düşey manyetik dipol frekans sondajı model eğrilerinin lineer
Transkript
düşey manyetik dipol frekans sondajı model eğrilerinin lineer
i ) u s E Y l r A N y E T l K I ) 1 F { . ) ; ,F I i i _ i i , / _ l i 5 SONDAJI Ii'()DEL ECIi 1 ].,ir;I(i ii 1 Ii L1}iEER SUZGE(: IE(rjiLI.-Ii:iLE H E S A P I , A I i I ' J I :1: : S a r nr ZLrlil-,i.ri_ Y r i I , : s e k i , i s a r : r , ,I ' c l i .leoflzil: l ' 1 i i h . A t i t r t ; i 1i m r l h l r i QQt'r ANKARA UNiVERSiTESi FEN Bi LiMLERi ENSTiTtJSiJ D U $ E Y M A N Y E T I K D r P O L F R E K A N SS O N D A J I M O D E L E G R i L E R i N i N L r i i E E R S U Z G E CT E O R E M i i L E H E S A P L A N M A S I sami ziiNeUr Y U K S E K L i S A N S T E Zi J E O F i Z T K M L I H E N D i S L i C i ANAB i L J.MDAL I Bu Te: )iot Pr ,: t . - r 3 . 1 -., Takclri i'J'J(j : :ECFLT 'l'arihrnde Alagrdaki Juri Taraf rndan 7rl(yetmi g) ,,!r oybiriiqi/Oyqoklu!u D,lC . Dr Ahrnet T i Ie BASKIJR Katiul Edilmiptir. Y, Doq. Dr & n r r r H l N D E V A T - iL, \/ ://,i "i 1 AZET DOSEY }',1ANYET1K DIPoL FREKANS SONDAJI MoDEL EG'RTLERTNTN Lt NEER SzGEq rEoREMl Saml t LE HESAPLANMAST zuNB0L Itnka.ra Unlwersltesl Fen BillmIeri EnstltOsO Jeoflzlk Anabllimdalr M0hendisIlQl Danr$nnn : Doq.Dr. Ahmet TuQrul 1SSO, Sayfa JOrl: BAPKUR : 88 Prof.Dn. All KEqELt Doq.Dn. AhrreL Tugrul BA,SOKUR Y. Doq. Dn . Oznur I'lt NDEVALLI D0gey manyeLlk dlpol frekans sondaJlnln CDI.IDFS) trporlsl frekans orLam:-ndakl MaxwelL denklemlerl-ne dayanrr. Bu metotda m.anyetlk alan bilegenlerl elektromanyeLik CEM-) wekf6r poLanslyell kullanriarak hesaplan:.r. ManyeLlk rrekt6r potanslyell MaxwelI denklemlerlnden L0retiIen dalga denkl eml nl saSl ar . Bu dal ga denkl eml nl n qd5z0mtJ sl L j ndl r 1k koordlnatlarda gekllnde verlllr. Blrer inLegral dekleml lfade edlLebllen alan bllegenlerl l"lneer sCrzgeg kuram:.ndan yara,rlanrlarak sayr sal oI arak hesapl anabl ll r. Hesapl amal ar (1946) rdan Anderson C1S7S): Korkealaakso ve Saksa alrnan bllglsayar prograrnr yardrmr yaptlm.t.gtrr. lIe FORTRAN dlltnde yazrlmrg program bu oldukqa hrzlr olarak alan bi I egenl er I ni n genl 1 k or an:. nl hesapl ar . Dl'{D FS model q-/o- ve, R.o oranrna eQnllerl hazrrlanmrgtrr. Model 96re 2L eQrll,eri Rzfo, e karpr Bu clz|lmigLir. +rtllnde lr=n"l metoLda arazi egri.Ierinln grafik yaklagrmla yorumu yaprlmrgtrr. Melodun ayrrmlrllElnln, Ost Labakanrn lleLkenllglne(ar), (o./or), lletkenllk kontrasLrna al:-cr -verlcl arasrndakl uzakrrgrn labaka CR,/D) we alr.cr-verLcl arastndakl uzaklrga Dl'tD FS nr n Schl umberger 96r OI mO5tor . kargr I agLr r:. I ma.sr yapllmr gLrr. karrnlrgrna (R) ba{lr sondal r oranr oldueu j 1e der ANAHTAR KELTMELER: Dogey m.anyetik dlpol frekans sondaJr CDMDFS), Maxwell denklemlerl, manyetik alan bllegenlerl el"ekLromanyet"ik CEld) , wekL()r poLanslyell, lineer stlzgeg kuramr ii ABSTRACT CAI,CULATION OF THE VERTICAL HAGNETIC DIPOLE F'REOUENCY SOUNDING HODEL CURVES BY USING THE LINEAR F'IITER THEORY Sami ZUNBUL Ankara University Graduate School of Natural and Applied Sciences Department of Geophyslcal Engineering Supervisor : Assoc. Prof . Dr Ahmet Tugrul BASOKUR 1990, Page: BB Jury :Prof.Dr. AIi KEQELT Assoc.Prof.Dr. Ahmet T. BA$OKUR Assoc. Prof . Dr . 6znur MiNDEvAtLI The theory of the Vertical Hagnetlc Dipole Frequency Sounding (VHD FS) ls based the in on Maxwell equatlons (EM) frequency domain. In this method; the Electrornagnetic potential, vector satisf ies the is EM rdave equation, derived from Maxwell equations. Solutlon this wave of equation can be solved in the cylindrical coordinate system. The EM field components can be represented as integral equations. The sample values of components these are calculated by us ing the I inear theory. f i lter The computations are performed by using program the computer (1985). written by Anderson (1979): Korkealaakso and Saksa The amplitude ratio of the field components can be computed program in FORTRAN programming by this language. The VMD FS model curves are prepared according conductivity to the (o /o ) ratios and the ratio of transmitter-rece iver zf distance with the to thickness amplitude of plotted The model curves were tR/D). Hr/H, on the and ordinate RTfo, on the absclssa. In this method, the graphical approach to the evaLuation of the f ield curves was carr ied out. The (o /q l, R/D ratio conductivity contrast and R are found as 2t parameters inf luencing the resolution of the method. The results the of VMD FS are compared with Schlumberger sounding resulLs obtai ned using the same parameters as of the V|'lD FS. K E Y W O R D S: V e r t i c a l Magnetic Dipole Frequency Sounding (VMD FS ), Maxwell equations, Electromagnetlc (EM) vector i a 1 , L i e a r f i l t e r theory. Potent tit IE$EKKUR Bu galrsmanrn 1le destek Prof. Dn. tglen g1zlml lqln H. JoloJl ve Jcloflzlk blr werl AKINr AYVACI ve A. hazrrlanmasrnda glrtslnde a, M0hendlsllgt Ba5kanlrgr borg bl1111.m.. Btlglsayar t rr&, . A. O. personellne' bana JeoloJl yardrmcr BOIOnO GOLI e, bllglsayarlarrndan Odasr yararlandrQrm t na rra deferll gerekll UOur Lezln Legekk0r0 ve a BASI(|'JRr Tu$rul sisterrLcrlndcn Bagkanlrgr meslektagrm Mohendlslerl g6roglerl ve KAYIRANT a, Dalre 6grencllerlnden bana ftklrlerl Ahmet DoC.Dn. hesaplanmasrnda egrllerln olan oIan, Turan verllerln Bllgi gerqekle'gnreslnda yararlandrgrm Kolu ve Jeoflzlk l CI NDAKI i.ER 1 G1R15. 1 2. MiiTorxJN nio{?lst.. ?1.7 Fr-ekans Or-lamrnda ? 'C. ?'-"2. E ve H lqln 4.3. EM Vek1-6r Potanslyell Maxwel I Dalga t)enkl emler.l. . . Denklr:mler'l -l ve Dalga Denklerni. 4 e.3.1. PeneLrasyon derlnllgl 3. HO}''OJEN VE TAEAKAT-I DUSY 3. 1. MANYET1K HomoJen BIR OnLam. 3.2. 1.1k1 Labakalr orLam 1q1n OrLam Manyetlk lqln Alan 4. MANYET1K 4.7. 1k1 4.2. T_ we T lntegralLerlnln otYardrmr IIe Sayrsal 4 . 3. ma.nyel-lk .....a5 Vekt-6r PoLanslyell Bllegenlerl- EM geklrdek ..-.."?6 Foknslyonu. ALAN Halka . . .3O BILESENLERIN1N SondaJr SAYISAL we Kargrlrklr KuplaJ Llneer Olarak HESABT . . .34 Oranlarr. S0zgeq .36 Kuramr HesaplanmasL .....-.4e kurulmast. ....A2 ZX{ANKS AJ-tprogram:lntegrallcrlnln 4. 3. l.Hesaplama Yardrrm Sayrsa.l ll-e Olarak To we T, Hesaplanm:rsl algorlLm^asr 5. D T . { DF R E A K A N S S O I I D A J I 5. 1. lkl Tabaka Model PararneLrelerlne 16 HesapJ.anmasl . hesabr 3.4. 4. 2.1.S0zge9 (Hz,,alr) 1a bllegenlerlnln n-Tabakalr ve s s A-Lan 81 legenlerlnln Tabakalr 5.5- OZERINDE DIPOI-. 1kl aLan ORTAM OrLam. 3. 1. 1. M.anyretlk 3. e. z .. MODEL EGRTLER1 Egrllerlnln G6re ... d.B ..4t} .,....54 S.,ndaJ lnceLennresl ....5S 6 T I ( t T A B A K A } . ' O D E LE G R T L E R T .-....64 7. DI{DFSVERTLERTNINYORUMU ......67 7.1 . lkl Tabakalr Graflk YoLla Ortama .Alt DMD FS Edrllerinln DeSerlendlrllnresl 7.2. Ornek Blr a. D T ' { DF R E K A N S$ i . { D A J I N I N A Y R I M L I L I G I . . . 8.1. Da5Ok l4etodun S. D{-{D FS I t elkenl 16e Sahl p ......6A ..,.77 O r L a m _ al r d a PeneLrasyonu. MODEL }.,OD€L EG{TlLERl 1 O. ......62 Derperlendlrnx.. EG{?ILERIN'N ...73 SCHLUMBERG{iR l LE KARSI LA.5|I RI LM.A.SI SOI.{DAJI 75 SONUCLAR u|6 KAYNAKLAR.,.... B7 EK-A SiHGELER : l ndoksi. yon B D sa)'t sr , kalrnlrdr Labakanrn :i. L E alan : Elektrik F siddt=i,i vek Ltir : Manyet-i k (vekt'cJr) yel r poLansr Ci =O havadak L - i=1 .,2.,3-, f : Frekans H : MarryeLj.k alan H= : ManyeLi.k i-{ : MarryeLik . 1 ( -t , SrCdel-r (vekt6r) alantrr dUsev brleseni aLanrn ;rai'a}' (radial) ' br IeSenr, , - l/2 : j". k ), otrla.nidaki I ail orLama vayrllm daiga sabit-i' L RC\) : EM geklrdek R : A1:.cr -verici € :Elekfriki 0) : ?nf (r :i. fonksiyonu, uzaklrk, arastndakipermiLtivile, aqr sa1 frekans , ileLkenlrdi orfamrn L 6 : PeneLrasyon derj.nlrQr \ : l nt-eqras)/orr paranrei,resi O :Ke},fr .i-t : Manyel- j, k skaler CSkin deplh) ' , forrksj.r'on, gegi r genJ. r l: , FUt urr br ri ml erde mI:s si stenri esas a I r n m r 5 t - . rr . i , 1. GIRIS Elektromanlctlk olan D0gey l.lanyetlk 6lg0len nlcellk genllk oranrdrr Max1-probe Dlpol frekansa ( alanrn d0gey rrekt6r potansl)rell CHz) CH" CDMD FS) bilegenlerlnln adr olarak, radlal) nanyet.lk bilegenlerl hesaplanabll gartlarr btrl pratlktekl Teorlk denklemlerlnden slnrr alan nretodun kullanrlarak uygun SondaJrnda manyetlk Bu yatay Maxurell denkrerntnln kargr SondaJr t drr vc sondaJlarrndan Frekans lH_/H.l>. Frekans potansl)reIl, CEI.D derlnllk elde ir. EM .rrekt6r EM edllen El-l dalga g6zorrcstyle altrnda alda edlleblllr. Vekt-6r potansl)rell H, blrer lntegral lntegral denkremlerl EM geklrdek Lernslr eden sonucunda hesapl anarak uygun f onksl )rcnu konrrolOslrcnu hesaplannug denklc'mi gekl lne bl llnen blrer olarak blr elde lfade deglgken 1l e fonkslyonunun karmagrk genl l k <>ranr sa.yr l*.n. Daha 1le ve Bu fonkslyonunun 6ncedan istenen sayrsal oran Hz yaprlarak d6nogomo sOzgeg fonksl)rcnu cdllen edlleblllr. getlrlleblllr. sozgeg geklrdek kullanrlarak aran I eI de edi I l r . yncdefl konvolosyonu bilegenlerl 2 . H E T O D U NT E O R I S i 2. l.Frekans Ortamrnda A o- genl lk Haxwell we ort faz denklemleri: ozere, olmak A=A.,ri slntlsoldal Sekllnde olarak deftgen lzotrop ortam asagrdakl dalga blClnlne elektromanlrctlk lgln glbl frekans tanrmlanrr sahlp zamanla CEID alanlara ortamrnda Maxwell cward alt homoJen denklemlerl" 1966): VxE+l c,rp'H=O I \b<l{-Co+1 a<^r)'E=O II V' E=O III V'H=O rv Burada; E = elekLr1k alan gtddetl H = rnanyetlk alan Slddqtl a = ortamrn al = ?nf lr= cttektedl manyetlk r . lletkenllQl acrsal e= elektrlkl 1f ade harmonlk CVolL/nD, (A/m), Cmho,zm), frekans, geglrgenl permlttlvlteyl 1k, t.l. E ve H I . .r I I rlal al. Iriln Oalge wr+ I I - Dsnf'riemlerl Maxrr,+l .l t eml *rr j lri:r clenkl t-r-ri:lsyonel , VxVx[ +1 t^lp, Vxl I =O (1J ( ')'\ VxVxl'l-C c'+ i e<l) VxL. =O .lenk.Lelrnleri elde edilir. rlenkl<lmlerinden i: ve vazt II. Max-well (1) kargrI:.klarr I1 r sa , L J-l VxVxl I +1 ap( o+ i e<o) ' H =O c4) eI de egiLliginden denkl gClre emf eri Vlt. *ktt VxVxA=W' edl I I r . yararLanarak denklernlerine V'E=O rre vre III. we V'H=O E ve H wek t.()r eI a-t'n IV. oldugundan Marwe}J dolayl nde"rt (5) =o (6) rlH*pt11=9 sek-lrnde (;l) ve VxVxL, +r c.+r( cr'+l cor)' E =O denk I eml er 1 C4) ve L litrn '.lenk I enrl er i ncl,e Y*r-.i n{: ve i tlt iqin dalga denklemler.l elde edllir C3) , l'l.tl .rrl.r I r)l ar-Al acllarrsllt 2.3.8M Vektor EM Leortde al anl ardan yasyonl di f eransi kullapm:rk Mallick III.|,laxwell 19BO). bir baska edrleblIir Denklemr qc)zmek rCih a oldukqa 1964). il lcl TI t <lertrei.Llrkler FIM LureLl I ebi. L ern t r x - - a: tn s i y e l trygun oJacakLrr (PaLra denklemine F:in vekLor C P h j . J .I j . p s DaIga prol>,iemlerr I c>nslyonlarrnr E ve Potaneiyeli vr} bi -l i n l r l * l cla ol ar ak i I t .1 AYI ',al)1 !l y,,ryi lt tl:i i ,l.r ti lJr: <terrk I em,l er rlrr denk I eml er i l l t r l r r x > lt . z ( _ ) r1 a n l l ,;l rtlr i'rlrrv..) tlDlA g6re r()tasyonel1 Bu dururnda V'E=O ve olciuQunclatr ifade 5ekl1nde manyeLik kaynaklar rCin, ('7) E =-i o/.rvxF oi ar ak Bu seCr I i r vekL6r Bur ada potansiyelinl F manyetlk I I . vekLc}r M.axvrel -I PoLansiyelidlr- denk 1 eml nde yer 1 ne yazarsak, Vxl l=-i '.lr;,( o+t v x ( H+ k 2 f- ) = o c.r€) Vxi' (8) (9) - denk ] emJ ni rot-asyonel bir 6te yazar e'der 1 z . eI de we deglldlr fonksiyondan bundan dolayr gi bl a$agrdaki sk al er il-l bi I (ll+kzl gdre t ureLl lebl l i r: (10) H = - k 2 F- v . C c11) (7) ve II. denklemlnl VxVxA=tZV' a-fa M:axwell eSl t,I 1 Ql nl yer.i.ne denkIe,mlnde de ) k eryf i H + k z F= - V - O yandan ak denk.I emi. ner C S) kuL lanarak. - I <,rprH = -i co;.rVxVxFH=VxVxF H=V V -r-€r denkleminl elde CT?') ( 11) ederiz. ( 1-?