Statistical Properties of Wave Chaotic Scattering and Impedance Matrices
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Transcript of 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
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Electromagnetic Coupling in Computer Circuits
connectors
cables
circuit boards
Integrated circuits
Schematic• Coupling of external radiation to computer circuits is a complex processes:
aperturesresonant cavitiestransmission linescircuit elements
• Intermediate frequency range involves many interacting resonances
• System size >> Wavelength
• Chaotic Ray Trajectories
• What can be said about coupling without solving in detail the complicated EM problem ?
• Statistical Description !(Statistical Electromagnetics, Holland and St. John)
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Z and S-MatricesWhat is Sij ?
V1
V2
VN1
S
V1
V2
VN1
outgoing incoming
S matrix
S (Z Z0 ) 1(Z Z0)
• Complicated function of frequency• Details depend sensitively on unknown parameters
Z() , S()
N- PortSystem
•••
V1
V1
VN
VN
N ports • voltages and currents, • incoming and outgoing waves
V1, I1
VN, IN
V1
V2
VN
Z
I1
I2
I2
voltage current
Z matrix
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Random Coupling Model
0
0.2
0.4
0.6
0.8
1
1.2
0 0.5 1 1.5 2 2.5
integrablechaotic A
s
chaotic B
3. Eigenvalues kn2 are distributed according
to appropriate statistics:
- Eigenvalues of Gaussian Random Matrix
sn (kn12 kn
2) / k2Normalized Spacing
2. Replace exact eigenfunction with superpositions of random plane waves
n limN Re2
ANak exp i knek x k
k1
N
Random amplitude Random direction Random phase
1. Formally expand fields in modes of closed cavity: eigenvalues kn = n/c
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Statistical Model of Z MatrixFrequency Domain
Port 1
Port 2
Other ports
Losses
Port 1
Free-space radiationResistance RR()ZR() = RR()+jXR ()
RR1()
RR2()
Zij ()j
RRi1/2(n )RRj
1/2(n)n
2 winw jn
2(1 jQ 1) n2
n
Statistical Model Impedance
Q -quality factor
n - mean spectral spacing
Radiation Resistance RRi()
win- Gaussian Random variables
n - random spectrum
Systemparameters
Statisticalparameters
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Model ValidationSummary
Single Port Case:
Cavity Impedance: Zcav = RR z + jXR
Radiation Impedance: ZR = RR + jXR
Universal normalized random impedance: z + j
Statistics of z depend only on damping parameter: k2/(Qk2)(Q-width/frequency spacing)
Validation:HFSS simulationsExperiment (Hemmady and Anlage)
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Normalized Cavity Impedance with Losses
Port 1 Losses
Zcav = jXR+(+j RR
Theory predictions forPdf’s of z=+j
Distribution of reactance fluctuationsP()
k2
Distribution of resistance fluctuationsP()
k2
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Two Dimensional Resonators
• Anlage Experiments• HFSS Simulations• Power plane of microcircuit
Only transverse magnetic (TM) propagate forf < c/2h
Ez
HyHx
h
Box withmetallic walls
ports
Ez (x, y)VT (x, y) / h
Voltage on top plate
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HFSS - SolutionsBow-Tie Cavity
Moveable conductingdisk - .6 cm diameter“Proverbial soda can”
Curved walls guarantee all ray trajectories are chaoticLosses on top and bottom plates
Cavity impedance calculated for 100 locations of disk4000 frequencies6.75 GHz to 8.75 GHz
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Comparison of HFSS Results and Modelfor Pdf’s of Normalized Impedance
Normalized Reactance Normalized Resistance
Zcav = jXR+(+j RR
Theory
Theory
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EXPERIMENTAL SETUPEXPERIMENTAL SETUP
Sameer Hemmady, Steve Anlage CSR
Eigen mode Image at 12.57GHz
8.5”
17” 21.4”
0.310” DEEP
Antenna Entry Point
SCANNED PERTURBATION
Circular Arc
R=25”R=25”
Circular Arc
R=42”R=42”
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
0. 31”
1.05”
1.6”
Perturbation
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0 1 2 30
1
2
k2 / (k 2Q) 7.6
k2 / (k 2Q) 4.2
k2 / (k 2Q) 0.8
-2 -1 0 1 20
1
2
k2 / (k 2Q) 0.8
k2 / (k 2Q) 4.2
k2 / (k 2Q) 7.6
Re Z / RR
(Im Z XR) / RR
