Unit : 2 NETWORK TOPOLOGY

Network Topology: Graph of a network, Concept of tree and co-tree, incidence matrix,

tie-set, tie-set and cut-set schedules, Formulation of equilibrium equations in matrix form,

Solution of resistive networks, Principle of duality.

Network Topology

Importance of the topological approach for solving the electric circuits

Node: It’s the point in the network to which two or more electric elements are connected

Branch: It’s the line segment, which represents the network elements or a combination

of network elements connected between the two nodes

Path: It’s the group of elements of the property that they can be traversed in such a way

that no node again passing possible

Loop: It’s the closed path in the network

Mesh: It’s an independent loop, which don’t any other loop in it.

R

L

C

branch

branch

branch branch

R

R

R

R

R

R

C

C L

L

L

C

C

L

L

C

a

a

a

a

a

a

a

a

a

a

a

a

a

b

b

b

b

b

b

b

b

b

b

b

b

b

c

c

c

Graph of a network

It’s the geometric figure, in which all the passive elements are represented by the

line segments, all the ideal voltage sources are represented by the short circuits, and all

the ideal current sources are represented by the open circuits, retaining all the nodes

Graph=Nodes(named by letters)+Branch(named by numbers)

Example:

R R R

L CV1 V2

C R

RLL

I

Oriented Graphs:

It’s the graph in which all the nodes are named by letters, all the branches are

named by numbers, with arbitrary assignment of directions are mentioned for each

branches

Procedure of oriented graph formation from the circuit

1. Name the nodes of the graph by letters

2. Name the branches by using numbers

3. Arbitrarily, take the direction of branch currents over the branches

Example:

R R R

L CV1 V2

a b

1

2

3

45

a b

cc

2

4 5

1 3

Concept of Tree and a Co-tree

Tree: It’s the sub-graph of the graph, in which no loops are present, formed by the

removal of some number of branches of the graph. Tree can have different number of

trees in the graph.

Twig(Tree branches): It’s the branch present in the tree of a graph

Chord(Links): It’s the branch of trees to be removed to form a tree

Number of Trees in a graph: |AAT| = |ArArT|

Twig, and Links relation in a tree:

Number of twigs: t=(n-1)

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

Co-tree: Co-tree of a given tree is the sub-graph of the graph, formed by the branches of

the graph, that are not in the given tree

Example:

a b c

def

1 2

3

4

5

6

7

a b c

def

1 2

46

7

a b c

def

1 2

4

5

6

a b c

def

3

4

5

6

7

a b c

def

2

35

6

7

a b c

def

1 2

35

6

Network Variable

It’s the independent variable on which values of all other elements depends on

It’s the assumed variable, specific to the method of analysis using for the analysis of the

circuit

Types of Network Variables

1. Current Variables:

Types:

1. Branch Currents : Currents in the branches of the circuit

2. Loop Currents : Currents in the loops of the circuit

Method providing the current variables: Mesh-current method of network analysis

2. Voltage Variables:

Types:

1. Node-to-Datum Voltages : It’s the voltage between the node and the reference

node(assumed of zero potential) in the circuit

2. Node-pair Voltages : It’s the voltage between two nodes in circuit

Ex.: Branch voltages

Concept of Incidence matrix

It’s the (nxb) matrix, which provides the complete information regarding the

branch connections and branch orientations to all nodes

All-incidence matrix: Alternate name for the incidence matrix

Procedure of constructing incidence matrix

1. Form the oriented graph of the graph

2. Arbitrarily, take the direction of branch currents over the branches

3. Take the branches along the rows, and take the nodes along the column

4. Arbitrarily, take the incoming branch currents to the named nodes as –ve (+ve),

and the outgoing branch currents from the node as the +ve(-ve)

Importance of Incidence Matrix

1. Provides the branch – node connection information(i.e., nodes between which the

branch is connected- information) in the circuit

2. Provides the orientation of branch currents in the circuit

3. Provides the direction of branch currents, since determinant of A is zero, and sum

of elements in each column of A is zero

4. Oriented graph, can be constructed using A or reduced incidence matrix Ar

5. Provides KCL expressions, at each node (from its each row)

Reduced Incidence Matrix, Ar

It’s the incidence matrix, in which the information of any one node is completely

not present. Usually, in the reduced incidence matrix, the reference node information is

going to be omitted

Importance of Reduced Incidence Matrix, Ar

1. Incidence matrix can be constructed from Ar

2. Oriented graph can be constructed from Ar

Provides the number of trees (i.e., |AAT| = |ArArT| possible in the graph

Formulation of equilibrium equations in matrix form – Incidence

matrix

Node to datum voltages and Matrix, A

Columns of A: Provides the relation between the branch voltages

e1 = -Va

e2 = Va-Vb

e3 = Va-Vc

e4 = Vc-Vb

e5 = Vb

e6 = Vc

So, Eb=ATVn

Node Transformation Equation: Eb=ATVn

Eb: Column matrix(bx1), of the branch voltages

AT: Transpose of A

Vn: Column matrix{(n-1)x1}, of the datum voltages

e1 -1 0 0

e2 1 -1 0

e3 1 0 -1 Va

e4 0 -1 1 Vb

e5 0 1 0 Vc

e6 0 0 1

KCL and Matrix, A:

