033259-1-F FINAL REPORT MICROPACKAGING FOR MM-WAVE CIRCUITS ARO Contract: DAAH04-95-1-0321 Linda P.B. Katehi May 1998 33259-1-F = RL-2457

Reporting period: May 1995 - April 1998 Students: Manos Tentzeris (student) Robert Robertson (student) Faculty: Linda P.B. Katehi Other Supporting Research Staff Michael Krumpholz Manuscripts Published or Submitted During the Reporting Period. 1. Linda P.B. Katehi, James F. Harvey and Emmanouil Tentzeris, "Time Domain Analysis Using Multiresolution Expansions", Book on Time Domain Techniques, Editor: A. Taflove, Willey. 2. E. Tentzeris, M. Krumpholz and L.P.B. Katehi, "Characterization of Shielded Microwave Circuit Components Using MRTD", submitted to IEEE Transactions on Microwave Theory and Techniques. 3. E. Tentzeris, R. Robertson, J.F. Harvey and L.P.B. Katehi," Stability and Dispersion Analysis of Battle-Lemarie Based MRTD Schemes," submitted to the to IEEE Transactions on Microwave Theory and Techniques. 4. E. Tentzeris, J.F. Harvey and L.P.B. Katehi, "Time Adaptive, Time Domain Techniques for the Design of Microwave Circuits," submitted in the Microwave and Guided Wave Letters. 5. E. Tentzeris, A. Cangellaris and L.P.B. Katehi, " Space/Time Adaptive Meshing and Multiresolution Time Domain Techniques," presented in the 1997 ACES Conference. 6. E. Tentzeris, R. Robertson, A. Cangellaris and L.P.B. Katehi, "Space and Time Adaptive Gridding Using MRTD" presented in the 1997 International Symposium on Microwave Theory and Techniques. 7. E. Tentzeris, J.F. Harvey and L.P.B. Katehi, "Time Adaptive Time Domain Techniques for the Design of Microwave Circuits," presented in the 1997 International Symposium on Microwave Theory and Techniques.

8. E. Tentzens, R. Robertson and L.P.B. Katehi, "PML Implementation for the BattleLemarie MRTD Schemes, presented in the 1998 ACES Conference. 9. L. Roselli E. Tentzeris and L.P.B. Katehi, "Nonlinear Circuit Characterization Using a Multiresolution Time Domain Technique," accepted for presentation in the 1998 International Symposium on Microwave Theory and Techniques. 10. E. Tentzeris and L.P.B. Katehi, "Space Adaptive Analysis of Evanescent Waveguide Filters," accepted for presentation in the 1998 International Symposium on Microwave Theory and Techniques. 11. E. Tentzeris, "Time Domain Numerical Techniques for the Analysis and Design of Microwave Circuits," Ph.D. Dissertation, The University of Michigan, April 1998. Honors and Awards * 1997 Best Paper Award by the International Microelectronics and Packaging Society (IMAPS) * First Prize in Symposium Paper Award Contest with Katherine Herrick for the paper "W-Band Micromachined Finite Ground Coplanar (FGC) Line Circuit Elements," IEEE MTT-S, Denver, CO, June 1997 Brief Description of Performed Research Complex antenna and circuit problems including their package on wafer require very intensive calculations due to the need to accurately simulate the underlying high-frequency effects and account for all the parasitic mechanisms. As part of this project, we have successfully applied a novel frequency domain scheme recently developed at the University of Michigan that allows for the very successful and computationally efficient solution of complex antenna problems. This technique has been applied to a variety of circuit and antenna problems and has demonstrated the capability to provide accurate solutions in a much more efficient ways than the conventional techniques. The whole idea in this approach is the use of wavelets in the expansion of the unknown functions. The use of wavelets allows for the computation of the values of the derivatives of the unknown field quantities in addition to the average values of the field. This allows for the development of novel space-adaptive schemes with unique capabilities. In previous years the Battle-Lemarie based MRTD Technique has been applied to a variety of homogeneous microwave problems and has exhibited significant savings in memory and 1-n tD ~IIVI UI

execution time. Nevertheless, the most important advantage of this new technique is its capability to provide space and time adaptive gridding without the problems the conventional FDTD is encountering. This is due to the use of two separate sets of basis functions, the scaling functions and wavelets and the capability to threshold the filed coefficients due to excellent conditioning of the formulated mathematical problem. This year a space/time adaptive gridding algorithm based on the MRTD scheme was proposed and applied to inhomogeneous waveguide problems. As examples, the propagation of a Gabor pulse in partially-filled parallel plate waveguide and a parallel plate filter was simulated and the S-parameters have been calculated for validation of the theory. Wavelets were placed only at locations where the EM fields have significant values, creating a spaceand time-adaptive dense mesh in regions of strong field variations, while maintaining a much coarser mesh elsewhere. A mathematically correct way of dielectric modeling has been presented and evaluated. The proposed adaptive gridding offers extra economy in memory by a factor of 30%-40% for the 2-dimensional case. This algorithm has been extended to three dimensions and has been applied to the numerical modeling of evanescent mode waveguide band-pass filters. To extend the capabilities of the Battle-Lemarie based MRTD scheme, an entire-domain algorithm has been proposed and applied to the numerical analysis of nonlinear circuits including diodes. The frequency spectrum of a mixer diode has been calculated and verified by comparison to reference data. In comparison to Yee's conventional FDTD scheme, the proposed scheme offers memory savings by a factor of 2-6 pre dimension maintaining a similar accuracy. Especially in the approximation of time derivatives, the use of entire domain expansion basis would require very high memory resources for the storage of the field values everywhere on the grid for the whole or a large fraction of the simulation time. This problem does not exist in the approximation of the spatial derivatives since the filed values on the neighboring spatial grid points have to be calculated and stored no matter what expansion basis are used. For that reason, Harr basis functions have been utilized and a time-adaptive time-domain technique based on intervalic wavelets has been proposed and applied to various types of circuit problems. The scheme has exhibited significant savings in execution time and memory requirements while maintaining a similar accuracy with conventional circuit simulators. Numerical experiments have shown that the use of an absolute threshold of 10-6 and a relative threshold of 5x10-4 offered an extra economy of 25%-35% in comparison to MRTD schemes based only on scaling functions.

Another research topic for 1997 was the investigation of the stability and the dispersion performance of entire-domain basis MRTD schemes for different stencil sizes and for 0 -resolution wavelets. Analytical expressions for the maximum stable time-step were derived for both scaling only and scaling-wavelet algorithms. It has been observed that larger stencils decrease the numerical phase error making it significantly lower that FDTD for low and medium discretizations. Stencil sizes greater than 10 offer a smaller phase error that FDTD even for high discretizations. The enhancement of wavelets further improves the discretization performance for discretizations closer to the Nyquist limit (2-3 cells/wavelength) making it comparable to that of much denser grids, though it decreases the value of the maximum time-step guaranteeing the stability of the scheme. In the following copies of the submitted/presented papers are included for further information.

Submitted to the IEEE Transactions on Microwave Theory and Techniques Characterization of Shielded Microwave Circuit Components Using the Multiresolution Time Domain Method (MRTD) Emmanouil Tentzeris, Michael Krumpholz, Linda P.B. Katehi Radiation Laboratory, Department of Electrical Engineering and Computer Science University of Michigan, Ann Arbor, MI 48109-2122 Abstract The recently developed Multiresolution Time Domain (MRTD) technique is applied to the modeling of shielded microwave circuit problems. The technique demonstrates excellent accuracy and efficiency in the calculations with savings of one order in computation times and of two orders of magnitude in memory compared to the conventional The enhancement of wavelets provides very efficient computations of the characteristic impedance and effective dielectric constant of a variety of printed lines operating in a shielded environment. I Introduction Despite the wealth of available codes for analysis and design in microwave frequencies, many problems in electromagnetics and specifically in circuit and antenna problems have been left untreated due to the complexity of the geometries and the inability of the existing techniques to deal with the requirements for large size and high resolution due to the fine but electrically important geometrical details. The straightforward use of existing discretization methods suffers from serious limitations due to the required substantial computer resources and urealistically long computation times. As a result, during the past thirty years the available techniques are almost incapable of dealing with the needs of technology leading into a quest for fundamentally different modeling approaches. Recently a new technique has been successfully applied [1,?, 2] to a variety of microwave problems and has demonstrated unparalleled properties. This technique is derived by the use of multiresolution analysis for the discretization of the time-domain Maxwell's equations. The multiresolution time domain technique (MRTD) based on Battle Lemarie functions has been applied to linear as well as nonlinear propagation problems and has demonstrated savings in 1

time and in memory by one and two orders of magnitude respectively. In addition to time and memory, the most important advantage of this new technique is its capability to provide space and time adaptive meshing without the problems encountered by the conventional FDTD [3]. The capability to provide adaptive meshing is connected with the use of two separate sets of basis functions, the scaling and wavelet functions, and the capability to threshold the field coefficients due to the excellent conditioning of the formulated mathematical problem. This advantage and capability of the technique is demonstrated herein by performing a space adaptive meshing. For the derivation of the MRTD scheme, the electromagnetic fields are represented by an expansion in cubic spline Battle-Lemarie scaling and wavelet functions [4], [5] with respect to space. For this type of basis functions, the evaluation of the moment method integrals is simplified due to the existence of closed form expressions in spectral domain and simple representations in terms of cubic spline functions in space domain. The use of non-localized basis functions cannot accomodate localized boundary conditions. To overcome this difficulty, the image principle is used to model perfect electric and magnetic boundary conditions. Pulse functions are used as expansion and test functions in time-domain. In this paper, a 2.5D MRTD scheme is proposed and applied to a variety of shielded of transmission line problems. Specifically, propagation constant, characteristic impedance and field patterns are derived for shielded and open transmission line structures and compared to FDTD results. II The 2.5D-MRTD scheme For simplicity, an overview of the 2.5D-MRTD scheme is presented for a homogeneous medium. The derivation is similar to that of Yee's FDTD scheme [?], which uses the method of moments with pulse functions as expansion and test functions. The magnetic field components are shifted by half a discretization interval in space and time-domain with respect to the electric field components. Using the approach of [6], Maxwell's curl equations for a homogeneous medium with the permittivity e and the permeability p can be written in the following form aEx _a9H - = +/3Hy (1) ay = -H- (2) 9Ez al ly (H, is the propagation constant and x y he electric and magnetic field components where /3 is the propagation constant and j = yf-i. The electric and magnetic field components 2

incorporated in these equations are expanded in a series of Battle-Lemarie scaling and wavelet functions in both x- and y-directions. For example, E, can be represented as: +00oo ES(x,y,t) = E kEiP2%,mhk (t) 1 +l/2 (X)Om(Y) k,l,m=-oo +oo + E E z.El+l/2,mhk(t) +l+l/2(X) 1i,m(y) i k,l,m=-oo +00 + E E kEL+/X,m hk(t),L+1/2 () kim(Y) i k,l,m=-oo +00 + E E k+l,mhk(t) li,1+)l/2() i)j,m(Y) (4) ij k,l,m=-oo where qm(x) = 0(5z- m) 8and li,m(x) = i( z - m) represent the Battle-Lemarie scaling and i-th order resolution wavelet function respectively in space and hk(t) represent rectangular pulses in time. kE^' and k+1/2H,4V with K = x, y, z and A, v = %, ' are the coefficients for the field expansions in terms of scaling and wavelet functions. The indices 1, m and k are the discrete space and time indices related to the space and time coordinates via x = IAx,y = mAy and t = kAt, where Ax,Ay are the space discretization intervals in x- and y-direction and At is the time discretization interval. For an accuracy of 0.1% the above summations are truncated to 18 terms. Upon inserting the field expansions, Maxwell's equations are sampled using pulse functions as time-domain test functions and scaling and wavelet functions as spacedomain test-functions. Following the procedure of [1], the 2.5D MRTD scheme is derived. As an example, let's consider the discretization of eq.(l). For simplicity, it is assumed that the fields have been expanded only in scaling functions summations. Wavelets can be added in a straightforward way. Applying the Galerkin's technique, the following difference equation is obtained for a homogeneous medium with the permittivity c, 1 m+8 t(k+E1 /2,m - kEl/2,m = ( z a(i')k+1/2H+l/2,i,+l/2) + 1 k+1/2HI+/2,m, (5) t A y i'=m —9 with the coeficients a(i') defined in [1]. The unit cell of the 2.5D-MRTD scheme is identical to the unit cell of the conventional Yee's FDTD scheme. However, due to the different field expansion functions, the field components in the two techniques have a different physical meaning. Deriving MRTD and FDTD using the method of moments, the field components have to be interpreted as field expansion coefficients. From the different field expressions, it is clear that the field expansion coefficients of the FDTD scheme represent the total field value 3

at a specific point, while the field expansion coefficients of the 2.5D-MRTD scheme represent a fraction of the total field. To calculate the total field at a space point, the field expansions are sampled with delta test functions in space and time domain. For example, the total electric field E.(xo, Yo, to) with (k - 1/2) At < to < (k + 1/2) At is calculated by E o(x., y, t) = J E(x, y, t) 6(x - x) 6(y - y,) (t - to) dx dydt 00 = Z kEI+ /2,m Oil'+/12(Xo) Om'(yo) * (6) 1,m'=-oo Extending the dispersion analysis of [1] from 3-dimensional to 2.5-dimensional space, the stability condition for the 2.5D-MRTD scheme results in 'At < (7) At 1.568c~/(~)2 + ()2 + ()2 (7) with the wave propagation velocity c. It is preferable to choose At at least 2.5 time less than the stability limit. In this way, much more linearity of the dispersion characteristics is achieved. III Applications of the 2.5D-MRTD scheme to Shielded Transmission Lines In this paper, the 2.5D-MRTD scheme is applied to the analysis of shielded stripline and microstrip lines to investigate propagation and coupling effects. Results for these shielded structures are presented and discussed separately below. A shielded stripline is a simplified version of a membrane microstrip shown in (Fig.la). The metallic shield has dimensions 47.6mm x 22.0mm and the central strip has length 11.9mm. The stripline is filled with air (c, = 1.). The analysis for the higher order propagating modes is straightforward. For the analysis using Yee's FDTD scheme, a 40 x 10 mesh was used resulting in a total number of 400 grid points. When the structure was analyzed with the 2.5D-MRTD scheme, a mesh 8 x 4 (32 grid points) was chosen reducing the total number of grid points by a factor of 12.5. In addition, the execution time for the analysis was reduced by a factor of 3 to 4. The time discretization interval was chosen to be identical for both schemes and equal to the 0.8 of the 2.5D-MRTD maximum At. For the analysis 3 = 30 was used and 5,000 time-steps were considered. 4

Mode TEM Shield TE1o Analytic values 1.4324 GHz 3.4615 GHz 8x2 MRTD 1.4325 GHz 3.4648 GHz Rel.Error 0.007% 0.095% 8x4 MRTD 1.4325 GHz 3.4641 GHz Rel.Error 0.007% 0.075% 16x4 MRTD 1.4325 GHz 3.4633 GHz Rel.Error 0.007% 0.052% 40x10 FDTD 1.4322 GHz 3.4585 GHz Rel.Error -0.014% -0.087% Table 1: Mode frequencies for 3 = 30 From (Table 1) it can observed that the calculated frequencies of the two first propagating modes for (3 = 30 by use of 2.5D-MRTD scheme are very close to the theoretical values, since the largest error is less than 0.1%. The relative error of the 2.5D-MRTD calculated frequencies is always positive, which corresponds to an overestimation of the resonant frequencies. This is exactly what has to be expected from the dispersion behavior of the MRTD schemes. The use of non-localized basis functions in the 2.5D-MRTD scheme causes significant effects. Localized boundary conditions are impossible to be implemented, so the perfect electric boundary conditions are modelled by use of the image principle. The perfect electric conductor is replaced by an open structure with electromagnetic fields characterized by even or odd symmetry. Odd symmetry is imposed to the electric field components that are tangential to the conductor in order to ensure zero electric field on the conductor and even symmetry for the magnetic field components that are tangential to the conductor. The non-localized character of the basis functions offer the opportunity to calculate the field values in any point of the discretization cells. The field values at the neighbooring cells can be combined appropriately by adjusting the scaling functions' values and by applying the image principle. For example, the total electric field Ez(xx, yo, to) with (k-1/2) At <to < (k+1/2) At is calculated by Eq.(6) by simply truncating the l,m summation from -12 to 12 for each index. That means that the summation based only at the 12 neighbooring cells from each side gives the total field component values with good accuracy. In (Fig.2-4), the value of the Ey field has been calculated and plotted for the 4 cells exactly below the strip by use of the 2.5D-MRTD scheme. The relative position of the strip is from 15 to 25. For the TEM mode the pattern obtained by use of the conventional FDTD scheme is plotted for comparison. For the shield 5

TE0o mode, the analytically calculated pattern has been added for reference. All results are normalized to the peak value. It can observed that the agreement of the MIRTD calculated field pattern with the reference data is very good for the shield TE0o mode, where the values are changing slowly (sinusoidally) (Fig.2). On the contrary, for the TEM mode where the edge effect is more prominent, the agreement is not good. In this case, wavelets of 0-Resolution are added in both directions to describe the higher spatial frequencies. It can be observed from (Fig.3) that the wavelet coefficients for the 8x4 grid have a significant contribution (> 10%) close to the stripline. Increasing the grid size from 8 to 16 to the strip direction and/or from 4 to 8 to the normal to the strip direction improves more the accuracy of the field representation (Fig.4). The characteristic impedance ZO for the TEM mode of the stripline is computed from the equation: Zo = V = c Eydy -, (8) I ~c~ Hdl where the integration paths C, and Cc are shown in (Fig.la). Since both of the schemes used in the analysis are discrete in space-domain, the above integrals are transformed to summations. For the FDTD summations, only one field value per cell is needed, due to the fact that pulse expansion functions which are constant for each cell are utilized. On the contrary, for the 2.5DMRTD summation the field values for a number of subpoints along the integration path have to be calculated, since the expansion functions are not constant for each cell. It can be observed from Table 2 that the accuracy of the calculation of the characteristic impedance is improved by increasing the number of subpoints per cell, at which the field values are calculated. An accuracy better than 1% is achieved if the field values are computed for more than 9 subpoints per cell along the integration path for the scheme including wavelets of 0-resolution to both directions. On the contrary, the value of Zo that is calculated from the scheme based only on scaling functions is oscillating, thus indicating that a denser mesh is required. The analytical value of the Zo is 95.58 2 [7]. The modification of the dimensions of the MRTD mesh (Table 3) shows that the accuracy of the calculation of the Zo by use of the MRTD is much better than that of the Yee's FDTD scheme with a 40x10 mesh (relative error -3.28%). A similar procedure is used for the analysis of the shielded coupled-stripline geometry of (Fig.lb) for the first even and odd mode. Both strips have a length of 11.9mm, the distances between them is 11.9mm, from the top and bottom PEC's are 11.0mm and from the left and right PEC's are 11.9mm. The structure is filled with air (c, = 1.). For the analysis with the conventional FDTD scheme, a 70 x 20 mesh resulted in a total number of 1400 grid pints. The 6

Subpoints/cell Z~c (Qf) Relative error Z.~ (Q) Relative error 3 80.56 -15.71 % 84.04 -12.07 % 5 94.46 -1.17 % 92.55 -3.17 % 7 99.06 +3.64 % 94.59 -1.04 % 9 101.44 +6.13 % 94.96 -0.65 % 11 97.56 +2.07 % 95.01 -0.60 % Table 2: Zo for different number of subpoints/cell (8x4 Grid). Zo (Q) Relative error Analyt. Value 95.58 0.0% 8x4 MRTD 95.01 -0.60% 8x8 MRTD 95.19 -0.41% 16x4 MRTD 95.71 0.14% 40x10 FDTD 92.44 -3.28%!.,! Table 3: Zo for different mesh sizes (11 subpoints/cell). same accuracy is achieved by an MRTD mesh 14 x 4 (56 grid points) resulting in an economy of memory by a factor of 25. The space distribution of the tangential-to-stripline E is plotted in logarithmic scale in (Fig.5) for the even mode. The 2-D MRTD technique is also used for the analysis of a shielded microstrip (Fig.lc) with width 9.9mm on a dielectric substrate with ~r = 10.65 and thickness 11mm. The microstrip is placed in the center of a rectangular shield 69.3mm x 44mm. The same accuracy for the characteristic impedance calculation (Theoretical Zo = 50 Ohms) is achieved by an FDTD mesh 140 x 80 and an MRTD mesh 28 x 20resulting in an economy in memory by a factor of 20. IV Conclusion A multiresolution time-domain scheme in 2 dimensions has been applied to the numerical analysis of shielded striplines and microstrips. The field patterns and the characteristic impedance have been calculated and verified by comparison to reference data. In comparison to Yee's conventional FDTD scheme, the proposed 2.5D-MRTD scheme offer memory savings by a 7

factor of 25 and execution time savings by a factor of about 4-5 maintaining a better accuracy for characteristic impedance calculations. This indicates memory savings of a factor 5 per dimension leading to two orders of memory savings in three dimensions. Compared to 2.5D-FDTD, 25 times less cells in MRTD require about 5 times less running time, thus the computation time per cell is increased by a factor of 5. This leads to computation time savings of more than one order for 3 dimensional structures. For structures, where the edge effect is prominent, additional wavelets have to be introduced to improve the accuracy when using a coarse MRTD mesh. V Acknowledgments This work has been made possible by a scholarship of the NATO science committee through the German Academic Exchange Service and by the U.S. Army Research Office. References [1] M.Krumpholz, L.P.B.Katehi, "MRTD: New Time Domain Schemes Based on Multiresolution Analysis", IEEE Trans. Microwave Theory and Techniques, vol. 44, no. 4, pp. 555-561, Ap ril 1996. [2] E.M.Tentzeris, R.L.Robertson, M.Krumpholz, L.P.B.Katehi, "Application of MRTD to Printed Transmision Lines", Proc. MTT-S 1996, pp. 573-576. [3] K.S.Yee, "Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media", IEEE Trans. Antennas Propagation, pp.302-307, May 1966. [4] G.Battle, "A block spin construction of ondelettes", Comm.Math.Phys., vol. 110, pp. 601-615, 1987. [5] P.G.Lemarie, "Ondelettes a localisation exponentielle", J.Math.Pures Appl., vol. 67, pp. 227-236, 1988. [6] S.Xiao, R.Vahldieck, "An Improved 2D-FDTD Algorithm for Hybrid Mode Analysis of Quasi-planar Transmission Lines", MWSYM 93, vol. 1, pp.421-424. [7] B.C.Wadell, "Transmission Line Design Handbook", pp. 136-137, Artech House, 1991. 8

LIST OF FIGURE CAPTIONS Figure 1: Printed Lines Geometries. Figure 2: Shield TElo Ey pattern. Figure 3: TEM Ey Pattern Components (8x8 Grid). Figure 4: TEM Ey Pattern Comparison (8x8 Grid). Figure 5: Tangential E-field Distribution (Shielded - Even Mode). 9

Cc I l I I I v I I i-.... --- —-------------- _____L^__I (a) (b) (c) Figure 1: Printed Lines Geometries. 10

TE 0 Mode 1 0.9 0.8 0.7 0.6 w 0.5 0.4 0.3 0.2 0.1 6 8 10 12 position Figure 2: Shield TE1o Ey pattern. 20 11

w,'30 J -- 0 10 20 30 40 50 60 70 Horizontal axis (x) Figure 3: TEM Ey Pattern Components (8x8 Grid). 80 12

c L I I LLJ I 20 10I I -20 0 20 40 60 80 100 x-axis Figure 4: TEM Ey Pattern Comparison (8x8 Grid). Figure 4: TEM Ey Pattern Comparison (8x8 Grid). 13

Figure 5: Tangential E-field Distribution (Shielded - Even Mode). 14

To be submitted to the IEEE MTT-T Journal STABILITY AND DISPERSION ANALYSIS OF BATTLE-LEMARIE BASED MRTD SCHEMES Emmanouil M. Tentzeris', Robert L. Robertson', James Harvey2 and Linda P.B. Katehil Radiation Laboratory, Department of Electrical Engineering and Computer Science University of Michigan, Ann Arbor, MI 48109-2122 2 Army Research Office, NC I Abstract The stability and the dispersion performance of the recently developed Battle-Lemarie MRTD schemes is investigated for different stencil sizes. The contribution of wavelets is enhanced and analytical expressions for the maximum allowable time step are derived. It is observed that larger stencils decrease the numerical phase error making it significantly lower than FDTD for low and medium discretizations. The addition of wavelets further improves the dispersion performance for discretizations close to the Nyquist limit, though it decreases the value of the maximum time-step guaranteeing the stability of the scheme. II Introduction Finite-Difference Time-Domain numerical techniques are widely used now-a-days for the analysis of various microwave geometries and for the modelling of EM wave propagation. Though many of them are very simple to implement and can be easily applied to different topologies with remarkable accuracy, they cause a numerical phase error during the propagation along the discretized grid. For example, the numerical phase velocity in the FDTD can be different than the velocity of light, depending on the cell size as a fraction of the smallest propagating wavelength and the direction of the grid propagation. Thus, a non-physical dispersion is introduced and affects the accuracy limits of FDTD simulations, especially of large structures. In addition, it is well-known that the finite-difference schemes in time and space domain require that the used time step should take values within an interval that is a function of the cell size. If the time-step takes a value outside the bounds of this interval, the algorithm will be numerically unstable, leading to a spurious increase of the field values without limit as the time increases. Though the stability and the dispersion analysis for the conventional Yee's FDTD algorithm has been thoroughly investigated, only a few results have been presented concerning MRTD schemes based on cubic spline Battle-Lemarie scaling and wavelet functions [2]. The functions of this family do not have compact support, thus the finite approximations of the derivatives are finite stencil summations instead of finite differences. In this paper, the effect of these stencils' size as well as of the enhancement of wavelets is investigated and comparison with 2nd-order and higher-order FDTD schemes displays difference in their respective behaviors. 1

III Stability Analysis Following the stability analysis described in [1], the MRTD [2] equations are decomposed into separate time and space eigenvalue problems. Assuming an expansion only to scaling functions (S-MRTD), the left-hand side time-differentiation parts can be written as an eigenvalue problem k+l/2H,-1/ 12= - jA kHi-/ 2 (1) At- () k+1/2H-1/2,j - 1/2H1/2,j = A (2) -+1E - A H,_/ (2),- kEi-,j = A k+1/2Ei,j (3) At In order to avoid having any spatial mode increasing without limit during normal time-stepping, the imaginary part of A, Imag(A), m ust satisfy the equation 2 2 -- < Imag(A) < A (4) At - 'A For each time step k, the instantaneous values of the electric and magnetic fields distributed in space across the grid can be Fourier-transformed with respect to the i- and j- coordinates to provide a spectrum of sinusoidal modes (plane wave eigenmodes of the grid). Assuming an eigenmode of the spatial-frequency domain with kx and k. being the x- and y- components of the numerical eigenvector, the field components can be written EIj = Ezeij(khlAZ+ktJAy H.j-_1/2 = H.oe1(k'I^A+kw(J-1/2)A^ HI-1/2,J = H,.ej(k.(l-1/2)Ax+kJA Substituting these expressions to (1)-(3) and applying Euler's identity, we get 4 1 - 1 n A2 = 4 1 2 =- ()2 ( E a(i')si(k(i) + l/2)A )2 + ()( E a(')in(k(j + 1/2)Ay))2] i'= j =O Thus, A is a pure imaginary, which can be bounded for any wavevector k = (ks, kv): n- 1 1 - 2 (E la(i')) ( )2 + ()2 < Imag(A) i'=0 2 - 1 15)2 11 1 < 2c (" la(i')]) Ax (AY)2, (5) i' =O where c = is the velocity of the light in the modeled medium. Numerical stability is maintained for every spatial mode only when the range of eigenvalues given by (5) is contained entirely within the stable range of time-differentiation eigenvalues given by (4). Since both ranges are symmetrical around zero, it is adequate to set the upper bound of (5) to be smaller or equal to (4), giving: 1 At < (6) c i-= o la(i')j) For Ax = Ay = A, the above stability criterion gives A A AtS-MRTD < CV n-I sS= (7) - Zi,=o la(i')l cv/ 2

It is known [3] that AtFDTD < (8) c (Ad= + (A~) which gives for Az = Ay = A AtFDTD < cV- (9) cv2 Equations (7)-(9) show that for same discretization size, the upper bounds of the time-steps of FDTD and S-MRTD are comparable and related through the factor s. The stability analysis can be generalized easily to 3D. The new stability criteria can be derived by the equations (7) and (9) by substituting the term V2X with v-3. More complicated expressions can be derived for the maximum allowable time-step for schemes containing scaling and wavelet functions. For example, the upper bound of the time-step for the 2D MRTD scheme with 0-resolution wavelets to the one (x-direction) or two directions (x- and y-directions) is given by tWoS-MRTD < — -- V/( Ei 1 a(i')l i Ibo(i')l + 4(Ei1 Ico(i')l)2) + 7 ( '1 a(')1)2 and AtWoWo-MRTD < 1 c + y I= a(i')I bo(i') + 4(Ei'= oco(i') )2 For Az=Ay=A, the above equations give AtWoS-MRTD,maz - 8WoS / with WoS = /,-. ('"n-~1 nb\(i')1 + 4(En"1-l fx -— x WS= /(Z =0 Ia(i')I i-.=o Ibo(i')I + 4(i,= co(i)2) + a Ia(J')l)2 and AtWoWo-MRTD,maz ' 8Wo Wo C A with n, —l nb-i ne-1 SWoWo = la(i')l E Ibo(i)l + 4( Ic(i')l)2 i'=0 i'=0 i'=0 It can be observed that the upper bound of the time step depends on the stencil size b, na, n,. This dependence is expressed through the coefficients ssswoS, wos, o which decrease as the stencil size increases. Figure 1 shows that sss practically converges to the value 0.6371 after na > 10 and swos s 0.4872 and swowO - 0.4095 for na = nb = fn > 10. IV Dispersion Analysis To calculate the numerical dispersion of the S-MRTD scheme, plane monochromatic traveling-wave trial solutions are substituted in the discretized Maxwell's equations. For example, the Ez component for the TM mode has the form 3

kE, I = Ezo.e(tk' A+tJA-wtAt) where k, and ky are the x- and y- components of the numerical wavevector and w is the wave angular frequency. Substituting the above expressions into the Equations (1)-(3), the following numerical dispersion relation is obtained for the TM mode for the S-MRTD Scheme after alge braic manipulation ( )]2( a(i')sin(k,(i + 1/2)Az))]2 i'-O n.-1 + [-( a(j.')sin(k,(j + l/2)Ay))] (10) j'=0 For square unit cells (Az=Ay=A) and wave propagating at an angle > with respect to x-axis (k, = k cos) and ky = k sin+), the above expression is simplified to A w At n.-I ^ sin(2 )]2 A = ( a(i') sin(k cost (i' + 1/2) A))2 P-=o nh-1 + (Z a(j') sin(k sin) (j' + 1/2) A))2 (11) j'=o This equation relates the numerical wavevector, the wave frequency, the cell size and the time-step. Solving this numerically for different angles, time step sizes and frequencies, the dispersion characteristics can be quantified. Defining the Courant number q = (cAt)/A and the number of cells per wavelength ni = AREAL/A and using the definition of the wavevector k = (2n)/ANUM the dispersion relationship can be written as n, —1 [- sin(7r q /ni)]2 = [ a(i') sin(7r u (2i'+ 1) cosq /nt)]2 q 1'=0 n.-i + [ a(j') sin(7 u (2j' + 1) sin /nl )]2 (12) j'=0 where u = AREAL/ANUM is the ratio of the theoretically give n to the numerical value of the propagating wavelength and expresses the phase error introduced by the S-MRTD algorithm. To satisfy the stability requirements, q has to be smaller than 0.45 (= 0.6371/v2) for the 2D simu lations. The above analysis can be extended to cover the expansion in scaling and 0-resolution wavelet functions in x-, y- or both directions. The general dispersion relationship is (CC + C2C2 + CC CC + 4C5 + C5C6)2 (C C2 + C2C3)2 + A B +-)4(CG1C2+ +C2C3)2(C4C +CC)2( + )2; = 1 (13) 4

Scheme C1 C2 C3 C4 Cs C6 55 0 0 0 $0 0 o WoS $0 0 0 #0 o _0 SWo 0 o0 #0 $0 o o0 WoWo # 0 O o 0 0o oo # Table 1: Coefficients Ci for Different MRTD Schemes with F = 1 - [(C1C2 + C2C3)]2-[( 5 +C )]2 - [7(C2C2 + C3C3 + C5C5 + C6C6] (14) A = 1 - -(CC + C2C2 + C5C5 + C6C6) B = 1 - (C2C2+C3C3+C4C4+ C5C5) (15). The Ci are defined by C, = E a(j')sin(k,(j' + 1/2)A) pAsin(wAt/2),=o At c = -As/n(wAt/2) E bo(i)sin(kv(i' + 1/2)A) =Asin(wAt/2) E =0 At C4 = - Ain(At/2),_ a(i')sin(k,(i' + 1/2)A) At n, C2 = Ai(WAt/2) co(i')sin(ki'A) At nb = - sin(wAt/2) E bo(i')sin(k.(i' + 1/2)A) (16) Eq.(13) can be applied to the dispersion analysis of SS (only scaling functions), WoS (0-resolution wavelets only to x-direction), SWo (0-resolution wavelets only to y-direction) and WoWo (0-resolution wavelets to both x- and y- directions) following Table 2. In case the Ci $ 0, it can be calculated by Eq.(16). The above equation is solved numerically by use of Bisection-Newton-Raphson Hybrid Technique for different values of na, nb, no ni, ' and q. Fig.(2)-(5) show the variation of the numerical phase velocity as a function of the inverse of the Courant number l/s=l/q for stencil sizes n. = nb = n, = 8,10,12,14. For each figure, three different discretization sizes are used: 10 cells/wavelength (coarse), 20 cells/wavelength (normal) and 40 cells/wavelength (dense). The results are compared to the respective values of conventional FDTD. It can be observed that the phase error for F.D.T.D. decreases quadratically. The variation of the phase error in M.R.T.D. exhibits some unique features. Though for any stencil size the numerical phase error for M.R.T.D. discretization of lOcells/A is smaller than that of the F.D.T.D. discretization of 40cells/A, the M.R.T.D. error doesn't behave monotonically [4]. It decreases up to a certain discretization value and then it starts increasing. This value depends on the stencil size and takes larger values for larger stencils. For example, this value is 5

between 10 and 20 cells/A for stencil equal to 10, between 20 and 40 cells/A for stencil=12 and very close to 40 cells/A for stencil=14 and can be used as a criterion to characterize the discretization range that the M.R.T.D. offers significantly better numerical phase performance than the F.D.T.D. The phase error caused by the dispersion is cumulative and it represents a limitation of the conventional FDTD Yee algorithm for the simulation of electrically large structures. It can be observed that the error of S-MRTD is significantly lower, allowing the modeling of larger structures. FDTD is derived be expanding the fields in pulse basis. As it is well known the Fourier transform of the pulse is a highly oscillating Si(x). On the contrary, the Fourier transform of the Battle-Lemarie Cubic spline is similar to a low-pass filter. That "smooth" spectral characteristic offers a much lower phase error even for very coarse (close to 3-4 cells/A) cells. By using a larger stencil na, the entire-domain oscillating nature of the scaling functions is better represented. Thus, smoother performance for low discretizations (Fig.(6)) and lower phase error for higher discretizations (Fig.(7)) is achieved as na increases from 8 to 12. Wavelets contribute to the improvement of the dispersion characteristics for even coarser cells (close to 2.2-2.4 cells/A) as it is demonstrated in Fig.(8)-(13). For discretizations above 4 cells/A the effect of the wavelets is negligible. (Fig.(1)) and (Fig.(13)) show clearly that the phase error has a minimum for a specific discretization (17 for = 10 and 25 for = 12). Fig.(14)-(17) show that for discretizations smaller than 30cells/A the choice of the Courant number affects significantly the dispersion performance which starts converging to the minimum numerical phase error (0.8 deg/A for na = nb = n, = 10 and 0.2 deg/A for ni = nb = n, = 12) for 1/q close to 10. On the contrary, the F.D.T.D. dispersion is almost independent of the Courant number (Fig.(18)-(19)). It has been claimed in [5] that the S-MRTD Scheme is slightly oscillating and its performance is only comparable with the 14th order accuracy Yee's scheme. Though this is true for the S-MRTD schemes with stencil size of 8, the comparison of the dispersion diagrams of Yee's FDTD scheme, Yee's 16th order (H.F.D.-16) and 22th order (H.F.D.-22) and S-MRTD and Wo-MRTD schemes with different stencils leads to interesting results. For comparison purposes, the values of At = Atmaz/5 and Atmax = 0.368112A1/c have been used and all the dispersion curves are substracted by the linear dispersion relation for ID simulations. Fig.(20) shows that the S-MRTD scheme with stencil 10 has a comparable performance to the 16th order Yee's scheme. The enhancement of the wavelets for the same stencil improves significantly the dispersion characteristics of the MRTD scheme increasing the dynamic range of w by approximately 90% and comparing favorable even to the 22th order Yee's scheme. This is expected due to the fact that the scaling+wavelet basis spans a larger ("more complete") subspace of R than the scaling functions alone. Both S-MRTD and Wo-MRTD schemes have identical numerical phase errors up to the point that the S-MRTD scheme starts diverging (Fig.(21)). As the stencil size of the Wo-MRTD scheme is increasing from 6 to 12 (Fig.(22)-(23)), the oscillatory variation of the phase error is diminishing to a negligible level generating an almost flat algorithm similar to the higher order Yee's ones. As a conclusion, due to the poor dispersion performance of the FDTD technique even for 10 cells/wavelength a normal to coarse grid is always required to avoid significant pulse distortions especially for the higherspatial-frequency components. MRTD offers low dispersion even for sparse grids very close to the Nyquist limit. V Conlusion The stability and the dispersion performance of the recently developed Battle-Lemarie MRTD schemes has been investigated for different stencil sizes and for 0-resolution wavelets. Analytical expressions for the maximum 6

stable time-step have been derived. Larger stencils decrease the numerical phase error making it significantly lower than FDTD for low and medium discretizations. Stencil sizes greater than 10 offer a smaller phase error than FDTD even for discretizations close to 50 cells/wavelength. The enhancement of wavelets further improves the dispersion performance for discretizations close to the Nyquist limit (2-3 cells/wavelength) making it comparable to that of much denser grids, though it decreases the value of the maximum time-step guaranteeing the stability of the scheme. VI Acknowledgments This work has been made possible by the U.S. Army Research Office. References [1] A.Taflove, "Computational Electrodynamics", Artech House, 1995. [2] M.Krumpholz, L.P.B.Katehi, "MRTD: New Time Domain Schemes Based on Multiresolution Analysis", IEEE Trans. Microwave Theory and Techniques, vol. 44, no. 4, pp. 555-561, Ap ril 1996. [3] K.S.Yee, "Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media", IEEE Trans. Antennas Propagation, pp.302-307, May 1966. [4] K.L.Shlager and J.B.Schneider, "Analysis of the Dispersion Properties of the Multiresolution Time-Domain Method", IEEE AP-S 1997 Proceedings, vol. 4, pp. 2144-2147, 1997. [5] W.Y.Tam, "Comments on "New Prospects for Time Domain Analysis"", IEEE Microwave and Guided Wave Letters, vol. 6, pp. 422-423, 1996. 7

Figure 1: Figure 2: Figure 3: Figure 4: Figure 5: Figure 6: Figure 7: Figure 8: Figure 9: Figure 10: Figure 11: Figure 12: Figure 13: Figure 14: Figure 15: Figure 16: Figure 17: Figure 18: Figure 19: Figure 20: Figure 21: Figure 22: Figure 23: Stability ParameteITD IURE CAPTIONS Dispersion Characteristics of S-MRTD for no=8. Dispersion Characteristics of S-MRTD for no= 10. Dispersion Characteristics of S-MRTD for na=12. Dispersion Characteristics of S-MRTD for na=14. Stencil Effect on the Dispersion Characteristics of S-MRTD. (Sparse Grid). Stencil Effect on the Dispersion Characteristics of S-MRTD. (Dense Grid). Wavelets Effect on the Dispersion Characteristics of MRTD for n.=8 (Coarse Grid). Wavelets Effect on the Dispersion Characteristics of MRTD for na=8 (Denser Grid). Wavelets Effect on the Dispersion Characteristics of MRTD for n==10 (Coarse Grid). Wavelets Effect on the Dispersion Characteristics of MRTD for na=10 (Denser Grid). Wavelets Effect on the Dispersion Characteristics of MRTD for n.=12 (Coarse Grid). Wavelets Effect on the Dispersion Characteristics of MRTD for na=12 (Denser Grid). Effect of the Courant Number on the Dispersion Characteristics of Wo - MRTD for na = nb = n = Effect of the Courant Number on the Dispersion Characteristi cs of Wo - MRTD for na = nb = n = Effect of the Courant Number on the Dispersion Characteristics of Wo - MRTD for no = nb = n = Effect of the Courant Number on the Dispersion Characteristics of Wo - MRTD for na = nb = nc = Effect of the Courant Number on the Dispersion Characteristics of FDTD (Coarse Grid). Effect of the Courant Number on the Dispersion Characteristics of FDTD (Denser Grid). Comparison of the Dispersion Performance of S-MRTD and Wo-MRTD with Different Higher Order I Details of Fig.(20). Comparison of the Oscillations of Wo-MRTD Scheme for Different Stencil Size. Details of Fig.(22). 8

