A PLANE WAVE MONTE CARLO SIMULATION METHOD FOR REVERBERATION CHAMBERS

Transcription

1 A PLANE WAVE MONTE CARLO SIMULATION METHOD FOR REVERBERATION CHAMBERS L. Musso *,**,***, V. Berat *, F. Caavero **, B. Demouli *** * Directio de la Recherche Techocetre Reault 1, Av. du Golf 7888 Guyacourt, Frace ** Dipartimeto di Elettroica Politecico di Torio C.so Duca degli Abruzzi Turi, Italy *** Laboratoire TELICE Uiversité Lille 1 UPRES A Villeeuve d Ascq, Frace Abstract - A Mote Carlo simulatio is proposed to represet the electromagetic eviromet of mode stirred reverberatio chambers. The proposed method is based o a plae wave model ad allows us to simulate both the poit statistics ad the spatial correlatio of the electromagetic distributio iside a ideal reverberatio chamber. By meas of this approach, it is possible to predict the electromagetic couplig to electrical devices. The applicatio of the proposed method to predict the electromagetic couplig to a trasmissio lie iside a reverberatio chamber is proposed, ad predictio results are compared with measuremets. I. INTRODUCTION Differet approaches have bee used to simulate the reverberatio chambers (RC) electromagetic (EM) eviromet. Determiistic approaches allowig the simulatio of EM fields distributio iside a RC have bee ivestigated by a Ray Tracig method [1], by the FDTD method [], ad by the Fiite Elemet method [3], for a two-dimesioal case. O the other had, a statistical approach has bee proposed i [4], describig how to geerate stochastic fields by usig a radom umbers geerator. This approach was applied i [5] to predict the couplig of fields to a electrical moopole by usig the method of momets. By this statistical approach, radom fields are geerated for each spatial poit followig ideal RC probabilistic distributios, ad a spatial correlatio is imposed o the radom geerator, i order to satisfy the spatial correlatio existig i a RC. Mea values of EM quatities over oe stirrer rotatio are foud as mea values over a set of radom fields superpositio. A differet approach is ivestigated i this work, which combies the plae waves itegral model for RC ad a Mote Carlo (MC) statistical approach. Drawig o Hill s plae wave model for a RC [6], the RC EM eviromet is recreated i this work by a superpositio of a fiite umber of radom plae waves. The statistical properties attributed to the fields by the stirrer rotatio are take ito accout by suitably modellig the probabilistic distributio of plae waves parameters: a radom umber geerator is adopted to produce such parameters. Mea values for EM quatities are the foud as mea values over a set of radom plae waves superpositio. This ovel approach allows a simple simulatio of the EM coditios of the ideal RC eviromet. It will be show that resultig fields poit statistics satisfy ideal RC statistics, ad that fields spatial correlatio itrisic to this approach aturally reproduce RC spatial correlatio. The iterest of this approach lies i the applicatio of the method to the predictio of EM quatities coupled to electrical devices. It will be show how to predict mea values of EM coupled quatities iside a RC by disposig of a set of resposes of the device to offedig radom plae waves. The applicability of the method is thus relied to the ability of predictig the respose of the device to offedig plae waves. For distributed devices, this method ca be readily applied, sice the theory of plae waves couplig to trasmissio lies is well established [7]. As a example, the case of iduced curret ito the termial load of a sigle wire over a groud plae will be ivestigated i this work. The work is orgaised as follows. Sectio II proposes the fudametals of the method: plae waves radom properties assumptios ad resultig fields probabilistic descriptio. Sectio III is focused o the applicatio of the method to predict coupled EM quatities ito electrical devices, ad discusses a complete ad three fast MC methods. Fially, i Sectio IV such methods are applied to the predictio of the iduced curret flowig i the termial load of a sigle wire trasmissio lie exposed to RC fields. II. PLANE WAVE MONTE CARLO METHOD FOR A REVERBERATION CHAMBER Ideal properties of the workig volume of a RC are modelled here by a superpositio of radom plae waves. A fiite umber of plae waves is cosidered ad the itegral represetatio cotaied i [6] is

