Analysis and Application for the Source-free R M LC Circuits

Xiang Gao^1*Yuhan Qian¹Shengpeng Li¹Wenjuan Li¹Yao Su²Yue Liu³

• ¹Aerospace Times FeiHong Technology Company Limited, Beijing, China
• ²Chinese Academy of Sciences Institute of Automation, Beijing, China
• ³Changchun University of Technology, Changchun, China

When a memristor circuit/network is designed, both memristors and conventional electronic components are inevitably required, particularly energy storage elements (e.g., capacitors and inductors). It has found that most existing studies focus on oscillatory phenomena generated by memristive systems, such as chaotic attractors, period-doubling oscillations, spiking and bursting oscillations. However, there is a notable lack of literature exploring and analyzing the energy exchange between these components, as well as the resulting oscillatory behaviors and outcomes arising from such interactions. It is well known that the unit of a memristor, like that of a resistor, is the ohm ($\Omega$). In general circuits, the energy exchange between resistors and energy storage elements can induce nonlinear behaviors such as step functions, damping phenomena, both of which stem from the energy exchange between resistors and capacitors/inductors. So, when a memristor exchanges energy with energy storage elements, will similar behaviors emerge? In this paper, to advance the theoretical completeness of memristive systems and take the classical HP memristor model as an example, four source-free circuit topologies integrating memristors with energy-storage elements are investigated deeply. Firstly, through mathematical modeling, the four circuits are all found to be governed by transcendental equations. Simulation results demonstrate memristor states ($R_0$​, $kR_d$​) and energy-storage elements collectively regulate response characteristics, damped oscillatory and decay behavior. The active power and apparent power curves reveal distinct energy exchange behaviors between components, differing fundamentally from conventional circuits. These findings demonstrate that due to the presence of memristors, such circuits cannot be employed for step response generation, but are exclusively applicable for energy memorization and dissipation. Secondly, two types of four-component source-free circuits are configured and analysis, the following conclusion are demonstrated: (1) capacitor and inductor provide energy to the system, while memristors exhibit hysteretic behavior, collectively and fundamentally co-modulating oscillation modes and attractor phenomenon; (2) The dual characteristics of memristors—memory capability and energy dissipation—making them essential candidates for implementing neural networks and AI systems. Finally, the application circuits were configured and explored. The study reveals that even within the same circuit, varying memristor placements can lead to distinct topological configurations and divergent nonlinear output behaviors.