0.7! I 1.. 0.65.0.55 -- WoWo Co 0.5 - - 0.45 - 0.4 -1. 0 2 4 6 8 10 12 na=nb=nc Figure 1: Stability Parameter s for MRTD. 9

Phase Error (Ste=8 vs. FDTD) 1/s Figure 2: Dispersion Characteristics of S-MRTD for na=8. Phase Error (Ste=10 vs. FDTD) S..... -- - - 2.5-: 2 E 0 la }1.5 0 a) 10 m MR10 — MR20 o MR40 FD20 -FD40 }K IK X --- —--------------------- )K )E 2oooooo o o o o o o \ "" I I 0.5 Itn 1 2 3 4 5 6 7 8 9 10 1/s - Figure 3: Dispersion Characteristics of S-MRTD for no=10. 10

Phase Error (Ste=12 vs. FDTD) 3 2 E G) m 0 CO 1 o a. MR10.5 -- MR20 o MR40 -- FD20 2 - - FD40.51 x.5 - o o o o I p o n5L -— ~ —'" ----. —K-".. ----"-...... [11N I ~ T 'T 3 I 0. 0 2 3 4 5 6 7 8 9 1/s 1 Figure 4: Dispersion Characteristics of S-MRTD for na=12. Phase Error (Ste=14 vs. FDTD) 1/s Figure 5: Dispersion Characteristics of S-MRTD for na=14. 11

Figure 6: Stencil Effect on the Dispersion Characteristics of S-MRTD (Sparse Grid). 10 15 20 25 30 35 40 45 50 Samples/Lambda Figure 7: Stencil Effect on the Dispersion Characteristics of S-MRTD (Dense Grid). 12

1 sol I I I I I I b 10o E 0 10 0 LLLU 0 = 50 a - na=8,nb=nc=0 na=8,nb=nc=4 K — na=8,nb=nc=8 na=8 nb=nc=12 FDTD I *,%" ^ I m - - wwwm --- — - - - n, 0 1 2 3 4 5 6 Samples/Lambda 7 8 9 10 Figure 8: Wavelets Effect on the Dispersion Characteristics of MRTD for n.=8 (Coarse Grid). it). I 9C 0 coJ v E 0 V 0 10 0 W a. I I II A II II 8 - na=8,nb=nc=O -- na=8,nb=nc=4 7 -- na=8,nb-nc=8 na=8 nb=-nc=12......FDTD 5... 4 -2-...................................... 15 20 25 Samples/Lambda 30 35 40 Figure 9: Wavelets Effect on the Dispersion Characteristics of MRTD for n.=8 (Denser Grid). 13

'-14 -D E a 12 CD '10 0 Ie,a. 6 0 2 4 6 8 10 12 Samples/Lambda Figure 10: Wavelets Effect on the Dispersion Characteristics of MRTD for na=10 (Coarse Grid). 10 III * -w 9F 8F - na=10,nb=nc=0 - - na=10,nb=nc=6 - - na=10,nb=nc=10. FDTD e0 E -- UJ OD 0 C:,,. 71 6 5 4 3h 2 1............. I I.................... I I I I - I nt Ih e-A 1-*I 10 15 20 25 30 35 40 45 50 Samples/Lambda Figure 11: Wavelets Effect on the Dispersion Characteristics of MRTD for na=10 (Denser Grid). 14

0 2 4 6 8 10 12 Samples/Lambda Figure 12: Wavelets Effect on the Dispersion Characteristics of MRTD for na=12 (Coarse Grid). 2,. I, i - - -- ~~ ---0 _ l —L.- -- A% 1.8 - na=lznD=nc=u - na=12,nb=nc=8 1.6.-.- na=12,nb=nc=12..... FDTD -c 1.4 -X 1.2 0.8.. e 0.8 -0.6-... 0.4 0.2 20 25 30 35 40 45 50 55 60 Samples/Lambda Figure 13: Wavelets Effect on the Dispersion Characteristics of MRTD for na=12 (Denser Grid). 15

rn is/r 9 B 8 7 co 10.0 E co 0 2 1-p CL 0 a, 6t 5 \ I- 1/q=1.1 -- 1/q=3 \ - 1/q=5 \ 1/q=10 I:.... 41 3 2 1 0 2 3 4 5 6 7 8 Samples/Lambda 9 10 11 12 Figure 14: Effect of the Courant Number on the Dispersion Characteristics of Wo - MRTD for no = nb = nc=10 (Coarse Grid). 0 la m E 0 -a 0 Ul v /) 0 r 30 35 40 Samples/Lambda 60 Figure 15: Effect of the Courant Number on the Dispersion na = nb = n,=10 (Denser Grid). Characteristics of Wo - MRTD for 16

D 10.0 E co -D U, 2 Co f.. 2 3 4 5 6 7 8 Samples/Lambda 9 10 11 12 Figure 16: Effect of the Courant Number on the Dispersion Characteristics of Wo - MRTD for n. = nb = n,=12 (Coarse Grid). 1 0.0 E L. 0 VC W.c a. 30 35 40 Samples/Lambda 60 Figure 17: Effect of the Courant n. = nb = nc=12 (Denser Grid). Number on the Dispersion Characteristics of Wo - MRTD for 17

20 15 0 vC 10 CL a. o 2 3 4 5 6 7 8 Samples/Lambda 9 10 11 12 Figure 18: Effect of the Courant Number on the Dispersion Characteristics Grid). 2 \ 1/q=1.1 1\ - 1/q-3 1.5 1/q —5 10 15 1/q=10 0. -0.5.. a." a0.5 10 15 20 25 30 35 40 45 50 55 Samples/Lambda Figure 19: Effect of the Courant Number on the Dispersion Characteristics Grid). of FDTD (Coarse 60 of FDTD (Denser 18

0." 0% PI,25F I I I,&..i + 0.21 - -TI -~ F.D.T.D. + H.F.D.-16 * H.F.D.-22 - - S-M.R.T.D. --— WO-M.RT.D + 0.15 0.1 F * 0.05 x 0 I 0.,. - +e +.~ — -' + I I I i I I / / 0 111ii9t91Q11 HI El H 11I11I9111H fWf! 99tI1fjI1.Ibjjb -0.0SF~ \ I \ I -A-.. V- I '1 5 1 - -- - II 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 ED*w Figure 20: Comparison of the Dispersion Performance of S-MRTD and Wo-MRTD with Different Higher Order Yee's Schemes. x1 a 0.05 0.1 0.15 0.2 0.25 Dt'w 0.3 Figure 21: Details of Fig.(20). 19

0. 25, 1 1 1 1 1 7 &..; I 0.2F -~ M.R.T.D.-6 * M.R.T.D.-8 - - M.R.T.D.-10 --— M.R.T.D.-l12.H.F.D.-22 0.15 F 0 0.11 I 0.05 F Ci — 4;Z;aN -A -- I I I I I,(s I 0 0.05 0.1 0.15 EXt*w 0.2 0.25 0.3 0.35 Figure 22: Comparison of the Oscillations of Wo-MRTD Scheme for Different Stencil Size. 0.02 -M.R.T.D.-6 0.015 *M.R.T.D.-8I — M.R.T.D.-10 0.0 * -M.R.T.D.-12........H.F.D.-22 0.005 __ _ _ __ _ _ _ 0j 0.15 ~Dt *w Figure 23: Details of Fig.(22). 20

To be submitted to the IEEE Microwave and Guided Wave Letters TIME ADAPTIVE TIME-DOMAIN TECHNIQUES FOR THE DESIGN OF MICROWAVE CIRCUITS Emmanouil M. Tentzeris1, James Harvey2, Linda P.B. Katehi1 Radiation Laboratory, Department of Electrical Engineering and Computer Science University of Michigan, Ann Arbor, MI 48109-2122 2 Army Research Office, NC Abstract A novel Time Adaptive Time-Domain Technique based on the Haar expansion basis is proposed and applied to various circuit problems. This scheme offers improved time resolution in comparison to conventional Time-Domain schemes (F.D.T.D.) while maintaining a similar accuracy with commercial circuit simulators. I Discussion on the Expansion Basis Choice for MRTD It is well known that the method of moments provides a mathematically correct approach for the discretization of integral and partial differential equations. Since it allows for the use of any complete and orthonormal set, the choice of an appropriate expansion set may lead to different time domain schemes. For example, the expansion of the unknown fields using pulse functions leads to Yee's FDTD scheme. In a MRTD scheme [?,?] the fields are represented by a two-fold expansion in scaling and wavelet functions with respect to time/space. Scaling functions guarantee a correct modelling of smoothly-varying fields. In regions characterized by strong field variations or field singularities, higher resolution is enhanced by incorporating wavelets in the field expansions. The major advantage of the use of Multiresolution analysis to time domain is the capability to develop time and space adaptive grids. This is due to the property of the wavelet expansion functions to interact weakly and allow for a spatial sparsity that may vary with time through a thresholding process. MRTD schemes based on cubic spline Battle-Lemarie scaling and wavelet functions have been successfully applied to the simulation of 2D and 3D open and shielded problems [?,?,?,?]. The functions of this family do not have compact support, thus the MRTD schemes have to be truncated with respect to space. Localized boundary conditions (PECs, PMCs etc.) and material properties are modelled by use of the image principle and of matrix equations respectively. However, this disadvantage is offset by the low-pass (scaling) and bandpass (wavelets) characteristics in spectral domain, allowing for an a priori estimate of the number of resolution levels necessary for a correct field modelling. In addition, the evaluation of the moment method integrals during the discretization of Maxwell's PDEs is simplified due to the existence of closed form expressions in spectral domain and simple representations in space domain. Dispersion analysis of this MRTD scheme shows the capability of excellent accuracy with up to 2 points/wavelength (Nyquist Limit). However, specific circuit problems may require the use of functions with compact support. Epsecially in the approximation of time derivatives, the use of entire domain expansion basis would require very high memory resources for the storage of the field values everywhere on the grid for the whole or a large fraction of the simulation time. This problem does not exist in the approximation of the spatial derivatives since the field values on the neighboring spatial grid points have to calculated and stored no matter what expansion basis are used. For that reason, Haar basis functions have been utilized and have led to [?]. As an extension to this approach, intervalic wavelets (Fig.l1) may be incorporated into the solution of SPICE-type circuits. Results from that new technique will be presented in this Chapter. 1

II Applications in SPICE problems For simplicity, the ID MRTD scheme will be derived. It can be extended to 2D and 3D in a straightforward way. In addition, only the 0-resolution of wavelets is enhanced. The Voltage and the Current are displaced by half step in both time- and space-domains (Yee cell formulation) and are expanded in a summation of scaling functions in space and scaling (X) and wavelet (tbo) components in time 00 00 V(zt) E E (iV v, (t) + i VO o,i (t)) m () m=-00 i=-oo 00 00 I(zt) = E (.I0.5Im-.o.5i-0._5(t) + i-0.5I-o.50,i-0.5(t)) Om-0.5(z), (1) m=-oo i=-oo00 where 4$i(t) = 0(t/At - i) and Oo,i(t) = o(t/At - i) represent the 0-order intervalic scaling and 0-resolution wavelet functions. The conventional notation kVm is used for the voltage component at time t = kAt and z = mAz, where At and Az are the time-step and the spatial cell size respectively. The notation for the current I is similar. Due to the finite-domain nature of the expansion basis, the Hard Boundary conditions (Open/Short Circuit) can be easily modeled. If a Short Circuit exists at the z = mAz, then both scaling and wavelet voltage coefficients for the m - cell must be set to zero for each time-step k. kV2 =^kV~ =0, k=0,l,2,... (2) Similarly, an Open Circuit at z = (m - 0.5)Az can be modeled by applying the conditions k-0o.5I-0.5 = k-O.5I 0.5 0, k = 0, 1, 2,... (3) The alternating nature of the 0-resolution wavelet function guarantees the double time-domain resolution of the MRTD scheme. Assuming that the voltage scaling and wavelet coefficients at m = Az for a specific time-step k, two values can be defined for the time span [(k - 0.5)At, (k + 0.5)At] of this time-step kvotall =- kVm + kVm~, t E [(k - 0.5)At, kAt] (4) kVtotal2 = kVm - V~, t E [kAt, (k + 0.5)At] (5) II.1 Distributed Elements II.1.1 Lossless Line The ideal transmission line (Fig.2) equations are given by dV dI dz = - dt d = - Cdi (6) dz - (6) where Ldis, Cdia are the distributed inductance and capacitance of the line. Inserting the expansions of Eq.(l) and applying the Method of Moments, the following MRTD equations are derived i (k+lV- ) = - (k+0.5I+0.5 - k+.5Im-0.5) (7) 2

t (k o k o) = (k+0.5sm+o., - k+o.5.-0.5) (8) Ldt (k+0.5 O. -0. - - o.05. = ) (1 -9) t (k+0.5m-0.5 k- 0.5-0.5) = -- (Vm k 1 (10) It can be observed that Eqs.(7) and (9) updating the scaling coefficients only are independent of the Eqs.(8) and (10) updating the wavelet coefficients. To create an efficient time adaptive algorithm, all four equations must be coupled. An efficient way is to apply the excitation in a physically correct manner. If the excitation has the time-dependence g(t) at the location z = meAz, then the scaling and wavelet coefficients for this cell have to take the values r(k+0.5)At kV = / t g(t) 4k(t) dt (c-0.5)At (k+0.5)At kV, = g(t) o,k(t) dt. (11) (k-0.5)At To validate this approach, the MRTD algorithm was applied to the simulation of a lossless transmission line with (Ldi,,Cdi.) = (20nH/m, 3nF/m) for a Gabor excitation and time-step dt = dtmaz/l.01. Fig.(3) which displays the Voltage Scaling and Wavelet Coefficients evolution at z = 200Az for the first 800 time-steps of the simulation, shows that the wavelet coefficients have the correct shape (significant values only at areas with significant scaling function values) and are close to the 12% of he respective scaling functions. Fig.(4) which compares the total voltage value at z = 200Az calculated by FDTD (Sc.ONLY) and MRTD (Sc.+Wav.) for the time-steps 357-362 demonstrates the ability of this MRTD scheme to double the conventional FDTD resolution in the time-domain by providing two values for each time-step. The fact that the wavelet coefficients take significant values only for a small number of time-steps allows for their thresholding by comparing them to a combination of relative to the respective value of the scaling coefficient (5.e-4) and absolute (1.e-6) thresholds. Fig.(5) proves that up to 60% of the maximum number of wavelet coefficients are necessary for an accurate simulation, offering an extra economy in memory by a factor of 20%. 11.1.2 Lossy Line The lossy transmission line (Fig.6) equations are derived by the ideal transmission line equations (Eq.(6)) adding the Conductor Loss Rdis and the Dielectric Loss Gdi, dV dl dz - Rdi.I -Lds d dI dV d -Gd V - Cdi.d (12) Following a procedure similar to the previous section, the following MRTD equations are derived k+1V = - C At (+0.5+O.5 - k-OIm-0.) + 2 t (k+0.5m+O.5 -k-0.5Im-0.5) k+1lVO - C At (k+0.5m+0.5 - k-0.5Imo-.) + 2) At (k+o.5ImO.5 — o0.5-0.5) 2 -(C C2)2 + ) v 2 Cn2 ko C ^2 k Xm (14) 3

(.5 - C4) -t (,+ At (k - k1 o ) 'At (kI'c kVc) ( At C3 C3 2 X -0k.5 5+ (C3 — 2 -0.5Im-0.5 X (16) 3 c3 with C1 = Cdi.Az, C2 = 0.5 GdiAZ At, C3 = LdiAz, C4 = 0.5 Rdi,Az At, For this type of transmission line, the equations giving the scaling and wavelet coefficients for voltage and current are coupled. Nevertheless, the condition (11) has to be applied in order to satisfy the physical boundary condition at the excitation cell(s). It has to be noted, that Eqs.(13)-(16) can be used only for lossy lines with low to medium Loss Coefficients. The threshold C2 < 4C7 for Gdia (C4 < 4C3 for Rdi,) gave satisfactory results for all simulations. For higher loss coefficients, the Loss can be modeled in an exponential way similar to [?]. For example, for large values of Rdis (C4 > C3), Eqs.(15)-(16) have to be replaced by the following uncoupled expressions k+O.SI.-o.5 = e -o.sm-o.5- e ( -k -_) (17) k —0.5 m0 —0.5 -- C3kV k-+. RAt~ = -o.RS.At VAt +_0o.5 C m - =,00.5.5e (V o _-,kV ) (18) Using this procedure, a termination layer similar to the FDTD widely used PML layer can be easily modeled. The Rdis, Gdi should have a spatial parabolic distribution with very high maximum value and they should satisfy the condition Gdi. = RdijLdi,/Cdis for each cell of the layer. In this way, one matched transmission line can be simulated by choosing the appropriate Rdi, Gdis that satisfy the specified numerical reflection coefficient (usually smaller than -80dB). For validation purposes, the propagation of a Gabor pulse along a lossy line with Rdis = 5M/m has been simulated and the scaling and wavelet voltage coefficients have been probed at the positions z = 140Az and z = 160Az. Data for the first 200 time-steps (At = 2At/3) have been plotted in Fig.(7). The maximum value of the wavelet coefficients (approximately 7% of the respective scaling coefficient) is smaller than that of the lossless line. By applying a thresholding procedure using an absolute threshold of 10-6 and a relative threshold of 5e -4, an extra economy of 29% is achieved, since only 60% of the voltage and 25% of the current wavelet coefficients take values above the thresholds throughout the simulation time (Fig.(8)). 11.2 Lumped Elements 11.2.1 Passive Elements Lumped Passive Elements such as Capacitors, Inductors and Resistors can be modeled in a similar way with the Distributed ones by numerically distributing them along one cell. For example, if one lumped Capacitor Clum is located at z = mAz along a lossy line with (Rdi4, Gdis, Ldis, Cdis), the voltage coefficients k+lVm, k+l1VmO will 4

still be given by Eqs.(13)-(14). The only difference is that the constant C1 will have the new value C1 = CttA with C.um Ctot = Cdit + iz. (19) 11.2.2 PN-Diode To model lumped active elements such as a PN-diode, their nonlinear equation has to be discretized after inserting the voltage and current expansions. The MRTD equations are not linear and require the use of numerical solvers for nonlinear systems. The combined Newton-Raphson/Bisection solver has provided stable solutions for PN-diode simulations with Io <.e - 10A, though sometimes diverges for larger values. The voltage scaling and wavelet coefficients for the diode cell are updated by inserting the voltage and current expansion in the equation IDIODE(V) = IO (eqVT _ 1) (20) adding the diode capacitance Cj to the Cdi, and applying the moments method, thus giving the nonlinear system for a diode positioned in parallel (C5 + Cdi,) kV, + C5 kVfo + (C- Cdi.) k1V,- C5 k-IVIO + (V-(0.5.- mo5 - k-.5I-o.5) + 0.5 At Cj(eT/q Io (^-_v-^-_vg~) + eT/V so (^.V +^v~O)) = 0 (21) -C5 kVm - (C5 - Cdi) kVm~o + C5 klV, - (C5 + Cum) k ~1VO + (U-0.5I 0.5 - k-o. 5I-o.s) + 0.5 At Cj(ekTq Io (,_-iV -^,_-V~) - ekT/q Io (,V +,V~)) = 0 (22) with C5 = 0.5 At Gium. (23) To validate the algorithm, the rectifier topology of Fig.(9) is analyzed using FDTD (Scaling Only) with At = Atmaz/4.4 and MRTD (Scaling+wavelets) with double time-step At = Atma,,/2.2. A lossless line with (Ldi9, Cdis) = (20nH/m, 3nF/m) and a PN-Diode with Io = 3pA are used in the simulation. The probed total voltage is plotted in Fig.(10) and the agreement is very good. The use of an absolute threshold of 10-6 and a relative threshold of 5e - 4 offers an extra economy of 35% for the MRTD algorithm. III Conclusion A Time Adaptive Time-Domain Technique based on intervalic wavelets has been proposed and applied to various types of circuits problems with active and passive lumped and distributed elements. This scheme exhibits significant savings in execution time and memory requirements while maintaining a similar accuracy with conventional circuit simulators. References [1] M.Krumpholz, L.P.B.Katehi, "MRTD: New Time Domain Schemes Based on Multiresolution Analysis", IEEE Trans. Microwave Theory and Techniques, vol. 44, no. 4, pp. 555-561, April 1996. [2] E.Tentzeris, M.Krumpholz and L.P.B. Katehi, "Application of MRTD to Printed Transmission Lines", Proc. MTT-S 1996, pp. 573-576. [3] R. Robertson, E. Tentzeris, M. Krumpholz, L.P.B. Katehi, "Application of MRTD Analysis to Dielectric Cavity Structures", Proc. MTT-S 1996, pp. 1861-1864. [4] E.Tentzeris, R.Robertson, M.Krumpholz and L.P.B. Katehi, "Application of the PML Absorber to the MRTD Technique", Proc. AP-S 1996, pp. 634-637. 5

[5] K.Goverdhanarn, E.Tentzeris, M.Krumphoiz and L.P.B. Katehi, "An FDTD Multigrid based on Multiresolution Analysis", Proc. AP-S 1996, pp. 352-355. [6] E.Tentzeris, R.Robertson, A.Cangellaris and L.P.B. Katehi, "Space- and Time- Adaptive Gridding Using MRTD", Proc. MTT-S 1997, pp. 337-340. 6

LIST OF FIGURE CAPTIONS Figure 1: Figure 2: Figure 3: Figure 4: Figure 5: Figure 6: Figure 7: Figure 8: Figure 9: Figure 10: 0-Order Intervalic Function Basis. Ideal (Lossless) Transmission Line. Voltage Coefficients. Comparison MRTD-FDTD. Fraction of Wavelets above Threshold. Lossy Transmission Line. Voltage Coefficients. Fraction of Wavelets above Threshold. Rectifier Geometry. Comparison MRTD-FDTD. 7

1/Dt I/Dt I I I I t t - 4 - - U I (k-O.5) Dt (k+O.5) Dt (k-O.5) Dt (k+O.5) jD -1/Dt it Figure 1: 0-Order Intervalic Function Basis. Ld VW -1 - m 00 I9 ds V 6M=NbI - z Figure 2: Ideal (Lossless) Transmssion Line. 8

0.5 0.5 -\I -i1 - 0.4- Scaing - - Wavelet 0.3 - 0.2 -0.1 0 -0.1 -0.2 -0.3 -0.4 -0.5..................... --- 0 100 200 300 400 500 Time-Step (=dtmax/1.01) 600 700 800 Figure 3: Voltage Coefficients. fl5.S I I I II I 0.48 0.46 0.44 0.42 0, 0.4 O / - MRD (Sc+Wav) /l/ L7. 3 3 3 35. 36 36. 31 3 3 0.38 0.36 0.34 0.32 nr %ll ~ v '57 357.5 358 358.5 359 359.5 360 360.5 361 361.5 362 Time-Step Figure 4: Comparison MRTD-FDTD. 9

Percentage of Non-zero Wavelets (x100%) 1. 0.5 0. Volt. Cur. Thr= le-6 (Eco: 20%) 500 Time-Step Figure 5: Fraction of Wavelets above Threshold. Rdis Ldis A AAK 'V - - - V ~-QI Gdis Cdis z U Figure 6: Lossy Transmission Line. 10

co -0.2 " ' "', j e ': t t I' 1 iI -0.4 1 -0.6 - i e i -1.. -0.8 i 0 20 40 60 80 100 120 140 160 180 200 Tre-Step Figure 7: Voltage Coefficients. Percentage of Non-zero Wavelets (x100%) 1.0 0.5 Volt. Cur. Thr= 1.-6 (Eco: 29%) 0.0 F 0 Time-Step Figure 8: Fraction of Wavelets above Threshold. 11

Rdis Ldis Gdis Z I fC dis Figure 9: Rectifier Geometry. 4,,! -Input 3 --- FDTD — MRTD 2 -_I 0) V~rAp 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Tife-Step (sdtax/2.2) Figure 10: Comparison MRTD-FDTD. 12

Presented to the 1997 ACES Conference Space/Time Adaptive Meshing and Multiresolution Time Domain Method (MRTD) Emmanouil Tentzerisl, Andreas Cangellaris2, Linda P.B. Katehi1 'Radiation Laboratory, EECS Department, University of Michigan, Ann Arbor, Ml 48109-2122, USA 2University of Arizona, Tucson, AZ, USA I Introduction Recently the principles of the Multiresolution Analysis have been successfully applied [1, 2] to the time-domain numerical techniques used for the analysis of a variety of microwave problems. New techniques have been derived by the use of scaling and wavelet functions for the discretization of the time-domain Maxwell's equations. The multiresolution time domain technique (MRTD) based on Battle Lemarie functions has been used for the simulations of planar circuits and resonating structures. The conventional FDTD absorbers (e.g. PML) have been generalized in order to analyze open planar structures. MRTD has demonstrated unparalleled savings in execution time and memory requirements (2 orders of magnitude for 3D problems). In addition to time and memory, MRTD technique can provide space- and time- adaptive meshing without the problems that the conventional FDTD variable grids are encountering (e.g. reflections between dense-coarse regions). This unique feature stems from the use of two separate sets of basis functions, the scaling and wavelets. Due to the excellent conditioning of the formulated mathematical problem, MRTD offers the capability to threshold the wavelet field coefficients. This advantage of the MRTD Technique is demonstrated herein by performing a space-/timeadaptive meshing. In this paper, a space-/time- adaptive meshing algorithm based on the MRTD scheme is proposed and validated for a specific waveguide problem. Wavelets up to the second resolution are placed only at locations where the EM fields have significant values. These locations are changing with the time as the pulse is propagating inside the waveguide and with the space as the pulse is approaching regions of discontinuities. The proposed algorithm offers the opportunity of a space-/time- adaptive mesh with variable resolution of the field representation. In this way, significant memory and execution time savings can be achieved in comparison to the conventional variable-mesh FDTD algorithms.

II MRTD Formulation Without loss of generality, the 2D-MRTD scheme for the TM. modes will be described herein. To derive the scheme equations, the field components are represented by a series of cubic spline Battle-Lemarie scaling and 1-order wavelet functions along the z-direction, while pulses are used for the time representation. Wavelets of higher-order can be included in a similar way. After inserting these series expansions in Maxwell's equations and sampling them with pulse functions in time and scaling/wavelet functions in space domain, we derive the following equations for the electric field: 1 1 m+ml mmm3 - L+1/2m) - - E b(i)k+1mm 1(k+lD+1/2,m kD 2,m) a(i)k+1/2 l/2,i+1/2 +)+1/2 H1/2,i+1/2) i=m-m2 i=m-m4 1 1 m+m3 m+m5 E(k+lD l/2,m -kDl+12m) ( b(i)k+l/2 Hl'l/2,i+1/2 + C(Z)k+l/2 H+2,i1/2) i=m-m4 i=m-me 1 1 e1+l Di (k+l1Dm+1/2 -) = ( a(i)k+l/2 Hi+/2,m+1/2) i=1-12 1 1+13 t +l+ l/2 - kD +/2) - "( EZ (i)k+1/2 HO/2,+1/2) t(k+lDm+l2kDm+l/2) ( ()k+l/2Hi+1/2,1+1/2) i=1-14 where kD I', kE, and kHyf with ~=q (scaling),+ (wavelets) are the coefficients for the electric flux, electric and magnetic field expansions. The indices 1, m and k are the discrete space and time indices, which are related to the space and time coordinates via x = lAx,z = mAz and t = kAt, where Axa,Az are the space discretization intervals in x- and z-direction and At is the time discretization interval. The coefficients a(i), b(i), c(i) are derived and given in [1]. For an accuracy of 0.1% the values ml = m5 = 8, m2 = m3 = m4 = m6 = 9 have been used. The indices 1i have to take similar values to achieve tha same accuracy in the summations. The use of non-localized basis functions in the 2D-MRTD scheme causes significant effects. Localized boundary conditions are impossible to be implemented, so the perfect electric boundary conditions are modelled by use of the image principle in a generic way. The implementation of the image theory is performed automatically for any number of PEC, PMC boundaries. The material discontinuities are represented in terms of scaling and wavelet functions resulting into a linear matrix equation as explained in [1, 3] where this technique was used in the modeling of anisotropic dielectric media. In addition, the total value of a field component at a specific point of the mesh is a summation of the contributions from the neighbooring non-localized scaling and wavelet functions. The field values at the neighbooring cells can be combined appropriately by adjusting the scaling and wavelet function values and by applying the image principle. The demand for the simulation of open structures led to the generalization of the perfectly matched layer (PML) technique [4], so as it can be used in the MRTD simulations. The conductivity is expanded in terms of scaling functions instead of pulse functions with respect to

space. The amplitudes of the expansion scaling functions follow the PML spatial conductivity distribution. In our simulations, the parabolic distribution was used, though the realization of other distributions (linear, cubic,...) is straightforward. For example, if we assume that the PML absorbing material (e, Ar, oE) extends to the z-direction, substituting D(i), z, t) = -b(')xz(x z, t)e-)t/( (1) and H(Y(x z, t) = f()(x, z, t)e-O)t/ (2) for i=-b, Vt, leads to the following equation: aDz - aftHy (3) Following a procedure similar to the one used for the derivation of the non-PML region equations, we get for D: components _O )0 At/Ek qx k+l1+Di /2,m e (m) k+l/2,m At _E m+ml m+ma - -e (mA / E a(i)k+1/2 Hl+l/2,i+1/2 + E b(i)k+l/2 H'+l/2, +/2) A i=m-m2 i=m-m4 g-tOx — ff )At/,k qTlfx k+lDt+l/2,m - = e (maZ) kt+lm A... m+m3 m+m5 e —()O5t/ E b(H)+11/2 H1+/2,i+1/2 + C()k/2 2,i+1/2) i=m-m4 i=m-m6 The finite-difference equations for D(+t) and H(+09)y are similar. For all simulations, a parabolic distribution of the conductivity a is used in the PML region (N cells): (mH) = EH(M)2 for m=0,l,..,N, (4) with CEaH the maximum conductivity at the end of the absorbing layer. As in [5], the "magnetic" conductivity aH is given by: E FH (mZ) = (mz) for m=0,1,..,N, (5) e A and the MRTD mesh is terminated by a perfect electric conductor (PEC) at the end of the PML region. This PEC is modelled by applying the image theory.

III Space/Time Adaptive Meshing The wavelet components' amplitudes have negligible values away from the discontinuities or at regions where the excitation pulse has not propagated yet. There are numerous ways of taking advantage of the above feature. The simplest one is to threshold the wavelet components to a fraction (usually < 0.1%) of the scaling component at the same cell (space adaptivity) for each time-step. All components below this threshold are eliminated from the subsequent calculations for the same time-step (time adaptivity). This procedure offers only a moderate economy in memory (factor close to 2). Also, this algorithm allows for the dynamic memory allocation in its programming implementation, while maintaining a low complexity. The above space-/time- adaptive meshing scheme is applied to the analysis of the partiallyloaded parallel-plate waveguide of (Fig.l) for the frequency range 0-22.5GHz. The waveguide is half-filled with air and half-filled with dielectric with r, = 2.56. An FDTD 16 x 640 (10240 cells) mesh and an MRTD 2 x 80 (160 cells) mesh (160 grid points with dx = 0.18Ao, dz = 0.3Ao - close to the Nyquist Limit for f = 22.5GHz) are used for the Time-Domain simulations (3,000 time-steps). The 160 grid points of the MRTD mesh express the number of the used scaling functions. The number of the wavelets is varying with time and depends on the predefined threshold. For consistency, the time step for both schemes is chosen to be equal to the 1/8 of the FDTD maximum At. The waveguide is excited with a Gabor function 0-22.5GHz along a vertical line for the FDTD simulation and for a rectangular region of 12 cells to the longitudinal direction (due to the non-localized character of the Battle-Lemarie scaling and wavelet functions) for the MRTD simulations. Other excitations (e.g.Gaussian) can be applied i a straightforward way. For both cases, a PML region of 16 cells and craE=0.4S/m absorbs the waves in the front and back open planes. The capability of the MRTD technique to provide space- and time- adaptive gridding is verified by thresholding the wavelet components to the 0.1% of the value of the scaling function at the same cell for each time-step. The accuracy achieved by using only the wavelets with values above the threshold is equal to what would be if wavelets were used everywhere. Though this number is varying in time, its maximum value is 36 out of a total of 160 to the z-direction (economy in memory by a factor of 52 instead of 32). In addition, execution time is reduced by a factor 4-5. For larger thresholds, the ringing effect due to the elimination of the wavelets deteriorates the performance of the algorithm. For example, using a threshold of 1% (13 out of a 160 wavelets to the z-direction) increases the error by a factor of 2.1. The results for the Reflection Coefficient for 10 GHz are validated by comparison to the theoretical value jIR = 0.231 (=(v.56-1.0)/(v.56+1.0)). MRTD gives the value 0.2296 and FDTD gives 0.2304 (similar accuracy). The normal electric field is probed at a distance 10 cells away from the source and is plotted in (Fig.2) in time-domain. Similar accuracy can be observed for

the FDTD and the MRTD meshes. Fig.3 demonstrates the space- and time-adaptive character of the meshing algorithm. It is clearly shownw that the wavelets follow the propagating exciation pulse before and after the incidence to the dielectric interface and can be omitted elsewhere. The location and the number of the wavelet coefficients with values above the threshold ("effective wavelets") are different for each time-step, something that creates a mesh with high resolution ("dense") in regions of strong field variations, while maintaining a much lower resolution ("coarse") for the rest cells. IV Conclusion A simple space- and time- adaptive meshing algorithm based on an MRTD scheme has been proposed and has been validated for a parallel-plate waveguide problem. The electric field value and the reflection coefficient have been calculated and verified by comparison to reference data. The proposed scheme exhibits memory savings by a factor of 52 in 2D, as well as execution time savings by a factor of 4-5, while maintaining a similar accuracy with Yee's conventional FDTD scheme. In addition, this algorithm doesn't increase the programming complexity and can be effectively extended to 3D problems. V Acknowledgments This work has been partially funded by ARO and by NSF. References [1] M.Krumpholz, L.P.B.Katehi, "MRTD: New Time Domain Schemes Based on Multiresolution Analysis", IEEE Trans. Microwave Theory and Techniques, vol. 44, no. 4, pp. 555-561, April 1996. [2] E.Tentzeris, M.Krumpholz and L.P.B. Katehi, "Application of MRTD to Printed Transmission Lines", Proc. MTT-S 1996, pp. 573-576. [3] R. Robertson, E. Tentzeris, M. Krumpholz, L.P.B. Katehi, "Application of MRTD Analysis to Dielectric Cavity Structures", Proc. MTT-S 1996, pp.. [4] E.Tentzeris, R.Robertson, M.Krumpholz and L.P.B. Katehi, "Application of the PML Absorber to the MRTD Technique", Proc. AP-S 1996, pp. 634-637. [5] J.-P. Berenger, "A Perfectly Matched Layer for the Absorption of Electromagnetic Waves", J. Computational Physics, vol. 114, pp. 185-200, 1994.

/ e/r=l.O er=2.56 4.8mm - * - Figure 1: Dielectric-loaded Waveguide. I I I X _ x T 0.8 0.6 0.4 o 1o 02 1 0 \ I -MRTD -- FDTD -- -----— L ----. -0.2 -0.4 - -0.6 -,.8 _ I i I I I I 0 1000 2000 3000 4000 Time-Step 5000 6000 7000 Figure 2: Normal E-field Time Evolution.!~! 5 ' Lti": ~;?i~:i;? I 4 I I( ~ I -, -..... I. --— L:* '.^si t before -; T7^-7777 - incidence 4E+ --- t- after..; t ~,- incidence Figure 3: Space-/Time- Adaptive Meshing Demonstration.

