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EE-304 Electrical Network Theory [Class Notes1] - 2013

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Electrical Network Topology, Electrical Network Graph Theory, Node, Branch, Twig, Link, Tree, Cotree

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EE-304 Electrical Network Theory [Class Notes1] - 2013

  1. 1. Network Topology and Graph Theory EE-304 ENT credits: 4 L{3} P{0} T{1} Lairenlakpam Joyprakash Singh, PhD Department of ECE, North-Eastern Hill University (NEHU), Shillong – 793 022 jplairen@nehu.ac.in August 8, 2013 1 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
  2. 2. Network Toplogy Terms & Definitions Introduction to Electrical Network Topology Terms and Definitions 2 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
  3. 3. Network Toplogy Terms & Definitions Introduction to Electrical Network Topology Terms and Definitions Circuit elements 2 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
  4. 4. Network Toplogy Terms & Definitions Introduction to Electrical Network Topology Terms and Definitions Circuit elements Node 2 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
  5. 5. Network Toplogy Terms & Definitions Introduction to Electrical Network Topology Terms and Definitions Circuit elements Node Branch 2 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
  6. 6. Network Toplogy Terms & Definitions Introduction to Electrical Network Topology Terms and Definitions Circuit elements Node Branch Path 2 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
  7. 7. Network Toplogy Terms & Definitions Introduction to Electrical Network Topology Terms and Definitions Circuit elements Node Branch Path Closed path or Circuit or Loop or Mesh 2 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
  8. 8. Network Toplogy Terms & Definitions Introduction to Electrical Network Topology Terms and Definitions Circuit elements Node Branch Path Closed path or Circuit or Loop or Mesh Topology, rather Electrical network topology 2 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
  9. 9. Network Toplogy Terms & Definitions Introduction to Electrical Network Topology Terms and Definitions Circuit elements Node Branch Path Closed path or Circuit or Loop or Mesh Topology, rather Electrical network topology - Graph and its types 2 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
  10. 10. Network Toplogy Terms & Definitions Introduction to Electrical Network Topology Terms and Definitions Circuit elements Node Branch Path Closed path or Circuit or Loop or Mesh Topology, rather Electrical network topology - Graph and its types Tree 2 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
  11. 11. Network Toplogy Terms & Definitions Introduction to Electrical Network Topology Terms and Definitions Circuit elements Node Branch Path Closed path or Circuit or Loop or Mesh Topology, rather Electrical network topology - Graph and its types Tree Twigs 2 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
  12. 12. Network Toplogy Terms & Definitions Introduction to Electrical Network Topology Terms and Definitions Circuit elements Node Branch Path Closed path or Circuit or Loop or Mesh Topology, rather Electrical network topology - Graph and its types Tree Twigs Co-tree 2 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
  13. 13. Network Toplogy Terms & Definitions Introduction to Electrical Network Topology Terms and Definitions Circuit elements Node Branch Path Closed path or Circuit or Loop or Mesh Topology, rather Electrical network topology - Graph and its types Tree Twigs Co-tree Links or Chords 2 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
  14. 14. Network Toplogy Terms & Definitions Network Topology: Terms and Definitions - I Circuit elements: 3 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
  15. 15. Network Toplogy Terms & Definitions Network Topology: Terms and Definitions - I Circuit elements: - The mathematical models of a two terminal electrical devices, 3 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
  16. 16. Network Toplogy Terms & Definitions Network Topology: Terms and Definitions - I Circuit elements: - The mathematical models of a two terminal electrical devices, - Completely characterized by its voltage-current relationship, 3 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
  17. 17. Network Toplogy Terms & Definitions Network Topology: Terms and Definitions - I Circuit elements: - The mathematical models of a two terminal electrical devices, - Completely characterized by its voltage-current relationship, - Can not be subdivided into other two-terminal devices. 3 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
