reverberation chamber modeling based on image theory

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Reverberation Chamber Modeling Based on ImageTheory: Investigation in the Pulse Regime

Emmanuel Amador, Christophe Lemoine, Philippe Besnier, Alexandre Laisné

To cite this version:Emmanuel Amador, Christophe Lemoine, Philippe Besnier, Alexandre Laisné. Reverberation Cham-ber Modeling Based on Image Theory: Investigation in the Pulse Regime. IEEE Transactions on Elec-tromagnetic Compatibility, Institute of Electrical and Electronics Engineers, 2010, 52 (4), pp.778-789.�10.1109/TEMC.2010.2049576�. �hal-00537133�

https://hal.archives-ouvertes.fr/hal-00537133

https://hal.archives-ouvertes.fr

REVERBERATION CHAMBER MODELING BASED ON IMAGE THEORY... 1

Reverberation Chamber Modeling Based onImage Theory: Investigation in the Pulse

RegimeEmmanuel Amador, Student, IEEE , Christophe Lemoine, Philippe Besnier, Member, IEEE ,

and Alexandre Laisné

Abstract—In this paper we propose a straight-forward 3D time domain model of a reverbera-tion chamber (RC) based on image theory. Thismodel allows one to describe the earliest mo-ments of an arbitrary waveform in an RC. Timedomain and frequency domain results from thismodel are analyzed and compared withmeasure-ments conducted in a reverberation chamber.

Index Terms—frequency domain, image the-ory, lowest usable frequency, model, pulseregime, reverberation chamber, time domainanalysis, transients.

I. IntroductionA. Time Domain Reverberation Chamber ModelingReverberation chambers (RC) are traditionally

used and studied in frequency domain to con-duct electromagnetic compatibility (EMC) mea-surements either for electromagnetic immunity(EMI) or electromagnetic susceptibility (EMS) as-sessments. Their study in the time domain andin the pulse regime becomes relevant with regardto high intensity radiated field testing. In suchimmunity tests to radar-like signals, the incidentwaveform may be strongly altered by the intrin-sic nature of the reverberation chamber (Fig. 1).Therefore a comprehensive view of an RC in thetime domain through modeling would be advan-tageous to study the transients and the effect ofloading on the functioning of the chamber to namesome essential questions about the pulse regime inan RC.Preliminary experimental investigations in [1],

[2] with a real radar source deliver some insightsinto the effect of loading on the levels obtainedfor EMI testing. Transients in an RC have alsobeen investigated in [3] from the viewpoint of modetuned operations.

E. Amador, C. Lemoine and P. Besnier are with theInstitute of Electronics and Telecommunications of Rennes(IETR), INSA of Rennes, Rennes 35043, France, mail:[email protected],A. Laisné is with the Centre d’Essais Aéronautique de

Toulouse (CEAT) of the Délégation Générale de l’Armement(DGA) in Toulouse, 31131 Balma, France.DRAFT VERSION, GO TO IEEEXPLORE FOR THE

FINAL VERSION

0 1 2 3 4 5 6 7 8x 10 5

0.20.1

00.10.2

Emitted signal

Am

pli

tud

ein

V

Time in s

0 1 2 3 4 5 6 7 8x 10 5

0.05

0

0.05

Received signal at posi tion AA

mp

litu

de

inV

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0 1 2 3 4 5 6 7 8x 10 5

0.05

0

0.05Received signal at posi tion B

Am

pli

tud

ein

V

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Fig. 1. Signal measured in a RC with a 40 µs sinusoidalpulse at 1 GHz (above) at two different positions A (middle)and B (bottom) separated by 30 cm (no stirring process).

Very little modeling work is available in theliterature to study transients. Use of transmissionline matrix (TLM) or finite difference time do-main (FDTD) simulations are possible solutions forsimulating an RC [4]–[7], but using a space-timediscretization of highly conductive and electricallylarge cavities induces severe limitations. Ratherthan using standard Maxwell equations, using theirasymptotic approximations may reveal the intrinsicbehavior of RC cavities. Optical techniques like raytracing (RT) discretize the environment and arenot able to simulate hundreds of reflections and socannot reproduce the reverberation phenomenon inthe chamber.

Image theory is a very straightforward approachthat allows one to simulate hundreds of reflectionsand so to reproduce a reverberation environment.In [8], the authors use image theory to modela rectangular shielded cavity. An asymptotic ap-proach for time domain simulations is used to

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extract the quality factor of the shielded cavity ata given frequency. To our knowledge it is the firstpaper about a time domain analysis of a RC basedupon an optical model. In another paper [9], theauthors investigate various approaches to modelan RC. A 2D and a 3D model based on imagetheory are proposed. But the hardware limitationsof the computer at the time did not allow them tosimulate enough energy to converge for a frequencyanalysis of the simulated cavities. Capabilities ofmodern computers give a second impetus to opticalapproaches for RCs.

B. Image Theory for a Reverberation ChamberOur model is based upon image theory [10]. In

[10], a representation of a rectangular waveguidethrough image theory is given. Our model is anadaptation of this model of a waveguide to thegeometry of a rectangular cavity. It consists inclosing the waveguide by adding two boundaryconditions. In our model, an elementary currentis placed in the cavity. The positions and theangular orientations of image-current created bythe reflective walls must be determined. This verystraightforward model does not directly involveMaxwell’s equations nor a spatial discretization ofthe environment. It fits the simple geometry ofa shielded cavity. It only uses the far-field1 andfree-space radiation approximation of millions ofelementary currents to represent a complex interac-tion between an elementary current and electricalconductor boundaries.The core of the simulation is an impulse re-

sponse. By convolving the impulse response witha chosen waveform, the model is able to simulatethe behavior of the RC with a particular waveform.By applying a Fourier transform on the impulseresponse, we can explore the frequency domain.The rough representation of the RC by a rectan-gular shielded cavity without a mode stirrer is tooelementary to obtain deterministic results. But ithas been shown that source stirring is equivalentto mechanical stirring [11]–[14] and so from a sta-tistical point of view taking randomly N receiverpositions is equivalent to taking N stirrer positions.In our model, the stirrer is not described but itsabsence will not affect the statistics of the results aslong as the receiver positions are taken randomly.This paper is dedicated to the presentation of

our model and its use for two applications. Aftera detailed explanation of our model, preliminaryelementary results are given to verify the physicalconsistency of our model. Two applications of ourmodel are presented, an estimation of the lowest

1Near field radiation are not simulated but can be easilyadded to the model

−→i

−→i

−→i

−→i′

−→i′ −→

i′

Perfect conductor

Fig. 2. Image theory applied to electrical currents.

usable frequency (LUF) and a statistical analysisof waveforms obtained in the pulse regime. Theresults obtained with this model are analyzed andcompared statistically with measurements made inthe RC in our laboratory at the Institute of Elec-tronics and Telecommunications of Rennes (IETR).

