# Modeling LLSF

1: EMA HIRF Background (IEL)
2: HIRF Certification
3: Methods of Modeling Cable Harnesses For LLSC Evaluation
4: Computational Electromagnetics: Methods of Modeling Aircraft for LLSF Evaluation

4.1 Introduction

4.1.1 Figure of Merit

4.2 Canonical Box Model

4.3 Power Balance Method and Statistical Electromagnetics

4.3.1 Cavity Statistics
4.3.2 Cumulative Distribution Functions
4.3.3 Field Homogeneity
4.3.4 Field Isotropy
4.3.5 Shielding Effectiveness and PWB
4.3.6 Angle of Incidence Variations

4.4 Evaluation of Absorption Losses in Boeing 707

4.4.1 Simplified Aircraft Model Setup
4.4.2 Shielding Effectiveness
4.4.3 Decay Time Constant and Q
4.4.4 Cabin Analysis

4.5 Summary and Conclusions
References

## 4.1        Introduction

For an open space problem, electromagnetic fields at any given location are deterministic and stable. It is easy to set up such a problem in EMA3D, and obtain the fields at any desired location. While the fields are computed in time domain, they can also be Fourier transformed into the frequency domain and analyzed.

It is difficult to model fields at higher HIRF frequencies due to some computational limitations and the nature of the FDTD codes. While a big advantage of FDTD is that a broadband result can be obtained in a single run, the limitations are listed below:

a)      Problem space, or the total number of cells in the problem space, is limited by the memory
required to perform computations. The total memory is a trade-off between cell size and
volume of the problem space: the smaller the cell size for a given volume, the more memory
is required.

b)      Cell size is related to the FDTD time step in such a way that a smaller cell size requires a smaller time step for the FDTD method, and therefore more total time steps are required to complete a
specific simulation time computation. This limits the highest resolvable frequency of the
computation, which is traditionally given by fmax = c / (cell size)/10, where c is the electromagnetic wave velocity in free space. Usually, with the help of some canonical studies, fmax = c / (cell size)/5 can be considered. Failure to adequately resolve the higher frequencies results from electromagnetic wave dispersion that impacts the accuracy of computational results.

c)      From a) and b), execution time for an electromagnetic model goes up by a factor of 16 with a
cell size reduction by a factor of 2. This means that as the physical size of the model increases, the computational upper frequency limit decreases.

However, regulations by the FAA require HIRF certification up to 18 GHz, and in some cases up to 40 GHz, and it is difficult to obtain deterministic CEM solutions above ~ 2 GHz. Testing is usually done to assess how the HIRF environment would affect a given aircraft and its equipment, however, these tests are difficult to perform, expensive, and it can be difficult to determine a worst case environment with the test performed. Testing also does not help with minimizing the costs of the design process.

In addition, as small changes in geometry are made inside a cavity, as in a typical design process (such as moving a cable by a few centimeters), the frequency response of the measured fields can change drastically at any given location.  Because of this, and also the fact that things move around on an aircraft and one aircraft of the same model is never exactly like another, it can be useful to describe fields statistically.

In this chapter, we investigate several CEM techniques for modeling electromagnetic fields inside cavities using EMA3D. We start with a simplest canonical model, and move on to a somewhat more sophisticated model of a Boeing 707.  The effects of absorption losses and aperture losses on the shielding effectiveness are studied. In addition, statistical studies of electric fields inside the canonical models are conducted at higher frequencies (up to 12 GHz), the conditions for considering the fields as statistical are also presented, and the power balance method is introduced for frequencies too high to be simulated using FDTD.

To illustrate how moving objects around can affect internal fields inside a cavity at higher frequencies, we construct a metal walled room with a metal desk, cabinet and conduit as shown in Fig. 4.1.  The room is 3.05 m x 3.05 m x 2.44 m in size.  A dipole source is located in front of the desk at a height of 0.65 m above the floor.  The fields were sampled at 567 points inside the room (see Fig. 4.2), with point 305 randomly selected as a typical test point.

Figure 4.1 Sample Problem Geometry.

For the purposes of this demonstration, the dipole source was oriented along the x-direction (out of the paper, parallel to the conduit).  A double exponential excitation source was used, such as that described in Ch. 3 but with frequency content up to 4 GHz.

