# Order-Reduction models for FIT

**Fundamentals:**

The discretization of electromagnetic devices with the Finite Integration Technique (FIT) leads in the general case, for an N-gridpoint discretization, to a linear system with about 6N unknowns. For many applications, this system can be solved in a fast and reliable manner in the time domain with a leap-frog approach, despite its large size of the order of millions. In the case of resonant structures however, the simulation time can be quite long, since the signal amplitude in the computational domain decreases very slowly. An alternative solution in frequency domain requires the solution of a large system for every desired frequency point, and is therefore also very time consuming.

Since it can be shown that the transfer properties of the structure can often be described through a much smaller number of parameters, the aim of the “Model Order Reduction” is to find a system of much smaller dimension which approximates the transfer behavior accurately enough.

**Model Order Reduction:**

The reduction of the model order is based on a formulation of the FIT system in the state space. The transfer function results from the large square system matrix, an excitation (input) matrix and an output matrix. The reduction takes place through the projection of the initial system on a Krylov subspace of smaller dimension. Usual methods herefor are Lanczos and Arnoldi. For the Lanczos method it can be shown that a direct relation exists between the reduced system and a Padú-Approximation of the transfer function (Padú Via Lanczos, PVL).

Besides the fast calculation of the transfer function, the reduced system provides a means to directly generate an equivalent circuit which approximately models the structure's behavior. This way, simulations of the coupling between the element and an external electric network can be readily achieved.

**Publications:**

- I. Munteanu, T. Wittig, T. Weiland, D. Ioan: FIT/PVL Circuit Parameter Extraction for General Electromagnetic Devices. IEEE Transactions on Magnetics, Volume 36, Issue 4, Part 1, July 2000, pp. 1421-1425.
- T. Wittig, I. Munteanu, R. Schuhmann, T. Weiland: Model Order Reduction with a Two-Step Lanczos Algorithm. Accepted for presentation at: Compumag 01, Evian, France.