2 approximated by a fiite sum of plae waves cotributios. Repeated trials of a superpositio of radom cotributios allow us the estimatio of EM quatities mea values accordig to a MC approach. The first step i performig a MC simulatio of a physical process is to establish the probabilistic distributios of parameters that must be radomly geerated. I this case, the parameters are the propagatio directio, field polarisatio, amplitude ad phase of the cotributig plae waves. I a isotropic eviromet, as i the case of the workig volume of a RC, plae waves are supposed to have o preferred propagatio directio ad o preferred field polarisatio. This meas that uiform distributios are chose for propagatio directio agles ad for the polarisatio agle, over the solid agle ad over π, respectively. Additioally, multiple scatterig pheomea iside a RC result i the fact that the phase of plae waves has o preferred value; thus, a uiform distributio is chose. Fially, costat amplitude is chose for all plae waves, equal to E 0, for coveietly matchig simulatios with experimets. Probabilistic distributios for plae waves parameters are summarised i Table 1. Table I Plae waves probabilistic distributios Parameter Distributio θ ϕ U 0,4π Propagatio directio Ω (, ) [ ] Polarisatio θ P U [ 0,π] Phase φ U [ 0,π] Amplitude E δ ( E ) I Table 1, U stads for uiform distributio. Accordig to the assumptios of Table 1, it is possible to aalytically determie the mea value ad variace for oe rectagular compoet of the electric field resultig i oe poit of the space by the cotributio of oe radom plae wave. For example, it ca be show that for the z compoet of the electric field i the origi of a Cartesia system, mea value ad variace of real ad imagiary parts are give by E 0 ( Re{ Ez0 }) = mea( Im{ Ez0} ) = 0 ( Re{ E }) var( Im{ E }) / 6 mea (1.a) var = (1.b) z 0 z0 = E0 The, by applyig the cetral limit theorem (CLT), the probabilistic distributio of real ad imagiary parts of the same field compoet give by the superpositio of radom plae waves ca be determied. The CLT states that for a great umber of cotributig plae waves, real ad imagiary parts of the resultig field compoet are distributed as a ormal distributio, whose mea value µ ad variace σ are give i the followig equatio. Re E 0 zrc zrc N µ = 0, σ = 6 () { E }, Im{ E } Equatio () allows us to characterise fields i oe spatial poit of a ideal RC workig volume. Fially, the isotropy ad homogeeity properties of fields iside the RC workig volume allow us to exted these results to ay electric field rectagular compoet at ay poit iside the volume. As a result, the distributios of the amplitudes of a geeric field rectagular compoet E i ad of the total electric field E tot at ay poit iside the chamber, ca be show to be i agreemet with RC statistics cotaied i [6], [8] ad [9]. Furthermore, as each plae wave cotributio cocers the etire workig volume, spatial correlatio for fields is also i agreemet with RC model cotaied i [6]. The iterest of the proposed approach is that it is possible to relate mea values for fields iside a RC to the costat chose plae waves amplitude E 0 ad to the umber of plae waves cosidered. I fact, startig from equatio () ad idicatig mea values i RC as, it is possible to obtai: E i, RC = E0 π /1 E i, RC = E0 / 3 E 15 = E0 16 π 3 tot, RC E, RC E0 tot = (3) Furthermore, the advatage of keepig E 0 costat is that it becomes a free parameter to be matched with measuremets of the mea field level iside the chamber, thus allowig for comparisos betwee simulatios ad measuremets. III. MONTE CARLO COMPLETE AND FAST METHODS FOR EM COUPLED QUANTITIES MEAN VALUES PREDICTION Accordig to the approach formulated i Sectio II, MC predictios of EM quatities coupled to electrical devices i RC ca be obtaied as superpositio of plae waves couplig cotributios. I particular, if x is a geeric coupled complex quatity (curret or voltage), its mea value i RC ca be computed as: x RC = Ε x PW, i (4) i= 1 where x RC is the coupled quatity i RC, x PW, i is the coupled quatity correspodig to the i-th radom plae wave, stads for mea value over a stirrer

