Electrical Network Graph Theory Topology

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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

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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

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Terms and Definitions Circuit elements

Node Branch Path

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Terms and Definitions Circuit elements Node

Branch Path

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Branch

Path

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Branch Path

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Branch Path

Closed path or Circuit or Loop or Mesh

Topology, rather Electrical network topology

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Branch Path

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Branch Path

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Branch Path

Tree

Twigs Co-tree Links or Chords

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Branch Path

Tree Twigs

Co-tree Links or Chords

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Branch Path

Tree Twigs Co-tree

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Branch Path

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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.

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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:

Branch:

Path:

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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:

Branch:

Path:

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Network Topology: Terms and Definitions - I

Circuit elements:

Node:

Branch:

Path:

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Network Topology: Terms and Definitions - I

Circuit elements:

Node:

Branch:

Path:

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Network Topology: Terms and Definitions - I

Circuit elements:

Node:

Branch:

Path:

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Network Topology: Terms and Definitions - I

Circuit elements:

Node:

Branch:

Path:

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Network Topology: Terms and Definitions - I

Circuit elements:

Node:

Branch:

Path:

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Network Topology: Terms and Definitions - I

Circuit elements:

Node:

Branch:

Path:

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Network Topology: Terms and Definitions - I

Circuit elements:

Node:

Branch:

Path:

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Network Topology: Terms and Definitions - I

Circuit elements:

Node:

Branch:

Path:

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Network Topology: Terms and Definitions - I

Circuit elements:

Node:

Branch:

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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.

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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]:

mesh.

Network:

Circuit:

Planar circuit:

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Network Topology: Terms and Definitions - II

Loop:

through branches and comes to the original/starting node,

- Also known as closed path or circuit.

Mesh1 [2]:

mesh.

Network:

Circuit:

Planar circuit:

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Network Topology: Terms and Definitions - II

Loop:

Mesh1 [2]:

mesh.

Network:

Circuit:

Planar circuit:

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Network Topology: Terms and Definitions - II

Loop:

Mesh1 [2]:

mesh.

Network:

Circuit:

Planar circuit:

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Network Topology: Terms and Definitions - II

Loop:

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:

Circuit:

Planar circuit:

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Network Topology: Terms and Definitions - II

Loop:

Mesh1 [2]:

mesh.

Network:

Circuit:

Planar circuit:

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Network Topology: Terms and Definitions - II

Loop:

Mesh1 [2]:

mesh.

Network:

Circuit:

Planar circuit:

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Network Topology: Terms and Definitions - II

Loop:

Mesh1 [2]:

mesh.

Network:

Circuit:

Planar circuit:

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Network Topology: Terms and Definitions - II

Loop:

Mesh1 [2]:

mesh.

Network:

Circuit:

Planar circuit:

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Network Topology: Terms and Definitions - II

Loop:

Mesh1 [2]:

mesh.

Network:

Circuit:

Planar circuit:

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Network Topology: Terms and Definitions - II

Loop:

Mesh1 [2]:

mesh.

Network:

Circuit:

Planar circuit:

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Network Topology: Terms and Definitions - II

Loop:

Mesh1 [2]:

mesh.

Network:

Circuit:

Planar circuit:

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Network Topology: Terms and Definitions - II

Loop:

Mesh1 [2]:

mesh.

Network:

Circuit:

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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.

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Network Topology: Terms and Definitions - III

Topology:

Graph:

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Network Topology: Terms and Definitions - III

Topology:

Graph:

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Network Topology: Terms and Definitions - III

Topology:

Graph:

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Network Topology: Terms and Definitions - III

Topology:

Graph:

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Network Topology: Terms and Definitions - III

Topology:

Graph:

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Network Topology: Terms and Definitions - III

Topology:

Graph:

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Network Topology: Terms and Definitions - III

Topology:

Graph:

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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 2 4

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An Electrical Network & its Graph - I

1 2 3 4 5 A R1 R2 R4 C R3 (a) 1 2 3 4 a b c d e f (b)

Figure 1 : (a) A circuit and (b) its graph. Note:

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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:

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An Electrical Network & its Graph - III

1 2 3 4 5 + − Vs Is R R1 I1 I2 R2 R3 I3 C IC L IL (a) 1 2 3 4 5 a b c f e d g (b) 1 2 3 4, 5 a b c f e d (c)

Figure 3 : (a) A circuit, (b) its directed graph and (c) simplified directed graph of (b).

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An Electrical Network & its Graph - IV

1 2 3 4 + − Vs Ivs R R1 I1 R2 I2 IIs C IC L IL (a) 1 2 3 4 a b c e d (b) 1 2 3 4 a b c e d (c)

Figure 4 : (a) A circuit, and (b),(c) its directed graphs.

Note:

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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:

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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:

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Network Topology: Terms and Definitions - IV

Tree:

Twigs:

Links or Chords:

Co-tree:

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Network Topology: Terms and Definitions - IV

Tree:

Twigs:

Links or Chords:

Co-tree:

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Network Topology: Terms and Definitions - IV

Tree:

Twigs:

Links or Chords:

Co-tree:

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Network Topology: Terms and Definitions - IV

Tree:

Twigs:

Links or Chords:

Co-tree:

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Network Topology: Terms and Definitions - IV

Tree:

Twigs:

Links or Chords:

Co-tree:

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Network Topology: Terms and Definitions - IV

Tree:

Twigs:

Links or Chords:

Co-tree:

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Network Topology: Terms and Definitions - IV

Tree:

Twigs:

Links or Chords:

- The branches that are removed from the graph while forming a

Co-tree:

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Network Topology: Terms and Definitions - IV

Tree:

Twigs:

Links or Chords:

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Network Topology: Terms and Definitions - IV

Tree:

Twigs:

Links or Chords:

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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 a 2 b 3 c d e f

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Summary and a Question:

Q. Does the following graph with branches a and e form a tree?

1 2 3 4 a b c f e d

+ The number of nodes in this subgraph is equal to that of the given graph. + But it has unconnected subgraphs and

moreovertotal number branches 6= n − 1(= 3). Therefore, it is not a tree.

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Summary and a Question:

1 2 3 4 a b c f e d

+ The number of nodes in this subgraph is equal to that of the given graph.

+ But it has unconnected subgraphs and moreovertotal number branches 6= n − 1(= 3). Therefore, it is not a tree.

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Summary and a Question:

1 2 3 4 a b c f e d

+ The number of nodes in this subgraph is equal to that of the given graph. + But it has unconnected subgraphs and

moreovertotal number branches 6= n − 1(= 3). Therefore, it is not a tree.

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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.

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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

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Network Toplogy Khublei Shibun!

Thank You!

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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)

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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 I2 R3 R4 I3 C IC L IL Is (a) 2 3 4 1, 5 (b)

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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 I2 R3 R4 I3 C IC L IL Is (a) 2 3 4 1, 5 (b)

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Problems for Practice - II

Note: Transformer gives a unconnected graph!

For example - 2: 1 2 3 4 5 6 + − Vs R1 I1 R2 I2 K C IC R3 I3 I (a) 1 2 6 3 4 5 (b)

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Problems for Practice - II

Note: Transformer gives a unconnected graph!

For example - 2: 1 2 3 4 5 6 + − Vs R1 I1 R2 I2 K C IC R3 I3 I (a) 1 2 6 3 4 5 (b)

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Few non-planar Graphs

1 2 3 4 (a) 1 2 3 4 5 6 (b)