Validation of simple estimates for average field strengths in complex cavities against detailed...

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Published in IET Science, Measurement and TechnologyReceived on 23rd May 2008Revised on 28th August 2008doi: 10.1049/iet-smt:20080064

Special Issue – selected papers from CEM 2008

ISSN 1751-8822

Validation of simple estimates for average fieldstrengths in complex cavities against detailedresults obtained from a 3D numerical model ofa carA.R. RuddleAdvanced Engineering Department, MIRA Limited, Watling Street, Nuneaton, Warwickshire CV10 0TU, UKE-mail: [email protected]

Abstract: Very simple estimates for average electric and magnetic field strengths over the passengercompartment of a car are compared with values obtained from 3D numerical simulations. The estimates arebased on the power balance method, assuming that the windows are electrically large and represent thedominant loss mechanism. A method to include the effects of simple glazing in electrically large windows isalso proposed and validated against numerical models. The results demonstrate that remarkably goodestimates can be obtained for internal sources at frequencies in the band 1–2 GHz. This approach is expectedto become more effective for larger cavities and higher frequencies. Average field estimates of this naturecould be useful in the assessment of human exposure to electromagnetic fields for vehicles and otherresonant environments that are not amenable to deterministic modelling because of their electrical size.

1 IntroductionAn application of electromagnetic field modelling that is ofincreasing importance is the assessment of human exposureto electromagnetic fields. Distant sources will result inrelatively uniform, plane wave illumination in openenvironments. However, many real-world exposureenvironments, such as vehicles and buildings, are complexresonant structures. In these environments, the local spatialfield distribution may be highly non-uniform, even when itoriginates from distant sources. Detailed measurement ofsuch field distributions is likely to be a lengthy andlaborious process, and is unable to provide advance warningof possible issues in the early phases of development, wheremitigation measures are generally more cost-effective toimplement.

Numerical simulation offers the potential to overcomesome of the limitations of measurements, by providingcomprehensive field distribution data at the design stage.Unfortunately, the practical deployment of 3D numerical

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modelling techniques is often hampered by issues such asthe need for very significant computational resources, or thelack of sufficiently detailed information concerning thegeometry of the exposure environment. This papertherefore outlines an investigation of an alternativemodelling approach, using the power balance method [1]to obtain estimates concerning the statistical properties ofthe field distributions within complex cavities. Thesestatistical measures are inevitably much more limited thanthe information that can be obtained from detailednumerical simulations, and rely on the assumption that thecavity is electrically large. However, power balancecalculations require very little information about thegeometry of the target environment, and their computingrequirements are insignificant (compared with 3D models).An approach for assessing the impact of glazing materialsin electrically large windows is also investigated.

The test case considered in this work is a passenger car,which is sufficiently compact for detailed numericalsimulation to be achievable up to low microwave

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frequencies with contemporary desk-top computers. In thiscase, therefore, it is feasible to validate the estimatesobtained using the power balance approach againstcorresponding statistics derived from detailed numericalmodels of a system of real-world complexity.

2 In-vehicle field exposureassessmentElectromagnetic field exposure guidelines [2, 3] are typicallyframed in terms of ‘reference levels’ and ‘basic restrictions’.The ‘reference levels’ relate to quantities such as electricand magnetic field strengths, whereas the ‘basic restrictions’are specified in terms of quantities such as current densityand specific absorption rate (SAR) levels induced in thebody by local electromagnetic fields. Assessing compliancewith the field reference levels should therefore be lessdifficult than determining induced current density or SARdistributions. Compliance with the field reference levels isassumed to guarantee compliance with the basic restrictionsfor non-localised exposures. Exceeding the reference levelsdoes not necessarily mean that the basic restrictions areexceeded, but the threat is deemed to be sufficiently highthat additional investigation to establish compliance withthe basic restrictions is warranted.

The reference levels are derived from the basic restrictionsby mathematical modelling and extrapolation from laboratorymeasurements. Safety margins are included to allow forvariations between individuals, and for local fieldenhancements arising from reflections in the environment.The reference levels are therefore intended to be spatiallyaveraged values over the entire body of the exposedindividual in non-localised, non-uniform exposures, butwith the important proviso that the localised basicrestrictions on exposure are not exceeded [2, 3].Consequently, the use of reference levels is not consideredto be appropriate [3] for the highly localised exposures thatresult from personal transmitters in contact with the body.

In many real-world environments, it may not be easy tospecify the locations of exposed humans. In a building, forexample, the occupancy scheme may well change over time.In vehicles, the occupant locations are more readilyidentifiable, but in this environment the occupants aresurrounded by significant reflectors in very close proximity.Thus, variations in occupant size, position and numberscould perhaps be of greater significance for in-vehicleexposures.

