Parametric oscillator

A parametric oscillator is a harmonic oscillator with time-dependent parameters ( natural frequency and damping [NB 1] ). An oscillator can be supplied in this way energy in order to increase the amplitude of the vibration. The method of energy supply is called parametric excitation, the motion parameters aroused or rheolineare vibration. One example is the momentum Get in a swing by periodically raising and lowering the center of gravity parallel to the suspension. [NB 2]

A feature of a purely parametric oscillation generated is that they, in contrast to a forced oscillation without an initial displacement from the equilibrium situation can not arise.

Technical systems with time-varying parameters can be found for example in turbomachinery and helicopter construction. Parametric oscillators are used in a number of technical systems, particularly in electrical engineering, for example in the construction of low noise amplifiers. Further they can be used for frequency conversion. An optical parametric oscillator can convert for example an incident laser wave into two lower frequency radiations.

• 4.1 Stability analysis by Hill
• 4.2 Floquet stability analysis by

• 5.1 applications
• 5.2 Principle of operation

Definition

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

The time-dependent functions and 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 excitation function called pumping function. The circuit or mechanism for changing the parameter is pump.

Characteristic of such excitation is that in an oscillator, which starts with an initial amplitude of zero, the amplitude is zero, because for the initial conditions are obtained always. However, since the gain functions even at the tiniest, unintended deflections, this case is not observed in reality.

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

Of particular practical interest is the case of resonance, in which the parameters change with double the natural frequency of the oscillator. Then the oscillator oscillates in phase-locked with the parametric variation, absorbing energy proportional to the energy, he already possesses. Without a mechanism that compensates for this increase, the amplitude of oscillation thus grows exponentially.

In systems with several degrees of freedom, the parameter matrix form and the dependent variables have to be summarized in a vector.

History

The first observations are by Michael Faraday, who described in 1831 Surface waves in a wine glass, which was suggested to "sing ". He noted that the oscillations of the wine glass of forces were excited at twice the frequency. In 1859, then reporting parameter Franz excited vibration generated in the string, by using a tuning fork to periodically change the tension of the string with a double resonance frequency. A description of parametric excitation as a general phenomenon was first written by Rayleigh in 1883 and 1887.

George William Hill met in 1886 in a special ODE with variable parameters when it detected faults experienced by the moon's orbit by the influence of the sun.

One of the first to the concept applied to electrical circuits, George Francis FitzGerald was in 1892 tried to stimulate oscillations in LC sections by changing the inductance of the resonant circuit with a dynamo as a pump.

Parametric amplifiers were first used in the years 1913 to 1915 for a radio transmitted telephone connection from Berlin to Vienna and Moscow. The potential of the technology for future applications was already at that time, recognized for example by Ernst Alexanderson. The first parametric amplifier operated by changing the inductance. Since then, other methods, such as the variable capacitance diode, Klystronröhren, Josephson junctions and optical methods have been developed.

Mathematical Description

Summary of parameters for an excitation function

We begin as the above differential equation:

To both time-dependent factors in the differential equation to a pumping function together, can first perform a transformation of variables to eliminate the rate- dependent term. We start from:

Arises After two dissipation and insertion into the original equation

With

The above equation, in which periodically changes is called Hillsche equation.

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

Where the constant of the damped oscillation frequency of the oscillator corresponds to, ie

The time-dependent function is called pumping function. Any type of parametric excitation thus can be described by the following differential equation always

Solution to a sinusoidal excitation frequency doubled

We consider the above differential equation

We assume that the pump function can be written as

Wherein the pumping frequency is approximately equal to half the oscillation frequency. This special case of the hill between differential equation is called the Mathieu differential equation. However, an exact match of the frequencies is not necessary for the solution, as the vibration adjusting the pump signal. After the set of Floquet the solution of the differential equation can be written as

The amplitudes and are time dependent. But usually applies to a parametric excitation that the amplitudes change slowly than the sine or cosine of the solution. In other words, the change in the oscillation amplitude is slower than the oscillation itself Substituting this solution into the differential equation, and retains only first order terms in, one obtains two coupled equations

In order to decouple this system of equations, a further transformation of variables can be performed

And thereby obtains the equations

With the constants

The constant is named upset. The equation for this is not dependent on. With a linear approximation can be shown that exponentially approaches the equilibrium point. In other words, the parametric oscillator coupled to a phase-locked to the pump signal. Substituting, whereby it is assumed that the coupling is established, the differential equation for the amplitude to

The solution of this equation is an exponential function. Thus, the amplitude of increases exponentially must apply accordingly

The largest growth of the amplitude is obtained for the case. However, the corresponding vibration of the untransformed variable does not need to grow. Its amplitude is given by the following equation:

You see, that's their behavior depending on whether is greater than, less or equal to the time integral of the velocity-dependent parameter.

