Circuit Synthesizable Guaranteed Passive Modeling for Multiport Structures

Transcription

1 Cicuit Synthesizable Guaanteed Passive Modeling fo Multipot Stuctues Zohaib Mahmood, Luca Daniel Massachusetts Institute of Technology BMAS Septembe-23, 2010

2 Outline Motivation fo Compact Dynamical Passive Modeling What is Passivity? Existing Techniques Rational Fitting of Tansfe Functions Results 2

3 Outline Motivation fo Compact Dynamical Passive Modeling What is Passivity? Existing Techniques Rational Fitting of Tansfe Functions Results 3

4 Motivation fo Model Geneation Electomagnetic Field Solve Cicuit Simulato (Time Domain Simulations) 4

5 Motivation fo Model Geneation Step 1: Field Solves OR Measuements Fequency Response Samples (H i ) S/Z-Paametes Step 2 : Samples Tansfe Function Matix Z H ( s) = Z 11 ( s). N1 ( s) Z Z 1N NN ( s). ( s) H (s) must be PASSIVE Cicuit Simulato (Time Domain Simulations) 5

6 Outline Motivation fo Compact Dynamical Passive Modeling What is Passivity? Existing Techniques Rational Fitting of Tansfe Functions Results 6

7 What is a Passive Netwo/Model? DEFINITION Passivity is the inability of a system (o model) to geneate enegy All physical systems dissipate enegy, and ae theefoe passive Fo numeical models of such systems, this is not guaanteed unless enfoced Passivity fo an impedance (o admittance) matix is implied by positive ealness. 7

8 Conditions fo Passivity Conditions fo Passivity (Hybid Paametes) H ˆ ( s) is passive iff: Hˆ( s) = Hˆ() s Condition 1 Conjugate Symmety Condition 2 Stability Condition 3 Non-negativity Hˆ ( s ) is analytic in R { s } > 0 ˆ ˆ Ψ ( jω) = H( jω) + H( jω) ± 0 ω Real impulse esponse All poles in left half plane Ψ( jω) Non-negative eigen values of foall ω 8

9 Conditions fo Passivity Conditions fo Passivity (Hybid Paametes) H ˆ ( s) is passive iff: Hˆ( s) = Hˆ() s Condition 1 Conjugate Symmety Condition 2 Stability Condition 3 Non-negativity Hˆ ( s ) is analytic in R { s } > 0 ˆ ˆ Ψ ( jω) = H( jω) + H( jω) ± 0 ω Real impulse esponse All poles in left half plane Ψ( jω) Non-negative eigen values of foall ω 9

10 Conditions fo Passivity Conditions fo Passivity (Hybid Paametes) H ˆ ( s) is passive iff: Hˆ( s) = Hˆ() s Condition 1 Conjugate Symmety Condition 2 Stability Condition 3 Non-negativity Hˆ ( s ) is analytic in R { s } > 0 ˆ ˆ Ψ ( jω) = H( jω) + H( jω) ± 0 ω Real impulse esponse All poles in left half plane Ψ( jω) Non-negative eigen values of foall ω 10

11 11 Manifestation of Passivity Multi Pot Case n n n n n n s X j s R s Z + = )] ( [ )] ( [ )] ( [ ) ( ) ( ) ( ) (,,1 1, 1,1 s R s R s R s R n n n n

12 Multi Pot Case Manifestation of Passivity [ Z ( s)] = [ R( s)] + j[ X ( s)] n n n n n n R R 1,1 n,1 ( s) ( s) R R 1, n, n( s) ( s) n -Fequency dependent eal matix must be positive semidefinite fo all fequencies -Popety of entie matix, cannot enfoce element-wise 12

13 What if passivity is not peseved -Cicuit simulation may blow-up -Simulato convegence issues -Results may become completely non-physical. o... passive model +... non-passive model (time domain simulation) 13

14 Outline Motivation fo Compact Dynamical Passive Modeling What is Passivity? Existing Techniques Rational Fitting of Tansfe Functions Results 14

