  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 also shown that the $Q$ of the effective medium
response is essentially equal to the $Q$ of an individual resonant particle.
Thus, by measuring the $Q$ of a single fabricated metamaterial particle, the
effective permeability or permittivity of a metamaterial can be calculated
simply and accurately without requiring complex simulations, fabrication, or
measurements. Experimental validation shows that the 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. This 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 $\tan \delta \approx 0.001$
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.

Optics