Kapitza-Inspired Stabilization of Non-Foster Circuits via Time Modulations

TL;DR

The paper addresses stabilization of non-Foster circuits, which are prone to instability despite their broadband potential. It develops a Kapitza-inspired mapping by recasting the non-Foster $L(t)C$ dynamics into the normal form $\phi''(t) + \frac{1}{L(t)C} \phi(t) = 0$ and linking it to Kapitza’s pendulum equation $\theta''(t) + [\alpha\Omega^2 \cos(\Omega t) - \omega_0^2] \sin \theta = 0$ in the small-angle limit. A direct Kapitza-like modulation $L(t) = \frac{1}{C[\alpha\Omega^2 \cos(\Omega t) - \omega_0^2]}$ can stabilize the circuit if the modulation frequency $\Omega$ exceeds $\Omega_{\mathrm{lim}}=\sqrt{2}\,\omega_0/\alpha$, but requires negative $L(t)$. Practically, the authors propose a positive-time modulation $L(t) = L_{eq} + \frac{c_1}{i_{L_{eq}}(t)}$ with a DC bias so that $L(t)>0$ while mimicking $L_{eq}<0$, yielding a stable resonance at $\omega_{eq} = 1/\sqrt{L_{eq}C}$ and enabling real-time frequency reconfiguration. The results introduce Vibrational Electromagnetics as a framework for applying time-modulations to complex media and show potential applications to transmission lines and space-time metamaterials, including real-time frequency reconfiguration.

Abstract

With his formal analysis in 1951, the physicist Pyotr Kapitza demonstrated that an inverted pendulum with an externally vibrating base can be stable in its upper position, thus overcoming the force of gravity. Kapitza's work is an example that an originally unstable system can become stable after a minor perturbation of its properties or initial conditions is applied. Inspired by his ideas, we show how non-Foster circuits can be stabilized with the application of external \textit{electrical vibration}, i.e., time modulations. Non-Foster circuits are highly appreciated in the engineering community since their bandwidth characteristics are not limited by passive-circuits bounds. Unfortunately, non-Foster circuits are usually unstable and they must be stabilized prior to operation. Here, we focus on the study of non-Foster $L(t)C$ circuits with time-varying inductors and time-invariant negative capacitors. We find an intrinsic connection between Kapitza's inverted pendulum and non-Foster $L(t)C$ resonators. Moreover, we show how positive time-varying modulations of $L(t)>0$ can overcome and stabilize non-Foster negative capacitances $C<0$. These findings open up an alternative manner of stabilizing electric circuits with the use of time modulations, and lay the groundwork for application of, what we coin \textit{Vibrational Electromagnetics}, in more complex media.

Figures (4)

• Figure 1: Connection between Kapitza's inverted pendulum and time-modulated non-Foster circuits. Vibrational resonances appear in both systems, making them stable.
• Figure 2: Current $i(t)$ in the non-Foster $L(t)C$ resonator. The modulated $L(t)$ is the one that mimics the behavior of Kapitza's inverted pendulum [eq. \ref{['L_kapitza']}]. (a) Unstable case. (b) Stable case. The red dotted line represents a normalized version of $\phi(t)$. Parameters: $\alpha=0.1$, $C=-10^{-3}$ F.
• Figure 3: Stabilization of the non-Foster $L(t)C$ resonator with a positive-valued time modulated $L(t)>0$. (a) Effect of varying the equivalent inductance $L_\mathrm{eq}$ (for $c_1 = 10$, $I = 3$ A). (b) Effect of varying the DC current source (for $c_1 = 10$, $L_\mathrm{eq} = -1$ H). In all cases, we have considered $C=-1$ F.
• Figure 4: Dynamic reconfiguration of frequency in the non-Foster $L(t)C$ resonator. Parameters: $C=-1$ F, $I = 5$ A, $c_1 = 10$ Wb.