Towards the generation of industrial bundles through a random process under realistic constraints

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1 Author manuscrpt, publshed n "EMC Europe th Internatonal Symposum on EMC jont wth 20th Internatonal Wroclaw Symposum on EMC, Wroclaw : Pologne (2010)" Towards the generaton of ndustral bundles through a random process under realstc constrants Charles Jullen (1)(2), Phlppe Besner (1) Unversté Européenne de Bretagne (1), INSA, IETR, UMR CNRS 6164, Rennes, FRANCE charles.jullen@nsa-rennes.fr Mchel Dunand (2) Dept. RTDSA (2) LABINAL Vllemur sur tarn, FRANCE Isabelle Junqua (3) Dept. DEMR/CEM (3) ONERA Toulouse, FRANCE hal , verson 1-15 Dec 2010 Abstract To get closer to the realty of aeronautcs wre harnesses, we propose to represent a bundle as a non-unform transmsson lne by dscretzng t nto several sectons. For each secton, postons of conductors are randomzed. However ths procedure s appled wth some geometrcal constrants thus resultng n a much more realstc bundle profle. Currents and voltages may be determned on each conductor. Then, we use statstcal tools to analyze our results. In order to save computaton tme, we also propose smplfed models for smulaton studes of the common mode current. Keywords-component; Cable Bundle; Common-mode current; Crosstalk; Multconductor Transmsson Lne; Statstcal; Random; Webull; Lnear Congruental Generator I. INTRODUCTION Embedded electrc and electronc functons play an ncreasng role n modern transportaton systems. Therefore the number of cable nterconnectons s stll very mportant even f RF technques or multplexng may coexst. These cables are very well known to be potental paths for electromagnetc nterference. Snce the late 80 s, mportant theoretcal work has been done to model mult-conductor harnesses. We may frst menton the work done to descrbe dstrbutons of multconductor nterconnectons based on the electromagnetc topology prncples [4-8]. Ths theory splts the complex problem nto sub-elementary problems and uses the equatons of transmsson lnes theory to handle calculaton of electromagnetc nteractons n bundles. Snce the late 90 s, modelng the effects of non-unformty n bundles to consder nterlacng has become essental to cope wth realstc geometres of cables bundles. Indeed, n aerospace or automotve context, elementary wres nsde harnesses whch can be up to several hundred meters long are rarely parallel to each other and rarely arranged n a precse order. The authors n [1] descrbe a way to model the randomness of the arrangement of the cable n a secton of wre bundle usng the RDSI algorthm. From our sde, we propose to descrbe the bundle as a successon of sectons. Each secton s defned by the geometry of conductors whch may be determned randomly by a dfferent method from [1], takng nto account a dsplacement between the conductors closer to realty. Our goal s to buld up a model whch s as realstc as possble. Ths model must consttute a reference for computaton of EM nteractons nsde bundles. Further studes wll be then dedcated to the development of less complex representaton when possble. After a presentaton of the two methods of bundle generaton (Secton II), we descrbe the case study [1] n Secton III. Then we mplement our model and compare t wth results obtaned n [1]. In ths case study, a voltage s appled on one nternal wre. Frst, the common mode current nduced on the bundle s determned (Secton IV.A). Then we focus on the crosstalk parameter between conductors of the bundle (Secton IV.B). The next step conssts n a statstcal analyss of results by lookng at ther dstrbuton functon and usng the probablty dstrbuton of Webull n Secton V. Fnally we show n Secton VI how a smplfed model can reduce the computng tme for studyng the common mode current wth lttle loss of accuracy. II. DIFFERENT MODELS OF INTERLACING CABLES A. Model wth RDSI algorthm The method developed n [1] conssts n: - cuttng the bundle of cable nto sectons respectng the constrant shown n [3] where the length of subsectons remans below λ/10, λ beng the wavelength. - takng a fxed geometry as n Fg. 1 (a) - nter-changng conductors numbers assocated wth these crcles n ths geometry. These numbers change randomly from one secton of the conductor to the next one accordng to a Gaussan dstrbuton. Consequently a smooth transton between two consecutve sectons s acheved. Per unt length L and C matrces are calculated once and approprate permutatons are acheved for each secton n order to evaluate currents and voltages on conductors. B. Model wth LCG algorthm The random generaton s done wth a pseudo-random algorthm: Lnear Congruental Generator (LCG) defned n: x n-1 = (a x + c) mod(m) (1) n

