Shielding Effectiveness of Three Dimensional Gratings using the Periodic FDTD Technique and CPML Absorbing Boundary Condition

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1 Sheldg Effectvee of Three Dmeoal Gratg ug the Perodc FDTD Techque ad CPML Aborbg Boudary Codto J.Ala Rode, J. Paul Sker, ad Steve L. Joh The Aeropace Corporato Chatlly, Va, 151, USA Abtract: Th paper exame the beeft of gratg thcke regard to ehacg EMI atteuato. Aaly performed ug the FDTD techque ug perodc boudary codto alog wth ew formulato of the CPML aborbg boudary. Furthermore, t demotrated that a mple wavegude below cutoff approxmato provde accurate reult for the ormal cdet plae wave cae. Keyword: CPML, FDTD, Gratg, Sheldg Effectvee (SE), EMI 1. Itroducto Metalzed gratg are ofte ued helded ecloure deg to allow vetlato or codut for coductor whle atteuatg electromagetc eergy. Whle may deg ue wavegude below cutoff prcple to ure adequate atteuato, cot may tme prohbt uch robute. The focu of th paper to exame the effectvee of gratg, grll, or vet array a the thcke of thee tructure are creaed.. CPML formulato wth the Perodc FDTD Perodc aaly code are uquely uted for problem volvg large array of aperture whch are llumated by plae wave. If oe aume that o ear feld ource are preet, th type of aaly ca provde good performace metrc for typcal ar vet or cable pa-through tructure. The oluto method th work the perodc plt-feld FDTD (SF-FDTD) a frt troduced by Rode, et al [1],[] ad documeted later [3]. Proper mplemetato of the PML aborbg boudary wth the plt feld mplemetato perodc FDTD (SF-FDTD) ha bee documeted prevouly [3]. I that work, the PML update equato wa accomplhed ug a plt feld approach. A more robut mplemetato may be accomplhed ug the recetly troduced covoluto PML (CPML). The advatage of corporatg the CPML wth SF-FDTD that the addtoal PML overhead traparet o far a the mplemetato of the plt feld update expreo ad gfcat atteuato of evaecet mode accomplhed. The frequecy doma expreo for Ampere law cludg CPML are gve by 1 θ φ P x + σp x = Q z Q y jω Q z c ( ω ) y z z c (1) 1 θ coφ P y + σp y = Q x Q z + jω Q z c ( ω ) z z x c () θ coφ θ φ P z + σp z = Q y Q x jω Q y + jω Q x c x y c c (3) 5 ACES

2 Report Documetato Page Form Approved OMB No Publc reportg burde for the collecto of formato etmated to average 1 hour per repoe, cludg the tme for revewg tructo, earchg extg data ource, gatherg ad matag the data eeded, ad completg ad revewg the collecto of formato. Sed commet regardg th burde etmate or ay other apect of th collecto of formato, cludg uggeto for reducg th burde, to Wahgto Headquarter Servce, Drectorate for Iformato Operato ad Report, 115 Jeffero Dav Hghway, Sute 14, Arlgto VA -43. Repodet hould be aware that otwthtadg ay other provo of law, o pero hall be ubject to a pealty for falg to comply wth a collecto of formato f t doe ot dplay a curretly vald OMB cotrol umber. 1. REPORT DATE 1 JAN 5. REPORT TYPE N/A 3. DATES COVERED - 4. TITLE AND SUBTITLE Sheldg Effectvee of Three Dmeoal Gratg ug the Perodc FDTD Techque ad CPML Aborbg Boudary Codto 5a. CONTRACT NUMBER 5b. GRANT NUMBER 5c. PROGRAM ELEMENT NUMBER 6. AUTHOR(S) 5d. PROJECT NUMBER 5e. TASK NUMBER 5f. WORK UNIT NUMBER 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) The Aeropace Corporato Chatlly, Va, 151, USA 8. PERFORMING ORGANIZATION REPORT NUMBER 9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 1. SPONSOR/MONITOR S ACRONYM(S) 1. DISTRIBUTION/AVAILABILITY STATEMENT Approved for publc releae, dtrbuto ulmted 11. SPONSOR/MONITOR S REPORT NUMBER(S) 13. SUPPLEMENTARY NOTES See alo ADM1846, Appled Computatoal Electromagetc Socety 5 Joural, Newletter, ad Coferece., The orgal documet cota color mage. 14. ABSTRACT 15. SUBJECT TERMS 16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF ABSTRACT UU a. REPORT uclafed b. ABSTRACT uclafed c. THIS PAGE uclafed 18. NUMBER OF PAGES 4 19a. NAME OF RESPONSIBLE PERSON Stadard Form 98 (Rev. 8-98) Precrbed by ANSI Std Z39-18

