Parametric oscillator

Harmonic oscillator whose parameters oscillate in time

One of the first varactor parametric amplifiers, invented at Bell Labs around 1958. This 4 stage amplifier achieved 10 dB gain at 400 MHz. Parametric amplifiers are used in applications requiring extremely low noise.

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.^[1][2][3] 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 ${\displaystyle \omega }$ and damping ${\displaystyle \beta }$.

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 ( ${\displaystyle \omega _{s},\omega _{i}}$).

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.^[4]

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".^[5] 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.^[6] Parametric oscillation was first treated as a general phenomenon by Rayleigh (1883,1887).^[7][8][9]

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.^[10][11] 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).^[12] 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.^[13] 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.^[13]

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.^[13] At the time it was invented microwaves were just being exploited, and the varactor amplifier was the first semiconductor amplifier at microwave frequencies.^[13] 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

${\displaystyle {\frac {d^{2}x}{dt^{2}}}+\beta (t){\frac {dx}{dt}}+\omega ^{2}(t)x=0}$

This equation is linear in ${\displaystyle x(t)}$. By assumption, the parameters ${\displaystyle \omega ^{2}}$ and ${\displaystyle \beta }$ depend only on time and do not depend on the state of the oscillator. In general, ${\displaystyle \beta (t)}$ and/or ${\displaystyle \omega ^{2}(t)}$ are assumed to vary periodically, with the same period ${\displaystyle T}$.

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 ${\displaystyle \beta }$, 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.^[1][2][3] 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

${\displaystyle q(t)\ {\stackrel {\mathrm {def} }{=}}\ e^{D(t)}x(t)}$

where ${\displaystyle D(t)}$ is the time integral of the damping coefficient

${\displaystyle D(t)\ {\stackrel {\mathrm {def} }{=}}\ {\frac {1}{2}}\int _{0}^{t}\beta (\tau )\,d\tau }$.

This change of variable eliminates the damping term in the differential equation, reducing it to

${\displaystyle {\frac {d^{2}q}{dt^{2}}}+\Omega ^{2}(t)q=0}$

where the transformed frequency is defined as

${\displaystyle \Omega ^{2}(t)\ {\stackrel {\mathrm {def} }{=}}\ \omega ^{2}(t)-{\frac {1}{2}}{\frac {d\beta }{dt}}-{\frac {1}{4}}\beta ^{2}(t)}$.

In general, the variations in damping and frequency are relatively small perturbations

${\displaystyle \beta (t)=\omega _{0}{\big [}b+g(t){\big ]}}$

${\displaystyle \omega ^{2}(t)=\omega _{0}^{2}{\big [}1+h(t){\big ]}}$

where ${\displaystyle \omega _{0}}$ and ${\displaystyle b}$ are constants, namely, the time-averaged oscillator frequency and damping, respectively. The transformed frequency can then be written in a similar way as

${\displaystyle \Omega ^{2}(t)=\omega _{n}^{2}{\big [}1+f(t){\big ]}}$