15 Designes way aound -- Analytic / Intuitive Appoaches RL/RC Netwos chaacteized at opeating fequency Develop RLC Netwo fom intuition 15

16 Numeical Appoaches Technique Pos Cons Pojection appoaches e.g. PRIMA [Odabasioglu 1997] Vecto Fitting [Gustavsen 1999] Pole discading appoaches [Mosey 2001] Petubation based appoaches [Talocia 2004, Gustavsen 2008] Optimization based appoaches [Suo 2008] Passivity peseved Efficient, Robust Passivity enfoced Passivity enfoced Passivity enfoced Does not wo with fequency esponse data. Passivity not peseved Highly estictive, non-passive pole-esidues ae discaded Two step pocess. Final models may lose accuacy and optimality Computationally expensive, fequency dependent constaints Ou Appoach: Enfoce passivity duing identification, using efficient optimization famewo 16

17 Numeical Appoaches Technique Pos Cons Pojection appoaches e.g. PRIMA [Odabasioglu 1997] Vecto Fitting [Gustavsen 1999] Pole discading appoaches [Mosey 2001] Petubation based appoaches [Talocia 2004, Gustavsen 2008] Optimization based appoaches [Suo 2008] Passivity peseved Efficient, Robust Passivity enfoced Passivity enfoced Passivity enfoced Does not wo with fequency esponse data. Passivity not peseved Highly estictive, non-passive pole-esidues ae discaded Two step pocess. Final models may lose accuacy and optimality Computationally expensive, fequency dependent constaints Ou Appoach: Enfoce passivity duing identification, using efficient optimization famewo 17

18 Outline Motivation fo Compact Dynamical Passive Modeling What is Passivity? Existing Techniques Rational Fitting of Tansfe Functions Results 18

19 Poblem Statement Given fequency esponse samples { ω i, H i } Seach fo optimal passive ational appoximation in the pole esidue fom Hˆ () s κ = + = 1 s R a D 19

20 Poblem Statement Given fequency esponse samples { ω i, H i } Seach fo optimal passive ational appoximation in the pole esidue fom Hˆ () s κ = + = 1 s R a D Fomulate as optimization poblem L : min H Hˆ ( s) 2 pq, 2 i i pq, L : min max H Hˆ ( s) i i Subject to: H ˆ ( s) PASSIVE 20

21 Poblem Statement Given fequency esponse samples { ω i, H i } Seach fo optimal passive ational appoximation in the pole esidue fom Hˆ () s κ = + = 1 s R a D Fomulate as optimization poblem L : min H Hˆ ( s) 2 pq, 2 i i pq, L : min max H Hˆ ( s) i i Subject to: H ˆ ( s) PASSIVE 21

22 Convex Optimization Poblems Non-convex poblems difficult to solve min H Hˆ ( s) pq, i i 2 Non-convex function (finding global minimum-extemely difficult) Subject to: ˆ ( ) : PASSIVE H s Must efomulate as convex optimization poblem Convex objective function Convex constaints Convex function (finding global minimum-easy) 22

23 Modeling Flow Step 1: Identify a COMMON set of STABLE poles -Vecto Fitting Algoithm [Gustavsen 1999] - Optimization Famewo [Suo 2008] Step 2: Use POLES fom step 1 to identify passive model min H Hˆ ( s) pq, i i 2 Subject to: ˆ ( ) : PASSIVE H s 23

24 Poblem Fomulation Hˆ ( jω ) κ R = + D jω a = 1 κ κc / 2 ˆ ω = 1 = 1 ( ) ˆ c = H j + H ( jω) + D κ κc /2 c c c c R R + ji R ji = R + R R R + c c c c + D = 1 jω a = 1 jω Ra ji a jω Ra + jia 24

