Name: modeling voltage and current on transmission lines
Uploaded: 2024-11-21T14:54:26.000Z
Duration: 1 min 26 s
Channel: JoVE

考虑一个表示线路段的电路。
从接收端测量电压和电流，单位长度的串联阻抗和并联导纳已定义。
利用基尔霍夫定律，并令 Δx 趋近于零取极限，可推导出两个线性微分方程。对电压方程求导后代入电流方程，可得到一个二阶齐次方程。
该解包含两个积分常数和一个传播常数。将其代入电压方程并整理分母，可得到含有特性阻抗的电流表达式。
通过从边界条件确定积分常数并识别双曲函数，可得到分布参数线路的ABCD参数。这些参数提供了任意位置的电流和电压相对于受端值的关系。
ABCD 参数也在发送端进行评估。传播常数是一个复数，当其与长度相乘时为无量纲量，有助于确定双曲函数。
这些参数适用于任何线路长度。在涉及短线路和中等长度线路的手工计算中，可以采用近似方法。

输电线微分方程

Transmission-Line Differential Equations

输电线是电力系统的重要组成部分。它们的特点是每单位长度的电阻 (R)、电感 (L) 和电容 (C) 具有分布特性。为了分析这些线路，需要采用微分方程来模拟线路上电压和电流的变化。

线段模型

表示长度为 Δ_x 的线段的电路有助于理解输电线参数。电压 V(x) 和电流 i(x) 是从接收端测量的。该线路每单位长度具有串联阻抗 z=R+jωL 和并联导纳 y=G+jωC。将基尔霍夫电压定律 (KVL) 和基尔霍夫电流定律 (KCL) 应用于此设置，推导出沿线电压和电流变化的关系。

当 Δ_x 趋近于零时，这些关系转变为线性、一阶、齐次微分方程:

<img alt="Equation1" src="/files/ftp_upload/16146/16146_Equation%201.svg" style="height: 30px;">
<img alt="Equation2" src="/files/ftp_upload/16146/16146_Equation%202.svg" style="height: 30px;">


这些微分方程反过来描述了电压和电流随传输线长度的变化。

求解这些微分方程需要确定传播常数 γ，它包含了串联阻抗和并联导纳的影响。沿线电压和电流的一般解包括两个积分常数，它们是使用接收端的边界条件来评估的。

特性阻抗Z_c是一个关键参数，反映了输电线的固有阻抗。利用微分方程的解，沿线电压和电流以双曲函数、cosh（余弦双曲）和sinh（正弦双曲）表示，这些数学函数用于模拟电压和电流分布。

<img alt="Equation3" src="/files/ftp_upload/16146/16146_Equation%203.svg" style="height: 14px;">
<img alt="Equation4" src="/files/ftp_upload/16146/16146_Equation%204.svg" style="height: 32px;">


ABCD 参数

ABCD 参数或输电线常数由这些双曲函数中得出。这些参数提供了一种矩阵表示，将线路上任何一点的电压和电流与接收端的值联系起来。ABCD 参数对于理解信号如何通过输电线传播至关重要。

<img alt="Equation5" src="/files/ftp_upload/16146/16146_Equation%205.svg" style="height: 32px;">

传播常数

传播常数 γ 是一个复数，其实部和虚部分别表示衰减和相移。当乘以线路长度时，该常数变为无量纲，并且对于评估描述电压和电流分布的双曲函数至关重要。

精确的 ABCD 参数适用于任何线路长度，并为输电线路行为提供精确解决方案。对于涉及短线和中长线的实际手工计算，可以使用近似值。这些参数全面分析了各种运行条件下的输电线路，确保高效可靠的电力传输。

考虑一个代表线路段的电路。  
从受电端测量电压和电流，其具有定义的串联阻抗和并联...

Education

教育

JoVE Core

Electrical Engineering

电气工程

Unit 1

第一单元

Steady-State Transmission Lines and Power Flows

稳态传输线与功率流

steady-state-transmission-lines-and-power-flows

KEY TERMS AND CONCEPTS

关键术语与概念

modeling voltage and current on transmission lines

Transmission lines carry electric power, and their resistance, inductance, and capacitance are spread out along the line. Because these properties are distributed per unit length, voltage and current change as they move from one end of the line to the other. Differential equations are used to describe that variation.

