Platform for Weakly Coupled Electro-Mechanical Resonators with Arbitrary Tunability ^†

Abstract

This paper presents a platform for weakly coupled mechanical and electrical resonators with arbitrary tunability. Two- and three-degree-of-freedom (DoF) weakly coupled resonator systems are demonstrated experimentally with this platform, showing excellent tuning and matching flexibility. Fully electrical coupling logic circuitry gives the proposed system better adjustability and lower susceptibility to fabrication tolerances compared to purely mechanical weakly coupled resonators. Due to its mother/daughter board structure and the simplicity of the coupling logic, the proposed system can easily be extended to any-DoF systems.

1. Introduction

In recent years, there has been a growing interest in weakly coupled resonators due to their enhanced sensitivity and common-mode resilience in contrast to single resonators [1,2,3]. However, conventional coupled resonators are limited in their adjustability after fabrication as they are purely mechanical resonators. The pre-mode localization effect resulting from micro-fabrication tolerance, such as mass, spring, and damping mismatch, can cause coupling system issues.

This work presents a fully tunable platform for coupling mechanical and electrical resonators with an arbitrary coupling factor, rather than relying solely on mechanical resonators. The electrical resonators can easily be tuned and matched to the mechanical resonators after fabrication. By adding more electrical/mechanical resonators and extending the coupling logic circuitry, the proposed structure can be extended to any type and any-degree-of-freedom (DoF) weakly coupled resonators. Compared to existing solutions for tunability, which require computational hardware [4], the proposed system reduces the complexity of setup and avoids data conversion and processing at runtime.

2. System Structure and Working Mechanism

A 2-DoF and 3-DoF coupled resonator system is demonstrated to show the effectiveness and flexibility of the proposed platform, which is derived from the serial RLC electrical equivalent. The platform is shown in Figure 1 [5]. Electrical or mechanical resonators $R_1$ and $R_3$ are connected on the daughter boards. Each resonator has an independent actuation circuit and readout circuits. To demonstrate the functionality, $R_1$ and $R_3$ are the equivalent circuit of the mechanical resonators. In future work, $R_1$ and $R_3$ will be replaced with actual mechanical resonators. All resonators are connected through the tunable coupling circuit on the mother board. Resonator $R_2$ is a tunable RLC electrical resonator made up of lumped components. The coupling logic is built with analog operation circuits [6], which realizes fully electrical coupling. It has a coupling factor $k_c/k$ ranging from 0 to 1.

3. Measurement Results and Discussion

Figure 2 shows the measurement results of a 2-DoF and 3-DoF weakly coupled resonator system. Figure 2a,b show the frequency responses of resonator $R_1$ with different coupling factors. The plots show that the distances in frequency between different modes widen when the coupling factor increases. Figure 2c shows the frequency responses of resonator $R_1$ when keeping the coupling factor at 0.1 and reducing the capacitance value in the RLC resonator $R_2$. Due to the perturbation in stiffness, corresponding frequency shifts and magnitude changes are observed.

The results show that the proposed platform system can couple individual resonators together with an arbitrary coupling factor, demonstrating excellent flexibility and tunability. Also, since the coupling logic is a combination of identical analog operation circuits, it is easy to extend to higher-dimensional DoF systems.