phase detector characteristic

Consider a classical phase detector implemented with analog multiplier and low-pass filter.

File:Multuplier phase detector in time domain.svg

Here $f^1(\backslash theta^1(t))$ and $f^2(\backslash theta^2(t))$ denote high-frequency signals, piecewise differentiable functions $f^1(\backslash theta)$, $f^2(\backslash theta)$ represent waveforms of input signals, $\backslash theta^\{1,2\}(t)$ denote phases, and $g(t)$ denotes the output of the filter.

If $f^\{1,2\}(\backslash theta)$ and $\backslash theta^\{1,2\}(t)$ satisfy the high frequency conditions (see {{cite journal|author1=G. A. Leonov |author2=N. V. Kuznetsov |author3=M. V. Yuldashev |author4=R. V. Yuldashev |year = 2011|title = Computation of Phase Detector Characteristics in Synchronization Systems|journal = Doklady Mathematics|volume = 84|issue = 1|pages = 586–590|doi = 10.1134/S1064562411040223|s2cid=119897026 |url=http://www.math.spbu.ru/user/nk/PDF/2011-DAN-Phase-detector-characteristic-Nonlinear-analysis-PLL.pdf}}{{cite book|author1=N.V. Kuznetsov |author2=G.A. Leonov |author3=M.V. Yuldashev |author4=R.V. Yuldashev |title=ISSCS 2011 - International Symposium on Signals, Circuits and Systems |chapter=Analytical methods for computation of phase-detector characteristics and PLL design |date=2011 |pages = 7–10|doi = 10.1109/ISSCS.2011.5978639|isbn=978-1-61284-944-7 |s2cid=30208667 }}) then phase detector characteristic $\backslash varphi(\backslash theta)$ is calculated in such a way that time-domain model filter output

:$$

g(t) = \int\limits_0^t f^1(\theta^1(\tau))f^2(\theta^2(\tau))d \tau

and filter output for phase-frequency domain model

:$$

G(t) = \int\limits_0^t \varphi(\theta^1(\tau) - \theta^2(\tau))d \tau

are almost equal:

:$g(t)\; -\; G(t)\; \backslash approx\; 0$

:File:Pd mult.svg

Consider a simple case of harmonic waveforms $f^1(\backslash theta)=\backslash sin(\backslash theta),$ $f^2(\backslash theta)=\backslash cos(\backslash theta)$ and integration filter.

:$\backslash sin(\backslash theta^1(t))\backslash cos(\backslash theta^2(t))\; =\; \backslash frac\{1\}\{2\}\backslash sin(\backslash theta^1(t)\; +\; \backslash theta^2(t))\; +\; \backslash frac\{1\}\{2\}\backslash sin(\backslash theta^1(t)\; -\; \backslash theta^2(t))$

Standard engineering assumption is that the filter removes

the upper sideband $\backslash sin(\backslash theta^1(t)\; +\; \backslash theta^2(t))$ from

the input but leaves the lower sideband $\backslash sin(\backslash theta^1(t)\; -\; \backslash theta^2(t))$

without change.

Consequently, the PD characteristic in the case of sinusoidal waveforms is

:$$

\varphi(\theta) = \frac{1}{2}\sin(\theta).

Consider high-frequency square-wave signals $f^1(t)\; =\; \backslash sgn(\backslash sin(\backslash theta^1(t)))$ and $f^2(t)\; =\; \backslash sgn(\backslash cos(\backslash theta^2(t)))$.

For this signals it was found{{cite journal|author = G. A. Leonov|year = 2008|title = Computation of phase detector characteristics in phase locked loops for clock synchronization|journal = Doklady Mathematics|volume = 78|issue = 1|pages = 643–645|doi = 10.1134/S1064562408040443| s2cid=108828295 }} that similar thing takes place.

The characteristic for the case of square waveforms is

:$$

\varphi(\theta) = \begin{cases}

1+\frac{2\theta}{\pi}, & \text{if }\theta \in [-\pi,0],\\

1-\frac{2\theta}{\pi}, & \text{if }\theta \in [0,\pi].\\

\end{cases}

Consider general case of piecewise-differentiable waveforms $f^\{1\}(\backslash theta)$, $f^2(\backslash theta)$.

This class of functions can be expanded in Fourier series.

Denote by

:$$

a^p_i=\frac{1}{\pi}\int\limits_{-\pi}^{\pi} f^p(x)\sin(ix)dx,

$b^p\_i=\backslash frac\{1\}\{\backslash pi\}\backslash int\backslash limits\_\{-\backslash pi\}^\{\backslash pi\}\; f^p(x)\backslash cos(ix)dx,$

$c^p\_i=\backslash frac\{1\}\{\backslash pi\}\backslash int\backslash limits\_\{-\backslash pi\}^\{\backslash pi\}\; f^p(x)dx,\; p\; =\; 1,2$

the Fourier coefficients of $f^1(\backslash theta)$ and $f^2(\backslash theta)$.

Then the phase detector characteristic is

:$$

\varphi(\theta) = c^1c^2 + \frac{1}{2}\sum\limits_{l=1}^{\infty}\bigg((a^1_la^2_l + b^1_lb^2_l)\cos(l\theta) + (a^1_lb^2_l - b^1_la^2_l)\sin(l\theta)\bigg).

Obviously, the PD characteristic $\backslash varphi(\backslash theta)$ is periodic, continuous, and bounded on $\backslash mathbb\{R\}$.

Modeling method based on this result is described in Patent RU 2011113212/08(019571)

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Category:Detectors

Category:Electronic circuits