Transmission Line Basics II - Class 6 Prerequisite Reading assignment: CH2 Acknowledgements: Intel Bus Boot Camp: Michael Leddige
Agenda 2 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)
Two Transmission Line Viewpoints 3 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 Line Concept 4 Power Frequency (f) is @ 6 Hz Wavelength (λ) is 5 1 6 m ( Over 3,1 Miles) Power Plant Consumer Home
PC Transmission Lines 5 Signal Frequency (f) is approaching 1 GHz Wavelength (λ) is 1.5 cm (.6 inches) Microstrip Integrated Circuit Stripline T Stripline Copper Trace PCB substrate W Cross Section of Above PCB Cross section view taken here Via FR4 Dielectric Micro- Strip Signal (microstrip) T Copper Plane W Ground/Power Signal (stripline) Signal (stripline) Ground/Power Signal (microstrip)
Key point about transmission line operation 6 Voltage and current on a transmission line is a function of both time and position. V I f f ( z, t ) ( z, t ) I 1 V 1 V 2 dz 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 I 2
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 7 - (a) - (b) - (c) - - (d)
Segment 2: Transmission line equivalent circuits and relevant equations 8 Physics of of transmission line structures Basic transmission line equivalent circuit?equations for transmission line propagation
E & H Fields Microstrip Case 9 How does the signal move Signal path from source to load? Y Z (into the page) X Remember fields are setup given an applied forcing function. (Source) The signal is really the wave propagating between the conductors Electric field Magnetic field Ground return path
Transmission Line Definition 1 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 Z V Propagation velocity is the speed with which signals are transmitted through the transmission line in its surrounding medium. v I c εr
Presence of Electric and Magnetic Fields H I I I I I I 11 E I V - - - - V V I I I V H V V I I 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.
T-Line Equivalent Circuit 12 General Characteristics of Transmission Line Propagation delay per unit length (T ) { time/distance} [ps/in] Or Velocity (v ) {distance/ time} [in/ps] Characteristic Impedance (Z ) Per-unit-length Capacitance (C ) [pf/in] Per-unit-length Inductance (L ) [nf/in] Per-unit-length (Series) Resistance (R ) [Ω/in] Per-unit-length (Parallel) Conductance (G ) [S/in] ll lr lc lg
Ideal T Line 13 Ideal (lossless) Characteristics of Transmission Line ll Ideal TL assumes: Uniform line lc Perfect (lossless) conductor (R ) Perfect (lossless) dielectric (G ) We only consider T, Z, C, and L. 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 1/1 th the signal rise time.
Signal Frequency and Edge Rate vs. Lumped or Tline Models 14 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.
T Line Rules of Thumb 15 So, what are the rules of thumb to use? May treat as lumped Capacitance Use this 1:1 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
Other Rules of Thumb 16 Frequency knee (Fknee).35/Tr (so if Tr is 1nS, Fknee is 35MHz) This is the frequency at which most energy is below Tr is the 1-9% edge rate of the signal Assignment: At what frequency can your thumb be used to determine which elements are lumped? Assume 15 ps/in
When does a T-line become a T-Line? When do we need to use transmission line analysis techniques vs. lumped circuit analysis? Wavelength/edge rate Tline 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 17
Equations & Formulas How to model & explain transmission line behavior
Relevant Transmission Line Equations 19 Propagation equation γ ( R jωl)( G jωc) α jβ Z α is the attenuation (loss) factor β is the phase (velocity) factor Characteristic Impedance equation ( R ( G jωl) jωc) In class problem: Derive the high frequency, lossless approximation for Z
Ideal Transmission Line Parameters 2 Knowing any two out of Z, T d, C, and L, the other two can be calculated. C and L are reciprocal functions of the line crosssectional dimensions and are related by constant me. ε is electric permittivity ε 8.85 X 1-12 F/m (free space) ε r i s relative dielectric constant µ is magnetic permeability µ 4p X 1-7 H/m (free space) µ r is relative permeability Z C v L ; C T ; Z 1 ; µε µ µ r µ ; C L T L d Z µε; ε r ε ε. T L ; C ; Don t forget these relationships and what they mean!
