### How Fast Can You Measure Strain?

### IS THERE A LIMIT TO THE FREQUENCY RESPONSE OF A CAPACITIVE STRETCH SENSOR?

Stretch sensors are an enabling technology for the next generation of wearables. Such smart wearables are intrinsically garment-like, can detect and measure muscle contraction, chest expansion, joint motion, and more. StretchSense makes precise and robust capacitive based stretch sensors that are integrated into fabric. These integrated sensing systems are used to develop smart garments that provide real-time measurement of human motion.

In this paper, we will seek to answer the question; is there a limit to the frequency response of a capacitive stretch sensor? This question is relevant for applications that involve high frequency and/or rates of strain. To answer it, we need to look at the composition and physics behind our capacitive sensors.

StretchSense sensors consist of thin layers of silicone rubber, a non-conducting dielectric material, flanked by stretchy, carbon filled, silicone electrodes. When they are stretched, their ability to hold an electric charge at a given voltage changes. This charge-holding capability, capacitance *C*, is the ratio of the charge stored on the sensor *Q*, divided by the voltage across its dielectric *V*:

Sensor capacitance is also related to sensor geometry:

ϵ_{0}, ϵ_{r}, *A*, and *t* represents the dielectric constant for vacuum, the relative dielectric constant for the dielectric material (silicone), the sensor area, and the dielectric thickness. Stretching increases sensor area *A*, and decreases its thickness *t*. Sensors can also be built up by adding additional layers of dielectric and electrode: in equation 2, the factor *n* represents the number of dielectric layers.

Thus, if we can measure the sensor’s capacitance electrically, using the relationship described in equation 1, we can directly relate this to its geometry through equation 2, and calculate stretch. We do this by feeding an electric charge to the sensor through an oscillatory (sinusoidal) signal from the dedicated sensor electronics to the stretchy electrodes that sandwich the silicone dielectrics. This signal is sent down the line at a fixed sensing frequency. For the sensor to be effective at measuring a vibration, the sensing signal must be substantially greater than the frequency of the mechanical vibration being monitored. However, there are limitations on how high this sensing frequency can be.

A simple electrical model demonstrates that charging any capacitor, including a StretchSense sensor, is not instantaneous, for it behaves as a resistor in series with a capacitor. If we were to charge a standard capacitor through a resistor R with a constant voltage difference of V0, the voltage rise across the capacitor V would be given by:

The factor *RC* in equation 3 is called the *time constant*. When the clock time *t* equals *RC*, the voltage across the capacitor will be 0.63 *V _{0}*. We can see from equation 3 that our capacitive sensor would take infinite time for it to fully charge at the constant voltage

*V*, and the product of capacitance and resistance will govern the rate of charging.

_{0}Our simple representation of the sensor as a single resistor and single capacitor in series is termed the *lumped parameter* model. This is easy to analyze and useful for characterizing capacitance for strain measurement. The electrical resistance in the carbon-filled silicone electrodes is substantially higher than for the metal electrodes of conventional capacitors. Thus, the resistance in the sensor is largely associated with the resistance in the electrodes (Figure 1), and we lump this electrode resistance together into a single equivalent resistance: *R _{S}*. For the capacitance, the total capacitance in the sensor is similarly lumped together into a single capacitance represented by

*C*.

The sensor is not a perfect capacitor; there will be some charge leakage that we account for in the lumped parameter model with the single leakage resistor *R _{P}*.

*Figure 1: Schematic representing a sensor layer that is composed of a silicone dielectric of capacitance C flanked by stretchy electrodes. The equivalent lumped parameter circuit is depicted on the right. Rs represents the series resistance of the electrode layers, and RP represents the resistance across the dielectric.*

This lumped capacitance model works well and forms the basis for stretchy sensor capacitance measurement as long as the sensing frequency is not too high. We can get a feeling for the limits on its validity by calculating the time constant for a sensor, RSC, and using the inverse of this number to obtain a ballpark estimate of the maximum sensing frequency, above which we would expect our lumped model to break down.

Typical values for sensor resistance and capacitance suggest a RSC time constant on the order of 10 microseconds for a standard 100 mm long sensor. So, from this we have an optimistic maximum sensing frequency (1/RSC) expectation of about 100 kHz. Realistically, the maximum sensing frequency will be lower than this, as stretching a sensor has the dual effect of increasing capacitance and resistance, thus substantially increasing the time constant.

There are also transient changes in resistance in carbon-filled silicone that could account for a further reduction in maximum sensing frequency. These changes involve resistance growing to several times its resting value at the start of stretch and relaxation. This transient resistance phenomenon is illustrated in another posting (refer Capacitive vs. Resistive Strain/Stretch Measurement). It is also the subject of our current research. Such transient changes in resistance make strictly resistive stretch sensing very difficult, if not impossible, to do. Capacitive sensing, as we practice it, is largely unaffected by these transient effects (refer Capacitive vs. Resistive Strain/Stretch Measurement).

Currently, we are getting all-round good fidelity of measurement on our standard sensors at sensing frequencies above 1 kHz. This is ideal as most wearable applications involve mechanical vibrations that are 10 Hz or less in frequency: well below the kHz range! And as sensor size reduces, so will capacitance and series resistance. Thus, the time constant will also reduce and maximum sensing frequency will increase. This is good, as higher frequency vibrations generally have much smaller displacements, and require smaller sensors that we can also produce.

To answer the question posed at the start: Is there a limit to the frequency response of a capacitive stretch sensor? The answer is yes; sensor size and electrode resistance will influence the limit; but for wearable applications, this should not be a point of concern.

A word of caution: There are also elastic wave phenomena that might influence stretch sensor measurement. The frequencies associated with the waves will be influenced by how the sensor is mounted (Is it touching the structure? How taut is it? Etc.) We’ll return to these effects in a later publication.

If you would like to learn more about the frequency limitations on capacitive stretch sensing, please have a look at the paper by Xu et al. [1].

## Reference

Xu, D., et al., *Sensing frequency design for capacitance feedback of dielectric elastomers*. Sensors and Actuators A: Physical, 2015. 232: p. 195-201.