In order to investigate the validity of field reference levelsin the vehicle environment, numerical models of a car andseveral occupants (represented by human-shapedhomogeneous dielectric bodies) have been used toinvestigate occupant SAR and empty vehicle internal fielddistributions for an internal source [4] and for a roof-mounted monopole [5] at 400 MHz. It is reported in [6]

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that average permittivity and conductivity values for thebody as a whole can be approximated as two-thirds of theirvalues for muscle tissue. The homogeneous humansimulants were therefore assigned isotropic electricalproperties with a relative permittivity of 38.65 and anelectrical conductivity of 0.55 S/m, based on measuredparameters for human muscle at 400 MHz [7].

The results of [4] suggest that whole-body average SARlimits are likely to be reached with an internal source whenthe radiated power is high enough to produce average fieldlevels over the interior of the empty vehicle that exceed thefield reference levels of [2] by a factor of 1.95. This factoris derived from the worst-case SAR values for a series ofeight occupancy configurations representing a driver withup to three passengers. A corresponding factor of 2.6 wasfound for the roof-mounted antenna scenario [5]. Thesevalues compare favourably with the results of numericalinvestigations based on anatomically detailed uprighthuman simulants under plane wave illumination, where thesafety factor provided by the general public field referencelevels is around 2.25 at 400 MHz [8].

Thus, there is some evidence to suggest that averageinternal field levels for empty cars may provide a usefulmeasure to assess human exposure threats. Approximatemethods that can provide reliable estimates for the averagefield levels in complex cavities could therefore be ofconsiderable benefit in assessing human exposure toelectromagnetic fields in vehicles and other resonantenvironments.

3 Reference data: 3D numericalmodelsThe reference data for this investigation were obtained from3D numerical models. These models were derived fromCAD geometry for the body-shell and doors of passengercars, as well as the major metallic parts in the passengercompartment (including the front and rear seat frames, andthe interior steering components). All of these parts wereapproximated as perfect conductors, and the structure waslocated in a free-space environment.

The simulations were carried out using a commercial 3D,full-wave field solver based on the transmission line matrixtechnique [9]. Models of this nature have shown goodcorrelation with spatial field measurements in the vicinityof a complete vehicle at 400 MHz [10], and withbroadband field measurements at points inside a completevehicle for frequencies up to 1 GHz [11].

Using these models, computed RMS electric and magneticfields at more than two million points were used to deriveaverage field levels over the interior of the empty vehicle.Results for frequencies of 1–2 GHz were obtained from avehicle model that was excited using a simple dipole

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located in the vicinity of the rear seat, for both vertical andtransverse horizontal (relative to the longitudinal axis of thevehicle) antenna orientations. The dipole was 11 cm longand 2 mm in diameter, with a source impedance of 50 V

and an arbitrary excitation voltage. As the input impedancechanges depending on position and orientation inside thevehicle, all results were normalised to represent a radiatedpower of 1 W. This ensures that all data sets can bedirectly compared, and readily scaled to any power level ofinterest.

The position of the dipoles inside the car is not easy todescribe because of the complexity of the surroundingstructure. In all cases, the antenna feed point wasmaintained at the same point, but this point can only bemeaningfully described relative to the vehicle structure. Thetransverse position was displaced by 4.5 cm from thelongitudinal axis, towards the driver’s side, but the verticaland longitudinal positions are more difficult to define asthere are no obvious references planes. Relative measuresbased on readily identifiable features of the vehicle structureare therefore the only alternative. The vertical position was80.5 cm below the highest point of the roof, whereas thelongitudinal position was 40.5 cm forward of the point onthe rear edge of the roof panel on the axis of the vehicle.

The average internal field values corresponding to aradiated power of 1 W CW are given in Table 1, whichalso includes data obtained at 400 MHz from a model ofanother car of similar size [4]. The latter was also excitedwith a different antenna model (a ‘zig-zag’ monopoleradiating against a small conducting block), and forlongitudinal horizontal polarisation only. Nonetheless, thelocation of the 400 MHz antenna was similar (i.e. off-axisand in the vicinity of the rear seats).

The results in Table 1 show that the average field valuesare at similar levels for frequencies above 1 GHz, and


do not differ markedly between the two sourcepolarisations. Although results for the two polarisationsdiffer by 15% at 1.4 GHz, the differences are ,10% at1.2 and 1.8 GHz, ,5% at 1.5 and 1.6 GHz and ,2.5%at 1 and 2 GHz. However, the values obtained at400 MHz are lower than those obtained at the higherfrequencies (by a factor of the order of 30%). The ratio ofthe average electric and magnetic fields is also within ,1%of the wave impedance of free space for frequencies of1–2 GHz.

Results obtained from simulations under plane waveillumination from the front of the vehicle are alsosummarised in Table 2, with the results normalised torepresent 1 W/m2 incident power density.

The results in Table 2 show that the average fieldvalues under plane wave illumination are different forthe two polarisations, with the average for the verticalsource lower than for horizontal except at 3 GHz.However, the relative difference between the averagevalues for the two polarisations decreases as the frequencyincreases above 1 GHz. This behaviour may be because ofchanges in the scattering from the vehicle structure as thefrequency changes. At 3 GHz, however, the differenceis ,3.5%. The ratio of the average internal electricand magnetic fields under plane wave illumination is within2% of the wave impedance of free space over the band1–3 GHz.

