A Method for Introducing Nonuniform Grids into the FDTD Solution of the Transmission-Line Equations Based on the Renormalization of the Per-Unit-Length Parameters Roberto B. Armenta and Costas D. Sarris Edward S. Rogers Sr. Department of Computer and Electrical Engineering, University of Toronto, ON M5S 3G4, Canada. rarmenta@waves.utornto.ca cds@waves.utoronto.ca Abstract: one of the challenging aspects of using the finite-difference time-domain (FDTD) method to solve the transmission-line (TL) equations is to choose a discretization grid with an adequate spatial resolution. In the standard formulation of the algorithm, a uniformly spaced discretization grid is used; nonetheless, different regions of the TL may require different levels of spatial resolution. To address this issue, this paper presents a nonuniform gridding scheme with an attractive feature: it preserves the simplicity of the standard FDTD update equations. It is shown that all the details of the nonuniform grid can be absorbed into effective per-unit-length parameters through a coordinate system transformation. By using the proposed step summing coordinate system transformation, a nonuniformly spaced grid can be mapped onto a uniformly spaced grid where a standard set of FDTD update equations can be applied. Keywords: finite-difference time-domain method and transmission lines.. Introduction When the finite-difference time-domain (FDTD) method is used to sove the transmission-line (TL) equations, a discretization grid with an adequate spatial resolution level must be carefully chosen. In the standard (2,p)-version of the algorithm, a uniform discretization grid is used. However, when dealing with position dependent per-unit-length parameters, different regions of the TL may require very different resolution levels. In such case, using a uniformly spaced discretization grid produces an unnecessary computational overhead. In the past, two different alternatives have been proposed to address this issue: the first one is to use a wavelet based multiresolution timedomain method (e.g., []-[2]); and the second one is to incorporate a nonuniform grid (e.g., [3]-[4]) into the original (2,2)-algorithm [5]. While many useful methods for introducing a nonuniform grid have been proposed before (see [6] for a summary), all of them introduce cumbersome modifications into the standard FDTD update equations. This paper proposes a method for introducing a nonuniform grid that preserves the simplicity of the standard (2,p)-algorithm on a uniform grid. It is shown that all the details of the nonuniform grid can be absorbed into effective or renormalized per-unit-length parameters through a one-dimensional coordinate system transformation. By using The notation (2,p) means second order accurate in time and p-th order accurate in space.
a coordinate system transformation, a nonuniformly spaced grid can be mapped onto a uniformly spaced grid where a (2,p)-set of standard FDTD update equations can be applied. The rest of this paper is divided into three sections: Section 2 briefly reviews the (2,p)-FDTD procedure for solving the TL equations; Section 3 introduces a one-dimensional coordinate system transformation that can be used to map a nonuniformly spaced grid onto a uniformly spaced one; and, in Section 4, these two are put together with a simple example. 2. The Standard (2,p)-FDTD Algorithm on a Uniform Grid For a general TL, the position and time dependent voltage and current functions V (z, t) and I(z, t) are related by V (z, t) I(z, t) = G(z) V (z, t) C(z) C(z) = R(z) I(z, t) L(z) L(z) I(z, t), () z V (z, t), (2) z where G(z) is the per-unit-length conductance, R(z) is the per-unit-length resistance, C(z) is the per-unit-length capacitance and L(z) is the per-unit-length inductance of the TL. In the standard (2,p)-FDTD algorithm, () and (2) are discretized on a uniformly spaced grid using second order accurate centered difference approximations in time and p-th order accurate centered difference approximations in space at points ((n+/2) t, i z) and (n t, (i+/2) z) for all n = 0,, 2,..., N T and i = 0,, 2,..., N Z. When a uniform discretization grid is used, the cell size z, which determines the spatial resolution of the grid, is usually selected so that λ min 20 z λ min 0 where λ min = z 0+L Z min {λ(z)} ; λ(z) = z=z 0 f max L(z)C(z). (3) In the above, f max is the maximum frequency excited by the source, and L Z is the z-axis length of the TL. Also, in the case of the (2,4)-algorithm, the time step t is picked according to λ min f max t/ z 6/7 (4) in order to satisfy stability requirements. Similar formulas apply to other high-order methods. Whenever z and t are picked according to (3) and (4), the uniform grid will be unnecessarily fine in regions where λ(z) is much larger than λ min. This well-known problem can be addressed by introducing a nonuniform discretization grid. Nonuniform grids are usually introduced in three steps. In step, a piecewise constant approximation of λ(z) is introduced. For this purpose, the TL s z-axis domain is divided into M nonoverlapping sections where the m-th section is contained within the interval [z m, z m ]. Then, λ(z) is approximated by a constant value λ m min = inside the m-th section. In step 2, all the sections where λ m min λ max = zm min z=z m {λ(z)} (5) M max m= {λm min} (6)
