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Dual log-spiral designs that represent inclusions at the elemental level, in the synthesis of metamaterials with negative values of permittivity (e) and permeability (µ), are analysed for their application in the X band frequency range based on the equivalent circuit model. The dual log-spiral resonator is useful when it comes to miniaturization for relevant applications. The extended LC model focuses on the impact of start angle in coupling factor and mutual inductance values. Estimation of Scattering parameters, bandwidth of operation and impedance of metamaterial samples is performed. Results from full wave simulation tests on this model are used to accurately design miniaturized artificial magnetic inclusion in a homogeneous material so as to have anomalous values of e and µ. Discussions on the effective role of Artificial Neural Network method as a potent optimization technique in design are also presented.