Cut Set and Tie Set

In network topology, analysing a circuit goes beyond just applying Kirchhoff's laws directly. The concepts of cut set and tie set provide a systematic, graph-theoretic approach to write independent KCL and KVL equations for any network, regardless of its complexity. These methods are especially important for GATE aspirants dealing with large networks.

Graph Theory Foundations

A network can be represented as a graph consisting of nodes (vertices) and branches (edges). For a graph with n nodes and b branches, a tree is a connected subgraph that includes all n nodes but contains no loops. A tree always has exactly (n - 1) branches. The remaining (b - n + 1) branches not in the tree form the co-tree and are called links or chords.

The number of independent KCL equations equals (n - 1), which corresponds to the number of tree branches. The number of independent KVL equations equals (b - n + 1), which is the number of links. These two numbers together define the structure of all independent equations needed to solve a network completely.

Cut Set Matrix

A cut set is a minimal set of branches whose removal disconnects the network graph into exactly two separate parts. The word minimal here is critical: removing any single branch from the cut set must reconnect the graph. For each tree branch, there exists one and only one fundamental cut set, which consists of that tree branch plus some links from the co-tree.

The cut set matrix (also called the Q matrix) is formed by writing one row for each fundamental cut set. Each column corresponds to a branch of the network. The entry is +1 if the branch is in the cut set and oriented same as the reference direction, -1 if opposite, and 0 if not in the cut set. The cut set matrix directly yields the independent KCL equations: Q multiplied by the branch current vector equals zero, written as Qib = 0.

Tie Set Matrix

A tie set (also called a loop or f-loop) is defined as a minimal loop. For each link in the co-tree, there exists exactly one fundamental tie set, formed by that link along with the unique path of tree branches connecting the two endpoints of that link. Every link defines exactly one fundamental loop in this way.

The tie set matrix (also called the B matrix) is formed with one row for each fundamental tie set. The entry is +1 or -1 based on orientation alignment with the loop, and 0 for branches not in the loop. The tie set matrix gives KVL equations: B multiplied by the branch voltage vector equals zero, written as Bvb = 0. Together, the Q and B matrices provide a complete solution framework for any network.

Mathematical Expressions

For a network graph with n nodes and b branches, the key relations are as follows. The number of tree branches equals (n - 1). The number of links equals (b - n + 1). These are also called the nullity and rank of the graph respectively. The cut set matrix Q has dimensions (n - 1) x b, and the tie set matrix B has dimensions (b - n + 1) x b. An important orthogonality property holds: Q multiplied by the transpose of B equals the zero matrix, confirming the independence of KCL and KVL sets.

The branch current vector ib and branch voltage vector vb satisfy: Q * ib = 0 (KCL) and B * vb = 0 (KVL). Once the tree branch voltages are known, link currents can be found. Once link voltages are known, tree branch currents follow. This forms the basis of systematic network analysis used in simulation tools.

Practical Understanding

The cut set and tie set approach is particularly useful when analysing networks with a large number of branches where direct loop or nodal analysis would become unmanageable. The method is also the theoretical basis for how circuit simulation programs like SPICE form their system equations. In GATE, questions typically ask to identify the number of independent loops, the rank of the incidence matrix, or to construct a specific row of the Q or B matrix.

Solved Example

Given:
A network graph with n = 4 nodes and b = 6 branches.
A tree is selected with 3 branches: b1, b2, b3.
Links (co-tree): b4, b5, b6.

Why this formula applies:
For a connected graph, number of tree branches = n - 1
Number of links = b - n + 1 (these define independent loops)

Formula:
Tree branches = n - 1
Links (tie sets) = b - n + 1
Cut sets = n - 1

Substitution:
Tree branches = 4 - 1 = 3
Links = 6 - 4 + 1 = 3

Calculation:
For link b4, fundamental tie set = {b4} + unique tree path connecting its nodes.
For tree branch b1, fundamental cut set = {b1} + links that cross the partition created by removing b1.

Final Answer:
Number of fundamental cut sets = 3
Number of fundamental tie sets = 3
Total independent equations = 3 (KCL) + 3 (KVL) = 6, sufficient to solve 6 unknowns.

Exam Tip: In GATE, the number of independent KVL equations equals the number of links = b - n + 1. Never confuse this with b - n. Also, Q * B^T = 0 is a standard identity tested in MCQs.

• A fundamental cut set contains exactly one tree branch and the links that cross the partition formed by its removal.
• A fundamental tie set contains exactly one link and the tree branches forming the unique path between the link's terminal nodes.
• The Q matrix has (n-1) rows and b columns; B matrix has (b-n+1) rows and b columns.
• KCL is encoded in Q * ib = 0; KVL is encoded in B * vb = 0.
• The orthogonality condition Q * B^T = 0 confirms the two equation sets are independent.

Quick Revision

• Tree: connected subgraph with all n nodes and (n-1) branches, no loops.
• Co-tree links = b - n + 1; these determine the number of independent KVL equations.
• Fundamental cut set: one tree branch + relevant co-tree links. Gives KCL equations via Q * ib = 0.
• Fundamental tie set: one co-tree link + tree path. Gives KVL equations via B * vb = 0.
• Orthogonality: Q * B^T = 0. This is a frequently tested identity in GATE.
• Trap: Do not confuse (b - n + 1) links with (b - n). The +1 accounts for the connected graph property.
• Total independent equations = (n-1) + (b-n+1) = b, matching the number of branch unknowns.