Oblique–Oblique Projection in TLM-MOR for High-$Q$Structures

The nonsymmetric properties of the matrix statement of the transmission-line matrix (TLM) method require the application of general Krylov subspace methods for its model-order reduction (MOR). However, the utilization of the most representative type of such general Krylov subspace methods, namely, the Arnoldi algorithm, is computational expensive. On the other hand, the other popular method, namely, the classical nonsymmetric Lanczos algorithm, requires the transpose of the TLM matrix in order to form the bi-orthogonal basis utilized in its application; hence, its algorithmic simplicity is also penalized and its computational complexity is increased. We present in this paper a novel scattering-symmetric (S-symmetric) algorithm, which is used for the oblique projection of the TLM system. The S-symmetric Lanczos algorithm generates a bi-orthogonal basis by means of a single sequence like the symmetric Lanczos procedure. Thus, it is faster and consumes less memory in comparison to the conventional nonsymmetric Lanczos algorithm. However, the dimension of the resulting reduced TLM matrix can still be too large. Therefore, rather than directly applying the conventional eigenvalue decomposition to it, a second projection of the TLM system is performed in order to extract only those eigenvalues and associated eigenstates that are the most influential on the system response in the desirable frequency band. Such an oblique-oblique projection approach provides for TLM-based MOR in the most computationally efficient manner. The advantages of the proposed TLM-MOR process are demonstrated through its application to the electromagnetic analysis of high-Q filters and a patch antenna