Download Coupling

Statistical Properties of Wave Chaotic Scattering and
Impedance Matrices
MURI Faculty:
MURI Students:
Tom Antonsen, Ed Ott, Steve Anlage,
Xing Zheng, Sameer Hemmady, James Hart
AFOSR-MURI Program Review
Electromagnetic Coupling in Computer
Circuits
Schematic
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)
• 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
Z and S-Matrices
What is Sij ?
V1, I1
V1 
N ports
• voltages and currents,
• incoming and outgoing
waves
V1 
•
•
VN, IN •
VN 
N- Port
System
VN 
Z matrix
V1 
V2 
 
 
 
VN 
voltage
I1 
I2 
Z  
 
 
I2 
current
S matrix
V1 
V1 
V  
V  
 2   S  2 
 
 
  
  
VN1
VN1 
outgoing
incoming
S  (Z  Z0 )1 (Z  Z 0 )
Z(w) , S(w)
• Complicated function of
frequency
• Details depend sensitively on
unknown parameters
Random Coupling Model
1. Formally expand fields in modes of closed cavity: eigenvalues kn = wn/c
2. Replace exact eigenfunction with superpositions of random plane waves
 2 N

n  lim N  Re
ak expikn e k  x   k 

AN
k 1


Random amplitude
Random direction
3. Eigenvalues kn2 are distributed according
to appropriate statistics:
Random phase
1.2
integrable
chaotic A
chaotic B
1
0.8
- Eigenvalues of Gaussian Random Matrix
Normalized Spacing
sn  (k
2
n1
2
n
 k ) / k
2
0.6
0.4
0.2
0
0
0.5
1
1.5
s
2
2.5
Statistical Model of Z Matrix
Frequency Domain
Port 1
Losses
Statistical Model Impedance
Other ports
Zij (w )  
RR1(w)
Port 2
RR2(w)
2
 w n win w jn
j
1/2
1/2
RRi (w n )RRj (w n ) 2


w (1 jQ1 )  w 2
n
n
System
parameters
Radiation Resistance RRi(w)
w2n - mean spectral spacing
Q -quality factor
Port 1
Free-space radiation
Resistance RR(w)
ZR(w) = RR(w)+jXR (w)
Statistical
parameters
wn - random spectrum
win- Gaussian Random variables
Model Validation
Summary
Single Port Case:
Cavity Impedance:
Radiation Impedance:
Zcav = RR z + jXR
ZR = RR + jXR
Universal normalized random impedance:
z r + jx
Statistics of z depend only on damping parameter:
(Q-width/frequency spacing)
Validation:
HFSS simulations
Experiment (Hemmady and Anlage)
k2/(Qk2)
Normalized Cavity Impedance with Losses
Theory predictions for
Pdf’s of z =r+jx
Port 1
Distribution of reactance fluctuations
P(x)
k2
Losses
Zcav = jXR+(r+jx RR
Distribution of resistance fluctuations
P(r)
k2
Two Dimensional Resonators
ports
h
Box with
metallic walls
Ez
Hx
Hy
Only transverse magnetic (TM) propagate for
f < c/2h
• Anlage Experiments
• HFSS Simulations
• Power plane of microcircuit
Voltage on top plate
Ez (x, y)  VT (x, y) / h
HFSS - Solutions
Bow-Tie Cavity
Curved walls guarantee all
ray trajectories are chaotic
Losses on top and bottom
plates
Moveable conducting
disk - .6 cm diameter
“Proverbial soda can”
Cavity impedance
calculated for
100 locations of disk
4000 frequencies
6.75 GHz to 8.75 GHz
Comparison of HFSS Results and Model
for Pdf’s of Normalized Impedance
Normalized Reactance
Normalized Resistance
Theory
Theory
x
Zcav = jXR+(r+jx RR
r
EXPERIMENTAL SETUP
Sameer Hemmady, Steve Anlage CSR
Circular
Arc
1.05”
1.6”
0. 31”
R=42”
Antenna Entry Point
Perturbation
0.310”
DEEP
8.5”
SCANNED
PERTURBATION
Circular
Arc
R=25”
17” 21.4”
Eigen mode Image at
12.57GHz
 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
Comparison of Experimental Results and Model
for Pdf’s of Normalized Impedance
r  ReZ / RR
x  (ImZ   X R ) / RR
Intermediate Loss
Low Loss
High Loss
2
2
k2 / (k 2Q)  7.6
Theory
k2 / (k 2Q)  7.6
k2 / (k 2Q)  4.2
k2 / (k 2Q)  0.8
1
0
0
1
r
2
3
k2 / (k 2Q)  4.2
1
0
-2
k2 / (k 2Q)  0.8
-1
0
x
1
2
Normalized Scattering Amplitude
Theory and HFSS Simulation
Actual Cavity Impedance:
Normalized impedance :
Universal normalized scattering coefficient:
Statistics of s depend only on damping parameter:
Theory predicts:
1
P( s , ) 
P ( s)
2 s
Uniform distribution in phase
Zcav = RR z + jXR
z r + jx
s = (z 1)/(z 1) = | s| exp[ i]
k2/(Qk2)
Experimental Distribution of Normalized
Scattering Coefficient
s=|s|exp[i]
1
5/
8
100
Im( s )
a)
ln
[P(|s|2)]
Distribution of Reflection Coefficient
3/
8
Theory
80
7/
8
/8
60
0
40
 /8
20
-0
0
11
 3/8
20
40
0
60
80
Re(s
)
100
1
Distribution independent of 
|s|2
Frequency Correlations in Normalized Impedance
Theory and HFSS Simulations
Zcav = jXR+(r+jx RR
RR = <(r(f1)-1)(r(f2)-1)>
XX = <x(f1)x(f2)>
RX = <(r(f1)-1 )x(f2)>
(f1-f2)
Properties of Lossless Two-Port Impedance
(Monte Carlo Simulation of Theory Model)
Eigenvalues of Z matrix
det Z  jX1  0
X1,2  XR  x1, 2 RR
1,2 
x1,2  tan  
 2 
2
Individually x1,2 are
Lorenzian distributed
Distributions same as
In Random Matrix theory
1
HFSS Solution for Lossless 2-Port
Joint Pdf for 1 and 2
Disc
2
Port #1:
Port #2:
1
Comparison of Distributed Loss
and Lossless Cavity with Ports
(Monte Carlo Simulation)
Distribution of resistance fluctuations
P(r)
r
Distribution of reactance fluctuations
P(x)
Zcav = jXR+(r+jx RR
x
Time Domain Model for Impedance Matrix
Frequency Domain
Z(w )  
jw


