Tie-Set Matrix

What is Tie-Set Matrix

A tie-set is a set of branches in a loop such that each loop contains one link of the graph and the remaining three branches. For a graph with n number of nodes and b number of branches, the number of tie-sets equals the number of loops formed in the graph, and the number of fundamental loops equals the number of links in the graph.

Number of tie-sets = b – n + 1

Tie-set Matrix Definition

Tie-set matrix, fundamental loop matrix, or circuit matrix is represented by [B]_ij and is defined as a rectangular matrix with b columns and as many rows as loops. A tie-set matrix provides the relation between branch currents and link currents. The tie-set matrix can have the following values:

• [B]_ij = 1; if branch j is in the loop i and orientations coincide
• [B]_ij = -1; if branch j is in the loop i and orientations do not coincide
• [B]_ij = 0; if branch j is not in the loop i

Procedure to Find Tie Set Matrix

Each tree will provide an individual tie-set matrix. To write the tie-set matrix, consider a tree of the graph as shown below. The number of fundamental loops equals the number of links in the graph.

The fundamental loops represented by ‘FL_n‘ and written along with the associated branches as shown below:

FL₁:[1, 2, 3]

FL₂:[2, 4, 5]

FL₃:[3, 4, 6]

The tie-set matrix for the above example can be written as:

Properties of Tie set matrix

The rows of the tie-set matrix represent the fundamental loops, tie-set currents, or loop currents. The tie-set matrix column represents the tree’s number of branches. The remaining properties of the tie-set matrix are as follows:

• A fundamental loop in a tree will contain only one number of links and the remaining twigs.
• The direction of the link current present decides the orientation of the loop current in that particular loop.
• The rank of the tie-set matrix is equal to the number of tie-sets, i.e., b-n+1.
• Tie-set matrix provides the relation between loop currents and branch currents.

Tie Set Matrix and KVL

Consider the same graph and tree used in the procedure to find a tie-set matrix. Let the branch voltages be V_b1, V_b2, V_b3, V_b4, V_b5, and V_b6 of respective branches. The tie-set matrix provides the fundamental KVL mesh equations as follows:

For fundamental loop 1, FL₁: V_b1 + V_b2 – V_b3 = 0

For fundamental loop 2, FL₂: V_b2+ V_b4+ V_b5 =0

For fundamental loop 3, FL₃: V_b3 + V_b4 + V_b6 =0

The above equations can be written in matrix form as,

[B]_ij [V_b]=0

Tie Set Matrix and KCL

Tie-set matrix is also used to write down the KCL equations. Consider the same graph and tree used for KVL equations as shown below. The branch currents I_b1, I_b2, I_b3, I_b4, I_b5, and I_b6 can be represented in terms of the fundamental loop currents i_L1, i_L2, and i_L3, as shown below:

From the matrix column,

I_b1= i_L1

I_b2 = i_L1 + i_L2

I_b3 = – i_L1 + i_L3

I_b4 = i_L2 + i_L3

I_b5 = i_L2

I_b6 = i_L3

In matrix form, the above equation can be written as,

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