A phase detector characteristic is a function of phase difference describing the output of the
phase detector
A phase detector or phase comparator is a frequency mixer, analog multiplier or logic circuit that generates a signal which represents the difference in phase between two signal inputs.
The phase detector is an essential element of the phase- ...
.
For the analysis of
Phase detector
A phase detector or phase comparator is a frequency mixer, analog multiplier or logic circuit that generates a signal which represents the difference in phase between two signal inputs.
The phase detector is an essential element of the phase- ...
it is usually considered the models
of PD in signal (time) domain and phase-frequency domain.
In this case for constructing of an adequate nonlinear mathematical model of PD in phase-frequency domain it is necessary to find the characteristic of phase detector.
The inputs of PD are high-frequency signals and the output contains a low-frequency error correction signal, corresponding to a phase difference of input signals. For the suppression of high-frequency component of the output of PD (if such component exists) a low-pass filter is applied. The
characteristic of PD is the dependence of the signal at the
output of PD (in the phase-frequency domain) on the difference of phases
at the input of PD.
This characteristic of PD depends on the realization of PD and the types of waveforms of signals. Consideration of PD characteristic allows to apply averaging methods for high frequency oscillations and to pass from analysis and simulation of non autonomous models of phase synchronization systems in time domain to analysis and simulation of autonomous dynamical models in phase-frequency domain
.
[
]
Analog multiplier phase detector characteristic
Consider a classical phase detector implemented with analog multiplier and low-pass filter.
Here
and
denote high-frequency signals,
piecewise differentiable functions
,
represent
waveforms
In electronics, acoustics, and related fields, the waveform of a signal is the shape of its graph as a function of time, independent of its time and magnitude scales and of any displacement in time.David Crecraft, David Gorham, ''Electronic ...
of input signals,
denote phases, and
denotes the output of the filter.
If
and
satisfy the high frequency conditions (see ) then phase detector characteristic
is calculated in such a way that time-domain model filter output
:
and filter output for phase-frequency domain model
:
are almost equal:
:
:
Sine waveforms case
Consider a simple case of harmonic waveforms
and integration filter.
:
Standard engineering assumption is that the filter removes
the upper sideband
from
the input but leaves the lower sideband
without change.
Consequently, the PD characteristic in the case of sinusoidal waveforms is
:
Square waveforms case
Consider high-frequency square-wave signals
and
.
For this signals it was found that similar thing takes place.
The characteristic for the case of square waveforms is
:
General waveforms case
Consider general case of piecewise-differentiable waveforms
,
.
This class of functions can be expanded in Fourier series.
Denote by
:
the Fourier coefficients of
and
.
Then the phase detector characteristic is
:
Obviously, the PD characteristic
is periodic, continuous, and bounded on
.
Modeling method based on this result is described in
[Patent RU 2011113212/08(019571)]
Examples
References
{{reflist
Detectors
Electronic circuits