TL/Finite Length. 2Vi. What happens when a traveling wave reaches the end of a transmission line?
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1 TL/Finite Length What happens when a traveling wave reaches the end of a transmission line? Vi Vi Vi Ir If It Zt Zt Thought Process: Transmission line with impedance Cut open Develop TEC for open transmission line Terminate with termination impedance Zt Determine current It in termination impedance Split up current into two components: Current If (Ii) due to forward traveling wave Current Ir due to reflection 9/9/5 EE6471 (KR) 35
2 TL/Finite Length/Reflection Coefficient Vi Ir If Termination current is the superposition of the current due to the forward traveling wave, and the current due to the reflection It Zt It Ir Ir Vi Z + Zt If Vi Z It Vi Zt Z Zt + Z Vi If Z Vi + Zt Ii Famous Telegrapher s Equation: Reflection coefficient kr kr can be complex (and f- dependent) but in practice it is desirable to keep kr constant and real kr Ir Ii Vr Vi Zt Zt + Z Z 9/9/5 EE6471 (KR) 36
3 TL/Finite Length/TL/Coefficients Vs Zs Zl TLIA Hx TLT Hx TLRl TLRs Definitions: TLIA(: TL Input Acceptance Coefficient TLT(: TL Output Transmission Coefficient TLRl(: TL Load-End Reflection Coefficient TLRs(: TL Source-End Reflection Coefficient Hx 9/9/5 EE6471 (KR) 37
4 TL/Finite Length/Coefficients/TLRl & TLRs Load-end reflection coefficient TLRl(: TLRl( Zl( Zl( + ( ( Definition of reflection coefficient kr applied to both load-end and source-end of transmission line Source-end reflection coefficient TLRs(: TLRs( Zs( Zs( + ( ( Reflection coefficients No reflection if Zt If Zt and are real: kr in a range [-1..+1] if Zt kr-1 if Zt kr+1 9/9/5 EE6471 (KR) 38
5 TL/Finite Length/Coefficients/TLT Output Transmission Coefficient TLT(: Vt TLT ( 1+ TLRl( Vi because 1+ TLRl 1+ Zt Zt + Z Z Zt + + Zt Zt + Zt Zt + Vt Vi Zt + Zt Ir It Vt Vi + ( 1 TLRl) q.e.d. Vi If Zt 9/9/5 EE6471 (KR) 39
6 TL/Finite Length/Coefficients/TLIA Input acceptance coefficient TLIA(: TLIA( Zs( ( + ( Fraction of the input voltage accepted by the transmission line 9/9/5 EE6471 (KR) 4
7 TL/Finite Length/Transfer Function Zs Transfer function S ( for Vs signals emerging from the TLIA TLT transmission line: Hx Zl Vt Hx TLRl S( TLIA( Hx( TLT ( TLRs Hx ( TLRl( Hx( TLRs( Hx( ) TLT ( ) S1( TLIA( Hx( p S S N ( TLIA( Hx( ( ) N TLRl( Hx( TLRs( TLT ( ( 1+ TLRl( ) TLIA( Hx( Vt( SN ( N 1 TLRl( Hx( TLRs( Vs( ( 9/9/5 EE6471 (KR) 41
8 TL/Finite Length/Transfer Function/Example Example: Reflections on a transmission line C14pF/m, L35nH/m. RGnegligible. Length1m. 5Ω. Tp_pul7ns/m. 5V. Zs1Ω. Zl1kΩ. TLIA.833. TLRl.99. TLRs S 99.8% S V V v_tl_end nf V t( nf) ns Case 1: Low source impedance with unterminated transmission line 17 9/9/5 EE6471 (KR) 4
9 TL/Finite Length/Transfer Function/Example v_tl_end nf V t( nf) ns Case : High source impedance with unterminated transmission line 17 Same example different termination resistors top: Zs5Ω. Zl1kΩ. (TLIA.91. TLRl.99. TLRs.818. S 9%) bottom: Zs5Ω. Zl1kΩ. (TLIA.5. TLRl.99. TLRs. S 99.5%) v_tl_end nf V t( nf) ns Case 3: Source-end terminated 17 9/9/5 EE6471 (KR) 43
