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

Network Topology and Graph TheoryEE-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 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph

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

TreeTwigsCo-treeLinks or Chords





Circuit elements

Node

Branch

Path


Topology, rather Electrical network topology- Graph and its types






Circuit elements

Node

Branch

Path



Tree

TwigsCo-treeLinks or Chords





Circuit elements

Node

Branch

Path



TreeTwigs

Co-treeLinks or Chords





Circuit elements

Node

Branch

Path



TreeTwigsCo-tree

Links or Chords





Circuit elements

Node

Branch

Path






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 commonconnection,

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

Branch:

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

- Represented by a line in the graph.

Path:

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




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:





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:





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:






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


Branch:



Path:








Branch:


- Represented by a line in the graph.Path:








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











Path:




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 nodesthrough 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 amesh.

Network:

- The interconnection of two or more circuit elements forms anelectical 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 thatno branch passes over or under any other branch.

1Engineering Circuit Analysis, 8e4 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph



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 nodesthrough branches and comes to the original/starting node,



Network:


Circuit:


Planar circuit:





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 nodesthrough branches and comes to the original/starting node,



Network:


Circuit:


Planar circuit:






- Also known as closed path or circuit.

Mesh1 [2]:


Network:


Circuit:


Planar circuit:








Network:


Circuit:


Planar circuit:







- A loop that does not contain any other loops within it,

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

Network:


Circuit:


Planar circuit:








Network:


Circuit:


Planar circuit:








Network:- The interconnection of two or more circuit elements forms anelectical network.

Circuit:


Planar circuit:









Circuit:- Network that contains at least one closed path,

- Every circuit is a network, but not all networks are circuits.Planar circuit:









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

Planar circuit:









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 thatno branch passes over or under any other branch.



Network Topology: Terms and Definitions - III

Topology:

- Deals with properties of networks which are unaffected when thenetwork is stretched, twisted, or otherwise distorted the size andthe shape,

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

Graph:

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

- 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

- 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.




Topology:- Deals with properties of networks which are unaffected when thenetwork is stretched, twisted, or otherwise distorted the size andthe shape,


Graph:

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







Topology:- Deals with properties of networks which are unaffected when thenetwork is stretched, twisted, or otherwise distorted the size andthe shape,


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





Network Toplogy Network Circuits & Their Graphs

Network Topology: An example

A circuit with topologically equivalent graphs:

12

3

4

+−Vs

Is

R1IR1

R2IR2

R3

IR3

C

IC

LIL

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

1 2 3

4

5 A

R1 R2

R4CR3

(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

1 2 3

4

5 A

R1IR1

R2IR2

R4

IR4

C

IC

R3

IR3

(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

12

3

4

5

+−Vs

Is

R

R1I1

R2I2

R3

I3

C

IC

LIL

(a)

12

3

4

5

a

b c

fe

d

g

(b)

12

3

4, 5

a

b c

fed

(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

12

3

4

+−Vs

Ivs

R

R1I1

R2I2

IIsC

IC

LIL

(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.



An Electrical Network & its Graph - V

1A

1 Ω

1 Ω

1 Ω

1 Ω

1 Ω

+−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

Tree:

- A connected subgraph having all the nodes of a graph withoutany 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 atree are termed as links or chords,

- Links are complement of twigs.

Co-tree:

- The graph constituted with links is known as co-tree.




Tree:- A connected subgraph having all the nodes of a graph withoutany 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



Twigs:


Links or Chords:



Co-tree:






- 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



Twigs:


Links or Chords:



Co-tree:









Twigs:

- The branches of a tree are known as twigs,Links or Chords:



Co-tree:









Twigs:- The branches of a tree are known as twigs,

Links or Chords:



Co-tree:










Links or Chords:


- Links are complement of twigs.Co-tree:










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













Co-tree:




Tree and Cotree

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:

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

12

3

4

a

b c

fed

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

+ But it has unconnected subgraphs andmoreover total number branches6= n− 1(= 3). Therefore, it is not a tree.


Network Toplogy References

Text Books & References

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


Network Toplogy Khublei Shibun!

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 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 agraph, we replace active elements by their internal resistancesto simplify analysis and computation.For example - 1:

1

23 4

5

+−Vs

R1

Is

R2I1

R3I2

R4

I3

C

IC

LIL

Is

(a)

23

4

1, 5

(b)



Problems for Practice - II

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

12 3 4

56

+−Vs

R1

I1

R2I2

K

CIC

R3

I3

I

(a)

12

6

34

5

(b)



Few non-planar Graphs

1

2

3

4

(a)

1 2

3

45

6

(b)


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

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