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Transmission Lines TEM Waves Tele 2060 Transmission Lines • Transverse Electromagnetic (TEM) Waves • Structure of Transmission Lines • Lossless Transmission Lines Martin B.H. Weiss Transmission of Electromagnetic Energy - 1 University of Pittsburgh Tele 2060 TEM Waves • Definition: A Wave in which the Electric Field, the Magnetic Field, and the Propogation Direction are Orthogonal Use the “Right Hand Rule” to Determine Relative Orientation Thumb of Right Hand Indicates Direction of Propagation Curved Fingers of Right Hand Indicates the Direction of the Magnetic Field The Electric Field is Orthogonal ω β • Electric Field: e(t,x) = Emaxsin( t- x) V/m ω β • Magnetic Field: h(t,x) = Hmaxsin( t- x) A/m β=2π/λ ω=2πf=2π/T x is Distance Martin B.H. Weiss Transmission of Electromagnetic Energy - 2 University of Pittsburgh Tele 2060 TEM Waves λ • Note that =vp/f Same as Distance = (Speed * Time) in Physics vp is the Phase Velocity In Free Space, vp=c, the Speed of Light ==E µε • Wave Impedance: Z0 H = 1 • Phase Velocity: vp µε Martin B.H. Weiss Transmission of Electromagnetic Energy - 3 University of Pittsburgh Tele 2060 Transmission Constants • Electric Permittivity of the Transmission Medium (ε) ε = ε ε r 0 Where ε is r The Relative Permittivity Also the Dielectric Constant Typically Between 1 and 5 -12 ε = 8.85*10 (Farads/meter) 0 • Magnetic Permeability of the Transmission Medium (µ) µ=µ µ r 0 Where µ is the Relative Permeability r -7 µ = 4 *10 Henry/meter 0 Martin B.H. Weiss Transmission of Electromagnetic Energy - 4 University of Pittsburgh Tele 2060 Model of a Transmission Line L∆xR∆x ii-∆i ∆i v C∆xG∆xv-∆v ∆x Martin B.H. Weiss Transmission of Electromagnetic Energy - 5 University of Pittsburgh Tele 2060 Circuit Parameters for Twisted Pair Cable = 120( 2s) Z0 ε ln d • r = µ (2s) • L (π )ln d = µε • C (2 s ) ln d •d = Diameter of Cable •s = Distance Between the Wires Martin B.H. Weiss Transmission of Electromagnetic Energy - 6 University of Pittsburgh Tele 2060 Circuit Parameters for Coaxial Cable = 60 D Z0 ε ln( ) • r d = µ (D) • L (2π)ln d =2µε • C ()D ln d •d = diameter of inner conductor •D = diameter of cable (O.D.) Martin B.H. Weiss Transmission of Electromagnetic Energy - 7 University of Pittsburgh Tele 2060 Comments on the Distributed Parameters • They are Dependent on Frequency • If Transmission Lines are Produced with Variable s, d, or D, then the Propagation Parameters Vary This Results in Transmission Irregularities Bending can Cause Changes in Propagation Characteristics • Temperature Can Affect the Local Value of ε and µ Martin B.H. Weiss Transmission of Electromagnetic Energy - 8 University of Pittsburgh Tele 2060 Balanced and Unbalanced Lines • Measure Capacitance Relative to Ground • Balanced Transmission Lines Capacitance Relative to Ground for All Conductors is Identical Twisted Pair Supports Balanced Transmission • Unbalanced Capacitance Relative to Ground for All Conductors is Different Coax Supports Unalanced Transmission • Twisted Pair can be Connected to Coax using a Balanced-to- Unbalanced Converter (Balun) Martin B.H. Weiss Transmission of Electromagnetic Energy - 9 University of Pittsburgh Tele 2060 Fields of Twisted Pair and Coax Martin B.H. Weiss Transmission of Electromagnetic Energy - 10 University of Pittsburgh Tele 2060 Transmission Line Constants • Phase Velocity • Propagation Coefficient • Characteristic Impedance Martin B.H. Weiss Transmission of Electromagnetic Energy - 11 University of Pittsburgh Tele 2060 Phase Velocity = c vp ε • r • For a Coaxial Line, LC=µ v = 1 So p LC Martin B.H. Weiss Transmission of Electromagnetic Energy - 12 University of Pittsburgh Tele 2060 Propagation Coefficient (γ) • Variation of Current or Voltage with Distance Along the Transmission Line γ - x I = ISe γ - x V = VSe x is the Distance Along the Transmission Line from the Source VS and IS are the Voltage and Current of the Source, Respectively • γ= [(R+jωL)(G+jωC)]1/2 or, γ= α+ jβ β= 2π/λ - Phase Shift