2 hal , verson 1-15 Dec 2010 where x n s the sequence of pseudorandom values (x 0 s the ntal value), m the module, a the multpler and c the ncrement. It s one of the oldest and most effcent algorthms to generate pseudo-random numbers. It s used mostly for random functons n complers as ANSI C, Mcrosoft vsual, GCC, Random class API for Java,.... Our model generates geometrc sectons of bundles where conductors are randomly placed nto a 2-D (x, z) plane (the bundle follows the y-axs) under some constrants. Placement s made by evaluatng the neghborhood of each conductor whch s randomly chosen. It may however be constraned by a noton of group forcng some conductors to stay nearby, as for example twsted par. Once ths partcular neghborhood s determned, we select a conductor randomly and compute also ts new coordnates by a random process. The placement must stll comply wth both, a crteron of proxmty whch can be adjusted to more or less condensed secton, and a constrant of non-overlappng wth respect to conductors already placed. Moreover, the conductor s placed n a mnmal radus of space. The mnmal radus of space s determned by dfferent technques of barycentrc calculatons and calculaton of maxmum dstance. Once all conductors placed, a fnal check enables to ensure that the bundle s at the correct heght above the reference ground plane. Usng ths process, we obtan a geometry as the one shown n Fg. 1 (b) and a whole bundle as llustrated n Fg. 2. The transton between two successve sectons s ensured by a randomly varyng parameter n the range lmted by the mechancal constrants of conductors nsde the bundle. The change of each conductor coordnates fulflls a consstency n the ntertwnng. These geometres are then used by a software developed at ONERA, called LAPLACE, to compute the R, L, C, G matrces (per unt length parameters of transmsson lnes) of each secton. These per unt length matrces whch entrely descrbe the bundle from an electromagnetc pont of vew are ncluded n the CRIPTE [4] software. Ths software solves the well known BLT equaton appled on mult conductor transmsson lne networks to fnally obtan currents and voltages on each elementary conductor [5-8]. Fgure 2. Three dmensonal vsualzaton of the bundle wth 100 cross sectons III. DESCRIPTION OF CASE STUDY IMPLEMENTATION OF OUR MODEL The cable bundle proposed n [1] s composed of 14 conductors each of dameter 1.8mm and wth 0.9mm thck PVC nsulaton. Ths cable bundle s 2m long and s placed above an alumnum plane (2.62x1.2m). Each conductor s loaded dfferently from 10Ω to 100kΩ symmetrcally or not. The conductor source s the conductor number 2 on whch a voltage generator loaded wth 50Ω s appled. Computatons are made at 0.31m (P1 poston), 0.81m (P2 poston) and 1.69m (P3 poston) far from the orgn on the whole cables bundle as n Fg. 3. Loads + Generator P1 P2 P3 Plan de masse Fgure 3. Schematc of the measurement setup Loads 1.8mm 0.9mm (a) (b) Fgure 1. Two dmensonal cross secton vew, artcle [1] modelng (a) and modelng wth LCG algorthm (b) Fgure 4. Two dmensonal cross secton vew of artcle [1]

3 The frst secton of the bundle defned n [1] and recalled n Fg. 4 s our ntal secton geometry for our own generaton process. Then we appled our LCG model prevously descrbed to generate 100 geometrc sectons of 0.02m, representng 2m of cable bundle. 40 dfferent bundles have been generated through ths random process. Fnally we computed currents/voltages on elementary conductors of these 40 generated bundles nduced by the voltage generator on conductor number 2 from 100Hz to 1GHz. IV. NUMERICAL RESULTS PHYSICAL ANALYSIS Common mode current [dba] Current n P1 Current n P2 x x Current n P3 hal , verson 1-15 Dec 2010 A. Analyss of the common mode current The objectve s, here, to compare our LCG algorthm to the one proposed n [1]. Therefore, for physcal analyss, we represent the maxmum, average and standard devaton of the common mode current for the 40 bundles modeled on the entre frequency range at the measurement pont P1 (Fg. 5). Common mode current [dba] Maxmum Mean Standard devaton * * Fgure 5. Accumulated maxmum, average and standard devaton common mode current from 40 smulatons at pont P1 Frst of all, we fnd the same behavor as n [1] wth smlar resonance frequences but wth slghtly hgher levels whch could be explaned by the numercal scheme used n our case to solve the BLT equaton. We note that n Fgure 5, the maxmum and average common mode current have qute the same magntude n low frequency, before the resonant doman. Ths s confrmed by a very low standard devaton 45 db lower than the average up to 50MHz. Wth ncreasng frequency, when the bundle becomes resonant, the standard devaton compared to the average s much more mportant. It shows that n that frequency range the common mode current depends on the bundle nternal geometry. Fg. 6 llustrates the varaton of the common mode current at ponts P1, P2 and P3 computed on one sample of the bundle among the 40 generatons. Fgure 6. Smulated common mode current at pont P1, P2 and P3 wth the same bundle realzaton In Fg. 6, we observe that the frequency response of the common mode current strongly depends on the measurement poston and the dstance from the source. We can note some symmetry of the measurement ponts P1 and P3 lookng at the resonance and antresonance frequences assocated to these dstances. For example, at 450MHz we fnd an antresonance n P3 and a resonance n P1. Agan the behavor of the curves s smlar to the one observed n [1]. B. Analyss of crosstalk In the followng descrpton, we are nterested n determnng the crosstalk between an arbtrary wre (2) and another wre (3) whch are randomly placed n the bundles descrbed above. The crosstalk parameter between conductor and conductor j s defned by: H(f) V j(f) V (f) = (2) where V j (f) s the nduced voltage on wre j whle V (f) s the appled voltage on wre. In our case, ths crosstalk has been evaluated between conductor 2 (on whch the voltage generator s appled) and conductor 3 for 16 generatons of bundles as n [1]. These results are presented n fgure 7. Crosstalk [db] Maxmum Mnmum Fgure 7. Near-end crosstalk between wre 3 and wre 2 for 16 generatons of bundles The two curves above and below represent the maxmum and mnmum results among the 16 generatons of bundles n

4 Fg. 7. We note that n low frequency up to 1MHz, we have a slope of 20dB/decade explaned by the nductve couplng between both conductors. Above 1MHz, the random poston of the conductors, the nature of the bundle and capactve effects become ncreasngly mportant whch causes more devaton n the frequency response. V. STATISTICS FOR COMMON MODE CURRENT In order to better understand the results, t s necessary to go through a statstcal analyss of the results of smulatons of dfferent bundles. Statstcal analyss must be the most general as possble, we use a dstrbuton law of Webull. The probablty densty functon of the Webull law s defned by: PDF Common mode current [A] x x Gaussan Webull hal , verson 1-15 Dec 2010 f(x) (b-1) (-a x ) = a b (x ) e (3) where a s a scale parameter, b, the shape parameter and X the random value under analyss. Raylegh, exponental or even Gaussan dstrbutons appear as partcular cases of ths Webull dstrbuton wth specfc values of a and b. The determnaton of these two parameters can be made by several methods. One of them s based on the maxmum lkelhood method and s detaled n [2,9,10]. By applyng ths method, we obtan a system of two equatons wth two unknowns: N + b N = 1 N = a ln(x ) + a b N b x = 1 N = 1 (x b ln(x )) = 0 where N s the number of smulatons and X are the current values for the smulatons to a specfc frequency. Fg. 8, 9 and 10 llustrate the pdf probablty dstrbuton functon (pdf) and the cumulatve dstrbuton functon (cdf) assumng a Gaussan law as n [1] and the proposed Webull law derved from our samples of computed common mode currents on the 40 generated bundles at respectvely 506MHz, 528MHz, 550MHz. These frequences have been chosen because they are respectvely located at the top, mddle and bottom of a resonance peak. All curves are normalzed for ncreased readablty of the fgures. PDF Common mode current [A] b x x Gaussan Webull Fgure 8. Analytcal probablty densty functons and the hstograms at 506MHz. The number of smulaton s 40. (4) Fgure 9. Analytcal probablty densty functons and the hstograms at 528MHz. The number of smulaton s 40. PDF Common mode