ECE 546 Lecture 13 Scattering Parameters

Transcription

1 ECE 546 Lecture 3 Scattering Parameters Spring 08 Jose E. Schutt-Aine Electrical & Computer Engineering University of Illinois jesa@illinois.edu ECE 546 Jose Schutt Aine

2 Transfer Function Representation Use a two-terminal representation of system for input and output ECE 546 Jose Schutt Aine

3 Y-parameter Representation I yvyv I yv yv ECE 546 Jose Schutt Aine 3

4 Y Parameter Calculations y I V V V 0 V 0 y I To make V = 0, place a short at port ECE 546 Jose Schutt Aine 4

5 Z Parameters V z I z I V z I z I ECE 546 Jose Schutt Aine 5

6 Z-parameter Calculations z V V I I I 0 I 0 z To make I = 0, place an open at port ECE 546 Jose Schutt Aine 6

7 H Parameters V h I h V I h I h V ECE 546 Jose Schutt Aine 7

8 H Parameter Calculations h V I I V 0 V 0 h I To make V = 0, place a short at port ECE 546 Jose Schutt Aine 8

9 G Parameters I gvg I V g V g I ECE 546 Jose Schutt Aine 9

10 G-Parameter Calculations g I V V V I 0 I 0 g To make I = 0, place an open at port ECE 546 Jose Schutt Aine 0

11 TWO PORT NETWORK REPRESENTATION Z Parameters V Z I Z I V Z I Z I Y Parameters I Y V Y V I Y V Y V - At microwave frequencies, it is more difficult to measure total voltages and currents. - Short and open circuits are difficult to achieve at high frequencies. - Most active devices are not short- or open-circuit stable. ECE 546 Jose Schutt Aine

12 Wave Approach Use a travelling wave approach V Ei Er V Ei Er Ei r I = E Ei E I r = Zo Zo - Total voltage and current are made up of sums of forward and backward traveling waves. - Traveling waves can be determined from standing-wave ratio. ECE 546 Jose Schutt Aine

13 Wave Approach a = E i Z o a = E i Z o b = E r Z o b = E r Z o Z o is the reference impedance of the system b = S a + S a b = S a + S a ECE 546 Jose Schutt Aine 3

14 Wave Approach b S = a=0 a S = b a a=0 S = S = b a b a a=0 a=0 To make a i = 0 ) Provide no excitation at port i ) Match port i to the characteristic impedance of the reference lines. CAUTION : a i and b i are the traveling waves in the reference lines. ECE 546 Jose Schutt Aine 4

15 S Parameters of TL S = S = S = S = R j LG j C Z c R j L G jc ( X ) X ( )X X Z Z c c Z Z X e l ref ref ECE 546 Jose Schutt Aine 5

16 S Parameters of Lossless TL Z c LC L C S = S = S = S = ( X ) X ( )X X X Z Z c c Z Z j l e ref ref If Z c = Z ref S = S = 0 j l S = S = e ECE 546 Jose Schutt Aine 6

17 N-Port S Parameters a i V i Z oi b S S a b S S a b S a n nn n b=sa If b i = 0, then no reflected wave on port i port is matched b i V i Z oi V i V i Z oi : incident voltage wave in port i : reflected voltage wave in port i : impedance in port i ECE 546 Jose Schutt Aine 7

18 N-Port S Parameters v Z o a+b () i Z o a-b () v=zi (3) Substitute () and () into (3) Z o a+b Z a-b Z Defining S such that b = Sa and substituting for b Z o U+Sa Z U-Sa o Z U+SU-S - SZ+ ZoU ZZoU Z o SZ ZS o U : unit matrix ECE 546 Jose Schutt Aine 8

19 N-Port S Parameters If the port reference impedances are different, we define k as k Z o Z o Z on v=k(a+b) and - i=k a-b and - k(a + b) Zk a - b ZS - - S = Zk + k Zk - k SZ - Z=k U+S U-S k ECE 546 Jose Schutt Aine 9

20 Normalization Assume original S parameters as S with system k. Then the representation S on system k is given by Transformation Equation - - S = k (U + S )(U - S ) kk + k k (U + S )(U - S ) kk - k - If Z is symmetric, S is also symmetric ECE 546 Jose Schutt Aine 0

