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

거리를 따라 보이는 전력선은 단상, 2선, 무손실 송전선으로 모델링할 수 있습니다.
직렬 인덕턴스와 션트 커패시턴스를 특징으로 하는 라인 섹션을 생각해 보십시오.
Kirchhoff의 법칙을 사용하여 전압과 전류에 대한 방정식을 작성하고 Delta x로 나눕니다. 델타 x가 0에 가까워지면 위치와 시간이 모두 변수이기 때문에 편도함수를 포함하는 방정식이 파생됩니다.
라플라스 변환은 초기 조건이 0이라고 가정하고 도함수를 하나의 변수로만 단순화하여 적용됩니다.
이를 미분하면 각각의 해를 가진 선형의 2차 동차 미분 방정식이 생성됩니다. 속도는 인덕턴스와 커패시턴스 값의 함수입니다.
역 라플라스 변환을 취하고 시간 이동을 적용하면 전압 및 전류 파동을 나타내는 함수가 생성됩니다. 이 표현식은 전방 및 후방 진행 파동을 나타냅니다.
상수를 계산하기 위해 해는 2차 방정식으로 대체됩니다.
양쪽의 지수 함수의 계수는 동일하며, 각각 순방향 및 역방향 전압 측면에서 순방향 및 역방향 전류와 특성 임피던스를 산출합니다.

송전선: 과도 동작

Traveling Waves: Lossless Lines

제공된 콘텐츠는 단상 무손실 전송선에서 진행파의 동작을 탐구합니다. 루프 인덕턴스 LH/m과 라인 간 커패시턴스 C F/m을 특징으로 하는 길이 Δx의 단상 2선 무손실 전송선으로 시작합니다. 이러한 매개변수는 직렬 인덕턴스 LΔx와 션트 커패시턴스 CΔx를 생성합니다.

<img alt="Figure1" src="/files/ftp_upload/16175/16175_figure_001.svg" style="height: 125px;">


선로상의 임의의 위치 x, 시간 t에서의 전압 v(x,t)와 전류 i(x,t)는 키르히호프의 전압 법칙(KVL)과 키르히호프의 전류 법칙(KCL)을 사용하여 표현됩니다.

<img alt="Equation1" src="/files/ftp_upload/16175/16175_PT_Equation_1v2.svg" style="height: 70px;">


Δx가 0에 가까워지면 편미분 방정식이 나타나 전압, 전류 및 시간과 위치에 대한 변화율 간의 관계를 포착합니다. 라플라스 변환은 이러한 편미분 방정식을 초기 조건이 0이라고 가정하여 상미분 방정식으로 변환하는 데 적용됩니다.

<img alt="Equation2" src="/files/ftp_upload/16175/16175_PT_Equation_2v2.svg" style="height: 70px;">


이 방정식의 해는 이동파의 속도가 전송선의 단위 길이당 인덕턴스와 커패시턴스에 따라 달라진다는 것을 보여줍니다.

<img alt="Equation3" src="/files/ftp_upload/16175/16175_PT_Equation_3v4.svg" style="height: 30px;">


이러한 솔루션은 전방 및 후방 파동을 설명합니다. 전방 파동은 양의 x 방향으로 이동하고 후방 파동은 음의 x 방향으로 이동합니다. 솔루션은 시간 이동을 통합하고 전압과 전류를 전방 및 후방 이동 파동의 합으로 설명하는 함수를 생성합니다. 이러한 파동은 전송선을 따라 반대 방향으로 이동하며, 라인의 인덕턴스와 커패시턴스의 영향을 받습니다. 이러한 표현식은 단위 길이당 인덕턴스와 커패시턴스의 함수인 라인의 특성 임피던스로 이어집니다.

<img alt="Equation4" src="/files/ftp_upload/16175/16175_PT_Equation_4v4.svg" style="height: 30px;">

이 매개변수는 파동이 선을 따라 어떻게 전파되고 각 끝의 종단과 상호 작용하는지 이해하는 데 중요합니다.

거리를 따라 보이는 전력선은 단상, 2선, 무손실 송전선으로 모델링할 수 있습니다.
시리즈로 특징 지어지는 라인 섹션을 고려하십시오 ...

Education

교육

JoVE Core

Electrical Engineering

전기공학

Unit 1

제1단원

Transmission Lines: Transient Operation

전송선로: 과도 동작

KEY TERMS AND CONCEPTS

주요 용어 및 개념

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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