The inclined windscreen aperture of the car presents anarea of around 0.5 m2 to the incoming 1 W/m2 planewave, suggesting a very rough estimate for the powercoupled into the passenger compartment that is of theorder of 0.5 W at most. The fact that the magnitudes ofthe average fields obtained for plane wave illumination areslightly lower than those obtained using 1 W internalsources therefore seems to be broadly consistent.

Table 1 RMS field strengths averaged over vehicle cabin for horizontal andvertical internal antennas at 1 W CW radiated power

Frequency, MHz RMS electric field, V/m RMS magnetic field, mA/m

Vertical Horizontal Vertical Horizontal

2000 22.5 22.9 59.4 60.7

1800 23.4 21.2 61.7 56.3

1600 23.4 24.5 61.9 64.9

1500 23.3 24.3 61.6 64.3

1400 21.7 24.8 57.3 65.9

1200 22.9 21.8 60.4 57.9

1000 22.4 22.7 58.2 59.5

400 – 17.4 – 45.0

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Table 2 RMS field strengths averaged over vehicle cabin for horizontal andvertical plane waves incident from front at 1 W/m2 CW power density

Frequency, MHz RMS electric field, V/m RMS magnetic field, mA/m

Vertical Horizontal Vertical Horizontal

3000 13.2 13.5 34.4 35.5

2400 15.2 13.7 39.6 36.2

2000 16.2 13.8 42.4 36.5

1400 16.7 14.0 43.7 37.3

1000 16.0 13.0 41.7 34.1

400 14.2 12.5 35.2 31.4

4 Approximation of car windowcouplingSince the 3D numerical models included only perfectconductors, the apertures provide the only source of loss forthese systems. Although there are many apertures in the carbody, most of these features are relatively small. Thewindows are therefore the dominant apertures and theprimary source of loss in the passenger compartment.

The power PT( f ) transmitted through an aperture of areaA at frequency f under incident power density S( f ) per unitarea is

PT f� �¼ TA( f ) S f

� �A (1)

where TA( f ) is an aperture transmission coefficient that, ingeneral, depends on the aperture geometry as well asthe frequency, angle of incidence and polarisation of theilluminating field. A method for estimating thetransmission coefficient of a rectangular aperture underuniform illumination at normal incidence is given in [12],where it is reported that the transmission coefficientsobtained for rectangular apertures are quite similar to thosefor square apertures of the same area. The transmissioncoefficient for a rectangular aperture with sides a and b [12] is

TA f� �¼

4abf 2

c2

ðp=20

ðp=20

a( f , u, f)b( f , u, f)� �2

sin uð Þdudf

(2)

where c is the speed of light and

a( f , u, f)¼sin paf sin (u) cos (f)=c� �paf sin (u) cos (f)=c

(3)

b( f , u, f)¼sin pbf sin (u) sin (f)=c� �pbf sin (u) sin (f)=c

(4)

Car windows are clearly not simple rectangles and the sizeand shape of these apertures differ widely between vehicles.

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Representative dimensions for typical car windows werederived from vehicle CAD data and used to estimatetransmission coefficients for rectangular apertures of similardimensions. Typical window areas for a passenger car are ofthe order of 1–1.1 m2 for the windscreen, a little less forthe rear window of a hatchback, perhaps 0.5 m2 for the rearwindow of a saloon car and around 0.3 m2 for each of theside windows.

The frequency-dependent transmission coefficientsobtained using (2)–(4) are illustrated in Fig. 1, based onrepresentative dimensions for the windows of a medium-sized saloon car. The values for the transmission coefficientare found to be within +6% of unity for frequencies aboveabout 300 MHz, and within +2% from 1 GHz. As thefrequency increases the aperture transmission coefficientconverges to unity. In this ‘electrically large’ regime, thetransmitted power depends only on the incident powerdensity, angle of incidence and the aperture area.

Figure 1 Normal incidence transmission coefficients foruniformly illuminated rectangular apertures withdimensions that are representative of saloon car windows


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The results of Fig. 1 suggest that car windows are probablysufficiently large to be regarded as electrically large at thefrequencies considered in the numerical simulationsreported here (i.e. 400–3 GHz). This approximation isexpected to become more reliable as the frequency increases.

5 Estimates for field populationsin over-moded cavities withelectrically large aperturesIn electrically large and complex cavities, the simultaneousexcitation of many modes is expected to result in fielddistributions with statistical properties that are relativelyinsensitive to variations in cavity shape and sourcepolarisation. This offers the possibility of making estimatesfor the statistics of the internal fields that are based onmuch more limited information about the geometry than isrequired for deterministic simulations.

The statistical approach has been applied to the analysis ofreverberation chamber fields [13] and the shieldingeffectiveness of electrically large cavities [1]. Thesetechniques have also been used to investigate the statisticsof the orthogonal components of electric fields coupled intoa rectangular cavity through rectangular apertures underexternal plane wave illumination [14] as part of a statisticalassessment of vehicle EMC risks.

