Network Analysis - ECE - 3rd Sem - VTU - Unit 2 - Network Topology - ramisuniverse,ramisuniverse

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, 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.

a

a

a

branch R

L

b

R a

b

b

C

a a

C

b

C

a

branc h

c

C L

b b

C

L

b

c

branc h

b

C a

L

R

a

c

L

b

R

a

R

a

R

b

R

a

b

b L

a

branch

b

Graph of a network  It’s the geometric figure, in which all the passive e lements 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 b y the open circuits, retaining all the nodes Graph=Nodes(named by letters)+Branch(named by numbers) Example: R

V1

R

L

V2

C

C

L

R

R

R

L

I

Oriented Graphs:

It’s the graph in which all the nodes are named by b y letters, all the branches are named by numbers, with arbitrary assignment of d irections 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 b ranches

Example: R

a

R

1

b

a

R

2

3 4

V1

L 4

b

2

C

5

V2

5

3

1 c

c

Concept of Tree and a Co-tree Tree: It’s the sub-graph of the graph, in which no loops are present, formed by b y 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 T

T

 Number of Trees in a graph: |AA | = |ArAr  |

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 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

1

b

2

5

7 6 f  a

1

c

a

d c

2

f 

c

a

1

3

b

c

b

5

c

2

e

4

f 

d

e a

3

b

6 d

f 

c

5

7

6 f 

2

6 d

e

7

d

4

a

6 e

2

5

6

4 e b

b

7

3

5

f 

1

3

4 e

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

d

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 b ranches 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 d eterminant 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 inciden ce 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  T

T

Provides the number of trees (i.e., |AA | = |ArAr  | 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,

T

Eb=A Vn T

Node Transformation Equation: E b=A Vn

E b: Column matrix(bx1), of the branch voltages T

A : 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, AI b=0 , I b: 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 -1 1 0 1

i5

=

0

i6 T

Basic Equtions: E b=A Vn, and AI b=0

Equilibrium equations with node to datum voltages as variables:

Consider the general branch of the network:

vg

ib-ig

Zb

ib

ib eb ig

vg= Total series voltage in the the branch ig = Total current source across the branch Z b= Total impedance of the branch i b=Branch current Y b= Total admittance of the branch Voltage and current relations in the general branch diagram: e b=vg+Z b(i b-ig) i b=ig+Y b(e b-vg)

For the network, with more number of branches: b ranches: E b=Vg+Z b(I b+Ig) I b=Ig+Y b(E b-Vg) E b: (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) Z b: (bxb) matrix of the branch impedances Y b: (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, AI b=0 A[Ig+Y b(E b-Vg)]=0

Substituting for Ib,

AIg+AY b(E b-Vg)=0 AIg+AY bE b-AY bVg=0 T

AIg+AY b(A Vn)-AY bVg=0

Substituting for Eb=ATVn

T

AIg+AY bA Vn-AY bVg=0 T

AY bA Vn=AY bVg-AIg T

AY bA = Yn and AY bVg-AIg=In So, In=YnVn

-1 n

or Vn=Yn I

---------------------1 ---------------------2

Eq 1 or 2: Set of o f (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 T

Number of Possible Trees in a graph:  N = AA ,

A: Incidence matrix

Concept of cut-set and cut-set schedule Cut-Set: It’s a set of branches of a connected conn ected 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, identified, by the possible tree for  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 cutsetts) = (n-1), where n is the number of nodes p resent 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 h as orientation opposite to the orientation of cutset 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 eleme nt in the cutset schedule is +1 and similarly for all other branch es Branches like, 4, 5, and 6 are not present, in the cut-set: so entries will be 0 s 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 equilibrium equations 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 T

Node Transformation Equation: E b=Q Vt

E b: Column matrix(bX1), of branch voltages Vt: Column matrix{(n-1)X1}, of twig voltages T

Q : 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, QI b=0,

I b: 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

T

Basic Equations: E b=Q Vt,

and

QI b=0

Equilibrium equations with tree twig voltages as variables: vg

ib-ig

Zb

ib

ib eb ig

T

QI b=0, and E b=Q Vt I b=Ig+Y b(E b-Vg) QI b = Q[Ig+Y b(E b-Vg)], (By multiplying multiplying both sides sides by Q) Q) 0 = QIg+QY bE b-QY bVg T

0=QIg+QY b(Q Vt)-QY bVg, Substituting for E b T

0=QIg+QY bQ Vt-QY bVg T

QY bQ Vt=QY bVg-QIg QY bVg-QIg=In, T

QY bQ Vt=In,

T

and

QY bQ =Yn ----------------- 1

and

YnVt=In

----------------2

Eq 1 or 2 : Set S et of (n-1) equilibrium equations with tree twig voltages as va riables 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 equilibrium equations equations in matrix form – Tie-set matrix KCL and Matrix, B

Columns of B: provides the relation between the branch current and the loop loo p current So,

i1=il1 i2=il2 i3=il3 i4=il4 i5=il2-il1 i6=il3-il2 i7=il4-il3 i8=il1-il4

So,

T

Ib=B Il

I b: Column matrix(bx1), of branch currents T

B : 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,

E b: Column matrix(bx1), of branch voltages

BEb=0,

T

Basic equations: I b=B Il, and

BE b=0

Equilibrium equations with loop currents as variables vg

ib-ig

Zb

ib

ib eb ig

E b=Vg+Z b(I b-Ig),

T

I b=B Il,

and

BE b=0

BE b=0 B[Vg+Z b(I b-Ig)]=0 BVg+BZ b(I b-Ig)=0 BVg+BZ bI b-BZ bIg=0 T

BVg+BZ bB Il-BZ bIg=0, T

BZ bB Il=BZ bIg-BVg

T

(Substituting I b=B Il)

Vl=BZ bIg-BVg So ,

Zl Il=Vl

and or

T

Zl=BZ bB -----------------1 -1

Il=Zl Vl

------------ ----2

Eq 1 and 2: Set S et of Equilibrium equations with loop currents as the indep endent 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 bran ches 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

Voltage

Branch current

Branch voltage

Mesh

 Node

Loop

 Node pair 

Loop current

 Node pair voltage

Mesh current

 Node voltage

 Number of loops Link 

 Number of nodes

Twig

Twig

Tie-set

Cut-set

Short circuit

Open circuit

Series circuit

Parallel circuit

Inductance

Capacitance

Resistance

Conductance

Thevenin’s Theorem

 Norton’s Theorem

KCL

KVL

Closing switch

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 nodevoltage 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: T

AI b=0 and AY bA Vn=AY bVg-AIg

T

QI b=0 and QY bQ Vt=QY bVg-QIg

T

BE b=0 and BZ bB Il=BZ bIg-BVg

E b=Vg+Z b(I b-Ig)

E b=A Vn

I b=Ig+Y b(E b-Vg)

E b=Q Vt I b=B Il

T

T

T