Presented to the IEEE MTT-S Microwave Conference 1997 SPACE- AND TIME- ADAPTIVE GRIDDING USING MRTD TECHNIQUE Emmanouil M. Tentzeris, Robert L. Robertson, Linda P.B. Katehi Radiation Laboratory, Department of Electrical Engineering and Computer Science University of Michigan, Ann Arbor, MI 48109-2122 Andreas Cangellaris Department of Electrical and Computer Engineering University of Arizona, Tucson, AZ Abstract- The MRTD scheme is applied to the analysis of waveguide problems. Specifically, the field pattern and the S-parameters of a dielectric-loaded parallel-plate waveguide are calculated. The use of wavelets enables the implementation of a space- and time-adaptive gridding technique. The results are compared to those obtained by use of the conventional FDTD scheme to indicate considerable savings in memory and computational time. I Introduction Recently a new technique has been successfully applied [1-4] to a variety of microwave problems and has demonstrated unparalleled properties. This technique is derived by the use of multiresolution analysis for the discretization of the time-domain Maxwell's equations. The multiresolution time domain technique (MRTD) based on Battle Lemarie functions has been applied to linear as well as nonlinear propagation problems. The PML absorbing boundary condition has been generalized in order to analyze open planar structures. MRTD has demonstrated savings in time and memory of two orders of magnitude. In addition, the most important advantage of this new technique is its capability to provide space and time adaptive gridding without the problems that the conventional FDTD is encountering. This is due to the use of two separate sets of basis functions, the scal ing and wavelets and the capability to threshold the field coefficients due to the excellent conditioning of the formulated mathematical problem. In this paper, a space/time adaptive gridding algorithm based on the MRTD scheme is proposed and applied to the waveguide problems. As an example, the propagation of a Gabor pulse in a partiallyfilled parallel-plate waveguide is simulated and the S-parameters are evaluated. Wavelets are placed only at locations where the EM fields have significant values, creating a space- and time- adaptive dense mesh in regions of strong field variations, while maintaining a much coarser mesh elsewhere. II The 2D-MRTD scheme For simplicity the 2D-MRTD scheme for the TM, modes will be used herein. To derive the 2D-MRTD scheme, the field components are represented by a series of cubic spline Battle-Lemarie [5] scaling and wavelet functions to the longitudinal direction in space and pulse functions in time. After inserting the field expansions in Maxwell's equations, we sample them using pulse functions in time and scaling/wavelet functions in space domain. As an example, sampling 9D,/t, = - OHy/Oz in space and time, the following difference equation is obtained 1

i /(k+lDz p0T __) I+ 1 +/2,m l+ l /2,m Ay. E a(()s+l/2 H +1/2,i+1/2 i=m-m2 m+m3 + b(i)k+1/2 iH+l/2,i+l/2) ' (1) i= m-m4 r- z * + 1 + 1+l/2,m - *ko+1/2,m) = m+ms Ay( E b(i)+l/2 E,+1/2,i+l/2 i=m-m4 m+ms + E c(i)k+ /2 H+1/2,,+/2), (2) i=m-m6 where kDrm and kHfi, with t=- (scaling),+ (wavelets) are the coefficients for the electric and magnetic field expansions. The indices l,m and k are the discrete space and time indices, which are related to the space and time coordinates via x = IAx,z = mAz and t = kAt, where Az,Az are the space discretization intervals in x- and z-direction and At is the time discretization interval. The coefficients a(i), b(i), c(i) are derived and given in [2]. For an accuracy of 0.1% the values ml = m5 = 8,m2 = m3 = m4 = m6 = 9 have been used. For open structures, the perfectly matched layer (PML) technique can be applied by assuming that the conductivity is given in terms of scaling and wavelet functions instead of pulse functions with respect to space [4]. The spatial distribution of the conductivity for the absorbing layers is modelled by assuming that the amplitudes of the scaling functions have a parabolic distribution. The MRTD mesh is terminated by a perfect electric conductor (PEC) at the end of the PML region. Usually, 8-16 cells of PML medium with orax=0.4S/m provide reflection coefficients smaller than -90 dB. In order to use a pulse excitation at z = mAz with respect to space and to obtain an excitation identical to an FDTD excitation, we decompose the pulse in terms of scaling wavveet o t) +4 +4 (ZE C()m+i + jE co(i)m+i) (3) i=-4 i=-4 where the coefficients cO,(i), c(i) are given in Table 1 for i > 0. For i < 0 it is cO(-i) = c+(i) and c. (i) = c,.(1 - i). EF(O,kAt) is the time dependence of the excitation. For Jil < 4, the above excitation components are superimposed to the field values obtained by the MRTD algorithm. For example, the total E'O' will be given by k,m+W E = EF (0, kA t) c (i) + Et km+i total km+ Due to the nature of the Battle-Lemarie expansion functions, the total field is a summation of the contributions from the non-localized scaling and wavelet functions. For example, the total electric field E (xo, zo, to) with (k- 1/2) At < to < (k + 1/2)At is calculated in the same way with [2, 3] by Li Ex(xoZo,to) = Z kE+l/2,m, 11+11/2(Zo),m'(Zo) l,m'r=-11 12,i + E kE1+1/2,m 4l +l /2 (o) Oi,rm (Zo) i l',m'=-l2,i where Om (x) = 0( ' -m) and ti,m(x) = 0i( m) represent the Battle-Lemarie scaling and i-resolution wavelet function respectively. For an accuracy of 0.1% the values li = 12,i = 4 have been used. There are many different ways to take advantage of the capability of the MRTD technique to provide space and time adaptive gridding. In DSP, thresholding of the wavelet coefficients over a specific timeand space- window (5-10 points) contribute significant memory economy, but increase the implementation complexity and the execution time. The simplest way is to threshold the wavelet components to a fraction (usually < 0.1%) of the scaling function at the same cell for each time-step. All components below this threshold are eliminated from the subsequent calculations. This is the simplest thresholding algorithm. It doesn't add any significant overhead in execution time, but it offers only a moderate (pessimistic) economy in memory (factor close to 2). Also, this algorithm allows for the dynamic memory allocation in its programming implementation. III Applications of 2D-MRTD The 2D-MRTD scheme is applied to the analysis of the partially-loaded parallel-plate waveguide of (Fig.1) for the frequency range 0-30GHz. For the 2

analysis based on Yee's FDTD scheme, a 16 x 800 mesh is used resulting in a total number of 14400 grid points. When the structure is analyzed with the 2D-MRTD scheme, a mesh 2 x 100 (200 grid points) is chosen (dx = 0.24Ao, dz = 0.4Ao for f = 30GHz). This size is based on the number of the scaling functions, since the wavelets are used only when and where necessary. The time discretization interval is selected to be identical for both schemes and equal to the 1/10 of the 2D-MRTD maximum At. For the analysis we use 8,000 time-steps. The waveguide is excited with a Gabor function 0-30GHz along a vertical line for the FDTD simulation and for a rectangular region for the MRTD simulations. In all cases, the front and back open planes are terminated with a PML region of 16 cells and a-,,=0.4S/m. The longitudinal distance between the excitation and the dielectric interface is chosen such that no reflections would appear before the Gabor function is complete. The capability of the MRTD technique to provide space and time adaptive gridding is verified by thresholding the wavelet components to the 0.1% of the value of the scaling function at the same cell for each time-step. It has been observed that the accuracy by using only a small number of wavelets is equal to what would be achieved if wavelets were used everywhere. Though this number is varying in time, its maximum value is 22 out of a total of 100 to the z-direction (economy in memory by a factor of 28-30). In addition, execution time is reduced by a factor 4-5. For larger thresholds, the ringing effect due to the elimination of the wavelets deteriorates the performance of the algorithm. For example, using a threshold of 1% (6 out of a 100 wavelets to the z-direction) increases the error by a factor of 2.5. The normal electric field is probed at a distance 10 cells away from the source and is plotted in (Fig.2) in time-domain. Comparable accuracy can be observed for the FDTD and the MRTD meshes. In addition, the reflection coefficient S11 is calculated by separating the incident and the reflected part of the probed field and taking the Fourier transform of their ratio (Fig.3). The results for 5 GHz (TEM propagation) are validated by comparison to the theoreti cal value obtained applying ideal transmission line theory [6] and are plotted at Table 2. The timeand space-adaptive character of the gridding is exploited in (Figs.4,5) which show that the wavelets follow the propagating pulses before and after the incidence to the dielectric interfaces and have negligible values elsewhere. The location and the number of the wavelet coefficients with significant values are different for each time-step, something that creates a dense mesh in regions of strong field variations, while maintaining a much coarser mesh for the other cells. IV Conclusion A space- and time- adaptive gridding algorithm based on a multiresolution time-domain scheme in two dimensions has been proposed and has been applied to the numerical analysis of a waveguide problem. The field pattern and the reflection coefficient have been calculated and verified by comparison to reference data. In comparison to Yee's conventional FDTD scheme, the proposed scheme offers memory savings by a factor of 5-6 per dimension maintaining a similar accuracy. The above algorithm can be effectively extended to three-dimension problems. V Acknowledgments This work has been funded by NSF and ARO. References [1] M.Krumpholz, L.P.B.Katehi, "New Prospects for Time Domain Analysis", IEEE Microwave and Guided Wave Letters, pp. 382-384, November 1995. [2] M.Krumpholz, L.P.B.Katehi, "MRTD: New Time Domain Schemes Based on Multiresolution Analysis", IEEE Transactions on Microwave Theory and Techniques, pp. 555-572, April 1996. [3] E.Tentzeris, M.Krumpholz and L.P.B. Katehi, "Application of MRTD to Printed Transmission Lines", Proc. MTT-S 1996, pp. 573-576. [4] E.Tentzeris, R.Robertson, M.Krumpholz and L.P.B. Katehi, "Application of the PML Absorber to the MRTD Technique", Proc. AP-S 1996, pp. 634-637. [5] I.Daubechies, "Ten Lectures on Wavelets", Philadelphia, PA: Society for Ind. and App. Math., 1992. [6] D.M.Pozar, "Microwave Engineering", pp. 94-96, Addison-Wesley, 1990. 3

Table 1: Excitation Decomposition Coeffs i 0 1 2 3 4 cO(i) 0.915 0.038 0.010 -0.009 0.005 co(i) -0.103 -0.103 0.121 -0.030 0.015 Table 2: Sll calculated by 2D-MRTD 511 (f) Relative error Analyt. Value [6] 0.4298 0.0% 16x800 FDTD 0.4283 -0.3% 2x100 MRTD 0.4360 +1.4% er = 1 I'V 4.8 mm 4 Figure 3: Sll values (Frequency-Domain). Frmqusncy [GH] T 500 Tme-Steps ^ --- 8 mm -- Figure 1: Dielectric-loaded Waveguide. '1 r --- —-r 0.8 I I I I I I I T 1000 Tire-Steps 0.8 0.6 0.4 0.2 0 -02 0.4 0.2 0 -0.2 -0.4 0.61 0.4I l3 X 0:2 u I o z:1 -0.21 - I I I I -- 16x800 FDTD - - 2x100 MRTD Figure 4: Adaptive Grid Demonstration. -0.6 - IRa 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 ime-Step 10000 Figure 2: Normal E-field (Time-Domain). 4

Presented to 1997 IEEE AP-S Symposium Time Adaptive Time-Domain Techniques for the Design of Microwave Circuits Emmanouil Tentzeris', James Harvey2, Linda P.B. Katehi' 'Radiation Laboratory, EECS Department, University of Michigan 2Army Research Office Abstract The recently developed MRTD schemes are used for the development of a time adaptive timedomain technique for circuit design. The new technique exhibits considerable savings in memory and computational times in comparison to the conventional FDTD scheme. I Introduction Significant attention is being devoted now-a-days to the analysis and design of various types of microwave circuits. The finite-difference-time-domain (FDTD) scheme is one of the most powerful numerical techniques used for numerical simulations. However, despite its simplicity and modeling versatility, the FDTD scheme suffers from serious limitations due to the substantial computer resources required to model electromagnetic problems with medium or large computational volumes. In addition, the FDTD scheme cannot provide the accuracy required for computer simulations of time-dependent electromagnetic interactions in electrically long regions or in regions which contain non-linear materials. Such simulations are very important for integrated device modelling, especially in relation to the design of non-linear photonic devices. To alleviate these problems hybrid combinations of FDTD with other numerical techniques and higher order FDTD schemes based on Yee's grid have been proposed. MRTD (MultiResolution Time Domain Method) [1, 2] has shown unparalled properties in comparison to Yee's FDTD. MRTD is not a new methodology. It is a correct and accurate generalization of the conventional discretization approaches. It provides the correct mathematical frame for solving problems in time domain and allows for the development of time/space adaptive grids. II Introduction to MRTD It is well known that the method of moments provides a mathematically correct approach for the discretization of integral and partial differential equations. Since it allows for the use of any complete and orthonormal set, the choice of an appropriate expansion set may lead to different time domain schemes. For example, the expansion of the unknown fields using pulse

functions leads to Yee's FDTD scheme. In a MIRTD scheme the fields are represented by a twofold expansion in scaling and wavelet functions with respect to time/space. Scaling functions guarantee a correct modelling of smoothly-varying fields. In regions characterized by strong field variations or field singularities, higher resolution is enhanced by incorporating wavelets in the field expansions. Wavelets are introduced only at specific locations, allowing for a time/space adaptive grid capability. MRTD schemes based on cubic spline Battle-Lemarie scaling and wavelet functions (Fig.l) have been successfully applied to the simulation of 2D and 3D open and shielded problems [1, 2, 3, 4]. The functions of this family do not have compact support, thus the MRTD schemes have to be truncated with respect to space. Localized boundary conditions (PECs, PMCs etc.) and material properties are modelled by use of the image principle and of matrix equations respectively. However, this disadvantage is offset by the low-pass (scaling) and band-pass (wavelets) characteristics in spectral domain, allowing for an a priori estimate of the number of resolution levels necessary for a correct field modelling. In addition, the evaluation of the moment method integrals during the discretization of Maxwell's PDEs is simplified due to the existence of closed form expressions in spectral domain and simple representations in space domain. Dispersion analysis of this MRTD scheme shows the capability of excellent accuracy with up to 2 points/wavelength (Nyquist Limit). However, specific circuit problems may require the use of functions with compact support. For that reason, Haar basis functions have been utilized and have led to [5]. As an extension to this approach, intervalic wavelets of higher order may be incorporated into the solution of SPICE-type circuits. Results from that new technique will be shown at the Conference. III Time Adaptive MRTD Scheme The major advantage of the use of Mutiresolution analysis to time domain is the capability to develop time and space adaptive schemes. This is due to the property of the wavelet expansion functions to interact weakly and allow for a spatial sparsity that may vary with time through a thresholding process. The adaptive character of this technique is extremely important for the accurate modelling of sharp field variations of the type encountered in beam focusing in nonlinear optics, etc. The use of the principles of the multiresolution analysis for adaptive grid computations for PDEs has been suggested by Perrier and Basdevant [6]. To understand the fundamental steps of such an adaptive scheme for Maxwell's hyperbolic system, let's consider Maxwell's equations in 2D (1 for space and 1 for time): a -6(z1) ia | t U= (E(z, t), H(z, t))T (1) At=At= -[ u(z)-1 0 o After manipulation, the above equation can be written as

MU= F TZhtD phT Z 1 =LO (2) where Zh,Th are half shift operators for space and time coordinates z,t and ZI, Tt are their Hermitian conjugates. Dt, Dz are difference operators given by: 1 8 9 1 8 9 t = ( E aat(i)T-i+ at(i)T-i), = -( az(i)Z-'+ E az(i)Z-), (3) i=- 9 i=-9 i=-9 where a,*, a, are the coefficients associated with the scalar and the wavelet functions respectively. At each time step we keep both the wavelet field values that are larger than a given threshold as well as the adjacent values. An adjacent wavelet field value is defined on the basis of the wavelet resolution level(s) incorporated in the solution. Recently, an efficient space/time adaptive meshing prosedure was proposed [7] for Battle-Lemarie expansion functions. In this paper, intervalic wavelets are used for the expansion of the fields (Fig.2). The adaptive mesh will be applied to a variety of circuit problems and results will be discussed during the presentation. IV Conclusion A Time Adaptive Time-Domain Technique based on intervalic wavelets has been proposed and applied to various types of circuits problems with lumped and distributed elements. This scheme exhibits significant savings in execution time and memory requirements while maintaining a similar accuracy with conventional circuit simulators. V Acknowledgments This work has been partially funded by NSF. References [1] M.Krumpholz, L.P.B.Katehi, "MRTD: New Time Domain Schemes Based on Multiresolution Analysis", IEEE Trans. Microwave Theory and Techniques, vol. 44, no. 4, pp. 555-561, April 1996. [2] E.Tentzeris, M.Krumpholz and L.P.B. Katehi, "Application of MRTD to Printed Transmission Lines", Proc. MTT-S 1996, pp. 573-576. [3] R. Robertson, E. Tentzeris, M. Krumpholz, L.P.B. Katehi, "Application of MRTD Analysis to Dielectric Cavity Structures", Proc. MTT-S 1996, pp.. [4] E.Tentzeris, R.Robertson, M.Krumpholz and L.P.B. Katehi, "Application of the PML Absorber to the MRTD Technique", Proc. AP-S 1996, pp. 634-637. [5] K.Goverdhanam, E.Tentzeris, M.Krumpholz and L.P.B. Katehi, "An FDTD Multigrid based on Multiresolution Analysis", Proc. AP-S 1996, pp.. [6] V.Perrier and C.Basdevant, "La decompositionene ondelettes periodiques: un outil pour 'analyse des champs inhomogenes. Theorie et algorithmes", La Recherche Aerospatiale, no.3, pp.53-67, 1989. [7] E.Tentzeris, R.Robertson, A.Cangellaris and L.P.B. Katehi, "Space- and Time- Adaptive Gridding Using MRTD", to be presented in the 1997 IEEE MTT-S, Denver, CO.

8 0 LL k 0 %^ "A AA (71 C, C, *w t X I t t a --.......-............ 04 ~o of 6 0 0 0 0 *~ d. A a -o. id ib 46 I Ssrno Leoo~ofl momos~ wavelit oq a........ I 14 ft K rrr 0 - - 0 0 0 a a a 1. W. a 0

Accepted for Presentation to the ACES Conference 1998 PML Implementation for the Battle-Lemarie Multiresolution Time-Domain Schemes Emmanouil Tentzeris, Rob Robertson, Linda P.B. Katehi Radiation Laboratory, EECS Department, University of Michigan, Ann Arbor, Ml 48109-2122, USA I Introduction The Multiresolution Time Domain (MRTD) Technique based on cubic-spline Battle Lemarie scaling and wavelet functions has shown successful application to a variety of microwave problems and has demonstrated unparalleled properties in terms of memory and execution time by one and two orders of magnitude respectively. This technique is used to model open and shielded propagation problems [1, 3] and non-linear optical applications [2]. In addition to time and memory, the most important advantage of this new technique is its capability to provide space and time adaptive meshing without the problems encountered by the conventional Finite Difference Time Domain(FDTD) [4] method. In this paper, an efficient non-split formulation of the PML absorber [5] for the Battle-Lemarie based MRTD scheme is presented. This formulation is validated and applied in the analysis of a two-dimensional parallel-plate waveguide geometry offering a numerical coefficient of reflection below -90dB. Additionally, examples for a three-dimensional patch antenna geometry are given. II Derivation of the MRTD equations for the PML layer Without loss of generality, the PML Absorber equations will be presented for a homogeneous medium for TM propagation in 2D. The Absorber formulation for TE propagation is straightforward. Assuming that the PML area is characterized by (c, o,) and electric and magnetic conductivities (aE, C), the TM equations can be written o -t +aEE = -E = (1) &E, OHy eo + aEE, = = (2) toy ( IO~ O + aHHy aX (3) Otax: - z

PML cells only to the z-direction are considered. Equations for PML cells in the x- and vrdirections can be derived in a similar was. For each point z of the PML area, the magnetic conductivity aH needs to be chosen as [5]: CE(Z) o H(Z) ( (4) co Ho for a perfect absorption of the outgoing waves. A parabolic spatial distribution of OE,H, oEH(z) = oa(1 -, with p=2 for 0 < z < 6 = PML thickness (5) is used in the simulations, though higher order distributions (e.g.Cubic p=3) can give similar results. The PML area is terminated with a PEC and usually has a thickness varying between 4-16 cells. The maximum value maE is determined by the designated reflection coefficient R at normal incidence, which is given by the relationship R= e-~ f 0C E()d - =e~ (6) The electric and magnetic field components incorporated in these equations are expanded in a series of Battle-Lemarie scaling and wavelet functions in both x- and z-directions. For example, Es can be represented as: +oo E,(x,z,t) - ' ~ EE 1+1/k2,m k (t) i+1/2 (2)4m(Z) k,l,m=-oo +00 + E E +1/2,mhk(t) ()+1/2() (7i,m(Z) i k,,m=-oo +oo + El/2,mhk(t) i,l+1I/2 (X) Om(z) i k,l,m=-oo +oo + Z Z kE+1/2mhk(t) 2i,l+l/2(X) Pj,=m(Z) (7) ij k,,m=-oo where km(;) = 0(^ - m) and Pi,m(X) = i1i( - m) represent the Battle-Lemarie scaling and i-th order resolution wavelet function respectively in space and -hk(t) represent rectangular pulses in time. kEJI and k+l/2HLm with K = x, y, z and p, v = kf, 4 are the coefficients for the field expansions in terms of scaling and wavelet functions. The indices 1, m and k are the discrete space and time indices related to the space and time coordinates via x = lAx,z = mAz and t = kat, where AL,Az are the space discretization intervals in x- and z-direction and At is the time discretization interval. For an accuracy of 0.1% the above summations are truncated to 16-24 terms. For simplicity, expansion only in scaling functions will be considered. Wavelets are implemented in a similar way. Upon inserting the field expansions, Maxwell's equations

are sampled [3] using pulse functions as time-domain test functions and scaling functions as space-domain test-functions and the following non-split formulation of the fields for the PML region is derived: k-+l -a At/co E~Z,' e ka.,,l+l/2,m k+l+l/ /2,m l k +1/2,m 1 m+8 - Z oa(i')k,+l/2r' H,'8) (A e-0 / 1 +12(, /2) il '=m-9 w e ieE b 'E 1 1+8 a 1/2 i'-9+1/2,m+1/2) i'-=1-9 k~~l/S- e -/H +1/2.A1+1/2,m+1/2 k_1/2.L'l+1/2,m+1/2 A-O'5 ~'t// ' m ti'i'+1/2 Az m14 i0=l-9 i' ---m-9 where the terms are given by Eq.(12). validate the proposed algorithm. A TM source with a Gabor time variation is excited close to one side of the waveguide. The benchmark MRTD solution with no reflections is obtained by simulating the case of a much longer parallel-plate waveguide of the same width to provide a reflection-free observation area for the time interval of interest. A quadratic variation in PML conductivity is assumed for all cases, with maximum theoretical reflection coefficient of 10-5 at normal incidence. Numerical reflection is observed for the frequency range [0, 0.9ffMi] (TEM propagation) where fTMI = - = 3.125 (GHz) is the cutoff frequency of the TM1 mode. It can be seen from Figs.(1)-(2) that for 8 PML cells and a_7 =0.4 S/m it is S11 <-65 dB and for 16 PML cells and aE=0.2 S/m the reflection is smaller than -91 dB. Thus, the non-split PML absorber can be used effectively in the simulation of antennas and active elements using MRTD. III Application of PML to the Analysis of Antenna Geometries MRTD can successfully model both planar circuits [6] and resonating structures [7]. Recently the techniques developed for the simulation of both structures are combined to model a threedimensional patch antenna geometry [8]. Full three-dimensional MRTD analysis is used, with PML expanded through three coordinate directions. The procedure to derive an equation for the

At PML cells along z a ax a0E _max max max FDTD (60 x 100 x 16) 1.3297. 10-'3s 6 3.0 3.0 3.0 MRTD(30 x 50 x 9) 1.6008. 10-'3s 2-6 3.0 3.0 11.53 MRTD (20 x 20 x 9) 1.3297 - 10-'3s 6-10 3.0 3.0 11.53 Table 1: Computational Parameters. three-dimensional MRTD scheme, with PML along all three coordinate directions is presented in [8]. The patch antenna used in our simulations has the dimensions 12.45mm x 16mm, with a microstrip line 20 mm long used as a feed. A Gaussian pulse 4 mm from the PML layer is used to excite the microstrip. The substrate has a thickness of 0.794 mm and a relative dielectric constant equal to 1. An FDTD mesh of 60 x 100 x 16 is compared to MRTD grids of 30 x 50 x 9 and 20 x 20 x 9, which exhibit savings of memory over FDTD on the order of 7.22 and 33 respectively. Note that these values do not include the PML layers. Figure 3 shows a comparison plot of calculated S11 data for the three cases listed above. Six cells of PML are added along the ~x, ~y and +z directions with CmEx = a E= 3.0 and aEz = 11.53 for all cases. The time discretization interval used for the MRTD 30 x 50 x 9 scheme is At = 1.6008 10-13s while the MRTD 20 x 20 x 9 scheme uses a time discretization interval of At1.42 14 10-~3s. FDTD uses a time discretization interval of At = 1.3297- 10-'3s. In all three cases the simulation is performed for 10000 time steps. This information is summarized in Table 1. Figure 4 shows a comparison of S11 data for different numbers of z-directed PML layers for an MRTD discretization of 30 x 50 x 9. Note that the S11 values correlate very well even for only 2 PML layers in the z-direction. Figure 4 shows a comparison of S11 data for different numbers of z-directed PML layers for an MRTD discretization of 20 x 20 x 9. Once again the values of Sl show good correlation. IV Conclusion An efficient PML absorber in non-split formulation is presented for the MRTD Scheme based on cubic spline Battle-Lemarie scaling functions. This absorber is used effectively to model an antenna geometry providing extremely small numerical reflections. In comparison to Yee's conventional FDTD scheme, the proposed MRTD scheme coupled with the PML absorber offer memory savings by a factor of 12-30 and execution time savings by a factor of about 3-5 maintaining a better accuracy for S-parameter calculations. For structures where the edge effect is prominent, additional wavelets can be used to improve the accuracy when using a

coarse MRTD mesh. V Acknowledgements This work was made possible by ONR contract N00014-95-1-1299 and ARO contract DAAH04 -95-1-0321. References [1] M.Krumpholz, L.P.B.Katehi, "MRTD: New Time Domain Schemes Based on Multiresolution Analysis", IEEE Trans. Microwave Theory and Techniques, vol. 44, no. 4, pp. 555-561, April 1996. [2] M.Krumpholz, L.P.B.Katehi, "MRTD Modeling of Nonlinear Pulse Propagation", to be published at the IEEE Trans. Microwave Theory and Techniques. [3] E.M.Tentzeris, R.Robertson, M.Krumpholz, L.P.B.Katehi, "Application of the PML Absorber to the MRTD Technique", Proc. AP-S 1996, pp. 634-637. [4] K.S.Yee, "Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media", IEEE Trans. Antennas Propagation, pp.302-307, May 1966. [5] J.-P. Berenger, "A Perfectly Matched Layer for the Absorption of Elecromagnetic Waves", J.Comput. Physics, vol. 114, pp. 185-200, 1994. [6] E. Tentzeris, M. Krumpholz and L.P.B. Katehi, "Application of MRTD to Printed Transmission Lines", Proc. MTT-S 1996, pp. 573-576. [7] R. Robertson, E. Tentzeris, M. Krumpholz and L.P.B. Katehi, "Application of MRTD Analysis to Dielectric Cavity Structures",Proc. MTT-S 1996, pp. 1840-1843. [8] R Robertson, E. Tentzeris,, and L.P.B. Katehi "Modelling of Membrane Patch Antennas using MRTD Analysis", Proc. AP-S 1997, pp. 126-129.

8 PML Cells - Non-split - TEM 1 1.5 2 2.5 3 Frequency (GHzJ Figure 1: 8 PML cels. 16 PML Cells - TEM - Non-Split -s=.2 — s=0.4 "-"0. -s=0.6.1 8=1.0 Ij"1. ItI 11.5 2 2.5 3 -1( V CD.. Frequency [GHz] Figure 2: 16 PML cells.

Figure 3: S1 comparison plots for a patch antenna 8 10 12 Frequency (GHz) 20 Figure 4: MRTD(30 x 50 x 9) S11 plot for varying PML layers in the z-direction

0 2 4 6 8 10 12 14 16 18 20 Frequency (GHz) Figure 5: MRTD(20 x 20 x 9) S11 plot for varying PML layers in the z-direction

Accepted for Presentation to the IEEE MTT-S Microwave Conference 1998 NONLINEAR CIRCUIT CHARACTERIZATION USING A MULTIRESOLUTION TIME DOMAIN TECHNIQUE (MRTD) Luca Rosellil, Emmanouil M. Tentzeris2, Linda P.B. Katehi2 1 University of Perugia, Perugia, ITALY 2 Radiation Laboratory, Department of Electrical Engineering and Computer Science University of Michigan, Ann Arbor, MI 48109-2122, U.S.A. Abstract- The MRTD scheme is applied to the modeling of nonlinear circuits. Specifically, the implementation of passive and active elements is discussed. The results are compared to those obtained by use of the commercial CADs to indicate considerable savings in memory and computational time. I Introduction Recently, the use of multiresolution analysis for the discretization of the time-domain Maxwell's equations has led to the development of the Multiresolution Time Domain Technique (MRTD). This technique has been applied to linear as well as nonlinear propagation problems and has demonstrated savings in time and memory of two orders of magnitude. In addition, the most important advantage of this new technique is its capability to provide a very effective way for space and time adaptive gridding without encountering the problems that the conventional FDTD has to resolve. In this paper, an algorithm to model nonlinear circuits using the MRTD scheme is proposed and applied to diode problems. As an example, the harmonic analysis of a diode enclosed in a metallic shield and terminated with lumped resistors is performed and a simple stripline mixer circuit using the same diode is analyzed. II The MRTD scheme To derive the MRTD scheme, the field components are expanded in a series of cubic spline BattleLemarie [1, 2] scaling and wavelet functions in space and pulse functions in time. The MRTD equations are derived by applying the Method of Moments to the Maxwell's equations after inserting the field expansions. For open structures, the perfectly matched layer (PML) technique can be applied by assuming that the conductivity is given in terms of scaling and wavelet functions instead of pulse functions with respect to space [3]. The MRTD mesh is terminated by a perfect electric conductor (PEC) at the end of the PML region. Unlike the FDTD, where the consistency with the image theory is implicit in the application of the boundary conditions, the entiredomain nature of the wavelet and scaling functions requires an explicit use of the boundary conditions. In particular, image theory has to be applied for the evaluation of the field component coefficients in the vicinity of Perfect Electric and Magnetic Walls. Due to the nature of the Battle-Lemarie expansion functions, the total field is a summation of the contributions from the non-localized scaling and wavelet functions. III Lumped Elements Similarly to LEFDTD technique [4], the basis of the algorithm is given by a particular interpretation of the current density term contained in the Curl(H) Maxwell's equation. Let's assume for the rest of the discussion that all the lumped elements are z-oriented. a, z * (V x — = Z. (V x H) +.i &t (1) 1

The current term can be considered as the superposition of two separate terms, one coming from the finite conductivity of the medium JO and the other coming from the presence of a lumped element J. Eq.(1) leads to the following general updating expression for the E-field S-MRTD (Scaling-functions-based) coefficients: +lE,n+ 2 lmn+ = i (7 xH) At + k+*4 J,m,n+4 where an ideal dielectric medium with a = 0 has been assumed. The discretization of the last term can be obtained by expressing the constitutive relationship of the related device in terms of electric field and current density (instead of V-I relation as usual). Since the field components are expanded in pulses in the time-domain, the time discretization of the J-E relation of the lumped devices is straightforward and similar to FDTD. III.1 Resistor Assuming that the resistor is z-oriented and a positive voltage (with respect to the z-axis) is applied, we have: Vz = -AzE, I, = J.AzAy Since the current flow due to a positive voltage is negative with respect to the z-axis, Ohm's Law can be written in the following form: _AzE, Ju" R y (2) RAxAy By discretizing equations 1 and 2 accordingly to the S-MRTD scheme and assuming that no current density is supported by the medium we obtain: B -— z k rz + +,m,n+ = C,m,n+ + 1+8 + Ch a(i)k+ +k,,Hf.+~i=l —9 m+8 j=m-9 CAoiy m a(j)k+~H1;+~,n+~ where Az E Az At 2RAxAy ' At 2RAxAy 111.2 Capacitor The I-V Law of the capacitor is: d( = (t) dt Expanding the E- and H- components in scaling functions in space and pulses in time and applying the Moments Method, the capacitor can be described by +liE m,n+i =k - Em,n+ + 1+8 + BA a (i)k+ H+ + - i=I-9 m+8 Z a()k+Ht, n+ BAy._m-9 j=m-9 where the coefficient B is given by: f +C Az B = At 111.3 Inductor The constitutive relation of the inductor is: I(t) = L V(t)dt Following the same procedure described for the resistor and the capacitor we obtain: C B k k m+E,n+ 7 k n+ A iE no + i=1 1 +8 + A Zx E a(i)k*+H?,,n+4 -i= —9 m+8 - A.y E a(j)k+iH+H n+ AAy j=mwhere the coefficients A, B, C are given by: A a zAt azat A= t +2LAxAy B= 2LAAy' = At III.4 Diode with Junction and Diffusion Capacitances According to the model adopted in [5], the equivalent circuit of the diode includes both the non linear junction and diffusion capacitances (Cdi(Vd) and Cj(Vd)) and the total current can be expressed as: Id = Ij + Icdi + IC, 2

with Ij = Jo (e -7- - l), Ic, = Cdi(Vd) d ICj = Cj(Vd) dt In the above equations K is the Boltzmann constant, T is the absolute temperature, Io is the inverse saturation current of the diode and ir is the ideality factor that will be omitted in the rest of the discussion. The two non linear capacitances, in turn, are modeled by the following equations: Cdi(Vd) = TdIJo (exVd - 1) Cj(Vd) = Cj(0) (1 -- C (Vd) = () (F3 + -m F2 < if Vd > Fco if Vd < Fco where Fc, F2, F3 are suitable coefficients, m is the doping profile coefficient (usually 0.5 for abrupt junction), o0 is the built-in voltage and Cj (0) is the static capacitance at Vd = 0. The current equations are discretized in a similar way with the other lumped elements and two E-field transcedental equations are derived for Vd < Fcr/o and Vd > Fc4o. These equations can be solved in an iterative procedure (e.g. Newton-Raphson algorithm). IV Applications of Nonlinear MRTD The modeled Schottky GaAs diode has the following parameter values: Io = 5.e - 11 A, r7 = 1.25, Rs = 13 A, Cj(0) =.29pF, rd = 0, m = 0.5, Fc =.5 For the analysis of the testing structure of Fig.(1), we have set up a mesh of 8 x 30 x 6 cells with a cell size equal to 30 x 60 x 30 pm (60 pm is A/10 at about 135 GHz). The same structure has been also analyzed, for comparison, with FDTD method. This analysis has been performed by adopting two different meshes: the same mesh described before and a doubled mesh with the dimension: 16 x 60 x 12 and A/10 at about 270 GHz. The structure has been excited at the center with an impressed current source window. A sine-wave with a frequency of 45 GHz has been used, while a probe at the center of the structure has been considered. Figures (2),(3) and (4) show the results obtained with the coarse FDTD, the finer FDTD, and the MRTD respectively. The MRTD simulation has adopted the same mesh used in the coarsest FDTD analysis. The good agreement between the FDTD simulation with the fine mesh and the MRTD one, together with the fairly different results obtained with the coarse mesh FDTD analysis, put at the evidence the capability of the MRTD to better predict the frequency behavior of this non linear circuit. In particular, it is evident that with a coarse mesh, MRTD, in contrast to FDTD, can detect the harmonic null due to the location of the probe in the middle of the structure (in this position, in theory, no even harmonic mode should be detected). Figure (5) shows the geometry of a stripline singleended mixer, which is analyzed by use of MRTD. The used Schottky diode has the characteristics described above and is zero biased for simplicity. The LO and RF excitation signals have frequencies 43 GHz and 45 GHz and powers 20 dBm and -20 dBm respectively. The left (short-circuited) stub with length 900 pm is used as an IF signal block and the right (open-circuited) stub with length 1640 pm blocks the LO/RF signals at the output section. For this configuration, MRTD gives a conversion loss of -8.1 dB. LIBRA, a commercial EM simulator, calculates the conversion loss at -8.8 dB. In addition, (Table 1) shows that the relative output power of the harmonics gets similar values for MRTD and LIBRA simulations. These results emphasize the inherent capability of MRTD to describe efficiently the nonlinear elements, which create a discrete but infinite spectrum. Moreover, the MRTD allows for a time adaptive scheme which offers significant computational profit due to the iterative algorithm for the solution of the nonlinear equations. It has to be pointed out that LIBRA can give reliable results only for quasistatic geometries such as Figure (5). On the contrary, MRTD can simulate efficiently structures with multimodal propagation without the huge memory requirements of the conventional FDTD schemes. 3

Table 1: Harmonics Power Distribution [dBm] Freq [GHz] 2 41 43 45 88 LIBRA -28.8 -56.1 -33.2 -40.1 -36.3 MRTD -28.1 -54.7 -31.4 -38.2 -34.7 V Conclusion An algorithm for the modeling of lumped elements with the MRTD scheme based on the Battle-Lemarie basis has been proposed and has been applied to the numerical analysis of a diode problem. The frequency spectrum has been calculated and verified by comparison to reference data. In comparison to Yee's conventional FDTD scheme, the proposed scheme offers memory savings by a factor of 2-6 per dimension maintaining a similar accuracy. VI Acknowledgments This work has been funded by NSF and ARO. The authors would like to thank Dr. F. Alimenti for the interesting and useful discussion about theoretical aspects concerning the multiresolution techniques and Prof. R. Sorrentino for the continuous support he gave to this work. References [1] M.Krumpholz, L.P.B.Katehi, "MRTD: New Time Domain Schemes Based on Multiresolution Analysis", IEEE Trans. Microwave Theory Tech., pp. 555-572, 1996. [2] E.Tentzeris, M.Krumpholz and L.P.B. Katehi, "Application of MRTD to Printed Transmission Lines", Proc. MTT-S 1996, pp. 573-576. [3] E.Tentzeris, R.Robertson, M.Krumpholz and L.P.B. Katehi, "Application of the PML Absorber to the MRTD Technique", Proc. AP-S 1996, pp. 634-637. [4] W.Sui, D.Christensen and C.Durney, "Extending the two-dimensional FDTD Method to Hybrid Electromagnetic Systems with Active and Passive Lumped Elements", IEEE Trans. Microwave Theory Tech., pp. 724 -730, 1992. [5] G.Massobrio and P.Antognetti, "Semiconductor Device Modeling with Spice 2/E", McGraw-Hill. Zo = 50 Q Excitation Diode 50LI 50 Figure 1: Diode Test Structure. I --- -100 o -.0 -3.40.0 -40.0 I 140 -40.0 -70.0.. 0 -ao. O40 6.0 *6. 0.0 116.0 140.0 166.0 1 0.0 Figure 2: FDTD coarse mesh. Figure 3: FDTD fine mesh. Figure 4: MRTD coarse mesh. LO RF Zo = 50 Q IF - I - 50 S.C..C. si Stub Stub - - P11 j50Q Figure 5: Mixer Geometry. 4

Accepted for Presentation to the IEEE MTT-S Microwave Conference 1998 SPACE ADAPTIVE ANALYSIS OF EVANESCENT WAVEGUIDE FILTERS Emmanouil M. Tentzeris, Linda P.B. Katehi Radiation Laboratory, Department of Electrical Engineering and Computer Science University of Michigan, Ann Arbor, MI 48109-2122, U.S.A. Abstract- The MRTD scheme is applied to the analysis of evanescent waveguide filters. Specifically, a space adaptive algorithm in 3 dimensions is implemented by thesholding the wavelet values. The results are compared to those obtained by use of the conventional FDTD to indicate considerable savings in memory and computational time. I Introduction The Space Adaptive Gridding [1], based on the application of the Multiresolution Analysis principles to the discretization of the time-domain Maxwell's equations [2, 3], has been employed in the analysis of linear and nonlinear structures. It has offered significant savings in memory and execution time requirements. The application of the wavelets improve the conditioning of the simulating algorithm and allow for a space adaptive grid by thresholding the wavelet coefficients. This adaptivity is useful especially in evanescent mode structures that require time-domain simulations for a large time span in order to take into consideration the slow wave propagation. In this paper, a space adaptive grid is applied for the analysis of evanescent-mode waveguide bandpass filters [4, 5, 6]. These structures have found many applications in satellite communication systems, as preselectors or in multiplexers, due to several advantages over the conventional coupled resonator filters, such as compactness and wide stopbands. The S-parameters of one specific geometry are calculated and compared to results obtained by the conventional FDTD. II The MRTD scheme The 3D-MRTD scheme can be derived by representing the field components as a series of cubic spline Battle-Lemarie scaling and wavelet functions in space-domain and pulse function in time. Applying the Method of Moments to the Maxwell's equations results in the MRTD equations. Generally, the features of the 3D-MRTD algorithm are similar to those of the 2D-MRTD algorithm. Nevertheless, there are some differences as far as it concerns the implementation of the excitation and of the PML absorber. In order to use a pulse excitation with respect to space at a specific grid point for a 2D geometry and to obtain an excitation identical to that used by FDTD, the pulse is decomposed in terms of scaling and wavelet functions on a square surface around the excitation point. For the 3D-MRTD algorithm, this decomposition takes place in a cubic volume around this point, since the excitation affects the amplitudes of the scaling and the wavelet function in all 3 directions. It has been observed that 4 cells along each direction around the excitation point provide an accurate representation of the source for most cases. The maximum allowable time step required for the stability of 3D-MRTD algorithms has to contain the effect of all three space discretizations. For a summation stencil of 9 terms per direction and for 0 -resolution wavelet expansion it takes the value 0.37 c Jl/(Ax)2 + 1/(Ay)2 + 1/(Az)2 where c is the velocity of light. For larger stencils, the maximum value of the time step takes lower values. 1

