Sensor is a device for registering a change called a stimulus. The word “sensor” originates from the Latin word “sensus’, that means “feeling” (something). The change registered by a sensor can be represented by an event (a very rapid change), or variation of an amount, value, a number, or appearance of something, which can be a medium, or an object.

In the case of registering an event, a sensor can just serves to state that an event has occurred. Additional sensor features can give characteristics of time and location of the event(s) that have been occurred. Time based sensor characteristics can also represent the frequency with which the events have occurred. The location based sensor characteristics can represent the distribution in space (shape, physical size, volume, or distance) of several events that have been occurred. In the case of registering a variation, the sensor can react on a change in the appearance, value, level, or a number of something in the medium or an object.

The reaction of the sensor on an stimulus generated by a medium or an object is called sensor response. The sensor’s response can be in a form of a direct action, provided by the sensor’s components, or a measurable physical quantity, called a signal. The sensor’s part interacting with the medium or object of interest is called an input, while the part that generates the response (action or signal) is called an output. Sensors registering changes in the same phenomena can use completely different principals for their operation.

The most advanced and sophisticated sensor designs on Earth can be found in the nature’s living organisms, such as human body, animals, microorganisms, and plants. For many years, such sensors are being studied. “Nature, conversely, has produced extraordinary sensory systems in biological species that exceed the capabilities of a broad range of man-made sensors. Understanding the physical, chemical, and biological processes that are responsible for these sensory abilities may produce a blueprint for replicating or reconstructing them in man-made devices. Research involving the mimicry of biological systems, called biomimetics, is a branch of biotechnology that abstracts good designs from nature to enable, design and develop new man-made materials and structures. This interdisciplinary field, which engages researchers from the field of biology, materials science, engineering, chemistry and physics, provides opportunities to develop new technologies by exploiting nature’s designs and achievements. The diversity and resourcefulness of biological sensing are enormous and largely unexplored.” [1].

While this important field is being explored, our focus is some of the sensors from the humongous variety of manmade sensors that have been and continue being developed by thousands of scientists and engineers around the world. There are several classifications of sensors currently used in the human life. Such classifications are done by sensor type, phenomena used, principle of operation, application field, etc. Besides virtual sensors [2,3] used in the informatics, statistical, and mathematical sciences, all sensors are based on phenomena that occur in the nature and belong to natural sciences, such as all fields of Physics, Chemistry, and Biology. However, the basic principles of all these sensors originate from Physics and, therefore, they are called Physical Sensors. The most common types of physical sensors based on direct physical phenomena are: temperature, electrical, electronic, optical, optoelectronic, magnetic, electromagnetic, mechanical, pressure, humidity, mechanical, etc.

Sensors can also vary by size (from several centimeters down to few nanometers) design and application fields. Sensors can be designed to register and or measure a change of only one value or several values, often called effluents or parameters. In case of registering changes in several parameters the sensor is an integrated sensor (when separate single sensors are assembled with a goal to be used together without an intention ever take them apart), which also can be often called a sensor system (when the separate sensors can be taken apart, for replacement, repair, or rearrangement). The Wikipedia (free online encyclopedia) gives the following list [4] of sensor types:

  1. Acoustic, sound, vibration
  2. Automotive, transportation
  3. Chemical
  4. Electric current, electric potential, magnetic, radio
  5. Flow, fluid velocity
  6. Ionizing radiation, subatomic particles
  7. Navigation instruments
  8. Position, angle, displacement, distance, speed, acceleration
  9. Optical, light, imaging, photon
  10. Pressure
  11. Force, density, level
  12. Thermal, heat, temperature
  13. Proximity, presence

The above classification is based on the type of registered phenomenon. Some sensors are continuously affected by the registered phenomenon and just are measuring changes of its value(s). Sensors, which are registering the instance of the phenomena of interest occurrence first, are called detectors.

A number of parameters common for most sensors is used to characterize sensing abilities of a particular sensor. There is a number of good books explaining their meaning and determinations [5], but there are also a lot of useful information on the internet describing the terminology and sensor characteristics [6,7]. Given below is a useful tutorial on sensors from National Instruments:

  1. Sensitivity
  2. Range
  3. Precision
  4. Resolution
  5. Accuracy
  6. Offset
  7. Linearity
  8. Hysteresis
  9. Response Time

Dynamic Linearity

1. Sensitivity

The sensitivity of the sensor is defined as the slope of the output characteristic curve (DY/DX in Figure 1) or, more generally, the minimum input of physical parameter that will create a detectable output change. In some sensors, the sensitivity is defined as the input parameter change required to produce a standardized output change. In others, it is defined as an output voltage change for a given change in input parameter. For example, a typical blood pressure transducer may have a sensitivity rating of
10 mV/V/mm Hg; that is, there will be a 10-mV output voltage for each volt of excitation potential and each mm Hg of applied pressure.