Two different variations in the geometry for the simulation were evaluated in order to emphasize some of the key aspects of the utility of the statistical methodology.  First a case was run where the geometry was as depicted in Fig. 4.1 and 4.2.  This case was labeled “no box” because another variation included a small metal box on top of the desk in the simulation.  This second case was labeled “box”.

Figure 4.2 Test point sampling within the room volume, “no box” case.

The simulated frequency domain data for the “no box” case, as one might expect, are quite erratic as shown in Fig. 4.3.  The “box” case added a small box to the room geometry as depicted in Fig. 4.4. Figure 4.5 illustrates the difference between the predicted results for the cases with and without the presence of the box.  It can be seen from this figure that the presence of the box has caused significant perturbations to the amplitude spectrum of the simulation.  However, the statistical distributions for both data sets still maintain the same character, as shown in Fig. 4.6 by the cumulative distribution functions for the “box” and “no box” cases.

The intent of this variation was to illustrate the fact that small changes in test geometry can result in large changes in localized response.  This imposes larger uncertainties on measurements since the values determined are very dependent on many details of the test geometry. Figure 4.3 Typical field squared variations at point 305 for the “no box” case. Figure 4.4 Room geometry for the “box” case. Figure 4.5 Difference of squared field component for “box” and “no box” cases. Figure 4.6 Cumulative distribution functions for the “box” and “no box” cases.

### 4.1.1        Figure of Merit

Electric field attenuation is the figure of merit for the LLSF aircraft certification. However, most of the HIRF research measurements, as well as the statistical electromagnetics framework which will be described in later parts of this chapter, use shielding effectiveness (SE) to evaluate HIRF. For the sake of consistency, we also use SE in our studies. The definition of SE and its relationship to aircraft attenuation are presented in Equation 4.1:

where Ec is the electric field inside the cavity, and Ei is the incident electric field.

In addition to the SE the quality factor Q and decay time constant τ are also studied.  Q is a measure of how well a cavity stores energy, and can be derived from extracting τ at different frequencies in a given cavity. Q can be obtained directly from τ :

(4.2) where ω is the angular frequency. Details about extracting τ are presented later in the chapter.

## 4.2        Canonical Box Model

A schematic of the canonical box model is shown in Fig. 4.7.  It consists of a 1.0 x 0.8 x 0.8 m3 rectangular box with a circular aperture. The aperture radius is r1 = 0.15 m, which means that it is in an electrically large regime above 500 MHz.  Two spheres inside the box act as absorbers and/or scatterers, each with a radius r2 = 0.1 m. The box is illuminated from the outside with a broadband Gaussian plane wave with frequency content up to 12 GHz, coupling inside the cavity at a 45° angle (elevation and azimuthal) through the aperture.  The model cell size is 5 mm, which gives the highest frequency resolution of  f = c / (cell size)/10 of 6 GHz.  However, we were able to check that the frequency resolution is good up to fmax = c / (cell size)/5 by running the one of the model cases at half the cell size and checking that the results are the same up to fmax. Therefore the reported results are shown for frequencies up to 12 GHz.

The line of 49 points inside the box from the lower right corner to the upper left are locations where the rectangular field components Ex(t), Ey(t), and Ez(t) were recorded. These locations are labeled 1 through 49. Eight different simulation cases were considered and compared to each other, and are listed in Table 4.1. The cases vary the conductivity of the spheres as well as the conductivity of the walls in order to observe the absorption and wall losses. There are two variations for the wall conductivity: perfect electrical conductor (PEC), and a composite material with conductivity σ  = 104 S/m resembling a typical carbon fiber composite. Figure 4.7 Canonical box model (top) and illumination waveform in time domain (bottom left) and frequency domain (bottom right).