  18. 18. Network Toplogy Terms & Definitions Network Topology: Terms and Definitions - I Circuit elements: - The mathematical models of a two terminal electrical devices, - Completely characterized by its voltage-current relationship, - Can not be subdivided into other two-terminal devices. Node: 3 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
  19. 19. Network Toplogy Terms & Definitions Network Topology: Terms and Definitions - I Circuit elements: - The mathematical models of a two terminal electrical devices, - Completely characterized by its voltage-current relationship, - Can not be subdivided into other two-terminal devices. Node: - A point at which two or more circuit elements have a common connection, 3 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
  20. 20. Network Toplogy Terms & Definitions Network Topology: Terms and Definitions - I Circuit elements: - The mathematical models of a two terminal electrical devices, - Completely characterized by its voltage-current relationship, - Can not be subdivided into other two-terminal devices. Node: - A point at which two or more circuit elements have a common connection, - The number of branches incident to a node is known as the degree of that node. 3 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
  21. 21. Network Toplogy Terms & Definitions Network Topology: Terms and Definitions - I Circuit elements: - The mathematical models of a two terminal electrical devices, - Completely characterized by its voltage-current relationship, - Can not be subdivided into other two-terminal devices. Node: - A point at which two or more circuit elements have a common connection, - The number of branches incident to a node is known as the degree of that node. Branch: 3 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
  22. 22. Network Toplogy Terms & Definitions Network Topology: Terms and Definitions - I Circuit elements: - The mathematical models of a two terminal electrical devices, - Completely characterized by its voltage-current relationship, - Can not be subdivided into other two-terminal devices. Node: - A point at which two or more circuit elements have a common connection, - The number of branches incident to a node is known as the degree of that node. Branch: - A single path, containing one circuit element, which connnects one node to any other node, 3 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
  23. 23. Network Toplogy Terms & Definitions Network Topology: Terms and Definitions - I Circuit elements: - The mathematical models of a two terminal electrical devices, - Completely characterized by its voltage-current relationship, - Can not be subdivided into other two-terminal devices. Node: - A point at which two or more circuit elements have a common connection, - The number of branches incident to a node is known as the degree of that node. Branch: - A single path, containing one circuit element, which connnects one node to any other node, - Represented by a line in the graph. 3 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
  24. 24. Network Toplogy Terms & Definitions Network Topology: Terms and Definitions - I Circuit elements: - The mathematical models of a two terminal electrical devices, - Completely characterized by its voltage-current relationship, - Can not be subdivided into other two-terminal devices. Node: - A point at which two or more circuit elements have a common connection, - The number of branches incident to a node is known as the degree of that node. Branch: - A single path, containing one circuit element, which connnects one node to any other node, - Represented by a line in the graph. Path: 3 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
  25. 25. Network Toplogy Terms & Definitions Network Topology: Terms and Definitions - I Circuit elements: - The mathematical models of a two terminal electrical devices, - Completely characterized by its voltage-current relationship, - Can not be subdivided into other two-terminal devices. Node: - A point at which two or more circuit elements have a common connection, - The number of branches incident to a node is known as the degree of that node. Branch: - A single path, containing one circuit element, which connnects one node to any other node, - Represented by a line in the graph. Path: - A set of elements that may be traversed in order without passing through the same node twice. 3 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
  26. 26. Network Toplogy Terms & Definitions Network Topology: Terms and Definitions - II Loop: 1Engineering Circuit Analysis, 8e 4 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
  27. 27. Network Toplogy Terms & Definitions Network Topology: Terms and Definitions - II Loop: - A close path or a closed contour selected in a network/circuit, 1Engineering Circuit Analysis, 8e 4 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
