Parametric oscillator

A parametric oscillator is a system that can vibrate, also known as an oscillator , with time-dependent parameters that change the natural frequency and damping . In this way, energy can be supplied to an oscillator in order to increase the amplitude of the oscillation. The method of supplying energy is called parametric excitation , while movement is called parameter-excited or rheolinear oscillation. One example is the swing of a swing by periodically raising and lowering the center of gravity parallel to the suspension.

A feature of a purely parametrically generated oscillation is that, unlike a forced oscillation , it cannot arise without an initial deflection from the rest position.

Technical systems with time-dependent parameters can be found in turbo-machine and helicopter construction, for example. Parametric oscillators are used in a number of technical systems, especially in electrical engineering , for example in the construction of low-noise amplifiers . They can also be used for frequency conversion. An optically parametric oscillator can, for example, convert an irradiated laser wave into two radiations of lower frequency .

definition

An oscillator with purely parametric excitation can be described by the following homogeneous linear differential equation :

${\displaystyle {\ddot {x}}+p_{1}(t){\dot {x}}+p_{2}(t)x=0}$

The time-dependent functions and are parameters of the system. The parameters have the property that they are real , do not depend on the state of the oscillator and change periodically . It can be shown that both parameters can be combined into a time-dependent excitation function. Such an excitation function is called a pump function. The circuit or mechanism that changes the parameters is called a pump. ${\displaystyle p_{1}}$ ${\displaystyle p_{2}}$

The characteristic of such an excitation is that in the case of an oscillator that starts with an initial amplitude of zero, the amplitude remains zero, because the initial conditions are always obtained . However, since the amplification works even with the tiniest, unintentional deflections, this case cannot be observed in reality. ${\displaystyle (x,{\dot {x}})=(0,0)}$ ${\displaystyle {\ddot {x}}=0}$

Therefore, the parametric excitation is often supplemented by a forced excitation, so that the differential equation becomes inhomogeneous. In addition to time-dependent parameters, you get an independent external force and thus a combined forced and parameter excitation.

${\displaystyle {\ddot {x}}+p_{1}(t){\dot {x}}+p_{2}(t){x}=p_{3}(t)}$

The simplest case of resonance , in which the parameters change with twice the natural frequency of the oscillator, is of particular practical interest . Then the oscillator oscillates phase-locked with the parametric change and absorbs energy proportional to the energy it already has. Without a mechanism to compensate for this increase, the amplitude of the oscillation thus increases exponentially . An impressive application example is the “swinging incense kettle” mentioned below in the cathedral of Santiago de Compostela , the end of the Camino de Santiago in northern Spain.

In systems with several degrees of freedom, the parameters have matrix form and the dependent variables are combined in a vector.

history

The first observations come from Michael Faraday , who in 1831 described surface waves in a wine glass that was made to "sing". He found that the vibrations of the wine glass were excited by forces with double frequency. In 1859 Franz Melde then generated parameter-excited vibrations in a string by using a tuning fork to periodically change the tension of the string at twice the resonance frequency . A description of parametric excitation as a general phenomenon was first written by Rayleigh in 1883 and 1887.

George William Hill came across a special DGL with variable parameters in 1877 when he determined disturbances that the lunar orbit experiences due to the influence of the sun.

One of the first to apply the concept to electrical circuits was George Francis FitzGerald , who in 1892 tried to stimulate oscillations in LC elements by using a dynamo as a pump to change the inductance of the oscillating circuit.

Parametric amplifiers were first used in 1913-1915 for a radio telephone connection from Berlin to Vienna and Moscow. The potential of the technology for future applications was recognized back then, for example by Ernst Alexanderson . The first parametric amplifiers worked by changing the inductance. Since then, other methods such as the capacitance diode , klystron tubes , Josephson junctions and optical methods have been developed.

