Fast PLL Optimization for High Q Resonators Without Frequency Sweeps

Introduction

The quality factor of a resonator, denoted by Q, is a reliable indication of its sensing performance, i.e. large signal amplification at resonance and low energy dissipation leaking to the environment. The Q-factor is defined as the ratio of the stored energy over the dissipated energy per vibration cycle:

\(Q = 2\pi \frac{E_{stored}}{E_{dissipated}}\)

The higher the Q, the lower the energy being dissipated by the Device Under Test (DUT), which in the frequency domain translates into a very narrow measurement bandwidth - or so-called resonator natural bandwidth, ∆f, expressed by:

\(\Delta f = \frac{f_0}{2Q}\)

where \(f_0\) is the resonance frequency. Consequently, operating high-Q resonators makes measurement very slow unless some active feedback is introduced. Even for pure sensor characterization (with no feedback required), handling such sensors can be time consuming or unreliable. In this blog, I will demonstrate a fast and simple method to guarantee optimal operating conditions that every Zurich Instruments user can implement, provided they have the PID/PLL option enabled on their lock-in amplifier. For an overview of PLL and its applications, take a look at our PLL White Paper.

Status Quo of Open-loop Characterization

Conventional Phase Locked Loop (PLL) optimization requires frequency sweeping to obtain resonator characteristics such as phase and amplitude as a function of frequency. In this way, precise determination of phase setpoint at resonance, Q-factor, and gain can be obtained in a so-called open-loop transfer function.

For high-Q resonator (> 10’000), however, especially in vacuum, at low temperature, or with sensors made of single crystal such as quartz or silicon, sweeping the drive frequency can take minutes to leave enough settling time at each frequency step. Moreover, this approach often requires sweeping smaller and smaller frequency ranges once the resonance is identified to get enough data points. This makes the overall process highly time-consuming, or even impossible, in case of resonance frequency drift. It’s like looking for a needle in a haystack. 

There have been some alternatives to this method; for instance, generating chirp burst as described in this blog; but again, for very narrow sensor linewidth, the frequency resolution requirement is very high and resource intensive. It also requires an Arbitrary Waveform Generator (AWG) that is not available on all instruments and usually works best for high-frequency resonators of several MHz or more.

Start the PLL with an Educated Guess

Instead of performing a frequency sweep in open-loop, one can close the loop from the beginning. In practice, this approach is justified by the fact that sensors are designed with specific applications in mind, such as in Atomic Force Microscopy (AFM) or measuring an acceleration. For instance, changing a cantilever in a Scanning Probe setup or characterizing various MEMS in a wafer for quality inspection will all exhibit similar characteristics within some standard deviation. 

One can therefore make a sensible initial guess about the sensor characteristics and enter those values into the PID advisor to get reasonable values for the P, I, and possibly D parameters. Since we don’t know the phase at resonance, let’s enter a PLL setpoint of -90° as the most likely lag introduced by the resonator. If there is some initial phase offset (at a frequency much below resonance), it’s important to subtract -90° from this initial phase. The slope from the gain should always be negative since the phase will decrease at resonance. The PLL loop can then be closed with a range big enough around the expected resonance frequency. Even with an initial error of ±20%, the PLL is still able to lock to the resonance. Please note that for this initial test, ‘internal PLL’ DUT model can also be chosen in the PID Advisor since Q is expected to be very high anyway.

For the purpose of this blog, using an encapsulated Quartz Tuning Fork (QTF) resonator, I entered \(f_0\) = 1.7MHz (actual value 1.84MHz) and a phase setpoint of -50° (actual value -87°) as can be seen on the screenshot below. Even with such a large initial error, the PLL could still lock after a drift from 1.7 to 1.84MHz. The PID error is of course large but can be addressed while the resonator can be operated in closed loop.

Phase at Resonance: Sweeping PID Setpoint

With the Plotter tool of LabOne, one can monitor the change in Amplitude R while entering different PLL setpoints by increment of a few degrees in the PLL tab. This can be automated using the Sweeper module. While keeping the PLL running, it is possible to sweep the PLL setpoint around its actual setpoint. Such sweep in closed loop can be much faster than the standard frequency sweep in open loop, because the feedback doesn’t let the resonator relax. In this way, there is no need to wait for any settling time inversely proportional to \(f_0 / 2Q\). This is the main reason why such a method is advantageous for high Q resonator.

