基尔霍夫定律的拓扑本质：桥接电路分析与拓扑物态的课程概念演化设计

The Topological Essence of Kirchhoff's Laws: A Conceptual Evolution Design for Bridging Circuit Analysis and Topological States

摘要: 基尔霍夫定律在电路课程中被普遍视为列写方程的代数起点，但其背后深刻的图论与拓扑约束内涵在教学中几乎未能得到体现。与此同时，拓扑电路等前沿研究方向在数学上存在同一性：电路网络的拉普拉斯矩阵与紧束缚模型的哈密顿量具备结构上的等价关系。基于这一共性，我们可以搭建起基础电路分析到拓扑物态物理的教学桥梁。本文重新挖掘基尔霍夫定律的拓扑本质，提出一条贯穿电路课程的概念演化主线：从关联矩阵与图拉普拉斯的内在联系出发，到节点导纳矩阵的结构化理解，再借助一维Su-Schrieffer-Heeger (SSH)拓扑电路模型将教学启发延伸至拓扑边缘态及拓扑保护思想。文章给出了四阶渗透式教学设计，并以SSH电路为综合探究模块，从理论推导、仿真实验到论证型实验报告，形成完整的课堂实施方案。基于认知学习理论的分析表明，这一设计有望显著深化学生对基尔霍夫定律约束本质的理解，在培养工程直觉和激发科研好奇心方面具备潜在优势。

Abstract: Kirchhoff’s laws are generally regarded as the algebraic starting point for writing equations in circuit courses, but the profound graph theory and topological constraints behind them are almost never reflected in teaching. Meanwhile, cutting-edge research directions such as topological circuits share a mathematical commonality: the Laplace matrix of circuit networks and the Hamiltonian of tightly bound models have a structural equivalence. Based on this commonality, we can build a teaching bridge from basic circuit analysis to the physics of topological states. This paper re-examines the topological essence of Kirchhoff’s laws and proposes a conceptual evolutionary thread running through the circuit course: Starting from the intrinsic connection between the incidence matrix and the graph Laplace, to a structured understanding of the nodal admittance matrix, and then extending the teaching inspiration to topological edge states and topological protection ideas with the help of the one-dimensional Su-Schrieffer-Heeger (SSH) topological circuit model. This article presents a four-stage pervasive teaching design, using the SSH circuit as a comprehensive inquiry module. It forms a complete classroom implementation plan, from theoretical derivation and simulation experiments to demonstrable experimental reports. Analysis based on cognitive learning theory indicates that this design has the potential to significantly deepen students’ understanding of the essence of Kirchhoff’s laws and constraints, and possesses potential advantages in cultivating engineering intuition and stimulating scientific curiosity.