D EPARTMENT /S EMESTER (ECE III SEM) NETWORK THEORY SECTION-D Manav Rachna University 1 Graph Theory in Circuit Analysis Suppose we wish to find the node voltages of the circuit below. We know how to do this by hand. For large-scale circuits, we may wish to do this via a computer simulation (i.e. PSpice). We will need to express this circuit in a standard form for input to the program. Manav Rachna University 2 Graph Theory in Circuit Analysis Whether the circuit is input via a GUI or as a text file, at some level the circuit will be represented as a graph, with elements as edges and nodes as nodes. For example, when entering a circuit into PSpice via a text file, we number each node, and specify each element (edge) in the circuit with its value and endpoints. Manav Rachna University 3 Graph of a Circuit Here is a graph of the circuit. It is simply the circuit without elements. We refer to the lines above as edges (and the nodes are nodes). The graph provides connectivity information. To actually solve the circuit using this graph, the types of elements forming the edges would need to be provided. a b c d e f g h 4 Trees and Co-Trees A tree is defined as any set of edges in a graph that touches every node without forming any closed paths. Also known as Hamiltonian path! Each tree has a co-tree, which is the set of edges not in the tree. a b c d e f g h Manav Rachna University 5 Cut Set A cut set is a minimal set of edges that, when broken, breaks the graph into two completely separate parts (two groups of nodes. Minimal means that a cut set cannot contain another smaller cut set that would break the graph into the same two parts. a b c d e f g h Manav Rachna University 6 Fundamental Cut Set Suppose I am given a tree. A fundamental cut set w.r.t. that tree is a cut set that only contains one branch of the tree. There may be many fundamental cut sets w.r.t. a given tree. a b c d e f g h 7 Finding Fundamental Cut Sets Systematically 1. Redraw the graph with the tree in a straight line. 2. For each tree edge, form its fundamental cut set as follows: 2a) that tree edge is a member of this fundamental cut set 2b) cut that edgewhat two groups of nodes are separated? 2c) the fundamental cut set also contains all edges in the co-tree that connect these two groups d a b Manav Rachna University 8 H OW IS THIS USED IN CIRCUIT ANALYSIS ? Identifying a tree for a circuit, and all of the fundamental cut sets that go with it, can be used in nodal analysis. Here are the steps simulation software may take to perform nodal analysis: 1.From user input, make a connectivity matrix (graph) and record the circuit element on each edge. 2.Choose a tree using the following guidelines: a)Place an edge in the tree if it contains a voltage source, or if the voltage over the edge controls a dependent source. b)Place an edge in the co-tree if it contains a current source, or if the current in the edge controls a dependent source. 3.Find all of the fundamental cut sets for this tree. n nodes yields n-1 fundamental cut sets Manav Rachna University 9 H OW IS THIS USED IN CIRCUIT ANALYSIS ? 4.Each fundamental cut set breaks the circuit into two pieces: two supernodes. Write a KCL equation for one supernode in each fundamental cut set (in terms of node voltages). The KCL equations for the two supernodes formed by a fundamental cut set will be the same. This is where the circuit element info comes into play. This yields n-1 equations in n node voltage variables. 