All Conditions for Synchronous Harmonic Natural Oscillations in LC Circuits

The full set of conditions for synchronous harmonic oscillations in multiple LC circuits are established. These are for discrete lossless linear systems in source-free response. An Electric Networks literature review indicates that even the very basic condition for synchronous harmonic free oscillations is rarely presented, which is: initial conditions as an eingenvector for the initial charges and no currents (for
${t} =0$
). It is demonstrated herein that there are more conditions: three in total; in addition, simulation results are presented to illustrate all these possibilities, in which synchronous harmonic oscillation can actually be experienced. It is also noted here, that reactance or LC networks are correct models in Physics (pulsed power systems, metamaterials and metal-dielectric composites) and for Engineering devices (electrotherapy, ignition and radio).