> denk I eml er i ni blrlestlrerek, v v.F -vF F +kzF+v .e = o e91Ll igi bulunur. fonksiyorr kosulunu Burada oldugundan kullanmak lYard ISAZ). skaler dolayr oldukga Bu durumda, p ( 13) A uygun fonksiyonu nin keyfi seclmirrde olacakLrr. (Vanyan blr LorenLz 1967 ve ul' =o vtt -k f' 'l r-:l ft dal ga. {Hel grdtier.: i 15-r rroltz) !i :.ser (11) denklemine dOnti3;ur. }-hrrye+-i k rlenlcl"emlnden, ?:1,-i t;u i.i= *.k 1t . 1 ci;-rr:rl. orir-unr-':'iVarryan Fur aeiak i 4' denr-tr e.in:,nr saglarligrnl I \'. M ; - ^ x r r ' e Il .alarr denl:l'eminde ci6] 1967, Fat.ra ve MaIl. 1ck f onk sl yonunun sk a I er 96r'eblI yerine mek igin, 1g8f,-'. da (11) da:. ga d*nl'l.r'tni tii yazr lrrsa, v'rktl'*V"{r)=O f ,p "u"\z'=c, we cira (.L4> egllJ-iginda:': . a ^k=4=r-, c) dugt: gr:l- rrJ r-bj i i :- " de yararlanarak, {3.7} Penelrasyon derinl (<5): idi ".3.1. ,iC-ir"r tJ;rlr.;;r derrklemirrder I: k seklinde yer' (r.ialE;l lt "rJ,lr: sayr sr) r= -J t^rp{ cr+l <rr) lanrmranrr. Yer ---:idugu 191rr ihmal degigLirme edlIebilece{inden" akrmlarr dalga qok saytsr koqtrk Ck) k =( _i urpr'>t/z olur. F- lqin dalga cJenkleml a$ad.rda.ki rl.rbi bir cc)zrlme :;ahj-pt-l r. -kz F=F Burada, ()E E k=a-i/-l tiicelikleri rwt birbirine /u s =fs=l - 191n eSit ( lvard daj-ga denklemlnl o E e ''-nl n dc)nogor. orup" a ve f]:-e*rr olarak, E .-Lf> Bu dur unrla. Ir CclzOmC, i e^rt *i<oL B gerqel ktlqormesl Hohma.nn l gaz) . ve n esas -(a+L ft)z =F e az () sekline sayr | Lanr ml anr r F=F bir l, o\r/z \A) sekl i nde karmagrk seklinde bir tlalgan:,n sayl oldugundarr z s6noml-r*nms,sLni brlyudrJkge gOslerlr. Herhangr brr fakLdruyle genlrginrn dalganrn EM azalm;rst o.l-ar ak Lanr mI anl r' . penel-rasyon Pont+tr'asyon Penetrasyon f I 2. Sekif da azaldrkqa anl a5r I acaQr orLamlara hassasdr dO5trk r. fnekanslardan alcak gereklr. yant inmesinde et.kln bir nrn EM frekanslara rol deQeri degisimj lncelendtginde gdrtllrJr. artt.rQr f rekansl dal gan:, n o olarak a.1. dalgalar o] arak sIra baQIr $ekiI 6 f rekansl-r r gi n Bunun o, ya yt1kse'k Uzere Sc'nr-rq yaprlabilmesi ve riepLh) skin ) g6sterllmekLedir. wa o de{erlerj. da f - dorlru 6, der i nl I {i L derinliginin (6 derinliQi (L]"' i*1"'=5os o=[ --f <t L. p o ) cleljnl.ere clal gaI ar r ise dr*ri n derl Buradan nI i k sr O orLamlara rondaj l. nr r) f rekans: nl n dogru deQiglirrime-sl de dalganrn yuksek derinIer-e ovnar. a) Seklf a- l. Penetrasyon derinllelnin olarak deElSiml. C6) f ve ya baSlr l/r: - 3. HOI'{O.IEN VE ]'ABAKALT t,,l:ZFjRI NI)E OR]'AM Bt [.. l)tJSEY MANYET1K DI POI- 3 . I . l - | , : ' r 1 , : ' . j e nO r t a r n ! Eksenl (Ie'*") ele La5ryan alalrm. sillndrrik z-y6nunde ktiq0k Bu koordinat nokt,a.sr vardrr dlpoJ-den oIsun. bunlar, bir Burada akrm f ctr poJ manyetik gdsterrildlgi gl bl yrJksekliQinde bir olasr we P, bl rincil kaynaklanan da h Pr,P. aI ternat.l 9 :-j.1'de Sekll, sisfemlnde yerl"ergLirllmlS ki, ya halka dipol noktaya t':r- blr drlzenlenml tig noklalarrd: manyellk adet r . t>l gum Btlyl e :rIan, -kR + F =C- | 18) A.Z I Pr(r,Q,'.-in) I T\ Sekil 3. i. Yeryuzunden yerl P ,P L23 h eptr ri 1 mi 9 ve P ytksekliQrnde bi r manyeti dl cum noklaL arr . b:. rk nok Laya d i pol ve i,r (Keller verrJir Seklinde v(-. F'rjschLne{ht C=I d;r.,'A.trCI d:l rnanye.Lrk mr:nrerrt-.-)ve R=.(-r' Maxwe,i l clenklern) denkl verkL6rel koor-ciirratlal'daki ve yc)nUnde r o}acakL:.n. durumda \{ar d r gd)z0mu ol-arak yal.n:.2 ayrrmak al rnmrgtrr. r nl n suretLyle peki l de s6z bi I ergeni denklemi, O gibi selc-r-iip, gerekli ,F dR =l ar her iki Laraf o" R ye mevc ut. aCrsl,na gdre (PaLr.a veril"ir ve slnrr saelayan $arllarrnr bul unabl Li r . Bundan ol_acagrndan numaralr nln bl 9lmde olan dolayr 1kl azF , -=2. orz b()lunerek arz azF = R--. , ozz F"=F fcnkslyona alrnarak duR F denkrem C6ZOIebl-Iir: - tam son! a. dj.fe'ranslyel fonkslyonu Lurewleri 1 ar\ z F=RCr) ZCz) oi.arak ve 1 987) : konusu c1s) lzleyen Q olmayacaQrndan oz" uygun z s;ilir:'lir-1k / ve yalnrz ve rrr.rm;rr:*I r 82F Er z*bl-Iegenl C( 1 5) denkle,mj.nde O" sadece a$aErdaki we, Hohmann qc5z0mri aFagr daki trl n::adece degrgim 1 aF z+__z+t=kzF arz Bu denklemln bin dalga (dzaq=O) dolayl. EzF skaler,:lalga iiurad;i ,ir , ^ clerrkla;rrnlnin o"lan rcl acaSr ndan sr f r r 1 g8O, Mal I i ck d:ricr.r kar5rlr$r Bu simeLriden EM ''-r'".'j"'' rrf <l*; t-:rli l t;rr rrdt'rl potansiyelde t Ur ewl- er 1 de bj r eml eri 19n8l. 62z azz C19) denklemincJe ye,ri n* yazt lrrsa, 1d"RIdRtrJzZ .+----.---+PaCeo) R denklemi elde de$lSkenlerine Larafrnda sadece Sekllnde 1945). z rR drz <ir dlr. Buna 1 dzR R drz de$l gken r, sadece sag ^Arfken (1Sas) 1 dR 7 cJzz rR dr z dzz gore yazLlabilir. gc)re; prensibine ayr-rlmasr denklemlerln Dlferansiyel edilir. ba$rrrslz ,.,1.",2 Z Burada, \ aylrrm denklemin sol t-araf rnda lse derrr CaO) sabltidir denkleml , (Arfken Buradan, t dzz z dzz ( a1) =x2+kz ldzRldR + R drz +71z=O rR dr veya r=}.r de$isken ddnuSumtini crzR 1 denk ] emJ er i nr (et> e +R:jO xr- {eA) d(Xr) eI de erder i z. denkl e r m in i n Cd)zumti r +z( k2+^.2)r'/2 clnslnden, Cee) clnslndendlr karma.5r k , dR + d(\r)2 R r I e qarparak yap.rp, CMcLachlan q()ztlmu genel Ct)zumu jse denklemi.nin Bu durumda 1934). ve CPaLra fonksiyonlarl Bessel l'la.]I i ck C1g) de'nkleminin 198O) : ,= f*foa\)e-zc\2+kz) "t *rrx)*z(\2*kz) J L ^"f ., cxr)d\ JO o c e3) Sekllnde veriIlr. havadakl CF ) o'l agaElCakl gibi $ek1I ve 3-1!cieki yerdekl yezrlabllir dipole vekL6r CF ) (PaLra ve Mallick aiL havadakl potanslyelleri 1980): -kn@ eo? F =(oPJo + | acx>*-^o"J c).r)dL <?4> o cn f ^(\r)d\ F =l BC\)e"r-J o 'J o ('e5) Burada, potarr:;i {e4) yel I , yansrtan denklemincleki k., ilk havianr n, k, orup, (e4) sabiLIer d*: ter"im yeri birlncil rr manyerl-i k ve. (2s) wekl.cJr frzel .l i kl er i n j denk]ermlel-.rncle srr-as-l iIe n 2>r/z =(L'+k r)orl gek I r ncierdi r ; AC ^) ve g a r L l a r : -r r d a . n hesapl anacak denk I em.l er i nden Lan,;:r.nsl yel ol duQurrdan k ve ma.nyetik derecede ma.nyeti ffi AF tz ilz saSr an1 r permeabiriteslyle f arkl oI an fonksi sirnlr yonl ardr r . sr nl r k 1 se Maxwel l Sar LI ar r rra a.I anl ar sr nr r da 96r- €', sr.ir ek L i z =O cla , <1olay:. $ar tl ar r yonL arr fonksi edi I en F_F ot sr nr r BC \) el de el ekLri z)t,z n =(),.2+k ve c KeL l. er yerinki oI madr gr ndan r 1 s66) . Bur ada qok blrbirlnden dol ayr ha wanr n 6nemll po=l:^=p ol arak alrnmrglrr. llr nr r i"ntegral edile.cek e kn R formu garLl arl nl uygul amadan t)nce Sommerfeld formolo bi rt nci I yardrmryle al anr n ifade olursa, k( r2 *='>tt" cr" *='>"' ft - e r -z(rrz*kz)''" J cr." +kz>1/2 J (Xr)d\ o ( a6) 3. 1'den SekrI ll=(r'2+ f=*n l')"t i ser P noklasr z manyet,ik formul gdr0lecegl de d!.. Burada i. qi ndi r . momenLine uzere Cz-h) ve P, P, Bdyl erce yerrderi sahip birincll P"r:z.rklrklari nc.:kLagr h iq:in, y{rksekl kayn;rk rctrl lCin Ch--z-) i Si nde l cla Sommetrf'eld u, KR e I (-( R c)"2*kz) lc() geklinde vektd)r verilir. Bundan potansiyeli agafr @. A rr o Daha (5nce uygul anr p e llJLn oo edllen Jo bi rbi \ n g' ri nl -^oh -n denk -l-emi ndek i yazL I abr- l i r . o l=-t1 l+Rrlr6.'-r-ro^i ]r"axr)d\ lqin nl n ve'rilen katsayr gclLCrreceSinden) +ACL) =BC).) o cLe-noh (?.4) anarak gi bi daki z=O f s (xr)d).' ler Q- -n ll I fade o o c.-no=t =L yararl A(\)=n ot BC\) : s:.nrr I ar r caa) 5artlarr eSi t.l enl rse: clenklemleri B( X) edrlir. elde ve ACA) denklemlerden Bu q(5zrJlLlr se, An-n A( A) - C- c' I nn+n oc, e-n()h I e\-hh -C- BC\) oIarak n bulunur. A(X) yerine denklemLerlnde wekL6r -r-n e o ()L ve (ZA) sabiLIerini tlCX) hava ve yazLlarak yer ve C25) manyeLlk lCin poLansl yelleri, s F -o =( t **-^ol=-nl* J L"o ^ ^o,^, .o ^o*^i .-^o(z*h)1.; J o (xr)dx ( 2S) F =C 1lo sekllnde flax I elde edl11r. ESer dlpoJ- yer).Oz0ne oI arak c 30) C\r)dX .tr"-toh.J Jn+n ooa a} r nr rsa., denleml) agagrdaki 1940, Slnha vekLd)r harradaki gekllde ve ColIeL indirllecek ifade 1973): edillr olursa potansl CPaira yani h=O yel 1 CCeS) ve MaIllck 1e, It+" 7.01 o+- F_C o ^ n-n *-""ot] nn-tn oor . J,rcxr ) dA ( 31) 3. 1. 1. Manyeti k Hesaplanmasr Homojen olacagrndan {.H ve ? Bi I egenl er i ni n Al an H) r ! ortam we do-I aYr s: Y] e t-ant ml anan clenk I eml yI e C31) ko=o clurumunda n=\ o wek L6r poLanslyeli, ['-+]" @ F=f,1 r '" -)ry @ 2X r =cJ -o ,\+n Sekllne d6n0gecekLir. CSeklI 3. e) C16) glbl Hohma.nn 1"918'7): - ^ z- . J e, - o asaEldakl .J o CrJ (\r)d\ C3a) CXr)d\ a Manyetik denkfemlnden cKoefoed hesaplanlr AF H=V9'F=VC o) alan ya.r ar I anar ak wd 1se bllepenleri LS7e, ( k =k o Ward c 33) &z g=J t azF ozF H=J &=' t =O) c34),C35) &zdr ve 11 (34) ve (35) manyeLik yararlanarak denklemlerinden alan H =2C z clii5ey ve r adyal bile5enleri, J -rrz eo J o (3?,) (Xr)d). @ H =2C r peklinde elde mcnyetrk J edilir kuvvet *-to=.J ( lYard (xr)d\ c 3e) 1 1967) . gizgtleri l t \ .: J. /.f .,/ Hr /H ttetksn Dugsy Sekil 3. e. D(rgey Manyetlk Cizgileri ve monyeti"k Dipole yorL-€on9uz orLcrn di.PoI CD|'{D) ai t 6lqCtm noktastndaki manyeLik kuwef alan bile5enleri i8 P I ot'lt az'l't 'a Seklf 3.3.1kl Labakalr nokLalarr 2.2.tki Tabakalr Sekit Burada olarak) B6yle 3.1'deki blllnIr orLamdaki fonksiyonlarr ve dipol P ). iki pL b6lgelerdeki 1 ' r od e { e r i n e dlpol e5if iqln olsun deQerleri kabul olarak blrincll Ozerinde orLam Labakalr yerleSLlrilmig nokfaya bOLOn manyetlk F=c oLarak bir bo5lugun blr CPi, P ,P 234 6l qum DMD OrLam: h ytiksekllginde 3,3). {rzer indeki ortam vekt()r Cpo=u. edilmistir. potanslyell: c 3a) e-kn/R CKeIIer OC bOlge Ft CSekif ve Chava,l Frischknecht . Cl =O,1 ,?) , ve 2. C15) 1S66). Labaka) denklemlyle lqln tk1 Labakalr potansiyei verllen c 3s) I .:! dal ga denk I emi ni 1966). Btr d*:nkl emin Csilindirik oldugu saQI amal r dr r sr I ir:cij ri k koordinaLlardaki icin ve Pafra utt, 1 orz 9ek I 1 ndedl r . + dr +CC\2+k dolayr ' =t1r'. &z-z denk -Iemi n Bu n:rt-larclakr genel I poLansiyeli 2)z) ( 4L) C41) denklemindeki dak 1 gi bi zlO 3.3'deki yonlarr Kel-Ier orLama vekL()r potanslyelinin YazL 1 abl ] i r : L91n, tiC birincil vekLc)r (1955), CBhattacharya a$agr Sekf l fonksi Oz'dt$::77 q6ztlmu , io veril.rr. I r {r (4o) I F=erJLz\fJ geklinde kar5r 1980): utr, dF,, r koordj Fr j sr-hk n'.:r.:l-it. ve si melriden MaIIick +- ( Kel I er ve aiL vekLdr potansiyeliyle toplamr Frischknecht Seklinde 1966) 2A -Ch+d)(z( -h igin, co f ( f ez ( \ z + k z )Ir ' / z t"=tLJ{o"cx> '2-'1'/2\ "2 d(X)e-z(A{-Ktr o z.( -'( h+d) }'rr.x.