Sensors come in different flavors. They can be inertial devices for motion detection, tehy can be based on photosensitive materials, they can absorb surrounding molecules to detect their concentration, and so on. Developments in micromachining and micro-/nano-electromechanical system (MEMS/NEMS) technologies led to miniaturization and low power consumption. Hence, multiple sensors based on different physical principles can fit into a smartphone or can be used for Internet of Things (IoT) applications.
The sensing element can be considered as a transducer that converts the conditions in the sensed environment into electrical signals. Finding the optimal working conditions requires the characterization of this conversion. Depending on the sensor and its working principle, various measurements such as its frequency response, quality factor, capacitance transients should be performed. For an active sensing element such as an inertial device, driving it at its optimal conditions maximizes the performance. Hence, closing feedback loops to adapt the drive is essential.
To perform these tasks, developing application-specific circuitry (ASICS) in parallel is often required, which makes the development process more cumbersome and expensive. To tackle this challenge, Zurich Instruments' lock-in amplifiers offer a toolset for both time and frequency domain measurements as well as feedback control loops in a single platform for both sensor characterization and control. This blog post covers the basics of these tools with tips and tricks for an efficient workflow. You can also watch the related webinar video with demonstrations of each tool.
Lock-in measurement plays a crucial role in recovering weak electrical signals provided by the sensors. Hence, we start by looking into the principles of the lock-in measurement in a nutshell. For detailed theory, please take a look at our white paper on the principles of lock-in detection and its related video.
To understand the importance of this technique, we can look at the typical noise floor of a measurement system. As shown in Figure 1, the white noise sets the baseline, 1/f noise starts to pick up at low frequencies, and several spikes occur from various sources. At low frequencies or DC, where 1/f noise is dominant, recovering a small signal is not a trivial task. The trick is then bringing the signal of interest to a frequency regime where the noise floor is minimal. This is done by modulating the drive signal of the sensor with a periodic sine wave at a fixed frequency (reference signal).
Figure 1: The typical noise floor with white noise setting the baseline, 1/f noise picks up at lower frequencies and various spikes come from different sources.
The lock-in amplifier demodulates this signal using the reference signal to capture its phase and amplitude by the following steps:
- Use an analog to digital converter (ADC) to carry the signal to the digital domain to achieve maximum accuracy.
- Mix the signal with the reference that is used for the modulation Known as dual-phase demodulation, it is done at two different phases simultaneously to recover both the amplitude and phase of the signal.
- Use a low-pass filter to clear the spurious signals and other harmonics.
These steps are automatically performed in Zurich Instruments' lock-in amplifiers, while the user keeps control of all the essential parameters. They include the ADC range, modulation frequency, and low pass filter settings. The filter bandwidth and order are critical to balance the signal-to-noise ratio and temporal resolution of the measurement. Lowering the filter bandwidth increases the signal quality in exchange for measurement speed.
In this section, we list the tools that come with Zurich Instruments' lock-in amplifiers by explaining their key aspects. These tools are included in the LabOne® user interface and programing libraries. For this demonstration, we use a quartz resonator (the resonance frequency is approx. 1.84 MHz, and the quality factor is approx. 10k) to mimic a proximity sensor. To measure the drive signal together with the resonator’s response, we also use a dual-channel instrument.
Figure 2: The setup we use to demonstrate the time and frequency domain tools.
The Scope gives direct access to the signal after the ADC digitizes it. It is a good practice to check the signal condition before performing the lock-in measurement. By performing FFT, it is possible to see the frequency components and determine spurious components. This information helps to set up the low-pass filter settings of the lock-in, such that the spurious components are filtered. Besides, the ADC range can be set up accordingly by observing the signal-to-noise ratio of the raw signal. The Scope can also access the DACs (digital to analog converter) of the signal output channels similarly.
The scope shot displayed below compares the reference signal (~1.84 MHz sine wave with 150 mV peak voltage) with the resonator’s response. It is weaker than the reference since the drive is far from the resonance. However, the signal-to-noise ratio is comparable thanks to the accordingly adjusted ADC range. A slight delay of a few ns in the resonator’s response is also visible, as indicated by the markers.
Figure 3: Lock-in tab (above): The instrument has multiple oscillators, demodulators, and output amplifiers. The Lock-in tab becomes the cockpit to control all these components. The blue rectangles indicate the components used during this measurement, and the arrow indicates the ADC range settings. Note that auto-ranging is available. Scope tab (below): the horizontal line indicates the trigger level and its hysteresis. Vertical and horizontal markers are helpful for analyzing the signals. Further math/analysis tools are also available such as hysteresis and peak fitting.