21 Dissipated Power Pd a U-S S a T T * * The dissipation matrix D is given by: T * D=U-S S Passivity insures that the system will always be stable provided that it is connected to another passive network For passivity () the determinant of D must be () the determinant of the principal minors must 0 be 0 ECE 546 Jose Schutt Aine

22 Dissipated Power When the dissipation matrix is 0, we have a lossless network SS T * U The S matrix is unitary. For a lossless two port: S S S S If in addition the network is reciprocal, then S S and S S S ECE 546 Jose Schutt Aine

23 ECE 546 Jose Schutt Aine 3 Lossy and Dispersive Line S S S S l e c o c o Z Z Z Z

24 Frequency-Domain Formulation* * J. E. Schutt-Aine and R. Mittra, "Scattering Parameter Transient analysis of transmission lines loaded with nonlinear terminations," IEEE Trans. Microwave Theory Tech., vol. MTT- 36, pp , March 988. ECE 546 Jose Schutt Aine 4

25 Frequency-Domain B ( ) S A( ) S A ( ) B ( ) S A( ) S A ( ) ECE 546 Jose Schutt Aine 5

26 Time-Domain Formulation ECE 546 Jose Schutt Aine 6

27 Time-Domain Formulation b() t s ()* t a () t s ()* t a () t b () t s ()* t a () t s ()* t a () t a () t () t b() t T() t g () t a () t () t b () t T () t g () t T() t i i () t i Z o Z () t Z o Z () t Z i Z () t Z i o o ECE 546 Jose Schutt Aine 7

28 Time-Domain Solutions a () t ' () ts(0) T() tg() t() tm() t () t () ts (0) T() tg() t tm() t () t ' a () t ' () ts(0) T() tg() t() tm() t () t () ts (0) T() tg() t tm() t () t ' ECE 546 Jose Schutt Aine 8

29 Time-Domain Solutions b() t s (0) a () t s (0) a () t M () t ' ' b () t s (0) a () t s (0) a () t M () t ' ' ' ' ' ' () t () t s(0) () t s(0) () t s(0) () t s(0) M () t H () t H () t M () t H () t H () t s ' ij (0) s (0) t ij H () t s ( t ) a ( ) ij ij j ECE 546 Jose Schutt Aine 9

30 Special Case Lossless Line s() t s() t 0 s() t s() t t l v M() t at l v M() t a t l v l a t T() t g() t () t at v l a t T() t g() t () t at v l b() t at v l b() t a t v Wave Shifting Solution ECE 546 Jose Schutt Aine 30

31 Time-Domain Solutions v () t a () t b() t v () t a () t b () t i () t i () t a() t b() t Z Z o o a() t b() t Z Z o o ECE 546 Jose Schutt Aine 3

32 Line length =.7m Z o = 73 v = 0.4 m/ns Simulations ECE 546 Jose Schutt Aine 3

33 Simulations Near End Volts Time (ns) Far End 6 4 Volts Time (ns) ECE 546 Jose Schutt Aine 33

34 Simulations Line length = 5 in L = 539 nh/m C = 39 pf/m R o = k (GHz) / Pulse magnitude = 4V Pulse width = 0 ns Rise and fall times = ns 5 Near End 6 Far End Volts lossy lossless Volts lossy lossless Time (ns) Time (ns) ECE 546 Jose Schutt Aine 34

35 N-Line S-Parameters* B =S A +S A B =S A +S A * J. E. Schutt-Aine and R. Mittra, "Transient analysis of coupled lossy transmission lines with nonlinear terminations," IEEE Trans. Circuit Syst., vol. CAS-36, pp , July 989. ECE 546 Jose Schutt Aine 35

36 Scattering Parameters for N-Line S =S =T Γ - ΨΓΨ -ΓΨΓΨ T - S =S = E E -ΓΨ-ΓΨΓΨ T o Γ = +EE ozohoh Zm -EEoZoHoH Zm o o o m o Ψ =W -l T = + EE Z H H Z EE - ECE 546 Jose Schutt Aine 36 -

