$Q$-Based Design Equations and Loss Limits for Resonant Metamaterials and Experimental Validation

Practical design parameters of resonant metamaterials, such as loss tangent, are derived in terms of the quality factor Q of the resonant effective medium permeability or permittivity. Through electromagnetic simulations of loop-based resonant particles, it is first shown that the Q of the effective medium response is essentially equal to the Q of an individual resonant particle. This implies that by measuring the Q of a single fabricated metamaterial particle, the effective permeability or permittivity of a meta-material can be estimated simply and accurately without complex simulations, fabrication, or measurements. Experimental validation shows that the frequency-dependent complex permeability analytically estimated from the measured Q of a single fabricated self-resonant loop agrees with the complex permeability extracted from S parameter measurements of a metamaterial slab to better than 20 %. This Q equivalence reduces the design of a metamaterial to meet a given loss constraint to the simpler problem of the design of a resonant particle to meet a specific Q constraint. The Q-based analysis also yields simple analytical expressions for estimating the loss tangent of a planar loop magnetic metamaterial due to ohmic losses. It is shown that tan delta ap 0.001 is a strong lower bound for magnetic loss tangents for frequencies not too far from 1 GHz. The ohmic loss of the metamaterial varies inversely with the electrical size of the metamaterial particle, indicating that there is a loss penalty for reducing the particle size at a fixed frequency.