For electrically large apertures, the average aperturetransmission cross-section (averaging over all angles ofincidence over a hemisphere) reduces to half the area [1].Assuming that the electric and magnetic fields are relatedby the wave impedance of free space, the power PD( f ) lostthrough the windows of a car at frequency f (where f isnot ,300 MHz) is approximately

PD f� �’ 1

2

kERMS f� �

l2

120p

Xi

Ai

2(5)

where ,ERMS( f ) is the RMS electric field strength averagedover the volume of the cavity and the factor 1/2 reflects theassumption that only half of the power is represented bythe angular spectrum of plane waves assumed to bepropagating towards any given aperture.

Using the power balance approach [1], the power radiatedinto the interior of the vehicle is equated with the power lostthrough the apertures. The average RMS internal electricfield strength is then related purely to the radiated powerPR( f ) and the window areas Ai via

kERMS f� �

l ’ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

240pPi Ai=2� �PR f

� �s(6)

This very simple result is only valid for electrically large cavitiesthat are dominated by electrically large apertures. For systems


with significant apertures that do not satisfy this criterion (e.g.cars below about 300 MHz) the simple half-area sum of (6)would need to be replaced by more complex analytical ([1, 12,15]) or numerical models for the average transmission cross-sections of resonant or sub-resonant apertures.

The statistical distributions for the orthogonal fieldcomponents are expected to be of Rayleigh form inelectrically large empty cavities, for which the amplitudeprobability and cumulative distribution functions arecharacterised by the mean value. The net fields are expectedto approach a Chi distribution with six degrees of freedom[13], which is characterised by the variance of the fieldcomponents. The variance is also expected to be related tothe average values of the RMS field over the cavity volume.

For the electric field, the population variance nE( f )2 isexpected to be [13]

vE f� �2¼

kERMS f� �

l2

6(7)

The amplitude probability distribution PfERMS( f )g forthe net electric field is therefore of the form

P ERMS f� ��

¼ 27ERMS f

� �5

kERMS f� �

l6e �3 ERMS fð Þ=kERMS fð Þl½ �

2� �

(8)

while the cumulative distribution CfERMS( f )g for the netelectric field is given by

C ERMS f� ��

¼ 1� e �3 ERMS fð Þ=kERMS fð Þl½ �2

� �

�X2

m¼0

3m

m!

ERMS f� �

kERMS f� �

l

" #2 m (9)

The corresponding population distributions for the magneticfield may be estimated in a similar manner.

6 Comparisons between simpleestimates and 3D numerical modelresultsThe area of the car windows was determined from the vehicleCAD data. For the car model excited with external planewaves and internal dipoles, the average internal field valuesestimated using (6) were 21.7 V/m for the electric field and57.6 mA/m for the magnetic field (for radiated power of1 W CW). These values are remarkably close to the resultsobtained from the detailed numerical models with internaldipole sources for frequencies in the band 1–2 GHz (Table 1).

The power coupled into the vehicle because of externalplane wave illumination is less easy to quantify, but the

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rough estimate of 0.5 W for a power density of 1 W/m2

incident from the front of the vehicle (because of thesloping windscreen aperture) would suggest average internalfield strength estimates of 15.3 V/m and 40.7 mA/m.These values are also of similar magnitude to the resultsobtained from 3D numerical simulations (Table 2).

The simple estimates for the average internal fields for thevehicle model used in the 400 MHz internal sourcesimulations were 23.6 V/m and 62.6 mA/m. These levelsare a little higher than the corresponding estimates for thecar used for all other investigations, because of the fact thatthe window area for the car model used with the 400 MHzinternal source was slightly smaller. The estimated averagefields are higher than the results obtained from thenumerical simulation; by 35% for the electric field, and39% for the magnetic field. As the window areas arebelieved to be large enough to satisfy the electrically largeapproximation at 400 MHz, it seems more likely that thevolume of the passenger compartment is perhaps too smallto support a sufficiently large number of modes to justifythe electrically large cavity assumption.

Sample amplitude probability and cumulative distributionsfor the net fields obtained from the 3D simulations withdipole sources are shown in Figs. 2–4, which also show thecorresponding Chi-6 distributions (8) and (9) based on theaverage RMS electric field magnitudes estimated using (6).

The distributions obtained from the numerical models arefairly similar between frequencies and between sourcepolarisations. The Chi-6 distributions based on the averagefield estimates are higher around the peaks of the amplitudeprobability distribution and lower at higher field levels.However, the vehicle interior is not strictly electrically large,

Figure 2 Amplitude distributions for electric field in metal-only vehicle model with internal dipole source at 1 GHz

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and the region sampled from the numerical model includesthe source as well as conducting structures such as thesteering wheel and front seat frames. Consequently, it is notsurprising that the distributions from the numerical vehiclemodels do not entirely conform to the theoretical ideal forthe electrically large and empty cavity.

