Editing Phase-locked loop (section)

===Time domain model of APLL===
The equations governing a phase-locked loop with an analog multiplier as the phase detector and linear filter may be derived as follows. Let the input to the phase detector be <math>f_1(\theta_1(t))</math> and the output of the VCO is <math>f_2(\theta_2(t))</math> with phases <math>\theta_1(t)</math> and <math>\theta_2(t)</math>. The functions <math>f_1(\theta) </math> and <math>f_2(\theta)</math> describe [[waveforms]] of signals. Then the output of the phase detector <math>\varphi(t)</math> is given by

:<math>\varphi(t) = f_1(\theta_1(t)) f_2(\theta_2(t))</math>

The VCO frequency is usually taken as a function of the VCO input
<math>g(t)</math> as
:<math>\dot\theta_2(t) = \omega_2(t) =
\omega_\text{free} + g_v g(t)\,</math>
where <math>g_v</math> is the ''sensitivity'' of the VCO and is expressed in Hz / V;
<math>\omega_\text{free}</math> is a free-running frequency of VCO.

The loop filter can be described by a system of linear differential equations
:<math>
\left\{
\begin{array}{rcl}
   \dot x & = & Ax + b \varphi(t), \\
   g(t) & = & c^{*}x,
\end{array}
\right.
\quad
x(0) = x_0,
</math>

where <math>\varphi(t)</math> is an input of the filter,
<math>g(t)</math> is an output of the filter, <math>A</math> is
<math>n</math>-by-<math>n</math> matrix,
<math>x \in \mathbb{C}^n,\quad b \in \mathbb{R}^n, \quad c \in
\mathbb{C}^n, \quad</math>. <math>x_0 \in \mathbb{C}^n</math> represents an initial state of the filter. The star symbol is a [[conjugate transpose]].

Hence the following system describes PLL
:<math>\left\{
\begin{array}{rcl}
 \dot x &= &Ax + b f_1(\theta_1(t)) f_2(\theta_2(t)),\\
 \dot \theta_2 &= & g_v c^{*}x + \omega_\text{free} \\
\end{array}
\right.
\quad
x(0) = x_0, \quad \theta_2(0) = \theta_0.
</math>
where <math>\theta_0</math> is an initial phase shift.