EE - 304 Electrical Network TheoryTie-Set(Loop) Matrix

North-Eastern Hill University

29.08.2016 joIp_rkas, ECE, NEHU, Shillong - 793 022

Tie-Set (Loop/Circuit) MatrixComplete Loop/Circuit Matrix, Bc

Complete loop matrix or simply circuit matrix is used to describe the way in whichthe branches constitute loops or circuits in a graph.

- To write this matrix in a systematic way, an arbitrary orientation is specified foreach loop using an ordered list of its branches and nodes.

- The total number of loops are determined using an exhaustive search.

There are two types of loop/circuit matrices:- Complete loop/circuit matrix, BC,- Loop/circuit/tie-set matrix, B.

For a graph having n nodes and b branches, the complete loop matrix Bc , is arectangular matrix with b columns and as many rows as there are loops.

AeL. je. siHh/L. J. Singh(nehu/NEHU) EE - 304 Electrical Network Theory 29.08.2016 1 / 14

Tie-Set (Loop/Circuit) MatrixComplete Loop/Circuit Matrix, Bc

Complete loop matrix or simply circuit matrix is used to describe the way in whichthe branches constitute loops or circuits in a graph.

- To write this matrix in a systematic way, an arbitrary orientation is specified foreach loop using an ordered list of its branches and nodes.

- The total number of loops are determined using an exhaustive search.

There are two types of loop/circuit matrices:- Complete loop/circuit matrix, BC,- Loop/circuit/tie-set matrix, B.

For a graph having n nodes and b branches, the complete loop matrix Bc , is arectangular matrix with b columns and as many rows as there are loops.

AeL. je. siHh/L. J. Singh(nehu/NEHU) EE - 304 Electrical Network Theory 29.08.2016 1 / 14

Tie-Set (Loop/Circuit) MatrixComplete Loop/Circuit Matrix, Bc

Complete loop matrix or simply circuit matrix is used to describe the way in whichthe branches constitute loops or circuits in a graph.

- To write this matrix in a systematic way, an arbitrary orientation is specified foreach loop using an ordered list of its branches and nodes.

- The total number of loops are determined using an exhaustive search.

There are two types of loop/circuit matrices:- Complete loop/circuit matrix, BC,- Loop/circuit/tie-set matrix, B.

For a graph having n nodes and b branches, the complete loop matrix Bc , is arectangular matrix with b columns and as many rows as there are loops.

AeL. je. siHh/L. J. Singh(nehu/NEHU) EE - 304 Electrical Network Theory 29.08.2016 1 / 14

Tie-Set (Loop/Circuit) MatrixComplete Loop/Circuit Matrix, Bc

Complete loop matrix or simply circuit matrix is used to describe the way in whichthe branches constitute loops or circuits in a graph.

- To write this matrix in a systematic way, an arbitrary orientation is specified foreach loop using an ordered list of its branches and nodes.

- The total number of loops are determined using an exhaustive search.

There are two types of loop/circuit matrices:

- Complete loop/circuit matrix, BC,- Loop/circuit/tie-set matrix, B.

For a graph having n nodes and b branches, the complete loop matrix Bc , is arectangular matrix with b columns and as many rows as there are loops.

AeL. je. siHh/L. J. Singh(nehu/NEHU) EE - 304 Electrical Network Theory 29.08.2016 1 / 14

Tie-Set (Loop/Circuit) MatrixComplete Loop/Circuit Matrix, Bc

Complete loop matrix or simply circuit matrix is used to describe the way in whichthe branches constitute loops or circuits in a graph.

- To write this matrix in a systematic way, an arbitrary orientation is specified foreach loop using an ordered list of its branches and nodes.

- The total number of loops are determined using an exhaustive search.

There are two types of loop/circuit matrices:- Complete loop/circuit matrix, BC,

- Loop/circuit/tie-set matrix, B.

For a graph having n nodes and b branches, the complete loop matrix Bc , is arectangular matrix with b columns and as many rows as there are loops.

AeL. je. siHh/L. J. Singh(nehu/NEHU) EE - 304 Electrical Network Theory 29.08.2016 1 / 14

Tie-Set (Loop/Circuit) MatrixComplete Loop/Circuit Matrix, Bc

Complete loop matrix or simply circuit matrix is used to describe the way in whichthe branches constitute loops or circuits in a graph.

- To write this matrix in a systematic way, an arbitrary orientation is specified foreach loop using an ordered list of its branches and nodes.

- The total number of loops are determined using an exhaustive search.

There are two types of loop/circuit matrices:- Complete loop/circuit matrix, BC,- Loop/circuit/tie-set matrix, B.

For a graph having n nodes and b branches, the complete loop matrix Bc , is arectangular matrix with b columns and as many rows as there are loops.

AeL. je. siHh/L. J. Singh(nehu/NEHU) EE - 304 Electrical Network Theory 29.08.2016 1 / 14

Tie-Set (Loop/Circuit) MatrixComplete Loop/Circuit Matrix, Bc

Complete loop matrix or simply circuit matrix is used to describe the way in whichthe branches constitute loops or circuits in a graph.

- To write this matrix in a systematic way, an arbitrary orientation is specified foreach loop using an ordered list of its branches and nodes.

- The total number of loops are determined using an exhaustive search.

There are two types of loop/circuit matrices:- Complete loop/circuit matrix, BC,- Loop/circuit/tie-set matrix, B.

For a graph having n nodes and b branches, the complete loop matrix Bc , is arectangular matrix with b columns and as many rows as there are loops.

AeL. je. siHh/L. J. Singh(nehu/NEHU) EE - 304 Electrical Network Theory 29.08.2016 1 / 14

Tie-Set (Loop/Circuit) MatrixComplete Loop/Circuit Matrix, Bc

That is, the complete loop matrix, Bc, of the graph of n nodes and b branches isBranches→

Circuits↓ 1 2 3 4 . . . b

Bc =

L1L2L3L4L5L6...

b11 b12 b13 b14 . . . b1bb21 b22 b23 b24 . . . b2bb31 b32 b33 b34 . . . b3bb41 b42 b43 b44 . . . b4bb51 b52 b53 b54 . . . b5bb61 b62 b63 b64 . . . b6b...

......

......

...

That is, the complete loop matrix, Bc, of the graph of n nodes and b branches isBranches→

Loops↓ 1 2 3 4 . . . b

Bc =

L1L2L3L4L5L6...

b11 b12 b13 b14 . . . b1bb21 b22 b23 b24 . . . b2bb31 b32 b33 b34 . . . b3bb41 b42 b43 b44 . . . b4bb51 b52 b53 b54 . . . b5bb61 b62 b63 b64 . . . b6b...

......

......

...

Orientation ofloops/circuits in the graphis considered to be arbitrary

while deriving Bc!

There will be b number ofcolumns in Bc!

Number of rows in Bcequals to that of possible

loops in the graph!

The elements of the complete loop/circuit matrix have the following values:

bij =

1, if the branch j is in the loop i and their orientations coincide,−1, if the branch j is in the loop i and their orientations are opposite,0, if the branch j is not in the loop i.

AeL. je. siHh/L. J. Singh(nehu/NEHU) EE - 304 Electrical Network Theory 29.08.2016 2 / 14

Tie-Set (Loop/Circuit) MatrixComplete Loop/Circuit Matrix, Bc

That is, the complete loop matrix, Bc, of the graph of n nodes and b branches isBranches→

Circuits↓ 1 2 3 4 . . . b

Bc =

L1L2L3L4L5L6...

b11 b12 b13 b14 . . . b1bb21 b22 b23 b24 . . . b2bb31 b32 b33 b34 . . . b3bb41 b42 b43 b44 . . . b4bb51 b52 b53 b54 . . . b5bb61 b62 b63 b64 . . . b6b...

......

......

...

That is, the complete loop matrix, Bc, of the graph of n nodes and b branches isBranches→

Loops↓ 1 2 3 4 . . . b

Bc =

L1L2L3L4L5L6...

b11 b12 b13 b14 . . . b1bb21 b22 b23 b24 . . . b2bb31 b32 b33 b34 . . . b3bb41 b42 b43 b44 . . . b4bb51 b52 b53 b54 . . . b5bb61 b62 b63 b64 . . . b6b...

......

......

...

Orientation ofloops/circuits in the graphis considered to be arbitrary

while deriving Bc!

There will be b number ofcolumns in Bc!

Number of rows in Bcequals to that of possible

loops in the graph!

The elements of the complete loop/circuit matrix have the following values:

bij =

1, if the branch j is in the loop i and their orientations coincide,−1, if the branch j is in the loop i and their orientations are opposite,0, if the branch j is not in the loop i.

AeL. je. siHh/L. J. Singh(nehu/NEHU) EE - 304 Electrical Network Theory 29.08.2016 2 / 14

Tie-Set (Loop/Circuit) MatrixComplete Loop/Circuit Matrix, Bc

That is, the complete loop matrix, Bc, of the graph of n nodes and b branches isBranches→

Circuits↓ 1 2 3 4 . . . b

Bc =

L1L2L3L4L5L6...

b11 b12 b13 b14 . . . b1bb21 b22 b23 b24 . . . b2bb31 b32 b33 b34 . . . b3bb41 b42 b43 b44 . . . b4bb51 b52 b53 b54 . . . b5bb61 b62 b63 b64 . . . b6b...

......

......

...

That is, the complete loop matrix, Bc, of the graph of n nodes and b branches isBranches→

Loops↓ 1 2 3 4 . . . b

Bc =

L1L2L3L4L5L6...

b11 b12 b13 b14 . . . b1bb21 b22 b23 b24 . . . b2bb31 b32 b33 b34 . . . b3bb41 b42 b43 b44 . . . b4bb51 b52 b53 b54 . . . b5bb61 b62 b63 b64 . . . b6b...

......

......

...

Orientation ofloops/circuits in the graphis considered to be arbitrary

while deriving Bc!

There will be b number ofcolumns in Bc!

Number of rows in Bcequals to that of possible

loops in the graph!

The elements of the complete loop/circuit matrix have the following values:

bij =

1, if the branch j is in the loop i and their orientations coincide,−1, if the branch j is in the loop i and their orientations are opposite,0, if the branch j is not in the loop i.

AeL. je. siHh/L. J. Singh(nehu/NEHU) EE - 304 Electrical Network Theory 29.08.2016 2 / 14

Tie-Set (Loop/Circuit) MatrixComplete Loop/Circuit Matrix, Bc

That is, the complete loop matrix, Bc, of the graph of n nodes and b branches isBranches→

Circuits↓ 1 2 3 4 . . . b

Bc =

L1L2L3L4L5L6...

b11 b12 b13 b14 . . . b1bb21 b22 b23 b24 . . . b2bb31 b32 b33 b34 . . . b3bb41 b42 b43 b44 . . . b4bb51 b52 b53 b54 . . . b5bb61 b62 b63 b64 . . . b6b...

......

......

...

That is, the complete loop matrix, Bc, of the graph of n nodes and b branches isBranches→

Loops↓ 1 2 3 4 . . . b

Bc =

L1L2L3L4L5L6...

b11 b12 b13 b14 . . . b1bb21 b22 b23 b24 . . . b2bb31 b32 b33 b34 . . . b3bb41 b42 b43 b44 . . . b4bb51 b52 b53 b54 . . . b5bb61 b62 b63 b64 . . . b6b...

......

......

...

Orientation ofloops/circuits in the graphis considered to be arbitrary

while deriving Bc!

There will be b number ofcolumns in Bc!

Number of rows in Bcequals to that of possible

loops in the graph!

The elements of the complete loop/circuit matrix have the following values:

bij =

1, if the branch j is in the loop i and their orientations coincide,−1, if the branch j is in the loop i and their orientations are opposite,0, if the branch j is not in the loop i.

AeL. je. siHh/L. J. Singh(nehu/NEHU) EE - 304 Electrical Network Theory 29.08.2016 2 / 14

Tie-Set (Loop/Circuit) MatrixComplete Loop/Circuit Matrix, Bc

Consider the following network and let’s find its Bc!

1 2 3

4

5 mA

5 mA

0.4ix

0.4ix

25µF

25µF25µF

10 kΩ

10 kΩ10 kΩ

27 kΩ

27 kΩ27 kΩ

2 H

2 H2 H

ix

Check what’s missing in the given circuit!- Assign node numbers- Assign branch current directions

Let’s now draw the graph of the given circuit!a. Identify arbitrary locations for 4 nodesb. Connect nodes for each branch in order

1 2 3

4

aa

bb cc

dd ee ff

Graph could have been drawn this waydepending on the location of nodes!Or this way!!