Mathematical description

Summary of the parameters to form an excitation function

We start with the differential equation above:

${\displaystyle {\ddot {x}}+\beta {\dot {x}}+\omega ^{2}x=0}$

In order to combine both time-dependent factors in the differential equation to form a pump function, a variable transformation can first be carried out in order to eliminate the speed-dependent term. We therefore set:

${\displaystyle x=q\,e^{-0,5\int _{0}^{t}\beta dt}}$

After deriving it twice and inserting it into the original equation,

${\displaystyle {\ddot {q}}+\Omega ^{2}q=0}$

With

${\displaystyle \Omega ^{2}=\omega ^{2}-{\frac {1}{2}}{\dot {\beta }}-{\frac {1}{4}}\beta ^{2}}$

The above differential equation, which changes periodically, is called Hill's differential equation . ${\displaystyle \Omega }$

The excitation is usually interpreted as a deviation from a time average

${\displaystyle \Omega ^{2}=\omega _{n}^{2}(1+f)}$

where the constant corresponds to the damped oscillation frequency of the oscillator, so ${\displaystyle \omega _{n}}$

${\displaystyle \omega _{n}^{2}=\omega _{0}^{2}-\beta _{0}^{2}}$

The time-dependent function is called the pump function. Every type of parametric excitation can always be described by the following differential equation ${\displaystyle f}$

${\displaystyle {\ddot {q}}+\omega _{n}^{2}(1+f(t))q=0}$

Solution for a sinusoidal excitation with double frequency

We consider the differential equation above

${\displaystyle {\ddot {q}}+\omega _{n}^{2}(1+f)q=0}$

We assume that the pump function can be written as ${\displaystyle f}$

${\displaystyle f(t)=f_{0}\sin(2\omega _{p}t)}$

where half the pumping frequency corresponds approximately to the oscillation frequency. This special case of Hill's differential equation is called a mathematical differential equation . However, an exact match of the frequencies is not necessary for the solution, since the oscillation adapts to the pump signal. According to Floquet's theorem , the solution of the differential equation can be written as ${\displaystyle \omega _{p}\approx \omega _{n}}$

${\displaystyle q=A\cos(\omega _{p}t)+B\sin(\omega _{p}t)}$

The amplitudes and are time-dependent. For a parametric excitation, however, it usually applies that the amplitudes change more slowly than the sine or cosine terms of the solution. In other words, the change in the oscillation amplitude happens more slowly than the oscillation itself. If you insert this solution into the differential equation and only keep first-order terms in , you get two coupled equations ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle f_{0}\ll 1}$

${\displaystyle 2\omega _{p}{\dot {A}}={\frac {f_{0}}{2}}\omega _{n}^{2}A-\left(\omega _{p}^{2}-\omega _{n}^{2}\right)B}$

${\displaystyle 2\omega _{p}{\dot {B}}=-{\frac {f_{0}}{2}}\omega _{n}^{2}B+\left(\omega _{p}^{2}-\omega _{n}^{2}\right)A}$

In order to decouple this system of equations, another variable transformation can be carried out

${\displaystyle A=r\cos \theta }$

${\displaystyle B=r\sin \theta }$

and thereby obtains the equations

${\displaystyle {\dot {r}}=r(\alpha _{\mathrm {max} }\cos 2\theta )}$

${\displaystyle {\dot {\theta }}=-\alpha _{\mathrm {max} }(\sin 2\theta -\sin 2\theta _{\mathrm {eq} })}$

with the constants

${\displaystyle \alpha _{\mathrm {max} }={\frac {f_{0}\omega _{n}^{2}}{4\omega _{p}}}}$

${\displaystyle \sin 2\theta _{\mathrm {eq} }={\frac {2\epsilon }{f_{0}}}}$

${\displaystyle \epsilon ={\frac {\omega _{p}^{2}-\omega _{n}^{2}}{\omega _{n}^{2}}}}$

The constant is called detuning. The differential equation for does not depend on . With a linear approximation it can be shown that the equilibrium point is approached exponentially . In other words, the parametric oscillator is phase-locked to the pump signal. If one sets what one assumes that the coupling has established itself, the differential equation for the amplitude becomes ${\displaystyle \epsilon }$ ${\displaystyle \theta }$ ${\displaystyle r}$ ${\displaystyle \theta }$ ${\displaystyle \theta _{\mathrm {eq} }}$ ${\displaystyle \theta (t)=\theta _{\mathrm {eq} }}$

${\displaystyle {\dot {r}}=\alpha r}$

${\displaystyle \alpha =\alpha _{\mathrm {max} }\cos(2\theta _{\mathrm {eq} })=\alpha _{\mathrm {max} }{\sqrt {1-(2\epsilon /f_{0})^{2}}}}$