Sensitivity Error

The sensitivity error (shown as a dotted curve in Figure 1) is a departure from the ideal slope of the characteristic curve. For example, the pressure transducer discussed above may have an actual sensitivity of 7.8 mV/V/mm Hg instead of 10 mV/V/mm Hg.

2. Range

The range of the sensor is the maximum and minimum values of applied parameter that can be measured. For example, a given pressure sensor may have a range of -400 to +400 mm Hg. Alternatively, the positive and negative ranges often are unequal. For example, a certain medical blood pressure transducer is specified to have a minimum (vacuum) limit of -50 mm Hg (Ymin in Figure 1) and a maximum (pressure) limit of +450 mm Hg (Ymax in Figure 1). This specification is common, incidentally, and is one reason doctors and nurses sometimes destroy blood pressure sensors when attempting to draw blood through an arterial line without being mindful of the position of the fluid stopcocks in the system. A small syringe can exert a tremendous vacuum on a closed system.
Ideal curve and sensitivity error

Figure 1. Ideal curve and sensitivity error. Source: J.J. Carr, Sensors and Circuits Prentice Hall.


Dynamic Range

The dynamic range is the total range of the sensor from minimum to maximum. That is, in terms of Figure 1, Rdyn = Ymax – l -Yminl.

3. Precision

The concept of precision refers to the degree of reproducibility of a measurement. In other words, if exactly the same value were measured a number of times, an ideal sensor would output exactly the same value every time. But real sensors output a range of values distributed in some manner relative to the actual correct value. For example, suppose a pressure of exactly 150 mm Hg is applied to a sensor. Even if the applied pressure never changes, the output values from the sensor will vary considerably. Some subtle problems arise in the matter of precision when the true value and the sensor’s mean value are not within a certain distance of each other
(e.g., the 1-s range of the normal distribution curve).

4. Resolution

This specification is the smallest detectable incremental change of input parameter that can be detected in the output signal. Resolution can be expressed either as a proportion of the reading (or the full-scale reading) or in absolute terms.

5. Accuracy

The accuracy of the sensor is the maximum difference that will exist between the actual value (which must be measured by a primary or good secondary standard) and the indicated value at the output of the sensor. Again, the accuracy can be expressed either as a percentage of full scale or in absolute terms.

6. Offset

The offset error of a transducer is defined as the output that will exist when it should be zero or, alternatively, the difference between the actual output value and the specified output value under some particular set of conditions. An example of the first situation in terms of Figure 1 would exist if the characteristic curve had the same sensitivity slope as the ideal but crossed the Y-axis (output) at b instead of zero. An example of the other form of offset is seen in the characteristic curve of a pH electrode shown in Figure 2. The ideal curve will exist only at one temperature (usually 25°C), while the actual curve will be between the minimum temperature and maximum temperature limits depending on the temperature of the sample and electrode.

Typical pH electrode characteristic curve showing temperature sensitivity

Figure 2. Typical pH electrode characteristic curve showing temperature sensitivity. Source: J.J. Carr, Sensors and Circuits Prentice Hall.

7. Linearity

The linearity of the transducer is an expression of the extent to which the actual measured curve of a sensor departs from the ideal curve. Figure 3 shows a somewhat exaggerated relationship between the ideal, or least squares fit, line and the actual measured or calibration line (Note in most cases, the static curve is used to determine linearity, and this may deviate somewhat from a dynamic linearity) Linearity is often specified in terms of percentage of nonlinearity, which is defined as:



Nonlinearity (%) is the percentage of nonlinearity
Din(max) is the maximum input deviation
INf.s. is the maximum, full-scale input

The static nonlinearity defined by Equation 6-1 is often subject to environmental factors, including temperature, vibration, acoustic noise level, and humidity. It is important to know under what conditions the specification is valid and departures from those conditions may not yield linear changes of linearity.