Coefficient α= N/x - Attenuation Coefficient (Nepers/m) N = ln (I/Is) Martin B.H. Weiss Transmission of Electromagnetic Energy - 13 University of Pittsburgh Tele 2060 Transmission Line Constants • Characteristic Impedance Assume Infinite Line No Reflection by Definition The Ratio of Maximum voltage to Maximum Current at any Point on the Line is Constant Independent of Position This is Known as Characteristic Impedance, Z0 • Group Velocity Ratio of the Difference in ω to the Difference in β ==γω ω So, Define vg γβ β In AM, vg is the Velocity of the Modulating Signal • Distortion Occurs When β Varies Rapidly with ω Martin B.H. Weiss Transmission of Electromagnetic Energy - 14 University of Pittsburgh Tele 2060 Lossless Line at RF • Definitions α=0 Distances, l, Measured from Load End Reasonable for Short Lengths of Good Cable • Characterization β j l Vi=VIe β -j l Vr=VRe Where VI and VR are the Voltages at the Load (l=0) The Same Can be Done for Current (I) V=Voltage at any Point=Vi+Vr I=Current at any Point=Ii+Ir Martin B.H. Weiss Transmission of Electromagnetic Energy - 15 University of Pittsburgh Tele 2060 Lossless Line at RF • Note the Following I=V/Z0 Therefore, I = (Vi-Vr)/Z0 The Negative Sign Reflects a Phase Change at the Load Because the Reflected Current is Flowing in the Opposite Direction Thus, the Load Current, IL = (VI-VR)/Z0 • At the Load, l is Zero + VL VVIR ZZ==− Thus, the Load Impedance is LIL VVIR0 Martin B.H. Weiss Transmission of Electromagnetic Energy - 16 University of Pittsburgh Tele 2060 Lossless Line at RF • Define the Reflection Coefficient Γ L=VR/VI Ratio of the Reflected to the Incident Wave − ZZL 0 Γ =+ • Note that, After a Bit of Algebra L ZZL 0 Martin B.H. Weiss Transmission of Electromagnetic Energy - 17 University of Pittsburgh Tele 2060 Observations on the Reflection Coefficient •ZL=Z0 Matched Impedances Γ L = 0 No reflection •ZL = 0 Short Circuit Load Γ L = -1 Inverted complete reflection •ZL is Infinite Open circuit Γ L=1 Normal complete reflection Martin B.H. Weiss Transmission of Electromagnetic Energy - 18 University of Pittsburgh Tele 2060 Standing waves • Requirements for occurence ≠ ZZL 0 ≠ ZZS 0 •Result Reflection at Load Reflection at Source • Analogy - String Vibration • Characteristics Amplitude of VSW Changes with Position Distance between Minima = λ/2 At any Point, the Transmitted Frequency is Present Thus, a VSW is Like an AM Signal, Except with Distance Martin B.H. Weiss Transmission of Electromagnetic Energy - 19 University of Pittsburgh Tele 2060 VSWR • Definition: VSWR=Vmax/Vmin Vmax is the maximum voltage of the standing wave Vmin is the minimum voltage of the standing wave 1 <<∞VSWR Γ • Note that L =(VSWR-1)/(VSWR+1) • This is Important Because VSWR is Readily Measureable Martin B.H. Weiss Transmission of Electromagnetic Energy - 20 University of Pittsburgh Tele 2060 Matching Networks • Used to Achieve Reflectionless Match • Passive Components: RLC • Stubs Pieces of Transmission Line Length, Location, and Termination are Critical Use Smith Chart to Calculate Length and Position of Stubs Martin B.H. Weiss Transmission of Electromagnetic Energy - 21 University of Pittsburgh Tele 2060 Time Domain Reflectometry • Recall Γ Matched: ZL=Z0 so L= 0 Γ Short circuit: ZL=0 so L=-1 Γ Open circuit: ZL= Infinte so L=1 • If a Line Fails Break: Open Circuit Short: Short Circuit Martin B.H. Weiss Transmission of Electromagnetic Energy - 22 University of Pittsburgh Tele 2060 Time Domain Reflectometry • Use Reflection Properties to Locate Fault • Operation: Send a Pulse with a Rapid Rise Time down the Line Watch for Reflection If there is No Reflection, the Line is Correctly Terminated If there is an Inverted Reflection, there is a Short Circuit If there is a Non-Inverted Reflection, there is an Open Circuit (Break) Measure Time from Reflection, and Compute the Distance to the Fault Based on vp Martin B.H. Weiss Transmission of Electromagnetic Energy - 23 University of Pittsburgh.
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