For this webinar we teamed up with Dr. Tomás Manzaneque, whose work focuses on the fabrication, modeling and interfacing of mechanical micro-resonators in different fields of application: examples are rheological sensing, ultra-sensitive mass sensing, passive conditioning of RF signals for the Internet-of-Things communications, and single-cell biopsy. His research interests branch out to the non-linear dynamics of mechanical resonators and the efficient actuation and detection of mechanical vibrations through piezo-electric materials. Zurich Instruments' lock-in amplifiers are an invaluable asset through high sensitivity, high resolution, and modularity that enables multiple signal detection and control at the same time. The signal generator of the lock-in amplifier drives the sensor, while its demodulator extracts the desired signal with adjustable bandwidth to optimize the measurement signal-to-noise ratio. The built-in temporal and spectral measurement tools in the LabOne® software help characterize the sensor and find the optimum drive conditions efficiently. The PLL and PID options of the lock-in amplifier can then track multiple eigenmodes of the resonator to capture rapid frequency shifts. During this webinar, we looked into the best practices when using lock-in amplifiers to characterize a sensing element, discussed the setup of feedback controllers, and considered the tradeoffs between open- and closed-loop operation.
The video of the webinar can be found here.
In the Q&A session, we addressed several questions which we answered below.
How did you choose the demodulator's low-pass filter parameters for your PLL measurements? Have you observed an effect of the filter order and bandwidth on the Allan Deviation?
In our observations, the bandwidth of the phase demodulator (BW-dem) does not influence the Allan deviation, as long as it is at least one order of magnitude larger than the bandwidth of the PLL transfer function (BW-PLL), which is set by the proportional constant of the PI controller. The theoretical explanation is found in . Therefore, the choice of BW-dem limits the maximum BW-PLL that can be set. In our experience, BW-dem has to be set to a significantly lower value than the resonance frequency. Otherwise, the 2f component arising from the mixing in the demodulator will leak through the low-pass filter, ruining the Allan deviation. Higher orders of the filter order allow for closer values of BW-dem to the resonance frequency. In our experience, filter order 8 allows for a BW-dem only 2 times smaller than the resonance frequency. If a filter of order 2 is used, BW-dem needs to be at least 20 times smaller than the resonance frequency to avoid the 2f leakage.
Can you comment on the open-loop versus closed-loop detection to characterize a MEMS / NEMS resonator in the non-linear regime?
Leeson's effect governs the difference between open-loop and closed-loop performance. In principle, the same effect should be valid for non-linear resonators, but this has not yet been verified experimentally to the best of my knowledge.
The Allan deviation in the open-loop configuration seems to decrease with increasing gate times. At which point does the Allan deviation not decrease anymore, is there a minimum?
Indeed, a minimum Allan deviation is found for large enough gate times. Fluctuations of the resonance frequency produce this as a result of variations in the temperature and other conditions.
Can the nanomechanical mass spectrometer detect where the particles land on the cantilever?
Yes. The nanomechanical mass spectrometer monitors the frequency shift on the vibrational modes upon landing particles. These vibrational modes have different mode shapes. Therefore, the frequency shift is a function of the added mass and the landing position. To separate these convoluted parameters, the frequencies of multiple modes are tracked simultaneously as explained by Dohn and collaborators . For larger particles, particle stiffness can also play a role .
Can the phase-locked loop be implemented around electrical signals (U, I) instead of a displacement?
Yes. We gave examples of optomechanical sensors during the webinar, where the dynamics of a mechanical system, e.g. displacement, are transduced to an optical modulation. A photodetector captures this modulation providing electrical signals that are demodulated via the lock-in amplifier. Direct electrical transduction methods such as resistive and capacitive coupling are alternatives to capture the mechanical motion as in a tuning fork cantilever. In all cases, the lock-in amplifier measures the phase of the resulting electrical signal to track the sensor's resonance. An overview of the phase-locked loop building blocks and possible configurations is available here.
How can we prevent cables from disturbing the measurement?
Cables and connectors can indeed introduce phase delays and parasitics impedance, which can be a challenge, especially for accurate impedance measurements. The MFIA Impedance Analyzer (or the MFLI Lock-in Amplifier with the MF-IA option) makes it possible to compensate these effects and maintain the measurement accuracy. You can read more about this functionality in this blog post. If the disturbance increases the measurement floor or adds spurs, the demodulator settings can be adjusted to reject them as much as possible. The included Labone Sweeper tool automatically adjusts these parameters for the best results when sweeping a system parameter.
How can the measurement noise be reduced for better precision and improved Allan deviation?
Extracting tiny signals of a sensor buried in noise is a challenging task and the fundamental use of a lock-in amplifier. Various parameters must be set based on the signal's properties to take full advantage of the lock-in detection. For instance, narrowing the demodulator's low-pass filter bandwidth and selecting a higher-order filter can help to reject noise and filter out spurs. The price of this is increased measurement time. Therefore, when configuring a lock-in amplifier, the key is finding the right tradeoff between the measurement noise and the measurement time. Zurich Instruments' lock-in amplifiers come with time- and frequency-domain signal characterization tools to help you in this process. This short video provides a number of tips to improve your lock-in amplifier measurements. In the case of closed-loop operation, both the demodulator's filter settings and PID controller parameters define the loop filter bandwidth. Broader loop filter bandwidth increases the reaction time but again reduces the noise rejection. Zurich Instrument's PLLs and PID controllers come with a PID Advisor that allows the user to simulate the transfer function of the entire setup and assist in designing the closed-loop operation.
You mentioned impedance measurements on slide 20. Can you give more details on this point?
One step of characterizing a sensor is accurately measuring its impedance at a given frequency. To monitor its response to a changing environment demands a precise instrument as well. In particular, small changes occur in capacitance in a few micro-seconds only with capacitively coupled devices, such as MEMS inertial sensors. The Zurich Instruments MFIA Impedance Analyzer can address all these aspects. For more information, take a look at this page or, even better, contact us to discuss your application.
Tomás, is there a postdoc position available in your group?
Thank you for your interest. There is no open postdoc position at this time, but candidates applying for individual fellowships such as Marie Skłodowska-Curie Actions are always welcome.
 Demir A. & Hanay M. S. Fundamental sensitivity limitations of nanomechanical resonant sensors due to thermomechanical noise. IEEE Sensors Journal, 20(4), 1947-1961 (2019).
 Dohn, S., Svendsen, W., Boisen, A. & Hansen, O. Mass and position determination of attached particles on cantilever based mass sensors. Rev. Sci. Instrum. 78, 103303 (2007).
 Malvar, O. et al. Mass and stiffness spectrometry of nanoparticles and whole intact bacteria by multimode nanomechanical resonators. Nat. Commun. 7, 13452 (2016).