Wire and Cable Networks: Experiment and Theory

1. Wire and Cable Networks: Experiment and Theory Lei Chen, TornikeGhustishvili Min Zhou, Ben Frazier, Edward Ott, Thomas Antonsen, Steven Anlage Acknowledgement: This work is supported by AFOSR/AFRL COE Grant #FA9550-15-1-0171

2. Question: How can one predict statistics of signals injected into networks of cables? • Conclusions: • The RCM has shortcomings. • A modified version of the RCM shows promise. • Cable networks can perfectly absorb at certain frequencies • Networks with different statistics can be constructed Motivation You are here

3. Networks of cables are prevalent in military and civilian systems

4. Coupling Model for a Network Network Graph Cable networks can be described by identifying nodes and bonds

5. Statistics of Impedance Matrices for Cavities Well Described By the RCM A Statistical model for the impedance of N-port system (cavity) in which the internal wave dynamics is chaotic is N x N matrix, radiation impedances of the ports. Statistical fluctuations Loss Parameter Mean spacing of resonances ID GRV’s Eigenvalues of a random matrix from GOE Average quality factor

6. Cavity and Network Impedance Analogy Formally Exact Cavity Impedance Formally Exact Network Impedance Open port voltages of eigenmodes cavity eigenmodes resonant frequencies resonant frequencies Random Plane Wave Assumption Leads to RCM If no Random “Plane Wave” Assumption Applies. Requires new statistics

7. Measuring Impedance of Microwave Networks Measure S11(w) for many realizations of the graph T-Junction Network Analyzer Coaxial Cable Apply the Random Coupling Model to the impedance data and see if there is a universally fluctuating impedance in the graph l Microwave Networks without Circulator Earlier work by L. Sirko, Polish Acad. of Sci., Warsaw

8. Characterizing Impedance Waves do not return to the port Resistance and Reactance of one realization, average over realizations and measured radiation impedance Port Re[Z] Im[Z] 50ohms 50ohms Port 50ohms Experimental Setup for measurement of radiation impedance Does the RCM work for Microwave Networks?

9. Statistics The RCM works well for this 1-port Microwave Network! Statistics of Normalized Impedance

10. 2-Port Network Statistics We see an inconsistency in the best-fit loss parameter for diagonal vs. off-diagonal elements The same inconsistency is seen in numerical simulations that mimic the experiment

11. Random Plane Wave Hypothesis? s4 Signals entering a node excite a standing wave on an adjacent bond. If the reflection coefficients are large the bond is isolated from the rest of the graph bond under test s1 s5 s2 s6 s3 waves on edge under consideration waves on adjacent edges Reflection coefficient take to be uniformly distributed

12. Statistics of “trapped modes” GRV Uniform RV P[V/<V>] P[V/<V>] V/<V> V/<V>

13. Applies to cable connecting two cavities s4 edge under test s1 s5 s2 s6 s3 Relative energy stored on line and cavities Energy stored on line Energy stored in a cavity gives trapped mode statistics

14. Representation of Network Impedance Matrix: Basic Equations Cables described by Telegraphist equation with series resistance and shunt conductance Junctions described by impedance matrices Z Some junctions identified as ports to network Normal modes with ports open circuited Currents entering junctions Voltages at junctions

15. Expand Driven Solution in Normal Mode Basis Driven voltages and currents Normal mode basis Mode energy Mode amplitudes Impedance defined in terms of normal mode open port voltages

16. Analysis of Networks (T. Ghustvili)

17. Voltage Statistics - PDF Procedure: Generate network with different numbers of nodes and bonds - Solve for 20k eigenmodes Normalize eigenmode energies Make histograms of V/<V> N = # of nodes B = # of bonds P(V) V/<V> P(V) Modified junctions P(V) V/<V> V/<V>

18. Correlations Between Nodes Depend on Separation N=8, B=3 ViVj kLijmod (2 pi) Lij - shortest path from i to j

19. Conclusion I Two effects cause voltage statistics to depart from simple RCM Trapped modes increase the likelihood of both low and high voltage values. Avoid resonances – Voltage statistics at different nodes in a network (or equivalently ports) have strong correlations that must be incorporated into an RCM-like model.

20. Chaotic Coherent Perfect Absorption (CPA) with Graph Experimental Data on Tetrahedral Graph with Variable Attenuator Only Zero-Crossing Eigenvalues are Shown CST Simulation of Tetrahedral Graph with Variable Attenuator Only Zero-Crossing Eigenvalues are Shown Each trace (color coded) are plotted at fixed frequency with varying attenuation.

21. At certain frequencies and attenuation, the absolute value of the S-matrix eigenvalue goes near zero. Some of the parameter configurations are shown on the left, and we construct the CPA signals using these corresponding eigenstates in the simulation model. Construct CPA Signal from Simulation Result Nearly 100% absorption of injected power is achieved. Reserve the phase of input signal 2

22. Graph With SymplecticSymmetry Antiunitary symmetry Rehemanjiang, A., Allgaier, M., Joyner, C.H., Müller, S., Sieber, M., Kuhl, U. and Stöckmann, H.J., 2016. Microwave realization of the gaussiansymplectic ensemble. Physical review letters, 117(6), p.064101. Circulator Circulator Connecting bonds Subgraph

23. CST Simulation for Eigenvalue Spacing Set up the proposed graph in CST, and solve the eigenvalues. The simulated spectral level-spacing distribution agree well with Wigner’s GSE prediction!

24. RCM Fit - requires different loss parameters GOE Fit GUE Fit

25. Planned Experiment Single Inverter Schematic Circuit Design Fabricated Board Chaotic Cavity SMA Connector Port 2 • Simple circuit (6 CMOS inverters in series) placed inside a wave chaotic cavity • Single chip (TI CD4069UBE CMOS hex Inverter), circuit powered by 9V battery • Measurements will investigate • Changes in impedance in both time and frequency domains • Increase in leakage current • Errors (inverter state changes) induced by RF coupling • Impacts with circuit both powered and unpowered, and with and without a driving antenna at port 1

26. Summary • Conclusion • The two-port Tetrahedral Graph shows non-universal statistical properties. (Lei Chen) • The theoretical explanation for non-universal statistical properties of the graph is being investigated (TornikeGhutishvili) • Probability distribution of normalized eigen-frequency spacing from proposed GSE graph agrees well with Wigner’s GSE prediction. (Lei Chen) • Impedance statistics of proposed GSE graph cannot be fit by the RCM GOE or GUE predictions. • Experiments on circuits in reverb chamber being initiated (Ben Frazier)

27. Phase Difference between Open & Short Bonds Still requires fine-tuning of the phase! • Phase difference should be 180 degree. • Phase difference should be frequency-independent. After adjustment Before Rehemanjiang, A., Allgaier, M., Joyner, C.H., Müller, S., Sieber, M., Kuhl, U. and Stöckmann, H.J., 2016. Microwave realization of the gaussiansymplectic ensemble. Physical review letters, 117(6), p.064101.

28. Experimental Impedance Statistics In total, 165() realizations are obtained by choosing 8 coaxial cables out of a set of 11 cables. ] ] ] ] ] ] ] ]