1 2

34

ab

cd e

f1

2 3

4

a

b

c

d

e f1

2 34

a

bc f

e

d

Or this way!

AeL. je. siHh/L. J. Singh(nehu/NEHU) EE - 304 Electrical Network Theory 29.08.2016 3 / 14

Tie-Set (Loop/Circuit) MatrixComplete Loop/Circuit Matrix, Bc

Consider the following network and let’s find its Bc!

1 2 3

4

5 mA

5 mA

0.4ix

0.4ix

25µF

25µF25µF

10 kΩ

10 kΩ10 kΩ

27 kΩ

27 kΩ27 kΩ

2 H

2 H2 H

ix

Check what’s missing in the given circuit!

- Assign node numbers- Assign branch current directions

Let’s now draw the graph of the given circuit!a. Identify arbitrary locations for 4 nodesb. Connect nodes for each branch in order

1 2 3

4

aa

bb cc

dd ee ff

Graph could have been drawn this waydepending on the location of nodes!Or this way!!

1 2

34

ab

cd e

f1

2 3

4

a

b

c

d

e f1

2 34

a

bc f

e

d

Or this way!

AeL. je. siHh/L. J. Singh(nehu/NEHU) EE - 304 Electrical Network Theory 29.08.2016 3 / 14

Tie-Set (Loop/Circuit) MatrixComplete Loop/Circuit Matrix, Bc

Consider the following network and let’s find its Bc!

1 2 3

4

5 mA

5 mA

0.4ix

0.4ix

25µF

25µF25µF

10 kΩ

10 kΩ10 kΩ

27 kΩ

27 kΩ27 kΩ

2 H

2 H2 H

ix

Check what’s missing in the given circuit!- Assign node numbers

- Assign branch current directions

Let’s now draw the graph of the given circuit!a. Identify arbitrary locations for 4 nodesb. Connect nodes for each branch in order

1 2 3

4

aa

bb cc

dd ee ff

Graph could have been drawn this waydepending on the location of nodes!Or this way!!

1 2

34

ab

cd e

f1

2 3

4

a

b

c

d

e f1

2 34

a

bc f

e

d

Or this way!

AeL. je. siHh/L. J. Singh(nehu/NEHU) EE - 304 Electrical Network Theory 29.08.2016 3 / 14

Tie-Set (Loop/Circuit) MatrixComplete Loop/Circuit Matrix, Bc

Consider the following network and let’s find its Bc!

1 2 3

4

5 mA

5 mA

0.4ix

0.4ix25µF

25µF

25µF

10 kΩ

10 kΩ

10 kΩ

27 kΩ

27 kΩ

27 kΩ

2 H

2 H

2 H

ix

Check what’s missing in the given circuit!- Assign node numbers- Assign branch current directions

Let’s now draw the graph of the given circuit!a. Identify arbitrary locations for 4 nodesb. Connect nodes for each branch in order

1 2 3

4

aa

bb cc

dd ee ff

Graph could have been drawn this waydepending on the location of nodes!Or this way!!

1 2

34

ab

cd e

f1

2 3

4

a

b

c

d

e f1

2 34

a

bc f

e

d

Or this way!

AeL. je. siHh/L. J. Singh(nehu/NEHU) EE - 304 Electrical Network Theory 29.08.2016 3 / 14

Tie-Set (Loop/Circuit) MatrixComplete Loop/Circuit Matrix, Bc

Consider the following network and let’s find its Bc!

1 2 3

4

5 mA

5 mA

0.4ix

0.4ix25µF

25µF

25µF

10 kΩ

10 kΩ

10 kΩ

27 kΩ

27 kΩ

27 kΩ

2 H

2 H

2 H

ix

Check what’s missing in the given circuit!- Assign node numbers- Assign branch current directions

Let’s now draw the graph of the given circuit!

a. Identify arbitrary locations for 4 nodesb. Connect nodes for each branch in order

1 2 3

4

aa

bb cc

dd ee ff

Graph could have been drawn this waydepending on the location of nodes!Or this way!!

1 2

34

ab

cd e

f1

2 3

4

a

b

c

d

e f1

2 34

a

bc f

e

d

Or this way!

AeL. je. siHh/L. J. Singh(nehu/NEHU) EE - 304 Electrical Network Theory 29.08.2016 3 / 14

Tie-Set (Loop/Circuit) MatrixComplete Loop/Circuit Matrix, Bc

Consider the following network and let’s find its Bc!

1 2 3

4

5 mA

5 mA

0.4ix

0.4ix25µF

25µF

25µF

10 kΩ

10 kΩ

10 kΩ

27 kΩ

27 kΩ

27 kΩ

2 H

2 H

2 H

ix

Check what’s missing in the given circuit!- Assign node numbers- Assign branch current directions

Let’s now draw the graph of the given circuit!a. Identify arbitrary locations for 4 nodes

b. Connect nodes for each branch in order

1 2 3

4

aa

bb cc

dd ee ff

Graph could have been drawn this waydepending on the location of nodes!Or this way!!

1 2

34

ab

cd e

f1

2 3

4

a

b

c

d

e f1

2 34

a

bc f

e

d

Or this way!

AeL. je. siHh/L. J. Singh(nehu/NEHU) EE - 304 Electrical Network Theory 29.08.2016 3 / 14

Tie-Set (Loop/Circuit) MatrixComplete Loop/Circuit Matrix, Bc

Consider the following network and let’s find its Bc!

1 2 3

4

5 mA

5 mA

0.4ix

0.4ix25µF

25µF

25µF

10 kΩ

10 kΩ

10 kΩ

27 kΩ

27 kΩ

27 kΩ

2 H2 H

2 H

ix

Check what’s missing in the given circuit!- Assign node numbers- Assign branch current directions

Let’s now draw the graph of the given circuit!a. Identify arbitrary locations for 4 nodesb. Connect nodes for each branch in order

1 2 3

4

a

a

bb cc

dd ee ff

Graph could have been drawn this waydepending on the location of nodes!Or this way!!

1 2

34

ab

cd e

f1

2 3

4

a

b

c

d

e f1

2 34

a

bc f

e

d

Or this way!

AeL. je. siHh/L. J. Singh(nehu/NEHU) EE - 304 Electrical Network Theory 29.08.2016 3 / 14

Tie-Set (Loop/Circuit) MatrixComplete Loop/Circuit Matrix, Bc

Consider the following network and let’s find its Bc!

1 2 3

4

5 mA

5 mA

0.4ix

0.4ix

25µF

25µF

25µF

10 kΩ

10 kΩ

10 kΩ

27 kΩ

27 kΩ

27 kΩ

2 H

2 H

2 H

ix

Check what’s missing in the given circuit!- Assign node numbers- Assign branch current directions

Let’s now draw the graph of the given circuit!a. Identify arbitrary locations for 4 nodesb. Connect nodes for each branch in order

1 2 3

4

a

a

b

b cc

dd ee ff

Graph could have been drawn this waydepending on the location of nodes!Or this way!!

1 2

34

ab

cd e

f1

2 3

4

a

b

c

d

e f1

2 34

a

bc f

e

d

Or this way!

AeL. je. siHh/L. J. Singh(nehu/NEHU) EE - 304 Electrical Network Theory 29.08.2016 3 / 14

Tie-Set (Loop/Circuit) MatrixComplete Loop/Circuit Matrix, Bc

Consider the following network and let’s find its Bc!

1 2 3

4

5 mA

5 mA

0.4ix

0.4ix25µF

25µF

25µF

10 kΩ10 kΩ

10 kΩ

27 kΩ

27 kΩ

27 kΩ

2 H

2 H

2 H

ix

Check what’s missing in the given circuit!- Assign node numbers- Assign branch current directions

Let’s now draw the graph of the given circuit!a. Identify arbitrary locations for 4 nodesb. Connect nodes for each branch in order

1 2 3

4

a

a

b

b c

c

dd ee ff

Graph could have been drawn this waydepending on the location of nodes!Or this way!!

1 2

34

ab

cd e

f1

2 3

4

a

b

c

d

e f1

2 34

a

bc f

e

d

Or this way!

AeL. je. siHh/L. J. Singh(nehu/NEHU) EE - 304 Electrical Network Theory 29.08.2016 3 / 14

Tie-Set (Loop/Circuit) MatrixComplete Loop/Circuit Matrix, Bc

Consider the following network and let’s find its Bc!

1 2 3

4

5 mA

5 mA

0.4ix

0.4ix25µF

25µF

25µF

10 kΩ

10 kΩ

10 kΩ

27 kΩ

27 kΩ

27 kΩ

2 H

2 H

2 H

ix

Check what’s missing in the given circuit!- Assign node numbers- Assign branch current directions

Let’s now draw the graph of the given circuit!a. Identify arbitrary locations for 4 nodesb. Connect nodes for each branch in order

1 2 3

4

a

a

b

b

c

c

d

d ee ff

Graph could have been drawn this waydepending on the location of nodes!Or this way!!

1 2

34

ab

cd e

f1

2 3

4

a

b

c

d

e f1

2 34

a

bc f

e

d

Or this way!

AeL. je. siHh/L. J. Singh(nehu/NEHU) EE - 304 Electrical Network Theory 29.08.2016 3 / 14

Tie-Set (Loop/Circuit) MatrixComplete Loop/Circuit Matrix, Bc

Consider the following network and let’s find its Bc!

1 2 3

4

5 mA

5 mA

0.4ix

0.4ix25µF25µF

25µF

10 kΩ

10 kΩ

10 kΩ

27 kΩ

27 kΩ

27 kΩ

2 H

2 H

2 H

ix

Check what’s missing in the given circuit!- Assign node numbers- Assign branch current directions

Let’s now draw the graph of the given circuit!a. Identify arbitrary locations for 4 nodesb. Connect nodes for each branch in order

1 2 3

4

a

a

b

b

c

c

d

d e

e ff

Graph could have been drawn this waydepending on the location of nodes!Or this way!!

1 2

34

ab

cd e

f1

2 3

4

a

b

c

d

e f1

2 34

a

bc f

e

d

Or this way!

AeL. je. siHh/L. J. Singh(nehu/NEHU) EE - 304 Electrical Network Theory 29.08.2016 3 / 14

Tie-Set (Loop/Circuit) MatrixComplete Loop/Circuit Matrix, Bc

Consider the following network and let’s find its Bc!

1 2 3

4

5 mA

5 mA

0.4ix

0.4ix25µF

25µF

25µF

10 kΩ

10 kΩ

10 kΩ

27 kΩ27 kΩ

27 kΩ

2 H

2 H

2 H

ix

Check what’s missing in the given circuit!- Assign node numbers- Assign branch current directions

Let’s now draw the graph of the given circuit!a. Identify arbitrary locations for 4 nodesb. Connect nodes for each branch in order

1 2 3

4

a

a

b

b

c

c

d

d

e

e f

f

Graph could have been drawn this waydepending on the location of nodes!Or this way!!

1 2

34

ab

cd e

f1

2 3

4

a

b

c

d

e f1

2 34

a

bc f

e

d

Or this way!

AeL. je. siHh/L. J. Singh(nehu/NEHU) EE - 304 Electrical Network Theory 29.08.2016 3 / 14

Tie-Set (Loop/Circuit) MatrixComplete Loop/Circuit Matrix, Bc

Consider the following network and let’s find its Bc!

1 2 3

4

5 mA

5 mA

0.4ix

0.4ix25µF

25µF

25µF

10 kΩ

10 kΩ

10 kΩ

27 kΩ

27 kΩ

27 kΩ

2 H

2 H

2 H

ix

Check what’s missing in the given circuit!- Assign node numbers- Assign branch current directions

Let’s now draw the graph of the given circuit!a. Identify arbitrary locations for 4 nodesb. Connect nodes for each branch in order

1 2 3

4

a

a

b

b

c

c

d

d

e

e

f

f

Graph could have been drawn this waydepending on the location of nodes!Or this way!!

1 2

34

ab

cd e

f1

2 3

4

a

b

c

d

e f1

2 34

a

bc f

e

d

Or this way!

AeL. je. siHh/L. J. Singh(nehu/NEHU) EE - 304 Electrical Network Theory 29.08.2016 3 / 14

Tie-Set (Loop/Circuit) MatrixComplete Loop/Circuit Matrix, Bc

Consider the following network and let’s find its Bc!

1 2 3

4

5 mA

5 mA

0.4ix

0.4ix25µF

25µF

25µF

10 kΩ

10 kΩ

10 kΩ

27 kΩ

27 kΩ

27 kΩ

2 H

2 H

2 H

ix

Check what’s missing in the given circuit!- Assign node numbers- Assign branch current directions

Let’s now draw the graph of the given circuit!a. Identify arbitrary locations for 4 nodesb. Connect nodes for each branch in order

1 2 3

4

aa

bb cc

dd ee ff

Graph could have been drawn this waydepending on the location of nodes!