Table 4.1 Canonical Box Simulation Cases

 Case No. Description 1 Empty cavity (spheres omitted), PEC walls 2 Sphere A omitted, sphere B σ = 0.01 S/m, PEC walls 3 Sphere A and B σ = 0.01 S/m, PEC walls 4 Sphere A PEC, sphere B σ = 0.01 S/m, PEC walls 5 Empty cavity (spheres omitted), composite walls (σ = 104 S/m, 1 mm thick) 6 Sphere A omitted, sphere B σ = 0.01 S/m, composite walls 7 Sphere A and B σ = 0.01 S/m, composite walls 8 Sphere A PEC, sphere B σ = 0.01 S/m, composite walls

The time-domain waveforms obtained at location 19 for all eight cases are shown in Fig. 4.8.  It is worth noting that as more losses are added to the model, the faster the time-domain waveforms decay. This is consistent with the fact that more losses lead to lower Q and shorter decay time.

Electric fields in frequency domain are presented in Fig. 4.9.  Here, the resonance peaks become lower and broader when losses are present, also indicating lower Q. All electric fields presented in the frequency domain are normalized to the source. The SE for each case as defined by equation 4.1 is presented in Fig. 4.10 at probe location 19.  The SE curves at that location look very similar for all eight cases, but with visibly higher resonances for Case 1, which has no absorption or wall losses. Figure 4.8 Time domain electric fields at location 19 for each simulation case. Ez overlays Ex and Ey. Figure 4.9 Frequency domain electric fields at location 19 for each simulation case. Figure 4.10 Shielding effectiveness at location 19 for each simulation case.

It is sometimes useful to look at time-domain waveforms that represent the total average power inside a cavity.  If the decay of these waveforms is exponential, it is possible to extract Q as a function of frequency after running the average power waveform through a 50 MHz bandwidth filter. This is particularly interesting because aircraft tests often use band limited Gaussian white noise (BLGWN) to determine electric fields and Q inside cavities.

In order to get the average power in frequency domain, we first take the raw time series waveforms at each location and Fourier-transform them into the frequency domain. Then the total squared electric field is normalized to the incident electric field as Etot2 = (Ex2 + Ey2 + Ez2)/Einc2 and location averaged to get the average cavity power, E2tot,av(f). A 50 MHz digital filter for various center frequencies is applied to E2tot,av(f). Then an inverse Fourier transform is taken to get the bandwidth limited time-domain waveforms.  Afterward, an exponential fit is applied to the bandwidth limited results to obtain the decay time τ, and equation 4.2 is used to calculate Q.  Figure 4.11 shows the time domain waveforms bandwidth limited at 6 GHz for each simulation case plotted on a natural log scale.

Some cases in Fig. 4.11 show an exponential decay (linear on a natural log scale) only up to approximately 1 μs, after which the time series appears to transition to another exponential decay with slower decay time.  Here, only the faster portions of the exponential decay were fitted to obtain τ.  We do not have a good explanation for the slower exponential decay portion of the waveforms at this time. In addition there is a direct energy coupling peak around 100 ns, which we also ignore for the fit (see, for example, case 3 in Fig. 4.11).

This procedure is analogous to real life frequency stirring, and is only effective when the cavity is overmoded. In this case, the cavity becomes overmoded above 1 GHz, and any averaging or bandwidth limiting below this frequency would not make physical sense.  Figure 4.12 presents the Q extracted for each simulation case in the frequency range from 1 to 12 GHz. It is quite obvious that the cases with shortest decay times have lower Q values and smoother Q curves.

For aperture coupling into an overmoded cavity it is possible to use a power balance method for the evaluation of the shielding effectiveness as described by Hill . This is beneficial at frequencies high enough where the FDTD method becomes too large for the available computing resources and when the fields inside the cavity are statistical. In the following section we explore the statistical conditions for the applicability of the power balance method, check whether it applies to our canonical box model, and compare the power balance method results to the FDTD results. Figure 4.11 Bandwidth limited time domain waveforms at 6 GHz. Figure 4.12 Cavity Q for each simulation case.

## 4.3        Power Balance Method and Statistical Electromagnetics

### 4.3.1        Cavity Statistics

For full application of the statistical theory to the power balance method (PWB), it is necessary that the field distributions within the cavity meet the reverberant criteria upon which the theory is based.  The main characteristics of these criteria include:

• The fields in the cavity should be isotropic, that is all field components should have equal amplitudes over an ensemble average
• The field distributions in the working volume of the cavity should be uniform, that is all locations should look the same over an ensemble average
• The ensemble mean of each electric field component should be zero
• The square of any field component should follow the Chi-square with two degrees of freedom statistics
• The total electric or magnetic field squared should follow the Chi-square with six degrees of freedom statistics
• The total field squared (E2 + H2) should follow the Chi-square with twelve degrees of freedom statistics

It has been theoretically demonstrated that the filed distributions within a reverberant closed cavity follow chi-square statistics .  A brief overview of the chi-square probability density function (PDF) and cumulative distribution function (CDF) equations for several cases will be presented here as background material.

