Qualification of tabulated scattering parameters

Transcription

1 Qualification of tabulated scattering parameters Stefano Grivet Talocia Politecnico di Torino, Italy IdemWorks s.r.l. 4 th IEEE Workshop on Signal Propagation on Interconnects 9 2 May 2, Hildesheim, Germany

2 Why S-parameters S-parameters are always defined Impedance or admittance may not S-parameters are normalized Good numerical properties in simulation S-parameters are easily measured Even at very high frequency, good reliability Standard format for S-parameters Touchstone files from measurement hardware All field solvers provide S-parameters on output Tabulated frequency data Intrinsic IP protection for vendors Do not disclose design details, but only I/O electrical properties Best way to represent broadband EM/circuit interactions The essence of Signal and Power Integrity Is this characterization complete? Yes, but 2

3 Scattering network functions A ( ) A ( s 2 ) s B ( ) B 2 ( s) s power waves complex frequency B B 2 = S S 2 ( s) ( s) S S 2 22 ( s) A ( s) A 2 Scattering matrix 3

4 4 S.Grivet-Talocia, Qualification of tabulated scattering parameters, IBIS Summit, 2 May 2, Hildesheim, Germany For circuits: real rational functions For lumped circuits: S-parameters are real rational functions Valid for all complex frequencies in the entire complex plane n n n n ij s b b s b s a s a a s S = L L ) ( C = ) ( scr scr scr scr scr scr s S R = reference resistance

5 Examples of S-parameter data Via array 2 ports Wiring harness 8 ports High-speed channel 8 ports 5

6 Examples of S-parameter data Scattering matrix entries, magnitude.8.6 S(,) x Scattering matrix entries, phase (degrees) 2 S(,) Frequency [Hz] x.8.6 Scattering matrix entries, magnitude S(,).8.6 Scattering matrix entries, magnitude S(,) x 8 Scattering matrix entries, phase (degrees) S(,) Frequency [Hz] x Scattering matrix entries, phase (degrees) S(,) Frequency [GHz] 6

7 From frequency to time-domain: impulse response Inverse Fourier transform ω = Im s h( t) + j = ω ω t S( j ) e dω 2π ω max ω n s jω DC σ σ = Re s Inverse Laplace transform h( t) = 2 πj σ + j σ j S( s) e st ds Complex s-plane 7

8 8 S.Grivet-Talocia, Qualification of tabulated scattering parameters, IBIS Summit, 2 May 2, Hildesheim, Germany Finding impulse responses + = ω ω π ω d e j S t h t j ) ( 2 ) ( Strategy : Discrete Fourier Transform (Fast Fourier Transform, FFT) Strategy 2: Fit a parametric model allowing analytic Fourier/Laplace inversion = = Δ 2 ) exp ( ) ( M m m M mk j j S M k h π ω ) ( ) ( ) exp( ) ( ) ( t S t u t p R t h S p j R j S N n n n N n n n δ ω ω = = + + Δ = N m j m π ω 2

9 Qualification process The limited information in the Touchstone file must describe the electrical behavior of the structure of interest have enough resolution: sampling cover a sufficient bandwidth fulfill fundamental passivity requirements causality energy gain, passivity Need a qualification methodology based on rigorous theoretical foundation allowing robust numerical implementation checking all above conditions 9

10 Passivity conditions * S ( jω) = S ( jω) Guarantees real-valued impulse response. Always assumed by construction ω = Im s S( jω) or max i { S( )} σ jω i ω max Energy condition: structure must not amplify signals. Sometimes called simply passivity condition ω n S( jω) is causal σ = Re s No anticipatory behavior in time-domain. Note: causality is a prerequisite for passivity!

11 Passivity: a ping-pong match Model: Load: B = S*A A = P*B A(s) B(s) Model One-port case The poor man s illustration of passivity: iterate through signal reflections Start with B= and A = Model hits signal: B = S*A Load hits signal: A = P*B = (P*S)*A Model hits signal: B = S*A = S*P*S*A A = Load hits signal: A 2 = P*B = (P*S) 2 *A N = And the winner is A N = (P*S) N *A A N

12 Passivity: a ping-pong match Model: Load: B = S*A A = P*B A(s) B(s) Model One-port case A N = N ( PS) A P < N, S < A remains bounded N N P =, S > A Blow-up! P is a reflection coefficient: for a passive load it does not exceed 2

13 Passivity: a ping-pong match Model: Load: B = S*A A = P*B A(s) B(s) Model One-port case Passivity requires that S( jω) for all frequencies! (not just the modeling bandwidth all means really all, from to Inf) 3

14 Passivity: what? A (s) B(s) Model A 2 (s) B2(s) In case of matrices, math is more complicated but visualization is simple and straightforward 4

15 Passivity: what? { singular values of S( jω) } ω S( s) is passive, Singular Values Frequency, GHz 5

16 Not all S-parameter models should be passive Small-signal characterization of a FET-based amplifier 6

17 A passive interconnect model All curves are below 7

18 Where do passivity violations come from? Data from measurement Improper calibration and de-embedding Human mistakes Measurement noise Data from simulation Poor meshing Inaccurate solver Bad models or assumptions on material properties Poor data post-processing algorithms Human mistakes Putting together results from two solvers 8

19 Non-passive data: so what? Non-passive data Fitting Accurate Non-passive model Model may blow-up during transient analysis Non-passive data Fitting Passivity enforcement Passive model Acceptable if passivity violations are small 9

20 Can we tolerate a passivity violation? YES Measured data CM filter from vendor VERTICAL ZOOM 2

21 Can we tolerate a passivity violation? NO Measured data Bad calibration by student 2

22 Can we fix passivity violations? Test case: measured cable Although the responses present no energy gain, an artificial clipping is evident at high frequencies. Is this clipping ok? 22