7 Impact of glazing on averagefield estimatesVehicle glazing is expected to become increasingly importantabove 1 GHz, where the glass is sufficiently thick to become

Figure 3 Amplitude distributions for electric field in metal-only vehicle model with internal dipole source at 2 GHz

Figure 4 Cumulative distributions for electric field in metal-only vehicle model with internal dipole source at 2 GHz


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resonant. Furthermore, the transmission properties ofdielectric layers also depend on the field polarisation andangle of incidence. Consequently, a more detailedcalculation is required in order to estimate average internalfields for cars at microwave frequencies.

Simple window glass could be approximated as a planarslab of thickness d formed from a single non-magneticmaterial of relative permittivity 1( f ) and zero conductivity.The effective transmittance Tp( f, d, u) for a particular fieldpolarisation p at an angle of incidence u (i.e. with theelectric field either parallel or perpendicular to the plane ofincidence) can then be obtained using

Tp f , d , u� �

¼tp1 f , u� �

tp2 f , u� �

exp �id f , d , u� ��

1þ rp1 f , u� �

rp2 f , u� �

exp �id f , d , u� ��

2

(10)

where the rpi( f, u) and tpi( f, u) represent the reflection andtransmission coefficients [16] corresponding to theparticular polarisation for the air-glass (denoted p1) andglass-air (denoted p2) interfaces and u is the angle ofincidence in air. The phase parameter d( f, d, u) is

d f , d , u� �

¼2pfd

ffiffiffiffiffiffiffiffiffiffiffi1 f� �q

c

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�

1

1 f� � sin2 uð Þ

s(11)

For frequencies where the glass is significant, (6) could bemodified to account of the associated reduction intransmittance by introducing frequency-dependent apertureweighting factors wk( f ) such that

kERMS f� �

l ’ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

240pPk

Ak=2� �

wk f� �� PR f

� �vuut (12)

where the factors wk( f ) depend on the thickness andelectrical properties of the glazing used for window k.

Assuming that the energy is distributed equally betweenthe two polarisations, the average transmission cross-sectionksk( f )l (averaged over all possible angles of incidence) foran electrically large dielectric-filled aperture is estimated from

ksk f� �

l ¼Ak

2p

ð2p

0

df

ðp=20

Tk? þ Tkjj

� �2

cos uð Þ sin uð Þdu

(13)

where the terms Tk? and Tkkrepresent the transmittances foraperture k with the field perpendicular and parallel to theplane of incidence, respectively. Consequently, theweighting factors of (12) are given by

wk f� �¼

ð p=2

0

Tk? þ Tkjj

� �cos uð Þ sin uð Þ du (14)

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The frequency dependence of the aperture weightingfactor obtained from (14) is illustrated in Fig. 5, for glass ofrelative permittivity 6 at a range of thicknesses that arerepresentative of vehicle glazing.

More sophisticated laminated and double-glazed windowpanels could also be treated in a similar manner, bydetermining the effective transmittances for the resultingmulti-layer system. The effects of finite losses could also beincluded by introducing a complex permittivity parameter,in which the real part represents the relative permittivityand the imaginary component reflects the conductivity. Inthis case, it would also be necessary to estimate the powerabsorbed by the glass. This approach could also be appliedto other types of dielectric boundary, such as the brick,concrete or plaster walls of buildings, provided that theelectrical properties data are available.

The Q factor for the leaky cavity can be estimated from thevolume and the total aperture transmission cross-section [1].For a passenger compartment of volume V, the Q factor istherefore

Q f� �’ 8pVf

cP

k

Akwk f� �� (15)

Determining the volume of the passenger compartment fromCAD data is not easy, because of the complexity of thestructure. However, a rough estimate of around 3 m3

(obtained from the 3D models) gives the results shown inFig. 6, which compare favourably with reported Q factorsestimated from antenna transmission measurements carriedout inside a complete car of similar size but differenttype [17].

Figure 5 Aperture weighting factors for glass of relativepermittivity 6 at thicknesses in the range 3–6 mm

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8 Comparison between estimatesfor dielectric-filled apertures and carmodels with glazingUsing the approximate approach outlined in Section 7, theexpected impact of lossless glass on the average internalelectric field strength has been estimated for a number ofglass thickness configurations (Fig. 7). These resultsinclude frequencies up to 10 GHz, which remain asignificant challenge for deterministic 3D numericalsimulations.

For the most realistic scheme (5 mm windscreen with allother panels 3 mm thick), the estimated changes are onlyaround 10% over the band 1–2 GHz. The results of

Figure 7 Estimated average internal electric field forperfectly conducting vehicle with lossless glass

Figure 6 Estimated Q factor for vehicle without glass, andwith glass of relative permittivity 6 at various thicknesses

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corresponding numerical simulations are also of similarmagnitude (Table 3). However, increases of up to 40% arepredicted from the simple model, depending on thefrequency and glass thickness. Thus, thicker glass isexpected to result in larger changes at lower frequencies.Consequently, Table 4 summarises estimates and resultsfrom numerical simulations for 1 cm thick glass. The latterare again of similar magnitude to the estimates.

