Difference between revisions 111804229 and 111804236 on dewiki

A '''parametric oscillator''' is a simple harmonic oscillator whose parameters
(its resonant frequency <math>\omega</math> and damping <math>\beta</math>)
are variable

:<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>.  The parameters 
<math>\omega^{2}</math> and <math>\beta</math> depend only 
on time and not on the state of the oscillator.  In general, 
<math>\beta(t)</math> and/or <math>\omega^{2}(t)</math> 
are assumed to vary periodically with the same period <math>T</math>.


Remarkably, 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, 
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 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.


(contracted; show full)would not couple (i.e., provide mutual positive feedback) 
between the <math>-\omega_{p}</math> and <math>+\omega_{p}</math>
components of <math>q(t)</math>.


==Parametric amplifiers==


The parametric oscillator equation can be extended by 
adding an external driving force <math>E(t)</math>

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

We assume that the damping <math>D</math> is sufficiently strong 
that, in the absence of the driving force <math>E</math>, the 
amplitude of the parametric oscillations does not diverge, 
i.e., that <math>\alpha t < D</math>.  In this situation, 
the parametric pumping acts to lower the effective
damping in the system.  For illustration, let the damping be constant 
<math>\beta(t) = \omega_{0} b</math> and assume that the 
external driving force is at the mean resonant frequency
<math>\omega_{0}</math>, i.e., 
<math>E(t) = E_{0} \sin \omega_{0} t</math>.
The equation then becomes

:<math>
\frac{d^{2}x}{dt^{2}} + b \omega_{0} \frac{dx}{dt} + 
\omega_{0}^{2} \left[ 1 + h_{0} \sin 2\omega_{0} t \right] x = 
E_{0} \sin \omega_{0} t
</math>

whose solution is roughly

:<math>
x(t) = \frac{2E_{0}}{\omega_{0}^{2} \left( 2b - h_{0} \right)} \cos \omega_{0} t
</math>

As <math>h_{0}</math> approaches the threshold <math>2b</math>, 
the amplitude diverges.  When <math>h \geq 2b</math>, the system enters
parametric resonance and the amplitude begins to grow exponentially,
even in the absence of a driving force <math>E(t)</math>.⏎
⏎

==Other relevant mathematical results==

If the parameters of any secoond-order linear differential equation
are varied periodically, [[Floquet analysis]] shows that the solutions 
must vary either sinusoidally or exponentially.

The <math>q</math> equation above with periodically varying <math>f(t)</math>
is an example of a [[Hill equation]].  If <math>f(t)</math> is a simple 
sinusoid, the equation is called a [[Mathieu equation]].

⏎
⏎
==History==

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".  Melde (1859)
generated parametric oscillations in a string by employing a tuning fork
to periodically vary the tension at twice the resonant frequency of the 
string.  Parametric oscillation in general was first treated by 
Rayleigh (1883,1887), whose papers are still worth reading today.

Parametric oscillators have been developed as low-noise amplifiers, 
especially in the radio and microwave frequency range.  Thermal noise 
is minimal, since a reactance (not a resistance) is varied.  Another
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 elementmethods have been developed 
since, e.g., the varactor diodes, [[klystron tube]]s, and Josephson junctions. ⏎
and [[optical parametric oscillator|optical methods]].


==References==

Faraday M. (1831) "On a peculiar class of acoustical figures; and on certain forms
assumed b 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", 
''Ann. Phys. Chem.'' (ser. 2), '''109''', 193-215.

Strutt JW (Lord Rayleigh). (1883) "On Maintained Vibrations", ''Phil. Mag.'', '''15''', 229-235.

Strutt JW (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", ''Phil. Mag.'', '''24''', 145-159.

Strutt JW (Lord Rayleigh). (1945) ''The Theory of Sound'', 2nd. ed., Dover.

Kühn L. (1914) ''Elektrotech. Z.'', '''35''', 816-819.

Pungs L. DRGM Nr. 588 822 (24 October 1913); DRP Nr. 281440 (1913); 
''Elektrotech. Z.'', '''44''', 78-81 (1923?); ''Proc. IRE'', '''49''', 378 (1961).

Mumford WW. (1961) "Some Notes on the History of Parametric Transducers", 
''Proc. IRE'', '''48''', 848-853.


==See also==

==External Links==* [[Optical parametric oscillator]]
* [[Optical parametric amplifier]]

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