Circuit Synthesizable Guaranteed Passive Modeling for Multiport Structures Zohaib Mahmood, Luca Daniel Massachusetts Institute of Technology BMAS September-23, 2010
Outline • Motivation for Compact Dynamical Passive Modeling • What is Passivity? • Existing Techniques • Rational Fitting of Transfer Functions • Results 2
Outline • Motivation for Compact Dynamical Passive Modeling • What is Passivity? • Existing Techniques • Rational Fitting of Transfer Functions • Results 3
Motivation for Model Generation Electromagnetic Field Solver Circuit Simulator (Time Domain Simulations) 4
Motivation for Model Generation Step 1: Field Solvers OR Measurements Frequency Response Samples (H i ) S/Z-Parameters 2 : Step Samples Transfer Function Matrix ( s ) ( ) ... ( ) Z s Z s H must be 11 1 N = ( ) . . . H s PASSIVE ( ) ... ( ) Z s Z s 1 N NN Circuit Simulator (Time Domain Simulations) 5
Outline • Motivation for Compact Dynamical Passive Modeling • What is Passivity? • Existing Techniques • Rational Fitting of Transfer Functions • Results 6
What is a Passive Network/Model? DEFINITION Passivity is the inability of a system (or model) to generate energy • All physical systems dissipate energy, and are therefore passive • For numerical models of such systems, this is not guaranteed unless enforced • Passivity for an impedance (or admittance) matrix is implied by ‘positive realness’. 7
Conditions for Passivity Conditions for Passivity ( Hybrid Parameters ) ˆ ( ) is passive iff: H s ˆ ˆ = ( ) ( ) H s H s ˆ ( ) is analytic in { } ℜ > 0 H s s Ψ ω = ˆ ω + ˆ ω ∀ ω † ( ) ( ) ( ) ± 0 j H j H j ⇔ Condition 1 – Conjugate Symmetry Real impulse response ⇔ Condition 2 – Stability All poles in left half plane Ψ j ω ⇔ ( ) Condition 3 – Non-negativity Non-negative eigen values of ω forall 8
Conditions for Passivity Conditions for Passivity ( Hybrid Parameters ) ˆ ( ) is passive iff: H s ˆ ˆ = ( ) ( ) H s H s ˆ ( ) is analytic in { } ℜ > 0 H s s Ψ ω = ˆ ω + ˆ ω ∀ ω † ( ) ( ) ( ) ± 0 j H j H j ⇔ Condition 1 – Conjugate Symmetry Real impulse response ⇔ Condition 2 – Stability All poles in left half plane Ψ j ω ⇔ ( ) Condition 3 – Non-negativity Non-negative eigen values of ω forall 9
Conditions for Passivity Conditions for Passivity ( Hybrid Parameters ) ˆ ( ) is passive iff: H s ˆ ˆ = ( ) ( ) H s H s ˆ ( ) is analytic in { } ℜ > 0 H s s Ψ ω = ˆ ω + ˆ ω ∀ ω † ( ) ( ) ( ) ± 0 j H j H j ⇔ Condition 1 – Conjugate Symmetry Real impulse response ⇔ Condition 2 – Stability All poles in left half plane Ψ j ω ⇔ ( ) Condition 3 – Non-negativity Non-negative eigen values of ω forall 10
Manifestation of Passivity • Multi Port Case = + [ ( )] [ ( )] [ ( )] Z s R s j X s × × × n n n n n n ( ) ( ) R s R s 1 , 1 1 , n ( ) ( ) R s R s , 1 , n n n 11
Manifestation of Passivity • Multi Port Case = + [ ( )] [ ( )] [ ( )] Z s R s j X s × × × n n n n n n -Frequency dependent real matrix ( ) ( ) R s R s must be positive semidefinite for all 1 , 1 1 , n frequencies ( ) ( ) R s R s -Property of entire matrix, cannot , 1 , n n n enforce element-wise 12
What if passivity is not preserved -Circuit simulation may blow-up -Simulator convergence issues -Results may become completely non-physical. o ... passive model +... non-passive model (time domain simulation) 13
Outline • Motivation for Compact Dynamical Passive Modeling • What is Passivity? • Existing Techniques • Rational Fitting of Transfer Functions • Results 14
Designers’ way around -- Analytic / Intuitive Approaches • RL/RC Networks characterized at operating frequency • Develop RLC Network from intuition 15