  28. 28. Network Toplogy Terms & Definitions Network Topology: Terms and Definitions - II Loop: - A close path or a closed contour selected in a network/circuit, - A path that may be started from a prticular node to other nodes through branches and comes to the original/starting node, 1Engineering Circuit Analysis, 8e 4 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
  29. 29. Network Toplogy Terms & Definitions Network Topology: Terms and Definitions - II Loop: - A close path or a closed contour selected in a network/circuit, - A path that may be started from a prticular node to other nodes through branches and comes to the original/starting node, - Also known as closed path or circuit. 1Engineering Circuit Analysis, 8e 4 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
  30. 30. Network Toplogy Terms & Definitions Network Topology: Terms and Definitions - II Loop: - A close path or a closed contour selected in a network/circuit, - A path that may be started from a prticular node to other nodes through branches and comes to the original/starting node, - Also known as closed path or circuit. Mesh1 [2]: 1Engineering Circuit Analysis, 8e 4 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
  31. 31. Network Toplogy Terms & Definitions Network Topology: Terms and Definitions - II Loop: - A close path or a closed contour selected in a network/circuit, - A path that may be started from a prticular node to other nodes through branches and comes to the original/starting node, - Also known as closed path or circuit. Mesh1 [2]: - A loop that does not contain any other loops within it, 1Engineering Circuit Analysis, 8e 4 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
  32. 32. Network Toplogy Terms & Definitions Network Topology: Terms and Definitions - II Loop: - A close path or a closed contour selected in a network/circuit, - A path that may be started from a prticular node to other nodes through branches and comes to the original/starting node, - Also known as closed path or circuit. Mesh1 [2]: - A loop that does not contain any other loops within it, - Any mesh is a circuit/loop but any loop/circuit may not be a mesh. 1Engineering Circuit Analysis, 8e 4 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
  33. 33. Network Toplogy Terms & Definitions Network Topology: Terms and Definitions - II Loop: - A close path or a closed contour selected in a network/circuit, - A path that may be started from a prticular node to other nodes through branches and comes to the original/starting node, - Also known as closed path or circuit. Mesh1 [2]: - A loop that does not contain any other loops within it, - Any mesh is a circuit/loop but any loop/circuit may not be a mesh. Network: 1Engineering Circuit Analysis, 8e 4 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
  34. 34. Network Toplogy Terms & Definitions Network Topology: Terms and Definitions - II Loop: - A close path or a closed contour selected in a network/circuit, - A path that may be started from a prticular node to other nodes through branches and comes to the original/starting node, - Also known as closed path or circuit. Mesh1 [2]: - A loop that does not contain any other loops within it, - Any mesh is a circuit/loop but any loop/circuit may not be a mesh. Network: - The interconnection of two or more circuit elements forms an electical network. 1Engineering Circuit Analysis, 8e 4 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
  35. 35. Network Toplogy Terms & Definitions Network Topology: Terms and Definitions - II Loop: - A close path or a closed contour selected in a network/circuit, - A path that may be started from a prticular node to other nodes through branches and comes to the original/starting node, - Also known as closed path or circuit. Mesh1 [2]: - A loop that does not contain any other loops within it, - Any mesh is a circuit/loop but any loop/circuit may not be a mesh. Network: - The interconnection of two or more circuit elements forms an electical network. Circuit: 1Engineering Circuit Analysis, 8e 4 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
  36. 36. Network Toplogy Terms & Definitions Network Topology: Terms and Definitions - II Loop: - A close path or a closed contour selected in a network/circuit, - A path that may be started from a prticular node to other nodes through branches and comes to the original/starting node, - Also known as closed path or circuit. Mesh1 [2]: - A loop that does not contain any other loops within it, - Any mesh is a circuit/loop but any loop/circuit may not be a mesh. Network: - The interconnection of two or more circuit elements forms an electical network. Circuit: - Network that contains at least one closed path, 1Engineering Circuit Analysis, 8e 4 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
