Frequency and phase measurement
An important electrical quantity with no
equivalent in DC circuits is frequency. Frequency measurement is
very important in many applications of alternating current, especially
in AC power systems designed to run efficiently at one frequency and one
frequency only. If the AC is being generated by an electromechanical
alternator, the frequency will be directly proportional to the shaft
speed of the machine, and frequency could be measured simply by
measuring the speed of the shaft. If frequency needs to be measured at
some distance from the alternator, though, other means of measurement
will be necessary.
One simple but crude method of frequency
measurement in power systems utilizes the principle of mechanical
resonance. Every physical object possessing the property of elasticity
(springiness) has an inherent frequency at which it will prefer to
vibrate. The tuning fork is a great example of this: strike it once and
it will continue to vibrate at a tone specific to its length. Longer
tuning forks have lower resonant frequencies: their tones will be lower
on the musical scale than shorter forks.
Imagine a row of progressively-sized
tuning forks arranged side-by-side. They are all mounted on a common
base, and that base is vibrated at the frequency of the measured AC
voltage (or current) by means of an electromagnet. Whichever tuning fork
is closest in resonant frequency to the frequency of that vibration will
tend to shake the most (or the loudest). If the forks' tines were flimsy
enough, we could see the relative motion of each by the length of the
blur we would see as we inspected each one from an end-view perspective.
Well, make a collection of "tuning forks" out of a strip of sheet metal
cut in a pattern akin to a rake, and you have the vibrating reed
frequency meter:
The user of this meter views the ends of
all those unequal length reeds as they are collectively shaken at the
frequency of the applied AC voltage to the coil. The one closest in
resonant frequency to the applied AC will vibrate the most, looking
something like this:
Vibrating reed meters, obviously, are not
precision instruments, but they are very simple and therefore easy to
manufacture to be rugged. They are often found on small engine-driven
generator sets for the purpose of setting engine speed so that the
frequency is somewhat close to 60 (50 in Europe) Hertz.
While reed-type meters are imprecise,
their operational principle is not. In lieu of mechanical resonance, we
may substitute electrical resonance and design a frequency meter using
an inductor and capacitor in the form of a tank circuit (parallel
inductor and capacitor). One or both components are made adjustable, and
a meter is placed in the circuit to indicate maximum amplitude of
voltage across the two components. The adjustment knob(s) are calibrated
to show resonant frequency for any given setting, and the frequency is
read from them after the device has been adjusted for maximum indication
on the meter. Essentially, this is a tunable filter circuit which is
adjusted and then read in a manner similar to a bridge circuit (which
must be balanced for a "null" condition and then read).
This technique is a popular one for
amateur radio operators (or at least it was before the advent of
inexpensive digital frequency instruments called counters),
especially because it doesn't require direct connection to the circuit.
So long as the inductor and/or capacitor can intercept enough stray
field (magnetic or electric, respectively) from the circuit under test
to cause the meter to indicate, it will work.
In frequency as in other types of
electrical measurement, the most accurate means of measurement are
usually those where an unknown quantity is compared against a known
standard, the basic instrument doing nothing more than indicating
when the two quantities are equal to each other. This is the basic
principle behind the DC (Wheatstone) bridge circuit and it is a sound
metrological principle applied throughout the sciences. If we have
access to an accurate frequency standard (a source of AC voltage holding
very precisely to a single frequency), then measurement of any unknown
frequency by comparison should be relatively easy.
For that frequency standard, we turn our
attention back to the tuning fork, or at least a more modern variation
of it called the quartz crystal. Quartz is a naturally occurring
mineral possessing a very interesting property called
piezoelectricity. Piezoelectric materials produce a voltage across
their length when physically stressed, and will physically deform when
an external voltage is applied across their lengths. This deformation is
very, very slight in most cases, but it does exist.
Quartz rock is elastic (springy) within
that small range of bending which an external voltage would produce,
which means that it will have a mechanical resonant frequency of its own
capable of being manifested as an electrical voltage signal. In other
words, if a chip of quartz is struck, it will "ring" with its own unique
frequency determined by the length of the chip, and that resonant
oscillation will produce an equivalent voltage across multiple points of
the quartz chip which can be tapped into by wires fixed to the surface
of the chip. In reciprocal manner, the quartz chip will tend to vibrate
most when it is "excited" by an applied AC voltage at precisely the
right frequency, just like the reeds on a vibrating-reed frequency
meter.
Chips of quartz rock can be precisely cut
for desired resonant frequencies, and that chip mounted securely inside
a protective shell with wires extending for connection to an external
electric circuit. When packaged as such, the resulting device is simply
called a crystal (or sometimes "xtal"), and its schematic
symbol looks like this:
Electrically, that quartz chip is
equivalent to a series LC resonant circuit. The dielectric properties of
quartz contribute an additional capacitive element to the equivalent
circuit, and in the end it looks something like this:
The "capacitance" and "inductance" shown
in series are merely electrical equivalents of the quartz's mechanical
resonance properties: they do not exist as discrete components within
the crystal. The capacitance shown in parallel due to the wire
connections across the dielectric (insulating) quartz body is real, and
it has an effect on the resonant response of the whole system. A full
discussion on crystal dynamics is not necessary here, but what needs to
be understood about crystals is this resonant circuit equivalence and
how it can be exploited within an oscillator circuit to achieve an
output voltage with a stable, known frequency.
