The Application of the Finite-Difference Time-Domain Method to EMC Analysis

Transcription

1 The Application of the Finite-Difference Time-Domain Method to EMC Analysis Stephen D Gedney University of Kentucky Department of Electrical Engineering Lexington, KY &s&r& The purpose of this paper is to provide a tutorial of the FDTD method and its application to the full-wave analysis of practical EMC problems The FDTD method is a versatile technique and a powerfbl tool that can allow for the efficient and accurate simulation of electromagnetic interaction and radiation for EMC analysis The presentation will begin with an introduction to the FDTD method based on the traditional Yee-algorithm For the analysis of radiating structures, various absorbing boundary conditions, including the novel Perfectly Matched Layer (PML) absorbing boundary condition, and their relative performances will be described Techniques of computing radiated fields from the time-dependent fields will also be discussed Included in this discussion will be the limitations of modeling complex structures using regular lattice grids, as well as the benefits of advanced FDTD algorithms that utilize irregular and unstructured grids Acceleration techniques and the efficient implementation of the FDTD method on high performance computers will also be of concern Finally, a number of practical EMC problems analyzed using the FDTD method will be presented Introduction Over the last decade, the Finite-Difference Time- Domain (FDTD) method has become one of the most widely used computational techniques for the full wave analysis of electromagnetic phenomena Its popularity can be accredited to the simplicity of the algorithm, while providing a robust and accurate analysis of electromagnetic fields [l] Since it is based on a volume discretization, three-dimensional material inhomogeneities are inherently modeled, allowing for the analysis of planar stratified medium, non-planar devices, devices on finite dielectric substrates and ground planes, and lossy conductors and dielectrics Lumped or hybrid elements including passive loads and active elements [2] can also be incorporated into the discrete Maxwell s equations Other recent advances in absorbing boundary conditions have provided for the accurate broad-band analysis of printed circuits and antennas in unbounded medium [ 11 Finally, both broad-band analysis and narrow-band characteristics of devices can be obtained with a single FDTD simulation In the last decade, the introduction of inexpensive highspeed memory, powerful vector processors, high-speed RISC processors, and the maturation of high-performance parallel computers has provided the economical and powerful computational resources that are necessary to perform the analysis of practical engineering applications via the FDTD method [3] Correspondingly, the FDTD method has received extraordinary popularity over the last 5 years If the FDTD algorithm is to be a truly viable design tool in industry, it will be necessary to obtain reasonable computational speeds on inexpensive workstations This will require advanced algorithms and acceleration techniques, as well as affordable highperformance computing THE FDTD ALGOFUTHM The FDTD algorithm is traditionally based central difference approximations of the temporal and spatial derivatives of Maxwell s curl equations: (1) By staggering the discrete electric and magnetic fields in space and time, this leads to a second order accurate approximation, from which an explicit update scheme can be derived For example, the explicit update expression for Er is: E, i++c (2) Given the initial condition for the fields, this recursive update scheme can be used to explicitly calculate the vector fields in the entire space in discrete time The explicit field update expression is stable, provided, the discrete time step satisfies the stability criterion At<llc (1/dr)*+(llA~Q~+(l/Az)~ Due to the explicit nature of the algorithm, the FDTD method is quite efficient computationally An orthogonal lattice with dimensions Nx x NV x NZ requires N = 6 NXh$N degrees of freedom to represent the three vector components of the electric and magnetic fields throughout the three-dimensional space The total memory required to store the fields and the update coefficients thus increases as O(N) The number of floating point operations per time iteration will also increase as O(N), while the total number of time iterations roughly increases 1 as O(p) Therefore, the number of floating point operations of the FDTD algorithm will increase as 0( Nj) Due to the structure of the explicit update O /96/$ EEE 117