Or this way!!

1 2

34

ab

cd e

f

1

2 3

4

a

b

c

d

e f1

2 34

a

bc f

e

d

Or this way!

AeL. je. siHh/L. J. Singh(nehu/NEHU) EE - 304 Electrical Network Theory 29.08.2016 3 / 14

Tie-Set (Loop/Circuit) MatrixComplete Loop/Circuit Matrix, Bc

Consider the following network and let’s find its Bc!

1 2 3

4

5 mA

5 mA

0.4ix

0.4ix25µF

25µF

25µF

10 kΩ

10 kΩ

10 kΩ

27 kΩ

27 kΩ

27 kΩ

2 H

2 H

2 H

ix

Check what’s missing in the given circuit!- Assign node numbers- Assign branch current directions

Let’s now draw the graph of the given circuit!a. Identify arbitrary locations for 4 nodesb. Connect nodes for each branch in order

1 2 3

4

aa

bb cc

dd ee ff

Graph could have been drawn this waydepending on the location of nodes!Or this way!!

1 2

34

ab

cd e

f

1

2 3

4

a

b

c

d

e f

12 3

4

a

bc f

e

d

Or this way!

AeL. je. siHh/L. J. Singh(nehu/NEHU) EE - 304 Electrical Network Theory 29.08.2016 3 / 14

Tie-Set (Loop/Circuit) MatrixComplete Loop/Circuit Matrix, Bc

Consider the following network and let’s find its Bc!

1 2 3

4

5 mA

5 mA

0.4ix

0.4ix25µF

25µF

25µF

10 kΩ

10 kΩ

10 kΩ

27 kΩ

27 kΩ

27 kΩ

2 H

2 H

2 H

ix

Check what’s missing in the given circuit!- Assign node numbers- Assign branch current directions

Let’s now draw the graph of the given circuit!a. Identify arbitrary locations for 4 nodesb. Connect nodes for each branch in order

1 2 3

4

aa

bb cc

dd ee ff

Graph could have been drawn this waydepending on the location of nodes!Or this way!!

1 2

34

ab

cd e

f1

2 3

4

a

b

c

d

e f

12 3

4

a

bc f

e

d

Or this way!

AeL. je. siHh/L. J. Singh(nehu/NEHU) EE - 304 Electrical Network Theory 29.08.2016 3 / 14

Tie-Set (Loop/Circuit) MatrixComplete Loop/Circuit Matrix, Bc

Now, let’s find all possible loops on this graph below to get Bc.

1 2 3

4

a

b c

fed

Consider the clock-wise orientation for each loop, the complete loop matrix, Bc, is

1 2 3

4

aa

bbcc

dd ee ff

a

b c

d e f

L2L4

L5L6L7

Loop 1: L1a,c,bLoop 2: L2b,e,dLoop 3: L3a,f,e,bLoop 4: L4a,c,e,dLoop 5: L5c,f,eLoop 6: L6a,f,dLoop 7: L7b,c,f,d

In Bc, there will be: b branches −→ b columnsall loops −→ as many rows as loops the graph has

There are only 7 possible loops that can be drawn on the given graph!Thus, Bc has only 7 rows.

BranchesLoops a b c d e f

Bc =

L1L2L3L4L5L6L7

1 -1 -1 0 0 00 1 0 1 1 01 -1 0 0 -1 11 0 -1 1 1 00 0 1 0 -1 11 0 0 1 0 10 1 1 1 0 1

AeL. je. siHh/L. J. Singh(nehu/NEHU) EE - 304 Electrical Network Theory 29.08.2016 4 / 14

Tie-Set (Loop/Circuit) MatrixComplete Loop/Circuit Matrix, Bc

Now, let’s find all possible loops on this graph below to get Bc.

1 2 3

4

a

b c

fed

Consider the clock-wise orientation for each loop, the complete loop matrix, Bc, is

1 2 3

4

a

a

b

b

c

c

d

d

e

e

f

f

a

b c

d e f

L2L4

L5L6L7

Loop 1: L1a,c,bLoop 2: L2b,e,dLoop 3: L3a,f,e,bLoop 4: L4a,c,e,dLoop 5: L5c,f,eLoop 6: L6a,f,dLoop 7: L7b,c,f,d

In Bc, there will be: b branches −→ b columnsall loops −→ as many rows as loops the graph has

There are only 7 possible loops that can be drawn on the given graph!Thus, Bc has only 7 rows.

BranchesLoops a b c d e f

Bc =

L1L2L3L4L5L6L7

1 -1 -1 0 0 00 1 0 1 1 01 -1 0 0 -1 11 0 -1 1 1 00 0 1 0 -1 11 0 0 1 0 10 1 1 1 0 1

AeL. je. siHh/L. J. Singh(nehu/NEHU) EE - 304 Electrical Network Theory 29.08.2016 4 / 14

Tie-Set (Loop/Circuit) MatrixComplete Loop/Circuit Matrix, Bc

Now, let’s find all possible loops on this graph below to get Bc.

1 2 3

4

a

b c

fed

Consider the clock-wise orientation for each loop, the complete loop matrix, Bc, is

1 2 3

4

a

a

b

b

c

c

d

d

e

e

f

f

a

b c

d e f

L2L4

L5L6L7

Loop 1: L1a,c,b

Loop 2: L2b,e,dLoop 3: L3a,f,e,bLoop 4: L4a,c,e,dLoop 5: L5c,f,eLoop 6: L6a,f,dLoop 7: L7b,c,f,d

In Bc, there will be: b branches −→ b columnsall loops −→ as many rows as loops the graph has

There are only 7 possible loops that can be drawn on the given graph!Thus, Bc has only 7 rows.

BranchesLoops a b c d e f

Bc =

L1

L2L3L4L5L6L7

1 -1 -1 0 0 0

0 1 0 1 1 01 -1 0 0 -1 11 0 -1 1 1 00 0 1 0 -1 11 0 0 1 0 10 1 1 1 0 1

AeL. je. siHh/L. J. Singh(nehu/NEHU) EE - 304 Electrical Network Theory 29.08.2016 4 / 14

Tie-Set (Loop/Circuit) MatrixComplete Loop/Circuit Matrix, Bc

Now, let’s find all possible loops on this graph below to get Bc.

1 2 3

4

a

b c

fed

Consider the clock-wise orientation for each loop, the complete loop matrix, Bc, is

1 2 3

4

a

a

b

bc

c

d

d

e

e f

f

a

b c

d e f

L2

L4L5L6

L7

Loop 1: L1a,c,b

Loop 2: L2b,e,d

Loop 3: L3a,f,e,bLoop 4: L4a,c,e,dLoop 5: L5c,f,eLoop 6: L6a,f,dLoop 7: L7b,c,f,d

In Bc, there will be: b branches −→ b columnsall loops −→ as many rows as loops the graph has

There are only 7 possible loops that can be drawn on the given graph!Thus, Bc has only 7 rows.

BranchesLoops a b c d e f

Bc =

L1

L2

L3L4L5L6L7

1 -1 -1 0 0 0

0 1 0 1 1 0

1 -1 0 0 -1 11 0 -1 1 1 00 0 1 0 -1 11 0 0 1 0 10 1 1 1 0 1

AeL. je. siHh/L. J. Singh(nehu/NEHU) EE - 304 Electrical Network Theory 29.08.2016 4 / 14

Tie-Set (Loop/Circuit) MatrixComplete Loop/Circuit Matrix, Bc

Now, let’s find all possible loops on this graph below to get Bc.

1 2 3

4

a

b c

fed

Consider the clock-wise orientation for each loop, the complete loop matrix, Bc, is

1 2 3

4

a

a

b

bc

c

d

d e

e

f

f

a

b c

d e f

L2L4

L5L6L7

Loop 1: L1a,c,bLoop 2: L2b,e,d

Loop 3: L3a,f,e,b

Loop 4: L4a,c,e,dLoop 5: L5c,f,eLoop 6: L6a,f,dLoop 7: L7b,c,f,d

In Bc, there will be: b branches −→ b columnsall loops −→ as many rows as loops the graph has

There are only 7 possible loops that can be drawn on the given graph!Thus, Bc has only 7 rows.

BranchesLoops a b c d e f

Bc =

L1L2

L3

L4L5L6L7

1 -1 -1 0 0 00 1 0 1 1 0

1 -1 0 0 -1 1

1 0 -1 1 1 00 0 1 0 -1 11 0 0 1 0 10 1 1 1 0 1

AeL. je. siHh/L. J. Singh(nehu/NEHU) EE - 304 Electrical Network Theory 29.08.2016 4 / 14

Tie-Set (Loop/Circuit) MatrixComplete Loop/Circuit Matrix, Bc

Now, let’s find all possible loops on this graph below to get Bc.

1 2 3

4

a

b c

fed

Consider the clock-wise orientation for each loop, the complete loop matrix, Bc, is

1 2 3

4

a

a

b

bc

c

d

d

e

e

ff

a

b c

d e f

L2

L4

L5L6L7

Loop 1: L1a,c,bLoop 2: L2b,e,dLoop 3: L3a,f,e,b

Loop 4: L4a,c,e,d

Loop 5: L5c,f,eLoop 6: L6a,f,dLoop 7: L7b,c,f,d

In Bc, there will be: b branches −→ b columnsall loops −→ as many rows as loops the graph has

There are only 7 possible loops that can be drawn on the given graph!Thus, Bc has only 7 rows.

BranchesLoops a b c d e f

Bc =

L1L2L3

L4

L5L6L7

1 -1 -1 0 0 00 1 0 1 1 01 -1 0 0 -1 1

1 0 -1 1 1 0

0 0 1 0 -1 11 0 0 1 0 10 1 1 1 0 1

AeL. je. siHh/L. J. Singh(nehu/NEHU) EE - 304 Electrical Network Theory 29.08.2016 4 / 14

Tie-Set (Loop/Circuit) MatrixComplete Loop/Circuit Matrix, Bc

Now, let’s find all possible loops on this graph below to get Bc.

1 2 3

4

a

b c

fed

Consider the clock-wise orientation for each loop, the complete loop matrix, Bc, is

1 2 3

4

a

a

b

bc

c

d

d e

e

f

f

a

b c

d e f

L2L4

L5

L6L7

Loop 1: L1a,c,bLoop 2: L2b,e,dLoop 3: L3a,f,e,bLoop 4: L4a,c,e,d

Loop 5: L5c,f,e

Loop 6: L6a,f,dLoop 7: L7b,c,f,d

In Bc, there will be: b branches −→ b columnsall loops −→ as many rows as loops the graph has

There are only 7 possible loops that can be drawn on the given graph!Thus, Bc has only 7 rows.

BranchesLoops a b c d e f

Bc =

L1L2L3L4

L5

L6L7

1 -1 -1 0 0 00 1 0 1 1 01 -1 0 0 -1 11 0 -1 1 1 0

0 0 1 0 -1 1

1 0 0 1 0 10 1 1 1 0 1

AeL. je. siHh/L. J. Singh(nehu/NEHU) EE - 304 Electrical Network Theory 29.08.2016 4 / 14

Tie-Set (Loop/Circuit) MatrixComplete Loop/Circuit Matrix, Bc

Now, let’s find all possible loops on this graph below to get Bc.

1 2 3

4

a

b c

fed

Consider the clock-wise orientation for each loop, the complete loop matrix, Bc, is

1 2 3

4

a

a

b

b

c

c

d

d e

e f

f

a

b c

d e f

L2L4

L5

L6

L7

Loop 1: L1a,c,bLoop 2: L2b,e,dLoop 3: L3a,f,e,bLoop 4: L4a,c,e,dLoop 5: L5c,f,e

Loop 6: L6a,f,d

Loop 7: L7b,c,f,d

In Bc, there will be: b branches −→ b columnsall loops −→ as many rows as loops the graph has

There are only 7 possible loops that can be drawn on the given graph!Thus, Bc has only 7 rows.

BranchesLoops a b c d e f

Bc =

L1L2L3L4L5

L6

L7

1 -1 -1 0 0 00 1 0 1 1 01 -1 0 0 -1 11 0 -1 1 1 00 0 1 0 -1 1

1 0 0 1 0 1

0 1 1 1 0 1

AeL. je. siHh/L. J. Singh(nehu/NEHU) EE - 304 Electrical Network Theory 29.08.2016 4 / 14

Tie-Set (Loop/Circuit) MatrixComplete Loop/Circuit Matrix, Bc

Now, let’s find all possible loops on this graph below to get Bc.

