Distributed strain sensing with notch filters

Distributed, multi-modal sensor networks are important in soft robotics for tasks like closed-loop control, state estimation, and large-area tactile sensing. However, soft systems and sensors typically lack the compute power and multiplexing capabilities of rigid systems, making sensor integration more cumbersom. Further, to deal with the more complicated state and dynamics compared with rigid devices, they typically need many more sensors to boot.

In this work, we introduced an approach that encapsulates resistive or capacitive soft-material sensors in variable-frequency notch filters, enabling distributed strain sensing over a shared bus with minimal wiring and automatic, high-fidelity sensor response decoupling, all without requiring complicated multiplexing or compensation approaches to tune, reconfigure, and disambigute sensors. Each sensor node is assigned a unique spectral band, and the entire network can be interrogated with a single frequency sweep. The result is a framework that supports sensor isolation, easy network reconfiguration, and heterogeneous sensing without requiring recalibration when sensors are added or removed. It works with sensors that vary their resistance or capacitance with the target signal. We developed the theory for both and demonstrated our findings using networks of capacitive strain gauges.

^†Copyright Notice: Figures and videos designated with ©2025 IEEE are reprinted, with permission, from Wertz, A., Shah, D., & Majidi, C., “Soft Transducers with Notch Filters for Spatially-Distributed Strain Sensing” IEEE Sensors Journal, Oct 2025 (Wertz et al., 2025). All other site content, homelab tutorials, and unmarked media are © 2025 Anthony Wertz and provided for educational purposes. I will try to call out the additional fun bonus material that may not have made it to a publication or somewhere else on this site.

Approach

A notch (band-reject) filter attenuates a narrow frequency band determined by its passive component values. By replacing one of those passive components with a strain-sensitive element, such as a soft capacitor, the notch frequency becomes a function of the sensing target (in the case of strain gauges, the applied strain). Each sensor on the network is designed to operate in a different frequency band, so their responses can be independently detected from a single chirp input.

While we look at both twin-T and a Bainter notch filters in the supplemental materials, in the paper we mainly focused on the Bainter notch topology, which offers several advantages:

• Robust notch depth: The zero quality factor $Q_z$ is proportional to the op-amp gain, so the notch depth is largely unaffected by component tolerances.
• Single sensing element: Only one strain-sensitive capacitor or resistor is needed per node, simplifying fabrication.
• Tunable sensitivity: Fixed resistors and capacitors can be added in series or parallel to the sensing element to independently tune the sensitivity and operating frequency.
• Accessible frequencies: Appropriate component selection yields notch frequencies in the low kilohertz range, easing data acquisition requirements compared to passive LC designs operating at hundreds of kilohertz to megahertz.

Device fabrication

Stretchable parallel-plate capacitors were fabricated by patterning oxidized eutectic gallium-indium (OGaIn) liquid metal paste electrodes onto double-sided VHB tape using laser-cut stencils. A small notch filter PCB was pressed onto the patterned VHB and connected to the electrodes using a conductive biphasic liquid metal ink. The assembled devices could be stretched to over 200% strain during operation.

Bonus: Although not presented in much detail in the published manuscript, the sensor network from the figure above was fabricated in in three stages using a direct-ink-write (DIW) Voltera V-One printer. Here’s a nifty video part of that process. Ultimately, because of the very low capacitance we were able to get from the printed capacitors, the twin-T didn’t function well in multi-node configurations. With the Bainter, fabrication didn’t need to be as precise because there was only one capacitor to fabricate, and we did not see the same downstream attenuation as we did with the very low capacitance (and thus very high resistance) twin-T filters.

Effective gauge factor

We derived the effective gauge factor, indicating the frequency response to strain, for several configurations. For the particular case of directly replacing the tuning filter capacitor in the Bainter filter with a strain-sensitive component, the effective gauge factor is:

where $G_c$ is the capacitive gauge factor and $\epsilon$ is the strain. Assuming a positive $G_c$, this nonlinear relationship shows the sensor has higher sensitivity at lower strains and decreasing sensitivity at higher strains. Importantly, fixed components can be added in parallel or series to tune the sensitivity independently of the sensor’s native gauge factor, giving the designer flexibility without changing the sensing element itself. The shapes of these response curves can differ quite a bit depending on the gauge factors (especially the sign) and whether or not you replace one or both of the capacitive and resistive filter elements. In the supplemental material we derive these relationships for both the twin-T and Bainter filters.

Single-sensor validation

The sensor was cyclically strained to 100% on an Instron and the notch frequency tracked via logarithmic chirps from 1 to 100 kHz. The sensor response was measured with a Salae Logic 8, and the measured frequency shift was fit to the derived model using the Levenberg-Marquardt algorithm, yielding a near-perfect fit with less than 0.5% strain estimation error. The fitted gauge factor was $G=0.95\pm 0.05$, close to the expected $G=1$ for a parallel-plate capacitor.

Here is a video of the twin-T sensor characterization using an Instron to measure tensile stress and strain. We used the twin-T design here, so the downstream attenuation due to the high-resistance network is very apparent in the transfer function.

Multi-sensor demonstration

A key strength of notch filter encapsulation is that sensors on the same network are decoupled from one another. To demonstrate this, three Bainter notch sensors with different zero-strain frequencies were placed on a single five-wire bus. All three notches were clearly detected from a single chirp. The center sensor was then progressively strained, shifting its notch frequency down until it overlapped with and passed through the first sensor’s notch, with no adverse effect on the network response.

Because each sensor node is buffered and spectrally isolated, networks can be reconfigured by simply connecting or disconnecting sensors without requiring recalibration or compensation algorithms. Heterogeneous networks mixing different sensor types (e.g., strain, temperature, force) are also supported, so long as each sensor transduces its signal of interest to a change in resistance or capacitance.

Bonus: We discuss the theoretical operational limitations below, but here are three Bainter nodes operating together between 1-100Hz sampling frequencies. We see even though the higher frequency measurements are noisier, it is still easy to pick out a good estimation of the notch frequency and, indirectly, the strain in this case.

Network design and theoretical limits

We derived design guidelines for building sensor networks, including procedures for sequential sensor placement and expressions bounding the maximum number of sensors in a given bandwidth. For a homogeneous network, the number of sensors that can fit in a bandwidth $[f_L,f_H]$ is:

where ${\Gamma }_1$ is determined by the maximum strain and gauge factor. With a margin based on the 3 dB bandwidth, this becomes:

where $\alpha =1/(2Q)$. In the limiting case of binary detection (one resolvable frequency point per sensor), a 1–100 kHz bandwidth can theoretically support hundreds to thousands of sensors depending on the quality factor and frequency resolution.

Summary

This work shows that encapsulating soft sensors in notch filters is a practical approach to distributed sensing with minimal wiring, easy reconfiguration, and tunable sensitivity. The framework is agnostic to the particular notch topology and sensing modality, making it broadly applicable to electronic skins, wearable devices, and soft robots. For a deeper look at the theory, including twin-T notch analysis, gauge factor derivations for all configurations, and detailed network design examples, see the manuscript and supplementary material.