = J c_nsc V_nsv and V = (4) E where I c_nsc is the vector o currents in the non-current source chords o G, J is the current sources vector, V _nsv is the vector o ranches voltage in the non-voltage source and E is the voltage sources vector. Using the aove-dependencies, this equation is derived: B Z B I c _ ns ( E Z J ) = B (5) Equation 5 is called the loop system o equations and matrix B Z B is the loop impedance matrix. Since presenting examples is a undamental element o the didactic process, a computational example is introduced elow that illustrates the prolems aove. he ormulation o structural matrices o the electric circuit, with the help o the Maple sotware program, is given elow. HE MAHEMAICAL PROGRAM, MAPLE here are a ew very good commercial programmer environments or mathematical computations and one o them is Maple. Maple is useul technical computing sotware or engineers, mathematicians, scientists and educators [2][]. aking into consideration a large quantity o literature and, especially, the availaility o ree examples on the Internet, the authors o this paper decided to use this program to solve prolems connected with circuit theory. Maple is a complete mathematical prolem-solving environment that supports a wide variety o mathematical operations, such as numerical analysis, symolic algera and graphics. It allows a choice o more than 4,000 commands ranging rom perorming asic 7

3 arithmetic and algera, to computations involving advanced topics such as tensor analysis, group theory, and more. Maple oers interactive mathematical visualisation, a user interace with typeset mathematics, word processing acilities, and a modern programming language, making it powerul and lexile or users in education, research and industry. Maple is a large and complicated program. But despite that, thanks to the examples, it is relatively easy to use to solve dierent assignments in the ield o circuit theory. Using Maple makes it possile to work in interactive mode. o carry out calculations, develop design sheets, teach undamental concepts or create complicated simulation models, Maple s computation engine can handle every type o mathematics. With Maple, interactive documents can e created. he sotware has two modes: Document mode and Worksheet mode. Document mode is designed or perorming calculations quickly. Worksheet mode is designed or interactive use through commands and programming using the Maple language. he example elow was derived y exploiting the Worksheet mode using ext mode (1-D Math). ILLUSRAIVE EXAMPLE Shown elow in Figure 1 is the variety o symols o the graphic elements o the electric circuit. Figure 1: a) Impedance Z k with current I k and voltage V k, ) Independent voltage source, c) Dependent voltage source, d) Independent current source, e) Dependent current source. ow consider linear circuit impedances, as well as independent and dependent voltage source and current sources. he schema o the electrical network and graph G representation o are shown in Figure 1. Branch circuits correspond to graph edges. Figure 2: a) he electrical network, ) Graph G representation o. Consider also the connected graph and its spanning tree. A su-graph o G is a spanning tree o G i the su-graph is a tree and contains all the vertices o G. he spanning tree o the graph o Figure 2) is shown in Figure 3). Edges o a spanning tree are called the ranches o. he ranches o the tree are marked with doule lines. For the given spanning tree o connected graph G, the co-spanning tree relative to is the su-graph o G induced y the edges that are not present in. he edges o a co-spanning tree are called chords. he co-spanning tree relative to the spanning tree o Figure 3) consists o these chords: c 1, c 2, c, c 8, c 9. he spanning tree was chosen so as to make voltages and currents - which control dependent voltage sources and current - to e in chords. Figure 3: he electrical network and its graph with a spanning tree and directed co-spanning tree. 8

