ELECTRONIC MEASUREMENTS: MEASUREMENT THEORY, CIRCUITS AND SENSORS 9788740332674, 8740332675

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Geert Langereis

Electronic Measurements Measurement Theory, Circuits and Sensors



GEERT LANGEREIS

ELECTRONIC MEASUREMENTS MEASUREMENT THEORY, CIRCUITS AND SENSORS

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Electronic Measurements: Measurement Theory, Circuits and Sensors 1st edition © 2020 Geert Langereis & bookboon.com ISBN 978-87-403-3267-4 Peer review: Peter de Wit, Fontys University of Applied Sciences, Eindhoven

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ELECTRONIC MEASUREMENTS

Contents

3 Electric Currents and Potentials

28

3.1

Electric currents and potentials

28

3.2

AC and DC signals

31

3.3

Voltage meters and current meters

35

3.4 Resistance

36

3.5 Summary

41

4

Network theory

42

4.1

Kirchhoff’s laws

43

4.2 Superposition

49

4.3

Norton and Thévenin source transformations

51

4.4

The Wheatstone bridge

54

4.5

Dimensioning a volt meter and current meter

56

4.6

Time dependent circuits

59

4.7 Summary

64

5

Basic Sensor Theory

66

5.1

Background literature

66

5.2

A sensor in the measurement chain

67

5.3

Selector part and transducing part

68

5.4 Quantities

69

5.5

Self-generating vs. modulating sensors

70

5.6

Sensitivity, offset and calibration

71

5.7 Reference

71

5.8

Drift and cross-sensitivity

72

5.9

Transfer curve and non-linearity

72

5.10

Motion artefacts

75

5.11

Types of sensors

75

5.12

The smart sensor

77

5.13 Summary

78

6

Sensor-Actuator Systems

79

6.1

From measured quantities to knowledge

79

6.2

How data flows

81

6.3

Sensor/actuator network concepts

83

6.4

Multivariate analysis

84

6.5

Network topologies

85

6.6 Trends

85

6.7

Trends on material and device level

86

6.8

Trends at system topology level

87

6.9 Summary

88

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ELECTRONIC MEASUREMENTS

Contents

7 Signal Conditioning and Sensor Read-out

89

7.1

Sensor Interfaces

89

7.2

The push button

90

7.3

Resistive sensors

93

7.4

Capacitive sensors

97

7.5

Photo diode

101

7.6

Signal conversion

102

7.7 Bandwidth

106

7.8

107

Long wires

7.9 Summary

112

8

ADC and DAC

113

8.1

From analog to digital and back

113

8.2

Analog to digital conversion

115

8.3

Digital to analog conversion

120

8.4 Summary

122

9

Bus Interfaces

123

9.1

The advantages of bus interfaces and networks

124

9.2

Buses optimized for sensor networks

125

9.3

Industrial buses

125

9.4 Summary

126

10 References

127



Appendix A Circuits, Graphs, Tables, Pictures and Code

131



Appendix B: Common Mode Rejection Ratio (CMRR)

135



Appendix C: A Schmitt Trigger for sensor level detection

138

Index

141

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ELECTRONIC MEASUREMENTS

Measurement Theory

1 MEASUREMENT THEORY This chapter introduces the theory of performing measurements. You will learn: • The role of a measurement in a design or research process, and the basic terminology that is used in validation methods, • How well-defined units are used to express quantitative measurement results, • The definition of different scales to express the relation between numbers, • The measurement chain in which a phenomenon in the physical world is transformed into a number for computing purposes, • The basic quantifiers to express the quality of relation between a measured value and the real value in the physical world. After finishing this chapter, the reader is able to • Recognize a structure in the steps needed to turn a value in the physical world into a numerical value in an electronic system like a computer, and can name the steps, • Apply units to physical measured values and can understand the various scales of measurement, • Apply the terms resolution, accuracy, precision, offset and calibration correctly. A measurement is an action to verify an assumption. The outcome is evidence in combination with the appropriate interpretation. Therefore, it is essential to understand all ins and outs of a measurement. What is measured? What may have influenced the measurement? This chapter gives some background in the common knowledge behind measurements.

