Event detection occurrence for planar piece-wise affine hybrid systems

Nonlinear behavioural model of charge pump PLLs

“…By considering the filter equations and Equations (8) and (2), we obtain Note that 0≤'≤1 and that 0≤t≤1 is a periodic sawtooth function of period T ref .…”

“…In this case, the PLL no longer works correctly and after only about 90 ms, the tuning voltage saturates at the lower allowed value. This behaviour can be roughly justified by noticing that the equation that governs the variation of the output frequency of the VCO (see Equation (8) in Appendix A), in the extreme case of a = 0 determines a dependence of dΦ vco dt on the integral of v w . By considering the linear model of the PLL, this introduces a second pole in the loop gain of the linear model, which is not compensated by the CP output filter in Figure 1 and this leads to an 'unstable' operation of the PLL.…”

“…2. The phase error, defined in Equation (2) Note that the simplest way to implement the VCO characteristic described by (8) is through a block that generates the signal…”

“…Equation (10) defines how the possibly periodic phase error function' is compared with the reference waveformt and how current is thus injected in the CP output filter. By considering the filter equations and Equations (8) and (2), we obtain…”

“…[1][2][3][4]) and papers (e.g. [5][6][7][8][9][10][11][12][13][14][15][16][17][18] and references therein) have been published on this subject. PLLs are inherently nonlinear circuits, clocked by an external reference signal: the dynamic behaviour that drives the PLL to switch from a pulling to a locking working condition, the fact that the PLL must remain locked against suitable frequency variations of the reference signal, and the determination of the effects owing to possible spurs introduced by the CP and by a fractional frequency divider (FD) [17,19,20] are key aspects of the PLL design.…”