1 2 3

4

a

b c

fed

Consider the clock-wise orientation for each loop, the complete loop matrix, Bc, is

1 2 3

4

a

a

b

b

c

c

d

d e

e f

f

a

b c

d e f

L2L4

L5L6

L7

Loop 1: L1a,c,bLoop 2: L2b,e,dLoop 3: L3a,f,e,bLoop 4: L4a,c,e,dLoop 5: L5c,f,eLoop 6: L6a,f,d

Loop 7: L7b,c,f,d

In Bc, there will be: b branches −→ b columnsall loops −→ as many rows as loops the graph has

There are only 7 possible loops that can be drawn on the given graph!Thus, Bc has only 7 rows.

BranchesLoops a b c d e f

Bc =

L1L2L3L4L5L6

L7

1 -1 -1 0 0 00 1 0 1 1 01 -1 0 0 -1 11 0 -1 1 1 00 0 1 0 -1 11 0 0 1 0 1

0 1 1 1 0 1

AeL. je. siHh/L. J. Singh(nehu/NEHU) EE - 304 Electrical Network Theory 29.08.2016 4 / 14

Tie-Set (Loop/Circuit) MatrixComplete Loop/Circuit Matrix, Bc

Now, let’s find all possible loops on this graph below to get Bc.

1 2 3

4

a

b c

fed

Consider the clock-wise orientation for each loop, the complete loop matrix, Bc, is

1 2 3

4

aa

bbcc

dd ee ff

a

b c

d e f

L2L4

L5L6L7

Loop 1: L1a,c,bLoop 2: L2b,e,dLoop 3: L3a,f,e,bLoop 4: L4a,c,e,dLoop 5: L5c,f,eLoop 6: L6a,f,dLoop 7: L7b,c,f,d

In Bc, there will be: b branches −→ b columnsall loops −→ as many rows as loops the graph has

There are only 7 possible loops that can be drawn on the given graph!Thus, Bc has only 7 rows.

BranchesLoops a b c d e f

Bc =

L1L2L3L4L5L6L7

1 -1 -1 0 0 00 1 0 1 1 01 -1 0 0 -1 11 0 -1 1 1 00 0 1 0 -1 11 0 0 1 0 10 1 1 1 0 1

AeL. je. siHh/L. J. Singh(nehu/NEHU) EE - 304 Electrical Network Theory 29.08.2016 4 / 14

Tie-Set (Loop/Circuit) MatrixComplete Loop/Circuit Matrix, Bc

1 2 3

4

a

b c

d e f

L2L4

L5L6L7

Consider the clock-wise orientation for each circuit, the complete circuit matrix, Bc, is

Circuit 1: L1a,c,bCircuit 2: L2b,e,dCircuit 3: L3a,f,e,bCircuit 4: L4a,c,e,dCircuit 5: L5c,f,eCircuit 6: L6a,f,dCircuit 7: L7b,c,f,d

There are only 7 possible circuits (closed paths) that can be drawn on the given graph!Thus, Bc has only 7 rows.

BranchesCircuits a b c d e f

Bc =

L1L2L3L4L5L6L7

1 -1 -1 0 0 00 1 0 1 1 01 -1 0 0 -1 11 0 -1 1 1 00 0 1 0 -1 11 0 0 1 0 10 1 1 1 0 1

AeL. je. siHh/L. J. Singh(nehu/NEHU) EE - 304 Electrical Network Theory 29.08.2016 5 / 14

Tie-Set (Loop/Circuit) Matrix, B

In a graph with n nodes and b branches, we know that there are t = n − 1 twigs in atree and l = b − t = b − n + 1 links in the co-tree.

Consider a tree of a graph:- Re-placement of each link one at a time in the tree forms a closed path or loop orcircuit. Thus, every link forms a unique loop, closed path or circuit called thefundamental loop (f-loop) or fundamental circuit (f-circuit) or tie-set, in short.

- The number of f-circuits is same as that of links in a given tree. Hence, thenumber of f -circuits or tie-sets is equal to l = b − n + 1.

- The reference direction of the f -loop is taken as that of the link.

AeL. je. siHh/L. J. Singh(nehu/NEHU) EE - 304 Electrical Network Theory 29.08.2016 6 / 14

Tie-Set (Loop/Circuit) Matrix, B

In a graph with n nodes and b branches, we know that there are t = n − 1 twigs in atree and l = b − t = b − n + 1 links in the co-tree.

Consider a tree of a graph:- Re-placement of each link one at a time in the tree forms a closed path or loop orcircuit. Thus, every link forms a unique loop, closed path or circuit called thefundamental loop (f-loop) or fundamental circuit (f-circuit) or tie-set, in short.

- The number of f-circuits is same as that of links in a given tree. Hence, thenumber of f -circuits or tie-sets is equal to l = b − n + 1.

- The reference direction of the f -loop is taken as that of the link.

AeL. je. siHh/L. J. Singh(nehu/NEHU) EE - 304 Electrical Network Theory 29.08.2016 6 / 14

Tie-Set (Loop/Circuit) Matrix, B

In a graph with n nodes and b branches, we know that there are t = n − 1 twigs in atree and l = b − t = b − n + 1 links in the co-tree.

Consider a tree of a graph:- Re-placement of each link one at a time in the tree forms a closed path or loop orcircuit. Thus, every link forms a unique loop, closed path or circuit called thefundamental loop (f-loop) or fundamental circuit (f-circuit) or tie-set, in short.

- The number of f-circuits is same as that of links in a given tree. Hence, thenumber of f -circuits or tie-sets is equal to l = b − n + 1.

- The reference direction of the f -loop is taken as that of the link.

AeL. je. siHh/L. J. Singh(nehu/NEHU) EE - 304 Electrical Network Theory 29.08.2016 6 / 14

Tie-Set (Loop/Circuit) Matrix, B

In a graph with n nodes and b branches, we know that there are t = n − 1 twigs in atree and l = b − t = b − n + 1 links in the co-tree.

Consider a tree of a graph:- Re-placement of each link one at a time in the tree forms a closed path or loop orcircuit. Thus, every link forms a unique loop, closed path or circuit called thefundamental loop (f-loop) or fundamental circuit (f-circuit) or tie-set, in short.

- The number of f-circuits is same as that of links in a given tree. Hence, thenumber of f -circuits or tie-sets is equal to l = b − n + 1.

- The reference direction of the f -loop is taken as that of the link.

AeL. je. siHh/L. J. Singh(nehu/NEHU) EE - 304 Electrical Network Theory 29.08.2016 6 / 14

Tie-Set (Loop/Circuit) Matrix, B

Example: In the graph shown below, we select a tree:

1 2 3

4

a

b c

fed

1 2 3

4

a

b c

fed

a

b c

L1b

edL2

a

b

e f

L3

In the given diagram on the left, we haveNumber of nodes : n = 4

Number of branches : b = 6Number of branches on tree : t = n − 1 = 3

Number of links : l = b − t= b − (n − 1)= b − n + 1 = 3

Therefore, number of f -circuits : 3f -loops/f -circuits/tie-sets are formed by

::::::::re-placing

:::::each

:::link

::::one

:::at

::a

::::time

:::in

:::the

:::::tree!

Possible tie-sets or f -loops or f -circuits are now given byf -circuits : L1a,b,c

: L2b,d,e: L3a,b,e,f

f -loops : L1a,b,c: L2b,d,e: L3a,b,e,f

tie-sets : L1a,b,c: L2b,d,e: L3a,b,e,f

Note: Orientation of f -loop/f -circuit/tie-set is that of the link.Note: Every link defines a fundamental loop/circuit/tie-set of the graph.

AeL. je. siHh/L. J. Singh(nehu/NEHU) EE - 304 Electrical Network Theory 29.08.2016 7 / 14

Tie-Set (Loop/Circuit) Matrix, B

Example: In the graph shown below, we select a tree: twigsa, b, e.

1 2 3

4

a

b c

fed

1 2 3

4

a

b c

fed

a

b c

L1b

edL2

a

b

e f

L3

In the given diagram on the left, we haveNumber of nodes : n = 4

Number of branches : b = 6Number of branches on tree : t = n − 1 = 3

Number of links : l = b − t= b − (n − 1)= b − n + 1 = 3

Therefore, number of f -circuits : 3f -loops/f -circuits/tie-sets are formed by

::::::::re-placing

:::::each

:::link

::::one

:::at

::a

::::time

:::in

:::the

:::::tree!

Possible tie-sets or f -loops or f -circuits are now given byf -circuits : L1a,b,c

: L2b,d,e: L3a,b,e,f

f -loops : L1a,b,c: L2b,d,e: L3a,b,e,f

tie-sets : L1a,b,c: L2b,d,e: L3a,b,e,f

Note: Orientation of f -loop/f -circuit/tie-set is that of the link.Note: Every link defines a fundamental loop/circuit/tie-set of the graph.

AeL. je. siHh/L. J. Singh(nehu/NEHU) EE - 304 Electrical Network Theory 29.08.2016 7 / 14

Tie-Set (Loop/Circuit) Matrix, B

Example: In the graph shown below, we select a tree: twigsa, b, e.

1 2 3

4

a

b c

fed

1 2 3

4

a

b c

fed

a

b c

L1b

edL2

a

b

e f

L3

In the given diagram on the left, we haveNumber of nodes : n = 4

Number of branches : b = 6Number of branches on tree : t = n − 1 = 3

Number of links : l = b − t= b − (n − 1)= b − n + 1 = 3

Therefore, number of f -circuits : 3f -loops/f -circuits/tie-sets are formed by

::::::::re-placing

:::::each

:::link

::::one

:::at

::a

::::time

:::in

:::the

:::::tree!

Possible tie-sets or f -loops or f -circuits are now given byf -circuits : L1a,b,c

: L2b,d,e: L3a,b,e,f

f -loops : L1a,b,c: L2b,d,e: L3a,b,e,f

tie-sets : L1a,b,c: L2b,d,e: L3a,b,e,f

Note: Orientation of f -loop/f -circuit/tie-set is that of the link.Note: Every link defines a fundamental loop/circuit/tie-set of the graph.

AeL. je. siHh/L. J. Singh(nehu/NEHU) EE - 304 Electrical Network Theory 29.08.2016 7 / 14

Tie-Set (Loop/Circuit) Matrix, B

Example: In the graph shown below, we select a tree: twigsa, b, e.

1 2 3

4

a

b c

fed

1 2 3

4

a

b c

fed

a

b c

L1

b

edL2

a

b

e f

L3

In the given diagram on the left, we haveNumber of nodes : n = 4

Number of branches : b = 6Number of branches on tree : t = n − 1 = 3

Number of links : l = b − t= b − (n − 1)= b − n + 1 = 3

Therefore, number of f -circuits : 3

f -loops/f -circuits/tie-sets are formed by::::::::re-placing

:::::each

:::link

::::one

:::at

::a

::::time

:::in

:::the

:::::tree!

Possible tie-sets or f -loops or f -circuits are now given byf -circuits : L1a,b,c

: L2b,d,e: L3a,b,e,f

f -loops : L1a,b,c: L2b,d,e: L3a,b,e,f

tie-sets : L1a,b,c: L2b,d,e: L3a,b,e,f

Note: Orientation of f -loop/f -circuit/tie-set is that of the link.

Note: Every link defines a fundamental loop/circuit/tie-set of the graph.

AeL. je. siHh/L. J. Singh(nehu/NEHU) EE - 304 Electrical Network Theory 29.08.2016 7 / 14

Tie-Set (Loop/Circuit) Matrix, B

Example: In the graph shown below, we select a tree: twigsa, b, e.

1 2 3

4

a

b c

fed

1 2 3

4

a

b

c

f

ed

a

b c

L1

b

edL2

a

b

e f

L3

In the given diagram on the left, we haveNumber of nodes : n = 4

Number of branches : b = 6Number of branches on tree : t = n − 1 = 3

Number of links : l = b − t= b − (n − 1)= b − n + 1 = 3

Therefore, number of f -circuits : 3

f -loops/f -circuits/tie-sets are formed by::::::::re-placing

:::::each

:::link

::::one

:::at

::a

::::time

:::in

:::the

:::::tree!

Possible tie-sets or f -loops or f -circuits are now given by

f -circuits : L1a,b,c: L2b,d,e: L3a,b,e,f

f -loops : L1a,b,c: L2b,d,e: L3a,b,e,f

tie-sets : L1a,b,c: L2b,d,e: L3a,b,e,f

Note: Orientation of f -loop/f -circuit/tie-set is that of the link.