II. Shielded Cavity ModelA. Image Theory

Image theory is generally introduced with elec-tric charges. Let a negative charge be placed ata distance d from an infinite perfectly conductingplane. This conductive plane is an anti-symmetricalplane, thus a positive charge is facing the negativecharge. The resulting field of the negative chargeand the plane is the field created by an electrostaticdipole with the two charges. Image theory can beapplied to moving charges. Fig. 2 sums up thedifferent possible configurations with an electriccurrent vector −→i . Fig. 3, inspired by [10], presentsa vertical and an horizontal view of the imagecurrents created by applying the construction rulespresented in Fig. 2 to an arbitrarily oriented cur-rent in a rectangular cavity. The real cavity (in boldline, in the middle) is surrounded by image cavities.Each image cavity contains an image current. Wedefine the order of an image current, i.e, the orderof an image cavity, as the number of reflectionsinvolved in its creation. The number of cavities fora given order n > 0, is given by

Nn = 4n2 + 2 (1)

and the total number of cavities till the order n isgiven by

Mn = 1 +n∑i=1

(4i2 + 2)

= 1 + 2n+ 2n(n+ 1)(2n+ 1)3 . (2)

The growth of Mn is therefore proportional to n3.1) Generation of the image currents: Let a rect-

angular cavity be of length l, width p, and heighth. A corner of this cavity is the origin of the rect-angular coordinates and the three main directions

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−→i

z

k = 0

k = 1

k = 2

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k = −2

k = −3

(b)

Fig. 3. Image cavities and image currents in an horizontal plane (k = 0) (a) and a vertical plane (j = 0) (b). The orderof each cavity is indicated.

Ox, Oy, and Oz are defined by the edges of thecavity.Let an elementary current be placed within this

cavity at the point A(x0, y0, z0), its angular ori-entation in the cavity is defined by a tilt angle αand an azimuthal angle β as presented in Fig. 4.Generating the elementary image current meansthat we have to determine the position and theangular orientation of every current created by thereflections with the walls and save the numberof reflections involved for each direction. A one-by-one image creation process can take days togenerate millions of sources. The first step is toidentify patterns in Fig. 3 to speed-up the imagecreation process. An elementary current can beidentified by the reflections involved in its creation.If we consider an nth order current created by ireflections along the axis Ox, j reflections alongthe axis Oy, k reflections along the axis Oz, wehave n = |i|+ |j|+ |k|2.

If we examine the horizontal plane (k = 0)represented in Fig. 3(a), we can identify a patternof four juxtaposed elementary currents (Fig. 3(a)).This pattern is duplicated in the whole horizontalplane to create the plane k = 0 of elementarycurrents.If we study the orientations of the elementary

currents in a vertical plane (Fig. 3(b)), we can dis-cern that the parity of k dictates the orientation ofthe current in the corresponding horizontal plane.There are only two different horizontal planes. We

2The numbers i, j and k can be negative if the reflectionsare in the decreasing direction along the respective axis.

already have the plane k = 0 and thus all theeven horizontal planes. We can easily derive theodd planes from an even plane, the positions ofthe currents along the axis Ox and Oy are thesame and the vertical positions of the elementarycurrents differ. The height of all the currents in thek = 0 horizontal plane is z0 and the height of all thecurrents in the horizontal plane k = 1 is 2l−z0. Thetilt angles are conserved but the azimuthal anglesof the currents are reversed. We add π radians tothe azimuthal angle of the currents of the planek = 0 to obtain the azimuthal angles of the currentscontained in an odd plane. These two horizontalplane configurations are then duplicated along theOz direction, the vertical positions of the generatedsources and the number of reflections along the axisOz are adjusted accordingly.2) Cavity loss: Image theory is applied with

perfectly conducting materials. To simulate a lossyrectangular cavity, we introduce three loss coef-ficients Rx, Ry, Rz corresponding to the threemain directions of propagation.3 Unlike in [8] wherecomplex reflection coefficients are used because thelosses in the cavity are a function of the con-ductivity of the walls, the loss coefficients in ourmodel include the losses created by one reflectionon a wall and by the lossy objects in the cavityfor a ray traveling along a given direction.Theseloss coefficients are average loss coefficient as if allthe losses in the cavity were from an absorbing

3In this paper, the three coefficients have the same valueR. In some situations it can be helpful to have three distinctcoefficients to simulate an open door or a lossy wall.

March 6, 2010 DRAFT


−→i

Ox

yz

A(x0, y0, z0)

−→ex

−→ey−→ez

α

β

l

h

p

Fig. 4. Angles and coordinates of the elementary dipole inthe cavity.

paint spread on the cavity walls. In this section wefocus on an elementary current a inside a nth orderimage-cavity. This elementary current is createdby i reflections along the Ox axis, j reflectionsalong the Oy axis, and k reflections along the Ozaxis. The attenuation associated to this elementarycurrent a is:

Ra = R|i|x R|j|y R

|k|z , with |i|+ |j|+ |k| = n. (3)

The magnitude of this current can be written:

Ia = I0 ·Ra, (4)

where I0 is the intensity of every current in the sys-tem if the walls are perfectly conducting. AssumingR = Rx = Ry = Rz, and if we neglect the crossproduct, the maximal amount of energy possibleEtot found in the system is proportional to:

Etot ∝ I20 +

∞∑i=1

(4i2 + 2)I20 ·R2i. (5)

As R < 1, the sum above converges and the systemdescribed by the model is stable.3) Channel impulse response simulation: The

intensity of the particular elementary current a, −→Iacan be written:

−→Ia = Ia · −→w = I0R

|i|x R|j|y R

|k|z · −→w , (6)

where −→w is the normalized vector along the di-rection of the considered elementary current. Thecurrent in the real cavity and all the image currentssimultaneously emit an elementary impulse f(t).The intensity of the elementary current a can bewritten:

Ia(t) = I0Ra ·f(t), with f(t) ={

1 if t = 0,0 otherwise.

