Name: wave speed on lossless transmission lines
Uploaded: 2024-11-21T14:54:41.000Z
Duration: 1 min 27 s
Channel: JoVE

沿街道可见的输电线可以建模为单相双线无损传输线。
考虑一段具有串联电感和并联电容的线路，其方向性由发送端指向接收端。
根据基尔霍夫定律，写出电压和电流的方程，并除以 Δx。当 Δx 趋近于零时，由于位置和时间均为变量，可推导出包含偏导数的方程。
假设初始条件为零，并将导数简化为仅一个变量，应用拉普拉斯变换。
对这些方程求导可得到线性的二阶齐次微分方程，各自具有相应的解。速度是电感和电容值的函数。
进行拉普拉斯逆变换并应用时间平移，可得到表示电压和电流波的函数。这些表达式代表了正向和反向传播的行波。
为了确定常数，将解代入二阶方程中。
通过等式两边指数函数的系数相等，可得到正向电流和反向电流，分别用正向电压、反向电压和特性阻抗表示。

输电线路:暂态运行

Traveling Waves: Lossless Lines

提供的内容探讨了单相无损传输线上行波的行为。从长度为 Δx 的单相双线无损传输线开始，其特征是环路电感 LH/m 和线间电容 C F/m。这些参数得出串联电感 LΔx 和并联电容 CΔx。

<img alt="Figure1" src="/files/ftp_upload/16175/16175_figure_001.svg" style="height: 125px;">

线路上任意位置x处、时间t时的电压v(x,t)和电流i(x,t)可以用基尔霍夫电压定律(KVL)和基尔霍夫电流定律(KCL)表示。

<img alt="Equation1" src="/files/ftp_upload/16175/16175_PT_Equation_1v2.svg" style="height: 70px;">

当 Δx 趋近于零时，偏微分方程出现，获得电压、电流及其相对于时间和位置的变化率的关系。应用拉普拉斯变换将这些偏微分方程转换为常微分方程，假设初始条件为零。

<img alt="Equation2" src="/files/ftp_upload/16175/16175_PT_Equation_2v2.svg" style="height: 70px;">

这些方程的解表明，行波的速度取决于传输线单位长度的电感和电容。

<img alt="Equation3" src="/files/ftp_upload/16175/16175_PT_Equation_3v4.svg" style="height: 30px;">

方程解描述了前向和后向的波;前向波沿正 x 方向传播，后向波沿负 x 方向移动。方程解结合时间移位，可以得出了将电压和电流描述为前向和后向行波之和的函数。这些波沿传输线向相反的方向移动，受线路电感和电容的影响。这些表达式得出线路的特性阻抗，它是单位长度电感和电容的函数。

<img alt="Equation4" src="/files/ftp_upload/16175/16175_PT_Equation_4v4.svg" style="height: 30px;">

该参数对于理解波如何沿线传播和与两端的终端相互作用至关重要。

街道上常见的输电线可以建模为单相双线无损传输线。  
考虑一段具有串联……特性的线路

Education

教育

JoVE Core

Electrical Engineering

电气工程

Unit 1

第一单元

Transmission Lines: Transient Operation

输电线路：暂态运行

KEY TERMS AND CONCEPTS

关键术语与概念

wave speed on lossless transmission lines

Traveling waves on a lossless transmission line are shaped by the line’s inductance and capacitance. The model starts with a single-phase, two-wire line of length Δx. It has loop inductance L H/m and line-to-line capacitance C F/m. A short section of the line can then be treated as a series inductance LΔx and a shunt capacitance CΔx.

Voltage v(x,t) and current i(x,t) are written for any position x and time t along the line. Kirchhoff’s Voltage Law, or KVL, is used for the voltage relation. Kirchhoff’s Current Law, or KCL, is used for the current relation. As Δx becomes very small, these relations turn into partial differential equations that link voltage and current to their changes in time and position.

Laplace transforms are then applied to these equations. This step assumes zero initial conditions and converts the partial differential equations into ordinary differential equations. The solutions show that wave speed depends on the inductance and capacitance per unit length of the transmission line. The voltage and current can be written as sums of forward and backward traveling waves.

The forward wave moves in the positive x-direction. The backward wave moves in the negative x-direction. A time shift appears in the solution, which helps describe how the waves travel along the line. These expressions also lead to the characteristic impedance, which depends on the line inductance and capacitance per unit length and helps describe how the waves interact with terminations at each end.

Wave Speed on Lossless Transmission Lines

wave-speed-on-lossless-transmission-lines

traveling waves: lossless lines

Traveling waves on a lossless transmission line are shaped by the line’s inductance and capacitance. The model starts with a single-phase, two-wire line of length Δx. It has loop inductance L H/m and line-to-line capacitance C F/m. A short section of the line can then be treated as a series inductance LΔx and a shunt capacitance CΔx. Voltage v(x,t) and current i(x,t) are written for any position x and time t along the line. Kirchhoff’s Voltage Law, or KVL, is used for the voltage relation.

boundary conditions: lossless lines

Transmission Line Reflections at Both Ends

Single-phase, two-wire, lossless transmission lines show how voltage and current waves reflect at both ends of a circuit. In this model, the line has a length, a surge impedance, and a wave velocity. These are set by the line’s inductance and capacitance per unit length. A home electrical wiring system can be treated this way in a simple circuit model. At the receiving end, the boundary condition says that the voltage equals the receiving-end impedance times the current. This relationship is...

bewley lattice diagram

Tracing Transient Waves with a Bewley Diagram

The Bewley lattice diagram organizes voltage-wave reflections in transmission-line transients. It helps students follow how a signal moves along a line and bounces back from each end. The diagram was developed by L. V. Bewley. The vertical axis shows transient time, and the horizontal axis shows the line position x. Diagonal lines show traveling waves. These lines make the wave motion easier to read because each segment marks a wave moving through the line. Each reflection is found by...

lossy lines and overvoltages

Signal Attenuation and Power Losses

Transmission lines with series resistance and shunt conductance can weaken signals, change wave shape, and waste power. These effects are called attenuation, distortion, and power losses. They matter in long power lines and other practical transmission systems. When constant resistance and conductance are present, the voltage and current equations change. The propagation constant shows that the wave includes both forward-traveling and backward-traveling parts. As the wave moves along the line,...

insulation coordination

BIL and Surge Protection for Power Equipment

Insulation coordination matches electric equipment insulation strength with protective device settings to guard against expected overvoltages. Engineers choose these levels using judgment and cost. The goal is to keep equipment safe while avoiding unnecessary expense. Equipment can usually survive short, high transient overvoltages. Repeated tests with the same waveform can still give inconsistent results. For that reason, standard impulse voltage waveforms are used in testing. These waveforms...

key terms and concepts

Learn how inductance and capacitance set wave speed on a lossless transmission line, and how forward and backward waves lead to characteristic impedance.

Wave Speed on Lossless Transmission Lines | JoVE K12

JoVE Core Electrical Engineering offers easy-to-understand animated videos that cover fundamental concepts of electric circuits along with circuit analysis techniques. It incorporates first-order and second-order circuits using operational amplifiers and circuit transformation concepts. The inclusion of practical design examples and real-world applications through scientist-in-action videos adds significant educational value, making complex concepts accessible.

electrical engineering

Wave Speed on Lossless Transmission Lines

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