Note: Every link defines a fundamental loop/circuit/tie-set of the graph.

AeL. je. siHh/L. J. Singh(nehu/NEHU) EE - 304 Electrical Network Theory 29.08.2016 7 / 14

Tie-Set (Loop/Circuit) Matrix, B

Example: In the graph shown below, we select a tree: twigsa, b, e.

1 2 3

4

a

b c

fed

1 2 3

4

a

b

c

fe

d

a

b c

L1b

edL2

a

b

e f

L3

In the given diagram on the left, we haveNumber of nodes : n = 4

Number of branches : b = 6Number of branches on tree : t = n − 1 = 3

Number of links : l = b − t= b − (n − 1)= b − n + 1 = 3

Therefore, number of f -circuits : 3

f -loops/f -circuits/tie-sets are formed by::::::::re-placing

:::::each

:::link

::::one

:::at

::a

::::time

:::in

:::the

:::::tree!

Possible tie-sets or f -loops or f -circuits are now given by

f -circuits : L1a,b,c: L2b,d,e: L3a,b,e,f

f -loops : L1a,b,c: L2b,d,e: L3a,b,e,f

tie-sets : L1a,b,c: L2b,d,e: L3a,b,e,f

Note: Orientation of f -loop/f -circuit/tie-set is that of the link.

Note: Every link defines a fundamental loop/circuit/tie-set of the graph.

AeL. je. siHh/L. J. Singh(nehu/NEHU) EE - 304 Electrical Network Theory 29.08.2016 7 / 14

Tie-Set (Loop/Circuit) Matrix, B

Example: In the graph shown below, we select a tree: twigsa, b, e.

1 2 3

4

a

b c

fed

1 2 3

4

a

b c

fed

a

b c

L1b

edL2

a

b

e f

L3

In the given diagram on the left, we haveNumber of nodes : n = 4

Number of branches : b = 6Number of branches on tree : t = n − 1 = 3

Number of links : l = b − t= b − (n − 1)= b − n + 1 = 3

Therefore, number of f -circuits : 3

f -loops/f -circuits/tie-sets are formed by::::::::re-placing

:::::each

:::link

::::one

:::at

::a

::::time

:::in

:::the

:::::tree!

Possible tie-sets or f -loops or f -circuits are now given by

f -circuits : L1a,b,c: L2b,d,e: L3a,b,e,f

f -loops : L1a,b,c: L2b,d,e: L3a,b,e,f

tie-sets : L1a,b,c: L2b,d,e: L3a,b,e,f

Note: Orientation of f -loop/f -circuit/tie-set is that of the link.

Note: Every link defines a fundamental loop/circuit/tie-set of the graph.

AeL. je. siHh/L. J. Singh(nehu/NEHU) EE - 304 Electrical Network Theory 29.08.2016 7 / 14

Tie-Set (Loop/Circuit) Matrix, BLet’s obtain the matrix!

Example - 1: In the graph shown below, we select a tree:

1 2 3

4

a

b c

fed

1 2 3

4

a

b c

fed

a

b c

L1b

edL2

a

b

e f

L3

Tree : twigsa, b, e

1 2 3

4

a

b c

fed

L1a, b, c

1 2 3

4

a

b c

fed

a

b c

L1

L2b, e, d

1 2 3

4

a

c

f

b

edL2

L3a, f , e, b

1 2 3

4

c

d

a

b

e f

L3

f -loops : L1a,b,c: L2b,e,d: L3a,f,e,b

f -circuits : L1a,b,c: L2b,e,d: L3a,f,e,b

tie-sets : L1a,b,c: L2b,e,d: L3a,f,e,b

Tie-Sets or f -Loops or f -Circuits are :

Tie-Sets or f -Loops or f -Circuits are :

Tie-Set or f -Loop or f -Circuit matrix is :

Tie-Set or f -Loop or f -Circuit matrix is :Branchesf -Loops a b c d e f

B =L1L2L3

-1 1 1 0 0 00 1 0 1 1 01 -1 0 0 -1 1

Branches

f -Circuits a b c d e f

B =L1L2L3

-1 1 1 0 0 00 1 0 1 1 01 -1 0 0 -1 1

Branches

Tie-Sets a b c d e f

B =L1L2L3

-1 1 1 0 0 00 1 0 1 1 01 -1 0 0 -1 1

AeL. je. siHh/L. J. Singh(nehu/NEHU) EE - 304 Electrical Network Theory 29.08.2016 8 / 14

Tie-Set (Loop/Circuit) Matrix, BLet’s obtain the matrix!

Example - 1: In the graph shown below, we select a tree: twigsa, b, e. Thus, theco-tree: linksc, d , f .

1 2 3

4

a

b c

fed

1 2 3

4

a

b c

fed

a

b c

L1b

edL2

a

b

e f

L3

Tree : twigsa, b, e

1 2 3

4

a

b c

fed

L1a, b, c

1 2 3

4

a

b c

fed

a

b c

L1

L2b, e, d

1 2 3

4

a

c

f

b

edL2

L3a, f , e, b

1 2 3

4

c

d

a

b

e f

L3

f -loops : L1a,b,c: L2b,e,d: L3a,f,e,b

f -circuits : L1a,b,c: L2b,e,d: L3a,f,e,b

tie-sets : L1a,b,c: L2b,e,d: L3a,f,e,b

Tie-Sets or f -Loops or f -Circuits are :

Tie-Sets or f -Loops or f -Circuits are :

Tie-Set or f -Loop or f -Circuit matrix is :

Tie-Set or f -Loop or f -Circuit matrix is :Branchesf -Loops a b c d e f

B =L1L2L3

-1 1 1 0 0 00 1 0 1 1 01 -1 0 0 -1 1

Branches

f -Circuits a b c d e f

B =L1L2L3

-1 1 1 0 0 00 1 0 1 1 01 -1 0 0 -1 1

Branches

Tie-Sets a b c d e f

B =L1L2L3

-1 1 1 0 0 00 1 0 1 1 01 -1 0 0 -1 1

AeL. je. siHh/L. J. Singh(nehu/NEHU) EE - 304 Electrical Network Theory 29.08.2016 8 / 14

Tie-Set (Loop/Circuit) Matrix, BLet’s obtain the matrix!

Example - 1: In the graph shown below, we select a tree: twigsa, b, e. Thus, theco-tree: linksc, d , f . We now derive Tie-Set Matrix, B, by placing back each linkone at a time on the tree.

1 2 3

4

a

b c

fed

1 2 3

4

a

b c

fed

a

b c

L1b

edL2

a

b

e f

L3

Tree : twigsa, b, e

1 2 3

4

a

b c

fed

L1a, b, c

1 2 3

4

a

b c

fed

a

b c

L1

L2b, e, d

1 2 3

4

a

c

f

b

edL2

L3a, f , e, b

1 2 3

4

c

d

a

b

e f

L3

f -loops : L1a,b,c: L2b,e,d: L3a,f,e,b

f -circuits : L1a,b,c: L2b,e,d: L3a,f,e,b

tie-sets : L1a,b,c: L2b,e,d: L3a,f,e,b

Tie-Sets or f -Loops or f -Circuits are :

Tie-Sets or f -Loops or f -Circuits are :

Tie-Set or f -Loop or f -Circuit matrix is :

Tie-Set or f -Loop or f -Circuit matrix is :

Branchesf -Loops a b c d e f

B =L1L2L3

-1 1 1 0 0 00 1 0 1 1 01 -1 0 0 -1 1

Branchesf -Circuits a b c d e f

B =L1L2L3

-1 1 1 0 0 00 1 0 1 1 01 -1 0 0 -1 1

Branches

Tie-Sets a b c d e f

B =L1L2L3

-1 1 1 0 0 00 1 0 1 1 01 -1 0 0 -1 1

AeL. je. siHh/L. J. Singh(nehu/NEHU) EE - 304 Electrical Network Theory 29.08.2016 8 / 14

Tie-Set (Loop/Circuit) Matrix, BLet’s obtain the matrix!

Example - 1: In the graph shown below, we select a tree: twigsa, b, e. Thus, theco-tree: linksc, d , f . We now derive Tie-Set Matrix, B, by placing back each linkone at a time on the tree.

1 2 3

4

a

b c

fed

1 2 3

4

a

b c

fed

a

b c

L1

b

edL2

a

b

e f

L3

Tree : twigsa, b, e

1 2 3

4

a

b c

fed

L1a, b, c

1 2 3

4

a

b c

fed

a

b c

L1

L2b, e, d

1 2 3

4

a

c

f

b

edL2

L3a, f , e, b

1 2 3

4

c

d

a

b

e f

L3

f -loops : L1a,b,c: L2b,e,d: L3a,f,e,b

f -circuits : L1a,b,c: L2b,e,d: L3a,f,e,b

tie-sets : L1a,b,c: L2b,e,d: L3a,f,e,b

Tie-Sets or f -Loops or f -Circuits are :

Tie-Sets or f -Loops or f -Circuits are :

Tie-Set or f -Loop or f -Circuit matrix is :

Tie-Set or f -Loop or f -Circuit matrix is :

Branchesf -Loops a b c d e f

B =L1L2L3

-1 1 1 0 0 0

0 1 0 1 1 01 -1 0 0 -1 1

Branchesf -Circuits a b c d e f

B =L1L2L3

-1 1 1 0 0 00 1 0 1 1 01 -1 0 0 -1 1

Branches

Tie-Sets a b c d e f

B =L1L2L3

-1 1 1 0 0 00 1 0 1 1 01 -1 0 0 -1 1

AeL. je. siHh/L. J. Singh(nehu/NEHU) EE - 304 Electrical Network Theory 29.08.2016 8 / 14

Tie-Set (Loop/Circuit) Matrix, BLet’s obtain the matrix!

Example - 1: In the graph shown below, we select a tree: twigsa, b, e. Thus, theco-tree: linksc, d , f . We now derive Tie-Set Matrix, B, by placing back each linkone at a time on the tree.

1 2 3

4

a

b c

fed

1 2 3

4

a

b

c

f

ed

a

b c

L1

b

edL2

a

b

e f

L3

Tree : twigsa, b, e

1 2 3

4

a

b c

fed

L1a, b, c

1 2 3

4

a

b c

fed

a

b c

L1

L2b, e, d

1 2 3

4

a

c

f

b

edL2

L3a, f , e, b

1 2 3

4

c

d

a

b

e f

L3

f -loops : L1a,b,c: L2b,e,d: L3a,f,e,b

f -circuits : L1a,b,c: L2b,e,d: L3a,f,e,b

tie-sets : L1a,b,c: L2b,e,d: L3a,f,e,b

Tie-Sets or f -Loops or f -Circuits are :

Tie-Sets or f -Loops or f -Circuits are :

Tie-Set or f -Loop or f -Circuit matrix is :

Tie-Set or f -Loop or f -Circuit matrix is :

Branchesf -Loops a b c d e f

B =L1L2L3

-1 1 1 0 0 00 1 0 1 1 0

1 -1 0 0 -1 1

Branchesf -Circuits a b c d e f

B =L1L2L3

-1 1 1 0 0 00 1 0 1 1 01 -1 0 0 -1 1

Branches

Tie-Sets a b c d e f

B =L1L2L3

-1 1 1 0 0 00 1 0 1 1 01 -1 0 0 -1 1

AeL. je. siHh/L. J. Singh(nehu/NEHU) EE - 304 Electrical Network Theory 29.08.2016 8 / 14

Tie-Set (Loop/Circuit) Matrix, BLet’s obtain the matrix!

Example - 1: In the graph shown below, we select a tree: twigsa, b, e. Thus, theco-tree: linksc, d , f . We now derive Tie-Set Matrix, B, by placing back each linkone at a time on the tree.

1 2 3

4

a

b c

fed

1 2 3

4

a

b

c

fe

d

a

b c

L1b

edL2

a

b

e f

L3

Tree : twigsa, b, e

1 2 3

4

a

b c

fed

L1a, b, c

1 2 3

4

a

b c

fed

a

b c

L1

L2b, e, d

1 2 3

4

a

c

f

b

edL2

L3a, f , e, b

1 2 3

4

c

d

a

b

e f

L3

f -loops : L1a,b,c: L2b,e,d: L3a,f,e,b

f -circuits : L1a,b,c: L2b,e,d: L3a,f,e,b

tie-sets : L1a,b,c: L2b,e,d: L3a,f,e,b

Tie-Sets or f -Loops or f -Circuits are :

Tie-Sets or f -Loops or f -Circuits are :

Tie-Set or f -Loop or f -Circuit matrix is :

Tie-Set or f -Loop or f -Circuit matrix is :

Branchesf -Loops a b c d e f

B =L1L2L3

-1 1 1 0 0 00 1 0 1 1 01 -1 0 0 -1 1

Branchesf -Circuits a b c d e f

B =L1L2L3

-1 1 1 0 0 00 1 0 1 1 01 -1 0 0 -1 1

Branches

Tie-Sets a b c d e f

B =L1L2L3

-1 1 1 0 0 00 1 0 1 1 01 -1 0 0 -1 1

AeL. je. siHh/L. J. Singh(nehu/NEHU) EE - 304 Electrical Network Theory 29.08.2016 8 / 14

Tie-Set (Loop/Circuit) Matrix, BLet’s obtain the matrix!