3 where ad = κ + 1 σ. (4) jωε + α P = E η exp( k[ θcoφx+ θφy+ coθz] (5) Q = H exp( k [ θ coφx+ θφy+ coθz] η. (6) I (4), the varable σ determe the atteuato rate of propagatg wave wth the PML, whle κ, α determe the atteuato of evaecet wave wth the PML. The bac formulato cot of plttg each of thee equato (ad ther E feld couterpart) to two part ad troducg a alterate tme grd uch that the prmary ad alterate grd are taggered tme by ½ tme tep. For each of the two tme grd, the ormalzed feld P x compoet update become ε r P x = P x σηβ ση ( 1 β ) P x ε + σηβ c r t c t a a a + 1 ησ θ φ ( 1 ) (,, 1 ) x 1 + Q Q Q Q + + ψ zj zj 1 yk yk κ Q z Q j z j xz y z z. (7) Smlar equato are mplemeted for the remag feld compoet ad thee are vald throughout the etre computatoal doma. Subequetly, ummato varable are troduced ad appeded to the A part of the plt feld equato [4] reultg from (1) through (4) ad are cremeted at each half tme tep a ψ a+ 1 a P x = P x + Q + Q + ψ z y (8) px c tµ r y κ z = +, =, = 1. + (9). (1) σ σ σ σ + εκ ε σ + εκ ε ψ 1 t t a b Q qx qx y a e b e z σ κ k α d t w Fgure 1 Geometry of th tudy, fte array of quare aperture.

4 3. Reult The bac geometry choe for th tudy gve Fgure 1. Show here a perodc array of quare aperture wth depth t, whch a varable the tudy. For th tudy, w=5. ml ad d=4 ml. A plae wave llumate the tructure from above at a agle θ to the urface ormal TM polarzato. I addto to FDTD mulato, aalytc reult for the tramo coeffcet are computed for the ftely th cae baed o well kow formula. Thee aalytc expreo for tramo ad reflecto are Z Zcoθ τtm =, τte = (11) Z + Z coθ Z coθ + Z where θ the agle of cdece wth repect to ormal, Z the mpedace of free pace, ad Z the heet mpedace of the ftely th cree whch gve by Golde [5] formula a Z te jzd d tm jzd coθ d = l, Z = l λ πw λ πw. (1) I (1), d the ceter-to-ceter pacg of the perodc grd, ad w the wdth of the metal member. Furthermore, wave-gude below cutoff theory ued order to apply (11) to the geeral cae wth thcke. The combato of (11) wth th wavegude atteuato (poed for the lowet order mode) provde a eay to ue formula for predctg thck urface heldg performace. Fgure llutrate the varato of tramo coeffcet wth varyg cree thcke a the depth goe from ftely th to ½ ch. Aalytc reult ad FEM reult are alo preeted here ad how good agreemet. Fgure 3 llutrate the effect of varyg the cdet agle from ormal cdece to 6 degree for TM (to urface) polarzato a prcple plae of the lattce. 4. Cocluo Wavegude below cutoff cocept do provde a good approxmato of the effect of addg depth to perodc lattce of aperture metallc gratg. A a reult, oe ca cofdetly ue uch formula for computg the approprate thcke or depth of uch geometre order to atta a dered EMI atteuato. The FDTD techque proved qute ueful for aalyzg thee tructure ad provded qute fat reult. The ue of the CPML aborbg boudary proved eetal to adequately atteuatg trog evaecet mode whch were preet for hgher agle of cdece. Whle the problem tuded here cotaed oly metallc materal, the formulato geeral ad may be appled to a wde array of problem.

5 Tram;o (db) e+9 4e+9 6e+9 8e+9 1e+1 Frequecy (Hz) FDTD Iftely Th Aalytc Iftely Th FDTD.5. Thck Aalytc.5. Thck FDTD.5. Thck Aalytc.5. Thck FEM.5. Thck FEM.5. Thck Fgure Sheldg Effectvee mprovemet wth varyg thcke at zero degree cdet Tram;o (db) e+9 4e+9 6e+9 8e+9 1e+1 Frequecy (Hz) Degree FDTD 3 Degree FDTD 3 Degree FEM 45 Degree FDTD 45 Degree FEM 6 Degree FDTD 6 Degree FDTD Fgure 3 Sheldg Effectvee varato wth agle of cdece for.5 ch gratg. Referece 1 Rode, J. A., Electromagetc Aaly of Complex Structure ug the FDTD Techque Geeral Curvlear Coordate, Ph. D. The, Uverty of Ketucky, Lexgto, KY, Rode, J. A., S. D. Gedey, M. P. Keler, J. G. Maloey, P. H. Harm, Tme-doma Aaly of perodc tructure at Oblque Icdece: Orthogoal ad Noorthogoal FDTD Implemetato, IEEE Tra. Mcrowave Theory ad Techque, Vol. 46, 1998, pp Alla Taflove, Advace Computatoal Electrodyamc, The Fte-Dfferece Tme-doma Method, Artech Houe, J. A. Rode ad S. D. Gedey, Covolutoal PML (CPML): A effcet FDTD mplemetato of the CFS-PML for arbtrary meda, Mcrowave ad Optcal Techology Letter, Vol 7,. 5 K. E. Golde, T. M. Smth, Smulato of a Th Plama Sheath by a Plae of Wre, IEEE Traacto Nuclear Scece, Vol NS-11, pp. 5-3, Ja 1964.