Parametric oscillator

A parametric oscillator is a driven harmonic oscillator in which the oscillations are driven by varying some parameter of the system at some frequency, typically different from the natural frequency of the oscillator. A simple example of a parametric oscillator is a child pumping a playground swing by periodically standing and squatting to increase the size of the swing's oscillations. The child's motions vary the moment of inertia of the swing as a pendulum. The "pump" motions of the child must be at twice the frequency of the swing's oscillations. Examples of parameters that may be varied are the oscillator's resonance frequency formula_1 and damping formula_2.
Parametric oscillators are used in several areas of physics. The classical varactor parametric oscillator consists of a semiconductor varactor diode connected to a resonant circuit or cavity resonator. It is driven by varying the diode's capacitance by applying a varying bias voltage. The circuit that varies the diode's capacitance is called the "pump" or "driver". In microwave electronics, waveguide/YAG-based parametric oscillators operate in the same fashion. Another important example is the optical parametric oscillator, which converts an input laser light wave into two output waves of lower frequency (formula_3).
When operated at pump levels below oscillation, the parametric oscillator can amplify a signal, forming a parametric amplifier (paramp). Varactor parametric amplifiers were developed as low-noise amplifiers in the radio and microwave frequency range. The advantage of a parametric amplifier is that it has much lower noise than an amplifier based on a gain device like a transistor or vacuum tube. This is because in the parametric amplifier a reactance is varied instead of a (noise-producing) resistance. They are used in very low noise radio receivers in radio telescopes and spacecraft communication antennas.
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.
History.
Parametric oscillations were first noticed in mechanics. Michael 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". Franz Melde (1860) 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 (1883,1887).
One of the first to apply the concept to electric circuits was George Francis FitzGerald, who in 1892 tried to excite oscillations in an LC circuit by pumping it with a varying inductance provided by a dynamo.
 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, 1916). These early parametric amplifiers used the nonlinearity of an iron-core inductor, so they could only function at low frequencies.
In 1948 Aldert van der Ziel pointed out a major advantage of the parametric amplifier: because it used a variable reactance instead of a resistance for amplification it had inherently low noise. A parametric amplifier used as the front end of a radio receiver could amplify a weak signal while introducing very little noise. In 1952 Harrison Rowe at Bell Labs extended some 1934 mathematical work on pumped oscillations by Jack Manley and published the modern mathematical theory of parametric oscillations, the Manley-Rowe relations.
The varactor diode invented in 1956 had a nonlinear capacitance that was usable into microwave frequencies. The varactor parametric amplifier was developed by Marion Hines in 1956 at Western Electric. At the time it was invented microwaves were just being exploited, and the varactor amplifier was the first semiconductor amplifier at microwave frequencies. It was applied to low noise radio receivers in many areas, and has been widely used in radio telescopes, satellite ground stations, and long-range radar. It is the main type of parametric amplifier used today. Since that time parametric amplifiers have been built with other nonlinear active devices such as Josephson junctions.
The technique has been extended to optical frequencies in optical parametric oscillators and amplifiers which use nonlinear crystals as the active element.
Mathematical analysis.
A parametric oscillator is a harmonic oscillator whose physical properties vary with time. The equation of such an oscillator is
This equation is linear in formula_5. By assumption, the parameters formula_6 and formula_2 depend only on time and do "not" depend on the state of the oscillator. In general, formula_8 and/or formula_9 are assumed to vary periodically, with the same period formula_10.
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 formula_2, 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 oscillators, 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 arc. 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, leads nowhere.
Transformation of the equation.
We begin by making a change of variables
where formula_13 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 formula_19 and formula_20 are constants, namely, the time-averaged oscillator frequency and damping, respectively.
The transformed frequency can be written in a similar way:
where formula_22 is the natural frequency of the damped harmonic oscillator
and
Thus, our transformed equation can be written
The independent variations formula_26 and formula_27 in the oscillator damping and resonance frequency, respectively, can be combined into a single pumping function formula_28. 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 formula_28 is sinusoidal, specifically
where the pumping frequency formula_31 but need not equal formula_22 exactly. The solution formula_33 of our transformed equation may be written
where the rapidly varying components have been factored out(formula_35 and formula_36) to isolate the slowly varying amplitudes formula_37 and formula_38. This corresponds to Laplace's variation of parameters method.