  37. 37. Network Toplogy Terms & Definitions Network Topology: Terms and Definitions - II Loop: - A close path or a closed contour selected in a network/circuit, - A path that may be started from a prticular node to other nodes through branches and comes to the original/starting node, - Also known as closed path or circuit. Mesh1 [2]: - A loop that does not contain any other loops within it, - Any mesh is a circuit/loop but any loop/circuit may not be a mesh. Network: - The interconnection of two or more circuit elements forms an electical network. Circuit: - Network that contains at least one closed path, - Every circuit is a network, but not all networks are circuits. 1Engineering Circuit Analysis, 8e 4 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
  38. 38. Network Toplogy Terms & Definitions Network Topology: Terms and Definitions - II Loop: - A close path or a closed contour selected in a network/circuit, - A path that may be started from a prticular node to other nodes through branches and comes to the original/starting node, - Also known as closed path or circuit. Mesh1 [2]: - A loop that does not contain any other loops within it, - Any mesh is a circuit/loop but any loop/circuit may not be a mesh. Network: - The interconnection of two or more circuit elements forms an electical network. Circuit: - Network that contains at least one closed path, - Every circuit is a network, but not all networks are circuits. Planar circuit: 1Engineering Circuit Analysis, 8e 4 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
  39. 39. Network Toplogy Terms & Definitions Network Topology: Terms and Definitions - II Loop: - A close path or a closed contour selected in a network/circuit, - A path that may be started from a prticular node to other nodes through branches and comes to the original/starting node, - Also known as closed path or circuit. Mesh1 [2]: - A loop that does not contain any other loops within it, - Any mesh is a circuit/loop but any loop/circuit may not be a mesh. Network: - The interconnection of two or more circuit elements forms an electical network. Circuit: - Network that contains at least one closed path, - Every circuit is a network, but not all networks are circuits. Planar circuit: - A circuit that may drawn on a plane surface in such a way that no branch passes over or under any other branch. 1Engineering Circuit Analysis, 8e 4 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
  40. 40. Network Toplogy Terms & Definitions Network Topology: Terms and Definitions - III Topology: 5 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
  41. 41. Network Toplogy Terms & Definitions Network Topology: Terms and Definitions - III Topology: - Deals with properties of networks which are unaffected when the network is stretched, twisted, or otherwise distorted the size and the shape, 5 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
  42. 42. Network Toplogy Terms & Definitions Network Topology: Terms and Definitions - III Topology: - Deals with properties of networks which are unaffected when the network is stretched, twisted, or otherwise distorted the size and the shape, - Not concerned with the particular types of elements appearing in the circuit, but only with the way in which branches and nodes are arranged. 5 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
  43. 43. Network Toplogy Terms & Definitions Network Topology: Terms and Definitions - III Topology: - Deals with properties of networks which are unaffected when the network is stretched, twisted, or otherwise distorted the size and the shape, - Not concerned with the particular types of elements appearing in the circuit, but only with the way in which branches and nodes are arranged. Graph: 5 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
  44. 44. Network Toplogy Terms & Definitions Network Topology: Terms and Definitions - III Topology: - Deals with properties of networks which are unaffected when the network is stretched, twisted, or otherwise distorted the size and the shape, - Not concerned with the particular types of elements appearing in the circuit, but only with the way in which branches and nodes are arranged. Graph: - A graph corresponding to a given network is obtained by replacing all circuit elements with lines. 5 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
  45. 45. Network Toplogy Terms & Definitions Network Topology: Terms and Definitions - III Topology: - Deals with properties of networks which are unaffected when the network is stretched, twisted, or otherwise distorted the size and the shape, - Not concerned with the particular types of elements appearing in the circuit, but only with the way in which branches and nodes are arranged. Graph: - A graph corresponding to a given network is obtained by replacing all circuit elements with lines. - Connected graph: A graph in which at least one path exists between any two nodes of the graph. If the network has a transformer as one of the element, then the resulted graph is unconnected 5 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