The size of the stencil affects significantly the dispersion characteristics of the used algorithm. Larger stencil for the summations including scaling functions coefficients improves the phase error performance for medium and high sampling rates (discretization size < A/10). Increasing the stencil size in summations of wavelet functions coefficients offers a better dispersion performance for lower sampling rates (between A/2.2 and A/5). In our simulations, the used stencil size has had the value of 9 for a phase error smaller than 1~ for most discretizations. The use of the non-localized Battle-Lemarie basis functions causes significant effects. Localized boundary conditions are impossible to be directly implemented, so perfect electric and magnetic boundary conditions are modelled by use of the image principle in a generic way. The implementation of image theory in 3 dimensions is performed automatically for any number of PEC, PMC boundaries. Due to the nature of the Battle-Lemarie expansion functions, the total field is a summation of the contributions from the non-localized scaling and wavelet functions in 3 directions. For example, the total electric field E.(xo,yo, zo, t) with (k - 1/2)At < to < (k + 1/2) At is calculated E Ez(xoo, Zo, to) = E kEIt+21/2,m,ng '+1/2 (0) Om' (Yo)On, (Zo)+ 1',m',n'=-1i 12,i k E 0PlZlmn' '+l/2(Xo) Om' (Yo) Pim1 (Zo) i l',m',n= —i2,i where,m(z) = 4( -m) and bi,,m(x) = 0bi( -m) represent the Battle-Lemarie scaling and i-resolution wavelet functions respectively. Only wavelets to zdirection have been included for simplicity. For an accuracy of 0.1% the values 11 = 12,i = 6 have been used. The purpose of a space adaptive grid is to use a coarse mesh and implement a local magnification by the selective use of wavelets. Wavelets are placed only at locations where the EM fields have significant values, creating a space- and time- adaptive dense mesh in regions of strong field variations without adding a significant computational overhead. There are many different ways to take advantage of the capability of the MRTD technique to provide space and time adaptive gridding. All of them rely on the fact that the wavelet values can be thresholded without affecting the accuracy of the algorithm. The simplest way is to threshold the wavelet components to a fraction (usually < 0.5%) of the scaling function coefficient at the same cell for each time-step. All components below this threshold are eliminated from the subsequent calculations. This procedure doesn't add any significant overhead in execution time (usually < 12%), but it offers only a moderate economy in memory requirements (round 28 - 35%). Comparison of the wavelet values over a specific space window of scaling neighboors (often equal to the stencil size) would offer a more significant economy in memory, but would demand more execution time. Another way of creating a space adaptive grid is to use an absolute threshold. This requires the knowledge of the spatial field distribution in advance, something that makes it inappropriate for simulations of complicated structures. Generally, in 3D cases where both memory and execution requirements are high, the first thresholding algorithm offers an optimized performance. III Applications of Nonlinear MRTD Without loss of generality, the space adaptive algorithm used in all simulations presented herein includes one resolution of wavelets only to the z- (longitudinal) direction. For validation purposes, this scheme has been used for the analysis of the testing structure of Figure (1). This filter geometry contains four bilateral E-plane fins in a single WR62 waveguide housing (15.799 mm x 7.899 mm). The thickness of the fins is t=0.9 mm and the gap width is w=3.1mm. The agreement of data obtained from the space adaptive grid for a relative threshold of 0.5% and those obtained by use of mode matching [4] is very good (Figure(2)). Another evanescent-mode Eplane finned waveguide bandpass filter geometry is shown in Figure(3). A WR90 waveguide (22.86 mm x 10.16 mm) is used at the input and output stages and a rectangular 2

waveguide with a crossection of 7.06 mm x 6.98 mm is used as the housing of the filter. Geometrical parameters of the filter take the values 11 = 12 =0.5 mm, 13 =7.75mm and 14=0.94mm. The width of the fins is chosen to be equal to the waveguide side length a = w =7.06 mm. The MRTD space adaptive grid is used to optimize the geometry. An 20x20x389 grid is used for the simulations and 85,000 time steps are considered. A Gabor pulse from 10-18 GHz is used as the excitation along a plane at z = 44. Front and back waveguides are terminated with 8 PML layers with R = 10-5. A relative threshold of 0.5% is employed and offers economy in memory at least by 32%. In the geometry under study, we have different electrical paths between the input and output ports; one (the main path) is constructed with the coupled TElo - TEo - TE1o modes, and the others (the subsidiary paths) are constructed with the coupled TElo - TEmo - TEo modes, where TEmo for m > 1 express the higher order evanescent modes. These modes play primarily an important role to produce a desired off-passband performance, but it also affects significantly the passband behavior. Therefore, we can not use the conventional synthesis method. The slow velocity of the evanescent waves, require the use of very dense grids of the conventional FDTD algorithm for a large number of time-steps (close to 150,000). For example, a grid of 90x20x778 has been used for 135,000 steps to provide comparable results. On the contrary, space adaptive MRTD algorithms can use coarse grids everywhere except from the areas that the evanescent modes have significant values. Localized use of wavelets in these regions offer the necessary grid magnification. This effect can be observed in Figure(4) that shows the wavelet coefficients amplitude for an arbitrary time step after the pulse has propagated along the whole structure. The results from the optimization (Figures (5)-(7)) show that as the used fins get wider and come closer, the S21 gets higher values without affecting the significantly the bandwidth of the filter. IV Conclusion A space adaptive 3D algorithm based on BattleLemarie scaling and wavelet functions has been applied in the numerical modeling of evanescent-mode waveguide bandpass filters. The S-parameters of one specific geometry are calculated and offer memory savings by a factor of 3-6 per dimension and execution time savings by a factor of 2.5 compared to results obtained by the conventional FDTD. V Acknowledgments This work has been funded by ARO. References [1] E.Tentzeris, R.Robertson, A.Cangellaris and L.P.B.Katehi, "Space- and Time- Adaptive Gridding Using MRTD", Proc. MTT-S 1997, pp.337-340. [2] M.Krumpholz, L.P.B.Katehi, "MRTD: New Time Domain Schemes Based on Multiresolution Analysis", IEEE Trans. Microwave Theory Tech., pp. 555-572, 1996. [3] E.Tentzeris, R.Robertson, M.Krumpholz and L.P.B.Katehi, "Application of the PML Absorber to the MRTD Technique", Proc. AP-S 1996, pp. 634-637. [4] J.Bornemann and F.Arndt, "Rigorous Design of Evanescent-Mode E-Plane Finned Waveguide Bandpass Filters", Proc. MTT-S 1989, pp. 603-606. [5] V.Labay and J.Bornemann, "A New Evanescent-Mode Filter for Densely Packaged Waveguide Applications", Proc. MTT-S 1992, pp.901-904. [6] K.Iguchi, M.Tsuji and H.Shigesawa, "Negative Coupling between TElo and TE2o Modes for Use in EvanescentMode Bandpass Filters and their Field-Theoretic CAD", Proc. MTT-S 1994, pp. 727-730. -I -I 0 z t Figure 1: Validation Structure. 3

Figure 2: Validation Data. I! I i I b b [ -- i [ r —! 12 1 1 4 1 Figure 3: Optimized Filter Geometry. Figure 5: S-Parameters for 11 = 14 =l.0mm, 12 =0.5mm, 13 =6.75mm. -1 1 cM 20 40 60 80 100 12 14016 180 I I I I I I I I I I I I I Figure 6: Parametric Variation of 12. r1 _mi nil ".5- -' "f- - - - - -:,- - -i - 4 ' - 0.0. 0 20 40 60 80 100 120 140 160 180 20D 0.5 0 20 40 60 80 100 120 140 160 180 200 z-axis Relative Position (in Az) Figure 4: Wavelet Coefficients Spatial Distribution (Az: cell size to the z-direction). FmqwInoy Figure 7: Parametric Variation of 13. 4

TIME-DOMAIN NUMERICAL TECHNIQUES FOR THE ANALYSIS AND DESIGN OF MICROWAVE CIRCUITS by Emmanouil M. Tentzeris A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Electrical Engineering) in The University of Michigan 1998 Doctoral Committee: Professor Linda P.B.Katehi, Chair Professor John Volakis Professor Andrew Yagle Assoc. Professor Kamal Sarabandi Dr. James Harvey

ABSTRACT TIME-DOMAIN NUMERICAL TECHNIQUES FOR THE ANALYSIS AND DESIGN OF MICROWAVE CIRCUITS by Emmanouil M. Tentzeris Chair: Linda P.B.Katehi This dissertation investigates the effects of the application of the principles of Multiresolution Analysis (MRA) to time-domain numerical techniques used for the analysis and design of microwave circuits. The improvement in the efficiency in terms of memory and execution time requirements is quantified and the inherent capability of MRA to create a mathematically consistent time/space adaptive gridding is exploited. Initially, various aspects concerning the popular finite-difference time-domain technique (F.D.T.D.) are investigated and a memory-efficient waveguide absorber based on analytical Green's functions is developed and applied to the optimization of a specific waveguide probe geometry. After reviewing the general principles of Multiresolution Analysis, novel time-domain schemes based on space-domain expansions in scaling and wavelet functions are derived. FDTD implementation schemes (excitation, hard/open boundary and dielectric interfaces) are extended to Multiresolution schemes based on entire-domain expansion basis, while

maintaining similar performance characteristics. These schemes offer the unique opportunity of a multi-point field representation per cell. Battle-Lemarie functions are used throughout the dissertation due to their special qualities. These Multiresolution Time-Domain Schemes in 2D are applied to the numerical analysis of shielded and open striplines and microstrips. The field patterns and the characteristic impedance are calculated and verified by comparison to reference data. In comparison to Yee's conventional FDTD scheme, the proposed 2.5D-MRTD scheme offer memory savings by a factor of 25 and execution time savings by a factor of about 4-5 maintaining a better accuracy for characteristic impedance calculations. The stability and the dispersion performance of the Battle-Lemarie MRTD schemes is investigated for different stencil sizes and for 0-resolution wavelets. Analytical expressions for the maximum stable time-step are derived in a way similar to the "magic step" of the FDTD algorithm. A dynamically changing space- and time- adaptive meshing algorithm based on a multiresolution time-domain scheme in two dimensions and on absolute and relativ e thresholding of the wavelet values is proposed and applied to the numerical analysis of various nonohomogeneous waveguide geometries offering additional memory economy. In the last Chapter, intervalic wavelets are added in the time-domain. This TimeAdaptive Time-Domain Technique is used for the analysis of various types of circuits problems with active and passive lumped and distributed elements. This scheme exhibits significant savings in execution time and memory requirements while maintaining a similar accuracy with the FDTD technique. ii

~ Emmanouil M. Tentzeris 1998 All Rights Reserved

"As you set out for Ithaka hope your road is a long one, full of adventure, full of discovery. Laistrygonians, Cyclops, angry Poseidon - don't be afraid of them: you'll never find things like that one on your way as long as you keep your thoughts raised high, as long as a rare sensation touches your spirit and your body. Keep Ithaka always in your mind. Arriving there is what you're destined for. But don't hurry the journey at all. Better if it lasts for years, so you're old by the time you reach the island, wealthy with all you've gained on the way, not expecting Ithaka to make you rich. Ithaka gave you the marvellous journey. Without her you wouldn't have set out. She has nothing left to give you now. And if you find her poor, Ithaka won't have fooled you. Wise as you have become, so full of experience, you'll have understood by then what these Ithakas mean. " K.Kavafis, "Ithaka" (1911) ii

To my parents; To my brother, John. ii

ACKNOWLEDGEMENTS The completion of this dissertation and my entire doctoral study at the University of Michigan would not have been possible without the support of many people. I would like to take this opportunity to express my sincere gratitude to all those who helped me in one way or another during these prolific and memorable years. My first sincere gratitude and deepest appreciation goes to my advisor, Professor Linda Katehi, since without her insight, expertise, encouragement and understanding, this research would have been a much more difficult and painful journey. Her enthusiasm and boundless energy for research and teaching is very infectious. I am also grateful for her continuous financial support and for providing me with the opportunity to attend numerous conferences to present my research. I would also like to thank my comittee members for their time and consideration. Many friends and colleagues at the Radiation Laboratory have contributed to this work through discussions. Special thanks go to Dr. Michael Krumpholz, from whom I have constantly benefited through a great deal of fruitful conversations. Also, I would like to mention Mr. Robert Robertson. My close work with him and the understanding that we have between us made our research better and our friendship stronger. Dr. Nihad Dib and Ms. Kavita Goverdhanam are deeply appreciated for their useful suggestions. In summary, all Rad Lab friends, too many to mention by name, created a very enjoyable environment iii

for work and study. I should mention that the theoretical work on the FG-CPW has been done in cooperation with Dr. George Ponchak from the NASA Lewis Research Center. I thank him for his time and for supplying us with the results which validated the developed theoretical analysis. I specially thank Professor Andreas Cangellaris at the University of Illinois at UrbanaChampagne for his comments on the development of an efficient Time and Space Adaptive Gridding Algorithm. The major party of my study and research at the University of Michigan was sponsored by the Army Research Office. The continuous enthusiasm of this office, and especially of Dr.Harvey, in my research work is cordially appreciated. Dr.George Eleftheriades, Messrs. John Papapolymerou, Hristos Anastasiou, Hristos Patonis, Loucas Louca, Nikos Zacharopoulos and Kostas Sarris with their dear friendship made difficult moments easier. In addition, my numerous friends from the Hellenic Students Association of the University of Michigan gave a nice flavor and broadened my outlook on life. I also express my deepest gratitude to my parents Markos and Irene Tentzeris, who sowed in me from the early years the seeds of aspiration to ascend to the highest levels in every aspect of life, especially in education and knowledge. Finally, I wish to thank my younger brother John, who has always believed in me throughout my entire education. iv

TABLE OF CONTENTS DEDICATION....................................................... ii DEDICATION......................................... ii ACKNOWLEDGEMENTS.............................. iii LIST OF TABLES..................................... viii LIST OF FIGURES.................................. ix CHAPTERS 1 Introduction................................... 1 1.1 Time-Domain Techniques........................ 1 1.2 Wavelets-Multiresolution Analysis..................... 3 1.3 Overview of the Dissertation...................... 5 2 The Finite Difference Time Domain Technique (F.D.T.D.) and its Applications in the Analysis and Design of Microwave Circuits and Waveguide Probes...................................... 8 2.1 Foundations of the Finite Diference Time Domain (F.D.T.D.) Technique..................................... 8 2.1.1 Overview of Numerical Absorbing Boundary Conditions.. 14 2.1.2 Excitation Topics.......................... 15 2.1.3 Linear Predictors............................ 17 2.2 Applications of F.D.T.D. to Planar Circuits................ 20 2.2.1 Open Circuit Design........................... 20 2.2.2 Viahole Analysis.............................. 21 2.2.3 Filter Design.......................... 22 2.2.4 Finite-Ground CPW Line Analysis.............. 23 2.3 Application of FDTD to Waveguide Structures........... 24 2.3.1 FDTD and Waveguide Probe Structures............. 24 2.3.2 Novel Absorber Description....................... 26 2.4 Conclusion........................................... 36 3 Development of New Time-Domain Schemes with Higher Order Basis Functions........................................ 54 v

3.1 Introduction............................... 3.2 Fundamentals on Nlultiresolution Analysis................. 3.3 How to construct a Wavelet Function................. 3.4 The 2D MIRTD scheme............................ 3.4.1 Modeling of Hard Boundaries............... 3.4.2 Modeling of Excitation...................... 3.4.3 Treatment of Open Boundaries - PML Absorber....... 3.4.4 Total Field Calculation....................... 3.5 Conclusions.................................... 4 Characterization of Microwave Circuit Components Using the Multiresolution Time Domain Method (MRTD)......... 4.1 Introduction................................ 4.2 The 2.5D-MRTD scheme....................... 4.3 Applications of the 2.5D-MRTD scheme to Shielded Transmission Lines.................................. 4.4 Validation of the MRTD-PML Split and Non-split Algorithms... 4.5 Application of PML to the Analysis of Open Stripline Geometries. 4.6 Conclusion................................ 5 Stability and Dispersion Analysis of Multiresolution Time-Domain Schemes 5.1 Introduction............................... 5.2 Stability Analysis............................ 5.3 Dispersion Analysis........................... 5.4 Conclusion................................ 6 Development of a Space- and Time-Adaptive MRTD Gridding Algorithm 54 )55 57 63 71 72 75 79 81 86 86 87 89 94 95 98 109 109 110 113 118 for the Analysis of 2D Microwave Dielectric Geometries... 6.1 Introduction....................... 6.2 The 2D-MRTD Nonhomogeneous scheme........ 6.2.1 The 2D-MRTD scaling and wavelets scheme. 6.2.2 The PML numerical absorber......... 6.2.3 The Excitation Implementation........ 6.2.4 The Modelling of Dielectrics.......... 6.2.5 Total Field Calculation............. 6.2.6 Time-Adaptive Gridding............ 6.3 Applications of 2D-MRTD............... 6.3.1 Air-Filled Parallel Plate Waveguide...... 6.3.2 Parallel-Plate Partially Filled Waveguide... 6.3.3 Parallel-Plate Five-Stage Filter........ 6.4 Conclusion............................ 131.... 131.... 132.... 132.... 133.... 134.... 135.... 136.... 137.... 138.... 138.... 139.... 140.... 141 7 Time cuits 7.1 7.2 7.3 Adaptive Time-Domain Techniques for the Design of Microwave CirIntroduction................................. Time Adaptive MRTD Scheme.................... Applications in SPICE problems.................... 7.3.1 Distributed Elements...................... 149 149 150 151 152 vi

7.3.2 Lumped Elements................................ 156 7.4 Conclusion...................................... 158 8 Summary of Achievements-Future Work....................... 164 BIBLIOGRAPHY..................................... 166 vii

LIST OF TABLES Table 3.1 Coefficients a(i), bo(i), co(i)............................ 69 3.2 Excitation Scaling Decomposition Coefficients............. 74 3.3 Excitation Wavelet Decomposition Coefficients............ 74 4.1 Mode frequencies for / = 30......................... 90 4.2 Zo for different number of subpoints/cell (8x4 Grid).......... 92 4.3 Zo for different mesh sizes (11 subpoints/cell).............. 93 4.4 Dominant mode frequency for / = 30................... 97 4.5 Zo for different mesh sizes................................ 98 5.1 Coefficients Ci for Different MRTD Schemes.............. 115 6.1 S11 calculated by 2D-MRTD........................ 142 viii

LIST OF FIGURES Figure 2.1 Yee's FDTD cell................................................3 37 2.2 Patch Geometry to be used as Open................................. 38 2.3 Reflection Coefficient of the Open............................... 38 2.4 Viahole Structure........................................... 39 2.5 S-Parameters of the Viahole............................... 39 2.6 E-Distribution across Top Viahole (Top), Middle Ground Plane (Bottom).. 40 2.7 E-Distribution across Bottom Viahole.................... 41 2.8 Coupled Line Filter Geometry.................................. 42 2.9 Coupled Line Filter S21........................................... 42 2.10 Coplanar waveguide with finite width ground planes (F.G.C.).......... 43 2.11 Normal H-Distribution (Log) for B=25 pm (up), 100 pm (bottom)...... 44 2.12 Waveguide probe structure........................... 45 2.13 Waveguide Test Structure........................... 46 2.14 Reflection coefficient for the TEo mode.............................. 47 2.15 Comparison of Green's Function ABC and PML.......................47 2.16 Validation Data for the Reflection Coefficient....................... 48 2.17 Reflection Coefficient for different Dielectric Thicknesses.............. 49 2.18 E- and H-field Distributions across the Probe Structure Symmetry Plane. 50 2.19 E- and H-field Distributions across the Coplanar Feedline Plane......... 51 2.20 Reflection Coefficient for different Patch Widths..................... 52 2.21 Reflection Coefficient for different Distances from the Top Surface ShortCircuit....................................................... 52 2.22 Experimental Validation for Sl1.......................... 53 2.23 Experimental Validation for S21.............................. 53 3.1 BL Cubic Spline Scaling - Spatial Domain....................... 82 3.2 BL Cubic Spline Wavelet - Spatial Domain.................... 82 3.3 BL Cubic Spline Scaling - Spectral Domain........................83 3.4 BL Cubic Spline Wavelet - Spectral Domain....................... 83 3.5 Image Theory Application for tangential-to-PEC E-field............... 84 3.6 Treatment of Wavelet Components of normal-to-PEC E-field.......... 85 4.1 Printed Lines Geometries.......................... 100 ix

4.2 Shield TElo Ey pattern............................... 101 4.3 TEM Ey Pattern Components (8x8 Grid)............................101 4.4 TEM Ey Pattern Comparison (8x8 Grid)........................ 102 4.5 Tangential E-field Distribution (Shielded - Even Mode)...............102 4.6 4 PML cells - Non-split formulation - Dense Grid................... 103 4.7 8 PML cells - Non-split formulation - Dense Grid............... 103 4.8 16 PML cells - Non-split formulation - Dense Grid.................. 104 4.9 Multimodal Propagation - Dense Grid......................... 104 4.10 TEM Propagation - Coarse Grid.......................... 105 4.11 Multimodal Propagation - Coarse Grid.......................... 105 4.12 Comparison of Sampled vs. Non-sampled PML........................106 4.13 Comparison of the Split and Non-split Formulations.................. 106 4.14 Open Single Stripline - Ey TEM Distribution..................... 107 4.15 Open Coupled Stripline Geometry....................... 107 4.16 Tangential E-field Distribution (Open - Even Mode).................. 108 4.17 Tangential E-field Distribution (Open - Odd Mode)............... 108 5.1 Stability Parameter s for MRTD...................... 119 5.2 Dispersion Characteristics of S-MRTD for na=8........................ 119 5.3 Dispersion Characteristics of S-MRTD for na=10................ 120 5.4 Dispersion Characteristics of S-MRTD for na=12................................120 5.5 Dispersion Characteristics of S-MRTD for na=14.......................121 5.6 Stencil Effect on the Dispersion Characteristics of S-MRTD (Sparse Grid).. 122 5.7 Stencil Effect on the Dispersion Characteristics of S-MRTD (Dense Grid).. 122 5.8 Wavelets Effect on the Dispersion Characteristics of MRTD for na=8 (Coarse Grid)................................................ 123 5.9 Wavelets Effect on the Dispersion Characteristics of MRTD for na =8 (Denser Grid)............................................ 123 5.10 Wavelets Effect on the Dispersion Characteristics of MRTD for na= 10 (Coarse Grid).......................................... 124 5.11 Wavelets Effect on the Dispersion Characteristics of MRTD for na= 10 (Denser Grid).......................................................124 5.12 Wavelets Effect on the Dispersion Characteristics of MRTD for na =12 (Coarse Grid)..................................... 125 5.13 Wavelets Effect on the Dispersion Characteristics of MRTD for na=12 (Denser Grid)....................................... 125 5.14 Effect of the Courant Number on the Dispersion Characteristics of Wo - MRTD for na = nb = n=10 (Coarse Grid)...................... 126 5.15 Effect of the Courant Number on the Dispersion Characteristics of Wo - MRTD for na = nb = nc=10 (Denser Grid)...................... 126 5.16 Effect of the Courant Number on the Dispersion Characteristics of Wo - MRTD for na = n= n =12 (Coarse Grid)..........................127 5.17 Effect of the Courant Number on the Dispersion Characteristics of W - MRTD for na = nb = nc=12 (Denser Grid)....................... 127 5.18 Effect of the Courant Number on the Dispersion Characteristics of FDTD (Coarse Grid)........................................128 x

5.19 Effect of the Courant Number on the Dispersion Characteristics of FDTD (Denser Grid).................................... 5.20 Comparison of the Dispersion Performance of S-MRTD and Wo-Ml Different Higher Order Yee's Schemes................ 5.21 Details of Fig.(5.20)......................... 5.22 Comparison of the Oscillations of Wo-MRTD Scheme for Different; 5.23 Details of Fig.(5.22)....................... 6.1 Time- and Space- adaptive grid.................. 6.2 Normal E-field (Time-Domain)...................... 6.3 Non-zero Wavelets' Number..................... 6.4 Adaptive Grid Demonstration (t=1000 steps)............ 6.5 Parallel-Plate Partially Filled Waveguide............... 6.6 S-Parameters of the Waveguide................. 6.7 Non-zero Wavelets' Number........................ 6.8 Adaptive Grid Demonstration (t=1000 steps)............ 6.9 Parallel-Plate Five-Stage Filter.................... 6.10 S-Parameters of the Filter....................... 6.11 Non-zero Wavelets' Number.................... 6.12 Adaptive Grid Demonstration (t=1000 steps)............. 7.1 0-Order Intervalic Function Basis.................. 7.2 Ideal (Lossless) Transmission Line................... 7.3 Voltage Coefficients......................... 7.4 Comparison MRTD-FDTD..................... 7.5 Fraction of Wavelets above Threshold............... 7.6 Lossy Transmission Line....................... 7.7 Voltage Coefficients......................... 7.8 Fraction of Wavelets above Threshold................ 7.9 Rectifier Geometry........................... 7.10 Comparison MRTD-FDTD....................... RTD with............ Stencil Size....... *..... ~.*. *.. * *.... ~... * *... * *..... *.. * * * *. * * * *..* *.. * * *. * * 1'2S 129 129 130 130 142 143 143 144 144 145 145 146 146 147 147 148 159 159 160 160 161 161 162 162 163 163 xi

CHAPTER 1 Introduction 1.1 Time-Domain Techniques With the advent of microwave circuits used in high-frequency communications, there is a compelling need to develop efficient and reliable full wave simulation techniques for the modeling process. Until 1990, the modeling of electromagnetic wave interactions was dominated by frequency-domain techniques. Apart from high-frequency asymptotic methods [1, 2], electromagnetic simulations involved setting up and solving frequency-domain integral equations [3, 4] for the phasor electric and magnetic currents induced on the surfaces of the geometries of interest. This Method of Moments (MoM) involves setting up and solving dense, full, complex-valued systems of tens of thousands of linear equations using direct or iterative techniques. Though MoM has been proven to be a very robust technique, it is plagued by significant computational burdens, when it is used at very large geometries. In addition, modeling of a new structure requires the reformulation of the integral equation, a task that may require the very difficult derivation of a geometry-specific Green's function. On the contrary, techniques based on the partial differential equation (PDE) solutions of the Maxwell's equations yield either sparse matrices (frequency-domain finite-element 1

methods) or no matrices at all (time-domain finite-difference or finite-volume methods). In addition, specifying a new geometry is reduced to a problem of mesh generation only. 'Thus, time-domain PDE solvers could provide a framework for a space/time microscope permitting the EM designer to visualize with submicron/subpicosecond resolution the dynamics of electromagnetic wave phenomena propagating at light speed within proposed geometries. Finite-Difference Time-Domain (FDTD) is a direct solution method for Maxwell's time dependent curl equations. It is based upon volumetric sampling of the unknown near-field distribution within and around the structure of interest over a period of time. No potentials are employed. The sampling is set below the Nyquist limit and typically more than 10 samples per wavelength are required. The time-step has to satisfy the stability condition. For simulations of open geometries, absorbing boundary conditions (ABC) are employed at the outer grid truncation planes in order to reduce spurious numerical reflection from the grid termination. In 1966, Yee [5] introduced the first finite-difference time-domain technique (FDTD) for the solution of Maxwell's curl equations. Interleaved positioning of the electric and magnetic field components provided a second-order accuracy of the algorithm. Taflove and Boldwin [6] presented the numerical stability criterion for Yee's algorithm and Mur [7] published the first numerically stable second-order accurate absorbing boundary condition (ABC) for the Yee's mesh. The perfectly Matched Layer (PML) ABC, introduced in 2D by Berenger in 1994 [8] and extended to 3D by Katz et al. [9], provides numerical reflection comparable to the reflection of anechoic chambers with values -40dB lower than the Mur ABC. The FDTD technique has been applied to various High-Frequency simulations with remarkable success. Taflove [10] and Umashankar [11] used FDTD to model scattering and compute near/far fields and RCS for 2D and 3D structures. Waveguide - Cavity struc 2

tures and microstrips were analyzed with FDTD by Choi [12] and Zhang [13] respectively. MIaloney [14] introduced the FDTD modeling of antennas and El-Ghazaly [15] applied this technique to picosecond optoelectronic switches. Toland et al. [16] published the first FDTD models of nonlinear devices (tunnel diodes and Gunn diodes) exciting cavities and antennas and Sui et al. [17]modeled lumped electronic circuit elements in 2D. Despite the numerous applications of FDTD, many practical geometries, especially in microwave and millimeter-wave integrated circuits (MMIC), packaging, interconnects, subnanosecond digital electronic circuits (such as multichip modules (MCNI)) and antennas used in wireless and microwave communication systems, have been left untreated due to their complexity and the inability of the existing techniques to deal with requirements for large size and high resolution. Multiresolution analysis based on the expansion in scaling and wavelet functions has demonstrated a capability to provide space and time adaptive gridding without the problems encountered by the conventional Finite Difference Time-Domain schemes. As a result, it could be used as a powerful foundation for the development of very efficient electromagnetic simulation techniques. 1.2 Wavelets-Multiresolution Analysis The term "wavelets" has a very broad meaning, ranging from singular integral operators in harmonic analysis to subband coding algorithms in signal processing, from coherent states in quantum analysis to spline analysis in approximation theory, from multiresolution transform in computer vision to a multilevel approach in the numerical solution of partial differential equations, and so on. Considering the characteristics of time-domain numerical techniques for the solution of Maxwells' equations, wavelets could be considered to be mathematical tools for waveform representations and segmentations, time-frequency anal 3

vsis and fast and efficient algorithms for easy implementation in both time and frequency domains. One of the most important characteristics of expansion to scaling and wavelet functions is the time-frequency localization. The standard approach in ideal lowpass ("scaling") and bandpass ("wavelet") filtering for separating an analog signal into different frequency bands emphasizes the importance of time localization. The Multiresolution Analysis (MRA), introduced by Mallat and Meyer [18, 19], provides a very powerful tool for the construction of wavelets and implementation of the wavelet decomposition/reconstruction algorithms. The sampling theorem can be used to formulate analog signal representations in terms of superpositions of certain uniform shifts of a single function called a scaling function. Stability of this signal representation is achieved by imposing the Riesz condition on the scaling function. Another important condition of an MRA is the nested sequence of subspaces as a result of using scales by integer powers of 2. In the case of cardinal B-splines [20], an orthonormalization process is used to produce an orthonormal scaling function and, hence, its corresponding orthonormal wavelet by a suitable modification of the two-scale sequence. The orthonormalization process was introduced by Schweinler and Wigner [21] and the resulting wavelets are the Battle-Lemarie wavelets, obtained independently by Battle [22] and Lemarie [23] using different methods. The only orthonormal wavelet that is symmetric or antisymmetric and has compact support (to give finite decomposition and reconstruction series) is the Haar [24] wavelet [25]. Nevertheless, these wavelets exhibit poor time-frequency localization. Throughout this Thesis, Battle-Lemarie and Haar scaling and wavelet functions will be used as an expansion basis for the E- and H- field components in space and time domain respectively, in order to derive an efficient and fast Multiresolution Time-Domain Scheme for the numerical approximation 4

of Maxwell's equations in a way similar to [26]. 1.3 Overview of the Dissertation Chapter 2 gives a general overview of the FDTD Technique. Excitation topics and ways of improving the algorithm performance are discussed separately. Next, FDTD is used in the analysis of various planar circuits and waveguide probe structures. A new waveguide absorber based on analytic modal green's functions is developed; it is characterized by a better performance in memory requirements than the PML absorber, while maintaining similar accuarcy. The scattering parameters of the probe structures are calculated and verified by comparison with FEM and experimental data. The effect of critical geometrical parameters on the probe performance are investigated and the probe behavior is optimized. Chapter 3 starts with a discussion on the need of development of novel time-domain schemes which would alleviate the serious memory and execution time limitations of the existing techniques. The basic principles of the Multiresolution Analysis as well as the technique of the construction of wavelet functions are presented. Analytical spectral expressions for the linear and cubic cardinal splines are derived as an example. The 2D MRTD algorithm based on Battle-Lemarie expansion basis is developed for a grid similar to that of the FDTD. Hard Boundaries, such as Perfect Electric Conductors, and arbitrary excitations are implemented in an automatic way. The principles of the PML absorber are extended in split and nonsplit formulations providing a very efficient absorber. Notes on the total field value calculation at every spatial point conclude this Chapter. In Chapter 4, the MRTD scheme is applied to the numerical analysis of 2.5D shielded and open striplines and microstrips. The field patterns and the characteristic impedance are caluclated and verified by comparison to reference data. Simulations display memory 5

savings by a factor of 23 and execution time savings by a factor of 4-5. For structures where the edge effect is prominent, additional wavelet resolutions have to be introduced to maintain a satisfactory performance while using a coarse MRTD grid. The non-split P.ML algorithm is evaluated for different cells sizes and its performance is comparable to that of the conventional FDTD PML absorber. Chapter 5 investigates the stability and the dispersion performance of MIRTD for different stencil (number of summation terms) sizes and for O-resolution of wavelets. Analytical expressions for the maximum stable time-step are derived for schemes containing only scaling functions or combination of scaling and wavelet functions. It is proved that larger stencils decrease the numerical phase error making it significantly lower than FDTD for low and medium discretizations. The addition of wavelets further improves the dispersion characteristics for discretizations close to the Nyquist limit, though it decreases the value of the maximum stable time-step. A mathematically correct way of dielectric modeling is presented and evaluated in the first part of Chapter 6. A dynamically changing space- and time- adaptive meshing MRTD algorithm based on a combination of absolute and relative thresholding of the wavelet values is proposed. Different thresholding implementations are evaluated by the application of the dynamically changing grid to the numerical analysis of various nonhomogeneous 2D waveguide structures. This scheme offers memory savings by a factor of 5-6 per dimension in comparison to FDTD. The direct application of the principles of the Multiresolution Analysis to the time domain is presented in Chapter 7. A Time Adaptive Time-Domain Technique based on Haar basis is proposed and applied to various types of circuits problems with active and passive lumped and distributed elements. The addition of the wavelets increases the resolution in 6

time. something that is very important especially in circuits with nonlinear devices such as diodes and transistors. This scheme exhibits significant savings in execution time and memory requirements while maintaining a similar accuracy with conventional circuit simulators. The Thesis closes with ideas for future work described in Chapter 8. 7

CHAPTER 2 The Finite Difference Time Domain Technique (F.D.T.D.) and its Applications in the Analysis and Design of Microwave Circuits and Waveguide Probes 2.1 Foundations of the Finite Diference Time Domain (F.D.T.D.) Technique Considering an area with no electric or magnetic current sources, the time-depen dent Maxwell's equations are given in differential form by Faraday's Law: aB =Vx E-Jm Ampere's Law: = Vx H-J Gauss's Law for the electric field: V D -0 8

Gauss's Law for the magnetic field: V.B=O Here, E is the electric field vector, D is the electric flux density vector, H is the magnetic field vector, B is the magnetic flux density vector, Je is the electric conduction current density, Jm is the equivalent magnetic conduction current density. In linear, isotropic nondispersive materials, B and D can be related to H and E, respectively, using the constitutive equations: B = pH D = EE (2.1) where p is the magnetic permeability and e is the electric permittivity. To account for the electric and magnetic loss mechanisms, an equivalent electric and magnetic current can be introduced Je = aE Jm = p'H (2.2) with a the electric conductivity and pi the equivalent magnetic resistivity. Combining Eqs.(2.1)-(2.2) with Maxwell's equations, we obtain _1 p' — Vx E- H (2.3) Ot OE 1H = -x I — E (2.4) ct E e The curl equations (2.3)-(2.4) yield the following system of six coupled scalar equations in the 3-D rectangular coordinate system (x, y, z): a9H - 1 OEy E, at Z- y p 9

OH, 1 (E- Ey OH I aE, aE, = ( p'H ) ot t ay Ox 9E, 1 9H, 9Hy Ot E Oy OXz Ey 1 OH, 9H, O - Ox - OE, _ OHy OH1 Ot - - ~ Ox- -oaEz) (2.5) at e ay Eq.(2.5) forms the basis of the FDTD numerical algorithm for general 3-D objects. The FDTD algorithm need not explicitly enforce the Gauss's Law relations. This occurs because they are theoretically a direct consequence of the curl equations. However, the FDTD space grid must be structured so that the Gauss's Law relations are implicit in the positions of the electric and magnetic field vector components in the grid and the numerical space derivative operations upon these vector components that model the action of the curl operator. The above system of equations can be reduced to 2-D assuming no variation in the z-direction. That means that all partial derivatives with respect to z equal zero and that the analyzed structure extends to infinity in the z-direction with no change in the shape or position of its transverse cross section. Eq.(2.5) will give in rectangular coordinates: OH~ 1 9Ez OHt 1 01i9E f (2.6) = (- p'H) (2.7) Ot A LOy OH 1 (E - ~x- - p p() (2-8) at ay t oEy = 1 OH, ( E 'Hz) (2.9) E 1 - aE) (2.10) at e Oy MEy 1 aOH~ ( aEY) (2.10) Ot E Ox OE~ - Hy 9(I OH OEz 1 - - aEz) (2.11) Eqs.(2.6),(2.7),(2.11) constitute the transverse magnetic (TMZ) mode; the rest the trans 10

verse electric (TEZ) mode 2-D equations. The TEZ and TAF modes are decoupled since they contain no common field vector components. These modes are completely independent for structures composed of isotropic materials or anisotropic materials having no off-diagonal components in the constitutive tensors. That means that they can exist simultaneously with no mutual interactions. Equations for 1-D cases can be derived in a similar way assuming no variation in the xor y-direction in excess to no variation in the z-direction. Yee [5] proposed a set of finite-difference equations for the time-dependent Maxwell's curl equations, solving for both electric and magnetic fields in time and space instead of solving for the electric field alone (or the magnetic field alone) with a wave equation. In this way, the solution is more robust and more accurate for a wider class of structures. In addition, both electric and magnetic material properties can be modeled in a straightforward manner. In Yee's discretization cell (Fig.2.1), E- and H- fields are interlaced by half space and time gridding steps. The spatial displacement is very useful in specifying field boundary conditions and singularities and creates finite-difference expressions for the space derivatives which are central in nature and second-order accurate. It has been proven that the Yee mesh is divergence-free with respect to its electric and magnetic fields, and thereby properly enforces the absence of free electric and magnetic charge in the source-free space being modeled. The time displacement (leapfrog) is fully explicit, completely avoiding the problems involved with simultaneous equations and matrix inversion. The resulting timestepping algorithm is non-dissipative; numerical wave modes propagating in the mesh do not spuriously decay due to a nonphysical artifact of the time-stepping algorithm. Denoting any function u of space and time evaluated at a discrete point in the grid and 11

at a discrete point in time as I iA x jA y,k A.Z.,lIAt) I lUijk where A~t is the time step and Ax, A~y, A.z1 the cell size to the x-, y- and. z-direction, the first partial space derivative of ut in the x-direction and the first time derivative of 71 are approximated with the following central differences respectively i9u lUi+112,j,k - lUi-1/2,j, k +O0[(zAX)2] c9U = 1+1/2UI,3-,k - 1-1/2Ui,,,k +O(~)](.2 Applying the above notation, the following FDTD equations are derived for 3-D geometries (i_ -,j,k A (i - EY.,klIEkk IE if _.,jk. = I 0. '-jk o! + \+ Z) y (2.14) 10+ 't-o'j'k-0/ '-0.5 t...0.5,3JA.0.5k + +~ At) -l L511k_ EkS~ + 1x +z = - A -, _,_k ('-o -OJIj-051,k p 5 a 1 —5 io-1...5j-....5,k lr0,k5 -k Az1k 12

