Statistical Properties of Wave Chaotic Scattering and Impedance Matrices - PowerPoint PPT Presentation

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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 ... – PowerPoint PPT presentation

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Title: Statistical Properties of Wave Chaotic Scattering and Impedance Matrices


1
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
2
Electromagnetic Coupling in Computer Circuits
3
Z and S-MatricesWhat is Sij ?
4
Random Coupling Model
1. Formally expand fields in modes of closed
cavity eigenvalues kn wn/c
5
Statistical Model of Z MatrixFrequency Domain
6
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 Experi
ment (Hemmady and Anlage)
7
Normalized Cavity Impedance with Losses
Theory predictions for Pdfs of z rjx
8
Two Dimensional Resonators
9
HFSS - SolutionsBow-Tie Cavity
Cavity impedance calculated for 100 locations of
disk 4000 frequencies 6.75 GHz to 8.75 GHz
10
Comparison of HFSS Results and Modelfor Pdfs of
Normalized Impedance
Normalized Reactance
Normalized Resistance
x
r
Zcav jXR(rjx) RR
11
EXPERIMENTAL SETUP Sameer Hemmady, Steve Anlage
CSR
Circular Arc R42
Antenna Entry Point
0.310 DEEP
8.5
Circular Arc R25
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

12
Comparison of Experimental Results and Modelfor
Pdfs of Normalized Impedance
13
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)
14
Experimental Distribution of Normalized
Scattering Coefficient
ssexpif
15
Frequency Correlations in Normalized
ImpedanceTheory and HFSS Simulations
(f1-f2)
16
Properties of Lossless Two-Port Impedance(Monte
Carlo Simulation of Theory Model)
17
HFSS Solution for Lossless 2-Port
Joint Pdf for q1 and q2
Disc
q2
Port 1
Port 2
q1
18
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(rjx) RR
19
Time Domain Model for Impedance Matrix
20
Incident and Reflected Pulsesfor One Realization
Incident Pulse
21
Decay of MomentsAveraged Over 1000 Realizations
Prompt reflection eliminated
Log Scale
Linear Scale
ltV2(t)gt1/2
ltV3(t)gt1/3
ltV(t)gt
22
Quasi-Stationary Process
2-time Correlation Function (Matches initial
pulse shape)
Normalized Voltage u(t)V(t) /ltV2(t)gt1/2
23
Histogram of Maximum Voltage
Vinc(t)max 1V
24
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
25
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
26
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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