Sample electric field amplitude distributions obtained fromnumerical simulations and simple estimates for a vehicle with1 cm thick lossless glass are compared in Fig. 8, for theelectric field at 2 GHz. The distributions obtained from thenumerical models with glazing are qualitatively more likethose obtained using the simple estimates than for the casewithout glazing.

Table 3 RMS electric fields averaged over vehicle cabin foran internal source – 5 mm thick windscreen, all otherwindows 3 mm thick

Frequency, MHz RMS electric field at 1 W CW, V/m

Simple estimate 3D numericalmodels

Vertical Horizontal

2000 24.2 23.9 24.7

1800 23.9 24.6 22.8

1600 23.6 23.8 25.7

1500 23.4 24.1 24.6

1400 23.3 22.1 24.8

1200 22.9 23.6 22.6

1000 22.7 21.8 22.5

Table 4 RMS electric fields averaged over vehicle cabin foran internal source – all windows 1 cm thick

Frequency, MHz RMS electric field, V/m at 1 W CW

Simple estimate 3D numericalmodels

Vertical Horizontal

2000 29.7 27.4 27.7

1800 29.0 27.1 26.7

1600 28.3 26.9 28.9

1500 27.9 27.6 27.8

1400 27.4 26.7 27.9

1200 26.5 27.1 26.0

1000 25.5 24.8 24.6


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9 DiscussionThe objectives of this work were to investigate whethersimple analytical calculations could be used to obtainreliable estimates for the average field strength over theinterior of a vehicle, and to assess the likely range ofvalidity of the approximations underlying this approach.

The transmission calculations based on aperturedimensions that are representative of car windows (Section4) suggest that the electrically large aperture approximationis probably reasonable for frequencies above about300 MHz for a medium-sized passenger car. Thisapproximation allows considerable simplifications to bemade in the power balance calculations used to estimate theaverage field strengths over the interior of the car withoutwindow glazing (Section 5). In the electrically large regime,the exact geometry of the windows, as well as the frequencyand polarisation, can be neglected for the empty apertureand the transmission characteristics are determined fromgeometrical optics. The simple closed-form expression thatresults therefore gives an estimate for the average internalfield strength that depends only on the power exciting thecavity and the total window area for the particular vehicleunder consideration. The geometrical optics approach alsofacilitates a relatively simple extension to take account ofthe impact of window glazing (Section 7).

For systems with physically larger window apertures, suchas trucks, buses, trains and buildings, the range of validity ofthe electrically large aperture approximation is expected topersist to lower frequencies than for the medium-sizedpassenger car that is considered in this work. Conversely,for systems with smaller apertures, such as passenger

Figure 8 Amplitude distributions for electric field in carwith 1 cm thick window glass due to internal source at2 GHz

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aircraft, the frequency above which this approximationbecomes valid will be higher than for the passenger car.The power balance approach can still be used to estimatethe average field over the interior of a large cavity withelectrically small apertures (as in [1]), but the calculationswill be more complicated because of the need to accountfor the more complex behaviour of resonant or sub-resonant apertures.

The estimates for the average internal fields outlined inSections 5 and 7 are found to provide a remarkably goodmatch the results obtained from detailed numericalsimulations of a car with a source inside the passengercompartment for frequencies above 1 GHz. However, thesimple approach overestimates the average internal fields foran unglazed car with a 400 MHz internal source. The400 MHz results obtained for the other vehicle under planewave illumination also look a little low compared with thehigher frequencies.

The transmission coefficients reported in Section 4 suggestthat the vehicle windows are probably large enough to justifythe electrically large approximation at 400 MHz (withinbetter than 4% of unity), although this approximation isclearly even better above 1 GHz. It therefore seems morelikely that the volume of the passenger compartment maynot be sufficiently large for the statistical arguments to befully valid this frequency. At 400 MHz, the overalldimensions of the cabin interior are only of the order of1.3 � 2 � 2.7 wavelengths. This increases to 3.3 � 5 � 6.7wavelengths at 1 GHz, rising to around 6.7 � 10 � 13.3wavelengths at 2 GHz.

Although good correlations between statistical estimatesand numerical simulation are reported in [14] at 224 MHz,the system that was investigated was of larger volume andmuch simpler geometry than the CAD-based car modelsused in this work. The structure used for numericalmodelling in [14] was a rectangular cavity with similardimensions to the interior of a car and equipped with sixlarge rectangular apertures, all with areas of similar order toa car windscreen. Furthermore, the interior of the structureused in [14] was empty, while the car models used togenerate the numerical results reported here contain themetallic elements of the front and rear seat frames, as wellas the steering gear.

The basis for (5) is an assumption that the number ofmodes propagating in the cavity is so large that the spatialfield distribution is approximately uniform and can berepresented as a superposition of plane waves propagatingin all directions and polarisations, with the powerdistributed equally between them [1]. Only those wavespropagating towards an aperture contribute to the lossthrough that aperture, and summing over all polarisationsand all angles of incidence over the 2p steradians on thecavity side of the aperture, these waves account for half ofthe total power. However, these assumptions become

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increasingly weak as the number of cavity modes falls to lowlevels.

