Capacitive vs. Resistive Strain/Stretch Measurement
Here at StretchSense, we measure strain by tracking changes in the capacitance of our soft sensors. Strain can also be measured by sensing change in electrical resistance. In this monograph, I will compare the two methods (resistance vs. capacitance strain sensing), show how each is calculated, and demonstrate why we believe capacitance to be the parameter of choice for measurement of human body strains.
Resistance strain gauges, commercially available from the early 1940s, are now predominantly fabricated from metal foil . Such strain gauges are now ubiquitous and used for characterizing small mechanical strains in rigid engineering structures, force (load) cells, scales for weighing, and the engineering testing of metals and rigid composites. The change in the gauge resistance is typically characterized using a Wheatstone bridge resistor network (figure 1).
Figure 1: a) Metal foil strain gauge schematic b) A quarter bridge circuit can be used to measure resistance change in the gauge.
Many engineering components (e.g. hoses, electrical conduits, machine suspension parts), things we wear (e.g. clothes, orthotic devices) or use as furniture (e.g. bedding, cushions), and our own skin (with modulus order of 1 MPA ), are soft and stretchy and routinely exposed to strains greater than 10 percent. A metal foil gauge would be too stiff for measuring strain on such soft and stretchy substrates.
Materials for measurement of large strains are based on elastomers such as silicone, acrylic, and natural rubber. These materials are electrical insulators, and since electricity must be involved in the measurement, they must be able to conduct electricity too. By adding particles of a conductive material such as carbon black or nickel to the uncured polymer, silicone can be turned into a conductor. At a sufficiently high volume fraction of conducting particles, agglomerating chains of filler will form (figure 2). Conduction through the composite will depend on chain overlap and electron tunneling between chains.
Figure 2: a) A conducting filler can be added to a polymer matrix until the percolation threshold is reached and b) it becomes a conductor.
Silicones filled with carbon can be used for compression contact sensors and wearable tensile strain sensors (e.g. smart gloves) . Such piezoresistive behavior can also be used for switching a charge on and off  for the control of static electric motors, energy harvesters, rubbery logic devices, and electromechanical oscillators. The switching behaviour has recently been demonstrated in a silicone filled with particles of nickel and used for a rubbery mechano-electrical oscillator.
For characterizing the resistance changes under large strains in ‘filled’ elastomers such as silicone, we cannot use a model based on small strain analysis. We start with a large strain model (Figure 3a).
We also assume the filled silicone is incompressible (e.g. volume is constant). When we do this, we get a ‘squared’ expression for strain ratio in the direction of stretch; thus, if the strain is doubled, we would expect resistance to increase by a factor of four.
Figure 3: a) Schematic of filled elastomer sensor. b) A carbon-filled silicone resistive sensor of length 50mm, width 10mm, and height 0.5 mm was subjected to a stretch test, stretched 30mm over 2 seconds, held for 8 seconds and then relaxed over 2 seconds, shows substantial transient behavior at the start and end of testing.
In figure 3b we depict an example resistance measurement using a carbon-filled silicone electrode that demonstrates substantial gains in resistance during stretching to a 30% increase in length and at relaxation. These large changes in resistance that presumably combine with the non-linear changes described in the model are not accounted for in the large strain model.
In summary, resistance measurement as a way to characterize large strains, while feasible, will be complicated by non-linear changes due to geometric effects and changes to resistivity. Perhaps the greatest concern has to do with highly non-linear dynamic changes in resistivity, illustrated experimentally in Figure 3. Resistivity by itself is also subject to environmental and subject-specific conditions (e.g. temperature, damage, etc.). Thus, frequent sensor calibration could be necessary, accompanied by a dynamic model of the mechanism that can be used to correct for the influence of the transient effects displayed in Figure 3.
Capacitive elastomer sensors can be assembled by sandwiching elastomer dielectric layers between elastomer electrodes that are filled with conducting particles. Figure 4 depicts a simple embodiment of such a sensor, fabricated in three layers with a dielectric between the electrodes. Capacitance for a simple stretch sensor will be proportional to strain.
Figure 4: Schematic of dielectric elastomer sensor: dielectric of thickness t flanked by stretchable carbon-filled silicone electrodes.
Algorithms for calculating sensor capacitance are based on a lumped-parameter equivalent electrical circuit for the sensor that includes a series resistance for the electrodes (Relectrodes) that flank the dielectric in series with a variable capacitance CDE in parallel with a resistance that accounts for leakage across the membrane of the dielectric layer(Rmembrane) (Figure 5):
Figure 5: Equivalent circuit for dielectric elastomer sensor.
Using a small electrical signal that is fed to the sensor, it is possible to estimate sensor capacitance CDE once the magnitude of the imaginary component of the impedance (Iim) is known: ; where ω is the signal frequency. The lumped resistance parameters can be estimated in real-time; there is no need for a priori knowledge of any stretch-dependent electrode resistance.
Using the same sensor, we depict the capacitance response to a 30 percent step change in strain (Figure 6). This can be compared with the measurement depicted in Figure 3, on a carbon-filled electrode The capacitance calculated in the sensor faithfully follows the strain vs. time plot.
Figure 6: A sensor that is stretched to 30% strain, held at that strain for 10 seconds and then relaxed over 2 seconds. We calculate capacitance for the whole sensor.
A key question remains: how does stretch influence relative permittivity? For it is through permittivity that we link changes in geometry with capacitance change. Dielectric permittivity is influenced by temperature, deformation, and the nature of the compliant electrode layers that flank the dielectric. A recent study has demonstrated that non-linearity in capacitance measurement can be attributed to stretch-influenced changes in the dielectric constant and that the sensor ends will affect the measurement. Thus, for good measurement fidelity, pre-calibration is recommended.
Like all models, the lumped-parameter one depicted in Figure 5 has its limitations. A capacitive sensor can more realistically be represented by an infinite number of smaller capacitors and resistors joined together in a transmission line as depicted in Figure 7, with each resistor and capacitor set acting as a low-pass filter. As sensing frequency is increased, the signal will be attenuated as it progresses along the sensor. It is, therefore, important to limit the sensing signal to frequencies below which the signal attenuation is negligible.
Figure 7: A capacitive sensor depicted as an electrical transmission line. At sufficiently low sensing signal frequency the entire length of the sensor can be represented as a single lumped capacitance and resistance.
The transmission line behaviour can enable direct localized sensing along the sensor, achieved by sending multiple signals, each at a different frequency along the same line so that each signal component is attenuated differently. Processing of the signal data can then be used to identify where along the sensor the material is being stretched or perhaps touched. This can potentially be used for multiple position sensing along a single sensor — and in two dimensions as well — enabled by laminating two sensor sheets one atop the other.
SUMMARY: RESISTANCE VS. CAPACITANCE SENSING
Electrical resistance measurement has clearly proved its worth for small strain sensing. However, for large strain sensing that involves filled elastomers, the measurement will be highly non-linear due to geometric influences and transient changes in resistivity. Capacitance measurement offers an opportunity for reliable, near linear strain measurement that is relatively unaffected by transient changes in electrode resistance.
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 The SR4, the world’s first commercial strain gauge (c1941), named for the inventors, E.E. Simmons (Caltech, 1937) and A.C. Ruge (MIT, 1938), and two others, was simple: the strain dependent resistance change of four tungsten filaments arranged in a diamond pattern was electrically measured. Reliability problems associated with wire gauges bonded to tested helicopter parts during the early 1950s led engineers at Saunders-Roe Ltd., a British aero- and marine-engineering company on the Isle of Wight to develop the first metal foil gauges.