Example - 1: In the graph shown below, we select a tree: twigsa, b, e. Thus, theco-tree: linksc, d , f . We now derive Tie-Set Matrix, B, by placing back each linkone at a time on the tree.

1 2 3

4

a

b c

fed

1 2 3

4

a

b c

fed

a

b c

L1b

edL2

a

b

e f

L3

Tree : twigsa, b, e

1 2 3

4

a

b c

fed

L1a, b, c

1 2 3

4

a

b c

fed

a

b c

L1

L2b, e, d

1 2 3

4

a

c

f

b

edL2

L3a, f , e, b

1 2 3

4

c

d

a

b

e f

L3

f -loops : L1a,b,c: L2b,e,d: L3a,f,e,b

f -circuits : L1a,b,c: L2b,e,d: L3a,f,e,b

tie-sets : L1a,b,c: L2b,e,d: L3a,f,e,b

Tie-Sets or f -Loops or f -Circuits are :

Tie-Sets or f -Loops or f -Circuits are :

Tie-Set or f -Loop or f -Circuit matrix is :

Tie-Set or f -Loop or f -Circuit matrix is :

Branchesf -Loops a b c d e f

B =L1L2L3

-1 1 1 0 0 00 1 0 1 1 01 -1 0 0 -1 1

Branchesf -Circuits a b c d e f

B =L1L2L3

-1 1 1 0 0 00 1 0 1 1 01 -1 0 0 -1 1

Branches

Tie-Sets a b c d e f

B =L1L2L3

-1 1 1 0 0 00 1 0 1 1 01 -1 0 0 -1 1

AeL. je. siHh/L. J. Singh(nehu/NEHU) EE - 304 Electrical Network Theory 29.08.2016 8 / 14

Tie-Set (Loop/Circuit) Matrix, BLet’s obtain the matrix!

Example - 1: In the graph shown below, we select a tree: twigsa, b, e. Thus, theco-tree: linksc, d , f . We now derive Tie-Set Matrix, B, by placing back each linkone at a time on the tree.

1 2 3

4

a

b c

fed

1 2 3

4

a

b c

fed

a

b c

L1b

edL2

a

b

e f

L3

Tree : twigsa, b, e

1 2 3

4

a

b c

fed

L1a, b, c

1 2 3

4

a

b c

fed

a

b c

L1

L2b, e, d

1 2 3

4

a

c

f

b

edL2

L3a, f , e, b

1 2 3

4

c

d

a

b

e f

L3

f -loops : L1a,b,c: L2b,e,d: L3a,f,e,b

f -circuits : L1a,b,c: L2b,e,d: L3a,f,e,b

tie-sets : L1a,b,c: L2b,e,d: L3a,f,e,b

Tie-Sets or f -Loops or f -Circuits are :

Tie-Sets or f -Loops or f -Circuits are :

Tie-Set or f -Loop or f -Circuit matrix is :

Tie-Set or f -Loop or f -Circuit matrix is :

Branchesf -Loops a b c d e f

B =L1L2L3

-1 1 1 0 0 00 1 0 1 1 01 -1 0 0 -1 1

Branchesf -Circuits a b c d e f

B =L1L2L3

-1 1 1 0 0 00 1 0 1 1 01 -1 0 0 -1 1

BranchesTie-Sets a b c d e f

B =L1L2L3

-1 1 1 0 0 00 1 0 1 1 01 -1 0 0 -1 1

AeL. je. siHh/L. J. Singh(nehu/NEHU) EE - 304 Electrical Network Theory 29.08.2016 8 / 14

Tie-Set (Loop/Circuit) Matrix, BLet’s obtain the matrix!

Example - 1: In the graph shown below, we select a tree: twigsa, b, e. Thus, theco-tree: linksc, d , f . We now derive Tie-Set Matrix, B, by placing back each linkone at a time on the tree.

1 2 3

4

a

b c

fed

1 2 3

4

a

b c

fed

a

b c

L1b

edL2

a

b

e f

L3

Tree : twigsa, b, e

1 2 3

4

a

b c

fed

L1a, b, c

1 2 3

4

a

b c

fed

a

b c

L1

L2b, e, d

1 2 3

4

a

c

f

b

edL2

L3a, f , e, b

1 2 3

4

c

d

a

b

e f

L3

f -loops : L1a,b,c: L2b,e,d: L3a,f,e,b

f -circuits : L1a,b,c: L2b,e,d: L3a,f,e,b

tie-sets : L1a,b,c: L2b,e,d: L3a,f,e,b

Tie-Sets or f -Loops or f -Circuits are :

Tie-Sets or f -Loops or f -Circuits are :

Tie-Set or f -Loop or f -Circuit matrix is :

Tie-Set or f -Loop or f -Circuit matrix is :

Branchesf -Loops a b c d e f

B =L1L2L3

-1 1 1 0 0 00 1 0 1 1 01 -1 0 0 -1 1

Branches

f -Circuits a b c d e f

B =L1L2L3

-1 1 1 0 0 00 1 0 1 1 01 -1 0 0 -1 1

BranchesTie-Sets a b c d e f

B =L1L2L3

-1 1 1 0 0 00 1 0 1 1 01 -1 0 0 -1 1

AeL. je. siHh/L. J. Singh(nehu/NEHU) EE - 304 Electrical Network Theory 29.08.2016 8 / 14

Tie-Set (Loop/Circuit) Matrix, B

Example - 2: We select a tree:

1 2 3

4

a

b c

fed

In the given diagram on the left, we haveNumber of nodes : n = 4

Number of branches : b = 6Number of branches on tree : t = n − 1 = 3

Number of links : l = b − t= b − (n − 1)= b − n + 1 = 3

Therefore, number of f -circuits : 3Tree : twigsa, b, d

1 2 3

4

a

b c

fed

L1a, b, c

1 2 3

4

fed

a

b c

L1

L2b, e, d

1 2 3

4

a

c

f

b

edL2

Orientation ofloop/circuitchanges withthat of the link!

L3a, f , d

1 2 3

4

bc

e

a

fdL4

Tie-Sets or f -Loops or f -Circuits are : Tie-Set or f -Loop or f -Circuit matrix is :tie-sets : L1a,b,c

: L2b,e,d: L3a,f,d

BranchesTie-Sets a b c d e f

B =L1L2L3

-1 1 1 0 0 00 1 0 1 1 01 0 0 1 0 1

AeL. je. siHh/L. J. Singh(nehu/NEHU) EE - 304 Electrical Network Theory 29.08.2016 9 / 14

Tie-Set (Loop/Circuit) Matrix, B

Example - 2: We select a tree: twigsa, b, d. Thus, the co-tree: linksc, e, f .

1 2 3

4

a

b c

fed

In the given diagram on the left, we haveNumber of nodes : n = 4

Number of branches : b = 6Number of branches on tree : t = n − 1 = 3

Number of links : l = b − t= b − (n − 1)= b − n + 1 = 3

Therefore, number of f -circuits : 3Tree : twigsa, b, d

1 2 3

4

a

b c

fed

L1a, b, c

1 2 3

4

fed

a

b c

L1

L2b, e, d

1 2 3

4

a

c

f

b

edL2

Orientation ofloop/circuitchanges withthat of the link!

L3a, f , d

1 2 3

4

bc

e

a

fdL4

Tie-Sets or f -Loops or f -Circuits are : Tie-Set or f -Loop or f -Circuit matrix is :tie-sets : L1a,b,c

: L2b,e,d: L3a,f,d

BranchesTie-Sets a b c d e f

B =L1L2L3

-1 1 1 0 0 00 1 0 1 1 01 0 0 1 0 1

AeL. je. siHh/L. J. Singh(nehu/NEHU) EE - 304 Electrical Network Theory 29.08.2016 9 / 14

Tie-Set (Loop/Circuit) Matrix, B

Example - 2: We select a tree: twigsa, b, d. Thus, the co-tree: linksc, e, f . Wenow derive Tie-Set Matrix, B, by placing back each link one at a time on the tree.

1 2 3

4

a

b c

fed

In the given diagram on the left, we haveNumber of nodes : n = 4

Number of branches : b = 6Number of branches on tree : t = n − 1 = 3

Number of links : l = b − t= b − (n − 1)= b − n + 1 = 3

Therefore, number of f -circuits : 3

Tree : twigsa, b, d

1 2 3

4

a

b c

fed

L1a, b, c

1 2 3

4

fed

a

b c

L1

L2b, e, d

1 2 3

4

a

c

f

b

edL2

Orientation ofloop/circuitchanges withthat of the link!

L3a, f , d

1 2 3

4

bc

e

a

fdL4

Tie-Sets or f -Loops or f -Circuits are : Tie-Set or f -Loop or f -Circuit matrix is :tie-sets : L1a,b,c

: L2b,e,d: L3a,f,d

BranchesTie-Sets a b c d e f

B =L1L2L3

-1 1 1 0 0 0

0 1 0 1 1 01 0 0 1 0 1

AeL. je. siHh/L. J. Singh(nehu/NEHU) EE - 304 Electrical Network Theory 29.08.2016 9 / 14

Tie-Set (Loop/Circuit) Matrix, B

Example - 2: We select a tree: twigsa, b, d. Thus, the co-tree: linksc, e, f . Wenow derive Tie-Set Matrix, B, by placing back each link one at a time on the tree.

1 2 3

4

a

b c

fed

In the given diagram on the left, we haveNumber of nodes : n = 4

Number of branches : b = 6Number of branches on tree : t = n − 1 = 3

Number of links : l = b − t= b − (n − 1)= b − n + 1 = 3

Therefore, number of f -circuits : 3

Tree : twigsa, b, d

1 2 3

4

a

b c

fed

L1a, b, c

1 2 3

4

fed

a

b c

L1

L2b, e, d

1 2 3

4

a

c

f

b

edL2

Orientation ofloop/circuitchanges withthat of the link!

L3a, f , d

1 2 3

4

bc

e

a

fdL4

Tie-Sets or f -Loops or f -Circuits are : Tie-Set or f -Loop or f -Circuit matrix is :tie-sets : L1a,b,c

: L2b,e,d: L3a,f,d

BranchesTie-Sets a b c d e f

B =L1L2L3

-1 1 1 0 0 00 1 0 1 1 0

1 0 0 1 0 1

AeL. je. siHh/L. J. Singh(nehu/NEHU) EE - 304 Electrical Network Theory 29.08.2016 9 / 14

Tie-Set (Loop/Circuit) Matrix, B

Example - 2: We select a tree: twigsa, b, d. Thus, the co-tree: linksc, e, f . Wenow derive Tie-Set Matrix, B, by placing back each link one at a time on the tree.

1 2 3

4

a

b c

fed

In the given diagram on the left, we haveNumber of nodes : n = 4

Number of branches : b = 6Number of branches on tree : t = n − 1 = 3

Number of links : l = b − t= b − (n − 1)= b − n + 1 = 3

Therefore, number of f -circuits : 3

Tree : twigsa, b, d

1 2 3

4

a

b c

fed

L1a, b, c

1 2 3

4

fed

a

b c

L1

L2b, e, d

1 2 3

4

a

c

f

b

edL2

Orientation ofloop/circuitchanges withthat of the link!

L3a, f , d

1 2 3

4

bc

e

a

fdL4

Tie-Sets or f -Loops or f -Circuits are : Tie-Set or f -Loop or f -Circuit matrix is :tie-sets : L1a,b,c

: L2b,e,d: L3a,f,d

BranchesTie-Sets a b c d e f

B =L1L2L3

-1 1 1 0 0 00 1 0 1 1 01 0 0 1 0 1

AeL. je. siHh/L. J. Singh(nehu/NEHU) EE - 304 Electrical Network Theory 29.08.2016 9 / 14

Tie-Set (Loop/Circuit) Matrix, B

Example - 2: We select a tree: twigsa, b, d. Thus, the co-tree: linksc, e, f . Wenow derive Tie-Set Matrix, B, by placing back each link one at a time on the tree.

