Problem Set #1. i z. the complex propagation constant. For the characteristic impedance:

Transcription

1 Problem Se # Problem : a) Using phasor noaion, calculae he volage and curren waves on a ransmission line by solving he wave equaion Assume ha R, L,, G are all non-zero and independen of frequency From hese soluions derive he formulas for he characerisic impedance and he phase velociy of he ransmission line v i Using phasors: i()( z jω L + R), v( z)( jω + G) z z v ombine: ( jω L + R)( jω + G) v z jkz v ± m () z A ± e : boh propagaing direcions where k j ( jω L + R)( jω + G) he complex propagaion consan For he characerisic impedance: jkz v A e jωl + R Z Noe ha for he negaive raveling wave: jkz i ( jk) A e jω + G jωl + R v Z i ω ω For he phase velociy: If we wrie k b ja, hen υ b Re k b) Using he volage and curren equaions above, derive he formulas for he reflecion and ransmission coefficiens a he inerface of wo ransmission lines wih characerisic impedances Z and Z The oal waves a he boundary on he side of he firs ransmission line are: v v + v and i i + i [ v v ] Z Due o coninuiy he volage on he side of he second ransmission line mus be he same as he volage on he side of he firs ransmission line The same also holds for he + v v v + v Z Z + Z Z curren Hence: Z Z v v + i i v v Z + Z Z + Z v + v + v The ransmied wave is: ( ) τ v + ( )

2 c) alculae he volage a poin A as a funcion of ime unil ns for he following sysem Assume ha he source produces a sep of V a ime 5Ω 5Ω A Ω ns 5Ω 3ns 5Ω 4ns 5Ω The reflecion and ransmission coefficiens are: Going from lef o righ: 5, τ , τ Going from righ o lef: , τ , τ We calculae some poins a node A: A ime : V()667 V A ime ns-: V(ns-)V()+667*84 V A ime ns+: V(ns+)V(ns-)+667**(-333)756 V oninuing manually quickly becomes oo cumbersome, and herefore we wrie Malab code (Problemcm) ha generaes he various propagaing waves in he sysem and keeps rack of he reflecions ha reach node A The resuls up o ime ns are shown below The Sequence enries show he pah of he wave before i reached node A (which is denoed as )

3 To verify he correc operaion of he code we run a simulaion of he sysem in Spice and compare The able below shows he (ime, volage) pairs for he breakpoins in he waveform a node A Noe ha only he volages immediaely afer he reflecion can be disinguished: Time (ns) Volage (mv) The simulaed waveform is shown below:

4 d) Using circui argumens derive he characerisic impedance and phase velociy of a lossless ransmission line wih muual inducance and capaciance, ha is excied in he odd mode Repea for he case of he even mode Hin: Find he equivalen inducance, capaciance seen by a conducor and subsiue in he formulas in (a): self L self muual L muual self L self Noe: In he W-elemen models L L self, L L muual, self + muual, muual

5 In he odd mode for he wo conducors: i -i, v -v Therefore: dv d( v v ) dv i self + muual self + muual self + d d d di di di v Lself + Lmuual ( Lself Lmuual ) L Lself Lmuual d d d The characerisic impedance and phase velociy are herefore: L Z,υ L In he even mode for he wo conducors: i i, v v Therefore: dv d( v v ) dv i self + muual self self d d d di di di v Lself + Lmuual ( Lself + Lmuual ) L Lself + Lmuual d d d The characerisic impedance and phase velociy are herefore: L Z,υ L ( ) muual Problem : a) The following seup is used o obain he odd- and even-mode TDRs for various componens: vin 5Ω vin lin 5Ω 5Ω lin ns nin nin The simulaion resuls for he various cases are conained in he following files: omponen r File FR4 channel FR4_channel_even / FR4_channel_odd Rogers channel Rogers_channel_even / Rogers_channel_odd 4 FR4 linecard race FR4_L_even / FR4_L_odd Assuming ha he componens are lossless, find he W-elemen models

6 Figure a_ The odd-mode ransien response for he FR4 channel is shown in he above Fig a_ The single-sided simulus a node vin and he ransien waveform a node lin are depiced The ineresing par of he second waveform is highlighed and is shown in more deail in Figs a_ and a_3 Figure a_

