AXIEM™, a 3D planar electromagnetic (EM) simulator from AWR Corp., was first announced in 2007 with innovative technology to solve large, complicated problems with unprecedented speed and accuracy. Released as an integral part of AWR's design environment, AXIEM provided designers with the ability to simulate large problems (more than 100,000 unknowns), send the layout of a circuit to AXIEM, simulate and seamlessly bring the results back into the circuit simulator.

The new 2010 release of AXIEM takes the speed, capacity and accuracy assets of the technology a step further by opening up the design space (literally). AXIEM 2010 has added post-processing capabilities for antenna analysis, including various field patterns, antenna currents, and, of course, feed impedances. It is now possible for designers to see the standard antenna parameters of interest, including gain and directivity, polarization, and power patterns. Planar antenna arrays are traditionally a formidable problem for planar EM simulators because of the large number of unknowns resulting from the meshing of the elements. AXIEM has the capability of solving problems with over 100,000 unknowns in minutes using compressed, iterative, matrix solution techniques. This feature is critical for large arrays, because it is simply not possible to solve these large problems with traditional direct matrix methods.

AXIEM's integration into AWR's design environment gives users a great deal of flexibility in how the tool is to be used. For example, portions of a circuit's layout can be automatically sent to AXIEM for EM analysis, and the resulting S-parameters can be automatically included in the circuit for further analysis. The method, called EM extraction, can be controlled in sophisticated ways. As such, AXIEM is ideal for planar antenna problems, as the designer can draw the antenna in the same layout as the rest of the circuit, have it automatically sent to AXIEM for analysis, and then have the results returned for further analysis.

**Simulation of Large Problems**

The bulk of the simulation time spent in a planar EM simulation is in solving the matrix. The matrix is formed by looking at the interactions between the mesh currents on the conductors of the circuit. Roughly speaking, if there are N meshes, the matrix created is a dense N × N matrix. Traditionally, it takes order N^{2} amount of time to create the matrix and order N^{3} amount of time to solve the matrix. The mathematical term "order of" means the following: If the number of cells is doubled, the fill time is four times longer (order N^{2}), and the solve time is eight times longer (order N^{3}). This is why the matrix solve dominates for large enough problems, roughly about 10,000 unknowns.

The problem for antenna designers is that a large array, for example 16 × 16 elements, can have well over 100,000 unknowns, and the N^{3} scaling of the solve time quickly renders the problem completely impractical to solve. Say it takes 10 seconds to solve 1000 unknowns at one frequency. The number of unknowns in the array is 100 times bigger (100,000 unknowns), and takes 10^{6} times longer to solve. Therefore, the solve time is 10^{7} seconds, which is about four months.

Fortunately, new iterative solution methods have been developed, where the matrix is compressed and solved by making repeated guesses or iterations to the solution. When properly applied, the methods can lead to matrix fill and solve times as fast as order Nln(N). The array problem mentioned in the previous paragraph can be solved in about 80 minutes per frequency point, which is a realistic simulation time for a practical design. The matrices that result from planar array problems typically have many blocks of elements that have about the same value. These blocks of elements correspond to groups of cells that are far away from one another. This is why the compressed, iterative methods work well.

The compression methods only work well if the matrix is properly conditioned. The mathematical term "conditioning of a matrix" refers to how easy it is to solve the matrix condition on a computer with finite precision arithmetic. The matrices generated in EM simulators are traditionally not well conditioned. It is therefore necessary to pre-condition the matrix. There are several ways to do this. A large part of AWR's 20 man-years of development investment in AXIEM has been devoted to perfecting the iterative solvers and pre-conditioners (for more information, read http://web.awrcorp.com/default.asp?docId=16033).

**Figure 1** An 8 × 16 patch array antenna with a corporate feed mesh (a) and close up for two elements in the array (b).

**Figure 1** shows an 8 × 16 patch array, which had a mesh of 96,000 elements. The total number of unknowns was 164,000. This array simulated in about 30 minutes per frequency point on an 8-core machine with 8 GB of RAM (AXIEM takes advantage of multi-core machines).

**Figure 2** Schematic of the antenna array and feed network.

**EM Extraction**

Planar antennas are normally coupled together with the associated feed network and drive circuitry. EM extraction is a technique where the antenna is coupled with the circuit in the schematic and its layout. The advantages are that the designer can see the antenna as part of the entire layout, the feed network of the antenna can be easily drawn and simulated, and the resulting S-parameter files from AXIEM are automatically incorporated back into the entire circuit simulation. **Figure 2** shows the schematic of an eight-element patch array and its associate Butler feed matrix. Note the element labeled "Extraction Block." This element can be configured to automatically send all, or part of, the layout of the schematic to AXIEM for simulation. The resulting S-parameters are used by the circuit for the final simulation. **Figure 3** shows the layout, and a 3D version of the layout.

**Figure 3** An eight-element array and Butler matrix feed.

**Antenna Measurements and Patterns**

AXIEM now supports a variety of antenna measurements, including the standard antenna pattern measurements for electric field as a function of sweep angles theta and phi. Both circular and polar antenna pattern measurements can be made. **Figure 4** shows the left hand and right hand polarization patterns in the principal planes for the 8 × 1 array fed by the Butler matrix. Two different scan angles are shown. Additionally, the designer can look at currents on the antennas, as well as the electric fields in various cross-sections of the circuit. This can be useful when trying to diagnose an impedance mismatch or phasing problem.

**Figure 4** 8 × 1 array antenna patterns for two different scan angles.

**Key 2010 AXIEM Features**

- Seamless integration with Microwave Office® and Analog Office® software
- Proprietary full-wave planar EM solver technology
- Advanced hybrid meshing technology
- Numerous source/excitations including auto-calibrated internal ports
- Parametric studies, optimization and tuning
- 3D visualization and animation
- Support for 64-bit PC platform and multi-core configurations
- Antenna capabilities, such as:
- Antenna measurements, including gain
- Patterns for linear, circular and elliptical polarizations
- Current visualization

**Conclusion**

Developed specifically to provide designers with a way to implement EM simulation as a practical part of the design flow rather than simply a back-end verification tool, AXIEM continues to fulfill its promise as a software product that offers cutting-edge technologies engineers need as they face the design challenges posed by today's complex wireless products. Highlighted in the new 2010 release of AXIEM are innovative antenna capabilities that are quickly becoming an indispensable tool for antenna designers.

**AWR Corp., El Segundo, CA (310) 726-3000, www.awrcorp.com RS No. 303**