The solution to this equation is an exponential function. So that the amplitude of increases exponentially, it must therefore apply ${\displaystyle q}$

${\displaystyle \omega _{n}{\sqrt {1-(f_{0}/2)}}<\omega _{p}<\omega _{n}{\sqrt {1+(f_{0}/2)}}}$

The greatest increase in amplitude is obtained for the case . However, the corresponding oscillation of the untransformed variable does not have to increase. Their amplitude is given by the following equation: ${\displaystyle \omega _{p}=\omega _{n}}$ ${\displaystyle x}$ ${\displaystyle R}$

${\displaystyle R(t)=r_{0}e^{\alpha t-{\frac {1}{2}}\int ^{t}d\tau \ \beta (\tau )}}$

You can see that their behavior depends on whether it is greater, less or equal to the time integral of the speed-dependent parameter. ${\displaystyle \alpha t}$

Illustration with Fourier components

Since the above mathematical derivation can seem complicated and tricky, it is often helpful to look at a clearer derivation. To do this, we write the differential equation in the form

${\displaystyle {\ddot {q}}+\omega _{n}^{2}q=-\omega _{n}^{2}f(t)q}$,

We assume that the pump function is a sinusoidal function of double frequency and that the oscillation already has a corresponding shape, i.e.

${\displaystyle f(t)=f_{0}\sin 2\omega _{p}t}$

${\displaystyle q(t)=A\cos \omega _{p}t}$

A trigonometric identity can be used for the product of the two sinusoidal functions, so that two pump signals are obtained.

${\displaystyle f(t)q(t)={\frac {f_{0}}{2}}A\left(\sin \omega _{p}t+\sin 3\omega _{p}t\right)}$

In Fourier space , the multiplication is a superposition of the Fourier transforms and . The positive amplification comes from the fact that the component of and the component of become an excitation signal with and analogously with opposite signs. This explains why the pump frequency has to be in the vicinity of twice the resonance frequency of the oscillator. A pump frequency that is very different would not couple, i.e. not result in positive feedback between the components and . ${\displaystyle {\tilde {F}}(\omega )}$ ${\displaystyle {\tilde {Q}}(\omega )}$ ${\displaystyle +2\omega _{p}}$ ${\displaystyle f}$ ${\displaystyle -\omega _{p}}$ ${\displaystyle q}$ ${\displaystyle +\omega _{p}}$ ${\displaystyle 2\omega _{n}}$ ${\displaystyle -\omega _{p}}$ ${\displaystyle +\omega _{p}}$

Stability and resonance

Exemplary stability map of a vibrating pendulum ( numerical solution)

The case in which the change in the parameters increases the amplitude of the oscillation is called parametric resonance. For applications it is often interesting whether an oscillation is stable . In the considered case of a harmonic oscillator, stable means that the energy and thus the oscillation amplitude does not diverge towards infinity. Stable vibrations are therefore bound, unstable ones unbound. The stability of a system can be illustrated in a stability map (see example on the right). Two methods of stability analysis are explained below.

Hill stability study

The starting point is an approach function of the form

${\displaystyle \psi =e^{\lambda t}z(t)}$

where the first factor contains an eigenvalue that characterizes the stability (see below) and the second factor is periodic with the parameter frequency. As a complex Fourier series , it has the following form

${\displaystyle z(t)=\sum _{n=-\infty }^{\infty }c_{n}e^{in\Omega t};\quad n=\dots ,-2,-1,0,1,2,\dots }$

The (periodic) system matrices are also developed in a Fourier series. The principle of harmonic balance leads to an eigenvalue problem with matrices of size [K (2N + 1) × K (2N + 1)] ( K = degrees of freedom, N = number of Fourier terms) with the eigenvalues ​​that are of interest for the stability analysis (the number of Eigenvalues ​​according to the matrix sizes). ${\displaystyle \lambda =\alpha +i\omega }$

${\displaystyle \det[\lambda ^{2}+\lambda \cdot {\overline {p_{1}}}+{\overline {p_{2}}}]=0}$

The size of the real part of the eigenvalue determines the stability.

