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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 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
Contents
- 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
- References
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 (
When operated at pump levels below oscillation, the parametric oscillator can amplify a signal, becoming a parametric amplifier (paramp). Varactor parametric amplifiers have been 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 ordinary 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 have been 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
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). The early paramps varied inductances, but other methods have been developed since, e.g., the varactor diodes, klystron tubes, Josephson junctions and optical methods.
The mathematics
This equation is linear in
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
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
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
where
and
Thus, our transformed equation can be written
The independent variations
Solution of the transformed equation
Let us assume that
where the pumping frequency
where we have factored out the rapidly varying components (
Substituting this solution into the transformed equation and retaining only the terms first-order in
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
where the decay constant
In other words, the parametric oscillator phase-locks to the pumping signal
Taking
whose solution is
The amplitude
The maximum growth rate of the amplitude occurs when
If the detuning
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
which represents a simple harmonic oscillator (or, alternatively, a bandpass filter) being driven by a signal
Assume that
Being off-resonance, the
Expressed in Fourier space, the multiplication
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 :
where
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.
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.
Mathematical equation
The parametric oscillator equation can be extended by adding an external driving force
We assume that the damping
whose solution is roughly
As
Advantages
1:It is highly sensitive
2:low noise level amplifier for ultra high frequency and microwave radio signal
3:The unique capability to operate as a wireless powered amplifier that doesn't require internal power source
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