wn- Gaussian Random variables
RR (w n )
n
wn
 wn2 wn2
w 2 (1 jQ1 )  w 2n
Statistical Parameters
Time Domain
wn - random spectrum
wn- Gaussian Random variables
2 2
 d 2

d
1
R
(
w
)
w
2
R
n
n wn d
I(t)
 2  2 n dt  w n Vn (t)   
wn
dt
dt

V(t)   Vn (t)
n
n 
wn
Q
Incident and Reflected Pulses
for One Realization
1
Incident Pulse
Prompt Reflection
1
Reflected Voltage
0.5
Incident Voltage
0.5
0
Delayed Reflection
0
f = 3.6 GHz
t = 6 nsec
-0.5
-0.5
-1
0
0.02
0.04
0.06
0.08
0.1
0.12
t [sec]
-1
0
0.02
0.04
0.06
t [sec]
0.08
0.1
0.12
Prompt reflection
removed by matching Z0
to ZR
Decay of Moments
Averaged Over 1000 Realizations
Prompt reflection eliminated
Log Scale
Linear Scale
<V2(t)>1/2
|<V3(t)>|1/3
<V(t)>
Quasi-Stationary Process
2-time Correlation Function
Normalized Voltage
u(t)=V(t) /<V2(t)>1/2
(Matches initial pulse shape)
t1 = 8.0  10-7
2.5
t1 = 5.0  10-7
t1 = 1.0  10-7
1.5
2
<u(t )u(t )>
2
1
1
0.5
0
-0.5
-1
-1.5 10-8 -1 10-8
-5 10-9
0
t -t
1 2
5 10-9
1 10-8
1.5 10-8
Histogram of Maximum Voltage
1000 realizations
Incident Pulse
120
1
100
Mean = .3163
80
Count
Incident Voltage
0.5
0
60
40
20
-0.5
0
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55
|V(t) |
-1
0
0.02
0.04
0.06
t [sec]
0.08
0.1
0.12
max
|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?)
Current
• Generalize to systems consisting of circuits and fields
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
Dielectric
Excitation port
-theoretically
-numerically
-experimentally
Test port
Bow-tie shaped cavity
Features:
Ray splitting
Losses
Additional complications to be added later
HFSS simulation courtesy J. Rodgers