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Transmission Line Basics

Transmission Line Basics. Purpose –. Acknowledgements: Intel Bus Boot Camp: Michael Leddige. Agenda. The Transmission Line Concept Transmission line equivalent circuits and relevant equations Reflection diagram & equation Loading Termination methods and comparison Propagation delay

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Transmission Line Basics

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  1. Transmission Line Basics Purpose – Acknowledgements: Intel Bus Boot Camp: Michael Leddige

  2. Agenda • The Transmission Line Concept • Transmission line equivalent circuits and relevant equations • Reflection diagram & equation • Loading • Termination methods and comparison • Propagation delay • Simple return path ( circuit theory, network theory come later) Transmission Lines Class 6

  3. Two Transmission Line Viewpoints • Steady state ( most historical view) • Frequency domain • Transient • Time domain • Not circuit element Why? • We mix metaphors all the time • Why convenience and history Transmission Lines Class 6

  4. Transmission Line Concept Power Frequency (f) is @ 60 Hz Wavelength (l) is 5 106 m ( Over 3,100 Miles) Power Plant Transmission Line Could be considered as Non-Transmission Line Consumer Home Transmission Lines Class 6

  5. Integrated Circuit Stripline T Microstrip PCB substrate Cross section view taken here W Via FR4 Dielectric Cross Section of Above PCB Copper Trace Signal (microstrip) Ground/Power Signal (stripline) T Copper Plane Signal (stripline) Ground/Power Signal (microstrip) W PC Transmission Lines Signal Frequency (f) is approaching 10 GHz Wavelength (l) is 1.5 cm ( 0.6 inches) Transmission Lines Class 6

  6. The major deviation from circuit theory with transmission line, distributed networks is this positional dependence of voltage and current! Must think in terms of position and time to understand transmission line behavior This positional dependence is added when the assumption of the size of the circuit being small compared to the signaling wavelength Key point about transmission line operation Voltage and current on a transmission line is a function of both time and position. Transmission Lines Class 6

  7. Examples of Transmission Line Structures- I • Cables and wires (a) Coax cable (b) Wire over ground (c) Tri-lead wire (d) Twisted pair (two-wire line) • Long distance interconnects Transmission Lines Class 6

  8. Common Examples on a PWB • 2 basic types • Stripline (a) • Microstrip (b) • Maybe embedded microstrip a b Transmission Lines Class 6

  9. Segment 2: Transmission line equivalent circuits and relevant equations • Physics of transmission line structures • Basic transmission line equivalent circuit • Equations for transmission line propagation Transmission Lines Class 6

  10. E & H Fields – Microstrip Case The signal is really the wave propagating between the conductors How does the signal move from source to load? Remember fields are setup given an applied forcing function. (Source) Transmission Lines Class 6

  11. Transmission Line “Definition” • General transmission line: a closed system in which power is transmitted from a source to a destination • Our class: only TEM mode transmission lines • A two conductor wire system with the wires in close proximity, providing relative impedance, velocity and closed current return path to the source. • Characteristic impedance is the ratio of the voltage and current waves at any one position on the transmission line • Propagation velocity is the speed with which signals are transmitted through the transmission line in its surrounding medium. Transmission Lines Class 6

  12. Presence of Electric and Magnetic Fields • Both Electric and Magnetic fields are present in the transmission lines • These fields are perpendicular to each other and to the direction of wave propagation for TEM mode waves, which is the simplest mode, and assumed for most simulators(except for microstrip lines which assume “quasi-TEM”, which is an approximated equivalent for transient response calculations). • Electric field is established by a potential difference between two conductors. • Implies equivalent circuit model must contain capacitor. • Magnetic field induced by current flowing on the line • Implies equivalent circuit model must contain inductor. Transmission Lines Class 6

  13. lR0 lL0 lG0 lC0 T-Line Equivalent Circuit • General Characteristics of Transmission Line • Propagation delay per unit length (T0) { time/distance} [ps/in] • Or Velocity (v0) {distance/ time} [in/ps] • Characteristic Impedance (Z0) • Per-unit-length Capacitance (C0) [pf/in] • Per-unit-length Inductance (L0) [nf/in] • Per-unit-length (Series) Resistance (R0) [W/in] • Per-unit-length (Parallel) Conductance (G0) [S/in] Transmission Lines Class 6

  14. lL0 lC0 Ideal T Line • Ideal (lossless) Characteristics of Transmission Line • Ideal TL assumes: • Uniform line • Perfect (lossless) conductor (R00) • Perfect (lossless) dielectric (G00) • We only consider T0, Z0, C0, and L0. • A transmission line can be represented by a cascaded network (subsections) of these equivalent models. • The smaller the subsection the more accurate the model • The delay for each subsection should be no larger than 1/10th the signal rise time. Transmission Lines Class 6

