Parametric oscillator
[1][2][3] The child's motions vary the moment of inertia of the swing as a pendulum.
The circuit that varies the diode's capacitance is called the "pump" or "driver".
Parametric excitation differs from forcing since the action appears as a time varying modification on a system parameter.
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".
[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.
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.
Since that time parametric amplifiers have been built with other nonlinear active devices such as Josephson junctions.
A familiar experience of both parametric and driven oscillation is playing on a swing.
is the natural frequency of the damped harmonic oscillator and Thus, our transformed equation can be written as The independent variations
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.
We proceed by substituting this solution into the differential equation and considering that both the coefficients in front of
Note, however, that this growth rate corresponds to the amplitude of the transformed variable
equation may be written in the form which represents a simple harmonic oscillator (or, alternatively, a bandpass filter) being driven by a signal
Pumping at a grossly different frequency would not couple (i.e., provide mutual positive feedback) between the
This effect is different from regular resonance because it exhibits the instability phenomenon.
Parametric excitation differs from forcing since the action appears as a time varying modification on a system parameter.
The classical example of parametric resonance is that of the vertically forced pendulum.
[14] The effect of friction is to introduce a finite threshold for the amplitude of parametric excitation to result in an instability.
[15] For small amplitudes and by linearising, the stability of the periodic solution is given by Mathieu's equation: where
term acts as an ‘energy’ source and is said to parametrically excite the system.
A practical parametric oscillator needs the following connections: one for the "common" or "ground", one to feed the pump, one to retrieve the output, and maybe a fourth one for biasing.
The parametric oscillator equation can be extended by adding an external driving force
In this situation, the parametric pumping acts to lower the effective damping in the system.
and assume that the external driving force is at the mean resonance frequency
, the system enters parametric resonance and the amplitude begins to grow exponentially, even in the absence of a driving force