2 operators, the FDTD algorithm is highly vectorizable, and scalable on multiprocessor computers [3] THEPERFECTLYMATCHEDLAYER BOUNDARYCONDITION One of the greatest challenges of applying the FDTD technique to open radiation problems has been the development of accurate and computationally efficient absorbing boundary conditions Very recently, the perfectly matched layer (PML) boundary condition, introduced by J P Berenger [4], has been used as a means to truncate FDTD lattices The PML boundary condition Fig 1-60 g -70 g -80 w g -90 a :[ : i I/ i I i ; j ; i i ; j i ; ; i 1 t i i i i i i i 1 Reflection error due to a 10 cell thick PML termination of a 50 Q microstrip line printed on a duroid substrate omax = 90 (m = 4) is a ldssy material boundary layer that is perfectly matched to the solution space This has been achieved using field splitting [4,5], coordinate stretching [6] techniques, or as a uniaxial medium [7,8] Subsequently, it has been shown that an arbitrarily polarized wave incident on the PML medium is perfectly transmitted, has the same phase velocity and characteristic wave impedance as the incident wave, while attenuating rapidly along the normal axis [4-6,8] It has been found that the PML medium can result in reflection errors as minute as -80 db to -100 db [4-6,8] over an extremely broad band, and can be applied to homogeneous media [4-&l, inhomogeneous media [5,8], as well as lossy and dispersive media [8] To demonstrate the effectiveness of the PML medium, Fig 1 illustrates the reflection loss due to a 50 R microstrip line printed on a duroid substrate terminating into a PML boundary The PML is 10 FDTD cells thick and had a fourth-order spatial variation along the normal axis (relative to the PML interface) In this example, better than -90 db reflection error is seen over a very large band, including down to DC As an example of the application of the PML, Fig 2 illustrates the reflection loss of a microstrip fed printed patch antenna For this simulation, a 10 cell PML layer was placed 3 cells and 10 cells from the edge of the patch antenna The difference in the result is negligible, demonstrating that the PML can be placed in close proximity of any discontinuities in the The PML boundary condition has made a significant impact on time-dependent PDE-based solutions to Maxwell s equations, specifically, since: 1) the extremities of the mesh can be placed within a few cells of a source or discontinuity without suffering spurious boundary reflection error - thus greatly reducing mesh dimensions, 2) accurate low frequency as well as high-frequency analysis of electronic packages (lossy and lossless) can now be performed in a single simulation - this also has applicability for areas such as electromagnetic compatibility (EMC) or the study of biological hazards due to electric and magnetic fields due to sources such as high power transmission lines Fig 2 IS1 11 of a microstrip fed patch antenna (superimposed) printed on a 3125 mil Duroid substrate(g = 22) computed via the FDTD method The FDTD lattice is terminated by a lo-cell thick uniaxial PML layer which is placed: * 3 cells from the edge of the patch and 5 cells above the patch, and + 10 cells from the edges of the patch 4 THEPGYALGORITHM L A number of explicit methods for the solutions of Maxwell s equations based on non-orthogonal structured [9,10] and unstructured methods [ 11,121 have been proposed The planar generalized Yee-algorithm (PGY) algorithm [13,14] has the advantage that it models threedimensional geometries with planar symmetry via a mesh that is fully unstructured along two-dimensional crosssections and structured along the third dimension The PGY algorithm is based on the discretization of Ampere s and Faraday s laws in their integral form by projecting the vector fields onto the edges of a dual, staggered unstructured grid By exploiting planar symmetry, significant memory savings can be realized Based on such a discretization of the fields, and employing a central difference approximation for the time-derivatives, an explicit time-marching solution for vector field updates 118

3 can be derived The matrices are highly sparse, although due to the unstructuredness of the grid they must be stored Fortunately, due to the regularity of the grid along one dimension, only the matrices due to one layer of cells are actually stored This greatly reduces the overall memory requirement The explicit update scheme is then expressed as a linear operator as where, 3 and B are coefficient vectors representing the flux density vectors normal to the grid faces, B and a are coefficient vectors representing the electric and magnetic field intensities along the primary and dual grid edges, respectively, De, A, and Ah are the sparse update matrices defined in [ 13,141, and AhP and A, are the sparse projection