1 2 3

4

a

b c

fed

In the given diagram on the left, we haveNumber of nodes : n = 4

Number of branches : b = 6Number of branches on tree : t = n − 1 = 3

Number of links : l = b − t= b − (n − 1)= b − n + 1 = 3

Therefore, number of f -circuits : 3

Tree : twigsa, b, d

1 2 3

4

a

b c

fed

L1a, b, c

1 2 3

4

fed

a

b c

L1

L2b, e, d

1 2 3

4

a

c

f

b

edL2

Orientation ofloop/circuitchanges withthat of the link!

L3a, f , d

1 2 3

4

bc

e

a

fdL4

Tie-Sets or f -Loops or f -Circuits are : Tie-Set or f -Loop or f -Circuit matrix is :tie-sets : L1a,b,c

: L2b,e,d: L3a,f,d

BranchesTie-Sets a b c d e f

B =L1L2L3

-1 1 1 0 0 00 1 0 1 1 01 0 0 1 0 1

AeL. je. siHh/L. J. Singh(nehu/NEHU) EE - 304 Electrical Network Theory 29.08.2016 9 / 14

Tie-Set (Loop/Circuit) Matrix, BRank of the tie-set matrix!

Example - 3: In the graph shown below on the left, we select a tree:

1 2 3

4

a

b c

d e f

1 2 3

4

a

b c

d e f

Tree: twigsb, c, e

1 2 3

4

a

b c

d e f

Co-tree: linksa, d , f

1 2 3

4

a

b c

fed

a

b c

L1b

edL2

c

e fL3

f -loops : L1a,c,b: L2b,e,d: L3c,f,e

Tie-Sets or f -Loops or f -Circuits are :

Tie-Set or f -Loop or f -Circuit matrix is :Branches

f -Loops a b c d e f

B =L1L2L3

1 -1 -1 0 0 00 1 0 1 1 00 0 1 0 -1 1

We now re-arrange columns in B with links first and twigs later as a, d , f , b, c, e.

Brancheslinks twigs

f -Loops a d f b c e

B =L1L2L3

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

Thus, B = [Bl, Bt] = [I, Bt]. And B is non-singular matrix which rank is (b − n + 1).

AeL. je. siHh/L. J. Singh(nehu/NEHU) EE - 304 Electrical Network Theory 29.08.2016 10 / 14

Tie-Set (Loop/Circuit) Matrix, BRank of the tie-set matrix!

Example - 3: In the graph shown below on the left, we select a tree: twigsb, c, e.Thus, the co-tree: linksa, d , f .

1 2 3

4

a

b c

d e f

1 2 3

4

a

b c

d e f

Tree: twigsb, c, e

1 2 3

4

a

b c

d e f

Co-tree: linksa, d , f

1 2 3

4

a

b c

fed

a

b c

L1b

edL2

c

e fL3

f -loops : L1a,c,b: L2b,e,d: L3c,f,e

Tie-Sets or f -Loops or f -Circuits are :

Tie-Set or f -Loop or f -Circuit matrix is :Branches

f -Loops a b c d e f

B =L1L2L3

1 -1 -1 0 0 00 1 0 1 1 00 0 1 0 -1 1

We now re-arrange columns in B with links first and twigs later as a, d , f , b, c, e.

Brancheslinks twigs

f -Loops a d f b c e

B =L1L2L3

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

Thus, B = [Bl, Bt] = [I, Bt]. And B is non-singular matrix which rank is (b − n + 1).

AeL. je. siHh/L. J. Singh(nehu/NEHU) EE - 304 Electrical Network Theory 29.08.2016 10 / 14

Tie-Set (Loop/Circuit) Matrix, BRank of the tie-set matrix!

Example - 3: In the graph shown below on the left, we select a tree: twigsb, c, e.Thus, the co-tree: linksa, d , f . We now derive Tie-Set Matrix, B, by placing backeach link one at a time on the tree.

1 2 3

4

a

b c

d e f

1 2 3

4

a

b c

d e f

Tree: twigsb, c, e

1 2 3

4

a

b c

d e f

Co-tree: linksa, d , f

1 2 3

4

a

b c

fed

a

b c

L1b

edL2

c

e fL3

f -loops : L1a,c,b: L2b,e,d: L3c,f,e

Tie-Sets or f -Loops or f -Circuits are :

Tie-Set or f -Loop or f -Circuit matrix is :Branches

f -Loops a b c d e f

B =L1L2L3

1 -1 -1 0 0 00 1 0 1 1 00 0 1 0 -1 1

We now re-arrange columns in B with links first and twigs later as a, d , f , b, c, e.

Brancheslinks twigs

f -Loops a d f b c e

B =L1L2L3

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

Thus, B = [Bl, Bt] = [I, Bt]. And B is non-singular matrix which rank is (b − n + 1).

AeL. je. siHh/L. J. Singh(nehu/NEHU) EE - 304 Electrical Network Theory 29.08.2016 10 / 14

Tie-Set (Loop/Circuit) Matrix, BRank of the tie-set matrix!

Example - 3: In the graph shown below on the left, we select a tree: twigsb, c, e.Thus, the co-tree: linksa, d , f . We now derive Tie-Set Matrix, B, by placing backeach link one at a time on the tree.

1 2 3

4

a

b c

d e f

1 2 3

4

a

b c

d e f

Tree: twigsb, c, e

1 2 3

4

a

b c

d e f

Co-tree: linksa, d , f

1 2 3

4

a

b c

fed

a

b c

L1

b

edL2

c

e fL3

f -loops : L1a,c,b: L2b,e,d: L3c,f,e

Tie-Sets or f -Loops or f -Circuits are :

Tie-Set or f -Loop or f -Circuit matrix is :Branches

f -Loops a b c d e f

B =L1L2L3

1 -1 -1 0 0 0

0 1 0 1 1 00 0 1 0 -1 1

We now re-arrange columns in B with links first and twigs later as a, d , f , b, c, e.

Brancheslinks twigs

f -Loops a d f b c e

B =L1L2L3

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

Thus, B = [Bl, Bt] = [I, Bt]. And B is non-singular matrix which rank is (b − n + 1).

AeL. je. siHh/L. J. Singh(nehu/NEHU) EE - 304 Electrical Network Theory 29.08.2016 10 / 14

Tie-Set (Loop/Circuit) Matrix, BRank of the tie-set matrix!

Example - 3: In the graph shown below on the left, we select a tree: twigsb, c, e.Thus, the co-tree: linksa, d , f . We now derive Tie-Set Matrix, B, by placing backeach link one at a time on the tree.

1 2 3

4

a

b c

d e f

1 2 3

4

a

b c

d e f

Tree: twigsb, c, e

1 2 3

4

a

b c

d e f

Co-tree: linksa, d , f

1 2 3

4

a

b

c

f

ed

a

b c

L1

b

edL2

c

e fL3

f -loops : L1a,c,b: L2b,e,d: L3c,f,e

Tie-Sets or f -Loops or f -Circuits are :

Tie-Set or f -Loop or f -Circuit matrix is :Branches

f -Loops a b c d e f

B =L1L2L3

1 -1 -1 0 0 00 1 0 1 1 0

0 0 1 0 -1 1

We now re-arrange columns in B with links first and twigs later as a, d , f , b, c, e.

Brancheslinks twigs

f -Loops a d f b c e

B =L1L2L3

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

Thus, B = [Bl, Bt] = [I, Bt]. And B is non-singular matrix which rank is (b − n + 1).

AeL. je. siHh/L. J. Singh(nehu/NEHU) EE - 304 Electrical Network Theory 29.08.2016 10 / 14

Tie-Set (Loop/Circuit) Matrix, BRank of the tie-set matrix!

Example - 3: In the graph shown below on the left, we select a tree: twigsb, c, e.Thus, the co-tree: linksa, d , f . We now derive Tie-Set Matrix, B, by placing backeach link one at a time on the tree.

1 2 3

4

a

b c

d e f

1 2 3

4

a

b c

d e f

Tree: twigsb, c, e

1 2 3

4

a

b c

d e f

Co-tree: linksa, d , f

1 2 3

4

a

b

c

fe

d

a

b c

L1b

edL2

c

e fL3

f -loops : L1a,c,b: L2b,e,d: L3c,f,e

Tie-Sets or f -Loops or f -Circuits are :

Tie-Set or f -Loop or f -Circuit matrix is :Branches

f -Loops a b c d e f

B =L1L2L3

1 -1 -1 0 0 00 1 0 1 1 00 0 1 0 -1 1

We now re-arrange columns in B with links first and twigs later as a, d , f , b, c, e.

Brancheslinks twigs

f -Loops a d f b c e

B =L1L2L3

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

Thus, B = [Bl, Bt] = [I, Bt]. And B is non-singular matrix which rank is (b − n + 1).

AeL. je. siHh/L. J. Singh(nehu/NEHU) EE - 304 Electrical Network Theory 29.08.2016 10 / 14

Tie-Set (Loop/Circuit) Matrix, BRank of the tie-set matrix!

Example - 3: In the graph shown below on the left, we select a tree: twigsb, c, e.Thus, the co-tree: linksa, d , f . We now derive Tie-Set Matrix, B, by placing backeach link one at a time on the tree.

1 2 3

4

a

b c

d e f

1 2 3

4

a

b c

d e f

Tree: twigsb, c, e

1 2 3

4

a

b c

d e f

Co-tree: linksa, d , f

1 2 3

4

a

b c

fed

a

b c

L1b

edL2

c

e fL3

f -loops : L1a,c,b: L2b,e,d: L3c,f,e

Tie-Sets or f -Loops or f -Circuits are :

Tie-Set or f -Loop or f -Circuit matrix is :Branches

f -Loops a b c d e f

B =L1L2L3

1 -1 -1 0 0 00 1 0 1 1 00 0 1 0 -1 1

We now re-arrange columns in B with links first and twigs later as a, d , f , b, c, e.

Brancheslinks twigs

f -Loops a d f b c e

B =L1L2L3

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

Thus, B = [Bl, Bt] = [I, Bt]. And B is non-singular matrix which rank is (b − n + 1).

AeL. je. siHh/L. J. Singh(nehu/NEHU) EE - 304 Electrical Network Theory 29.08.2016 10 / 14

Tie-Set (Loop/Circuit) Matrix, BRank of the tie-set matrix!

Example - 3: In the graph shown below on the left, we select a tree: twigsb, c, e.Thus, the co-tree: linksa, d , f . We now derive Tie-Set Matrix, B, by placing backeach link one at a time on the tree.

1 2 3

4

a

b c

d e f

1 2 3

4

a

b c

d e f

Tree: twigsb, c, e

1 2 3

4

a

b c

d e f

Co-tree: linksa, d , f

1 2 3

4

a

b c

fed

a

b c

L1

b

ed

L2

c

e f

L3

f -loops : L1a,c,b: L2b,e,d: L3c,f,e

Tie-Sets or f -Loops or f -Circuits are :

Tie-Set or f -Loop or f -Circuit matrix is :

Branchesf -Loops a b c d e f

B =L1L2L3

1 -1 -1 0 0 00 1 0 1 1 00 0 1 0 -1 1

We now re-arrange columns in B with links first and twigs later as a, d , f , b, c, e.

Brancheslinks twigs

f -Loops a d f b c e

B =L1L2L3

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

Thus, B = [Bl, Bt] = [I, Bt]. And B is non-singular matrix which rank is (b − n + 1).

AeL. je. siHh/L. J. Singh(nehu/NEHU) EE - 304 Electrical Network Theory 29.08.2016 10 / 14

Tie-Set (Loop/Circuit) Matrix, BRank of the tie-set matrix!

Example - 3: In the graph shown below on the left, we select a tree: twigsb, c, e.Thus, the co-tree: linksa, d , f . We now derive Tie-Set Matrix, B, by placing backeach link one at a time on the tree.

1 2 3

4

a

b c

d e f

1 2 3

4

a

b c

d e f

Tree: twigsb, c, e

1 2 3

4

a

b c

d e f

Co-tree: linksa, d , f

1 2 3

4

a

b c

fed

a

b c

L1

b

ed

L2

c

e f

L3

f -loops : L1a,c,b: L2b,e,d: L3c,f,e

Tie-Sets or f -Loops or f -Circuits are :

Tie-Set or f -Loop or f -Circuit matrix is :

Branchesf -Loops a b c d e f

B =L1L2L3

1 -1 -1 0 0 00 1 0 1 1 00 0 1 0 -1 1

We now re-arrange columns in B with links first and twigs later as a, d , f , b, c, e.