See also : method of harmonic balance

Floquet stability study

Another possibility for determining the stability limits is the stability analysis according to Floquet . With 2K given linearly independent real initial conditions (K = degrees of freedom) , the DGL is integrated numerically, in simple cases also analytically over a parameter period, and from the values ​​obtained a 2 K × 2 K transfer matrix is generated, whose conjugate complex eigenvalues ​​stability or Identify instability (so-called floquish eigenvalue problem ). In the stability map shown on the right, the Floquet method is applied to the DGL of the undamped vibrating pendulum or the math DGL. Periodic solutions are present on the boundary lines between the stable and the unstable area (shown here in yellow). In the stability range, the solution tends to zero after a certain time. It can also be seen for this DGL that if there is no damping for the stability analysis, in addition to the excitation with double and single natural frequency, the excitation with 2/3 (no longer recognizable here 2/4, 2/5 etc.) of the natural frequency still has a certain amount has mathematical meaning. ${\displaystyle \Phi }$

See also : Floquet's theorem

Parametric amplifiers

Applications

Parametric oscillators as low-noise amplifiers ( English Low Noise Amplifier ) occur particularly in the radio and microwave range. An oscillating circuit with a capacitance diode is excited by changing its capacitance periodically. YAG - waveguide in the microwave technique to work on the same principle.

Advantages of using parametric amplifiers are

• their high sensitivity
• their low thermal noise because a reactance (and not a resistance) is changed

Working principle

A parametric amplifier is operated as a frequency mixer. The amplification of this signal mixture is reflected in the amplification factor of the output. The weak input signal is mixed with the strong oscillator signal and the resulting signal is used in the subsequent receiver stages.

Parametric amplifiers also work by changing the vibration parameters. It can be understood intuitively for an amplifier with variable capacitance as by means of the following relations. The charge on the capacitor is

${\displaystyle Q=C\cdot V}$

and therefore the voltage across the capacitor

${\displaystyle V=Q/C}$

If a capacitor is charged until the voltage corresponds to that of the weak input signal and then the capacitance of the capacitor is reduced, for example by moving the plates of a plate capacitor further apart, the applied voltage increases and thus the weak signal is amplified. If the capacitor is a capacitance diode, the plates can be moved, i.e. the capacitance can be changed, by simply applying a time-dependent voltage. This driving voltage is also called the pump voltage.

The resulting output signal contains different frequencies that correspond to the sum and difference of input signal and output signal , i.e. and . ${\displaystyle f_{1}}$ ${\displaystyle f_{2}}$ ${\displaystyle f_{1}+f_{2}}$ ${\displaystyle f_{1}-f_{2}}$

In practice, a parametric oscillator needs the following connections:

• Ground connection
• Pump voltage
• output
• partly a fourth to set the parameters

A parametric amplifier also needs an input for the signal to be amplified. Since a capacitance diode only has two connections, it can only be used in conjunction with an LC network. This can be implemented as a transimpedance amplifier , as a traveling wave tube amplifier or with the help of a circulator .

"Rocking up": the swinging censer

An elementary example: The "swinging incense kettle" of the cathedral of Santiago de Compostela hanging from the church ceiling is "rocked" by a team of so-called "botafumeiros" to parametric resonance, using the principle of "double frequency": Always at zero crossing the pendulum length of the boiler is systematically shortened "by pulling it up".

literature

• Ludwig Kühn: About a new radio telephone system . In: Electrotechnical Journal . tape 35 , 1914, pp. 816-819 .
• WW Mumford: Some Notes on the History of Parametric Transducers . In: Proceedings of the IRE . tape 48 , no. 5 , 1960, pp. 848-853 , doi : 10.1109 / JRPROC.1960.287620 .
• L. Pungs: The control of high frequency currents using iron chokes with superimposed magnetization . In: ETZ . tape 44 , 1923, pp. 78-81 .
• L. Pungs: Comments on the History of Parametric Transducers . In: Proceedings of the IRE . tape 49 , no. 1 , 1961, pp. 378 , doi : 10.1109 / JRPROC.1961.287827 . See correspondence . In: Proceedings of the IRE . tape 49 , no. 1 , 1961, pp. 349-381 , doi : 10.1109 / JRPROC.1961.287827 .
• Jeffery Cooper: Parametric Resonance in Wave Equations with a Time-Periodic Potential . In: SIAM Journal on Mathematical Analysis . tape 31 , no. 4 , January 2000, p. 821-835 , doi : 10.1137 / S0036141098340703 .

