Talk:Charge-pump phase-locked loop

The pull-in range (capture range, acquisition range) of CP-PLL
F. Gardner, in the conclusion of his pioneering work, 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."

By analogy with the Egan conjecture on the pull-in range of type 2 APLL, Amr M. Fahim noted about CP-PLL that "in order to have an infinite capture range, an active filter must be used for the loop filter".

B. Razavi noted regarding the CP-PLL that "We wish to develop a PLL that locks regardless of the initial value of the output frequency. Such a PLL would have an acquisition range equal to the VCO tuning range, with no limitations imposed by the loop bandwidth" and "The PFD concept eliminates the trade-off between the acquisition range and the loop bandwidth." , and that "the combination of a PFD and a charge pump offers two important advantages ... 1) the capture range is only limited by the VCO output frequency range".

In the book it is noted that "The charge-pump PLL has two poles at the origin (type-II) in its open-loop transfer function. It locks faster and its static phase error is zero if mismatches and leakages are negligible. Moreover, its capture range is only limited by its VCO tuning range".

nk (talk) 05:54, 3 October 2020 (UTC)

Gardner approximation
Gardner’s analysis is based on the following approximation: time interval on which PFD has non-zero state on each period of reference signal is
 * $$t_p = |\theta_e|/\omega_{\rm ref},\ \theta_e = \theta_{\rm ref} - \theta_{\rm vco}.$$

Then averaged output of charge-pump PFD is
 * $$i_d = I_p \theta_e/2\pi$$

with corresponding transfer function
 * $$I_d(s) = I_p\theta_e(s)/2\pi$$

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

\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 some sense, the loop operates on sampled basis and not as a straightforward continuous-time circuit. A sampled system almost always has more stability problems than arise in continuous-time systems. In particular, ananalog, second-order PLL is unconditionally stable for any value of loop gain, but the sampled equivalent will go unstable if the gain is made too large."

--Marat Yuldashev 12:21, 8 June 2020 (UTC)

Hold-in range of CP-PLL
Gardner's estimate of local stability (hold-in range):

\omega_{\rm ref} > \frac{1}{4}\left(K_{\rm vco}I_pR + \sqrt{(K_{\rm vco}I_pR)^2 + 4\frac{K_{\rm vco}I_p}{C}}\right) $$ plus VCO overload condition

\omega_{\rm ref} > K_{\rm vco}I_pR $$ According to Gardner "for any practical circuit" VCO overload condition dominates local stability one. — Preceding unsigned comment added by Maratyv (talk • contribs) 20:58, 4 June 2020 (UTC)