matrices defined in [13,14] The explicit field updates are performed using (3), leading to an efficient computational technique that is second-order accurate for the simulation of the time-varying fields, and is stable, providing the time step satisfies the stability criterion [13,14] The principal advantage of techniques based on unstructured grids, is that highly complex geometries can be easily modeled through the use of commercially available CAD tools and automatic mesh generation Furthermore, unstructured meshing allows for a more accurate representation of the field interaction complex boundaries, which are often encountered in EMC analysis, since the edges of the mesh can be conformed to irregular surfaces = 04 mm) A total of 4000 time stens were used (At = 025 ps) On 16 processors of an i&c/860, the PGY algorithm required 769 CPU-s, as compared to 864 CPU-s required by the FDTD algorithm The comparison between the two is excellent, and quite good compared with [ 151 As a second illustration, lets consider the analysis of a Thermal Conduction Module (TCM) interconnect structure used for the IBM 3090 [ 17,181 A TCM interconnect consists of multilayered mesh planes Sandwiched between each mesh plane are two layers of orthogonal signal lines Furthermore, the lines are interconnected between layers through rectangular vias In this study, the PGY-algorithm is used to study the propagation characteristics of the signal lines, the cross-coupling of lines, and the properties of the vias Figure 5 illustrates two orthogonal signal lines between two mesh planes The mesh planes are embedded in a dielectric slab of thickness TOD View Side View The FDTD and PGY methods have been used to analyze numerous EMC related problems These software tools provide the ability to efficiently model broad band signals interacting within complex structures with inhomogeneous material profiles The FDTD method is best suited for analyzing structures with geometries that are separable in a Cartesian coordinate system The PGY algorithm is better suited for modeling structures with complex geometries due to its modeling flexibility and CAD interface In this section, both techniques are employed to study a few interesting applications As a first example, consider the cylindrical via through a ground plane illustrated in Fig 3 This geometry has been studied using a number of methods [ 151 and serves as a benchmark problem The via connects two 50 Q microstrip lines and is composed of a cylindrical post passing through a cylindrical hole in a ground plane The magnitude of the S-parameters of the cylindrical via computed using both a parallel FDTD algorithm [3] and the PGY algorithm These results are compared in Fig 4 The unstructured grid describing the via consisted of 223,616 hexahedral cells The FDTD lattice had a dimension of 93 x 133 x 49 (AX = AJJ = mm, & 779 Fig 3 g Fig 4 Geometry of a cylindrical via through a PEC ground plane i i : ; G -40 ~~~I~~I~~~I~~,,ll,,,ii : : f (GHz) Magnitude of the S-parameters for the cylindrical via through a ground plane computed using the planar generalized Yesalgorithm

4 Fig 5 1 Side View m mhm I m I Signal line within a planes Perpendicular TCM intercomect mesh Line Coupling characteristic impedance and capacitance per unit length were extracted as 2 = 455R (45452) Cf= go,, =227f lm ( f Im) where the values in the parenthesis are measured results from [ 151 Figure 6 presents the S-parameters of the two orthogonal lines, where Sll represents the reflection loss, and S21 represents the insertion loss, or the through power on the excited line S31 and S4l represen the power coupled to either end of the orthogonal line The next example is a coupled line transmission line structure Specifically, Fig 7 illustrates three adjacent microstrip lines printed on a 20-mil alumina substrate backed by a ground plane The line widths are 20 mils, and each is separated by 20 mils The lines have a 90 mitered bend, as illustrated in the figure The inner line (line 1) is excited by a 2 V pulse with a 30 ps rise and fall time and a 70 ps duration The voltage source has a 50 s1 internal impedance, and is modeled as a lumped load [ 131 The remaining ports are terminated into 50 Sz loads to ground The fields excited in this structure where simulated using the PGY algorithm An unstructured mesh of roughly 180,000 hexahedral cells was used The entire simulation required 7,000 time steps (& = 01 ps), and took 1,000 seconds on 16 processors of an ipsc/860 It also took 990 seconds on 2 processors of an SGI Power Challenge Fig 8 illustrates the near end and far end voltages recorded for this structure Significant coupling to line 2 is observed, and only a small amount of coupling to