Fundamental Cut Set Matrix - Electrical Network Graphs

This is my fourth article on an electrical network graph. The article is about the cut-set matrix and how it is useful in determining branch voltages and node voltages. As the name implies cut set is a set of branches that when removed from the graph results in an unconnected graph.

Key Questions:

What is a cut set and a fundamental cut set?
Procedure for obtaining cut sets of a graph and writing matrix elements
Fundamental cut set matrix and KVL
Fundamental cut set and KCL

Cut Set and Fundamental Cut Set:

Cut set: It is a set of branches that when cut or removed from a graph, separates the graph into two groups of nodes or the graph splits into two isolated graphs. It is a minimum set of branches of a connected graph, such that if this set of branches is removed, it will reduce the rank of the graph. And also results in an unconnected graph.

Cut set defines the minimum set of branches C, if this set of branches removed from the graph results in an unconnected graph.

If anyone branch from a set of branches C is removed, we still get a connected graph.

Fundamental cut set: It contains only one twig and one or more links.

Procedure For Obtaining Fundamental Cut Set:

As discussed above fundamental cut set contains only one twig and one or more links. To obtain a fundamental cut set follow these steps.

Select an oriented graph
Select a tree and mark the links as well with the dotted line
The number of twigs is equal to the number of cut sets
Remove one twig and necessary links to get the fundamental cut set
Repeat the above procedure for all twigs

Fig 1 graph and its tree

Fig 2 Fundamental Cut Sets

Have a closer look at fig 2.

F-cut set C1 is obtained by removing a twig 5 and a link 3. The direction or orientation of C1 is in the same direction of twig 5
F-cut set C2 is obtained by removing twig 6 and links 4 and 3. The orientation of C2 is the same as the direction of twig 6
F-cut set 3 C3 is obtained by removing twig 1 and links 2 & 4. The orientation of C3 is the same as the direction of twig 1

Fundamental Cut Set Matrix (Q):

It describes the branches contained in the cut set and their orientation.

In the cut-set matrix, each row represents a cut set and columns represent branch voltages.

Order of Fundamental Cut Set Matrix:

The number of fundamental cut sets is equal to the number of twigs that is N-1.

B is the total number of branches.

Order of fundamental cut matrix becomes (N-1)xB

Elements of Fundamental Cut Set:

Suppose the graph has 'b’ branches, N nodes, (N-1) twigs in a tree and (N,-1) fundamental cut sets. Let 'i’ be the rows and 'j’ be the columns of the cut-set matrix.

aij = +1 for the twig of selected fundamental cut set
aij = +1 if the links have the same current direction as that of the selected twig
aij = -1 if the links have opposite current direction as that of selected twig
aij = 0 if the twigs and are not a part of the selected fundamental cut set

Procedure For Writing Fundamental Cut Set Matrix:

Consider first f-cut set C1

Branch 5 (twig) = +1
Branch 3 (link) = -1

Consider second f-cut set C2

Branch 6 (twig) = +1
Branch 4 (link) = -1
Branch 3 (link) = +1

Consider third f-cut set C3

Branch 1 (twig) = +1
Branch 2 (link) = +1
Branch 4 (link) = +1

$Branches\Rightarrow \\ Cutset \Downarrow \begin{bmatrix} 0 & 0 & -1 & 0 & +1 & 0\\ 0 & 0 & +1 & -1 & 0 & +1\\ +1& +1 & 0 & +1 & 0& 0 \end{bmatrix}$

Fig 3 Fundamental Cut Set Matrix

Fundamental Cut Set Matrix and KCL

Q.Ib = 0

Where Q = cut set matrix

Ib = branch current vector

$\begin{bmatrix} 0 & 0 & -1 & 0 & +1 & 0\\ 0 & 0 & +1 & -1 & 0 & +1\\ +1& +1 & 0 & +1 & 0& 0 \end{bmatrix}.\begin{bmatrix} i_{1}\\ i_{2}\\ i_{3}\\ i_{4} \\ i_{5} \\ i_{6} \end{bmatrix} = 0$