are put together into a single entity called the main grid region, which receives a coarse uniform discretization grid. Once the main grid region has been defined, finer grids are assigned to the remaining TL sections, which are called the refined grid regions. Finally, in step 3, a special set of FDTD update equations is generated to deal with the refined grid regions. The alternative nonuniform gridding approach proposed here consists of keeping the first two steps and replacing the last one with a coordinate transformation that maps the nonuniform grid generated in step 2 from (t, z)-space to (t, z)-space. If chosen as described in the next section, the coordinate transformation can be used to map a nonuniform grid in (t, z)-space to a uniform grid in (t, z)-space, where the problem can be solved using a standard (2,p)-set of update equations. 3. The Step Summing Coordinate System Transformation To define a coordinate transformation z z, we need to specify a function z = f( z). Substituting z = f( z) into () and (2) yields V ( z, t) I( z, t) = Ḡ( z) V ( z, t) C( z) C( z) = R( z) I( z, t) L( z) L( z) I( z, t), (7) z V ( z, t), (8) z Ḡ( z) = G( z)f ( z), C( z) = C( z)f ( z), R( z) = R( z)f ( z) and L( z) = L( z)f ( z). Observe that (7) and (8) have the same form as () and (2). This property which is known as metric invariance implies that (7) and (8) can be solved numerically in the (t, z)-domain by using a standard (2,p)-set of equations provided that the renormalized per-unit-length parameters are used in the computations. So, now the crucial question is: what choice of f( z) will achieve the desired mapping? First of all, f( z) must satisfy two key properties:. it must offer a one-to-one mapping z z, and 2. it must be continuous with continuous first derivatives. The first property is required so that the voltages and currents are uniquely defined in the (t, z)- domain. The second property is required so that no artificial discontinuities are introduced into the renormalized per-unit-length parameters (which could cause spurious reflections). The set of functions that satisfy the above two properties is very large; nonetheless, a simple coordinate transformation that has been found to be very versatile is presented next. For convenience, let us work with the inverse relation z = f (z) instead of z = f( z). Since a one-to-one mapping is required, this does not cause problems. Consider the coordinate transformation z = f (z) = z + ϕ(z), ϕ(z) = a m = λ max λ m min M a m d m ϕ m (z) where (9a) m= 0, d m = z m z m + 20 z; (9b)
ϕ m (z) = 2 + ( ) π 2 cos (z (z m + 0 z)), z m 0 z z z m + 0 z d m, z > z m + 0 z. (9c) Parameters λ max and λ m min were previously defined in (5) and (6). Also, z represents the cell size of the uniform grid in (t, z)-space. This parameter is usually chosen so that λ max 20 z λ max 0. (0) Once z has been chosen, the total number of cells N Z which must be the same in both spatial domains can be determined from { f (z M ) f } (z 0 ) N Z = round. () z To understand how the proposed coordinate system transformation achieves the desired mapping, observe that (9) yields z = z in the main grid region (i.e. where a m = 0). This implies that the main grid will have identical cell sizes in both spatial domains. On the other hand, in the refined grid regions (i.e. where a m > 0), the coordinate transformation adds a step function ϕ m (z) that effectively contracts distances (by a factor /(a m +)) as we go from the z-domain to the z-domain. This spatial contraction is what allocates a higher density of grid points in the refined grid regions even though a uniform grid is used in the (t, z)-space. When solving a TL problem using (9), three important practical issues must be kept in mind. First, in order to apply the transformation, we must know z = f( z) at a uniformly spaced set of points z i = i z for all i =, 2,..., N Z. While it is possible to analytically invert (9) to obtain z i = f( z i ), the easiest way to obtain the z i s is through an interpolation routine that computes them from the data of a plot of f (z) vs. z. Second, in order to renormalize the per-unit-length parameters, it is necessary to obtain the renormalization factor f ( z i ). Once the z i s have been obtained by interpolation, it is possible to obtain f ( z i ) from f ( z i ) = d dz f (z). (2) z=zi A closed form expression for the derivative of f (z) can be easily obtained from (9). The third and most important issue is the stability of the resulting FDTD scheme in the (t, z)-domain. As the per-unit-length parameters are renormalized, the stability criterion will be different from that of the conventional (2,p)-FDTD scheme. It can be shown that, for the (2,4)-set of update equations, the proposed time stepping procedure is stable as long as 7λ max f max t 6 z S, S = z 0+L Z min z=z 0 { } d dz f. (3) (z) Similar formulas will apply for higher order cases. Moreover, since renormalizing the per-unitlength parameters does not affect the dual structure of the TL equations, the proposed approach does not suffer from late time instability problems [7].