Rows of A: provides the relations between the branch currents

Example: -i1+i2+i3=0

-i2-i4+i3=0

-i3+i4+i6=0

So, AIb=0 , Ib: Column matrix(bx1), of the branch currents

i1

i2

-1 1 1 0 0 0 i3

0 -1 0 -1 1 0 i4 = 0

0 0 -1 1 0 1 i5

i6

Basic Equtions: Eb=ATVn, and AIb=0

Equilibrium equations with node to datum voltages as variables:

Consider the general branch of the network:

ib

vgib-ig

Zb

ib

ig

eb

vg= Total series voltage in the branch

ig = Total current source across the branch

Zb= Total impedance of the branch

ib=Branch current

Yb= Total admittance of the branch

Voltage and current relations in the general branch diagram:

eb=vg+Zb(ib-ig)

ib=ig+Yb(eb-vg)

For the network, with more number of branches:

Eb=Vg+Zb(Ib+Ig)

Ib=Ig+Yb(Eb-Vg)

Eb: (bx1) matrix of branch voltages

Vg: (bx1) matrix of source voltages in the branches (Includes the initial capacitor voltages

in loop analysis)

Ig: (bx1) matrix of the source currents in the branches (Includes the initial inductor

currents in the nodal analysis)

Zb: (bxb) matrix of the branch impedances

Yb: (bxb) matrix of the branch admittances

Yn: {(n-1)x(n-1)} matrix of the node admittances

Vn: Column matrix {(n-1)x1}, of the node to datum voltages

In: Column matrix {(n-1}x1}, of the source currents

Consider, AIb=0

A[Ig+Yb(Eb-Vg)]=0 Substituting for Ib,

AIg+AYb(Eb-Vg)=0

AIg+AYbEb-AYbVg=0

AIg+AYb(ATVn)-AYbVg=0 Substituting for Eb=ATVn

AIg+AYbATVn-AYbVg=0

AYbATVn=AYbVg-AIg

AYbAT = Yn and AYbVg-AIg=In ---------------------1

So, In=YnVn or Vn=Yn I-1n ---------------------2

Eq 1 or 2: Set of (n-1) equilibrium equations. Solving the equilibrium equations, gives the

node to datum voltages, and from them the branch voltages and branch currents can be

found out

Number of Possible Trees in a graph: N = AAT, A: Incidence matrix

Concept of cut-set and cut-set schedule

Cut-Set: It’s a set of branches of a connected graph, whose removal causes the graph to

become unconnected into exactly two connected sub-graphs. Any of the branches of the

cutest, if restored destroys the separation property of the two sub-graphs

Consider the Oriented graph, of the network. (1,2,3) group of branches forms the

cut-set A. By removing these branches these branches, the graph becomes unconnected

and two sub-graphs are formed. One sub-graph is node a and branches (1,2,3). The other

sub-graph contains nodes b, c, d and branches (4,5, 6).

Single isolated node : It is also considered as a connected sub-graph

By replacing any one of the branches of the cut-set, the two sub-graphs gets

connected. Other cut-sets in the graph: B(1,5,6), C(3,4,6), D(2,4,5)

Fundamental Cut-sets: These are the cut-sets, which are minimum in number required

to be identified for the network solution. These can be identified, by the possible tree for

the graph

Example: A(1,2,3), B(1,5,6), C(3,4,6) : Because, each contains only one (the mandatory

requirement to become a fundamental cut-set) tree twig, present in the tree considered for

writing the fundamental cut-sets (In the example: considered tree: (3,4,5)Number of

Fundamental Cut-sets = Number of branches in the tree (Considered for writing the cut-

setts) = (n-1), where n is the number of nodes present in the graph

Orientation of the Cut-set: It’s the same direction as the direction of the tree branch,

which is present within it.

Cut-set Schedule: It provides the relation between the tree-branch voltages and all the

other branch voltages. Elements in the cut-set schedule, are filled using the orientation of

the particular cut-set

Example: Cut-set A( 1,2, 3): Branch 1 has orientation opposite to the orientation of cut-

set A, so the element in the cut-set schedule is written as -1.

