Parametric Amplifier
Math
Abstract
This work presents a theoretical analysis of non-degenerate parametric amplifiers, which achieve signal amplification through time-varying reactive elements without introducing thermal noise. We derive the fundamental current-voltage relationships for parametric circuits and systematically eliminate variables to obtain the signal admittance expression. This result demonstrates how parametric coupling creates effective negative conductance proportional to pump power, enabling amplification when the negative conductance exceeds the circuit’s passive losses.
Keywords
Parametric amplifier
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Today we will be discussing the non-degenerate parametric amplifier. I will be borrowing the derivation from [1], [2], and [3].
Devices with nonlinear susceptance can be used to create parametric amplifiers and oscillators. A well known example is the reverse-biased pn junction, which has nonlinear charge-voltage characteristics arising from voltage-dependent capacitance. Such a nonlinear behavior results in frequency mixing among the following components:
- signal,
- idler,
- pump.
The energy is transferred from the pump wave to the weaker signal and idler waves. This is the operating principle of the parametric amplifier.
Circuit equivalent of a parametric amplifier is shown in Figure 1 or in its alternative form in Figure 2.
Nonlinear Capacitance
Consider a capacitor which has a voltage-dependent capacitance. Such dependence naturally arises from the pn junction diode. \[\begin{equation} \mathcal{C} = C(1+\alpha v) \label{eq:cap-linear}, \end{equation}\] where \(C\) is the capacitance at zero voltage and \(\alpha\) represents the linear dependence of the capacitance on the voltage. The charge on the capacitor is then \[\begin{equation} q(t) = C\left(1+\alpha v(t)\right)v(t)=Cv(t) + a v^2(t) \label{eq:charge-nonlinear}, \end{equation}\] where \(a=\alpha C\). The current through the capacitor is
\[\begin{equation} i(t) = \frac{dq(t)}{dt} = C\frac{dv(t)}{dt} + 2a v(t)\frac{dv(t)}{dt} \label{eq:current-nonlinear}. \end{equation}\] We will assign the angular frequencies \(\omega_1\), \(\omega_2\), and \(\omega_3\) for the signal, idler and pump waves, respectively. In Figure 1 we have chosen the sign of the currents such that the total voltage across the nonlinear capacitance is the sum of the voltages across the signal, idler and pump circuits. The voltage across the nonlinear capacitor is given by \[\begin{align} v(t) &= v_1(t) + v_2(t) + v_3(t) \\ &= V_1\cos(\omega_1 t + \phi_1) + V_2\cos(\omega_2 t + \phi_2) + V_3\cos(\omega_3 t + \phi_3) \label{eq:voltage-components} \end{align}\]
The relations between \(\omega\)’s are easier to see in the alternative form of the circuit in Figure 2. Imagine that the idler circuit is disconnected. The capacitor will have a tank circuit on the left with the resonant frequency \(\omega_1\) and the signal circuit will have a tank circuit on the right with the resonant frequency \(\omega_3\). The idler circuit will be tuned to beat at the difference frequency \(\omega_2=\omega_3 - \omega_1\), or equivalently:
\[\begin{equation} \omega_3 = \omega_1 + \omega_2 \label{eq:frequency-relation} \end{equation}\]
Now let’s define the individual frequencies of the signals. It is done with the typical trick of disabling all but one of the voltage/current sources. One other ingredient is the observation that a parallel \(L-C\) tank circuit becomes a short circuit at frequencies far separated from the resonant frequency. As we look from the signal side into the circuit in Figure 1 at a frequency of \(\omega_1\), the idler and pump circuits become short circuits making \(C\) and \(C_1\) appear in parallel. The same argument applies to the idler side and the pump side of the circuit.
Therefore the angular frequencies in Eq. \(\ref{eq:voltage-components}\) satisfy
\[\begin{equation} \omega_k = \frac{1}{\sqrt{L_k(C_k + C)}} \label{eq:resonant-frequency2}, \end{equation}\] where \(k=1,2,3\).