The generating function for the Chi-square distributions is:

(4.3) where Γ is the usual gamma function for integral argument, υ is the number of degrees of freedom (DOF) and x is a Chi-square variate that is the square or sum of the squares of a field component or combination of field components with standard normal distributions.  For the cases to be described here, the gamma function is given by: Important cases for the PDF and CDF equations for the case of the electric field are provided below.

Electric field component (2-DOF; Ex example)

(4.4) (4.5) Total electric field (6-DOF)

(4.6) (4.7) where E2 is the square of the total electric field ( E2 = Ex2 + Ey2 + Ez2).

Some of the useful measures inherent to the Chi-square statistics include:

Mean: The mean value (μ) the Chi-Square distribution with υ-DOF is equal to the number of degrees of freedom for the distribution.

(4.8) Standard Deviation:  The standard deviation for the Chi-Square distribution with υ-DOF is equal to the square root of two times the number of degrees of freedom for the distribution.  It is a measure of the spread of the distribution.

(4.9) Variance:  Variance is a parameter that measures how dispersed a random variable’s probability distribution is.  It is equal to the square of the standard deviation.

(4.10) Coefficient of Variation:  The measure is the ratio of the standard deviation to the mean.

(4.11) Any of these measures can provide a metric for estimation of how well a particular data set matches the ideal chi-square statistics.  Other measures of conformance have been proposed, such as the Kolmogorov-Smirnov goodness-of-fit test .  However, this test can only tell you that a data set belongs to the Chi-square distribution; it cannot tell you that it does not.  Consequently we generally fall back on a more subjective evaluation based on the coefficient of variance which is expected to be a value of 1.0 for the square of a field component and 0.577 for the total electric field.

### 4.3.2        Cumulative Distribution Functions

The CDFs presented in Fig. 4.13 come from the canonical box data sets.  They compare the predicted and theoretical PDF curves for the square of the individual components of the electric field .  The distributions have been normalized by the square of the source field. The data used for these plots were generated by extracting the frequency domain data values in a specified bandwidth around a center frequency from broadband excitation.  This is termed frequency stirring (as opposed to mechanical stirring achieved with a stirring paddle rotated to a set of unique orientations).  The center frequency for the CDFs presented here is 11 GHz with an inclusion of data values 25 MHz on either side of the center frequency for a bandwidth limiting of 50 MHz. Figure 4.13 11 GHz band-limited cumulative distribution functions for E2x(f), E2y(f), E2z(f), all normalized to the incident field. E2x(f) and E2y(f) overlay.

In order to have statistically uniform fields, however, the condition of: E2x(f) = E2y(f) = E2z(f) has to be met.  In all cases E2x(f) ≈ E2y(f), indicating that the fields are indeed uniform in x and y, but not in z. Here, Case 4 comes closest to meeting the field uniformity condition. Case 4 does not have wall losses, and only one of the spheres contributes to the absorption losses, whereas the second sphere, which is PEC, scatters the fields, contributing to the field uniformity. Thus we can conclude that the fields inside our canonical cavity are not quite uniform.

The analysis is extended to 6-DOF with the evaluations of the CDFs for the total electric field shown in Fig. 4.14.  Comparing the calculated 6-DOF CDFs to the theoretically derived CDFs, it is obvious that only Cases 4 and 8 have very good fits, again likely due to the scattering of the PEC sphere in both cases. Figure 4.14 11 GHz band-limited cumulative distribution functions of total normalized field, compared with theoretical 6-DOF Chi-squared curves.

### 4.3.3        Field Homogeneity

Applying the metric that the coefficient of variation (COV) should be 0.577 from Equation 4.11 for the total electric field case as a measure of field homogeneity, it is seen that only cases 4 and 8 have a very good comparison with the COV metric. Otherwise all of the other cases appear to have reasonably good overlays. This is shown in Fig. 4.15. For the plots in Fig. 4.15, the data were frequency stirred and presented for all 49 measured locations, which is analogous to both frequency and mode stirring. The same frequency stirring technique was applied as described in the previous sections.