>axJ i qi n ru l' ( f I r '. = c f I { o c x > ez ( ^ z + k t )z" t LJ L'= \ 1 l - s oc \ r ) d \ l ) J o c 4e) Burada, eLde C=Ida/An edllecek olan Ca6) yer olarak ve \ ya baElmlr notasyonlarrnr ya.z:-lbillr verllen deplgLlrme c\2+kr2) glbi Qt,Q.,Q",Q. de'nklemiyIe yararlanrp, alrp u. we 05 CKeller FarLlarrndan fonkslyonlardrr. Sommerfeld akrmlarrnl ihmal t/2 =vr., cr.2+k"2>r/2 =v", da kul.Lanarak srnrr C4e> lnLegra.linden ederek k o =O *,t=t oLpo. denklemlerl ve Frlschknechtlg66): a5agrdakl .:. cn ,- ' rn z r I _Iz'l F -c I 1 J l. L F =c | J " ^tl . ( ., xr ) crA [*^= +@2( \).L J o t"=. J [+". t. =. f [*r. re (\)e-^'l. I J l c\r )ct\ .' ^, .u r=*e.(r.>*-'.=] ' Jo( \r ) c!\ ^, .u ,=f. Jo(xr ) dx i'r3) q5, | 6- 2, O, {1366) 3 da 6 1 w e 6' 5 veriLen a.l-anlarrn Labaka dayanarak elde fonksiyonlarl Tanjansiyel sonuglandrrrlabilir. C1366) edilen bu surek1i s,lnr r s:-nrr de sarllarr KeIler- atr 1 = - AF z=-(-h+d) gartlarrnda.n 1?tzoz F=F , 2s&zoz da prensibint+ AF aF z=-h 1rk Manyet"i.k da, AF z=OdaF=F,i=- irriscirknechtve- ElekLrrk agagrdaki 56zO ve: KeIler gegerken srnlrlarrnr edllen we Frischknecht , AF F =F e'azoz. ( 44) veri 5ekI i nde kullanrlarak ( 43) I mi 5t..i r . elde edjlern rle'nkl erml er j ( 44) ve .ait- Qi fonksjyonlartl)a a5:edrdakr denk I em si stc.m.l . ! Cramer C45) denklem denklern yerine aI arak g()re kuralrna gclzulUp 7. sisteminde, slstemlndeki yazr.l-rrsa, 2. d. denklemclen (45) denklem bul unabilir. fonksiyorrlarr denkJeme gdre Q. sist-emi AC Qr=Q. Ceki lip daha 3. basiL dir. Bu denklemde bir hal , h*ra r ' , + ( v , - \ 1e t ' h @ . .A ) = - 2 \ e - ' ^ ' h "-u, (h+d ) -e.( (h+d ) =0 (h*d )*y'oc \) evr x> e-vz er( tJ "-t' (h+d ) (h+d ) :v205(\) vl (h+d ) -vlr4(A) ,rLr(;\.) e 6v1 e-vz - (^.+vl) c46) gekline sisLerninl verllmlstlr. d6n0;tir. Cramer Bu kuralrna lineer q6re homoJen qdzCtmu ise olmayan izleyen denklem bigimde, 23 -(\-v t -v e I I )e -r." e I-' o r" 1 (h+d) 1' e Vr(h+d) v e - rz ( r. -\) v (h+d) JC v (h+d) -v -V 2 I k 1 2' e I ( h+d ) -v (h+d) .,X I -AJe Lv t _e-Vz ( h+d ) .-v, 1' -Ve1 _., -L/\-T-V I' V e Fvl (h+d) ( h+d ) _v evr ( h+d ) I t ( -vh ^*', _) e I [t I -v v(h+d) € r -AJe (v -v ./h e 1 I f-, LlI (h+d)"-v, 2' V e 2 (h+d) (\+v "t, (h+d) (h+d) 2' v e 1J -v 1 ( h + d )' ll ' )*-'rh- . JJ . . r e - u . ( h + d ) e - v , ( h + d ) ( w , - \ ) e - r^,l) -l -'a^._g )-, -AJE LV rr l-' L l o e o -€,\.e o 1' -ve -v (h+d) t2 D =Z\e-)'h"t. 72 ( h+d )v vz ( h+d ) e -\)e'r" (v rr F. I v (h+d) -vel v (h+d) - X h- ' - + v evl ( h+d | .-u, evl ( h+d ) -v, evr ( h+d ) ( h+d )a\e-\h ( 4a) lrr'r, D 2I D (v -l i ( v t o' 3 +v -) '2w .iv tztt )(,\tv .)--(r, -A hr o -v e 4 2' 2 ( h + d )' v - r r' t e 2' e / rl rt +' A s \ _ |. it h -t-rl \ .r ' u -v D l- 1 I (_) -v (h+d) (hrd) 1' e \ / = (v D Hawadaki ( h + d )' P - v 2' r, *^) --?u nd a., aX**( ' (\)= +w )CX+v 12tl wekL(1r potansi yonl ar1 nr n )-Cv -v yel I eri , ( h+d )ar..-).h ( 5f)) -v) 1Z -/\Je J(v --2v 1 (51) d gerekli igin olan Qr.,Q. eOzUmu, -(v --\)(w +v tt CrC X) =CzC\) Cv )+(v +\)(v )-(v -v 21t +\)(w +v 21 tI )(v *-v )e z -2v d 1 --2).ir -}.)e 2A ( 5a) ol arak \ -d4..: I 1 e fonksi -v -v 2- aa )e -rr o _]Ih f) =cA_a: -rr -(\+v o Vr(h+d) I 2l -z\c v e ( 1l!,1) cl ,\J€' el de edi I i r CK e l l " e r ve Fri schknecht 1966). 'i5 orl.am iein Labakalr 3.e. 1.lki m a n y t ' : tl k aI.ln t-'iI o<orrl rr l nr rl tresabr : i qi rr F Keller aI r p o1 ar ak r=F r=F ro r; Oz r-ttnundeln f'c>nksi Yonl arl nr r) Qrvu 6" havadak C1966) we Frischknecht polansi vek t.r1r i t.) "Yl :lonra yel i r:i dan, ()J F -Clf-r^ :o f | + LR o J R(\)e-'r"(z+2h)J t ^ . ., O o ] i ki Labakal F r . r r . r , J a I;:CX) SekIinde yazrlrr. qeki:Cek f'onksi yonu (v )+(v -v )-(w 2t -v +v 2t 1t R(\)= +X)(v +v 11 RC^,d,o,f) lki bilegenleri tabakalr CH z ve H ) r 2l +\)e-2vr )Cw -\)e-zvr fonksi parametrelerin ilgiIi seklinde A )Cw Cek i r dek Lanr ml anmr S L.l"r . ol ar ak I qr n orLant r oIup -\)(v 1w s3) h=o orLamda yonu RC\) fonksiyonudur. igin olursa, hesaplanacak d manyetik C34) rre alan (35) denkl emleri nden yararlanarak' fl =' az ( a*=" JL c lL -R a2 H-- " a-.a. 4@ tL .@ ( f r lLc l L- R f - x= J + lRcx)e o oJ -\= + rl R c x ) e , oJ J o ( \r)"^]] c 54) ( \r) ( 55) "^]) :r6 c0 -1327 tt iC I | -.-- +-+ zlRsRT () , 3zr tJ =c l-+ ' LR" ol ar ak bul unur f^' koyar r g = Cl - '.LR3 3.3. .Al an Bi I esenl eri vd Koefoed CSekil 3.4) vekfdr or t.amrja trer' frangi ( lla) o(\r)d\] r q(l) (\r)dX I o.l.ur OrLam tcin n*TabakaIr (|j7) ) d^] bj, 1e5enJ" er i , RC\)J *, \ ) J eI cle edi l mr 9 ( l'j['i ) J Labak al r i ki ak .l\2 J [*^' I ( ^r rrlA I @ J sek j i nde " \ '' "t - _ l 1 al an k manrreli ,t H=C Rf ),)e, _\ R( \) e "'',J ( I.r z=O br r- nok Ladak i I ,z ^' ( Koef oed Vekf 0r wd 1 97e) PoLansi yel i ve ManYert.i k : (1972)'de potansiyell n-Labakalr bl r orLam I cr n " 0 f ,r F = = C L - O-- f J R(\,d,? .,f) .-\=Jocxr)d\ ( 60) $eki I 3.4. n Labakalr di pol . ortam Qzerindeki d(tgey manyeLik ;::s j f arle.:i yl e vrl ve:rr.l mi;rLi r'. ilc'l,ker:.l jk sayls-rrlr r-)t.abakalr Rurada (:-=J,r3,3,. de$errler.idir' q*:kirdek f J t r ra c i a k i orL;rm bagr nt.r sr ndan iqln hesapl trtC\) =R o,n Bu baQrnL-r-da, t a n,n 1,5'7?) . l, L t ' ir-1,i. a1anr, ikrnci t) -zd l+V K rrh L .Je r v L C\) =O t k z = i Z n L 'tO o ' "f L C34) durumunda) y.ine;leme (\Je-td.t, +R L- I -- u =(\z*k?)t'z Laba.kaIr n gC5sLermek Lerdi r . sayL s:. nr i-t,n R t.abaka R(\,,J.,s,f) yerytiztrndekr indis v v kl arl CX) birinci i ncji :-; cle t.abak a vd n verilen biqjmde CKoef oed anr r ()l{.tp ,r) 1,,:rrksiycnrr 121 eyen kal rnlr v€i L1r Lahaka d. ve orLam ! C35) iqin v. i,k =( v -v.).2(v r hesaplanacak alan +w.) i" k yararlanarak denklemlerinden manyetik k bilegenlerini Cz=O of ursa, @ t Hz = clLReJ."oJ * [ ^t *ax,d, o,r) J cxr)d\l @ ?f.-t I{ ==cl | \2 r LJ R(\,d,o,f) r L J (\r)d\l t (62) J n (61) .:,cr el d€t edr I : l- ( tloc'I t:r-Ll .''f1 :l !l'7e) g:ekirr:Jek Si nha ve R(X), fonksiyc:rru yansr modunclak i k aLsayr rna yukarr n dakr l,.alrlriral,r gl br -frarrsvers rndan laraf C1!)73) Col leL 'cle. (1972) l(r:r+fc:ed vd Sek I r ncle sr r(. lI) ortaln t . , ]n t a k L a t;er'aLle*r (1'F-") FlIekt.r'rk gl bi aSaQr dak i Lanr ml;rnmr 5t.r r: N-Y K o1 LAJ=- ( O..J ,, TEN+y ot Burada, Y +N M+l M Lanh(u d M M ) ( 64) Y_N MMN{-YLanhcud) MI|{+1MM x-i),1t:, I --l\ ",. ri.rr*.1 ) , NN N =Ll /ir"r,u, Itll -.2., (l=L,\+K) l|L{ K , =L L@UO Ml|ll 2-1./2 _ 1,/?. ) tf=o,t,2,...,!) ( 65) Y , ( trli-) 1 r . :L I t t-al:aka Ortt-{r t,"'italt.t.;"kr llk rJr"rrl l etnl nr.1{Jr) N ( 64) r q-i rr e,:f mek r*l rle. Lrtr cie(lerr Ar ,:lr rr,l;n ti hers;r pI a n r r' , y 6 j r I n{i detnk 1 elni nde baQ:rrt. I ikj Her anr.r. oI ar a l1 lit:.:;apf yi l-rr:l emel- i cic'rJr+ri rti t, crl --rr' .e k solrllq l: orr;lt';; L- , er5i i. st.rt lug I aI iIc: ver.iIeri ur et.i r . 3.4" (}nceki R(\) p:arameLr'€].1 eri [-r-r ayrr Karmagl k hesaplanlr. par'amelreL eri degisik ol"arak ayr'-i ol arak t'abraka RC^) , o]an baQl i frekansa ve sayl sl Labaka Cd , o.) , yop f onksj brr f ' c . : r r k sj y r . - t r r t j Cek i r ,lek . { vL] o. ,' X t n t r r f u r n k s j -y c r n u ] Ci n de{err f rekans bi r tl X, fonksiyonlr f orrksi yonudur nj n ba{rnt.r:sr ylneleme b15lumcle gekrr<leP- EM : FonksiYonu EM Cekirderk almakt-adrr. deQerler \zBc x> =*-t"RC y) fonksiyonunun arLtrkga fonksiyon buyumekt-e ve kaymakLadrr yapr 1 mr gLr r . yat-ay dugey eksen (Sekil vr:nurrrun t.abaka] boyunca orf am k r sr mI ar r-nl n elavr arrr pI ar r gdlst.er i. .I.rnr.:k i.e-.,-lt t- . ve da j ncelenmesi gd)re vt:l r. Fr ek ans boyunca eksen 3.5. r o da 3.6. uzerr ndeki $ek r | 3. 7. ydnde y()ne ). reel 3,b I er i de{er neglaLif negatif $ekiI geki, rdek olan fonksiyonu paramelrelere ilgili bi qi mde i zJ- eyen lonksi girig suzgece $eklinde doQru Cek i rdek ve satral ve c de J} -'it.'lr'lr; {:.rlaf .1}. j n f5qrlsLn rJt:''Qr'-;} Ildt'k j ek:,c:t) lroyunt-a yat.ay s i k I I qe de{i t"7 ned€n ol makt.;icir r l'a{.I } t1) ge"k r.Je t:lu5lry kayrn;r:',rlra ' (Kos*f',rerd rl{irf i :; I n l'rr I' d*r wcl 1 A ' 7 ? ) . n f *1O llz I f -1 CX) llz +r I I u I \ i \ I I \ I I -2- f -1 LIOO ltz \i\ i I I Ii i - r.v I -j--r-r-r--i-i--r 0,0 f = 1 O Hz tlz f=1@ 1- qqo r.c Seki I 3-5 - o =O. 01 fslooo Hz mho/m Ca ) ve o =O. OO1 mho,'m Cb ) tl sahi.p degerine reel f onksi yonunun homojen k:, sr ml arl .)rf amlar iiz-er j nder Qekirdek gore de$i si mi rrr n f rekansa 0.0 n I tI r I N &. I t- 1 I I 'l-i'-:--l-"'r .J 0.0 ] I ) -I -l I - 1 . 0- -tI -,]I -1 - n n --l I -l -l I _t il n -l I I I I I t -iI J - 1 6 . C-l I -'lt -1 i i 1z -20.Cil rTtTrTITrl_t qqqq -.. Fr i-t rritrrr trr r ir-r-:-:-r---Tl--T--r-"-T-T-l U (.) ro -m l-.- v1 cl rlll Y =I n ( 7 / x ) Seki I 3.6. o =O. 01 mho/'m homo3 en --+ L:j r or Larn uzelr i nde 1 ( -.--) r t:e.i f onk si yonrtnun Qek i r dek fl'-'1";ttr.;;r k:S:nilartnltl C- -----_) sanaf ( f = . 1O ' ; e 1 OO Hz.) C a) , ( i' -1 ()(;O ..,i, j r.)',-){)i)lI.:.) ds:gi Si mi . ve .l$rtl { bi n I o - | : i | I K r: rl r^tr\,_ n, --:- -l I-- r---r a, a.r n :l n.0 -t " - l l \ \ \/ \/ d -1(X) ^' Y=l n(t />\> c- a -O. 01 rl <l -1 OO o h, d =:!'o n, -O. ()! o n,),r'lh, al hl,o/ n. il _1 . 0 o o Seki L 3. 7. qqoqoooooQ 4 u rr d O l ve 1OO f=1O orLam uzerinde Tabakalr qekirdek reel(--) fonksiyonunun iqin sanal C -----) k r s r m l a r r n t t ) d a v r a l ' r l .5 . 1a r 1 . Hz ve 4 " l'lANY[:T1Fi: AL-AN B I L.-l::LNt..-t:ti:1Ni N SAy I SAt- Hi:5AE{i qr-:nt:l bin denk,l .rnrl .+ri y.l e f'ra.iiy1 ..t ver j -ier) lresal--.r n r yapma.k dc)rrr-r5umti yap:. I r p:, mar:yet,i i <;i rr L-:u B=R,rd (i-1) k al an al r nar ak , ver l{r . . . 1 2, ' r r r z de;er 5l. rlrr }>i I e5enl f eri , ] j t =c[L \2 ro'i,,F', | *R(g) - Ij" ir)t.rit;lnl L:i I t:Serrl er j nt rr say.r s;il de.rtk I elml +t.d.) - H,=c[ ('i,.) vrr" * I o t" R ( q ) -"r ( q Bo) a o l 53 J Bt (66) J .c(t F L a t"=.[ Seklinde elde ,. o J s'' R(E) J.(oB) on edilir CKeIIer ve (67> ] Frischkne'chL 1966). denk I eml- er de, @ TocA,B,D ,k )= T-cA,B,D,k I L ' )= | *an, gt | *an, gt J.cge> J--o J J^cgB) i- dg dg ( aP) c 6s) Bu tg .:J (.t ) R z+17 A= - L).---t F=--r 6 o -, _ 1/2 d>:-L -) .f o i=1 2 ? <Dpq k u I l a.rrr I ar ak nolasyonlarr Sekilde agagrdaki 1966 Si nha ve H r1 )l iA,B,D,k rJ ! ' 0,.'] daki H /H zr oranl, (:.72:) T r , . A r B r D . r l::, .) LL Yukarr daki edi .1" i- r . H /H uzere zlo kuramr hesabr Anderson yardr h t - s a p r l a l r n r ; j i " , - : )C ; l l r g : , I a c a k L r r . T T ve T de vt: t. anmasl n:- inLegralleri OL (1979)'dan mr yl e eml- erden hesapl oI ar ak Ru qa,l r grnada C79..7?) ve denkl oranrnrn say:- saJ. I j nee+r' suzgeq Lrr r FrischknechL ve T o C A , E , D i , l , :. . ) - 1 / f J 3 z=O gerek Li rmek Ledi r. vd o z=h=O inLegrallerinin Koefoed I T (A,B,D t sonra el de anlagrlaca$r f ":t r"= t L n IF=o oI ar ak ( Kel I er L basi daha. 1 973): +- H =C l_ L6. " r err I 1',1 |L K O ^ 3 Bu iglemlerden b.i I esenl edi 1j r el- de we> C o I l e L r' -Cl-z gi bi aI an Ckonwol yar:rrlanrlarak usyon gekt i ncle) 36 4.1.lkr Kaynak ( 1oo;:) ol arak halka brr-inin ile alrcr yarrgaprnrn ikisi 1 954) . de Bu bi ri nci I 1t<i halkalr halkada c 1 lg r i l r r r ' . 