Robust stability analysis of a phase-locked loop voltage-controlled oscillator circuit

Phase-locked loop circuits are widely used for communications applications such as frequency synthesizers for missile tracking. These systems often become unstable due to the vibrations and large g-forces experienced during a missile launch. We consider a phase-locked loop circuit modified by an additional RC low-pass filter for improved noise stability. To analyze the robust stability of this circuit, a parameter space analysis was performed with the components of the RC filter and the circuit gain K as adjustable parameters. The analysis shows the resistor and capacitor values that lead to stable circuit behavior for given values of gain and desired noise attenuation. In addition, we reduced the characteristic equation of the system to an affine polynomial by holding R constant, and evaluated its stability using the well-known edge theorem. Finally, we used the mapping theorem to obtain sufficient conditions for the stability of the original multilinear polynomial