3 rotatio, ad Ε is the expected value. The MC complete method cosists i estimatig the expected value i (4) by takig the arithmetic mea value of several (let us say m) simulatio trials of the sum i the right-had term of (4). This meas that m simulatios are required. I this case, each of the m simulatio results (i.e. each of the m sums of the right term of (4)) correspod to oe positio of a virtual stirrer. If plae waves parameters are chose as described i Table 1, ad plae waves amplitude is chose accordig to equatio (3), matchig betwee simulatio ad measuremets made i a real chamber ca be obtaied. If we are able to umerically simulate the respose of a device to oe offedig plae wave, i.e. x PW, i, equatio (4) ca be used i a MC simulatio method to predict mea value of the respose i RC. The icoveiece of this method is i the large computig time required by the m simulatios. The possibility of simplifyig the geeral method of equatio (4) is ivestigated i the followig. If we are lookig for mea value of the received power, which is proportioal to the squared magitude of coupled curret (voltage), (4) ca be writte as = Ε P RC Ε xpw, i xpw, i (5) i= 1 where the expected value i the last term of (5) is take with respect to the squared amplitude of a coupled quatity due to oe radom plae wave. It ca be easily show that the last equality i (5) is valid for ay complex radom quatity x PW, i that has zero mea value for the real ad imagiary parts. This last assumptio ca be cosidered true for ay liear electrical device. Last equality i (5) gives thus a meas to simplify the complete method of equatio (4) by replacig the m simulatios described above by simply simulatios (the expected value i the last term of (5) is estimated as mea arithmetic value over plae waves cotributios), ad multiplyig the result times the umber of simulatios. We will call this procedure fast1 MC method. A importat physical implicatio of (5) is that it equals mea value i MSRC ad mea values over plae wave icidece times, provided that plae waves amplitude is chose accordig to mea fields values iside the MSRC as i (3). The same result was obtaied by a differet approach (see equatio (39) i [6]). To obtai a equivalet result for mea value of iduced curret amplitude predictio, it is ot sufficiet to take the square root of (5). Two differet solutios ca be used to obtai a equivalet fast method. Such solutios are show by (6) ad their validity domais are discussed i the followig. Ε { x } x PW, i = Ε PW, i i= 1 Ε π x PW, i = Var Re PW, i i= 1 ( { x }) (6.a) (6.b) It ca be show [10] that (6.a) is valid oly whe real ad imagiary parts of the quatity x PW, i are ormally distributed with zero mea ad equal variace, which is ot true i a geeral case 1. I cases where this assumptio is verified, (6.a) equals mea value i RC ad mea values over plae wave icidece times. Of more geeral validity is (6.b), which is valid for ay radom complex quatity x PW, i which has zero mea value ad equal variace for real ad imagiary parts. I coclusio, four differet MC methods are proposed. The complete method i (4) ca be used to predict ay EM quatity iside a RC (amplitude or squared amplitude); mea values over oe stirrer rotatio as well as values for sigle stirrer positios ca be predicted by this method (this implies that also maximal values ca be predicted). Three fast methods for the predictio of oly mea values are also proposed. For squared amplitudes mea values predictios, fast1 method i (5) ca be used if the predicted quatity has zero mea value for real ad imagiary parts over radom plae waves. For amplitudes mea values predictios, fast method i (6.a) ca be used for quatities whose real ad imagiary parts are ormally distributed with zero mea ad equal variace over radom plae waves icidece; fast3 method i (6.b) ca be used for quatities whose real ad imagiary parts have zero mea ad equal variace over radom plae waves icidece. IV. MONTE CARLO PREDICTION OF THE COUPLED CURRENT FLOWING IN A TRANSMISSION LINE The iterest of the MC predictio method discussed i sectios II ad III lies ito the ability of predictig EM coupled quatities to a electrical device excited by a offedig plae wave. Oe case of iterest is coupled curret to multicoductor trasmissio lies, for which couplig theory is well established [7]. A applicatio example is cosidered i this sectio, cosiderig a sigle lossless trasmissio lie over a groud plae. The differet MC methods discussed i Sectio III have bee applied to the predictio of the iduced curret flowig i the termial load of a sigle wire ruig over a groud plae whe it is tested iside a RC. For 1 It is helpful to otice that ormal distributio is required here for x PW, i ad ot for x PW, i which is always ormally distributed accordig to the CLT.