1ki j.krncil c > r a n t "n d a n zi yade CmuLual (Keller halkaLr induklenen volLaj, I empedans empedans veya Z=Y/1. olarak Burada 1966). verici c5rrUne k r : l ,l a n r l . r r - kavramLarr ise gbz al an LopJ.amJ,ar.rn:.n, I r kl r kargrlrklr (Wait redi I mi s sj.sLemler kar5r we FrischknechL haLkalarrn davranrr alan.larrn coupling) igin bu nor mal j z.e eger halkalardan mesafe, gibi dipol ol arak sisLemLer fanrmLanrr olarak ve- kuplaj met.odl ar r nda daha. briyukse ni ce,l j.k birincrl al ana kargrl:"kIr kaLrndan maLemaLik 5icJclerLi Lerimlerj. sorrcla.j lral kanl n bi r k uquk LraJka ar-as-rnciaki beS mef odda a-l.rnclrQrnda i nrtrjk tl f ()ranlarl: Krrpla3 t.agr yarr akr m l . :r r l I a n r l c l r g r verici her vc. F.argrlrktr Soncla.lI Halk:l halkaya V al:.cr uygulanan akrnrdrr. Alrcr duzenl 1954). duzenl enmesi nde rre en EM deri-nl-ik emeler ver-rci cok halkalarrn kul l-anr l an sondajLarrnda gunl ardrr 1. Sistcrfn: YaLay a. Sisfem: Dik 3. Srsfem: Dugey 4.SisLem: Dugey (Sekil_ 4 birbirine si sLem gogunlukla wardr 06re r kullanrlan ( l{ai t bu 4. 1): eS-dtizlemli CHorizonLal cop-l anar) (Perpendicular), eg-duzlemLi e5 eksenli Cwertj (verlical cal coplanar), coaxial). 37 ttzayda .'.i:r l:r:st. k r . r t r :al . j r 9 j r'r Z gekl:-nde z-,p z (' (Ke-l-ler yaz.rlabilir karg:. 1r klr ' de a.9agr dak i gr bi dik kuplaj birincil alanr ise iqin srfrr verici olduQundan (74> 1z,p C73) dogrulLusundaki ve toplam H r,a we {74) ) alanlar, Loplamr i{, ve denklemlerirrcle Lopl am al anl a.rr dogrul Lul ardaki z,e halkalar =H zH ) yazLlabilir. Cl{ r . oran:' , OfI alanlarrn Frischknechl c73) yd)ndeki (Z/Z r v e r i I m - rg t i ve =H /H birbirine yaLay kargrl":-klr Keller icirr halkalar Olzz,p e. SisLemdeki bobinin g6sLerir'. e5--dUzlein,l-i Z2o Burarcla 1S66). FrischknechL aLanr yat.ay oranl ) ve birincif kuplaj CZ/Z i I gi I i al:rrr nc"i I biri halkada.ki z.,p dogrulLusundaki Seklinde hal k ar ar duzl eml ,i =*Cr,R3 =H 1. Sistemde ( 1 966) ( e$ Zo al r cr karStlrkll arasrndaki ol ar- ak , Z t{ -i I i r . c.l ,Jr:Qr-rrrrlan) {?.-,9Oo 5 t c l d t : . t .j gdster uyl e ,sembol hal k aI a r SekIinde, gc)sLermekt.edi r . birinci I ve H. z ve ikincil" ^n t . sisrrv vATAv r:q-r.li_rzr-nvr.i (Ht)RIZt NTl\I-, CoPT-ANAR) ln @' h -Y - - - . - - - - - 1 : ) 2 . i.----+--R--; oix ffi -; Li/- (\/ERTI(]AL nn $ekil .r. sisrEv DU$Ev es..rxseNr-i |( +. (\/ERTI(:AL COAXIAL) i i , ,i I I - J- i i (7ri) .ril I DMD F r - r ra d a L : ' r t t : n t l ; r t - t - !r halka kullarrrlan sonclajlarrnda halka lki duzenl emel er i 4.1. r ' - r la. t ' ; , ' l ' CC)PLANAR) rrrm v*--i ;'v ffi -----r- <prnpr:NDTcLTLAR) :. sisrEv p\isev rq-ocizr-evr-i ^ \l/ f---l--x*; lrlmrrfn SISTEM r ,p i - - ;r i t r r : . i I rqr-n derrJe'rl r:'r'r , ql II --- t - 2 t, _1\.,__ 11-) - = C . ' F - -l - z,F, -j(_-l L:: -ll [- l \ tt ) ,) ^2 r.5 ( T!-.) 21=,f1::Q ol drri!unr-lan , ( 79) =-C,,R3 II ;-el., I j titlt: ol up:, H -O orl at'r Lkiirr-i-L I E{,)') r.l) ; tlerlr,:.t- I r:t r :rY. c H .,JT z,a () ,k T (A,ii.[r,1.. ii r.t, 1.rl,i (.A,f,,ll 6n lill rirr tlL r':l t1 i' | .1 i 1: ,. i ' - - r lr i i- f:i ) F J l r c J e r r r kl . ; t n ] r . l 72".2-- () kar'5r l.: kl.r cl<+rrk I eml erri ndern l:,;rr'l: nr.l.r l . krr;:l a.j or anl - L^l - K _ ^ 3 ) ()t! cC,/63)T C.A,B,D =-BUT(.A.B.D.1,. L [-rt-r1 i-rriurr reel ve FrischknechL sistemleri igin ayrrntrlr Ke]ler birgi ve ntn, we sanal 4.2 ve CoI I eL ortam i 9i n arryle Ker] I ,ar- ve Ig73 I . we rT. de g(5st-eritmigLi,r ve lyait 1s5s). kargrlrklr Sinha kuptaj ve coll c1966) denkl eml eri eL Laraf i ncel rnclan n Rt ytr (Keller Dider oranlarr c1gz3), vu= s i s L e ; n r le r e a m p . 1i l i i d l e r r n i we 4.3 1966 (74) B; L krsrmf Frischknechf ( 73) orant ve Homojen Sekir ) T, i.Si nha 1966). gc.:re clegigimi ) ,K -C,'R3 .rjr, rn {g.ll ''tL t: -- z2o! ..','4.) vr., ) ,k Olr =1 *R3T cA,R,r),k aiL i"i"\.t i,, j ri har.ck r:t l e, LzJ,= Fr j.scl:knecfrL '-.r:.i..,;tr1.,r l I " r r . . tn r -.C,,R3+CC,,63)T (A,R,D .z- ,:,I ,.ii-..ri.' vel bobi n hakkrnda wai t. veri encll gi ncie c1 s5s), lmi5t.i lf z/Hr r. .'io N N I E l f :[ (! Y Y .^ tJ J a a o -o,f-,- \ ,, \-1--.t I .i-,.'V,--,=-,---'-l olz).56 I9rOIr2 ; SekiI 4.A. Homojen ort.am tizel-ii-iclt- T. v€. Ti. :;i=t.em igin kargrlrkl r kuplaj \/e] orarrj,rlr-l nt n rr.',:l sana.l (KelIer k:.srm.l-arr ve Fr-i,schl:nechL 1966'clan alrnmr 5Lr r) . t. o N N o.5 Seki I 4.3. Homoj en yer yUzu uzer i ncle igin kargrlrkli krrprlaj Cl{ai t 1955 clen al r rrrn: slt r ) . rzr? T. or ii rll T I . si sleml- er amp} j. t-tici.l er I r{ ) ( 7..,,'. .) (t !t --ii-- ( 7.,'7. ) t i.r r ' . 1r j r r Q t . r k o J a y i - t k l ; l g r : r t - r i i e l ) . i l - 1I C8 4 ) clenkI eml er i I'l /ll zr or anl yuk:rr': rJaki 1--BeT n - 1^1 3 i I l r . : l ' : 1 " :, [ J r . J r ' : r r j . .t , i I '.r ( il3) gi bi i ' r r . , : . . ; Ir ; i \rrj k < - . n r _ ar lr a k traGr trt,.t r:i;r yr:,t l I r.:a ciak r aga{r H I ;rnabi,l i r' : '-| 1 t{ ( "r. I* r H T ( A. B. D ) lh=o ()IL T CA,[f,D z=A kargrlr kIr A'-l ciaki Yukarl kuplaj ,k 'F':l ( a:i) ,K ) hesapl oI ar ak sayr sal uzere.: <la J,, hes.abr sayr sal oranlarlnln 96r itJ eceQi cle emi er-deD denkl i nLegral 1eri ni n )'1 ve T, anmasi, n: gerektirmekledir. (7e) clenklemlerin sonug kuplaj k uI I anl I a.r ak Korke'alaakso gC5rulebilir. DMD frekans sorrclajr oranlarr (Z/'Zo) el cle ve ol cJtrQu egiL Saksa ndan edi I ebi I ecedi 1S86). cli r- t>u I ncc:1 endt Qr nde , denk 1 eml er i Ca5) birbirine qtkarrlacak kar6ilrkl:" ve r" mo<lel rrr Buradan e$rilerinrn (7./7-i'),rt Y* (, lii rrh:: :l-t?7S oranl ve -aF I vr'I 4.4.T itrtrt;l .rji,'t -i I .r fiayl :i.'rl Oi ar ;rl. Y ' r t - r l i n tI llr.tr ;r lnt i rrj l'r l,i trr''.t',i' l:.tt;'. it a)I l J e r : ; a 1 r) ; r . r r r r ; , rr: .: 4 . a ' : . 1 . S l r ; r g : + g K r r r r r . in r a s . r Bu bdl timrle, k aLsayr I ar r nr n d$nti:; nas;r l gaJ r 5r I acakLr r. C e + kj r c l e k Hanke.l n i:,i r fc_'rnksr y(rnun uml er- i yl e: hesapl arrcir g: i l me-'yr,: lldst.er r:-l ma}.- uzel'e Lamsayr <\uzge'( i i nee-r k ( )' der- c:,.-eden tt. II:;liker-l dtlt-rr,i5umtt, (l] = KCb) o.larak LanrmIanl Besse.l f onksi birlikLe Bu degigkenin deQigkerr ( b\) \d)., BrrraclaCok k ar gr n k C X) karma5tk J", a. sayLcia b >o ( B(i) derreceden araSLl (Koefoed 1A7?, dtSnugum degrgkeni b)O dol ayi si -v1.e fonksiyonlarr cldnugunil er i yl e ^ rmac:t vd ve srf r r r rrct Cekj rdek Seklinde sonucunda, i rrLegrasyr-ln yonuyl f onksi Anclerson gerqel K( b) ci ns e; 1S79). b.i i' sayr ger gei bi r olabilir. sayr sal sr-izgeg k urmak i gi n (86)'de x=J,';Cb) , y=J r,(7 /X) ddnU5umrj kul larrr.l arak doQr u-l Lusuncla l'rer i ki r. I.k{ \) Hankel denklem -. k ( _A ) . r Lanrmlam.rglard:r ddnugurn o1 masr na II yonr,rdur. j-ni parameLres I I yarrl s j met-r' j ** el de ve Cher erLmek et.mek I g j r-r) i 1r: r;arpr I ar'ak , i kr Lrr.t apsi S rlt+nk -l emi n +"j CD x\ x,,- g c:| K( J -s i nLerqral bu formtrna i nlegr sahip ) L lilv ( l ) e r r r k . le m C 8 7 ) edi I i r . i ncel oI.duQu gClruIebilir : Girj.9 fonksiyonu, / / endi Oj rrcJe, konvolUsyon lineer tl inLegrali (Papaoulis Br-r 1S62). af de , *K( (> .*) t **-YJ Konwolusyon bilinen : Qr k:. 9 c.*-v:.|, Leoreminden bul unabj fonksiyonlarrn aza.l"an girig-qrkrp c797e) osyon genel nden fonksiyonrarrnln sonuqlar suzgeq urelen da mumktn Anderson oldugunu c1g7g) clgzg) ve bl linen suzgeql suzgeqler yanl yl e s1ra, en i erlcre hlzl r cek j r ciek suzgeqboyunurr belirt-mjs.lercljr. agagrdak:, ve oLarak ke_vf i edilmesi bx_r hrzrr edi I en Bunun i 9i n Anclerson yanrLlarryIe konvorve dolayr suzgeq oIu5an er de gc)stermiSlerdir. suzgeg kul-Lanrrarak gaIrgmalarrnda I eri uyumlu 1g6a). bi r fonksiyonlarrnclan s6nen krsa]Lrlamsrnrn qifLi cPapaouris 6nemlidir. yapLrkrarr ordukga sebeplerden oldukca i nLegral eciile,biie,ce'Qini fonksiyonudur yararlanrlarak Hedef I enen seqimi vd yonu, fonksiyonlar:" 1i r. Koerfoed kul"lanarak f onksi Suzgeg ^J girip-9rkrp yanr Lr oIarak e'l de x-\' (- (. l L^l denkleminjn k Ce-Y) konvol J clenkJ" emi inl.egral r1 I x-y , ., le f I K(e iki Bu Hankel '14 dAnr ( (ir- n rl1\lthl r< iirnr r kul I anar ai 1| g j Lr \,/r'l {:ryl'r F : y z fr : | s t tT r;r'r; l r'1' 1 k r.tr'tnrt:'t.ltr r -b .^'', l '^.'J' 7 f-^'* J LD/\.,)ON yat.ay _2 ve'rlik bobinler lCi.n benzer fonksj.yonlara (I97A> wd Koefoed yapfrklanr kurarkern suzgeq olarak Chrzlr tisLel segmiqlerdir. adr mlar aSaQr daki gdre C197?)'e Koefoed kurmadaki suzgeq sonra seqimi.nden Fonksiyonlarln diQer /zlo, ( 8{r) lda) fonksiyonlar azalan) 7 JD <lrr'. l:)O galrgmalarrrrda yukarrdaki ./1c., ( tJtJ J t J a)O, . (:a). (-o/\, z ) Burada i!j7q) Arrt-ler-snr-: t b A )d tr= ( -L, o 19n5: sekildedir: 7 . en k0Cuk sabiL hale bir Kuramsal deQerinden absis Burada (197Sj) en fonksi Ax ile deSerine drneklenerek kullanrlacak 6rnekleme yonl arl nl n absis buyuk aralrgr Ornekleme getirilir. Anderson gi ri g-qr kr 5 stizgecin herbi ri kadar sayrsal kurulmas;tnda aralrdrnr Ax=J i:( 1O) ,/7\ - 5j,3=O. 2 seqmisLir. her bi r'i elde Bunun i ci n sebebi ise i nl-er pol asyon edi lebilercegi ni girig hat.asr g6sLerebilmek ve rrr n qrkrg fonksiyonlarlnrn 1 O-d cian iqindir. k uqtik ol ar ak .1 s "+J Ksr-:f'Oeqiv,lclgTil)Y.r1-'itl.l.rlrl"'rr'"r':l'-.ririrrlr-l,ir rryg(trr ()lar'al., t 5 r I ' r r : k i r * m < .a: r a l r g r n r en . , . l r j i r .rr. ; , 1 e r - c l i .l . eme Olnekf r:lur !.Imunrla el r,le e,li I en ()1:rcaE-r nl ( ,.;ebep 1O) /6 Ax=ln( l'Jer iki f azl a aLrnlr, apclrnda.n spekLrumu, dOnu5umu allnrnlF hat.ay:r karmasrk sr f rrlal'a c\rnekleme ip]emine uygun grriS bag,l anmasr yl €) krrrt.trlLtnabiI r;tkt1-; edj I i r ' e1 de spektrumu 1' adrntda iSlemint-len se'q:ilert=k de$e'ri ba5langrq (Koef oecl vd inir blJ]utrr'1r spekLrumuna b6>lul)rne btr i:r"rttr.t'+t tJ$nrrlE;tJlnit a] lnnll 5 suzg6?q yant t, f onk:;i yonunun Bu adrmda 'eyrrk fonksiyr;p11r1 $rn'=klerren ddnu$umu BOyt ece l-rr-r,yuk bel i r f mi 91 er di r - oJ acaQr nr-) e. daha 'l' irt-t'"rli' t.;l rir Ii:!.r at' .l Ii liaLal 1 OO-2OO kat- iken "l'it L : , r . r l e : t j r " lr l + t i ' i ' r h ; r ara.llgt-nrl) se,;.i I me:;i Ax'ln(-io';.1il ve 197e Ra5oktrr 1 EA4) . 3. suzgeq I Sinc-ya.n:. orneklenmi$ spekLrr-rmr: ile, siec si nc( y) =si n( ny/Ly:- qarpll spekLrumu 4. e;lmek 'inc j qj n,3. yat-alr ve sjnc*yanrL dr k ve 4 . 5' de' Or- nek bulunabiLi.r =yanl L al r n1 r . bobi f az ar ak c'r.L nl_ er yani {Bagokur suzgeq yonunrln, 1Sa4) Koef oed i qr r.r vrl (797e) ha.zr r I a.rran gara.f ikleri i I mr gi er cli r' . e l de+ k af say: I ar r nr spekLrumunun siizgeg spektrrrmu g()st.er f onksj 2. adr nrdak.r siizr;eg /(ny/LY) arak acitmdakr Four i err dClnugunru spekt.rtlmr-: ' L6jrs aYrrk ! den aL L nml $ suzgec Sek j i I '.-r-j n 4- 4 vt3 '{ L-r Kl':efcrecl ayrI nt,r ] I Ct karnlr ve Brrtra n€iqat-if yakl.rSLrQrnr kaymasr rregaLi rJ(l1*:, absjs f bir bi r Bi5yI ece t>rneklenmig Sek i I de scJnuml yc>l olduQu sLnc (Bagokur Ax=i . : ,J l i c - y a . n t enmesi srJzgercit-, ), I-i LlZeri Lru ne at'al yonttrrrlr) yatda t {r ,lr r' . b,j r , - : I r * l l ' -I, e m e r m P t . o i i l. sr f r r i l-yrJrlrr rme'trj tr yer'Ie5t.i yaLay i-rtzl t I ', r Ir asi Cr>k p.rozit.if emg: 6rntakl f onksi i kr g*?qer) rrfrl'rl.'rtr C sa<i] amak ni nokLalarL 1qa4 L s,()t)l.I{, cle+cler.le+rtric' Ax) :,(7Ol- /7O \,it:)lll.rllll l.i I i,..irr Burarla f onl':s,i yonr-rn nok t.aI ar I n.l n , Ir<>zisyonundaki , I,,Irlli f o r r i . 