Circuit Synthesizable Guaranteed Passive Modeling for Multiport - - PowerPoint PPT Presentation

Circuit Synthesizable Guaranteed Passive Modeling for Multiport Structures

Zohaib Mahmood, Luca Daniel Massachusetts Institute of Technology BMAS September-23, 2010

2

Outline

• Motivation for Compact Dynamical Passive Modeling
• What is Passivity?
• Existing Techniques
• Rational Fitting of Transfer Functions
• Results

3

Outline

• Motivation for Compact Dynamical Passive Modeling
• What is Passivity?
• Existing Techniques
• Rational Fitting of Transfer Functions
• Results

4

Motivation for Model Generation

Circuit Simulator (Time Domain Simulations) Electromagnetic Field Solver

5

Motivation for Model Generation

Step 1: Field Solvers OR Measurements Frequency Response Samples (Hi) S/Z-Parameters Circuit Simulator (Time Domain Simulations)

          = ) ( ... ) ( . . . ) ( ... ) ( ) ( : 2

1 1 11

s Z s Z s Z s Z s H Matrix Function Transfer Samples Step

NN N N

 

) (s H 

must be

PASSIVE

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Outline

• Motivation for Compact Dynamical Passive Modeling
• What is Passivity?
• Existing Techniques
• Rational Fitting of Transfer Functions
• Results

7

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’.

8

Conditions for Passivity Conditions for Passivity (Hybrid Parameters)

Condition 1 – Conjugate Symmetry Real impulse response Condition 2 – Stability All poles in left half plane Condition 3 – Non-negativity Non-negative eigen values of forall

†

ˆ ˆ ( ) ( ) ˆ ( ) is analytic in { } ˆ ˆ ( ) ( ) ( ) H s H s H s s j H j H j ω ω ω ω = ℜ > Ψ = + ∀ ± ⇔ ⇔ ⇔

( ) jω Ψ ω

ˆ ( ) is passive iff: H s

9

Conditions for Passivity Conditions for Passivity (Hybrid Parameters)

Condition 1 – Conjugate Symmetry Real impulse response Condition 2 – Stability All poles in left half plane Condition 3 – Non-negativity Non-negative eigen values of forall

†

ˆ ˆ ( ) ( ) ˆ ( ) is analytic in { } ˆ ˆ ( ) ( ) ( ) H s H s H s s j H j H j ω ω ω ω = ℜ > Ψ = + ∀ ± ⇔ ⇔ ⇔

( ) jω Ψ ω

ˆ ( ) is passive iff: H s

10

Conditions for Passivity Conditions for Passivity (Hybrid Parameters)

Condition 1 – Conjugate Symmetry Real impulse response Condition 2 – Stability All poles in left half plane Condition 3 – Non-negativity Non-negative eigen values of forall

†

ˆ ˆ ( ) ( ) ˆ ( ) is analytic in { } ˆ ˆ ( ) ( ) ( ) H s H s H s s j H j H j ω ω ω ω = ℜ > Ψ = + ∀ ± ⇔ ⇔ ⇔

( ) jω Ψ ω

ˆ ( ) is passive iff: H s

11

Manifestation of Passivity

• Multi Port 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

    

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Manifestation of Passivity

• Multi Port Case
• Frequency dependent real matrix

must be positive semidefinite for all frequencies

• Property of entire matrix, cannot

enforce element-wise

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

    

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What if passivity is not preserved

• Circuit simulation may blow-up
• Simulator convergence issues
• Results may become completely

non-physical.

• ... passive model

+... non-passive model (time domain simulation)

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Outline

• Motivation for Compact Dynamical Passive Modeling
• What is Passivity?
• Existing Techniques
• Rational Fitting of Transfer Functions
• Results

15

Designers’ way around -- Analytic / Intuitive Approaches

• RL/RC Networks characterized at operating frequency
• Develop RLC Network from intuition

16

Numerical Approaches

Our Approach: Enforce passivity during identification, using efficient optimization framework

Technique Pros Cons

Projection approaches e.g. PRIMA [Odabasioglu 1997] Passivity preserved Does not work with frequency response data. Vector Fitting [Gustavsen 1999] Efficient, Robust Passivity not preserved Pole discarding approaches [Morsey 2001] Passivity enforced Highly restrictive, non-passive pole-residues are discarded Perturbation based approaches [Talocia 2004, Gustavsen 2008] Passivity enforced Two step process. Final models may lose accuracy and optimality Optimization based approaches [Suo 2008] Passivity enforced Computationally expensive, frequency dependent constraints

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Numerical Approaches

Our Approach: Enforce passivity during identification, using efficient optimization framework