1.1 WHY DO WE MEASURE? The direct purpose of a measurement is to get knowledge about a system or phenomenon. Roughly speaking, a measurement can be part of two processes: • The first one is a research process where we are creating knowledge about our world. We try to understand the world, and to do this, we are building models. Measurements are needed to verify these models. This is the scientific approach.

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Measurement Theory

• The second process in which we need measurements is the product creation process. The design of a product is done based on specifications. In that case, we must know if our initial products, or prototypes, meet these specifications. In some cases, we have to check if we meet a certain standard, for example an emission standard. We need the measurements to convince the customer that the product does what we promise. So the measurements help us to guide our design process: this is the engineering approach. This is summarized in Figure 1. It is the process where measurements, simulations and models form a sequence to gain knowledge in a scientific, documented and reproducible way. In fact, with a measurement, we compare reality with our understanding or interpretation of reality. The interpretation of reality, in a simplified representation, is called a model. The measurement compares reality to a model.

Figure 1: The relation between measurements, models and a simulation

The measurement always contains an intended dataset (the responses that come from the phenomenon we are looking for), but is normally disturbed by noise. Noise is the unintended content of the data. It may disturb our conclusions if we are not aware of the noise. The outcome of the measurement, interfered with noise, is our input for interpretation. Based on the outcome we may validate the model (the scientific cycle) or validate the reality (the product design cycle). Note that also in product design, we may first need to update the model, before it is accurate enough to validate the product. Also engineering needs the scientific approach. In product engineering, there may be three conclusions:

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Measurement Theory

• The product meets the specs and can be finalized • The product does not meet the specs, so we have to change the design • The product does not meet the specs, but because of budget and time constraints, we decide to change the specifications Measuring can have two meanings: • In a first definition, we can define measuring as the verification of the structure and values of a model (so qualitative and quantitative model verification). This is mainly done in a scientific research process. • In a more narrow scope, we can define measuring as the quantitative determination of a value. This is the step needed to collect and verify data as part of both the engineering and scientific process. In measurement theory, we are more interested in the second definition of measuring.

1.2 TERMINOLOGY IN EXPERIMENTAL RESEARCH METHODS Scientists, especially social scientists, may look slightly different at a problem compared to an engineer. Although there is fundamentally no big difference between an engineering physical model and a (statistical) model for a real world problem, the paradigms of scientists and engineers are different (Bartneck and Rauterberg 2007): • Scientists try to understand and model the world without affecting it • Engineers try to change the world by making new solutions. As a result, the experimenter may look different at the experiment and use different vocabulary. In experimental research, for example as elaborated in the book by Andy Field (Field 2013), we distinguish: • Correlational research or cross-sectional research when we only observe relations without affecting it. In this case we study the natural world. This can be done in several ways, for example by (Martin and Bateson 1993) 1. Taking a snapshot of many variables at a single time (probing) or 2. Measuring variables in time (longitudinal research) • Experimental research where we manipulate a variable to see how it affects a system. Also this can be done in two ways: 1. Different groups take part in each experimental condition - between groups, between subjects or independent design or 2. A single group or person is used to try several inputs - within-subject or repeated-measures design. 9

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Measurement Theory

In experimental and correlational research, we speak about variables as the observed quantities: • A variable that is the cause of a reaction is the independent variable (or predictor variable) and • A variable that is assumed to be the reaction is the dependent variable (or outcome variable). Where the words between brackets are more appropriate for correlational research and the first words for experimental research where the input is manipulated deliberately.

1.3 QUANTITIES AND UNITS With a measurement, we always measure a quantity (Dutch: grootheid) by comparing the quantity with a unit (Dutch: eenheid): —ƒ–‹–› ൌ —„‡” ή ‹– (1)

For example, if we say the length of an object is 5m, then the length is the quantity, and the value is five times the unit “meter”. More background on quantities and some classifications can be found in chapter 5 about sensor theory. All existing quantities can be expressed in a set of basic quantities. The Bureau International des Poids et Mesures (BIPM) has defined such a set as the International System of Units (Système International d’Unités), with the international abbreviation SI Units. The seven base units are summarized in Table 1.

