Parametric oscillator - AbsoluteAstronomy.com

A parametric oscillator is a harmonic oscillator

Harmonic oscillator

In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force, F, proportional to the displacement, x: \vec F = -k \vec x \, where k is a positive constant....

whose parameters oscillate in time. For example, a well known parametric oscillator is a child pumping a swing by periodically standing and squatting to increase the size of the swing's oscillations. The varying of the parameters drives the system. Examples of parameters that may be varied are its resonance frequency

and damping

.

Parametric oscillators are used in many applications. The classical varactor parametric oscillator will oscillate when the diode's capacitance is varied periodically. The circuit that varies the diode's capacitance is called the "pump" or "driver". In microwave electronics, waveguide

Waveguide (electromagnetism)

In electromagnetics and communications engineering, the term waveguide may refer to any linear structure that conveys electromagnetic waves between its endpoints. However, the original and most common meaning is a hollow metal pipe used to carry radio waves...

/YAG based parametric oscillators operate in the same fashion. The designer varies a parameter periodically in order to induce oscillations.

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. For example, the Optical parametric oscillator

Optical parametric oscillator

An optical parametric oscillator is a parametric oscillator which oscillates at optical frequencies. It converts an input laser wave into two output waves of lower frequency by means of second order nonlinear optical interaction. The sum of the output waves frequencies is equal to the input wave...

converts an input laser

Laser

A laser is a device that emits light through a process of optical amplification based on the stimulated emission of photons. The term "laser" originated as an acronym for Light Amplification by Stimulated Emission of Radiation...

wave into two output waves of lower frequency (

).

Parametric resonance occurs in a mechanical system when a system is parametrically excited and oscillates at one of its resonant frequencies. Parametric excitation differs from forcing since the action appears as a time varying modification on a system parameter. This effect is different from regular resonance because it exhibits the instability

Instability

In numerous fields of study, the component of instability within a system is generally characterized by some of the outputs or internal states growing without bounds...

phenomenon.

History

Michael Faraday

Michael Faraday, FRS was an English chemist and physicist who contributed to the fields of electromagnetism and electrochemistry....

(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 resonance frequency of the string. Parametric oscillation was first treated as a general phenomenon by Rayleigh

John Strutt, 3rd Baron Rayleigh

John William Strutt, 3rd Baron Rayleigh, OM was an English physicist who, with William Ramsay, discovered the element argon, an achievement for which he earned the Nobel Prize for Physics in 1904...

(1883,1887), whose papers are still worth reading today.

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 (Ernst Alexanderson

Ernst Alexanderson

Ernst Frederick Werner Alexanderson was a Swedish-American electrical engineer, who was a pioneer in radio and television development.-Background:...

, 1916). The early paramps varied inductances, but other methods have been developed since, e.g., the varactor diodes, klystron tubes, Josephson junctions and optical methods

Optical parametric oscillator

The mathematics

This equation is linear in

. By assumption, the parameters

and

depend only on time and do not depend on the state of the oscillator. In general,

and/or

are assumed to vary periodically, with the same period

.

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 provided by

, 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

Harmonic oscillator

s, in which the amplitude grows linearly in time regardless of the initial state.

A familiar experience of both parametric and driven oscillation is playing on a swing. Rocking back and forth pumps the swing as a driven harmonic oscillator, but once moving, the swing can also be parametrically driven by alternately standing and squatting at key points in the swing. This changes moment of inertia of the swing and hence the resonance frequency, and children can quickly reach large amplitudes provided that they have some amplitude to start with (e.g., get a push). Standing and squatting at rest, however, goes nowhere.

Transformation of the equation

We begin by making a change of variables

where

is a time integral of the damping

This change of variables eliminates the damping term

where the transformed frequency is defined

In general, the variations in damping and frequency are relatively small perturbations

where

and

are constants, namely, the time-averaged oscillator frequency and damping, respectively. The transformed frequency can be written in a similar way:

where

is the natural frequency of the damped harmonic oscillator

and

Thus, our transformed equation can be written

Remarkably, the independent variations

and

in the oscillator damping and resonance frequency, respectively, can be combined into a single pumping function

. The converse conclusion is that any form of parametric excitation can be accomplished by varying either the resonance frequency or the damping, or both.