10 TL Part /Overview Transmission Lines High Frequency Mechanisms in Transmission Lines Skin-Effect Proximity Effect Terminations Transmission Lines on PCBs Equations 9/9/5 EE6471 (KR) 44
11 TL/Skin Effect At low frequencies, current density inside a conductor is uniform. At high frequencies, it isn t. Conductor carrying high frequency currents: Current flow primarily on the surface of a conductor Phenomena is called skin effect Current density falls off exponentially with depth into the conductor J ( d) J e d δ J d with skin depth δ ρ π f µ ρ : µ : material resistivity permeability 9/9/5 EE6471 (KR) 45
12 TL/Skin Effect Skin effect Current density falls off exponentially with depth into the conductor Modelling: Current flows uniformly in an outer shell of the conductor with thickness δ. Skin depth is a material property (not a function of conductor shape) For most transmission lines, skin effect is the reason for their lossy nature fs ρ πµ r How to tackle skin effect problems Litz wire Planar conductors for f<fs skin effect negligible. RRdc for f>fs skin effect. Resistance increases with square root of frequency 9/9/5 EE6471 (KR) 46
13 TL/Skin Effect conducting area: A r π r A π r ( ) δ π ( δ δ ) for high frequencies δ<<r: Resistance (per unit length): 1 Rhf pul r fµρ π A π rδ R pul ρ A δ r 9/9/5 EE6471 (KR) 47
14 TL/Skin Effect/Example Example: AWG4 Transmission Line C4pF/m, L4nH/m, Rdc8mΩ/m wire radius AWG4: r53µm skin effect frequency fs67khz δ_cu ( f( fi) ) µ meter Plots top right: skin depth bottom right: effective resistance per unit length f( fi) Meg Hz R_potl ( f( fi) ) pul R Rdc + pul Rhf pul /9/5 EE6471 (KR) f( fi) Me g Hz 1
15 TL/Skin Effect/Propagation Constant A Example: AWG4 Transmission Line. Three regions: Low frequency: RC TL behaviour (distortion) Mid frequency: LC TL behaviour (no distortion, just delay) High frequency: Distortion because of skin effect Re( A_potl ( p( fi) )) Im( A_potl ( p( fi) )) f( fi) Meg Hz 9/9/5 EE6471 (KR) 49 1
16 TL/Skin Effect/Example Example: Transmission of a pulse over a long AWG4 Telephone Line C4pF/m, L4nH/m, R8mΩ/m Tp4µs/km Length of transmission line: 1km 6 6 v1 j 1V vo j 1V t() j ns Signal distortion on an RLC Transmission Line due to skin effect 9/9/5 EE6471 (KR) 5
17 TL/Proximity Effect Proximity Effect Current distribution in a conductor is affected by currents in adjacent conductors Like the skin effect, the proximity effect leads to a larger effective resistance at high frequencies Much harder to quantify (use tables, graphs, field solvers) For same current direction increase in resistance is modest (even if conductors almost touch) For opposite current direction proximity effect can be many times higher than skin effect (depending on distance of conductors) Take proximity effect into account whenever conductors are brought closer together than about 3 times their diameter H field Same Current Direction H field Opposite Current Direction 9/9/5 EE6471 (KR) 51
18 TL/Special Case Cl Cl Cl Cl Equally Spaced Capacitive Loads Frequently encountered in large bus formations (e.g. memory modules) n capacitive loads are of equal value and spaced evenly over the length of the transmission line applicable if effective length of rising edge exceeds spacing between capacitive loads 9/9/5 EE6471 (KR) 5 Z ' Tp' C L C L n Cl + length n Cl + length