These plots demonstrate the variability of the COV metric as a function of the center frequency for the extracted data sets.  As expected, the COV metric converges on the theoretical value as the center frequency for the band width limited selected data is increased, indicating an improved correspondence with the chi-square statistics at higher frequencies.  The reasons for this are fairly obvious.  As the frequency increases, in general so does the mode density until the cavity makes a transition from non-reverberant to reverberant behavior.  This transition point is generally referred to as the lowest usable frequency (LUF).  The LUF for a closed unloaded cavity is approximately 3 times the lowest resonant frequency for the smallest dimension of the cavity.  The actual LUF is dependent on many factors including the real number of modes present (ideally > 60), losses, etc.  For the canonical problem reviewed here, it is clear that the LUF is around 3 GHz or ~5 times the estimated LUF of 560 MHz. Figure 4.15 50 MHz band-limited COV as a function of frequency for all eight cases. The calculated COV is in blue, and should be compared to the theoretical limit for 6-DOF (red).

### 4.3.4        Field Isotropy

A final look at the reverberation metrics, a measure of isotropy, is shown in Fig. 4.16.  To develop these plots, the mean values of the square of the field over a frequency band width of 50 MHz around the same set of center frequencies as used for Fig. 4.15 were computed.  Then the mean values of these frequency value means were determined from all spatial locations and the results plotted.  This was done for each electric field component as well as for the total electric field.  For comparison purposes, the total electric field result was divided by 3 based on the reverberant assumption that all field components should have equal amplitude.

For statistically uniform fields, these curves should converge.  None of the cases converge exactly, though cases 5-8 look best. Figure 4.16 50 MHz band-limited mean of the squares for each field component, together with the 1/3 mean of the squared of the total field, as a function of frequency.

### 4.3.5        Shielding Effectiveness and PWB

A measure of the shielding effectiveness of regions of an aircraft is of high importance for manufacturers and their certification process. Calculations of the SE of the canonical box investigated here are provided in Fig. 4.17. The SE was calculated according to Equation 4.1, but using ensemble average fields, i.e., 50 MHz frequency stirred and location averaged. We also compare the SE curves with those derived from the power balance (PWB) method, which is applicable only when the fields inside the cavity can be considered reverberant. From the COV calculations shown in Fig. 4.15 we know that the LUF for this problem is 3 GHz, which means that the PWB method is applicable only above 3 GHz.

The PWB method is described in detail by Hill . Here we only mention that in an aperture excited reverberant cavity Pt = Pd, where Pt is the power transmitted through the aperture(s) and Pd is the power dissipated inside the cavity. The power can be dissipated by four basic mechanisms: aperture losses, absorption losses, wall losses and antenna losses. There are no antennas in the model, but all cases have one or more of the aperture, absorption, and wall losses. Then the total dissipated power for this problem is Pd = Pap + Pabs + Pwall, where Pap is the power dissipated due to aperture losses, Pabs is dissipated power due to absorption losses, and Pwall is due to the wall losses.

For an aperture coupled cavity the power density inside the cavity, Sc, is defined as

(4.12) Where s t is the transmission cross section, l is the wavelength, V is the volume of the cavity, and Si is incident power density. Then the SE based on the PWB method can be written as

(4.13) Here, Q is defined as

(4.14) Here, is the aperture leakage cross section, is the absorption cross section, S is the cavity surface area, and δ is the skin depth of the wall.

Using reference , we can estimate that for the conductivity of 0.01 S/m the absorption cross section of one lossy sphere is approximately 0.018 m2. When both spheres are lossy, such as in Cases 3 and 7, the total absorption cross section is doubled. Table 4.2 summarizes the loss mechanisms for the various simulation cases of our canonical model, together with aperture leakage and absorption cross sections.

