Impedance-Based Closed-Form Expressions to Calculate Recoverable Energy and Corresponding Q of Single-Port Radiators

In this article, the recoverable energy and the corresponding quality factor Q of a single-port radiator are determined based on first principles. The final equations obtained express the recoverable energy and its Q solely in terms of the traditional frequency-domain impedance of the radiator. These closed-form expressions are very straightforward and combine two terms. The first term involves the derivative with respect to frequency of the reactance. This term is known since long to be related to the reactive energy of a radiating system. The second term is new and involves the product of the resistance and a time duration that is characteristic for the considered radiator. This second term shows two very important properties. First, it makes the total expressions semidefinite, and second, it is negligible for small lossless radiators. It is proved that for structures with highly separated resonances, the concept of recoverable energy is strongly linked to the calculation procedure for Q based on the frequency derivative of the impedance. Finally, for the first time, recoverable energy is calculated for a fabricated antenna, based on measurements.