A line-section model helps show how this works. In the model, a small length, Δx, is used to represent the line. Voltage V(x) and current i(x) are measured from the receiving end. The line also includes a series impedance, z = R + jωL, and a shunt admittance, y = G + jωC, per unit length.

Kirchhoff’s Voltage Law and Kirchhoff’s Current Law are applied to the small line section. As Δx becomes very small, the results turn into linear, first-order, homogeneous differential equations. These equations describe how voltage and current change along the transmission line.

Solving the equations gives the propagation constant, γ. This complex value combines the effects of series impedance and shunt admittance. Its real part shows attenuation, and its imaginary part shows phase shift. The product of γ and the line length is dimensionless and is used in the hyperbolic functions that describe the line behavior.

The general solutions for voltage and current include two integration constants. These constants are found from boundary conditions at the receiving end. The solutions also define the characteristic impedance, Zc, which reflects the intrinsic impedance of the transmission line. Voltage and current can then be written with cosh and sinh functions.

From these hyperbolic expressions, the ABCD parameters, also called transmission line constants, are obtained. They give a matrix form that relates the voltage and current at any point on the line to the values at the receiving end. The exact ABCD parameters work for any line length, while short- and medium-length lines may use approximations for hand calculations.

Modeling Voltage and Current on Transmission Lines

modeling-voltage-and-current-on-transmission-lines

transmission-line differential equations

Transmission lines carry electric power, and their resistance, inductance, and capacitance are spread out along the line. Because these properties are distributed per unit length, voltage and current change as they move from one end of the line to the other. Differential equations are used to describe that variation. A line-section model helps show how this works. In the model, a small length, Δx, is used to represent the line. Voltage V(x) and current i(x) are measured from the receiving end.

lossless lines

Lossless Lines and Voltage Behavior

Lossless transmission lines in electrical engineering are described by a purely imaginary propagation constant and a resistive characteristic impedance. Their ABCD parameters show an equivalent pi circuit with an imaginary series impedance and a shunt admittance. When the product of beta and line length is less than pi, the line has inductive series impedance and capacitive shunt admittance, which keeps it lossless. Wavelength is the physical length over which voltage or current phase changes...

maximum power flow and line loadability

Transmission Line Power Limits and Loadability

Transmission line power limits and loadability depend on voltage, angle, and line impedance. For lossy transmission lines, ABCD parameters in phasor form are used to relate sending-end and receiving-end voltages and currents. This matrix relation makes it possible to find the receiving-end current and then calculate the complex power delivered to the load. From the complex power, the real and reactive power components can be determined. For a lossless line, the equations become simpler, and...

the power flow problem and solution

Power Flow Analysis in Electric Networks

Power flow analysis helps determine how real and reactive power move through an electric network. It uses the system’s single-line diagram, which shows the buses, transmission lines, transformers, and loads. The analysis focuses on balanced three-phase steady-state operation. Each bus in the system has four key variables: voltage magnitude V k, phase angle δ k, real power P k, and reactive power Q k. Two of these four variables are treated as inputs. The power flow program solves for the...

control of power flow

Voltage and Reactive Power Control

Voltage and reactive power control help keep a power system stable and efficient. In this topic, generator settings, switched equipment, and transformer taps work together to guide how power moves through the network. A generator can be modeled with a Thevenin equivalent circuit under balanced steady-state conditions. The main terms are terminal voltage Vt, excitation voltage Eg, power angle δ, and positive-sequence synchronous reactance Xg. From this model, the generator current and the...

fast decoupled and dc powerflow

Power Flow Shortcuts for Grid Contingencies

Fast decoupled power flow and DC power flow give engineers faster ways to study a power system during contingencies such as generator outages or transmission line failures. These methods help operators get quick power flow solutions, which is important for real-time system adjustments. Fast decoupled power flow simplifies the Jacobian matrix, the set of equations used in power flow calculations, by neglecting some elements. The method drops J2 and J3 and assumes voltage magnitudes stay close...

key terms and concepts

See how distributed R, L, and C lead to transmission-line differential equations, propagation constant γ, characteristic impedance Zc, and ABCD parameters.

Modeling Voltage and Current on 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

Modeling Voltage and Current on Transmission Lines

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