**The Random Coupling Model: A User's Guide (updated 4 June, 2014) **

Steven M. Anlage, Thomas Antonsen, James Hart, Sameer Hemmady, Jen-Hao Yeh, John Rodgers, Gabriele Gradoni, Edward Ott, Xing Zheng

Physics and ECE Departments, University of Maryland

__Research funded by the AFOSR-MURI and DURIP programs, with continuing support from ONR__

This page provides an overview of the Random Coupling Model (RCM) and its use in predicting HPM effects. The RCM is a method for making statistical predictions of induced voltages and currents for objects and components contained in complicated (ray-chaotic) enclosures and subjected to RF fields. It is based on simple universal predictions of wave chaos theory and is quantitatively supported by random matrix theory. The system-specific (non-universal) aspects of the problem are quantified by means of the radiation impedance of the "ports" involved in the problem, as well as prominent short orbits. Please take a look at our papers, presentations, Frequently Asked Questions (FAQs), and caveats below. We hope to make this model useful and accessible to all interested parties, so please give us your feedback.

A comprehensive paper describing the Random Coupling Model for the engineering community has been published as IEEE Trans. Electromag. Compat. **54**, 758-771 (2012). A recent review article on the Random Coupling Model: Gabriele Gradoni, *et al*., Wave Motion**51**, 606-621 (2014). pdf

A power-point presentation that provides an Overview of the Random Coupling Model

We have also developed a first-principles model of fading based on the Random Coupling Model.

Biniyam Taddese's Ph.D. thesis "SENSING SMALL CHANGES IN A WAVE CHAOTIC SCATTERING SYSTEM AND ENHANCING WAVE FOCUSING USING TIME REVERSAL MIRRORS"

**FAQ's about the Random Coupling Model**:

How do I know my enclosure is chaotic for ray orbits? Most enclosures __are__ ray-chaotic (its hard to make them otherwise!) The
“soda can effect” is a good operational test - in other words the impedance of the cavity is a strong function of frequency and details of the internal configuration of the components. See also Fig. 8.2 and related discussion in Sameer Hemmady's Ph.D. thesis.

**Suppose I don't know the Q, volume, Z _{Rad} precisely, is this a problem? **
Most likely no. The predicted PDFs are not highly sensitive to the loss parameter. Z

**I have a wide-band incident signal. Is the RCM still valid?** Yes. First of all, the loss parameter is usually not a very strong function of frequency because it depends on the fixed geometry of the system as well as the fact that the quality factor is a smoothly varying function of frequency with small fluctuations (see Fig. 7 in the paper of Barthelemy, Legrand and Mortessagne, Phys. Rev. E **71**, 016205 (2005). This shows a nice example of how the Ohmic losses vary with frequency in a ray-chaotic microwave enclosure.) Secondly, the frequency dependence of the radiation impedance accounts for the variation of the coupling of the signal in/out of the enclsoure. Rather than assuming the loss parameter is constant, one could extend the RCM to include the variation of the loss parameter k^{2}/(Dk^{2} Q) with frequency to make a more accurate estimate of the PDF of induced voltages from the broadband excitation.

**What is the minimum number of modes for the RCM model to be applicable?** The higher the mode number the better (for best results use overmoded cavities). Generally you would need more than about 50 propagating modes below your lowest frequency of interest. We have empirical evidence that the RCM "degrades gracefully" in the limit of low frequencies. We hope to work on an extension of the RCM to lower frequencies in the future.

**How do I know if my enclosure is overmoded?** A general rule-of-thumb would be to look at the ratio of the maximum to minimum transmitted power at a given frequency for your measured cavity ensemble. The enclosure (cavity) is overmoded if this ratio is more than about 20 dB in magnitude. See also Fig. 8.2 and related discussion in Sameer Hemmady's Ph.D. thesis.

**What about cross-talk between ports? **This is included in the RCM.
See section IV of the paper Phys. Rev. E **74 **, 036213 (2006).

**Does a port have to be on the surface of the enclosure? **No, it can be inside and away from the walls of the enclosure. In fact, this is often the most interesting case because you would like to know the statistics of induced voltages on an electronic component inside the enclosure.