The Numeric tab and Plotter display the data stream from the demodulator (see here to learn more about LabOne’s data stream concept). To capture a single value at a time, you can use the numeric tab; to capture a time trace, you can use the Plotter. For a quick validation of the demodulated data or parameter adjustment, the Plotter is the perfect tool. Multiple demodulated data streams and other auxiliary signals can be visualized on a single plot and compare
The snapshot of the Plotter in Figure 4 displays both the amplitude and phase of the resonator’s response. A rapid change occurs in both demodulated amplitude and phase upon contact (literally tapping on it) with the free-running resonator, mimicking a proximity sensor. Of course, this is not the desired behavior from a resonator, and we will look at this in the last section.
Figure 4: The time trace of both the amplitude and phase of the resonator’s response. The blue arrows indicate the contact times. The Plotter can overlay multiple traces, even with different units. In this case, the vertical axis shows the values and the unit that corresponds to the active trace only.
The DAQ module is a powerful tool to capture the demodulated data stream. It differs from the Plotter such that it captures the demodulated data for a given number of samples (or duration) shot by shot. It can perform triggered measurements, align the timestamps of different data streams, take the FFT of each shot, and build up images (see here for more details). This module is commonly used to capture the step response of the sensor. For example, the automation of gyroscope characterization using a test table is possible by triggering the DAQ module at each rotation to measure the gyro’s step response. Such a transient measurement can also be performed upon a change in the drive signal.
We demonstrate this with our resonator setup, where we initially drive the resonator at the resonance (1.82 MHz sine wave with 100 mV peak voltage). To capture its response when the drive is closed, we can set a trigger that looks for the demodulated amplitude. The trigger is set for the negative edge with its level below the amplitude observed while the resonator is driven. Once we stop driving, this high-quality resonator decays to its ground state over several milliseconds. This transient is often referred to as ring-down measurement and used to quantify the quality factor.
Figure 5: Step response of the resonator upon stopping the drive. The gray horizontal line is the trigger level used for this measurement. X1 marker indicates the moment the drive stopped, and X2 marker indicates the time that it took to amplitude decayed by 1/e. This time interval is referred to as the decay time, which is 3 ms in our case. The quality factor can then be calculated by Q = πx(Resonance Frequency)x (decay time), which amounts to ~17.3k.
The Parametric Sweeper allows varying the drive parameters systematically to characterize the sensor’s response. It can function as a frequency response analyzer (also referred to as Bode analyzer) by sweeping the drive frequency, or it can be configured to sweep other parameters such as DC bias voltage. Data from different sources (demodulators, auxiliary inputs, and more) can be displayed simultaneously. Its XY Mode allows displaying Nyquist plots and I-V curves as well.
During a sweep, the signal properties such as its amplitude and signal-to-noise ratio vary. It is essential to adapt the Lock-in filter settings to the varying signal for the best accuracy and precision. Hence, the Sweeper offers auto bandwidth functionality. The settling time, harmonic suppression, and averaging are the other adjustable parameters. Besides, the Sweeper comes with presets that fit typical applications, e.g., frequency response analysis, noise density characterization, and impedance measurements for an efficient workflow.
To characterize the resonator, we configure the Sweeper as a frequency response analyzer and sweep the drive signal over a broadband, where we expect the resonance to coincide. We observe both the amplitude and phase simultaneously. Once the resonance is identified, we narrow the frequency band for better resolution. For linear resonators, a fit can then be applied to determine the peak position, full with half maximum, quality factor, and phase at the resonance, as shown in Figure 6.
Figure 6: A frequency sweep performed to characterize the frequency response at the resonance. The math tools offer a resonance fit in both phase and amplitude curves. The fit parameters, as well as the fit error, are provided. From the phase fit -which typically gives more reliable results (0.02 % fit error)- we see that the quality factor of the resonator amounts to ~17.3k matching perfectly with the ring-down measurement.
The Chirp FFT provides the frequency response of the device with both high spectral and temporal resolution. It differs from the Sweeper (used as frequency response analyzer) by exciting a broad frequency band with a chirp pulse in a single shot instead of sweeping point by point. The Scope is then triggered to capture the chirp pulse. The key to the success of this method is performing leakage-free FFT on the chirp signal. This requires the frequency components of the chirp signal to match precisely to the frequency grid of the discrete Fourier transform. To achieve that, we use UHF-AWG (arbitrary waveform generator) option to generate the chirp signal with a defined number of samples and sampling rate. For further information, please take a look at this blog post.
For our resonator, we define a chirp signal using the AWG with a frequency span of 3 MHz. It also provides a precise marker signal to trigger the measurement. Scope samples with the matching number of points and performs the FFT to capture the frequency response of the resonator less than a second. In this configuration, the spectral resolution amounts to a few Hz only over a few MHz band. Since the temporal resolution is a fraction of seconds, fast perturbations on the system can be monitored. Images can be formed with one axis being the frequency and the other being the time. Note that LabOne allows using multiple tools simultaneously as we did here with the AWG and Scope.
Figure 7: (Top) A chirped pulse in the time domain. (Bottom) Chirp FFT spectrum of the resonator showing the resonance and the sidebands in the log scale. (Inset) A zoomed-in view of the resonance peak in a resolution of a few Hz.