8. Hysteresis

A transducer should be capable of following the changes of the input parameter regardless of which direction the change is made; hysteresis is the measure of this property. Figure 4 shows a typical hysteresis curve. Note that it matters from which direction the change is made. Approaching a fixed input value (point B in Figure 4) from a higher value (point P) will result in a different indication than approaching the same value from a lesser value (point Q or zero). Note that input value B can be represented by F(X)1, F(X)2, or F(X)3 depending on the immediate previous value—clearly an error due to hysteresis.

Ideal versus measured curve showing linearity error

Figure 3. Ideal versus measured curve showing linearity error. Source: J J Carr, Sensors and Circuits Prentice Hall

Hysteresis curve

Figure 4. Hysteresis curve. Source: J.J. Carr, Sensors and Circuits Prentice Hall.

9. Response Time

Sensors do not change output state immediately when an input parameter change occurs. Rather, it will change to the new state over a period of time, called the response time (Tr in Figure 5). The response time can be defined as the time required for a sensor output to change from its previous state to a final settled value within a tolerance band of the correct new value. This concept is somewhat different from the notion of the time constant (T) of the system. This term can be defined in a manner similar to that for a capacitor charging through a resistance and is usually less than the response time.

The curves in Figure 5 show two types of response time. In Figure 5a the curve represents the response time following an abrupt positive going step-function change of the input parameter. The form shown in Figure 5b is a decay time (Td to distinguish from Tr, for they are not always the same) in response to a negative going step-function change of the input parameter.
Rise Time

Fall Time

Figure 5. (a) Rise-time definition; (b) fall-time definition. Source: J.J. Carr, Sensors and Circuits Prentice Hall.

10. Dynamic Linearity

The dynamic linearity of the sensor is a measure of its ability to follow rapid changes in the input parameter. Amplitude distortion characteristics, phase distortion characteristics, and response time are important in determining dynamic linearity. Given a system of low hysteresis (always desirable), the amplitude response is represented by:

F(X) = aX + bX2 + cX3
+ dX4 + ••• + K (6-2)

In Equation 6-2, the term F(X) is the output signal, while the X terms represent the input parameter and its harmonics, and K is an offset constant (if any). The harmonics become especially important when the error harmonics generated by the sensor action fall into the same frequency bands as the natural harmonics produced by the dynamic action of the input parameter. All continuous waveforms are represented by a Fourier series of a fundamental sinewave and its harmonics. In any nonsinusoidal waveform (including time-varying changes of a physical parameter). Harmonics present will be that can be affected by the action of the sensor.

Quadratic Error

Cubic Error

Figure 6. Output versus input signal curves showing (a) quadratic error; (b) cubic error. Source: J.J. Carr, Sensors and Circuits Prentice Hall.

The nature of the nonlinearity of the calibration curve (Figure 6) tell something about which harmonics are present. In Figure 6a, the calibration curve (shown as a dotted line) is asymmetrical, so only odd harmonic terms exist. Assuming a form for the ideal curve of F(x) = mx + K, Equation 6-2 becomes for the symmetrical case:

F(X) = aX + bX2 + cX4 + ••• + K (6-3)

In the other type of calibration curve (Figure 6b), the indicated values are symmetrical about the ideal mx + K curve. In this case, F(X) = -F(-X), and the form of Equation 6-2 is:

F(X) = aX + bX3 + cX5 + ••• + K (6-4)

A very important parameter in sensor operation is determination of the noise. Given below is a list of senor parameter definitions and an example of sensor noise determination:

  1. Sensor/Instrument Specification Definitions:

Range – is the maximum and minimum value range over which a sensor works well. Often sensors will work well outside this range, but require special or additional calibration. e.g. salinity sensors deployed in salinity of a few PPT in an estuary which is below where the PSU scale is defined (2 PSU).  However, generally if you try to operate a sensor outside its range, it will not work (give a constant output at the max, significantly change sensitivity or give erratic results) or be damaged e.g. a 130 m pressure sensor deployed at 200 m depth.

Accuracy – how well the sensor measures the environment in an absolute sense.  That is how good the data is when compared with a recognized standard.  e.g. a temperature sensor accurate to 0.001º C is expected to agree within 0.001º C with a temperature standard such as a triple-point-of-water cell or the temperature measured by a PRT standardized in recognized calibration standards or by another sensor with the same accuracy calibrated properly.  This is what you want to compare results with other observations.

