Difference between revisions 111804260 and 111804264 on dewiki

A '''parametric oscillator''' is a simple [[harmonic oscillator]] whose parameters (its resonant frequency <math>\omega</math> and damping <math>\beta</math>) vary in time in a defined way

:<math>
\frac{d^{2}x}{dt^{2}} + \beta(t) \frac{dx}{dt} + \omega^{2}(t) x = 0
</math>

This equation is linear in <math>x(t)</math>.  By assumption, the parameters 
(contracted; show full)A familiar experience of parametric oscillation is playing on a swing. By alternately raising and lowering their center of mass (changing their moment of inertia and, thus, the resonant frequency) at key points in the swing, children can quickly reach large amplitudes provided that they have some amplitude to start with (e.g., get a push).  Doing so at rest, however, goes nowhere.

==Transformation of the equation==

We begin by making a change of variables

:<math>
q(t) \
equiv \stackrel{\mathrm{def}}{=}\   e^{D(t)} x(t)
</math>

where <math>D(t)</math> is a time integral of the damping

:<math>
D(t) \equiv \stackrel{\mathrm{def}}{=}\   \frac{1}{2} \int^{t} d\tau \ \beta(\tau)
</math>

This change of variables eliminates the damping term

:<math>
\frac{d^{2}q}{dt^{2}} + \Omega^{2}(t) q = 0
</math>
 
where the transformed frequency is defined

:<math>
\Omega^{2}(t) = \omega^{2}(t) - 
\frac{1}{2} \left( \frac{d\beta}{dt} \right) - \frac{1}{4} \beta^{2}
</math>

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

:<math>
\beta(t) = \omega_{0} \left[b + g(t) \right]
</math>

:<math>
\omega^{2}(t) = \omega_{0}^{2} \left[1 + h(t) \right]
</math>

where <math>\omega_{0}</math> and <math>b\omega_{0}</math> are constants, namely, the time-averaged oscillator frequency and damping, respectively.  The transformed frequency can be written in a similar way

:<math>
\Omega^{2}(t) = \omega_{n}^{2} \left[1 + f(t) \right]
</math>

where <math>\omega_{n}</math> is the [[natural frequency]] of the damped harmonic oscillator

:<math>
\omega_{n}^{2} \equiv \stackrel{\mathrm{def}}{=}\   \omega_{0}^{2} \left( 1 - \frac{b^{2}}{4} \right)
</math>

and 

:<math>
\omega_{n}^{2} f(t) \equiv \stackrel{\mathrm{def}}{=}\   \omega_{0}^{2} h(t) - 
\frac{1}{2\omega_{0}} \left( \frac{dg}{dt} \right)
- \frac{b}{2} g(t) - \frac{1}{4} g^{2}(t)
</math>

Thus, our transformed equation can be written

:<math>
(contracted; show full)-\left( \frac{f_{0}}{2} \right) \omega_{n}^{2} B + 
\left( \omega_{p}^{2} - \omega_{n}^{2} \right) A
</math>

We may decouple and solve these equations by making another change of variables

:<math>
A(t) \
equiv \stackrel{\mathrm{def}}{=}\   r(t) \cos \theta(t)
</math>

:<math>
B(t) \equiv \stackrel{\mathrm{def}}{=}\   r(t) \sin \theta(t)
</math>

which yields the equations

:<math>
\frac{dr}{dt} = \left( \alpha_{\mathrm{max}} \cos 2\theta \right) r
</math>

:<math>
\frac{d\theta}{dt} = -\alpha_{\mathrm{max}} 
\left[\sin 2\theta - \sin 2\theta_{\mathrm{eq}} \right]
</math>

where we have defined for brevity

:<math>
\alpha_{\mathrm{max}} \equiv \stackrel{\mathrm{def}}{=}\   \frac{f_{0} \omega_{n}^{2}}{4\omega_{p}}
</math>

:<math>
\sin 2\theta_{\mathrm{eq}} \equiv \stackrel{\mathrm{def}}{=}\   \left( \frac{2}{f_{0}} \right) \epsilon
</math>

and the detuning

:<math>
\epsilon \equiv \stackrel{\mathrm{def}}{=}\   \frac{\omega_{p}^{2} - \omega_{n}^{2}}{\omega_{n}^{2}}
</math>

The <math>\theta</math> equation does not depend on <math>r</math>, and linearization near its equilibrium position <math>\theta_{\mathrm{eq}}</math> shows that <math>\theta</math> decays exponentially to its equilibrium

:<math>
\theta(t) = \theta_{\mathrm{eq}} + 
\left( \theta_{0} - \theta_{\mathrm{eq}} \right) e^{-2\alpha t}
</math>

where the decay constant 

<math>\alpha \equiv \stackrel{\mathrm{def}}{=}\   \alpha_{\mathrm{max}} \cos 2\theta_{\mathrm{eq}}</math>.

In other words, the parametric oscillator phase-locks to the pumping signal <math>f(t)</math>.

Taking <math>\theta(t) = \theta_{\mathrm{eq}}</math> (i.e., assuming that the phase has locked), the <math>r</math> equation becomes

:<math>
\frac{dr}{dt} = \alpha r
</math>

whose solution is <math>r(t) = r_{0} e^{\alpha t}</math>; the amplitude of the <math>q(t)</math> oscillation diverges exponentially.  However, the corresponding amplitude <math>R(t)</math> of the ''untransformed'' variable <math>x \equiv \stackrel{\mathrm{def}}{=}\   q e^{-D}</math> need not diverge

:<math>
R(t) = r(t) e^{-D} = r_{0} e^{\alpha t - D}
</math>

The amplitude <math>R(t)</math> diverges, decays or stays constant, depending on whether <math>\alpha t</math> is greater than, less than, or equal to <math>D</math>, respectively.  

(contracted; show full)er common use is frequency conversion, e.g., conversion from audio to radio frequencies.  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 (Alexanderson, 1916).  The early paramps varied inductances, but other methods have been developed since, e.g., the varactor diodes, [[klystron tube]]s, Josephson junctions and [[optical parametric oscillator|optical methods]].
 A practical parametric oscillator needs two three connections. One for common, one to feed the pump, and one to retrieve the oscillation and maybe a fourth one for biasing. A parametric amplifier needs a fifth port for the seed. Since a varactor diode has only two connections it can only be a part of a LC network with four [[eigenvector]]s with nodes at the connections. This can be implemented as a [[transimpedance amplifier]], a [[Travelling wave tube amplifier|traveling wave amplifier]] or by means of a [[circulator]].

==References==

Faraday M. (1831) "On a peculiar class of acoustical figures; and on certain forms assumed by a group of particles upon vibrating elastic surfaces", 
''Phil. Trans. Roy. Soc. (London)'', '''121''', 299-318.

Melde F. (1859) "Über Erregung stehender Wellen eines fadenförmigen Körpers", 
(contracted; show full)* [[Harmonic oscillator]]
* [[Optical parametric oscillator]]
* [[Optical parametric amplifier]]

[[Category:Oscillators]]
[[Category:Amplifiers]]
[[Category:Dynamical systems]]
[[Category:Ordinary differential equations]]