Parametric oscillator

We begin by making a change of variables

${\displaystyle q(t)\ {\stackrel {\mathrm {def} }{=}}\ e^{D(t)}x(t)}$

where ${\displaystyle D(t)}$ is a time integral of the damping

${\displaystyle D(t)\ {\stackrel {\mathrm {def} }{=}}\ {\frac {1}{2}}\int ^{t}d\tau \ \beta (\tau )}$.

This change of variables eliminates the damping term

${\displaystyle {\frac {d^{2}q}{dt^{2}}}+\Omega ^{2}(t)q=0}$

where the transformed frequency is defined

${\displaystyle \Omega ^{2}(t)=\omega ^{2}(t)-{\frac {1}{2}}\left({\frac {d\beta }{dt}}\right)-{\frac {1}{4}}\beta ^{2}}$.

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

${\displaystyle \beta (t)=\omega _{0}\left[b+g(t)\right]}$

${\displaystyle \omega ^{2}(t)=\omega _{0}^{2}\left[1+h(t)\right]}$

where ${\displaystyle \omega _{0}}$ and ${\displaystyle b}$ are constants, namely, the time-averaged oscillator frequency and damping, respectively.

The transformed frequency can be written in a similar way:

${\displaystyle \Omega ^{2}(t)=\omega _{n}^{2}\left[1+f(t)\right]}$,

where ${\displaystyle \omega _{n}}$ is the natural frequency of the damped harmonic oscillator

${\displaystyle \omega _{n}^{2}\ {\stackrel {\mathrm {def} }{=}}\ \omega _{0}^{2}\left(1-{\frac {b^{2}}{4}}\right)}$

and

${\displaystyle \omega _{n}^{2}f(t)\ {\stackrel {\mathrm {def} }{=}}\ \omega _{0}^{2}\left\{h(t)-{\frac {1}{2\omega _{0}}}\left({\frac {dg}{dt}}\right)-{\frac {b}{2}}g(t)-{\frac {1}{4}}g^{2}(t)\right\}}$.

Thus, our transformed equation can be written

${\displaystyle {\frac {d^{2}q}{dt^{2}}}+\omega _{n}^{2}\left[1+f(t)\right]q=0}$.

The independent variations ${\displaystyle g(t)}$ and ${\displaystyle h(t)}$ in the oscillator damping and resonance frequency, respectively, can be combined into a single pumping function ${\displaystyle f(t)}$. The converse conclusion is that any form of parametric excitation can be accomplished by varying either the resonance frequency or the damping, or both.

Let us assume that ${\displaystyle f(t)}$ is sinusoidal, specifically

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

where the pumping frequency ${\displaystyle \omega _{p}\approx \omega _{n}}$ but need not equal ${\displaystyle \omega _{n}}$ exactly. The solution ${\displaystyle q(t)}$ of our transformed equation may be written

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

where the rapidly varying components have been factored out(${\displaystyle \cos \omega _{p}t}$ and ${\displaystyle \sin \omega _{p}t}$) to isolate the slowly varying amplitudes ${\displaystyle A(t)}$ and ${\displaystyle B(t)}$. This corresponds to Laplace's variation of parameters method.

Substituting this solution into the transformed equation and retaining only the terms first-order in ${\displaystyle f_{0}\ll 1}$ yields two coupled equations

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

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

These equations may be decoupled and solved by making another change of variables

${\displaystyle A(t)\ {\stackrel {\mathrm {def} }{=}}\ r(t)\cos \theta (t)}$

${\displaystyle B(t)\ {\stackrel {\mathrm {def} }{=}}\ r(t)\sin \theta (t)}$

which yields the equations

${\displaystyle {\frac {dr}{dt}}=\left(\alpha _{\mathrm {max} }\cos 2\theta \right)r}$

${\displaystyle {\frac {d\theta }{dt}}=-\alpha _{\mathrm {max} }\left[\sin 2\theta -\sin 2\theta _{\mathrm {eq} }\right]}$

where for brevity the following are defined

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

${\displaystyle \sin 2\theta _{\mathrm {eq} }\ {\stackrel {\mathrm {def} }{=}}\ \left({\frac {2}{f_{0}}}\right)\epsilon }$

and the detuning

${\displaystyle \epsilon \ {\stackrel {\mathrm {def} }{=}}\ {\frac {\omega _{p}^{2}-\omega _{n}^{2}}{\omega _{n}^{2}}}}$.