1 :j, _ y r : ' r t t - t r - r t . hl le- rr L +n CC sabj"LLir Lamsayr yalllt de{t'r)e.r.i belirt-miSlr:rdj-r. olan ( . ' r r l . + t r r li . r ii r r c i - y a r l ] cler$erlerinin ;rbsj.s ; i lr,:,'l eyF:r'6rl' C)l arak $] ardrr. pozit.if (I97;li v.l iti gdruImu5LUr. vo,n!t Srnc ycnrt AcsLs Sekl I 4.4 I. (a) suzgec-i Ve n fI. si rrc sisi-ern1€:rr - (i)) y a r ) 1 1 . . : { r j 1 .r - t t i''rrr hrzrrlarrmrs ( Ktrt-:i r.rr",'i 1 97?,) . t 47 SekrI 4. 5. Surzgeql er i n Ca-f.sis{-em, f az spek t_r r-rmu b-I ]. si si- em) CKoef oed r:{r j- I er j, . wd 1 g7A) . "+8 ZIiANKS 4-a yard: mr Al t.pr r:)r7r.iml I rr{-eg.al I err i ^i r'r s:ryr :'a i Anclerrson C1 S75) denkl emleriyle veri ve Hanker 1 " dereceden hesapl amr gtr r . hesabr geqen t rja C1 979) ai f program -J ve bu yayr nl anan To -* T, gritfiklr:rr Rrr f aydal anar ak donuF Um(rnr.i al an - l e + r - in , i n ZI"|ANKS sayr sar ri I ecekt-i r. ll nJ U 0l o N rJ 'DO ^' o.o -10 sizqc6 It Apeiei *' -{ 0 A) ol N .l a J I Seki I 4. 6. .to ve yanr Lr ,J, Hanke.l C.Anderson c16nrr5(imtr l g7g) o c l C r n u Su n r u arrnmrgt.rr). i nt.egra] ( f.l!]) kat,say-r l ar- r rrr ar- I nl" n k at.sayr 1 ar r ndarr Hank e] ve krrllanar.ak F l ; r . n ke r lgzsrdar-: suzgeg e .ir-rzg€)q -l f onk si yonl yarclrmryl gergeklegLi g a I : - . : n r ; r s rr r c i ; r ( B R ) ar ar: .i . 6! trar cAnclerson qal : 5macia adr rie-:-,rlrlar.rrlr.r<. r ci,+nkienrl t-r-rrri donugtimunu yan: I gdsterilmekLedir adl r yapt.r {r int.ergrai sek i I suzg€)ci ni n r\nde,r son l en (lr ;erak -l- il t, s u z - . i l r : , , 1- I)l n Sl NC 4'3 4 . -3. 1 l{e:saJ:,l ali.r ta, A 1 r l t _ r ri t . r n ; r : , I ve+ T, int.eqralle;r'i yetridt.:rr v ; t . z t ) " : r r; 1 J 4 r : ] u l s ; r , CN r T.,r R) = (lJ | r tr'- u'= R( g) gt I F( g) 'Jt | ,,,J Bu i nf egral denkl. eml err lri n ol dugun r I g$s L er mek o( gF) rig ( CtO'i (: Qt -) gB) tig { de bi rer k t>nvol risrrr.;rr i r-rt.egrai I i 9 .i n x=.i it( B) , y=i tr(t zg) c l ( 5 n u g r i m r - r n t ik r r . l I a n a r a k , c e-*T.rco") = ec *-t> e-zye,*-yJ.,c e*-v--lcly -** | *'Trc *") = - clr ( I :r) we ( '33) T^ lnLegrali () i cc .-"> *-'Y..*-vJ, c ,-x-Yto" denk I eml er i i ncel end i gi nde ; igin, gi rl g suzgeq g-rkrg f o r . r ks i y o n u f onk si yonu= fonksiyonu - R Ce * v ) € ] - z v **--Y.i = **t ( ou'-Y) , ]*t ( ge) c s3) FN l" t" r nt.t':gt;aLi i r " . ir r gi r i 5 :suzger; sayr sal krrnvol yapabi Li-.iy<>nf ormr:nda ro(ir=) ncl) olacaklrr. ( S4) j Burada anmr S oI an we ( 95) kalsayr werilen CAndersrcn 1979) o,1 suzgeq sayr saJ (s4) [r...rr] Fo,rCi) kat.sayl g e r e r kl i konwoltisyon L ar r dr r . r i r . x=/.riC B) Bu nedeni fonksi tizere cB)=l 2. fS-A.. \ L /_ . =t, 1 F( o . l ) L R C e ^ . - " ) l z B J, C9 1 i yonuna i zl eyen yaprlabilir : c 96) e: gi r-r y v(+ CS O ) oI mak t.oplamtyle dnceden sa.yr sal gekirdek bi r B) O ve daha R Cj - i ) sadece apmas:- nl anrnasL rrin ( 95) ver *l Lamsay:lar, N T =l RCi-i, herhangi biqimcle i nt.ergr al l er bu lerj inLegral [t,.i-tr] hesapl uygul C':j:l) o1 ur-sa, [F.{i-i)] sa.yr sa1 erindeki I arr n ac'ak denk l eml er i f onk si yonunun denkleml C'l;J:) ve l melr i qi n yaz.tI rrci)=f *.', hesapl =. €r*Ti(e*) fr-rrrksiyonu nr ( e*"v) et-YJ o-ldugr"r gt5riJIebilrr. hesabr R Ce - Y ) e - " Y f onk si ycrnr:= r;rkr5 Sekl inde = | orrk si Y,:nr: 51 Br-trad;r F (o,t)r ' I er cjah:l r)rtr:eden A,rrder-son hes;rplar)ml F sr.)zgt+g kal.sayrlar.r ve N (-283 N_-)1, l2 deSerlr:ridir. veriIen N, ve tolerans kullanrlan) ol arak ekcle maksimum olan) a5:- r-r CA.-x) Uzere brt usLeL absis olomatik olarak yl e amal ara n -27<Ar<31 , olmalrdrr. ZHANKS a1Lprogramr inLegrallerinin yardrmr sayrsal bir geqmek donuSum bjri orLama eri Brl i 9i n olacak de{igkeni (sabit. iqin Ccrr=O. Ol To ilehesaplanmrg dederl ant r . i=7,2,...,e43 herhangi Homojen oIarak baglr hesapl (5nune ZFIANKS'da kullanrLan cloneiSumtinurr o1arak fonksiyonuna al tprogramt hesapl i =Nr, .,N, kaychr-rlmrS =TOL gekirdek r ndan Laraf i s.r, deferler CAItprogramda aral;-Qr Harrkel Lamsay-r rgin ZHANKS seqilmisLir. ilgi1i aiL 1en absis Sekilde B, N, kullanrlan veri aI Lprogramda of rna.k deQerri ve d-rr'. C1()75) mho,'mJ v'ij Tn 4.1'de Qizelge gc)rulebilir. Karmagrk 1. dereceden ZHANKS Hankel kaLsayrlarr qaErrldrgrnda Bu bir nedenl be=Ilek er daha bloQunda di ser ve (Common i l gi l i egLi ren hesapJ-anrr. Hank el srizgeg uygun absi s ol. ar ak ZHANKS i1k karmagl hesaplanan sforager aLan saklanmakLadrr. a.ralrklr diSer ve olarak sr r asr nda eS N E H = 1) de{erleri sonr a anmasl Larafrndan Cparamet.re fonksiyon(r olarak skalacia al Lprogram hrzlr gergekl dahj.l i hesapl ClogariLmik t>rnekl enmig) oldukca ddnu$umu f onk si yonunun degerleri qekirdek nda., DATA deyiminde Cek i r clek genel ddnusumunu a.l t,programr O. fonksiyonlarlnln. cekirdek k clegigkenler Block)saklarr.rr-" ci6nu5umunu yer r ne F - C e k j r r J r = k f r . > r r k s " iy o n u n r t n I qrn gr:'Ltrnlek ' da A n d e r s c - ' r r C1 9 7 9 ) i nL..:-qral ,l er-i ni r-r prcrgr-;amcla , dalr;r ,J '( vr:r i f mi st-i r' . &rrce ( ()5) d e n k i c . r n iy l e T vri sonr a yapt.r k t.alr hesal-'r nr sayr sal T., i 1r+ yarclr mr al Lpro€lram,r ZIJANKS ui.rl a y t . r r ' r t " rL r k t t l . i : e r r rm r Br-r a.l t.. pt-ogramln gr+r'el.. k.r] maz. h e r s a p : J A t ' t r l l . - r : ni rt Lekl'ar 1 aI.)a 1 t;tr , vet'i ^ 'n3 )-/t5 HT zO T yer j ne ba$rnt--rsrnda 4.1! den de karmaprk ol acaklr r. nedenl Bu t- T., ve t* oranr n er b:u oranl T r: ---,1 \r.J Zgr i r r : s a p J " a n - r -r c)ranl zr HrrH. oldu{undan 1"" aI I er j- i nLeqr --_ uH bi r er karmaslk da ampl j L(icl(i, t /2 - I ri. f'az-t itzer. g6ru1e,ce$i sayr H ,/H konttl arak -? z [[o*' " ,,u )l r J + f c m cH / t I , ] ' ] r L t- v.,/ z i se, r rr",'H /flmCtl./HJf al-H"/If ) ?=ArcLan [Rec gek I i nde L \'\C' of acak Lr r . Bu qal r gmada Hz/lf or anl nl n arnpl i Ludu j l e' " Homojq?n ve i1gilenilecekt.ir. c i e -{ e ; r i ll H z ., ' H r l I n i n ver i nr:el" errecek r-rekansa Li r' , Labakal.r haQlr olan oftamdaki rieQi5inri r:Je amp:liLut-i al ,rnacnk FI J' Ct ztri gr- 4. 'l int-e-gra]lerirrj t) rr fer'k:rris vr-j i r i , J r . r k . =i y o 1 1 : : a y t - s . r ( R J : y e r k . : . r l 5 r re-"n-rL vt-> sanal k r gr nrl a r I ('n =(lt. 01 , l r + , - ' ) e ri r r e + n1|lr_:-'rn salr.r I. I I I r : n r oj .e n i l ' t ; t'r bj r or t.am i ci n) Iil(It) tit(I0) ff(T]t lfl(Tl) ') 0.104t:'r, ;t.61b639 0.,14'1b47 I )r )1.: 0.,1'i6't'/0 l . $ t 1 r J 4 l 0.) 09t09 2.382t69 q 0.ltb.yJ0 ) , . t 2 ) 0 4 2 0.4;1609;lI . b3'i/9 0.4l0L;{'U i . 3:il90:i 0.In'100 1.8'1i,4'J3 ":.0,0 ti .0 0.3'i16'Ji t . l i r { i t 0 { 0 .I 3 l 5 t 5 I . 6 2 i 8 3 7 ()',i'J t J 0. 0 0 . 3 8 6 1 1 r i I . 0 l I 0 t , 0.I3{,yJ4 I . 4',/ 0..1/,,6:Jil0 . 9 0 0 6 3 , 10.I t [;l*J I .351677 40.0 0.:iL6',J(,90 . g t ) l l ; 0 0.l44Y% 1.2i70J6 i0 .0 0.3t'i;177 0. r;7'J:107 0. l:;0/ti3 I l A d ? ? r 6().(, 0.r.tt4(]') o.i'/it)(]'; 0.I'J5090 ].000084 0 . 9 1 4l B1 7 00. 0 . 3 221J 4 0.,l9J.1l.i v . I JO't.,(,. u (),0 0.lJ(l9200 {:rtt30? 0 . 1 6 1 0 4 { 0.tr44292 I6 0 . : l l l B l I j 0. t6:i0j9 0.7860 9 0. 0 0.;,19;1,10i )( ) 0 . 0 0.tf tb0l,6 u . - 1 ' i l . r J u J 0.164581 0.73639 I I ;10 .0 0 . b;.i 0. 2699i 4 0. 2;j0.j t:j 0.I 66b71 0.655732 t i 0 . 0 $.'i'J() 0.li{6641r 0.249'196 0.158([/9 0.'.t6'r'.j23 2 0 0 . 0 0. tJ,t-t 0.;lt:j196 0.l3;l/4d 0. t676n 0.460970 n 0 . a 0 . t t 2 0 . 1 8 9 9 6 9 ('.084I32 0.165{l{ 0.3486"t6 0.334678 300 .0 I .03'J 0.I59069 0.0i090J 0 .l 6 l l n (,.027 350.0 ).ili 0 .I i t 3 6 l tfb U . I JOJTJ 0.292i08 4 0 0 . 0 t . l 9 ; 1 0 . t : J 5 u l 0 . 0 0 9l 8i 44 0 . 1 i , l 3 1 5 0.2581 '(tl 500.0 | 0 . l n t r 4 - 0 . 0 i 3 9 5 8 0.14:,s58 0.206493 b00. 0 | . 4 b 0 0 . 0 9 1 4 ( ; 5-0.02ri0 l5 0.l37,l7l 0 .1 5 8 5 7 5 '/00.0 tl.07i63!, -0.0:r6i96 0.l79Jt0 0 . 1 3 9 5 5 3 l.\'/ / 8 0 0 . 0 I . hrlb 0 . 0 6 r 5 7 4- 0 . 0 4 1 7 7 9 n t ) l ' r . } { 0 .i I 6 9 0 8 9 0 0 . 0 ) . 7 t i \ 0 . 0 59t 0 u -0.().'14;'51 0.09861 5 -0.0462t4 I 00{i.r) | il,)r: 0.107767 0 . 0 8 3 5 4 7 0.0t:37d I200 . 0 :.1.()Ll; 0.0'ta7 4l -u.045i.1{ 0.a9ii'7) 0.050833 I 500.0 2.309 0.014;180-0.0437'i0 0.0d0 I 57 0 . 0 3 8i 91 2(t3P.6 z. ouo 0.0005ti3 -('.03651 I 0.0605'J 0.016851 2 i 0 0 . 0 ;1.980 -0.00b2 l7 -0.0;193970.0,164050.005978 -0.009:;02- ().0:3405 0.0360:1 0.00(t2:,ti 3()iJ0,0 3 5 0 0 . 0 3.i2rj -0.010904-0.0i 8i89 0.028?95-0.002808 ' 0 . 0 1 1 r 7 5-0.0t4t81 0.|'t'tni5 -0.004330 4000.0 ., t'.: t -0.0l 0i97 -0. 0094,1 i000.0 I 0.01{5s8 -0.0051 72 -0.009284 -0. 6000.0 4.6t1 006 i {? 0.0098s -0.0048?7 7000.0 ,t.987 -0.007936-0.0040c3 0.006861-0.00{I 6I -0.Q06'/43-(J,(t0"1/t;(t(,.004f,$ -A.00-:477 8()(J0.0 J , . 1 J l -0. 9000. 0 i. 6i:r 00574t - 0 . 0 0 1 9 , t { 0.00359i -0.002875 .: (rI l -0. i 0000.0 004-q21-0.001'i9'/ 0.002848- U . V V Z J / J -0.00371 I 2000.0 b.529 I 566 I -0.0007d8 0.00t3J3-0.00 '/.300 -0,002{.00 - 0 . 0 0 0I I4 0 . 0 0 l u l -0.001 1i000.0 039 -0.00t 6i7 -0.c00;10,10.000617-0.000572 20000. 0 O.'lt;7 :5000.0 9.424 - 0 . 0 0t I7 9 -0,000) 23 0.ItJ03.1/-0.000359 -0 30000.0 t0.:J;lJ -0.000895 ,000cd2 0.000'276-0.000252 -0.00061r:i- 0 .0 0 c c 5 5 0.000202-0.0001 3:('00.0 l l . l J U 85 40000.0 t l-9;10 -0.0005/3 -0.00004 I 0.000t!1 -0.000 I 43 lu.0 l:1.0 li.0 /,r Lr, tetJ l . L V J l 1 , , /\ 't.I I l\r' r< 5 . D M D F R E K A N SS O N D A J I H O D E L E G R I L E R I (197.3.) L j c , : .IIr + L ve C1 S a O ) Saksa proclrdln.ry, 1i: 3 f l i r i r . , . i [ r , . . ] . i r i 1, . 1 . .. . - ' i..ri . , . , K o r k e " e l . e . : i . l : j , ( - , qel t,:rraf r ndan m r 1 d € > lr i gi n di pr.:l tr bi i e$.el-r} erden kLlpl-aJ kar-Ftlrkl.t yat',.it m.t F)rO,-lFerrll ()l .rrr gerekli hesap>l l anarak oranl a-rtnl anmt;i-"r di I j nde FORTRAN t-. vAX --VM-Ssi sLemi no* uygui amasr \ / 4 . 1 1 1I m l s t . l r .l i z wf-: i ( 'r tr ;11--r'wi pr()r]ralnl l-:tl l an Ri t gi c a. u. bi j 91 Sa)1ar' sosciajr f rek.arrs ya::r 1 q -j- l ) rr 1 9.l ern Da i - R:;k . ) r Pr ogr amda karma5r p a . r a m e l - . t -e l e ' r , 5i: i 1 er i I hclsabt ^?^ K ISC 1d tylar: am1:] i t rlrlr.i c)r a nl- l) 1 n I l t'H r.:',Jri l c't' Mocle.l e{r i nt . e g r a l I e r i r r , r n ol atr sayl hazrrlarl)I1l sek.l inde Model * ' i..: To bi r k h e s a p l , al)mak t-adr r' . l H= / H ,I ] r+n bj1e;enleri alan vc' r ' edr Sc5Zu DMD sa(lece i ls: vei'r bj r r s t s t ' e r r r l -e r r amaki'a':ir da i.ri:.',;i lres.rl:l ,rmirkt-arl-rr" buLtirr hesapl yukarl nlaltyel.jk I' rja al wf:r 1-r'o:l;rtr:; kr"rl I.tl)ar-3k' bile9*;nleI'iti.i yarar r;a 1 l 5mada i::i. tabak proqram bt: trl ar:rk kayrrak E)"{ alan z-arnan r.rrt.anrrrrrl;li Bu I en i sl-i ri I fri ) 11i s;v;'. iresa;r) C1Sa6) vf-' K r * ' r ' ls ; r 1 : r a k - " . ( ] qeli5tirilr'n ndan larafr Yt'' liir; r I , ' r c l . tk 1 , r r - i fiazr Lrli ol arak, dayalr programJarlna bi ltti s:iy:lr (1!J"29,) Andersotr .ii r.r,Jl'i i t+l i , ntt-"'j'+l sonrJa.j r frekans l-.ii t ek af -.tt. r . Lrr-sa1rl 'arr-r r k +n pr r:'qF 'anla ,-l-r.ri i errr gunI ar d.l r- : fabaka -). i.l et.k.