(7)The orientation of the current at this position isgiven by a tilt angle α, defined by the angle between

−→w and −→ez and an azimuthal angle β defined bythe angle between −→w − (−→w · −→ez)−→w and −→ex. Theelectrical field created by the elementary currenta and received at a reception point P within thereal cavity can be written4:

−→Ea(t) = −ωµdhI0Raf(t− ta)

4πdasin θa

cos θa cosφa · −→ucos θa sinφa · −→v ,

− sin θa · −→w(8)

with: −→u = Rα,β · −→ex−→v = Rα,β · −→ey−→w = Rα,β · −→ez

(9)

where dh is the length of the elementary dipole,da is the distance between the position of theelementary current a and the reception point P ,ta is the time of arrival at the reception point, andθa and φa are angular coordinates of the point Pin the local spherical coordinate system attached tothe elementary current a. −→u , −→v , and −→w define thelocal rectangular basis attached to the elementarycurrent. Rα,β is the rotation matrix5 that changesthe rectangular basis (−→ex,−→ey ,−→ez) into the local basis(−→u ,−→v ,−→w ) and c is the celerity.

From the E-field expression (8) in the localrectangular coordinate system attached to the ele-mentary current, we can deduce the expression inthe rectangular coordinate system attached to thesimulated cavity:

−→Ea(−→ex,−→ey,−→ez) = R−1

α,β ·−→Ea(−→u ,−→v ,−→w )

= R−α,β ·−→Ea(−→u ,−→v ,−→w ) . (11)

The channel impulse response (CIR) is given byadding the contribution of every current in oursystem. If M is the total number of currents in oursystem, from (8) we can deduce three CIRs corre-sponding to the three rectangular components:

sx,y,z(t) =M∑i=0

−→Ei(t) · −−−→ex,y,z. (12)

The CIR can be convoluted with a chosen signalto simulate the waveform obtained at the positionP in the shielded cavity. By applying a Fouriertransform on the CIR, the frequency domain of thecavity can be studied.

4(8) is valid for a dipole radiation pattern (− sin θ). Oneshould note that any 3-D radiation pattern can be employed.

5Rα,β represents a rotation of an angle α around a unitaryvector −→e β = − cosβ−→ex + sinβ−→ey ,

Rα,β =(

cos2 β+(1−cos2 β) cosα − cos β sin β(1−cosα) sin β sinα− cos β sin β(1−cosα) sin2 β+(1−sin2 β) cosα cos β sinα

− sin β sinα − cos β sinα cosα

)(10)

March 6, 2010 DRAFT


One should note that the quantity C = −ωµdhI04π

equals 1 V in our simulations. It means thatthe quantity ωdh is constant and independent ofthe frequency. This approximation is adequate ifthe power radiated by the antenna used for themeasurements is constant for the considered band-width. As our model cannot pretend to be deter-ministic, obtaining absolute values is not necessary.The E-fields in this paper are expressed in V.m−1

but the values are arbitrary. However, when thelevels obtained by simulation are compared to mea-sured levels, we introduce an invariant correctionfactor (or bias for values in dB).

B. Assumptions

Using image theory to model a shielded cavitymeans that we omit the energy diffracted by thewall edges in the cavity. We only consider theenergy reflected by the different walls. This op-tical approach is validated if the dimensions thecavity involved are substantially bigger than thewavelength6. In these conditions the geometricallaws of optics can be applied to the image currentemissions. This model uses far-field radiation only.The interaction between the emitting antenna andthe device under test in a reverberation chamberimplies more often far field radiation than near fieldradiation. Near-field radiations are neglected butthey can easily be added to (8) to suit a particularconfiguration. The radiating current does not haveany physical dimension, but this limitation can beeasily bypassed by juxtaposing emitting currentsto simulate a radiating line or an antenna in thecavity.

C. Discussion: Applicability and Limitations

1) Applicability: Our model was designed tomeasure the effects of a parameter on the wave-forms obtained in the cavity. The parameters ofour model that can be modified are:• the dimensions of the cavity, our model is able

to simulate any rectangular cavity,• the waveform, our model is able to reproduce

the behavior of the cavity for any arbitrarysignal,

• the loading of the cavity, it can be simulatedby precisely adjusting the loss coefficients,

• the directivity of the emitting source, it canbe tuned by modifying the radiation of theelementary currents or by creating an arrayantenna with elementary currents.

6However, results presented in III-B show a good ade-quacy between measurements and simulations even at lowfrequency.

2) Limitations: The model presented in thispaper is designed for an empty shielded cavity. Toinclude a stirrer or an object in the cavity, the envi-ronment must be discretized and this straightfor-ward model would become a complex ray-tracingmodel. Our approach here is to preserve the sim-plicity of the model. The presence of a lossy objectinside the cavity can be integrated by reducing theloss coefficients. Because the exploitation of thismodel is statistical, the effect of this approximationon the time-domain response is reduced. The me-chanical stirring process can be replaced by movingthe emitter or the receiver (source stirring) [11]–[14]. In the frequency domain, however, electronicstirring provides an efficient stirring process, aslong as the frequencies considered are independent[15]. Our model aims to represent roughly thebehavior of an RC and the simulations presentedin this paper will be analyzed statistically andcompared to measurements conducted in a real RC.

If the model is relatively straightforward, thememory usage of the algorithm can be a problem.Image-currents are generated numerically and theirpositions and various attributes7 are stored in amatrix. The number of sources is given by (2).In reality the main parameter of the simulation isnot the maximum order but the duration of thesimulated time-window LT . It means that we onlyneed the image-currents within a radius c·LT . Thisfiltering can save a lot of memory if the RC is notcubic.

Table I sums up the memory usage for a giventime window. On a 64 bits platform with 32 GB ofmemory, we manage to reach a time-window of 3µs.

TABLE ILength of the simulated time-windows and memory

usage for the RC at the IETR.

LT Number of currents M Memory usage500 ns 1.1×106 90 MB1 µs 8.8×106 700 MB3 µs 238×106 19 GB10 µs 8.8×109 700 GB100 µs 8.8×1012 700 TB

Measurements in our RC at the IETR haveshown that 99% of the energy of an impulse re-sponse is contained in 20 µs when the cavity isempty. This means that the model we proposecan only simulate the first moments of our empty

7For each current the attributes are: the rectangular co-ordinates, the number of reflections along each axis, tilt andazimuth angles, the amplitude and the phase of the currentfor array antenna.