Table 4.2 Losses in Each Simulation Case

 Case No. Losses Leakage Cross Section (m2) Absorption Cross Section (m2) 1 Aperture 0.0353 0.00 2 Aperture, absorption 0.0353 0.018 3 Aperture, absorption 0.0353 0.036 4 Aperture, absorption 0.0353 0.018 5 Aperture, wall 0.0353 0.00 6 Aperture, absorption, wall 0.0353 0.018 7 Aperture, absorption, wall 0.0353 0.036 8 Aperture, absorption, wall 0.0353 0.018

The wall losses depend on the skin depth of the wall material and introduce a frequency dependence, as can be seen in Fig. 4.17 for Cases 5-8. Figure 4.17  Ensemble average shielding effectiveness as a function of frequency (blue curve), compared with PWB method in reverberant regime (red curves).

While it is easy to calculate all of the losses in our canonical model, it would be an impossible task for an actual aircraft. However, we can calculate the total absorption cross section inside a cavity using a reverberation chamber approach. In a reverberant cavity, the absorption cross section can be defined as 

(4.15) where Qlis the Q for a loaded cavity, and Qu is for an unloaded cavity. From equations 4.1 and 4.13 we can solve for Ql and Qu in terms of ensemble averaged electric fields: Then the absorption cross section can be written as

(4.16) Obtaining the absorption cross section according to Equation 4.16 for Cases 2, 3, 4, and 6, 7, 8, we end up with values within 16% of those listed in Table 4.2. We then used these values to calculate the SE using the PWB method defined by Equations 4.13 and 4.14, and the results are shown in Fig. 4.17. We can conclude that the PWB method works well for the predictions of SE even when the absorption losses are not well known.

### 4.3.6        Angle of Incidence Variations

The following evaluations were performed for angles of incidence (AOI) of 0°, 20°, 30° and 80° as depicted in Fig. 4.18 to help characterize the effect that the AOI of the source wave has on the response of the model.  The previous cases applied a somewhat arbitrary AOI, as depicted in Fig. 4.7. The new AOI k-vectors were all located in the yz plane with the angles measured from the y-axis.  The field orientation was along the z-axis.

All plots include 50 MHz bandwidth frequency averaging and also spatial averaging over the 49 probe locations.  Case 2 was selected as a baseline for this study.

Figure 4.19 provides the AOI comparison for the COV for the square of the total electric field as compared with the theoretical value of 0.577.  Comparing these results with the Case 2 plot in Fig. 4.15 shows that there is significant impact on the response with changing AOI.  The 60° AOI case is the only one that is comparable to the baseline arbitrary incidence case of Fig. 4.15.  The implication is that the average of a number of AOIs may be required to achieve good statistics. Figure 4. 18 Cases for Angle of Incidence Study Figure 1.9 50 MHz band-limited COV for Case 2 with varied AOI (blue), compared with 6-DOF theoretical limit (red).

Figure 4.20 shows the AOI variation of the square of the electric field averaged over a 50 MHz frequency band width at center frequencies spaced at 25 MHz over the entire frequency range for the analysis.  These results were also averaged over the 49 spatial locations.

The effect of changing the AOI is even more prominent in these plots.  This is not surprising given that the incident field for all the AOI variations only had y– and z-components with the x-component only arising from scattering inside the cavity. Figure 4.20  50 MHz band-limited mean of the squares for each field component, together with the 1/3 mean of the squared of the total field, as a function of frequency.

The conclusion to be drawn from the results in Figure 4.20 is that it is important to illuminate the test object with a field vector that has equal xyz-components (on average).

Figure 4.21 shows the variation in the SE evaluations as a function of the source wave AOI.  These results should be compared with Case 2 of Fig. 4.17. Again, better correspondence with the theoretical value is achieved for the more arbitrary AOI case except for the 0° incidence case. We can now conclude that for this canonical problem, the PWB method can be applied even when the fields inside the cavity are not perfectly homogeneous or isotropic. Figure 4.21 50 MHz and location averaged shielding effectiveness for case 2 with varied angle of incidence (blue curves), compared with theoretical values (red curves).

## 4.4        Evaluation of Absorption Losses in Boeing 707

It is generally not possible to obtain all the material losses inside an aircraft. Therefore, a technique needs to be developed where absorption losses for LLSF evaluation can be reasonably approximated and compared with experimental measurements.