4. Transmssion-Line Example: a Bragg Reflector For demonstration purposes, consider the TL structure shown in Fig.. The TL is lossless with piecewise constant per-unit-length parameters given in Table. This structure is the TL equivalent of a Bragg reflector with two dielectric slabs. The goal in this example is to compute the reflection coefficient at z = 8.622 cm assuming that absorbing boundaries are present at z = z 0 and z = z 5. This problem was solved in the (t, z)-domain by using a (2,4)-set of update equations and a uniform discretization grid with the following parameters: z =.268 mm = λ max /.82, λ max = c 0 /f max, f max = 20 GHz, t = 0.345 ps, N Z = 424, N T = 32768. Both z and N Z were picked according to (0) and (). Observe that if we were to solve the problem using a uniform discretization grid in the (t, z)-domain with z = λ max /.82, the condition in (3) would be violated (since λ max = 5λ min ). To excite the TL, a Gaussian pulse of the form ( ( ) ) t 2 t0 f S (t) = exp, t 0 = 3T S, T S = 2f max T S was added at every time step to the voltage node located at z = 6.340 cm. The voltage waveform was sampled at the node located at z = 8.622 cm in order to compute the reflection coefficient from a discrete Fourier transform of the incident and reflected pulses. The amplitude of the computed reflection coefficient is shown in Fig. 2 together with the corresponding analytic solution. 2.68 cm.268 cm 4.058 cm.268 cm 4.058 cm TL Conductors 2 3 4 5 z 0 = 0 z z 2 z 3 z 4 z 5 = L Z Figure : TL example under consideration. The TL has been divided into five sections (labeled with underlined numbers) with per-unit-length parameters given in Table. Table : Per-unit-length parameters of the TL example. z m L m C m µ 0 ε 0 2 µ 0 25ε 0 3 µ 0 ε 0 4 µ 0 25ε 0 5 µ 0 ε 0 m = Section Number L m = Per-Unit-Length Inductance C m = Per-Unit-Length Capacitance µ 0 = 4π0 7 H/m ε 0 = 8.8542 0 2 F/m c 0 = / µ 0 ε 0
Reflection Coefficient Amplitude (db) 5 0!0!20!30!40!50 FDTD Solution Analytic Solution!60 0 2 3 4 5 6 7 Frequency (GHz) 8 9 0 (a) 9e:;ec<=on?oe::=c=en< @AB;=<ude Error (db)!0!20!30!40!50!60!70!80!90!00!0!20 0 2 3 4 5 6 7 Frequency (GHz) 8 9 0 (b) Figure 2: Plots of (a) the computed reflection coefficient amplitude and (b) the error in the estimate of the reflection coefficient amplitude. 5. Conclusions In summary, a method for introducing a nonuniform grid into the FDTD solution of the TL equations has been presented. A unique feature of this method is that it uses a coordinate transformation to introduce localized grid refinements. This allows the method to preserve the simplicity of the standard (2,p)-FDTD update equations. The presented coordinate transformation automatically selects an appropriate grid density for each region of the TL based on the TL s material properties. Finally, while the presented computed results were obtained using a (2,4)-set of update equations, the proposed approach easily incorporates other high-order FDTD schemes. References [] S. Grivet-Talocia and F. Canavero Wavelet-Based High-Order Adaptive Modeling of Lossy Interconnects, IEEE Trans. Electromagn. Compat., vol. 43, pp. 47-484, 200. [2] S. Barmada and M. Raugi, Transient Numerical Solutions of Nonuniform MTL Equations with Nonlinear Loads by Wavelet Expansion in Time or Space Domain, IEEE Trans. Circuit Sys. I, vol. 47, pp.78-90, 2000. [3] P. Monk, Subgridding FDTD Schemes, ACES Jrnl., vol., pp. 37-46, 996. [4] K. Xiao, D. J. Pommerenke, and J. L. Drewniak, A Three-Dimensional FDTD Subgridding Method with Separate Spatial and Temporal Subgridding Interfaces, in Proc. Intl. Symp. Electromagn. Compat., EMC 05, pp. 578-583, 2005. [5] J. A. Roden, C. R. Paul, W. T. Smith and S. D. Gedney, Finite-Difference, Time-Domain Analysis of Lossy Transmission Lines, IEEE Trans. Electromagn. Compat., vol. 38, pp. 5-24, 996. [6] A. Taflove and S. Hagness (ed.), Computational Electrodynamics: the Finite-Difference Time- Domain Method. Boston: Artech House, 2005, ch.. [7] R. Schuhmann and T. Weiland, A Stable Interpolation Technique for FDTD on Non- Orhtogonal Grids, Intl. Jrnl. Numer. Model., vol., pp. 299-306, 998.