| 有限长有损线路一端开路时传输线方程的解析解 |
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| 引用本文：李云阁1,张娜娜2,曾垂杨2,王乐之3,翟章良2.有限长有损线路一端开路时传输线方程的解析解[J].电网与清洁能源,2022,38(9):1~9 |
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| 基金项目:国家自然科学基金项目（52077167） |
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| 中文摘要:通常使用传输线方程来分析输电线路上的电磁暂态过程。文章讨论了在线路一端开路、另外一端施加阶跃直流电压时传输线方程的解析解，讨论中考虑线路电阻。文章利用了2种方法：基于分离变量法求解偏微分方程、基于留数法进行拉普拉斯反变换。研究所得解析解是以三角函数表示的级数，各次谐波为驻波，可转换为前、反行波形式。2种技术路线截然不同的方法得出完全相同的公式。 |
| 中文关键词:输电线路 传输线方程 解析解 分离变量法 拉普拉斯变换 |
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| Analytical Solution to Telegraph Equation for Finite Lossy Transmission Lines with One End Open-Circuited |
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| Abstract:The telegraph equation is often used to analyze the electromagnetic transients on transmission lines. The paper discusses the analytical solution to the equation with difficult boundary conditions that the line is excited by a step voltage source at one end and is open-circuited at the other end. The resistance of the line is taken into account. Two methods are employed. The method of separation of variables is used to solve the partial differential equation， and residues are calculated to conduct invers Laplace transform. The analytical solution is in the form of a trigonometric series. Each of its harmonic components is a standing wave， which can be converted into the familiar superposition of a forward and a backward traveling waves. Though the two methods follow completely different technical routes， they result in exactly the same formula. |
| keywords:transmission line telegraph equation analytical solution the method of separation of variables Laplace transform |
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