A THEORETICAL MODEL OF A LOSSY DIELECTRIC SLAB FOR THE CHARACTERIZATION OF RADAR SYSTEM PERFORMANCE SPECIFICATIONS

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1 THEORETCL MODEL OF LOSSY DELECTRC SLB FOR THE CHRCTERZTON OF RDR SYSTEM PERFORMNCE SPECFCTONS Grgor L. Charvat, MSEE Prof. Edward J. Rothwll, PhD Dartmnt of Eltrial and Comutr Enginring Mihigan Stat Univrsit Enginring Building East Lansing, M 4885 BSTRCT Som radar aliations ruir a sstm to auir rang rofil or S ntwork analr data through a loss diltri lar to masur somthing bhind that loss diltri lar. t is oftn diffiult to sif th dnami rang ruirmnts of suh a sstm du to th flash of initial rfltd transmittd nrg from th loss diltri lar. t is also diffiult to dtrmin th most fftiv arhittur for suh a sstm, suh as uls F, ultra-widband imuls, FMCW, or anothr mor oti arhittur. n this ar a thortial modl is dvlod of a loss diltri lar, a radar transmittr and rivr, and a standard radar targt on th othr sid of th loss diltri lar. Th thortial rsults from this modl rovid insight into th dnami rang ruirmnts for an radar sstm that must auir rang rofil data or S ntwork analr data through a loss diltri of an rmabilit, rmittivit, and ondutivit at an mirowav or RF frun rang in ordr to masur somthing bhind that loss diltri lar. Kwords: Radar Sattring, Radar Sstm Prforman, Radar Sstm Sifiations, Masurmnt Through a Loss Diltri, Sattring from a Loss Diltri. ntrodution Whn dvloing sifiations and diding on arhittural ruirmnts for a tst sstm (radar, S ntwork analr, t) that must masur through a loss diltri it is oftn diffiult to did on suh issus as dnami rang and radar mod. n this ar, a thortial lanar modl of a loss diltri, air, and mtal sht is dvlod (s figur ). Using this modl, thortial rang rofils ar ratd to haratri dnami rang ruirmnts for suh a tst sstm whn it is ruird to masur a mtal sht as a radar targt on th oosit sid of a loss diltri and air lar. thortial modls ar wll known [,, 3, 4, 5]. n aml of rvious work in this ara would b in [6], whr a mtal lindr insid of a loss diltri bo was imagd thortiall, thn imagd in th lab using ral data. n this as, th lab rsults losl mathd that of th thortial. n ordr to k this thortial stud ratial, it was didd to rsnt th thortial data in th form of rang rofils. Whr, a radar (or S ntwork analr) hir signal from 5 MH to 3 GH was simulatd, and th tim domain rang rofil rsults of th sattrd fild rsntd. Suh rang rofil rsults ar asil rogniabl to anon in th fild. Th gomtr of this thortial stud is dsribd in stion. Stion 3 outlins th roblm solving stratg. Stion 4 rsnts th satial frun domain Fourir transform airs usd throughout th drivation. Stion 5 sts through ah st of th thortial roblm bing solvd. Stion 6 rsnts thortial rsults in th form of rang rofil data. Stion 7 summaris th rsults and disusss futur work. ll rfrns ar indiatd in stion 8.. Gomtr Th gomtr of this thortial stud is shown in figur. This is an infinitl lanar roblm, and thus ah of th lars ar infinit in th lan. Th lin sour is infinit in th dirtion and loatd at,. Rgion btwn b t ontains th infinit lan loss diltri of finit thiknss. Rgion btwn t is fr sa. Rgion 3 at is fr sa. Rgion 4 btwn d b is also fr sa. Th last lar ovring d is an infinit lanar rft ltri ondutor (PEC) for us as th mtal sht radar targt. Th thortial modl rsntd in this ar was solvd basd on lanar wavguid and iruit thor. Problm solving rodurs and onts for solving suh

2 jk ( k ), ( k, ) dk π () Whr: k vtor magnti otntial satial frun domain vtor magnti otntial satial frun 5. Solving th Problm Figur : Gomtr. 3. Problm Solving Stratg This lanar roblm is solvd b utiliing th tim harmoni Mawll s uations, and satial frun domain analsis. Thr ar svral sts ruird for solving suh a roblm, ths ar outlind blow:. Dfin th lin sour.. Modl th roblm as filds in trms of vtor otntials and find th wav uations. 3. Convrt to th satial frun domain and find th ordinar diffrntial uations. 4. Find th solutions to th ordinar diffrntial uations in ah rgion. 5. l th boundar onditions. 6. Solv ah of th onstants, n uations and n unknowns. 7. Solv for th filds in rgion Tak th invrs satial frun Fourir transform to gt th tim harmoni solution. 9. Tak th invrs Fourir transform of th tim harmoni solution to gt th tim domain radar rang rofil solution. 4. Satial Frun Domain Fourir Transform Pairs s indiatd in st 3 of stion 3, th tim harmoni wav uations must b onvrtd to th satial frun domain b Fourir analsis. Shown blow ar th Fourir transform airs for this ross, ths will b rfrrd to throughout this ar: jk ( k, ) (, ) d () Th first st in solving this roblm is in dfining th lin sour aording to th gomtr in figur : J (, ) ˆ δ ( h) δ ( ) Calulat th surfa urrnt dnsit: K h ( ) lim J (, ) d ˆ δ ( )d h (3) (4) Whr: h t sour hight abov th loss diltri Th filds in trms of magnti vtor otntial funtions: E H H jω (, ) (, ) (, ) Dfin th wav uation for fr sa rgions, 3, 4: (5) k (6) Dfin th wav uation for loss diltri rgion : k (7) Whr th wav numbrs in uations 6 and 7 ar: k for rgions, 3, 4 (8) ω ε