37 Scattering Parameter Matrices E o : Reference system voltage eigenvector matrix E : Test system voltage eigenvector matrix H o : Reference system current eigenvector matrix H : Test system current eigenvector matrix Z o : Reference system modal impedance matrix Z m : Test system modal impedance matrix ECE 546 Jose Schutt Aine 37

38 Eigen Analysis * Diagonalize ZY and YZ and find eigenvalues. * Eigenvalues are complex: i = i + ji W(u) e uj u e uj u e nu j n u ECE 546 Jose Schutt Aine 38

39 Solution V m =EV I m =HI V ( x) = W( x) AW( x) m - I ( x) =Z W( x) AW( x) m m Z m = Λ EZH - - m Z c =E ZmH =E ΛmEZ ECE 546 Jose Schutt Aine 39

40 Solutions a () t () t s' (0) T() t g () t () t M () t ( ts ) ' (0) ( ts ) ' (0) () ts' (0) T tg() t tm() t a () t () t s' (0) T () t g () t () t M () t ( ts ) ' (0) ( ts ) ' (0) () ts' (0) Ttg () () t tm() t ECE 546 Jose Schutt Aine 40

41 Solutions ' - - ' ' t t s t s t s t s ( ) - - ( ) (0) - ( ) ' (0) ( ) (0) ( ) (0) - - ' ' ' t t s t s t s t s () -- () ' (0) - () (0) () (0) () (0) b() t s (0) a () t s (0) a () t M () t ' ' b () t s (0) a () t s (0) a () t M () t ' ' ECE 546 Jose Schutt Aine 4

42 Solutions v () t a () t b() t v () t E a () t b() t m o v () t a () t b () t v () t E a () t b () t m o V (z) V (z) z=0 z=l... V 3 (z) L, C ECE 546 Jose Schutt Aine 4

43 Lossless Case Wave Shifting s () t s () t ( t- ) M t a t m () ( - ) M t a t m () ( - ) m a t T t g t t a t m () () () () ( - ) a t T t g3 t 3 t a t m () () () () ( - ) b t a t m () ( - ) b t a t m () ( - ) ECE 546 Jose Schutt Aine 43

44 Solution for Lossless Lines (t m ) (t m ) (t m ) (t mn ) a i (t m ) a (t m ) a (t m ) a n (t mn ) ECE 546 Jose Schutt Aine 44

45 Why Use S Parameters? Y-Parameter Test line: Zc, l Y e l Z ( e ) c Z c : microstrip characteristic impedance : complex propagation constant l : length of microstrip Y can be unstable short Reference Line Zo S-Parameter Test Line Test line: Zc, l ( e ) S l e Z c Z c Z o Z o Reference Line S is always stable Zo ECE 546 Jose Schutt Aine 45

46 Choice of Reference Z Z c c Z Z ref ref Z c R j L G jc Z ref is arbitrary What is the best choice for Z ref? At high frequencies Z c L C Thus, if we choose Z ref L C j LCd S e X o S 0 ECE 546 Jose Schutt Aine 46

47 Choice of Reference S-Parameter measurements (or simulations) are made using a 50-ohm system. For a 4-port, the reference impedance is given by: Z o = Z: Impedance matrix (of blackbox) S: S-parameter matrix Z o : Reference impedance I: Unit matrix o o S ZZ I ZZ I o Z I S I S Z ECE 546 Jose Schutt Aine 47

48 Reference Transformation Method: Change reference impedance from uncoupled to coupled system to get new S-parameter representation Z o = Uncoupled system Z o = as an example Coupled system ECE 546 Jose Schutt Aine 48

49 Choice of Reference Z o = using as reference.5 S - Linear Magnitude Harder to approximate using S Z o = S - 50 Ohm S - Zref Easier to approximate (up to 6 GHz) as reference Frequency (GHz) ECE 546 Jose Schutt Aine 49

50 Choice of Reference using.5 S - Linear Magnitude Z o = as reference Harder to approximate S Z o = using as reference S - 50 Ohm S - Zref Easier to approximate (up to 6 GHz) Frequency (GHz) ECE 546 Jose Schutt Aine 50