5.Set one node voltage to zero volts (ground) and solve. Manav Rachna University 10 B ASIC C ONCEPTS OF THE G RAPH T HEORY Definition: In a connected graph G of v vertices the subgraph T that satisfies the following properties is called a tree. TT is connected TT contains all the vertices of G TT contains no circuit, TT contains exactly v-1 number of edges. In every connected graph G there exists at least one tree. Manav Rachna University 11 B ASIC C ONCEPTS OF THE G RAPH T HEORY Let G have p separated parts G 1, G 2,..., G p, that is G=G 1 G 2 ... G p, and let T i be a tree in G i, i=1,2,...,p, then,T=T 1 T 2... T p is called a forest of G. DEFINITION: The complement of a tree is called a co-tree and the complement of a forest is called a co-forest. The edges of a tree or a forest are called branches and the edges of a co-tree or co- forest are called chords (links). Manav Rachna University 12 B ASIC C ONCEPTS OF THE G RAPH T HEORY 9 possible trees and corresponding co-trees: v1v1 v2v2 v3v3 v4v4 v5v5 e1e1 e2e2 e3e3 e4e4 e5e5 e6e6 Manav Rachna University 13 B ASIC C ONCEPTS OF THE G RAPH T HEORY DEFINITION: Let G be a graph and let r and be respectively the number of branches and chords of G, then r and are called respectively the rank and the nullity of the graph. THEOREM: Let G have v vertices, e edges and p connected parts, then its rank and nullity are given respectively by r=v - p and =e v + p Manav Rachna University 14 B ASIC C ONCEPTS OF THE G RAPH T HEORY DEFINITION: Let G be a connected graph and let T and T be tree and co-tree respectively, that is G=T T. Let a chord e T and its unique tree path (a path which is formed by the branches of T) define a circuit. This circuit is called the fundamental circuit (f-circuit) of G. All such circuits defined by all the chords of T are called the fundamental circuits (f-circuits) of G. If G is not connected, then the f-circuits are defined with respect to a forest. Manav Rachna University 15 B ASIC C ONCEPTS OF THE G RAPH T HEORY Note that the number of f-circuits is given by the nullity of G and that, with respect to a chosen tree T of G, each f-circuit contains one and only one chord. Manav Rachna University 16 B ASIC C ONCEPTS OF THE G RAPH T HEORY v1v1 v2v2 v3v3 v4v4 v5v5 e1e1 e2e2 e3e3 e4e4 e5e5 e6e6 Consider the following graph e7e7 e8e8 f-circuits: c f1 ={e 3,e 1,e 2 } c f2 ={e 6,e 8,e 4,e 5 } c f3 ={e 7,e 8,e 4,e 5 } Nullity of G =e-v+p=8-6+1=3 v6v6 Manav Rachna University 17 B ASIC C ONCEPTS OF THE G RAPH T HEORY DEFINITION: The cut-set of a graph G is the subgraph G x of G consisting of the set of edges satisfying the following properties: The removal of G x from G reduces the rank of G exactly by one. No proper subgraph of G x has this propery. If G is connected then the first property in the above definition can be replaced by the following phrase. The removal of G x from G separates the given connected graph G into exactly two connected subgraphs. Manav Rachna University 18 B ASIC C ONCEPTS OF THE G RAPH T HEORY Consider the following graph and the following set of edges G 1 ={e 1,e 2 } G 2 ={e 4,e 6,e 7 } G 3 ={e 2,e 3,e 4,e 8 } G 4 ={e 2,e 3,e 6 } v1v1 v2v2 v3v3 v4v4 v5v5 e1e1 e2e2 e3e3 e4e4 e5e5 e6e6 e7e7 e8e8 v6v6 Cut-set is also a cut-set is not a cut-set, because a subset of G 4 is cut-set is not a cut-set, because the removal of G 3 from G results in three connected subgraphs Manav Rachna University 19 B ASIC C ONCEPTS OF THE G RAPH T HEORY DEFINITION: Let G be a connected graph and let T be its tree. The branch e t T defines a unique cut-set (a cut-set which is formed by e t and the chords of G). This cut-set is called the fundamental cut-set (f-cutset) of G. All such cut-sets defined by all the branches of T are called the fundamental cut-sets (f-cutsets) of G. If G is not connected then the f-cutsets are defined with respect to a forest. Manav Rachna University 20 B ASIC C ONCEPTS