Q-Factor of the Resonators With Frequency Dependent Negative Resistor

The frequency dependent impedance of Frequency Dependent Negative Resistor (FDNR) or D-element is calculated using frequency dependent open-loop gain of the operational amplifiers violet and finite crossover frequency. The general analytical result for the frequency dependent impedance of FDNR gives approximative formula for the Q-factor if FDNR is part of a resonance circuit. The presented analytical result is illustrated by an experiment with a contemporary low noise and high speed operational amplifiers (Op Amp). The purpose of this study is to clarify which factors determines the Q-factors of resonators with FDNR. The derived formula for the Q-factor reveals the novelty of our result – the Q-factor is determined by the nonideality parameters of Op Amps, finite static open loop gain, and frequency dependent open-loop gain determined by the finite crossover frequency. The measurement reveals the frequency dependence of a simple resonance filter. The derived formulas can be used for precise calculation of active filters with FDNR which perhaps are the main application of D-elements. The derived analytical formulas reveal how FDNR can be optimized in order resonators with them to have Q-factor comparable with the quartz resonators which opens widest applications in this direction.