A parametric oscillator is a
driven harmonic oscillator in which the oscillations are driven by varying some parameter of the system at some frequency, typically different from the
natural frequency
Natural frequency, also known as eigenfrequency, is the frequency at which a system tends to oscillate in the absence of any driving force.
The motion pattern of a system oscillating at its natural frequency is called the normal mode (if all pa ...
of the oscillator. A simple example of a parametric oscillator is a child pumping a
playground swing by periodically standing and squatting to increase the size of the swing's oscillations.
[ Note: In real-life playgrounds, swings are predominantly driven, not parametric, oscillators.] The child's motions vary the
moment of inertia
The moment of inertia, otherwise known as the mass moment of inertia, angular mass, second moment of mass, or most accurately, rotational inertia, of a rigid body is a quantity that determines the torque needed for a desired angular accele ...
of the swing as a
pendulum
A pendulum is a weight suspended from a pivot so that it can swing freely. When a pendulum is displaced sideways from its resting, equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward th ...
. The "pump" motions of the child must be at twice the frequency of the swing's oscillations. Examples of parameters that may be varied are the oscillator's resonance frequency
and damping
.
Parametric oscillators are used in several areas of physics. The classical
varactor parametric oscillator consists of a semiconductor
varactor diode connected to a
resonant circuit or
cavity resonator
A resonator is a device or system that exhibits resonance or resonant behavior. That is, it naturally oscillates with greater amplitude at some frequencies, called resonant frequencies, than at other frequencies. The oscillations in a resonator ...
. It is driven by varying the diode's capacitance by applying a varying
bias voltage. The circuit that varies the diode's capacitance is called the "pump" or "driver". In microwave electronics,
waveguide
A waveguide is a structure that guides waves, such as electromagnetic waves or sound, with minimal loss of energy by restricting the transmission of energy to one direction. Without the physical constraint of a waveguide, wave intensities de ...
/
YAG-based parametric oscillators operate in the same fashion. Another important example is the
optical parametric oscillator
An optical parametric oscillator (OPO) is a parametric oscillator that oscillates at optical frequencies. It converts an input laser wave (called "pump") with frequency \omega_p into two output waves of lower frequency (\omega_s, \omega_i) by mean ...
, which converts an input
laser
A laser is a device that emits light through a process of optical amplification based on the stimulated emission of electromagnetic radiation. The word "laser" is an acronym for "light amplification by stimulated emission of radiation". The ...
light wave into two output waves of lower frequency (
).
When operated at pump levels below oscillation, the parametric oscillator can
amplify a signal, forming a parametric amplifier (paramp).
Varactor parametric amplifiers were developed as
low-noise amplifiers in the radio and microwave frequency range. The advantage of a parametric amplifier is that it has much lower noise than an amplifier based on a gain device like a
transistor
upright=1.4, gate (G), body (B), source (S) and drain (D) terminals. The gate is separated from the body by an insulating layer (pink).
A transistor is a semiconductor device used to Electronic amplifier, amplify or electronic switch, switch ...
or
vacuum tube
A vacuum tube, electron tube, valve (British usage), or tube (North America), is a device that controls electric current flow in a high vacuum between electrodes to which an electric potential difference has been applied.
The type known as ...
. This is because in the parametric amplifier a
reactance is varied instead of a (noise-producing)
resistance. They are used in very low noise radio receivers in
radio telescope
A radio telescope is a specialized antenna and radio receiver used to detect radio waves from astronomical radio sources in the sky. Radio telescopes are the main observing instrument used in radio astronomy, which studies the radio frequency ...
s and
spacecraft communication
A spacecraft is a vehicle or machine designed to fly in outer space. A type of artificial satellite, spacecraft are used for a variety of purposes, including communications, Earth observation, meteorology, navigation, space colonization, ...
antennas.
Parametric resonance occurs in a mechanical system when a system is parametrically excited and oscillates at one of its resonant frequencies. Parametric excitation differs from forcing since the action appears as a time varying modification on a system parameter.
History
Parametric oscillations were first noticed in mechanics.
Michael Faraday
Michael Faraday (; 22 September 1791 – 25 August 1867) was an English scientist who contributed to the study of electromagnetism and electrochemistry. His main discoveries include the principles underlying electromagnetic inducti ...
(1831) was the first to notice oscillations of one frequency being excited by forces of double the frequency, in the crispations (ruffled surface waves) observed in a wine glass excited to "sing".
Franz Melde
Franz Emil Melde (March 11, 1832 in Großenlüder near Fulda – March 17, 1901 in Marburg) was a German physicist and professor. A graduate of the University of Marburg under Christian Ludwig Gerling, he later taught there, focusing primar ...
(1860) generated parametric oscillations in a string by employing a tuning fork to periodically vary the tension at twice the resonance frequency of the string. Parametric oscillation was first treated as a general phenomenon by
Rayleigh (1883,1887).
One of the first to apply the concept to electric circuits was
George Francis FitzGerald
Prof George Francis FitzGerald (3 August 1851 – 22 February 1901) was an Irish academic and physicist who served as Erasmus Smith's Professor of Natural and Experimental Philosophy at Trinity College Dublin (TCD) from 1881 to 1901.
FitzGera ...
, who in 1892 tried to excite oscillations in an
LC circuit
An LC circuit, also called a resonant circuit, tank circuit, or tuned circuit, is an electric circuit consisting of an inductor, represented by the letter L, and a capacitor, represented by the letter C, connected together. The circuit can a ...
by pumping it with a varying inductance provided by a dynamo.
