Electronics Engineering And Circuit Design

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.