5: Introduction to Transmission Lines
- Page ID
- 88494
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 5.1: Introduction to Transmission Lines - Distributed Parameters
- This page covers the transmission of electrical signals through various types of transmission lines, such as coaxial cables. It introduces a generic transmission line model that focuses on distributed parameters like inductance and capacitance, highlighting their relationship to the properties of conductors. The text emphasizes the connection between these parameters and presents a simplified model of an ideal lossless transmission line, omitting resistance and conductance for initial analysis.
- 5.2: Telegrapher's Equations
- This page explains the application of Kirchhoff’s Laws to transmission lines, detailing the voltage and current at input and output. It presents equations for voltage drop across an inductor and current through a capacitor. By applying Kirchhoff’s Voltage Law (KVL), it derives the telegrapher's equations, which describe the relationship between spatial and temporal changes in voltage and current, illustrating signal propagation along transmission lines.
- 5.3: Transmission Line Equation
- This page explains the telegrapher's equations, which model voltage and current in transmission lines. It shows that both signals travel at phase velocity \(v_p = \frac{1}{\sqrt{LC}}\) and establishes the characteristic impedance \(Z_0 = \sqrt{\frac{L}{C}}\). Two independent solutions for voltage and current are derived, with relationships for signals moving in opposite directions: \(V^{+}/I^{+} = Z_{0}\) and \(V^{-}/I^{-} = -Z_{0}\).
- 5.4: Transmission Line Examples
- This page covers the stripline, detailing its structure with a central conductor and ground planes separated by a dielectric. It explains the parallel capacitance between the conductor and the ground planes, providing an equation for capacitance per unit length. Additionally, it connects this to wave propagation speed and characteristic impedance, and concludes by comparing approximate versus exact impedance calculations while discussing the limitations of initial electric field assumptions.
- 5.5: Exciting a Line
- This page covers the behavior of a semi-infinite transmission line under a DC voltage source, explaining the connection between source voltage, internal impedance, and the line using voltage divider principles. It justifies the selection of 300 Ω twin-lead and 75 Ω coaxial cables for TV signals, emphasizing their impact on power transfer and signal loss.
- 5.6: Terminated Lines
- This page examines a finite terminated transmission line's behavior by analyzing the steady-state conditions of forward and reverse waveforms. It highlights how total voltage and current depend on these waves and load impedance, introducing reflection coefficients and input impedance's role in generator-line interaction. The page concludes with a transformation method to facilitate voltage and current calculations along the transmission line.
- 5.7: Bounce Diagrams
- This page covers bounce diagrams, essential for transmission line engineers to visualize voltage wave behavior. It details the reflection of voltage waves between source and load, emphasizing key calculations like reflection coefficients. It also explores the analysis of voltage across a load resistor over time and position, deriving the voltage divider equation and demonstrating wave propagation.
- 5.8: Cascaded Lines
- This page explores the application of bounce diagrams for analyzing cascaded transmission line issues, particularly in relation to Ethernet interfaces and impedance mismatches (50 Ω vs. 75 Ω). It covers voltage reflection and transmission coefficients, illustrating the behavior at junctions. Additionally, it analyzes a "Charged Line" problem with a transmission line connected to a load resistor, detailing the dynamics of voltage changes upon switching and the effects of reflections.