7 Figure a_ We see ha he reflecion coefficien from Fig a_ is 5 The ravel ime on he is from Figs a_ and a_3: 6399ns 35ns 6745ns Therefore we ge he equaions: L Z Z, Z L + Z (a) Z, + Z L + Z l l L L (a) υ l Solving Eqs (a) & (a) we ge: + L Z 33 nh and l 345 pf l L We similarly look a he even-mode ransien response of he FR4 channel and ge: ns 349ns and 67ns 5 Eqs (a) and (a) also hold for he even-mode case and solving for he inducance, capaciance we ge: + L Z 46 nh l 3 pf l L From problem d) we know ha in he W-elemen models:

8 L + L L L Lself 378 nh (a3a) L L L Lmuual 488 nh (a3b) + self + muual 88 pf (a3c) muual -58 pf (a3d) The following ables summarize he resuls for he res of he circuis: Rogers channel: L L L /L (ns) (nh) (pf) (ns) (nh) (pf) (nh) L (nh) / (pf) (pf) FR4 linecard race: L (ns) (nh) (pf) (ns) L (nh) (pf) L /L (nh) L (nh) / (pf) (pf) b) The in problem (a) is he cascade of a 3 FR4 linecard race and a 6 FR4 channel erminaed a R5Ω alculae and draw he odd- and even-mode ransien waveforms a he node lin unil ime ns From par (a) we can immediaely find he characerisic impedances and ravel imes for he wo componens 358e 9 For he linecard: Z L 49 4Ω 335e L 3 54e 358e 9 335e 9 53ns 398e 9 Z L 6 6Ω 83e L 3 54e 398e 9 83e 9 5ns 33e 9 For he channel: Z H 49 Ω 345e H 6 54e 33e 9 345e 9 5ns

9 46e Ω 3e H 6 54e 46e 9 35e 9 3ns Z H We presen he firs few poins in he case of he odd-mode exciaion: Le 6, 4, 3 be he reflecion coefficiens and τ + 994, τ + 996, be he ransmission coefficiens when he wave ravels from lef o righ In he opposie direcion we have: 6, 4, 6 τ +, τ + 4 A ns: V(ns)5+(-6)x5497 V A 36 ns: V(36ns)V(ns)+994*5*(-4)*6495 V A 4 ns: V(4ns)V(36ns)+ 994*5*(-4)*6*(-4)*6 495V ec I becomes quickly obvious ha manual calculaion is impossible, and herefore we use he Malab program Problembm I gives he various waves ha ravel in he sysem a any poin in ime and updaes he volage a lin, if necessary The parameer Vepsilon defines he minimum volage ampliude for which a wave is no ignored For epsilonnv, we ge he resul: Wih he same program we ge he following resuls in he case of he even-mode exciaion (Vepsilon uv)

10 Exercise: Verify he resuls by running a Spice simulaion wih ideal ransmission lines c) The in problem (a) is now he cascade of a FR4 linecard race of lengh 5, a backplane via of lengh cm and a nelco channel of lengh 7 All componens are assumed o be lossless The odd and even mode TDR resuls are conained in he files TDR_Prc_oddr and TDR_Prc_evenr Using hese simulaions, deermine he W- elemen models for each of he componens vin 5Ω vin lin 5Ω 5Ω lin ns nin L nin nin3 NELO nin4 nin nin nin3 nin4 We sar wih he node lin which is he inpu o he 5Ω ransmission line Firs we look a he odd-mode waveforms A 3ns we ge he firs reflecion from he inpu of he FR4 linecard race From he disconinuiies we ge: L ns 35ns and also he ravel ime: L 836ns

11 + L L L Hence: L L Z 35 nh and L l L 33 pf L ll LL LL Also: Z L 494Ω L From he waveform n which is he inpu o he FR4 linecard race, we ge: ns 3677ns and also he ravel ime: 5ns + Hence: L Z L 87 nh and l 34 pf l L L Also: Z 373Ω From he waveform n which is he inpu o he backplane via, we ge: NELO ns 85ns and also he ravel ime: NELO 3ns + NELO NELO Hence: L NELO Z 367 nh and l NELO NELO NELO LNELO l NELO 8 pf NELO For he even case in a similar manner: ns 35ns L 96, L 837ns 5 + L L L Hence: L L Z 3995 nh, L l L 87pF L ll LL LL Also: Z L 66Ω L ns 3678ns 49, 6ns Hence: L Z L 33 nh, l pf l L L Also: Z 549Ω

