Correct Answer

⚙️ TE • Technical Concept & PrincipleElectricalPower System

Option A

$b \times (n-1)$

Quick Summary:

The bus incidence matrix (often denoted by $A$ ) relates the branches of a network to its nodes. Since one node is chosen as the reference (datum) node, the dimension of the reduced incidence matrix is $b \times (n-1)$ , where $b$ is the number of branches and $n$ is the total number of nodes.

⚙️TETechnical SolutionConcept & Principle

💡 Explanation

🔢 Key Formulas

$A = [A_{ij}]_{b \times (n-1)}$ — The reduced bus incidence matrix where $A_{ij}$ represents the connection of branch $i$ to node $j$ .

⚙️ Working Principle

In graph theory applied to electrical networks, the full incidence matrix has dimensions $n \times b$ . By eliminating the row corresponding to the reference node to satisfy KCL, we obtain the reduced incidence matrix of dimension $(n-1) \times b$ . The question asks for the dimensions as $(branches) \times (nodes)$ , which is $b \times (n-1)$ .

📌 Key Points

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The full incidence matrix has dimensions $n \times b$ .
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Removing the reference node row reduces the dimension to $(n-1) \times b$ (transpose is $b \times (n-1)$ ).
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The matrix contains entries 1, -1, or 0 based on branch orientation and node connection.
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It is used for formulating network equations and transformation to primitive networks.

✅ Advantages

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Provides a systematic way to construct the Y-bus matrix.
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Facilitates the formulation of nodal admittance equations.

❌ Disadvantages / Limitations

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Requires identifying a reference node, which may change based on the study.
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Can become sparse for large networks, though this is often an advantage for computation.

🛠️ Applications / Uses

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Power flow analysis
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Short circuit studies
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Network topology processing

📄 Additional Information

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The full incidence matrix is typically defined as $n \times b$ (rows as nodes, columns as branches). However, the question specifies $b$ as branches and $n$ as nodes in the format $b \times (n-1)$ , which represents the transpose of the standard reduced incidence matrix.
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Option B, C, and D do not represent the standard reduced dimensions used in power system analysis.

📊 Diagram / Illustration

✅

A is correct — The dimension of the bus incidence matrix, excluding the reference node, is $b \times (n-1)$ .

Core Concepts Used

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Graph Theory Power System Network Topology Reduced Incidence Matrix

💡 EXAM TIP

Always remember that one node must be designated as the ground/reference node for nodal analysis to result in a non-singular system of equations.

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