The RMS Model Cannot Capture PLL Small-Signal Instability

In this letter, the instantaneous differential equations of a simple grid-following converter connected to an infinite bus are derived. It is shown that the Root Mean Square model that assumes steady-state relationships between voltages and currents is unable to capture Phase-Locked Loop instability. This claim is backed with time-domain simulations, eigenvalue analysis, and by benchmarking the regions of attraction in the phase plane.