This paper describes a method for analyzing active impedance, i. e. equivalent resistance and equivalent reactance, of a narrow-band transistor Colpitts crystal oscillator. This oscillator, employing an AT-cut resonator filter, has a very narrow-band width and an achievement of extremely low phase-noise characteristics is expected. The analysis proposed is based on an algebraic formula, which employs a nonlinear approximation for transistor gm, and a simplified circuit model. Calculated results are compared with the experimental results in the frequency characteristics of the oscillator active impedance with changing the driving signal current. Good agreement between the calculation and experimental results shows that the proposed technique is suitable for designing Colpitts crystal oscillators with resonator filters. In addition we apply this technique to the analysis of dual-mode crystal oscillators.

A highly stable oven-controlled crystal oscillator (OCXO) with low phase-noise characteristics has been developed using a dual-mode SC-cut quartz crystal oscillator. The OCXO uses a conventional oven-control system for coarse compensation and a digital-correction system, which uses B-mode signal in an SC-cut resonator as a temperature sensor, for fine compensation. Combining these two forms of compensation greatly improves the stability of the C-mode frequency without requiring a double-oven system. The experimental results indicated that the frequency stability of the proposed OCXO, including the frequency-temperature hysteresis, is ten times better than that of a conventional, free-running OCXO. The results also indicated that the proposed OCXO has good frequency retraceability and low phase-noise characteristics.