In this section, we summarize an investigation of absorption losses numerically using EMA3D. For a full treatment of the problem see . We start with a simplified model of an empty Boeing 707 aircraft and illuminate it with a broadband plane wave at broadside incidence, horizontal polarization, with spectral content up to 1.2 GHz. SE is evaluated in three aircraft cavities, namely, the cabin, cockpit, and avionics bay. We then start adding lossy materials, which consist of randomly spaced cubes with low conductivity, to investigate how the SE changes as a function of the number of absorbing bodies, in the frequency range from 100 MHz to 1.2 GHz. We also observe how the statistical properties of electric fields degrade as absorption losses are added. In addition to SE, τ and Q are extracted as a function of frequency and the number of absorbers. Numerical results are then compared with experimental data . We note that there are various discrepancies between the simplified model and the aircraft itself, as well as some uncertainties, which will be described below. Therefore, we use experimental data here only for reference, and we do not attempt to match the modeling results to experimental results.

### 4.4.1        Simplified Aircraft Model Setup

A simplified model of an empty (no absorption losses) Boeing 707 aircraft  is presented in Fig. 4.22. The interior is mostly empty except for the floor, three bulkheads (two in the cockpit, one in the tail), and four avionics boxes in the avionics bay below the cockpit floor. The skin, floor, bulkheads, and avionics boxes are all metal. Figure 4.22 Simplified Boeing 707 model. The plane wave is incident from broadside with a horizontal polarization.

There were some uncertainties about the Boeing 707 aircraft that need to be pointed out. From , there may be a metallic layer on the cockpit windshield that was not characterized, and therefore not implemented into the model. The cabin seats of the aircraft were absent during testing, and all seats are absent in the model. The exact geometry of the aircraft during test was not well known, thus some discrepancies may arise between the aircraft and the model. The coupling mechanisms between the cockpit, cabin, and avionics bay were uncertain, however, significant coupling was observed during testing. Exact antenna and probe positions of the test were not described in the test report.

A broadband Gaussian plane wave electric field source was used for external excitation. The angle of incidence was 90° (broadside), with horizontal polarization and frequency range between 100 MHz and 1.2 GHz.

For numerical results, a principal field component Ex was obtained at three representative locations inside the cockpit, six inside the cabin (three forward, three aft), and three inside the avionics bay. While other field components also exist inside the aircraft, only Ex is obtained for comparison with measurements. The antennas were not modeled. Additionally, to compare the numerical results with experimental data as presented in , 50 MHz frequency stirring was applied to raw numerical data. Thus only the 50 MHz frequency stirred results will be presented here for SE, time-decay constant t, and quality factor Q.

After running and analyzing the empty Boeing 707 model, absorption losses are gradually added to the cockpit, cabin, and the avionics bay. Absorption losses are represented in the model as semi-randomly spaced cubes with the side length σ = 0.5 m and conductivity s = 0.01 S/m. Another four simulations were completed with a different number of absorbing bodies each, as summarized in Table 4.3.

Table 4.3 Absorbing bodies in each aircraft cavity for five different simulations

 Simulation Number No. AbsorbersCockpit No. AbsorbersAvionics Bay No. AbsorbersCabin 1 0 0 0 2 1 5 25 3 2 10 50 4 4 21 100 5 6 21 150

### 4.4.2        Shielding Effectiveness

The SE values were obtained after Fourier transforming time-domain data at each location, then 50 MHz bandwidth averaging the frequency domain data.  Then data were location averaged inside each cavity (i.e., averaged three locations inside the cockpit, three locations in the avionics bay, and six locations in the cabin). This method of data processing was chosen so that the numerical results could be consistently compared with the experimental results from . The SE was evaluated according to Equation 4.1.

Since aperture losses are the only losses present in the empty Boeing 707 model, the SE is less than 5 dB above 500 MHz, 15 to 30 dB lower than what is measured experimentally . As absorbing bodies are gradually added to the model, the SE gradually increases, eventually making absorption the dominant loss mechanism inside the aircraft. This is shown in Fig. 4.23.

### 4.4.3        Decay Time Constant τ and Q

The Qs are shown in Fig. 4.24 for a different number of absorbers inside each aircraft cavity, and again compared with experimental results. Figure 4.23  Shielding effectiveness for a different number of absorbing bodies inside the cockpit, avionics bay, and cabin. Figure 4.24  Q and τ for a different number of absorbing bodies inside the cockpit, avionics bay, and cabin.

