ÖZET Bu çalışmada, kısaca FDTD olarak tanımlanan Zamanda Sonlu Farklar (Finite- Difference Time-Domain) yöntemi ayrıntılı olarak incelenmiştir. FDTD yöntemi FD olarak bilinen sonlu farklar yönteminin 1966 yılında Yee tarafından Maxwell denklemlerine uyacak şekilde zaman domeni için genişletilmesiyle birlikte ortaya atılmış, özellikle 1980'lerin ortalarında bilgisayarların hız ve kapasitelerindeki hızlı artışla birlikte elektromagnetik problemler için en çok kullanılan yöntemlerden biri haline gelmiştir. Çalışma sırasında öncelikle FDTD yönteminde ortam modelleme, sayısal dispersiyon, zamanda ve konumda ayrıklaştırma, hata analizi gibi konular incelenmiş, daha sonra da konumda sınırlı sayıda hücre kullanılması nedeniyle oluşan ya pay sınırlardaki yansımaların giderilmesi için uygulanan sınır koşulları üzerinde durulmuştur. Sonsuza giden sınırları simüle eden birinci-derece ve ikinci-derece Mur, Higdon, DBC (Dağıtıcı Sınır Koşulu), Liao ekstrapolasyonu gibi açık sınır koşullan teorik olarak açıklandıktan sonra, 1994 yılında Berenger tarafından ortaya atılan PML (Mükemmel uyumlu Tabaka) yöntemi anlatılmıştır. Uygulamalar sırasında mikroşerit hatlardaki karakteristik empedans, efektif dielektrik sabiti ve yansıma katsayısı FDTD yöntemi ile sayısal olarak bu lunmuş ve bu değerler yardımıyla açık sınır koşullarının etkileri birbiri ile karşılaştırılmıştır. Bu işlemlerin sonucunda gerekli optimizasyonlar yapıldığı taktirde PML yönteminin en güvenilir sonuçları verdiği görülmüştür. Ancak PML yöntemi FDTD algoritmasında ihtiyaç duyulan hafıza miktarını arttırmakta ve programın çalışmasını da yavaşlatmaktadır. Bu yüzden, çok hassas analizlere gerek duyulmuyorsa, incelenen yapıya ve probleme uygun olan başka bir açık sınır koşulu kullanılabilir. En son olarak FDTD yönteminin farklı problemlere ve yapılara ne şekilde uygulanacağına dair örnek olması için alçak geçiren filtre, çeyrek dalga empedans transformatörü, mikroşerit küple devresi gibi düzlemsel mikroşerit yapılar FDTD ile incelenmiştir. v iii SUMMARY THE FINITE-DIFFERENCE TIME-DOMAIN METHOD AND ABSORBING BOUNDARY SIMULATIONS In this study the Finite-Difference Time-Domain (FDTD) method is investigated. FDTD is widely regarded as one of the most popular full-wave computational electromagnetics (EM) algorithm. It was first investigated in 1966 by Yee [1] and only a few studies have appeared until the end of 1970s because;. Large computer storage and high speed requirements have not been supplied until the mid of 1980s.. FDTD, as introduced by Yee was not suitable for most of the EM problems such as antenna simulations and RCS calculations [27] etc.. Source implementations and open boundary simulations have not been han dled in early FDTD, as introduced by Yee. Yee [1] cliscretized Maxwell's two curl equations directly in time and spatial do mains and put them in iterative form. In Yee formulation the physical volume of interest is divided into cubic reference cells where characreristics of the medium are defined by three parameters; permittivity (e), conductivity (cr) and permeabil ity (fi). Within the reference cell three electric and three magnetic field compo nents are located at different locations in such a way to minimize the computation duty after the discretization of two curl equations by using central-difference ap proach (or by taking up to the second order terms in their Taylor's expansion). Besides the differences in the locations of six field components, there is also a half time step difference between electric and magnetic field components. This is called a leap-frog computation. During FDTD simulation electric field com ponents are calculated at each cell at time instants t = 0,At,2At,3At,... etc., but magnetic field components are calculated at t = At/2,3At/2, 5At/2,... etc., having a At/2 time difference between electric, and magnetic field components. Therefore, in electromagnetic analysis, synchronization in both time and spatial domains are needed for E and H field or V and / calculations at a fixed point in FDTD volume. This is fortunately accomplished by simple cell and time averag ing processes. Since 1980, hundreds of publications about FDTD have appeared related to the algorithmic improvements as well as its applications to broad range of complex electromagnetic problems. Major improvements may be groupped in: ix. Narrow and broad band source simulations and their injections in given FDTD cells.. Open boundary simulations which extend the ability of FDTD algorithms to