6 µHz Frequency Resolution in MEMS Resonance Tracking with the VHFLI Lock-in Amplifier

Introduction

Micro-electromechanical systems (MEMS) resonators are at the heart of modern sensing technologies, from inertial navigation and timing references to environmental sensing and RF filtering. Their performance often depends on the ability to track resonance frequency shifts with extremely high precision, sometimes under harsh operating conditions and noisy environments.

Achieving sub-millihertz resolution in frequency tracking is challenging: thermal drift, environmental perturbations, phase noise, and measurement bandwidth all limit performance. Thanks to the state-of-the-art analog and digital performance of the VHFLI Lock-in Amplifier from Zurich Instruments, this is readily available for a wider range of applications than ever before. In this blog, we demonstrate how this new lock-in amplifier enables 6 µHz frequency resolution when tracking a MEMS resonator using a phase-locked loop (PLL), which has been validated under real experimental conditions together with our customer at ONERA, the French Aerospace Lab.

This level of performance opens new possibilities for ultra-stable sensing and precision metrology.

Frequency Tracking and Sensor Sensitivity

Oscillators and resonators are widely used as frequency references and sensing elements. Physical perturbations – changes in mass, temperature, pressure, or acceleration – shift their resonance frequency. As a consequence, the ability to track these frequency shifts with high resolution allows:

• The detection of minute environmental or sample changes, such as atomic resolution in scanning probe microscopy
• Improved long-term stability, using injection locking, for instance
• Enhanced sensitivity in sensing applications
• Better characterization of noise and drift mechanisms, by plotting the Allan variance

To achieve optimal performance, the resonator is typically driven at resonance and tracked using a phase-locked loop (PLL) that continuously locks to its instantaneous frequency, as explained in details in this technical white paper.

VHFLI Lock-in Amplifier for High-Precision Phase-Locked Loop (PLL)

The VHFLI Lock-in Amplifier is a digital lock-in amplifier designed for precision measurements from DC to 50 MHz, with an optional extension up to 200 MHz. It combines signal generation, demodulation, and PID control in a single instrument, simplifying the implementation of advanced feedback schemes.

Key capabilities relevant to MEMS resonance tracking include:

• A high dynamic reserve of 120 dB and a noise floor as low as 3 nV/sqrt(Hz)
• Multi-demodulator architecture (up to 8) with independent phase, harmonics, and oscillators between the driving signal and detection
• Integrated PLL and PID feedback loops (up to 4) to control frequency, amplitude, or DC offset
• Seamless workflows via the LabOne software environment

These features allow stable resonance tracking while maintaining high sensitivity and adjustable measurement bandwidth. The PLL was tested on an ultra-stable resonator developed by ONERA, controlled in temperature with a natural bandwidth at room temperature of $$\Delta{f} = \frac{f_{0}}{Q} \approx 60 mHz$$ After identifying the resonance frequency \(f_0 = 8.9 kHz\) and Q-factor \(Q=150'000\) using the sweeper integrated in LabOne, the PLL was configured from its PID advisor recommended settings. The PID advisor takes \(f_0,  Q\) and the target closed-loop bandwidth as input parameters to calculate the optimal P and I coefficients. With a closed-loop feedback bandwidth set to 10 Hz using the PID Advisor, we could steadily track the resonance and analyse its noise distribution from the time trace in the Plotter, as can be seen in Figure 1.

At this level of resolution, the histogram is discretized in steps corresponding to the minimal frequency resolution in a feedback loop of 6 µHz, as stated as in the official specifications of the instruments. The white noise follows a normal distribution with a standard deviation of 36 µHz centered around the resonance frequency of the sensor.

Applications Enabled by µHz-Level Resolution

High resolution frequency control is a never-ending quest that Zurich Instruments has pursued first with the HF2LI Lock-in Amplifier, more than a decade ago, and now with the VHFLI as its successor. From a quick Google scholar search, close to 1000 peer-reviewed papers related to MEMS research have already been published using the HF2LI, listed in three main research areas:

Precision sensing

• Mass detection and adsorption monitoring
• Pressure and gas sensing
• Temperature and drift characterization

Inertial and timing devices

• Gyroscope bias stability monitoring
• Frequency reference stabilization

Fundamental research

• Noise characterization in resonant systems
• Material and damping studies
• Opto- and nano-mechanical experiments

The VHFLI can therefore expand the great work enabled by the worldwide community of HF2LI users, especially in MEMS and sensors research, and bring it to the next level with improved usability and performance.

Conclusion

Tracking MEMS resonators with micro-hertz resolution is essential for next-generation sensing and precision instrumentation. Using the VHFLI Lock-in Amplifier, we demonstrated 6 µHz frequency resolution in real operating conditions with a PLL-based resonance tracking setup.

By combining low-noise detection, integrated feedback control, and a flexible software environment, the VHFLI enables researchers and engineers to push the limits of frequency stability and measurement sensitivity.

This performance paves the way for improved MEMS sensors, enhanced metrology techniques, and deeper insight into resonant system dynamics.

Acknowledgements

I would like to thank existing users at Onera, The French Aerospace Lab, for providing the sensor and test environment to validate the method.