High LossLow Loss Intermediate Loss
Comparison of Experimental Results and Modelfor Pdf’s of Normalized Impedance
Theory
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Normalized Scattering AmplitudeTheory and HFSS Simulation
Theory predicts:
P( s ,)1
2Ps ( s )
Uniform distribution in phase
Actual Cavity Impedance: Zcav = RR z + jXR
Normalized impedance : z + jUniversal normalized scattering coefficient: s = (z)/(z) = | s| exp[ i]Statistics of s depend only on damping parameter: k2/(Qk2)
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Experimental Distribution of Normalized Scattering Coefficient
0 20 40 60 80 1000
20
40
60
80
100
a))Im(s
1
-1
0
1-1
0 )Re(s
s=|s|exp[i]
Theory
|s|2
ln [P(|s|2)]Distribution of Reflection Coefficient
Distribution independent of
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Frequency Correlations in Normalized ImpedanceTheory and HFSS Simulations
Zcav = jXR+(+j RR
RR = <((f1)-1)((f2)-1)>
XX = <(f1)(f2)>
RX = <((f1)-1 )(f2)>
(f1-f2)
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Properties of Lossless Two-Port Impedance(Monte Carlo Simulation of Theory Model)
Eigenvalues of Z matrix
det Z jX 1 0
X1,2X
R
1, 2R
R
1,2 tan1,2
2
Individually 1,2 are Lorenzian distributed
1
2
Distributions same asIn Random Matrix theory
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HFSS Solution for Lossless 2-Port
2
1
Joint Pdf for 1 and 2
Port #1:
Port #2:
Disc
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Comparison of Distributed Lossand Lossless Cavity with Ports
(Monte Carlo Simulation)
Distribution of reactance fluctuationsP()
Distribution of resistance fluctuationsP()
Zcav = jXR+(+j RR
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d2
dt2 2nddt
n2
Vn(t)
1
RR(n)n2wn
2
n
ddt
I(t)
Time Domain
V( t) Vn (t)n
n n
Q
wn- Gaussian Random variables
Time Domain Model for Impedance Matrix
Z() j
RR (n)n
n2 wn
2
2(1 jQ 1) n2
n
Frequency Domain
Statistical Parameters
wn- Gaussian Random variables
n - random spectrum
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Incident and Reflected Pulsesfor One Realization
-1
-0.5
0
0.5
1
0 0.02 0.04 0.06 0.08 0.1 0.12
Inci
dent
Vol
tage
t [sec]
Incident Pulse
-1
-0.5
0
0.5
1
0 0.02 0.04 0.06 0.08 0.1 0.12
Ref
lect
ed V
olta
ge
t [sec]
Prompt Reflection
Delayed Reflection
Prompt reflection removed by matching Z0 to ZR
f = 3.6 GHz = 6 nsec
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Decay of MomentsAveraged Over 1000 Realizations
<V(t)>
<V2(t)>1/2
|<V3(t)>|1/3
Linear Scale Log ScalePrompt reflection eliminated
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Quasi-Stationary Process
-1
-0.5
0
0.5
1
1.5
2
2.5
-1.5 10-8 -1 10-8 -5 10-9 0 5 10-9 1 10-8 1.5 10-8
<u(
t 1)u
(t2)>
t1-t
2
t1 = 1.0 10-7
t1 = 8.0 10-7
t1 = 5.0 10-7
Normalized Voltageu(t)=V(t) /<V2(t)>1/2
2-time Correlation Function(Matches initial pulse shape)
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-1
-0.5
0
0.5
1
0 0.02 0.04 0.06 0.08 0.1 0.12
Inci
dent
Vol
tage
t [sec]
Incident Pulse
Histogram of Maximum Voltage
0
20
40
60
80
100
120
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55
Cou
nt
|V(t) |max
1000 realizations
Mean = .3163
|Vinc(t)|max = 1V
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Progress
• Direct comparison of random coupling model with -random matrix theory -HFSS solutions -Experiment
• Exploration of increasing number of coupling ports
• Study losses in HFSS • Time Domain analysis of Pulsed Signals
-Pulse duration-Shape (chirp?)
• Generalize to systems consisting of circuits and fields
Current
Future
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Role of Scars?
• Eigenfunctions that do not satisfy random plane wave assumption
Bow-Tie with diamond scar
• Scars are not treated by either random matrixor chaotic eigenfunction theory
• Semi-classical methods
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Future Directions
Bow-tie shaped cavity
Dielectric
Test port
Excitation port
• Can be addressed
-theoretically
-numerically
-experimentally
Features:Ray splittingLosses
HFSS simulation courtesy J. RodgersAdditional complications to be added later