# Milestones

## 2004 Annual Report

#### ANNUAL PROGRESS SUMMARY

### Reporting Period: 1 September 2003 to 31 August 2004

#### OBJECTIVES

The objective of this project is to develop novel and accurate numerical techniques to model complex antenna feeds, feed networks, and active antenna elements for analyzing/ designing/optimizing multifunctional, conformal antennas/arrays involving complex/ nonlinear materials/devices.

Robust schemes for modeling feeds and distribution networks are critical to synthesizing advanced antenna/array systems. Indeed, while simplified feed models permit the calculation of reasonably accurate radiation patterns, they fail for all but the simplest systems when input impedances and mutual coupling (S-parameters) are to be accurately predicted. In practice, most antennas are fed by coaxial lines, waveguides, or microstrip elements and their analysis is complicated by the presence of baluns and tuning circuits. Simplified models present in most antenna analysis packages often entirely fail to accurately account for the geometric details of the feed and use nonvariational equations to estimate input impedances.

Direct integration of nonlinear circuitry into antennas/arrays often permits feed/ distribution networks and packages to be simplified and reduced in size and weight. However, one of the principal challenges in designing active antenna arrays involves the prediction of, and compensation for, electromagnetic interference and signal distortion caused by the nonlinear behavior of the active devices, which may result in spurious, unpredictable, and undesirable antenna responses. Current computational tools capable of modeling nonlinear packages are often low-order finite difference based and fail to capture nonlinear effects with accuracies that the antenna engineer has become accustomed to when using integral equation methods.

To overcome the problems described above, we will:

- Subtask 1: Develop accurate antenna feed models for the time- and frequency-domain finite element and integral equation based simulation techniques. Develop stable and variational formulations for the calculation of the input and mutual impedances or S-parameters that fully characterize the mutual coupling in conformal arrays.
- Subtask 2: Implement models for passive and active circuit elements in time-domain finite element field solvers to enable active antenna element design optimization;
- Subtask 3: Develop broadband macromodels for the portions of a feed network that do not interact with the active elements and their stable incorporation in the nonlinear field solvers used for active array modeling and simulation;
- Subtask 4: Develop efficient simulation techniques that are compatible with fast finite element and integral equation field solvers and thus can be used to enable efficient multi time-scale transient simulation of active antenna arrays with uncompromised electromagnetic accuracy;
- Subtask 5: Inherent and central to each subtask is a validation and verification effort involving the application of the developed technology to existing antenna/array models and government-provided antennas and arrays.

### STATUS OF EFFORT/ACCOMPLISHMENTS/NEW FINDINGS

**1. Antenna Feed Modeling in the Frequency and Time Domains**

Robust schemes for modeling feeds and distribution networks are critical to synthesizing advanced antenna systems. While simplified feed models permit the calculation of reasonably accurate radiation patterns, they fail for all but the simplest systems when input impedances and mutual coupling (S-parameters) are to be accurately predicted. In practice, most antennas are fed by coaxial lines and/or microstrip elements. Simplified models present in most antenna analysis packages often entirely fail to accurately account for the geometric details of the feed and use non-variational equations to estimate input impedances. During this performance period, we developed an accurate method to model coaxial feeds in both frequency and time domains. This model is directly based on the full-wave analysis of electric and magnetic fields in the feed structures, instead of using the voltage and current concepts. This model involves a reference surface, often chosen to be close to the coaxial opening, and then represents the total field as the superposition of the incident and reflected waves (including higher-order modes excited by the structures in the vicinity of the feed). An exact boundary condition can then be derived, which can be incorporated into the finite element solution of the antenna problem. As a result, the input for the numerical analysis is the incident field in the feed line and the output is the reflection coefficient (S11), from which the input impedance can be readily calculated. In the case for multiple antennas, which is of particular interest in this project, a full S-parameter matrix can be computed by exciting each antenna. This model is applicable to almost all antenna feeds. Except for numerical discretization, the model is exact and its numerical solution is error controllable. More important, since this model calculates the reflected waves in the feed lines, one can incorporate the effect of feed structures, such as corporate feed, into the antenna array analysis. We have tested and validated this model on a few problems. The development of this model in the time domain is especially important because it provides an essential component for the modeling of broadband antennas using time-domain techniques.

