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Optimisation techniques are currently being used to design antennas with Q values close to theoretical limits. In order to evaluate how closely these designs approach the limits, it is necessary to calculate their Qs from measured or simulated data. Many standard techniques are accurate for large Qs, but yield values significantly below the actual Q for low Q cases. In this study, equations are derived to bracket the actual value of Q, and the issues involved in attempting to determine an exact value are discussed. Both time-averaged and time-varying energies are considered. The derivations and conclusions are different from those previously published. Data from electromagnetic simulations of design examples are used to illustrate.

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Circuit models for the lowest-order modes radiating from within an arbitrarily-shaped envelope have been developed based on the exact circuits for spherical modes and the radar scattering from a conductor of the arbitrary shape. These circuits predict lower bounds on the Q of small antennas with any combination of coupling between modes. Circuit parameters are tabulated for a range of shapes, and the procedure for computing parameters for other shapes is explained. The bounds on a small circularly-polarized cylindrical antenna illustrate an application. Predictions for scattering by the arbitrary envelope filled with a high or material serve to validate the procedure.

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