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ELECTRONIC MEASUREMENTS

Measurement Theory

Quantity

Symbol

Unit

Length

L

m

The meter is the length of the path travelled by light in vacuum during a time interval of 1/299 792 458 of a second

Time

T

s

The second is the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium 133 atom

Mass

M

kg

The kilogram is equal to the mass of the international prototype of the kilogram

Electric Current

I

A

The ampere is that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed 1 meter apart in vacuum, would produce between these conductors a force equal to 2×10-7 newton per meter of length

Temperature

T

K

The kelvin is the fraction 1/273.16 of the thermodynamic temperature of the triple point of water

cd

The candela is the luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency 540×1012 Hz and that has a radiant intensity in that direction of 1/683 watt per steradian

N

mole

The mole is the amount of substance of a system which contains as many elementary entities as there are atoms in 0.012 kilogram of carbon 12

Α

rad (deg)

The length of an arc of a unit circle is numerically equal to the measurement in radians of the angle that it subtends

sr

The steradian is a dimensionless unit, the quotient of the area subtended and the square of its distance from the center

Luminosity

Amount of Substance

Iv

Help units: Angle

Solid Angle

Ω

Table 1: SI Quantities and units

In electrical engineering, we are using some of the SI derived units that are defined by the BIPM as well. These are shown in Table 2.

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Quantity

Measurement Theory

Symbol

Unit

SI-Units

Current

I

A

A

Potential Difference (= Voltage)

U

V

kg⋅m2s−3A−1

Resistance

R

Ω

kg⋅m2s−3A−2

Capacitance

C

F

kg−1m−2s4A2

Frequency

f

Hz

s-1

Table 2: Derived SI quantities and units as used for electrical engineering

A consequence of using standard units is that with some phenomena we get huge or extremely small numbers. Therefore, we can scale the quantities by using a prefix for the quantity symbol. The prefixes are in Table 3. To get an idea of how big the range of numbers is, the documentary “Powers of 10” from 1977 is very illustrative (YouTube n.d.). A more modern flash-based tool is also available (The scale of the universe n.d.).

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ELECTRONIC MEASUREMENTS

Measurement Theory

p

n

Μ

m

unit

K

M

G

T

10-12

10-9

10-6

10-3

1

103

106

109

1012

Table 3: Prefixes used to scale the units

While doing calculations, it is strongly advised to use units as a second check for the correctness of the used equation. For example, we may remember the equation ଵ

• ൌ ଶ ƒ– ଶ(2)

which gives the distance s as a function of time t for a constant acceleration a, where the start time and place are both zero. When we fill in an acceleration, let’s say 9.8m/s2, and a time of 10s, we find ଵ

୫

• ൌ ଶ ͻǤͺ ୱమ ሺͳͲ•ሻଶ ൌ ͶͻͲ. (3)

Because the unit of the result is in meters (the s2 units cancel out), we have a first check that the formula is probably correct.

1.4 MEASUREMENT SCALES Measurements are done on a certain scale. This means that the result of a measurement can result into expression like “a is not b”, “a is bigger than b”, or “a is 10.3 times the unit b”. The scale is how numbers are arranged along a line. We distinguish the scales as indicated in Table 4. The different measurement scales are sometimes also referred to as the levels of measurements (Field 2013).

