On the PLL spectral purity problem

It is well known that the spectral purity of a PLL's VCO is degraded by phase detector products. An analytical solution of this problem is given here for the PLL containing an analog multiplier type phase detector with sinusoidal inputs. The theory is applied to a PLL with a proportional plus integral loop filter. The results are compared to those obtained experimentally and by an approximate method that ignores loop dynamics.