  15. Signal Frequency and Edge Rate vs. Lumped or Tline Models In theory, all circuits that deliver transient power from one point to another are transmission lines, but if the signal frequency(s) is low compared to the size of the circuit (small), a reasonable approximation can be used to simplify the circuit for calculation of the circuit transient (time vs. voltage or time vs. current) response. Transmission Lines Class 6

  16. T Line Rules of Thumb So, what are the rules of thumb to use? May treat as lumped Capacitance Use this 10:1 ratio for accurate modeling of transmission lines Td < .1 Tx May treat as RC on-chip, and treat as LC for PC board interconnect Td < .4 Tx Transmission Lines Class 6

  17. Other “Rules of Thumb” • Fknee = 0.35/Tr (so if Tr is 1nS, Fknee is 350MHz) • Is the frequency at which most energy is below • Tr being the 10-90% edge rate of the signal • Assignment: At what frequency can your thumb be used to determine which elements are lumped? • Assume 150 ps/in Transmission Lines Class 6

  18. When do we need to use transmission line analysis techniques vs. lumped circuit analysis? Tline Wavelength/edge rate When does a T-line become a T-Line? • Whether it is a bump or a mountain depends on the ratio of its size (tline) to the size of the vehicle (signal wavelength) • Similarly, whether or not a line is to be considered as a transmission line depends on the ratio of length of the line (delay) to the wavelength of the applied frequency or the rise/fall edge of the signal Transmission Lines Class 6

  19. Equations & Formulas How to model & explain transmission line behavior

  20. Relevant Transmission Line Equations Propagation equation is the attenuation (loss) factor  is the phase (velocity) factor Characteristic Impedance equation In class problem: Derive the high frequency, lossless approximation for Z0 Transmission Lines Class 6

  21. L 0 = = Z ; T L C ; 0 d 0 0 C 0 T 0 = = C ; L Z T ; 0 0 0 0 Z 0 1 = = me v ; C L ; 0 0 0 me m = m m e = e e ; . r 0 r 0 Ideal Transmission Line Parameters • Knowing any two out of Z0, Td, C0, and L0, the other two can be calculated. • C0 and L0 are reciprocal functions of the line cross-sectional dimensions and are related by constant me. • e is electric permittivity • e0= 8.85 X 10-12F/m (free space) • eri s relative dielectric constant • m is magnetic permeability • m0= 4p X 10-7H/m (free space) • mr is relative permeability Don’t forget these relationships and what they mean! Transmission Lines Class 6

  22. Base equation Parallel Plate Approximation • Assumptions • TEM conditions • Uniform dielectric (e) between conductors • TC<< TD; WC>> TD • T-line characteristics are function of: • Material electric and magnetic properties • Dielectric Thickness (TD) • Width of conductor (WC) • Trade-off • TD; C0 , L0 , Z0  • WC; C0 , L0 , Z0  To a first order, t-line capacitance and inductance can be approximated using the parallel plate approximation. Transmission Lines Class 6

  23. Improved Microstrip Formula • Parallel Plate Assumptions + • Large ground plane with zero thickness • To accurately predict microstrip impedance, you must calculate the effective dielectric constant. From Hall, Hall & McCall: Valid when: 0.1 < WC/TD < 2.0 and 1 < er < 15 You can’t beat a field solver Transmission Lines Class 6

  24. Improved Stripline Formulas • Same assumptions as used for microstrip apply here From Hall, Hall & McCall: Symmetric (balanced) Stripline Case TD1 = TD2 Valid when WC/(TD1+TD2) < 0.35 and TC/(TD1+TD2) < 0.25 Offset (unbalanced) Stripline Case TD1 TD2 Transmission Lines Class 6

  25. Refection coefficient • Signal on a transmission line can be analyzed by keeping track of and adding reflections and transmissions from the “bumps” • Refection coefficient • Amount of signal reflected from the “bump” • Frequency domain r=|S11| • If at load or source the reflection may be called gamma (GL or Gs) • Time domain r is only defined a location • The “bump” • Time domain analysis is causal. • Frequency domain is for all time. • We use similar terms – be careful • Reflection diagrams – more later Transmission Lines Class 6

  26. Reflection and Transmission Incident 1+r Transmitted r Reflected Transmission Lines Class 6

  27. A: Terminated in Zo Zs - Zo Zo Zo r = = 0 Zo Vs + Zo Zo B: Short Circuit Zs - 0 Zo r = = - Zo 1 Vs + 0 Zo C: Open Circuit Zs ¥ - Zo Zo r = = 1 Vs ¥ + Zo Special Cases to Remember Transmission Lines Class 6

