Enforcement Passivity. Frequency Data. Wenliang Tseng, Sogo Hsu, Frank Y.C. Pai and Scott C.S. Li. Asian IBIS Summit, Taipei, Taiwan November 12, 2010

Transcription

1 Enforcement Passivity of S-parameter S Sampled Frequency Data Wenliang Tseng, Sogo Hsu, Frank Y.C. Pai and Scott C.S. Li Asian IBIS Summit, Taipei, Taiwan November 12, 2010

2 Agenda Causality and passivity fundamentals Enforcing passivity for sampled frequency data S-parameter rational function approximation A new format of S-parameter rational function Summary FOXCONN Page - 2

3 S-parameter Model IBIS S-parameter Touchstone file (1) From measurement (2) From EDA tool synthesis IBIS # Hz S RI R 50 Touchstone file description S ij (f) 1.000e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e-4 S-parameter raw data 1.962e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e Frequency S ij (f k ) f k frequency f max FOXCONN Page - 3

4 Impact of Non-Causality Non-transition Input Causality violation Correct response Non-causal S-parameterS FOXCONN Page - 4

5 Impact of Non-Passivity Correct response Non-passive S-parameterS failure to converge! FOXCONN Page - 5

6 Causality & Passivity Conditions Causality Theorem Theorem: An LTI system is causal if and only if all the elements response matrix h(t) are vanishing for t<0,, i.e., h( t) 0, t 0. (t) h ij of its impulse Passivity Theorem Theorem: An scattering matrix S(s) represents a passive linear system iff * * 1. S( s ) S ( s) where * is the complex conjugate operator 2. each element of S(s) is analytic in Re{ s} 0 H 3. [ I S ( s) S( s)] 0. for all. FOXCONN Page - 6

7 Causality & Passivity Conditions Theorem: If h(t) admits a Fourier transform, the following facts are equivalent. 1. h( t) 0, t H ( j ) is the limit, as 0, of a function H (s) defined in Re{ s} 0 and here analytic and of polynomial growth. 3. H ( j ) F{ h( t)} satisfies Kramers-Kronig Kronig relations. Theorem: If an LTI system is passive, then it is also causal. FOXCONN Page - 7

8 Passivity Condition 1 & 2 Passivity Condition 1: S( ) S * ( ) S( ) S1( ) js2( ) S1( ) S1( ) S2( ) S2( ) even function odd function Passivity Condition 2: S( ) satisfies Kramers-Kronig Kronig relations. S( ) S1( ) js2( ) S 1 ) pv S ( ') d ' 2 1( ' S 1 ) pv S ( ') d ' 1 2( ' S 1 ( ) IFT s ( ) even function t 1 t equivalent to S 2 ( ) FT odd function t s 2( 1 t t) sgn( t) s ( ) FOXCONN Page - 8

9 Passivity Condition 3 Check the eigenvalue of Eigenvalue of [I-S H S] I-S H S Check the singular value of S H S U V H [ I S S] 0 U 1 SV Singular value of S frequency Leads to Passivity violation FOXCONN Page frequency Leads to Passivity violation

10 Enforcing Passivity for S-Parameter Raw Data Currently the methodologies of enforcing passivity focused on Passivity Condition 3 only. Simple method (E.D. Campbell, IEEE ICCE 2010) Singular value of S 1 raw data Enforcing Passivity Assign singular value to 1! Perturbation method (M.S. Nakhla, IEEE MTT-S 2005) Singular value of S 1 raw data 1 2 frequency Enforcing Passivity Assign singular value by optimization! FOXCONN Page frequency

11 Enforcing Passivity Singular Value Results Simple Method Perturbation Method Raw data Enforcing passivity Raw data Enforcing passivity FOXCONN Page - 11

12 Enforcing Passivity S-Parameter Results Simple Method Perturbation Method Raw data Enforcing passivity Raw data Enforcing passivity FOXCONN Page - 12

