论拉格朗日方程与基尔霍夫定律相互演绎  

On the Mutual Deductions of Lagrange’s Equation and Kirchhoff’s Law

作　　者：陈希有[1] 郭源博[1] 齐琛[1] 李冠林[1] 章艳[1] CHEN Xiyou;GUO Yuanbo;QI Chen;LI Guanlin;ZHANG Yan(School of Electrical Engineering,Dalian University of Technology,Dalian 116023,Liaoning Province,China)

机构地区：[1]大连理工大学电气工程学院,辽宁省大连市116023

出　　处：《中国电机工程学报》2022年第5期2036-2044,共9页Proceedings of the CSEE

基　　金：国家自然科学基金项目(51877025);中央高校基本科研业务费资助(DUT19JC07)。

摘　　要：概述拉格朗日方程在电路理论中的研究现状。论述拉格朗日方程与基尔霍夫定律的相互演绎关系。在用拉格朗日方程演绎KVL时,提出选择独立广义坐标的专用树法;指出电路中拉格朗日方程的若干特点;在用KVL演绎拉格朗日方程时,无需写出能量函数。用网络图论原理证明了电路中的拉格朗日方程是一组独立的KVL。在用基尔霍夫定律演绎拉格朗日方程时,无须计算能量函数及其偏导数。这种相互演绎关系说明,拉氏方程包含了基尔霍夫定律,电路的动态行为同样遵循其他能量系统的动态行为,电路与其他能系统在拉氏方程层面上是统一的。通过这种相互演绎,可以提高对电路与其他能量系统共同规律的认识。文中的电路例子,可以为相关应用背景提供参考。The research of Lagrange’s equations in circuit theory was reviewed. Then, the mutual deductive relationship between Lagrange’s equation and Kirchhoff’s law were discussed. When deducing Kirchhoff’s law by Lagrange’s equation, a proper tree for choosing independent generalized coordinates was proposed, and some characteristics of Lagrange’s equation in circuit were pointed out. When using Kirchhoff’s voltage law(KVL) to deduce Lagrange’s equation,there was no need to write the energy function. It was proved that the Lagrange’s equations in the circuit were a set of independent KVL equations by taking the network graph theory. This mutual deductive relationship shows that the Lagrange’s equation contains Kirchhoff’s law, the dynamic behaviors of the circuit also follow the dynamic behaviors of other energy systems, and the circuit and other energy systems are unified at the meaning of Lagrange’s equation. Through this mutual deduction, the understanding of the common laws of circuits and other energy systems can be greatly improved. The representative circuit examples in the article can provide direct reference for different application backgrounds.

关 键 词：拉格朗日方程 基尔霍夫定律 能量函数 专用树 电路教学 

分 类 号：TM13[电气工程—电工理论与新技术]