  28. Assignment – Building the SI Tool Box Compare the parallel plate approximation to the improved microstrip and stripline formulas for the following cases: Microstrip: WC = 6 mils, TD = 4 mils, TC = 1 mil, er = 4 Symmetric Stripline: WC = 6 mils, TD1 = TD2 = 4 mils, TC = 1 mil, er = 4 Write Math Cad Program to calculate Z0, Td, L & C for each case. What factors cause the errors with the parallel plate approximation? Transmission Lines Class 6

  29. Transmission line equivalent circuits and relevant equations • Basic pulse launching onto transmission lines • Calculation of near and far end waveforms for classic load conditions Transmission Lines Class 6

  30. RS RL VS VL RL VL VS = + RL RS Review: Voltage Divider Circuit • Consider the simple circuit that contains source voltage VS, source resistance RS, and resistive load RL. • The output voltage, VL is easily calculated from the source amplitude and the values of the two series resistors. Why do we care for? Next page…. Transmission Lines Class 6

  31. Solving Transmission Line Problems The next slides will establish a procedure that will allow you to solve transmission line problems without the aid of a simulator. Here are the steps that will be presented: • Determination of launch voltage & final “DC” or “t =0” voltage • Calculation of load reflection coefficient and voltage delivered to the load • Calculation of source reflection coefficient and resultant source voltage These are the steps for solving all t-line problems. Transmission Lines Class 6

  32. TD Rs A B Vs Zo Rt 0 Vs (initial voltage) t=0, V=Vi Z0 Rt Vi Vf VS VS = = + + Rt Z0 RS RS Determining Launch Voltage Step 1 in calculating transmission line waveforms is to determine the launch voltage in the circuit. • The behavior of transmission lines makes it easy to calculate the launch & final voltages – it is simply a voltage divider! Transmission Lines Class 6

  33. TD Rs A B Vs - Rt Zo rB Zo = Rt 0 Vs + Rt Zo (initial voltage) Vreflected = rB (Vincident) VB = Vincident + Vreflected t=0, V=Vi (signal is reflected) t=2TD, r t=TD, V=Vi + (Vi ) r r (r V=Vi + (Vi) + )(Vi ) B B A B Voltage Delivered to the Load Step 2: Determine VB in the circuit at time t = TD • The transient behavior of transmission line delays the arrival of launched voltage until time t = TD. • VB at time 0 < t < TD is at quiescent voltage (0 in this case) • Voltage wavefront will be reflected at the end of the t-line • VB = Vincident + Vreflected at time t = TD Transmission Lines Class 6

  34. Voltage Reflected Back to the Source Rs A B Vs Zo rA rB Rt 0 Vs TD (initial voltage) t=0, V=Vi (signal is reflected) t=2TD, r t=TD, V=Vi + (Vi ) r r (r V=Vi + (Vi) + )(Vi ) B B B A Transmission Lines Class 6

  35. - Rs Zo rA = + Rs Zo Vreflected = rA (Vincident) VA = Vlaunch + Vincident + Vreflected Voltage Reflected Back to the Source Step 3: Determine VA in the circuit at time t = 2TD • The transient behavior of transmission line delays the arrival of voltage reflected from the load until time t = 2TD. • VA at time 0 < t < 2TD is at launch voltage • Voltage wavefront will be reflected at the source • VA = Vlaunch + Vincident + Vreflected at time t = 2TD In the steady state, the solution converges to VB = VS[Rt / (Rt + Rs)] Transmission Lines Class 6

  36. Problems • Consider the circuit shown to the right with a resistive load, assume propagation delay = T, RS= Z0 . Calculate and show the wave forms of V1(t),I1(t),V2(t), and I2(t) for (a) RL= and (b) RL= 3Z0 Transmission Lines Class 6

  37. Step-Function into T-Line: Relationships • Source matched case: RS= Z0 • V1(0)= 0.5VA, I1(0)= 0.5IA • GS = 0, V(x,) = 0.5VA(1+ GL) • Uncharged line • V2(0)= 0, I2(0)= 0 • Open circuit means RL=  • GL =  / = 1 • V1()= V2()= 0.5VA(1+1)= VA • I1()= I2 ()= 0.5IA(1-1)= 0 Transmission Lines Class 6

