A Phase and Amplitude Tunable Quadrature $LC$ Oscillator: Analysis and Design

This paper presents a new analytical approach to extract closed form equations for phase and amplitude imbalances raised from mismatches between two LC-tanks in quadrature oscillators. For more generality, we considered different coupling factors for the two coupled oscillators. This makes our analysis so general that we could design a phase and amplitude tunable quadrature oscillator accordingly. We show that choosing appropriate inversely proportional coupling factors makes it possible to have exactly zero phase and amplitude errors. Choosing so does not have any impacts on phase noise; indeed, the trade-off between phase noise and phase error in coupled quadrature oscillators has been broken here. We show that using differential stages for tuning tail currents of oscillating stages can somehow implement the inversely proportional coupling factors. Tuning the phase and amplitude errors, the frequency does not change and power consumption remains constant too; these are the advantages of the proposed circuit. The theoretical results and proposed quadrature oscillator are evaluated and confirmed through simulations using TSMC 0.18 model technology on a 5-GHz quadrature oscillator with current consumption of 4.2 mA at 1.8 V supply voltage.