Orientation of the branch 2: same as that of the cut-set A(1,2,3) – the element in the cut-

set schedule is +1 and similarly for all other branches

Branches like, 4, 5, and 6 are not present, in the cut-set: so entries will be 0s in their place

Fundamental Cut-set Matrix, Q

Cut-set elements can be written in the form of the matrix called the fundamental cut-set

matrix

Tree: It’s written by removing sufficient number of branches from the graph, so that the

graph does not contain any loops within it. Links are the removed branches. Twigs are

the branches of tree

Formulation of equilibrium equations in matrix form – Cut-set matrix

Twig voltages and Q matrix

Columns of Q: gives the relationship between the twig voltages and the branch voltages

So, e1=v2-v1, e2=v1, e3=v1-v3, e4=v3, e5=v2, e6=v3-v2

Node Transformation Equation: Eb=QTVt

Eb: Column matrix(bX1), of branch voltages

Vt: Column matrix{(n-1)X1}, of twig voltages

QT: Transpose matrix of Q

e1 -1 1 0

e2 1 0 0

e3 1 0 -1 V1

e4 0 0 1 V2

e5 0 1 0 V3

e6 0 -1 1

KCL and Matrix, Q

Rows of Q: gives the relation between the branch currents ( satisfying the KCL)

-i1+i2+i3=0

-i2+i5-i6=0

-i3+i4+i6=0

Therefore, QIb=0, Ib: Column matrix(nx1), of branch currents

i1

i2

-1 1 1 0 0 0 i3

1 0 0 0 1 -1 i4 = 0

0 0 -1 1 0 1 i5

i6

Basic Equations: Eb=QTVt, and QIb=0

Equilibrium equations with tree twig voltages as variables:

ib

vgib-ig

Zb

ib

ig

eb

QIb=0, and Eb=QT Vt

Ib=Ig+Yb(Eb-Vg)

QIb = Q[Ig+Yb(Eb-Vg)], (By multiplying both sides by Q)

0 = QIg+QYbEb-QYbVg

0=QIg+QYb(QTVt)-QYbVg, Substituting for Eb

0=QIg+QYbQTVt-QYbVg

QYbQTVt=QYbVg-QIg

QYbVg-QIg=In, and QYbQT=Yn ----------------- 1

QYbQTVt=In, and YnVt=In ----------------2

Eq 1 or 2 : Set of (n-1) equilibrium equations with tree twig voltages as variables

If Vt is known, Vbs can find out. Branch currents can be find out, if the elements in the

network are known

Concept of tie-set and tie-set schedule

Tie-Set: It’s the collection of branches forming a loop.

Fundamental Tie-sets: Minimal number of tie-sets needed to identify the network, can

be written by selecting one possible tree of the graph ( with the condition that each tie-set

branches must form a close loop and contain at least one tree twig, and can be more than

one also)

By adding one link to the tree, a loop is formed and by adding other links to the

graph, remaining loops are formed, then such a loop is the fundamental loop.

Number of fundamental loops = Number of links

So, the branches of the fundamental loop: forms the fundamental tie-set

Number of fundamental tie-sets = Number of links

Tie-set Matrix(Loop Incidence Matrix), B (or M): Provides the relationship between

the loop currents and the branch currents

Tie-set Schedule: It’s the schedule, which gives the relationship between the loop

currents and the branch currents

Direction of tie-set loop current: Same as that of the link present within it

Elements in the Tie-set schedule and Tie-set matrix:

If branch is not present in the tie-set: 0

If the branch is present and the direction is same as that of the loop current: 1

If the branch is present and the direction is opposite to that of the loop current: -1

Example: Tree considered for writing the tie-set: (5,6,7,8)

Dotted lines represents the links

Thick lines represents the twigs

Branch currents: i1, i2, i3, i4

Loop currents: il1, il2, il3, il4

Tie-set schedule and the tie-set matrix is as below:

Formulation of equilibrium equations in matrix form – Tie-set matrix

KCL and Matrix, B

Columns of B: provides the relation between the branch current and the loop current

So, i1=il1

i2=il2

i3=il3

i4=il4

i5=il2-il1

i6=il3-il2

i7=il4-il3

i8=il1-il4

So, Ib=BTIl

Ib: Column matrix(bx1), of branch currents

BT: Transpose of B

Il: Column matrix(mx1), of loop currents

m: Number of independent loops

i1 1 0 0 0

i2 0 1 0 0

i3 0 0 1 0

i4 0 0 0 1 il1

i5 -1 1 0 0 il2

i6 0 -1 1 0 il3

i7 0 0 -1 1 il4

i8 1 0 0 -1

Branch Voltage and Matrix B:

