The paper on which this article is based was originally presented at the 2016 IEEE EMC Symposium, where it received recognition as the Best Symposium Paper. It is reprinted here with the gracious permission of the IEEE.
Statistical model-based EMC design is evolving for complex electronic networks installed in enclosed spaces typical of spacecraft, aircraft and automobiles. The process starts with prediction of the electromagnetic field strength in each approximately reverberant EM field domain in the vehicle. At high frequencies and in electrically large domains, both temporal and spatial field response is highly sensitive to small geometric details which are uncertain within normal manufacturing tolerances; and which are not repeatable experimentally. Given this uncertainty, it is not practical to model design details explicitly. For design purposes, it is only meaningful to predict the statistical time average and space-average mean electric field, and the maximum expected field strength.
Bremner  has shown how a modal expansion basis for each reverberant EM field can be used to predict the first two statistical moments of the response – mean and variance – from the physical EM parameters of each subsystem. With selection of a suitable probability density function   , it is then possible to predict both mean and maximum expected response in each domain. Bremner showed that a Log Normal distribution for the total energy of a reverberant enclosure provides a good estimate of maximum predicted electric field response in three different size reverberant volumes. In the EMC design process, the predicted maximum expected electric field is normally used to define the radiation immunity test specification for all electronic units and network systems operating in that domain.
In the common event that certain critical electronic units or subsystems cannot be easily “test qualified” or re-designed to operate in the maximum expected EMC environment, it often becomes necessary to design EM field attenuation measures, typically in the form of shielding enclosures, avionics boxes, etc. The EMC design problem is to define the amount of EMC attenuation required for the critical electronic unit(s) –
typically quantified as the shielding effectiveness. To do this, electronic system designers need a simple means to estimate the EM field induced currents (both differential and common mode) in power lines and signal (transmission) lines – for both external cables and traces on printed circuit boards. At high frequencies and for electrically large shielding enclosures, the spatiotemporal EM field response will be sensitive to uncertain geometric details. In the presence of this inherent uncertainty, the only statistically meaningful model is one that seeks to predict the mean and maximum expected response in wire conductors.
This paper introduces a simple statistical power balance model for prediction of the maximum expected current in a wire conductor inside an aperture enclosure. It calculates both the statistical mean and variance of the immission entirely parametrically – i.e., from only the physical design parameters of the problem. Familiar probability density functions are then used to predict the maximum expected immission for design purposes. The statistical power balance model requires minimal EMC design information and solves orders of magnitude faster than existing numerical models, making it ultimately viable – in principle – for scaled-up, full system-level modeling.
Foundations of Statistical EMC Modeling
Hill  has laid the foundations for statistical power balance modeling of EM fields in reverberant, multi-modal domains. While full wave simulation tools are an alternative model-based design approach, Trout   and others  have shown that deterministic models do not reliably estimate the predicted EM field at a point, without statistical reduction to account for uncertainties in the EM source field and uncertainty in EM parameters of the reverberant domain.
Junqua, et al.  and others  have shown how Hill’s statistical power balance approach could be applied to a quite general network of reverberant (multi-modal) electromagnetic subsystems. Junqua’s approach has been applied to modeling the EM environments in connected cavities, typical of a space launch vehicle payload fairing . Encouraging results were obtained for the statistical mean (space-time average) electric field in each subsystem. However, for space vehicle EMC design purposes, it is also important to be able to predict the maximum expected EM field strength.
Kovalevsky et al.  have recently shown that the statistical energy analysis model can predict the mean and variance of EM response for an aperture cavity in a reverberation chamber. The authors suggest that the predicted statistical moments can be used with a “double Rayleigh distribution” reported by others to predict the maximum expected field strength in the aperture cavity. However, no results are presented to show validation of this aspect of the model.
This paper extends the foregoing research work to validate statistical power balance modeling approach for prediction of the statistical mean and maximum expected current induced in both the shielding enclosure and in a wire conductor located within an aperture cavity, as shown in Figure 1. The statistical model results are compared with S-parameter measurements and full wave EM simulation results for configurations similar to those studied by Tait  and by Rajamani et al. .
As for reverberation chambers we can adequately quantify the electric field by one of three parameters – the total electromagnetic energy U (or energy density W = U/V ) of the reverberant field; or the total magnitude of the electric or magnetic field strength ET, HT; or the magnitude of a vectorial component of the electric or magnetic field; eg., Ex, Ey, Ez. The total electromagnetic energy  is
For a statistically uniform electric field in a well-stirred reverberation chamber, it has been shown  that the power balance principle applies
That is, the power input from an external source Pin must be balanced by the total power dissipated by all losses, where Q is the cavity Q factor (η is the corresponding damping loss factor η=1/Q.)
Subsystem Energy Definitions
The total energy of the reverberation chamber is
where e is the permittivity of air, VR is the volume of the chamber and |ET ,R|2 is the space-time average of the mean-squared total electric field vector in the chamber.