25 Poblem Fomulation Hˆ ( jω ) κ R = + D jω a = 1 κ κc / 2 ˆ ω = 1 = 1 ( ) ˆ c = H j + H ( jω) + D κ κc /2 c c c c R R + ji R ji = R + R R R + c c c c + = 1 jω a = 1 jω Ra ji a jω Ra + jia D seies/paallel inteconnection of fist ode netwos seies/paallel inteconnection of second ode netwos esistive/conductive netwo 25

26 Poblem Fomulation Hˆ ( jω ) κ R = + D jω a = 1 κ κc / 2 ˆ ω = 1 = 1 ( ) ˆ c = H j + H ( jω) + D κ κc /2 c c c c R R + ji R ji = R + R R R + c c c c + = 1 jω a = 1 jω Ra ji a jω Ra + jia D seies/paallel inteconnection of fist ode netwos Passivity Conditions: seies/paallel inteconnection of second ode netwos esistive/conductive netwo Condition 1 Conjugate Symmety: Enfoced by constuction Condition 2 Stability: Enfoced duing pole-identification Condition 3 Non-negativity: Enfoced on the building blocs 26

27 Passivity Conditions κ κc / 2 D = 1 = 1 ˆ( ) ˆ ( ) ˆ c H jω = H jω + H ( jω) + A sufficient condition fo passivity: ˆ ( ), ˆ c H jω H ( jω), D passive Hˆ( jω) passive ˆ ( ) 0, ˆ c RH jω ± RH ( jω) ± 0, D ± 0 RHˆ( jω) ± 0 27

28 Real-only poles ˆ H ( jω ) = R jω a ˆ a R H ( j ) j ω = ω + a ω + a a Passivity Conditions ˆ( ) ˆ ( ) ˆ c H jω = H jω + H ( jω) + ωr ˆ a R H ( jω ) 0 R = ± R ± 0 ω κ κc / 2 D = 1 = 1 28

29 Real-only poles ˆ H ( jω ) = R jω a ˆ a R H ( j ) j ω = ω + a ω + a a Passivity Conditions ˆ( ) ˆ ( ) ˆ c H jω = H jω + H ( jω) + ωr ˆ a R H ( jω ) 0 R = ± R ± 0 ω κ κc / 2 D = 1 = 1 - Diect Matix D ± 0 29

30 Passivity Conditions Real-only poles Complex poles ˆ c RH ( jω ) ± 0. ˆ H ( jω ) = ˆ a R H ( j ) j ω = ω + a ω + a ˆ a R H ( jω ) 0 R = ± R ± 0 ω ω 0 c c c c 2 c c c c ( R R) + ω ( R + R). CPR( ω) = Ra R Ia I Ra R Ia I ± 0, ω. c c c c.lim CPR( ω) Ra RR Ia IR ± 0 c c c c. lim CPR( ω) Ra RR + Ia IR ± 0 ω ˆ( ) ˆ ( ) ˆ c H jω = H jω + H ( jω) + R jω a a κ κc / 2 D = 1 = 1 ωr - Diect Matix D ± 0 30

31 Passivity Conditions Real-only poles Complex poles ˆ c RH ( jω ) ± 0. ˆ H ( jω ) = ˆ a R H ( j ) j ω = ω + a ω + a ar H jω R 2 2 ω + a R ˆ ( ) = ± 0 ± 0 ω 0 c c c c 2 c c c c ( R R) + ω ( R + R). CPR( ω) = Ra R Ia I Ra R Ia I ± 0, ω. c c c c.lim CPR( ω) Ra RR Ia IR ± 0 c c c c. lim CPR( ω) Ra RR + Ia IR ± 0 ω ˆ( ) ˆ ( ) ˆ c H jω = H jω + H ( jω) + R jω a κ κc / 2 D = 1 = 1 ωr - Diect Matix D ± 0 Linea Matix Inequalities (Extemely efficient) Unnowns 31