line 3 is observed Fig GHz S-Parameters representing the interaction of the orthogonal signal lines between the mesh planes 14 mm, and sr = 95 (Alumina) The mesh planes have a pitch p = 05 mm, and a thickness of w = 01 mm The separation distance between the mesh planes is d = 06 mm, and the signal line is placed a distance of h = 02 mm from the mesh plane also has a thickness w = 01 mm The PGY-based model consisted of a mesh plane with 11 X 1 1 cells, situated in free space The signal line was excited by an ideal voltage source and was terminated in a matched load (emulated by the ABC) A low frequency SUMMARY The FDTD method has been found to provide an extremely useful tool for the analysis of EMC related problems Over the last 5 years, extensive application of the FDTD method to practical engineering problems has been observed This is specifically attributed to the availability of economical high-performance computers, as well as the introduction of advanced FDTD algorithms and tools 131 [41 REFERENCES A Taflove, Ed, Finite Difference Time Domain M&hods for Electrodynamic Analyses,, Artech House, NY, 1995 M Pike&May, A Taflove and J Baron, FD-TD modeling of digital signal propagation in 3-D circuits with passive and active loads, IEEE Transactions on Microwave Theory and Techniques, vol 42, pp , Aug 1994 S D Gedney, Finite-difference time-domain analysis of microwave circuit devices on high performance vector/parallel computers, IEEE Transactions on Microwave Theory and Techniques, vol 43, pp , October 1995 J-P Berenger, A perfectly matched layer for the absorption of electromagnetic waves, Journal of Computational Physics, October 1994 D S Katz, E T Thiele, and A Tatlove, Validation and extension to three-dimensions of the Berenger PML absorbing boundary condition for FD-TD meshes, IEEE 120

5 ,,,( El [71 I WI [16l II171 Microwave and Guided Wave Letters, vo14, no8, p 26% 270, August 1994 W C Chew and W H Weedon, A 3D perfectly matched medium from modified Maxwell s equations with stretched coordinates, Microwave and Optical Technology Letters, vol 7, pp , September 1994 Z S Sacks, D M Kingsland, R Lee, and J F Lee, A perfectly matched anisotropic absorber for use as an absorbing boundary condition, IEEE Trans on Antennas andpropagation, ~0143, pp , Dec 1995 S D Gedney, An Anisotropic PML Absorbing Media for FDTD Simulation of Fields in Lossy Dispersive Media, Electromagnetics, in press R Holland, Finite-difference solution of Maxwell s equations in generalized nonorthogonal coordinates, IEEE Transactions on Nuclear Science, vol NS-30, pp , June 1983 J-F Lee, R Palendech and R Mittra, Modeling threedimensional discontinuities in waveguides using non orthogonal FD-TD algorithm, IEEE Trans on Microwave Theory and Techniques, vol 40, pp , Feb 1992 [Ill N Madsen, Divergence preserving discrete surface integral methods for Maxwell s equations using nonorthogonal unstructured grids, Technical Report #UCRL-JC , LLNL, February 1992 P21 V Shankar, A Mohammadian and W Hall, A timedomain, finite-volume treatment for the Maxwell equations, Electromagnetics, vol 10, pp , January-June 1990 [I31 S Gcdney, F Lansing and D Rascoe, A generalized Yeealgorithm for the analysis of MMIC devices, IEEE Transactions on Microwave Theory and Techniques, in press 1141 Stephen D Gedney and Faiza Lansing, A parallel planar generalized Yee-algorithm for the analysis of microwave circuit devices, International Journal on Numerical Modeling (Electronic Networks, Devices and Fieldr), Vol 8, pp , May-August 1995 W D Becker, P Harms, and R Mittra, Time-domain electromagnetic analysis of interconnects in a computer chip package, IEEE Transactions OR Microwave Theory and Techniques, vol 40, pp , December 1992 B J Rubin, An electromagnetic approach for modeling high performance computer packages, IBM J on Research and Development, vol 34, pp , July 1990 A Cangellaris, M Gribbons and J Prince, Electrical characteristics of multichip module interconnects with perforated reference planes, IEEE Transactions on Components, Hybrids, and Manufacturing Technology, vol 16, pp pp 113-l 18, February 1993 / I60mils /l Fig 7 Coupled lines with 90 mitered bend printed on a 20 mil substrate The printed lines are 20 mils wide, and are separated by 20 mils Each line is terminated in a 50 R load at the near and far ends Line 1 is excited by a voltage source (near end) Line 2 is the center line, and line 3 is the outer line 12,,,,,,,,,,,,,,,/,,),(,,,,,,,, 08, ~& i ; i ; f > y r $ 2 _ (a) o2 bi- *,,1,,,,1,,1,1,,111,,,~1,,,, t (PS) 121 Fig 8 0 Near end (a) and Far end (b) line voltages of the coupled lines in Fig 5 excited by a 2 V trapezoidal pulse with 50 R matched load and a 30 ps rise time and fall time and 70 ps duration