A parametric oscillator is a driven harmonic oscillator in which the oscillations are driven by varying some parameters of the system at some frequencies, typically different from the natural frequency 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 of the swing as a pendulum. 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. 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/YAG-based parametric oscillators operate in the same fashion. Another important example is the optical parametric oscillator, which converts an input laser 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 or vacuum tube. 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 telescopes and spacecraft communication 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 (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".Faraday, M. (1831) "On a peculiar class of acoustical figures; and on certain forms assumed by a group of particles upon vibrating elastic surfaces", Philosophical Transactions of the Royal Society (London), 121: 299-340. Franz Melde (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.Melde, F. (1860) "Über Erregung stehender Wellen eines fadenförmigen Körpers" [On the excitation of standing waves on a string], Annalen der Physik und Chemie (2nd series), 109: 193-215. Parametric oscillation was first treated as a general phenomenon by Rayleigh (1883,1887).Strutt, J.W. (Lord Rayleigh) (1883) "On maintained vibrations", Philosophical Magazine, 5th series, 15: 229-235.Strutt, J.W. (Lord Rayleigh) (1887) "On the maintenance of vibrations by forces of double frequency, and on the propagation of waves through a medium endowed with periodic structure", Philosophical Magazine, 5th series, 24: 145-159.Strutt, J.W. (Lord Rayleigh) The Theory of Sound, 2nd. ed. (N.Y., N.Y.: Dover, 1945), vol. 1, pages 81-85.
One of the first to apply the concept to electric circuits was George Francis FitzGerald, who in 1892 tried to excite oscillations in an LC circuit by pumping it with a varying inductance provided by a dynamo.See: FitzGerald, George F. (29 January 1892) "On the driving of electromagnetic vibrations by electro-magnetic and electrostatic engines," The Electrician, 28: 329-330. Reprinted: George Francis Fitzgerald with Joseph Larmor, ed., The Scientific Writings of the Late George Francis Fitzgerald (London, England: Longmans, Green, & Co., 1902; Dublin, Ireland: Hodges, Figgis, & Co., 1902), pp. 277–281. Reprinted: (Anon.) (11 February 1892) "Physical Society, January 22," Nature, 45: 358-359. 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, 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 could amplify a weak signal while introducing very little noise. In 1952 Harrison Rowe at Bell Labs 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. 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 telescopes, satellite ground stations, and long-range radar. 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 oscillators and amplifiers which use nonlinear crystals as the active element.
Mathematical analysis A parametric oscillator is a harmonic oscillator 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 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 oscillators, 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
where and are constants, namely, the time-averaged oscillator frequency and damping, respectively. The transformed frequency can then be written in a similar way as ,
where is the natural frequency of the damped harmonic oscillator
and .
Thus, our transformed equation can be written as .
The independent variations and in the oscillator damping and resonance frequency, respectively, can be combined into a single pumping function . The converse conclusion is that any form of parametric excitation can be accomplished by varying either the resonance frequency or the damping, or both.
Solution of the transformed equation Let us assume that is sinusoidal with a frequency approximately twice the natural frequency of the oscillator:
where the pumping frequency but need not equal exactly. Using the method of variation of parameters, the solution to our transformed equation may be written as
where the rapidly varying components, and have been factored out to isolate the slowly varying amplitudes and
We proceed by substituting this solution into the differential equation and considering that both the coefficients in front of and must be zero to satisfy the differential equation identically. We also omit the second derivatives of and on the grounds that and are slowly varying, as well as omit sinusoidal terms not near the natural frequency, as they do not contribute significantly to resonance. The result is the following pair of coupled differential equations:
This system of linear differential equations with constant coefficients can be decoupled and solved by eigenvalue/eigenvector methods. This yields the solution
where and are the eigenvalues of the matrix
and are corresponding eigenvectors, and and are arbitrary constants.
The eigenvalues are given by
If we write the difference between and as and replace with everywhere where the difference is not important, we get .
If then the eigenvalues are real and exactly one is positive, which leads to exponential growth for and This is the condition for parametric resonance, with the growth rate for given by the positive eigenvalue
Note, however, that this growth rate corresponds to the amplitude of the transformed variable whereas the amplitude of the original, untransformed variable can either grow or decay depending on whether is an increasing or decreasing function of time,
Intuitive derivation of parametric excitation The above derivation may seem like a mathematical sleight-of-hand, so it may be helpful to give an intuitive derivation. The equation may be written in the form
which represents a simple harmonic oscillator (or, alternatively, a bandpass filter) being driven by a signal that is proportional to its response .
Assume that already has an oscillation at frequency and that the pumping has double the frequency and a small amplitude . Applying a trigonometric identity for products of sinusoids, their product produces two driving signals, one at frequency and the other at frequency .
Being off-resonance, the signal is attenuated and can be neglected initially. By contrast, the signal is on resonance, serves to amplify , and is proportional to the amplitude . Hence, the amplitude of grows exponentially unless it is initially zero.
Expressed in Fourier space, the multiplication is a convolution of their Fourier transforms and . The positive feedback arises because the component of converts the component of into a driving signal at , and vice versa (reverse the signs). This explains why the pumping frequency must be near , twice the natural frequency of the oscillator. Pumping at a grossly different frequency would not couple (i.e., provide mutual positive feedback) between the and components of .
Parametric resonance