On the Theory of Quadrature Oscillations Obtained Through Parallel $LC$ VCOs

This paper presents a theory of parallel quadrature voltage-controlled oscillators, which are widely used to generate in-phase and quadrature signals in communication systems. The theory is developed without making any hypothesis on the circuit topology and allows us to thoroughly analyze the synchronized oscillations, as well as the effect of unavoidable parameter mismatches, by means of closed-form expressions for the amplitude and phase of oscillations and their orbital stability. The stability of oscillations is assessed analyzing the relevant eigenvalues, which are obtained in a closed form. The validity of the presented results has been verified through circuit simulations and experiments.