**Does the RCM work for pulsed excitations? **Yes, we have developed the theory for the time-domain. The long-term behavior of a cavity excited with a pulse has been studied in detail (Phys. Rev. E. 79 016208 (2009)). We have also examined the contributions of short-orbits to the impedance (Phys. Rev. E **80**, 041109 (2009), see below), and this will influence the short-time behavior of excited systems. Experimental verification is largely complete (Phys. Rev. E 81, 025201(R) (2010)). We have investigated the sensitivity of time-domain signals to the scattering properties of ray-chaotic enclosures (see the YouTube movie). We have also developed a new sensor paradigm based on the sensitivity of wave scattering in such systems to small perturbations (Appl. Phys. Lett. **95 **, 114103 (2009)).

**What happens if there is a port in the enclosure that is overlooked? **
It is incorporated in the model through the scattering it produces, as well as
modifications to the Q and cavity volume.

**How do I identify the presence of a new object inside my enclosure? **The addition of a new object to the enclosure can be detected by a change in the loss parameter (since the object takes up a certain electromagnetic volume and will have some degree of loss), by the creation or destruction of short-orbit trajectories, or by modification of the radiation impedance of a nearby port.

**How many ports do I need to include in the RCM to describe my enclosure?** We believe that the most important ports to explicitly include are those that are actively adding energy to the system, and those that represent the sensitive/susceptible object(s) in the system. On a related note, one might also consider adding single ports that represent classes of more-or-less identical objects in the enclosure. The present code (Terrapin RCM Solver v1.0) considers only 2-port systems. An extension of this code to a higher number of points can be arranged through Sameer Hemmady.

**Suppose I don't have an ensemble of enclosures, can I still use the RCM? **
Yes, it is often a good approximation to use a frequency average to
substitute for an ensemble average. In addition, our work on removing the influence of short-orbits (Phys. Rev. E **80**, 041109 (2009), Phys. Rev. E 81, 025201(R) (2010), see below) allows one to use single-realizations and smaller ranges of frequency averaging to uncover the universal statistical fluctuations. This means that predictions of induced voltage PDFs will be improved if basic knowledge of short orbits in the system is also available.

I am worried about using a single Q-value. Suppose Q varies from mode to mode?
Ray-chaotic systems tend to show small fluctuations of Q with mode number.
Also, remember that what counts is the loss parameter k^{2}/(Dk^{2} Q), not Q by itself.
See also the "wide-band" FAQ above.

**What about antenna polarization for 3-D enclosures?** This effect is included in the radiation impedance of the ports of interest.

**Does the RCM take into account field variations associated with the presence of a wall?** Yes. The presence of a wall is included in the radiation impedance of the port located near the wall, edge, or corner of a structure.

**Does the short-orbit extension of the RCM take into account multiple short-orbits produced by waves that bounce off of the port?** Yes, these multiple bounce short orbits are naturally included through consideration of the impedance.

**Can the RCM work for multiple enclosures conected by apertures?** Yes, the RCM has been extended to study the statistics of fields in a cascade of connected reverberant systems. See the paper of Gradoni, Phys. Rev. E **86**, 046204 (2012).

**Random Coupling Model: CAVEATS
and additional details **

**What could possibly go wrong? **

If you need to predict the outcome of a specific measurement in a specific situation, then the RCM cannot help The RCM provides only statistical predictions. The 'extended RCM' now includes short-orbit system specific information, in addition to the system specific port information.

If there are strong periodic contributions to the ray dynamics (e.g. short periodic orbits from parallel planes), these will lead to deviations from RCM predictions. Such orbits are now included in the extended RCM. However, scars, cusps, caustics, singularities, and perhaps “Freak Waves” can produce large local enhancements of electromagnetic fields, and they fall outside of the Random Coupling Model.

What is the low-frequency limit of the model? Rough rule of thumb: Apply the RCM to mode #50 and greater. One should have at least 3 wavelengths along each dimension of the enclosure. This is still an open question.

One __failure mode__ of the RCM arises when there is poor coupling to the enclosure (i.e. |S_{Rad,11}|~1 or |S_{Rad,22}|~1). The poor coupling makes it very difficult to de-embed the antenna properties from the impedance data. When S_{Rad} approaches 1 in magnitude, the Z_{Rad} will become very large in magnitude and sensitive to small errors in S_{Rad}. This propagates into large values and large errors for Z_{Cav}. This is likely to cause deviations in the induced voltage PDF.