Parametric amplifiers (paramps) were first used in 1913-1915 for radio telephony from Berlin to Vienna and Moscow, and were predicted to have a useful future (
Ernst Alexanderson, 1916).
[Alexanderson, Ernst F.W. (April 1916]
"A magnetic amplifier for audio telephony"
''Proceedings of the Institute of Radio Engineers'', 4: 101-149. These early parametric amplifiers used the nonlinearity of an iron-core
inductor
An inductor, also called a coil, choke, or reactor, is a passive two-terminal electrical component that stores energy in a magnetic field when electric current flows through it. An inductor typically consists of an insulated wire wound into a c ...
, so they could only function at low frequencies.
In 1948
Aldert van der Ziel pointed out a major advantage of the parametric amplifier: because it used a variable reactance instead of a resistance for amplification it had inherently low noise.
A parametric amplifier used as the
front end of a
radio receiver
In radio communications, a radio receiver, also known as a receiver, a wireless, or simply a radio, is an electronic device that receives radio waves and converts the information carried by them to a usable form. It is used with an antenna. Th ...
could amplify a weak signal while introducing very little noise. In 1952 Harrison Rowe at
Bell Labs
Nokia Bell Labs, originally named Bell Telephone Laboratories (1925–1984),
then AT&T Bell Laboratories (1984–1996)
and Bell Labs Innovations (1996–2007),
is an American industrial research and scientific development company owned by mul ...
extended some 1934 mathematical work on pumped oscillations by Jack Manley and published the modern mathematical theory of parametric oscillations, the
Manley-Rowe relations.
The
varactor diode invented in 1956 had a nonlinear capacitance that was usable into microwave frequencies. The varactor parametric amplifier was developed by Marion Hines in 1956 at
Western Electric
The Western Electric Company was an American electrical engineering and manufacturing company officially founded in 1869. A wholly owned subsidiary of American Telephone & Telegraph for most of its lifespan, it served as the primary equipment ma ...
.
At the time it was invented microwaves were just being exploited, and the varactor amplifier was the first semiconductor amplifier at microwave frequencies.
It was applied to low noise radio receivers in many areas, and has been widely used in
radio telescope
A radio telescope is a specialized antenna and radio receiver used to detect radio waves from astronomical radio sources in the sky. Radio telescopes are the main observing instrument used in radio astronomy, which studies the radio frequency ...
s, satellite
ground station
A ground station, Earth station, or Earth terminal is a terrestrial radio station designed for extraplanetary telecommunication with spacecraft (constituting part of the ground segment of the spacecraft system), or reception of radio waves f ...
s, and long-range
radar
Radar is a detection system that uses radio waves to determine the distance (''ranging''), angle, and radial velocity of objects relative to the site. It can be used to detect aircraft, Marine radar, ships, spacecraft, guided missiles, motor v ...
. It is the main type of parametric amplifier used today. Since that time parametric amplifiers have been built with other nonlinear active devices such as
Josephson junctions.
The technique has been extended to optical frequencies in
optical parametric oscillator
An optical parametric oscillator (OPO) is a parametric oscillator that oscillates at optical frequencies. It converts an input laser wave (called "pump") with frequency \omega_p into two output waves of lower frequency (\omega_s, \omega_i) by mean ...
s and amplifiers which use
nonlinear crystals as the active element.
Mathematical analysis
A parametric oscillator is a
harmonic oscillator
In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force ''F'' proportional to the displacement ''x'':
\vec F = -k \vec x,
where ''k'' is a positive const ...
whose physical properties vary with time. The equation of such an oscillator is
:
This equation is linear in
. By assumption, the parameters
and
depend only on time and do ''not'' depend on the state of the oscillator. In general,
and/or
are assumed to vary periodically, with the same period
.
If the parameters vary at roughly ''twice'' the
natural frequency
Natural frequency, also known as eigenfrequency, is the frequency at which a system tends to oscillate in the absence of any driving force.
The motion pattern of a system oscillating at its natural frequency is called the normal mode (if all pa ...
of the oscillator (defined below), the oscillator phase-locks to the parametric variation and absorbs energy at a rate proportional to the energy it already has. Without a compensating energy-loss mechanism provided by
, the oscillation amplitude grows exponentially. (This phenomenon is called parametric excitation, parametric resonance or parametric pumping.) However, if the initial amplitude is zero, it will remain so; this distinguishes it from the non-parametric resonance of driven simple
harmonic oscillator
In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force ''F'' proportional to the displacement ''x'':
\vec F = -k \vec x,
where ''k'' is a positive const ...
s, in which the amplitude grows linearly in time regardless of the initial state.
A familiar experience of both parametric and driven oscillation is playing on a swing.
[ Rocking back and forth pumps the swing as a driven harmonic oscillator, but once moving, the swing can also be parametrically driven by alternately standing and squatting at key points in the swing arc. This changes moment of inertia of the swing and hence the resonance frequency, and children can quickly reach large amplitudes provided that they have some amplitude to start with (e.g., get a push). Standing and squatting at rest, however, leads nowhere.
]
Transformation of the equation
We begin by making a change of variable
:
where is the time integral of the damping coefficient
:.
This change of variable eliminates the damping term in the differential equation, reducing it to
:
where the transformed frequency is defined as
:.
In general, the variations in damping and frequency are relatively small perturbations
: