Continuous time linear model of the second order CP-PLL and Gardner's conjecture

Gardner's analysis is based on the following approximation:^[2] time interval on which PFD has non-zero state on each period of reference signal is

${\displaystyle t_{p}=|\theta _{e}|/\omega _{\rm {ref}},\ \theta _{e}=\theta _{\rm {ref}}-\theta _{\rm {vco}}.}$

Then averaged output of charge-pump PFD is

${\displaystyle i_{d}=I_{p}\theta _{e}/2\pi }$

with corresponding transfer function

${\displaystyle I_{d}(s)=I_{p}\theta _{e}(s)/2\pi }$

Using filter transfer function ${\displaystyle F(s)=R+{\frac {1}{Cs}}}$ and VCO transfer function ${\displaystyle \theta _{\rm {vco}}(s)=K_{\rm {vco}}I_{d}(s)F(s)/s}$ one gets Gardner's linear approximated average model of second-order CP-PLL

${\displaystyle {\frac {\theta _{e}(s)}{\theta _{\rm {ref}}(s)}}={\frac {2\pi s}{2\pi s+K_{\rm {vco}}I_{p}\left(R+{\frac {1}{Cs}}\right)}}.}$

In 1980, F. Gardner, based on the above reasoning, conjectured that transient response of practical charge-pump PLL's can be expected to be nearly the same as the response of the equivalent classical PLL^[2]: 1856 (Gardner's conjecture on CP-PLL^[7]). 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^[8]: 6 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 ${\displaystyle t=0}$. The PFD state ${\displaystyle i(0)}$ is determined by the PFD initial state ${\displaystyle i(0-)}$, the initial phase shifts of the VCO ${\displaystyle \theta _{vco}(0)}$ and Ref ${\displaystyle \theta _{ref}(0)}$ signals.

The relationship between the input current ${\displaystyle i(t)}$ and the output voltage ${\displaystyle v_{F}(t)}$ for a proportionally integrating (perfect PI) filter based on resistor and capacitor is as follows

${\displaystyle {\begin{aligned}v_{F}(t)=v_{c}(0)+Ri(t)+{\frac {1}{C}}\int \limits _{0}^{t}i(\tau )d\tau \end{aligned}}}$

where ${\displaystyle R>0}$ is a resistance, ${\displaystyle C>0}$ is a capacitance, and ${\displaystyle v_{c}(t)}$ is a capacitor charge. The control signal ${\displaystyle v_{F}(t)}$ adjusts the VCO frequency:

${\displaystyle {\begin{aligned}{\dot {\theta }}_{vco}(t)=\omega _{vco}(t)=\omega _{vco}^{\text{free}}+K_{vco}v_{F}(t),\end{aligned}}}$

where ${\displaystyle \omega _{vco}^{\text{free}}}$ is the VCO free-running (quiescent) frequency (i.e. for ${\displaystyle v_{F}(t)\equiv 0}$), ${\displaystyle K_{vco}}$ is the VCO gain (sensivity), and ${\displaystyle \theta _{vco}(t)}$ is the VCO phase. Finally, the continuous time nonlinear mathematical model of CP-PLL is as follows

${\displaystyle {\begin{aligned}{\dot {v}}_{c}(t)={\tfrac {1}{C}}i(t),\quad {\dot {\theta }}_{vco}(t)=\omega _{vco}^{\text{free}}+K_{vco}(Ri(t)+v_{c}(t))\end{aligned}}}$

with the following discontinuous piece-wise constant nonlinearity

${\displaystyle i(t)=i{\big (}i(t-),\theta _{ref}(t),\theta _{vco}(t){\big )}}$

and the initial conditions ${\displaystyle {\big (}v_{c}(0),\theta _{vco}(0){\big )}}$. This model is a nonlinear, non-autonomous, discontinuous, switching system.