(2.16) EY- E,j ry- f _ at \ N | '.k \ l E' k I a t)k,k 10.5 Hio05 k0o.5 105 Ht0.5,k-0.5 _ 105 io.5,jo.5,k - o.5 H-i0.5 ijo.5 s k az. ) (2.17) 2c-,j,k ^ \ 2/ 2,j-0 5 -, t + ) WE ^..-0.5 + l____ + \1 1 2~. 2e.,1 + a2,, / 10.5 o.j j.,k-..50.5 -5o.s,5,ko.s -.5o.s 10.5 t 0o.k-0.5 H,- o0.5s.s L Ax y A (2.18) where Ci,j,k and P jk are the electric and magnetic loss coefficients for the (i, j, k)-cell. The notation la Uibjckd = l-aUi-b,j-c,k-d is used for compactness.It can be observed that a new value of a field vector component at any space lattice point depends only on its previous value and the previous values of the components of the other field vectors at adjacent points. Therefore, at any given time step, the value of a field vector component at p different lattice points can be calculated simultaneously if p parallel processors are employed, something that demonstrates the fact that the FDTD algorithm is highly parallelizable. The exponential decay of propagating waves in certain highly lossy media is so rapid that the standard Yee time-stepping algorithm fails to describe. Holland [27] has proposed an exponential time-stepping. For example, for large values of a, the field component E: is given by E t -0.s,5k e 'jt/,,'j,k I Ei o, k - (ek - (s 5 io.ojk 0,j,k ( H k - H. fz Hy -,Hk,j,\Ho.2 o.5,i- o.s k ~ 10.5,-0.5,y-0o.5 Ak I,-o.5j,ko.5 H -.5J, o.5. Ay AZ 13

instead of -Eq.(2.16). Stability analysis [57] has shown that the upper bound for the FDTD time step for a homogeneous region of space (er, r) is given by C ()2 + (y)2 + ()2 for 3-D simulations and At < for 2-D simulations. Lower values of upper bounds are used in case a highly lossy material or a variable grid is employed. Discretization with at least 10-20 cells/wavelength almost guarantee that the FDTD algorithm will have satisfactory dispersion caharacteristics (phase error smaller than 5~/A for time step close to the upper bound value). 2.1.1 Overview of Numerical Absorbing Boundary Conditions It is very common for the geometries of interest to be defined in "open" regions where the spatial domain of the computed EM fields is unbounded in one or more coordinate directions. Since no computer can store an unlimited amount of data, the field computation domain must be limited in size. The computation domain must be large enough to enclose the structure of interest, and a suitable absorbing boundary condition (ABC) on the outer perimeter of the domain must be used to simulate its extension to infinity. ABC's cannot be directly obtained from the numerical algorithms for Maxwells' curl equations defined by the Yee's finite-difference systems. This is due to the fact that these systems employ a central spatial difference scheme that requires knowledge of the field one-half cell to each side of an observation point. Central differences cannot be implemented at the outermost lattice planes, since by definition there exists no information concerning the fields at points one-half space cell outside of these planes. Backward finite differences are generally of lower accuracy 14

for a given space discretization, so they cannot be used as a reliable solution. Several approximate ABC's have been proposed [28, 29, 30, 31]. In our FDTD simulations, 1st and 2nd order Mur ABC [7], coupled with Mei-Fang Superabsorption [32] for complicated structures, have been used to terminate open domains due to their simplicity and versatility. Reflection coefficients close to -60dB have been achieved for a wide range of incidence angles and frequencies. For waveguide structures a new ABC based on Green's functions has been developed. Reflection coefficients obtained by the recently developed PML [8] have been used as a reference for the validation of the novel ABC. 2.1.2 Excitation Topics The first source to be modeled in FDTD was a plane wave incoming from infinity [5]. The plane wave source is very useful in modeling radar scattering problems, since in most cases of this type the target of interest is in the near field of the radiating antenna, and the incident illumination can be considered to be a plane wave. The hard source [33] is another common FDTD source implementation. It is set up simply by assigning a desired time function to specific electric or magnetic field components in the FDTD space lattice. In this way, it radiates a numerical wave having a time waveform corresponding to the source function. This numerical wave propagates symmetrically in both directions from the source point. However, this way of excitation has some drawbacks. As time-stepping is continued to obtain either the sinusoidal steady state or the late-time impulse response, the reflected - from the discontinuities - numerical wave eventually returns to the source grid location. Since the total electric field is specified at the excitation point without regard to any possible reflected waves, the hard source causes a spurious, nonphysical retroreflection of these waves toward the structure of interest, failing 15

to simulate the propagation of the reflected wave energy. A simple way to avoid this problem is to remove the source from the algorithm after the pulse has decayed essentially to zero and apply instead the regular Yee update. However, this approach cannot be used for continuous source waveforms where the source remains active even after reflections propagate back to it. It has been observed that much less error occurs for hard sources in 2-D and 3-D than in 1-D because the hard sources in 2-D and 3-D intercept and retroreflect much smaller fractions of the total energy in the FDTD grid. Collinear arrays of hard-source field vector components in 3D can be useful for exciting waveguides and strip lines. The total field excitation eliminates the retroreflection problems of the hard source. A proper field component is simply added to the field values given by the regular FDTD equations. Let's consider for example Eq.(2.16) for rij,k = 0 and no field variation to the z-direction 'xrx, A t (1 o..sjo.5,k.5-o.st-o.j 5,k 1 E =E0.5,Jk = i-Y. Ejk In the total field implementation of the source, one time dependent term is added to the field component of interest. Calling for simplicity this term As, Ex component at the excitation cell is updated by Ex 5At / o.s s-o.5,jok - 1o.5 H-5,j^o.5k + it-0.5tjtk == 'E-0.5ik +-Ei,j,k Ay If the circuit and the position where the source is applied allow a conductance current to flow, this term actually can be seen as an impressed conductance current density given by A8s = jn+l/2 ECZ On the other side, if a conductance current cannot flow, and thus only a displacement current can exist (e.g. the excitation of an empty cavity by applying a point source in 16

the middle), it actually works as if an additional term added to the ~E component. The modified discretized Maxwell equation can be written as: Zt l o.sijo.5 k - lo.s5 -o5,j -o.5s k Ei_-o.s (,, - IHEio.5,jk- AS- ty - = Ei,j,k Ay That corresponds to the following analytical expression aEx ds(t) _ H, At dt ay Thus, the term added to the field component is the derivative of the waveform we want to obtain. As a conclusion, if the circuit allows a conductive current density to be supported, the desired waveformmust be simply added to the field component at the location of the source; if only a displacement current can be supported by the structure, the derivative of the desired waveform must be added instead. In the FDTD simulations reported in this Chapter, a gaussian pulse (nonzero DC content) was used as the excitation of the microstrip and stripline structures.The Gabor function s(t) = e-((t-t~)/(pw))2 sin(wt) (2.20) where pw = 2. * V6 f, to = 2pw, w = 7r(fmni + fmax), was used as the excitation of the waveguide structures, since it has zero DC content. By modifying the parameters pw and w, the frequency spectrum of the Gabor function can be practically restricted to the interval [fmin,fmar]. As a result, the envelope of the Gabor function represents a gaussian function in both time and frequency domain. 2.1.3 Linear Predictors It is very common, especially for high-speed circuit structures, to use a cell size A that is dictated by the very fine dimensions of the circuit and is almost always much finer than 17

needed to resolve the smallest spectral wavelength propagating in the circuit. As a result, with the time step At bound to A by numerical stability considerations, FDTD simulations have to run for tens of thousands of time steps in order to fully evolve the impulse responses needed for calculating impedances, S-parameters or resonant frequencies. One popular way to avoid virtually prohibitive execution time has been to apply contemporary analysis techniques from the discipline of digital signal processing and spectrum estimation. The strategy is to extrapolate the electromagnetic field time waveform by 10:1 or more beyond the actual FDTD time window, allowing a very good estimate of the complete system response with 90% or greater reduction in computation time. The class of linear predictors or autoregressive models (AR) is the most popular time series modeling approach due to the fact that an accurate estimate of the AR parameters can be derived by solving a set of linear equations. Though Prony's method [34] uses a sum of deterministic exponential functions to fit the data, the AR approach constructs a random model to fit a statistical data base to the second-order. Let's consider the FDTD impulse response p + 1 equally spaced time samples after at time-step n f{n j-+ l fIn+p li,j,kA Jij,k'".. li,j,k This time series is said to represent the realization of an AR process of order p if it satisfies the following relationship f,i,,k = -a ik- -alf,,k + q(n) where the constants a1,...,ap are the AR parameters to be determined from the previous values of f and q(n) is a white noise process whose variance has to be calculated before carrying out the extrapolation of f. Once the AR coefficiemts have been determined, the above equation permits the prediction of a new value of the time series from p known 18

previous values. Numerous different approaches for the evaluation of ai have been proposed. Three of them of the most widely used: the covariance method, the forward-backward method and the nonlinear predictor. The covariance method involves setting up and solving a p x p linear system of equations cff(1,1) cff(1,2)... Cff(l,p) al cff(l,0) Cff(2,1) Cff(2,2)... Cff(2,p) a2 cff(2,0) Cff(p,l) cff(p,2)... cff(p,p) ap cff(p,O) where c f are the covariances defined by N-1 cff(a, b) = - (f I n-a + i M+n-b) N-p Z 1 j n=p The above matrix can be solved with Cholesky decomposition. The order p of the model is very critical. The use of low order AR model causes the extrapolated waveform to attenuate quickly in a nonphysical manner. However, a high-order model can cause divegence problems in some cases because of statistical instabilities introduced by the large order. A coomon way to estimate p is the use of the Akaike Information criterion [34]. Forward and backward prediction methods avoid these problems by working directly with the time-domain data, rather than calculating the covariance functions of the data. It solves the following (p + 1) x (p + 1) linear system r(O,O)... r(O,p) 1 ep r(l,0)... r(l,p) aI 0 r(p,0)... r(p,p) ap 0 where for 0 < a, b < p, N-p r( (a, I (f M+p+l-bf fM+p+l-a + f M+l+b fIM+l+a =r(a, 1~ i,j,k I k + f 1=1 19

p ep= E alr(0, ) 1=o Marple [35] reported favorable results for the forward-backward method versus existing popular AR approaches such as the Burg and the Yule-Walker algorithms. It provided more accurate spectra and its order was much lower (close to 10% - 15% of the order of the other approaches). In addition, the forward-backward method is sufficiently robust and fast, though it's slightly less stable than the covariance methods. 2.2 Applications of F.D.T.D. to Planar Circuits 2.2.1 Open Circuit Design The F.D.T.D. is initially applied in the design of a patch to be used as an open for the frequency range from 0-6 GHz (Fig.2.2). The dielectric constant of the substrate is ~r = 5.46 and the dielectric thickness is 0.5 mm. The feeding microstrip line (104.86875 mm) _f t1&t-t0 )2 is excited by applying horizontally the Gaussian pulse e( pw ) with pw = 8.333 * 10-11, dt = 2.9 10-13sec, to = 3pw. The excitation is on for t = 0,.., tab time-steps with tab = 6pw/At. During this period, a PEC (perfect electric conductor) is placed behind the source at the vertical to the propagation plane. After t becomes larger than tab, this PEC is replaced with a 1st order Mur's absorber and the results converge after 30,000 time steps. After numerical experimentation, it is observed that the smallest vertical distance the top-plane 1st order Mur absorber can work efficiently equals to 30 times the substrate thickness. The front and the side absorbers are placed at a distance 49.35 mm and 7.7425 mm away from the patch respectively. In addition, the resonant frequency of the patch antenna should be such that it would not cause any problems for the operating frequency range. As a result, the almost square shape of the patch is maintained, but the dimensions 20

have to be appropriately modified. After using a mesh with cell size dx = O.1mm dy = 0.20375mm. dz = 1.23375mm. the optimum performance patch dimensions are found to be: 7.4025 mm (length) x 7.335 mm (width). (Fig. 2.3) demonstrates that the performance of the open is almost perfect since the reflection coefficient is larger than 0.97 for the whole frequency range. 2.2.2 Viahole Analysis The viahole transition between two bended microstrips (Fig.2.4) is another geometry analyzed with F.D.T.D. The two microstrip lines are sandwiched on a dielectric substrate with Er = 7 and the ground plane is placed in the middle of their distance. The top stripline is excited by applying horizontally a Gaussian pulse 0-20GHz. The discretization cell has dimensions 10um x 50pm x 50pm and the time step is 31ps. A forward-backward predictor based on the first 4,300 steps with order p = 27 is employed to shorten the computation time of the 18,000 steps. The S-parameters are calculated (Fig.2.5). (Fig.2.6-2.7) showing the total E-field distribution along the top and bottom microstrip planes as well as along the ground plane at frequency 10 GHz, demonstrate the capability of the F.D.T.D. technique for an accurate spatial mapping of EM energy. Knowledge of the electric field intensity over a microwave circuit is extremely useful in directly identifying microwave circuit problems such as the existence of substrate modes, circuit radiation, device to device coupling. With tighter control over line lengths and losses that may be derived from electric field intensity (and phase), it may be possible to reduce the number of iterations during the design of MMIC's. Also, with a map of the electric field intensity above the substrate it would be possible to define low electric field regions around a device that could be used for placement of more circuitry, thus saving valuable chip real-estate. 21

2.2.3 Filter Design (Fig.2.8) displays the geometry of a three stage coupled line filter fabricated on Duroid (er=10.8, h=635 jm). All dimensions are in j2m. The bandpass filter has a measured insertion loss of 2.0 dB in the passband from 8.0GHz to 10.5GHz and provides better than -25dB rejection at 12GHz. (Fig.2.9) shows that good agreement is achieved between measurements and FDTD calculations. The FDTD cell was chosen to be 52.9 um for the vertical direction, 100 gm for the propagation direction abd 25 gm for the direction normal to propagation. The time step is chosen to be 73 fsec to satisfy the stability criterion. These choices result in a grid with 140x234x448 cells. The Ist-order Mur's ABC is applied to the boundaries of the computational domain and superabsorber is enhanced at the input and output planes. For wideband S-parameter extraction, a Gaussian pulse of 100 psec is used as the vertical microstrip exciation. The source is applied 5 cells inside the feedline in the propagation direction. Two simulations of pulse propagation along the microstrip line are made: one simulation for the filter and one for a 50 Q microstrip through-line. The filter simulation gives the sum of the incident and the reflected waveforms and the through-line simulation gives only the incident waveform. By subtracting the incident from the total waveform, the reflected waveform at the input port is derived, which permits the calculation of the reflection coefficient Sn. The transmission coefficient S21 is given by the ratio of the Fourier transforms of the transmitted and the incident waveforms. The field probes are located at distances far enough from the filter discontinuities to eliminate evanescent waves. 22

2.2.4 Finite-Ground CPW Line Analysis Coplanar waveguide with finite width ground planes (F.G.C.) (Fig.2.10) is characte rized through measurements and F.D.T.D. to determine the optimum ground plane width. It is found that the characteristics (attenuation, effective permittivity) of the Finite Ground Coplanar Line are not dependent on the ground plane width if it is greater than twice the center conductor width, but less than Ad/8 to keep the radiation losses and dispersion small. Also, the field distribution plots show that the power that propagates along the F.G.C. is concentrated on the surface of the substrate and the magnitude of this power is inversely dependent on the ground plane width. For small finite ground plane, there exists a significant amount of power on the surface of the substrate outside of the ground planes.This is demonstrated by the distribution of the normal-to-strip magnetic field Hy for lines with ground plane widths of B=25 and B=100 gam (Fig.2.11) and S = W = 25pm on Si wafers of er=11.9 and of thickness of 400 /im. The field is approximately twice as strong for the narrower ground plane, and decays away from the outer edge of the ground plane. As a conclusion, coplanar waveguide with a finite ground plane width as small as twice the center strip width may be used without adversely affecting the attenuation and permittivity of the lines. The 2.5-D FDTD algorithm is used in the simulations. The dimensions of the Yee's cell are chosen to be 2.5/um for the direction parallel to the coplanar line and 25pm for the normal direction. The time step is 7.45 ps and the 1st-order Mur's ABC is applied to the top, left and right boundaries of the computational domain. The top absorber is placed at a distance equal to 15 times the dielectric thickness and the side absorbers at a distance equal to 7 times the gap of the coplanar line. A delta function with even (odd) symmetry is used for the excitation of the horizontal electric field across the gaps. The propagation 23

constant used in the simulations has the value 100. 2.3 Application of FDTD to Waveguide Structures 2.3.1 FDTD and Waveguide Probe Structures Significant attention is being devoted now-a-days to the analysis and design of waveguideprobes [36] -- [53]. Many different configurations of waveguide probes are used either to sense the modal propagation inside the waveguides or to mount active elements inside cavities. The common design objective is to maximize the coupling between the probe and the waveguide over the widest possible frequency range. The characterization of waveguideprobes demands an accurate calculation of the scattering parameters over a wide band of frequencies. In this Section, FDTD is used in the RF characterization of diode mounting and waveguide probe structures. The waveguide probe geometry analyzed in this section is shown in (Fig.2.12). The probe is fed by a shielded coplanar line and has the shape of a patch. It is inserted into the waveguide through a slot and it is supported by a dielectric substrate which is not connected to any waveguide wall. The dimensions of the probe as well as the thickness and the dielectric constant of the substrate are of critical importance to achieve broadband coupling and low reflection loss. Usually more than one mode are excited inside the rectangular waveguide, making the numerical simulation tedious when using the conventional absorbing boundary conditions (ABC's) [7], [32]. These ABC's specify the tangential electric field components at the boundary of the mesh in such a way that waves are not reflected. For TEM structures the waves will be normally incident to the boundaries of the mesh, thus requiring a simple approximate absorbing boundary condition, Mur's first order absorbing boundary condition 24

[7]. The assumption of normal incidence is not valid for the fringing fields propagating tangential to the walls. For this reason, for non-TEM structures the superabsorption boundary condition [32] is used in conjunction with Mur's absorber for better accuracy. This combination results in an improvement with respect to the reflection coefficient. However, despite the use of superabsorber, when the frequency range of interest becomes large, significant reflections occur, even if there is only one propagating mode. To overcome this difficulty, numerous approaches have been proposed. The technique of diakoptics [40], initially developed for TLM [41] and later for FDTD [42], used in conjunction with the modal Green's function has been successfully applied to TLM [43], [44], [45]. In the analogous FDTD approach [46], the fields are decomposed into incident and reflected wave amplitudes ("TLM" approach) and the characteristic impedance is used for the calculation of the reflected wave amplitudes. A similar absorber based on a circuit (voltage-current) approach has been proposed by F. Moglie et al. [47]. Due to the field decomposition, both of these approaches are characterized by higher memory and execution time requirements than the conventional FDTD absorbers. In contrast to these approaches, the Diakoptics technique is derived directly from Maxwell's equations following an approach similar to [48] and only total field values are used. The absorber proposed is based on the analytic Green's functions of the waveguide modes. These Green's functions are used to calculate the tangential electric (for TE modes) and magnetic (for TM modes) field components located at the boundary of the mesh. The tangential fields one cell away from the boundary are decomposed into modes and for each mode the tangential field at the boundary is calculated by taking the convolution of the mode amplitude and the Green's function for this mode with respect to time. For simplicity, we consider only TE propagating modes, while the approach for the TM propagating modes 25

is dual and straightforward. A similar approach based on numerical Green's functions has been presented in [49]. This approach requires the numerical evaluation of each mode's Green's function that is obtained by running an FDTD simulation for each mode and/or the application of the FD2 principles. On the contrary, the proposed absorber evaluates analytically the Green's functions by applying the Inverse Fourier transform to the well-known expressions in frequency domain. Thus, similar accuracy is obtained without a significant computational overhead. 2.3.2 Novel Absorber Description For the sake of simplicity in the presentation, we consider only TEZ,n modes, propagating in the z-direction, and assume that the waveguide cross-section is located on the xyplane. For the tangential magnetic field adjacent to the boundary of the mesh at k = n -0.5, eqs.(2.13),(2.14) for non-lossy material are simplified to, At EY + - I,n- 'E y 112xt. _IE. lnI,j+l/2,,nz- (2.21) l+l/2Hi'j+l/2'n~-0.5-1-l1/2Hi'+l/2'n-0.5 = -- A-1 ----z (2.21) 1/2i+l/2,jn-0.5 -1-1/2H1+l /2,j,n-0.5 -. (2.22) The absorber is used to calculate the tangential electric field components at the boundary of the mesh (k = nz) from the tangential electric field components one cell away from the boundary plane (k = nz - 1). The tangential magnetic field components HF j. and Hyfn,-o.5 are updated using eqs.(2.21)-(2.22) and depend both on the values of the electric field components calculated by Yee's FDTD scheme and on the values of the electric field components calculated by the absorber. Using eq.(2.15), the normal magnetic field components at k = n, H1+/2,j+1/2,n, may be calculated from E+l/2, and Ej+l/ Thus, for the TEzn modes, the normal magnetic field components are also determined so 26

that the reflection from the boundary is minimized. A similar argument can be used for the position of the absorber for the TMl,, modes. In order to derive the absorber based on the analytic Green's functions, we start with the wave equation in cartesian coordinates 1 d2F _ 2 2 92 1 a2 VF - t2 = (2 + + - 2 2 )F = (2.23) where F indicates the tangential electric field components Ex(x, y, z, t), Ey(x, y, z, t) and c represents the velocity of light. The tangential electric field components in the waveguide can be written as oo oo Ex(x, y, z, t) = Emr (z,) cos(3,mx) sin(13y,,y) (2.24) m=O n=l oo oo E(x, y, z, t) = E EEY,n(z, t) sin(,x,mx)cos(3y,ny) (2.25) m=l n=O where 1m r nr s,m - py,n = -, (2.26) a b m, n E N, axb are the waveguide cross section area and En(z, t) and Ey, (z,t) are the modal coefficients given by 2(2 -m,0) ba b s IX) E t 2(2,) Ij E(z y z, t) cos( ) sin( y)dxdy (2.27) E,( t) - 2(2 - no) EY(x,,, t) sin( m ) cos( y)dxdy (2.28) In eqs. (2.27) and (2.28), Sm,o is the Kroenecker delta given by 1 for m = O 6m,O = 0 for m 5 0 27

In view of the above, eq.(2.23) yields F z92 t) (, 2 + 2 + F z t)1 O (2.29) = M +,m C r n +2 oqt-2 m t) where Fm,n(z, t) = E (z, t). Applying the Fourier transformation (Fm,n(Z,) =.F{Fm,n(Zt)}) with the angular frequency w=2rf, the wave equation is transformed into frequency domain, and eq.(2.29) yields OZ2- X=,m + (yn-+2 Fm,n(Z ( w) = 0 (2.30) OZ ^ C2 Following a procedure analogous to [48] and assuming a given amplitude Fm,((nz - 1)z, w) of the TE', mode at k = -n - 1, we obtain Fmn(ZW) = - 2 (iFmn((nzA - L ) + 2 Fzmm,n(2(nz- 1 )Az, w) + m,(Oz) zt-) 2 (z ezm2IZ) -,=(nz -l, )A (2.31) with { V/~2 - mn for w > Wc,mn z,mn = (2.32) n- for < wc,mn where Wcmn = c(/(3,m)2 + (/y,n )2 is the cutoff frequency of the TE n mode. The function Fm,n(z,w) has exponentially increasing and decreasing solutions with respect to z for w < wc,mn. The exponentially increasing solutions have to vanish for z -- oo for w < Wc,mn, thus eq.(2.31) yields mn( - F,)n(Z,w) Fmn((nz- 1)Az^, Z) = zmn Oz, - (2.33) Pz,mn 2 Z=(n-)z ' Z=(nZ -l)A2; 28

and Fm,n(Z.,) = GTE (z - (n~ - 1)Az,) Fm,n((n.- l)A,). (2.34) where GTE. (z,) = e-J3z,mnz is the Green's function for the TE,, modes. By satisfying eq.(2.33), Fmn((nz - 1)Az,w) results in an outward propagating solution with respect to z for w < Wc,mn only. Thus, computation of Fm,n(Z,w) according to eq.(2.34) requires no backward propagating solution. Applying the convolution theorem [50], eq.(2.34) in time-domain reduces to +00 Fm,n(Z t) = / GTEi (z - (n - 1)Az, t - t') Fm,n((n, - 1)Az, t') dt' 2.35) -00 where GTE2 (zt) = F Gl{GTEm,(z,a)}. As a result, the tangential electric field components at the boundary of the mesh at k=nz are expressed in the form: +00 Fm,n(nzAz,t) = J GTEZ n(AZt - t') Fm,n((nz - 1)Az,t') dt'. (2.36) -00 Following a procedure similar to [63], Fm,n((nz - 1)Az, t') can be expanded in a series of triangle basis functions in time-domain. Inserting this expansion in eq.(2.36) and sampling Fm,n(nz Az, t) using delta functions with respect to time, we obtain 00 Fm,n(nzAz, lAt) = E -lGTEn,, Fm,n((nz - 1)Az, l'At), (2.37) i1=-00 where the discrete FDTD Green's function IGTE4, may be calculated analytically by +00 1 +00 IGTEZ, = J GTE(Z(Az, l\t- x) g(x) dx = 2- GTi,,(Azw) (w) ejwlAt dw (2.38) -00 -00 and x = t'- 'At. The triangle basis function is given by: 1-I ^|t for Ixl < At g(x) = 0 for ixi > At 29

and its Fourier transform is: [gsin(.) 12 ) (w) =.{g(t)} = Adt [ l t (2.39) L 2 J Due to causality, we have iGTE&, =0 for I < 0 (2.40) and as a result, Fm,n(nAz, lAt) = E I-I'GTEr Fm,n((nz - 1)Azl 'At), (2.41) I'=-00 which represents the mathematical formulation of the Diakoptics technique. As an example, let's consider the TE1t, mode. For the y-component IE.n of the tangential electric field at k = nz, eqs.(2.25) and (2.41) yield E^n= 1E IGTEoz El ((nz - 1)Az, 'at) sin(riax/a), (2.42) 11=-00 where EO((nz - 1)Az, I'At) may be calculated from eq.(2.28). The discrete FDTD Green's function 1-i'GTEZ is given by +00 -I' GTEi = 2 GTE 0(AZ, ) g(w) ej(-')Atdw (2.43) -00 with g(w) given by eq.(2.39) and GTE (Az,w) = e-Plo, (2.44) where 3z,1o is calculated by eq.(2.32) for m = 1,n = 0. Absorber Evaluation To validate the absorber presented herein, we calculate the magnitude of the reflection coefficient in frequency domain for the waveguide structure shown in (Fig.2.13). The xyplane of the waveguide at z = 0 is short-circuited and the ABC is utilized to calculate the 30

electric field components in the xy-plane at z = 2880Az. The waveguide cross-section is 47.6mm x 22mm and the cell size is given by Ax = 4.76mm, Ay = 1.1mm and A- = 0.4mm. We use a mesh of the size 10 x 20 x 2880 and run the simulation for 25000 time-steps. All conductors are assumed to be perfect electric conductors. We simulate the wave propagation for frequencies between 3.1GHz and 7.4GHz so that three different modes are excited, TE1,0, TEQo and TEo,I. To accommodate the presence of these three modes, we use a superposition of three Gaussian pulses multiplied with the corresponding mode patterns at z = 2840Az to provide the correct excitation. For the calculation of the reflection coefficient p, we use the formula P = E ' (2.45) where Et is the tangential electric field probed at z = 2860Az and Eref is the tangential electric field probed at the same position of a semi-infinitely long waveguide (no effect from reflections from the ABC) with the same cross section. The semi-infinite length of the waveguide is approximated by 6700Az and the tangential electric field is probed again at z = 2860Az. The evaluated ABC is replaced by a PEC. The length of this reference waveguide is chosen such as no reflections from the PEC plane return to the probe position for the 25000 steps of simulation. The absorber based on the analytic Green's function is compared to the 1st-order Mur's ABC coupled with the superabsorption condition. The effective dielectric constant [32] for the superabsorber is chosen to 0.407. For practical applications, the infinite summation in eq.(2.41) has to be approximated by a finite number of terms T. This approximation corresponds to a truncation of the discrete FDTD Green's function according to tGTE&, = 0 for I > T, (2.46) 31

where T represents the length of the discrete FDTD Green's function with respect to time. We obtain Fm,n(n,,t) = I-I'GTEn, Fm,n(nz - I 1') (2.47) I'=l-T and eq.(2.42) can be written as,IEY = E -,GT E1,o(n, - 1,1') sin(riAx/a). (2.48) 1'=1-T The reflection coefficient is minimized if we truncate the discrete FDTD Green's function at its zeros. In (Fig.2.14), results for the reflection coefficient for the TE20Q mode are shown for three different values of T, 616, 1127 and 2646. The graph for the Ist-order Mur ABC with the Superabsorption condition is symbolized with (sup). The larger the length T of the discrete FDTD Green's function, the more effective the absorber becomes. For T = 2646, the amplitude of the reflection coefficient is less than -40dB for almost the whole frequency range. Thus, the ABC based on the analytic Green's function is effective in a much wider frequency range than the super-absorbing 1st-order Mur ABC. This is true even when we improve the performance of the superabsorbing 1st-order Mur ABC by applying it to each waveguide mode separately. Similar results were observed for the reflection coefficient for the TEQo and TEO'1 modes. The PML absorber [8] achieves a comparable behavior for a wide frequency range. For example, the length T = 2646 of the discrete TEf0o Green's function offers a reflection coefficient very close to that of a PML layer of 4 cells with R = 10-5 (Fig.2.15) and T = 4161 has similar performance with a PML layer of 8 cells with R = 10-5. Generally, considering larger values of the length T is equivalent to increasing the number of the PML cells. Nevertheless, the memory requirements of the proposed absorber are much lower than the memory requirements for the PML absorber. For each mode, the convolution of eq.(2.42) 32

requires the storage of the T terms of the modal Green's function and of the T previous values of the mode amplitude at the = (n - 1)Az. Thus, the extra memory requirement of the Green's function absorber is 2 x T real numbers per mode. A PML layer of N cells to the z-direction requires M = 6 x N x n: x niy new variables, where n: x ny is the grid size for the waveguide cross-section. Generally M > 2 x T, especially for large grids. Due to the details of the waveguide probe structure analyzed in the next section, the waveguide cross-section grid has a size of 477 x 220 cells. That means that even a PML layer of 4 cells to the z-direction requires the storage of M = 2,518,560 new variables!! Using an absorber based on Green's functions with length T = 2646 for the TE1O, T = 2238 for the TEO and T = 2412 for the TEO,1, only 14,592 new variables have to be stored (0.58% of the PML memory requirements). As a result, the Green's function-based ABC offers a significant economy in memory while maintaining similar accuracy with the PML absorber. Waveguide Probe Structure Characterization The FDTD technique coupled with the proposed waveguide absorber is used in the RF characterization of the waveguide probe geometry shown in (Fig.2.12). The probe in the shape of a rectangular patch is fed by a shielded 50M coplanar line and is inserted into the waveguide through a slot. The dielectric substrate carrying the probe is not connected to any waveguide wall. This type of probe can be used as a coupler to a rectangular waveguide or as a diode mounting structure. The dimensions of the probe as well as the substrate thickness and the dielectric constant of the substrate are of critical importance in optimizing coupling to the waveguide. In our simulations, we try to optimize the thickness of the dielectric substrate carrying a probe which is 3.6mm wide. The dielectric constant of the substrate is assumed to be 33

er=12 (GaAs). The width of the dielectric substrate entering the waveguide is 5.8mm and its thickness is limited to less than 2mm. The probe is designed to feed a VWR-187 rectangular waveguide and for this reason, excitation is provided on the coplanar feedline by a Gabor function which covers the frequency range of 3.1 GHz to 7.4 GHz. For the simulated frequency range, three different modes are excited inside the waveguide. TE1o, TEZ, and TEO,1, with the cutoff frequencies 3.15 GHz, 6.30 GHz and 6.82 GHz respectively. The mesh used in the FDTD simulation consists of 480x477x52 cells with a time step of At = 0.31425ps. The simulation runs for 20,000 time steps to achieve converging results. The absorber discussed previously is used to absorb simultaneously all propagating modes of the waveguide for the simulated frequency range. To characterize the probe performance for different dielectric thicknesses, the magnitude of the reflection coefficient IS11| for the dominant TE1Z0 mode is calculated. For validation purposes, the calculated results are compared to data derived by the FEM (Finite Element Method) assuming a probe width of 3.6mm and a dielectric thickness of 2.0 mm (See (Fig.2.16)). For the FDTD simulation, the waveguide absorber based on the Green's functions for the three propagating waveguide modes is used at the terminal plane. For the FEM simulation, an artificial absorber depending on frequency and angle of incidence is applied to terminate the waveguide. For the whole operating frequency range (3.1-7.5 GHz) the performance of both absorbers is comparable and the results show very good agreement. The dimensions of the shield of the coplanar feedline are chosen to be 5.8mm x 3.8mm, such as only the CPW dominant mode can propagate and the field patterns are not disturbed by the side walls in the frequency range of the simulation. In this way, the superabsorption condition can be applied effectively at the input plane of the feedline. The performance of the probe has been evaluated for three different dielectric thicknesses 34

2.0mm, 1.2mm and 0.0 mm, with the last value corresponding to a microwave probe printed on a dielectric membrane [52]. Results in terms of the reflection coefficient are shown in (Fig.2.17). As it can be observed from this figure, the value of the reflection coefficient reduces over a large frequency range and shows symmetrical behavior round the center design frequency as the dielectric thickness approaches zero. The electric field (E) and magnetic field (H) distributions for zero dielectric thickness are plotted for t = 6,000 time steps across the probe structure symmetry plane (Fig.2.18) and across the coplanar feedline plane (Fig.2.19) and represent the transmitted and the reflected energy respectively. The reflection coefficient of the Si-membrane printed probe has been calculated for four different patch widths 3.6mm, 9.8mm, 11.4mm and 13.0mm and the results are shown in (Fig.2.20). From this figure, it can be concluded that the width of 9.8mm offers the most symmetrical behavior for the frequency of operation. The reflection coefficient for widths larger than 9.8mm is much smaller than that of 3.6mm for most of the simulated frequencies except a small region round 4.6 GHz. Nevertheless, the widths of 11.4mm and 13.0mm offer no significant improvement over the width of 9.8mm. Another geometry parameter of the Si-membrane printed probe that has been investigated is the distance of the probe patch from the short circuit of the waveguide. Lengths of 8.8mm, 10.4mm, 12.0mm and 13.6mm have been used and the results are plotted in (Fig.2.21) it can be noticed that the value of 12.0mm offers the best performance in terms of the value and the bandwidth of the reflection coefficient. The FDTD results derived by using the absorber presented in Section 111.2 have been validated by comparing to experimental data. The probe has dimensions 13.2mm x 4.3mm on a dielectric substrate with thickness 2.1mm, width 28.7mm and Er=13. The probe has been inserted in a WR229 waveguide and is located at a distance of 14.7mm from the top 35

surface short circuit. For the FDTD absorber there have been used T=28'71 time steps. The performance of the probe has been evaluated for the frequency range of 3.3-4.6GHz and the results are shown in (Figs.2.22-2.23). The agreement between the FDTD and the experimental results is good especially in the frequency range of the optimum performance of the probe. The abrupt variation in S21 observed for the higher frequencies in the experiment is maybe due to calibration or other reasons related to the experimental setup. 2.4 Conclusion The finite-difference time-domain method has been used to analyze planar circuits and waveguide probe structures. For the analysis, a waveguide absorber based on analytic Green's functions has been developed. This absorber is characterized by a better performance in accuracy and computational efficiency than the super-absorbing 1st-order Mur ABC and by a better performance in memory requirements than the PML absorber. The scattering parameters of the probe structures have been calculated and the results have been verified by comparison with FEM and experimental data. The influence of critical geometrical parameters on the probe performance has been investigated and optimized. 36

H i, j-/2, k-l/2 k-1/2 EX Hz i- 2, j, k i-/2, j-/2, k - I - - - - ------ / z Figure 2.1: Yee's FDTD cell. Figure 2.1: Yee's FDTD cell. 37

Figure 2.2: Patch Geometry to be used as Open. a) o.a 0 0 4 -| c 1) 1 2 3 4 5 Frequency [GHz] Figure 2.3: Reflection Coefficient of the Open. 6 38

Figure 2.4: Viahole Structure. 0.9 0.85 0.8 0.75 0.7 E aU. 0.65 0.6 0.55 0.5 fA AC I I I I -------------------------------------------- - S11 -- S21 - I I I i! - I I I I I I I! 2. 8 1 1 1 0 2 4 6 8 10 12 14 16 Frequency [GHz].Figure 2.5: S-Parameters of the Viahole. 18 20 39

Figure 2.6: E-Distribution across Top Viahole (Top), Middle Ground Plane (Bottom). 40

Figure 2.7: E-Distribution across Bottom Viahole. 41

-- 3800 -4 — 1600-' 100 1 250 450 100 50 Figure 2.8: Coupled Line Filter Geometry. 0 5 10 15 Frequency [GHz] Figure 2.9: Coupled Line Filter S21. 42

B-~- W~~ S-o B -D --- —— w -O-S-w H ~ Figure 2.10 Coplanar waveguide with finite width ground planes (F.G.C.). Figure 2.10: Coplanar waveguide with finite width ground planes (F.G.C.). 43

m b..w 0 b i.......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... e4-............................................................................................................................................................................................................................................................................................................ 14-01........................................

Perfect Electric Conductor x Probe Waveguide z Feedlot Figure 2.12:- Waveguide probe structure. 45

40 20 0 -20 -40 -60 -80 - - -100 40 20 0 -20 -40 -60 -80 -100 Figure 2.11: Normal H-Distribution (Log) for B=25 jim (up), 100 jim (bottom). I1 6

( X v -10 -20 m - 'D I-, J -30 -40 c a) 0 0 a)........... su p T=616 l........................................................ _.... -- T=1127 T=2646? d...'L -.... '\.'.................................................................................. -..../.........:...........,' /: '..................... -..........I I I~................................................................................. '.'-...................... '.: '....~...............'.". '~''''''' 5 ~~ ~~~ ~~ I~~ ~~ ' ~~ ~~ ~~ ~~ ~~ ~~r ~~ r~~ ~: ' ~~ tri -I I I -601 -70 '.IiI I 3.5 4 4.5 5 5.5 Frequency (GHz) 6 6.5 7 Figure 2.14: Reflection coefficient for the TEzo mode. 5 5.5 Frequency [GHz] Figure 2.15: Comparison of Green's Function ABC and PML. 47

P.E.C. Excitation A.B.C. Is —A Id x y y z=O z=284( ) Az z=2880Az Figure 2.13: Waveguide Test Structure. 48

3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 Frequency [GHz] Figure 2.17: Reflection Coefficient for different Dielectric Thicknesses. 49

I H I at t - 6000 * dt on const-X plane 30 E20 E N10 0 I I 1 60 1 80 20 iso 180 200 0 20 40 60 8o0 YlU0 1 20 140 I E | at t - 6000 * dt on const-X plane 30 0 -50 -100 0 -50 -100 i20 N 10 1 I 140 160 180 Z00 0 20 40 60 80 ~[mO] 1 20 Ir fi-m Figure 2.18: E- and H-field Distributions across the Probe Structure Symmetry Plane. 50

5 5.5 Frequency [GHz] Figure 2.17: Reflection Coefficient for different Dielectric Thicknesses. 51

Figure 2.20: Reflection Coefficient for different Patch Widths. -10 '- -15 CD -20 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 Frequency[GHz] Figure 2.21: Reflection Coefficient for different Distances from the Top Surface Short-Circuit. 52

Cl 2 --v 133 3.4 3.6 3.8 4 4.2 4.4 Frequency [GHz] Figure 2.22: Experimental Validation for S1l. 0 -0.013 -0.02 I i i I I I )K X - F.D.T.D. )3 Experiment -0.03 m -0.04 -0.05 cn -0.05 -0.06. -0.07 - -0.08 - -0.09 F -0.1 I I 3.4 3.6 3.8 4 Frequency [GHz] 4.2 4.4 Figure 2.23: Experimental Validation for S21. 53

CHAPTER 3 Development of New Time-Domain Schemes with Higher Order Basis Functions 3.1 Introduction Significant attention is being devoted now-a-days to the analysis and design of various types of printed components for microwave applications. To understand high-frequency effects and incorporate them into the design process, there is a compelling need to implement full-wave solutions during the modeling process. There has been a variety of full wave techniques developed for this purpose, with many of them available commercially. Despite the wealth of available codes, many problems in electromagnetics and specifically in circuit and antenna problems have been left untreated due to the complexity of the geometries and the inability of the existing techniques to deal with the requirements for large size and high resolution due to the fine but electrically important geometrical details. The straightforward use of existing discretizatization methods suffers from serious limitations due to the required substantial computer resources and urealistically long computation times. As a result, during the past thirty years the available techniques are almost incapable of dealing with the needs of technology leading into a quest for fundamentally different 54

modeling approaches. The use of multiresolution analysis in time domain has shown that Yee's FDTD scheme can be derived by applying the method of moments for the discretization of Maxwell's equations [51] using pulse basis functions for the expansion of the unknown fields. The use of scaling and wavelet functions as a complete set of basis functions is called multiresolution analysis and demonstrates that Multiresolution Time-Domain (MRTD) schemes are generalization s to Yee's FDTD and can extend the capabilities of the conventional FDTD by improving computational efficiency and substantially reducing computer resources. 3.2 Fundamentals on Multiresolution Analysis A mutiresolution analysis consists of a sequence of successive approximation spaces Vj. More precisely, the closed subspaces Vj satisfy... V2 C V1 C Vo C V- C V-2 C... (3.1) with U Vj = L2(R) (density) (3.2) jEZ n Vj = {0} (separation) (3.3) jEz There exist many ladders of spaces satisfying the above conditions that have nothing to do with "multiresolution"; the multiresolution aspect is a consequence of the additional requirement f(x) E Vj *- f(2Jx) E Vo (scaling) (3.4) That is, all the spaces are scaled versions of the central space Vo. Another feature that we 55