Substituting this solution into the transformed equation and retaining only the terms first-order in formula_39 yields two coupled equations
These equations may be decoupled and solved by making another change of variables
which yields the equations
where for brevity the following are defined
and the detuning
The formula_49 equation does not depend on formula_50, and linearization near its equilibrium position formula_51 shows that formula_49 decays exponentially to its equilibrium
where the decay constant
In other words, the parametric oscillator phase-locks to the pumping signal formula_28.
Taking formula_56 (i.e., assuming that the phase has locked), the formula_50 equation becomes
whose solution is formula_59; the amplitude of the formula_33 oscillation diverges exponentially. However, the corresponding amplitude formula_61 of the "untransformed" variable formula_62 need not diverge
The amplitude formula_61 diverges, decays, or stays constant, depending on whether formula_65 is greater than, less than, or equal to formula_13, respectively.
The maximum growth rate of the amplitude occurs when formula_67. At that frequency, the equilibrium phase formula_51 is zero, implying that formula_69 and formula_70. As formula_71 is varied from formula_22, formula_51 moves away from zero and formula_74, i.e., the amplitude grows more slowly. For sufficiently large deviations of formula_71, the decay constant formula_76 can become purely imaginary since
If the detuning formula_78 exceeds formula_79, formula_76 becomes purely imaginary and formula_33 varies sinusoidally. Using the definition of the detuning formula_78, the pumping frequency formula_83 must lie between formula_84 and formula_85 in order to achieve exponential growth in formula_86. Expanding the square roots in a binomial series shows that the spread in pumping frequencies that result in exponentially growing formula_86 is approximately formula_88.
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 formula_86 equation may be written in the form
which represents a simple harmonic oscillator (or, alternatively, a bandpass filter) being driven by a signal formula_91 that is proportional to its response formula_86.
Assume that formula_93 already has an oscillation at frequency formula_71 and that the pumping formula_30 has double the frequency and a small amplitude formula_39. Applying a trigonometric identity for products of sinusoids, their product formula_97 produces two driving signals,
one at frequency formula_71 and the other at frequency formula_99
Being off-resonance, the formula_101 signal is attenuated and can be neglected initially. By contrast, the formula_71 signal is on resonance, serves to amplify formula_86, and is proportional to the amplitude formula_104. Hence, the amplitude of formula_86 grows exponentially unless it is initially zero.
Expressed in Fourier space, the multiplication formula_106 is a convolution of their Fourier transforms formula_107 and formula_108. The positive feedback arises because the formula_109 component of formula_28 converts the formula_111 component of formula_33 into a driving signal at formula_113, and vice versa (reverse the signs). This explains why the pumping frequency must be near formula_114, twice the natural frequency of the oscillator. Pumping at a grossly different frequency would not couple (i.e., provide mutual positive feedback) between the formula_111 and formula_113 components of formula_33.
Parametric resonance.
Parametric resonance is the parametrical resonance phenomenon of mechanical perturbation and oscillation at certain frequencies (and the associated harmonics). This effect is different from regular resonance because it exhibits the instability 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 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 Mathieu's equation:
where formula_119 is some perturbation from the periodic solution. Here the formula_120 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.
Parametric amplifiers.
Introduction.
A parametric amplifier is implemented as a mixer. 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.
Charge formula_121 in a capacitor obeys:
therefore the voltage across is
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 "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.
Mathematical equation.
The parametric oscillator equation can be extended by adding an external driving force formula_124:
We assume that the damping formula_126 is sufficiently strong that, in the absence of the driving force formula_127, the amplitude of the parametric oscillations does not diverge, i.e., that formula_128. In this situation, the parametric pumping acts to lower the effective damping in the system. For illustration, let the damping be constant formula_129 and assume that the external driving force is at the mean resonance frequency formula_19, i.e., formula_131. The equation becomes
whose solution is roughly
As formula_134 approaches the threshold formula_135, the amplitude diverges. When formula_136, the system enters parametric resonance and the amplitude begins to grow exponentially, even in the absence of a driving force formula_124.
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 formula_86 equation above with periodically varying formula_28 is an example of a Hill equation. If formula_28 is a simple sinusoid, the equation is called a Mathieu equation.