Current-Voltage Relations
Now we take the voltage expression Eq. \(\ref{eq:voltage-components}\) and substitute it in Eq. \(\ref{eq:current-nonlinear}\) and reorganize the terms:
\[\begin{eqnarray} i(t) &=&-\omega_1 CV_1 \sin(\omega_1 t + \phi_1) - \omega_1 a V_2 V_3 \sin(\omega_1 t + \phi_3 - \phi_2)\nonumber\\ && -\omega_2 CV_2 \sin(\omega_2 t + \phi_2) - \omega_2 a V_1 V_3 \sin(\omega_2 t + \phi_3 - \phi_1)\nonumber\\ && -\omega_3 CV_3 \sin(\omega_3 t + \phi_3) - \omega_3 a V_1 V_2 \sin(\omega_3 t + \phi_1 + \phi_2) \label{eq:total-current0}. \end{eqnarray}\] Let’s label the terms in the above equation as follows:
\[\begin{equation} i_1(t) = -\omega_1 CV_1 \sin(\omega_1 t + \phi_1) - \omega_1 a V_2 V_3 \sin(\omega_1 t + \phi_3 - \phi_2) \label{eq:current-1} \end{equation}\]
\[\begin{equation} i_2(t) = -\omega_2 C V_2 \sin(\omega_2 t + \phi_2) - \omega_2 a V_1 V_3 \sin(\omega_2 t + \phi_3 - \phi_1) \label{eq:current-2} \end{equation}\]
\[\begin{equation} i_3(t) = -\omega_3 CV_3 \sin(\omega_3 t + \phi_3) - \omega_3 a V_1 V_2 \sin(\omega_3 t + \phi_1 + \phi_2) \label{eq:current-3} \end{equation}\]
Equations \(\ref{eq:current-1}\)–\(\ref{eq:current-3}\) combine to give the total current:
\[\begin{equation} i(t) = i_1(t) + i_2(t) + i_3(t) \label{eq:total-current} \end{equation}\] We want to convert Eqs. \(\ref{eq:current-1}\)–\(\ref{eq:current-3}\) to the form of the current-voltage relations using the components of the voltage in Eq. \(\ref{eq:voltage-components}\). We will use the following trigonometric identity:
\[\begin{eqnarray} \sin(\omega_1 t + \phi_3 - \phi_2) &=& \sin(\omega_1 t +\phi_1+\phi_3 - \phi_2 - \phi_1)\nonumber\\ &=&\sin(\omega_1 t +\phi_1)\cos(\phi_3 - \phi_2 - \phi_1) + \cos(\omega_1 t +\phi_1)\sin(\phi_3 - \phi_2 - \phi_1)\nonumber\\ &=&\frac{1}{V_1 \omega_1} \left[ \dot v_1(t)\cos(\phi_3 - \phi_2 - \phi_1) + \omega_1 v_1(t)\sin(\phi_3 - \phi_2 - \phi_1)\right] \label{eq:trig-identity} \end{eqnarray}\]
This conversion gives:
\[\begin{eqnarray} i_1(t) &=& C\dot v_1(t)+ \frac{a V_1 V_3}{V_1}\left[\cos(\phi_3 - \phi_2 - \phi_1)\dot v_1(t)- \omega_1 v_1(t)\sin(\phi_3 - \phi_2 - \phi_1)\right] \label{eq:current-1-rewritten}\\ i_2(t) &=& C\dot v_2(t)+ \frac{a V_3}{V_2}\left[\cos(\phi_3 - \phi_2 - \phi_1)\dot v_2(t)- \omega_2 v_2(t)\sin(\phi_3 - \phi_2 - \phi_1)\right] \label{eq:current-2-rewritten}\\ i_3(t) &=& C\dot v_3(t)+ \frac{a V_1 V_2}{V_3}\left[\cos(\phi_3 - \phi_2 - \phi_1)\dot v_3(t)+ \omega_3 v_3(t)\sin(\phi_3 - \phi_2 - \phi_1)\right] \label{eq:current-3-rewritten} \end{eqnarray}\]
Admittance
Moving to the frequency domain, Eqs. \(\ref{eq:current-1-rewritten}\)–\(\ref{eq:current-3-rewritten}\), we get the admittances \(Y_k\) (k = 1,2,3) as we look into the circuit from point \(A_1-A_2\), \(A_2-B_2\), and \(B_1-B_2\), respectively:
\[\begin{eqnarray} Y_1 &=& \frac{I_1(i\omega)}{V_1(i\omega)} = i\omega_1 C + i\omega_1 a \frac{V_2 V_3}{V_1} \exp [i(\phi_3 - \phi_2 - \phi_1)] \label{eq:admittance-1}\\ Y_2 &=& \frac{I_2(i\omega_2)}{V_2(i\omega_2)} = i\omega_2 C + i\omega_2 a \frac{V_1 V_3}{V_2} \exp [i(\phi_3 - \phi_2 - \phi_1)] \label{eq:admittance-2}\\ Y_3 &=& \frac{I_3(i\omega)}{V_3(i\omega)} = i\omega_3 C + i\omega_3 a \frac{V_1 V_2}{V_3} \exp [-i(\phi_3 - \phi_2 - \phi_1)] \label{eq:admittance-3} \end{eqnarray}\] To see the total admittance from the point of view of the signal, for example, we need to add the parallel admittances of the \(G_s\), \(G_L\), \(L_1\), \(C_1\), \(G_1\). It is a bit tricky since one may be inclined to think that the \(L_1-C_1\) tank will have \(0\) admittance at \(\omega_1\). However, we have shown in Eq. \(\ref{eq:resonant-frequency2}\) that the resonance frequency is shifted by \(C\). Let’s calculate the admittance of the \(L_1-C_1\) tank at \(\omega_1\):
\[\begin{equation} Y_{L_1-C_1} = \frac{1}{i\omega_1 L_1} + i\omega_1 C_1=\frac{1-\omega_1^2 L_1 C_1}{i\omega_1 L_1}=\frac{\omega_1^2 L_1 C_1}{i\omega_1 L_1}=-i \omega_1 C_1 \label{eq:admittance-L1C1}, \end{equation}\] which neatly cancels the first term in Eq. \(\ref{eq:admittance-1}\). All there is left to do is to add the parallel admittances of the \(G\)’s, \(G_T = G_s + G_L + G_1\) for the signal circuit.
The current-voltage relations for the three circuits are given by
\[\begin{eqnarray} I_s(i\omega) &=& \left\{G_T + i\omega_1 a \frac{V_2 V_3}{V_1} \exp [i(\phi_3 - \phi_2 - \phi_1)]\right\} V_1(i\omega) \label{eq:current-voltage-1}\\ 0 &=& \left\{G_2 + i\omega_2 a \frac{V_1 V_3}{V_2} \exp [i(\phi_3 - \phi_2 - \phi_1)]\right\} V_2(i\omega) \label{eq:current-voltage-2}\\ I_P(i\omega) &=& \left\{G_3 + i\omega_3 a \frac{V_1 V_2}{V_3} \exp [-i(\phi_3 - \phi_2 - \phi_1)]\right\} V_3(i\omega) \label{eq:current-voltage-3} \end{eqnarray}\]
Here \(I_s(i\omega)\) and \(I_P(i\omega)\) are the Fourier transforms of the input signal and pump currents, respectively.
By eliminating \(V_2\) and \(V_3\) from Eq. \(\ref{eq:current-voltage-1}\) using Eqs. \(\ref{eq:current-voltage-2}\) and \(\ref{eq:current-voltage-3}\), we obtain the admittance of the signal circuit.
\[\begin{equation} Y_s = G_T - G = G_T - \frac{\omega_1\omega_2 a^2}{G_2 G_3} \frac{|I_P(i\omega)|^2}{\left|1 + \frac{\omega_2\omega_3}{G_2 G_3} a^2 V_1^2\right|^2} \label{eq:signal-admittancex} \end{equation}\]
The negative conductance emerges as a result of the nonlinear capacitance driven by the pump wave at \(\omega_3\). If \(V_1\) satisfies the condition,
\[\begin{equation} \frac{\omega_2\omega_3}{G_2 G_3} a^2 V_1^2 < 1 \label{eq:stability-condition} \end{equation}\]
the negative conductance is independent of the signal input and the linear parametric amplification is realized.
Power Gain
The ratio of the power delivered to the load \(G_L\) to the input power to the source \(G_s\) is the gain in power.
\[\begin{equation} \mathcal{G} = \frac{G_L V_1^2}{|I_s|^2/4G_s}=\frac{4G_s G_L}{|Y_s|^2}. \label{eq:power-gain-definition} \end{equation}\]
References
[1]
H. Heffner and G. Wade, “Gain, bandwidth and noise characteristic of the variable parameter amplifier,” Journal of Applied Physics, vol. 29, pp. 1321–1331, 1958, doi: 10.1063/1.1723440.
[2]
National Institute of Informatics, “Parametric amplifiers and oscillator.” National Institute of Informatics, Japan, 2024 [Online]. Available: https://www.nii.ac.jp/qis/first-quantum/e/forStudents/lecture/pdf/noise/chapter11.pdf. [Accessed: 2024]
[3]
W. J. Robertson, “Parametric amplifier design,” PhD thesis, The Ohio State University, 1959 [Online]. Available: http://rave.ohiolink.edu/etdc/view?acc_num=osu1400145234