Resolution – the ability of a sensor to see small differences in readings.  e.g. a temperature sensor may have a resolution of 0.000,01º C, but only be accurate to 0.001º C.  That is you can believe the size of relative small changes in temperature, which are smaller than the accuracy of the sensor.  Resolution in often controlled by the quantization in digitizing the signal

– e.g. one bit is equal to 0.0005º C.  This is not a function of the sensor, but the sampling process.

Repeatability – This is the ability of a sensor to repeat a measurement when put back in the same environment.  It is often directly related to accuracy, but a sensor can be inaccurate, yet be repeatable in making observations.

Drift – This is the low frequency change in a sensor with time.  It is often associated with electronic aging of components or reference standards in the sensor.  Drift generally decreases with the age of a sensor as the component parts mature.  A smoothly drifting sensor can be corrected for drift.  e.g. Sea Bird temperature sensors that are drifting about 1 mºC/yr (and have been smoothly changing for several years) allow one to correct for the drift and get more

accurate readings.  Drift is also caused by biofouling that can’t be properly corrected for, but we often try.

Hysteresis – A linear up and down input to a sensor, results in an output that lags the input e.g. you get one curve on increasing pressure and another on decreasing.  Many pressure sensors have this problem, for better ones it can be ignored. It is often seen in a CTD when the pressure reading on deck after recovery is different from the reading before it is deployed. It is not a problem with the response time of the sensor, but is an inherent property of some sensors that is undesirable.  In a CTD it also may be a temperature sensitivity problem.

Stability – is another way of stating drift.  That is, with a given input you always get the same output.  Drift, short and long term stability are really ways of expressing a sensor’s noise as a function of frequency.  Sometimes this is expressed as guaranteed accuracy over a certain time period.  Drift is often a problem with pressure sensors under high pressure.  All sensors drift with

time – hence the standardization of PRTs in triple-point-of-water and gallium melt cells.

Response time – a simple estimate of the frequency response of a sensor assuming an exponential behavior. We will discuss in more detail below.

Refer to Sea Bird temperature specification sheets:

Self heating – to measure the resistance in the thermistor to measure temperature, we need to put current through it.  Current flowing through a resistor causes dissipation of heat in the thermistor, which causes it to warm up, or self-heating.  This is especially important in temperature measurement.  If the water velocity changes, the amount of advective cooling will change, and the temperature sensed will change as a function of velocity – anemometer effect.

Settling time – the time for the sensor to reach a stable output once it is turned on. Therefore, if you are conserving power by turning off the sensors between measurements, you need to turn on the power and wait a certain time for the sensor to reach a stable output.

Voltage required – the voltage supply range over which the sensor has a stable output. Too low a voltage and the sensor doesn’t work properly, too high and something burns out.

Current drawn – what the sensor draws at each voltage. Generally this varies inversely with voltage (the sensor draws a nearly constant power – current times voltage).  Therefore, you need to supply a known voltage and current.  If you do this from a battery, beware that the voltage decreases as the battery discharges, and the current rises. Also, the amount of power you can get out of a battery changes with the amount of current drawn.  Therefore, when you

calculate power requirements, you need to know the current over the full voltage range to get power and battery requirements.  We will discuss this in more detail later in a lab.

Output – a voltage range e.g. 0 to 5 volts for an input range of 0 to 30º C, or a frequency modulated sine wave, or a square wave of frequency range 6 to 12 kHz, etc.

Fit of calibrations to equation – thermistors are log devices, so fit as an inverse polynomials of log of sensor output (frequency), which change from K to Celsius by the 273.15

Sensor Noise Estimation: – Besides digitizing noise, there is another limitation in a measurement due to the inherent noise in a sensor itself.  Today we have the technology to reduce the digitizing interval or digitizing noise to below the sensor noise, so the limiting factor in a measurement is generally the physics of the sensor itself.  Ideally, one would put the sensor in a noise free environment and measure the spectra of the sensor’s output to get the sensor

noise.  However, this would only work if the sensor’s noise was not related to signal level, and if you could find a noise free environment.