13 Transient Response Correct Response By Sample method By Perturbation method FOXCONN Page - 13

14 Rational Function Approximation # Hz S RI R e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e-5 S-parameter e e e e-5 raw e-4 4 data 9.901e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e Frequency Touchstone file description S(f) Rational Function Approximation by Vector Fitting Algorithm S(f k ) f k frequency f max f ( s) N n 1 rn s p n d se FOXCONN Page - 14

15 Rational Function Passivity Conditions If S(s) is a rational function, the passivity condition 1 is satisfied. If S(s) is a rational function of s, the passivity condition 3 implies the analyticity of S(s) for Re{ s} 0. Passivity Theorem for rational function Theorem: A rational function of scattering matrix S(s) represents a passive linear system iff H [ I S ( s) S( s)] 0. for all. FOXCONN Page - 15

16 Perturbation Method with Rational Function Enforcing passivity: S S S M Rm s p m 1 m D Perturbation of eigenvalue Rm R P m m( R R ) P m m s p s p (by B. Gustavsen,, 2010 IEEE Trans. on Advanced Packaging) Target S 0 i i 1 for i 1...n m m T m Singular value of S 1 raw data Enforcing Passivity Assign singular value by perturbation of eigenvalue! 1 2 frequency FOXCONN Page - 16

17 Enforcing Passivity for S-Parameter Rational Function Singular values S-parameter Raw data Enforcing passivity Raw data Enforcing passivity FOXCONN Page - 17

18 Transient Response Correct Response Enforcing passivity FOXCONN Page - 18

19 Proposed Common-Pole Rational Function Standard File S Rr N n M i, j Rc m Rc m * i, j Di, j j Ei, j ( ) n 1 s prn m 1 s pcm s pcm * R Zo1 Zo2 Zo3 Zo4! reference resistor of ports Common pole Real N pr1 pr2 Complex M Re( 1) prn! Common pole definition! Real pole definition! Real pole data! Complex pole definition pc Re( pc 2) Re( pcm )! Complex pole real part data Im( pc 1 ) Im( pc 2 ) Im( pc )! Complex pole imagery part data M i j! Element definition, i is row and j is column DC D ij! DC element definition E E ij! E element definition Real residue N! Real residue definition 1 2 Rr N ij Rr! Real residue data ij Rr ij Complex residue M! Complex residue definition 1 2 Re( Rc ij ) M Re( Rc ) Re( Rc )! Complex residue real part data ij ij 2 Im( Rc ) M Im( Rc ) Im( )! Complex residue imagery part data 1 ij ij Rc ij FOXCONN Page - 19

20 Advantage of Rational Function Standard File Present S-parameter on full spectrum. Causality and passivity guarantee after enforcing passivity. Size reduction of S-parameter raw data. Type No. of Sampled data No. of port No. of pole-residue Size of Touchstone file Size of proposed file Size reduced to Via M 6.88K 0.41% Connector M K 3.63% Trace M 1.35M 1.28% FOXCONN Page - 20

21 Summary The traditional enforcing passivity for S-parameter raw date are not sufficient to satisfy the passivity conditions. Using enforcing passivity rational function to fit S-parameter raw data could be causality and passivity guarantee on full spectrum. The proposed rational function standard file has the advantage in size reduction of S-parameter sampled frequency data. FOXCONN Page - 21

22 Reference [1] Piero Triverio, S. Grivet-Talocia, M.S. Nakhla, F.G. Canavero and R. Achar, "Stability, Causality, and Passivity in Electrical Interconnect Models," IEEE TRANS. on ADVANCED PACKAGING, VOL. 30, NO. 4, NOVEMBER [2] B. Gustavsen and A. Semlyen, Rational approximation of frequency domain responses by vector fitting, IEEE Trans. Power Delivery, vol. 14, no. 3, July 1999, pp [3] Altan Odabasioglu, Mustafa Celik and Lawrence T. Pileggi,"PRIMA: Passive Reduced-Order Interconnect Macromodeling Algorithm," IEEE TRANS. on COMPUTER-AIDED DESIGN OF INTEGRATED CIRCUITS AND SYSTEMS,, VOL. 17, NO.8, Aug FOXCONN Page - 22

23 Thank you for your attention. FOXCONN Page - 23