current [A] x x Gaussan Webull Fgure 10. Analytcal probablty densty functons and the hstograms at 550MHz. The number of smulaton s 40. The Webull dstrbuton law overlaps farly well on the numercal results on all fgures. The Webull dstrbuton law wth approprate parameters s statstcally preferred to explan the results of smulatons. For the three frequences, we have these parameters: a=0.77 and b=2.05 for 506 MHz a=0.84 and b=1.74 for 528 MHz a=0.81 and b=1.79 for 550 MHz Snce the Webull law s a two-parameter functon t appears that some other dstrbuton laws appears to be specfc cases of ths functon. Thus, a Raylegh law s n fact a Webull law wth parameters a=0.78 and b=2. So wth these estmatons of a and b for 506 MHz, the dstrbuton s probably well approxmated wth the Raylegh law. We have also performed a Kolmogorov-Smrnov test wth a Massey crteron corrected for Webull Law [2] to check the valdty of the law to the data representaton. For a rsk threshold set at 1%, f we look at the maxmum dfference between the theoretcal and expermental dstrbuton functon, we get a value below ths maxmum dstance wth ths threshold of rsk. We can therefore conclude that the Webull dstrbuton law could be an acceptable canddate for ths process.

5 hal , verson 1-15 Dec 2010 VI. SIMPLIFIED MODELS We saw n the prevous paragraphs that the computaton of common mode currents s based on the evaluaton of RLCG matrces. Here, each of the 40 generated bundles s descrbed by 100 geometres (100 cross sectons). In other words, 4000 geometres must be generated and therefore matrces computed. For all models/smulatons on ths entre set of 40 bundles, the computaton lasts 8h to determne all the RLCG matrces. The calculatons are performed on a computer wth a 3.4GHz processor and 1GB of RAM. One way to decrease ths computng tme s to smplfy the geometrc model to reduce the number of RLCG matrces to compute. The frst possble smplfcaton conssts n generatng multple unform bundles whch sngle secton s randomly created. We obtan for 40 bundles, only 160 matrces. The second model conssts n reducng the 14 conductors to one equvalent conductor whch dameter s equvalent to the whole bundle dameter wth a thck delectrc equal to one of the14 conductors, as shown n Fg. 11 (a). (a) Fgure 11. Two dmensonal cross secton vew, wth equvalent model (b) and wth RDSI model (a) A. Common mode current Fg. 12 shows the average current mode common computed: - on the 40 bundles wth non-unform dscretzaton as n secton III, - on the 40 bundles dscretzed by a sngle random geometry - and the equvalent conductor. For ths equvalent conductor, the extremtes loads are supposed to be the 14 ntal loads n parallel. The equvalent source generator to apply s derved from a Thevenn equvalent model. (b) Common mode current [dba] Unform bundle x x Non-unform bundle Equvalent Bundle Fgure 12. average common mode current The average common mode computed on the bundle dscretzed by 100 sectons and the average common mode current computed on 40 sngle geometres are very close to each other n low frequency. Wth ncreasng frequency, we can note that there are some dscrepances whch can be explaned by the low number of unform bundles ncluded n the average and by the effects of the nterfaces between two sectons not ncluded n the unform bundle. However the devaton s rather small (about 20dB below the two averages), ths model represents farly well the bundles dscretzed by 100 sectons. The calculaton tme for 40 smulatons s about 1 hour that s 1/10th of the ntal computaton tme. The result s almost equvalent. Wth the equvalent conductor model, we fnd smlar behavor ncludng the same resonance frequences. We retan most nformaton but wth patterns of more mportant antresonance due to the dealzaton of our model n the resoluton of the BLT equaton [5-8]. The calculaton tme s 5 mnutes on the same computer used prevously. So dependng on the nature of nformaton sought, we can reduce the computng tme n usng equvalent models for common mode currents. B. Crosstalk current It would nevertheless be nterestng to see f such a smplfed model s applcable to evaluate the currents on one of the 14 conductors of the bundle. Fg. 13 and Fg. 14 represent maxmum, average and standard devaton of the current on a conductor free, for the ntal randomzed secton bundle and the equvalent randomzed unform bundle respectvely.