Technique Pros Cons

Projection approaches e.g. PRIMA [Odabasioglu 1997] Passivity preserved Does not work with frequency response data. Vector Fitting [Gustavsen 1999] Efficient, Robust Passivity not preserved Pole discarding approaches [Morsey 2001] Passivity enforced Highly restrictive, non-passive pole-residues are discarded Perturbation based approaches [Talocia 2004, Gustavsen 2008] Passivity enforced Two step process. Final models may lose accuracy and optimality Optimization based approaches [Suo 2008] Passivity enforced Computationally expensive, frequency dependent constraints

18

Outline

• Motivation for Compact Dynamical Passive Modeling
• What is Passivity?
• Existing Techniques
• Rational Fitting of Transfer Functions
• Results

19

Problem Statement

• Given frequency response samples

1

ˆ ( )

k k k

R H s D s a

κ =

= + −

∑

{ }

i i H

, ω

Search for optimal passive rational approximation in the pole residue form

20

Problem Statement

• Given frequency response samples

2 2 ,

ˆ : min ( )

i p q i

L H H s −

∑

,

ˆ : min max ( )

i p q i

L H H s

∞

− PASSIVE s H ) ( ˆ

1

ˆ ( )

k k k

R H s D s a

κ =

= + −

∑

{ }

i i H

, ω

Search for optimal passive rational approximation in the pole residue form

• Formulate as optimization problem

Subject to:

21

Problem Statement

• Given frequency response samples

2 2 ,

ˆ : min ( )

i p q i

L H H s −

∑

,

ˆ : min max ( )

i p q i

L H H s

∞

− PASSIVE s H ) ( ˆ

1

ˆ ( )

k k k

R H s D s a

κ =

= + −

∑

{ }

i i H

, ω

Search for optimal passive rational approximation in the pole residue form

• Formulate as optimization problem

Subject to:

22

Convex Optimization Problems

2 ,

ˆ min ( )

i p q i

H H s −

∑

ˆ ( ) : PASSIVE H s

Subject to:

• Non-convex problems difficult to solve
• Must reformulate as convex
• ptimization problem

– Convex objective function – Convex constraints Non-convex function

(finding global minimum-extremely difficult)

Convex function

(finding global minimum-easy)

23

Modeling Flow

Step 1: Identify a COMMON set of STABLE poles Step 2: Use POLES from step 1 to identify passive model

• Vector Fitting Algorithm

[Gustavsen 1999]

• Optimization Framework

[Suo 2008]

2 ,

ˆ min ( )

i p q i

H H s −

∑

ˆ ( ) : PASSIVE H s

Subject to:

24

Problem Formulation

/ 1 1 /2 1 1 2 1

ˆ ( ) ˆ ˆ ( ) ( )

c r c r

k r c k k k k r c c c c k k k k k k k k

H j j H j H j j a j a a j a j a a j j j

κ κ κ κ κ

ω ω ω ω ω ω ω

= = = = =

= + − = + +   ℜ ℜ = + + + +   − −ℜ − ℑ − ℑ − ℑ + ℑ ℜ  

∑ ∑ ∑ ∑ ∑

k r c c c c k k k k k

R D D R R R R R D

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Problem Formulation

/ 1 1 /2 1 1 2 1

ˆ ( ) ˆ ˆ ( ) ( )

c r c r

k r c k k k k r c c c c k k k k k k k k

H j j H j H j j a j a a j a j a a j j j

κ κ κ κ κ

ω ω ω ω ω ω ω

= = = = =

= + − = + +   ℜ ℜ = + + + +   − −ℜ − ℑ − ℑ − ℑ + ℑ ℜ  

∑ ∑ ∑ ∑ ∑

k r c c c c k k k k k

R D D R R R R R D

series/parallel interconnection

• f first order networks

series/parallel interconnection

• f second order networks

resistive/conductive network

26

Problem Formulation

/ 1 1 /2 1 1 2 1

ˆ ( ) ˆ ˆ ( ) ( )

c r c r

k r c k k k k r c c c c k k k k k k k k

H j j H j H j j a j a a j a j a a j j j

κ κ κ κ κ

ω ω ω ω ω ω ω

= = = = =

= + − = + +   ℜ ℜ = + + + +   − −ℜ − ℑ − ℑ − ℑ + ℑ ℜ  

∑ ∑ ∑ ∑ ∑

k r c c c c k k k k k

R D D R R R R R D

series/parallel interconnection

• f first order networks

series/parallel interconnection

• f second order networks

resistive/conductive network

Condition 1 – Conjugate Symmetry: Enforced by construction Condition 2 – Stability: Enforced during pole-identification Condition 3 – Non-negativity: Enforced on the building blocks