The ${\displaystyle \theta }$ equation does not depend on ${\displaystyle r}$, and linearization near its equilibrium position ${\displaystyle \theta _{\mathrm {eq} }}$ shows that ${\displaystyle \theta }$ decays exponentially to its equilibrium

${\displaystyle \theta (t)=\theta _{\mathrm {eq} }+\left(\theta _{0}-\theta _{\mathrm {eq} }\right)e^{-2\alpha t}}$

where the decay constant

${\displaystyle \alpha \ {\stackrel {\mathrm {def} }{=}}\ \alpha _{\mathrm {max} }\cos 2\theta _{\mathrm {eq} }}$.

In other words, the parametric oscillator phase-locks to the pumping signal ${\displaystyle f(t)}$.

Taking ${\displaystyle \theta (t)=\theta _{\mathrm {eq} }}$ (i.e., assuming that the phase has locked), the ${\displaystyle r}$ equation becomes

${\displaystyle {\frac {dr}{dt}}=\alpha r}$

whose solution is ${\displaystyle r(t)=r_{0}e^{\alpha t}}$; the amplitude of the ${\displaystyle q(t)}$ oscillation diverges exponentially. However, the corresponding amplitude ${\displaystyle R(t)}$ of the untransformed variable ${\displaystyle x\ {\stackrel {\mathrm {def} }{=}}\ qe^{-D(t)}}$ need not diverge

${\displaystyle R(t)=r(t)e^{-D(t)}=r_{0}e^{\alpha t-D(t)}}$

The amplitude ${\displaystyle R(t)}$ diverges, decays, or stays constant, depending on whether ${\displaystyle \alpha t}$ is greater than, less than, or equal to ${\displaystyle D(t)}$, respectively.

The maximum growth rate of the amplitude occurs when ${\displaystyle \omega _{p}=\omega _{n}}$. At that frequency, the equilibrium phase ${\displaystyle \theta _{\mathrm {eq} }}$ is zero, implying that ${\displaystyle \cos 2\theta _{\mathrm {eq} }=1}$ and ${\displaystyle \alpha =\alpha _{\mathrm {max} }}$. As ${\displaystyle \omega _{p}}$ is varied from ${\displaystyle \omega _{n}}$, ${\displaystyle \theta _{\mathrm {eq} }}$ moves away from zero and ${\displaystyle \alpha <\alpha _{\mathrm {max} }}$, i.e., the amplitude grows more slowly. For sufficiently large deviations of ${\displaystyle \omega _{p}}$, the decay constant ${\displaystyle \alpha }$ can become purely imaginary since

${\displaystyle \alpha =\alpha _{\mathrm {max} }{\sqrt {1-\left({\frac {2}{f_{0}}}\right)^{2}\epsilon ^{2}}}}$.

If the detuning ${\displaystyle \epsilon }$ exceeds ${\displaystyle f_{0}/2}$, ${\displaystyle \alpha }$ becomes purely imaginary and ${\displaystyle q(t)}$ varies sinusoidally. Using the definition of the detuning ${\displaystyle \epsilon }$, the pumping frequency ${\displaystyle 2\omega _{p}}$ must lie between ${\displaystyle 2\omega _{n}{\sqrt {1-{\frac {f_{0}}{2}}}}}$ and ${\displaystyle 2\omega _{n}{\sqrt {1+{\frac {f_{0}}{2}}}}}$ in order to achieve exponential growth in ${\displaystyle q}$. Expanding the square roots in a binomial series shows that the spread in pumping frequencies that result in exponentially growing ${\displaystyle q}$ is approximately ${\displaystyle \omega _{n}f_{0}}$.

The above derivation may seem like a mathematical sleight-of-hand, so it may be helpful to give an intuitive derivation. The ${\displaystyle q}$ equation may be written in the form

${\displaystyle {\frac {d^{2}q}{dt^{2}}}+\omega _{n}^{2}q=-\omega _{n}^{2}f(t)q}$

which represents a simple harmonic oscillator (or, alternatively, a bandpass filter) being driven by a signal ${\displaystyle -\omega _{n}^{2}f(t)q}$ that is proportional to its response ${\displaystyle q}$.