### 4.4.4        Cabin Analysis

Given that the passenger cabin is the largest cavity in the aircraft, and with fewest discrepancies between the model and the test, we choose the cabin for more detailed analysis. Figure 4.25(a) shows how the cabin SE varies with the number of absorbers at three different frequencies: 150 MHz, 750 MHz, and 1.15 GHz, and Fig. 4.25(b) shows how the cabin Q varies with the number of absorbers for the same frequencies. For each frequency, SE increases with the number of absorbers, while Q decreases. This is consistent with what has been observed experimentally inside a reverberation chamber loaded with salt water bottles .

A degradation of statistical uniformity with the added absorption losses is also observed, as shown in Fig. 4.26. The COV diverges from 1 (see 2-DOF COV) as more absorbing bodies are added to the cabin, as shown for two different frequencies, 750 MHz and 1.15 GHz. 750 MHz is chosen as the lowest frequency at which the empty aircraft cabin can support enough modes to be considered reverberant, and 1.15 GHz is chosen as the highest frequency available for analysis. It is clear from Fig. 4.26 that at both frequencies, the statistical uniformity degrades as more than 50 absorbing bodies are added. Figure 4.25 Cabin SE and Q as a function of the number of absorbing bodies at three different frequencies. Figure 4.26 Coefficient of variance inside the cabin as a function of the number of absorbing bodies.

## 4.5        Summary and Conclusions

In this chapter we looked at cavity fields statistics for an aperture excited cavity, at the PWB method for a cavity with not perfectly uniform fields, and the role of absorption losses inside an aircraft. From the canonical box model we discovered that even though the fields inside are not completely reverberant, i.e., mostly homogeneous but not completely isotropic, the PWB method can still be applied above the LUF to predict the shielding effectiveness. Figure 4.17 shows that the SE predicted using the PWB method compares well with the SE calculated from the FDTD modeled fields. It would be valuable, however, to check the validity of this method more rigorously using actual aircraft experimental data, however, it is difficult to obtain the data needed for such comparisons. As of the time of this writing we have none available to us.

From the Boeing 707 evaluation of absorption losses it was found that the SE increases and Q decreases with the increasing number of absorbers. There were various discrepancies between the physical aircraft and the model: landing gear, engines, seams, slots, and most avionics boxes were not modeled, as they were unknown, and the cockpit windshield may have been coated with a conductive material. The results from this study, however, were expected and consistent with previous observations . It was also observed that the statistical uniformity inside the aircraft cavities degrades with the increasing number of absorption losses. This implies that the statistical uniformity degrades with increasing SE. This raises a question as to whether an aircraft with enough losses to provide SE as high as that measured for the Boeing 707 can ever be considered reverberant, and if so, in what frequency range. One may be able to approximate absorption losses inside a reverberant aircraft using the PWB formalism, together with SE measurements for different aircraft, and classify absorption losses for each aircraft type. However, in a non-reverberant environment, as in the case of the Boeing 707, absorption losses are much more difficult to quantify, thereby making accurate implementation of absorption losses for CEM models more challenging.

## References

.   Hill DA (2009) Electromagnetic Fields in Cavities. John Wiley & Sons, Inc., Hoboken, New Jersey

.   Ted Lehman, “A Statistical Theory of Electromagnetic Fields in Complex Cavities” AFRL Interaction Note 494, May 1993.

.   Kolmogorov-Smirnov goodness-of-fit test

.   Hallbjorner P et al (2005) Extraction Electrical material Parameters of Electrically Large Dielectric Objects From Reverberation Chamber Measurements of Absorption Cross Section. IEEE Trans Electromagn Compat. doi: 10.1109/temc.2005.847391

.   Kitaygorsky J et al (2013) Parametric Evaluation of Absorption Losses and Comparison of Numerical Results to Boeing 707 Aircraft Experimental HIRF Results. UWB SP 10.

.   Johnson DM et al (1997) Phase II demonstration test of the electromagnetic reverberation characteristics of a large transport aircraft. http://www.dtic.mil/dtic/tr/fulltext/u2/a376368.pdf. Accessed Dec. 2010 