March 6, 2010 DRAFT


0 5 10 15 20 25 300.015

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plitu

de

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40

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%of

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ener

gyre

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Fig. 5. Measured channel impulse response at a givenreception point in the empty IETR RC (top) and percentageof the total energy of the impulse response vs. time (bottom).

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R=0.9904

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R=0.9944

R=0.9934

Fig. 6. Measured cumulated power with one absorber vs.time and simulations with different values of R. In this casethe value R = 0.9924 fits the measurement.

cavity. We can note that 60% of the energy iscontained in the first 3 µs (Fig. 5) when the RC isempty. Moreover this restriction is less problematicin the pulse regime and/or with a loaded cavity. Ifthe length of the pulse is smaller than the time-window, we can reasonably expect that the maxi-mum levels would appear within the time-window.If we simulate a loaded cavity, the CIR is rapidlyreduced and most (if not the total amount) of theenergy involved in the system would be simulated.

D. R Coefficient1) Estimation: The loss coefficient used in our

model is empirically determined by using a mea-surement of the CIR with a given loading. Weused the following methodology. First, we measure

8.7 m

3.7 m

2.9

m

−→i

P (6, 2.5, 1)A(1, 2, 1)

Ox

yz

Fig. 7. Configuration of the simulated cavity.

a CIR8. We use an arbitrary waveform genera-tor (Tektronix R© AWG 7052) to create the short-est impulse possible (200 ps, 2.5 GHz of band-with). The antennas used are generally a pairof wide band horn antennas or discone anten-nas. The signal is averaged to increase the SNRand recorded on a digital storage oscilloscope(Tektronix R© TDS6124C at 40 GS/s). We computethe square of the signal and we make severalsimulations with different values of R. In order tocompare the simulations with the measurement,both the simulations and the measurement arenormalized so that the cumulative power after3 µs equals 1. These simulations are comparedwith measurements and we identify graphically thecorrect value of R. Fig. 6 shows how the value ofR is chosen, the idea is to find the value of R(among 501 values between 0.95 and 1) that fitsthe most the measurement. In this case the valueR = 0.9924 fits the measurement made in ourRC with one absorber. Practically R values higherthan 0.995 are used to simulate an empty cavity.Values under 0.995 are used to simulate a loadedcavity. A theoretical approach to estimate R andthe approach evoked above are presented with moredetails in appendix A. As the coefficient R takes inaccount the loss in the cavity, the relation betweenR and the quality factor Q is discussed in appendixB.

III. ResultsA. Preliminary Results

In this section we present preliminary results ofour model. We have chosen to simulate a rectan-gular cavity similar to the RC of our laboratory.Its dimensions are 8.7× 3.7× 2.9 m. The emittingcurrent is placed at the point A in the left partof the cavity (Fig. 7). The results presented hereare simulations conducted in a particular reception

8The effect of the electronic stirring induced by this wide-band measurement implies the the CIRs measured or simu-lated have roughly the same profile in different locations, asingle measurement is enough to estimate R.

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Fig. 8. Channel impulse responses (linear values) alongthe three rectangular components of the E-field, simulationmade over 3µs, with R = Rx = Ry = Rz = 0.995.

point P in the right part of the cavity or in acomplete horizontal plane above the emitter (planez = 2 m).Fig. 8 presents the CIR of the simulated cavity

at the reception point P with an emitting currentoriented along the Oz axis. The amplitude of thesignal received along the axis Oz is greater thanthe two other rectangular components. This resultcan be understood when we take account of therespective positions of the emitter and the receiver.Due to the very regular geometry of our cavity andthe orientation of the elementary current, all theelementary image-currents are oriented along theaxis Oz and thus the polarization along this axis isfavored. The fast Fourier transforms of these CIRs(Fig. 9) show that for low frequencies, the cavity isundermoded. For frequencies lower than 200 MHz,we can clearly identify the resonant frequencies ofour cavity.Above 300 MHz the Fourier transforms exhibit

fast fading. A statistical analysis of the Fouriertransforms along the three rectangular components(Fig. 10) for independent [15] frequencies between 1and 2 GHz shows that the rectangular componentsof the E-field follow Rayleigh distributions [16].Fig. 10 shows that the component along the Oz axisis more scattered. As expected, the polarization atthe reception is mostly the same as the emitter.Without an arbitrary-shaped stirrer, the E-fieldin the simulated cavity is mainly oriented alongthe direction of the source, in this case alongthe axis Oz (vector (0, 0, 1)). By changing theorientation of the elementary current along thevector (1, 1, 1), the three rectangular componentsare almost equally scattered.Fig. 11 shows the response obtained along the

three rectangular components with a rectangular

0 50 100 150 200 250 300 350 400 450 500200

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Fig. 10. Probability density function of FFT(Ex), FFT(Ey)and FFT(Ez) over 300 MHz around 1100 MHz (1500 sam-ples).

monochromatic pulse signal at 1 GHz, the lengthof the pulse τ is 200 ns. The complexity of thesignals simulated is typical of pulse signals in areverberation chamber.

Fig. 12 shows the propagation of the power heldby a τ = 100 ns long pulse at 500 MHz in an hor-izontal plane of the cavity. This simulation madeof 12000 receiving points shows that the emittedpulse is reflected by the walls of the cavity. Wecan notice that the total power in the cavity growsduring the pulse emission and fades away after. TheFourier transform in an horizontal plane exhibitsthe characteristic cavity modes one should expectat a given frequency. Fig. 13 shows the component

March 6, 2010 DRAFT


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5Ez

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V/m

Fig. 11. E-field received (linear values) along the three rect-angular components for a monochromatic pulse of τ = 200ns at 1 GHz. Simulation made over 3µs, with R = Rx =Ry = Rz = 0.995.