CHAPTER 3 Development of New Time-Domain Schemes with Higher Order Basis Functions 3.1 Introduction Significant attention is being devoted now-a-days to the analysis and design of various types of printed components for microwave applications. To understand high-frequency effects and incorporate them into the design process, there is a compelling need to implement full-wave solutions during the modeling process. There has been a variety of full wave techniques developed for this purpose, with many of them available commercially. Despite the wealth of available codes, many problems in electromagnetics and specifically in circuit and antenna problems have been left untreated due to the complexity of the geometries and the inability of the existing techniques to deal with the requirements for large size and high resolution due to the fine but electrically important geometrical details. The straightforward use of existing discretization methods suffers from serious limitations due to the required substantial computer resources and unrealistically long computation times. As a result, during the past thirty years the available techniques are almost incapable of dealing with the needs of technology leading into a quest for fundamentally different 56

modeling approaches. The use of multiresolution analysis in time domain has shown that Yee's FDTD scheme can be derived by applying the method of moments for the discretization of Maxwell's equations [56] using pulse basis functions for the expansion of the unknown fields. The use of scaling and wavelet functions as a complete set of basis functions is called multiresolution analysis and demonstrates that Multiresolution Time-Domain (MRTD) schemes are generalization s to Yee's FDTD and can extend the capabilities of the conventional FDTD by improving computational efficiency and substantially reducing computer resources. 3.2 Fundamentals on Multiresolution Analysis A multiresolution analysis consists of a sequence of successive approximation spaces Vj. More precisely, the dclosed subspaces Vj satisfy... V2 C V C VO C V-i C V-2 C... (3.1) with U V, = L2(R) (density) (3.2) jEZ Vj = {0} (separation) (3.3) jEz There exist many ladders of spaces satisfying the above conditions that have nothing to do with "multiresolution"; the multiresolution aspect is a consequence of the additional requirement f(x) E Vj -+ f(2J) e Vo (scaling) (3.4) That is, all the spaces are scaled versions of the central space VO. Another feature that we 57

require from multiresolution analysis is the invariance of VO under integer translations f() E Vo +- f(x- n) E VoVn E Z (3.5) Because of Eq.(3.4), this implies that if f(x) E Vj, then f(x - 2Jn) E Vj for all n E Z. Finally, we require that there exists ' E Vo such that {4o,n, n E Z} is an orthonormal basis in VO, where for all j,n E Z,~bj,n(z) = 2-j/2 (2-jz - n). As a result, {i,n, n E Z} is an orthonormal basis for Vj for all j E Z; that is, < qjm,qj,n >= 6m,n, m,n E Z (3.6) where 6 notates the Kronecker symbol m=1 m n bm,n = 0 elsewhere Throughout this Chapter, there will be used the following notations for the inner product and norm for the space L2(R): < f,g >= j f(x)g(x)dx C-oo 00 Ilfll = Ilf112 =< f,f >1/2 The basic idea of the multiresolution analysis is that whenever a collection of dosed subspaces satisfy Eqs.(3.1)-(3.5), then there exists an orthonormal wavelet basis {j,n, n E Z} of L2(R), Itj,n(x) = 2-j/2+(2Jx - n), such that for all f in L2(R), Pj-lf = Pif + A < f, ij,k > bj,k (3.7) LkEZ where Pj is the orthogonal projection onto Vj. For every j E Z, define Wi to be the orthogonal complement of Vj in Vj_i. We have Vj-l = Vj 6 ~ H- (3.8) 58

with p 27r-periodic and jp(()l=1. In particular, we can choose p(e) = poeimp with m E Z, Ipoj = 1, which corresponds to a phase change and a shift by m for W. We will use this freedom to define? = 7gn-l,n, gn = (-1)nh-n+l (3.23) n The orthonormality condition of Eq.(3.6) can be relaxed. It is sufficient to require that the 4(x - k) constitute a Riesz basis of Vo; that means that they span Vo and for all (ck)kEZ E L2(Z) with Ek ICk 2 < oo it holds AE Ickl2 < || Ck(X - k)112 < BE ICk2 (3.24) k k k or equivalently 0 < (2r)-lA < 5E 1|( + 27r1)12 < (2r)-lB < oo (3.25) where A > 0,B < oo are independent of the c,. Supposing that b E L2(R) satisfies Eq.(3.25) and defining Vj = Span{fjk; k E Z}, then njez = 0. Also, if () is bounded for all ( and continuous near ( = 0, with q$(0) $ 0, then UjEz Vj = L2(R). One Riesz basis which satisfies these criteria, satisfies the density and separation qualities of the multiresolution analysis. Chui [?] has proven that {(f(x - k): k E Z} is an orthonormal family if and only if 21r Ek=- 1q(~ + 21r/)2 = 1, VX E R. This is a very useful criterion for the orthonormality of a specific scaling family. We can therefore construct an orthonormal basis O' for Vo by defining $ = (27r)-l/2[5 I(e + 27r)12]-1/2 (E) (3.26) Clearly, El 1A1(~ + 21rl)2 = (2r)-1, which means that the ql((x - k) are orthonormal. Finally, (0 = ei/2 mo&(/2 + r) 1'(~/2) (3.27) 59

with mo () = mo()[E 1i( + 27r)l12]1/2 [I 1X(24 + 2r/)l1]-1/2 (3.28) I I or equivalently (x)= (-1) n h1n+l'1 (- n) (3.29) n with mo() = E h e-i h. The Battle-Lemarie wavelets [22, 23] are associated with multiresolution analysis ladders consisting of spline function spaces. A B-spline with knots at the integers is considered the original scaling function. The zero order cardinal B-spline No is the characteristic function of the unit interval [0,1) 1 0< x < 1 No(x)= { 0 elsewhere For m > 1, the m-th order cardinal B-spline Nm is defined recursively by the following convolution: 00oo Nm,() = Nm-l( - t)No(t)dt -00 = Nm-,(x-t)dt (3.30) Jo with the Fourier transform mm(w) = (21r)-l/2ei 2si/2 ) m+l where e = 0 if m is odd and e = 1 if m is even. For even m, X = Nm is symmetric around x = 1/2, for odd m, around x = 0. Except for m = 0 the scaling functions constitute a Riesz basis, but they are not orthonormal. To apply the orthonormalization of Eq.(3.26), Daubechies [25] has shown that N( + 2m+2 d2m+1 cot( 2E lNm,(2x + 2rk)l2 = -!) C cot(X) k=-oo (2m + 1)! dX2"+1 60

The result of the orthonormalization is that support of the o' = R = support of the 4' for all the Battle-Lemarie wavelets. The "orthonormalized" '- has the same symmetry axis as (. The symmetry axis of 4 always lies at x = 1/2. (For m even, 4' is antisymmetric around this axis, for m odd, 4 is symmetric). Even though the supports of <pA and, equal the whole R, >-L and 4 still have very good (exponential) decay I, P(x)I <,Ce ^, x E R The Battle-Lemarie wavelets based on the m-th order cardinal B-splines belong to Ck with k < m - 1 and have m vanishing moments: f dx xl 4(x) = 0 for 1=0,1,..,m for 41) bounded for I < m. It is impossible for orthonormal 4, to have exponential decay and to belong to CO, with all derivatives bounded, unless 4 0. As a result, to achieve fast (exponential) decay, only a finite number k of derivatives can be continuous. The decay rate decreases as k increases. On the contrary, the Meyer wavelet, which is C~~, decays faster than any inverse polynomial, but not exponentially fast. In the general case, q - = Nm, the q5 satisfies f dx4(x) = 1 and 2n+l 2-2 ( (2x- n-1 + j), m = 2n = even 2n+2 2-2n-1 +2 2 (2X - n-1 +j), m = 2n + 1 = odd If we choose X> to be the 0-th order cardinal spline, 1 0 < x < 1 s(x)w= < 0 elsewhere 61

and we follow the previous steps, we end up with the Haar basis 1 0 < x < 1/2 +(x)-= -1 -1/2 < x < 1/2 0 elsewhere No orthonormalization is needed since b is orthogonal to its translations. Choosing the piecewise linear spline (m=l) as the scaling function, 1-Ixi o < 1x<1 (x)= - 0 elsewhere it satisfies +(x) = 0.5X(2x + 1) + q(2x) + 0.5b(2x - 1) (3.31) and its Fourier transform is +(g) = (2)-/2i'/2 sin/2)2 () (27')-12 — /2( 2 It can be observed that I2 1 1 27r E |(~ + 27r)l2 + cos = (1 + 2cos2(/2)) IEZ Since X is not orthogonal to its translates, it is needed to apply the orthogonalization trick described above. The orthonormalized scaling function is given by 4sin 2(5/2) () /(27r) / - 1/2[ + 2COS2(~/2)]1/2 The -L is not compactly supported unlike X itself. The corresponding mo is l() = cos2(/2) [1 + 2cos 2(/2)] 1/2 MO 1 + 2cos2(~) i and the wavelet 7 is given by 1 - - 2sin2(/4) 1/2 (3.32) = ( ) ei/2 sin2(/4) [1 + 2son(-(4)] 2 /2) (3.32) =-3e?7Ji/2 2(/ [1 + 2sin2(E/4) /) (3) /e sin2(T/4) (1 + 2cos2(~/2))(1 + 2CoS2((/4)) J(3/ 2 ) 62

The choice of the scaling function for the development of the new Time-Domain NMRTD scheme is the cubic cardinal spline (m = 3). After orthonormalization, the spectral expressions of the scaling and the wavelet functions are ( (2)1/2(sin()4 1 (334) 2/1-sin(+ sin4() - 4sSin6(3) and 2(~ + 2r) () =ei/2' + 2+2) q$(Q/2) = 0o() (3.35) The Cubic Spline Battle-Lemarie Scaling and Wavelet functions are plotted in (Figs.3.1 -3.2) in Spatial Domain and in (Figs.3.3-3.4) in Spectral Domain. 3.4 The 2D MRTD scheme For simplicity, the 2D MRTD scheme is analyzed for a homogeneous lossless medium with the permittivity c and the permeability,u. Assuming no variation along the y-direction, the Maxwell's equations for the two-dimensional TMZ mode [62] can be written as: OEx 10OHy =E -(3.36) at e az _Hy 1 6E, OZE y= (E - -) (3.37) t td Ox Oz dEz 1 a (3.38) dt e ax To derive the 2D MRTD Scheme, the electric and magnetic field components incorporated in these equations are expanded in a series of Battle-Lemarie scaling and wavelet functions in both x- and z-directions and in pulse functions in time. +00 E(x, z,t) = kE 2m hk(t) k1-1/2(x) m(Z) k,l,m=-oo 63

E,(x, Z, t) Hy (x, III t) +00 k,I,m=-oo +00 k,l,m=-00 +00 k,l,m=-00 +00 +00 k,l,m=-00 +00 kIl,m=-oo +00 k,l,m=-oo +00 kIMs=-oo +00 k,l,n=-oo +00 kl, oo +00 k,l,m=-oo +00 2rz _1 Ld Ld L1 -1/2,m + 00 CkE-l. xPx, _1iz- mp TX=O P,=x +oo 2rx-z1 S SkCi.xl/z hkt) hk(t) /1/() ( rx,=O P, =0 +00 2rz"rz -1,Vr.2px krz (t)or.Tr z kk p h + 0011m12X kz__t SkEt)17 hk(t) O1 (x /m21/2(Z) I km-1/2i +00 2rz - z1 S k E E1oj;rzzpz hkt kI(t) /,r (x)tk,z=O P,=O +00 2rx -1 C C kEz' Vlrxpx O h (t) or (X) O-1/2(Z) l,-1/2,mI/21/,P m r =O px =O +00 2 rx,rz 1 S S k+1/2H1.l/x2r,mpl/ hk+1/2or- (t) or-~i/, (x),$ E E Imr-1/2 IPx M~p\~J l-1/2,pz( T rx,rz=O px,pz=O +00 2(39-1 kc+112H Y900z 'pz- 12 hk+112r(t) 01-112(X) Vrz ~p(Z) r-.=O pz=O +oo 2r-T - k+112H Ys.'"rz h (t) or-T rw 1-1/2',PMX O1/2 k+1/2 1-112,p.T(X) Om-1/2(z) r. =O P.T =0 +00 2 rx,rz -1 (3-39) where,(x) = -( '- n) and -O',((x = 2r/2 O~(2([ x - n] - p) represent the BattleLemarie scaling and r-resolution wavelet function respectively. The expressions of the scaling and the 0-resolution wavelet in the spectral domain are given in Eqs.(3.34)-(3.35).Since higher resolutions of wavelets are shifted and dilated versions of the 0-resolution, their domain will be a fraction of that of the 0-resolution wavelet; thus there are going to be more than one higher resolution wavelet coefficients for each MRTD cell. Specifically, for the the arbitrary r-resolution and for the n-cell to the x-direction, there exist 2' wavelet 64

coefficients located at: = n + -, p = 0.., r - 1. This is the reason for the summation of the p terms for each resolution r in the expansion of Eq.(39). kEm' and k+l/2H^ with K = x, y, z and,u, v = 4, $ are the coefficients for the field expansions in terms of scaling and wavelet functions. The indices 1, m and k are the discrete space and time indices related to the space and time coordinates via x = lAx,z = mAz and t = kAt, where Ax,Az are the space discretization intervals in x- and z-direction and At is the time discretization interval. For an accuracy of 0.1% the above summations are truncated to a finite number of terms determined by the dispersion and stability requirements (typically between 22-26). The time-domain expansion function hk(t) is defined as hk(t)= h( t - k) (3.40) with the rectangular pulse function 1 for Ixi < 1/2 h(t) = 1/2 for Ixl = 1/2 0 for Ix! > 1/2 The magnetic field components are shifted by half a discretization interval in space and time-domain with respect to the electric field components (leap-frog). Upon inserting the field expansions, Maxwell's equations are sampled using pulse functions as time-domain test functions and scaling/wavelet functions as space-domain testfunctions. For the sampling in time-domain, the following integrals are utilized +00 J hk(t)hk'(t)dt = 5k,k' A (3.41) -00 and 65

Jhk(t)k t/2t dt = 6k,k' - 6k,k'+l -00 where 6k,k' is the Kroenecker symbol, (3.42) 1 bk,k' - 0 for k = k' for k 5 k' Sampling in space-domain is obtained by use of the orthogonality relationships f or the scaling and for the wavelet functions [25] +00 J (>m(x),m(x)dx =,,a +00 Jf (m(x)OM,pi(x)d = 0, V r,p -00 (3.43) (3.44) and +00 Jf,r (X)tm',,(x)dx = 6r,, e5m,ml' m,m' Ax -00 (3.45) The integrals containing derivatives can be approximated by the following expressions: with +00 '+ ( -1 j m(x) m d+ l/2( d a(i)6m+i,m, +00 -oo t=-na a(i) = 1 [(0 ) ~ sin[ (i + 1/2)]dr, t=rnd,r,l (3.46) (3.47) (3.48) 66

with d(i, p) =- 2-/2m(. ) 10o(/2)i sin[ (i + 0.5 + p/2r + 1/2+)] d, (3.49) p() a, dx ) E (p);+ (3.50) -00 =-onc,r,l with cr(i, ) = f 2-/2m()g) lo(/2) sin[ (i + 0.5- p/2 - 1/2+1)] d, (3.51) +00 a( 4),+ lrix2, J ),p (x) a dx br,r2 (i, P, P2)6m+i,m' (3.52) -00 t=-nb,rl,r2,1 with brj,r2(i, Pi,P2) = J 'o(/2'r )I |ko(g/2r2)l [sin[ (i+ 1/2+p2/2r2-p1 /2'r +1/2r2+1-1/2rl+1)]d (3.53) For the rest of the MRTD Technique description, an expansion only in a series of scaling and 0-resolution wavelet functions will be considered. Hints for the enhancement of additional wavelet resolutions will be presented where needed. Since for the 0-resolution (r = 0) there is only one wavelet coefficient per cell (p = 0), the p symbols will be omitted from the definition of the b, c, d coefficients, which will be given by J OM$,(x) ax. CO()M+iM (3. 54) -00 '=-ncO,l with co(i) = f~~ j m() iJo()|I C sin[ i] dC, (3.55) 7/x0 9i () (3.56) | )~m+/2(X) mdx d. E do(i)mj+ijm (3.56) — o i=-nd,O,l 67

with 0 ^ JO do(i) = j m(g) j1o() I ~ sin(gi + 1) d = co(i + 1) (3.57) Thus, eq.(3.56) can be written as or2 nc,O,2 — 1 J 'm() m+1/2( z dx co(i + )6m+ -,m (3.58) — 00 t=-nc,O,jl-1 fm(x) dx bo(i)6m+i,m, (3.59) -00 t=-nb,0,0,1 with bo(i) = bo,o(i) = I0(o()2 sin[ (i + 1/2)]d, (3.60) with a(i), bo(i), co(i) given in Table (3.1) [26]. Due to symmetries in the integrals for the 0-resolution, the coefficients satisfy the conditions: a(-l - i) = -a(i), bo(-l - i) = -bo(i) and co(-i) = -co(i) for i < 0. Hence, the stencil lengths have to be: nb,o,o,i = nb,,0,2 - 1 = nb and nc,o,1 = nc,o,2 = nc. These conditions are not general and do not hold for any other arbitrary resolution. The stencil size is determined by the dispersion requirements. It has to be noted that the Battle-Lemarie scaling function has exponential decay; thus, the coefficients a(i) for i > 12 are not zero, but their value is negligible (< 10-4). After applying the Galerkin technique to Eqs.(3.36)-(3.38), the following MRTD equations are derived: k+ - k i-1/2,j __ 1 t ^ — - a( )k+i/2i_-1/2,j+j'+1-1/2.'=-na 1 1 2 — n o nc + co(j')k+l/2 i-1/2,j+j'-/2) jt=-nC At - - -- A ( ~ J)k+l/2i-1/2,j+j +1-1/2 68

1 a(i) bo(i) CO(i) 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1.29161604157839 -0.155978843323672 5.9606303324687290E-02 -2.929157759806890E-02 1.5362399457426780E-02 -8.184462325283712E-03 4.3757585552354830E-03 -2.342365356649461E-03 1.25287771 7042020E-03 -6.7 16635068590737E-04 3.5835069074489797E-04 -1.93132166S4715780E-04 1.019327767057869E-04 -5.613943183518454E-05 2.834596805928539E-05 -1.700348604873522E-05 2.47253977327429 0.9562282774123074 0.1660591600788887 9.392437777679437E-02 3.141444475216036E-03 1.349356908709108E-02 -2.858941810094752E-03 2.778680514115529E-03 -1.129446 167303586E-03 7.071507309377701E-04 -3.491267305845643E-04 1.952711419194906E-04 -1.021304423384722E-04 5.531259273864269E-05 -2.947330468694831E-05 1.572110653438641E-05 0. -4.659 725-793402785E-02 5.453939813583327E-02 -3.699957746974982E-02 2.057449098775452E-02 -1.115303180864957E-02 5.976877725279031E-03 -3.202621363952005E-03 1.714086849566890E-03 -9.1 76508438494196E-04 4.91 1754748072018E-04 -2.629253013538502E-04 1.407386855875626E-04 -7.533840689573666E-05 4.033146235099674E-05 -2.159462850665844E-05 I Table 3.1: Coefficients a(i), bo(i), co(i) 69

t /,- i'-I/ 2,j _ A t k+1 E~L"j -k k-..o2 _ A t k+lE.f-1 - kEz'12 At kEz'Oko - kEz"Vo~ k+E 3 1/2 — t -1 At k~jEz~koo- kEz'0O46 i,3-12 i~j-1/2 At.7 =-nc -( Z CO(j')k+1/2HtYJ+I+0 -/ + S bo~')kl/2~'jJ 2,i.4 1/2) 1 3 -n 1b 1 - -( 5 c0(j')HYk+1/2 EAZ1 2-1/2,i+1j-+1-1/.7 =-nb + S b~i')kC/2Hj' -1K2H' +1 1/2 EAXi~'+-1/2,.i 1+-1/2 + b o(iI)k+1/2HY''Vo~l2.12 + S o~i)k+/2H/2,j -111/2) - -( 5 co(i') k+1/2HfY"cke EAX i+i'+1-1/2,ji-1/2 + >z o(i')k+1/2H:4y 1/2 -1/2 1 nac + iii bo a)ki/2Hk:414/2H,2 41/2 na= CO~* ii (i)k12')kE4"O..11 ncc (3.61) At k+1/2 i-if 2,.j-1/2-k12i12 12 1 1: a(j') k-1/2,"j+.4if + =/-n 1 1l nc -( 5 a(j') kE L',"PO, nc + >i: bo (i'p)kEzjio112 nc COUI)k F"'OV'o 1)] i-1/2,'+j/-.1 =-nc 3 70

H2Y, o HYI~o k+l/2 i-1/2,j-1/2- k-1/2 i-1/2.j-1/2 At nc + E co(i)k+-l,j-/2) t-=-nc 4k+l/2 i-/2,j-l/2 k-1/2 i-1/2,j-1/2 At nbri-1 +E bo(iZ)kEz+i,,-l/2) i=-nb = l[ ( E a(i')kEto-1/2 t =-na nc nb - 1 - ( E /o(j2') kE, + -E bo(j)kE-12+ j=-nc j'=-nb 1 1 nc = -f CO(i)kE, 21,/2 t i=-nc nc nb- 1 z- ( E co(j') kExi1/2, jj + E bo(j )kE-l/2j+j' E 3 =-nc 3 =-nb (3.62 The indices i, j and k are the discrete space and time indices related to the space and time coordinates via x = iAx,z = jAz and t = kAt, where Ax,Az are the space discretization intervals in x- and z-direction and At is the time discretization interval. The values of the stencil lengths n,, nb, depends on the accuracy and dispersion requirements. 3.4.1 Modeling of Hard Boundaries Unlike the FDTD where the consistency with the image theory is implicit in the application of the boundary conditions, for MRTD schemes based on entire-domain functions this theory must be applied explicitly in the locations of Perfect Electric (PEC) or Magnetic Conductors (PMC). The total value of a field component at a specific cell is affected by a theoretically infinite - practically finite - number of neighbooring cells due to the fact that the basis functions extend from -oo to oo. Some of these neighbors may be located on the other side of the conductor. This effect is taken into consideration by applying the image theory (Fig.3.5). In this way, the physical boundary condition of zeroing-out the E-field tangential to the PEC is automatically satisfied. For example, even symmetry is applied for the normal-to-PEC electric field components and odd symmetry for the parallel-to-PEC. Image Theory can be implemented automatically for an arbitrary number of hard boundaries. 71

The time-domain numerical techniques are modeling the real space by creating a discrete numerical grid. Sometimes, this mesh does not coincide with the electrical one and MRTD is one example. The enhancement of wavelets on MRTD requires a special treatment of the wavelet components of the normal-to-PEC electric field. Assuming a vertical PEC in (Fig.3.6), the electrical domains (I) and (II) are isolated from each other. That means that one wavelet component value of the normal electric field EXACTLY ON the PEC would create a non-physical electrical coupling. Thus, TWO wavelet components, one located infimitesimally left of the PEC and the other infinitesimally right of the PEC, have to be defined in order to satisfy the electrical isolation condition. The H-field component that is parallel to the PEC has to be treated in a similar way. The rest components of the Eand H-field have to be zeroed-out on both sides of the PEC, so one value is sufficient. In FDTD the interleaved positioning of the field components on the Yee's cell (which are the same with the scaling functions components on the MRTD's cell) requires that the normalto-PEC E-field component is located half cell size away from the conductor. In this way, the definition of only one field component per cell is sufficient. 3.4.2 Modeling of Excitation Without loss of generality, the modeling of the excitation for the 2D and 2.5D MRTD is presented. The 3D algorithm is a direct extension of the 2D. For simplicity, only 0-resolution wavelets are used. In order to apply a point (pulse) excitation P(xo, zo) for xo = mAx, z, = nAz, the pulse has to be decomposed in terms of scaling and wavelet functions * 00 00 P(x0, Zo) == Z Z c, ) ((m + ~ l,-) k(n + lz,-) lx,4k=-o lz,4=-o 72

+ Z co C~(lrQ lZIJ) 4(m + 1 i(n + ) + Z S C r, Z,/p) '(m + ) p(n + 1,) 00 00 + S, cS (, Iz,,) ~(m + lx,p);'(n + lz,. (3.63) 1=, =-00 lzp=-o00 with rm+0.5 rn+0.5 cqsq(lX,", I =o Jm-0.5 5 k(m + 1,O) 4(n + IzO) dz dx c 0(lx,, lz,,) = J J (m + t,) q(n + lz,p) dz dx m-0.5 n-0.5 rm+O.5 tn+O.5 c,(l,, lz,, ) = I-.5 (m + lx,/) /(n + Iz,O) dz dx rm+0.5 rn+0.5 c,O(lx, = J0 ( (m + l,lp) O((n + lz,,) dz dx. (3.64) [m-0.5 f+-0.5 Practically, the summations of Eq.(3.63) can be truncated to a finite number of terms. Usually 6-8 terms on each side of the excitation point per direction can offer an accuracy of representation close to 0.1%. In case the neigboring scaling or wavelet functions are located outside the computational domain (e.g. m + l, > nx or m + lx,o < 0 for a domain [0, n*] to the x-direction), image theory has to be applied for their translation inside the computational grid. If there is no discontinuity (hard boundary or dielectric interface) in the summation interval of Eq.(3.63), the double integrals of Eq.(3.64) can be split in two single integrals rm+0.5 n+0.5 c(l z) = (m + 1:,p) dx j05 q(n + Iz,, ) dz = c(1(:,,) c((lz,o) /m+0.5 rn+0.5 c1p(lx",, 1z) = ] d(m + lx,) dx J?1(n + Izl^) dz = co(lx,O~) cp(l,,) Jm-0.5 Jn-0.5 fm+0.5 rn+0.5 co,(ltx,,, lz) = J iP(m + +,Os) dx J q$(n + lz,,) dz = cp(l,,p) c(1(lz,,) Jm-0.5 + n-0.5 rm+~ n+0.5 c,(lx,)= lzJm) = /(m + lx,p) dx J (n + lz,,/,) dz = c(l,,) c(,) Jm-0.5 gn-0.5 with co, c~ given in Tables (3.2)-(3.3). 73

Due to the symmetries of the Battle-Lemarie scaling and wavelet functions, the decomposition coefficients have to satisfy the following conditions cs(t) = c((-l ), = -1, -2,.. cV(l4,) = c~(1 - l), 1 = 0, -1, -2,.. For each time-step, the excitation scaling and wavelet components have to be superimposed to the respective field values obtained by the MRTD algorithm in order to provide a transparent source similar to that described in Ch.2. -Ep,total kEC I km+lx,,n+lz, = kEm+l, ~,n+l,~ + Cq~(lx,~, Iz,) Ekji,total - kEip km+lK,,n+l,,~, km+lx,,,,n+iz,O + CpL(lx, Iz,. ) kE p0,total = kE I k m+=,,n+1,q kEm+$.,,n+i, + cpO(lx,,, lz,O) For the 2.5D-MRTD algorithm that requires impulse excitation in time-domain, the above superposition takes place only for the first time step (t=O). Nevertheless, for the 2D-MRTD it has to be repeated throughout the number of time-steps that the excitation is on. The I4 0 1 2 3 4 5 6 c0(l4) 0.91507 0.03820 0.00963 -0.00863 0.00502 -0.00268 0.00141 Table 3.2: Excitation Scaling Decomposition Coefficients I4 1 2 3 4 5 6 7 cO(l4) -0.10250 0.12115 -0.02975 0.01501 -0.00598 0.00298 -0.00139 Table 3.3: Excitation Wavelet Decomposition Coefficients 74

appropriate number of the time-steps will depend on the time dependence of the excitation (Gaussian, Gabor,...). Arbitrary excitation spatial distributions f(x, Z) for an area [xl = mjAXz x2 = m2Az] x [z1 = nlA/,z2 = n2Az] can be modeled in a similar way. The spatial distribution has to be sampled with scaling and wavelet functions, giving the new decomposition coefficients fm2+0.5 jrn2+0.5 c(lx:,+,lz, ) = 0.f(x,z) q(ml + 1,) q$(nl + l,,) dz dx Jm-0.5 dnl —0.5 rm2+0.5 r/n2+0.5 C=((lxzlz, ]) = f(x,z) 0(ml + l,) (nl + Iz,,) dz dx Jml-0.5 Jnl-0.5 7m2+0.5 rn2 +0.5 c(Ix, l,) = -05 f(x,z) k(mi + I,) (nI + lz,,)dz dx m2+-0.5 /n-+0.5 /om2+o.s /n2+0.5 c,(lx,, lzi) 0. -5f(x, ) =(m + l,,()) ((nl + lz,,) dz dx Jmi-0.5 Jni-0.5 For most simulations the choice of -8 ~< Ix,, x,k < (m2 - ml) + 8 and -8 < Iz,,, I,1' < (n2 - n) + 8 offer an accuracy close to 0.1%. 3.4.3 Treatment of Open Boundaries - PML Absorber As it was discussed in Ch.2, for all discrete-space full wave techniques a special treatment should be given to geometries of interest defined in "open" regions where the computational grid is unbounded in one or more directions. Since the computational domain is limited in space by storage limitations, an appropriate boundary condition should be implemented to effectively simulate open space and satisfy the radiation condition. Berenger [8] proposed the Perfect Matched Layer (PML) Absorber, which is based upon splitting the E- and H- field components in the ABC area and assigning artificial electric and magnetic loss coefficients. On the condition that these loss coefficients satisfy the PML relationship for each point of the absorber area, this nonphysical absorbing medium has a wave impedance less sensitive to the angle of incidence and frequency of outgoing waves than the preexisting 75

absorbers.In this Subsection, the non-split and split extensions of the PML absorber for the 2D-TAM Battle-Lemarie MRTD are discussed. Their performance is going to be validated in Ch.4. Assuming that the PML area is characterized by (e0, 0) and electric and magnetic conductivities (oE,aH), the TMZ equations can be written by adding an extra term to Eqs.(3.36)-(3.38) Eo t + aEE = (3.65) ( t 8z eo t + EEz (3.66) dt dx aH, 9Ez aEx / fo -= E a- Hy (3.67) +T H dx dz Without loss of generality, PML cells only along the z-direction are considered. The extension to the x- and y- directions is straightforward. For each point z of the PML area, the magnetic conductivity aH needs to be chosen as [8]: aE(Z) - H(z) (3.68) -- (3.68) Eo /o for a perfect absorption of the outgoing waves. A parabolic spatial distribution of aE,H, E.() =,aH(1- )P, with p=2 for O < z < 6 (3.69) is used in the simulations, though higher order distributions (e.g.Cubic p=3) can give similar results. The PML area is terminated with a PEC and usually has a thickness varying between 4-32 cells. The maximum value 0,ax is determined by the designated reflection coefficient R at normal incidence, which is given by the relationship _ 2~'u ax6 R=e f = e coc(p+l) (3.70) In MRTD, the PML area can be modeled by discretizing the above equations in a similar way to the non-conductive area described in the beginning of Section 3.4 and split and non-split formulations can be derived. 76

Similarly, the PML equations for the TEZ can be written as, OH, OEy ~ Ot + HH a = - Ot Oz 1o 10 +nHH, = t Oxz OEy OH, OH1 co -t + a EEy =- + E Ot Ox Oz (3.71) (3.72) (3.73) Split Formulation Following the approach of [8], Hy is split in two subcomponents, HyX, Hyz and Eqs.(3.65)(3.67) are written as, o) ExHyx + Hyz4 at + aE(z)E = - (3.74) at az cOE Ez OHyI + Hyz ~- Ox (3.75) OH t = f (3.76) O t Ox OHyz _E_ Ot + H(z)Hyz = - (3.77) For the sake of simplicity in the presentation and without loss of generality, the fields Ex, Ez, HBy, Hyz are expanded in terms of scaling functions only in space domain and pulse functions in time domain. By applying Galerkin's technique [26, 53], the following split PML equations are obtained rx,4> _ —~fE~l -AtC px,4>4> k+l Ei-1/2Ij = e E i_ k 2j + (e-EAt/o - 1) Az CE na-1 a)k+/2-"-1/2,j+j'+l-1/2 + k+l2/2-11i,2,j+j' +1-1/2) il=-na pEZ,44 EZ,7+ k+l ij-1/2 k ij-1/2 na-I At V ( yz, y4 + >Z a(i')(k+l,/2Hi/+i+ -1/2,j-1/2 + k+1/2H +i+l -/2,j-/2) = i'/=-na k+x/2 H 1 2ryx'4 +l/2i-1/2,-1/2 k-l/2i-1/2,j-1/2 77

no-1 +.( a(i')kE+,31/2) I -n + yz,1 _ 1/22o ty, k+l/2 i-l1/2,j-1/2 - e H Ak-l/2 i - 1/2, j-1/2 1 /2 na-1l 1 Ca or Atlg 7 -1/2, a()k /2,j+j. (3.78) Az a3 =-na Exponential time-stepping is being used for the field components affected by the PMIL conductivities aE, OH. Due to the entire-domain nature of the Battle-Lemarie scaling functions, the PML conductivity must be sampled by them over at least 12 cells (6 cells per side), rj+6 E,H= =J] crE,H(z)j(Z)dz (3.79) JjE-6 Image theory is applied to extend the conductivity layer outside the terminating PEC's. The presented formulation follows the idea introduced by Berenger for the FDTD. Nevertheless, an efficient non-split form of the PML equations does not demand extra memory for the storage of two Hy subcomponents per cell. Non-split Formulation Substituting in Eqs.(3.65)-(3.67) [54]: Ei(x, z, t) = E~(x, z, t)e-E(z)t/lo (3.80) and Hj(x, z, t) = Hj(x, z, t)e-)H(z)t'Lo (3.81) for i=x,z and j=y leads to the following system of equations: eo O -aH (3.82) at oz OEz OHy et -S = (3.83) Edt dx 78

0o - At = aOx Oz (3.84) Discretizing Eqs.(3.82)-(3.84) and inserting Eqs.(3.80)-(3.81) yields the unsplit formulation of the fields for the PML region: k+ Z,Oo k+l i,j-1/2 k+l/2 i-1/2,j-1/2 = e - At/ko Ex' k i-1/2,j - e05eAt/o At ( a(i )k+l/2H "2, 1 /2) -- ) /2.-r i-1/2,j+j'+1-1/2) co Az=h =-na + e jo5s l/A t/CO At ( x E a(i')k+l/2Hy4.'44.11/2, ) =- ea~ l-1/2A t/o m, k- l -1/2, j-1/2 + eO.Sa-l/2At/ At co (1 fla a(^ )kE^+i,,-1/2 - z E ia(j')kE^ 21,'1) j t-=na j'=-na (3.85) where the terms aE,H are given by Eq.(3.79). 3.4.4 Total Field Calculation Due to the nature of the Battle-Lemarie expansion functions, the total field is a summation of the contributions from the non-localized scaling and wavelet functions. For example, the total electric field Ez(xo, zo, to) with (k - 1/2) At < to < (k + 1/2) At, (i - 1) Ax < xo < i Ax and (j - 1/2) Az < zo < (j + 1/2) Az is calculated in the same way with [26, 53] by 11 Ex(x,Zo,to) = 1 ki+it 1/2,j+j i+i-l/2(Xo) ~j+jI(Zo) i',j=-ll rmax 12,r 2rZ -1 + +i E E kE /2,j+j, i+i-1/2(Xo) +j(Zo) rz=O i',j'=-12r Pz=O 79

rmarx 13r 2rx -1 + E kEi+ _ l/2,j+j*, '+ii-1/2.p(Xo) +jX'(o) rx=O i',j'=-13,r P=O rmax 14, r 2rx' z - 1 EC E k t+i_-1/2,j+J,?+i,_ 1/2,p(o) 3jrz (-o ) rx,rz=O jl,j'=-l4,r Ppz=O (3.86) where ^n(x) = <(fZ( - n) and -np() = 27/2 4o(2r[z - n] - p) represent the BattleLemarie scaling and r-resolution wavelet function respectively and rmax is the maximum wavelet resolution used in this area of the computational domain. It has been observed that the values 11 = 12,o = 13,0 = 14,0 = 10 and 12,1 = 13,1 = 14,1 = 6 offer accuracy close to 0.5% for most simulations incorporating the first two wavelet resolutions. For the cases of narrow strips with very sharp field discontinuities, the summation limits must increase up to 15-20 terms per direction. The fact that the MRTD is based on entire-domain basis functions with varying values along each cell offers the unique opportunity of a multi-point field representation per cell. The neighboring scaling and wavelet coefficients can be combined in an appropriate way to calculate the total field value for more than one interior cell points. In this way, MRTD creates a mesh with much larger density than that offered by the nominal number of the cells without increasing the memory requirements. This additional density is very useful in the calculation of the characteristic impedance of planar lines, where even a small field variation can cause a perturbation of the impedance value by 5 - 10Q. On the contrary, FDTD is based on pulse basis functions that have a constant value for each cell, offering a single-point field representation. 80

3.5 Conclusions After reviewing the general principles of Multiresolution Analysis, novel time-domain schemes based on expansions in scaling and wavelet functions (MRTD) have been derived. FDTD implementation schemes (excitation, hard/open boundary and dielectric interfaces) have been extended to Multiresolution schemes based on entire-domain expansion basis, while maintaining similar performance characteristics. These schemes offer the unique opportunity of a multi-point field representation per cell. Battle-Lemarie functions are used throughout the dissertation due to their special qualities. 81

BL Cubic Spline Scaling - Spatial Domain 1 -x CQ. X -4.53 Q. 10 -8 -6 -4 -2 0 2 4 6 8 x Figure 3.1: BL Cubic Spline Scaling - Spatial Domain. BL Cubic Spline Wavelet - Spatial Domain -1I I i, i I i,, I -10 -8 -6 -4 -2 0 2 4 6 8 x Figure 3.2: BL Cubic Spline Wavelet - Spatial Domain. 82

BL Cubic Spline Scaling - Spectral Domain.-( -. Q. C 0 0 0 CL.0.) Q. -10 -8 -6 -4 -2 0 2 4 6 8 ksi Figure 3.3: BL Cubic Spline Scaling - Spectral Domain. BL Cubic Spline Wavelet - Spectral Domain -10 -8 -6 -4 -2 0 2 4 6 8 ksi Figure 3.4: BL Cubic Spline Wavelet - Spectral Domain. 83

TANGENTIAL TO THE P.E.C. E-FIELD I- Is, II I - I Field at the P.E.C = 0 P.EC. I I I I I I i I I I I I t $ I I I I I i I. J Figure 3.5: Image Theory Application for tangential-to-PEC E-field. 84

I"m m m mm I rn-cell I I I I I *o I Imm mEmw m R.E.C. I (rn~1) -cellI, I EwI I E~. I I = - — M Il r m m - m m m U I I I *I ~ I I I* E I R.E.C. I I I ~- I I ES I I *M ~M~ m m m d * Scaling.TD. M.R.T.D.-W~avelet M.R.T.D.C F.D. Figure 3.6:- Treatment of Wavelet Components of normal-to-PEC E-field. 85

CHAPTER 4 Characterization of Microwave Circuit Components Using the Multiresolution Time Domain Method (MRTD) 4.1 Introduction Recently, the Battle-Lemarie based MRTD technique has been successfully applied [26, 53, 54] to a variety of microwave problems and has demonstrated unparalleled properties. When applied to linear as well as nonlinear propagation problems, it has exhibited MRTD schemes based on other entire-domain expansion basis can be developed in a similar way by calculating the appropriate summation coefficients. The use of Battle-Lemarie basis allows for a more simplified evaluation of the moment method integrals is simplified due to the existence of closed form expressions in spectral domain and simple representations in terms of cubic spline functions in space domain. The use of non-localized basis functions cannot accomodate localized boundary conditions. To overcome this difficulty, the image principle is used to model perfect electric and magnetic boundary conditions. Pulse functions are used as expansion and test functions in time-domain. In this Chapter, a 2.5D MRTD scheme is developed and applied to a variety of shielded and open of transmission line problems. Specifically, propagation constant, characteristic impedance and field patterns are derived 86

for shielded and open transmission line structures and compared to FDTD results. For the treatment of open boundaries, Berenger's PML principles [8] have been extended in split and non-split form, so as they can be used for entire-domain basis MRTD schemes. 4.2 The 2.5D-MRTD scheme For simplicity, an overview of the 2.5D-MRTD scheme is presented for a homogeneous medium. The derivation is similar to that of the 2D-MRTD scheme in CH.3, which uses the method of moments with pulse functions as expansion and test functions. The magnetic field components are shifted by half a discretization interval in space and time-domain with respect to the electric field components. Using the approach of [55], Maxwell's curl equations for a homogeneous medium with the permittivity e and the permeability, can be written in the following form e+ t =p (4.1) at Oy E-T = H x (4.2) dt t dE = dH dilx (4.3) at a y (4.3) where / is the propagation constant and j = V.-I. The electric and magnetic field components incorporated in these equations are expanded in a series of Battle-Lemarie scaling and wavelet functions in both x- and y-directions. For example, Ex can be represented as: +00 E(x, y, t) = l/2,m hk(t) q-1/2(x) qm(y) kl,m=-oo +00 +oo 2ry -1 + E E E kE1' hk(t) -1/12(X) 'py(Y) k,l,m=-oo ry=O py=O +oo +oo + sum 0 2"-T-x1 Ex?Irx' hkx X(9) M(y) E,m=-po k- 1r-/2,m r-1=hk(t /2,p( k,l,m=-oo r,:=O 87

