An exact formula for the maximum VCO sweep rate of a PLL

In many phase-locked loop (PLL) applications, the natural pull-in mechanism is too slow and unreliable, and it must be accelerated. By adding an externally-generated ramp to its control voltage input, the PLL voltage controlled oscillator (VCO) frequency can be swept towards the input reference frequency in an attempt to speed up the pull-in process. This popular acquisition aid has a significant limitation when it is used in a second-order, Type II PLL. If the applied ramp voltage has a slope magnitude greater than (alternatively, less than) some value R_m, the PLL state can (alternatively, cannot) sweep past the desired lock point, resulting in a phase lock failure (alternatively, success). In general, the maximum sweep rate magnitude R_m can be computed by using a numerical integration-and-search procedure that is described in the PLL literature. A special case exists for a second-order, Type II PLL that incorporates a triangular-characteristic phase detector (with logic-level binary input signals). For this case, it is possible to develop an exact, closed-form expression for R_m, the main result of this paper. For a range of loop parameters most often used in applications, R_m values are computed by using the exact formula, and these are used in two ways. First, they are used to validate the previously-mentioned numerical integration-and-search procedure. Second, they are compared to maximum sweep rate values computed for a PLL that utilizes a sinusoidal phase detector to show that the triangular-phase-detector PLL can be swept significantly faster than the sinusoidal-phase-detector loop.