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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

• 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 CircuitsSystem

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 ?2. Replace exact eigenfunction with superpositions of random plane waves

Random amplitude

Random direction

Random phase

3. Eigenvalues kn2 are distributed according

to appropriate statistics:

- Eigenvalues of Gaussian Random Matrix

Normalized Spacing

Random Coupling Model1. Formally expand fields in modes of closed cavity: eigenvalues kn = wn/c

Losses

Statistical Model Impedance

Other ports

RR1(w)

Port 2

Radiation Resistance RRi(w)

System

parameters

RR2(w)

Dw2n - mean spectral spacing

Port 1

Q -quality factor

Free-space radiation

Resistance RR(w)

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

wn - random spectrum

Statistical

parameters

win- Gaussian Random variables

Statistical Model of Z MatrixFrequency DomainModel 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

Experiment (Hemmady and Anlage)

Losses

Zcav = jXR+(r+jx) RR

Distribution of reactance fluctuations

P(x)

Distribution of resistance fluctuations

P(r)

Dk2

Dk2

Normalized Cavity Impedance with LossesTheory predictions for

Pdf’s of z=r+jx

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 ResonatorsCurved 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 CavityCavity impedance calculated for

100 locations of disk

4000 frequencies

6.75 GHz to 8.75 GHz

Theory

Comparison of HFSS Results and Modelfor Pdf’s of Normalized ImpedanceNormalized Reactance

Normalized Resistance

x

r

Zcav = jXR+(r+jx) RR

1.6”

0. 31”

Perturbation

Eigen mode Image at 12.57GHz

EXPERIMENTAL SETUP

Sameer Hemmady, Steve Anlage CSR

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

Low Loss

High Loss

Theory

r

x

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

Uniform distribution in phase

Normalized Scattering AmplitudeTheory and HFSS SimulationActual 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)

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

Experimental Distribution of Normalized Scattering Coefficients=|s|exp[if]

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)

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)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

wn- Gaussian Random variables

Statistical Parameters

wn - random spectrum

Time Domain

wn- Gaussian Random variables

Time Domain Model for Impedance MatrixDelayed Reflection

f = 3.6 GHz

t = 6 nsec

Prompt reflection removed by matching Z0 to ZR

Incident and Reflected Pulsesfor One RealizationIncident 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 = 5.0 10-7

t1 = 1.0 10-7

Quasi-Stationary Process2-time Correlation Function

(Matches initial pulse shape)

Normalized Voltage

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

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

• Can be addressed

-theoretically

-numerically

-experimentally

Features:

Ray splitting

Losses

Additional complications to be added later

HFSS simulation courtesy J. Rodgers

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