030601-8-T An AWE Implementation for Electromagnetic Analysis J. Gong and J.L. Volakis National Aeronautics and Space Administration Langley Research Center Hampton, VA 23681-0001 July 1996 30601-8-T = RL-2427

Report #030601-8-T NASA Langley Grant NAG 1-1478 Grant Title: Report Title: Simulation of Conformal Spiral Slot Antennas on Composite Platforms An AWE Implementation For Electromagnetic Analysis Report Authors: J. Gong and J.L. Volakis Primary University Collaborator: Primary NASA-Langley Collaborator: University Address: Date: John L. Volakis Fred Beck Telephone: (804) 864-1829 Radiation Laboratory Department of Electrical Engineering and Computer Science Ann Arbor, Michigan 48109-2122 Email: volakis@umich.edu Phone: (313)764-0500 July 1996

An AWE Implementation for Electromagnetic FEM Analysis Jian Gong and John L. Volakis Radiation Laboratory University of Michigan Ann Arbor, MI 481098-2122 jxgunmi(Cunmich.edu volakis(tiumich.edu Abstract Although full wave electromagnetic systems are large and cumbersome to solve, typically only a few parameters, such as input impedance, S parameters, and far field pattern, are needed by the designer or analyst. A reduced order modeling of these parameters is therefore an important consideration in minimizing the CPU requirements. The Asymptotic Waveform Evaluation(AWE) method is one approach to construct a reduced order model of the input impedance or other useful electromagnetic parameters. We demonstrate its application and validity when used in conjunction with the finite element method to simulate full wave electromagnetic problems. 1 Introduction The method of Asymptotic Waveform Evaluation (AWE) provides a reducedorder model of a linear system and has already been successfully used in VLSI and circuit analysis to approximate the transfer function associated with a given set of ports/variables in circuit networks [1, 2, 3]. The basic idea, of the method is to develop an approximate transfer function of a given linear system from a limited set of spectral solutions. Typically, a Pade expansion of the transfer function is postulated whose coefficients are then determined 1

)y matching the Pade representation to the available spectral solutions of the complete system. In this paper we investigate the suitability of the AWE method for approximating the response of a given parameter in full wave simulations of radiation or scattering problems in electromagnetics. Of particular interest is the use of AWE for evaluating the input impedance of the antenna over a given l)andwidth from a knowledge of the full wave solution at a few (even a single) frequency points. Also, the method can be used to fill-in a backscattering pattern with respect to frequency using a few data samples of that pattern. Below we first describe the recasting of the FEM system for application of the AWE. We then proceed to describe the AWE method and demonstrate its application, accuracy and efficiency in computing the input impedance of a shielded microstrip stub. 2 Theory of Asymptotic Waveform Evaluation 2.1 FEM System The application of the finite element method to full wave electromagnetic solutions amounts to generating a linear system of equations by extremizing the functional [4] =< V x E,- V7 x E > -k2 < E, 3E > +b.t. (1) where <,> denotes an inner-product and b.t. is a possible boundary term whose specific form is not required for this discussion. Also, the dyadics (a -27T UJ and 3 are material related coefficients, k - - is the wavenumber and A c w is the corresponding operating frequency with c being the speed of light. A discretized form of (1) incorporating the appropriate boundary conditions is [5] (Ao + kAA + k2A) {X} = f} (2) where A, denote the usual square (sparse) matrices and {f} is a, column matrix describing the specific excitation. 2

Clearly (2) can be solved using direct or iterative methods for a. given value of the wavenumber. Even though Ai is sparse, the solution of the system (2) is computationally intensive and must be repeated for each k to obtain a, frequency response. Also, certain analyses and designs may require both temporal. and frequency responses placing additional computationa.l burdens and a repeated solution of (2) is not a.n efficient approach in generating these responses. An application of AXWE to achieve an approxima.tion t-o these responses is an a.ttractive alternative. Below we formulate AWE ill conjunction with the finite element method (2), for modeling antenna and microwave circuit problems. For these problems, the excitation column {'} is a, linear function of the wavenumber and can therefore be stated as {f } - k {.f1} (3) with {f} being independent of frequency. This observation will be specifically used in our subsequent presentation. 2.2 Asymptotic Waveform Evaluation To describe the hbasic idea of AWE in conjunction with the FEM, we begin by first expanding the solution {X} in a Taylor series about k0 as {X} = {o0} + (k - ko) {X1} + (k - ko) {X2} +... +(k - ko)l {X,} + 0 t (k - ko)1+'} (4) where {X0} is the solution of (2) corresponding to the wavenumber ko. By introducing this expansion into (2) and equating equal powers of k in conjunction with (3), after some manipulations, we find that, {Xo} = 0oAo1 {f.} {X} = A1 [{If,}-A1 {Xoj }-2koA2 {Xo}] {X2} = -Ao [Al {X1} + A2({Xo} + 2ko {X1})] (5) {X} = -Ao [A1 {X } + A2({X-2 } + 2ko {X })] with Ao = Ao + koA, + ko2A2 (6) 3

