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.

This paper describes a circuit analysis technique that includes all circuit elements used in transistor Colpitts quartz crystal oscillators. This technique is applied to a quartz crystal oscillator that has a tank circuit for selecting the oscillation frequency. The results obtained with this technique are compared with SPICE simulation results. Good agreement in the results clearly shows the validity of our technique.

This paper describes a simulation technique of start-up characteristics that considers a nonlinearity of the drive level of quartz crystal resonators. A nonlinear resonator model for SPICE where the resonant resistance varies with the voltage added to a resonator is proposed. In an examination using a transistor Colpitts oscillator, the simulation using this technique agreed with the experimental results very well.

A VLSI algorithm for division in GF(2m) with the canonical basis representation is proposed. It is based on the extended Binary GCD algorithm for GF(2m), and performs division through iteration of simple operations, such as shifts and bitwise exclusive-OR operations. A divider in GF(2m) based on the algorithm has a linear array structure with a bit-slice feature and carries out division in 2m clock cycles. The amount of hardware of the divider is proportional to m and the depth is a constant independent of m.