Parallel Plate Approximation 21 Assumptions TEM conditions Uniform dielectric (ε ) between conductors T C << T D ; W C >> T D T-line characteristics are function of: Material electric and magnetic properties Dielectric Thickness (T D ) Width of conductor (W C ) Trade-off T D ; C, L, Z W C ; C, L, Z To a first order, t-line capacitance and inductance can be approximated using the parallel plate approximation. C C ε W C ε * PlateArea d T D T C Base equation W C ε T D T D L µ W C T D Z 377 W C F m F m µ r ε r Ω 8.85 ε r.4 π W C T D T D µ r W C pf m µh m
Improved Microstrip Formula Parallel Plate Assumptions Large ground plane with zero thickness To accurately predict microstrip impedance, you must calculate the effective dielectric constant. Z εe F 87 εr 1.41 ln 5.98T.8W C D T εr 1 εr 1 F.217 ε 2 12T D 2 1 WC 2 W C.2 ( ε r 1) 1 T D C ( r 1) < 1 W C for > 1 T D for W T C D T C ε WCT D W C T C From Hall, Hall & McCall: T D Valid when:.1 < W C /T D < 2. and 1 < r < 15 You can t beat a field solver 22
Improved Stripline Formulas 23 Same assumptions as used for microstrip apply here ε W C T C T D1 T D2 From Hall, Hall & McCall: Symmetric (balanced) Stripline Case T D1 T D2 Z sym 6 4( TD1 TD1) ln εr.67π (.8W C T C ) Valid when W C /(T D1 T D2 ) <.35 and T C /(T D1 T D2 ) <.25 Offset (unbalanced) Stripline Case T D1 >T D2 Z offset 2 Z Z sym sym (2A, W (2A, W C C, T, T C C, εr) Z, εr) Z sym sym (2B, W C (2B, W, T C C, T You can t beat a field solver, εr) C, εr)
Refection coefficient 24 Signal on a transmission line can be analyzed by keeping track of and adding reflections and transmissions from the bumps (discontinuities) Refection coefficient Amount of signal reflected from the bump Frequency domain ρsign(s11)* S11 If at load or source the reflection may be called gamma (Γ L or Γ s ) Time domain ρ 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
Reflection and Transmission 25 Incident 1ρ Transmitted ρ Reflected Reflection Coeficient Transmission Coeffiecent ρ Zt Zt Z Z τ ( 1 ρ ) "" "" τ 1 τ 2Zt Zt Z Zt Zt Z Z
Special Cases to Remember 26 A: Terminated in Zo Vs Zs Zo Zo ρ Zo Zo Zo Zo B: Short Circuit Vs Zs Zo ρ Zo Zo 1 C: Open Circuit Vs Zs Zo ρ Zo Zo 1
Assignment Building the SI Tool Box 27 Compare the parallel plate approximation to the improved microstrip and stripline formulas for the following cases: Microstrip: W C 6 mils, T D 4 mils, T C 1 mil, ε r 4 Symmetric Stripline: W C 6 mils, T D1 T D2 4 mils, T C 1 mil, ε r 4 Write Math Cad Program to calculate Z, Td, L & C for each case. What factors cause the errors with the parallel plate approximation?
Transmission line equivalent circuits and relevant equations 28 Basic pulse launching onto transmission lines Calculation of of near and far end waveforms for classic load conditions
Review: Voltage Divider Circuit 29 Consider the simple circuit that contains source voltage V S, source resistance R S, and resistive load R L. The output voltage, VL is easily calculated from the source amplitude and the values of the two series resistors. V S R S Why do we care for? Next page. R L V L V S R L R L V L R S
Solving Transmission Line Problems 3 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: 1.Determination of launch voltage & final DC or t voltage 2.Calculation of load reflection coefficient and voltage delivered to the load 3.Calculation of source reflection coefficient and resultant source voltage These are the steps for solving all t-line problems.
Determining Launch Voltage 31 TD Vs Vs Rs A Zo B Rt (initial voltage) t, VVi Z V i V S R t V f V S Z R S R t R S 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!
