Complementary variational formulation of Maxwell's equations in power series form
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Maxwell's equations in power series form have been formulated as complementary variational integrals. It has been found that each kth-order static-like fields can be formulated as two variational integrals. One of these integrals yields an upper bound to the stationary value while the other integral closes in from below. The general theory guarantees that the exact stationary value is always between the two integral values.;An illustrative example of a parallel-plate capacitor is discussed. It is shown that the zero-order trial field could be used to estimate the D.C. capacitance. Recognizing that the two variational functionals are "zero-order electric eneries," the D.C. capacitance was formulated as a quantity proportional to the functional value. Also, some general procedures were suggested whereby one can use the higher-order fields to estimate the frequency dependence of the capacitance.;The applicability of the dual extremum principles appears to include many problems of interest in electrical engineering. The continuing trend of miniaturizing circuit components allows circuit theory to be applied at tens of giga-Hertz and even more because of the small dimensions of the circuit compared to the wavelength. This seems to imply that in many microwave problems, useful information can be obtained by regarding the problems as almost static. These problems can then be recast as two variational integrals yielding both upper and lower bounds. Finally, it seems possible that numerical techniques, such as finite element methods, can be developed based on the two complementary functionals; this could lead to significant advantages over the existing methods of analyzing microwave problems.