Asymmetric Entrainment Structure of Pierce Oscillator Circuit

Experimental study is presented to uncover entrainment property of Pierce oscillator circuit (i.e., one of the most standard circuits for crystal oscillators) subject to sinusoidal external forcing. In contrast to the well-known Arnold tongues observed in a wide class of forced nonlinear systems, asymmetric shape was observed as an entrainment region of the circuit in two-dimensional parameter space spanned by the external forcing strength and frequency. Computation of the impedance curve revealed that the asymmetric shape is due to positive (inductive) or negative (capacitive) reactance of the Pierce circuit, which facilitates or inhibits the circuit oscillations. We extend this insight into two coupled Pierce circuits and show that, to efficiently achieve their synchronization, it is advantageous to drive a slow oscillator circuit by a fast circuit rather than to drive a fast circuit by a slow circuit. Our study provides a guideline on an optimal network configuration for coupled crystal oscillators, which may find applications in distributed digital clocks and wireless communication systems.