Fig 4 KCL and Cut Set Matrix

-i3 + i5 = 0

+i3 - i4 + i6 = 0

i1 + i2 + i4 = 0

Fundamental Cut Set Matrix and KVL

Vb = QT.Vt

Where Vb = branch voltage vector

QT = transpose of the cut-set matrix

Vt = tree branch voltage vector

$\begin{bmatrix} v_{1}\\ v_{2} \\ v_{3} \\ v_{4} \\ v_{5} \\ v_{6} \end{bmatrix}=\begin{bmatrix} 0& 0 & +1\\ 0& 0 & +1\\ -1& +1 &0 \\ 0& -1 &+1 \\ +1& 0 & 0\\ 0& +1 & 0 \end{bmatrix}.\begin{bmatrix} v_{t1}\\ v_{t5} \\ v_{t6} \end{bmatrix}$

Fig 5 KVL and Cut Set Matrix

v1 = vt6

v2 = vt6

v3 = vt5 - vt1

v4 = vt6 - vt5

v5 = vt1

v6 = vt5

Tie Set Matrix | Fundamental Loop Matrix - Electrical Network Graph

Tie Set Matrix| Fundamental Loop Matrix

This is my third article on electrical network graphs. In this article I am going to explain tie set matrix. Tie set matrix is also known as fundamental loop matrix, circuit matrix, and is denoted by letter B. It is the set of branches that gives us a closed path. It helps us to determine the branch currents.

Key Questions:

Explain tie set or fundamental loop
Explain tie set matrix
Procedure for writing tie set matrix
Evaluate branch voltages with the help of tie set matrix using KVL
Evaluate branch currents with the help of tie set matrix using KCL

Tie Set Matrix And Loop Currents:

Draw Graph o the above circuit.

Tie Set: Tie set is a set of branches contained in a loop. Each loop consists of a tree and a link or chord.

As you know a tree doesn't contain any closed path or loop. The link is added in a tree in order to join two nodes to form a closed path. This closed path is called fundamental loop. See Fig 1.

Fig 1 Graph, tree and tie sets

$\\branches \Longrightarrow Loops \Downarrow \begin{bmatrix}+1& 0&+1&0&+1&-1\\-1& +1&0&0&0&0\\+1&0&0&+1&+1 &0\end{bmatrix}$

Fig 2 Tie set matrix for the fundamental loops in Fig 1

Nodes = N = 4

Branches (twigs) = B = N-1 = 3

Links (chords) = B-(N-1) = 3

There is a graph
We select a tree from the given graph
Now mark all possible fundamental loops
In order to draw fundamental loop select a link such that it forms a closed path
Every link defines a fundamental loop
By convention the direction of current in the fundamental loop is same as that of the link current direction
The number of fundamental loops is equal to the number of links in cotree. In the above graph there are three fundamental loops

Tie Set Matrix: Tie set matrix is a two dimensional array in which loop currents and branch currents are tabulated. It is the matrix that helps to find out the branch currents.

The row in tie set matrix refers to the fundamental loop currents.

The column in tie set matrix refers to the direction of current in respective branches.

Fundamental loop 1 l1:

Fig 3(a) Fundamental loop 1

By convention the direction of current in f-loop is same as the direction of current in the link (dotted line) that is branch 3
Write down the tie set matrix ( row 1) for fundamental loop 1
Check branch current direction, if it is in the same direction as that of fundamental loop current direction then write +1 for that branch.
If the branch current is in the opposite direction as that as fundamental loop current direction then write -1 for that branch
Write 0 if the branch is not the part f fundamental loop
Branch 1 and branch 5 currents are in the same direction as f-loop while branch 6 current is in opposite direction

Fundamental loop 2 l2:

Fig 3(b) Fundamental loop 2

First mark the f-loop 2 direction of current. The direction of current in f-loop 2 is same as the direction of current in the link (dotted line) that is branch 2
Check branch current directions. In branch 1 current is in opposite direction while in branch 2 current is in the same direction as f-loop 2

Fundamental loop 3 l3:

Fig 3(c) Fundamental loop 3

Loop current is in the direction of link current
Check branch currents branch 1 and 5 currents are in the same direction as that of f-loop 3

Tie Set Matrix and KVL:

Each row corresponds to KVL equations, written for fundamental loop. Fundamental loop is added to the tree to form a closed loop
So there are N-1 KVL equations
Each column corresponds to branch currents in terms of link currents

v1+v3+v5-v6 = 0

-v1+v2 = 0
v1+v4+v5= 0

BVb=0
Where B is the tie set matrix, Vb is the branch voltage vector.