The specific mode density NS( f ) [18] gives an indicationof the number of modes that account the majority of theenergy in the cavity, and depends on the frequency, Qfactor and volume

NS f� �¼

8pVf 3

c3Q f� � (16)

An estimate for the specific mode density for the passengercar example is therefore obtained from

NS f� �’ f 2

c2

Xk

Akwk f� ��

(17)

Estimates for the specific mode density are illustrated inFig. 9, for a car with the glazing configurations consideredin Tables 3 and 4, and for the case without glass.

The results below 1 GHz are expected to becomeincreasingly unreliable, as the simple estimate for theaverage internal field is known to be overestimated byaround 35% at 400 MHz, compared with 14% at worst(and generally better than 5%) for 1–2 GHz. For largervehicles, or an estate car of similar size, the volume, andhence the number of modes, could be greater. However,the number of modes for the car is likely to be small below500 MHz.

The specific mode density estimates at 2 GHz are 76 with1 cm thick glass, and 142 for the case without glass. Thisimplies that for the higher Q case (with glass), a greaterproportion of the energy is associated with a smaller

Figure 9 Specific mode density estimated for passengercompartment of car with and without window glass

4The Institution of Engineering and Technology 2008

number of modes, rather than that fewer modes are excited.At 1 GHz, these values are much smaller, at 26 with 1 cmglass and 36 without glass. Nonetheless, simple estimatesfor the average internal fields at 1 GHz still compare verywell with the results obtained from the simulations withthe source inside the cabin. The internal structuresprobably help to increase the number of modes in the 3Dmodels at the higher frequencies.

The field amplitude distributions derived from the carmodels have longer tails than the theoretical Chi-6distributions for the electrically large and empty cavity,with the result that the amplitude probability distributionpeaks are also lower. The complexity of the real car modelis such that it would be extremely tedious to exclude thoseregions that are very close to metallic structures from thedata sets used to derive the internal field statistics. Thefields in these regions are among the highest values foundin the simulation results, for both internal and externalsources. In addition, the results obtained from models withinternal dipoles also include these sources. The disparitiesbetween the Chi-6 distributions projected from the averagefield estimates and the field populations extracted from thenumerical simulations for frequencies in the band 1–2 GHz can probably be ascribed to the inclusion of thesehigh-field regions in the numerical results. The simulationof a dipole at 2 GHz inside a vehicle with 1 cm thickglazing gave the closest correlation with the Chi-6distribution based on the estimated average internal field.This configuration is also associated with the highest Q value.

The tails of the field distributions would be of greater interestfor the assessment of EMC immunity risks, since the electronicmodules and their interconnecting wiring are generally locatedaround the periphery of the passenger compartment and withinstructures such as the front seats and door cavities. For humanfield exposure, however, it is the field over the empty volumeof the passenger compartment that is of interest, andcurrently available results suggest that the average fieldstrength may provide a useful measure for assessing such risks.

10 ConclusionsA simple approach for estimating the average field strengthinside a partial cavity with electrically large apertures, basedpurely on the area of the apertures and the power radiatedinside, has been investigated as a possible alternative tonumerical simulation of vehicles. A method for extendingthese estimates to include the effects of glazing has alsobeen proposed.

The effectiveness of these methods has been evaluatedagainst detailed 3D numerical models of the metal parts ofa car, with and without glazing. For internal sources inthe band 1–2 GHz, the average field values obtained fromthe numerical models are found to be within 14% of thecorresponding estimates, and more commonly much closerthan this. For the cases with glazing, 78% of the 3D model


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results are within 5% of the estimates and the largestdeviation is 9%. Without glazing, 64% of the 3D modelresults are within 5% of the estimates, whereas 92% of theresults are within 10%.

This is a remarkably good correlation, given the enormousdifference in the computational effort and the informationrequirements. Simple estimates for the Q factor of a caralso compare favourably with published results derived fromantenna transmission measurements inside a complete car.

Under plane wave illumination, differences between theaverage internal fields for horizontal and verticalpolarisation persist up to 2.4 GHz in the results from 3Dsimulations, although the trend is for this disparity todecrease as the frequency increases. The power coupled intothe interior is less easy to quantify for the external source,but a rough estimate is broadly consistent with the averagefields obtained from the numerical models at 3 GHz andthe predictions obtained using the simple method.

The approximations employed to estimate the average fieldstrength are expected to become more reliable for largerstructures and higher frequencies, which are also often lessamenable to deterministic numerical computation. Thisapproach may therefore offer a very convenient way toassess likely compliance with field reference levels for a classof exposures arising from sources radiating insideelectrically large partial cavities. Possible applications couldrange from larger road vehicles, such as trucks and busses,through to other resonant environments such as ships,trains, aircraft and buildings.