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0.2

0.25

0.3

0.35

0.4

0.45

Fig. 12. Total power (linear values) in the plane z = 2 m fora pulse signal of length τ = 100 ns at 500 MHz at differentinstant (t = 15 ns, t = 100 ns, t = 1µs). The elementarycurrent tilt angle is π/4 and its azimuth is π/4. Simulationare made over 1µs, with R = Rx = Ry = Rz = 0.995.

along the Oz axis of the E-field. First, we notethat the boundary conditions are respected, in

yin

m

(b) - Plane z = 2 m, f =81 MHz

x in m

yin

m

0 1 2 3 4 5 6 7 8

0

0.5

1

1.5

2

2.5

3

3.5 24681012141618

x 10 5

(d) - Plane z = 2 m, f =263 MHz

x in m

yin

m

0 1 2 3 4 5 6 7 8

0

0.5

1

1.5

2

2.5

3

3.52

4

6

8

10

12

14x 10 5

(a) - Plane z = 2 m, f =44 MHz

x in m

yin

m

0 1 2 3 4 5 6 7 8

0

0.5

1

1.5

2

2.5

3

3.5

0.5

1

1.5

2

x 10 4V/m

yin

m

(c) - Plane z = 2 m, f =88 MHz

x in m

yin

m

0 1 2 3 4 5 6 7 8

0

0.5

1

1.5

2

2.5

3

3.5

1

2

3

4

5

x 10 5

V/m

V/m

V/m

Fig. 13. Fourier transform of the Ez component of theelectric field (linear values) in the plane z = 2 m at differentfrequencies. The elementary current tilt angle is π/4 and itsazimuth is π/4. The TE110 cavity mode (a), the TE410 cav-ity mode (b), the TE220 cavity mode (c), and a combinationof different cavity modes around the LUF (d) are presented.

particular, the Ez component of the E-Field isnull along the walls. The first three pictures inFig. 13 show typical cavity modes. We have beenable to identify all the modes by using the resonantfrequencies formula for a rectangular cavity [10]with l = 8.7 m, p = 3.7 m, and h = 2.9 m:

fxyz = c

2

√(xl

)2+(y

p

)2+( zh

)2,with x, y, z ∈ N.

(13)The bottom picture in Fig. 13 is around 260 MHz(around the LUF of our cavity), different cavitymodes are combined and the distribution of thepower is not regular.

These preliminary results are consistent with ob-

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0 200 400 600 800 1000 12000

10

20

30

40

50

60

70

80

90

100

Frequency in MHz

reje

ct

rate

in%

500 ns1 µs2 µs3 µs

Fig. 14. Reject rate with the AD GoF test for the Rayleighdistribution for rectangular components of the E-field withN = 150 for different lengths of CIR, and an empty cavity(R = 0.998).

servations made in a real RC and with rectangularcavity theory.

B. Statistical Frequency Domain ExplorationIn this section, we simulate the chamber of our

laboratory without loading. Usually the calibrationof an RC is done without loading. By choosing tosimulate the case of an empty cavity, we choose theworst case scenario in terms of convergence of theCIR and we will be able to compare the simulationswith measurements that have been made duringthe past years. As presented in Section II-C2, wecannot simulate a full CIR if the cavity is notloaded. To perform a study of the frequency do-main, we use Fourier transforms of truncated CIRs.The length of the simulated CIR affects directly thefrequency response. When the frequency responseis very short, the fast Fourier transform of the CIRwidens the resonances. When the CIR is longerthe widening is less pronounced. This wideningis more important at lower frequencies. As a re-sult, the modes are artificially combined at lowfrequencies as if the quality factor Q is relativelylow. In order to study the statistic properties ofthe rectangular components of the E-field, we usethe Anderson-Darling (AD) goodness of fit test(GoF). First we calculate N CIRs in the cavityby moving the receiver. A fast Fourier transformis applied to these CIRs giving a N size samplefor every independent frequency [15]. Each sampleat a given frequency is tested with the AD GoFtest for Rayleigh distribution with Stephen’s values[17]. The null hypothesis H0 is accepted whenthe population composed of the N observationsfollows a Rayleigh distribution. On the contrary thealternative hypothesisH1 favors a distribution that

0 200 400 600 800 1000 12000

10

20

30

40

50

60

70

80

90

100

Frequency in MHz

Reje

cti

on

rate

Measurements

Ex

E y

E z

Fig. 15. Rejection rate in % with Anderson-Darling good-ness of fit test for the Rayleigh distribution with N = 100.Results from E-field measurements (along a rectangularcomponent) and from simulated E-field (three rectangularcomponents) with R = Rx = Ry = Rz = 0.998.

is not Rayleigh distributed. The level of significanceα = 0.05 means that the test will fail to recognize5% percent of the Rayleigh distributions. Figure 14shows the effect of the length of the CIR on thestatistics obtained for the rectangular componentsof the E-field. We can notice that the Rayleighdistribution is largely accepted after 400 MHz whenthe CIR is only 500 ns long. Measurements haveshown that the Rayleigh distribution is largelyaccepted only after 900 MHz with these tests.With 2 or 3 µs long CIRs, the simulations fit themeasurements (fig. 15). The results obtained withour model are compared with results extractedfrom measurements made in our RC [17]. Thesemeasurements were performed with an isotropicthree-axis field probe (Hi6005) and amplifiers. Thefield was measured for 30 independent positionsof probe and at least 30 independent positions ofstirrer. The samples are then grouped in series ofN = 100 independent samples [17]. Fig. 15 showsthe rejection rate of the AD GoF statistical testfor different frequencies for a measured rectangularcomponent of the E-field and for three rectangularcomponents simulated in an empty cavity. Forfrequencies under 300 MHz, the null hypothesis H0is massively rejected. This hypothesis is more andmore accepted when the frequency increases. Therejection rates from the measurements and our sim-ulations for the considered frequency range are verysimilar. These results are encouraging, globally itseems that the frequency domain behavior of oursimulated cavity fits the behavior of our real RCeven if we cannot simulate a full CIR.

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0 500 1000 1500 2000 2500 3000 3500 40000

50

100

150

200

250

300

350

400N

um

ber

of

cum

ula

ted

reje

cts

Frequency in MHz

Ex

Ey

Ez

250 MHz

260 MHz

270 MHz

Fig. 16. Cumulated number of rejected tests for thethree rectangular components of the E-field. The elementarycurrent tilt angle is π/4 and its azimuth is π/4.

