Electrical Network, Graph Theory, Incidence Matrix, Topology

7/29/2019 Electrical Network, Graph Theory, Incidence Matrix, Topology

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Incidence Matrix, AiEE-304 ENT credits: 4 L{3} P{0} T{1}

Lairenlakpam Joyprakash Singh, PhD

Department of ECE,

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

August 8, 2013

1 / 1 7 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Incidence Matrix
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Graph Theory Incidence Matrix

Graph Theory: Incidence Matrix

Incidence matrix provides information like

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1 Which branches are incident at which nodes,

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1 Which branches are incident at which nodes,2 What are the orientations of branches relative to the nodes.


G
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Types:


G T I M
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Types:

1 Complete incidence matrix, Ai,


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Types:

1 Complete incidence matrix, Ai,

2 Reduced incidence matrix, A.


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Complete Incidence Matrix, Ai

A complete incidence matrix of a connected graph with 4 branchesand 5 nodes is given by:

Ai =

branchesnodes

a b c d

1 a11 a12 a13 a142 a21 a22 a23 a24

3 a31 a32 a33 a344 a41 a42 a43 a445 a51 a52 a53 a54


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A complete incidence matrix of a connected graph with 4 branchesand 5 nodes is given by:

Ai =

branchesnodes

a b c d

1 a11 a12 a13 a142 a21 a22 a23 a24

3 a31 a32 a33 a344 a41 a42 a43 a445 a51 a52 a53 a54

In a matrix, Ai, with n rows and b columns, an entry, aij , in the ith

row and jth column has the following values:

aij =

1, if the branch j is incident to and oriented away from the node, i,1, if the branch j is incident to and oriented towards the node, i,

0, if the branch j is not incident to the node, i.


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An complete incidence matrix of a graph with n = 3 and b = 4

Ai =

branchesnodes

a b c d

1 a11 a12 a13 a142 a21 a22 a23 a243 a31 a32 a23 a24

1 2

3

a

b

cd




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Ai =

branchesnodes

a b c d

1 a11 a12 a13 a142 a21 a22 a23 a243 a31 a32 a23 a24

1 2

3

a

b

cd

The complete incidence matrix of the above graph with matrixelement values is

branches

nodes

a b c d

Ai =

1

2

3


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Ai =

branchesnodes

a b c d

1 a11 a12 a13 a142 a21 a22 a23 a243 a31 a32 a23 a24

1 2

3

aa

b

cd


branchesnodes

a b c d

Ai =

1

2

3

1

0

1


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Ai =

branchesnodes

a b c d

1 a11 a12 a13 a142 a21 a22 a23 a243 a31 a32 a23 a24

1 2

3

a

bb

cd


branchesnodes

a b c d

Ai =

1

2

3

1 1

0 1

1 0


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Ai =

branchesnodes

a b c d

1 a11 a12 a13 a142 a21 a22 a23 a243 a31 a32 a23 a24

1 2

3

a

b

ccd


branchesnodes

a b c d

Ai =

1

2

3

1 1 0

0 1 1

1 0 1


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Ai =

branchesnodes

a b c d

1 a11 a12 a13 a142 a21 a22 a23 a243 a31 a32 a23 a24

1 2

3

a

b

cdd


branchesnodes

a b c d

Ai =

1

2

3

1 1 0 1

0 1 1 0

1 0 1 1


Network & Graph Network, Graph, Ai, and A
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Example - I

1 2 3

4

+Vs

Is

R

R1I1 R2I2

R3

I3

C

IC

LIL

(a)


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Example - I

1 2 3

4

+Vs

Is

R

R1I1 R2I2

R3

I3

C

IC

LIL

(a)

12

3

4

a

b c

d e f

(b)

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

The complete incidence matrix of the above graph is


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Example - I

1 2 3

4

+Vs

Is

R

R1I1 R2I2

R3

I3

C

IC

LIL

(a)

12

3

4

a

b c

d e f

(b)



Ai =


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Example - I

1 2 3

4

+Vs

Is

R

R1I1 R2I2

R3

I3

C

IC

LIL

(a)

12

3

4

aa

b c

d e f

(b)