Passivity Conditions:

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• A sufficient condition for passivity:

Passivity Conditions

1 / 1 2

ˆ ˆ ˆ ( ) ( ) ( )

c r

r c k k k k

H j H j H j

κ κ

ω ω ω

= =

= + +

∑ ∑

D ˆ ˆ ˆ ( ), ( ), passive ( ) passive

r c k k

H j H j k H j ω ω ω ∀ ⇒ D

ˆ ˆ ˆ ( ) 0, ( ) 0, ( )

r c k k

H j H j k H j ω ω ω ℜ ℜ ∀ ⇒ ℜ D ± ± ± ±

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• Real-only poles

Passivity Conditions

2 2 2 2 2 2

ˆ ( ) ˆ ( ) ˆ ( )

r k r k r r k k r r k k r r k k r k

H j j a a H j j a a a H j a ω ω ω ω ω ω ω ω = − = − − + + ℜ = − ⇒ +

r k r r k k r r k k

R R R R R ± ±

1 / 1 2

ˆ ˆ ˆ ( ) ( ) ( )

c r

r c k k k k

H j H j H j

κ κ

ω ω ω

= =

= + +

∑ ∑

D

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• Real-only poles
• Direct Matrix

Passivity Conditions

2 2 2 2 2 2

ˆ ( ) ˆ ( ) ˆ ( )

r k r k r r k k r r k k r r k k r k

H j j a a H j j a a a H j a ω ω ω ω ω ω ω ω = − = − − + + ℜ = − ⇒ +

r k r r k k r r k k

R R R R R ± ±

D ±

1 / 1 2

ˆ ˆ ˆ ( ) ( ) ( )

c r

r c k k k k

H j H j H j

κ κ

ω ω ω

= =

= + +

∑ ∑

D

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• Real-only poles
• Direct Matrix
• Complex poles

( ) ( )

2

ˆ ( ) 0. . ( ) 0 , . .lim ( ) .lim ( )

c k c c c c k k k k c c k k c c k k

H j CPR a a a a CPR a a CPR a a

ω ω

ω ω ω ω ω ω

→ →∞

ℜ ⇒ = −ℜ ℜ ℑ ℑ −ℜ ℜ ℑ ℑ ∀ ⇔ ⇒ −ℜ ℜ ℑ ℑ ⇒ −ℜ ℜ − + + − ℑ ℑ +

c c c c k k k k c c k k c c k k

R R R R R R R R ± ± ± ±

Passivity Conditions

2 2 2 2 2 2

ˆ ( ) ˆ ( ) ˆ ( )

r k r k r r k k r r k k r r k k r k

H j j a a H j j a a a H j a ω ω ω ω ω ω ω ω = − = − − + + ℜ = − ⇒ +

r k r r k k r r k k

R R R R R ± ±

D ±

1 / 1 2

ˆ ˆ ˆ ( ) ( ) ( )

c r

r c k k k k

H j H j H j

κ κ

ω ω ω

= =

= + +

∑ ∑

D

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( ) ( )

2

ˆ ( ) 0. . ( ) 0 , . .lim ( ) .lim ( )

c k c c c c k k k k c c k k c c k k

H j CPR a a a a CPR a a CPR a a

ω ω

ω ω ω ω ω ω

→ →∞

ℜ ⇒ = −ℜ ℜ ℑ ℑ −ℜ ℜ ℑ ℑ ∀ ⇔ ⇒ −ℜ ℜ ℑ ℑ ⇒ −ℜ ℜ − + + − ℑ ℑ +

c c c c k k k k c c k k c c k k

R R R R R R R R ± ± ± ±

• Real-only poles
• Direct Matrix
• Complex poles

Passivity Conditions

2 2 2 2 2 2

ˆ ( ) ˆ ( ) ˆ ( )

r k r k r r k k r r k k r r k k r k

H j j a a H j j a a a H j a ω ω ω ω ω ω ω ω = − = − − + + ℜ = − ⇒ +

r k r r k k r r k k

R R R R R ± ±

D ±

1 / 1 2

ˆ ˆ ˆ ( ) ( ) ( )

c r

r c k k k k

H j H j H j

κ κ

ω ω ω

= =

= + +

∑ ∑

D

Unknowns

Linear Matrix Inequalities (Extremely efficient)

