Remarks on the Adler´s equation

The phase equation of oscillators driven by weak signals at a frequency multiple of the free-running oscillation frequency is studied. This equation, describing the behavior of many injection locked circuits, extends the applicability of the well-known Adler´s equation, which is limited to forcing signals close to the free-running frequency. The exact solution of the phase equation in time domain is derived for pulling operation and its utility in the calculation of the frequency spectrum of the system response is shown with reference to a widely used divide-by-2 frequency divider. Finally, a comparison of the results obtained by presented formulas with the results obtained by numerical integration and by SPICE simulations based on BSIM3 models is presented.