# The electrostatic characterization of an n-element planar array using the singularity expansion method

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In this paper, the singularity expansion method (SEM) is used to describe the electrostatic charge distribution on an array of thin linear antennas placed in a uniform electric field. The SEM, which has primarily been used to analyze transient scattering problems, decomposes the electromagnetic interaction process into various quantities such as singularities and modes. Using the SEM, the step plane wave induced transient current on the array is expanded in terms of its singularities (poles) in the Laplace transform (complex frequency domain.) The continuity equation is applied to the induced current expression to obtain the transient charge. The electrostatic charge distribution on the array is found by using the final value theorem on the transient charge expression. It is well known that the SEM factorization of a single linear element reveals that a single pole exists in the fundamental resonance region (near omega L/c = pi, where L is the length of the scatterer). For a two-element array, two poles are observed in the fundamental resonance region. This trend continues such that an n-element array has n poles in the fundamental resonance region. Associated with each pole is a unique modal current and corresponding charge distribution. For example, one of the two fundamental resonance region poles of the two-element array produces half-wavelength sinusoidal current distributions whose directions are the same on one scatterer but opposite on the other. The remaining fundamental resonance region pole produces half-wavelength sinusoidal current distributions whose directions are the same on both scatterers. Corresponding to each mode is a coupling coefficient which determines how much a particular mode couples into the response. A generalization of these results for an n-element array will be given. Furthermore, the electric polarizability is derived in terms of the SEM electric charge description. The value of this research lies in the elegance and strength of the SEM to factor a problem into various quantities which depend on different variables of the problem. By using the SEM to analyse the n-element planar array, a much deeper comprehension of the fundamental aspects of the electrostatic interaction process is achieved.