Note for Network Theory - NT by Itz Ashu

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NETWORK THEORY  BEES2211                    Prepared by Dr. R.Behera Ms. R.Pradhan Department of Electrical Engineering, I.G.I.T, Sarang, Dhenkanal                

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MODULE 1 NETWORK TOPOLOGY 1. Introduction: When all the elements in a network are replaces by lines with circles or dots at both ends, configuration is called the graph of the network. A. Terminology used in network graph:(i) Path:-Asequence of branches traversed in going from one node to another is called a path. (ii) Node:-A nodepoint is defined as an end point of a line segment and exits at the junction between two branches or at the end of an isolated branch. (iii) Degree of a node:- It is the no. of branches incident to it.       2-degree3-degree Tree:- It is an interconnected open set of branches which include all the nodes of the given graph. In a tree of the graph there can’tbe any closed loop. (v) Tree branch(Twig):- It is the branch of a tree. It is also named as twig. (vi) Tree link(or chord):-It is the branch of a graph that does not belong to the particular tree. (vii) Loop:- This is the closed contour selected in a graph. (viii) Cut-Set:- It is that set of elements or branches of a graph that separated two parts of a network. If any branch of the cut-set is not removed, the network remains connected. The term cut-set is derived from the property designated by the way by which the network can be divided in to two parts. (ix) Tie-Set:- It is a unique set with respect to a given tree at a connected graph containing on chord and all of the free branches contained in the free path formed between two vertices of the chord. (x) Network variables:- A network consists of passive elements as well as sources of energy . In order to find out the response of the network the through current and voltages across each branch of the network are to be obtained. (xi) Directed(or Oriented) graph:- A graph is said to be directed (or oriented ) when all the nodes and branches are numbered or direction assigned to the branches by arrow. (xii) Sub graph:- A graph said to be sub-graph of a graph G if every node of is a node of G and every branch of is also a branch of G. (xiii) Connected Graph:- When at least one path along branches between every pair of a graph exits , it is called a connected graph. (iv)

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Incidence matrix:- Any oriented graph can be described completely in a compact matrix form. Here we specify the orientation of each branch in the graph and the nodes at which this branch is incident. This branch is called incident matrix. When one row is completely deleted from the matrix the remaining matrix is called a reduced incidence matrix. (xv) Isomorphism:- It is the property between two graphs so that both have got same incidence matrix. B. Relation between twigs and linksLet N=no. of nodes L= total no. of links B= total no. of branches No. of twigs= N-1 Then, L= B-(N-1) C. Properties of a Tree(i) It consists of all the nodes of the graph. (ii) If the graph has N nodes, then the tree has (N-1) branch. (iii) There will be no closed path in a tree. (iv) There can be many possible different trees for a given graph depending on the no. of nodes and branches. (xiv) 1. FORMATION OF INCIDENCE MATRIX:• This matrix shows which branch is incident to which node. • Each row of the matrix being representing the corresponding node of the graph. • Each column corresponds to a branch. • If a graph contain N- nodes and B branches then the size of the incidence matrix [A] will be NXB. A. Procedure:=1. (i) If the branch j is incident at the node I and oriented away from the node, In other words, when =1, branch j leaves away node i. =-1. In other (ii) If branch j is incident at node j and is oriented towards node i, words j enters node i. (iii) If branch j is not incident at node i. =0. The complete set of incidence matrix is called augmented incidence matrix.   Ex-1:- Obtain the incidence matrix of the following graph.                                                                                                                         a             1           b            2     c                                                                                                                                                   5        4         3 

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  Node-a:- Branches connected are 1& 5 and both are away from the node. Node-b:- Branches incident at this node are 1,2 &4. Here branch is oriented towards the node whereas branches 2 &4 are directed away from the node. Node-c:- Branches 2 &3 are incident on this node. Here, branch 2 is oriented towards the node whereas the branch 3 is directed away from the node. Node-d:- Branch 3,4 &5 are incident on the node. Here all the branches are directed towards the node. So, branch Node 1         2         3         4         5  1      1         0        0         0         1   [ ]=2    ‐1         1        0         1         0  3     0        ‐1        1         0         0  4     0        0        ‐1         ‐1       ‐1                    B. Properties:(i) Algebraic sum of the column entries of an incidence matrix is zero. (ii) Determinant of the incidence matrix of a closed loop is zero. C. Reduced Incidence Matrix :If any row of a matrix is completely deleted, then the remaining matrix is known as reduced Incidence matrix. For the above example, after deleting row, we get, [Ai’]= Ai’ is the reduced matrix of Ai . Ex-2: Draw the directed graph for the following incidence matrix. Branch Node 1 1 -1 2 0 3 1 4 0 2 0 -1 1 0 3 -1 0 0 1 4 1 -1 0 0 5 0 0 -1 1 6 0 -1 1 0 7 1 0 0 -1