CHAPTER 4 Characterization of Microwave Circuit Components Using the Multiresolution Time Domain Method (MRTD) 4.1 Introduction Recently, the Battle-Lemarie based MRTD technique has been successfully applied [31, 58, 59] to a variety of microwave problems and has demonstrated unparalleled properties. When applied to linear as well as nonlinear propagation problems, it has exhibited MRTD schemes based on other entire-domain expansion basis can be developed in a similar way by calculating the appropriate summation coefficients. The use of Battle-Lemarie basis allows for a more simplified evaluation of the moment method integrals is simplified due to the existence of dosed form expressions in spectral domain and simple representations in terms of cubic spline functions in space domain. The use of non-localized basis functions cannot accommodate localized boundary conditions. To overcome this difficulty, the image principle is used to model perfect electric and magnetic boundary conditions. Pulse functions are used as expansion and test functions in time-domain. In this Chapter, a 2.5D MRTD scheme is developed and applied to a variety of shielded and open of transmission line problems. Specifically, propagation constant, characteristic impedance and field patterns are derived 88

to < (k + 1/2) At, Ax < xo < iAx and (j - 1/2) Ay < yo < (j + 1/2) Ay is calculated by E(xo,Yo, to) = J fEx(xy, t)6(x-Xo)6(y-yo)6(t-to)dx dydt 00 i+i-1/2,j+j i+i'-l/2(Xo) j+j'(Yo) i'.J =-oo kE ' +il/2,j+j i+i'-1/2(~o) 4j+j'(Yo) ~ (4.6) Extending the dispersion analysis from 2D to 2.5D space, the stability condition for the 2.5D S-MRTD scheme results in At < 1 (4.7) C(^)2 + ( )2) al ja(il)2 + ()2 with the wave propagation velocity c. It is preferable to choose At at least 1.2-2.5 time less than the stability limit. In this way, much more linearity of the dispersion characteristics is achieved. 4.3 Applications of the 2.5D-MRTD scheme to Shielded Trans* 4 mission Lines First, the 2.5D-MRTD scheme is applied to the analysis of shielded stripline and microstrip lines to investigate propagation and coupling effects. Results for these shielded structures are presented and discussed separately below. A shielded stripline is a simplified version of a membrane microstrip shown in (Fig.4.la). The metallic shield has dimensions 47.6mm x 22.0mm and the central strip has length 11.9mm. The stripline is filled with air (Er = 1.). The analysis for the higher order propagating modes is straightforward. For the analysis using Yee's FDTD scheme, a 40 x 10 mesh was used resulting in a total number of 400 grid points. When the structure was analyzed with the 2.5D-MRTD scheme, a mesh 8 x 4 (32 grid points) was chosen reducing 89

Mode TEM Shield TElo Analytic values 1.4324 GHz 3.4615 GHz 8x2 MRTD 1.4325 GHz 3.4648 GHz Rel.Error 0.007% 0.095% 8x4 MRTD 1.4325 GHz 3.4641 GHz Rel.Error 0.007% 0.075% 16x4 MRTD 1.4325 GHz 3.4633 GHz Rel.Error 0.007% 0.052% 40x10 FDTD 1.4322 GHz 3.4585 GHz Rel.Error -0.014% -0.087% Table 4.1: Mode frequencies for,3 = 30 the total number of grid points by a factor of 12.5. In addition, the execution time for the analysis was reduced by a factor of 3 to 4. The time discretization interval was chosen to be identical for both schemes and equal to the 0.8 of the 2.5D-MRTD maximum At. For the analysis /3 = 30 was used and 5,000 time-steps were considered. From (Table 4.1) it can observed that the calculated frequencies of the two first propagating modes for /3 = 30 by use of 2.5D-MRTD scheme are very close to the theoretical values, since the largest error is less than 0.1%. The relative error of the 2.5D-MRTD calculated frequencies is always positive, which corresponds to an overestimation of the resonant frequencies. This is exactly what has to be expected from the dispersion behavior of the MRTD schemes. The non-localized character of the basis functions offers the opportunity to calculate the field values in any point of the discretization cells. The field values at the neighbooring cells 90

can be combined appropriately by adjusting the scaling functions' values and by applying the image principle. For example, the total electric field E.(XO, yo, to) with (k - 1/2) At < to < (k + 1/2) At is calculated by Eq.(4.6) by simply truncating the i' j' summation from 11 = -12,..,12 for each index. That means that the summation based only at the 12 neighbooring cells from each side gives the total field component values with good accuracy. In (Fig.4.2-4.4), the value of the Ey field has been calculated and plotted for the 4 cells exactly below the strip by use of the 2.5D-MRTD scheme. The relative position of the strip is from 15 to 25. For the TEM mode the pattern obtained by use of the conventional FDTD scheme is plotted for comparison. For the shield TE10 mode, the analytically calculated pattern has been added for reference. All results are normalized to the peak value. It can observed that the agreement of the MRTD calculated field pattern with the reference data is very good for the shield TEo1 mode, where the values are changing slowly (sinusoidally) (Fig.4.2). On the contrary, for the TEM mode where the edge effect is more prominent, the agreement is not good. In this case, wavelets of 0-Resolution are added in both directions to describe the higher spatial frequencies. It can be observed from (Fig.4.3) that the wavelet coefficients for the 8x4 grid have a significant contribution (> 10%) close to the stripline. Increasing the grid size from 8 to 16 to the strip direction and/or from 4 to 8 to the normal to the strip direction improves more the accuracy of the field representation (Fig.4.4). The characteristic impedance ZO for the TEM mode of the stripline is computed from the equation: V _ c, Eydy I C-, dl (4.8) where the integration paths C, and Cc are shown in (Fig.4.la). Since both of the schemes used in the analysis are discrete in space-domain, the above integrals are transformed to 91

Subpoints/cell ZOC (Q) Relative error ZO (Q) Relative error 3 80.56 -15.71 % 84.04 -12.07 % 5 94.46 -1.17 % 92.55 -3.17 % 7 99.06 +3.64 % 94.59 -1.04 % 9 101.44 +6.13 % 94.96 -0.65 % 11 97.56 +2.07 % 95.01 -0.60 % Table 4.2: Zo for different number of subpoints/cell (8x4 Grid). summations. For the FDTD summations, only one field value per cell is needed, due to the fact that pulse expansion functions which are constant for each cell are utilized. On the contrary, for the 2.5D-MRTD summation the field values for a number of subpoints along the integration path have to be calculated, since the expansion functions are not constant for each cell. It can be observed from (Table 4.2) that the accuracy of the calculation of the characteristic impedance is improved by increasing the number of subpoints per cell, at which the field values are calculated. An accuracy better than 1% is achieved if the field values are computed for more than 9 subpoints per cell along the integration path for the scheme including wavelets of 0-resolution to both directions. On the contrary, the value of Zo that is calculated from the scheme based only on scaling functions is oscillating, thus indicating that a denser mesh is required. The analytical value of the Zo is 95.58 Q [56]. The modification of the dimensions of the MRTD mesh (Table 4.3) shows that the accuracy of the calculation of the Zo by use of the MRTD is much better than that of the Yee's FDTD scheme with a 40x10 mesh (relative error -3.28%). A similar procedure is used for the analysis of the shielded coupled-stripline geometry of (Fig.4.1b) for the first even and odd mode. Both strips have a length of 11.9mm, the 92

Z, (i) Relative error Analyt. Value 95.58 0.0% 8x4 MRTD 95.01 -0.60% 8x8 MRTD 95.19 -0.41% 16x4 MRTD 95.71 0.14% 40x10 FDTD 92.44 -3.28% Table 4.3: Zo for different mesh sizes (11 subpoints/cell). distances between them is 11.9mm, from the top and bottom PEC's are 11.0mm and from the left and right PEC's are 11.9mm. The structure is filled with air (er = 1.). For the analysis with the conventional FDTD scheme, a 70 x 20 mesh resulted in a total number of 1400 grid pints. The same accuracy is achieved by an MRTD mesh 14 x 4 (56 grid points) resulting in an economy of memory by a factor of 25. The space distribution of the tangential-to-stripline E is plotted in logarithmic scale in (Fig.4.5) for the even mode. The 2-D MRTD technique is also used for the analysis of a shielded microstrip (Fig.4.1c) with width 9.9mm on a dielectric substrate with Er = 10.65 and thickness 11mm. The microstrip is placed in the center of a rectangular shield 69.3mm x 44mm. The same accuracy for the characteristic impedance calculation (Theoretical Zo = 50 Ohms) is achieved by an FDTD mesh 140 x 80 and an MRTD mesh 28 x 20resulting in an economy in memory by a factor of 20. 93

4.4 Validation of the MRTD-PML Split and Non-split Algorithms The extension of the popular PML absorber [8] principles for MRTD applications has been presented in CH.3. In this Section, the numerical performance of this absorber is investigated for 4-32 cells and for different cell sizes (A/10-A/2.5). Specifically, propagation constant, characteristic impedance and field patterns are derived for open transmission lines and compared to 2D results. A parallel-plate waveguide of width d=48 mm, terminated at both ends by PML, is used to validate the described algorithm. A TMZ line source with a Gabor time variation is excited close to the one side of the waveguide. The benchmark MRTD solution with no reflections is obtained by simulating the case of a much longer parallel-plate waveguide of the same width to provide a reflection-free observation area for the time interval of interest. A quadratic variation in PML conductivity is assumed for all cases, with maximum theoretical reflection coefficient of 10-5 at normal incidence. Two frequency ranges are investigated,[0, 0.9fTM1] (TEM propagation) and [0, 0.9f7TM2] (TEM + TM, propagation), where fATMn = n = 3.125 n (GHz) is the cutoff frequency of the TMn mode. The time-step is chosen to be 0.637 of the Courant limit according to the stability analysis of Ch.5. For the TEM propagation frequency range, it can be seen from Figs.(4.6)-(4.8) that for dense grids (Cell Size = Xma/,10) even 8 PML cells offer a numerical reflection close to -80 dB. Different values of theoretical maximum reflection ranging from 10-5 to 10-8 don't change significantly the numerical performance of the absorber (variation of 4-5 dB's). When 16 PML cells are used, the spurious reflection is below -100 dB for the whole frequency range. Similar conclusions can be drawn for the multimodal propagation (TEM + TMA) in 94

Fig.(4.9). It can be observed that 8 and 16 PML cells cause a numerical reflection close to -70 dB and below -100 dB respectively. For coarse grids with cell sizes close to the Nvquist limit (Cell Size = Ama,,,/2.5), the behavior of the PML layer changes. The Large cell size causes retrospective reflections between the lossy cells and the numerical reflections from the absorber increase. Thus, a larger number of cells is required to obtain an acceptable reflection coefficient. Fig.(4.10)-(4.11) show that at least 32 cells are needed for reflection around -50 dB for the high frequencies. Again, the reflection at lower frequencies is negligible (below -100 dB's). It should be emphasized that the loss coefficients assigned to each cell must be given by Eq.(3.79); that implies that the conductivity profile must be sampled with the scaling and wavelet functions that have a significant value in the PML layer. For all simulations, scaling (and wavelet) functions located up to 6 cells away from the PML layer are used for the sampling. When this procedure is not applied and the loss coefficients get the point value of the loss distribution at each cell (FDTD approach), the PML performance gets worse as it is displayed at Fig.(4.12). It should be noted that the performances of the split and the non-split formulations are almost identical as it is displayed in Fig.(4.13). 4.5 Application of PML to the Analysis of Open Stripline Geometries The PML non-split algorithm presented in Section 3.3.2 can be easily extended for the 2.5D and the 3D MRTD algorithms incorporating scaling and wavelet functions maintaining the same performance characteristics. For each resolution added to the scheme, the conductivity must be sampled with an appropriately positioned wavelet function. It was observed that S11 changes only by 1-1.6 dB after the enhancement of multiple resolutions. 95

In this section, the 2.5D MRTD scheme is applied to the analysis of open single and coupled striplines to investigate propagation and coupling effects. In all simulations only wavelets of the 0-resolution are used for both directions, since the value of the higher resolution fields is negligible (smaller than 1%). First, the 2.5D MRTD scheme is applied to the analysis of the open stripline for the first (quasi-TEM) propagating mode. The analysis for the higher order propagating modes is straightforward. The central strip has a length of 23.8mm and the distances from the top and bottom are 5.5mm and 16.5mm respectively. The structure is filled with air (Er = 1.). The PML absorber is applied for 4 cells to the left and the right sides of the structure and the maximum theoretical reflection is Rma.=le-7. For the analysis using Yee's FDTD scheme, a 42 x 28 mesh is used resulting in a total number of 1176 grid points. Analyzing the structure with the 2D-MRTD scheme, a mesh 12 x 4 (48 grid points) is chosen to reduce the total number of grid points by a factor of 24.5. In addition, the execution time for the analysis is reduced by a factor of 4 to 5. The time discretization interval is chosen to be identical for both schemes and equal to 1/10 of the 2D-MRTD maximum At. For the analysis,3 = 30 is used and 20,000 time-steps are considered. From (Table 4.4) it can observed that the calculated frequencies of the dominant propagating mode for (3 = 30 by use of 2D-MRTD scheme is very close to the theoretical values, since the largest error is less than 0.1%, for mesh sizes much smaller than those used for the conventional FDTD simulations. In (Fig.4.14), the pattern of the Ey field just below the strip has been calculated and plotted by use of the 2D-MRTD scheme. The pattern obtained by use of the conventional FDTD scheme is plotted for comparison. Since the edge effect is prominent, a mesh 12 x 8 (96 grid points) with scaling functions and wavelets of 0-resolution is used for the MRTD 96

Mode TEM Rel.Error Analytic values 1.4324 GHz 0.000% 12x4 MRTD 1.4329 GHz 0.035% 12x8 MRTD 1.4325 GHz 0.007% 42x28 FDTD 1.4321 GHZ -0.021% Table 4.4: Dominant mode frequency for 3 = 30 simulation. The characteristic impedance Zo for the quasi-TEM mode of the stripline is computed from Eq.(4.8). For the FDTD summations, only one field value per cell is needed, due to the fact that pulse expansion functions which are constant for each cell are utilized. On the contrary, for the 2D-MRTD summation the field values for a number of subpoints along the integration path have to be calculated, since the expansion functions are not constant for each cell. (Table 4.5) shows that the accuracy of the calculation of the characteristic impedance is improved by increasing the number of subpoints per cell, at which the field values are calculated. An accuracy better than 1% is achieved if the field values are computed for more than 9 subpoints per cell along the integration path. (Table 4.5) shows the calculated values of the characteristic impedance Zo. A similar procedure was used for the analysis of the open coupled-stripline geometry of (Fig.4.15) for the dominant even and odd modes. Both strips have a length of 23.8mm, the distances between them is 23.8mm, from the top PEC 16.5mm and from the bottom PEC 5.5mm. The MRTD-PML layer has a thickness of 4 cells (23.8mm) with maximum reflection Rmax==le-7 and starts exactly at the edge of the striplines. The structure is filled with air (Ec = 1.). For the analysis with the conventional FDTD scheme, a 65 x 20 mesh 97

ZO (Q) Relative error Analyt. Value 56.83 0.0% 12x4 MRTD 57.24 +0.72% 12x8 MRTD 57.09 +0.46% 42x28 FDTD 54.96 -3.29% Table 4.5: ZO for different mesh sizes. resulted in a total number of 1300 grid points. The same accuracy is achieved by an MRTD mesh 20 x 4 (80 grid points) resulting in an economy of memory by a factor of 16.25. The space distribution of the tangential-to-stripline E is plotted in logarithmic scale in (Fig.4.16) for the even mode and in (Fig.4.17) for the odd mode. 4.6 Conclusion A multiresolution time-domain scheme in 2D has been applied to the numerical analysis of shielded and open striplines and microstrips. The field patterns and the characteristic impedance have been calculated and verified by comparison to reference data. In comparison to Yee's conventional FDTD scheme, the proposed 2.5D-MRTD scheme offer memory savings by a factor of 25 and execution time savings by a factor of about 4-5 maintaining a better accuracy for characteristic impedance calculations. This indicates memory savings of a factor 5 per dimension leading to two orders of memory savings in three dimensions. Compared to 2.5D-FDTD, 25 times less cells in MRTD require about 5 times less running time, thus the computation time per cell is increased by a factor of 5. This leads to computation time savings of more than one order for 3 dimensional structures. For structures, where the edge effect is prominent, additional wavelets have to be introduced to improve the accuracy 98

when using a coarse MRTD mesh. A non-split PML absorber has been evaluated and its performance is similar to that of the toconventional FDTD Split PL absorber (reflections close to -100 dB). 99

Cc I I - -- t --- —---- ______c^___I (a) (b)...................... aid 0 $.0 (c) Figure 4.1: Printed Lines Geometries. 100

TE10 Mode 2 4 6 8 10 12 14 16 position Figure 4.2: Shield TElo Ey pattern. Horizontal axis (x) Figure 4.3: TEM Ey Pattern Components (8x8 Grid). 101

40 x-axis Figure 4.4: TEM Ey Pattern Comparison (8x8 Grid). Figure 4.5: Tangential E-field Distribution (Shielded - Even Mode). 102

Sampled - TEM - 4 cells - U10 1.5 Freq [GHz] Figure 4.6: 4 PML cells - Non-split formulation - Dense Grid. Sampled - TEM - 8 cells - L110 1.5 Freq [GHz] Figure 4.7: 8 PML cells - Non-split formulation - Dense Grid. 103

Sampled - TEM - 16 cells - L/10 co V. -120 55 1 1.5 2 2.5 Freq [GHz] - Non-split formulation - Dense Grid. Figure 4.8: 16 PML cells TEM+TM1 Propagation - Cell Size Lmax/10 ) 1 2 3 4 5 Freq [GHz] Figure 4.9: Multimodal Propagation - Dense Grid. 104

TEM Propagation - Cell Size Lmax/2.5 co VV C') m Vcn) 0 0.5 1 1.5 2 2.5 Freq [GHz] Figure 4.10: TEM Propagation - Coarse Grid. TEM+TM1 Propagation - Cell Size Lmax/2.5 3 0 1 2 3 4 5 Freq [GHzJ Figure 4.11: Multimodal Propagation - Coarse Grid. 105

Sampled - TEM - 16 cells - /10 u la -t. -1. S5 1.5 2 2.5 Freq [GHz] Non-split formulation - Dense Grid. Figure 4.8: 16 PML cells - TEM+TM1 Propagation - Cell Size Lmax/10 3 4 5 Freq [GHz] Propagation - Dense Grid. Figure 4.9: Multimodal 106

250 V w I 200 i I I 150- I' 100 50 - I 50 0 50 100 150 Position Figure 4.14: Open Single Stripline - 200 250 Ey TEM Distribution. - 119,0 mm --- 4 cells 0.1 S/m max ^ ccnduc. | 11 23.8 23.8 23.8 Figure 4.15: Open Coupled Stripline Geometry. 107

Figure 4.16: Tangential E-field Distribution (Open - Even Mode). -:!N:!.F~~~~~kN-0.-;*,-D Figure 4.17: Tangential E-field Distribution (Open - Odd Mode). 108

CHAPTER 5 Stability and Dispersion Analysis of Multiresolution Time-Domain Schemes 5.1 Introduction Discretized Time-Domain numerical techniques are very popular in the analysis of various microwave geometries and for the modelling of EM wave propagation. Though many of them are very simple to implement and can be easily applied to different topologies with remarkable accuracy, they cause a numerical phase error during the propagation along the discretized grid [57]. For example, the numerical phase velocity in the FDTD can be different than the velocity of light, depending on the cell size as a fraction of the smallest propagating wavelength and the direction of the grid propagation. Thus, a non-physical dispersion is introduced and affects the accuracy limits of FDTD simulations, especially of large structures. In addition, it is well-known that the finite-difference schemes in time and space domain require that the used time step should take values within an interval that is a function of the cell size. If the time-step takes a value outside the bounds of this interval, the algorithm will be numerically unstable, leading to a spurious increase of the field values without limit 109

as the time increases. Though the stability and the dispersion analysis for the conventional Yee's FDTD algorithm has been thoroughly investigated, only a few results have been presented concerning MRTD schemes based on cubic spline Battle-Lemarie scaling and wavelet functions [26]. The functions of this family do not have compact support, thus the finite approximations of the derivatives are finite-stencil summations instead of finite differences. In this paper, the effect of these stencils' size as well as of the enhancement of wavelets is investigated and a comparison with 2nd-order and higher-order FDTD schemes exhibits the differencesin their respective behaviors. 5.2 Stability Analysis Following the stability analysis described in [57], the MRTD [26] equations are decomposed into separate time and space eigenvalue problems. Assuming a 2D expansion only to scaling functions (S-MRTD) similar to Eqs.(3.36)-(3.38) of CH.3, the left-hand side time-differentiation parts can be written as an eigenvalue problem 1/2H -1/2 - 1/2j1/2 =A ki /2 (5.1) k+1/2H_,/2 - k-1/2Hi_1/2, = A kH, (5.2) At At ~ ij-1/2 k+lE, - kE, — ) - A k+1/2Ei. (5.3) At 5 In order to avoid having any spatial mode increasing without limit during normal timestepping, the imaginary part of A, Imag(A), must satisfy the equation 2 2 - < Imag(A) <. (5.4) For each time step k, the instantaneous values of the electric and magnetic fields distributed in space across the grid can be Fourier-transformed with respect to the i- and j- coordinates 110

to provide a spectrum of sinusoidal modes (plane wave eigenmodes of the grid). Assuming an eigenmode of the spatial-frequency domain with kh and ky being the x- and y- components of the numerical eigenvector, the field components can be written EJ = Eo ej(kxIAx+kyJAy HiJ-i/2 = Hxoej(kxIAx+ky(J-1/2)AY HIy/2,J H= yo (k{(I-1/2)Ax+kyJAY Substituting these expressions to (5.1)-(5.3) and applying Euler's identity, we get a4 1 -1 n = — [ )2( a(i')sin(k:(i + 1/2)Ax))2 + ( a(j)sin((j + 1/2)Ay))2] i'=O j'=O Thus, A is a pure imaginary, which can be bounded for any wavevector k = (ks, ky): a- 1 1 - 2c ( la(i')) ( )2 + (y)2 Imag(A) n. -1 < 2c (E a(i')) + (.(y)2 where c = 7 is the velocity of the light in the modeled medium. Numerical stability is maintained for every spatial mode only when the range of eigenvalues given by (5.5) is contained entirely within the stable range of time-differentiation eigenvalues given by (5.4). Since both ranges are symmetrical around zero, it is adequate to set the upper bound of (5.5) to be smaller or equal to (5.4), giving: t < (5.6) c (Ena-1 a(i')l) A, + For Ax = Ay = A, the above stability criterion gives A A AtS-MRTD < c/ = sSS (57) <C,= Ena-1 la(i')i c+/ It is known [5] that AtFDTD 1, (5.8) (C r2 + Ay)2 111

which gives for Ax = Ay = A A AtFDTD < c X (5.9) Equations (5.7)-(5.9) show that for same discretization size, the upper bounds of the timesteps of FDTD and S-MRTD are comparable and related through the factor s. The stability analysis can be generalized easily to 3D. The new stability criteria can be derived by the equations (5.7) and (5.9) by substituting the term /2 with v/3. More complicated expressions can be derived for the maximum allowable time-step for schemes containing scaling and wavelet functions. For simplicity and without loss of generality, it is assumed that the stencil size is equal for all three summations (na = nb = nc = n). The upper bound of the time-step for the 2D MRTD scheme with 0-resolution wavelets to the one (x-direction) or two directions (x- and y-directions) for Az=Ay=A is given by A ttWoS-MRTD,max ' SWoS7 with 2 SWos = V/3(E' fal)2 + (Ei, Ibol)2 + 2(ZE, Icol)2 + (Ei, la + bol)v(EZ, la - bol)2 + 4(EZ' coi)2 and A AtWoWo-MRTDa,max SWWoW - with 2 SWoO =Wo V/2(Ei lal)2 + 2(E,, Ibol)2 + 4(Ei, IcoI)2 + 2(Ei la + bol)V/(Ei, a bol)2 + 4(Z, col)2 where the notation n — Exl = Z Ix(k')l k' k'=O has been used. 112

It can be observed that the upper bound of the time step depends on the stencil size na, rib, n7. This dependence is expressed through the coefficients sss,s'o1 ^i siwoo which decrease as the stencil size increases. (Figure 5.1) shows that sss practically converges to the value 0.6371 after na > 10 and swos 0.3433 and swoWo P 0.2625 for na = nb = nr > 10. The expression of sss can be easily derived by the expressions of sUs and sWOo by zeroing out the effect of b0oc0. 5.3 Dispersion Analysis To calculate the numerical dispersion of the S-MRTD scheme, plane monochromatic traveling-wave trial solutions are substituted in the discretized Maxwell's equations. For example, the Ez component for the TMZ mode has the form kEI,J = Ezoej(kxIAx+kyJAy-wkAt) where k, and ky, are the x- and y- components of the numerical wavevector and w is the wave angular frequency. Substituting the above expressions into Eqs.(3.36)-(3.38), the following numerical dispersion relation is obtained for the TMZ mode for the S-MRTD Scheme after algebraic manipulation 1 wLAt na-l [ —sin( )]2 [ (E a(i')sin(k(i +l 1/2)Ax))]2 1 na-1 + [-y( 1 a(j')sin(ky(j + 1/2)Ay))]2 (5.10) For square unit cells (Ax=Ay=A) and wave propagating at an angle es with respect to x-axis (k, = k cosq and ky = k sin+q), the above expression is simplified to 113

[At sin( )]2 ) = ( a(i') sin(k coso (i'+ 1/2) A))2 il=O na-1 + ( a(j') sin(k sin (j'+ 1/2) A))2 (5.11) j'=O This equation relates the numerical wavevector, the wave frequency, the cell size and the time-step. Solving this numerically for different angles, time-step sizes and frequencies, the dispersion characteristics can be quantified. Defining the Courant number q = (cAt)/A and the number of cells per wavelength ni = AREAL/A and using the definition of the wavevector k = (2r)/\ANUM the dispersion relationship can be written as na-1 [- sin(7r q /nl)]2 = [ a(i') sin(ir u (2i'+ 1) cos, /ni)]2 q i'=O na-1 + [ E a(j') sin(r u (2j' + 1) sine /nI)]2 (5.12) j'=o where u = AREAL/ANUM is the ratio of the theoretically given to the numerical value of the propagating wavelength and expresses the phase error introduced by the S-MRTD algorithm. To satisfy the stability requirements, q has to be smaller than 0.45 (= 0.6371/V2) for the 2D simulations. The above analysis can be extended to cover the expansion in scaling and O-resolution wavelet functions in x-, y- or both directions. The general dispersion relationship is (C1CI + C2C2 + C4C4 + C5C5)+ ()2[ + C2C2C3) ~f E A B +(A )4(C1C2+ C2C3)2(C4C5 + C5C6)2( + )2 = 1 (5.13) 1 +C(5.13) A BF with F= - [(C1C2 + C2C3)]2 (C4C5 + CsC6)]2 114

Scheme C1 C2 C3 C4 Cs C6 l4oS #0 0 0 o0 #0 #0 SWo $0 #0 o0 o0 0 0 WoW o # 0 #0 0 #0 0 #0 Table 5.1: Coefficients Ci for Different MRTD Schemes - [(C2C + C3C3 + C5C5 + C6C6] (5.14) A = 1 - (CC +C2C2 + C5C5+Cs6C6) B = - (C2C2 C3C3 + C4C4 + C5) (5.15) The Ci are defined by At = -Asin(wAt/2) a(J')sin(k(j' + 1/2)A) j1=O At nc C2 = - Ai(At/2) co(j')sin(kj'A) IuAsin(wAt/2) j, At nb C3 - in(wAt/2) bo(j')sin(ky(j'+ 1/2)A) 115= At a 4 = - A (At/2) a(i')sin(k,(i' + 1/2)A) Yasin(wAtI2) i,=0 C2= co(i')sin(ki'A),Asin(wAt/2) =0 o')i(ki) At C3 =- Ai (A//2)) bo(i')sin(kx(i' + 1/2)A) (5.16) Eq.(5.13) can be applied to the dispersion analysis of SS (only scaling functions), W0S (0-resolution wavelets only to x-direction), SW0 (0-resolution wavelets only to y-direction) and WoWo (O-resolution wavelets to both x - and y- directions) following Table (5.1). In case the Ci $ O, it can be calculated by Eq.(5.16). The above equation is solved numerically by use of Bisection-Newton-Raphson Hybrid 115

Technique for different values of na, nb, nc ni, 0 and q. (Figs.5.2-5.5) show the variation of the numerical phase velocity as a function of the inverse of the Courant number 1/s=l/q for stencil sizes n, = nb = n, = 8,10, 12,14. For each figure, three different discretization sizes are used: 10 cells/wavelength (coarse), 20 cells/wavelength (normal) and 40 cells/wavelength (dense). The results are compared to the respective values of conventional FDTD. It can be observed that the phase error for F.D.T.D. decreases quadratically. The variation of the phase error in M.R.T.D. exhibits some unique features. Though for any stencil size the numerical phase error for M.R.T.D. discretization of 10cells/X is smaller than that of the F.D.T.D. discretization of 40cells/A, the M.R.T.D. error doesn't behave monotonically [58]. It decreases up to a certain discretization value and then it starts increasing. This value depends on the stencil size and takes larger values for larger stencils. For example, this value is between 10 and 20 cells/A for stencil equal to 10, between 20 and 40 cells/) for stencil=12 and very close to 40 cells/A for stencil=14 and can be used as a criterion to characterize the discretization range that the M.R.T.D. offers significantly better numerical phase performance than the F.D.T.D. The phase error caused by the dispersion is cumulative and it represents a limitation of the conventional FDTD Yee algorithm for the simulation of electrically large structures. It can be observed that the error of S-MRTD is significantly lower, allowing the modeling of larger structures. FDTD is derived be expanding the fields in pulse basis. As it is well known the Fourier transform of the pulse is a highly oscillating Si(x). On the contrary, the Fourier transform of the Battle-Lemarie Cubic spline is similar to a low-pass filter. That "smooth" spectral characteristic offers a much lower phase error even for very coarse (close to 3-4 cells/A) cells: By using a larger stencil n,, the entire-domain oscillating nature of the scaling functions 116

is better represented. Thus, smoother performance for low discretizations (Fig.5.6) and lower phase error for higher discretizations (Fig.5.7) is achieved as na increases from 8 to 12. Wavelets contribute to the improvement of the dispersion characteristics for even coarser cells (close to 2.2-2.4 cells/A) as it is demonstrated in (Figs.5.8-5.13). For discretizations above 4 cells/A the effect of the wavelets is negligible. (Fig.5.11) and (Fig.5.13) show clearly that the phase error has a minimum for a specific discretization (17 for na = 10 and 25 for na = 12). (Figs.5.14-5.17) show that for discretizations smaller than 30cells/A the choice of the Courant number affects significantly the dispersion performance which starts converging to the minimum numerical phase error (0.8 deg/A for na = nb = nc = 10 and 0.2 deg/A for na = nb = nc = 12) for 1/q close to 10. On the contrary, the F.D.T.D. dispersion is almost independent of the Courant number (Figs.5.18-5.19). It has been claimed in [59] that the S-MRTD Scheme is slightly oscillating and its performance is only comparable with the 14th order accuracy Yee's scheme. Though this is true for the S-MRTD schemes with stencil size of 8, the comparison of the dispersion diagrams of Yee's FDTD scheme, Yee's 16th order (H.F.D.-16) and 22th order (H.F.D.-22) and S-MRTD and Wo-MRTD schemes with different stencils leads to interesting results.For comparison purposes, the values of At = Atmax/5 and Atmax = 0.368112A1/c have been used and all the dispersion curves are substracted by the linear dispersion relation for ID simulations. (Fig.5.20) shows that the S-MRTD scheme with stencil 10 has a comparable performance to the 16th order Yee's scheme. The enhancement of the wavelets for the same stencil improves significantly the dispersion characteristics of the MRTD scheme increasing the dynamic range of w by approximately 90% and comparing favorable even to the 22th order Yee's scheme. This is expected due to the fact that the scaling+wavelet basis spans 117

a larger ("more complete") subspace of R than the scaling functions alone. Both S-IMRTD and Wo-MRTD schemes have identical numerical phase errors up to the point that the S-MRTD scheme starts diverging (Fig.5.21). As the stencil size of the XVo-MRTD scheme is increasing from 6 to 12 (Figs.5.22-5.23), the oscillatory variation of the phase error is diminishing to a negligible level generating an almost flat algorithm similar to the higher order Yee's ones. As a conclusion, due to the poor dispersion performance of the FDTD technique even for 10 cells/wavelength a normal to coarse grid is always required to avoid significant pulse distortions especially for the higher-spatial-frequency components. MRTD offers low dispersion even for sparse grids very close to the Nyquist limit. 5.4 Conclusion The stability and the dispersion performance of the recently developed Battle-Lemarie MRTD schemes has been investigated for different stencil sizes and for O-resolution wavelets. Analytical expressions for the maximum stable time-step have been derived. Larger stencils decrease the numerical phase error making it significantly lower than FDTD for low and medium discretizations. Stencil sizes greater than 10 offer a smaller phase error than FDTD even for discretizations close to 40 cells/A. The enhancement of wavelets further improves the dispersion performance for discretizations close to the Nyquist limit (2-3 cells/wavelength) making it comparable to that of much denser grids, though it decreases the value of the maximum time-step guaranteeing the stability of the scheme. 118

L O co >-0.5.0 (n 2 4 6 8 10 12 Stencil na = nb = n Figure 5.1: Stability Parameter s for MRTD. Phase Error (Ste=8 vs. FDTD) I I I 1 2 3 4 5 6 7 8 9 10 1/s Figure 5.2: Dispersion Characteristics of S-MRTD for na-=8. 119

Phase Error (Ste=10 vs. FDTD) 3. — "o.0 E Co -) LL 0 V) CO,: n 1/s N Figure 5.3: Dispersion Characteristics of S-MRTD for na=10. Phase Error (Ste=12 vs. FDTD) r ol ax 0 LL. co 0. u3 en s) * MR10 2.5 -- MR20 o MR40 - FD20 2- -- FD40.5 I.5t ', K x. --- T --- "- i- - 2 Y _ rr~^ o o ''^ g <, g '"- - - - 0 --- - -- ~ --- " T - - r ' - ~ - - ' - - - ~ 0 ) [ 1 3 4 5 6 7 8 9 10 1/s Figure 5.4: Dispersion Characteristics of S-MRTD for na=12. 120

Phase Error (Ste=14 vs. FDTD) 32.. '... 1.8 1.6 1.4 Ca.~ E 1.2 a) 0 - 1 0 ~ 0.8 u. 0.6 0.4 0.2 / 3 MR10 / x -- MR20 0 MR40 FD20 -- FD40 \.. —...... ql t 3 X - nI D 1 2 3 4 5 6 7 8 9 10 1/s Figure 5.5: Dispersion Characteristics of S-MRTD for na=14. 121

Phase Error (Ste —IO vs. FDTD).0 E 0 0 -c I1/s Figure 5.3:6 Dispersion Characteristics of S-MRTD for n.=1LO. Phase Error (Ste=12 vs. FDTD) -WMRIO - - MR2O 0 MR40 -~FD20 - FD40 E 0 I0 w X W X [ I - 00 5 [ - - - E 0 I -- 1 2 3 4 5 6 7 8 9 10 Figure 5.4: Dispersion Characteristics of S-MRTD for rzla=12. 122

1 g( }. It I I I I I I OJV x' 100,,0 E -J -j a) 0 i, CU = 50 a. - na=8,nb=nc=0 - - na=8,nb=nc=4 - - na=8,nb=nc=8 na=8 nb=nc=12 FDTD........................ I. OI 0 I 1 2 3 4 5 6 7 8 9 10 Samples/Lambda Wavelets Effect on the Dispersion Characteristics of MRTD for na=8 (Coarse Grid). Figure 5.8: I. I. I V co la u1) V0 0 w Ca) a. 9 8 na=8,nb=nc=0 -- na=8,nb=nc=4 7 - - na=8,nb=nc=8 na=8 nb=nc=12 6 - 6 FDTD 4 3 A.. - 10 15 20 25 30 35 40 Samples/Lambda Figure 5.9: Wavelets Effect on the Disperistics of MRTD for na=8 (Denser Grid). 123

6 Samples/Lambda Figure 5.10: Wavelets Effect on the Dispersion for na=10 (Coarse Grid). Characteristics of MRTD 11 n..i m I u 9 8 I6 I — <cn 5 o 4 2 1 - na=10,nb=nc=O - - na=10,nb=nc=6 na=10,nb=nc=10 FDTD -. -.A '-"I.... Il I- -I I ~ i i I 10 15 20 25 30 Samples/Lambda 35 40 45 50 Figure 5.11: Wavelets Effect on the Dispersion Characteristics of MRTD for na=10 (Denser Grid). 124

'-14 -.0 E - 12 10 E0 2 8 -f - 4 - 2 0 Figure 5.12: 6 Samples/Lambda Wavelets Effect on the Dispersion for na=12 (Coarse Grid). Characteristics of MRTD 2i 1 1 1 1 1 1 1 I 1.8 - na=12,nb=nc=0 - - na=12,nb=nc=8 - - na=12,nb=nc=12... FDTD 1.6 - a 1.4 l0.0 E. 1.2 a, 0.8 ' 1 0 o0.8 a c0 - 0.6 l. F F 0.4 - 0.2 I- _ - -..*.. I I 20 25 30 35 40 Samples/Lambda 45 50 55 60 Figure 5.13: Wavelets Effect on the Dispersion Characteristics of MRTD for na=12 (Denser Grid). 125

1 -1 1I I, i,, I I I 9 8 7 0 D E 0 -J L. a) 0) (0 ~1 6 5 1/q=1.1 1/q=3..1/q=5 \ 1/q=10...-.' —... —. ---. =....,. —..,... — 4 3 2 1 0 - - - - 2 3 4 5 6 7 8 Samples/Lambda 9 10 11 12 Figure 5.14:.o 0.5 E 0 -i ". — 2 O-0.5 10 Figure 5.15: Effect of the Courant Number on the Dispersion Characteristics of Wo - MRTD for na = nb = nc=10 (Coarse Grid). 30 35 40 Samples/Lambda I Effect of the Courant Number on the Dispersion Characteristics of Wo - MRTD for n, = nb = nc=10 (Denser Grid). 126

in.-.I I I I I -- I — I 9 8 -7 0 'a E 6 0 I 5 i " 4 o3 a,) Cu 3 2 1 0 - 1/q=1.1 -- 1/q=3. - 1/q=5 \ 1/q=10. *.....,_............... -... - =.M I, t I,. I * 2 3 4 5 6 7 8 Samples/Lambda 9 10 11 12 Figure 5.16: 1 -0' 0.5 a, -J 0 0 b iw - C: - I O -0.5 10 Figure 5.17: Effect of the Courant Number on the Dispersion Characteristics of Wo - MRTD for na = nb = n,=12 (Coarse Grid). 30 35 40 Samples/Lambda Effect of the Courant Number on the Dispersion Characteristics of Wo - MRTD for n, = nb = n,=12 (Denser Grid). 127

a) | 20 -E 0 -I D 15 -U) 10 -a 5 0 -2 Figure 5.18: 2 1.5 - 10 E C 1 1 -0) o 0.5 (/3 a. e 3 4 5 6 7 8 9 10 11 12 Samples/Lambda Effect of the Courant Number on the Dispersion Characteristics of FDTD (Coarse Grid). 10 15 20 25 30 35 40 45 50 55 60 Samples/Lambda Figure 5.19: Effect of the Courant Number on the Dispersion Characteristics of FDTD (Denser Grid). 128

0.o - - P-9; I T I I I I ^ - | r..-, + 0.2 - F.D.T.D. + H.F.D.-16 * H.F.D.-22 -- S-M.R.T.D...- Wo-M.R.T.D. + 0.15 - 0.1 + /!' +. III x 0.05 + +fr 01 O II I I aImimim iiim iimiiimmmwo -0.05 - \ \ I \ I \j -0.1 - -.. - I 11 SIIII 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Dt*w Figure 5.20: 8 6 4 2 x 0 a Comparison of the Dispersion Performance of S-MRTD and Wo-MRTD with Different Higher Order Yee's Schemes. 0.05 0.1 0.15 0.2 Dt *w Figure 5.21: Details of Fig.(5.20). 0.25 0.3 129

x * 0.1 0.05 C -. — '~ ----0.05 i 0 0.05 0.1 Figure 5.22: Comparison of the ferent Stencil Size. 0.02 l l -.R.T.D.-6 0.015 -.015 M.R.T.D.-8 -- M.R.T.D.-10 0.01 01 -- M.R.T.D.-12... H.F.D.-22 0.005 0 / \ 0.35 Dt*w Oscillations of Wo-MRTD Scheme for Dif 0.05 0.1 0.15 0.2 Dt*w Figure 5.23: Details of Fig.(5.22). 130