  38. Step-Function into T-Line with Open Ckt • At t = T, the voltage wave reaches load end and doubled wave travels back to source end • V1(T)= 0.5VA, I1(T)= 0.5VA/Z0 • V2(T) = VA, I2 (T)= 0 • At t = 2T, the doubled wave reaches the source end and is not reflected • V1(2T)= VA, I1(2T)= 0 • V2(2T) = VA, I2(2T)= 0 Transmission Lines Class 6

  39. Waveshape:Step-Function into T-Line with Open Ckt This is called “reflected wave switching” Transmission Lines Class 6

  40. Problem 1b: Relationships • Source matched case: RS= Z0 • V1(0)= 0.5VA, I1(0)= 0.5IA • GS = 0, V(x,) = 0.5VA(1+ GL) • Uncharged line • V2(0)= 0, I2(0)= 0 • RL= 3Z0 • GL = (3Z0 -Z0) / (3Z0 +Z0)= 0.5 • V1()= V2()= 0.5VA(1+0.5)= 0.75VA • I1()= I2()= 0.5IA(1-0.5)= 0.25IA Transmission Lines Class 6

  41. Problem 1b: Solution • At t = T, the voltage wave reaches load end and positive wave travels back to the source • V1(T)= 0.5VA, I1(T)= 0.5IA • V2(T) = 0.75VA , I2(T)= 0.25IA • At t = 2T, the reflected wave reaches the source end and absorbed • V1(2T)= 0.75VA , I1(2T)= 0.25IA • V2(2T) = 0.75VA , I2(2T)= 0.25IA Transmission Lines Class 6

  42. Waveshapes for Problem 1b Note that a properly terminated wave settle out at 0.5 V Transmission Lines Class 6

  43. Transmission line step response • Introduction to lattice diagram analysis • Calculation of near and far end waveforms for classic load impedances • Solving multiple reflection problems Complex signal reflections at different types of transmission line “discontinuities” will be analyzed in this chapter. Lattice diagrams will be introduced as a solution tool. Transmission Lines Class 6

  44. Zo V(load) V(source) Vs Rs TD = N ps 0 Vs Rt r r load source V(load) Time V(source) 0 a A’ N ps A b B’ 2N ps c 3N ps B d C’ 4N ps e 5N ps Lattice Diagram Analysis – Key Concepts • Diagram shows the boundaries (x =0 and x=l) and the reflection coefficients (GL andGL ) • Time (in T) axis shown vertically • Slope of the line should indicate flight time of signal • Particularly important for multiple reflection problems using both microstrip and stripline mediums. • Calculate voltage amplitude for each successive reflected wave • Total voltage at any point is the sum of all the waves that have reached that point The lattice diagram is a tool/technique to simplify the accounting of reflections and waveforms Transmission Lines Class 6

  45. r r load source V(load) V(source) 0 Vlaunch 0 Time N ps Vlaunch V(load) V(source) Zo Vlaunch rload Vs Rs TD = N ps 0 Vs Vlaunch(1+rload) Rt Time 2N ps Vlaunch rloadrsource Vlaunch(1+rload +rload rsource) 3N ps Vlaunch r2loadrsource Vlaunch(1+rload+r2loadrsource+ r2loadr2source) 4N ps Vlaunch r2loadr2source 5N ps Lattice Diagram Analysis – Detail Transmission Lines Class 6

  46. Transient Analysis – Over Damped Transmission Lines Class 6

  47. Assume Zs=25 ohms Zo V(load) V(source) Zo =50ohms 2 v Zs Vs=0-2 volts TD = 250 ps 0 Vs æ ö Zo 50 = = = ç ÷ V Vs ( 2 ) 1 . 3333 initial + + Zs Zo 25 50 è ø r = 1 r = - 0 . 3333 load source V(load) Time V(source) - - Zs Zo 25 50 r = = = - 0 . 33333 0 source + + 1.33v Zs Zo 25 50 0v - ¥ - Zl Zo 50 r = = = 1 500 ps 1.33v load + ¥ + Zl Zo 50 1.33v 2.66v 1000 ps Response from lattice diagram -0.443v 3 1500 ps 2.22v -0.443v 2.5 2 Volts 1.77v 1.5 2000 ps 0.148v Source 1 0.5 Load 2500 ps 1.92 0 0.148v 0 250 500 750 1000 1250 1500 1750 2000 2250 Time, ps 2.07 Transient Analysis – Under Damped Transmission Lines Class 6

  48. Two Segment Transmission Line Structures Transmission Lines Class 6

  49. Assignment • Consider the two segment transmission line shown to the right. Assume RS= 3Z01 and Z02= 3Z01 /3. Use Lattice diagram and calculate reflection coefficients at the interfaces and show the wave forms of V1(t),V2(t), and V3(t). • Check results with PSPICE Transmission Lines Class 6

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