6 hal , verson 1-15 Dec 2010 Current [dba] Maxmum Mean x x Standard devaton Fgure 13. Accumulated maxmum, average and standard devaton of 40 smulatons non-unform bundle current on wre 3 Current [dba] Maxmum Mean x x Standard devaton Fgure 14. Accumulated maxmum, average and standard devaton of 16 smulatons unform bundle current on wre 3 We note that the ntal bundle and the unform bundle gve dfferent results for average current. There could be 2 reasons for that. Frst the number of smulated unform bundles may be not hgh enough. Second the nherent non unform structure of the actual bundle may explan ths dfferent behavor. These hypothess wll be nvestgated n a near future. However, a low standard devaton wth respect to average appears n Fg. 13 for frequences below 50MHz. It ndcates the smlarty of results n low frequency whatever the non-unform bundle acheved. Therefore a smple model would stll be vald n a quas-statc regme. In Fg. 14, we can observe antresonance frequences whch are specfc of the smplfed model of unform bundles. These results enable to glmpse the lmts of ths smplfed model compared to the phenomena that we want to observe. VII. CONCLUSION In ths paper, we have presented a model of random bundles based on the random placement of conductors n space wthn the constrants of nterface between sectons. Ths model has been developed n order to match as closely as possble the fne detals of the geometry of ndustral bundles. It s ntended to be a reference model before valdatng further possble smplfcaton. We have frst carred out a comparson wth the RDSI model developed by authors of [1]. Ths model s based on the modfed number of conductors from secton to secton and therefore could appear as a smpler model. It turns out that we fnd common mode currents comparable to those of the artcle [1]. In that very specfc case, t s even possble to further smplfy further the way of evaluatng the common mode current. I may be ndeed evaluated wth unform bundles. Lookng at crosstalk results, our model and the RDSI model are comparable. A fnal concluson would requre further nvestgaton and expermental studes. Addtonal work s also necessary to establsh smplfed models to correctly reproduce crosstalk phenomena. [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] REFERENCES Shshuang Sun, Gepng Lu, James L. Drewnak and Davd J. Pommerenke, Hand-Assembled Cable Bundle Modelng for Crosstalk and Common-Mode Radaton Predcton, IEEE Trans. Electromagnetc Compatblty, vol. 49, pp , August 2007 C. Lemone, Contrbuton to the statstcal analyss of measurements data n mode-strred reverberaton chamber. Applcatons for the evaluaton of strrng effcency and measurements uncertanty n the context of radofrequences and EMC., Thess, Insttut d'electronque et de Télécommuncatons de Rennes, pp , July 2008 P.Besner, Etude des couplages Electromagnétques sur des Réseaux de lgnes de Transmsson non Unformes à l'ade d'une Approche Topologque, Thess, January 1993 L.Paletta, Demarche Topologque pour l'étude des couplages électromagnétques sur des systèmes de câblages ndustrels de grande dmenson, Thess, September 1998 C.E Baum, T.K. Lu, F.M. Tesche, On the Analyss of General Multconductor Transmsson Lne Networks Interactons Notes, Notes 350, January 1988 C.E Baum, Electromagnetc Topology for the Analyss and Desgn of Complex Electromagnetc System, Fast Electrcal and Optcal Measurements, Vol. 1, pp , Martnus Njhoff, Dordrecht, 1986 P. Degauque, A. Zeddam, Compatblté Electromagnétque 1, Ed. Hermes, pp C. R. Clayton, Analyss of Multconductor Transmsson Lnes, Ed. Wley-Interscence, pp , 1994 A.Papouls, S. Unnkrshna Plla, Probablty, Random Varables and Stochastc Processes, Ed. Mc Graw Hll, 2002 W.Webull, A Statstcal Dstrbuton Functon of Wde Applcablty, ASME Journal of appled mechancs, Transactons of the Amercan Socety Of Mechancal Engneers, pp , September 1951

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