=n1 i rJi (51 sayl:l a j t ct t ic (Nt'lM)' trit irr'"1 wr-r't'- j ' - r r2 < - ' :rlr r l ; r f I l..ri:ai.:rl)l r't<l r I r') i ii'il t ti€ En aI Lt an uste ( SI GMACM) , M. Labak anr n I .a: l r n l , r k l a r r D(M),CM=1 ,2,. Pr ogr amda CFICM)) rc5latj.f i q.i nde E( M) =1 Kt Ml =1jI GM,A,( i"1).,51 , dorlr u r I eLk enl i k or a nl ar r i .1et-k *arrJi {i .1i r t . a t ' r ; a ka ,M: DCN)=O dr Labak al ar a ai L dj erl ek Lr r k permerabilife manyeLik i qi n ol duSu v$ . ) , sabi t. sab.r-Li er' de{erler,i vakr,rm I re oI ar ak et),i t. a1 r nm: 5l ardr r . Hazr-rIanan 7 QlIz.-4OkHz deQeri arasr seg:ilmesinin gel i r . drnek seLrebi frekanslarr-n Hesapl bir segi l mi p yapllmrgLrr. iqin gereken nda ol up Frekans bu metodun adr geqen a . m a . la r . l n qrkt.r edril-erinde model Qizelge yapr. 1 dr gr 5. ltde frekans b:rrrdr hesapl amal ar 47 arall qr nr n IOHz- endUsLride aralrkLa kuiLanr a1 lnmas.rndan bi 1 gi sayar werilmi9tir. f r ekans pr ogr aml na Okllz mrnda ileri ai 't. 55 M o d e l^ e- = d r' r ] e r i n i I rD' r- o =dc.im T A T . J A I i T 5I A Y I 3 I ] . L E I I ( E I l L I I ( O R A N I t I E f J E R L g f I{ T A T I A I ( A } 1 A LI I I T . I I ( L A R I ( I 1 ) ^* Al.ICI-VtiRICI ^AAAA^rr^ crklr- ? 5r rreQi . IJST TATjAI(A\]INIL;TKEIIt.IGI 100,00000 0.000 t t U T U N P A f { A f 1 E ( R I l l . U RA L T f i l N ll:,AXLlril Iar:, (nr)= k u L l a n r . l -a n hazr r 1 anmasr nda 0.0100 I .00000 I 00.000 UStL Ir0ttRU ^^ 300.o H Z / H R A t l f ' L t T U 0 . F A Z t rS 6 E R L E R I ___!11: :i1lit lo.0 I:.0 15.0 :5.0 f,0.0 35.0 40.0 50.0 60.0 7O.O B O. 0 90 .0 100.0 150.0 t00.0 300.0 350-0 400.0 :r00.0 600.o 700.0 300.0 900.0 1000.0 1t00.0 1:00 .0 -000.0 t500.0 :1000.0 3500,0 4000.0 5000.0 0000.0 7000.0 f )0 0 0 . 0 9000.0 I O000.0 t;r000.0 I 5000.0 30000,0 :15000.0 300c0.0 : ; ' , 0 ( ) i.)( \ 400')0.0 l:rc)I -i4'J 145:t.07! I ::,).i.4ri 100(r-5f14 9I B.0rl:l B:0.7I9 '/)5.7 /5 649./47 60t .549 563.698 459-441 410.936 3 5 5 . U BI 3I8.3I0 21o.576 269.0: I ?51.64r, _1 _0.188 0.:t06 0 . l:t1 0 - ?.67 0.:l9fl 0.3:16 0 . :l:,3 0.:177 0. -1?t , . LO6 .788 . iifr:i I4i. trlB I ' . . : ' ..Jr 5 0 tI2.J40 I 00.6:9 9l .8nt B5.073 1.061'; -J - 30',) . 1 ./ ; l : , r ) :1.93C) 3. ilrrS . l''/l) 71..t77 6^ -')/6 60. l:;c, 5:J.0b3 :i0.3Ut 43.946 4l - 0,.15 35,5"0 :]l.t.lr)3 :t9.0(,0 :l('.')0:i 2it , li'/ 68.4 6i -:) '/1.7 73.C, '/:' .O 76-3 7t.o 77.fi '/4.8 rio.4 Bt.3 01.l 83. 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O 90. o I00 .0 I50 .0 ? o o .o ?50.0 3 0 0 .o 3 5 0 .O 400.o 500.0 600.o 700.0 800.0 900.0 looo.o 1200.0 I 500.0 t000.0 2500. o 3000.o 3500.0 4000.0 5 0 0 0. 0 6 0 0 0 .o 7 0 0 0 .o 0 0 0 0 .o 9 0 0 0. 0 I0000.0 I 2000.0 15000.o 20000.0 :5000 .0 30000.0 33000.0 40000.0 12?.7 113.9 1:5.O l2:;.6 l:15.3 t24.7 I 24.0 l 1 t 3- I l:t .4 ' ll9.B llB.4 ll7 .2 Il6.r tl5.r 113,5 111.6 109.4 107.9 lOfr.7 1O5.7 105.0 IO3.7 102.8 lo?.0 101.3 IOO.6 100.I 99.0 9 7. 4 94.7 91.7 88.4 8{.8 80.8 72.3 64.0 55.8 45.2 4l .3 35.3 25.7 L7.O ll.6 9.9 e.7 a a 6. t 0, 0:,6 0.043 0.0',15 -0.00I -0. oil? -0. O3A -0. 05l "0.0(lI -o.o'/7 -o.o3fJ -0.0i)5 -0. IoI -0.106 -0.109 -0.115 -0.I21 -0. r30 -0. 137 -0. I {5 -O.l:r4 -0.162 -0.100 -0.199 -0.?lB *0.23S -0.238 -0.279 -0.3?3 -0 .391 -O.5r5 '.0,6s3 -O. fl08 -0.987 -1.197 -1.'/66 -?.723 -r.739 -1 I .913 50. ?fi8 9.737 4.955 4.794 21.791 -60.46{ 95.897 -q q21 37.50? Atlr)? IH':/IT: ',.16.6 40.0 44. I 50. I 53.r1 36.7 0.9 0.8 o.7 0.6 frll.0 64,0 f,(r. 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(,fJGO I .i:l, l]BB I l l ' J- ' ) 5 0 100.6:i9 ,l .8f19 85.073 ?9.i'/B 61.9/6 6C. lli(r .0050 l.Ofr4'l ) -ll0It(t :.1 .9RO4 3.:648 3.769? 4 . : 11 4 8 1-')470 53.0:;3 5.9t 00 I I .09:; 15 . 5.J0 - I 7 . f , 0 0I B. 4393 ! 1 l a 39.0fi0 :5 . 90:; ::.If,7 ll.lii05 t I .91]03 il ) ).Ft. Aflt(ffl1./ilR) 10.0 l't.0 15.0 20.o :-1.097 :.331 1 1 0. 0 :t (.ilr,./|A> ih7. -.4'^r..t9G -'11.()ll -,.]{.J.603 -34.87G 1 .q . l -'t':r 40.0 30 .0 rio.0 70-(, 00.0 ?o. o 100.o I:t0.0 l:i0.0 '.100.0 il:i0. o 300-0 350.0 400.0 :t00.0 600.0 '/oo.o o00.0 900, o t ooo.0 t:00 - 0 ll,O0-O i . 1 0 0 0 .o :.::i00.0 3000.0 3500.0 4000.0 :;000.0 (rO00.0 7000.0 rlo00.0 ..)o00.0 l0()00.0 1:r000.0 l:;000. 0 20000.0 ?i';000.0 :t0000.0 3:;000. o 40000.0 *:)Q.773 I ,)a I .985 .896 . tl l.3 . (r9{ -.att: L')'l - : 1 4 . 0 r ,I ":t! -21'/ -l I .69r) -.'rl\ - oaa 1 t1 1"a- -18.148 - | - ) .360 .341 . J05 -:t6B I .l:13 t.103 ' | t ) - L6.77A -)(t.641 -1b./iu ^.ti).98:l '-17 . 63:) *l$_4f,4 -1'J.366 . - ' , : o. J 4 7 -1i.346 l - 1{t0 r . 13.) 1.084 l.o0i 0.930 0.863 o.Bl3 0.765 0.634 0.615 0. lli:9 Q- 4 { , 7 0.4:t6 O.3i:,1 0.27.) 0.:t33 0. 2lt2 o.2:.t4 0.2tf 0.314 '-'.1 / . o37 -.J{.:rb:r -:]7.436 "40.01i -'ra.J1b -46.0f 0 -4.J. I {t(l -51't.c)3tl 'it 4 . ?t:tj ' J / . / L L " 3 { l. 3 U J -4:i.031. - : 1 , ) .6 6 0 -j/ -11'6 * : . 1 4 .l G 6 A'r\AAAA^^A^A*^^Il^t^iA1*lA^i*^*A^A*^AAAAi**A*A*AAl*^l**^*AAt*A*A^Ailt,tAAAAAlA^^ 59 5 l.rl,..tj..' t: l llil il*lltrlrj.:.i'trr:l;r.J br r r rr.-:i ve kal r nl r dl 1 ' re k a n s o-lacaQr aSa,irdaki aqrkLrr f rekarrs aral Ru karakt.errrttirr, e0rr lerinin rleLkenl -rl r cr -verici vF CD) s'.)rr(laJ1 l.;ri.>akanrr) ikinc.i 1OHz"-4OkHz (or,o.), -i$r a r a : ; r n . l a k -j uzakl r Qa paramei.rerle;r.: ba$Ir =1 r,)O, r:Ot'iIe>ri homojerr lr' F'=3OOm, kesi5me daeki g.t bi oI clr-rQu w,a 2 R,/D=1 viF:ndiqlncje eqrrlerle de$i Si mi ni nasagr ol dugu o. oo1 O. OOOJ (.r t o 2l t'l ri. fr-) , g()r€' g()r ul mu9Lur . R/D=2 =600 llz I - 103 =6000 Hz- l - , ^{4_ ' I =.1.O() , r l , . r r ' J l . ; , it n r t r t t t e{rr1€.r'ir) or'a nokLalarrnrn oJ;rrak 2. .r-r t, cie'Qi5LrQr rr =O i)O1 , t ) . ( . ) \ ) {r 1 rr ^') t:rl .tr;rL :rlrillt_r ,flr.lri I , . l r . . r t l ri , , i m l t l l 1 1 r ' ) q t - | I 1 r r ' e't.kt :,t v€J (gek r I I VF: 5. 1 1()2 o. 01 ,lti degi Si m1 S€ki r vt: rl r:I.1rt{r.t R,'D-i R,'D=1 tl CR) olarak eQr i. I er.i ni n r Qr rrcia mode1 2t (.v Labaka 5eki I de.lj r. <t /c, mod.:l P,rr;rltt-rl tt:lc:titrt- I rr,,''l''trrttts':.i : DMf) b,r$Jr f.l,,i. ! l'rr'Fl r.trllrr',.1.tr . '-.:. 1 '-=-.-0. ot 1a -i-' -r-i-:-r-:;f---T-;-ir:rx--rrr-nrnT---r l,l S' 2e7 k i 1 5 . 1 q /o =1 OO, 102 ic', r r-e R=3OO m. R/D=7 i .-r-nr-. ic' we 1(,--r5 ?,, ct I cleQi 5k etr. .'i., l l. I I ,\ -il lJ -\ . .l \1._ S€'kj I 5. a" c r . / { r - = - 1C ) ( ) , R , = 3 O O r n . R z - l ) . . l , 2 , 4 , g , 1 n , ; zaf_-l <., tletl:5k,:lr 5.2.a,b SekiI v€i c derQe.r'i ncle aynr qakr gr5rr-il rir 5t r {r paramet.r'erl ol arak q t.ut-trl <iu{u deeerl eri Egrr 30(] 5 tJU() 40(] .L 100c) Gior) q hazrrlandr$r -werici kendi sa{a nokLal.arrnda e$ri o =O. 01 , 1ztl oranr egrilerrin de{i grnr+digi deQi gr mr ni n I erin o /o Sek i I 5. 4 =4,2O,1OO, honrc'j c-n aga{: dak r raQmerr ayn-r il*-lie;l-kenl erJr 1 r .i i':, rnoderl CR)! rn j l *:k rrr I'r gi bi mrzda 5. 3. a' rlak i birbi absis ri ile absi s d.r cakrsml.gt-rr. yapr i ncel I an Rr'D oranrl)a 5me emede sairip rrr:k t-.al ar.1 nr rl kor"rLr-asLr l: esi al ,rn1f) olarak kesiSme ik eQri e(Jrrler ska.las.rnda R.r'D=Z,4,8 gr ubunda, egr i gd:r'tilmUSLUT'. t:r:]rll saQl andr {r Rzfo^ de{erler"i e:Qri homojen ni n we c frck:rrrs Qak r Sman:- n de{iSmesine R./f) uzaklrk kayd: rdr{r sola gdlr u1 ur . haz:. r.I anan aras.l,ndaki aral,arr.nda yada .i I tt b,i r-L:'j r -i rJ-r , 5.3.a,b Sekil i ncel- enmi sLi r . et.k i si I 1, 60() aLrcr Flf'r.r. No 50 gakr 5Lr {r 2a D d edril,=rin o,/o ve al r ndr 150 uzer i ndek i boyunca R o'l arak 300 alrnrp grubunda : - a l - r j F , e r d r i - 1e l j r - r D 1C)0 ayr)l "rrl, 5. e) . =1 OO /o K olarak er-e CSe'kil 6l =O. (;1 , a'21 Rzyl=2 inc*rler:clitlirrcje,eSrilt+re: ttr.. <Lr' i t-t ar' .r n.l nok t.;-:..1. f:.'., I v t lO 75 300 l 100 400 2 150 600 3 200 800 4 250 1000 5 t_ -r- ! 10 \\ N \\ \\\ .1 \ Tr-\ f 1 t_ -1' \- 10 t 'lo' Fre t=/"r=100, 1O t_ -t- -r- 10' 1oj R/D=4, c'.=0-01, R ve D degisken. 2 .10 o. N r /o LLI tr .-{ \< 10 2 10 J Fre 10' 'lot 5. 3. R,'D=Z 10' Fre . R/D=2 (300/rs0) S6k i I 1ot o l a t : a . k 5 e = 1 i, I d e r R w e , R/n=2 (r00l50), I) deQer'.1 r:r"i de-:tlj 5kerr l:a] € - , t . k1 I e ; m e C l rg i egr i lerin ayrrmlrl:-{r y()nde arLrr:.cr bir' z/ t7 a r t i . t k r -. r artl rnr ve b < > J . ri - 9 i i r Ko6t.rast. art't.r kg:a I , 1i t l r ttzak J a5ma:'r r.Llar;ik bel i rgi n eQriden homojen ayrl g d z l e r : r n i 5 i , . rr " gost.erdiEi kr r*r lma bir 0 <laha ,ja homoje;rr e$riden erin e$ri-l yanr nil;:" bunUn ama yapacagl et-ki kolayirk-la sr5ylenebilir. Sonug incelenmesi ile gu j let.kenli{i ve a.lrcr e{ri l erin f rekar:s kaymasrna r. ol ar ak ordinatLa lHz -verjci absis asrpda l ki o /o ve 2{ ,/H l, r' model R/D br: ya,:ia sr:nuql s,:l a dro ara a.I r nar ak ESAS -? .^. k Icl de tir=(;i;i rrri karakt.t=rj e$r'i I er'.i oranlar"r absisde saqa genel ar- r rl.l. 11 l-abak;trll Ir rrJrn bc2yuncd Egrilerin Labaka gr tipl R1rinci arasrnciaki,rzakl olmakLadrr. r. eQr r dak i varrlmrpt,rr. kanrya skal sebep clegr gmemekLedi dayal yukar: o " La r a k 5t'rI..l i ncJ*: 1 l r a z r r ' - la n a c a k L r r . lo' 10 l_ N T oC L .-:: '\--v--; ti.) ' !l ':i) I i_) lli-' $bk: I FA J- +. o_..O. o1 , t2t e$rilerjn (.t /'(j -4,?(,t, tromojen 1OC,' €:qr.iyle v{:r {-- F . . I - _ 'i.. , 4 , -) ri rlt.,iti.:rI i hirL.+Srne" r.rr:>kl.al;rr-r ti r+ 6 . 1 [ { 1 T A t i r t , l AI ' 1 o D H LE , : { r 1 L E F : ] t+QriJerinin s()r)(.r,:ttrrti.tnrr.rlel an1 a5l gtlre na kont.rasi-: I ml 5't-r r . sr->r-rdai.r CDMD FS) -}r.trar'.rk R,'D hazr iki <>rarr.r v? r'l anmasl t.arbal.-.r rnci,l;:1. e{rilo+l ,i jk rrl 'rr:a{t ' : 1i 1 ' r r r l f t-ekans tis;i:i i ;. Tai-.,it' modelt: Haztr'lanan lrazrl-laprnrgLrr iir+t'k{it1l tlyL]r.rr) nl n i. k dtr5ey i r t ' 1 ' 1 6 ' ' 1 1 1 ri>' :ir cl?"cr1 manyei gc)re Rr-tna ' " r . r 1 - t tt r ' t t r l t t . ' r b61 ulrtr.ir-, c5ncek i Ri I EK-A'da Erllerj verjlmigt..ir. al r nr p deQi.pke--n ol,ar ak ar t. r:rt.ama f abakalr ..5rnerk t-i.pik En egrilerini gOsl..ermekt,edir. qizgi R/'D ile, ciriz eEr'r gruburrriar-r da l'romoj <;n L kesi5rnekfe<lj ve de$eri o1 maktadr r. Ru da ke>si 5me nokLasr e:cl1r'i si deri R"D deQeri nclaki Yirksek ef kisin,:len j 1*+ i>.i r'1.:$tt|i.5 {:l rn-;r'it rta biiyuk kair -rl l t-aki eQri-l r;r('lttrril-r F!,,'f) e:r- ar da ai.-,sis nr">kt.asl ve. ai.'=i s (frek.rtrsLarcla dol.ayr 1 5 n e m s i ? 