Frequency-amplitude relations for cross-coupled voltage-controlled oscillator

Dependence of frequency on amplitude and control bias is considered for the cross-coupled voltage-controlled oscillator. Closed form expressions are derived for frequency of oscillation as a function of amplitude, for positive and negative control bias voltages. Theory of nonlinear ordinary differential equations is utilized to show that the capacitance-voltage relation is the main cause of frequency shift with amplitude. Furthermore, the case of small amplitudes relative to control voltage is analyzed, and a closed form expression is derived for dependence of frequency on amplitude. This relation is then verified using the concept of effective capacitance. The effective capacitance approach is also used to extend the analysis to large voltage swings. Dependence of frequency on tuner control voltage is calculated for both bias polarities. Implications of the aforementioned equations for voltage-controlled oscillator performance are discussed. Numerical calculations and simulations are used to compare and verify the closed form equations, showing good agreement. Copyright (c) 2012 John Wiley & Sons, Ltd.