CHAPTER 6 Development of a Space- and Time-Adaptive MRTD Gridding Algorithm for the Analysis of 2D Microwave Dielectric Geometries 6.1 Introduction In CH.4, the MRTD Technique has been applied to a variety of homogeneous microwave problems and has exhibited significant savings in memory and execution time. Nevertheless the most important advantage of this new technique is its capability to provide space and time adaptive gridding without the problems that the conventional FDTD is encountering. This is due to the use of two separate sets of basis functions, the scaling and wavelets and the capability to threshold the field coefficients due to the excellent conditioning of the formulated mathematical problem. In this Chapter, a space/time adaptive gridding algorithm based on the MRTD scheme is proposed and applied to nonhomogeneous waveguide problems. As an example, the propagation of a Gabor pulse in a partially-filled parallel-plate waveguide is simulated and the S-parameters are evaluated. Wavelets are placed only at locations where the EM fields have significant values, creating a space- and time- adaptive 131

dense mesh in regions of strong field variations, while maintaining a much coarser mesh elsewhere. The modification of the 2D MRTD algorithm to incluse dielectric variation is presented and wavelet thresholding approaches are compared and evaluated. 6.2 The 2D-MRTD Nonhomogeneous scheme 6.2.1 The 2D-MRTD scaling and wavelets scheme For simplicity the 2D-MRTD scheme for the TM, modes will be used herein. To derive the 2D-MRTD scheme, the field components are represented by a series of cubic spline Battle-Lemarie [25] scaling and wavelet functions up to the rmax-resolution to the longitudinal direction in space and pulse functions in time in a similar way to CH3. Due to the entire domain basis functions, D of one cell is related to E values all over the neighboring cells. To circumvent this problem, the CurlD equations have to replace the CurlE equations and then the E-coefficients have to be calculated from the D-coefficients in a mathematically correct way. After inserting the field expansions in Maxwell's equations, we sample them using pulse functions in time and scaling/wavelet functions in space domain. As an example, sampling OD/9tt, = - OHy/Oz in space and time, the following equations are obtained ma i m~ t(k+lDi+1/2,m - kDl/2,m) =- - t a(k+/ 2,m+i+1/2 i=-m: rmax 2r- 1 Mr,p + E: aK rp (rk+l/ /2,m+i+(2p+l)/2r+1) (61) r=O p=0 i=-_m Dr,p 1 k+1 DPIX DtIplx a ( kl l+/2,m+(2p1+l)/2r/+ k 1+1/2,m+(2p/+l)/2r'+l 1 m 1 mr',pt A E a + +1/2,m+i+1/2 t=-mZ 'y 132

'Pr -,p rmax 2r-1 V r,p + Z a pP(i)k+l/2H jl+l/2,m+i+(2p+l)/2r+) (6.2) r=O p=O or,pl i=-m r 'p 'mr,p where kDa, and kHI/ with g=O (scaling),4t,p (wavelets of r-resolution at the p-position of the cell) are the coefficients for the electric and magnetic field expansions. The indices 1, m and k are the discrete space and time indices, which are related to the space and time coordinates via x = lAxz = m/z and t = ckAt, where Ax,Az are the space discretization intervals in x- and z-direction and At is the time discretization interval. The coefficients a(i), aOo',(i), a00,(i)o (i) are derived in a similar way to CH.3. For an accuracy of 0.1% the values m = 10 - 12 and m', = m = mo,' = 8 - 12 have been used when only the -10 - 1 do,o -o,o O-resolution of the wavelets was applied. 6.2.2 The PML numerical absorber For open structures, the perfectly matched layer (PML) technique can be applied by assuming that the conductivity is given in terms of scaling and wavelet functions instead of pulse functions with respect to space [61]. The PML is characterized by artificial electric and magnetic conductivities aE and aH, which satisfy the relationship drH erE rD -H - E - = D (6.3) lo E co for each cell with constitutive parameters (E,C0o). The spatial distribution of the magnetic conductivity for the absorbing layers is modelled by assuming that the amplitudes of the scaling and the wavelet functions have a polynomial distribution (linear, parabolic,...) and by sampling this distribution with scaling and wavelet functions (Eq.(3.79)). It has been observed that the parabolic distribution is the most computationally efficient. For multidielectric non-magnetic structures, the electric conductivity is given from the above relationship assigning the appropriate c. The MRTD mesh is terminated by a perfect electric 133

conductor (PEC) at the end of the PML region. Following a procedure similar to (3.4.3) the equations for the D, scaling components in the PML region are given by A = _at/ -a t/co - 0.5 At/c " /\, kl i-1/2,j /2, - D( a(j )k+1/2 H /2,+j'+ l/2 J =-na rmax 2r-1 Tr,p +rC E Ek+112HY ",(6.4) + E a<p(i)k+/2 -1/2',+j'+(2p+l)/2+).4) r=O p=0 j=-m 3 -mr,p Similar equations can be obtained for the wavelet equations. For the simulations presented in this paper, there are used 24 cells of PML medium with CH for designated Rmax=l.e-6 for MRTD (coarser mesh) and for Rma,=l.e-7 for FDTD (4 times/dimension finer mesh) which provide reflection coefficients in the region of-80 to -90 dB. 6.2.3 The Excitation Implementation In order to implement an excitation EF(t) at z = mAz and to obtain an excitation identical to an FDTD excitation (pulse excitation with respect to space), the space pulse is decomposed in terms of scaling and wavelet functions. Cck rmax 2r-1 0Cr,p kEZC EF(k\t)( E C(i)m+i + E E Crp(i)M+i) (6.5) i= —, r=O p=O i=-C-r,p where the coefficients cs(i),cg,(i), c^,,(i), c,2(i) (wavelets of 0- and 1-resolutions) are given in Chapter 3. For i < 0 it is c(-i) = c+(i), co,1(i) = COo,(-1 - i), c,, (i) = c1,2 (-1-i) and c,),2 (i) = co,p, (-1-i). The above excitation components are superimposed to the field values obtained by the MRTD algorithm for the same time step. For example, the total kE +i will be given by kE+itotal = EF(kAt) co(i) + Em+i * (6.6) It has been observed that the minimum limits of the summations for an accuracy of 0.1% are (, = Co, =4 and (, o = Ci, =3 for the first 2 resolutions of the wavelets (rmax=l). 134

Similar accuracy can be observed when the scaling function at the excitation cell is set equal to the value of the excitation function EF(t). No superposition is used and the field scaling and wavelet values elsewhere are given by the MRTD equations. The wavelet coefficients are excitated through the coupling of the discretized MRTD equations. In the following numerical simulations the latter excitation technique was employed since it adds significantly smaller computational overhead. 6.2.4 The Modelling of Dielectrics Starting from the constitutive relationship D = cE for the total electric field at one mesh point and sampling the scaling and wavelet components with a similar way to (A) we reach the following equations for D: 14 rmax 2r-1 llr,p kD, =E C( kj +E Z C d. )Etij+j (6.7) j=- r=O p=O j'=-lPr,p and ()E rmax 2r-1 'rr,p kD, E C zd ' )kEi + ) (6.8) j= r= p=P=0 j'=-r,p where Ckd( j) = e 'Ck(Z)Ok+j'(Z)dZ (6.9) 00oo ~Or,p ) 00 (Z)?pr,p Cd(J) = k(z)'k+j'(z)dz (6.10) oo +,, d) = | ^' kPz (6.11) Ci, p,d(J) = __r ek (z)Ck+j,(Z)dz (6.12) Or P (J)- ' z'00 k kz Equations (6.7) and (6.8) can be written in a compact form [D] = [[E] (6.13) 135

For geometries with dielectrics varying from air (Er=1) to Si (Er=2.56), it was observed that the above summations can be truncated for /=/oo o =6 when only one resolution of wavelets (rnax=O) was used. Also, the integrals can be approximated by finite summations of 10 cells on each side of the central cell (k-cell). Due to the orthogonality relationhip between the scaling and the wavelet functions, for uniform dielectrics (constant E throughout the integration domain) these integrals are simplified to cod- = bi,o, cd = c 0 and p(i) = ir p io and [] becomes a diagonal matrix. For structures containing dielectric discontinuities, none of these integrals have a zero value. In this case, the whole geometry has to be preprocessed before the initialization of the time loop and coefficients codc, cpc, ckr,d have to be assigned to any cell (m,n) and included in the matrix [l]. For each cell the amplitude of these coefficients is compared to the amplitude of the self-term c^d(i). If all coefficients are below a threshold (usually < 0.1%), they are set to zero and this cell is exempted from the following inversion, otherwise it is included in a new submatrix. This submatrix has significantly smaller dimensions than [e] (usually < 10%) and contains only cells close to dielectric discontinuities. The inverse of this matrix is used for the calculation of the E from the D values for each time step. The inversion takes place only once, thus it adds only negligible computational overhead to the algorithm. 6.2.5 Total Field Calculation Due to the nature of the Battle-Lemarie expansion functions, the total field is a summation of the contributions from the non-localized scaling and wavelet functions. For example, the total electric field E(x,, zo, to) with (i - 1) Ax < xo < i Ax, (j -1/2)Az2 < zo < (j + 1/2) Az and (k - 1/2) At < to < (k + 1/2) At is calculated in the same way with 136

Similar accuracy can be observed when the scaling function at the excitation cell is set equal to the value of the excitation function EF(t). No superposition is used and the field scaling and wavelet values elsewhere are given by the MRTD equations. The wavelet coefficients are excited through the coupling of the discretized MRTD equations. In the following numerical simulations the latter excitation technique was employed since it adds significantly smaller computational overhead. 6.2.4 The Modelling of Dielectrics Starting from the constitutive relationship D = cE for the total electric field at one mesh point and sampling the scaling and wavelet components with a similar way to (A) we reach the following equations for D. lx rma 2 —1 /Or,p kDf = E C1(j')kEj+j + E E E C, d()k (6.7) j=- r=O p=O j'= —l,, and CO rmax 2r-1 C~r,p kDr' E c'' (j )kEj+j + c (')k E+., (6.8) j,=- r — p=- j'=-(r,p where Cd(j') = | >k(Z)bk+j(z)dz (6.9) -oo C'p ) = Ck(z),p (z)dz (6.10) oo00 C i,, pid( ) k= '' (z)Ck+j (z)dz (6.11) -00 C+,,p = J _',P(z)tk'Pj(z)dz (6.12) Equations (6.7) and (6.8) can be written in a compact form [D] = [[] (6.13) 137

1/10000 of the peak of the excitation time-domain function). This comparison is repeated for each time-step (time adaptiveness). All components below this threshold are eliminated from the subsequent calculations. Thl is is the simplest thresholding algorithm. It doesn't add any significant overhead in execution time (usually < 10%), but it offers only a moderate (pessimistic) economy in memory (factor close to 2). Also, this algorithm allows for the dynamic memory allocation in its programming implementation by using the appropriate programming languages (e.g. C). The principles of the dynamically changing time- and space-adaptive grid are demonstrated in (Fig.6.1). A pulse is propagating from the left to the right in a partially filled parallel plate waveguide. For t=0, the wavelets are localized at the excitation area. They follow the propagating pulse (t before the incidence to the dielectric interface) creating a moving dense subgrid. After the pulse has been split in reflected and transmitted pulses, the wavelets increase the grid resolution only around these pulses. Elsewhere the wavelet components have negligible values and are ignored. 6.3 Applications of 2D-MRTD 6.3.1 Air-Filled Parallel Plate Waveguide The 2D-MRTD scheme is applied to the analysis of an air-filled parallel-plate waveguide with width 4.8 mm. The front and back open planes are terminated with a PML region of 22 cells and aa calculated for designated Rma,=l.e-7. The waveguide is excited with a Gabor function 0-30GHz along a vertical line next to the PML region. A Gabor excitation is propagating for a distance of 2,000 mm. For the analysis based on Yee's FDTD scheme, a 4 x 1120 mesh is used resulting in a total number of 4480 grid points. When the structure is analyzed with the 2D-MRTD scheme, a mesh 2 x 160 (320 grid points) is 138

chosen (dx = 0.24Ao, d-Z = 0.42Ao for f = 30GHz). This size is based on the number of the scaling functions. since the wavelets are used only when and where necessary. The time discretization interval is selected to be identical for both schemes and equal to the 1/4 of the 2D-MRTD maximum At. For the analysis we use 3,000 time-steps. The longitudinal distance is chosen such that no reflections would appear before the Gabor function is complete and the schemes dispersion performance can be evaluated. The normal electric field EI is probed at three different locations and the results are plotted in (Fig.6.2) showing only minimal dispersion. The capability of the MRTD technique to provide space and time adaptive gridding is verified by thresholding the wavelet components to the maximum of the 0.01% of the value of the scaling function at the same cell for each time-step and the absolute threshold of 105. The use of the absolute threshold enhances the efficiency of the algorithm for very small field values. It has been observed that the accuracy by using only a small number of wavelets is equal to what would be achieved if wavelets were used everywhere. Though this number is varying in time, its maximum value is 122 out of a total of 320 to the z-direction (economy in memory by a factor of 5.1 to the z-direction) as it can be observed from (Fig.6.3). In addition, execution time is reduced by a factor 3-4. The principle of the space-adaptive grid is exhibited at (Fig.6.4) which represents the Ex field distribution at t=1000 time steps. The wavelets have a significant value only at the region of the propagating pulse, thus creating a locally dense grid. 6.3.2 Parallel-Plate Partially Filled Waveguide The second structure analyzed with the MRTD algorithm was the geometry of (Fig.6.5). A Gabor pulse 0-30 GHz is propagating from the left (air region) to the right (region 139

with ~r = 2.56). PML regions of 16 cells with aEa calculated for designated Rmaz=leterminate the grid and wavelets of 0-resolution are used in the longitudinal direction. The reflection coefficient S11 is calculated by separating the incident and the reflected part of the probed field and taking the Fourier transform of their ratio (Fig.6.6). Similar accuracy can be obtained for a 4x640 FDTD grid and a 2x80 MRTD grid with relative threshold 0.01% and absolute threshold 10-3. The maximum number of wavelets used is 36 (Fig.6.7) offering an economy in memory by a factor of 6.53 in comparison to the FDTD simulations for the longitudinal (z) direction. The results for 5 GHz (TEM propagation ) are validated by comparison to the theoretical value obtained applying ideal transmission line theory [62] and are plotted at Table (6.1). The time- and space-adaptive character of the gridding is exploited in (Fig.6.8) which show that the wavelets follow the reflected and the transmitted pulses after the incidence to the dielectric interface and have negligible values elsewhere. The location and the number of the wavelet coefficients with significant values are different for each time-step, something that creates a dense mesh in regions of strong field variations, while maintaining a much coarser mesh for the other cells. 6.3.3 Parallel-Plate Five-Stage Filter The last structure analyzed with the variable grid is the 5-stage filter of (Fig.6.9). A gabor function 0-4GHz is propagating from the left to the right. The input and output stages have er = 12.5 and the intermediate stages have Er = 50.5 (stages with d- = d5 = 0.5mm and d3 = 2mm ) and Er = 1. (stages with d2 = d4 = 14.mm). The total length to the longitudinal direction is 600 mm and to the vertical 4.8 mm. PML regions of 16 cells with aE ax calculated for designated Rmax = l.e-7 terminate the grid and wavelets of 0-resolution 140

are used to the longitudinal direction. The structure is analyzed by using an FDTD grid of 8x1600 cells, a scaling only MRTD grid of 2x400 cells and an adaptive (scaling+wavelets) MRTD grid of 2x200 cells. The relative threshold has the value of 0.01% and the absolute threshold equals to 10-4. The maximum number of wavelets required during the 3,000 time steps of the simulation is 102 (Fig.6.10), offering an economy by 37.25% in comparison to the scaling only grid and by a factor of 6.37 in comparison to the FI)TD scheme for the direction of wavelet expansion (z-direction). The accuracy in the calculation of the Sparameters is similar for all three schemes as it can be observed from (Fig.6.11). Again, the time- and space- adaptive character of the proposed gridding is demonstrated in (Fig.6.12) with the E, field space distribution for t=1000 time steps. 6.4 Conclusion A dynamically changing space- and time- adaptive meshing algorithm based on a multiresolution time-domain scheme in two dimensions and on absolute and relative thresholding of the wavelet values has been proposed and has been applied to the numerical analysis of various nonohomogeneous waveguide geometries. A mathematically correct way of dielectric modeling has been presented and evaluated. The field distributions and the S parameters have been calculated and verified by comparison to reference data. In comparison to Yee's conventional FDTD scheme, the proposed scheme offers memory savings by a factor of 5-6 per dimension maintaining a similar accuracy. The above algorithm can be effectively extended to three-dimension problems. 141

S11 (i2) Relative error Analyt. Value [62] 0.4298 0.0% 4x640 FDTD 0.4283 -0.3% 2x80 MRTD 0.4360 +1.4% T I m e Table 6.1: 511 calculated by 2D-MRTD " I N'I i I I I I i n i i ' I 1 ' i l i I incidence i...... ' I ft r -— W --- —" h —, I, '"|e r r~ i i incid enc Space --—. 1* Figure 6.1: Time- and Space- adaptive grid. 142

1 I T I I - I 0.8 - Ex(1,70) - Ex(1,100) - - Ex(1,130) 0.6 II I I I I I I I I II II I.I 11 I i I I I I I I I I Ij II II. I i 11, - I I I I 0.4 F au 0.2 A rt __ __________Z-_______-________________________ — 0 -' II Ii \' I -';'i i -0.2 - I '' II II --- I I i II -0.4 - I0.61 0 500 1000 1500 Time-Step 2000 2500 3000 Figure 6.2: Normal E-field (Time-Domain). V) 4 -co I *0 f1t 1500 Time-Step Figure 6.3: Non-zero Wavelets' Number 143

I'l x w 0! i I I I, i.5 -I0 I I-I I I I -n -V0 O 20 40 60 80 z-position 100 120 140 160 0.015 > 0.01 w 0.005 I,...._ 1- I I - I AL I iI I I I i 0 20 40 60 80 100 120 140 160 80 z-position 160 Figure 6.4: Adaptive Grid Demonstration (t=1000 steps). — er=.O e0r.56 I I I II 4.8 mmi 44111 Figure 6.5: Parallel-Plate Partially Filled Waveguide. 144

0.4 I i i i - MRTD 0.35 - - FDTD 0.3 0.25 - 0.2 -0.15 - 0.1 -0.05 0 I I I I I I I I 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Freq [GHz] Figure 6.6: S-Parameters of the Waveguide. 200 300 400 -500 600 700 800 Time-Step Figure 6.7: Non-zero Wavelets' Number. 145 900 1000

4-.x ULJ 1 — 1 -1 I I I I I I 0 10 20 30 40 50 60 70 8C z-position D8 I I II I I I 06- n D4 - ' )2 -I I -tL- _I _ _ - iIiJ O.C O0.( xO. -0.( O w~ V~ 0 10 20 30 1.5 40 z-position 50 60 70 80. I I I I I I I.. I I... I. I. I I! I I I I I I I I I I I I O0 0........ 6.A............... i 10 20 30 40 z-position 50 60 70 80 Figure 6.8: Adaptive Grid Demonstration (t=1000 steps). h 4.-. j - _.d i dl d2 d3 d4 d5 Figure 6.9: Parallel-Plate Five-Stage Filter. 146

P. r ~ I T 1 VI -- 7- I - -5 IV,-15 CYJ CU) -20 -25 I T I I I -- --- - --- - I I I I I I, 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Freq[GHz] I -~ MRTD 2x40 Sc (gr:800) - - MRTD 2x20 w t=0. 1 % (gr:482) - -FDTD 8x1 600 (gr: 12800) I -.an, I I I I 0 0.5 1 1.5 2 2.5 Freq[GHzJ 3 3.5 4 4.5 5 Figure 6.10: S-Parameters of the Filter. Ci) U1) 0 - 500 1000 1500 2000 2500 Figure 6.11:- Non-zero vwRvelets' Number. 3000 14 7

1 I I I I I I I I I I I 0 x w -11 III I I I I I I - I 0 20 40 60 80 100 I z-position 120 140 160 180 200 0.1 I I I I I I I I I I I I I I I I I I x w IIIIIIBIIIIIIJII[II~LhUliJLjf6 k - 120 140 160 180 200 80 100 z-position 1 (il v- - - - - - 0 20 40 60 80 100 120 140 160 180 200 z-position Figure 6.12: Adaptive Grid Demonstration (t=1000 steps). 148

CHAPTER 7 Time Adaptive Time-Domain Techniques for the Design of Microwave Circuits 7.1 Introduction In the previous Chapters, MRTD schemes based on cubic spline Battle-Lemarie scaling and wavelet functions have been successfully applied to the simulation of 2D and 3D open and shielded problems [26, 53, 65, 61]. The functions of this family do not have compact support, thus the MRTD schemes have to be truncated with respect to space. Localized boundary conditions (PECs, PMCs etc.) and material properties are modelled by use of the image principle and of matrix equations respectively. However, this disadvantage is offset by the low-pass (scaling) and band-pass (wavelets) characteristics in spectral domain, allowing for an a priori estimate of the number of resolution levels necessary for a correct field modelling. In addition, the evaluation of the moment method integrals during the discretization of Maxwell's PDEs is simplified due to the existence of closed form expressions in spectral domain and simple representations in space domain. Dispersion analysis of this MRTD scheme shows the capability of excellent accuracy with up to 2 points/wavelength (Nyquist Limit). However, specific circuit problems may require the use of functions with 149

compact support. Epsecially in the approximation of time derivatives, the use of entire domain expansion basis would require very high memory resources for the storage of the field values everywhere on the grid for the whole or a large fraction of the simulation time. This problem does not exist in the approximation of the spatial derivatives since the field values on the neighboring spatial grid points have to calculated and stored no matter what expansion basis are used. For that reason, Haar basis functions have been utilized and have led to [66]. As an extension to this approach, intervalic wavelets (Fig.7.1) may be incorporated into the solution of SPICE-type circuits, especially those containing active elements (PN Diodes,...). Results from this new finite-domain expansion basis will be presented in this Chapter. 7.2 Time Adaptive MRTD Scheme The major advantage of the use of Mutiresolution analysis to time domain is the capability to develop time and space adaptive schemes. This is due to the property of the wavelet expansion functions to interact weakly and allow for a spatial sparsity that may vary with time through a thresholding process. The adaptive character of this technique is extremely important for the accurate modelling of sharp field variations of the type encountered in beam focusing in nonlinear optics, etc. The use of the principles of the multiresolution analysis for adaptive grid computations for PDEs has been suggested by Perrier and Basdevant [67]. To understand the fundamental steps of such an adaptive scheme for Maxwell's hyperbolic system, let's consider Maxwell's equations in 2D (1 for space and 1 for time): a9f 0 az )~1T au= A [i = -( i X = (E(z,t),H(zt))T, (7.1) -After manipulation, the above equation can be written as After manipulation, the above equation can be written as 150

ETtDt TtD Muf = = O ZtDZ uZtDt where Zh, Th are half shift operators for space and time coordinates z,t and Zh, Tt are their Hermitian conjugates. Dt, Dz are difference operators given by: 1 8 9 8 9 Dt = ( a#t (i)T-i+ a at(i)T- ), D = -z( ~E aoz(i)Z-i+ E. z(i)Z-i) i=-9 =-9 i=-9 'i=-9 (7.2) where ao, aV, are the coefficients associated with the scalar and the wavelet functions respectively. At each time step we keep both the wavelet field values that are larger than a given threshold as well as the adjacent values. An adjacent wavelet field value is defined on the basis of the wavelet resolution level(s) incorporated in the solution. Recently, an efficient space/time adaptive meshing procedure was proposed [68] for Battle-Lemarie expansion functions. In this chapter, intervalic O-order wavelets are used for the expansion of the fields and a simple thresholding procedure is employed. The adaptive mesh is applied to a variety of circuit problems and results are discussed in the next section. 7.3 Applications in SPICE problems For simplicity, the 1D MRTD scheme will be derived. It can be extended to 2D and 3D in a straightforward way. In addition, only the O-resolution of wavelets is enhanced. The Voltage and the Current are displaced by half step in both time- and space-domains (Yee cell formulation) and are expanded in a summation of scaling functions in space and scaling (X) and wavelet (y0) components in time v(,t) = E E O^^W+^^W) <^) 00 00 00 00 V(z t)= SE E (i~v+,i(t) + v;o-,o,,(t)) 5(~) I(z, (i-o.5Im)_o0.5i-.5(t) + i-.0 ImO0.,0,-0.5(t)) 4m-0.5(Z) (7.3) 7=-oo00 t=-00 151

where,i(t) = p(t/At - i) and vo,-(t) = 4'o(t/At - i) represent the 0-order intervalic scaling and 0-resolution wavelet functions. The conventional notation klVm is used for the voltage component at time t = kAt and z = mAz, where At and Az are the time-step and the spatial cell size respectively. The notation for the current I is similar. Due to the finite-domain nature of the expansion basis, the Hard Boundary conditions (Open/Short Circuit) can be easily modeled. If a Short Circuit exists at the z = mAz, then both scaling and wavelet voltage coefficients for the m - cell must be set to zero for each time-step k. kVm= = kVm~ = 0, k = 0,1,2,... (7.4) Similarly, an Open Circuit at z = (m - 0.5)Az can be modeled by applying the conditions k-0.5In-0.5 = k-0.5/m-0.5 = 0, k = 0, 1, 2,... (7.5) The alternating nature of the 0-resolution wavelet function guarantees the double timedomain resolution of the MRTD scheme. Assuming that the voltage scaling and wavelet coefficients at m = Az for a specific time-step k, two values can be defined for the time span [(k - 0.5)At, (k + 0.5)At] of this time-step kVtoa = kV + kV, t [(k- 0.5)At, kAt] (7.6) kVt~tl2 = kV, -kVO t E [kAt, (k + 0.5)At]. (7.7) 7.3.1 Distributed Elements Lossless Line The ideal transmission line (Fig.7.2) equations are given by dV dI d = -Ldis dz dt dI dV dz -C dt (7.8) d- ~ 152

where Ldis,Cdis are the distributed inductance and capacitance of the line. Inserting the expansions of Eq.(7.3) and applying the Method of Moments, the following MIRTD equations are derived At (k+l V - kV) = - (k+o.51+0. - k+O.5i-0o.5) (7.9) (k+ -kV ) = - (k+o.I+ o.5 - k+) (7.10) L (k+o.5-0.5 k-0 51-0.5) = - (kV - kV-1) (7.11) Li (k+o05Io 0.5 — 0.5k~ —0.5) = -i (k+( -kV -1) (7.12) It can be observed that Eqs.(7.9) and (7.11) updating the scaling coefficients only are independent of the Eqs.(7.10) and (7.12) updating the wavelet coefficients. To create an efficient time adaptive algorithm, all four equations must be coupled. An efficient way is to apply the excitation in a physically correct manner. If the excitation has the timedependence g(t) at the location z = meAz, then the scaling and wavelet coefficients for this cell have to take the values r (k+0.5)At kVtO = g(t) Ok (t) dt J(k-0.5)At r(k+0.5)At kVA = / g(t) 'Ok) dt. (7.13) Je(k-0.5)At To validate this approach, the MRTD algorithm was applied to the simulation of a lossless transmission line with (Ldis, Cdis) = (20nH/m, 3nF/m) for a Gabor excitation and timestep dt = dtmae/1.01. (Fig.7.3) which displays the Voltage Scaling and Wavelet Coefficients evolution at z = 200zz for the first 800 time-steps of the simulation, shows that the wavelet coefficients have the correct shape (significant values only at areas with significant scaling function values) and are close to the 12% of the respective scaling functions. (Fig.7.4) which compares the total voltage value at z = 200Az calculated by FDTD (Sc.ONLY) and MRTD (Sc.+Wav.) for the time-steps 357-362 demonstrates the ability of this MRTD 153

scheme to double the conventional FDTD resolution in the time-domain by providing two values for each time-step. The fact that the wavelet coefficients take significant values only for a small number of time-steps allows for their thresholding by comparing them to a combination of relative to the respective value of the scaling coefficient (5.e-4) and absolute (l.e-6) thresholds. (Fig.7.5) proves that up to 60% of the maximum number of wavelet coefficients are necessary for an accurate simulation, offering an extra economy in memory by a factor of 20%. Lossy Line The lossy transmission line (Fig.7.6) equations are derived by the ideal transmission line equations (Eq.(7.8)) adding the Conductor Loss Rdis and the Dielectric Loss Gdis dV dl -d -RdisI - Ldisd dz dt dI dV dz -GdisV - Cdis dt (7.14) Following a procedure similar to the previous section, the following MRTD equations are derived k+lVm = k+l Vo~ = k+0.5Im-0.5 = rO o = k+0.5rm-0.5 - (C1 - C2) + C2 At (k+O.5sm+o. - k-0.5 m-0.5) C2 At (k+o.5I+o. -0.5 I-05+ C) At (k+ o.5I0+o0.5 k-0.5k -0.5) 22m.+~ -o.5+o - k-oo-0.5) 1 2 (2C2 2) + (7.16) j 2V- 2o -1" (7.25) 2 k + 04 1 1 (C2 - ( -+AV C At (o kV ) (3 —04) + 0-) k-o0.5I-o.5 - 2 k-0.5I-0.5 (7.17) 2 (7.3) 3 5 3 C4 C (C3 ) + c44 2 o 7.8 -2... k-o. 5!_o0. 5 - 2 ~ -o.22_.5 (7.17) C3 C2 C3 _ C4)2 + C2154 154

with C1 = CdsAz, C2 = 0.5 Gdi2Az At, C3 = LdisAz, C4 = 0.5 RdisAZ At. For this type of transmission line, the equations giving the scaling and wavelet coefficients for voltage and current are coupled. Nevertheless, the condition (7.13) has to be applied in order to satisfy the physical boundary condition at the excitation cell(s). It has to be noted, that Eqs.(7.15)-(7.18) can be used only for lossy lines with low to medium Loss Coefficients. The threshold C2 < 4C1 for Gdis (C4 < 4C3 for Rdis) gave satisfactory results for all simulations. For higher loss coefficients, the Loss can be modeled in an exponential way similar to [61]. For example, for large values of Rdis (C4 > C3), Eqs.(7.17)-(7.18) have to be replaced by the following uncoupled expressions -RdisAt -O.5Rdi,At At k+O.5. e L —0.5I. - e Ld ( kVm ) (7.19) C3 k+o.-do.s=eRlt -o.5Rdi'At At ' ^o IL = r^to dis ( eo t (7.20) k+0-5 e Ld, k-o0.5m-0.5 e - C3 Using this procedure, a termination layer similar to the FDTD widely used PML layer can be easily modeled. The Rdis, Gdis should have a spatial parabolic distribution with very high maximum value and they should satisfy the condition Gdis = RdisLdis/'Cdis for each cell of the layer. In this way, one matched transmission line can be simulated by choosing the appropriate Rdis, Gdis that satisfy the specified numerical reflection coefficient (usually smaller than -80dB). For validation purposes, the propagation of a Gabor pulse along a lossy line with Rdis = 5Q/m has been simulated and the scaling and wavelet voltage coefficients have been probed at the positions z = 140Az and z = 160Az. Data for the first 200 time-steps (At = 2At/3) have been plotted in (Fig.7.7). The maximum value of the wavelet coefficients 155

(approximately 7% of the respective scaling coefficient) is smaller than that of the lossless line. By applying a thresholding procedure using an absolute threshold of 10-6 and a relative threshold of 5e - 4, an extra economy of 29% is achieved, since only 60' of the voltage and 25% of the current wavelet coefficients take values above the thresholds throughout the simulation time (Fig.7.8). 7.3.2 Lumped Elements Passive Elements Lumped Passive Elements such as Capacitors, Inductors and Resistors can be modeled in a similar way with the Distributed ones by numerically distributing them along one cell. For example, if one lumped Capacitor Cum is located at z = mZz along a lossy line with (Rdis, Gdis, Ldis, Cdis), the voltage coefficients k+1V,, k+lV will still be given by Eqs.(7.15)-(7.16). The only difference is that the constant C1 will have the new value C1 = CtotAz with Cium Ctot = Cdist + aCz (7.21) PN-Diode To model lumped active elements such as a PN-diode, their nonlinear equation has to be discretized after inserting the voltage and current expansions. The MRTD equations are not linear and require the use of numerical solvers for nonlinear systems. The combined Newton-Raphson/Bisection solver has provided stable solutions for PN-diode simulations with IO < i.e - 10A,though sometimes diverges for larger values. The voltage scaling and wavelet coefficients for the diode cell are updated by inserting the voltage and current 156

expansion in the equation IDIODE(V) = IO (eqV/kT - 1) (7.22) adding the diode capacitance Cj to the Cdis and applying the moments method, thus giving the nonlinear system for a diode positioned in parallel (C5 + Cdis) kV, + C5 kVm~ + (C5 - Cdis) k-lV 0 At - Cs k-lV~ +- - k-0.5. - k.5I-o.5) C5 k-le0 + ~-o.m+0.5 + 0.5 At CI(ekT/q Io (k-iV-V-iv0O ) + ekT/q Io (kV +kV~)) = 0 (7.23) -C5 kVm~ - (C5- Cdis) kVm~ + Cs k-11Vr At f -(C5 + Clum) k-1Vm + (k-0.5Im+.5 - k-0.5-0.5) + 0.5 At Cj(ekTI I0 (k-lVmPhi - l- k/) _ ekT/q Io (kVm+kV)) 0 (7.24) with Cs = 0.5 At Gum. (7.25) To validate the algorithm, the rectifier topology of (Fig.7.9) is analyzed using FDTD (Scaling Only) with At = Atmax/4.4 and MRTD (Scaling+wavelets) with double time-step At = Atma,/2.2. A lossless line with (Ldis,Cdis) = (20n1/m,3nF/m) and a PN-Diode with Io = 3pA are used in the simulation. The probed total voltage is plotted in Fig.(10) and the agreement is very good. The use of an absolute threshold of 10-6 and a relative threshold of 5e - 4 offers an extra economy of 35% for the MRTD algorithm. 157

7.4 Conclusion A Time Adaptive Time-Domain Technique based on intervalic wavelets has been proposed and applied to various types of circuits problems with active and passive lumped and distributed elements. This scheme exhibits significant savings in execution time and memory requirements while maintaining a similar accuracy with the conventional FDTD technique. 158

A <k 1/Dt 1/Dt I I t t m - T 0 m — (k-0.5) Dt (k+0.5) Dt (k-0.5) Dt (k+0.5) I Dt I -1/Dt Figure 7.1: 0-Order Intervalic Function Basis. L dis % I QQ IT dis V z I Figure 7.2: Ideal (Lossless) Transmission Line. 159

0> D 0 100 200 300 400 500 600 700 800 Time-Step (=dtmax/1.01) Figure 7.3: Voltage Coefficients. Jl e.. im..I.. 0.48 0.46[ 0.44[ 0.42 0 0) Ad 0.4 0 *: // X / -\MRTD(Sc+Wav) // \\ /. --- —|. \ / - - FDTD (Sc) \ " 3575 3 0.38 0.36 1 0.34 F 0.32.3 L I I I I I I. I I I 357 357.5 358 358.5 359 359.5 360 360.5 361 361.5 362 Time-Step Figure 7.4: Comparison MRTD-FDTD. 160

Percentage of Non-zero Wavelets (x100%) 1. 0.5 00. Volt. Cur. Thr= le-6 (Eco: 20%) Time-Step Figure 7.5: Fraction of Wavelets above Threshold. Rdis Ldis V v7 Vt L 'E 'QI >% E. Gdis Cdis Cis ] z U Figure 7.6: Lossy Transmission Line. 161

0) 0 0 20 40 60 80 100 120 140 160 180 Time-Step 200 Figure 7.7: Voltage Coefficients. Percentage of Non-zero Wavelets (xl 00%) 1.0 0.5 Volt. Cur. Thr= 1.e-6 (Eco: 29%) 0.0 M: 0 Time-Step Figure 7.8: Fraction of Wavelets above Threshold. 162

dis dis Gdis V I z Cdi dis Figure 7.9: Rectifier Geometry. CD Figure 7.10: Comparison MRTD-FDTD. 163

CHAPTER 8 Summary of Achievements-Future Work The goal of this dissertation was to develop efficient time-domain numerical techniques for the analysis and design of microwave circuits. To achieve it, the principles of Multiresolution Analysis were employed and novel time-domain schemes based on field expansions in scaling and wavelet functions were derived (MRTD). These new schemes offer memory savings by a factor of 5 per dimension and execution time savings by a factor of 4-5 while maintaining a similar accuracy to the conventional FDTD technique. They also feature an inherent capability of a dynamically changing space- and time- adaptive gridding algorithm by thresholding the wavelet values in a mathematically correct way. In addition, they offer the unique opportunity of a multi-point field representation per cell. Results from 2D and 2.5D simulations prove the validity and the versatility of the MRTD schemes. Nevertheless, future work on the MRTD schemes should include the study of the effects of the enhancement of arbitrary wavelet resolutions for schemes based on entire-domain and finite-domain expansion basis. In addition, different functions should be tested and their performance for different geometries should be evaluated. As it was displayed in CH.6,7 dynamic gridding is achieved by a simple thresholding algorithm. It might be useful to perform a systematic study on the relationship between different thresholding schemes and 164

the error of calculations in order to guarantee a predefined accuracy and justify an a-priori choice of a relative and an absolute threshold. 165

BIBLIOGRAPHY 166

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REPORT DOCUMENTATION PAGE Form Noe1 OMB NO. 0704-0188 Public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comment regarding this burden estimates or any other aspect of this collection of information, including suggestions for reducing this burden to Washington Headquarters Services. Directorate for information Operations and Reports. 1215 Jefferson nvir~ Hinhwav i RillitA 1 ^04dA Arlinntnn VA: d09909-AI nnr~ tn th~ r^fHl^a nf K >nnnamant antl Rlf Qir~t Panarwnrk RAi(iiftion PrniAQQt A7n <nIl< Wnchinntnn rsr':rnren - aV, ID I,11 wC'lnT, u.v,*L v-,c ' rl,,,,tJrll V*M, ercew-&.3, an1lU iltJ un I vianagmit arlu uuLIU, '*JIIIL.. *VUUU1 tuI v —v Iol, Yvasningionlul ClU tqJ. 1. AGENCY USE ONLY (Leave blank) 2. REPORT DATE 13. REPORT TYPE AND DATES COVERED 5/1/98 Final Report, May 1995-April 1998 4. TITLE AND SUBTITLE 5. FUNDING NUMBERS Micropackaging for Mm-Wave Circuits DAAH04-95-1-0321 P-33939-EL 6. AUTHOR(S) Linda P.B. Katehi 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) 8. PERFORMING ORGANIZATION University of Michigan REPORT NUMBER Dept. of Electrical Engineering and Computer Science 033259-1-F 1301 Beal Avenue Ann Arbor, Michigan 48109-2122 9. SPONSORING / MONITORING AGENCY NAMES(S) AND ADDRESS(ES) 10. SPONSORING/ MONITORING U.S. Army Research Office AGENCY REPORT NUMBER ATTN: Dr. James F. Harvey Electronics Division P.O. Box 12211 Research Triangle Park, NC 27709-2211 11. SUPLEMENTARY NOTES The views, opinions and/or findings contained in this report are those of the author(s) and should not be construed as an official Department of the Army position, policy or decision, unless so designated by other documentation. 12a. DISTRIBUTION / AVAILABILITY STATEMENT 12b. DISTRIBUTION CODE Approved for public release; distribution unlimited. 13. ABSTRACT (Maximum 200 words) Complex antenna and circuit problems including their package on wafer require very intensive calculations due to the need toaccurately simulate the underlying high-frequency effects and account for all the parasitic mechanisms. As part of this project, we have successfully applied a novel frequency domain scheme recently developed at the University of Michigan that allows for the very successful and computationally efficient solution of complex antenna problems. This technique has been applied to a variety of circuit and antenna problems and has demonstrated the capability to provide accurate solutions in much more efficient ways than in the conventional techniques. The whole idea in this approach is the use of wavelets in the expansion of the unknown functions. The use of wavelets allows for the computation of the values of the derivatives of the unknown field quantities in addition to the average values of the field. This allows for the development of novel space-adaptive schemes with unique capabilities. i 14. SUBJECT TERMS 15. NUMBER IF PAGES 16. PRICE CODE 17. SECURITY CLASSIFICATION 18. SECURITY CLASSIFICATION 19. SECURITY CLASSIFICATION 20. LIMITATION OF ABSTRACT OR REPORT OF THIS PAGE OF ABSTRACT UNCLASSIFIED UNCLASSIFIED UNCLASSIFIED UL NSN 7540-01-280-5500 Standard Form 298 (rev. 2-89) Prescribod by ANSI Sid 239-18 29,11- 1 02