Numerical Approaches Technique Pros Cons Projection approaches Passivity Does not work with frequency e.g. PRIMA preserved response data. [Odabasioglu 1997] Vector Fitting Efficient, Robust Passivity not preserved [Gustavsen 1999] Pole discarding approaches Passivity Highly restrictive, non-passive [Morsey 2001] enforced pole-residues are discarded Perturbation based approaches Passivity Two step process. Final models [Talocia 2004, Gustavsen 2008] enforced may lose accuracy and optimality Optimization based approaches Passivity Computationally expensive, [Suo 2008] enforced frequency dependent constraints Our Approach: Enforce passivity during identification, using efficient optimization framework 16
Numerical Approaches Technique Pros Cons Projection approaches Passivity Does not work with frequency e.g. PRIMA preserved response data. [Odabasioglu 1997] Vector Fitting Efficient, Robust Passivity not preserved [Gustavsen 1999] Pole discarding approaches Passivity Highly restrictive, non-passive [Morsey 2001] enforced pole-residues are discarded Perturbation based approaches Passivity Two step process. Final models [Talocia 2004, Gustavsen 2008] enforced may lose accuracy and optimality Optimization based approaches Passivity Computationally expensive, [Suo 2008] enforced frequency dependent constraints Our Approach: Enforce passivity during identification, using efficient optimization framework 17
Outline • Motivation for Compact Dynamical Passive Modeling • What is Passivity? • Existing Techniques • Rational Fitting of Transfer Functions • Results 18
Problem Statement { } ω , i H • Given frequency response samples i κ Search for optimal R ∑ ˆ ( ) = + k H s D passive rational − s a approximation in the = 1 k k pole residue form 19
Problem Statement { } ω , i H • Given frequency response samples i κ Search for optimal R ∑ ˆ ( ) = + k H s D passive rational − s a approximation in the = 1 k k pole residue form • Formulate as optimization problem ∑ 2 − ˆ − ˆ : min ( ) L H H s : min max ( ) L H H s ∞ 2 i i , p q , p q i i ˆ ( ) H s PASSIVE Subject to: 20
Problem Statement { } ω , i H • Given frequency response samples i κ Search for optimal R ∑ ˆ ( ) = + k H s D passive rational − s a approximation in the = 1 k k pole residue form • Formulate as optimization problem ∑ 2 − ˆ − ˆ : min ( ) L H H s : min max ( ) L H H s ∞ 2 i i , p q , p q i i ˆ ( ) H s PASSIVE Subject to: 21
Convex Optimization Problems • Non-convex problems difficult to solve Non-convex function (finding global minimum-extremely difficult) ∑ 2 − ˆ min ( ) H H s i , p q i ˆ ( ) : PASSIVE H s Subject to: Convex function (finding global minimum-easy) • Must reformulate as convex optimization problem – Convex objective function – Convex constraints 22
Modeling Flow -Vector Fitting Algorithm Step 1: Identify a COMMON set of [Gustavsen 1999] STABLE poles - Optimization Framework [Suo 2008] ∑ 2 Step 2: Use POLES from step 1 ˆ − min ( ) H H s to identify passive model i p q , i ˆ ( ) : PASSIVE H s Subject to: 23
Problem Formulation κ R ∑ ˆ ( ω = + k ) H j D ω − j a = 1 k k κ κ / 2 ∑ ∑ r c = ˆ ω + ˆ ω + r c ( ) ( ) H j H j D k k = = 1 1 k k κ κ ℜ + ℑ ℜ − ℑ /2 r c c c c R R j R R j R ∑ ∑ r c = + + + k k k k k D ω − ω −ℜ − ℑ ω − ℜ + ℑ r c c c c j a j a j a j a j a = = 1 1 k k k k k k k 24
Problem Formulation κ R ∑ ˆ ( ω = + k ) H j D ω − j a = 1 k k κ κ / 2 ∑ ∑ r c = ˆ ω + ˆ ω + r c ( ) ( ) H j H j D k k = = 1 1 k k κ κ ℜ + ℑ ℜ − ℑ /2 r c c c c R R j R R j R ∑ ∑ r c = + + + k k k k k D ω − ω −ℜ − ℑ ω − ℜ + ℑ r c c c c j a j a j a j a j a = = 1 1 k k k k k k k series/parallel interconnection series/parallel interconnection resistive/conductive of first order networks of second order networks network 25
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