OF THE G RAPH T HEORY Note that the number of fundamental cut-sets is given by the rank of G and with respect to a chosen tree T of G, each fundamental cut-set contains one and only one branch. Manav Rachna University 21 B ASIC C ONCEPTS OF THE G RAPH T HEORY Consider the following graph v1v1 v2v2 v3v3 v4v4 v5v5 e1e1 e2e2 e3e3 e4e4 e5e5 e6e6 e7e7 e8e8 v6v6 f-cutsets: x f1 ={e 1,e 3 } x f2 ={e 2,e 3 } x f3 ={e 4,e 6,e 7 } x f4 ={e 5,e 6,e 7 } x f5 ={e 8,e 6,e 7 } Manav Rachna University 22 M ATRICES OF O RIENTED G RAPHS v1v1 v2v2 v3v3 v4v4 v5v5 v6v6 v7v7 v8v8 e1e1 e2e2 e3e3 e4e4 e5e5 e6e6 e7e7 e8e8 e9e9 e 10 e 11 The edge e 1 which has an orientation from vertex v 1 to vertex v 2 simply indicates that any transmission from v 1 to v 2 along e 1 is assumed to be positive. Any transmission from v 2 to v 1 along e 1 is assumed to be negative. Manav Rachna University 23 M ATRICES OF O RIENTED G RAPHS DEFINITION: Let e and v represent respectively the number of edges and vertices of a graph G. The incident matrix having v rows and e columns is defined as Manav Rachna University 24 M ATRICES OF O RIENTED G RAPHS v1v1 v2v2 v3v3 v4v4 v5v5 v6v6 v7v7 v8v8 e1e1 e2e2 e3e3 e4e4 e5e5 e6e6 e7e7 e8e8 e9e9 e 10 e 11 Incident Matrix: Property: Any column of contains exactly two nonzero entries of opposite sign. Manav Rachna University 25 M ATRICES OF O RIENTED G RAPHS Property: The determinant of any square submatrix of order q (1 q v) of is either one of the following values: 1, -1, 0. Now, consider a graph G of p connected parts: Manav Rachna University 26 M ATRICES OF O RIENTED G RAPHS DEFINITION: For a connected graph G, the matrix , obtained by deleting any one of the rows of the incidence matrix is called the reduced incident matrix. Note that since any column of contains exactly two nonzero entries of opposite sign, one can uniquely determine the incident matrix when the reduced incident matrix is given. Manav Rachna University 27 M ATRICES OF O RIENTED G RAPHS Property: Any square submatrix i of order v- 1 of the reduced incidence matrix of G is nonsingular if and only if the columns of i correspond to the branches of a tree T of G. Example v1v1 v2v2 v3v3 e1e1 e2e2 e3e3 i : nonsingular Manav Rachna University 28 M ATRICES OF O RIENTED G RAPHS In a graph G let k be the number of circuits and let an arbitrary circuit orientation be assigned to each one of these circuits. DEFINITION: The circuit matrix for a graph G of e edges and k circuits is defined as Manav Rachna University 29 M ATRICES OF O RIENTED G RAPHS Consider the following graph e3e3 v1v1 v2v2 v3v3 v4v4 e1e1 e2e2 e4e4 e5e5 e6e6 c 6 c 5 c 4 c 3 c 2 c 1 Manav Rachna University 30 M ATRICES OF O RIENTED G RAPHS Let b i represent the row of B that corresponds to circuit c i. The circuits c i,...,c j are independent if the rows b i,... b j are independent. DEFINITION: The f-circuit matrix B f of a graph G with respect to some tree T is defined as the circuit matrix consisting of the fundamental circuits of G only whose orientations are chosen in the same direction as that of defining chords. Manav Rachna University 31 M ATRICES OF O RIENTED G RAPHS The fundamental circuit matrix B f of a graph G with respect to some tree T can always be written as Manav Rachna University 32 M ATRICES OF O RIENTED G RAPHS Consider the following graph e3e3 v1v1 v2v2 v3v3 v4v4 e1e1 e2e2 e4e4 e5e5 e6e6 Manav Rachna University 33 M ATRICES OF O RIENTED G RAPHS THEOREM: If the column orderings of the circuit and incident matrices are identical then Manav Rachna University 34 M ATRICES OF O RIENTED G RAPHS Consider the following graph e3e3 v1v1 v2v2 v3v3 v4v4 e1e1 e2e2 e4e4 e5e5 e6e6 c 6 c 5 c 4 c 3 c 2 c 1 Manav