  46. 46. Network Toplogy Terms & Definitions Network Topology: Terms and Definitions - III Topology: - Deals with properties of networks which are unaffected when the network is stretched, twisted, or otherwise distorted the size and the shape, - Not concerned with the particular types of elements appearing in the circuit, but only with the way in which branches and nodes are arranged. Graph: - A graph corresponding to a given network is obtained by replacing all circuit elements with lines. - Connected graph: A graph in which at least one path exists between any two nodes of the graph. If the network has a transformer as one of the element, then the resulted graph is unconnected - Directed or Oriented graph: A graph that has all the nodes and branches numbered and also directions are given to the branches. 5 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
  47. 47. Network Toplogy Terms & Definitions Network Topology: Terms and Definitions - III Topology: - Deals with properties of networks which are unaffected when the network is stretched, twisted, or otherwise distorted the size and the shape, - Not concerned with the particular types of elements appearing in the circuit, but only with the way in which branches and nodes are arranged. Graph: - A graph corresponding to a given network is obtained by replacing all circuit elements with lines. - Connected graph: A graph in which at least one path exists between any two nodes of the graph. If the network has a transformer as one of the element, then the resulted graph is unconnected - Directed or Oriented graph: A graph that has all the nodes and branches numbered and also directions are given to the branches. - Subgraph: The subset of a graph. If the number of nodes and branches of a subgraph is less than that of the graph, the subgraph is said to be proper. 5 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
  48. 48. Network Toplogy Network Circuits & Their Graphs Network Topology: An example A circuit with topologically equivalent graphs: 1 2 3 4 + −Vs Is R1 IR1 R2 IR2 R3 IR3 C IC L IL 1 2 3 4 1 2 3 4 i) A Circuit ii) its graph iii) directed graph 1 2 3 4 1 2 3 4 1 2 3 4 Three topologically equivalent graphs of figure ii). 6 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
  49. 49. Network Toplogy Network Circuits & Their Graphs An Electrical Network & its Graph - I 1 2 3 4 5 A R1 R2 R4CR3 (a) 1 2 3 4 a b c d e f (b) Figure 1 : (a) A circuit and (b) its graph. Note: The maximum number of branches possible, in a circuit, will be equal to the number of nodes or vertices. There are at least two branches in a circuit. 7 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
  50. 50. Network Toplogy Network Circuits & Their Graphs An Electrical Network & its Graph - II 1 2 3 4 5 A R1 IR1 R2 IR2 R4 IR4 C IC R3 IR3 (a) 1 2 3 4 a b c d e f (b) Figure 2 : (a) A circuit and (b) its directed graph. Note: Each of the lines of the graph is indicated a reference direction by an arrow, and the resulted graph is called oriented/directed graph. 8 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
  51. 51. Network Toplogy Network Circuits & Their Graphs An Electrical Network & its Graph - III 1 2 3 4 5 + −Vs Is R R1 I1 R2 I2 R3 I3 C IC L IL (a) 1 2 3 4 5 a b c fe d g (b) 1 2 3 4, 5 a b c fed (c) Figure 3 : (a) A circuit, (b) its directed graph and (c) simplified directed graph of (b). Note: The active element branch is replaced by its internal resistance to simplify analysis and computation. 9 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
  52. 52. Network Toplogy Network Circuits & Their Graphs An Electrical Network & its Graph - IV 1 2 3 4 + −Vs Ivs R R1 I1 R2 I2 IIsC IC L IL (a) 1 2 3 4 a b c ed (b) 1 2 3 4 a b c e d (c) Figure 4 : (a) A circuit, and (b),(c) its directed graphs. Note: The active elements are excluded from the graph to simplify analysis and computation. 10 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
  53. 53. Network Toplogy Network Circuits & Their Graphs An Electrical Network & its Graph - V 1 A 1 Ω 1 Ω 1 Ω 1 Ω 1 Ω +− 1 V (a) (b) (c) Figure 5 : (a) A circuit, and its- (b) simplified graph and (c) directed graph. Note: When voltage source is not in series with any passive element in the given network, it is kept in the graph as a branch. 11 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
  54. 54. Network Toplogy Terms & Definitions Network Topology: Terms and Definitions - IV Tree: 12 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
  55. 55. Network Toplogy Terms & Definitions Network Topology: Terms and Definitions - IV Tree: - A connected subgraph having all the nodes of a graph without any loop. 12 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
