# Ring-Down Method for Rapid Determination of High Q-factor Resonators

With the miniaturization of circuits and components such as MEMS and now also NEMS (Nano-Electro-Mechanical Systems), the variety and quality of macro-fabricated devices have dramatically increased, leading to packaged or embedded systems with particularly sensitive electromechanical properties. The basic properties, such as **resonance frequencies, Q-factor and dissipative power** requires more and more sophisticated measuring tools with high input bandwidth for measurements in the hundreds of MHz and Q factor near the million.

The most basic method to recover the mechanical properties of a resonator is to sweep the driving frequency around the resonance to deduce f0 and Q from a Lorentzian fit. But the higher the Q factor, the longer it takes to reach equilibrium after each frequency step, which typically requires a **relaxation time of about Q/f0**. High Q factor at low resonance frequency is the worst case scenario, but even with, let say a Q factor of 1 million and a resonator at 1MHz, this corresponds to 1s settling time per data point. For a 300 points sweep and at least zooming 3 times in the region of interest, this will take at least 15 min to evaluate a single resonator, often more. For high-throughput measurements in industrial environment, this is simply not possible.

Here we will describe and compare the **ring-down method for a single measurement**, of a few second span, to precisely determine the Q-factor of a resonator. This method is particularly valuable for high-throughput measurements or for tracking change in Q factor under various conditions (temperature, strain, etc...).

## Q factor from a frequency sweep (reference measurement)

Within the LabOne user interface of the UHFLI Lock-in Amplifier, Sweeper, Plotter, Oscilloscope trace and Software trigger provide all measuring tools, that are necessary to capture the decay time of the resonance and even transient time of the demodulator itself. For the sake of comparison, let's first measure the Quality factor of a 1.8MHz quartz resonator through a frequency sweep.

From the measured resonance (click to enlarge) and by pulling the cursors to the Full With at Half Maximum (FWHM), we get the so-called natural bandwidth of the resonator, Δf, the Q-factor is then defined as: Q = f0/Δf

For our particular resonator we get **Q = 1842/0.098 = 18795 ** with this method, which may be fairly accurate but also tedious to get. Alternatively the Q factor can also be measured from the slope in the phase at the inflection point (same method but on a different signal).

## Q factor from the ring-down method

An alternative method to measure the Q-factor of a resonator is to measure the ring-down time. Upon switching off the drive signal of a resonator and because of dissipation, the free-decaying resonance amplitude follows an **exponential law: A(t) = A0.exp(-t/τ)**, where the time constant, τ , in the exponential argument is directly proportional to the Q-factor. White noise excitation is usually the easiest way to excite all resonant modes, but a harmonic excitation in the area around the resonance is fine too. A short impulse or stop in the driving excitation will suffice to observe the decay time of the amplitude oscillation. An important point to keep in mind is the actual bandwidth, or time constant, of the demodulator used. As a simple illustration of the effect of time constant (or demodulation bandwidth) to the measurements, we show below several decay curves for various Time Constant, TC = 30 / 20 / 10 / 5 / 1 ms (in red) when the oscillations drop from almost 7mV to zero:

Clearly the chosen bandwidth has to be large enough so that the measurements doesn't interfere with the decay time of the resonator. otherwise one measures the lock-in time constant instead.

For our measurement, we use TC=1ms to make sure the demodulator does not influence the decay time. For comparison purpose, it is possible to record several demodulated signals simultaneously from the same input signal but with different TC , which also helps found the best TC experimentally.

The figure above (click to see parameter details) shows a trace recorded with the Software Trigger tool of LabOne. The trigger was set to 3mV of the resonator amplitude with falling edge. From the recorded waveform, we can extract the exponential decay time τ, which corresponds to the time to reach A/e, where A is the Amplitude at the insert of the decay and e is the Euler number. For the same resonator as measured previously from the frequency sweep, τ = 3.20ms. The Quality factor is then :

*Q = π.f0.τ*

which translate into** Q = 3.14*1842*3.2 = 18508** for this second method. The difference between the 2 methods is less than 2%. The software trigger displays the envelope ring-down (demodulated amplitude), which is a nice and smooth curve to measure the decay time from. To convince oneself that the actual signal is indeed independent from the demodulator setting, a fast oscilloscope trace with pre-trigger can capture the complete event. The raw transient signal ringdown will look noisier but display all data points with the same decay time, regardless of demodulator settings:

Click on image to see more details. The internal Oscilloscope of the UHFLI can sample data up to 1.8GS/s at full ADC speed. For triggering, any of the raw scope signals, demodulated signals or trigger inputs can be used, thus allowing measurement of almost any transient phenomena.

### Conclusion

From the Software trigger, corroborated with the fast Oscilloscope trace, which are available for all UHFLI Lock-in Amplifier users, we have shown how to rapidly determine the Q factor of a resonator, without making a complete frequency sweep. Other fast determination of resonator properties will be addressed in future blogs.