Voltage Delivered to the Load TD Vs Rs A B Vs Zo Rt 32 (initial voltage) t, VVi t2td, VVi B ρ (Vi) A ρ (ρ B )(Vi ) (signal is reflected) ttd, VVi B ρ (Vi Rt ρ Zo V Β reflected ρ Β (V incident ) Rt Zo V B V incident V reflected Step 2: Determine V B in the circuit at time t TD The transient behavior of transmission line delays the arrival of launched voltage until time t TD. V B at time < t < TD is at quiescent voltage ( in this case) Voltage wavefront will be reflected at the end of the t-line V B V incident V reflected at time t TD
Voltage Reflected Back to the Source 33 Vs Vs Rs A ρ A Zo ρ B B Rt (initial voltage) TD t, VVi t2td, VVi ρ B (Vi) ρ A (ρ B )(Vi ) (signal is reflected) ttd, VVi ρ B (Vi )
Voltage Reflected Back to the Source 34 Rs ρ Zo V Α reflected ρ Α (V incident ) Rs Zo V A V launch V incident V reflected Step 3: Determine V A 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. V A at time < t < 2TD is at launch voltage Voltage wavefront will be reflected at the source V A V launch V incident V reflected at time t 2TD In the steady state, the solution converges to V B V S [R t / (R t R s )]
Problems Solved Homework 35 Consider the circuit shown to the right with a resistive load, assume propagation delay T, R S Z. Calculate and show the wave forms of V 1 (t),i 1 (t),v 2 (t), and I 2 (t) for (a) R L and (b) R L 3Z V S R S I 1 V 1 Z,Τ l I 2 V2 R L
Step-Function into T-Line: Relationships 36 Source matched case: R S Z V 1 ().5V A, I 1 ().5I A Γ S, V(x, ).5V A (1 Γ L ) Uncharged line V 2 (), I 2 () Open circuit means R L Γ L / 1 V 1 ( ) V 2 ( ).5V A (11) V A I 1 ( ) I 2 ( ).5I A (1-1) Solution
Step-Function into T-Line with Open Ckt 37 At t T, the voltage wave reaches load end and doubled wave travels back to source end V 1 (T).5V A, I 1 (T).5V A /Z V 2 (T) V A, I 2 (T) At t 2T, the doubled wave reaches the source end and is not reflected V 1 (2T) V A, I 1 (2T) V 2 (2T) V A, I 2 (2T) Solution
Waveshape: Step-Function into T-Line with Open Ckt 38 Current (A) I A.75I A.5I A I 1 I 2 V S R S I 1 V 1 Z,Τ l I 2 V2 Open.25I A Voltage (V) V A.75V A.5V A Τ 2Τ 3Τ 4Τ V 1 V 2 Time (ns) This is called reflected wave switching.25v A Τ 2Τ 3Τ 4Τ Time (ns) Solution
Problem 1b: Relationships 39 Source matched case: R S Z V 1 ().5V A, I 1 ().5I A Γ S, V(x, ).5V A (1 Γ L ) Uncharged line V 2 (), I 2 () R L 3Z Γ L (3Z -Z ) / (3Z Z ).5 V 1 ( ) V 2 ( ).5V A (1.5).75V A I 1 ( ) I 2 ( ).5I A (1-.5).25I A Solution
Problem 1b: Solution 4 At t T, the voltage wave reaches load end and positive wave travels back to the source V 1 (T).5V A, I 1 (T).5I A V 2 (T).75V A, I 2 (T).25I A At t 2T, the reflected wave reaches the source end and absorbed V 1 (2T).75V A, I 1 (2T).25I A V 2 (2T).75V A, I 2 (2T).25I A Solution
Waveshapes for Problem 1b 41 Current (A) I A.75I A.5I A I 1 I 2 V S R S I 1 V 1 Z,Τ l I 2 V2 R L.25I A Τ 2Τ 3Τ 4Τ Time (ns) Voltage (V) V A.75V A.5V A.25V A Τ 2Τ 3Τ I 1 I 2 Note that a properly terminated wave settle out at.5 V Solution Solution 4Τ Time (ns)
Transmission line step response 42 Introduction to to lattice diagram analysis Calculation of of near and far end waveforms for classic load impedances Solving multiple reflection problems Complex signal reflections at at different types of of transmission line discontinuities will be be analyzed in in this chapter. Lattice diagrams will be be introduced as as a solution tool.