4 Because only current chords are o interest to the method discussed here, only those signals were marked. he chord currents vector Ic is as ollows: I1 I 2 I c_nsc I c = = I () J J8 J9 Wanted unknown chord currents are: c_nsc [ I I I ] I = (7) 1 2 he voltages and currents which control dependent sources are expressed respectively y means o chord currents and independent voltages and current sources. In the case o the electric circuit here, there is: V = Z + (8) 5I5 ZI Because the currents o the ranches o the spanning tree are linear, the (8) equation can e changed to: V = Z ( α + Z I (9) 5 I 2 I 2) Shown elow is how Maple is used to perorm mathematical transormations. he linalg packages (Maple lirary) contain unctions that help users to work with matrix and linear system equations. An equation can e created in Maple y any method which leads to the construction o algeraic (or any other) equations. he restart command causes the Maple kernel to clear internal memory so that it then acts (almost) as i it has just egun. > restart;with(linalg): Equations o the relations o voltage-current or dependent sources. > E[10]:=k*V; E 10 := k V > J[9]:=alpha*Ic[2]; J 9 := α Ic 2 > V:=Z[5]*(Ic[2]-J[9])-Z[]*Ic[]; V := Z 5 Kirchho s voltage law equation around loop with chord c 1 is written as: > KVL1:=Z[1]*Ic[1] = E[1]-k*V; KVL1 := Z 1 Ic 1 = E 1 k ( Z 5 ) Kirchho's voltage law equation around loop with chord c 2 is written as: > KVL2:=(Z[2]+Z[4]+Z[5]+Z[7])*Ic[2]+Z[7]*Ic[]+(Z[4]+Z[7])*J[8]-(Z[5]+Z[7])*J[9] =E2+k*V; KVL2 := ( Z 2 + Z 4 + Z 5 ) Ic 2 + ( Z 4 ) J 8 ( Z 5 ) α Ic 2 = E2 + k ( Z 5 ) Kirchho's voltage law equation around loop with chord c is written as: > KVL3:=Z[7]*Ic[2]+(Z[3]+Z[7]+Z[])*Ic[]-Z[7]*J[8]+(Z[3]+Z[7])*J[9] = -E[3]; KVL3 := Z 7 Ic 2 + ( Z 3 + Z ) Z 7 J 8 + ( Z 3 ) α Ic 2 = E 3 Loop system o equations: > set_eqn:= [KVL1,KVL2,KVL3]: 9

5 Determining loop impedance matrix B Z B and matrix B ( E Z J ) > BZB:=genmatrix(set_eqn,[Ic[1],Ic[2],Ic[]],''); uses the ollowing: Z 1 k Z 5 ( 1 α ) k Z BZB := 0 ( Z 5 ) α + Z 2 + Z 4 + Z 5 k Z 5 ( 1 α ) Z 7 + k Z 0 ( Z 3 ) α Z 3 + Z > B:=eval():BE_ZJ:=matrix(3,1,B);Icns:=matrix(3,1,[Ic[1],Ic[2],Ic[]]); E 1 Ic 1 BE_ZJ := E2 ( Z 4 ) J 8 Icns := Ic 2 E + 3 Z 7 J 8 hen, explore the properties o the loop impedance matrix and ind vector o currents I c_nsc in the non-source chords. Remaining currents and voltages in the electric circuit can e calculated rom equations (1) and (3). COCLUSIOS he authors have shown in this paper how the mathematics sotware program, Maple, can e used eectively to ormulate structural matrices o the loop system o electric circuit equations. Without perorming many intermediate computations, and with the help o Maple, the loop impedance matrix or linear circuits was ound. Similarly, using the Maple program, an elegant system o equations and node equations can e ormulated. he Maple code demonstrated aove is relatively simple. It is easy to draw and animate the solution, too. he symolic, numerical and graphical eatures o mathematical programs allow students to carry out mathematical analysis o circuits without time-consuming calculations and to explore easily the ehaviour o a system. REFERECES 1. Bernhard, J., Activity-ased education in asic electricity and circuit theory, 2. Char, B.W., Maple User Manual. Waterloo, Canada: Maple Inc. (2009). 3. Chua, L., Desoer, C. and Kuh, E., Linear and onlinear Circuits. USA: McGraw-Hill Book Company (1987). 4. Dąrowski, A.M., Mathematical programmes in teaching o circuit theory. Proc. XVI BSE 2002, Gliwice, Poland, (2002). 5. Dąrowski, A.M., Computer programmes in teaching o electrical engineering. Proc. XVII BSE2003, Istena- Zaolzie, Poland, (2003).. Monagan, M., Geddes, K. and Heal, K., Maple Introductory Programming Guide. Waterloo, Canada: Maple Inc. (2009). 7. Swamy, M..S. and hulasiraman, K., Graphs, networks, and algorithms. ew York: Wiley Interscience (1981). 70

QUESTION BANK SUBJECT: NETWORK ANALYSIS (10ES34)

QUESTION BANK SUBJECT: NETWORK ANALYSIS (10ES34) NOTE: FOR NUMERICAL PROBLEMS FOR ALL UNITS EXCEPT UNIT 5 REFER THE E-BOOK ENGINEERING CIRCUIT ANALYSIS, 7 th EDITION HAYT AND KIMMERLY. PAGE NUMBERS OF