EE3901/EE5901 Sensor Technologies

Revision and summary

Revision and summary

Sensors

A sensor is a device that observes a measurand $x$ to produce a measurement $y=f(x)$ where $f$ is the transfer function.

Properties of sensors include:

• Sensitivity: the slope of the transfer function $s=\frac{df}{dx}$. For electrical sensors, the units of sensitivity are always (electrical quantity) / (physical quantity).
• Resolution: the smallest change in $x$ that can be detected.
• Span: the range of measurands that can be measured.
• Accuracy: how well the measurement reflects the true measurand.
• Precision: how repeatable the measurements are (i.e. the standard deviation of the measurement). Precision is often limited by noise.
• Bias: a constant offset in measurement, i.e. $y=f(x)+b$ where $b$ is bias.
• SNR: the ratio between power in the true measurement and power in the noise.
• Dynamic Range (DR): the ratio between the largest and smallest values that can be measured. Typically the smallest value arises due to a noise floor.
• Bandwidth: the passband region of a sensor’s frequency response, usually defined to be the region where the response is within 3 dB of its maximum.

Propagating uncertainty

Suppose you are performing a calculation $\boldsymbol{y}=f(\boldsymbol{x})$ where $\boldsymbol{x}$ is subject to uncertainty. Let $\Sigma_{\boldsymbol{x}}$ be the covariance of $\boldsymbol{x}$ and let $J$ be the Jacobian of $f$ evaluated at a specific point $\boldsymbol{x}=\boldsymbol{x}_{0}$.

Then the covariance of $\boldsymbol{y}$ is

In the case that $y$ is a scalar and there are no correlations between elements of $\boldsymbol{x}$, then this formula simplifies to

Sensor fusion

The Kalman filter is a method of fusing data from multiple sensors to update a state estimate. It is a two step method with a prediction step (based on a mathematical model of our expected state evolution) and an update step (where information from measurements is incorporated to improve the estimate). In the absence of measurements the prediction step can be run by itself, but since there is no feedback the error will always grow larger over time.

There are 3 common types of Kalman filter:

1. The standard Kalman filter is an optimal estimator for linear systems with linear measurement models.
2. The extended Kalman filter can handle non-linear system and non-linear measurement models. It works by linearising the equations about the current state estimate with a Taylor series approximation. Hence it requires that the equations are differentiable and generally “well behaved” (i.e. the derivative does not change too rapidly).
3. The unscented Kalman filter can also handle non-linear systems and measurements, but does not require derivatives of these models. Hence it can be used when the models are too complex to differentiate (e.g. when they are computer programs). It works by running an ensemble of state estimates through the models then mixing these together to estimate the resulting covariance. The trade-off is increased computational effort since the models are evaluated multiple times.