Solution of the transformed equation

Let us assume that

is sinusoidal, specifically

where the pumping frequency

but need not equal

exactly. The solution

of our transformed equation may be written

where we have factored out the rapidly varying components (

and

) to isolate the slowly varying amplitudes

and

. This corresponds to Laplace's variation of parameters method.

Substituting this solution into the transformed equation and retaining only the terms first-order in

yields two coupled equations

We may decouple and solve these equations by making another change of variables

which yields the equations

where we have defined for brevity

and the detuning

The

equation does not depend on

, and linearization near its equilibrium position

shows that

decays exponentially to its equilibrium

where the decay constant

.

In other words, the parametric oscillator phase-locks to the pumping signal

.

Taking

(i.e., assuming that the phase has locked), the

equation becomes

whose solution is

; the amplitude of the

oscillation diverges exponentially. However, the corresponding amplitude

of the untransformed variable

need not diverge

The amplitude

diverges, decays or stays constant, depending on whether

is greater than, less than, or equal to

, respectively.

The maximum growth rate of the amplitude occurs when

. At that frequency, the equilibrium phase

is zero, implying that

and

. As

is varied from

moves away from zero and

, i.e., the amplitude grows more slowly. For sufficiently large deviations of

, the decay constant

can become purely imaginary since

If the detuning

exceeds

becomes purely imaginary and

varies sinusoidally. Using the definition of the detuning

, the pumping frequency

must lie between

and

in order to achieve exponetial growth in

. Expanding the square roots in a binomial series shows that the spread in pumping frequencies that result in exponentially growing

is approximately

Intuitive derivation 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

equation may be written in the form

which represents a simple harmonic oscillator (or, alternatively, a bandpass filter) being driven by a signal

that is proportional to its response

.

Assume that

already has an oscillation at frequency

and that the pumping

has double the frequency and a small amplitude

. Applying a trigonometric identity

Trigonometry

Trigonometry is a branch of mathematics that studies triangles and the relationships between their sides and the angles between these sides. Trigonometry defines the trigonometric functions, which describe those relationships and have applicability to cyclical phenomena, such as waves...

for products of sinusoids, their product

produces two driving signals,
one at frequency

and the other at frequency

Being off-resonance, the

signal is attentuated and can be neglected initially. By contrast, the

signal is on resonance, serves to amplify

and is proportional to the amplitude

. Hence, the amplitude of

grows exponentially unless it is initially zero.

Expressed in Fourier space, the multiplication

is a convolution of their Fourier transforms

and

. The positive feedback arises because the

component of

converts the

component of

into a driving signal at

, and vice versa (reverse the signs). This explains why the pumping frequency must be near

, twice the natural frequency of the oscillator. Pumping at a grossly different frequency would not couple (i.e., provide mutual positive feedback) between the

and

components of

Parametric resonance

Parametric resonance is the parametric

Parameter

Parameter from Ancient Greek παρά also “para” meaning “beside, subsidiary” and μέτρον also “metron” meaning “measure”, can be interpreted in mathematics, logic, linguistics, environmental science and other disciplines....

al resonance

Resonance

In physics, resonance is the tendency of a system to oscillate at a greater amplitude at some frequencies than at others. These are known as the system's resonant frequencies...

phenomenon

Phenomenon

A phenomenon , plural phenomena, is any observable occurrence. Phenomena are often, but not always, understood as 'appearances' or 'experiences'...

of mechanical excitation

Excitation

-Science:* The excited state of an atom* The excitatory postsynaptic potential* The excitation provided with an electrical generator or alternator-Agitation:*Excitement...

and oscillation

Oscillation

Oscillation is the repetitive variation, typically in time, of some measure about a central value or between two or more different states. Familiar examples include a swinging pendulum and AC power. The term vibration is sometimes used more narrowly to mean a mechanical oscillation but sometimes...

at certain frequenc

Frequency

Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency...

ies (and the associated harmonic

Harmonic

A harmonic of a wave is a component frequency of the signal that is an integer multiple of the fundamental frequency, i.e. if the fundamental frequency is f, the harmonics have frequencies 2f, 3f, 4f, . . . etc. The harmonics have the property that they are all periodic at the fundamental...

s). This effect is different from regular resonance because it exhibits the instability

Instability

In numerous fields of study, the component of instability within a system is generally characterized by some of the outputs or internal states growing without bounds...

phenomenon.