Ai =

1

0

1

0


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Example - I

1 2 3

4

+Vs

Is

R

R1I1 R2I2

R3

I3

C

IC

LIL

(a)

12

3

4

a

bb c

d e f

(b)



Ai =

1 1

0 1

1 0

0 0


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Example - I

1 2 3

4

+Vs

Is

R

R1I1 R2I2

R3

I3

C

IC

LIL

(a)

12

3

4

a

b cc

d e f

(b)



Ai =

1 1 0

0 1 1

1 0 1

0 0 0


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Example - I

1 2 3

4

+Vs

Is

R

R1I1 R2I2

R3

I3

C

IC

LIL

(a)

12

3

4

a

b c

dd e f

(b)



Ai =

1 1 0 1

0 1 1 0

1 0 1 0

0 0 0 1



E l I
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Example - I

1 2 3

4

+Vs

Is

R

R1

I1 R2

I2

R3

I3

C

IC

LIL

(a)

12

3

4

a

b c

d ee f

(b)



Ai =

1 1 0 1 0

0 1 1 0 1

1 0 1 0 0

0 0 0 1 1



E l I


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Example - I

1 2 3

4

+Vs

Is

R

R1I1

R2I2

R3

I3

C

IC

LIL

(a)

12

3

4

a

b c

d e ff

(b)



Ai =

1 1 0 1 0 0

0 1 1 0 1 0

1 0 1 0 0 1

0 0 0 1 1 1



I id M t i A d A
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Incidence Matrices: Ai and A

Complete incidence matrix, Ai, has the order n b and satifies the followingproperties:



I id M t i A d A
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i) The algebraic sum of elements in any column ofAi is zero.



I id M t i A d A
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i) The algebraic sum of elements in any column ofAi is zero.ii) Given the incidence matrix A, the corresponding graph can easily be

constructed since Ai is a complete mathematical replica of the graph.



Incidence Matrices A and A
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iii) The determinant ofAi of a closed loop is zero.



Incidence Matrices: A and A
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Incidence matrix or Reduced incidence matrix, A:

1 If any row is removed from the incidence matrix Ai with nxb dimension, theremaining matrix is known as a reduced incidence matrix, A.Where n is the total number of nodes and b the total number of branches inthe given graph.



Incidence Matrices: A and A
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2 The order of the reduced incidence matrix A is (n 1) b, i.e. it has (n 1)rows and b columns.



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3 Normally, the row corresponding to the reference node is deleted to form areduced incidence matrix, A, from an incidence matrix, Ai.



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3 Normally, the row corresponding to the reference node is deleted to form areduced incidence matrix, A, from an incidence matrix, Ai.

Ai =

1 1 0 1 0 00 1 1 0 1 0

1 0 1 0 0 10 0 0 1 1 1

A =

1 1 0 1 0 00 1 1 0 1 0

1 0 1 0 0 1

Incidence Matrix, Ai Reduced incidence matrix, A



Number of possible trees in the graph Fig 1(b)
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Number of possible trees in the graph, Fig. 1(b)Let us considered the Figure 1 and write the incidence matrix, Ai and its reduced incidence matrix,A as

Ai =

1 1 0 1 0 00 1 1 0 1 0

1 0 1 0 0 10 0 0 1 1 1

A = 1 1 0 1 0 0

0 1 1 0 1 01 0 1 0 0 1

The transpose of the reduced incidence matrix, A is then given by

AT =

1 0 11 1 00 1 1

1 0 00 1 0

0 0 1

And now we have,

AAT =

1 1 0 1 0 00 1 1 0 1 0

1 0 1 0 0 1

1 0 11 1 00 1 1

1 0 00 1 00 0 1

=

3 1 11 3 11 1 3

Therefore the possible number of trees that can be constructed from the graph in Figure 1(b) is

AAT = 3

3 1

1 3

(1)1 11 3

+ (1)1 31 1

= 3(9 1) + (3 1) 1(1 + 3)

Although 6C3 = 20 combinations exist for tree formation, the p ossible combnations areAAT = 24 4 4 = 16 only.