4 the simulatio, each plae wave couplig cotributio was computed by meas of the covetioal theory of fields couplig to trasmissio lies [7]. Oe further cosideratio must be made cocerig computatio of radom plae waves cotributios. Accordig to physical properties of RCs, idepedet frequecies of excitatio of the chamber correspod to idepedet exited modal structures, ad, i a plae wave model, correspod to differet plae waves patters. This meas that for a proper MC simulatio, a differet patter of radom plae waves should be geerated for each idepedet frequecy. However, as plae waves are radom ad idepedet, the solutio of keepig the same radom patter for the etire frequecy rage ca be adopted, thus reducig computatioal time. I the followig, a 50 cm log sigle wire ruig at a height of 3 cm above the chamber floor is cosidered both for measuremet ad simulatio. Several experimetal trasmissio lie devices were tested i two differet chambers [11], ad experimetal results compared with simulatio results. At first, the iduced curret was predicted accordig to the complete method of equatio (4). The complete simulatio was made by illumiatig the lie 144 times by packets of 0 radom plae waves, ad the mea value of oe electric field rectagular compoet was measured iside the chamber ad used to set the value for the simulated plae waves amplitude, accordig to equatio (3). For the purpose of compariso, the mea value of the iduced curret i the lie termiatio was calculated over 144 stirrer positios. Fig. 1 shows the compariso betwee predicted ad measured mea values ad maximal values of the iduced curret amplitude as a fuctio of frequecy; Fig. shows cumulative desity fuctios for predictio ad measuremet of curret amplitude, both compared with a ideal χ distributio. I L (dba) L=50 cm, h=3 cm, Z =Z =50 Ohm L R Max( I L ) < I L > measuremet predictio f (Hz) Figure 1: MC complete method: amplitude of iduced curret at the trasmissio lie termial load mea curret value as a fuctio of frequecy Cumulative desity fuctio L=50 cm, h=3cm, Z =Z =50 Ohm L R measuremet predictio χ d.o.f. f=1e+009 Hz KS test (95%): passed χ test (r.s.l. 0.05%): passed I refereced to mea Figure : MC complete method: amplitude of iduced curret at the trasmissio lie termial load cumulative desity fuctio at a sigle frequecy Results of Fig. 1 ad show a good agreemet betwee predictio ad measuremet. Furthermore, Fig. shows that the coupled curret follows a χ distributio, both for predictio ad measuremet results. Visual agreemet for cumulative desity fuctio as well as statistical goodess of fit tests are passed. Oce validated the complete method by compariso with measuremet, fast methods of equatios (5) ad (6) ca be validated with respect to the complete method. The same trasmissio lie of the previous case was aalysed with fast methods. This time, a costat plae wave amplitude was chose as a fuctio of frequecy, sice we are ot attemptig to compare simulatio results to measuremets. Plae waves amplitude was chose equal to 1 V/m for the etire frequecy rage. Results i Fig. 3 show the compariso betwee the fast1 method for the predictio of mea value of squared curret amplitude ad the complete method, as i equatio (5). Results for the complete method where obtaied by estimatig expected value of equatio (4) as mea value over 1500 trials, where each simulatio was carried out by superposig 0 plae waves cotributios. Results for the fast1 method where obtaied by estimatig expected value of equatio (5) as mea value of 1500 trials. Simulatio results obtaied with the two methods are withi the radom simulatio ucertaity ad validate theory. Results i Fig. 4 show the compariso betwee the fast ad fast3 methods for the predictio of mea value of curret amplitude ad the complete method, as i equatios (6.a) ad (6.b). Simulatios of 1500 times 0 plae waves cotributios for the complete method ad of 1500 plae waves cotributios for the two fast methods were carried out.

