A first linear mathematical model of second-order CP-PLL was suggested by
F. Gardner in 1980. and then refined by N. Kuznetsov et al. in 2019. The closed form mathematical model of CP-PLL taking into account the VCO overload is derived in. These mathematical models of CP-PLL allow to get analytical estimations of the hold-in range (a maximum range of the input signal period such that there exists a locked state at which the VCO is not overloaded) and the pull-in range (a maximum range of the input signal period within the hold-in range such that for any initial state the CP-PLL acquires a locked state).). Following Gardner's results, by analogy with
the Egan conjecture on the pull-in range of type 2 APLL, Amr M. Fahim conjectured in his book that in order to have an infinite pull-in(capture) range, an
active filter must be used for the loop filter in CP-PLL (Fahim-Egan's conjecture on the pull-in range of type II CP-PLL).
Continuous time nonlinear model of the second order CP-PLL Without loss of generality it is supposed that trailing edges of the VCO and Ref signals occur when the corresponding phase reaches an integer number. Let the time instance of the first trailing edge of the Ref signal is defined as t = 0. The PFD state i(0) is determined by the PFD initial state i(0-), the initial phase shifts of the VCO \theta_{vco}(0) and Ref \theta_{ref}(0) signals. The relationship between the input current i(t) and the output voltage v_F(t) for a proportionally integrating (perfect PI) filter based on resistor and
capacitor is as follows : \begin{align} v_F(t) = v_c(0) + Ri(t) + \frac{1}{C}\int\limits_0^t i(\tau)d\tau \end{align} where R>0 is a resistance, C>0 is a capacitance, and v_c(t) is a capacitor charge. The control signal v_F(t) adjusts the VCO frequency: : \begin{align} \dot\theta_{vco}(t) = \omega_{vco}(t) = \omega_{vco}^{\text{free}} + K_{vco}v_F(t), \end{align} where \omega_{vco}^{\text{free}} is the VCO free-running (quiescent) frequency (i.e. for v_F(t)\equiv 0), K_{vco} is the VCO gain (sensivity), and \theta_{vco}(t) is the VCO phase. Finally, the continuous time nonlinear mathematical model of CP-PLL is as follows : \begin{align} \dot v_c(t) = \tfrac{1}{C}i(t), \quad \dot\theta_{vco}(t) = \omega_{vco}^{\text{free}} + K_{vco} ( Ri(t) + v_c(t) ) \end{align} with the following discontinuous piece-wise constant nonlinearity : i(t) = i\big(i(t-), \theta_{ref}(t), \theta_{vco}(t)\big) and the initial conditions \big(v_c(0), \theta_{vco}(0)\big). This model is a nonlinear, non-autonomous, discontinuous, switching system.
Discrete time nonlinear model of the second-order CP-PLL The reference signal frequency is assumed to be constant: \theta_{ref}(t) = \omega_{ref}t = \frac{t}{T_{ref}}, where T_{ref}, \omega_{ref} and \theta_{ref}(t) are a period, frequency and a phase of the reference signal. Let t_0 = 0. Denote by t_0^{\rm middle} the first instant of time such that the PFD output becomes zero (if i(0)=0, then t_0^{\rm middle}=0) and by t_1 the first trailing edge of the VCO or Ref. Further the corresponding increasing sequences \{t_k\} and \{t_k^{\rm middle}\} for k=0,1,2... are defined. Let t_k . Then for t \in [t_k,t_k^{\rm middle}) the \text{sign}(i(t)) is a non-zero constant (\pm1). Denote by \tau_k the PFD pulse width (length of the time interval, where the PFD output is a non-zero constant), multiplied by the sign of the PFD output: i.e. \tau_k = (t_k^{\rm middle} - t_k)\text{sign}(i(t)) for t \in [t_k,t_k^{\rm middle}) and \tau_k = 0 for t_k=t_k^{\rm middle} . If the VCO trailing edge hits before the Ref trailing edge, then \tau_k and in the opposite case we have \tau_k > 0, i.e. \tau_k shows how one signal lags behind another. Zero output of PFD i(t) \equiv 0 on the interval (t_k^{\rm middle},t_{k+1}): v_F(t) \equiv v_k for t \in [t_k^{\rm middle},t_{k+1}) . The transformation of variables (\tau_k,v_k) to p_k = \frac{\tau_k}{T_{\rm ref}}, u_k=T_{\rm ref} ( \omega_{\rm vco}^{\text{free}} + K_{\rm vco}v_k ) - 1, allows to reduce the number of parameters to two: \alpha = K_{\rm vco}I_pT_{\rm ref}R, \beta = \frac{K_{\rm vco}I_pT_{\rm ref}^2}{2C}. Here p_k is a normalized phase shift and u_k+1 is a ratio of the VCO frequency \omega_{\rm vco}^{\text{free}} + K_{\rm vco}v_k to the reference frequency \frac{1}{T_{\rm ref}}. Finally, the discrete-time model of second order CP-PLL without the VCO overload : \begin{align} & u_{k+1} = u_k +2\beta p_{k+1},\\ & p_{k+1} = \begin{cases} \frac{-(u_k + \alpha + 1) + \sqrt{(u_k + \alpha + 1)^2 - 4\beta c_k}}{2\beta}, \quad \text{ for } p_k \geq 0, \quad c_k \leq 0, \\ \frac{1}{ u_k + 1} -1 + ( p_k \text{ mod }1), \quad \text{ for } p_k \geq 0, \quad c_k > 0, \\ l_k-1, \quad \text{ for } p_k 1, \end{cases} \end{align} where : \begin{align} c_k = (1 - ( p_k \text{ mod }1))( u_k +1) - 1, S_{l_k} = -( u_k - \alpha + 1 ) p_k + \beta p_k^2, l_k = \frac{1 - (S_{l_k} \text{ mod }1)}{ u_k + 1}, d_k = (S_{l_k} \text{ mod }1) + u_k. \end{align} This discrete-time model has the only one steady state at (u_k=0,p_k=0) and allows to estimate the hold-in and pull-in ranges. == References ==