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The theory of characteristic modes (TCM) is presented for arbitrarily shaped 3-D structures including perfect electric conductors (PECs) and homogeneous penetrable objects. It is shown that by properly expressing the weighting operator of the generalized eigenvalue equation in terms of the integral operators related to the radiated fields, TCM can be formulated directly for the surface integral equation formulation of the problem. This avoids symmetrization or other modifications of the equations. The eigenvalues are shown to be related to radiated, reactive, and dissipated power, and the corresponding far-field patterns are orthogonal. The previously introduced TCM formulations for PEC are special cases of this generalized TCM.

Resonances, i.e., extrema of the eigenvalues of characteristic modes for closed perfectly conducting objects are investigated. The characteristic modal solutions based on the electric, magnetic, and combined field integral operators (EFIO, MFIO, and CFIO) are studied and compared with analytical solutions for a sphere. All these formulations are found to capture both external (radiating) and internal (cavity) resonances predicted by the analytical expressions. At the internal resonances, the eigenvalues obtained with the EFIO- and MFIO-based approaches are not correct, and the corresponding modes are nonunique. These solutions also exhibit a strong duality between the electric (TM) and magnetic (TE) type modes. A connection is found between the external and internal resonances and the condition numbers of the matrices. The modal expansion of the CFIO-based solution is correct, even though it also experiences the nonuniqueness of the EFIO- and MFIO-based solutions.