C. LUF

The LUF is a characteristic constant of an RC[18]. Above the LUF, the modes involved in thecavity are combined and the fields in the cavityare stochastic. In particular, the components of theelectric field follow a Rayleigh distribution [16]. Wedeveloped an original approach to determine theLUF with this model. It consists in calculating NCIRs in the cavity by moving the receiver. A fastFourier transform is applied to these CIRs givinga N size sample for every independent frequency.[15]. Each sample is tested with the AndersonDarling statistical test [17] with Stephen’s values.The level of significance chosen is α = 0.01, itmeans that the test is very severe and thus thegraphical analysis of the results will be easier.The results for a cavity matching the dimen-

sions of the RC in our laboratory are presented inFig. 16. The functions of the cumulative numberof rejected tests for each rectangular componentexhibit the same tendencies. Under the LUF, thenull hypothesis is massively rejected and the slopeis important. Around 200 MHz, the null hypothesisis more accepted and the slope of the cumulativenumber of rejects decreases. Above 500 MHz thenull hypothesis is largely accepted and the slopeequals α. These three distinct trends were noticedin [17]. If we trace the two asymptotes, for lowfrequencies and for high frequencies, they intersectaround the LUF. The LUF of our RC is around 250MHz. The results presented in Fig. 16 show thatthe LUF given by our model is around 260 MHzand is in agreement with the value commonly usedin our chamber.

Fig. 17. Experimental configuration.

D. Levels for Immunity Testing in the Pulsed ModeTo conduct immunity testing, one should know

the mean and the maximum levels a device undertest (DUT) will receive during the test [18]. Wewant to know if our model is able to predictthe levels and the average waveforms for a givenloading and a given signal. Our RC is loaded withabsorbers (Fig. 17) and the emitting antenna andthe DUT are both wide-band discone antennas.The first step is to measure an impulse channelresponse. The duration of the CIR gives an esti-mation of the loss coefficients used in the model.We found R = Rx = Ry = Rz = 0.98. Then thepulse signal (τ = 300 ns at 1 GHz) is emitted.The receiving antenna is moved in the RC. Wemade 50 measurements in our RC. The situationis reproduced in the model, using 50 simulations in50 arbitrary positions in our virtual cavity. Fromboth the measurements and the simulations, weextract the envelopes of the signals and we computethe average power of the levels received and themaximum power for every time-step. The levelsobtained by simulations in dB are corrected byadding a constant bias to fit the measurementsof the mean and the max. Fig. 18(a) shows agood agreement between the measurements andthe simulations. Fig. 18(b) shows that the ratiomax/mean is around 6.5 dB in both the simulationsand the measurements. This value is characteristicof an observation over N = 50 [19], [20]. In linearvalues, Fig. 19, we can note that measured andsimulated averages of the power are very similar.Our model simulates with a good accuracy thewaveforms obtained in the pulse regime.

IV. ConclusionA. Main Results

With very simple assumptions and straightfor-ward calculations, the model we propose is able tostatistically reproduce the behavior of an RC. Fre-quency domain simulations show properties similar

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0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8x 10 6

60

50

40

30(a)

Pow

er

ind

Bm

Time in s

mean (meas. )

mean (sim.)

max (meas. )

max (sim.)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8x 10 6

2

4

6

8

10

12(b)

dB

Time in s

max/mean (meas. )

max/mean (sim.)

Fig. 18. (a) Maximum and mean power (in dBm) frommeasurements and simulations. Maximum/mean ratio (indB) from measurements and simulations for a 300 ns longpulse at 1 GHz.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8x 10 6

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1x 10 4

Pow

er

inW

Time in s

mean (meas. )

mean (sim.)

Fig. 19. Normalized mean power from measurements andsimulations for a 300 ns long pulse at 1 GHz.

to a real RC. Time domain simulations exhibitsimilar behavior and levels. This model is able toexplore both the frequency domain and the timedomain. It can be helpful to conduct immunitytesting in the pulse regime in an RC as well asrapidly study the influence of various parameters(dimensions, loading, directivity) in the frequencydomain.

B. General Discussion and Modifications NeededIt may be advantageous to use this model to

extract general considerations about the effect ofthe loading on the performance of the RC andabout the transients. For instance, it may be help-ful to understand how the RC stores the energy

emitted by an antenna and how the directivityof the emitting source vanishes. This model basedon CIRs and convolutions can also be helpful tosimulate rapidly time-reversal experiments.

Further developments of the model are to be con-sidered. We are currently trying to add a stirringmethod to our model in order to add a degreeof freedom to our simulations and to apply thebattery of statistical methods developed in the pastyears to compare our model to real RCs mea-surements. We are currently working on a stirringprocess. Our first approach would be to simulatean amount of M situations, in which the positionof the emitting elementary dipole is different. Byapplying the superposition theorem, a stirring stepcould be a combination of N simulations amongM .Another approach could be to change the dipoleradiation pattern by a more directive pattern andto simulate a paddle with five or six of thesedirective sources rotating around a chosen axis.

Appendix AEstimation of the loss coefficient R

In the following, we assume that the CIR enve-lope is an exponential function. The rule of con-servation of energy dictates the amount of powerradiated by every image current in the system. Atthe order i, the (4i2 +2) cavities are approximatelyplaced on a sphere of radius id, where d is a dimen-sion of the reverberation chamber. The fraction ofpower emitted by these sources that would reachthe initial cavity after a time delay t = id/c isapproximately (4π/(4i2 +2)). We can approximatethe incoming simulated E-field from the (4i2 + 2)cavities of order i in the initial cavity by:

|Es(i)| ≈ Es0Ri ≈ Es0R

t cd ≈ Es0e

t cd lnR (14)

We can express the envelope of the measured E-field with an exponential profile:

|Em(t)| = Em0e−t/τ , (15)

With (14)and (15) we can approximate a relationbetween the “time to live” parameter τ and R. Ifthe dimensions of the rectangular cavity consideredare l, p, h and that h is the smallest dimension,the time between two reflections is at least h/c.Therefore:

τ ≈ − h

c lnR (16)

and because R = 1− ε with ε� 1,

τ ≈ − h

c ln(1− ε) ≈h

cε≈ h

c(1−R) . (17)

We can deduce an approximative value of R:

R ≈ 1− h

cτ. (18)

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From the measurements, we find τ = 6 µs whenthe cavity is empty. If we apply this formula withh = 2.9 m, we find R ≈ 0.998. This rough approachgives good results.The method we used to determine R consists in

finding the R value that minimizes the error be-tween simulations and measurements. We generatea great amount of simulations with R varying from0.95 to 1. By computing the power delay profilefrom a CIR measurement, we have:

Pm(t) = Pm0e−2t/τ , (19)

the energy measured Em(t) received till t is givenby:

Em(t) =∫ t

0Pm(t)dt = 1

2Pm0τ(

1− e−2t/τ).