handle antenna radiation simulations and radar cross-section (RCS) cal culations.. Both frequency and time domain near- to- far field transformations based on Huygen's equivalent source principle. where effords are still needed for further modifications. Some of the applications may also be groupped as;. The analysis of planar microstrip structures,. Specific Absorption Rate (SAR) calculations in human tissues near electro magnetic sources,. Mutual effects of hand-held receivers and human head,. Antenna simulations and RCS calculations,. Analysis of waveguiding structures,. The simulation of ground-penetrating radars,. The simulation of microwave ovens. In this study, PML which is the most, efficient and available ABC, is implemented numerically for 3D rectangular FDTD volume. Although PML is effective in absorbing guided waves of all angular distribution, its implementation is quite difficult. Only a few studies related to the application of PML to 3D-FDTD [18], [17], [19], [22] have appeared in the literature up to day. For this reason, this study aims to focus on not only FDTD analysis of complex structures, but also 3D PML implementations. Three dimensional (3D) FDTD mesh suggested by Yee [1] is shown in Section 2.1 (see figure 2.1). In a lossy source-free medium, Maxwell's curl equations are given as 9H - - ^o-jT = -V xE-crE (la) BE.^. = Vx// (lb)where £q, (iç,, a are permittivity, permeability and conductivity respectively, and they are discretized as A/ H?(iJ,k) = H^iiJ.k) - -- [E*(i,j,k) - E^(i,j,k- 1)] At [E:(i,j,k)-E?(i,j-l,k)} (2a) Hj{i%j,k) = H*-/i,j,k) - -£L [E,*) - E:(i,j,k- 1)] (2b) ff?(», j,*) = Hr/i,j,k) - -^- [££(*,**) - ^(i, j - 1,*)] A* [JEJ(i,i,Ar)-^(i-l,j,*)] (2c) /z0A:r 2e-jrAj 2A* Wj'-*> = 2^ME``{i'j'k) 2A/ 2Aİ [H?(i,j,k)-H:(i-lJ,k)] {2e + aAt)Ax 2At + WT^Wzm''hk)-H:i''i'k-l)] (2e) In these equations h = n + 1/2 and e = £o£r- Because of the 3D-FDTD mesh structure, electric fields are calculated at the integer multiplicants of time and magnetic fields are calculated at the fractional multiplicants of time. As it is common in all numerical methods there exist unexpected reflections from the boundaries of the FDTD computation space unless the boundaries are cov ered with appropriate boundary conditions. In order to remove the effects of these reflections the methods listed below are used in FDTD applications: xi. Boundaries are covered with PEC (perfectly electric conductor) by assigning zero value to the tangential electic field components during time simulations.. Boundaries are covered with PMC (perfectly magnetic conductor) by as signing zero value to the tangential magnetic fields components during time simulations.. Field values of the cells which are adjacent to the boundary inside FDTD are assigned to field values of the cells which are adjacent to the boundary outside (symmetry condition).. Radiation condition called as absorbing boundary condition (ABC) is ap plied at the boundary planes. In this thesis, different ABCs are theoretically explained and numerically com pared with each other according to their efficiencies and accuracies in FDTD applications. First-order MUR ABC: With the help of first-order approximation of the three dimensional wave equation first-order MUR ABC [2] at x - 0 boundary is given as (dr - %ldt)Et = 0 (3a) where E% represents the tangential electric field component relative to the bound ary wall. The discretized form of Eq.(3a) will then be £,`+1(0,j,A-) = £r(U,Â0 + ^^(İ^HW,*) - £T(0,i,*)) (3b) For example, the tangential electric field components at x - 0 boundary are Ey and Ez. Eq.(3b) tells that, Ey(i,j, k) and Ez(i,j, k) for all (j, k) will be related to the current (n) and one past (n - 1) time instant values of themselves and to the nodes one inside. By doing this, first-order Mur condition is satisfied. Second- order MUR ABC: With the help of second-order approximation of the three dimensional wave equation second-order MUR ABC [2] at x = 0 boundary is given as ( PML(o`.o,'.o.o) y f! Xi),o) PML«j`,o,;,o`.%) PML(o.ao>0'`) PML(oa.o»a`o,) Figure 1. 2D FDTD and PML structure yf PML J-JL ¦+-E, O H, H` © H` Figure 2. Right-upper part of 2D FDTD+PML structure PML is applied to two-dimensional FDTD as shown in Figure 1. FDTD equations are applied in the inner part and PML equations are used in the PML region. For the right-upper part of the Figure 1 (see Figure 2) in PML region (/ > IL and J > JL in Fig.3) the field equations are gj(i)Ai gar(l)At Eny+i{i,j) = e~ i) (l- e tr*(i+l/2)At CO ')