Editing Phase-locked loop (section)

===Phase domain model of APLL===
Consider the input of PLL <math>f_1(\theta_1(t))</math> and VCO output
<math>f_2(\theta_2(t))</math> are high frequency signals. Then for any piecewise differentiable <math>2\pi</math>-periodic functions
<math>f_1(\theta)</math> and <math>f_2(\theta)</math> there is a function <math>\varphi(\theta)</math> such that the output <math>G(t)</math> of Filter
:<math>
\left\{
\begin{array}{rcl}
 \dot x &= &Ax + b \varphi(\theta_1(t) - \theta_2(t)), \\
 G(t) &= &c^{*}x,
\end{array}
\right.
\quad
x(0) = x_0,
</math>
in phase domain is asymptotically equal (the difference <math>G(t)- g(t)</math> is small with respect to the frequencies) to the output of the Filter in time domain model. 
<ref>{{cite journal
 | author = G. A. Leonov, N. V. Kuznetsov, M. V. Yuldashev, R. V. Yuldashev
 | year = 2012
 | title = Analytical method for computation of phase-detector characteristic
 | journal = IEEE Transactions on Circuits and Systems II: Express Briefs
 | volume = 59
 | issue = 10
 | pages = 633–637
 |url=http://www.math.spbu.ru/user/nk/PDF/2012-IEEE-TCAS-Phase-detector-characteristic-computation-PLL.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://www.math.spbu.ru/user/nk/PDF/2012-IEEE-TCAS-Phase-detector-characteristic-computation-PLL.pdf |archive-date=2022-10-09 |url-status=live
| last2 = Kuznetsov
 | last3 = Yuldashev
 | last4 = Yuldashev
 | doi = 10.1109/TCSII.2012.2213362
 | s2cid = 2405056
 }}</ref>
<ref>{{cite book
 | author = N.V. Kuznetsov, G.A. Leonov, M.V. Yuldashev, R.V. Yuldashev
 | year = 2011
 | pages = 7–10
 | doi = 10.1109/ISSCS.2011.5978639| last2 = Leonov
 | last3 = Yuldashev
 | last4 = Yuldashev
 | title = ISSCS 2011 - International Symposium on Signals, Circuits and Systems
 | chapter = Analytical methods for computation of phase-detector characteristics and PLL design
 | isbn = 978-1-61284-944-7
 | s2cid = 30208667
 | chapter-url = https://zenodo.org/record/889367
 }}</ref> 
Here function <math>\varphi(\theta)</math> is a [[phase detector characteristic]].

Denote by <math>\theta_{\Delta}(t)</math> the phase difference
:<math>\theta_{\Delta} = \theta_1(t) - \theta_2(t).</math>
Then the following [[dynamical system]] describes PLL behavior

:<math>
\left\{
\begin{array}{rcl}
 \dot x &= &Ax + b \varphi(\theta_{\Delta}),\\
 \dot \theta_{\Delta} &= & - g_v c^{*}x + \omega_{\Delta}. \\
\end{array}
\right.
\quad
x(0) = x_0, \quad \theta_{\Delta}(0) = \theta_{1}(0) - \theta_2(0).
</math>

Here <math>\omega_{\Delta} = \omega_1 - \omega_\text{free}</math>; <math>\omega_1</math> is the frequency of a reference oscillator (we assume that <math>\omega_\text{free}</math> is constant).

====Example====
Consider sinusoidal signals

:<math>f_1(\theta_1(t)) = A_1 \sin(\theta_1(t)), \quad f_2(\theta_2(t))
= A_2\cos(\theta_2(t))</math>

and a simple one-pole [[RC circuit]] as a filter. The time-domain model takes the form

:<math>
\left\{
\begin{align}
 \dot x &= -\frac{1}{RC}x + \frac{1}{RC}
A_1A_2\sin(\theta_1(t)) \cos(\theta_2(t)),\\[6pt]
 \dot \theta_2 &= g_v c^{*}x + \omega_\text{free}
\end{align}
\right.
</math>

PD characteristics for this signals is equal<ref>A. J. Viterbi, ''Principles of Coherent Communication'', McGraw-Hill, New York, 1966</ref> to

:<math>
 \varphi(\theta_1 - \theta_2) = \frac{A_1 A_2}{2}\sin(\theta_1 - \theta_2)
</math>

Hence the phase domain model takes the form

:<math>
\left\{
\begin{align}
 \dot x &= -\frac{1}{RC}x + \frac{1}{RC}\frac{A_1 A_2}{2}\sin(\theta_{\Delta}),\\[6pt]
 \dot \theta_{\Delta} &=  - g_v c^{*}x + \omega_{\Delta}.
\end{align}
\right.
</math>

This system of equations is equivalent to the equation of mathematical pendulum

:<math>
\begin{align}
x & = \frac{\dot\theta_2 - \omega_2}{g_v c^*} = \frac{\omega_1 - \dot\theta_{\Delta} - \omega_2}{g_v c^*},\\[6pt]
\dot x & = \frac{\ddot\theta_2}{g_v c^*},\\[6pt]
\theta_1 & = \omega_1 t + \Psi,\\[6pt]
\theta_{\Delta} & = \theta_1 -\theta_2,\\[6pt]
\dot\theta_{\Delta} & = \dot\theta_1 - \dot\theta_2 = \omega_1 - \dot\theta_2,\\[6pt]
& \frac{1}{g_v c^*}\ddot\theta_{\Delta} - \frac{1}{g_v c^* RC}\dot\theta_{\Delta} - \frac{A_1A_2}{2RC}\sin\theta_{\Delta} = \frac{\omega_2 - \omega_1}{g_v c^* RC}.
\end{align}
</math>