### Understanding Stretch Sensor Sensitivity

People are always asking us to make sensitivity as high as possible, but as with everything in sensor-land, this has trade-offs! To help you navigate these challenges we’re going to take a deep dive into sensor sensitivity. We’ll walk through the process of how to calculate basic sensitivity, and explore trade-offs when integrating sensors into real word applications. Get in touch with us here if you would like to learn more about sensor sensitivity.

### LET’S START STRETCHING

For the following analysis, we’re just going to be talking about uniaxial stretch in a sensor: That is stretching a long narrow sensor in free space. In this case, the extension – capacitance relationship is pretty much linear (as shown in Figure 1 below). Also, the area of the sensor remains constant during stretch. Things get a bit more complicated when the sensor is in a more complex state of stress. For example, when stretching the sensor in more than one direction because it’s mounted in a smart textile or against a body.

*Figure 1: Graph showing capacitance vs extension for a uniaxially stretched silicone stretch sensor (Data from StretchSense Silicone Sensor Evaluation Kit VERSION 5.0 160427)*

### SOME QUICK AND DIRTY MATH

Sensitivity is defined as the amount of change in capacitance for a change in the extension. Put another way, sensitivity is the slope of the capacitance-extension curve (remember Figure 1?). We can write this mathematically with the formulation below where S is the sensitivity, C is the capacitance, and L is the length of the sensor.

To calculate sensitivity, we start with the capacitance. Capacitance relates to geometry according to the Parallel Plate equation below where C is the sensor’s capacitance, A is the surface area, D is the thickness, εo is the absolute permittivity, and εr is the relative permittivity of the dielectric layer.

The next step is to rearrange the capacitor equation in terms of length and stretch. We do this by substituting the length, width, and thickness relations shown in Figure 2 into the capacitor equation to get the formula below. L is the length, Lo is the original length, W is the width, Wo is the original width, D is the thickness, Do is the original thickness, and λ is the stretch in a given direction defined as (current size / original size).

In the special case of uniaxial tension in a volumetrically incompressible solid λy= λz so we can rewrite the equation as.

Differentiating with respect to length L we get.

Which is our sensitivity equation! Remember, this is a very quick rundown on how to calculate sensitivity for back of the envelope analysis. Each sensing project is different and may be much more complex. Use with caution! And contact our team at sales@stretchsense.com to learn more about how sensitivity applies to your specific project.

### UNDERSTANDING THE IMPACT OF SENSITIVITY

Before we analyse the impact of sensitivity it is helpful to understand the process of measurement which is shown in Figure 3 schematically in a 3 step process:

- First measure capacitance with an interrogation circuit.
- Then trace a line horizontally from the vertical axis until it intercepts the sensor curve.
- Finally trace a line vertically down until interception with the horizontal axis, which is your sensor reading!

This process seen in the below schematic includes the effects of noise. The dotted lines intercepting the capacitance axis represent the noise band in pF of the sensing electronics. A sensor with higher sensitivity has a smaller error band due to noise in the circuit. Increasing sensitivity will increase the signal to noise ratio of a sensor, and thus the accuracy.

*Figure 3: Schematic showing the effects of sensitivity on sensor measurement accuracy in the presence of noise*

### IMPROVING SENSITIVITY

From the sensitivity equation it can be seen that the sensitivity (in pF/mm) of the sensor is not related to the length of the sensor. Sensitivity can however be increased by:

- Increasing the width (Wo) of the sensor
- Introducing more layers into the sensor structure, thereby increasing the effective width of the sensor
- Creating thinner dielectric layers, thereby decreasing the distance between the flexible electrode layers (D)
- Utilising dielectric layers with higher relative permittivity (εr)

### DESIGN TRADE-OFFS

It is desirable to maximize the sensitivity of a sensor to gain the most accuracy. However, we have to weigh up the cost vs the potential benefits.

- Increasing width increases stiffness and is limited by the available space in the design.
- Adding layers increases size, stiffness, manufacturing complexity and cost.
- Thickness can be changed but normally is already minimised up to the limit of manufacturing constraints.
- Changing the dielectric constant will change the mechanical and manufacturing properties of the sensor.

When trying to gain more sensitivity we have to make a call based on what we’re trying to achieve. A glove, for example, has different requirements to a compression shirt. Remember sensitivity is not the only game in town. We can get improvements in accuracy by other means, for example reducing interrogation noise with filtering, shielding, or using higher specification electrical components. Each method comes with its own pros and cons.

### GET IN TOUCH

Stretch sensors are a great prototyping tool for technologists. The benefits of being wireless, soft, and lightweight make them ideal for embedding into clothing or attaching directly to the body. These characteristics make it possible to explore new applications that rely on precise body motion information. If developing wearable applications interests you, or you would like to learn more about stretch sensor sensitivity, drop us a line!