1 + < , . , ,' r > , 1 i r r r - l c > r r t r u t . U 1 1 ,a-{riierr ke,sJ"5ttre usL't.aba.kar)l.n rnc>ciel n':rklai rg-ir: i ecjilr+n Bu r. Labaka. rJe$eri orcij.nal- pozi"syr:nundarr deger lerr j nde f ark.1 r kuqiik g6sLerir'. elcler il.e daha bulunabil+cegini sr biiyuk kersikl orl-.ama aj.t. eEl-i i ki Daha '-;rrr,;b':-) (Mar; sorrtlajr uzere rler. defer'lerinln eLki egr i si c l r ( I i - : t . l i l <I eQr i gi-rst-.erj lmi5lercii gr)r-ril er:r>rJi {t\r',in1 R,,f) deSiStirilmesiyl.e r;'i z.rgiler Ie; er t/anr Honrojen rJeger.inin er{ri) ise f rr:k;rns DMD moclel VI . haz-r r I arran -.1 OO , (, /(r zt ol ar ak , c ' r r c l ir r a t , rrl.rqj t-lrn ) al..,sis t."rL:akanrt) h o n r o _ l, . > l ' r V a r r r t . - ,-it il"! j Ir.t Ir 55 Qi.zelge 6 , l. =O. 01 a parametreleri. Model R=3Ot)m mho-'m T O V'rdel Rz'D ,'.j 2r l , ' 3 , ' ? - ) ,2 , 3 , 4 ' 6 , B ' 1 6 r ,3/2 ,2,3, :.,3/2,2,3,4,6,8,16 eo L,3/2,2,3,4,4,4,16 5() L,3/2,2,3,4,6,4,16 t ,2, 4, l O, Ztl, 50, 1OO AT 4 , 6 , A, 10 1/1 lOO F-.a. i No II III IV L ,3/?, 2 , 3 , 4 , 6 , B , 16 6 t,fT B - VII L,2,4, rO,2(*r,50,100 VIII 1 , 2 , 4 , r .O , 2 ( ] , 5 0 , 1 0 0 !X L ,e,4,1O,20,50,1OO () L , 2, 4, 10 , 20 , 50 , 10c) 4 t,e,4,10,40,50,1oo t , 2 , 4 , I ( : ) ,a ( - ) :, j O , f O O A1 XII L /:t XI I I e,a RzD=1 b 'TrJksek irek.r11'-l;;'ri;r eqr i oldugu ile b:irleSmemjS; qOk i nr;e c l r r r l r m r l a r . l r r - . R t r r t r t n s r + l - t e . l l i .1 . . l . r . r l . ; ' C t Jr l t r d r l n r l a n r : - l r .;:ar .yl : ; r - tt . i . a t t gI-tlbu 6ne+ml i c , : .a1t - a k , homojen egriyl n:l.n :;:iciec,e t;.i aiL egri)er Ol ar ak: e aslrnda ke'nl j k l-.:rl:,:ikarrr tr r:r-:i (Sinha e$r'I I E'r€) benz€:r -J poZi IJMD f'rekans brl iki kr-:'ni.ras{ arasrncla (]r-.'k e$r 'i 1(+ri rr p ol tnzt:ri' r tra l ba(ii , cler i rlJ i rJi ne Lta'31 -! Fek i I rle t.at-rak':el : k - e ' : ; i5 t n ' . " ' : ' - 1I n a } ' , l p t-tl rJr-1$r;nt; L979). Labakalar-tn }r:rti;-r sahi kesr 5niekLe nokt.;r.rla ) r r r p l r 1 r y 1 t r p r t 1j r I e t cJ,5st..erir1er. ,le nrlmaraf VI I O. 1-1 OO c;.ar)l na oz/'(7r ent i I A7g) ' rrila a f . rn a n orarrlrrln crr/o, f arkl, r er a-yI)r gosLermekLedir Bu R.'D=?, el e i:lt { . . r L : , . 1 . i , ,na. t ( . : ' r l € ',i, r r i l r - j l r . : ' r ' i r r i ' l ' r s t t : r ' t n e k t ' t ' d i r ' c i r : - . C 1 j 5 t - . i t 3i k i nc.kt-ASl da CSrnek oI arak l ki nci lalrrrlr')r t ..i:,.ii..rlti tr' l,lrrtrI,lF'rr u z : e l - i r i c l e , [ i 4 l ; r e t , L i ] * : r -e : ; , j , , 'r1. li l r o , 1 r n : r : -t , l r r m r > r - i ( ) ]e g r j lrrrntr-rlrli] f'r(il.: d;l-rt yuzeyden syonda sondajr sclnl.rc.a gc5re c:]an 3,4 der.inijklerrinr+ hr-:mC'jr-:r'r '-''1r' i r:if?r) v€iril"e,r-j.t-rir', yaprIrr fSinha {)r Lama ba$)r k: r r I ma yr-.rlrlrll i-,+l<.nitll-t. 1979-t' 7 , , n H D f - S V n E : i . l . . f R i F lNi Y n l i l J I ' 1 L J fjti, liiina T ' c t b " r [a, l:i 7,1.11.i Y , : : , I1 ; t D e Q e r 1 e n d i r i l r l e s i jeof Di,Jer r::l drJQu y()rLlmunda ara?-j yaprllabrl-ir t.eor-ik CSini'ra mc:d€iJ- €:gI'lf (Jf ar'a,k , d.'y'nl 1-rekans olacak ba5] C anql lroyunca r:trarf j.k d':' t:rjrrirr nat. lizt=rine kayclrrr.l nl<rdel. e$n,i-ni n 1r. saO deder.l Qak r ;;rna sarji apdr k l-arr \()n!.a bir usf t.eki t-abakanr n e g € + r t . > rt-. a k bir ise frkans X deQerine, i l et.kenl j ,lt+ iS.lernden a.razr F bi r sd<: rtrorle-L e{ri 5,."k'r eQrisrvJe e a k l S . rn c - a . y a I . . a , : i a r l arr ska1alrl:r sonra' a[):^i s t-eorik rtr . surdtil-itl r'rc:i.-l,-acla R2{ ,t , I saltip devJelirre si ar azi de(Ji pt.r ri I,=r'r.'k) er -i Kayd.r-rma rS1€iml t.araf van.1 L Flu yerIeSLirj.irr. otlr isi absi | . lll_r'ti, qak r 5acak eri deder] C a 1 : , si s eEr r r)l-'arrr iionio jen Cizllir. t'rl''.)hrr:-) c - : r r - r r : ; -a I ^ ? 1 . 1r - , i 1 a r . r l . . r n j i . +1zi-trli(r llen.l i k (-Maxr .l r - l t , . r l l r .l:r,,l,.t r , j i t - t I t n t - s i y,-rl I a Rtr /r5ni..etnrl-:, 1S7g). s.=kllde sc)n,la.j f t-u+I.:;itr= DMD wi-'r iI t.:I irttl) -,'t'a..'-.t me*t.ocllat ciak.i i : r r . c - l ir i a i - . r , l i r . crrdl egrrsjnin iromoj t-n r.r tlr ..1f i. 1,. [,]lr i l. u'r' i tr i ir f S DMn : izik gj bi vL5r'-il.r--rinj Ai.i; ise i di , X o=i ii r" haClt nt.t:t a f t , - . r -' v t r *2KF ;rr asl . i L -i lri i j rrm,.'k t etli | 1 + ' < i r f . l ; t r ; ' t i . j I j r' r. Ilrt r r , l " i n l .i ('l-1i r'rlr;r. r - t ; : . , . 1I r i . i , r t ' r ) t : . , 1 r - ) r i 1 r , :( / , ve i . ' t - : la v r : : a i al7(l^) . at i\';'..i F i ri r a , J a . k' v,r't'i I t't'1 rrrlc,t, i ' , . t 1r l r r - n i , i 1 j t ' . ilk Br.r r;,Jr-,:ml{,irr'rl{:'n \c:rir'.-r ,.,r'cli rr;rl t.-;ir al') lre.l-ap,i nrl.rrl erlt;it-t , - - 1 t r 'lir : r i (A us|Leki t,.rl:.rkar)ln rr;r i k i G r ' r . t i . r rm l r)del Lab;rkar)ln .:lt'arlr r't-"'-t j ii, l"' .al,.,i ,., w:rl;i r .l e L k e n l i i : i " 2 uyt?(rh kultanrl-1-t . i1e Alarzi el de trrt'. jq:l i r-r-rt i l: ciI'rsi rrrl.:r't R it;in eLrnek 1l:ir-rt;i R.'I) is;e. eQr .i I er tn<>,.-i*:J Gr- ubr.r R rrtl:r < ; ; a l r, 5 i , , t r t J m a s l r:Qril.erinin mocl,al ..r-V.il i'res;r1:Ianabj I ir. CD) :iec i I en o.1 ar:rk tl()kt,.-r Fil,-ti kull.arrrlarak rQr cr'.yi r tJr f..aDl.:r .r,i I l , a b ; r l . : . : . r l. t erJrilerr) kal rnl e.i I ',r,i ii-^t rr.lrlr-tr]rl l)lli k.ttllin;1 brrl rrnal:i. l.i r'. ct deCler.i 2- i k i .l+r-r .:r1.a.nr E,3er' ':'ii rJr:r i 51e'ml r..l :,ljrcjr,rr rtl +rt:i. ( r:'zcji re..'tr,;lr i l Ert.kerrl i k 'f':lzl La,bal. aya a :;ahr ;ii t t.al:'ak al ar a k a] r n,J,.r I'. v€: ( .:>i flfra I ir hesaplanahi de,]erleri ; 1v l - r r i s':' 1-, l!=J/gr. 7 . ! . t r r r r e f , :B i r De,lerlendi rme; Ar a.z-j la U t n kar5r z s.>r'rria.jr rl e{r aclklandlgr 7. 1-b) sa$ i.araf r i.araflarlnrn X=2. 7OO, OOO li -. ':i . tIt r .J il ^r t,l -r {J ;f . * . i ^t t ^Y j.sini n gibi yapllal-.;iIir. eSris.i CakJ St nca)/a qnkr ve 5'LtrJ1 el bj l' itzer Homc,j'-6 absj incje s CS.3kil 7. t*c) i J 5lj-Iif rrg! iylr+rn,.ie.lt-r ;(trl)r'-r i.l k ft'":katr:; r5nq:eki l,'i!l r-imde e{risi cf*:.i.i 1 j i'--r itr b , : : , ' y ' , . . r t ' t ' : : ,( ' : ' 1 r k - . r , w c ltr' I 1 t r ' . - . / , r f 6 _ . r . ; , ( .l e , . J i , . l i _ i . r:.i: :r.t-'r:ra. DMD mr:d+ll kafiar) brrl rrnr:r. vF:' .r.L a rtart e 7-1*a) Csekil F = : i . C ) O C )o ] a r a k r.t( Br.l ak de9.:r-lenclirmes.i , It6-s.1F,lar-rtt-. qizilerr: I o1 arak br.t ar;rzj o1 ar c)r negi L r ( l f . 1s . t Ii{ri c*'l-tak I er j 1: b ' - lr "4r-1 1 - t q l! . a r : l ; : F?-9t-). i-)(-)a) rr'l ,itl,,rri Or.nrr.j},.le,r.r ry 'i-r. r)l t . ' r i. r I : L L krr-r.t nr;r'rt1!,..'iqltlr-iq;il ttrrii.rr t:l 10 : 2 i I r rJrrrrl 10r I TTlrrrq 10' I1 IiTlrrl tn IL' 10t rn ru J <n,l lu X R'Ig :.]*,'k,il 7.1 r Ir iir:t;r,, jr-11 r-r)t i i el 2 10 10, 5 9OOx tfl(rr'lFl '!n 4 10 t -t,1 . r t r . l s - 1 " :y); : t ( - l ; i u l ' ( . 1 . 1 nat.Cla 'i. (.;irrll i l (rr':tr)r r)!l.i i1r.: g.rkrSLrrll.m;rst : s ; i t J ] . r r r < 1 r$. r r r c i a n D=1 OO r n r > 1a r a k edriler- CSekil . lresapl ve anl 7.?--b) A irrr rri rl F(')t'llFI "'r'tr t lc-r rJr':t) ("iai..r5trra yapt1rt. Rr'D=3 <rl ;rn krlllarr:-larak q'1 ? F:.'lt'3 i L i j t l r , l l . it ) i t ' r . l t ' t - 1 ol du{tl R=3OOrn r . nti',rle'I tllrli;r.t . . z r ' . ' . ' 1 . tr - { J t ' t : j kr.rl I ar.r:-Iaral-- ?.-a) uvr-lrrr'rol;rrrr l:jcrr',il l+fllf (f r'rlbr t If mh(-'''rll mr-rrJt-*l ol'rrak l:ul trr-irrr' 10, -rL. -r--10' N I o_ tu C 1 10 t 10 R'for 3 10 10' "+ r0 ? 10 2 io ;**:i i i -; . ;.. A--(lt iti.rtr lttrrrit-:l 1n J rt vn r i I ,:r , a irnv 5 t,'-" i-j Gt' rrl:r-r , 4 , 1 1i. I r . r r . rnrrile'] 7t l'r,''r' -/arrn J{-ll\"lLa\,.i Il\Ll.1 p;tr trr*-,1.r1 t J1a. r - l a i i l . r t r r t r trlr.irrrfn l I i*' v€r uzal: i;i,", l..rlt ori.;rn,la lrir Cize1r;r3 'r.l ,rttrlj.4i f-r,i.-. -k,i'. e + l r r j I t . : r i I r r| p .ivrInli.rLrL1l r-rl-<rlrr clr.'dqi;;i m.i nr n egrrlt:t-ilide )set gc)r.e i_y€r br:,).jrc1inlegi-l{r orarrj ar i.t.rkr,a r : I r -r t i r r r r r r l a $ek i .l cle H::d'i '.i. f[';]' trra i''i:'ri'zelrrI) goru-l oldudu oran1 }::aQlr A r + q t 'j j F r , r - i n r. arl j 1 *cpr1i s$vl af-aSi enerbi 1 1 1 'r, i r ' , I i r t ' t l l r ( ) r il r 'i.' c:l ml:.i.t ", r \ . r 7 . - zIk a 5 t . t . ' i 1t el,ll i rlttr irntln{j;r ' r:(Ja rrl li rri',,-a ha';''.a< ,l i, t- ' rr! iri',r n:'"lr i,'l r-r I irr:lt-- A-VT lIi 'ii ,-tL.-'r,l iL rl;ir; aI'ir) 6ririr-trJu .virk sek tll .:lr.ll'i-ttlt,:i;. ill- IJf)r-I l-;irL:il' j tr(ir,tt orror='lOC'i i 1 el.kr.r-r'l i 1,, kc-:n1..r-asl.rr)1n m € . k i - . ? . j 1r . I . l a.y, Irn.l III,'rr1f prlr-l''l trilliar';ik A €5gi'iir-:t irr t-rkCa ()larak I - - ' ft . - ' l t ()l (..il;rJu B krr-rllna] qll-Lrbri i . e . l r r r l . I t . I . r l t t " t rt - ' r F : r l t = ? - | i - rn.rdp-l r>{ri1r"'r'i il.',:''1l ;'."4 i nc e,l er)cii eil rzr: L;una hesa.Fr,l anab I rr-ir -l !tC)sLq+r'dj.k1eri gbri.rlmekit-d,l ar;rkqa tal:.rk., csa,'ttr c I O r - ( t l n t . - . i :l.- * : c l i l ' . ali.l., I {.i fr .t r l:4/:-r., l)lt(t i. |-r;,zrrlAtr.lrr ()t'a.Irl rtl r) Rz D L , ' - r n i . - r- r : . 1 t eqr ..) l){.'la} ar-a\) t - . r la r - a l ' 1 , . r r ' . rniii..r.:rnrnel rr i vr:r lor,'<t^) 6:q.es alrn.rr-ain fi. i ,t I, , ' " i . 1 1 . ' .11" . , 1 i . . . l l r i l l t 1 rlt,l"it'r l.-t,:nl1'ast.lna j I *,1-i.*t rl j tli nt.r ir;r,'1) I rncrcir,ll ayrrrnl \/f-'1"l.'t etJj 1 nrei. t.eil r r ol .a kolri.t :,^ .t, . "' .r.lt" kc,nlrr|t,r, ilo1 1-I ei<arr-l iJt'rt.i.jrrl.jk tlti ! ) c , . t t ( + 1 . 1 ' 4 i : i y ( r l r 1 . li - i l - F yprln i$.i rl-) t.':.* .\r-.'Vr Tt tl I 11.,Lr ll-^L** 8. Dl'1D Fii:Ei:.r\l'15 l 'l I 72 ti. 1 l.i-'t:i 1 l.al:i;rka t-'r: r ki ,ira:.;.lrldaki uzakllk qc>s:;t.errrrrek{...+rli r'- i sr:: :ryf i rnl I I I k 1a kri(,iik F,f) r>r-:,r,r . : ij l , - l , : .I 1 1 1 1 1 r 1 . 11 . i r . t.t i . : , r - r ; ' r tnl r a l I a d t 2tli..l-r k't.;,tt}. :.1 t^\, ! i . r i r - ; r r 1- vFaI't'^l " , ' r J rt l c ' t t l r r r lrr.- ' l ir'jrrr', i,irJr;t'Jri e t ' 1i l r r r i i i l l |rrtyrrl q , l lI n t r : : . . 1( l u r i . . t l T t ( I J t r j . i , , , : ; L i ; - r <:rar)1.n1 Ru'I-l f)::Ililttt '', - li-: ,,, v'.. h:r.t'.lrl.:,rr.rt) qrubLr ayrlmIrl.lqln 6?r.larr t-: i.il , - l e - : f l . i : .i ,t i l ' . : t . , : l CRJ ijr.r e+rJli rnc*rlendi,]rnd€i clr.rrurncia vt: qrttlrrl l r r . r . - l l€ f e $ r i fJ. 1:). CSekjl ir, , : i r - , - 1 , ' \ ' ' 1 . t '1l i . ' L rr'l ry1;.,,;1 litJrtrlr.l ( ) l , 1 1 - .{1J l r l l ill.lt-rltrttlli'ia t-. z -a , \ f a ' , l l _ . . r0 - ? r'^' J t'J T 1 l_:*,1". r 'l IJ, I (-/ . Cl i-t1 nrltrt. t?1 r rl .;1; ;;.p r-f,,-. , llt , t - t ^ . - -\-; ,| /'(r a I j )l.t i-, ';(), f li-';. eri t I ,.;t-, s'l ,:li-. ()(). i,li,r t . , ( j , -1 R i r . - 1 r..' r i : ; i r r . t r € : ' r = lri ,' ' 1 r . i t l - : r i . ii()t tnr'r 1 '. lr i " 73 t_i. i i ,]Je li,r]ri Jr Or i.arn.larcl:'r [)rr5r,rk I i et.teni Met.r:rcJtl|r Penet.r ;1:'vr')r)(t: i".ahakanr r) Llst rJegeri ner bir ve c'l dr-lQrr sahrfJ i-abakalr t claha r)nce duSuk t.ai-rak anr n A. 2) o.ro e+cli1 en A- V; model r=1 egr ir:;t. i rrc--. sahi;r ol citrk$a gug yapt.r kl ar t nokLaL arcla Schl umber ger DMD enebi i r r . FS hem de sonda j r ncla i^) inc:e s c ' n d a ' i1 a r r n d : r FS ,-1r-r:;;irk saplad: r.tml:erger Labakayr saplayamaml i l et. k r:rrl vltl* -;€ti< aYn.l 9 i . r r ' ' - l ' rr . i rle'' nl t-l a 1 3ij7) sahadar rir rSr:! Ca] r s.mal ar k.i ar t ori..atrl;it',:l"r fji ulrltr r'e araS't-r rrlr kl.rrr Schl erl rle' q d , ? - . r l n i lc l n m e < l i I tl Munclr',2 Br-t trs{- qt- tlpl al. I l)l r) s:r,hi 1, qc.