**Electromagnetic simulation has dramatically improved the design of a wide range of wireless systems by making it possible for engineers to simulate their operation and predict their performance without the need for building and testing of prototypes.**

Electromagnetic simulation has dramatically improved the design of a wide range of wireless systems by making it possible for engineers to simulate their operation and predict their performance without the need for building and testing of prototypes. But while a design concept can generally be simulated in far less time than it can be built or tested, simulation by itself still frequently requires modeling of a large number of alternatives without any assurance of achieving an optimized design.

Researchers at the Computational Electromagnetics Research Laboratory at McMaster University, Hamilton, Ontario, have addressed this challenge by developing an efficient method for optimizing the design of wireless systems based on sensitivity analysis of the scattering parameters (S-parameters). The basic idea is that the number of simulations required to identify an optimized design can be reduced by determining the design sensitivity with respect to its shape and material parameters. This makes it possible to iterate to an optimized design in much less time than was required in the past.

Electromagnetic simulation takes only a small fraction of the time and expense involved in building and testing wireless systems. Simulation also provides more information than physical experiments by yielding results at every point in the solution domain, far exceeding the results that can be achieved with physical measurements. The simulation itself provides an analysis of the system such that the results of each simulation provide insight into only a single point in the design space. To obtain a more complete picture of the design space, a series of simulations must normally be carried out. Optimization streamlines the process of exploring the design space.

**New Method Reduces Optimization Time**

Researchers Natalia Nikolova, Ying Li, Yan Li and Mohammed Bakr of the Computational Electromagnetics Laboratory at McMaster University have made a significant advancement in this area by developing a sensitivity analysis method that specifically targets wireless systems. "The new approach greatly reduces the number of simulations required to optimize the design by calculating the sensitivity or derivatives of the S-parameters with respect to changes in design variables," Nikolova said. "It uses a fast MATLAB algorithm to directly calculate the derivatives of the systems coefficients with respect to shape and material parameters. It represents a considerable advancement over the conventional sensitivity-analysis method which involves finding the difference between two system states, the nominal unperturbed state and the perturbed state in which a variable has been changed."

Traditional approaches usually require that at least one simulation be performed for each parameter, often requiring many calculations. Typical perturbations include changes in the dimensions of an object or changing the material properties such as conductivity, permittivity or permeability.

The researchers have addressed this challenge by developing a new method based on the fact that when a model is excited by a waveform by properly manipulating its spectral components, the time waveform is the same in forward and backward time. This makes it possible to compute the derivatives of the scattering matrix with respect to all the design parameters. This approach eliminates the need to perform time-consuming adjoint simulations. The sole requirement for the electromagnetic analysis software is the ability to export the field solution at user-defined points. The S-matrix and its derivatives with respect to all design parameters can be obtained from a single simulation. The overhead of the sensitivity calculation is negligible compared to the computational intensity of the full-wave simulation.

The researchers based their new method on XFDTD finite difference time domain (FDTD) electromagnetic simulation software from Remcom, Inc., State College, Pennsylvania, because of its ability to quickly and reliably turn complicated geometries into accurate electromagnetic meshes. This ability has been extended in XFDTD v6.2 with the addition of an advanced meshing algorithm that makes meshing of certain difficult geometry features possible. Adaptive meshing capabilities reduce solution times while maintaining high levels of accuracy by automatically adjusting the mesh to provide more cells in areas with high transients and reducing cells in areas where there is less variation. In addition, the use of a parallel computational code allows for multiple computers to be connected in order to perform calculations faster as well as use larger workspaces.

**Single Resonator Filter Validation Study**

The researchers have validated their method by optimizing several example designs with respect to shape parameters. They matched the results of their XFDTD self-adjoint sensitivity analysis against the more traditional and computationally intensive parameter perturbation and finite differencing of S-parameters. The first structure that they used to validate the method was a single resonator filter shown in Figure 1. The size of the computational domain is 60 x 1' x 2000 cells. The excitation is a sine wave modulated by a Gaussian pulse for a band-limited spectrum from approximately 3 to 5 GHz. Five current-density excitation points are uniformly distributed to form a half-sine modal distribution.

Figure 2 shows the derivatives of the real part of S11 with respect to the width of the spectrum in the frequency band from 3.5 to 4.5 GHz. The curve generated by the new method closely follows the conventional method estimates.

**H-plane Filter Validation Study**

The H-plane filter shown in Figure 3 was previously modeled by using an in-house method which solved an additional adjoint problem. Then, in a single XFDTD simulation, the researchers obtained the S-parameters and their derivatives with respect to all design variables. The grid of size 56 by 1 by 301 is uniform. The excitation is a Gaussian modulated sine with a spectrum from 5 to 10 GHz. Five current sources are placed uniformly across the port conforming to the half-sine modal distribution. Figure 4 shows the derivative of the insertion loss |S21| with respect to the width of the septum (W4) for a parameter sweep at 7.0 GHz. The agreement between the finite difference sensitivity curves and the new method is very good.

The memory requirements of the sensitivity analysis are mostly due to the field waveforms recorded at the perturbation points. The number of these points is roughly equal to the number of cells surrounding a perturbation surface or line. XFDTD can export the field recorded at user-defined points or probes, which is usually done through text files. XFDTD updates these text files after every iteration of the time-stepping algorithm, resulting in very small memory requirements. The memory and read/write time requirements can be practically eliminated if the sensitivity solver is integrated with the electromagnetic simulator for direct data exchange through random access memory (RAM).

**Versatile and Computationally Efficient Method**

The researchers have developed a versatile and computationally efficient method for S-parameter sensitivity analysis with transient EM solutions. The method is based on the finite-difference E-field vector wave equation and the second-order discrete sensitivity analysis approach. It is developed for, but not limited to, structured-grid field solutions such as those provided by FDTD-based simulators. With the new method, the S-parameter matrix and its derivatives with respect to all design parameters are obtained through a single system analysis with XFDTD. The overhead of the sensitivity computation is negligible compared to the computational intensity of the time-domain full-wave simulation.

Finally, the new method is simple to implement because adjoint simulations are not needed. This makes the new algorithm readily applicable to the solution of microwave design problems, which could greatly benefit from the sensitivity information such as optimization, device modeling, tolerance and yield analyses.

"In summary, our approach to S-parameter sensitivity analysis is far more computationally efficient than most other methods," Nikolova said. "Traditional methods require electromagnetic system analyses for each design variable while our approach can provide the same information more efficiently using one electromagnetic simulation."

For more information, contact Remcom, 315 South Allen St., Suite 222, State College, PA 16801; (814) 861-1299, (888) 773-6266 toll-Free in U.S. and Canada; info@remcom.com; www.remcom.com

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