5 < I L > (dba) MC simulatio predictio results MC complete (1500*0 pws) MC fast1 (1500 pws) The ucertaity associated with simulatio results was also estimated accordig to the classical statistical theory of iferece o mea value ad variace estimatio. Ucertaity for simulatio results of Fig. 4 are proposed i Fig. 5. Results are proposed as the total 95% cofidece iterval ( 95% ) of mea estimated values, ad must be iterpreted i the followig way: mea values results (Fig. 4) ± 95% / (Fig. 5) give the 95% cofidece iterval of simulatio results Figure 3: MC complete ad fast1 methods: predictio of the mea squared amplitude of coupled curret -7 MC complete MC fast MC fast3 MC simulatio predictio results 95% (db) % cofidece iterval i MC simulatio results MC complete (1500*0 pws) MC fast (1500 pws) MC fast3 (1500 pws) < I L > (dba) Figure 4: MC complete, fast ad fast3 methods: predictio of the mea amplitude of coupled curret. Differeces betwee "fast" ad "fast3" methods are outlied i the text Results show that the fast3 method predictio accordig to eq. (6.b) matches the complete method predictio, while fast method predictio accordig to eq. (6.a) does t match the complete method predictio for all frequecies. To evaluate the correctess of hypothesis discussed i Sectio III ad layig at the basis of equatios (6), iduced curret statistical properties, with respect to a radom icidet plae wave, were ivestigated by a umerical approach. Results of this aalysis show that real ad imagiary parts of coupled curret have ideed zero mea value ad equal variace, but the probabilistic distributio of real ad imagiary parts are frequecy depedet (see also [1]). Thus, fast3 method of equatio (6.b) is applicable, while fast method of equatio (6.a) is ot geerally applicable, except for frequecies where real ad imagiary parts approach a Normal distributio. As show i Fig 4, for frequecies far from resoace ad ati-resoace of the lie, where Normal distributio is better approached, the fast results better match complete method results Figure 5: Ucertaity of MC complete, fast ad fast3 methods (1500 trials of 0 plae waves superpositio for complete method ad 1500 plae waves cotributios for fast ad fast3 methods) Ucertaity results i Fig. 5 are obtaied with a large umber of simulatio trials. This was made to strogly reduce ucertaity ad thus evidece as much as possible the differeces i fast ad fast3 method results i Fig. 4. Ucertaities for more realistic trials umbers are reported i Fig. 6, where complete method is applied with 50 trials of 0 plae waves cotributios ad fast ad fast3 methods with 50 plae waves cotributios. 95% (db) % cofidece iterval i MC simulatio results MC complete (50*0 pws) MC fast (50 pws) MC fast3 (50 pws) Figure 6: Ucertaity of MC complete, fast ad fast3 methods (50 trials of 0 plae waves superpositio for complete method ad 50 plae waves cotributios for fast ad fast3 methods)

6 Results i Fig.6 evidece that acceptable ucertaity is obtaied by complete ad fast3 methods, eve for a reduced umber of plae waves cotributios. V. CONCLUSIONS A Mote Carlo simulatio method based o a superpositio of radom plae waves has bee proposed to represet reverberatio chamber coditios. The geeratio of radom plae waves parameters has bee discussed ad it has bee show that the resultig fields agree with the reverberatio chambers fields statistics. Additioally it has bee show how to relate mea fields values measured i reverberatio chamber ad costat plae waves amplitude used i