(20)The energy from the simulations is given by:

Es(t) =∫ t

0Ps(t)dt = 1

2Ps0τs(R)(

1− e−2t/τs(R)),

(21)where τs(R) is the “time to live” parameter in oursimulations as a function of the loss coefficient R.The energy functions are scaled after t0 = 3µsbecause the levels of the simulations with differentvalues of R are different. We obtain:

Em(t0) = AEs(t0), (22)

where A is the scaling factor.And we try to find the value of R that minimizesthe error Err:

Err =∫ t0

0|Em(t)−AEs(t)|dt. (23)

Both methods are equivalent. The problem withthe first method is that we need to express thenumber of reflections in unit of time. With ourchamber it works well if we consider the smallestdimension. But our chamber is very long (8.7 ×2.7 × 2.9 m) and we can easily understand thatthe reflections along the height of our chamberare more numerous than the two other dimensionsand so are mainly responsible for the power decayobserved. The second method that involves numer-ous simulations of a particular chamber, takes inaccount the dimensions of the chamber and maybe more accurate.

Appendix BDiscussion on the relation between R and

Q

The purpose of this appendix is to establish alink between the quantity R defined in this paperand the quality factor Q. Theoretical expressions ofthe quality factor of an RC integrate the conductiv-ity of the materials used in an RC and the dimen-sions of the considered RC [21], [22]. The quality

0.986 0.988 0.99 0.992 0.994 0.996 0.998 1

18

16

14

12

10

8

6

4

2

0

x 10 5

R

-1/Q

4 absorbers

2 absorbers

3 absorbers

Empty RC

1 absorber

Perfect electric conductor cavity

Fig. 20. Quantity −1/Q vs. R for different loading andlinear fitting. The slope A ≈ 0.013.

factor extracted from measurements includes thelosses from the cavity and every object inside.As our loss coefficient R integrates all the lossesof our system, we can certainly derive a relationbetween R and Q. Measurements of the qualityfactor of our RC were conducted from 200 MHz to2 GHz. CIR measurements were done with a 200 pslong impulsion giving a 2.5 GHz bandwidth. Theantennas used were two identical wide band hornantennas (1-18 GHz). The values of the qualityfactor Q used here are an averaging of Q(f) with ffrom 1 GHz to 2 GHz. We used four identical piecesof rectangular absorbers. Fig. 20 shows that theremust be a linear relation between the quantities Rand −1/Q:

R ≈ 1− 1AQ

. (24)

We found A ≈ 0.013. This value of A foundempirically is relevant for our cavity only. Valuesof A may vary with the dimensions of the cavity,the conductivity of the walls, and the various lossyelements found in an RC.

It can be interesting to find an approximativeexpression of A. If we assume the exponentialprofile of the CIR, the quality factor Q at thefrequency f0 is given by the time domain expressionof the quality factor Q = 2πf0τ [23], if we consider(18) and (24), we can write:

h

cτ≈ 1

2πAf0τ, (25)

A ≈ c

2πf0h. (26)

With h = 2.9 m and f0 between 1 and 2 GHz,the average value of A obtained is 0.011 whichis very similar to the value obtained with themeasurements of the quality factor in our cavity.

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Acknowledgments

This work was supported by the French Min-istry of Defence DGA (Délégation Générale del’Armement), with a Ph.D. grant delivered to Em-manuel Amador and a“REI” grant 2008 34004delivered to IETR (INSA de Rennes). The authorswould like to thank Jérôme Sol for his assistanceduring the measurements.

References

[1] O. Lunden and M. Backstrom, “Pulsed power 3 GHzfeasibility study for a 36.7 m3 mode stirred reverber-ation chamber feasibility study for a 36.7 m3 modestirred reverberation chamber,” in ElectromagneticCompatibility, 2007. EMC 2007. IEEE InternationalSymposium on, July 2007, pp. 1–6.

[2] ——, “Absorber loading study in foi 36.7 m3 modestirred reverberation chamber for pulsed power mea-surements,” in Electromagnetic Compatibility, 2008.EMC 2008. IEEE International Symposium on, Aug.2008, pp. 1–5.

[3] L. Arnaut, “Time-domain measurement and analysis ofmechanical step transitions in mode-tuned reverbera-tion: Characterization of instantaneous field,” Electro-magnetic Compatibility, IEEE Transactions on, vol. 49,no. 4, pp. 772–784, Nov. 2007.

[4] C. Bruns and R. Vahldieck, “A closer look at rever-beration chambers - 3-D simulation and experimen-tal verification,” Electromagnetic Compatibility, IEEETransactions on, vol. 47, no. 3, pp. 612–626, Aug. 2005.

[5] M. Höijer, A.-M. Andersson, O. Lunden, and M. Back-strom, “Numerical simulations as a tool for optimizingthe geometrical design of reverberation chambers,” inElectromagnetic Compatibility, 2000. IEEE Interna-tional Symposium on, vol. 1, 2000, pp. 1–6.

[6] A. Coates, H. Sasse, D. Coleby, A. Duffy, and A. Or-landi, “Validation of a three-dimensional transmissionline matrix (TLM) model implementation of a mode-stirred reverberation chamber,” Electromagnetic Com-patibility, IEEE Transactions on, vol. 49, no. 4, pp.734–744, Nov. 2007.

[7] G. Orjubin, F. Petit, E. Richalot, S. Mengue, andO. Picon, “Cavity losses modeling using lossless FDTDmethod,” Electromagnetic Compatibility, IEEE Trans-actions on, vol. 48, no. 2, pp. 429–431, May 2006.

[8] D.-H. Kwon, R. Burkholder, and P. Pathak, “Ray anal-ysis of electromagnetic field build-up and quality factorof electrically large shielded enclosures,” Electromag-netic Compatibility, IEEE Transactions on, vol. 40,no. 1, pp. 19–26, Feb 1998.