32 Convex fomulation minimize RH R Hˆ( jω ) + IH IHˆ( jω ) c R, R, D subject to D ± 0 R ± 0 = 1,..., κ 2 2 i i i i i c c c c Ra R R + Ia IR ± 0 = 1,..., κ c c c c Ra RR Ia IR ± 0 = 1,..., κ whee Hˆ ( jω ) i κ κc = 1 ω = 1 c c c R R = + + D j a jω a c Linea Matix Inequalities (Extemely efficient) 32

33 Final Optimization Poblem Samples { ω i, H i }, Desied numbe of poles (N) Find N Stable poles a a minimize RH R Hˆ( jω ) + IH IHˆ( jω ) c R, R, D i 2 2 i i i i i subject to: D± 0, R ± 0 = 1,..., κ c c c c Ra R R + Ia IR ± 0 = 1,..., κ c c c c c Ra RR Ia IR ± 0 = 1,..., κ c Convex Optimization Poblem c c ˆ R R H ( j ) ω = + + c j a jω a a κ κ = 1 ω = 1 c, R, R, D D Cicuit Module (Netlist/ VeilogA Dynamical Model) 33

34 Outline Motivation fo Compact Dynamical Passive Modeling What is Passivity? Existing Techniques Rational Fitting of Tansfe Functions Results 34

35 Results: LINC Powe Amplifie Bloc diagam of the LINC powe amplifie achitectue 35

36 Results: LINC Powe Amplifie Bloc diagam of the LINC powe amplifie achitectue Layout of the Wilinson Combine 36

37 Results: LINC Powe Amplifie Bloc diagam of the LINC powe amplifie achitectue Layout of the Wilinson Combine -Numbe of poles 10 -Time [Matlab] 2 seconds -Eo < 0.7% EM Field Solve (dots) Tansfe Function Matix Z11( s)... Z1N ( s) H() s =... ZN 1( s)... ZNN ( s) Passive Modeling Algoithm (solid lines) Compaing eal and imaginay pats of the impedance paametes fom EM field solve (dots) and ou passive model (solid lines) 37

38 Results: LINC Powe Amplifie Hamiltonian Matix Based Passivity Test Model is passive if the associated Hamiltonian matix has no puely imaginay eigen value Tansfe Function Matix Z11( s)... Z1N ( s) H() s =... ZN 1( s)... ZNN ( s) Zoomed-in eigen values of the associated Hamiltonian matix fo the identified model of Wilinson combine 38

39 Results: LINC Powe Amplifie Bloc diagam of the LINC powe amplifie as simulated inside the cicuit simulato Z11( s)... Z1N ( s) H() s =... ZN 1( s)... ZNN ( s) Passive Tansfe Matix 39

40 Results: LINC Powe Amplifie Bloc diagam of the LINC powe amplifie as simulated inside the cicuit simulato Z11( s)... Z1N ( s) H() s =... ZN 1( s)... ZNN ( s) Passive Tansfe Matix LINC PA Nomalized Input Voltage (v in ) Nomalized Output Voltage (v out ) 40

41 Example: 8-Pot Powe/Gnd Distibution Gid QUALITY CHECK (Impedance matix) - Hamiltonian Matix Based Passivity Test Passed -Numbe of poles 20 -Time [Matlab] 74 seconds 41

42 Example: 4-Pot Inducto Aay QUALITY CHECK (Impedance matix) - Hamiltonian Matix Based Passivity Test Passed -Numbe of poles 23 -Time [Matlab] 72 seconds 42

43 Compaison Stuctue Wilinson Combine Numbe of Pots Numbe of Poles [Suo 2008] Time 1 (seconds) This Wo [Matlab] Powe Distibution Gid Coupled RF inductos NA NA Laptop: Coe2Duo 2.1GHz, 3GB, Windows 7 43

44 Conclusions Summaized how to develop models fom feq. esponse Poposed a Convex Optimization based modeling algoithm Demonstated odes of magnitude impovement ove simila techniques Pesented an example demonstating system level simulation of analog cicuits 44

45 THANK YOU 45