3 k ε for rgion (8) ω Rgion 3, : (4) Whr: ω π f f radar frun.5656e 7 (H/m) rmabilit of fr sa ε 8.854E (F/m) rmittivit of fr sa σ ε ε ε r j oml rmittivit ω σ ε r of diltri rgion ondutivit in S/m of th loss diltri rlativ rmittivit of th loss diltri Rgion, t : Rgion, b t : Rgion 4, d b : Whr is dfind as: j 3 j 4 5 j 6 7 j (5) (6) (7) ± k k (8) Whr th following must b obd du to hsial ralit: R { } > and m { } < Tak th satial Fourir transform of th wav uations 6 and 7 with rst to using uation. Thus, th wav uations bom th ordinar diffrntial uations (ODE s): Whr and ( k, ) for rgions, 3, 4 (9) ( k, ) for rgion () k ar dfind as: k () k k () Using uation, th satial Fourir transform was takn of th lin sour surfa urrnt dnsit uation 4, rsulting in: k (3) Th solutions to th ODE s 9 and ar wll known lan wav funtions in rtangular oordinats, and an b found in tts suh as [7]. From th lan wav funtions in [7] w hav th following solutions of th ODE s for ah rgion: Whr is dfind as: ± k k (9) Whr th following must b obd du to hsial ralit: R { } > and m { } < Using uation 5, and rgion dndnt ODE s 4 through 7 for th satial frun vtor otntials, al th boundar onditions at th intrfa of ah lar. Whr, th tangntial ltri and magnti filds ar ontinuous. This involvs numrous siml algbrai sts whih will not b rsntd hr. Th rsult of all boundar onditions solvd is a st of 7 uations and 7 unknowns, whr th n s ar th unknowns: () 3 j 3 () jt jt jt jt () jt jt jt jt [ ] [ ] (3) jb jb jb jb (4) jb jb jb jb [ ] [ ] (5)

4 jd jd (6) 6 7 W must algbraiall maniulat uations through 6 to find a solution for onl sin w ar onl intrstd in solving for th filds in rgion 3 (s rgion 3 ODE uation 4). f wr wantd to solv for othr rgions, w would hav to find mor of th solutions to th n s. Thus, th solution to was found to b: () Whr: H th Hankl funtion of th nd kind of ordr. nd, for this as: r n aling th idntit uation 3 to uation 3, th rsult is th tim harmoni vtor magnti otntial funtion for rgion 3: j j(t) Y Y (7) (), ) H ( k ) 4π j ( Whr Y is an arbitrar onstant uation, ontaining man variabls and a rsult of algbrai maniulation of uations through 6: Y jt jt j( t b) j( t b) X X X X (8) Whr X is an arbitrar onstant uation, ontaining man variabls and a rsult of algbrai maniulation of uations through 6: jb j( b d ) [ ] jb j( b d ) [ ] X (9) Substitut uation 7 into uation 4 to find th satial frun vtor otntial funtion (ODE) for rgion 3: j j t ( ) Y Y (3) Tak th invrs satial Fourir transform b substituting uation 3 into uation : j( t) Y jk dk 4π j Y (33) Thus, uation 33 is th solution to all th tim harmoni filds in rgion 3. n ordr to find sifiall th ltri or magnti filds, siml lug uation 33 into uation 5. disrt uadratur intgration routin suh as thos found in MTLB ould b utilid to valuat th intgral in uation 33. Th tim domain imuls rsons of th loss diltri lanar roblm is found b taking invrs Fourir transform of th tim harmoni ltri fild in uation 5. Whr, radio wavs ar inidnt on th loss diltri, and sattrd off th diltri, air, and th infinit lanar PEC on th othr sid (s figur ). 6. Thortial Data n ordr to k this thortial stud on th ratial sid of things, it was didd to us th rsults from stion 5 to rat thortial radar rang rofil data to s what a radar sstm or S ntwork analr might t to masur whn rsntd with a gomtr suh as that shown in figur. j(t) Y jk dk π j Y (3) Euation 33 was modifid so as to ignor th rinial sour ontribution of th inidnt fild du to th lin sour. This is don b siml taking out th Hankl funtion, th rsult bing: l th following Fourir transform idntit from [4] to uation 3: π j h jk dk () H ( kr) (3) (, ) j( t) Y jk dk 4π j Y (34)