$\begin{bmatrix} +1 & 0 & +1 & 0 &+1 & -1\\ -1 &+1 & 0 & 0 & 0 & 0\\ +1& 0 & 0 & +1 & +1 &0 \end{bmatrix}.\begin{bmatrix} v_{1}\\ v_{2} \\ v_{3} \\ v_{4} \\ v_{5} \\ v_{6} \end{bmatrix} = 0$

Fig 4 BVb=0

Tie Set Matrix and KCL:

$\begin{bmatrix} i_{1}\\ i_{2} \\ i_{3} \\ i_{4} \\ i_{5}\\ i_{6} \end{bmatrix}=\begin{bmatrix} +1 & -1 & +1\\ 0 & +1 & 0\\ +1& 0& 0\\ 0& 0& +1\\ +1& 0 & +1\\ -1 & 0& 0 \end{bmatrix} . \begin{bmatrix} I_{1}\\ I_{2}\\ I_{3} \end{bmatrix}$

Fig 5 BT.IL

i1=I1-I2I1+I3

i2=I2

i3=I1

i4=I3

i5=I1+I3

i6=-I1

IB = BT.IL

Where IB = branch current vector

IL = loop current

BT = transpose of tie set matrix

Conclusion:

I cover this topic in all aspects. I tried to make this topic as simple as possible, easy to understand and clear. I rechecked all the equations and calculations, but there's always probability of mistakes. So if you find any mistake, please comment. If you have any question you can ask freely. I know this is such a tedious method for analysing a network. It usually helpful for complex networks.

Incidence Matrix - Electrical Network Graph

Incidence Matrix - Graph Theory For Electrical Networks

This is my second article on electrical network graphs. In previous article I explain how to draw a network graph. This article is related to incidence matrix (Aa) of a graph. An incidence matrix is a convenient way of expressing an oriented graph in form of matrix. The entities of Incidence matrix gives information about which branch is connected to which node and its orientation. If the incidence matrix is given then you can easily convert it into graph. The incidence matrix is helpful for evaluation of current flowing through each branch. The article is all about this.

Key Questions:

What is incidence matrix?
Properties of incidence matrix
Reduced incidence matrix
How to get an incidence matrix from a given graph?
How to draw a graph from a given incidence matrix?
How to get KCL equations from incidence matrix?
How to get KVL equations from Incidence matrix?

Incidence Matrix (Aa):

A directed graph can be expressed in a compact matrix form. The branches are connected with each other with the help of nodes. A directed branch connected to a node is called incidence. It represents the orientation of the branches and the number of branches incident to a node. The number of branches incident to a node is called degree of node. We can redraw the graph if the incidence matrix is given. It is the coefficient matrix of KCL equations. The complete set of incidence matrix is called augmented incidence matrix.

Properties of incidence matrix:

Algebraic sum of the column entries of an incidence matrix is zero.
The determinant of a closed loop is 0
The determinant of incidence matrix of a tree is +1/-1

Order of incidence matrix: If there are 'n’ nodes and 'b’ branches in a network graph, then incidence matrix have 'n’ rows and 'b’ columns. So order of the incidence matrix in nxb.

Elements of the incidence matrix:

If the matrix with n rows and b columns then each entry of the matrix is aij , where i is the number of row and j is the number of columns.

The element aij = 1, if the branch j incident at node i, and leaves away from node i
The element aij = -1, if the branch j incident at node j, and enters the node i
The element aij = 0, if branch j is not the part of node i

Reduced incidence matrix (A): When any one row is completely deleted from the matrix then this is called reduced incidence matrix. The order of this matrix is (n-1)xb. This reduction results from mathematical manipulation.

Example#1: Obtain incidence matrix of the following graph.

Fig 1 Example

The above graph contain 4 nodes and 7 branches. Let's start writing Incidence matrix by considering each node separately.

Node a: branches 1 and 5 are oriented away from node a and branch 3 is incident at node a

Node b: branches 6 and 7 are oriented away from node b and branches 1 and 4 incident at node b

Node c: branch 3 is oriented away from node c and branches 2,5 and 6 incident at node c

Node d: branches 2 and 4 are oriented away from node d and branch 7 is incident at node d

Order of matrix:
There are 4 nodes, so there will be 4 rows. There 7 branches, so there will be 7 columns. Order is 4x7

$\begin{bmatrix} +1 &0 & -1 &0 & +1 & 0 &0 \\ -1 &0 & 0 &-1 &0 & +1 &+1 \\ 0 &-1 &+ 1 &0 & -1 & -1 &0 \\ 0 &+1 & 0 & +1 & 0 & 0 & -1 \end{bmatrix}$

Fig 2 Incidence Matrix of the graph in Fig 1

Example#2: Obtain reduced incidence matrix from the augmented incidence matrix which is evaluated in the above example

