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Existing methods for solving the low-frequency breakdown problem associated with full-wave solvers rely on low-frequency approximations, which has left a number of research questions to be answered. The conductors are also generally treated as perfect conductors and the dielectric loss is not considered. In this work, a rigorous method that does not utilize low-frequency approximations is developed to eliminate the low frequency breakdown problem for the full-wave finite-element based analysis of general 3-D problems involving inhomogeneous lossless and/or lossy dielectrics and nonideal conductors. This method has been validated by the analysis of realistic on-chip circuits at frequencies as low as dc. Furthermore, it is applicable to both low and high frequencies. In this method, the frequency dependence of the solution to Maxwell's equations is explicitly and rigorously derived from dc to high frequencies. In addition to eliminating the low-frequency breakdown, such a theoretical model of the frequency dependence can be used to understand how the field solution, in a complicated 3-D problem with both lossless/lossy inhomogeneous dielectrics and nonideal conductors, should scale with frequency and at which frequency full-wave effects become important.