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Sensor Theory

A sensor is an element that picks up a quantity (or a property) and converts it to the electrical domain. With sensors, we see the common terminology of sensitivity, offset, bias, drift, calibration, transduction, resolution, saturation, and hysteresis. Sometimes the word ”sensor” is used for the whole smart sensor which includes biasing, signal conditioning, a calibration procedure, a microcontroller, a bus interface and a package. In this chapter, we use the convention of a sensor being the transducer only.

Background literature

There are excellent books about sensors, their applications and the corresponding read-out electronics, also available for free on the Internet. Some of the better examples are:

  • R.S.C. Cobbold, Transducers for biomedical measurements: principles and applications1)
  • S. Middelhoek and S. Audet, Silicon sensors, microelectronics and signal processing2)
  • Jacob Fraden, Handbook of Modern Sensors - Physics, Designs, and Applications3)
  • Chapter 3 of the Linear Circuit Design Handbook by Analog Devices4), contains sensor descriptions from the perspective of the read-out electronics.
  • Analog Devices, Practical Design Techniques for Sensor Signal Conditioning, 19995)
  • Paul P.L. Regtien, Sensors for Mechatronics6)
  • Sander Struijk, Sensors and Actuators - Introduction to Sensors, course of the Technical University of Eindhoven7)

The measurement chain

The terminology on this page refers to the section about the measurement chain. The hardware needed to measure a quantity is represented in the block schematic of figure 1. What is measured is a physical quantity. The measurement is done in the following stages:

  • The coupling network is how the sensor-head receives the quantity. For most sensors this is the mounting method, but this can also represent an acoustical coupling in case of a microphone. It determines efficiency and cross-sesnitivity artefacts.
  • The sensor-head is the transducer that converts information from a physical domain to the electrical domain.
  • Transduction can not be done without proper biasing: meaning creating the electronic setting point and making an electronic signal from the transducer. In addition, there is a first analog pre-processing immediately after the electronic signal is made available. Biasing plus pre-processing is represented here in a single block called signal conditioning.
  • The signal conditioning shapes the signal for proper analog to digital conversion.

Fig. 1: The measurement chain

Selector part and transducing part

If we consider a sensor as a device that converts information from a domain of interest into a useful electronic signal, this conversion takes place in two steps. First, there is a selector part and next a transducer part as indicated in figure 2.

  • In the selector part, the information of interest (the quantity to be measured) is converted into a form that is measurable by the transducer.
  • The transducer part is an electronic device where the quantity of interest is coupled to a certain parameter

So, as the transducer part is non-selective, the selector must introduce the desired selectivity8). A remark about terminonology has to be made. The selector part and actuator part together make up the sensor head, which is in fact a transducer. This sensor head is part of the sensor, a word that normally refers to the whole product that includes the housing, signal conditioning, AD-Converter and bus system. In the scope of this chapter, when we talk about sensors it is about the the sensor head.

Fig. 2: A sensor has a selector and a transducer part, and converts information from a domain to the electrical domain

For many chemical sensors, this two-level interpretation in transducer part and selector part is quite clear. Commonly observed chemical selectors are selective membranes and modified surfaces. Examples of chemical transducers are sensitive transistors or capacitive transducers. Physical sensors normally have a less defined separation between those two parts. For measuring temperature or moisture, selectivity is created by taking a material where one of the material constants changes with temperature or water absorption respectively. Next, a transducer part is chosen, which converts the modulated material constant into a change of electrical resistance or capacitance.

Fig. 3: Dia06 Fig. 4: Dia08 Fig. 5: Dia09


The information a sensor probes is in fact a (physical) quantity. There are many different classifications of quantities9). A quantity can have a direction or not. In that case it is a vector or a scalar respectively. A quantity can be in indicator of a static variable or a dynamic variable, in which case it is a state variable or a rate variable. Another classification is whether the quantity is associated with an energetic phenomenon (a variable) or not (a property). Examples of variables are voltages, pressure, velocity, while examples of properties are length, mass, elasticity. According to measurement theory, we can classify variables as a cause or as an effect. Causes are inputs of the system, referred to as an independent variable, and effects are responses of the system on the cause, referred to as dependent variables. The relation between independent and independent variables is in the physics of the system. It can be related by physical phenomena, material properties, or system design such as geometry. This relation is in fact the design of the sensor. Most of the quantities can be either dependent or independent: the denomination depends on the system. The final classification is whether a quantity includes the form of the measured object. An intensive quantity is not dependent on the mass and size of the measured object, an extensive quantity does depend on the form. As a result, resistivity is an intensive material property and resistance an extensive device property. Temperature can be called an intensive variable.