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Measurement Theory

Categorial levels of measurement (distinct categories) Binary scale

Is a nominal scale with only two points

For example dead or alive. SPSS calls variables for this scale Nominal

Nominal scale

Nominal variables are those whose outcomes are categorical, not meaningfully put into numbers

Red and blue marbles in statistics, male and female labels in statistics, Boolean operators. SPSS calls variables for this scale Nominal

Ordinal scale

The scale indicates whether things are equal, larger or smaller than each other. Ordinal variables are those that are naturally ordered

Sorting objects from large to small without being interested in the absolute size. Likert scales in statistics (“Strongly agree”, “Agree”, etc.). SPSS calls variables for this scale Ordinal

Continuous levels of measurement (distinct scores) Interval scale

An expansion of the ordinal scale: now the sizes of the differences are known, the absolute reference not: we cannot say that 20°C is two times as warm as 10°C

The Celsius scale. SPSS calls related variables Scalar

Ratio scale

As interval scales, but now with an absolute reference level: now we can work with ratios, like 2m is two times as long as 1m

Electric current and measurements. SPSS calls variables also Scalar

Cardinal scale

Is a ratio scale where the reference is a generally accepted standard

Ampere, meter

Derived scale

When a number relates to the ratio of standard units

Capacitance in A⋅sec/V

length related

Table 4: The levels of measurements represent several measurement scales

When doing electronic measurements of currents and voltages, we normally deal with ratio scales.

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Measurement Theory

1.5 THE MEASUREMENT CHAIN The hardware needed to measure a quantity is represented in the block schematic of Figure 2.

Figure 2: The measurement chain

First of all, the physical quantity to be measured is not one to one coupled to the sensor selector part. For example, to measure temperature, there is the packaging of the sensor and some air or liquid shield between the object with the temperature of interest and the sensor material. Another example is the glue or screw with which a strain gauge is attached to an object: these connecting materials may affect the measurement. In sound recordings, there is the influence of the room which modulates the transfer from the sound source to the microphone. These interfering structures are called the coupling network and they are between the physical quantity parameter and the sensor. The sensor-head is the transducer that converts information from a physical domain to the electrical domain. The transducer has to be read out by an electronic circuit. This circuit normally has three functions: • Biasing of the sensor element, which is the creation of a setting point. For example, a resistive element has to be biased with a current in order to convert a change in resistance to a change in voltage. This is needed because resistance as such cannot be processed electronically, but a voltage can be treated as a signal • To bring the output signal to a level that is optimized for post-processing like the analog to digital converter input stage. This is called signal leveling • After bringing the sensor output to an appropriate voltage level, we may discover some filtering is needed. For example, it is wise to remove 50 Hz noise from the signal before sampling with an analog to digital converter. We call this stage the analog signal conditioning. Once the signal is pre-conditioned, it can be sampled by an analog to digital converter (ADC) and fed to a microprocessor. The microprocessor can be in the same smart sensor housing. In the digital domain we can do some additional signal processing and the conversion to a bus protocol like USB, SPI or I2C.

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Measurement Theory

Because the intention of the measurement is to do something with the data, there must be an output. This can be a dedicated display in a data analysis system or by a feedback circuit back into the system.

1.6 MEASUREMENT DEVICE TERMINOLOGY The terminology of Table 5 is important in measurement devices.

Word Range

In Dutch Bereik

Explanation Minimum / Maximum value that can be measured

Resolution

Resolutie

Minimum difference that can be measured. This can be because of digitization, but also because there is a noise floor or physical phenomenon that makes it impossible to measure smaller quantities

Accuracy

Nauwkeurigheid

How close a measured value is to the actual (true) value. Can be expressed as percentage of the full scale

Precision

Precisie

How close the measured values are to each other. Means good reproducibility

Afwijking

Systematic difference in measured and real value, This can be confusing: in the sensor response we can speak of an offset (the y-axis zero-crossing), but when talking about measurement equipment we can also say there is an offset when there is a bias between measured value and the true value

Kalibratie

a set of operations that establish, under specified conditions, the relationship between the values of quantities indicated by a measuring instrument or measuring system and the corresponding values realized by standards

Offset

Calibration

Table 5: Measurement device terminology

Be careful: calibration is comparing with the standard, and does not include adjustment. The Dutch word ijken is calibration with respect to the law for commercial use of a tool.

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Measurement Theory

1.7 SUMMARY, AND WHAT IS NEXT? In Figure 3 the sequence of measuring a quantity is represented by three steps. These are basically the same as the measurement chain introduced in Figure 2.

