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

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1 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

2 Ovrviw of Prsntation Motivation Gomtr Problm solving stratg Spatial frqunc domain Fourir transform pairs Solving th problm Thortical Data Conclusions and futur work

3 Motivation Thr ar svral motivating factors bhind this rsarch: o To undrstand th disprsion ffcts. o To dvlop tst sstm spcifications. o Th ultimat goal is to masur things on masurmnt sstms such as: 3

4 Gomtr broad sid of th barn scnario 4

5 Problm Solving Stratg 1. Dfin th lin sourc.. Modl th problm as filds in trms of vctor potntials and find th wav quations. 3. Convrt to th spatial frqunc domain and find th ordinar diffrntial quations. 4. Find th solutions to th ordinar diffrntial quations in ach rgion. 5. ppl th boundar conditions. 6. Solv ach of th constants, n quations and n unknowns. 7. Solv for th filds in rgion Tak th invrs spatial frqunc Fourir transform to gt th tim harmonic solution. 9. Tak th invrs Fourir transform of th tim harmonic solution to gt th tim domain radar rang profil solution. 5

6 Spatial Frqunc Fourir Transform Pairs ~ ( ) # jk k, z (, z) d # 1 $ # ~ ( ) ( ) jk, z k, z dk Whr: vctor magntic potntial ~ spatial frqunc domain vctor magntic potntial k spatial frqunc 6

7 Solving th Problm Th first stp in solving this problm is in dfining th lin sourc according to th gomtr in figur 1: Calculat th surfac currnt dnsit: r K r J (, z) I ˆ ( z h) ( ) h+# ( ) lim J(, z) dz I ˆ %( )dz #$ h# Whr: h t sourc hight abov th loss dilctric Th filds in trms of magntic vctor potntial functions: E r j# (, z) H 1 µ (, z) z H z 1 µ (, z) 7

8 Solving th Problm Whr: f f radar frqunc Dfin th wav quation for fr spac rgions, 3, 4: + k Dfin th wav quation for loss dilctric rgion 1: + k Whr th wav numbrs in quations 6 and 7 ar: k for rgions, 3, 4 µ k µ for rgion 1 µ E 7 (H/m) prmabilit of fr spac 8.854E 1 (F/m) prmittivit of fr spac # # # j r $ compl prmittivit of dilctric rgion 1 r rlativ prmittivit of th loss dilctric conductivit in S/m of th loss dilctric 8

9 Solving th Problm Tak th spatial Fourir transform of th wav quations with rspct to. Thus, th wav quations bcom th ordinar diffrntial quations (ODE s): Whr p and & ' $ % ' z & ' $ % ' z + p + q # ~ # ~ ( k, z) ( k, z) q ar dfind as: p k k q k for rgions, 3, 4 for rgion 1 k Th spatial Fourir transform was takn of th lin sourc surfac currnt dnsit quation, rsulting in: ~ k I 9

10 Solving th Problm Th solutions to th ODE s ar wll known plan wav functions in rctangular coordinats: Rgion 3, z Rgion, t z Rgion 1, Rgion 4, b z d z t ~ b c 1 ~ ~ ~ jpz + jpz c + c3 + jqz c4 + c5 + jpz c6 + c7 jpz jqz jpz Whr p is dfind as: p ± k k Whr th following must b obd du to phsical ralit: R { } > p and Im { p} < Whr q is dfind as: q ± k k Whr th following must b obd du to phsical ralit: R { q } > and Im { q} < 1

11 Solving th Problm Th rsult of all boundar conditions solvd is a st of 7 quations and 7 unknowns, whr th c s ar th unknowns: n c + 1 c c3 µ I jp c 1 + c c 3 c + jpt jpt + jqt + c3 c4 + c5 jqt + jpt jpt + jqt jqt [ c ] q[ c c ] p c c 4 + jqb jqb + jpb + c5 c6 + c7 jpb + jqb jqb + jpb jpb [ c ] p[ c c ] q c c + jpd jpd 6 + c7 11

12 Solving th Problm Thus, th solution to c 1 was found to b: c 1 µ I jp & $ % + jp(t) 1' Y Y # Y & q $ % & p $ % + jqt + jqt ' + ' jq( t' b) ' jq( t' b) 1' X # X 1' X # X + jpb jp( b d ) [ + ] + jpb jp [ ] ( b d ) p X q Th spatial frqunc vctor potntial function (ODE) for rgion 3: I jp t ~ µ ' + ( ) jp % & 1 Y Y $ # jpz 1

13 Solving th Problm Tak th invrs spatial Fourir transform: 1 µ ) + jp t # & # jpz + jk I ( ) 1 Y dk * # jp ' ( Y $ % ppl th following Fourir transform idntit: () dk H ( k ) 1 $ # # jp z# h p jk r () Whr: H th Hankl function of th nd kind of ordr. nd, for this cas: r z z µ I µ I ( 4 j 4$ j () In appling th idntit:, z) H ( k z ) # jp( z# t) + jk + dk # p 1# Y Y 13

14 Thortical Data t -5 ft b ft d -1 ft obsrvation point {.1 ft, z.1 ft} Th frqunc swp for th tim harmonic rsults was from 5 MHz to 3 GHz. Logarithmic rang profil Ral valud rang profil 14

15 Conclusions and Futur Work thortical modl of a broad sid of th barn cas was implmntd. Dnamic rang rquirmnts for a radar or S11 ntwork analzr wr dtrmind using this modl. Futur work will includ using a targt othr than an infinit PEC plan will b studid. 15

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

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