On the DC-Settling Process of the Pierce Crystal Oscillator in Start-Up

Abstract

This brief presents an analytical treatment of the initial motional current <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${i}_{m}{(}{0}{)}$ </tex-math></inline-formula> of the Pierce crystal oscillator (XO), which is critical for determining its start-up time <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${T}_{s}$ </tex-math></inline-formula> . Specifically, the limitation of energy injection to boost <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${i}_{m}{(}{0}{)}$ </tex-math></inline-formula> is studied, and the dc-settling process (DCS) at the dawn of the start-up for stimulating <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${i}_{m}{(}{0}{)}$ </tex-math></inline-formula> is shown. The phases of the DCS process are evaluated under different operation regions, and the theoretical implications are confirmed by detailed simulations. A design guide for an optimal DCS start-up is also proposed, allowing <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${i}_{m}{(}{0}{)}$ </tex-math></inline-formula> -boosting via simple voltage and timing controls.