Mode dynamics in fourth-order LC oscillators

We present a complete analysis of single and concurrent modes in fourth-order LC-voltage-controlled oscillators (VCOs), which are increasingly applied in dual-band communication systems. We give a procedure based on the averaging method that simplifies the derivation of the abridged equations, which are derived without resorting to a change of co-ordinates. The amplitudes of the oscillatory modes in steady state and in transient are found in explicit form. Conditions for the stability of the single and concurrent modes are derived, which apply to any active one-port dual-band LC-VCO and allow one to predict the nonlinearities ensuring the occurrence of a stable concurrent mode. Numerical and experimental results show a good accuracy of the presented formulas.