Another __failure mode__ of the RCM arises if the losses in the system are NOT uniformly distributed. If losses are highly localized (e.g. an electrically large aperture that allows many ray trajectories to exit the system), the assumption of uniform loss may be violated, and the statistical predictions of the RCM may not be strictly valid. Another way to look at it, the properties of the system will be dominated by short orbits (which are non-universal and non-statistical) because the longer "chaotic" orbits will exit the system through the localized loss mechanism (e.g. an electrically large aperture).

**When do you NOT want to use this model?
**

Enclosure Q ~ 1 or less. No reverberation, no chaos, very lossy. The impedance does not fluctuate. It reduces to the radiation impedance.

Enclosure size **NOT** much larger than wavelength l. A direct numerical solution of such a problem is not sensitive to details and should be employed rather than the RCM.
Rule of thumb for validity of the RCM: cube root of enclosure volume > about 3 wavelengths.
Remember: dielectrics inside the enclosure increase it's effective size.

**Random Coupling Model: Does it work in 3 Dimensions?
**

YES! Our results are not in any way predicated on the peculiar 2D “bow-tie” cavity employed in some of our early verification work,
nor on it's shape, thickness, or the antenna configuration, coupling or position.
The theory is well established and works in 3D
situations. In addition to our own 3D verification results, several other groups have performed demonstrations of RCM-like results in 3D :
**Sandia ** (Warne, *et al *.):
Demonstrated the utility of Z_{Rad} in removing the effects of coupling in 3D. They
independently discovered that the 3D statistics are governed by a single loss parameter.
**ONERA ** (Parmentier, *et al *.)
demonstrated the equivalence of the S_{Rad} and <S> in 3D.
They also independently discovered an S-variance ratio relation in 3D data
analogous to the Hauser-Feshbach relation of nuclear physics (see the paper: Phys. Rev. E **73**, 046208 (2006)).

**Loss Parameter: Does it depend on dimensionality?**

The general expression for the loss parameter is k^{2}/(Dk^{2} Q), where k is the wavenumber, Dk^{2} is the mean spacing between squared wavenumbers, and Q is the typical Q-factor of the modes (see above). In 2D cavities the loss parameter can be written as k^{2}*A*/(4pQ) (where *A* is the area of the cavity), whereas in 3D it can be written as k^{3}*V*/(2p^{2}Q) (where *V* is the volume of the cavity), using the Weyl formula for mean spacings of the corresponding closed systems. A general discussion of different ways to determine the loss parameter for a given system is presented in this appendix of Sameer Hemmady's Ph.D. thesis.

Random Coupling Model Publications:

Theory:

Xing Zheng, Thomas M. Antonsen Jr., Edward Ott, "**Statistics of Impedance and Scattering Matrices in Chaotic Microwave Cavities: Single Channel Case**," Electromagnetics **26**, 3 (2006). pdf. This and the following paper are the seminal papers on the Random Coupling Model.

Xing Zheng, Thomas M. Antonsen Jr., Edward Ott, "**Statistics of Impedance and Scattering Matrices of Chaotic Microwave Cavities with Multiple Ports**," Electromagnetics **26**, 37 (2006). pdf.

Xing (Henry) Zheng, Ph.D. thesis, "**Statistics of Impedance and Scattering Matrices in Chaotic Microwave Cavities: The Random Coupling Model**," University of Maryland, 2005.

Xing Zheng, Sameer Hemmady, Thomas M. Antonsen Jr., Steven M. Anlage, and Edward Ott, "**Characterization of Fluctuations of Impedance and Scattering Matrices in Wave Chaotic Scattering**," Phys. Rev. E **73**, 046208 (2006). pdf This paper also includes experimental verification of the raw-S and raw-Z variance ratios.

James Hart, Thomas M. Antonsen, Jr., Edward Ott, "**Scattering a pulse from a chaotic cavity: Transitioning from algebraic to exponential decay**," Phys. Rev. E **79**, 016208 (2009). pdf The first paper on the time-domain version of the random coupling model.

James A. Hart, T. M. Antonsen, E. Ott, "**The effect of short ray trajectories on the scattering statistics of wave chaotic systems**," Phys. Rev. E **80**, 041109 (2009). pdf This paper presents the Extended Random Coupling Model.

