R12 Equivalent Resistance Reduction Steps

Authors

Vasiliy S. Znamenskiy, Borough of Manhattan Community College, CUNYFollow
Rafael Niyazov, The Borough of Manhattan Community College, CUNYFollow

Document Type

Image or Illustration

Publication Date

Fall 10-3-2025

Abstract

We present a concise, step-by-step reduction of a symmetric resistor network to its equivalent resistance 𝑅₁₂, emphasizing visual reasoning over algebraic brute force. By redrawing the circuit to expose symmetry, we identify equipotential node pairs joined by ideal conductors; the “bottom” bridging resistor is shown to carry zero current and is removed from consideration. The remaining topology reduces by two repeated series combinations, where each left/right pair of equal resistors collapses to 2R. The final network is a three-branch parallel combination (2𝑅∥𝑅∥2𝑅), yielding the closed-form result 𝑅₁₂=𝑅/2. The procedure illustrates three transferable techniques for introductory circuit analysis: (i) topology-preserving schematic rearrangement, (ii) symmetry arguments to eliminate redundant elements, and (iii) systematic series–parallel simplification using conductance additivity. Beyond computing 𝑅₁₂, the sequence serves as a teaching exemplar that cultivates intuition about equipotential nodes, current paths, and economy of representation, preparing students for more advanced methods such as nodal analysis and bridge balance conditions.

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