JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B.Tech II NETWORK ANALYSIS AND TRANSMISSION LINES Previous Question Paper (August/September - 2022 - R18)

A basic tie-set (or fundamental loop) is formed by selecting a tree of a connected graph and adding exactly one link (chord) to the tree. The resulting closed path is a tie-set. The basic tie-set matrix (B) relates the branch currents to the link currents, where each row represents a tie-set and each column represents a branch. Entries are 1 if the branch is in the tie-set and oriented in the same direction as the link, -1 if in the opposite direction, and 0 if not present.

A basic cut-set (or fundamental cut-set) is formed by selecting a tree and removing all tree branches except one. The cut-set partitions the graph into two disjoint sets of nodes. The basic cut-set matrix (Q) relates the branch voltages to the tree-branch voltages. Each row represents a cut-set and each column a branch. Entries are 1 if the branch is in the cut-set and oriented in the same direction as the tree branch, -1 if opposite, and 0 if not present.

To formulate the fundamental cut-set matrix, follow these steps:

1. Identify a spanning tree for the given graph. A spanning tree must include all nodes and contain no loops.
2. Label the remaining edges as links (chords).
3. For each tree branch, define a cut-set by removing that branch and identifying the minimum set of edges that, if removed, would disconnect the graph into two parts.
4. Assign a direction to the cut-set corresponding to the direction of the tree branch.
5. Construct the matrix Q where rows represent the fundamental cut-sets and columns represent all branches (tree branches and links).
6. Fill the matrix: +1 if the branch is in the cut-set and aligned with the tree branch, -1 if opposite, and 0 if not in the cut-set.

The resulting matrix will be in the form [I | Q_L], where I is the identity matrix corresponding to the tree branches, and Q_L represents the links.