Likewise, the total energy of the aperture cavity is
The total energy of the wire conductor is
where μ is the permeability of air surrounding the wire, Aw, Lw are the cross section area and length of the wire and |HT ,W |2 is the mean-squared total magnetic field vector in the chamber.
System Power Balance Equations
For multiply-connected reverberant subsystems, there is a power balance equation for each that will include energy transmitted to connected systems, as shown in Figure 2.
In the field of vibro-acoustics, Lyon  has shown that the net power flow 〈Pij 〉 between any two reverberant energy subsystems (i.j) – averaged over an ensemble of uncertain parameters – is proportional to the difference in their modal energy levels
where ni (ω)=ΔNi /Δω is the modal density and hij (ω) is a frequency-dependent coupling loss factor. Writing the power balance equations for each of the three subsystems in the canonical model yields a power balance matrix which can be solved for the average reverberant energy levels 〈UR 〉, 〈UC 〉, 〈UW 〉
Loss Factor Notation
Previous work on statistical power balance modeling used Q as the quantity describing energy dissipation or damping and effective cross section 〈σij 〉 in units of area, to quantify strength of coupling energy transmission. In this model, we prefer to use dimensionless loss factor notation for each. The damping or dissipation loss factor ηj is simply the inverse of the Qj factor. Likewise, there are simple linear conversions between coupling loss factors ηij and coupling effective section (area). The loss factor notation has some diagnostic advantages, as will be seen from the analysis presented in this article.
Coupling loss factors defined in this way satisfy the reciprocity relationship 
Parametric Variance Model
For reverberant wave fields in perfectly square or rectangular domains, Lyon  provides a non-parametric variance formulation based on a Poisson distribution of natural frequency spacings. For more realistic structures with quite general and irregular boundaries, Weaver  has shown that the natural frequency spacings are more correctly described by the Gaussian Orthogonal Ensemble (GOE) from random matrix theory. Weaver provides a non-parametric variance formulation for the energy density of a reverberant wavefield excited by one or more point excitations. The relative variance r2[U]=Var[U]〈U〉2 takes the form
where L and N are respectively the number of receiver and source positions used to calculate cavity energy. The “modal overlap” m is the principal term describing the uncertainty in resonance frequency. Modal overlap is a function of the frequency, the mean damping loss factor ηj, and the modal density (nj) of the reverberant enclosure 
mj (ω) = ω nj (ω) ηj (ω), where nj = Vj ω3/π2c3
The parameter K is a measure of the spatial variance of the cavity mode shapes ψr(x) which asymptotes to a value defined by wavefield dimensionality 
Probability Density Functions
Hill  has shown that the probability density function (PDF) in a directly excited reverberant EM field is Rayleigh distributed. The maximum expected field strength for any given probability can be scaled on the independent component mean squared level
σ2 = 〈|S31|2〉/2.
Bremner  has shown that the space-averaged electric field – which is proportional to the energy of the EM field – is close to a Log Normal distribution. The maximum expected energy level can be estimated from the mean energy level 〈UR〉 provided by the power balance model (7) and the Weaver GOE relative variance (9), transformed to Log[U] statistics.
In the closely related field of vibro-acoustics, Lyon  and Langley  have shown how the energy variance can be estimated for each subsystem in a power balance matrix model. The implication is that the probability density function – required to predict the maximum expected field strength – will be the same as for the single, directly excited reverberant subsystem. That is, the electric field in the aperture cavity – which drives the S31 immission in the wire conductor – will be Rayleigh distributed, as for the outer reverberation chamber.
However, Holland and St. John  have reported that a Lehman distribution is a better PDF for the current in a wire within an externally-illuminated enclosure. Other researchers have recently shown that the PDF of the electric field in a nested cavity may have a “double Rayleigh” distribution  . This important PDF is evaluated in what follows.
The authors are engaged in a program to compare both numerical models and statistical models with measurements taken in a reverberation chamber at NASA Kennedy Space Center, shown in Figure 3.
The reverberation chamber (RC) is an ETS-Lindgren DEI RF enclosure Model 26. Its dimensions are height 2.31 meters, width 3 meters and depth (length) 5.035 meters. The walls are lined with copper mesh.
The aperture cavity is a cardboard box covered inside and outside with aluminum foil. The box dimensions are height 0.665 meters, width 0.67 meters and depth (length) 0.98 meters. Apertures of different sizes were tested, ranging from a small circular aperture of 0.20m radius to a large square aperture 0.35m x 0.35m.
The transmit antenna (Tx1) is an ETS-Lindgren 3115 Double-Ridged guide antenna. An identical second antenna was used as reference antenna (Rx2) for measurement of the Q factor of the reverberation chamber.
The immission of the wire conductor (Rx3) was measured as an S parameter normalized by the power input to the source antenna, as documented for similar experiments by Tait  and Rajamani . The S parameters were measured with a vector network analyzer (VNA); Anritsu model 37369D.
