# Modal approaches for FIT

**Fundamentals:**

In the high-frequency technology, the components' behavior is often described by means of the so-called scattering parameters. A powerful technique for their calculation is a combination between the Finite Integration Technique (FIT) and the leap-frog-type time discretization, in which the structure is excited at one port and the output signal at another port is monitored, with the aid of special “waveguide boundary conditions” If the excitation is appropriately chosen as a broad-band signal, then the FFT of the signals provides broad-band S-parameters through a single calculation.

In many cases, this method is very efficient from both accuracy and performance points of view (state oft the art). If resonances of high Q-factors dominate the signal's behavior, then the computing time increases considerabley, because the energy stored in the computational domain decreases very slowly. Premature interruption of the calculation leaves some “trapped” energy components within the computation domain and leads to errors in the S-parameters.

In contrast to the time-domain method, the frequency-domain approach is characterized by the fact that a system of equations with complex unsymmetric matrix needs to be solved for every frequency point of interest. The use of such “direct” frequency-domain methods has therefore a quite limited applicability.

**Modal approach:**

The idea of the modal analysis is to take advantage of the weak point of the time-domain approach – the appearance of strong resonances. To this aim, the sought field solution is written as a series development of the structure's eigensolutions (see figure), and the problem is reduced to the determination of the weighting factors (series coefficients). Due to the problem's linear character and to the mode orthogonality, the weighting factors are easy to determine through the evaluation of an inner product. Once the eigensolutions are determined, this low-computational-complexity evaluation can be performed in as many frequency points as desired, yielding thus broad-band results.

An important feature is that, by using closed boundary conditions the eigenvalue problem remains real (for loss-free structures), and that typically only a small number of modes within the interesting frequency range needs to be determined.

The focus of the research is set on the extension of these methods to lossy problems, and to the coupling between the modal analysis and error estimation/correction approaches.

The method has many common features with approaches in the area of “Order Reduction”.

**Publications on this theme:**

- M. Dohlus, R. Schuhmann, T. Weiland: Calculation of Frequency Domain Parameters Using 3D Eigensolutions. Int. Journal of Numerical Modelling, Special Issue, Vol.12, 1999 , pp. 1-68.
- R. Schuhmann, P. Hammes, S. Setzer, B. Trapp, T. Weiland: A Modal Approach for the Calculation of Scattering Parameters in Lossfree and Lossy Structures Using the FI-Technique. Proc. of the 16th Annual Review of Progress in Applied Computational Electromagnetics (ACES 2000), Monterey, USA (2000), pp. 249-254.