  56. 56. Network Toplogy Terms & Definitions Network Topology: Terms and Definitions - IV Tree: - A connected subgraph having all the nodes of a graph without any loop. - Thus, a tree is a subgraph that has the following properties: - It must consist of all nodes of a complete graph. - For a graph having n number of nodes, the tree of the given graph will have n − 1 branches. - There exists one and only one path between any pair of nodes. - A tree should not have any closed path. - The rank of a tree is (n − 1). This is also the rank of the graph to which the tree belongs. 12 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
  57. 57. Network Toplogy Terms & Definitions Network Topology: Terms and Definitions - IV Tree: - A connected subgraph having all the nodes of a graph without any loop. - Thus, a tree is a subgraph that has the following properties: - It must consist of all nodes of a complete graph. - For a graph having n number of nodes, the tree of the given graph will have n − 1 branches. - There exists one and only one path between any pair of nodes. - A tree should not have any closed path. - The rank of a tree is (n − 1). This is also the rank of the graph to which the tree belongs. Twigs: 12 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
  58. 58. Network Toplogy Terms & Definitions Network Topology: Terms and Definitions - IV Tree: - A connected subgraph having all the nodes of a graph without any loop. - Thus, a tree is a subgraph that has the following properties: - It must consist of all nodes of a complete graph. - For a graph having n number of nodes, the tree of the given graph will have n − 1 branches. - There exists one and only one path between any pair of nodes. - A tree should not have any closed path. - The rank of a tree is (n − 1). This is also the rank of the graph to which the tree belongs. Twigs: - The branches of a tree are known as twigs, 12 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
  59. 59. Network Toplogy Terms & Definitions Network Topology: Terms and Definitions - IV Tree: - A connected subgraph having all the nodes of a graph without any loop. - Thus, a tree is a subgraph that has the following properties: - It must consist of all nodes of a complete graph. - For a graph having n number of nodes, the tree of the given graph will have n − 1 branches. - There exists one and only one path between any pair of nodes. - A tree should not have any closed path. - The rank of a tree is (n − 1). This is also the rank of the graph to which the tree belongs. Twigs: - The branches of a tree are known as twigs, Links or Chords: 12 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
  60. 60. Network Toplogy Terms & Definitions Network Topology: Terms and Definitions - IV Tree: - A connected subgraph having all the nodes of a graph without any loop. - Thus, a tree is a subgraph that has the following properties: - It must consist of all nodes of a complete graph. - For a graph having n number of nodes, the tree of the given graph will have n − 1 branches. - There exists one and only one path between any pair of nodes. - A tree should not have any closed path. - The rank of a tree is (n − 1). This is also the rank of the graph to which the tree belongs. Twigs: - The branches of a tree are known as twigs, Links or Chords: - The branches that are removed from the graph while forming a tree are termed as links or chords, 12 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
  61. 61. Network Toplogy Terms & Definitions Network Topology: Terms and Definitions - IV Tree: - A connected subgraph having all the nodes of a graph without any loop. - Thus, a tree is a subgraph that has the following properties: - It must consist of all nodes of a complete graph. - For a graph having n number of nodes, the tree of the given graph will have n − 1 branches. - There exists one and only one path between any pair of nodes. - A tree should not have any closed path. - The rank of a tree is (n − 1). This is also the rank of the graph to which the tree belongs. Twigs: - The branches of a tree are known as twigs, Links or Chords: - The branches that are removed from the graph while forming a tree are termed as links or chords, - Links are complement of twigs. 12 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
  62. 62. Network Toplogy Terms & Definitions Network Topology: Terms and Definitions - IV Tree: - A connected subgraph having all the nodes of a graph without any loop. - Thus, a tree is a subgraph that has the following properties: - It must consist of all nodes of a complete graph. - For a graph having n number of nodes, the tree of the given graph will have n − 1 branches. - There exists one and only one path between any pair of nodes. - A tree should not have any closed path. - The rank of a tree is (n − 1). This is also the rank of the graph to which the tree belongs. Twigs: - The branches of a tree are known as twigs, Links or Chords: - The branches that are removed from the graph while forming a tree are termed as links or chords, - Links are complement of twigs. Co-tree: 12 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
