Statistical Properties of Wave Chaotic Scattering and Impedance Matrices

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Statistical Properties of Wave Chaotic Scattering and Impedance Matrices. MURI Faculty: Tom Antonsen, Ed Ott, Steve Anlage, MURI Students: Xing Zheng, Sameer Hemmady, James Hart. AFOSR-MURI Program Review. Schematic.

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### Statistical Properties of Wave Chaotic Scattering and Impedance Matrices

MURI Faculty: Tom Antonsen, Ed Ott, Steve Anlage,

MURI Students: Xing Zheng, Sameer Hemmady, James Hart

AFOSR-MURI Program Review

Schematic

• Coupling of external radiation to computer circuits is a complex processes:

apertures

resonant cavities

transmission lines

circuit elements

• Intermediate frequency range involves many interacting resonances

• System size >>

Wavelength

• Chaotic Ray Trajectories

Integrated circuits

cables

connectors

circuit boards

• What can be said about coupling without solving in detail the complicated EM problem ?

• Statistical Description !

(Statistical Electromagnetics, Holland and St. John)

Electromagnetic Coupling in Computer Circuits

N- Port

System

N ports

• voltages and currents,

• incoming and outgoing waves

S matrix

Z matrix

Z(w) , S(w)

• Complicated function of frequency

• Details depend sensitively on unknown parameters

voltage

current

outgoing

incoming

Z and S-MatricesWhat is Sij ?

Random amplitude

Random direction

Random phase

3. Eigenvalues kn2 are distributed according

to appropriate statistics:

- Eigenvalues of Gaussian Random Matrix

Normalized Spacing

Random Coupling Model

1. Formally expand fields in modes of closed cavity: eigenvalues kn = wn/c

Port 1

Losses

Statistical Model Impedance

Other ports

RR1(w)

Port 2

System

parameters

RR2(w)

Dw2n - mean spectral spacing

Port 1

Q -quality factor

Resistance RR(w)

ZR(w) = RR(w)+jXR (w)

wn - random spectrum

Statistical

parameters

win- Gaussian Random variables

Statistical Model of Z MatrixFrequency Domain
Model ValidationSummary

Single Port Case:

Cavity Impedance: Zcav = RR z + jXR

Radiation Impedance: ZR = RR + jXR

Universal normalized random impedance: z = r + jx

Statistics of z depend only on damping parameter: k2/(QDk2)

(Q-width/frequency spacing)

Validation:

HFSS simulations

Port 1

Losses

Zcav = jXR+(r+jx) RR

Distribution of reactance fluctuations

P(x)

Distribution of resistance fluctuations

P(r)

Dk2

Dk2

Normalized Cavity Impedance with Losses

Theory predictions for

Pdf’s of z=r+jx

ports

h

Box with

metallic walls

Only transverse magnetic (TM) propagate for

f < c/2h

Ez

Hy

Hx

Voltage on top plate

• Anlage Experiments

• HFSS Simulations

• Power plane of microcircuit

Two Dimensional Resonators

Curved walls guarantee all ray trajectories are chaotic

Losses on top and bottom plates

Moveable conducting

disk - .6 cm diameter

“Proverbial soda can”

HFSS - SolutionsBow-Tie Cavity

Cavity impedance calculated for

100 locations of disk

4000 frequencies

6.75 GHz to 8.75 GHz

Theory

Theory

Comparison of HFSS Results and Modelfor Pdf’s of Normalized Impedance

Normalized Reactance

Normalized Resistance

x

r

Zcav = jXR+(r+jx) RR

1.05”

1.6”

0. 31”

Perturbation

Eigen mode Image at 12.57GHz

EXPERIMENTAL SETUP

Circular Arc

R=42”

Antenna Entry Point

0.310” DEEP

8.5”

Circular Arc

R=25”

SCANNED PERTURBATION

21.4”

17”

•  2 Dimensional Quarter Bow Tie Wave Chaotic cavity
• Classical ray trajectories are chaotic - short wavelength - Quantum Chaos
• 1-port S and Z measurements in the 6 – 12 GHz range.
• Ensemble average through 100 locations of the perturbation

Intermediate Loss

Low Loss

High Loss

Theory

r

x

Comparison of Experimental Results and Modelfor Pdf’s of Normalized Impedance

Theory predicts:

Uniform distribution in phase

Normalized Scattering AmplitudeTheory and HFSS Simulation

Actual Cavity Impedance: Zcav = RR z + jXR

Normalized impedance : z = r + jx

Universal normalized scattering coefficient: s = (z -1)/(z +1) = | s| exp[ if ]

Statistics of s depend only on damping parameter: k2/(QDk2)

1

5p/8

3p/8

a)

7p/8

p/8

0

Distribution of Reflection Coefficient

ln [P(|s|2)]

- p/8

-1

- 3p/8

Theory

1

-1

0

Distribution independent of f

|s|2

s=|s|exp[if]

Zcav = jXR+(r+jx) RR

RR = <(r(f1)-1)(r(f2)-1)>

XX = <x(f1)x(f2)>

RX = <(r(f1)-1 )x(f2)>

Frequency Correlations in Normalized ImpedanceTheory and HFSS Simulations

(f1-f2)

Eigenvalues of Z matrix

q2

Individually x1,2 are

Lorenzian distributed

Distributions same as

In Random Matrix theory

q1

Properties of Lossless Two-Port Impedance(Monte Carlo Simulation of Theory Model)
HFSS Solution for Lossless 2-Port

Joint Pdf for q1 andq2

Disc

q2

Port #1:

Port #2:

q1

Comparison of Distributed Lossand Lossless Cavity with Ports(Monte Carlo Simulation)

Distribution of resistance fluctuations

P(r)

Distribution of reactance fluctuations

P(x)

r

x

Zcav = jXR+(r+jx) RR

Frequency Domain

wn- Gaussian Random variables

Statistical Parameters

wn - random spectrum

Time Domain

wn- Gaussian Random variables

Time Domain Model for Impedance Matrix

Prompt Reflection

Delayed Reflection

f = 3.6 GHz

t = 6 nsec

Prompt reflection removed by matching Z0 to ZR

Incident and Reflected Pulsesfor One Realization

Incident Pulse

Decay of MomentsAveraged Over 1000 Realizations

Prompt reflection eliminated

Log Scale

Linear Scale

<V2(t)>1/2

|<V3(t)>|1/3

<V(t)>

t1 = 8.0  10-7

t1 = 5.0  10-7

t1 = 1.0  10-7

Quasi-Stationary Process

2-time Correlation Function

(Matches initial pulse shape)

Normalized Voltage

u(t)=V(t) /<V2(t)>1/2

1000 realizations

Incident Pulse

Mean = .3163

Histogram of Maximum Voltage

|Vinc(t)|max = 1V

Progress

• Direct comparison of random coupling model with

-random matrix theory P

-HFSS solutions P

-Experiment P

• Exploration of increasing number of coupling ports P

• Study losses in HFSS P

• Time Domain analysis of Pulsed Signals

-Pulse duration

-Shape (chirp?)

• Generalize to systems consisting of circuits and fields

Current

Future

Role of Scars?

• Eigenfunctions that do not satisfy random plane wave assumption

• Scars are not treated by either random matrix

or chaotic eigenfunction theory

• Semi-classical methods

Bow-Tie with diamond scar

Future Directions

-theoretically

-numerically

-experimentally

Features:

Ray splitting

Losses