Rachna University 35 M ATRICES OF O RIENTED G RAPHS In a graph G let x be the number of cut-sets having arbitrary orientations. Then, we have the following definition. DEFINITION: The cut-set matrix for a graph G of e edges and x cut-sets is defined as Manav Rachna University 36 M ATRICES OF O RIENTED G RAPHS Consider the following graph and its seven possible cut-sets e3e3 v1v1 v2v2 v3v3 v0v0 e1e1 e2e2 e4e4 e5e5 e3e3 v1v1 v2v2 v3v3 v0v0 e1e1 e2e2 e4e4 e5e5 e6e6 x 1 x 2 x 3 x 4 x 5 x 6 x 7 Manav Rachna University 37 M ATRICES OF O RIENTED G RAPHS DEFINITION: The f-cutset matrix A f of a graph G with respect to some tree T is defined as the cut-set matrix consisting of the fundamental cut- set of G only whose orientations are chosen in the same direction as that of defining branches. Manav Rachna University 38 M ATRICES OF O RIENTED G RAPHS The fundamental cut-set matrix A f of a graph G with respect to some tree T can always be written as Manav Rachna University 39 M ATRICES OF O RIENTED G RAPHS Consider the following graph e3e3 v1v1 v2v2 v3v3 v0v0 e1e1 e2e2 e4e4 e5e5 e3e3 v1v1 v2v2 v3v3 v0v0 e1e1 e2e2 e4e4 e5e5 e6e6 Manav Rachna University 40 M ATRICES OF O RIENTED G RAPHS THEOREM: If the column orderings of the circuit and cut-set matrices are identical then Manav Rachna University 41 M ATRICES OF O RIENTED G RAPHS Consider the following graph e3e3 v1v1 v2v2 v3v3 v0v0 e1e1 e2e2 e4e4 e5e5 e3e3 v1v1 v2v2 v3v3 v0v0 e1e1 e2e2 e4e4 e5e5 e6e6 Manav Rachna University 42 M ATRICES OF O RIENTED G RAPHS THEOREM: In a graph G let the fundamental circuit and cut-set matrices with respect to a tree to be written as Manav Rachna University 43 FUNDAMENTAL POSTULATES Now, Let G be a connected graph having e edges and let be two vectors where x i and y i, i=1,...,e, correspond to the across and through variables associated with the edge i respectively. Manav Rachna University 44 FUNDAMENTAL POSTULATES 2. POSTULATE Let B be the circuit matrix of the graph G having e edges then we can write the following algebraic equation for the across variables of G 3. POSTULATE Let A be the cut-set matrix of the graph G having e edges then we can write the following algebraic equation for the through variables of G Manav Rachna University 45 FUNDAMENTAL POSTULATES 2. POSTULATE is called the circuit equations of electrical system. (is also referred to as Kirchoffs Voltage Law) 3. POSTULATE is called the cut-set equations of electrical system. (is also referred to as Kirchoffs Current Law) Manav Rachna University 46 F UNDAMENTAL C IRCUIT & C UT - SET E QUATIONS Consider a graph G and a tree T in G. Let the vectors x and y partitioned as where x b (y b ) and x c (y c ) correspond to the across (through) variables associated with the branches and chords of the tree T, respectively. Then and fundamental circuit equation fundamental cut-set equation Manav Rachna University 47 E XAMPLE In the following figure, an oriented graph with a tree, indicated in heavy lines, is given. Obtain the f-circuit and f-cutset matrices Obtain the reduced incident matrix, , (take (a) as reference Explain how you would derive the matrix A from and show this derivation on the matrix a b c d Manav Rachna University 48 S ERIES & P ARALLEL E DGES Definition: Two edges e i and e k are said to be connected in series if they have exactly one common vertex of degree two. eiei ekek v0v0 Manav Rachna University 49 S ERIES & P ARALLEL E DGES Definition: Two edges e i and e k are said to be connected in parallel if they are incident at the same pair of vertices v i and v k. eiei ekek vivi vkvk n+1 edges connected in series or in parallel? Manav Rachna University 50 End of Section D Manav Rachna University 51