  63. 63. Network Toplogy Terms & Definitions Network Topology: Terms and Definitions - IV Tree: - A connected subgraph having all the nodes of a graph without any loop. - Thus, a tree is a subgraph that has the following properties: - It must consist of all nodes of a complete graph. - For a graph having n number of nodes, the tree of the given graph will have n − 1 branches. - There exists one and only one path between any pair of nodes. - A tree should not have any closed path. - The rank of a tree is (n − 1). This is also the rank of the graph to which the tree belongs. Twigs: - The branches of a tree are known as twigs, Links or Chords: - The branches that are removed from the graph while forming a tree are termed as links or chords, - Links are complement of twigs. Co-tree: - The graph constituted with links is known as co-tree. 12 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
  64. 64. Network Toplogy Terms & Definitions Tree and Cotree Given a Graph: 1 2 3 4 a b c d e f Tree Twigs of tree Links of cotree 1 2 3 4 a b c d e f {a,b,d} {c,e,f} 1 2 3 4 a b c d e f {a,d,f} {c,b,e} 13 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
  65. 65. Network Toplogy Terms & Definitions Summary and a Question: Q. Does the following graph with branches a and e form a tree? 1 2 3 4 a b c fed 14 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
  66. 66. Network Toplogy Terms & Definitions Summary and a Question: Q. Does the following graph with branches a and e form a tree? 1 2 3 4 a b c fed
  67. 67. The number of nodes in this subgraph is equal to that of the given graph. 14 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
  68. 68. Network Toplogy Terms & Definitions Summary and a Question: Q. Does the following graph with branches a and e form a tree? 1 2 3 4 a b c fed
  69. 69. The number of nodes in this subgraph is equal to that of the given graph.
  70. 70. But it has unconnected subgraphs and moreover total number branches = n − 1(= 3). Therefore, it is not a tree. 14 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
  71. 71. Network Toplogy References Text Books & References M. E. Van Valkenburg Network Analysis, 3/e. PHI, 2005. W.H. Hayt, J.E. Kemmerly, S.M. Durbin Engineering Circuit Analysis, 8/e. MH, 2012. 15 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
  72. 72. Network Toplogy References Text Books & References M. E. Van Valkenburg Network Analysis, 3/e. PHI, 2005. W.H. Hayt, J.E. Kemmerly, S.M. Durbin Engineering Circuit Analysis, 8/e. MH, 2012. M. Nahvi, J.A. Edminister SchuamâĂŹs Outline Electric Circuits, 4/e. TMH, SIE, 2007. A. Sudhakar, S.S. Palli Circuits and Networks: Analysis and Synthesis, 2/e. TMH, 2002. 15 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
  73. 73. Network Toplogy Khublei Shibun! Thank You! Any Question? 16 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
  74. 74. Home Assignment Graph and Incidence Matrix Problems for Practice: Graph and Incidence Matrix 1. Classify whether each of the following graphs as planar or nonplanar. 2. Find the number of possible trees for each graph and draw all possible trees. 1 2 3 4 (a) a b c d (b) 1 2 3 4 5 (c) 17 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
  75. 75. Home Assignment Graph and Incidence Matrix Problems for Practice - II Note: While replacing all elements of the network with lines to form a graph, we replace active elements by their internal resistances to simplify analysis and computation. For example - 1: 1 2 3 4 5 + −Vs R1 Is R2 I1 R3 I2 R4 I3 C IC L IL Is (a) 18 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
  76. 76. Home Assignment Graph and Incidence Matrix Problems for Practice - II Note: While replacing all elements of the network with lines to form a graph, we replace active elements by their internal resistances to simplify analysis and computation. For example - 1: 1 2 3 4 5 + −Vs R1 Is R2 I1 R3 I2 R4 I3 C IC L IL Is (a) 2 3 4 1, 5 (b) 18 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
  77. 77. Home Assignment Graph and Incidence Matrix Problems for Practice - II Note: Transformer gives a unconnected graph! For example - 2: 1 2 3 4 56 + −Vs R1 I1 R2 I2 K C IC R3 I3 I (a) 19 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
  78. 78. Home Assignment Graph and Incidence Matrix Problems for Practice - II Note: Transformer gives a unconnected graph! For example - 2: 1 2 3 4 56 + −Vs R1 I1 R2 I2 K C IC R3 I3 I (a) 1 2 6 3 4 5 (b) 19 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
  79. 79. Home Assignment Graph and Incidence Matrix Few non-planar Graphs 1 2 3 4 (a) 1 2 3 45 6 (b) 20 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph

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