Vector Quantity has direction Scalar Quantity has no direction
State variable Quantity is static Rate variable Quantity is dynamic
Variable Quantity is associated with an energetic phenomenon Property Quantity is not associated with an energetic phenomenon
Dependent variable Quantity is response or effect Independent variable Quantity is cause
Extensive Size and mass included Intensive Size and mass not included
Tab. 1: Classifications of quantities

These definitions of classes of quantities make an interesting link to a generalized method of modelling across the domains. In lumped element modelling, we define extensive variables as state variables. Examples are temperature in the thermal domain, location in the mechanical domain, and charge in the electrical domain. The time variation of the state variables are rate variables and referred to as flows. Examples are heat flow, velocity and electric current. The third quantity we need for our models are the efforts, which are the cause of a flow and appear to be identical to intensive variables. Temperature, force and potential in the examples respectively.

Extensive variable State variable
$\partial / \partial t$ Extensive variable Rate variable = flow
Intensive variable effort
Tab. 2: Relation of quantities to lumped element terminology

Self generating vs. modulating sensors

If a sensor is seen as a transducer of information from one domain to another, two types can be distinguished. The first are sensors that convert energy from one domain to another. As a result, the output signal will be zero when no input is present because the only energy applied is the energy of the signal itself. This is called a self-generating or direct transducer10). Examples are the thermocouple and the dynamo. The second group of transducers consists of devices to which energy is applied by a source, which is subsequently modulated by a physical or chemical quantity. These are modulating transducers11), examples are the pH sensing transistor (referred to as ISFET) and the thermistor temperature sensor.

Fig. 6: Dia10 Fig. 7: Dia11 Fig. 8: Dia12 Fig. 9: Dia13 Fig. 10: Dia14 Fig. 11: Dia15 Fig. 12: Dia16 Fig. 13: Dia17

Sensitivity, offset and calibration

Generally, a sensor should give an output signal as a function of an input signal, related by a certain sensitivity parameter. If a linear relation is assumed as represented in figure 14, two things are important: the slope of this relation and the intercept at zero input. The operational model of a sensor device is a mathematical description that links the input uniquely to the output signal: the transfer curve. Normally, this requires the characterization of a slope and a y-axis intercept, either by calibration or complete determination of the model. When according to the model a guaranteed zero output is observed at zero input a one-point calibration will be sufficient, else at least a two point calibration must be performed. Also in some specific cases where either the offset or the slope is known (or more or less constant), we can suffice with a one point calibration. An example is a pH sensor which inherently has a slope of 59 mV/pH: only the offset has to be calibrated with a single calibration in a reference liquid.

Fig. 14: A linear sensor transfer curve

To come back to the self-generating and modulating sensors, the self-generating transducers have no output signal at zero input. In that case, there will be no offset (intercept in figure 14) and only the slope has to be known, for example by a one point calibration. On the other hand, transducers of the modulating type have a non zero y-axis intercept, and so the reference is often not well defined.

Fig. 15: Dia18 Fig. 16: Dia20


The reference defines the quantitative meaning of the offset. A reference can be a reference electrode in an electrchical probe, the definition by calibration of one g in an accelerometer, or the comparison with meting ice in a thermometer. The dependency on stable references and calibration are the key problems in sensor system applications. With modulating transducers, the offset can sometimes be eliminated by measuring the output with respect to another element that is not sensitive to the measured quantity12) 13). In that case, a zero output means that the conditions at the measuring device are equal to that at the other device. This is a relative measurement with which common undesired signals, like unstable references, can be eliminated. Relative measurements with respect to a second sensor are referred to as differential measurements. An often-used differential set-up is the Wheatstone bridge14). The advantage of bridge set-ups is that the output voltage can be set to zero at a desired sensor output by adjusting the trimming element. In addition, interfering signals that are common to the branches are being eliminated intrinsically.

