In this present communication we studied the frequency dependence of the effective electromagnetic parameters of left-handed and related meta-materials (NIMs) of the split ring resonator (SRR) and wire type. By varying the width of inserted material as well as the host material, we also calculate the effective parameters of NIM like effective permittivity and effective permeability. Using these parameters, we calculate the effective refractive index and impedance of the considered medium. Our results reveal that the negative index of materials can be achieved when the thickness of the host materials should equal to the thickness of the inserted materials.