Discrete time nonlinear model of the second-order CP-PLL

Time intervals of the PFD dynamics

The reference signal frequency is assumed to be constant: ${\displaystyle \theta _{ref}(t)=\omega _{ref}t={\frac {t}{T_{ref}}},}$ where ${\displaystyle T_{ref}}$, ${\displaystyle \omega _{ref}}$ and ${\displaystyle \theta _{ref}(t)}$ are a period, frequency and a phase of the reference signal. Let ${\displaystyle t_{0}=0}$. Denote by ${\displaystyle t_{0}^{\rm {middle}}}$ the first instant of time such that the PFD output becomes zero (if ${\displaystyle i(0)=0}$, then ${\displaystyle t_{0}^{\rm {middle}}=0}$) and by ${\displaystyle t_{1}}$ the first trailing edge of the VCO or Ref. Further the corresponding increasing sequences ${\displaystyle \{t_{k}\}}$ and ${\displaystyle \{t_{k}^{\rm {middle}}\}}$ for ${\displaystyle k=0,1,2...}$ are defined. Let ${\displaystyle t_{k}<t_{k}^{\rm {middle}}}$. Then for ${\displaystyle t\in [t_{k},t_{k}^{\rm {middle}})}$ the ${\displaystyle {\text{sign}}(i(t))}$ is a non-zero constant (${\displaystyle \pm 1}$). Denote by ${\displaystyle \tau _{k}}$ 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. ${\displaystyle \tau _{k}=(t_{k}^{\rm {middle}}-t_{k}){\text{sign}}(i(t))}$ for ${\displaystyle t\in [t_{k},t_{k}^{\rm {middle}})}$ and ${\displaystyle \tau _{k}=0}$ for ${\displaystyle t_{k}=t_{k}^{\rm {middle}}}$. If the VCO trailing edge hits before the Ref trailing edge, then ${\displaystyle \tau _{k}<0}$ and in the opposite case we have ${\displaystyle \tau _{k}>0}$, i.e. ${\displaystyle \tau _{k}}$ shows how one signal lags behind another. Zero output of PFD ${\displaystyle i(t)\equiv 0}$ on the interval ${\displaystyle (t_{k}^{\rm {middle}},t_{k+1})}$: ${\displaystyle v_{F}(t)\equiv v_{k}}$ for ${\displaystyle t\in [t_{k}^{\rm {middle}},t_{k+1})}$. The transformation of variables^[9] ${\displaystyle (\tau _{k},v_{k})}$ to ${\displaystyle p_{k}={\frac {\tau _{k}}{T_{\rm {ref}}}},u_{k}=T_{\rm {ref}}(\omega _{\rm {vco}}^{\text{free}}+K_{\rm {vco}}v_{k})-1,}$ allows to reduce the number of parameters to two: ${\displaystyle \alpha =K_{\rm {vco}}I_{p}T_{\rm {ref}}R,\beta ={\frac {K_{\rm {vco}}I_{p}T_{\rm {ref}}^{2}}{2C}}.}$ Here ${\displaystyle p_{k}}$ is a normalized phase shift and ${\displaystyle u_{k}+1}$ is a ratio of the VCO frequency ${\displaystyle \omega _{\rm {vco}}^{\text{free}}+K_{\rm {vco}}v_{k}}$ to the reference frequency ${\displaystyle {\frac {1}{T_{\rm {ref}}}}}$. Finally, the discrete-time model of second order CP-PLL without the VCO overload^[4][6]

${\displaystyle {\begin{aligned}&u_{k+1}=u_{k}+2\beta p_{k+1},\\&p_{k+1}={\begin{cases}{\frac {-(u_{k}+\alpha +1)+{\sqrt {(u_{k}+\alpha +1)^{2}-4\beta c_{k}}}}{2\beta }},\quad {\text{ for }}p_{k}\geq 0,\quad c_{k}\leq 0,\\{\frac {1}{u_{k}+1}}-1+(p_{k}{\text{ mod }}1),\quad {\text{ for }}p_{k}\geq 0,\quad c_{k}>0,\\l_{k}-1,\quad {\text{ for }}p_{k}<0,\quad l_{k}\leq 1,\\{\frac {-(u_{k}+\alpha +1)+{\sqrt {(u_{k}+\alpha +1)^{2}-4\beta d_{k}}}}{2\beta }},\quad {\text{ for }}p_{k}<0,\quad l_{k}>1,\end{cases}}\end{aligned}}}$

where

${\displaystyle {\begin{aligned}c_{k}=(1-(p_{k}{\text{ mod }}1))(u_{k}+1)-1,S_{l_{k}}=-(u_{k}-\alpha +1)p_{k}+\beta p_{k}^{2},l_{k}={\frac {1-(S_{l_{k}}{\text{ mod }}1)}{u_{k}+1}},d_{k}=(S_{l_{k}}{\text{ mod }}1)+u_{k}.\end{aligned}}}$

This discrete-time model has the only one steady state at ${\displaystyle (u_{k}=0,p_{k}=0)}$ and allows to estimate the hold-in and pull-in ranges.^[6]

If the VCO is overloaded, i.e. ${\displaystyle {\dot {\theta }}_{\rm {vco}}(t)}$ is zero, or what is the same: ${\displaystyle (p_{k}>0,u_{k}<2\beta p_{k}-1)}$ or ${\displaystyle (p_{k}<0,u_{k}<\alpha -1)}$, then the additional cases of the CP-PLL dynamics have to be taken into account.^[5] For any parameters the VCO overload may occur for sufficiently large frequency difference between the VCO and reference signals. In practice the VCO overload should be avoided.

Nonlinear models of high-order CP-PLL

Derivation of nonlinear mathematical models of high-order CP-PLL leads to transcendental phase equations that cannot be solved analytically and require numerical approaches like the classical fixed-point method or the Newton-Raphson approach.^[10]