The Random Coupling Model: A User's Guide (updated 16 November, 2006)
Steven M. Anlage, Thomas Antonsen, James Hart, Sameer Hemmady, Edward Ott, Xing Zheng
Physics Department, University of Maryland
Research funded by the AFOSR-MURI and DURIP programs
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 attack. 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. 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.
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)
This matlab code generates an ensemble of normalized 2x2 impedance (z) matrices for a given value of the loss parameter k2/(Dk2 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 (SRad) and 2x2 ray-chaotic cavity S data (SCav) 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 (SRad), and generates the eigenvalues of the 2x2 SCav (EigScav.txt) and ZCav (EigZcav.txt) matrices. Example input files are the ensemble of 2x2 z matrices with loss parameter k2/(Dk2 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.
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.
Suppose I don't know the Q, volume, ZRad precisely, is this a problem? Most likely no. The predicted PDFs are not highly sensitive to the loss parameter. ZRad tends to be slowly varying in frequency. It is most likely OK if you are away from antenna resonances. One possible extension of the RCM (suggested by David Dietz) is to include the possibility of a statistical variation of the loss parameter k2/(Dk2 Q) and Zrad, thus translating into a slightly different PDF of induced voltages.
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 k2/(Dk2 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.
What about cross-talk between ports? Included. 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. 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. Experimental verification is in progress.
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 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.
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 k2/(Dk2 Q), not Q by itself. See also the "wide-band" FAQ above.
What about antenna polarization for 3-D enclosures? 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.
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.
If there are strong periodic contributions to the ray dynamics (e.g. short periodic orbits from parallel planes), these can lead to deviations from RCM predictions. Scars, 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: Mode #50 or greater.
When do you NOT want to use this model?
Enclosure Q ~ 1 or less. No reverberation, no chaos, very lossy.
Enclosure size NOT much larger than wavelength l. Direct numerical solution not sensitive to details. Rule of thumb: enclosure > about 5-10 l. 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 verification work, nor on it's shape, thickness, or the antenna coupling. 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 ZRad 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 SRad 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 k2/(Dk2 Q), where k is the wavenumber, Dk2 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 k2A/(4pQ) (where A is the area of the cavity), whereas in 3D it can be written as k3V/(2p2Q) (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 , formerly known as: cond-mat/0408327 .
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 , formerly known as: cond-mat/0408317 .
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, formerly known as. cond-mat/0504196 . This paper also includes experimental verification of the raw-S and raw-Z variance ratios.
Experimental Verification:
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 cond-mat/0512131 .
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 nlin.CD/0506025 .
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 cond-mat/0606650 .
S. Hemmady, Ph.D. thesis, "A Wave-Chaotic Approach to Predicting and Measuring Electromagnetic Field Quantities in Complicated Enclosures," University of Maryland, 2006.
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
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.