The MFIA Impedance Analyzer or the MFLI Lock-in amplifier with the MF-IA option can both perform impedance measurements. Thanks to the capability of simultaneous voltage and current read of this platform, 2 and 4 terminal measurements are possible. Also, it has built-in device-under-test (DUT) modules such that the required impedance parameter can be determined conveniently, such as conductance and susceptance. The software and the internal calibration provide 0.05 % basic accuracy for the measurement. The MFITIF text fixture allows studying SMD components out of the box with this accuracy. If additional cabling is needed, the parasitic can be suppressed using the compensation routine provided with the software. The time and frequency domain tools described above can be used for impedance measurements as well for maximum flexibility. For further details on how to achieve the best impedance results, you can take a look at this blog post.
Figure 8: SMD mounted components such as a photodiode can be connected to the MFIA using the MFITF text fixture.
To demonstrate the capabilities of this platform, we use an SMD mounted photodiode and connect it to the MFIA, as shown in Figure 8. The photodiode has a capacitance in the order of 24 pF at 1 MHz with a dependency on the DC bias voltage. To characterize this dependency, we perform a transient capacitance measurement, such that the DC bias is varied from 0 to -1 mV as a square wave with 1 ms duty cycle. The impedance analyzer is configured for 4 terminal measurements with Rp || Cp (parallel resistance and capacitance) DUT model. To capture the transient, we use the DAQ Module and trigger the measurement at each negative edge of the square pulse. Thanks to the high precision and temporal resolution of the instrument, we resolve a 6.5 fF capacitance step that only takes a few µs.
Figure 9: The Impedance Analyzer tab (above) provides the user interface to perform impedance measurements. It allows defining the equivalent circuit model and displays the measurement results. The capacitance transient (below) of a photodiode upon a 1 mV DC bias step is captured using the DAQ module. X1-X2 markers show the transient time, and Y1-Y2 markers show the capacitance step.
The measurements that we preformed previously were in the open-loop configuration where the sensor was free-running. We observed this by interacting with the resonator while logging its phase and amplitude with the Plotter. The resonator’s response changed significantly upon contact and even settled to an arbitrary state after the perturbation.
Closed-loop control is required to force the resonator to oscillate at its resonance even under such a strong perturbation. The first step is to make sure that the resonator is always driven at its resonance. This is done by a phase-locked loop (PLL), where the measured phase is compared with a set value, and the result is used in a PID (proportional, integral, and differential gains) controller to adjust the drive signal frequency. The set value is the phase that we measured using the Sweeper before.
Forcing the resonator to have the same amplitude is the second step. A second PID controller takes care of this by acting upon the drive amplitude. The setpoint for the amplitude is the value determined by the sweep. See Figure 10 for a simplified diagram of this Amplitude Gain Control (AGC) system.
Figure 10: Automatic gain control scheme. A PLL controls the frequency of the output signal and makes sure that the sensor is driven at resonance. A PID stabilizes the sensor's amplitude to a setpoint.
For both PLL and AGC, we need to set up feedback controllers and configure PID controllers. The PID and PLL options offered with the Zurich Instruments lock-in amplifiers come with a PID advisor that can simulate the closed-loop transfer function for common DUT models (see here for further details). An important parameter for the advisor is the closed-loop target bandwidth, which determines how fast the feedback loop works. In other words, the loop can only recover perturbations within this bandwidth. For an efficient workflow, the following steps can be followed:
- Characterize the response of the resonator at its resonance (we did that already with the Bode analysis). This provides both the phase and amplitude setpoints required for the AGC.
- Use the PID advisor using the resonator frequency DUT model. This requires the resonance frequency and quality factor that we have characterized before and the target bandwidth. Copy the designed parameters from the advisor to the PLL and activate it.
- For the second PID controller, use the advisor again with the resonator amplitude DUT model. The target bandwidth of this second PID should be narrower than the PLL. Copy the designed parameters from the advisor to the PID and activate it.
The effect of the AGC on the resonator amplitude and frequency can be monitored using the Potter. For both signal traces, we observe a steady-state behavior where the standard deviations are as low as 21 µV for the amplitude and 240 mHz for the frequency. The perturbations are recovered much faster, and the resonator always comes back to its initial state. If we again consider the resonator as a proximity sensor, it has a higher sensitivity and larger temporal resolution. You can find the formal description of this method applied to vibratory gyroscopes in this application note.
Figure 11: The plotter tool can be used in parallel with the PLL and PID controllers to set up and adjust feedback loops.
Zurich Instruments' lock-in amplifiers provide a flexible toolset to characterize and control sensors. All of the tools listed can run simultaneously to build up complex schemes required by the application. Tools such as the DAQ module, powered by an intuitive user interface and programming libraries, ease the integration of the instrument in existing setups. This toolset can eliminate the time required to develop ASICS and make sensor development applications faster.