Parametric resonance occurs in a mechanical system when a system is parametrically excited and oscillates at one of its resonant frequencies.Parametric resonance takes place when the external excitation frequency equals to twice the natural frequency of the system. Parametric excitation differs from forcing since the action appears as a time varying modification on a system parameter. The classical example of parametric resonance is that of the vertically forced pendulum.

For small amplitudes and by linearising, the stability of the periodic solution is given by :

where

is some perturbation from the periodic solution. Here the

term acts as an ‘energy’ source and is said to parametrically excite the system. The Mathieu equation describes many other physical systems to a sinusoidal parametric excitation such as an LC Circuit where the capacitor plates move sinusoidally.

Introduction

A parametric amplifier is implemented as a mixer

Frequency mixer

In electronics a mixer or frequency mixer is a nonlinear electrical circuit that creates new frequencies from two signals applied to it. In its most common application, two signals at frequencies f1 and f2 are applied to a mixer, and it produces new signals at the sum f1 + f2 and difference f1 -...

. The mixer's gain shows up in the output as amplifier gain. The input weak signal is mixed with a strong local oscillator signal, and the resultant strong output is used in the ensuing receiver stages.

Parametric amplifiers also operate by changing a parameter of the amplifier.
Intuitively, this can be understood as follows, for a variable capacitor based amplifier.

Q [charge in a capacitor] = C x V

therefore

V [voltage across a capacitor] = Q/C

Knowing the above, if a capacitor is charged until its voltage equals the sampled voltage of an incoming weak signal, and if the capacitor's capacitance is then reduced (say, by manually moving the plates further apart), then the voltage across the capacitor will increase. In this way, the voltage of the weak signal is amplified.

If the capacitor is a varicap diode, then the 'moving the plates' can be done simply by applying time-varying DC voltage to the varicap diode. This driving voltage usually comes from another oscillator — sometimes called a "pump".

The resulting output signal contains frequencies that are the sum and difference of the input signal (f1) and the pump signal (f2): (f1 + f2) and (f1 - f2).

A practical parametric oscillator needs the following connections: one for the "common" or "ground", one to feed the pump, one to retrieve the output, and maybe a fourth one for biasing. A parametric amplifier needs a fifth port to input the signal being amplified. Since a varactor diode has only two connections, it can only be a part of an LC network with four eigenvectors with nodes at the connections. This can be implemented as a transimpedance amplifier, a traveling wave amplifier or by means of a circulator

Circulator

A circulator is a passive non-reciprocal three- or four-port device, in which microwave or radio frequency power entering any port is transmitted to the next port in rotation...

Mathematical equation

The parametric oscillator equation can be extended by adding an external driving force

We assume that the damping

is sufficiently strong that, in the absence of the driving force

, the amplitude of the parametric oscillations does not diverge, i.e., that

. In this situation, the parametric pumping acts to lower the effective damping in the system. For illustration, let the damping be constant

and assume that the external driving force is at the mean resonance frequency

, i.e.,

. The equation becomes

whose solution is roughly

approaches the threshold

, the amplitude diverges. When

, the system enters parametric resonance and the amplitude begins to grow exponentially, even in the absence of a driving force

Advantages

1:It is highly sensitive.

2:low noise level amplifier for ultra high frequency and microwave radio signal.

Other relevant mathematical results

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

The

equation above with periodically varying

is an example of a Hill equation. If

is a simple sinusoid, the equation is called a Mathieu equation.

External articles

Elmer, Franz-Josef, "Parametric Resonance". unibas.ch, July 20, 1998.
Cooper, Jeffery, "Parametric Resonance in Wave Equations with a Time-Periodic Potential". SIAM Journal on Mathematical Analysis, Volume 31, Number 4, pp. 821–835. Society for Industrial and Applied Mathematics, 2000 .
"Driven Pendulum: Parametric Resonance". phys.cmu.edu (Demonstration of physical mechanics or classical mechanics. Resonance oscillations set up in a simple pendulum via periodically varying pendulum length.)

The source of this article is wikipedia, the free encyclopedia. The text of this article is licensed under the GFDL.

History

The mathematics

Transformation of the equation

Solution of the transformed equation

Intuitive derivation of parametric excitation

Parametric resonance

Introduction

Mathematical equation

Advantages

Other relevant mathematical results

See also

Further reading

External articles