Sensors & Transducers

Chief Noter

July 25, 2021
7 min read
Electronics Resistance Strain Stress

Sensors and transducers are used to measure conditions in a system. This is used to monitor the
performance:

Is the system working?
How well is the system working?
Is the system working safely?
Is the system likely to fail anytime soon?

A sensor is a device that measures a quantity and converts it into a signal – generally a voltage.

A transducer, on the other hand, is a device that converts one form of energy into another, for
example heat into electrical potential in a thermocouple.

The two most common measurements to take are temperature and position.

Measuring Temperature
Resistive Temperature Detector Networks
Measuring Position
Summary

Measuring Temperature

There are multiple ways of measuring and processing temperature.

Thermostats

A thermostat is a very simple temperature sensor. It is not capable of measuring the temperature
on any scale, only one or two fixed values – it will not tell you the temperature, it will merely tell
you when a certain temperature has been reached.

thermostat, thermostat diagram, temperature sensor

Basic thermostats consist of a bimetallic strip, the top and bottom have different thermal
expansion coefficients. This means that as temperature changes, they contract and expand at
different rates, curling the strip towards or away from a pole.

Resistance Temperature Detectors (RTDs)

These are more complicated, outputting a linear resistance-temperature profile:

$R=R_0+\beta(T-T_0)$

The initial temperature/resistance and β are constants that depend on standards. These are fixed
and specified for the RTD standard (of which there are two – PT100 and PT1000).

PT100 RTD, RTDs, resistance temperature detectors, RTD circuit symbol

Because they output a linear graph, they are extremely useful up to about 850°C. They work
because the resistance of platinum varies fairly linearly with temperature – however there are
slight variations, and so they are better for a large temperature range.

Thermistors

A thermistor is a semiconductor with temperature-dependent resistance. The output is not linear,
but it is more accurate over a small temperature range than RTDs.

Thermistor, NTC thermistor graph, negative temperature coefficient resistor graph

The majority of thermistors are negative temperature coefficient (NTC). This means that as the
temperature increases, their resistance decreases as the semiconductor releases more electrons
for conduction.

$\frac{R}{R_0}=e^{B(\frac{1}{T}-\frac{1}{T_0})}$

$B$ is a component constant – typically about 4000K.

Thermocouple

These are the most accurate temperature measuring devices and are technically transducers not
sensors. They work on the basis that an emf is induced in a wire that experiences a temperature
gradient.

$V_{ad}=V_{ab}+V_{cd}$

If the top and bottom wires are the same, then the voltage induced in each is equal but opposite.
In this case, the output voltage is zero:

$V_{ab}=-V_{dc} \quad \quad V_{ad}=0$

Therefore, two dissimilar wires need to be used with different thermoelectric coefficients.

Tables are used to find the values, but generally thermocouples and supporting data are given for
ambient temperature 0 °C. This is generally not, the case, so corrections need to be made using
interpolation form data.

When ambient temperature is zero degrees, the voltage readings are so small that the results are
inaccurate. In this case, they need to be amplified.

Resistive Temperature Detector Networks

RTDs and thermistors both output a resistance. A detector network then has to be sued to convert
this signal into a practical output.

Calibration

The device being tested is put into the circuit shown, and the temperature is varied. The current
and voltage are measured as the temperature changes and are plotted on an I-V graph.

This is generally only used to calibrate a sensor in a lab, not in any system, as the output (the
graph) cannot really be used as an input to anything.

Constant Current

Connecting a temperature resistance device to a constant current source outputs a varying
voltage across the device:

$V_{out}=I(R_0+\Delta R)$

A voltage output is very helpful; however, these setups are not very precise. This is because the
resistance in the temperature sensor actually causes a slight change in temperature, affecting the
results. The power dissipated as heat is given as:

$P_{heat}=I^2(R_0+\Delta R)$

Potential Divider

An alternative to the constant current detector network is a potential divider:

Taking $R_1 = R_0 + \Delta R$ :

$V_{out}=V_{in}\frac{R_2}{R_0+\Delta R+R_2}$

Again, there is the issue of the resistance affecting the temperature. The power dissipated by this heat increase is:

$P_{heat}=I^2R=(\frac{V_{in}}{R_0+\Delta R+R_2})^2(R+\Delta R)$

Wheatstone Bridge

$V_a=V_{in}\frac{R_4}{R_3+R_4} \quad \quad V_b=V_{in}\frac{R_2}{R_1+R_2}$

$V_{out}=V_{in}(\frac{R_4}{R_3+R_4}- \frac{R_2}{R_1+R_2})$

This may look daunting, but it is just two potential dividers in parallel:

Simplified wheatstone bridge, how a wheatstone bridge works, double potential divider

When the two voltages (a and b) equal each other, the output voltage is zero. In this case, the
bridge is said to be balanced:

$\frac{R_3}{R_4}=\frac{R_1}{R_2}$

There are two versions of the Wheatstone bridge: manual and automatic:

Manual Wheatstone bridges have a variable resistor in position 4. When the resistance
across the thermistor changes, the variable resistor is adjusted until the output voltage is
zero (the bridge is balanced). The resistance on the variable resistor is known and
compared to the previous value.
Automatic Wheatstone bridges do not have a variable resistor in any position, so move out
of balance as the temperature changes. The output voltage is then measured and
interpreted as the temperature reading.
If the three fixed resistors all have resistance R, then the output voltage is given as:
$V_{out}=V_{in}\frac{\Delta R}{4R+2\Delta R}$

The Power dissipated in the resistors is:

$P_{heat}=I^2R=(\frac{E}{2R+\Delta R})^2(R+\Delta R)$

Measuring Position

Measuring position can vary from being incredibly simple (using buttons or switches), to slightly
more complex (potentiometers), to vastly complicated encoders.

Potentiometers

These can be linear or circular:

$\lambda=\frac{x}{X}$

$V_{out}=\lambda V_{in}=V_{in}\frac{x}{X}$

Circular potentiometer, rotary potentiometer

$\lambda=\frac{\theta}{\Theta}$

$V_{out}=\lambda V_{in}=V_{in}\frac{\theta}{\Theta}$

Potentiometers are good, but the contacts wear out over time. There are non-contact alternatives,
but these are more expensive.

Encoders

An encoder can be used to convert motion into a binary output.

This rotary encoder has three tracks, so the binary output will have three bits. Because there are
three tracks, there are 2³ = 8 possibilities. Therefore, the resolution of this encoder is an eighth of
the circle:

$Resolution=\frac{360}{2^3}=45^\circ$

$Resolution=\frac{Range}{2^M}$

Where $M$ is the number of bits (tracks).

A linear encoder has one less division, as the ends do not join up. Therefore:

$resolution=\frac{Range}{2^M-1}$

The absolute error in an encoder is given as ± half the resolution.

Strain Gauges

A strain gauge consists of a folder wire as shown above. When a load is applied, the length of
wire changes. Since resistivity is a material property and does not change, the change in length
leads to a change in resistance:

$\frac{dR}{R}=G\frac{dI}{I}=G\varepsilon$

$G$ is a gauge factor
$\varepsilon$ is the elastic strain

When the strain is very small (and Hooke’s Law applies), the relative change in $R$ is almost
proportional to the stress. However, to be more precise:

$\ln \frac{R_1}{R_2}=G\ln\frac{I_1}{I_2}$

Load Cell

A load cell is more or less a big strain gauge. It works on the same principal, but is configured to
deflect a known amount for a specific applied load. A Wheatstone bridge detector network is
required to get a voltage output:

All resistors are strain gauges/load cells
Those in red are in tension
Those in grey are in compression

When $G\varepsilon$ is small:

$V_{out}=\frac{1}{4} V_{in}G\varepsilon$

A sensor is a device that measures a quantity and converts it into a signal
A transducer is a device that converts one form of energy into another
Temperature can be measured using a thermostat, thermistor, RTD, or thermocouple
RTDs and thermistors both output a resistance, so require a temperature detector network – this could be a resistor network, potential divider, constant current network or wheatstone bridge
Position is measured using potentiometers or encoders
- Specific position transducers also exist, for example strain gauges or load cells

Sensors & Transducers

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