12 ns 854ns NELO 48, NELO 3ns 5 + NELO NELO Hence: L NELO Z 3848 nh, l NELO NELO NELO LNELO l NELO 54pF NELO To find he W-elemen parameers we use Eqs (a3a)-(a3d) and we ge: L L LL + LL L LL LL L L 364 nh, L 37 nh L L L + L L L L pf, -3 pf L + L L L L L 6 nh, L 7 nh + pf, -pf NELO NELO LNELO + LNELO NELO LNELO LNELO L L 358 nh, L 34 nh NELO NELO NELO + NELO NELO NELO NELO 67 pf, -3pF d) The following seup is used o measure he S-parameers of he The A sources produce sinusoids of equal/opposie phases in he case of even/odd mode operaion The ampliude of he source signals is V The is a FR4 channel same o he one used in Problem (a), only now he channel has losses The A simulaion resuls for he even/odd modes can be found in he files Sparam_FR4_evenac and Sparam_FR4_oddac, respecively Using hese, find he complee W-elemen model for he channel

13 vin 5Ω vin lin 5Ω 5Ω ns lin nin nin nou nou 5Ω 5Ω Noe : As in many pracical channels, you can assume ha R o, and G o, G o, This means ha you have o find he parameers R o,, R s,, R s,, G d,, G d, Noe : The resisance per uni lengh a a frequency f is given by: Rodd / even( f ) Ro + Rs, odd / even The conducance per uni lengh a a frequency f is given by: Godd / even( f ) Go + Gs, odd / even where: Rs, odd Rs, Rs,, R s, even Rs, + Rs,, G d, odd Gd, + Gd, and Gd, even Gd, Gd, f f We sar by deermining he D resisance R A D, we see ha he ampliude of he wave a node nin is 5V in boh odd and even mode In boh modes he circui a D is simplified as shown: 5Ω nin R D V 5Ω We have: RD R 58Ω R + D D and herefore R R D 6 Ω l To find he frequency-dependen parameers we make he assumpion ha a high L 33e frequencies Z, 49Ω L Noe 345e

14 also ha a backwards propagaing wave from righ o lef sees a reflecion coefficien 5 49 L a he beginning of he Le 'a' be he aenuaion consan of he ransmission line Then we can wrie down he oal wave a node nou as he sum of muliple reflecions: 4 Vo Vinc exp al ( + L) + Vinc exp 3al L ( + L ) + Vinc exp 5al L ( + L ) + (d) where V inc V ( ) 99V he iniially ransmied wave on he channel I is easy o see from (d) ha all erms afer he firs one can be ignored Hence: ( ) ( ) V ( nou) Vinc exp al ( + L) exp al a ln (d) l V ( nou) Bu we also know ha a Im [ j ( jωl + R)( jω + G) ] (d) where RR odd and GG odd are compued as described in he problem saemen From he odd-mode waveform a node nou we ge he following vecors of frequencies and corresponding volage magniudes: f[e9 5e9 3e9 35e9 4e9 45e9 5e9]; vou[ ]; Using MSE fiing and Eqs (d) & (d) we ge ha R s,odd 7e-4 Ω/m/sqr(Hz) and G d,odd 8e-Ω/m/Hz; Similarly in he odd mode case we have: f[e9 5e9 3e9 35e9 4e9 45e9 5e9]; vou[ ]; I should be noed here ha L 46e Z, 639Ω L 3e ( ) ( ) ( ) K Therefore (d) changes o: a ln l V ( nou) (d3) Using MSE we ge: R s,even 97e-4 Ω/m/sqr(Hz) and G d,even 3e-Ω/m/Hz; Hence we ge for he W-elemen parameers: Rs, even + Rs, odd 7e e 4 Rs, Rs, 835e 4 Ω / m / Hz Rs, even Rs, odd 97e 4 7e 4 Rs, 85e 4 Ω / m / Hz Gd, even + Gd, odd 3e + 8e Gd, Gd, 55e Ω / m / Hz Gd, even Gd, odd 3e 8e Gd, 5e Ω / m / Hz Exercise: Verify he assumpions made above ha he characerisic impedances a high frequencies are approximaely he same as in he lossless case