Drift and cross-sensitivity

The reason that references and calibration data are not fixed normally comes from interfering phenomena like ageing, temperature, moisture and motion. Especially in sensor applications on the human body using flexible low-wheight materials, the effects of permanent changes like corrosion and the absorption of moisture is dramatic. Changes in the calibration data (sensitivity and offset) due to permanent changes in the sensor and due to settling effects are called drift. Drift can only sometimes be cancelled out by a differential measurement or by mathematical anticipation using a known settling curve. Sensors are said to have a cross sensitivity with another quantity: meaning that they are not only sensitive for the intended quantity, but also for an interfering quantity. In some cases, the cross sensitivity can be eliminated in a differential set-up, but for some properties this is quite hard in practice.

Transfer curve and non-linearity

In some cases, the sensor dependency on the input quantity is not linear. As a result, the sensor transfer curve is not as linear as we saw in figure 14. For example, in figure 17 a sensor response is sketched which is highly non-linear: the sensor saturates for higher inputs. In fact, in this specific case it is not the sensor itself that saturates, but an incorrectly designed read-out circuit.

Fig. 17: A sensor with a highly non-linear curve can be defined over a certain input range

However, we can work with such a nonlinear transfer curve by defining it only in a given input range. As a result, there is also a limited output range. In between, the response is still non-linear: the sensitivity is dependent on the input level. We need mathematical methods to calculate the input from the measured output or we need a compensation network.

If the non-linearity over a specified input range is accaptable, we may approximate the sensor still with an input independent sensitivity. To be able to indicate how non-linear a sensor is, we need a number. In figure 18 it is indicated how we can define the level of non-linearity.

Fig. 18: A sensor transfer curve with non-linearity

Non-linearity is defined as \begin{equation} \mathtt {Non-linearity}=\frac{\mathtt {MaxDev}}{\mathtt {FullScale}}\cdot 100\% \end{equation}

with MaxDev the maximum deviation over a certain sensor range (for example in Volts) and FullScale the full scale range (also in Volts if MaxDev was in Volts).

Another phenomenon which is the result of a non-linear effect is hysteresis. When a transfer curve of a sensor when the quantity is going from a low level to a high level is different from the curve when going from a high level to a low level, we speak of hysteresis. This is especially seen with magnetic sensors because hysteresis is a magnetic property as shown in is seen in figure 19.

Fig. 19: Hysteresis in a magnetic material

Fig. 20: Dia22 Fig. 21: Dia23 Fig. 22: Dia24 Fig. 23: Dia25 Fig. 24: Dia28 Fig. 25: Dia30 Fig. 26: Dia31 Fig. 27: Dia32

Motion artefacts

Motion of the human body has similar spatial and temporal characteristics as some physiological properties of interest. This means that motion of the human body is distributed over the chest, head and extremities, just like the (electro)physiological data of interest. In addition, motion frequencies ranging from the sub-Hertz to the tens or hundreds of Hertz regime, are in the same frequency band as breathing, heart rate, etc. This means also that for signal pre-processing by filtering, we cannot simply suppress a disturbing frequency band. Such interferences of signals due to motion are referred to as motion artifacts, and are notoriously hard to eliminate for sensors around the human body.

In this subsection, we have seen how drift in the offset, slope and reference of a sensor result into the need for smart system solutions. Normally, it is a combination of calibration and differential topologies that is needed to suppress artifacts up to an acceptable level. These topics translate directly into aspects of signal robustness and reliability that will be essential in safety critical systems.

Types of sensors

The separation of sensors into a selector part and a transducer part, as explained with figure 2 in the previous subsection, makes us aware that several propertiess can be measured with different transducer elements. Imagine a selector element consisting of a compressible material to transform ‘touch’ or ‘pressure’ into a material deformation. To convert the deformation information into an electrical signal, several different types of transducers are possible. We could use an electrical resistance measurement, an electrical capacitive measurement, or an optical read-out system. When classifying the transducer parts as electrical components, the number of options is limited. The most common are:

  • Resistive sensors - like strain gauges, the Pt100 temperature sensor, and the LDR light sensor
  • Capacitive sensors - like accelerometers, some proximity sensors and some pressure sensors
  • Inductive and other magnetic sensors - like AMR sensors for position, rotation and speed
  • Piezoelectric sensors
  • Semiconductor sensors - like the NTC temperature sensor, photodiodes and phototransistors

For monitoring electrophysiological signals like ECG, EEG and EMG, there is transduction as well. Although the signal of interest is already electrical (electricity in the heart, nerves and muscles), there is a transition from the ion-world to the electron-world. This transition results in all types of sensor problems related to motion artefacts, calibration, referencing, etc. To understand them, we should know more about electrochemistry.

