Orthogonality Properties of Characteristic Modes for Lossy Structures

Abstract

Orthogonality of the characteristic modes with respect to the weight operator of the generalized eigenvalue equation, and in the far field, is investigated in the case of lossy conducting and dielectric objects. Linking the weight operator to radiated power is shown to provide orthogonal far fields in the lossless case. In the lossy case, both the orthogonality of the characteristic far fields and the weight operator orthogonality of the modal currents are satisfied with respect to the Hermitian inner products only for sufficiently symmetric geometries, such as a sphere. For irregular lossy shapes, independently of the symmetry of the formulation, the far-field orthogonality can be obtained only with respect to the symmetric (non-Hermitian) product. The weight operator orthogonality can be satisfied with (complex) symmetric formulations, but again only with respect to the symmetric product. Since the symmetric products are not related to any physical power quantity, the modes do not form a (radiated) power orthogonal set in the lossy case. Hence, for lossy structures CMs do not satisfy their classical definition and they need to be redefined.

Integral Operator-Based Characteristic Mode Theory for Conducting, Material, and Lossy Structures

“…Symmetrization, however, is not necessary and the same solutions are obtained with the asymmetric VIO, given in (95), and W = R V rad (96) [41]. In [42] an alternative power-based VIO formulation is proposed for dielectric-magnetic bodies.…”

“…If the object is both dielectric and magnetic, also the VIO (95) is an asymmetric operator. In the lossless case symmetrized VIO ( 12) is real and symmetric, and the GEE can still be expressed with its real and imaginary parts [41].…”

“…We note that even if both X reac and R diss are real-symmetric, or Hermitian, their sum, including multiplication with the imaginary unit i, is at most complex-symmetric. Thus, in the lossy case, excluding special symmetric objects such as a sphere, the eigenvectors are not generally power orthogonal [14], [41].…”

“…The far-field orthogonality is equal to the radiated power orthogonality [11], [12], [41]. Hence, in the lossless case this orthogonality is satisfied with the Hermitian inner product…”

“…In the lossy case the orthogonality relations are satisfied with symmetric products [41], [69], defined without the complex conjugate. Since the symmetric products do not agree with any physical power quantity, for lossy structures, the CMs do not form a radiated power orthogonal set of functions and their radiated fields are not Hermitian orthogonal [41], [69]. 10 VII.…”