23 Causality qualification Much more tricky 23

24 Causality: definitions Time-domain Frequency-domain IN H( jω) OUT Hilbert transform Kramers-Krönig dispersion relations IN OUT time time U ( ω ) = V ( ω ) = pv π pv π + + dω V ( ω ) ω ω dω U ( ω ) ω ω Note: no delay extraction here H( jω ) = U( ω ) + jv( ω ) 24

25 Causality: definitions Time-domain Frequency-domain IN H( jω) OUT Hilbert transform Kramers-Krönig dispersion relations IN OUT time + dω H ( jω ω H ( jω) = pv ) jπ ω time Note: no delay extraction here Self-consistency All samples are strongly related 25

26 Causality check via dispersion relations Δ ( jω ) = reconstruction error H( jω ) pv jπ true + H( jω ) ω reconstruction dω ω Causality violation ω But in practice reconstruction is difficult because data are:. known only up to a maximum frequency Ω + + Ω Ω TRUNCATION ERROR 2. tabulated DISCRETIZATION ERROR Errors may hide causality violations... or point out bogus ones! Identification of causality violations must account for these errors. 26

27 Causality check (ideal) H( jω) Hilbert transform H REC ( jω) 27

28 Causality check (ideal) H( jω) Hilbert transform H REC ( jω) Never verified numerically! Causal H ( jω) = REC H( jω) Non-causal H REC ( jω) H( jω) 28

29 Causality check H( jω) Generalized Hilbert transform H REC ( jω) Δ( jω) Causal H REC ( jω) H( jω) < Δ( jω) Error bar Non-causal H REC ( jω) H( jω) > Δ( jω) 29

30 n = Truncation error E ( ω ω ) q q (jω ) = pv H( jω ) β( jω ) dω T ( ) n n n ω jπ ( ω ω = ω ω ) q q Analytic bound ω > Ω - -2 n=4 n=8 n=2 Truncation error can be arbitrarily reduced T n (ω) -3-4 n Full control of truncation effects! Normalized frequency (percentage of Ω) 3

31 Causality check results Error bar Reconstructed data Raw data Anchor point (no error) Computed by IdEM R29b Causality violations 3

32 Causality check results Causality violations Computed by IdEM R29b 32

33 Causality check error Difference between reconstructed and raw data E db ( jω) = 2log H REC ( jω) H Δ( jω) ( jω) H Δ Numerical resolution ( error bar ) H REC 33

34 Causality check error 3 25 Maximum Causality Check Error (db) S(,) 2 Non-causal ( >db ) Causal ( <db ) Frequency [GHz] Computed by IdEM R29b 34

35 Where do causality violations come from? Data from measurement Improper calibration and de-embedding Human mistakes Measurement noise Data from simulation Poor meshing Inaccurate solver Bad models or assumptions on material properties Poor data post-processing algorithms Human mistakes Putting together results from two solvers 35

36 Non-causal data: so what? Non-causal data Fitting Stable poles Bad accuracy Model is not representative of the real structure Non-causal data Fitting Good accuracy Unstable poles Model cannot be simulated (will blow-up) 36

37 Cable test case again Several causality violations are detected. Such violations are very large and spread throughout the entire band, especially for the diagonal responses of the S-matrix (return losses at all ports) 37

38 Trying to extract a macromodel Due to the data inconsistencies, it is impossible to obtain an accurate model after the fitting process. Note that the model accuracy is very poor, especially for the responses where the largest causality violations have been detected. 38

39 Effects of causality violations Causality violations in the data affect the fitting. The approximation error does not decrease even if a very large number of poles is used. Output model is stable passive causal not accurate 39

40 Effects of causality violations Removing the stability condition during the fitting (allowing poles in the right half plane), the fitting converges with good accuracy, but the final model is unstable. Max Err:.6 Output model is accurate not stable not causal not passive S. Grivet-Talocia and P. Triverio, Modeling and Simulation of High-Speed Interconnects: Approaches, Challenges and Solutions, SPI 2 Tutorial, 9 May 2, Hildesheim, Germany 4

41 4 S.Grivet-Talocia, Qualification of tabulated scattering parameters, IBIS Summit, 2 May 2, Hildesheim, Germany Sampling: a trivial case C ) ( ω α ω = s s s s S R = reference resistance L L R LC 2 = = α ω 2nH 5pF = = L C

42 Sampling: a trivial case S 2, magnitude C L R C L = 5Ω = 5pF = 2nH Frequency, GHz 42

43 Sampling: a trivial case S 2, magnitude C L R C L =Ω = 5pF = 2nH Frequency, GHz 43

44 Sampling: a trivial case S 2, magnitude C L S 2, magnitude R =. Ω C L = 5pF = 2nH Frequency, GHz Frequency, GHz 44

45 Sampling: a trivial case How do we expect a solver will interpret the blue samples? S 2, magnitude Frequency, GHz 45

46 Detecting undersampled data via causality check undersampled well sampled Causality Error (db), Check n.2 causality violation detected Causality Error (db), Check n.2 no causality violations detected Frequency x Frequency x 9 46

47 Conclusions Tabulated S-parameter data may hide serious issues Passivity (energy gain) violations Easily checked at individual frequencies (singular value test) Causality violations Can be detected using Generalized Hilbert Transform Theoretically sound Robust numerical implementation Bad or insufficient sampling Also detectable via Generalized Hilbert Transform If not detected (and corrected) Any of these issues will lead to problems in simulation With any tool or method (see SPI 2 tutorial) 47

48 Thank you