Revision 112126462 of "Benutzer:Debenben/Parametrischer Oszillator" on dewiki

{{Importartikel}}
Ein '''parametrischer Oszillator''' ist ein [[harmonischer Oszillator]], dessen Parameter (die Resonanzfrequenz <math>\omega_0</math> sowie die Dämpfungskonstante <math>\beta</math>) eine zeitabhängige Funktion sind. 

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

Diese Gleichung ist linear in <math>x(t)</math>. Die Parameter <math>\omega_0^2</math> und <math>\beta</math> sind dabei nur zeitabhängig und hängen nicht von dem Zustand ab, in dem sich der Oszillator befindet. Üblicherweise wird angenommen, dass <math>\beta(t)</math> und/oder <math>\omega_0^2(t)</math> periodisch sind, und sich mit der gleichen periode <math>T</math> verändern.

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 oscillator]]s, 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.

==Transformation of the equation==

We begin by making a change of variables

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

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

:<math>
D(t) \equiv \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 \omega_{0}^{2} \left( 1 - \frac{b^{2}}{4} \right)
</math>

and 

:<math>
\omega_{n}^{2} f(t) \equiv \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>
\frac{d^{2}q}{dt^{2}} + \omega_{n}^{2} \left[1 + f(t) \right] q = 0
</math>

Remarkably, the independent variations <math>g(t)</math> and <math>h(t)</math> in the oscillator damping and resonant frequency, respectively, can be combined into a single pumping function <math>f(t)</math>.  The converse conclusion is that any form of parametric excitation can be accomplished by varying either the resonant frequency or the damping, or both.

==Solution of the transformed equation==

Let us assume that <math>f(t)</math> is sinusoidal, specifically

:<math>
f(t) = f_{0} \sin 2\omega_{p}t
</math>

where the pumping frequency <math>2\omega_{p} \approx 2\omega_{n}</math> but need not equal <math>2\omega_{n}</math> exactly.  The solution <math>q(t)</math> of our transformed equation may be written

:<math>
q(t) = A(t) \cos \omega_{p}t + B(t) \sin \omega_{p}t
</math>

where we have factored out the rapidly varying components (<math>\cos \omega_{p}t</math> and <math>\sin \omega_{p}t</math>) to isolate the slowly varying amplitudes <math>A(t)</math> and <math>B(t)</math>.  This corresponds to Laplace's variation of parameters method.

Substituting this solution into the transformed equation and retaining only the terms first-order in <math>f_{0} \ll 1</math> yields two coupled equations

:<math>
2\omega_{p} \frac{dA}{dt} = 
\left( \frac{f_{0}}{2} \right) \omega_{n}^{2} A - 
\left( \omega_{p}^{2} - \omega_{n}^{2} \right) B
</math>

:<math>
2\omega_{p} \frac{dB}{dt} = 
-\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 r(t) \cos \theta(t)
</math>

:<math>
B(t) \equiv 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 \frac{f_{0} \omega_{n}^{2}}{4\omega_{p}}
</math>

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

and the detuning

:<math>
\epsilon \equiv \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 \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 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.  

The maximum growth rate of the amplitude occurs when <math>\omega_{p} = \omega_{n}</math>.  At that frequency, the equilibrium phase <math>\theta_{\mathrm{eq}}</math> is zero, implying that <math>\cos 2\theta_{\mathrm{eq}}=1</math> and <math>\alpha = \alpha_{\mathrm{max}}</math>.  As <math>\omega_{p}</math> is varied from <math>\omega_{n}</math>, <math>\theta_{\mathrm{eq}}</math> moves away from zero and <math>\alpha < \alpha_{\mathrm{max}}</math>, i.e., the amplitude grows more slowly.  For sufficiently large deviations of <math>\omega_{p}</math>, the decay constant <math>\alpha</math> can become purely imaginary since

:<math>
\alpha = \alpha_{\mathrm{max}} 
\sqrt{1- \left( \frac{2}{f_{0}} \right)^{2} \epsilon^{2}}
</math>

If the detuning <math>\epsilon</math> exceeds <math>f_{0}/2</math>, <math>\alpha</math> becomes purely imaginary and <math>q(t)</math> varies sinusoidally.  Using the definition of the detuning <math>\epsilon</math>, the pumping frequency <math>2\omega_{p}</math> must lie between <math>2\omega_{n} \sqrt{1 - \frac{f_{0}}{2}}</math> and <math>2\omega_{n} \sqrt{1 + \frac{f_{0}}{2}}</math>.  Expanding the square roots in a binomial series shows that the spread in pumping frequencies that result in exponentially growing <math>q</math> is approximately <math>\omega_{n} f_{0}</math>.

==Intuitive understanding 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 <math>q</math> equation may be written in the form

:<math>
\frac{d^{2}q}{dt^{2}} + \omega_{n}^{2} q = -\omega_{n}^{2} f(t) q
</math>

which represents a simple [[harmonic oscillator]] (or, alternatively, a [[bandpass filter]]) being driven by a signal <math>-\omega_{n}^{2} f(t) q</math> that is proportional to its response <math>q</math>.  

Assume that <math>q(t) = A \cos \omega_{p} t</math> already has an oscillation at frequency <math>\omega_{p}</math> and that the pumping <math>f(t) = f_{0} \sin 2\omega_{p}t</math> has double the frequency and a small amplitude <math>f_{0} \ll 1</math>.  Their product <math>q(t)f(t)</math> produces two driving signals,
one at frequency <math>\omega_{p}</math> and the other at frequency <math>3 \omega_{p}</math>

:<math>
f(t)q(t) = \frac{f_{0}}{2} A 
\left( \sin \omega_{p} t + \sin 3\omega_{p} t \right)
</math>

Being off-resonance, the <math>3\omega_{p}</math> signal is attentuated and can be neglected initially.  By contrast, the <math>\omega_{p}</math> signal is on resonance, serves to amplify <math>q</math> and is proportional to the amplitude 
<math>A</math>.  Hence, the amplitude of <math>q</math> grows exponentially unless it is initially zero.

Expressed in Fourier space, the multiplication <math>f(t)q(t)</math> is a convolution of their Fourier transforms <math>\tilde{F}(\omega)</math> and <math>\tilde{Q}(\omega)</math>.  The positive feedback arises because the <math>+2\omega_{p}</math> component of <math>f(t)</math> converts the <math>-\omega_{p}</math> component of <math>q(t)</math> into a driving signal at 
<math>+\omega_{p}</math>, and vice versa (reverse the signs).  This explains why the pumping frequency must be near <math>2\omega_{n}</math>, twice the natural frequency of the oscillator.  Pumping at a grossly different frequency 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 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 was first treated as a general phenomenon 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 methods have been developed since, e.g., the varactor diodes, [[klystron tube]]s, Josephson junctions and [[optical parametric oscillator|optical methods]].

==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", 
''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.

<pre>
==See also==

* [[Harmonic oscillator]]
* [[Optical parametric oscillator]]
* [[Optical parametric amplifier]]

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

</pre>