The wire conductor is a 1mm radius copper wire of length 60 mm. The wire is connected to the top aluminum surface of the aperture cavity via a 50 ohm SMA connector. The aluminum conducting surface of the cavity forms an effective ground plane such that the electromagnetic reception of the wire conductor is somewhat similar to that of a traveling wave antenna.
Planned measurements with a newly commissioned mode stirrer were not completed in time for this paper. As an interim measure, a smaller ensemble of S-parameter measurements were made with the wire conductor in nine different positions in the aperture cavity, as shown in Figure 4.
Full Wave Numerical Simulation
A full wave numerical simulation of the test configuration was also conducted, using CST Studio frequency domain solver as shown in Figure 5.
Numerical model results for S31 in Figure 6 do not completely overlay with test results at specific frequencies. The agreement improves if both data sets are spatially averaged over nine conductor locations and are frequency band averaged.
This common observation   is one motivator for use a statistical power balance model; it is only statistically meaningful to compare the mean and maximum expected value of EM fields in enclosures. The other motivation is that the statistical power balance model solves in seconds and could be scaled up for EMC assessment of full system-level electronic networks; whereas the full wave numerical simulation – even for this simple canonical – took eight hours to solve on a desktop computer.
Additional simulations were performed with the source antenna oriented away from the aperture cavity and located at different positions in the RC. Since the frequency-band averaged S31 results did not change significantly, it was concluded that there are no strong “direct field” components in the measured S31 responses.
Statistical Model Results
The estimated Q factors for each of the three subsystems is shown Figure 7.
The modal overlap mj is calculated from the estimated Q factor and the corresponding modal density for each reverberant sub system. Results are shown in Figure 8.
This is a significant result. A modal overlap equal or greater than approximately unity ensures that the non-parametric variance models are valid. The modal overlap of the aperture cavity asymptotes to unity because losses from the wire conductor – acting as an antenna – control Q at low frequencies. The guided electromagnetic field on the wire conductor has a low Q since it is typically connected to an approximately matched, load impedance.
Energy in Aperture Cavity
The power balance model was used to predict the electromagnetic field strength in the cavity, with a 20mm radius circular aperture. The result in Figure 9 demonstrates that model predictions for the mean total energy are in satisfactory agreement with test results reported by Kovalevsky .
EM Pick-up (Immission) in Wire Conductor
The statistical model was used to predict the electromagnetic power pick-up in the wire conductor connected to a reference load of 50 ohm. Comparison of the S31 parameter from source antenna power input to wire conductor power output is shown in Figure 10. The statistical model (ensemble average) is in good agreement with the mean of the same S31 computed with a full wave numerical model.
For completeness, the results for S21 – for power output at a well- matched reference antenna in the RC – is also shown in Figure 10.
Shielding Effectiveness of the Aperture Enclosure
The dB difference between S31 and S21 in Figure 10 (bottom) is a measure of the shielding effectiveness  of the imperfect enclosure with aperture.
The relative SE* for the large aperture enclosure compared with the small aperture enclosure is plotted in Figure 11. Note that there is effectively zero shielding effectiveness with the large aperture. The statistical model provides a good prediction of ensemble mean SE. (ie. frequency band averaged) numerical model solution.
Observed Probability Density Functions
The energy level in the aperture cavity – proportional to <|S31|2 > – exhibits an approximately log normal distribution, which exhibits skewness for the small aperture case, as shown in Figure 12.
However the mean-squared electric field – proportional to |S31|2 – appears to deviate from the expected Rayleigh statistics. The histograms in Figure 13 show better agreement with the Double Rayleigh distribution  than Rayleigh statistics; Chi-squared with 2 dof .
The normalized cumulative density function (CDF) from  was therefore used to predict the maximum expected immission at the wire conductor
Maximum Expected Response
The Double Rayleigh CDF yields a P97.5 maximum expected level – for both small and large apertures – of
The resulting statistical model prediction of both mean and maximum expected immission in the wire conductor is shown superimposed on the 9 position ensemble of numerical model results in Figure 14; for both large aperture (top) and small aperture (bottom).
These preliminary results are encouraging. Further investigation is warranted on the robustness of the PDF assumptions being made.
A simple statistical power balance model has been introduced for predicting both the mean and the maximum expected voltage induced in a wire conductor shielded from a strong electromagnetic field by an enclosure. Preliminary comparisons with experiments and full wave numerical modeling results are promising. The model can predict the shielding effectiveness of the imperfect (aperture) enclosure and the maximum expected current induced in a wire conductor in the shielding enclosure, even at low frequencies where modal overlap is less than unity.
While this is a very simple configuration, it is important to note that the statistical power balance model requires minimal EMC design information and solves orders of magnitude faster than existing numerical models. As such, this “canonical” model demonstrates that statistical EMC modeling is potentially viable for scaled-up, full system-level modeling of complex electronic networks in spacecraft, aircraft and automobile.
The authors gratefully acknowledge funding for this work by NASA Kennedy Space Center under Special Projects contracts, including Contract No. NNK15LA78P.
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