## Compensating delay-induced phase shifts for high-frequency resonators

Phase-Locked Loop (PLL) or other PID controllers on nano-mechanical resonators rely on clean phase signals that exhibit large slope, or significant phase shift, around the resonance frequency. For really high-frequency NEMS and opto-mechanical systems above a few tens or hundreds of MHz, any small delay can have significant effect on the phase slope and therefore PLL response. Basically any 1 m of additional cable length between drive signal output and detection signal input corresponds to about 3 ns additional delay $$\delta t$$, or expressed in terms of phase: $$\phi=2\pi f \delta t$$ is about 2-deg phase offset @ 100 MHz. Since this behavior is also frequency dependent, any frequency sweep will exhibit additional slope on top of the regular phase behavior intrinsic to the system under investigation.

## Uncompensated frequency sweep

In practice, such a frequency sweep over large frequency span will exhibit the following behavior:

Figure 1: Uncompensated frequency sweep (click to enlarge). Green=phase, violet=amplitude.

In particular, we can see from this sweep that above about 1 MHz, the phase deviation starts to become non-negligible and increase with frequency where several phase wraps (going over 180°) appear above 50 MHz typically (click for more details).

## How does compensation work on the UHF platform?

Compensation delays can be obtained with additional circuitry but are usually system-dependent and thus, not flexible enough to be implemented straightforwardly. With the UHFLI, a compensation delay can be directly added to the Signal Input in units of the sampling clock. Here is a schematic representation inside the UHFLI of the added delay time between the actual frequency oscillators (NCO) and the lock-in output measurements for phase and amplitude.

Figure 2: Compensation delay schematic

Such scheme allows for both DAC and ADC intrinsic delay compensation coming from the instrument itself (also compensated after a standard calibration run) and extrinsic delay due to long cable and parasitic at the DUT. For precise delay compensation a tuning of the delay needs to be performed for each particular set-up and is described in the next paragraph.

## Convenient delay compensation in LabOne

Now that we have seen where the delay is coming from and where to compensate for it inside the UHFLI, we can focus on how to optimize it to the best operating conditions using the LabOne user interface. Since this is not a standard feature particular to a dedicated tab inside LabOne , it is located in the ZI Labs tab and can be manually tuned. Please note that after a new calibration run it returns to its original value.

Figure 3: Wide range frequency sweeps for manual delay compensation (click to enlarge)

In Figure 3, several frequency sweeps were taken for various delays (as seen in the History sub-tab) to reach an almost flat phase response over the entire frequency span. In this example the 16 MHz resonator used is sensitive to a small phase shift but for a 100 MHz resonator this effect is even more dramatic. Please not that phase unwrap was selected in the sweep settings under Advanced Modes -> Option: Phase Unwrap.

## Resonant phase determination

Now that the delays are properly compensated, we can zoom in again to the frequency of interest around the resonance. For the sake of arguments, again the 3 phase behaviors, namely, under-compensated (red), over-compensated (green) and just compensated (blue) are displayed in the following figure.

Figure 4: Resonance frequency sweep before and after compensation

In this scenario, the PLL would still be able to lock on any of the 3 phase behavior, we do have some offset but the slope is still within an acceptable range. What is interesting to note is that the phase at resonance, for the just compensated curve, is now centered at zero and that the phase slope is only due to the sensor characteristics and not to other parasitic, which are definitely ideal conditions for a very robust PLL lock.

## Real case scenario: a 110 MHz opto-nanomechanical resonator

The 16 MHz quartz resonator used above was a good candidate for a simple demonstration of compensating delays but in newly developed sensors at much higher frequency and with weaker detection scheme, the delay compensation can be the condition sine qua non for phase lock. Here is the anti-resonance response of a 110 MHz opto-mechanical sensor, which is optically actuated (and controlled via the UHFLI drive output) and optically detected (photo-detector at the UHFLI input)

Figure 5: Anti-resonance sweep of a nano-optomechanical system before and after compensation

From the uncompensated phase slope (red), it is clear that no PLL can lock since the slope contribution due to the time-delay induced phase shift is far greater than the slope contribution from the sensor. In such case the PLL would drift away to the upper or lower limit. On the green curve, the time delays are compensated and while still not particularly strong, the PLL allows for resonance tracking. Please note that for anti-resonance tracking the sign of the PI-gain of the PLL feedback loop has to be reversed because the phase slope changes sign (positive) compared to a resonance curve (negative).

## Acknowledgement

I am grateful to Sadik Hafizovic for pointing me to the relevant tab and for drawing the first sketch of the delay schematic :) and to Lucien Schwab for testing this on a real-world experiment at the Laboratoire Matériaux et Phénomènes Quantiques (MPQ) in the group of Ivan Favero in Paris.