A Probe-Based Harmonic Balance Method to Simulate Coupled Oscillators

The probe-based harmonic balance (HB) method is a well-known and largely used tool to compute the steady state behavior of autonomous circuits (oscillators). In this paper, the method is extended to analyze coupled oscillators, where the working frequencies and conditions, i.e., pulling and locking modes, have great relevance. It is shown that probe insertion can be considered as a specific matrix-bordering technique applied to the Jacobian matrix of the HB method. It transforms the original coupled autonomous system in a non-autonomous one, forcing the circuit to lock to the probes themselves. In this context, a novel approach to find the steady state solution is introduced. This approach exploits the properties of the power exchanged among the probes and the coupled oscillators. Possibly, more than one steady state solution with the oscillators working in pulling or locking modes can be found.