The physics of sensing

• Temperature affects resistance. Many metals exhibit a close to linear response characterised by a material property called the temperature coefficient of resistance (TCR), which has units of $\text{K}^{-1}$. Precisely:
  $$\frac{\Delta R}{R_{0}}=\text{TCR}\times\left(T-T_{0}\right)$$
  where $\Delta R$ is the change in resistance, $T$ is temperature, and $R_{0}$ is the resistance at temperature $T_{0}$. The overall resistance is $R=R_{0}+\Delta R$. This effect is used to create resistive temperature detectors (RTDs).
• Semiconductors have non-linear temperature dependence (which may have positive or negative slope). This effect is used to create thermistors.
• Light shining on a semiconductor can create electron-hole pairs. This raises the conductivity of the semiconductor (an effect used to create light dependent resistors). In the presence of a driving force it can also generate a current (an effect used to create photodiodes). The driving force can be intrinsic (e.g. created by the depletion region of a p-n junction) or extrinsic (e.g. an applied voltage).
• Mechanical stress can influence resistance, an effect called piezoresistance. There is typically a linear relationship between resistance and strain. This is the mechanism behind strain gauges.
• In certain materials, mechanical stress can induce electric polarisation and vice-versa, an effect called piezoelectricity. Specifically, normal or shear stress can induce polarisation along various axes, depending upon the material’s crystal structure. Piezoelectric strength is described by coefficients $\left[d_{ij}\right]$ which have units of C/N.
• A temperature gradient along a conductor creates a voltage, an effect called thermoelectricity. However, for practical use, the two ends of the conductor must be brought to the same place so the voltage can be measured. This requires two different materials with different Seebeck coefficients, so that the thermoelectric voltage in one direction is not cancelled by an equal but opposite voltage in the return path. This effect is used to create thermocouples, which are robust temperature sensors capable of measuring very high or low temperatures.
• Materials have an intrinsic property called permittivity (units of F/m). This property governs the capacitance created when the material is placed between two nearby electrodes. Changes in permittivity (by physically moving the material or by changing its properties) create changes in capacitance. Permittivity varies with temperature, humidity, frequency, and other parameters.
• Capacitance is directly proportional to overlap area of the electrodes, and inversely proportional to the separation distance. An example of varying the separation distance is in MEMS accelerometers where an electrode is deflected by accelerations.
• Charges moving in a magnetic field experience a force at right angles to both the field and the direction of movement. This effect is used to create Hall effect sensors, which are sensors of magnetic field strength.
• Materials have an intrinsic property called permeability (units of H/m). This property governs the inductance created when a coil of wire is wrapped around the material. Changes in permeability (usually by physically moving the material) create changes in inductance. Permeability is also dependent on temperature, frequency, and other parameters. An example type of sensor using variation in permeability is the LVDT.
• A time-varying magnetic field induces a current in nearby conductors. This effect is used to create eddy current sensors, which induce a current in a conductive test sample and measure the resulting changes in resistance or inductance.

Interface circuits

• Resistance and impedance can be measured by applying a voltage and monitoring the current or vice-versa. The simplest approach is the “two wire” circuit where the excitation current/voltage is applied to the sensor using the same wires by which the measurement is performed. However, if the wires are long enough then their impedance will affect the results. Hence an alternative is the “four wire” circuit where a current is injected using one set of wires and the voltage measured using another set.
• The Wheatstone bridge circuit is used to detect small changes in resistance. It consists of two voltage dividers in parallel. In the case of an AC power source it can also be used to detect small changes in capacitance or inductance. Temperature compensation can often be achieved by placing non-active sensors in the bridge layout in such a way that the temperature effects cancel out.
• A differential amplifier is any circuit that measures a voltage difference between two points, i.e.
  $$V_{0}=A_{d}\left(V_{1}-V_{2}\right),$$
  where $A_{d}$ is the differential mode gain. In practice there will be common mode gain as well:
  $$V_{0}=A_{d}\left(V_{1}-V_{2}\right)+A_{c}\left(\frac{V_{1}+V_{2}}{2}\right).$$
  The common mode rejection ratio ($\text{CMRR}=\frac{A_{d}}{A_{c}}$) should be large.
• One type of differential amplifier is the instrumentation amplifier. These are ICs designed to have very high CMRR and also very high input impedance.
• Differential amplifiers or instrumentation amplifiers are needed for many sensors or interface circuits that produce a voltage: e.g. voltage divider circuits, bridge circuits, circuits that measure current by the voltage drop across a resistance or impedance, piezoelectric sensors, thermoelectric sensors, Hall effect sensors, etc.
• Another interface circuit for capacitive and inductive sensors is to use the sensor to control the frequency of an oscillator. Digital counter circuits can precisely detect small differences in frequency.
• For various types of capacitive sensor, a linear response can be obtained with a single op-amp. Use an inverting amplifier circuit and place the sensor either in the input or the feedback path depending upon whether the impedance is directly or inversely proportional to the measurand.
• Capacitance can also be measured by transferring charge between known and unknown capacitors.
• A transimpedance amplifier is a current-to-voltage amplifier. It is used for sensors whose electrical response is a current as opposed to a voltage, such as a photodiode. However, it is challenging to achieve high gain and high bandwidth simultaneously. If a high gain and bandwidth are required, then use a multi-stage design with a transimpedance amplifier feeding a normal voltage amplifier.
• Active pixel sensors (APS) are used in cameras and optical spectrometers to interface with an array of photodiodes. Common circuit designs consist of 3 or more CMOS transistors that implement buffering and read-out circuitry alongside each photodiode. During image capture, the generated photocurrent discharges the photodiode’s inherent capacitance, resulting in a change in voltage.