>k ve t-rMir ar,r ri yapmr sl ar dr r . iteLkenl-ik.i DMD f r eakans sd;rl rlj .;al I 5malarrnda arazi sonciaS I ar r CduSiik t,abaka.l o] acalJl hem v€-'reC€,qi 'is1 *ai'r.i 1r gr)r'u1 rrrii5t iir" . iletkenliEe dUSrtk c;(lr i1,:l' jrr clr:Q:' 51- l" r r- i er ek j g i benzed D, nt tr:' Y^Jrll,Jr'\;trtl.r-' t - r - j t -j Rtt or anr R,'D Er.r dz,liretr,-e) r . si-l gr ubrrna i ar :lq;nuCl c:I dugu OO 5o6r.1g olarak i yi annlanll r;f sl.I .1l.k i 5;.i rir: i:). yr-rr'llnrrt Cyuksek iIetkenliQe ( $ek i I vr} E..f.a -:l i - a l : : ; r k ; 1r r r r r i i : . .1 .ikir gt"k-rJt-1.- rgra.f.ik agrk.ian,l.i{,r r:ldrlkCa meLc';ciun ($el.'i j jmr:;{.rr ara5t.trl all ol aral tn ilelkr,'rrl clugrlk kalrnlrQr,CeSigfir-ifr-:rek et.kisi f).=1 f) uzerinde, orLam r:l rlrtCrrr OOOO, i OO, I o*/t|^=I ayrrl el ,Ekt- r- t k sc)rlrJ{^ t)nr);) ozrii r errf I i r r r - , i ' 'ai i ; r r c i a k j 74 .-.-'-...-....-'--.-; i f I -J I I -1 I I r'*'lo I lJ -l I I I -l |...1 -i- I I -t - I I .'1 l l ..,] i -.1 I I I -.1 I ! I l I IC -.) -Tf -T -aJ Sekf l 8. a. ^- or=O. OOO! mho/m, R-3OO b- n , D, d e E l g k e n a.=O' OOO1 mho/m, R=1@ am/or-IOOOO,1OO,1 m , D 2 m, C-37. 5,50, fr,7.O0,15O,aOO, a^/o^-LAQQO, dcgf gken ; Dr=1 O 1OO,t; D.=5 m, C -33, 50, 75, 1OO m). 3OO m). 75 9. DT''DFS MODEL EGRILERINTN SC}ILT'MBERGER SO}.IDAJI lrr]nFl EGRiLERi tLE KARSILASIRILMASI Bu bt5lOmde rlq Labakalr sondaJr model kargr I agLrrr efrllerlnin modellere DMD alt model FS Schlumberger egrileri iIe yaprlmr gLrr. I nasr Model A q = O. OOO1 ,, D=1()m t gpr= mho/m 1OOOO ftn) o.=O. o7. Cp"=1OO) 4 Ir,lode} p= 2 - 37.5 50 - 78 - o"= 1 6p"= l'lodel A sondaJr modell e.erllerl yansrtan rrerllmlsLi. Sektr 100 - 150-aoo-300 1) esas arrna.ra.k g. 1 sekir DldD FS g.1 rastranmayan yoksek R./Dr=3o olan) 1. Egrl de hazrrranan g6stertrmcktedlr. egrlleri de lncerendlglnde 6zdlrenqrt Labakanrn etkrsl r*$r i I er I nde agr k qa 96r ttl mekLedi r . schlumberge.r $ekll DMD Fs c1o m Ayn:. g.2. a da egrltertnde karrnlrgrndakl schrum.berger sondaJr No 76 10' 10! itl d 1oI a arl o Sekf I 9. 1. l'4odeJ A tsas alrnarak SondaJr nodel e$rllerl CC-- -)lkitabaka, snAB/2 o hazr rlanan Schlumborger C3 tabakal"r ortanr) egrlsl). ? () T7 f'lodrl a * O. 01 r. 1OO m 1OO Cltt) Cp.= 1 urho/n B 5432Ir 50 rn - O. 1 - O.OO5 - - o= 7. - -p2= L e 5t 10 eOO 6 5 1 3 ? 2 O.5 O.2 Model O.OOI 1OOO ---> I O. OlCp = Schlunbar6tcr sondaJr g6sterltnektedlr. DMD FS 3. tabakanrn 3. Bunun ve etklsl yanr sr'ra Scilumberger bellrglndlr. o"/or)1 sondaJr vG 9- 2. a Fektl or/or(1 lncolend-ldllnda egrllerlnde nodel e.grllarl c,grf lcrl 9. e. Dl'lD hazrrlanan alrnarak rndeL So&II g6r0lrpnrektedlr. rpdel 1OO) Gtsas B I'bdel oldu$u rrodal FS ve b dc olan a{.klsl tabakanrn No Sch.S- l.todal o"= Egrl DMD e$rllerinde F:S Egrrl No e e { 10 -' Fr rk.nr 10. b , 6 a. lor ' -g:r'A8/2t Sekf f 9.a. iiodel (a)(( e{rllerl C- -) Labaka -)horrcJen tabe.ka nodel lkl rre Schlunberger lkt hazrlanan alrnarak B esas sondrJr model ! t-A8/2 e$rllerl) DMp npdcl l''s nodel er$ri ' egrl}orl.) nodel e$rllerl (b) CC-*--) 79 Hodel D= I c r = O . O 1 mho/m 1OO m C 1Pr= I 1OO fl'n) o =O. OO1, =O. OOO5, =O. OOO25 p2 =1OOO, =2OOO, 2 =4OOO Model D_ z Egri No 10 - a5 - 50 - 100 - o.o ar= O.O1 gp"= liodel parametreler C dekl DltD FS ve Schlumberger g6sferllmektedlr. kalrnlrgr sahlp ragmen 56z konusu gd)r0lme'mekLedir. $ekil 9.4. a daki numaralr 9.4. c egrl dekl ve 1 dururndadrr C$ekll egrller IIe Cakrgnrg gakrgma $ekfl 3 numaralr egri lle 9.4. b dekl 9.4. d). ve egrilor 9.4 e. olan FS aynr farklr durumdadrr Sekll 3 numaralr model Labaka kontnast,r S.4. de tabakanrn DMD Schlumberger srra numaralr bu grup lleLkenlik Bunun yanr Sekll ve =O. O25 hazrrlanan 9.3 Sekll edllen =O. O5, blrblrl alrnarak kontrastr elde or/o^=O.L, esas efrileri tlet-kenllk kalrnlrklarrna 9.3). model degfstlrilerek egrllerlnde olmasrna 1OO) Cseklf eSrllerlnde lncslendlglnde 9.4, b efrl birbiriyle dekl lte e $ekil Cakrsmr$ s Lodal D - 111 o = O. Ot 1OO m n*loln C*1 Cp = 1OO Ctn) o =O.OO1, 2 =1000, p '2 L23 Model Egrt No D=10-50-1OO 2 a"= O. O1 qp"= 1OO) t{odel D = 1OO n o = O. Ol I I mho/n C-2 = 1OO Ctt) Cp 'i a =O. OOs 2 =aOOO, P ,2 I 2 -----) 3 Sch. Model D=10-50-1OO 2 ar= O, Ol 7p"= 1OO) Sond. Egri No 81 10 z0 L o tlt 10 0 ! f r I o c $ekll 9.3. l'lode1 C nedel eSrllerl c----)lkI esas alrnarak CC-.-.-.-) tabaka spdel hazrrlanan Dil{D FS honroJen model efrlsl). egri f32, Model C-3 cr,- O. 01 mho/m Cp = 1OO ftn) D-= 1OO m ltt o -O. OO25 2 'p2 =400(), L23 D_= 10 - 50 - 1OO 2 Model Eerl o"= O. 01 6pr= 1OO) Model D = 1@ itt a = O. 01 m o =t 22 No D mho./rn Cp = 1OO Cln) Ce =L) I234 D_= 5 - 10 - aS - 50 m ? Model Eerl No Model No I234 o= O.Ol a9 Cp= lletkenllk kalrnlrgrnl.n hazrrlanan 9- 5. DMD FS ve g6steri de altrnda yer R,zD_=6O, 2 egrllerlnde 1OO) kontrastlnln deglgken oIduSu l mektedl r . ytksek 30,1a,6 olan oldukqa sablt Model Schlumberger alan EQrl RrDr=3 D model oldugu esas 1". lIeLke.nllkIl CdUSok tabakalarln eLkllerl bellrglndlr. alrnarak egrllerl ol duSu D" gekil tabakanr n 5zdLrenqLl) DiltD FS sekll 9.4. llodel c-1 hazrrlanan ca), c-e cb), Schf urnbrger c-3 cc) sondaJl Qsas alrnarak rrodel egrllerl(d). r-A.8/2 Sekf l 9.5. l-lodel eErllerl D esas (a) Schlumberger alrnarak CC- sondaJr hazrrlanan - -) homoJen nodel DMD rrodel egrl]erl FS efrl) (b). model ve 85 Sonug olarak, sahlp ince eQrllerlnde usf rasLra.nmrgtrr. e,frl I erl nde konLrastr a.yrr ml r 1r k deglldir. sondaJr or/or)t sondaJr olup metod her lkl Dl'{D schlumberger FS da lyl vrs durum Dt'{D Fs t I etkenl AB4=5OOO ortamrarda nazaran m) . o./or)1 ordukga konusu DhtD oldukga FS ve y(rksek Bunun alrcr-verici agrrrm i k oldulu s6z vermigtir. FS sondaJr durumrarda ayrr.nlrlrk kurranrlan Lahml n edl I nrekf edl r . Dl,{D ortamlarda sonuglar sondaJrna etkisine CO"/Cr)L) bu kullanrlan tabakalr Ozdirence) azaL makta tabakalr CR=3OOn DMD Fs schtumberger oldugu or/orlL egrilerinde nda sahlp - (y0ksek schrumberger e{rllerLnde sondaJrnda k0gOkt0r ileLkenllge rnrecegl o"/orlL ol dukga oldugu Schlumberger <a5) ra0men saptanamamaktadrr Schlumberger ol dukga (R/D tabakalartn rasLrannamasrna e.prllerrnde srra dOSOk iletkenliSe yanr uzaklr{r uzaklrfrndan A)rr ca oldugu lyt driSrlk zaruln sonugrar 86 1O. SOf.ftCLAR \ e HorrcJen bl I cSenl erl tabakalr hesaplannaslnln blllnrnektedtr- DldD FS potansl)€Ilnden vektor lntegra.l Ilneer denkleml suzgeg yez:-lmrg olan g()re oranlarrna tkl tabakalr hazrnlanan nrodel yaklagr.rala )mumunun sondaJr e$rllerl sondaJr Schlumberger lletkenltse (y0ksek 6zdirence) 2-6 t sonuqlar wermlgtlr- e$rllerlndekl oldukga arasrnda oldulu OLe yandan, ayrrmlrlrk dOSOk saptanam.amaktadr olmakta r. g6rolnogtor- arazl eQrllerlnln, graflk oranr anlagrlmrgtrr. , R/D oranr ve sondaJrna or/or(.1 Schlumberger orror)1 durumunda eErllerl.ne llet-kenllk d0g0k nazaran, oldukga durumlarda ye sonucunda, ortamlarda sahlp R Schlumberger kar$rlagLrrrlmasr Dl,{D frekans R.zu-, oranr- olacagr oldufu kolay R/D ve or/o, DMD FS eQrllerlnln lle g6re yararlanrlarak dr., o"/o^ g6rolmo$tor. oldugu FS egrllerlnden DltfD FS ayrrmlrlrgrnrn bagrmlr uygun Dl,'lD oldukga dlllnde paranetrelere e$rtlerln olup, ait FORTRAN hangl hazrrlanmasrnrn ortama hesabr sayrsal yaprLmrgtrr. lle eQrilerlnln aragtrrrlnrg Blrer hesaplanmlgtrr. yararla.nrlarak program:- de blle5enlerl bllegenlerinln alan kuramr.ndan DMD FS nodel ya-ylnl ardan alan yararlanrlarak bllglsayar hazrrlanaca$r gerekll 191n kullanarak olduQu uygun oldukga yel I potansl rrckt6r EM nl n alan manyretlk lqin ortam rae hassas DlrD FS nazaran konLrastr a7 KAYNAI.II.AF: ANDERSONT tl,l-. 1975. Irnproved for drqttal frlte,rs eval uat i ng Four i er and Hankel transform rnterlrals. U.S.6,S. Rep. U.S.6.S-Gd-75-O12e avai I es N TI S rep, PB-242-BCtCI/ LbtC,223 p. A N D E R S O N , l . l .L . t979. N u m e ri c a l i nt'eqrat i on of rel atecl Hankel transforms of orders (i and 1 by acJaptrve digital f i I t e r i n g . G e o p h y s r c s 4 4 , L 2 8 7 - -t 3 C ) 5 . ARFKEN, G. Ac aderni c BASOKUR, A. T. Ankar a 1985. Pr ess, f.lathematical London. f.lethods 1'144. D0gey El ekt r i tr B H A T A C H A R Y A ,B - K . , two layer earth. for Sanda.jr . Ptrysi,:ists. yayrnr , TPAO 1955. Electromagnetrc induction J. Geop. Research 60(3) , 275-ZBB- KELLER, c. v- ve FRTSCHKNECHT, F. c- , Methods in Geophysical prospecting. York. 1966. pergamon rn a Electrical press, New KOEFOED, o-, GHosH, D. p., poLMAN, c. J., rg72. computatlon of type curves for EM depth sounding r,ith a horizontal transmitting corl by means of digital linear frl..er Geophysi cal prospect ing 1"4, 229-24I . K O R K E A L A A K S O ,J . ve SAKSA, p., 1986. Calcuiation of EM field components in frequency and trne domain using dipole source excitation an layered earth model. Technical Research Centre of Finland, Research Notes 5g7. MCLACHLAN, N. V. , 1934 . Bessel Oxford Uni- press, London. MUNDRY, E. ve BLOHM, E. K., using vertical magnetic 35, 110-123. PAPAOLILiS, A., A1_'pljcatjonPATI1A, H. P, ve ?. Elsevler ]987. dipole. Functrons for Frequency geophysical Engineers, EM sound;.rrg prospectlng 1962. The Fourjer Integra) M c G r a w H i 1 l L . r e s s , n - e v Y cr k . MALLICii, X., Press, l-ondori . 19t0. Geo:ri,uncirirg arrd I'rtrrclpl aa PI{ILLIPS, H - B,'. ,1964. Vektorel A.U. Fen Fak. YayLnl, Ankara. Anaiiz. ERGUN, Qerviri:A.N. SINHA, A. K. , 1979. Maxiprobe wide-band A new EMR-16 : tnultifrequency ground EM system. Current Research Part B, Geological Survey of Canada, Paper 79-IB, 23-26 SINHA, A. K-, 1980A study misorientation effects in Geoexploration IB, III-I33. SINHA, A. K. oscillating conducting 7 3-25 - ve S-, 1973. COLLET, L. placed magnetic dipoles Geological Survey earth. VANYAN, L. L., Consultant 1967Bureau, Electromagnetic New York. EM fileds of rnultilayer over a of Canada, Paper Soundinq. Depth WAIT, J. R. , Mutual 1954. coupling of wire loops homogeneous ground. Geophysics 19,290-296. L/AfT, J. over and sounding. of topographic EM roultifrquency lying R., I955. Mutual electromagnetic coupling a homogeneous ground. Geophysics 20,630-637. WARD, S. H., In Mining 1967. EM Theory for Geophysical Geophysics, V-2, S-8.G., Tulsa- L I A R D , S . H . v e H O H M A N N ,c . W . , l_987. EM Geophyslcs-Theory Vol -1. S.E-G- Tu1sa. of on .i loops Application Methods in : Appli.ed EK-A A_I n F ; F b t- !t -r{ h - r tN o .i -\J -&o N -d r.L{ _ N N r-1 i: T t- x tw t. l- tI r 'i I l- a) ( ,t-t/ zi t)dLrry -.1 N b A - iI h r{ l, n -l l.\ N o F{ b N .\ uo '"1.t FI N O -l N b C) r{ b I ( tt j / zt_l)aurv //\l N r l\ - TTT r-i.r 11 (f r{ l.\ tl tt ;\ f,I i i- I r{ r O 4 F t il () (r .--.> tl N N : F. h Nv It- I I / /// I f-i'-rr.l--r-r* C) ,* I ..: C) / / \ l I ' , . ti li _ Il / / lt | \dlll\/ :_t_l )Ltr' r\/ rl { * ,h i, I r\ r{ b, f'l h r+-l N A_IV t -l x N o O I N b _{ do ; N N o FI N O : l-r--/tN 9r ( tr r /zt r\A, \Jr-1 / .- r1l-.lJV b e : r) n i.l I T x N tv. Cf, : O t\ N -$ () ; r{ -)Nor N -l N ,/, ,/ O -l / h [-r l-T--r-T-1- N a) (-rI i1zll) o r-r,ry A-VI n n x N t FI x N O rl x N -t{ r.l-a N & N (, rl o h N U o tz o r{ N ( rr t ,/zt r\rJrr LI'UUV \ J I - J/ b B _ VII n r<) t LX IW f-N L G I L ! el h N "-1 I \ @ N -i rH N C) ri - |.\ .N rl X -l- 't) I {.) 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