Gabriele Gradoni, Thomas M. Antonsen, Jr., and Edward Ott, “**Impedance and power fluctuations in linear chains of coupled wave chaotic cavities**,” Phys. Rev. E **86**, 046204 (2012). pdf.

T. M. Antonsen, G. Gradoni, S. M. Anlage E. Ott, “**Statistical Characterization of Complex Enclosures with Distributed Ports**,” proceedings of the 2011 IEEE International Symposium on Electromagnetic Compatibility, pp. 220-225. pdf

G. Gradoni, Jen-Hao Yeh, T. M. Antonsen, S. M. Anlage, E. Ott, “**Wave Chaotic Analysis of Weakly Coupled Reverberation Chambers**,” proceedings of the 2011 IEEE International Symposium on Electromagnetic Compatibility, pp. 202-207. pdf

Jen-Hao Yeh, Thomas M. Antonsen, Edward Ott, Steven M. Anlage, “**First-principles model of time-dependent variations in transmission through a fluctuating scattering environment**,” Phys. Rev. E (Rapid Communications) **85**, 015202 (2012). pdf An application of the RCM to model fading in communications.

Gabriele Gradoni, Jen-Hao Yeh, Bo Xiao, Thomas M. Antonsen, Steven M. Anlage, Edward Ott , “**Predicting the statistics of wave transport through chaotic cavities by the Random Coupling Model: a review and recent progress**,” Wave Motion**51**, 606-621 (2014). pdf

Experimental Tests and Verification of the RCM:

Sameer Hemmady, Xing Zheng, Thomas M. Antonsen, Edward Ott, and Steven M. Anlage, "**Universal Statistics of the Scattering Coefficient of Chaotic Microwave Cavities**," Phys. Rev. E **71**, 056215 (2005). pdf

Sameer Hemmady, Xing Zheng, Edward Ott, Thomas M. Antonsen, and Steven M. Anlage, "**Universal Impedance Fluctuations in Wave Chaotic Systems**," Phys. Rev. Lett. **94**, 014102 (2005). pdf

S. Hemmady, X. Zheng, T.M. Antonsen, E. Ott, S.M. Anlage, "**Universal Properties of 2-Port Scattering, Impedance and Admittance Matrices of Wave Chaotic Systems**," Phys. Rev. E **74 **, 036213 (2006). pdf.

Sameer Hemmady, Xing Zheng, Thomas M. Antonsen Jr., Edward Ott and Steven M. Anlage, "**Aspects of the Scattering and Impedance Properties of Chaotic Microwave Cavities**," Acta Physica Polonica A **109**, 65 (2006). pdf.

S. Hemmady, J. Hart, X. Zheng, T.M. Antonsen, E. Ott, S.M. Anlage, "**Experimental Test of Universal Conductance Fluctuations by means of Wave-Chaotic Microwave Cavities,” **Phys. Rev. B **74**, 195326 (2006). pdf. The RCM applied to a classical analog of quantum transport.

Jen-Hao Yeh, James Hart, Elliott Bradshaw, Thomas Antonsen, Edward Ott, Steven M. Anlage, “**Universal and non-universal properties of wave chaotic scattering systems**,” Phys. Rev. E **81**, 025201(R) (2010). pdf

Jen-Hao Yeh, James Hart, Elliott Bradshaw, Thomas Antonsen, Edward Ott, Steven M. Anlage, “**Experimental Examination of the Effect of Short Ray Trajectories in Two-port Wave-Chaotic Scattering Systems**,” Phys. Rev. E 82, 041114 (2010). pdf

S. Hemmady, Ph.D. thesis, "**A Wave-Chaotic Approach to Predicting and Measuring Electromagnetic Field Quantities in Complicated Enclosures**," University of Maryland, 2006. This is also available on Amazon.com as a book entitled "The Random Coupling Model."

T. Firestone, M.S. Thesis, "**RF Induced Nonlinear Effects in High-Speed Electronics**," University of Maryland, 2004.

S. Hemmady, T.M. Antonsen, E. Ott, S.M. Anlage, "**Statistical Prediction and Measurement of Induced Voltages on Components within Complicated Enclosures: A Wave-Chaotic Approach,” **IEEE Trans. Electromag. Compat. **54**, 758-771 (2012). pdf This paper offers an accessible introduction to the RCM for the engineering community.