Expressions (5) are referred to as the system moments whereas (6) is the system at the prescribed wavenumber (ko). Although an explicit inversion of Ao1 may be needed as indicated in (5), this inversion is used repeatedly and can thus be stored out-of —core for the implementation of AWE. Also, given that for input impedance computations we are typically interested in the field value at one location of the computational domain, only a. single entry of {Xi(k)} need be considered. say (the pth entry) XP(k). The above moments can then be reduced to scalar form and the expansions (5) become a scalar representation of X'(k) about the corresponding solution at ko. To vield a. more convergent expression, we can instead revert to a Pade expansion which is a conventional rational function. For transient analysis the Pade expansion can be cast by partial fraction decomposition [3, 6] into q 'i XyP() - + k-lP -(7) where Xq) is the limiting value as k tends to infinity. This is a qth order representation and is suitable for time/frequency domain transformation. As can be realized, the residues and poles (r, and k0 + k ) in (7) correspond to those of the original physical system and play important roles in the accuracy of the approximation. As can be expected a higher order expansion with more zeros and poles can provide an improved approximation. The accuracy of AWE relies on the prediction of the dominant residues and poles located closest to k0 in a complex plane. Its key advantage is that for many practical electromagnetic problems only a few poles and zeros are needed for a sufficiently accurate representation. For a hybrid finite element - boundary integral system, the implementation of AWE; is more involved because the fully populated boundary integral submatrix of the system has a. more complex dependence on frequency. In this case we may instead approximate the full submatrix with a spectral expansion of the exponential boundary integral kernel to facilitate the extraction of the system moments. This approach does increase the complexity in implementing AWE. However, AWE still remains far more efficient in terms of CPU requirements when compared to the conventional approach of repeating the system solution at each frequency. 4

3 Numerical Implementation As an a)pplica.tion of AWE to a. full wave electromagnetic simulation, we consider thie evaluation of the input impedance for a microstrip stub shielded in a mletallic rectangular cavity as shown in figure 1. The stub's input impedance is a strong function of frequency from 1-3 GHz and this example is therefore a good demonstration of AWiE's capability. The shielded cavity is 2.38cm x 6.00cm x 1.06cmn in size and the microstrip stub resides on a 0.35cm thick substrate having a dielectric constant of 3.2. The stub is 0.79cmn wide and A/2 long at 1.785 GHz and we note that the back wall of the cavity is terminated by a metal-backed artificial absorber having relative constants of,. = (3.2, -3.2) and t, = (1.0, -1.0). As a, reference solution, the frequency response of the shielded stub was first computed from 1 to 3 GHz at 40 MHz intervals (,50 points) using a full wave finite element solution. To demonstrate the efficacy and accuracy of AWE we chose a single input impedance solution at 1.78 GHz in conjunction with the 4th order and 8th order AWE in (7) to approximate the system response. Clearly the chosen number of poles or order of the expansion leads to different accuracies. As seen in Figure 2, the 4th order AWE representation is in agreement with the real and reactive parts of the reference input impedance solution over a 56% and 33% bandwidth, respectively. Surprisingly, the 8th order AWE representation recovers the reference solution over the entire 1-3 GH z band for both the real and reactive parts of the impedance. However, the CPU requirements for the 4th and 8th order approximations are nearly the same except for a few more matrix-vector products needed for the higher order expansion. The number of these operations are of the same order as that of the AWE expansion and are much smaller than the size of the original numerical system. We conclude that the AWE representation is an extremely useful addition to electromagnetic simulation codes and packages for computing wideband frequency responses using only a few samples of the system solution. References [1] S. Kumashiro, R. Rohrer and A. Strojwas, "Asymptotic waveform evaluation for transient analysis of 3-D interconnect structures," IEEE Trans. 5

Computer-Aided Design of Integrated Circuits and Systems, Vol. 12, No. 7, pp 988-996 Jul, 1993 [2] L. Pillage and R. Rohrer, "AWE: Asymptotic Waveform Estimation," Techln.nical Resport, SRC-CMU Research Center for Computer-Aided Design, Carnegie-Mellon University, 1988 [3] E. Chiprout and M. Nakhla, "Asymptotic waveform evaluation and moment matching for interconnect aina-lysis," Norwell. Kluwer Acad. Pubs, 1994 [4].J. L. Volakis, A Chatterjee and J. Gong, "A class of hybrid finite element methods for electromagnetics: a review," J. Electromagn. Waves Application.s Vol. 8, No. 9/10, pp. 1095-1124, 1994 [5] J. Gong, J.L. Volakis, A.C. Woo and H. G. Wang, "A Hybrid Finite Element-Boundary Integral Method for the Analysis of Cavity-backed Antennas of Arbitrary Shape," IEEE Tran.s Antenna and Propagat., Vol. 42, No. 9. pp 1233-1242, 1994 [6] Joseph Lehner, "Partial fraction decompositions and expansions of zero," Trans. Amer. Math. Soc., Vol. 87, pp 130-143, 1958 6

1.06 cm...... 2.38 cm, Figure 1: Illustration of the shielded microstrip stub excited with a current probe. 55 - 50 - 45, 40// E 035- / -,, sa/ 30 / E / 25 - 20 15O 10 -- -__-..., 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 frequency in GHz Figure 2: Impedance calculations using regular FEM frequency analysis for a shielded open microstrip stub shown in figure 1. Solid line is the real part and the dashed line denotes the imaginary part of the solutions. These computations are used as reference for comparisons. 7

(a) (b) Figure 3: Results of the 4th and 8th order AWE implementation using a single point expansion at 1.78 GHz. (a) Real part and (b) imaginary part of the Input impedance 8