Illustration with Fourier components

Since the above mathematical derivation can seem complicated and tricky, it is often helpful to create a more descriptive derivation to consider. We write the differential equation in the form

We assume that the pumping function is a sinusoidal function of twice the frequency, and that the oscillation already has a corresponding shape, ie

For the product of two sinusoidal functions can use a trigonometric identity, so that we obtain two pump signals.

In Fourier space, the multiplication is a superposition of the Fourier transform and. The positive reinforcement is because the component and the component of an excitation signal to be used with or in analogy with the opposite sign. This explains why the pump frequency must be in the vicinity of, twice the resonant frequency of the oscillator. A pump frequency which differs greatly, would not pair, so not in a positive feedback between the components and result.

Stability and resonance

In the case in which the change of the parameter is increased, the amplitude of vibration is referred to as parametric resonance. For applications, it is often interesting to see whether an oscillation is stable. In the considered case of a harmonic oscillator stable means that the energy and hence the oscillation amplitude does not diverge to infinity. Stable oscillations are bound, therefore, unstable unbound. The stability of a system can be in a stability map to illustrate (see exemplary illustration at right). The following are two methods for stability analysis is described.

Stability study by Hill

The starting point is a function of the form approach

The first factor involves an eigenvalue of the stability features (see below) and the second factor periodically with the frequency parameter is. Being a complex Fourier series, he has the following form

The ( periodic ) system matrices are also developed in a Fourier series. The principle of the harmonic balance results to an eigenvalue problem with matrices of size [K (2n 1) × K ( 2N 1 )] ( K = degrees of freedom, N = the number of Fourier terms ) to the interest for the stability viewing eigenvalues ​​(the number of eigenvalues ​​corresponding to the Matrizengrößen ).

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

Stability study by Floquet

Another possibility for determining the stability limits the stability study after Floquet. The DGL is given at 2K linearly independent real initial conditions (K = degrees of freedom) numerically integrated in simple cases analytically on a parameter period and from the obtained values ​​generates a 2K × 2K transfer matrix, which normally complex conjugate eigenvalues ​​of stability or instability characterize (so-called floquetsches eigenvalue problem). In the shown stability map on the right the Floquet method is applied to the differential equation of the undamped Rüttelpendels or the Mathieu differential equation. On the boundary lines between stable and unstable area (here shown in yellow ) are periodic solutions. In the region of stability, the solution goes to zero after a certain time. It can be seen for this DGL further that in the absence of damping for stability consideration in addition to the excitement with double and single natural frequency of the excitation with 2/3 ( unrecognizable 2/4, 2/ 5, etc. here ) the natural frequency for a certain has mathematical meaning.

Parametric Amplifier

Applications

Parametric oscillators as low-noise amplifiers (English Low Noise Amplifier ), are mostly found in the radio and microwave range. A resonant circuit with capacitance diode is excited by their capacitance is varied periodically. YAG waveguide in microwave technology work on the same principle.

Advantages of using parametric amplifiers are

• Their high sensitivity
• Their low thermal noise, because a reactance (not a resistance ) is changed

Principle of operation

A parametric amplifier is operated as a frequency mixer. The gain of this signal mixing is reflected in the gain of the output. The weak input signal is mixed with the strong local oscillator signal and the resultant signal is used in the subsequent receiver stages.

Parametric amplifiers also work by changing the oscillation parameters. It is intuitive to understand for an amplifier with variable capacity as determined by the following relations. Is the charge of the capacitor

And therefore, the voltage applied to the capacitor

When a capacitor is charged until the voltage to that of the weak input signal corresponds to, and then the capacitance of the capacitor is reduced, for example, by the plates of a plate capacitor are further apart, thus increasing the applied voltage, and thus amplifies the weak signal. When the capacitor is a variable capacitance diode, then the movement of the plates, that is done, a change in capacitance by simply applying a time-dependent voltage. This driving voltage is also called pump voltage.

The resulting output signal contains different frequencies corresponding to the sum and difference of input signal and output signal, ie, and.

Practical needs a parametric oscillator so the following connections:

• Ground connection
• Pump voltage
• Output
• Partially a fourth for setting the parameters

A parametric amplifier requires an additional input for the signal to be amplified. Since a varactor diode has only two ports, it can only be used in conjunction with an LC network. This can be realized as a transimpedance amplifier, a traveling wave tube amplifiers or by means of a circulator.