6: AC Steady-State Transmission
- Page ID
- 88495
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 6.1: Introduction to Phasors
- This page explores the shift from digital to analog signals in transmission lines and the growing need for engineers skilled in RF signals. It introduces a transmission line system driven by a sinusoidal source and the use of phasors to simplify the analysis of oscillating waveforms. Phasors, representing complex quantities, facilitate the manipulation of sinusoidal functions.
- 6.2: A/C Line Behavior
- This page covers phasor applications in transmission lines and steady-state analysis of electrical circuits. It revisits the Telegrapher's Equations by substituting time-dependent variables with phasors, leading to voltage solutions that incorporate wave propagation. The relationship between phase velocity, wavelength, and frequency is underscored.
- 6.3: Terminated Lines
- This page provides an overview of transmission line dynamics, focusing on steady-state conditions and traveling waves (positive and negative) associated with load impedance. It defines the reflection coefficient based on load and characteristic impedances and presents key equations relating voltage, current, and reflection coefficients.
- 6.4: Line Impedance
- This page covers line impedance (Zs) in transmission lines, addressing the challenge of unknown phasor values. It presents a method to derive Zs through variable substitution and division of equations, where Zs is defined as the voltage to current ratio along the line.
- 6.5: Crank Diagram
- This page explains the use of crank diagrams for analyzing voltage and current in transmission lines, detailing how to plot complex quantities and reflection coefficients on the complex plane. It illustrates the oscillation of voltage magnitudes and the rotation and length changes of corresponding vectors along the line. The text also highlights cases that simplify impedance calculations and demonstrates the formation of standing wave patterns due to angular changes.
- 6.6: Standing Waves/VSWR
- This page covers standing wave patterns in transmission lines, detailing the relationship between voltage and distance concerning wavelength. It explains how voltage maximums and minimums arise, leading to a Voltage Standing Wave Ratio (VSWR). The distinction between actual wavelength and voltage feature distance is clarified as half a wavelength. Furthermore, the page addresses impedance behavior and phasor alignment at the maximum and minimum voltage points.
- 6.7: Bilinear Transform
- This page examines bilinear transforms in complex analysis, highlighting the link between the reflection coefficient (r_s) and normalized impedance (Z_s/Z_0). It emphasizes the utility of these transformations for manipulating complex impedances within the complex plane, including a geometric perspective where the inverse of a line corresponds to a circle. This approach allows for simpler calculations of impedance values through rotations in the transformed space.
- 6.8: The Smith Chart
- This page covers the Bilinear Transform's application in mapping coordinates from the Zs plane to the rs plane, illustrating the transformation with examples involving circles of varying diameters. It also explains how complex impedance transforms into the Smith Chart, showing a non-uniform mapping where the real axis becomes a straight line, and the half-plane maps into a circle. The Smith Chart, developed for transmission line analysis, effectively represents these transformations and mappings.
- 6.9: Introduction to Using the Smith Chart
- This page covers the Smith Chart's applications, focusing on measurement and calculation of reflection coefficients and VSWR. It presents a simplified "mini Smith Chart" for ease of understanding, explains the relationship between VSWR and reflection coefficient, and discusses the importance of using wavelength as a distance measurement.
- 6.10: Simple Calculations with the Smith Chart
- This page covers the use of a Smith Chart to compute load impedance \(Z_L\) in transmission lines, detailing the parallel resistor and inductor calculations and conversion between impedance and admittance. It explains plotting on the Smith Chart, finding the reflection coefficient, and calculating input impedance \(Z_{in}\).
- 6.11: Power
- This page explains how to determine power in sinusoidal circuits through the use of phasors for voltage and current, highlighting the importance of the phase angle. It describes the process of converting phasors to time functions for calculating average power and uses trigonometric identities in the calculations. The page underscores the significance of mastering these power calculations for effective system performance analysis, noting that students often find the subject challenging.
- 6.12: Finding ZL
- This page covers the applications of the Smith Chart in analyzing voltage standing wave ratio (VSWR) and load impedance through standing wave patterns. It explains deriving normalized load impedance from voltage maxima and minima measurements on a transmission line and discusses the relationship between VSWR, reflection coefficient, and load characteristics.
- 6.13: Matching
- This page explores matching load impedances in transmission lines through capacitors to achieve resonance. It details how a capacitor can cancel inductive reactance, leading to a zero reflection coefficient and a VSWR of 1.0. Additionally, it examines a scenario with a mismatched 25Ω resistor and proposes finding a point along the line to add a negative reactance. The text also suggests using admittance for more straightforward solutions.
- 6.14: Introduction to Parallel Matching
- This page covers matching circles in transmission line theory, focusing on a 25 Ω load and its admittance. It explains how moving 0.10 λ from the load creates a matching circle for optimal performance. Various positions along the line and their corresponding admittances are detailed. The addition of a shunt capacitor or inductor for impedance matching is illustrated, highlighting the significance of these adjustments in maximizing power transfer.
- 6.15: Single Stub Matching
- This page addresses the challenges of using discrete inductors or capacitors for high-frequency impedance matching due to losses and value limitations. It proposes using transmission lines as matching stubs for variable reactance by adjusting length. The process involves impedance transformation to admittance on the Smith Chart and finding adjustments to cancel imaginary impedance components. Practical examples demonstrate how to determine the optimal stub length for effective impedance matching.
- 6.16: Double Stub Matching
- This page explains double stub matching, a technique for impedance matching using two adjustable stubs to achieve desired admittance. It outlines the process of navigating the Smith Chart to transition between a load and a matching circle while maintaining a constant real part and adjusting the imaginary part. The adjustments are made by moving distances on the line and using the stubs, ultimately facilitating an effective match.
- 6.17: Odds and Ends
- This page addresses the cascaded line problem with two transmission lines of different characteristic impedances. It details the systematic analysis using the Smith Chart to find the normalized impedance at their junction. The page highlights renormalization methods and the calculation of input impedance after utilizing a quarter wave matching section to reduce reflections from impedance mismatches, demonstrating effective connection between lines of varying impedances.