Rows of B: provides the branch voltage relation in the graph

e1-e5=e8=0

e2+e5-e6=0

e3+e6-e7=0

e4+e7-e8=0

So, BEb=0, Eb: Column matrix(bx1), of branch voltages

Basic equations: Ib=BTIl, and BEb=0

Equilibrium equations with loop currents as variables

ib

vgib-ig

Zb

ib

ig

eb

Eb=Vg+Zb(Ib-Ig), Ib=BTIl, and BEb=0

BEb=0

B[Vg+Zb(Ib-Ig)]=0

BVg+BZb(Ib-Ig)=0

BVg+BZbIb-BZbIg=0

BVg+BZbBTIl-BZbIg=0, (Substituting Ib=BTIl)

BZbBTIl=BZbIg-BVg

Vl=BZbIg-BVg and Zl=BZbBT -----------------1

So , Zl Il=Vl or Il=Zl-1Vl ------------ ----2

Eq 1 and 2: Set of Equilibrium equations with loop currents as the independent variables

Solving, equilibrium equtions, loop currents are obtained and from that branch currents

can be find out. If the elements of the branches are known, then the branch currents can

be find out

E shift and I shift

E-shift:

Normal nodal analysis, can not be applied if any one branch in the circuit,

contains an ideal voltage source, since the source current due to this branch to the

particular node is indeterminate.

At that time, the ideal voltage has to be shifted to the other branches that are

connected in series with it, without changing the characteristic of the network

I-shift:

Loop analysis can not be applied, when an ideal current source is present in any

branch of the network, since the voltage drop in the branch containing the ideal current

source is indeterminate.

At that time, the ideal current source is to be shifted to the parallel lines with each

of the branches forming the close loop with the branch having the ideal current source

Principle of Duality

Duality: It’s the similarity between the two quantities

Dual networks: Two networks are said to be the dual networks, if the mesh current

equations of one network are similar to the node voltage equations of the other network

R=1/G, L=C, C=R , and V(t)=i(t) : When LHS are of one network and RHS are of

another network - the networks are said to be dual to each other

Duality diagrams

Table:

Table

Quantity Dual Quanity

Current

Branch current

Mesh

Loop

Loop current

Mesh current

Number of loops Link

Twig

Tie-set

Short circuit

Series circuit

Inductance

Resistance

Thevenin’s Theorem

KCL

Closing switch

Voltage

Branch voltage

Node

Node pair

Node pair voltage

Node voltage

Number of nodes

Twig

Cut-set

Open circuit

Parallel circuit

Capacitance

Conductance

Norton’s Theorem

KVL

Open switch

Dual Networks

Procedure:

1. Place a node, inside each loop of the given network, and name it by a letter like a,

b and others. Place one extra node o, outside the network and call as the datum

node

2. Draw lines from node to node through the elements in the original network

traversing only one element at a time

3. Arrange the nodes marked in the original network on a separate space in the

paper, for drawing dual network

4. To each element, traversed in the original network, connect its dual element

between the corresponding nodes

Example:

Dual Graph

Dual graph: Graphs are the dual, when the equations written for one by using the mesh-

current analysis method, identical to the equations written for another by using the node-

voltage analysis method

Number of brnches: Same in the original network and its dual network

Number of twigs: Number of twigs in the original network is equal to the number of

links in its dual network

Number of independent loops in the graph = Number of node-pairs in its dual graph

Procedure:

1. Mark 5 node-pairs or 6 nodes on the paper. Sixth node is datum node or the

reference node. Nodes: a, b, c, d, and e Datum node: o

2. Assign each of the 5 nodes to teach of the meshes in the graph

3. For each mesh, note the tie-set and draw the corresponding cut-set in the dual

graph

Example: For loop a, the branches forming the tie-set are 1 and 4. So, in the dual

graph, at node a, draw two branches, one between a and b and the other between a

and o. Similarly, the other branches are connected in the dual graph

4. Orientations of the branches in the dual graph: When the mesh a is tranced in

clockwise direction, the orientation of branch 1 is divergent from node a and the

orientation of the branch 4 is convergent towards node a and hence, the orientations

of branches 1 and 4, are marked on the dual graph. Similarly, the orientations of the

other branches are marked

Example;

Formulae:

Eb=Vg+Zb(Ib-Ig) Eb=ATVn AIb=0 and AYbATVn=AYbVg-AIg

Ib=Ig+Yb(Eb-Vg) Eb=QTVt QIb=0 and QYbQTVt=QYbVg-QIg

Ib=BTIl BEb=0 and BZbBTIl=BZbIg-BVg