simulatio. The proposed method ca be used to predict electromagetic couplig to electrical devices, provided that their respose to a offedig plae wave ca be umerically computed; i particular the method is advatageous whe this last computatio ca be obtaied at a low cost. The method has bee applied to predict the curret iduced ito a sigle wire ruig over a groud plae iside a reverberatio chamber. I this case trasmissio lie theory was used to predict curret iduced by a offedig plae wave. A good agreemet was obtaied betwee predictio ad measuremet results. The proposed method allows us the predictio both of mea ad maximum values of coupled electromagetic quatities over oe stirrer rotatio. To reduce computatioal time, three faster methods have bee proposed to predict oly mea values over oe stirrer rotatio. Basig o simple statistical assumptios about real ad imagiary parts of electromagetic coupled quatities distributio, mea value for amplitude (curret or voltage) ad squared amplitude (power) of such quatities ca be predicted by faster methods. Ucertaity associated with Mote Carlo simulatio results was also addressed i this work. Ucertaity decreases as the umber of simulatio trials icreases, ad it has bee show that, for the reasoable umber of 50 plae waves cotributios, the ucertaity is of the order of ± 1. 5 db, that is of the same order of measuremet ucertaity iside a reverberatio chamber. VI. REFERENCES [1] D. H. Kwo, R. J. Burkholder ad P. H. Pathak, "Ray aalysis of electromagetic field built-up ad quality factor of electrically large shielded eclosures", IEEE Tras. Electromag. Compat., vol. 40, 1, pp. 19-6, Feb [] M. Höojer, A. M. Adersso, O. Lude ad M. Bäckström, "Three-dimesio fiite differece timedomai aalysis of reverberatio chambers", Proc. 4 th Europea Symposium o Electromag. Compat., Vol. 1, pp , Brugge, Belgium, Sep [3] C. F. Butig, "two dimesioal fiite elemet aalysis of reverberatio chambers; the iclusio of a source ad additioal aspects of aalysis", Proc. of IEEE It. Symp. o EMC, pp. 19-4, Seattle, USA, August [4] J. M. Ladbury, Motecarlo simulatio of reverberatio chambers, Natioal Istitute of Stadards ad Techology, Boulder Iteral Note, CO, Oct. 4-9, [5] T. H. Lehma, G. J. Freyer, M. O. Hatfield, J. M. Ladbury, G. H. Koepke, Verificatio of fields applied to a EUT i a reverberatio chamber usig umerical modelig, Proc. of 1998 IEEE It. Symp. o EMC, pp. 8-33, Dever, USA, August 4-8, [6] D. A. Hill, Plae wave itegral represetatio for fields i reverberatio chambers, IEEE Tras. Electromag. Compat., Vol. 40, pp , August [7] C. R. Paul, Aalysis of multicoductor trasmissio lies, Wiley series i microwave ad optical egieerig, New York, [8] J. G. Kostas ad B. Boverie, Statistical model for a mode-stirred chamber, IEEE Tras. Electromag. Compat., vol. 33, pp , Nov [9] T. H. Lehma, E. K. Miller, The elemetary properties of electromagetic fields i complex cavities, Proc IEEE At. ad Propag. Soc. It. Symp., Vol.3, pp , Jue 4-8, [10] A. Papoulis, Probability, Radom Variables, ad Stochastic Processes, Third Ed., McGraw-Hill Iteratioal Editios, [11] L. Musso, B. Demouli, F. Caavero, V. Berat, Susceptibility of a Trasmissio Lie i two Reverberatio Chambers, i Proc. 001 Rev. Chamb. A. Chamb. Ad OATS Users Meetig, Seattle, WA, USA, Jue 4-6, 001. [1] D. Bella, S. Pigari, A Probabilistic model for the respose of a electrically short twi-coductor trasmissio lie drive by a radom plae wave field, IEEE Tras. Electromag. Compat., vol. 43, pp , May 001.