Assume that ${\displaystyle q(t)=A\cos \omega _{p}t}$ already has an oscillation at frequency ${\displaystyle \omega _{p}}$ and that the pumping ${\displaystyle f(t)=f_{0}\sin 2\omega _{p}t}$ has double the frequency and a small amplitude ${\displaystyle f_{0}\ll 1}$. Applying a trigonometric identity for products of sinusoids, their product ${\displaystyle q(t)f(t)}$ produces two driving signals, one at frequency ${\displaystyle \omega _{p}}$ and the other at frequency ${\displaystyle 3\omega _{p}}$

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

Being off-resonance, the ${\displaystyle 3\omega _{p}}$ signal is attenuated and can be neglected initially. By contrast, the ${\displaystyle \omega _{p}}$ signal is on resonance, serves to amplify ${\displaystyle q}$, and is proportional to the amplitude ${\displaystyle A}$. Hence, the amplitude of ${\displaystyle q}$ grows exponentially unless it is initially zero.

Expressed in Fourier space, the multiplication ${\displaystyle f(t)q(t)}$ is a convolution of their Fourier transforms ${\displaystyle {\tilde {F}}(\omega )}$ and ${\displaystyle {\tilde {Q}}(\omega )}$. The positive feedback arises because the ${\displaystyle +2\omega _{p}}$ component of ${\displaystyle f(t)}$ converts the ${\displaystyle -\omega _{p}}$ component of ${\displaystyle q(t)}$ into a driving signal at ${\displaystyle +\omega _{p}}$, and vice versa (reverse the signs). This explains why the pumping frequency must be near ${\displaystyle 2\omega _{n}}$, twice the natural frequency of the oscillator. Pumping at a grossly different frequency would not couple (i.e., provide mutual positive feedback) between the ${\displaystyle -\omega _{p}}$ and ${\displaystyle +\omega _{p}}$ components of ${\displaystyle q(t)}$.

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 ${\displaystyle Q}$ in a capacitor obeys:

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

therefore the voltage across is

${\displaystyle V=Q/C.}$

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 "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.

The parametric oscillator equation can be extended by adding an external driving force ${\displaystyle E(t)}$:

${\displaystyle {\frac {d^{2}x}{dt^{2}}}+\beta (t){\frac {dx}{dt}}+\omega ^{2}(t)x=E(t)}$.

We assume that the damping ${\displaystyle D}$ is sufficiently strong that, in the absence of the driving force ${\displaystyle E}$, the amplitude of the parametric oscillations does not diverge, i.e., that ${\displaystyle \alpha t<D}$. In this situation, the parametric pumping acts to lower the effective damping in the system. For illustration, let the damping be constant ${\displaystyle \beta (t)=\omega _{0}b}$ and assume that the external driving force is at the mean resonance frequency ${\displaystyle \omega _{0}}$, i.e., ${\displaystyle E(t)=E_{0}\sin \omega _{0}t}$. The equation becomes

${\displaystyle {\frac {d^{2}x}{dt^{2}}}+b\omega _{0}{\frac {dx}{dt}}+\omega _{0}^{2}\left[1+h_{0}\sin 2\omega _{0}t\right]x=E_{0}\sin \omega _{0}t}$

whose solution is roughly

${\displaystyle x(t)={\frac {2E_{0}}{\omega _{0}^{2}\left(2b-h_{0}\right)}}\cos \omega _{0}t}$.

As ${\displaystyle h_{0}}$ approaches the threshold ${\displaystyle 2b}$, the amplitude diverges. When ${\displaystyle h\geq 2b}$, the system enters parametric resonance and the amplitude begins to grow exponentially, even in the absence of a driving force ${\displaystyle E(t)}$.

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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 ${\displaystyle q}$ equation above with periodically varying ${\displaystyle f(t)}$ is an example of a Hill equation. If ${\displaystyle f(t)}$ is a simple sinusoid, the equation is called a Mathieu equation.