Sun K. Hong, Biniyam T. Taddese, Zachary B. Drikas, Steven M. Anlage, Tim D. Andreadis, “**Focusing an arbitrary RF pulse at a distance using time-reversal techniques**,” J. Electromag. Waves and Apps. **27**, 1262-1275 (2013). pdf

Jen-Hao Yeh, Thomas M. Antonsen, Edward Ott, Steven M. Anlage, “**First-principles model of time-dependent variations in transmission through a fluctuating scattering environment**,” Phys. Rev. E (Rapid Communications) **85**, 015202 (2012). pdf

Jen-Hao Yeh, Edward Ott, Thomas M. Antonsen, Steven M. Anlage, “**Fading Statistics in Communications - a Random Matrix Approach**,” Acta Physica Polonica A, 120, A-85 (2012). pdf

Zachary B. Drikas, Jesus Gil Gil, Hai V. Tran, Sun K. Hong, Tim D. Andreadis, Jen-Hao Yeh, Biniyam T. Taddese and Steven M. Anlage, “**Application of the Random Coupling Model to Electromagnetic Statistics in Complex Enclosures**,” IEEE Trans. Microwave Theory Tech., in press.

Additional Information about the RCM:

The Anlage Statistical Methods Meeting presentation can be downloaded here.

The Hemmady MURI Final Review Meeting presentation can be downloaded here.

**Computer Code for implemetation of the RCM**. (Note that this code has been upgraded, and the new "Terrapin RCM Solver v1.0" is available by special request). Note that Sameer Hemmady has developed a 2.0 version of the Terrapin RCM Solver. Please contact him directly at:

**TechFlow Scientific - A Division of TechFlow Inc.**

*2155 Louisiana Blvd. NE, Suite 3200*

*Albuquerque NM 87111 USA*

*Email: **shemmady@techflow.com*

This matlab code generates an ensemble of normalized 2x2 impedance (z) matrices for a given value of the loss parameter k^{2}/(Dk^{2} Q), called "Ktwiddle" in the code. Generating this large number of matrices and finding their eigenvalues is much much faster in matlab than mathematica. The code assumes the system is time-reversal symmetric (GOE). It runs in Matlab 6.5, and generates a file for input into the program NormtoSZ.nb below.

The first mathematica analysis code (StoNorm.nb) takes the measured 2x2 radiation S data (S_{Rad}) and 2x2 ray-chaotic cavity S data (S_{Cav}) and finds the eigenvalues of the normalized impedance (z, EigZnorm.txt) and scattering (s, EigSnorm.txt) matrices. This code is written in Mathematica 5.0. Example input files are Srad.txt and Scav.txt. This cavity data set is just one rendition of the cavity. Ordinarily one wants to analyze a large number of renditions (~100) of the ray-chaotic cavity to compile statistics for the resulting normalized s and z matrices. This code essentially allows you to find the "hidden" universal statistical properties of your enclosure, by removing the non-universal coupling.

The second mathematica analysis code (NormtoSZ.nb) takes the ensemble of normalized 2x2 z matrices generated form the MatLab code above, along with the measured or calculated 2x2 radiation S matrix (S_{Rad}), and generates the eigenvalues of the 2x2 S_{Cav} (EigScav.txt) and Z_{Cav} (EigZcav.txt) matrices. Example input files are the ensemble of 2x2 z matrices with loss parameter k^{2}/(Dk^{2} Q) = 1.5 (RMTZ_1_5loss2.txt) and Srad.txt. This code essentially allows you to predict the statistical properties of the raw S and Z of your cavity, given the loss parameter of the enclosure and the radiation properties of the 2 ports.

**The revised (26 July, 2006) "Terapin RCM Solver v1.0" User's Guide is available here. **

Please direct questions and comments to 'anlage "at" umd.edu'

Link to Prof. Anlage's research web site

Link to the University of Maryland MURI'01 web site

Link to the Maryland / ONR Center for Applied Electromagnetics web site

This work is supported by the DoD MURI for the study of microwave effects under AFOSR Grant F496200110374, as well as AFOSR DURIP Grants FA95500410295 and FA95500510240, and the ONR / Maryland Center for Applied Electromagnetics, Task A2, Grant No. N000140911190.