Investigation of modes in double-tuned LC-VCOs

We present the results of an investigation of the oscillation modes settling in transformer-based double-tuned multi-frequency LC-VCOs, which are widely used in multiband communications systems. These are complex fourth-order systems with two nonlinearities, which exhibit stable single-frequency harmonic modes and a two-frequency quasi-harmonic mode. The amplitudes of oscillation modes in steady-state are derived in explicit form by using the harmonic balance method, whose applicability as an analytical solving method has been extended to these systems. Then, we derive the equations describing the first-order nonlinear dynamics (averaging equations), and the relative eigenvalues in explicit form. This allowed us to analyze the modes for stability determining conditions under which they are actually realized. The results found are confirmed by Spice simulations.