Network & Graph Possible trees of a network

Drawing all possible trees of the network shown in 1(a)
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Drawing all possible trees of the network shown in 1(a)

1 2 3

4

+

Vs

Is

R

R1

I1

R2

I2

R3

I3

C

IC

LIL

(a) A Circuit

12

3

4

a

b c

fed

(b) The graph

Possible trees of the above graph [Trees 1 - 4]:

12

3

4

a

b c

fed

(c) Tree 1

12

3

4

a

b c

fed

(d) Tree 2

12

3

4

a

b c

fed

(e) Tree 3

12

3

4

a

b c

fed

(f) Tree 4



Drawing all possible trees
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Drawing all possible trees . . .

Possible trees of the given graph [Trees 5 - 10]:

12

3

4

a

b c

fed

(g) Tree 5

12

3

4

a

b c

fed

(h) Tree 6

12

3

4

a

b c

fed

(i) Tree 7

12

3

4

a

b c

fed

(j) Tree 8

12

3

4

a

b c

fed

(k) Tree 9

12

3

4

a

b c

fed

(l) Tree 10



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Possible trees of the given graph [Trees 11 - 16]:

12

3

4

a

b c

fed

(m) Tree 11

12

3

4

a

b c

fed

(n) Tree 12

12

3

4

a

b c

fed

(o) Tree 13

12

3

4

a

b c

fed

(p) Tree 14

12

3

4

a

b c

fed

(q) Tree 15

12

3

4

a

b c

fed

(r) Tree 16

1 0 / 1 7 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Incidence Matrix


Summary:
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Summary:

In the given graph, we have the number of nodes and the branches are n = 4 and b = 6 respectively.Hence, total number of twigs and links in a tree are, t = n 1 = 3, and l = b t = b n + 1 = 3.Possible trees of the given graph Fig. 1(b) are only:

Possible trees(16 Nos.) drawn above may be written as:

Tree 1 : Twigs{a, b, d},Links{c, e, f}Tree 2 : Twigs{a, b, e},Links{c, d, f}Tree 3 : Twigs{a, c, e},Links{b, d, f}Tree 4 : Twigs{a, c, f},Links{b, d, e}Tree 5 : Twigs{a, b, f},Links{c, d, e}

.

.

.

.

.

.

.

.

.Tree 16 :Twigs{b, e, f},Links{a, c, d}


Network & Graph

Exercise:
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Q. Draw a complete graph from the following incidence matrix.

A =

1 1 0 1 0 01 0 1 0 0 1

0 1 1 0 1 0


Network & Graph

Exercise:
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A =

1 1 0 1 0 01 0 1 0 0 1

0 1 1 0 1 0

Answer: Creating a complete Incidence Matrix, Ai, from the givenreduced incidence matrix, A, we get

Ai =

1 1 0 1 0 01 0 1 0 0 1

0 1 1 0 1 00 0 0 1 1 1


Network & Graph

Exercise:
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A =

1 1 0 1 0 01 0 1 0 0 1

0 1 1 0 1 0


Ai =

1 1 0 1 0 01 0 1 0 0 10 1 1 0 1 00 0 0 1 1 1

And its graph

1 23

4

a

OR

1 2

3 4

a


Network & Graph

Exercise:
http://goforward/http://find/http://goback/


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A =

1 1 0 1 0 01 0 1 0 0 1

0 1 1 0 1 0


Ai =

1 1 0 1 0 01 0 1 0 0 1

0 1 1 0 1 00 0 0 1 1 1

And its graph

1 23

4

a

b

OR

1 2

3 4

a

b


Network & Graph

Exercise:
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A =

1 1 0 1 0 01 0 1 0 0 1

0 1 1 0 1 0


Ai =

1 1 0 1 0 01 0 1 0 0 1

0 1 1 0 1 00 0 0 1 1 1

And its graph

1 23

4

a

b c

OR

1 2

3 4

a

bc


Network & Graph

Exercise:
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A =

1 1 0 1 0 01 0 1 0 0 1

0 1 1 0 1 0


Ai =

1 1 0 1 0 01 0 1 0 0 1

0 1 1 0 1 00 0 0 1 1 1

And its graph

1 23

4

a

b c

dOR

1 2

3 4

a

b

c

d


Network & Graph

Exercise:


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A =

1 1 0 1 0 01 0 1 0 0 1

0 1 1 0 1 0


Ai =

1 1 0 1 0 01 0 1 0 0 1

0 1 1 0 1 00 0 0 1 1 1

And its graph

1 23

4

a

b c

d eOR

1 2

3 4

a

b

c

d

e


Network & Graph

Exercise:
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A =

1 1 0 1 0 01 0 1 0 0 1

0 1 1 0 1 0


Ai =

1 1 0 1 0 0

1 0 1 0 0 10 1 1 0 1 00 0 0 1 1 1

And its graph

1 23

4

a

b c

d e fOR

1 2

3 4

a

b

c

d

e

f


Network & Graph

Exercise:
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A =

1 1 0 1 0 0

1 0 1 0 0 10 1 1 0 1 0


Ai =

1 1 0 1 0 0

1 0 1 0 0 10 1 1 0 1 00 0 0 1 1 1

And its graph

1 23

4

a

b c

d e fOR

1 2

3 4

a

b

c

d

e

f


Network & Graph Incidence Matrix and KCL

Incidence Matrix and KCL
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Incidence Matrix and KCL:

Kirchhoffs current law (KCL) of a graph can be expressed in terms of

the reduced incidence matrix as AiIb = 0 where Ib represents branchcurrent vectors I1, I2, I3, I4, I5, and I6.



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Incidence Matrix and KCL:

Kirchhoffs current law (KCL) of a graph can be expressed in terms of

the reduced incidence matrix as AiIb = 0 where Ib represents branchcurrent vectors I1, I2, I3, I4, I5, and I6.

For example:

1 0 1 0 1 0 0 01 1 0 0 0 0 0 1

0 0 1 1 0 1 0 00 1 0 1 0 0 1 0

I1I2

I3I4I5I6

=

0

000



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A complete incidence matrix Ai and a graph constructedfrom it are given below

Ai =

1 0 1 0 1 0 0 0

1 1 0 0 0 0 0 10 0 1 1 0 1 0 00 1 0 1 0 0 1 00 0 0 0 1 1 1 1

If branch currents of branches a,b,c,d,e,f,g,h areI1, I2, I3, I4, I5, I6, I7 and I8 respectively, then usingAIb = 0 we have

1

2

34

5

ab

cd

e f g

h

1 0 1 0 1 0 0 01 1 0 0 0 0 0 1

0 0 1 1 0 1 0 00 1 0 1 0 0 1 0

I1

I2I3I4I5I6I7I8

=

0000

Branch current equations: from above equation- after applying KCL at nodes 1, 2, 3 and 4in the graph-

I1 + I3 I5 = 0I1 + I2 + I8 = 0I3 + I4 + I6 = 0I2 I4 + I7 = 0

I1 + I3 I5 = 0I1 + I2 + I8 = 0I3 + I4 + I6 = 0I2 I4 + I7 = 0



Incidence Matrix and Branch Voltages


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Incidence Matrix and Branch Voltages:If branch voltages of branches a, b, c, d, e, f, g and h in the ciruit areVa, Vb, Vc, Vd, Ve, Vf, Vg and Vh while nodes-to-reference voltages aree1, e2, e3, e4, e5, e6, e7 and e8 respectively, then the branch voltagevector, V, and the node voltage vector e are given by

V =

VaVbVcVdVe

VfVgVh

and e =

e1e2e3e4e5

e6e7e8

Then branch voltages of the last graph are then given by [AT][e] = [V]

1 1 0 00 1 0 11 0 1 00 0 1 1

1 0 0 00 0 1 00 0 0 10 1 0 0

e1e2e3e

4e5e6e7e8

=

VaVbVcVdVeVfVgVh


Network & Graph 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 & Graph 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. EdministerSchuams Outline Electric Circuits, 4/e.TMH, SIE, 2007.

A. Sudhakar, S.S. Palli

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


Network & Graph Khublei Shibun!


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Thank You!

Any Question?

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Electrical Network, Graph Theory, Incidence Matrix, Topology

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