Sweep results are shown in the screenshot below where we can clearly identify a maximum in Amplitude, when the PLL setpoint matches the actual resonance phase. At the same time, we can see the linear dependance of the frequency around the setpoint and this information will help us extract a precise value for the Q factor (see next section).

Fast and Precise Determination of Q Factor

The last missing piece of information for a complete characterization of the resonator and optimization of the PLL is the value of the Q factor. This calculation can be done from the phase slope at resonance. Normally, this phase slope measurement is done from a phase vs frequency sweep (open-loop), but in our case, we have it from a frequency vs phase setpoint sweep (closed-loop). In both cases, we can rely on the following formula, at resonance [1]:

  \(Q = \frac{f_0}{2} (\frac{d\phi}{df})\)

In Figure 2, a linear fit was made, which needs to be inverted and converted into radian/Hz to be used in the equation above. For this resonator, a value of Q = 19’593 was computed.

To verify the obtained quality factor, we can also perform ring-down measurements, similar to what has been addressed in an earlier blog. We are already at resonance, and with the correct phase setpoint, we get the maximum amplitude. For this ring-down measurement, we need to stop the PLL, and eventually switch off the driving signal. This data is best captured with the DAQ module of LabOne, using Amplitude R as the Trigger signal with a Negative Edge and Level below the actual measured amplitude. Once the DAQ is armed with enough Data Transfer rate and a Time Constant well below the decay time, the drive can be turned off. Acquisition results for the same resonator are showed on Figure 3 with a log scale for the Amplitude axis, hence demonstrating a nice exponential decay.

With this method, the Q factor can be extracted from the formula \(Q = \pi f_0 \tau\) where the decay time \(\tau\) = 3.35 ms is determined from Figure 3. The decay time corresponds to the time for which the following condition is met: \(A_0 / A(\tau) = e\), the Euler constant. Such a calculation gives Q = 19’376 in good accordance with the value obtained from a PID setpoint sweep.

Final PLL Optimization

Now that we have precisely measured important sensor characteristics, it is easy to make the final PID optimization step using the PID Advisor. Selecting the DUT model Resonator Freq, we can enter precise value for f_0 and Q and enter the desired target bandwidth, which is a trade-off between speed and resolution or acceptable level of error for the PLL. We can also enter the precise setpoint value as measured from the PID setpoint sweep and close the loop. Final noise results are shown in the plotter comparing initial PLL noise with suboptimal gain and setpoint values versus the fully optimized setup.

Use Cases and Limitations

The method described above has been tested in several labs and is particularly relevant for very large Q resonator where frequency sweeps take too long to get relevant parameters, as mentioned at the beginning.

Such a technique is particularly relevant for sensors operating at Low Temperature as Atomic Force Microscope [2] or for resonator cavity with very high finesse.

The main assumption is that there are no multiple resonances within the frequency span. Otherwise, a narrower span must be selected.

If the initial guess is not possible or is not enough to get the PLL to lock, then a quick identification of the frequency range of interest can be obtained using either

1. Thermal noise spectrum: this is readily available as a core module of LabOne called Spectrum or an averaged FFT of the Scope. This method works best for sensors with low stiffness (since they can be thermally activated more easily).
    
2. Low resolution frequency sweep: this can perform as a fast sweep since a complete characterization of the resonance is not needed. A rough estimation of the area of interest is enough, where the resonance is located, also identified with a shift in phase. Such a value can then become the initial guess.

Conclusion

Operating high-Q resonators is challenging, and parameters optimization is critical. The standard frequency sweep method can be long and fastidious, especially if accurate values need to be determined. In this blog, I demonstrated a simple method to optimize feedback from a PID setpoint sweep, with higher precision for resonance phase and Q factor determination in a minimal amount of time. In turn, such an accurate number leads to minimized phase noise error.

The same procedure described in this blog can be used for sensor characterization, even if no feedback is required for the applications. Using the PLL to quickly identify the resonance and measure high-Q factor can indeed be useful for fast characterization.

Acknowledgements

I would like to thank Prof. Hans-Josef Hug from EMPA in Zurich for his valuable insights regarding this fast PLL tuning method applied to the field of Noncontact-AFM.

Reference

[1] J. M. Lehto Miller et al, Effective quality factor tuning mechanisms in micromechanical resonators, Applied Physics Reviews 5, 041307 (2018)

[2] Y. Feng et al, Magnetic force microscopy contrast formation and field sensitivity, J Magn Magn Mater, vol. 551, p. 169073, 2022, doi: 10.1016/j.jmmm.2022.169073.