Lattice Diagram Analysis Key Concepts The lattice diagram is a tool/technique to simplify the accounting of reflections and waveforms Diagram shows the boundaries (x and xl) and the reflection coefficients (G L and G L ) 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 Vs Vs V(source) Zo Rs TD N ps V(load) Time V(source) V(load) N ps 2N ps 3N ps 4N ps 5N ps A B ρsource a e b c d Rt ρload A B C 43
Lattice Diagram Analysis Detail V(source) ρ source ρ load V(load) 44 V launch Time V launch N ps V launch ρ load 2N ps V launch (1ρ load ) Time V launch ρ load ρ source V launch (1ρ load ρ load ρ source ) 3N ps V launch ρ 2 loadρ source 4N ps V launch (1ρ load ρ 2 loadρ source ρ 2 loadρ 2 source ) Vs Vs V(source) Zo Rs TD N ps V(load) Rt V launch ρ 2 loadρ 2 source 5N ps
Transient Analysis Over Damped Vs V(source) Zo Zs TD 25 ps V(load) source.2 Time V(source) V(load) 2 v 5 ps.8v ρ 1.8v.8v ρ load v Assume Zs75 ohms Zo5ohms Vs-2 volts V ρ ρ initial source load Zo 5 Vs (2).8 Zs Zo 75 5 Zs Zo 75 5.2 Zs Zo 75 5 Zl Zo 5 1 Zl Zo 5 45 1 ps.16v 1.6v Response from lattice diagram 15 ps 2 ps 1.76v.16v.32v 1.92v Volts 2.5 2 1.5 1.5 Sour ce Load 25 ps 2 5 5 75 1 125 Time, ps
Transient Analysis Under Damped 2 v Vs V(source) Zo Zs TD 25 ps V(load) Assume Zs25 ohms Zo 5ohms Vs-2 volts 46 ρ. 3333 ρ source load 1 Time V(source) V(load) 1.33v v 5 ps 1.33v 1.33v V initial ρ ρ source load Zo Vs Zs Zo Zs Zo Zs Zo Zl Zo Zl Zo 5 (2) 25 5 25 5.33333 25 5 5 1 5 1.3333 1 ps -.443v 2.66v Response from lattice diagram 15 ps 2.22v -.443v 3 2.5 2 2 ps.148v 1.77v Volts 1.5 1 Source 25 ps 1.92.148v.5 Load 25 5 75 1 125 15 175 2 225 2.7 Time, ps
47 Two Segment Transmission Line Structures Vs Rs Zo1 Rt Zo2 X X a b c d e f g h i j k l 3 3 2 2 2 2 4 2 1 2 1 3 1 2 1 2 2 1 1 1 1 1 1 1 ρ ρ ρ ρ ρ ρ T T Z Rt Z Rt Z Z Z Z Z Z Z Z Z Rs Z Rs Z Rs Z V v o o o o o o o o o o o o o o s i 2 3 3 2 4 1 2 3 3 2 4 1 2 2 ht i k it h j g i f h dt e g et d f b e c d a c at b v a i ρ ρ ρ ρ ρ ρ ρ ρ ρ h f d c A C d c a B a A l k i g e b C i g e b B e b A ' ' ' A B C A B C 1 ρ 2 ρ 3 ρ 4 ρ 3 T 2 T TD TD TD 3TD 2TD 4TD 5TD
Assignment Consider the two segment transmission line shown to the right. Assume R S 3Z 1 and Z 2 3Z 1. Use Lattice diagram and calculate reflection coefficients at the interfaces and show the wave forms of V 1 (t), V 2 (t), and V 3 (t). Check results with PSPICE Previous examples are the preparation V S I 1 R S Z 1,Τ 1 Z 2,Τ 2 V 1 l 1 I 2 V2 l 2 V 3 I 3 Short 48