51 Choice of Reference S - Linear Magnitude using 0.3 S - 50 Ohm Z o = as reference S - Zref Harder to approximate S 0.5 using 0. Z o = as reference Frequency (GHz) Easier to approximate (up to 6 GHz) ECE 546 Jose Schutt Aine 5

52 Choice of Reference Z o = using as reference. 0.8 S3 - Linear Magnitude Easier to approximate S3-50 Ohm S3 - Zref using S3 0.6 Z o = as reference Harder to approximate Frequency (GHz) ECE 546 Jose Schutt Aine 5

53 . Choice of Reference 0.3 S Magnitude Zref=Zo Zref=80 ohms Zref=00 ohms S Frequency (GHz) ECE 546 Jose Schutt Aine 53

54 . Choice of Reference 0.85 S Magnitude Zref=Zo Zref=80 ohms Zref=00 ohms 0.8 S Frequency (GHz) ECE 546 Jose Schutt Aine 54

55 Modeling of Discontinuities. Tapered Lines. Capacitive Discontinuities ECE 546 Jose Schutt Aine 55

56 Tapered Microstrip General topology of tapered microstrip with dw :width at wide end, dn: width at narrow end, lw: length of wide section, ln : length of narrow section, lt: length of tapered section. ECE 546 Jose Schutt Aine 56

57 Tapered Line Analysis Using S Parameters* u () t s ()* t u () t s ()* t w () t ( j) ( j) j j j w () t s ()* t u () t s ()* t w () t ( j) ( j) j j j * J. E. Schutt-Aine, IEEE Trans. Circuit Syst., vol. CAS-39, pp , May 99. ECE 546 Jose Schutt Aine 57

58 Tapered Transmission Line Small End Excitation at small end Wide End Excitation at small end Volts Volts Time (ns) Time (ns) 5 Small End Excitation at wide end 5 Wide End Excitation at wide end Volts Volts Time (ns) Time (ns) ECE 546 Jose Schutt Aine 58

59 Tapered Transmission Line Varying tapering rate Near End.0564 mils/in.8 mils/in.57 mils/in Far End.0564 mils/in.8 mils/in.57 mils/in volts volts Time (ns) Time (ns) ECE 546 Jose Schutt Aine 59

60 Capacitive Load Z o Z o Z o C.5 Near End -- C=4 pf.5 Far end -- C=4 pf.5.5 Volts 0.5 Volts Time (ns) Time (ns) ECE 546 Jose Schutt Aine 60

61 Capacitive Load Z o Z o Z o C.5 Near end -- C=40 pf.5 Far end -- C=40 pf Volts Volts Time (ns) Time (ns) ECE 546 Jose Schutt Aine 6

62 Multidrop Buses Z sl Z stub Z o - Stubs of TL with nonlinear loads - Reduce speed and bandwidth - Limit driving capabilities ECE 546 Jose Schutt Aine 6

63 Transmission Lines with Capacitive Discontinuities ECE 546 Jose Schutt Aine 63

64 Capacitive Discontinuity V V V i r t V T E V r c c i V V E V V V Z R R Z i r i r t o o ECE 546 Jose Schutt Aine 64

65 Scattering Parameter Analysis u () t s ()* t u () t s ()* t w () t ( j) ' ( j) j j j u () t u () t u () t ' " j j j w t w t u t ' " j() j() j() ECE 546 Jose Schutt Aine 65

66 V olt V ol t Capacitive Loading 4 Near End 4 Far End 3 3 s s Ti me ( ns) Ti me ( ns) ECE 546 Jose Schutt Aine 66

67 Capacitive Loading loading period 600 mils 300 mils 50 mils 4 3 capacitive loading pf pf 4 pf volts 0 volts Time (ns) Time (ns) Computer-simulated near end responses for capacitively loaded transmission line with l = 3.6 in, w = 8 mils, h = 5 mils. Pulse parameters are Vmax = 4 V, tr = tf = 0.5 ns, tw = 4 ns. Left: Varying P with C = pf. Right: Varying C with P = 300 mils. ECE 546 Jose Schutt Aine 67