[9] S. Baranowski, D. Lecointe, M. Cauterman, and B. De-moulin, “Use of 2D models to characterize some fea-tures of a mode stirred reverberation chamber,” vol.9-13. International Symposium on ElectromagneticCompatibility, Sorrento, Italy, september 2002, pp.381–386.

[10] R. Harrington, Time-Harmonic ElectromagneticFields. New York: McGraw-Hill Book Company,1961, pp. 103–105.

[11] Y. Huang and D. Edwards, “A novel reverberatingchamber: the source-stirred chamber,” Sep 1992, pp.120 –124.

[12] C. Monteverde, G. Koepke, C. Holloway, J. Ladbury,D. Hill, V. Primiani, and P. Russo, “Source stirringtechnique for reverberation chambers; experimental in-vestigation,” Sept. 2008, pp. 1 –6.

[13] G. Cerri, V. Primiani, C. Monteverde, and P. Russo,“A theoretical feasibility study of a source stirring re-verberation chamber,” Electromagnetic Compatibility,IEEE Transactions on, vol. 51, no. 1, pp. 3 –11, Feb.2009.

[14] J. Kunthong and C. Bunting, “Statistical characteriza-tion of the 900MHz and 1800MHz indoor propagationusing reverberation source stirring technique,” in An-tennas and Propagation Society International Sympo-sium, 2009. APSURSI ’09. IEEE, June 2009, pp. 1–4.

[15] C. Lemoine, P. Besnier, and M. Drissi, “Estimatingthe effective sample size to select independent measure-ments in a reverberation chamber,” ElectromagneticCompatibility, IEEE Transactions on, vol. 50, no. 2,pp. 227–236, May 2008.

[16] D. Hill, “Plane wave integral representation for fields inreverberation chambers,” IEEE Transactions on Elec-tromagnetic Compatibility, vol. 40, no. 3, pp. 209–217,Aug 1998.

[17] C. Lemoine, P. Besnier, and M. Drissi, “Investigationof reverberation chamber measurements through high-power goodness-of-fit tests,” Electromagnetic Compat-ibility, IEEE Transactions on, vol. 49, no. 4, pp. 745–755, Nov. 2007.

[18] “IEC 61000-4-21: Electromagnetic compatibility(EMC) - part 4-21: Testing and measurementtechniques - reverberation chamber test methods,”IEC, Tech. Rep., 2003.

[19] J. Ladbury, G. Koepke, and D. Camell, “Evaluation ofthe NASA Langley research center mode-stirred cham-ber facility,” NIST, Technical Note 1508, January 1999.

[20] M. Höijer, “Maximum power available to stress ontothe critical component in the equipment under testwhen performing a radiated susceptibility test in the re-verberation chamber,” Electromagnetic Compatibility,IEEE Transactions on, vol. 48, no. 2, pp. 372 – 384,May 2006.

[21] D. Hill, “A reflection coefficient derivation for the Q ofa reverberation chamber,” Electromagnetic Compatibil-ity, IEEE Transactions on, vol. 38, no. 4, pp. 591–592,Nov 1996.

[22] P. Corona, G. Ferrara, and M. Migliaccio, “A spectralapproach for the determination of the reverberatingchamber quality factor,” Electromagnetic Compatibil-ity, IEEE Transactions on, vol. 40, no. 2, pp. 145–153,May 1998.

[23] N. Hodgson and H. Weber, Optical Resonators, Funda-mentals, Advanced Concepts and Applications, 1st ed.London, Great Britain: Springer-Verlag, 1997, p. 142.

Emmanuel Amador (S’10) receivedthe Diplôme d’Ingénieur degree fromInstitut National des Télécommunica-tions (Télécom INT), Evry, France in2006. He received his M.Sc. in electri-cal engineering from Laval University,Quebec City, QC, Canada in 2008. Heis currently a Ph.D. student in elec-tronics at the Institute of Electronicsand Telecommunications of Rennes(IETR), INSA of Rennes, France.

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Christophe Lemoine receivedthe Diplôme d’Ingénieur degreefrom Ecole Nationale Supérieurede l’Aéronautique et de l’Espace(SUPAERO), Toulouse, France, in2004. He received a Master degree infinancial risk management in 2005,and then was pursuing the Ph.D.degree in electronics from INSA ofRennes, France (defended in 2008).His current research interest at the

Institute of Electronics and Telecommunications of Rennes(IETR), Rennes, France, includes new theoretical andexperimental approaches of mode-stirred reverberationchambers for EMC, propagation channels and antennameasurement applications.

Philippe Besnier (M’04) receivedthe diplôme d’ingénieur degree fromEcole Universitaire d’Ingénieurs deLille (EUDIL), Lille, France, in 1990and the Ph.D. degree in electronicsfrom the university of Lille in 1993.Following a one year period at ON-ERA, Meudon as an assistant scien-tist in the EMC division, he was withthe Laboratory of Radio Propagationand Electronics, University of Lille,

as a researcher at the Centre National de la RechercheScientifique (CNRS) from 1994 to 1997. From 1997 to 2002,he was the Director of Centre d’Etudes et de Recherchesen Protection Electromagnétique (CERPEM) : a non-profitorganization for research, expertise and training in EMC,and related activities, based in Laval, France. He co-foundedTEKCEM in 1998, a private company specialized in turn keysystems for EMC measurements. Since 2002, he has beenwith the Institute of Electronics and Telecommunications ofRennes, Rennes, France, where he is currently a researcher atCNRS heading EMC-related activities such as EMC model-ing, electromagnetic topology, reverberation chambers, andnear-field probing.

Alexandre Laisné AlexandreLaisné was born in Villedieu-les-pôeles, France, in 1976. Hereceived the Master of Electricaland Electronic Engineering fromStrathclyde University, Glasgow,in 1999, as well as the ElectricalEngineering diploma and the Ph.D.in Electronics from the NationalInstitute of Applied Sciences ofRennes, respectively in 1999 and

2002. After research works at Rutherford AppletonLaboratory, United Kingdom, in 2002 and the Instituteof Electronics and Telecommunications of Rennes in 2003,he joined the Centre d’Essais Aéronautique de Toulouse(Direction Générale de l’Armement) in 2004. He is currentlyHead of the Numerical Simulation Department. His researchinterests include numerical simulation, EMC, ReverberationChambers, High Intensity Radiated Fields, lightning, SARand Hazards of Electromagnetic Radiation to Ordnance.

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reverberation chamber modeling based on image theory

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