19 TL/Termination Short Lines (l<lr/6) Termination required for damping bitstream_sampled j bitstream_rlc j j 13 Transmission Lines Termination to eliminate reflections 8.9 v_tl_end nf V t( nf) ns 17 9/9/5 EE6471 (KR) 53
20 TL/Terminations/End Termination End termination Driver connects directly to TL All reflections damped by termination resistor Rt (TLRl) Received voltage is equal to the Rt Cl transmitted voltage (S 1%) Short rise time Drawbacks: High power dissipation Imbalanced load (difficult to drive) Assumptions: Rt. Hx 1. TLIA 1. TLRl. Tr TL end Tr driver +. Cl S S ( ( 1+ TLRl( ) TLIA( Hx( 1 TLRl( Hx( TLRs( ( 1+ TLRl( ) 1 ( TLIA( Hx( 9/9/5 EE6471 (KR) 54
21 TL/Terminations/End Termination/Split Split End termination (Rt1 Rt) Advantages: Balanced power dissipation Easier to drive For CMOS, HCMOS Rt1Rt Rt1 Rt Cl Assumptions: Signal is dc-balanced (equal 1 s and ) Pd Pd Rt1 Rt1 + Pd Pd Rt Rt 4 Worst case: Static signal (1 or )... Pd Rt1max Pd Rt max (assuming that resistance of TL is negligible) 9/9/5 EE6471 (KR) 55
22 TL/Terminations/End Termination/AC Biased AC Biased End termination Time constant large vs signal period Rt Advantages: Lower average power consumption Lower static power consumption Disadvantage Difficult to drive if signal is not dcbalanced Rt Ct Cl Assumptions: Signal is dc-balanced (equal 1 s and ) Pd Rt 1 4 Static signal (1 or )... Pd Rt1 9/9/5 EE6471 (KR) 56
23 TL/Terminations/End Termination/Bifurcation Bifurcation Rt No reflections Difficult to implement ( vs ) Rt Cl Rt Cl 9/9/5 EE6471 (KR) 57
24 TL/Terminations/End Termination/Daisy Chain Diasy Chain Configuration Keep stubs as short as possible Minimise capacitive load Multiple stubs: Space equally Rt Cl 9/9/5 EE6471 (KR) 58
25 TL/Terminations/Source Termination Source termination All reflections damped at the source side by source termination resistor Rt (TLRs) Advantages: Lower average drive currents Disadvantages Output impedance of driver often not tightly specified Daisy-chaining not recommended Rt Assumptions: Rt. Hx 1. TLIA.5. TLRl1 Cl S ( ( 1+ TLRl( ) TLIA( Hx( 1 TLRl( Hx( TLRs( Tr TL end (. Z Cl) driver + Tr S ( TLIA( Hx( ( 1+ TLRl( ) 1 9/9/5 EE6471 (KR) 59
26 TL/Terminations/Source Termination Rt Cl Source termination Driving signal cut in half (TLIA.5) Driving signal propagates down TL Reflection at load side (TLRl1) Reflected signal travels back Reflected signal damps out at the source termination (TLRs) v1 v v3 v4 v1 v v3 v4 Tp Tp t t t t 9/9/5 EE6471 (KR) 6
27 TL/Terminations/Microstrip Equations d w h Example: Microstrip on FR4 εr4.5. oz copper Microstrip Equations Useful approximations Use numeric TL tools for improved accuracy Tp pul 87Ω 5.98h ln εr w + d ns εr +.67 meter _microstrip_approx( h, w, d, 4.5) 1 _microstrip( h, w, d, 4.5) Tp_microstrip_approx_pul( εr) pico s mmeter εr 9/9/5 EE6471 (KR) w h Tp_microstrip_pul(.5 inch,.1 inch, ounce, εr) pico s m meter
28 TL/Terminations/Stripline Equations w d b Stripline Equations Useful approximations Use numeric TL tools for improved accuracy 6Ω εr 1.9b ln.8w + d Tp pul 3.35 ns meter εr 9/9/5 EE6471 (KR) 6
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