$\begin{bmatrix} +1 &0 & -1 &0 & +1 & 0 &0 \\ -1 &0 & 0 &-1 &0 & +1 &+1 \\ 0 &-1 &+ 1 &0 & -1 & -1 &0 \\ 0 &+1 & 0 & +1 & 0 & 0 & -1 \end{bmatrix}\Rightarrow \begin{bmatrix} +1 &0 & -1 & 0 &+1 & 0 & 0\\ -1& 0 & 0 & -1 & 0 & +1 &+1 \\ 0 & -1 & +1 & 0 & -1 & -1 & 0 \end{bmatrix}$

Fig 3 Reduced Incidence Matrix

Example#3: Obtain network graph from the given incidence matrix.

$\begin{bmatrix} +1 & +1 & 0 & -1 & -1 & 0\\ 0&0 &+1 & 0 & +1 &0 \\ -1& -1 & 0 &0 & 0 &-1 \\ 0 & 0 &-1 & +1 & 0 & +1 \end{bmatrix}$

Fig 4 Incidence Matrix

Fig 5 Graph drawn from incidence matrix

How incidence matrix is helpful for determining KCL equations?

First of all we pick any one node as reference node. By picking a node as reference all of its elements becomes 0.

Label any node as datum/ground node while label the remaining node as a,b,c
Label the branches from 1 to b

Aai = 0

Where Aa is incidence matrix, it is the coefficient matrix of KCL equation. i is the branch current vector. There are 6 branches and we have to determine current through each branch.

From the given graph in fig 5 and incidence matrix in fig 4, evaluate KCL equations

Consider node d as a reference/ground node. So row 4 becomes 0.

Apply KCL at node a, node b and node c.

Aai = 0

$\begin{bmatrix} +1 & +1 & 0 & -1 & -1 & 0\\ 0 &0 &+1 & 0 & +1 & 0\\ -1& -1 & 0 & 0 & 0 & -1\\ 0& 0 & 0 & 0 & 0 & 0 \end{bmatrix}\cdot\begin{bmatrix} i_{1}\\ i_{2}\\ i_{3} \\ i_{4}\\ i_{5}\\ i_{6} \end{bmatrix}=0$

Fig 6

i1+i2-i4-i5= 0 equation 1

i5+i3 = 0 equation 2

i6+i4-i3= 0 equation 3

So there are (n-1) independent KCL equations. From above 3 equations evaluate branch currents.

i1 = i4 + i5 - i2

i2 = i4 + i5 - i1

i3 = - i5

i4 = i3 - i6

i6 = i3 - i4

How incidence matrix is helpful for determining KVL equations?

For determining branch voltages, consider anyone node as reference or ground node.

Label any node as datum/ground node while label the remaining node as a,b,c
Label the branches from 1 to b

v = AaT . e

Where v is the branch voltage vector, AaT is the transpose matrix of Incidence matrix Aa and e is the node voltages.

From the given graph in fig 5 and incidence matrix in fig 4, evaluate KVL equations

Fig 7 branch voltages

Select node d as reference node and all entries of row 4 becomes 0.

The network has 6 branches and 6 independent KVL equations

v = AaT . e

$\begin{bmatrix} v_{1}\\ v_{2} \\ v_{3} \\ v_{4} \\ v_{5} \\ v_{6} \end{bmatrix} = \begin{bmatrix} 1 &0 & -1 &0 \\ 1 &0 &-1 &0 \\ 0& +1 &0 &-1 \\ -1 & 0 &0 &+1 \\ -1 & +1 & 0 &0 \\ 0& 0 & -1 & +1 \end{bmatrix}\cdot\begin{bmatrix} e_{a} \\ e_{b} \\ e_{c}\\ e_{d} \end{bmatrix}$

Fig 8

v1 = ea - ec equation 4

v2 = ea - ec equation 5

v3 = -eb equation 6

v4 = ea. equation 7

v5 = eb- ea equation 8

v6 = -ec. equation 9

You can write the above branch voltage equations either by using graph or by solving matrix multiplication.

Isomorphic Graphs:

If two graphs have same incidence matrix then they said to be isomorphic. If same incidence matrix it means that they have same number of nodes (rows) and branches (columns).

Electronics Engineering And Circuit Design

Fundamental Cut Set Matrix - Electrical Network Graphs

Cut Set and Fundamental Cut Set:

Procedure For Obtaining Fundamental Cut Set: