Electrical Network Graph Theory Topology

7/29/2019 Electrical Network Graph Theory Topology

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Network Topology and Graph Theory

EE-304 ENT credits: 4 L3 P0 T1

Lairenlakpam Joyprakash Singh, PhD

Department of ECE,

North-Eastern Hill University (NEHU),Shillong – 793 [email protected]

August 8, 2013

1 / 2 0 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph

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Network Toplogy Terms & Definitions

Introduction to Electrical Network Topology

Terms and Definitions


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Circuit elements


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Circuit elementsNode


&

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Branch


N T T & D

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Branch

Path



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Branch

Path

Closed path or Circuit or Loop or Mesh



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Branch

Path


Topology, rather Electrical network topology



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Branch

Path



- Graph and its types



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Branch

Path




Tree



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Branch

Path




TreeTwigs



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Branch

Path




TreeTwigsCo-tree



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Branch

Path




TreeTwigsCo-tree

Links or Chords



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Network Topology: Terms and Definitions - I

Circuit elements:



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Circuit elements:

- The mathematical models of a two terminal electrical devices,



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Circuit elements:

- The mathematical models of a two terminal electrical devices,- Completely characterized by its voltage-current relationship,



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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.



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Circuit elements:


Node:



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Circuit elements:


Node:

- A point at which two or more circuit elements have a commonconnection,



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Circuit elements:


Node:


- The number of branches incident to a node is known as thedegree of that node.



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Circuit elements:


Node:



Branch:



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Circuit elements:


Node:



Branch:

- A single path, containing one circuit element, which connnectsone node to any other node,



N k l d D fi I

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Circuit elements:


Node:



Branch:


- Represented by a line in the graph.



N k T l T d D fi i i I

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Circuit elements:


Node:



Branch:


- Represented by a line in the graph.Path:



N t k T l T d D fi iti I

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Circuit elements:


Node:



Branch:


- Represented by a line in the graph.Path:

- A set of elements that may be traversed in order without passingthrough the same node twice.



N t k T l T d D fi iti II

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Network Topology: Terms and Definitions - II

Loop:

1

Engineering Circuit Analysis, 8e4 / 2 0 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph


N t k T l T d D fi iti II

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Loop:- A close path or a closed contour selected in a network/circuit,

1



Net o k To olog Te s and Definitions II

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- A path that may be started from a prticular node to other nodesthrough branches and comes to the original/starting node,

1



Network Topology: Terms and Definitions II

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- Also known as closed path or circuit .

1




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Mesh1 [2]:

1




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Mesh1 [2]:- A loop that does not contain any other loops within it,

1




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- Any mesh is a circuit/loop but any loop/circuit may not be amesh.

1




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Network:

1




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Network:- The interconnection of two or more circuit elements forms an

electical network.

1




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electical network.

Circuit:

1




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electical network.

Circuit:

- Network that contains at least one closed path,

1




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electical network.

Circuit:

- Network that contains at least one closed path,- Every circuit is a network, but not all networks are circuits.

1




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p gy




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.


electical network.

Circuit:


Planar circuit:

1




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p gy




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.


electical network.

Circuit:


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.1



Network Topology: Terms and Definitions - III

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p gy

Topology:




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p gy

Topology:

- Deals with properties of networks which are unaffected when the

network is stretched, twisted, or otherwise distorted the size andthe shape,




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Topology:



- Not concerned with the particular types of elements appearing inthe circuit, but only with the way in which branches and nodesare arranged.




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Topology:




Graph:




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Topology:




Graph:- A graph corresponding to a given network is obtained byreplacing all circuit elements with lines.




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Topology:





- Connected graph: A graph in which at least one path existsbetween any two nodes of the graph. If the network has atransformer as one of the element, then the resulted graph isunconnected




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Topology:






- Directed or Oriented graph: A graph that has all the nodesand branches numbered and also directions are given to thebranches.




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Topology:




Graph:- A graph corresponding to a given network is obtained by

replacing all circuit elements with lines.


- Directed or Oriented graph: A graph that has all the nodesand branches numbered and also directions are given to thebranches.

- Subgraph: The subset of a graph. If the number of nodes andbranches of a subgraph is less than that of the graph, thesubgraph is said to be proper.


Network Toplogy Network Circuits & Their Graphs

Network Topology: An example

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A circuit with topologically equivalent graphs:

12

3

4

+

−V s

I s

R1I R1

R2I R2

R3

I R3

C

I C

LI L

12

3

4

12

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).



An Electrical Network & its Graph - I

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1 2 3

4

5 A

R1 R2

R4C R3

(a)

12

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 equalto the number of nodes or vertices.

There are at least two branches in a circuit.



An Electrical Network & its Graph - II

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1 2 3

4

5 A

R1I R1R2I R2

R4

I R4

C

I C

R3

I R3

(a)

12

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 anarrow, and the resulted graph is called oriented/directed graph .



An Electrical Network & its Graph - III

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12

3

4

5

+

−V s

I s

R

R1I 1

R2I 2

R3

I 3

C

I C

LI L

(a)

12

3

4

5

a

b c

f e

d

g

(b)

12

3

4, 5

a

b c

f ed

(c)

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

Note:

The active element branch is replaced by its internal resistance tosimplify analysis and computation.



An Electrical Network & its Graph - IV

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12

3

4

+

−V s

I vs

R

R1I 1

R2I 2

I I sC

I C

L

I L

(a)

12

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 analysisand computation.

1 0 / 2 0 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph


An Electrical Network & its Graph - V

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1A

1Ω

1Ω

1Ω

1Ω

1Ω

+ −1V

(a) (b) (c)

Figure 5 : (a) A circuit, and its- (b) simplified graph and (c) directedgraph.

Note:

When voltage source is not in series with any passive element in thegiven network, it is kept in the graph as a branch.



Network Topology: Terms and Definitions - IV

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Tree:




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Tree:

- A connected subgraph having all the nodes of a graph without

any loop.




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Tree:


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.




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Tree:







Twigs:




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Tree:







Twigs:

- The branches of a tree are known as twigs ,




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Tree:







Twigs:


Links or Chords:




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Tree:







Twigs:


Links or Chords:

- The branches that are removed from the graph while forming atree are termed as links or chords ,




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Tree:







Twigs:


Links or Chords:


- Links are complement of twigs.




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Tree:




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 towhich the tree belongs.

Twigs:


Links or Chords:



Co-tree:




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Tree:




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 towhich the tree belongs.

Twigs:


Links or Chords:



Co-tree:

- The graph constituted with links is known as c o-tr ee .



Tree and Cotree

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Given a Graph:1 2 3

4

a b

c d ef

Tree Twigs of tree Links of cotree

12 3

4

a b

c d ef

a,b,d c,e,f

12 3

4

a b

c d ef

a,d,f c,b,e



Summary and a Question:

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

12

3

4

a

b c

f ed




Q D h f ll i h i h b h d f ?

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12

3

4

a

b c

f ed

The number of nodes in this subgraph isequal to that of the given graph.




Q D th f ll i h ith b h d f t ?

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12

3

4

a

b c

f ed

The number of nodes in this subgraph isequal to that of the given graph.

But it has unconnected subgraphs andmoreover total number branches

= n− 1(= 3). Therefore, it is not a tree.


Network Toplogy References

Text Books & References

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M. E. Van ValkenburgNetwork Analysis, 3/e .

PHI, 2005.

W.H. Hayt, J.E. Kemmerly, S.M. DurbinEngineering Circuit Analysis, 8/e .MH, 2012.


Network Toplogy References

Text Books & References

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M. E. Van ValkenburgNetwork Analysis, 3/e .

PHI, 2005.

W.H. Hayt, J.E. Kemmerly, S.M. DurbinEngineering Circuit Analysis, 8/e .MH, 2012.

M. Nahvi, J.A. EdministerSchuamâĂŹs Outline Electric Circuits, 4/e .TMH, SIE, 2007.

A. Sudhakar, S.S. Palli

Circuits and Networks: Analysis and Synthesis, 2/e .TMH, 2002.


Network Toplogy Khublei Shibun!

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Thank You!

Any Question?


Home Assignment Graph and Incidence Matrix

Problems for Practice: Graph and Incidence Matrix

1 Classify whether each of the following graphs as planar or

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1. Classify whether each of the following graphs as planar ornonplanar .

2. Find the number of possible trees for each graph and draw allpossible trees.

1

2

3

4

(a)

ab c

d

(b)

1 2

3

4

5

(c)



Problems for Practice - II

Note: While replacing all elements of the network with lines to form a

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Note: While replacing all elements of the network with lines to form agraph, we replace active elements by their internal resistances

to simplify analysis and computation.For example - 1:

1

23 4

5

+

−V s

R1

I s

R2I 1

R3I 2

R4

I 3

C

I C

LI L

I s

(a)




Note: While replacing all elements of the network with lines to form a

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Note: While replacing all elements of the network with lines to form agraph, we replace active elements by their internal resistances

to simplify analysis and computation.For example - 1:

1

23 4

5

+

−V s

R1

I s

R2I 1

R3I 2

R4

I 3

C

I C

LI L

I s

(a)

23

4

1, 5

(b)




Note: Transformer gives a unconnected graph!

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Note: Transformer gives a unconnected graph!For example - 2:

12 3 4

56

+

−V s

R1

I 1

R2I 2

K

C I C

R3

I 3

I

(a)




Note: Transformer gives a unconnected graph!

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Note: Transformer gives a unconnected graph!For example - 2:

12 3 4

56

+

−V s

R1

I 1

R2I 2

K

C I C

R3

I 3

I

(a)

12

6

34

5

(b)



Few non-planar Graphs

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1

2

3

4

(a)

1 2

3

45

6

(b)


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Electrical Network Graph Theory Topology

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