Brancheslinks twigs

f -Loops a d f b c e

B =L1L2L3

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

Thus, B = [Bl, Bt] = [I, Bt]. And B is non-singular matrix which rank is (b − n + 1).

AeL. je. siHh/L. J. Singh(nehu/NEHU) EE - 304 Electrical Network Theory 29.08.2016 10 / 14

Tie-Set Matrix, B, and KVL

Kirchhoff’s Voltage Law (KVL) of a graph can be applied to the f -loops to obtain aset of linearly independent equations.

1 2 3

4

a

b c

d e f

1 2 3

4

a

b c

d e f

Tree: twigsb, c, e

1 2 3

4

a

b c

d e f

Co-tree: linksa, d , f

1 2 3

4

a

b c

d e f

L1

L2 L3

Tie-set 1 : Va − Vc − Vb=0Tie-set 2 : Vb + Ve + Vd=0Tie-set 3 : Vc + Vf − Ve=0

Expressing the above equations in matrix form as 1 -1 -1 0 0 00 1 0 1 1 00 0 1 0 -1 1

= 0

VaVbVcVdVeVf

Thus, BVB = 0 where VB is a column vector of branch voltages.

AeL. je. siHh/L. J. Singh(nehu/NEHU) EE - 304 Electrical Network Theory 29.08.2016 11 / 14

Tie-Set Matrix, B, and KVL

Kirchhoff’s Voltage Law (KVL) of a graph can be applied to the f -loops to obtain aset of linearly independent equations. For this, we first select a tree as shown below.

1 2 3

4

a

b c

d e f

1 2 3

4

a

b c

d e f

Tree: twigsb, c, e

1 2 3

4

a

b c

d e f

Co-tree: linksa, d , f

1 2 3

4

a

b c

d e f

L1

L2 L3

Tie-set 1 : Va − Vc − Vb=0Tie-set 2 : Vb + Ve + Vd=0Tie-set 3 : Vc + Vf − Ve=0

Expressing the above equations in matrix form as 1 -1 -1 0 0 00 1 0 1 1 00 0 1 0 -1 1

= 0

VaVbVcVdVeVf

Thus, BVB = 0 where VB is a column vector of branch voltages.

AeL. je. siHh/L. J. Singh(nehu/NEHU) EE - 304 Electrical Network Theory 29.08.2016 11 / 14

Tie-Set Matrix, B, and KVL

Kirchhoff’s Voltage Law (KVL) of a graph can be applied to the f -loops to obtain aset of linearly independent equations. For this, we first select a tree as shown below.If Va, Vb, Vc , Vd , Ve and Vf are the branch voltages of branches a, b, c, d , e andf of the given graph, the KVL equation for each tie-set can be written as

1 2 3

4

a

b c

d e f

1 2 3

4

a

b c

d e f

Tree: twigsb, c, e

1 2 3

4

a

b c

d e f

Co-tree: linksa, d , f

1 2 3

4

a

b c

d e f

L1

L2 L3

Tie-set 1 : Va − Vc − Vb=0Tie-set 2 : Vb + Ve + Vd=0Tie-set 3 : Vc + Vf − Ve=0

Expressing the above equations in matrix form as 1 -1 -1 0 0 00 1 0 1 1 00 0 1 0 -1 1

= 0

VaVbVcVdVeVf

Thus, BVB = 0 where VB is a column vector of branch voltages.

AeL. je. siHh/L. J. Singh(nehu/NEHU) EE - 304 Electrical Network Theory 29.08.2016 11 / 14

Tie-Set Matrix, B, and KVL

Kirchhoff’s Voltage Law (KVL) of a graph can be applied to the f -loops to obtain aset of linearly independent equations. For this, we first select a tree as shown below.If Va, Vb, Vc , Vd , Ve and Vf are the branch voltages of branches a, b, c, d , e andf of the given graph, the KVL equation for each tie-set can be written as

1 2 3

4

a

b c

d e f

1 2 3

4

a

b c

d e f

Tree: twigsb, c, e

1 2 3

4

a

b c

d e f

Co-tree: linksa, d , f

1 2 3

4

a

b c

d e f

L1

L2 L3

Tie-set 1 : Va − Vc − Vb=0Tie-set 2 : Vb + Ve + Vd=0Tie-set 3 : Vc + Vf − Ve=0

Expressing the above equations in matrix form as 1 -1 -1 0 0 00 1 0 1 1 00 0 1 0 -1 1

= 0

VaVbVcVdVeVf

Thus, BVB = 0 where VB is a column vector of branch voltages.

AeL. je. siHh/L. J. Singh(nehu/NEHU) EE - 304 Electrical Network Theory 29.08.2016 11 / 14

Tie-Set Matrix, B, and KVL

Kirchhoff’s Voltage Law (KVL) of a graph can be applied to the f -loops to obtain aset of linearly independent equations. For this, we first select a tree as shown below.If Va, Vb, Vc , Vd , Ve and Vf are the branch voltages of branches a, b, c, d , e andf of the given graph, the KVL equation for each tie-set can be written as

1 2 3

4

a

b c

d e f

1 2 3

4

a

b c

d e f

Tree: twigsb, c, e

1 2 3

4

a

b c

d e f

Co-tree: linksa, d , f

1 2 3

4

a

b c

d e f

L1

L2 L3

Tie-set 1 : Va − Vc − Vb=0Tie-set 2 : Vb + Ve + Vd=0Tie-set 3 : Vc + Vf − Ve=0

Expressing the above equations in matrix form as 1 -1 -1 0 0 00 1 0 1 1 00 0 1 0 -1 1

= 0

VaVbVcVdVeVf

Thus, BVB = 0 where VB is a column vector of branch voltages.

AeL. je. siHh/L. J. Singh(nehu/NEHU) EE - 304 Electrical Network Theory 29.08.2016 11 / 14

Tie-Set Matrix, B, and KVL

Kirchhoff’s Voltage Law (KVL) of a graph can be applied to the f -loops to obtain aset of linearly independent equations. For this, we first select a tree as shown below.If Va, Vb, Vc , Vd , Ve and Vf are the branch voltages of branches a, b, c, d , e andf of the given graph, the KVL equation for each tie-set can be written as

1 2 3

4

a

b c

d e f

1 2 3

4

a

b c

d e f

Tree: twigsb, c, e

1 2 3

4

a

b c

d e f

Co-tree: linksa, d , f

1 2 3

4

a

b c

d e f

L1

L2 L3

Tie-set 1 : Va − Vc − Vb=0Tie-set 2 : Vb + Ve + Vd=0Tie-set 3 : Vc + Vf − Ve=0

Expressing the above equations in matrix form as 1 -1 -1 0 0 00 1 0 1 1 00 0 1 0 -1 1

= 0

VaVbVcVdVeVf

Thus, BVB = 0 where VB is a column vector of branch voltages.

AeL. je. siHh/L. J. Singh(nehu/NEHU) EE - 304 Electrical Network Theory 29.08.2016 11 / 14

Tie-Set Matrix, B, and Branch Currents

Branch currents can be expressed as a linear combination of link currents/loop cur-rents using B matrix as IB = B′IL where IB represents branch current matrix and ILthe loop current matrix.

1 2 3

4

a

b c

d e f

1 2 3

4

a

b c

d e f

Tree: twigsb, c, e

1 2 3

4

a

b c

d e f

Co-tree: linksa, d , f

1 2 3

4

a

b c

d e f

L1

L2 L3

iaibicidieif

=

1 0 0−1 1 0−1 0 10 1 00 1 −10 0 1

IL1

IL2

IL3

Writing branch currents in terms of loop/link currents,ia = IL1

ib = −IL1 + IL2

ic = −IL1 + IL3

id = IL2

ie = IL2 − IL3

if = IL3

AeL. je. siHh/L. J. Singh(nehu/NEHU) EE - 304 Electrical Network Theory 29.08.2016 12 / 14

Tie-Set Matrix, B, and Branch Currents

Branch currents can be expressed as a linear combination of link currents/loop cur-rents using B matrix as IB = B′IL where IB represents branch current matrix and ILthe loop current matrix.

1 2 3

4

a

b c

d e f

1 2 3

4

a

b c

d e f

Tree: twigsb, c, e

1 2 3

4

a

b c

d e f

Co-tree: linksa, d , f

1 2 3

4

a

b c

d e f

L1

L2 L3

iaibicidieif

=

1 0 0−1 1 0−1 0 10 1 00 1 −10 0 1

IL1

IL2

IL3

Writing branch currents in terms of loop/link currents,ia = IL1

ib = −IL1 + IL2

ic = −IL1 + IL3

id = IL2

ie = IL2 − IL3

if = IL3

AeL. je. siHh/L. J. Singh(nehu/NEHU) EE - 304 Electrical Network Theory 29.08.2016 12 / 14

Tie-Set Matrix, B, and Branch Currents

Branch currents can be expressed as a linear combination of link currents/loop cur-rents using B matrix as IB = B′IL where IB represents branch current matrix and ILthe loop current matrix.

1 2 3

4

a

b c

d e f

1 2 3

4

a

b c

d e f

Tree: twigsb, c, e

1 2 3

4

a

b c

d e f

Co-tree: linksa, d , f

1 2 3

4

a

b c

d e f

L1

L2 L3

iaibicidieif

=

1 0 0−1 1 0−1 0 10 1 00 1 −10 0 1

IL1

IL2

IL3

Writing branch currents in terms of loop/link currents,ia = IL1

ib = −IL1 + IL2

ic = −IL1 + IL3

id = IL2

ie = IL2 − IL3

if = IL3

AeL. je. siHh/L. J. Singh(nehu/NEHU) EE - 304 Electrical Network Theory 29.08.2016 12 / 14

Tie-Set Matrix, B, and Branch Currents

Branch currents can be expressed as a linear combination of link currents/loop cur-rents using B matrix as IB = B′IL where IB represents branch current matrix and ILthe loop current matrix. i.e.

1 2 3

4

a

b c

d e f

1 2 3

4

a

b c

d e f

Tree: twigsb, c, e

1 2 3

4

a

b c

d e f

Co-tree: linksa, d , f

1 2 3

4

a

b c

d e f

L1

L2 L3

iaibicidieif

=

1 0 0−1 1 0−1 0 10 1 00 1 −10 0 1

IL1

IL2

IL3

Writing branch currents in terms of loop/link currents,ia = IL1

ib = −IL1 + IL2

ic = −IL1 + IL3

id = IL2

ie = IL2 − IL3

if = IL3

AeL. je. siHh/L. J. Singh(nehu/NEHU) EE - 304 Electrical Network Theory 29.08.2016 12 / 14

Tie-Set Matrix, B, and Branch Currents

Branch currents can be expressed as a linear combination of link currents/loop cur-rents using B matrix as IB = B′IL where IB represents branch current matrix and ILthe loop current matrix. i.e.

1 2 3

4

a

b c

d e f

1 2 3

4

a

b c

d e f

Tree: twigsb, c, e

1 2 3

4

a

b c

d e f

Co-tree: linksa, d , f

1 2 3

4

a

b c

d e f

L1

L2 L3

iaibicidieif

=

1 0 0−1 1 0−1 0 10 1 00 1 −10 0 1

IL1

IL2

IL3

Writing branch currents in terms of loop/link currents,ia = IL1

ib = −IL1 + IL2

ic = −IL1 + IL3

id = IL2

ie = IL2 − IL3

if = IL3

AeL. je. siHh/L. J. Singh(nehu/NEHU) EE - 304 Electrical Network Theory 29.08.2016 12 / 14

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.

D. Roy Choudhury,Networks and Systems,New Age Publishers, 1998.

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

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

M. E. Van ValkenburgNetwork Analysis, 3/e.PHI, 2005.

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

D. Roy Choudhury,Networks and Systems,New Age Publishers, 1998.

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

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

AeL. je. siHh/L. J. Singh(nehu/NEHU) EE - 304 Electrical Network Theory 29.08.2016 13 / 14

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.

D. Roy Choudhury,Networks and Systems,New Age Publishers, 1998.

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

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

M. E. Van ValkenburgNetwork Analysis, 3/e.PHI, 2005.

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

D. Roy Choudhury,Networks and Systems,New Age Publishers, 1998.

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

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

AeL. je. siHh/L. J. Singh(nehu/NEHU) EE - 304 Electrical Network Theory 29.08.2016 13 / 14

Khublei Shibun!

Thank You!Any Question?

AeL. je. siHh/L. J. Singh(nehu/NEHU) EE - 304 Electrical Network Theory 29.08.2016 14 / 14