Variational-principle formulation of the two-term theory for arrays of cylindrical dipoles

For an array of identical, parallel, coplanar cylindrical dipoles, in which only one element is driven, the possibility of describing all of the current distributions accurately as linear combinations of only two qualitatively simple functions is considered. By constructing a variational-principle description of the array, a linear system consisting of two integral equations coupled to a set of algebraic equations is derived. The solution of this system gives the best possible representation of the currents that can be achieved using only two terms, in the sense that the variational-principle functional is stationary. For the circular array, it is shown that the algebraic part of the system can be solved explicitly and exhibits the expected behavior at resonance. Motivated by the two-term theory introduced by King in 1959 and improved and expanded by King et al. in 1968, simple approximate formulas are proposed for the two functions and some preliminary numerical results are presented. Finally, the relationship between the variational-principle formulation of the two-term theory and the formulation of King et al. is briefly discussed