Crystals, as resonant elements, typically
have much higher "Q" (quality) values than tank circuits built
from inductors and capacitors, principally due to the relative absence
of stray resistance, making their resonant frequencies very definite and
precise. Because the resonant frequency is solely dependent on the
physical properties of quartz (a very stable substance, mechanically),
the resonant frequency variation over time with a quartz crystal is
very, very low. This is how quartz movement watches obtain their
high accuracy: by means of an electronic oscillator stabilized by the
resonant action of a quartz crystal.
For laboratory applications, though, even
greater frequency stability may be desired. To achieve this, the crystal
in question may be placed in a temperature stabilized environment
(usually an oven), thus eliminating frequency errors due to thermal
expansion and contraction of the quartz.
For the ultimate in a frequency standard
though, nothing discovered thus far surpasses the accuracy of a single
resonating atom. This is the principle of the so-called atomic clock,
which uses an atom of mercury (or cesium) suspended in a vacuum, excited
by outside energy to resonate at its own unique frequency. The resulting
frequency is detected as a radio-wave signal and that forms the basis
for the most accurate clocks known to humanity. National standards
laboratories around the world maintain a few of these hyper-accurate
clocks, and broadcast frequency signals based on those atoms' vibrations
for scientists and technicians to tune in and use for frequency
calibration purposes.
Now we get to the practical part: once we
have a source of accurate frequency, how do we compare that
against an unknown frequency to obtain a measurement? One way is to use
a CRT as a frequency-comparison device. Cathode Ray Tubes typically have
means of deflecting the electron beam in the horizontal as well as the
vertical axis. If metal plates are used to electrostatically deflect the
electrons, there will be a pair of plates to the left and right of the
beam as well as a pair of plates above and below the beam.
If we allow one AC signal to deflect the
beam up and down (connect that AC voltage source to the "vertical"
deflection plates) and another AC signal to deflect the beam left and
right (using the other pair of deflection plates), patterns will be
produced on the screen of the CRT indicative of the ratio of
these two AC frequencies. These patterns are called Lissajous figures
and are a common means of comparative frequency measurement in
electronics.
If the two frequencies are the same, we
will obtain a simple figure on the screen of the CRT, the shape of that
figure being dependent upon the phase shift between the two AC signals.
Here is a sampling of Lissajous figures for two sine-wave signals of
equal frequency, shown as they would appear on the face of an
oscilloscope (an AC voltage-measuring instrument using a CRT as its
"movement"). The first picture is of the Lissajous figure formed by two
AC voltages perfectly in phase with each other:
If the two AC voltages are not in phase
with each other, a straight line will not be formed. Rather, the
Lissajous figure will take on the appearance of an oval, becoming
perfectly circular if the phase shift is exactly 90o between
the two signals, and if their amplitudes are equal:
Finally, if the two AC signals are
directly opposing one another in phase (180o shift), we will
end up with a line again, only this time it will be oriented in the
opposite direction:
When we are faced with signal frequencies
that are not the same, Lissajous figures get quite a bit more complex.
Consider the following examples and their given vertical/horizontal
frequency ratios:
The more complex the ratio between
horizontal and vertical frequencies, the more complex the Lissajous
figure. Consider the following illustration of a 3:1 frequency ratio
between horizontal and vertical:
. . . and a 3:2 frequency ratio
(horizontal = 3, vertical = 2):
In cases where the frequencies of the two
AC signals are not exactly a simple ratio of each other (but close), the
Lissajous figure will appear to "move," slowly changing orientation as
the phase angle between the two waveforms rolls between 0o
and 180o. If the two frequencies are locked in an exact
integer ratio between each other, the Lissajous figure will be stable on
the viewscreen of the CRT.
The physics of Lissajous figures limits
their usefulness as a frequency-comparison technique to cases where the
frequency ratios are simple integer values (1:1, 1:2, 1:3, 2:3, 3:4,
etc.). Despite this limitation, Lissajous figures are a popular means of
frequency comparison wherever an accessible frequency standard (signal
generator) exists.
- REVIEW:
- Some frequency meters work on the
principle of mechanical resonance, indicating frequency by relative
oscillation among a set of uniquely tuned "reeds" shaken at the
measured frequency.
- Other frequency meters use electric
resonant circuits (LC tank circuits, usually) to indicate frequency.
One or both components is made to be adjustable, with an accurately
calibrated adjustment knob, and a sensitive meter is read for maximum
voltage or current at the point of resonance.
- Frequency can be measured in a
comparative fashion, as is the case when using a CRT to generate
Lissajous figures. Reference frequency signals can be made with a
high degree of accuracy by oscillator circuits using quartz crystals
as resonant devices. For ultra precision, atomic clock signal
standards (based on the resonant frequencies of individual atoms) can
be used.
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