2. Development of a Time Domain Integral Equation (TDIE) Based Antenna Analysis Tool

A new full-wave time-domain methodology for analyzing coupling of electromagnetic transients into platform-mounted cable feed networks was developed. To enable the analysis of large-scale and geometrically intricate systems, a parallel implementation of the time-domain adaptive integral method (TD-AIM) (A.E. Yilmaz et al., *APS Digest*, 543-546, 2003), viz. the time domain counterpart of the frequency-domain AIM procedure, is used. The algorithm requires (per processor) CPU resources for analyzing phenomena spanning time steps on processors using auxiliary Cartesian point sources. This accelerated parallel solver is far more efficient than classical time-domain integral-equation solvers, the CPU requirements of which scale as , where is the numbers of surface-cable unknowns. The solver approximates all conductor surfaces by triangular patches and uses Rao-Wilton-Glisson basis functions to approximate their current distribution. Here, we extended the previously developed TD-AIM solver with a view to perform accurate analysis of platform-antenna feed network (cable) interactions. While cylindrical surfaces can be used to model wires without introducing new solver capabilities, doing so invariably leads to fine spatial discretizations and elevated unknown counts; this avenue for analyzing cables therefore should be avoided at all cost except where dictated by the local geometry being analyzed, e.g., near complex junctions. Thus, a thin-wire-like approximation is used, instead, to model the electromagnetic interactions with cables of subwavelength radius. To accurately model time-domain common-mode current variations, surface-wire junctions are modeled in a manner that ensures current continuity while avoiding fictitious charge accumulation at the junctions. A decomposition of the wire currents into differential-mode (transmission-line) and common-mode components is used to facilitate the modeling of electromagnetic coupling to and emissions from the wires as well as the effect of the coupled radiation on the functionality of the driver and receiver electronic circuits attached at the ends of the cable. The code resulting from this effort was applied to the analysis of a modulated feeder for a pair of log-periodic monopole arrays mounted on the trailing wing of an aircraft.

3. Domain Decomposition Methodologies for FEM-based Modeling of Planar, Multilayered, Signal Distribution Networks

A domain-decomposition assisted finite element modeling methodology is under development for expedient and robust electromagnetic modeling of the planar multi-layered signal distribution network of complex antenna arrays. The proposed methodology is aimed at the following two major obstacles associated with the finite element-based modeling of such networks. The first concerns the meshing of the signal distribution structure, a task that is hindered by the presence of multiple layers of metallization of fairly disparate layouts. The second concerns the convergence of finite element solvers, when dealing with structures exhibiting large variation in feature sizes from sub-wavelength features (e.g., the cross-sectional dimensions of the conductors and the substrate thickness) to multiple wavelengths.

The simple geometry depicted in Fig. 1 will be used to describe of the proposed domain decomposition methodology. The multi-layered structure considered contains four layers of metallization. The top-most layer contains two strip conductors used as the input and output for a bandpass filter. Also present in the top layer are two conducting pads used for via connections to the third metallization layer. The second layer is a solid ground plane. Shown in the plane are via holes through which the vias from the top layer connect to the third layer. The third metallization layer contains the single stripline resonator used in the filter. The fourth layer is a solid ground plane. Also shown are eight pins used to strap together the two ground planes.

__Boundary I (Domain I)__

**Figure 1**. An integrated bandpass filter. Shown on the right is the via hole through the top ground plane. The circular metallic boundary of the via hole and the via form a section of a coaxial cable.

For the purposes of electromagnetic analysis the structure is decomposed into two domains. Domain I is the volume occupied by the stripline structure bounded above and below by the two ground planes. Domain II involves the top microstrip structure (i.e., the first metallization layer), bounded below by a ground plane and above by a planar boundary used for truncation of the computational domain. Absorbing (radiation) boundary conditions are imposed on such a boundary for the proper modeling of the unbounded nature of the structure under investigation.

It is immediately recognized that, instead of using a single mesh for the discretization of the entire structure, two meshes, one for Domain I and one for Domain II can be utilized. In this manner mesh generation complexity is simplified considerably, since mesh generation in each domain is constrained only by the geometric attributes of the structure contained in the domain for which the mesh is constructed.

With a mesh constructed for each domain, the next issue concerns the way the finite element models for the two domains are interfaced (coupled) for the electromagnetic analysis of the entire structure. For the purpose of this paper, and considering the narrowband nature of the filter structure considered, it is clear that the two domains interact only through the two via holes in the ground plane. In other words, it is assumed that, for the bandwidth of interest to the analysis of this filter, the thickness of the ground plane is larger than the skin depth in the metal; hence, the fields on either side of the ground plane remain uncoupled everywhere except for the regions occupied by the via holes. The via hole region is shown enlarged in Fig. 1. Also indicated in the figure are Boundaries I and II, associated with the end cross sections of the coaxial cable formed by the via and the conducting wall of the via hole. Since the tangential electric and magnetic fields on the end cross sections can be expanded in terms of the modes of the circular coaxial waveguide, it is straightforward to develop, making use of the orthogonality properties of the modes and transmission line theory, two-port relationships between the coefficients of the modal expansions of the fields on the two boundaries. These relationships serve as the coupling relations between the finite element solutions in the two domains.

For the general case where the interface boundaries are such that they do not lend themselves to the application of modal field-based coupling relationships, Robin-Robin boundary conditions may be used toward the implementation of a robust iterative solution of the multi-domain finite element model.

Both cases are currently under investigation and results from their computer implementation will be presented in our next report.

**4. Fast Multipole Algorithm for Low-Frequency and Large Object Problems**