11 AcknowledgmentsThe work outlined above was carried out as part of SEFERE,a collaborative research project supported by the UK’sTechnology Strategy Board (contract reference TP/3/DSM/6/I/15266) and the Engineering and PhysicalSciences Research Council (grant EP/D033187/1). Theproject consortium includes MIRA Ltd (coordinator),ARUP Communications, BAE Systems Ltd, HaradaIndustries Europe Ltd, Jaguar Cars, UK National PolicingImprovements Agency, University of Sheffield and VolvoCar Corporation (Sweden). Further information can befound on the project website (www.sefere.org).

12 References

[1] HILL D.A., MA M.T., ONDREJKA A.R., RIDDLE B.F., CRAWFORD M.L.,JOHNK R.T.: ‘Aperture excitation of electrically large, lossycavities’, IEEE Trans. Electromagn. Compat., 1994, 36, (3),pp. 169–178

[2] ICNIRP: ‘Guidelines for limiting exposure to time-varying electric, magnetic, and electromagnetic fields (upto 300 GHz)’, Health Phys., 1998, 74, (4), pp. 494–522


[3] 1999/519/EC: ‘Council recommendation of 12th July1999 on the limitation of exposure of the general publicto electromagnetic fields (0 Hz to 300 GHz)’, Off. J. EurCommun., 1999, L199, pp. 59–70

[4] RUDDLE A.R.: ‘Computed SAR distributions for theoccupants of a car with a 400 MHz transmitter on therear seat’. Proc. 18th Zurich Int. EMC Symp. Munich,Germany, September 2007, pp. 37–40

[5] RUDDLE A.R.: ‘Impact of passenger distribution oncomputed electromagnetic field exposure for vehicleswith on-board transmitters’. Proc. EMC Europe Workshopon EMC of Wireless Systems, Rome, Italy, September2005, pp. 411–414

[6] DURNEY C.H., MASSOUDI H., ISKANDER M.F.: ‘Radiofrequencyradiation dosimetry handbook’ Brooks Air Force Base,Technical Report, USAFSAM-TR-85-73 (4th edn.), October1986

[7] GABRIEL C., GABRIEL S.: ‘Compilation of the dielectricproperties of body tissues at RF and microwavefrequencies’ Brooks Air Force Base Technical Report, AL/OE-TR-1996-0037, June 1996

[8] MCKINLAY A.F., ALLEN S.G., COX R., ET AL.: ‘Review of thescientific evidence for limiting exposure toelectromagnetic fields (0–300 GHz)’, Doc. NRPB, 2004, 15,(3), pp. 142–145

[9] JOHNS D.P., SCARAMUZZA R., WLODARCZYK A.J.: ‘Micro-stripes –microwave design tool based on 3D-TLM’. Proc. 1st Int.Workshop on Transmission Line Matrix (TLM) modeling –theory and applications, Victoria, Canada, August 1995,pp. 169–177

[10] RUDDLE A.R.: ‘Validation of predicted 3D electromagneticfield distributions due to vehicle mounted antennas againstmeasured 2D external electric field mapping’, IET Sci. Meas.Technol., 2007, 1, (1), pp. 71–75

[11] RUDDLE A.R., FERRIERES X., PARMANTIER J.-P., WARD D.D.:‘Experimental validation of time-domain electromagneticmodels for field coupling into the interior of a vehiclefrom a nearby broadband antenna’, IEE Proc. A, Sci.Meas. Technol., 2004, 151, (6), pp. 430–433

[12] KOCH G.F., KOLBIG K.S.: ‘The transmission coefficient ofelliptical and rectangular apertures for electromagneticwaves’, IEEE Trans. Antennas Propag., 1968, 16, (1), pp. 78–83

[13] KOSTAS J.G., BOVERIE B.: ‘Statistical model for a mode-stirred chamber’, IEEE Trans. Electromagn. Compat., 1991,33, (4), pp. 366–370

[14] KONEFAL T., MARVIN A.C., DAWSON J.F., ROBINSON M.P.: ‘A statisticalmodel to estimate an upper bound on the probability of

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failure of a system installed on an irradiated vehicle’, IEEETrans. Electromagn. Compat., 2007, 49, (4), pp. 840–848

[15] BUTLER C.M., RAHMAT-SAMII Y., MITTRA R.: ‘Electromagneticpenetration through apertures in conducting surfaces’,IEEE Trans. Electromagn. Compat., 1978, 20, (1), pp. 82–93

[16] LIDDEL H.M.: ‘Computer-aided techniques for the designof multilayer filters’ (Adam Hilger, Bristol, 1981), pp. 7–8

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[17] HEDDEBAUT M., GRANSART C., RIOULT J., DENIAU V.: ‘WxANcommunication effectiveness inside vehicle bodies, inpresence of passengers’. Proc. EuroEMC Workshop onEMC of Wireless Systems, Rome, Italy, September 2005,pp. 118–121

[18] PRICE R.H., DAVIS H.T., WENAAS E.P.: ‘Determination of thestatistical distribution of electromagnetic-field amplitudes incomplex cavities’, Phys. Rev. E, 1993, 48, (6), pp. 4716–4729


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