Consider the case where several sensors are measuring the same geophysical signal, s(t), and that in addition to this signal each sensor sees its own inherent noise, ni(t). Therefore, the signal that we record from sensor i is,


xi(t) = s(t) + ni(t)


The cross-covariance function then becomes

                                   Rxixj(τ) = {xi(t) xj(t-τ)}

= {[s(t) + ni(t)][s(t-τ) + njt-τ)]}

= {s(t) s(t-τ)} + {s(t) njt-τ)} +

{ni(t) s(t-τ)} + {ni(t) nj(t-τ)}


where {  } is the expected value of the quantity or the integral of the quantity in brackets over all time divided by the integral of time.  Assuming that the signal and the noise are uncorrelated,

e.g. {n(t) s(t)} = 0 and that the noise between the sensors as also uncorrelated {ni(t) ni(t)} = 0, then


Rxixj(t) = {s(t) s(t-τ)} + δij(ni(t) nj(t-τ)}


Transforming, the cross spectrum is



⌠ ∞

Cij =  ⎮Rij(t) e-ιωτ  dτ  = S(ω) + δijNij



where S is the transform of the signal and N is the transform of the noise.  For the case of only two sensors, 1 and 2,


C11(ω) = S(ω) + N11(ω)

 C12(ω) = S(ω)

C22(ω) = S(ω) + N22(ω)


The co-spectrun extracts the signal from the noise of the sensors and can be used to improve the signal to noise when it is poor with only one sensor.


If we define N as the sum of the two noises,


                                                           N(ω) = N11(ω) + N22(ω)

                                                                     = [C11(ω) – S(ω)] + [C22(ω) – S(ω)]

                                                                     = C11(ω) + C22(ω) – 2 C12(ω)


Consider the difference between the two signals,


d(t) = x1(t) – x2(t),


then the auto-covariance of this difference function is


R(τ) = {d(t) d(t-τ)}

= {[x1(t) – x2(t)] [x1(t-τ) – x2(t-τ)]}

= {x1(t) x1(t-τ)} – {x1(t) x2(t-τ)}

– {x2(t) x1(t-τ)} + {x2(t) x2(t-τ)}


Again assuming that the cross terms are zero as above, and transforming, the spectrum of the difference is


D(ω) = S(ω) + N11(ω) – S(ω) – S(ω) + S(ω) + N22(ω)

= N11(ω) + N22(ω) = N(ω)


Therefore the difference of two signals measuring the same geophysical signal and different uncorrelated noises is just the sum of the noises in the two sensors.  The noise also contains any calibration errors, etc. in our normalization of the sensor’s data.  It is obvious that the same

exercise can be carried out with three sensors, and cross spectral techniques used separate the

three individual sensor noise spectra. It should be clear that this noise spectra is the limiting quantity in any spectral measuring process i.e. we cannot get any better than our sensors.


An example of this is seen in spectra on the following page. Several temperature spectra are shown from moored temperature sensors in various locations on the west coast of the US. The record taken in the Strait of Juan De Fuca shows a spectrum which tends to level out at high

frequencies, and this level is above he digitizing noise level discussed above.  Since this mooring had dual sensors at that depth, a cross spectrum and a spectrum of the difference could be taken and the sum of the noise levels of the sensors estimated.  It was then assumed that the instrument noise of both sensors contributed equally to the results, so the noise spectrum was divided by 2 and is shown to be the cause of the spectrum leveling out.  This noise cannot be reduced without changing sensors to ones with lower noise level.  In this case we need not have sampled as often. A 30 cph (3 minute) sample would match the sampling interval and sensor noise to the environmental signal.  This reduced sample rate will have the effect of reducing the required data storage space, or allow the experiment to be lengthened.



[1] Bio-Inspired Sensors, by Paul Wolf. https://www.aps.org/about/governance/task-force/counter-terrorism/wolf.cfm

[2] Yang Li, Ioannis Pandis, and Yike Guo. Enabling Virtual Sensing as a Service. Informatics 2016, 3(2), 3. http://www.mdpi.com/2227-9709/3/2/3/htm

[3] Artashes Mkhitaryan at al. A framework for dynamic sensory substitution. Intelligent Robots and Systems (IROS 2014), 2014 IEEE/RSJ International Conference.


[4] List of sensors https://en.wikipedia.org/wiki/List_of_sensors

[5] Joseph J. Carr John M. Brown. Introduction to Biomedical Equipment Technology, Third Edition

Copyright: 1998 , ISBN: 0-13-849431-2.

[6] Sensor Terminology , http://www.ni.com/white-paper/14860/en/

[7] Jim Irish, WHOI, 10 Feb 2005 Ocean Instrumentation, Course 13.998 Lecture on Instrumentation Specifications.