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Convex formulation

2 2 , , 1 1

ˆ ˆ minimize ( ) ( ) subject to 1,..., 1,..., 1,..., where ˆ ( )

| | | |

c r

i i i i i i r c c k k c c c k k c r c k k k k

H H j H H j k a a k a a k H j j a j a

κ κ

ω ω κ κ κ ω ω ω

= =

ℜ −ℜ + ℑ − ℑ ∀ = −ℜ ℜ + ℑ ℑ ∀ = −ℜ ℜ ℑ ℑ ∀ = = + + − − −

∑ ∑ ∑ ∑

r c k k

R R D r k c c k k c c k k r c k k

D R R R R R R R D ± ± ± ±

Linear Matrix Inequalities (Extremely efficient)

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Final Optimization Problem Convex Optimization Problem

1 1

ˆ ( )

c r

r c k k k k

H j j a j a

κ κ

ω ω ω

= =

= + + − −

∑ ∑

r c k k

R R D

Samples , Desired number of poles (N)

{ }

i i H

, ω

(Netlist/ VerilogA ) Circuit Module Dynamical Model −

2 2 , ,

ˆ ˆ minimize ( ) ( ) subject to: 0, 1,..., 1,..., 1,...,

| | | |

i i i i i i r c c k k c c c k k c

H H j H H j k a a k a a k ω ω κ κ κ ℜ −ℜ + ℑ − ℑ ∀ − = −ℜ ℜ + ℑ ℑ ∀ = −ℜ ℜ ℑ ℑ ∀ =

∑ ∑

r c k k

R R D r k c c k k c c k k

D R R R R R ± ± ± ±

k

Find N Stable poles a

k

a

, , ,

k

a

r c k k

R R D

34

Outline

• Motivation for Compact Dynamical Passive Modeling
• What is Passivity?
• Existing Techniques
• Rational Fitting of Transfer Functions
• Results

35

Results: LINC Power Amplifier

Block diagram of the LINC power amplifier architecture

36

Results: LINC Power Amplifier

Block diagram of the LINC power amplifier architecture Layout of the Wilkinson Combiner

37

Results: LINC Power Amplifier

11 1 1

( ) ... ( ) ( ) . . . ( ) ... ( )

N N NN

Transfer Function Matrix Z s Z s H s Z s Z s     =       

EM Field Solver (dots) Passive Modeling Algorithm (solid lines)

Comparing real and imaginary parts of the impedance parameters from EM field solver (dots) and our passive model (solid lines)

Block diagram of the LINC power amplifier architecture Layout of the Wilkinson Combiner

• Number of poles – 10
• Time [Matlab] – 2 seconds
• Error < 0.7%

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Results: LINC Power Amplifier

11 1 1

( ) ... ( ) ( ) . . . ( ) ... ( )

N N NN

Transfer Function Matrix Z s Z s H s Z s Z s     =       

Zoomed-in eigen values of the associated Hamiltonian matrix for the identified model of Wilkinson combiner

Hamiltonian Matrix Based Passivity Test

Model is passive if the associated Hamiltonian matrix has no purely imaginary eigen value

39

Results: LINC Power Amplifier

11 1 1

( ) ... ( ) ( ) . . . ( ) ... ( )

N N NN

Z s Z s H s Z s Z s     =        Block diagram of the LINC power amplifier as simulated inside the circuit simulator Passive Transfer Matrix

40

Results: LINC Power Amplifier

11 1 1

( ) ... ( ) ( ) . . . ( ) ... ( )

N N NN

Z s Z s H s Z s Z s     =       

Normalized Input Voltage (vin) Normalized Output Voltage (vout)

Block diagram of the LINC power amplifier as simulated inside the circuit simulator Passive Transfer Matrix

LINC PA

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Example: 8-Port Power/Gnd Distribution Grid

QUALITY CHECK (Impedance matrix)

• Hamiltonian Matrix Based

Passivity Test – Passed

• Number of poles –20
• Time [Matlab] –74 seconds

42

Example: 4-Port Inductor Array

QUALITY CHECK (Impedance matrix)

• Hamiltonian Matrix Based

Passivity Test – Passed

• Number of poles –23
• Time [Matlab] –72 seconds

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Comparison Structure Number

• f Ports

Number

• f Poles

Time1 (seconds) [Suo 2008] This Work [Matlab]

Wilkinson Combiner 3 10 83 2 Power Distribution Grid 8 20

• NA-

74 Coupled RF inductors 4 23

• NA-

72

1Laptop: Core2Duo 2.1GHz, 3GB, Windows 7

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Conclusions

• Summarized how to develop models from freq. response
• Proposed a Convex Optimization based modeling algorithm
• Demonstrated orders of magnitude improvement over

similar techniques

• Presented an example demonstrating system level

simulation of analog circuits

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THANK YOU