Difference between revisions 111804244 and 111804253 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)\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==

Thise 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 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 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.

⏎
⏎
==See also==

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

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