Figure 3: The complete chain from a sensor via biasing to DA conversion

First of all, there is the sensor or transducer. We will see in chapter 5 about sensor theory that the sensor: • • • •

Converts a physical parameter to modulation of an electronic component May be non-linear Will have an offset that may drift, so we have to calibrate May be frequency dependent, and so has a certain bandwidth.

Next, there is a biasing circuit. A biasing circuit makes the step from the sensor to a voltage that can be sampled. This will be discussed in more detail in chapter 7 about signal conditioning and sensor read-out. We will see there are two purposes: • To make a voltage output out of the modulated electronic device (sensor) • To filter the signal to prepare it for long cables and A to D conversion Finally, there is a Analog to Digital Conversion as will be explained in chapter 8 about ADC and DAC. Analog to digital conversion must satisfy: • A good capture of the amplitude of the signal • An appropriate sampling frequency according to the Nyquist rate In fact, there is a fourth final step. The measurement information serves a certain goal. It has a communicative value for a designer or researcher. We must represent the information in an unambiguous way that underpins the conclusion of the measurement. This communicative purpose of measurements as evidence in a design process is that important that it is represented in Appendix A.

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ELECTRONIC MEASUREMENTS

Measurement Errors

2 MEASUREMENT ERRORS The learning goals for this chapter are • Introducing the different types of errors, • Explain how numbers can be used to express the value and nature of errors in a set of measurements, • Explain what happens when we do basic math (addition, subtraction, multiplication and division) on two measurements that have an uncertainty each. After studying this chapter, the reader is able to • Understand the difference between systematic, random, absolute and relative errors, • Understand and calculate accuracy and precision, • Calculate the mean and standard deviation, and understand the meaning of these two values with the measurement of multiple values, • Determine the proper rounding an representation in digits of a value given a certain error margin, and can do this after mathematical operations on erroneous values.

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ELECTRONIC MEASUREMENTS

Measurement Errors

All measurements will have errors. Either random errors or systematic errors. These errors have to be represented well in writing down the value of the quantity. We must also be aware of how errors propagate through the system.

2.1 TYPES OF ERRORS The previous chapter introduced Figure 2 as the basic measurement chain. This figure indicates two major insertion points of noise. However, that does not mean these are the only points where measurement errors occur. We distinguish errors that are caused by the system and errors due to the environment. • Systematic errors have a source within the system. For example, a calibration error of one of the measurement devices may give a bias error, which is a systematic error. Another example is the drift of a sensor resulting into an unexpected offset in the measurement. Systematic errors can be minimized by improving the measurement system. As a rule of thumb, we can say that a calibration with a ten times more accurate measurement method is needed. • Random errors are also called noise. They cannot be minimized by measuring more accurately because they have an external source: they don’t reproduce. Random errors are caused by inherently unpredictable fluctuations in the readings of a measurement tool or in the experimenter’s interpretation of the reading. Random errors are in many cases normally distributed, so the size of the error can be minimized by taking more measurements. Although most random errors have an external source, some specific random errors originate from within the system. An example is quantization noise in an analog to digital conversion, which gives a uniformly distributed noise. Another example is the noise in the electronics of the measurement tool itself. The two insertion points of noise in Figure 2 represent both environmental noise. The first source is in the measurement domain. This can for example be a motion artefact. An example of noise originating from after the transduction is electronic 50Hz noise after bad shielding. Besides the classification of errors into random and systematic errors, we can also speak about absolute and relative errors. • The absolute error is the difference between the measured value and the real value. For example, if we measure 1002Ω and we know the measured resistor is actually 989Ω, then the absolute error is 13Ω

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Measurement Errors

• In a relative error, the absolute error is normalized as (MeasuredValue– RealValue)/RealValue. For example, (1002Ω–989Ω) / 989Ω≈0.013 (1.3%) which comes without a dimension. The quantization error as mentioned before, is also observed as rounding errors when reading a value from a display. The last digits are not represented, so for example 14.3476 can be written as 14.3 while introducing an absolute error of 0.0476. Some errors are the result of transducers that are non-linear, these are nonlinearity errors. These can be expressed as a non-linearity number in percent. Non-linearity is explained in section 5.9. Errors can be reduced or compensated in some situations. This is partially explained in the chapter about Sensor/Actuator systems in section 6.3 about sensor/actuator network concepts. The most common methods are: • • • • • •

Feedback Stimulus-response measurement Differential measurement Compensation (feed-forward) Multivariate analysis Averaging.

2.2 ACCURACY AND PRECISION Consider a multimeter that has a reading of 1.000341V. This is a high precision reading, but we do not know whether it is accurate (correct). The words accuracy and precision are sometimes mixed up, but have completely different meanings. The most important mathematical tools we have are the average reading of a set of measurements and the standard deviation of the readings. The question is how they relate to accuracy and precision. Accuracy is defined as how close our average is to the “real” value. So, after defining the average (or mean) as ߤൌ

σ೙ ೔సభ ௫೔ ௡

(4)

the accuracy becomes |x0−P|, with x0 the true value.

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ELECTRONIC MEASUREMENTS

Measurement Errors

Precision indicates the variation on the measurements and can therefore be expressed in terms of the standard deviation ߪ௡ ൌ ට

మ σ೙ ೔సభሺ௫೔ ିఓሻ

௡ିଵ

.(5)

It can be understood why we use a root-mean-square for determining the precision: • Noise, tolerances and variances can result in positive and negative numbers: these would cancel out in an average, but the error is mathematically captured by adding squares • Relates to electrical power (remember that P=U˜I=U2/R, so in fact we compare powers). As shown in Figure 4, the accuracy is the proximity of measurement results to the true value (“trueness”). It relates to the systematic error which can only be reduced if we determine the offset by a method with better accuracy and compensate for it. Precision is the repeatability, or reproducibility of the measurement. It is determined by the random errors in the measurement (which can be reduced by taking more measurements) and by the resolution of measurement system.

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ELECTRONIC MEASUREMENTS

Measurement Errors

Figure 4: Accuracy and precision shown in the frequency of occurrence of measurements

In experimental research we distinguish (Field 2013): • Validity is whether an instrument actually measures what you think it is (what we would call a cross sensitivity from an engineering perspective). We distinguish • Criterion validity when you can compare it to a real objective value • Concurrent validity when data is recorded with respect to established criteria or a known dataset • Predictive validity when the data can be used to predict new values at a later stage • Content validity when the data covers the full range of the construct, so no influences are overlooked • Reliability is whether an instrument can be interpreted consistently across different situations: whether it reproduces in a test-retest reliability So validity maps to accuracy (trueness) and reliability to precision.

2.3 EXAMPLE In an example we take eight measurements of the resistance of a single resistor as shown in Table 6. What can we say about the resistance R?

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ELECTRONIC MEASUREMENTS

Measurement Errors

Measurement

Value found

1

1002Ω

2

960Ω

3

1047Ω

4

1010Ω

5

913Ω

6

986Ω

7

1037Ω

8

955Ω

Table 6: An example of eight measurements of a single resistor

First of all, the average value, or mean value is equal to (1002+...+955)÷8≈989Ω. So the best estimate for R is about 989Ω. But, how accurate is this number? Both wordings precision and accuracy determine the error (uncertainty) in the measurement.

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Measurement Errors

The standard deviation for the R in the example is ߪ௡ ൌ ට

ሺଵ଴଴ଶିଽ଼ଽሻమ ା‫ڮ‬ାሺଽହହିଽ଼ଽሻమ ଼ିଵ

ൎ ͳ͸Ǥͻȳ.(6)

This means that 95% of the measurements is between the average and plus/minus two sigma: 989Ω±2×16.9Ω.