After crossing the electrochemical barrier, we can pick up electrophysiological signals with conductive electrodes on top of the skin. These electrodes can be integrated with a textile using conductive yarns for wearable applications15). In other prototypes, the electrodes are placed in the fixed world, for example a car seat or a bed16). It is not absolutely necessary to have a galvanic contact to the human body: a capacitive coupling may be sufficient because for electrophysiological signals we are not interested in the DC value. The principle of using an insulated electrode for capacitive measurements of electrophysiological signals was first demonstrated by Richardson in 196717). Only recently, the technology was integrated into textiles18) 19) 20) 21).

The smart sensor

Fig. 28: A sensor with all signal processing, signal conditioning and self-calibration inside, is referred to as smart sensor

Sensor Technology TOC

1) , 14)
R.S.C. Cobbold, Transducers for biomedical measurements: principles and applications, John Wiley & sons, New York, 1974
2) , 8) , 10) , 11)
S. Middelhoek, and S. Audet, Silicon sensors, microelectronics and signal processing, Academic Press Limited, London, 1989
Jacob Fraden, Handbook of Modern Sensors - Physics, Designs, and Applications, Springer, 2010,
Analog Dialogue: Linear Circuit Design Handbook, by Analog Devices,
Analog Devices, Practical Design Techniques for Sensor Signal Conditioning, 1999,
6) , 9)
Paul P.L. Regtien, Sensors for Mechatronics, Elsevier, 2012
Sander Struijk, Sensors and Actuators - Introduction to Sensors, course of the Technical University of Eindhoven,
W. Olthuis, S. Böhm, G.R. Langereis and P. Bergveld, ‘Selection in system and sensor’, in: Mulchandani A and Sadik O A, Chemical and biological sensors for environmental monitoring, Washington D.C., Oxford University Press, pages 60-85, 2000
W. Olthuis, G.R. Langereis and P. Bergveld, ‘The merits of differential measuring in time and space’, Biocybernetics and Biomedical Engineering, 21(3), pages 5-26, 2001
M. Catrysse, R. Puers, C. Hertleer, L. Van Langenhove, H. van Egmond, D. Matthys, Towards the integration of textile sensors in a wireless monitoring suit, Sensors and Actuators A 114, pages 302–311, 2004
M. Ishijima, Monitoring of electrocardiograms in bed without utilizing body surface electrodes, IEEE Transactions on Biomedical Engineering, Vol. 40, No. 6, June 1993
P.C. Richardson, The insulated electrode: A pasteless electrocardiographic technique, 20th Conf. on Engineering in Medicine and Biology, p 15.7, 1967
M. Ouwerkerk, F. Pasveer and G.R. Langereis, ‘Unobtrusive sensing of psychophysiological parameters: some examples of non-invasive sensing technologies’, in J.H.D.M. Westerink, M. Ouwerkerk, T.J.M. Overbeek, W.F. Pasveer, B. de Ruyter, Probing Experience, From Assessment of User Emotions and Behaviour to Development of Products, Philips Research Book Series , Vol. 8, Springer, pages 163-193, 2008
T. Linz, L. Gourmelon, G.R. Langereis, Contactless EMG sensors embroidered onto textile, in BSN2007 - Body Sensor Networks, 4th International Workshop on Wearable and Implantable Body Sensor Networks, Aachen, Germany, 26-28 March 2007
L. Gourmelon and G.R. Langereis (2006), ‘Contactless sensors for surface electromyography’, in IEEE-EMBC ’06, 28th Annual International Conference IEEE Engineering in Medicine and Biology Society (EMBS), New York, August 30 – September 3 2006
G.R. Langereis, L. de Voogd-Claessen, A. Sipilä, C. Rotsch, A. Spaepen, T. Linz, ConText Contactless sensors for body monitoring incorporated in textiles, Portable '07, IEEE Conference on Portable Information Devices, Orlando, Florida, March 25-29, 2007
theory/sensor_technology/st4_sensor_theory.txt · Last modified: 2018/10/09 17:11 by glangereis