Effect of adding coils to dual-input twin-T quartz circuit

We derived an exact frequency equation and examined the behaviors of the effective Q factor and effective resistance for the case in which coils are added to a dual-input twin-T quartz circuit whose null frequency can be changed by adjusting the applied voltage ratio. In the circuit construction, two sets of T-circuits are used in which a quartz resonator is placed in the shunt branch and the output of each circuit settles in one terminal. The changeable frequency range exceeds the limit determined by the electromechanical coupling factor of the piezoelectric resonator used without any decrease in the effective Q factor and any increase in the effective resistance, within the entire changeable-frequency range when the resonance frequency of a coil and a capacitor in the series branch of the twin-T circuit is set to almost the same value as the resonance frequency of the resonators used in the twin-T circuit. The derived null-frequency equation and effective Q factor agreed with the experimental results for AT-cut quartz resonators.