Web links

• Franz-Josef Elmer: Parametric Resonance . unibas.ch, July 20, 1998.
• Solving the Mathieu equation with Matlab
• Video: Parametric excitation of vibrations . Institute for Scientific Film (IWF) 1987, made available by the Technical Information Library (TIB), doi : 10.3203 / IWF / C-1627 .

Remarks

1. ↑ The parameter of the first derivative is not synonymous with the presence of dissipation , even if it is often referred to as attenuation
2. ↑ Note: In reality, swings are often not only stimulated parametrically, see:
   William B. Case: Two ways of driving a child's swing . Archived from the original on September 25, 2013. Retrieved February 3, 2013.
   William B. Case: The pumping of a swing from the standing position . In: American Journal of Physics . 64, 1996, pp. 215-220.
   P. Roura, JA Gonzalez: Towards a more realistic description of swing pumping due to the exchange of angular momentum . In: European Journal of Physics . 31, 2010, pp. 1195-1207.

Individual evidence

1. ↑ Kurt Magnus: Vibrations: An introduction to the physical principles and the theoretical treatment of vibration problems. 8., revised. Edition, Vieweg + Teubner, 2008, Chapter 4, ISBN 3-8351-0193-5 .
2. ↑ Klaus Knothe, Robert Gasch : Structural Dynamics : Volume 2: Continua and their discretization. Springer, 1989, chapter 12, ISBN 3-540-50771-X .
3. ^ Archive of Applied Mechanics - March 1995, Volume 65, Issue 3, pp 178-193; Modal treatment of linear, periodically time-variant equations of motion ; doi: 10.1007 / BF00799297
4. ↑ Wolfgang Demtröder: Experimentalphysik 1: Mechanik und Wärme Springer, 2008, Chapter 11.7, ISBN 3-540-79294-5 .
5. ↑ Ludwig Bergman, Clemens Schaefer: Mechanics, Relativity, Warmth . Walter de Gruyter, 1998, ISBN 3-11-012870-5 , p. 618 ( limited preview in Google Book search).
6. ↑ M. Faraday: On a Peculiar Class of Acoustical Figures; and on Certain Forms Assumed by Groups of Particles upon Vibrating Elastic Surfaces . In: Philosophical Transactions of the Royal Society of London . tape 121 , 1831, pp. 299-340 , doi : 10.1098 / rstl.1831.0018 .
7. ↑ F. Melde: About the excitation of standing waves of a thread-like body . In: Annals of Physics . tape 187 , no. 12 , 1860, p. 513-537 , doi : 10.1002 / andp.18601871202 .
8. ↑ Lord Rayleigh: On maintained vibrations . In: Philosophical Magazine Series 5 . tape 15 , no. 94 , 1883, pp. 229-235 , doi : 10.1080 / 14786448308627342 .
9. ↑ Lord Rayleigh: On the maintenance of vibrations by forces of double frequency, and on the propagation of waves through a medium endowed with a periodic structure . In: Philosophical Magazine Series 5 . tape 24 , no. 147 , 1887, pp. 145-159 , doi : 10.1080 / 14786448708628074 .
10. ^ JWS Rayleigh: The Theory of Sound . Vol. 1, 2nd. ed., Dover, New York 1945, pp. 81-85.
11. ↑ Klaus Knothe, Robert Gasch: Structural Dynamics: Volume 2: Continua and their discretization. Springer, 1989, chapter 12.4, ISBN 3-540-50771-X .
12. ↑ Sungook Hong: Wireless: From Marconi's black box to the Audion . MIT Press, 2001, ISBN 0-262-08298-5 , pp. 165 ( limited preview in Google Book search).
13. ↑ EFW Alexanderson, SP Nixdorff: A Magnetic Amplifier for Radio Telephony . In: Proceedings of the Institute of Radio Engineers . tape 4 , no. 2 , April 1916, p. 101-120 , doi : 10.1109 / JRPROC.1916.217224 .
14. ↑ FM Arscott: Periodic Differential Equations; An Introduction to Mathieu, Lamé, and Allied Functions . The Macmillan Company, 1964, Chapter VII: Hill's Equation , pp. 141 ff .
15. ↑ H. Schlichting: The swinging incense kettle, in: Spectrum of Science (Special Physik.Mathematik.Technik 3/14), "Natural Laws in the Coffee Cup " , September 2014, p. 80