5 Pratial thortial rsults for th ltri fild in rgion 3 wr thn alulatd. Looking at figur, th valus for th gomtr wr hosn to b: t -5 ft b ft d - ft obsrvation oint {. ft,. ft} B aling uation 33 to uation 5, th tim harmoni ltri fild in rgion 3 was found. Th urrnt sour valu was usd. Th intgral in uation 33 was valuatd using th MTLB oml uadratur intgration funtion uadv. Frun dndnt rmittivit and ondutivit aramtrs wr utilid from [8]. Th frun sw for th tim harmoni rsults was from 5 MH to 3 GH. Th invrs Fourir transform was thn takn of th tim harmoni ltri fild rsulting in th imuls rsons of th sattring from th loss diltri, air, PEC lard roblm. Ths imuls rang rofil rsults ar shown in figurs and 3. Figur shows th ral valud tim domain rang rofil data of th imuls sattring off th loss diltri, air, PEC lard roblm. t is lar from figur that th wall aars at ns as td b th gomtr. t is also lar from figur that th PEC aars at a littl mor than ns, as td from a wav travling round tri through a diltri rgion, into air, rflting off a PEC, and bak again. Th loations of th loss diltri and th PEC in round tri tim ar indiatd in figur. and involvd with dtrmining sstm sifiations. Shown mor larl in this log magnitud lot, th wall aars at ns as td b th gomtr. nd also mor larl, th PEC aars at a littl mor than ns, as td from a wav travling round tri through a diltri rgion, into air, rflting off a PEC, and bak again. Th loations of th loss diltri, th bak sid of th loss diltri, and th PEC ar indiatd in figur 3. Figur 3: Log magnitud tim domain rang rofil rsults. From th rsults shown hr it an b onludd that if a radar sstm or S ntwork analr orating in th 5 MH to 3 GH rang wr to dtt a vr larg mtal lat on th oosit sid of a vr larg loss diltri, saratd b air btwn th mtal lat and th loss diltri using this gomtr, thn that radar sstm would nd a minimum dnami rang of 8.7 db. 7. Conlusions and Futur Work Figur : Ral valud tim domain rang rofil rsults. Figur 3 shows th log magnitud tim domain rang rofil data of th imuls sattring off th loss diltri, air, PEC lard roblm. This lot is of artiular intrst for anon in th fild of radar imaging This thortial stud has rsntd a thniu for modling th disrsiv ffts of a loss diltri. Ths wr shown for th as of an infinit PEC radar targt, whr in ordr to masur th sattring from suh a PEC sht on th othr sid of a loss diltri, a radar sstm orating in th frun rang of 5 MH to 3 GH must hav a minimum dnami rang of 8.7 db. Futur work will inlud modifing this thortial stud to inlud mor oml shas suh as ornr rfltors and lindrs. Suh rsults hav th otntial for furthring th undrstanding of radar sattring of radar targts on th oosit sid of a loss diltri. Suh rsults will b usful in dtrmining th dnami rang, radar mod, and frun sifiations of futur masurmnt sstms that ar ruird to masur somthing on th oosit sid of a loss diltri.

6 8. REFERENCES [] R. Collins, Fild Thor of Guidd Wavs. Nw York: EEE Prss, 99. []. shimaru, Eltromagnti Wav Proagation, Radiation, and Sattring. Nw Jrs: Prnti-Hall, 996. [3] J.. Kong, Eltromagnti Wav Thor. Hobokn, NJ: John Wil and Sons, 99. [4] W. C. Chw, Wavs and Filds in nhomognous Mdia. Nw York: EEE Prss, 999. [5] L. B. Flson, and N. Maruvit, Radiation and Sattring of Wavs. Nw York: EEE Prss, 994. [6] M. Shaht, E. J. Rothwll, C. M. Colman. Tim-Domain maging of Objts Within Enlosurs. EEE Transations on ntnnas and Proagation, Vol. 5, No. 6, Jun, [7] C.. Balanis, dvand Enginring Eltromagntis. Nw York: John Wil and Sons, 989. [8] U. B. Halab, K. Masr, and E. Kausl, Proagation Charatristis of Eltromagnti Wavs in Conrt, Thnial Rort. D-7387, Marh 989.

Electromagnetics Research Group A THEORETICAL MODEL OF A LOSSY DIELECTRIC SLAB FOR THE CHARACTERIZATION OF RADAR SYSTEM PERFORMANCE SPECIFICATIONS

Electromagnetics Research Group A THEORETICAL MODEL OF A LOSSY DIELECTRIC SLAB FOR THE CHARACTERIZATION OF RADAR SYSTEM PERFORMANCE SPECIFICATIONS Elctromagntics Rsarch Group THEORETICL MODEL OF LOSSY DIELECTRIC SLB FOR THE CHRCTERIZTION OF RDR SYSTEM PERFORMNCE SPECIFICTIONS G.L. Charvat, Prof. Edward J. Rothwll Michigan Stat Univrsit 1 Ovrviw of