2.4 TOLERANCE With the previous example, we took eight measurements of the same resistor. The systematic error (accuracy) is the result of the measurement tool which was the same with all eight measurements. There was also a random error (precision limitation) due to noise in the measurement. A similar experiment could be done with eight different resistors from the same batch. These should have a similar resistance, but there will be variation in the resistor values due to fabrication processes. This random variation is indicated by the tolerance. Sometimes the 2V range is used to define a tolerance. The 2V range is used instead of a single V because 95% of the measurement points fall within that range, which has become more or less standard. The tolerance is the permissible limit of variation in an object. The production process is optimized until all components are with specification (within the tolerance limits), or sometimes devices outside the specification range are discarded.

2.5 THE EFFECT OF TAKING MORE MEASUREMENTS Most errors have a normal distribution, meaning it follows the probability density curve of Gauss ˆ୬ ሺšሻ ൌ

ଵ

஢ξଶ஠

భ ౮షಔ మ ቁ మ ಚ

ି ቀ

‡

(7)

with the standard deviation V and the average P. The Gauss curve was already visible in Figure 4. By taking sufficient measurements, for example N, we can determine the shape in the Gauss curve. The location of this peak corresponds to the average P and the width of the curve to the standard deviation V. For a reasonable number of measurements (N>15), 95% of the measurements lies between P−2V and P+2V. The standard deviation V decreases with the square root of N, and so the precision increases with the square root of N. We can now see that with random errors, the precision can be increased by taking more measurements. For systematic errors, this averaging does not help, we still have the same offset in the value of P.

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ELECTRONIC MEASUREMENTS

Measurement Errors

For a systematic error of zero (P=0), we can say that the random error is equal to ±2V. When endlessly repeated measuring, the real value x0 is equal to the average P. When measured N times, the formula for the real value x0 with a probability of 95% is š଴ ൌ Ɋ േ

ଶ஢

ξ୒

.(8)

The systematic error can be approximated by |x0−P| for sufficient high N. However, because we do not know the real value x0, we have to use the independent reference (calibration) measurement that has a ten times higher accuracy.

2.6 SIGNIFICANT DIGITS The accuracy of a measured value is represented in the number of significant digits (‘meaningful digits’). So from the number of digits we can recognize the accuracy of the number. The number of significant digits is the total number of digits without noticing the comma, where a zero on the left side does not count. For example, 6.34 has three significant digits. This means that the real value lies between 6.335 and 6.345 and 0.2 has one significant digit. Note that 0.02 also has only one significant digit because the leading zeroes are not significant! • The value of 3000m lies between 2999.5m and 3000.5m. • The value of 3km lies between 2.5km and 3.5km. • When a value is measured with a certain instrument, the accuracy can be denoted explicitly, e.g. a force can be measured as 23.4N±0.3N. Once the standard deviation V of a measurement is known, we can use that for the representation of the number • Take the highest power of ten smaller than V/2: • For example, when V=0.03o V/2=0.015oaccuracy=0.01 • For example, when V=0.01o V/2=0.005oaccuracy=0.001 • Round to a multiple of this: • For example, when when V =0.03 and ym=8.314, then accuracy=0.01 and ym must be written as ym=8.31 • Last digit 5 round to even number (avoid bias) • For example: V=0.03 and ym=8.315, then accuracy=0.01 and ym=8.32 • For example: V=0.03 and ym=8.345, then accuracy=0.01 and ym=8.34 • When more than 1 decimal goes away than round in one step. • For example: V =0.3 and ym=8.345, then accuracy=0.1 and ym=8.3

25

ELECTRONIC MEASUREMENTS

Measurement Errors

In case of a calculation, do not round the intermediate results. Otherwise, you are summing up errors. Errors can be represented as relative errors (as a percentage). Take care of the exact meaning: • Absolute error: d=5.19±0.06mm is equivalent to the • Relative error with respect to the measured value: d=5.19mm±1.2% but also • Relative with respect to a full scale (for example of 200mm): d=5.19±0.03%

2.7 ERROR PROPAGATION In the measurement chain (or in our model), the reading may be the result of a mathematical operation on two input variables. For example, the length of a bar may be the sum of the first part plus a second part. Or, as another example, the output of a sensor is the product of the quantity to be measured times the sensitivity of the sensor. The question is what happens to the error of the output if both values (length 1 and length 2, or sensitivity and quantity) have noise and uncertainty. There are some basic rules to determine the error propagation under mathematical operations for the ‘worst-case’ estimation:

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ELECTRONIC MEASUREMENTS

Measurement Errors

• If two quantities are added or subtracted, the individual absolute uncertainty is added in the result • If two quantities are multiplied or divided, the percentages of uncertainty are added to get the percentage of uncertainty in the result • When finding the square root of a quantity, we divide the percentage of uncertainty by two. For squaring, the percentage uncertainty is multiplied by two. Note that when dealing with error propagation one has to handle random errors and systematic errors strictly separate. In case of a systematic error one has to take the sign into account with a difference or quotient of quantities. And, also with systematic errors, one has to subtract the errors (absolute respectively relative) with a sum or product of quantities. In case of a calculation, for example on a calculator, we normally take a simple approach for determining the number of digits: • With a product or quotient the number of significant digits of the result is equal to the smallest number of significant digits of the original numbers. • For example: R=U/I=21.3/0.2061=103.3478893740902ΩoR=103Ω • With an addition or subtraction the number of digits after the comma is equal to the smallest number of digits after the comma of the original numbers. • For example: I=2.5+0.357=2.9A This is summarized in Table 7.

Add or subtract

Multiply or divide

Error

Absolute errors add up

Relative errors add up

Number of digits

Lowest number of decimals

Lowest number of digits

Table 7: Propagation of errors in a worst-case approach

2.8 SUMMARY This chapter explained that errors can be either random or intrinsic to the system. Both types of errors have to be dealt with separately. The mathematics is introduced to express the errors in quantitative numbers using averages and standard deviations.

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ELECTRONIC MEASUREMENTS

Electric Currents and Potentials

3 ELECTRIC CURRENTS AND POTENTIALS This chapter has the overall goal to give the required theory of electricity. It is needed to understand a sensor as an electronic component and as a basis for the next chapter on electronic networks. To be more specific: • Introduce the concepts of electric current, potential difference and electric charge, • Explain the difference between alternating and direct currents, and become aware of small signal and large signal behavior, • Introduce the concept of resistance and to calculate the equivalent resistance of resistors in parallel and in series. After this chapter, the reader can • Calculate the equivalent resistance of a network of resistors, • Understand the difference between the alternating and direct current component of a signal. Most measurements are done using electronic tools. Electricity is based on electric charge and the movement of electric charge. The interaction between the availability of charge (potential) and the motion of charge (current) is determines the electronic behavior. In this chapter these basic concepts of electronics are discussed.

3.1 ELECTRIC CURRENTS AND POTENTIALS According to Coulomb’s law, an electric charge qA will encounter a force when facing another charge qB. This force is equal to ଵ

۴ ൌ ସ஠க

బ

୯ఽ ୯ా ୰మ

‫( ܚ܍‬9)

with r the distance between the two charges and ϵ0 the permittivity of vacuum being 8.85˜10−12F/m. Charges are expressed in Coulombs, where the elementary charge (the charge of a single electron) equals 1.602214˜10−19C. The unity vector er, pointing from source qA to source qB, defines force F as a vector. (Note: bold variables indicate vectors). Another interpretation of Coulomb’s law is to say that charge qA is subject to a force when it is in the electric field of charge qB. This electric field can be defined by normalizing to the charge qA: 28

ELECTRONIC MEASUREMENTS

Electric Currents and Potentials

۴

۳ ൌ ୯ (10) ఽ

Once there is a field, we can define electric potential U as ଶ

ଵଶ ൌ െ ‫׬‬ଵ ۳ ή †‫( ܛ‬11)

which is the effort it takes to move a charge along a line from 1 to 2 in steps ds. The dotvectorproduct is used to include only the component of the trajectory that is parallel to the vector E. In a homogenous field with strength E, and discarding vector directions, the potential difference U over a distance x becomes  ൌ െ ή š.(12)

A consequence of Gauss law (which is one of the Maxwell equations that are not explained in this book), is that there is no electric field inside an electric conductor. As a result, equation (12) shows there is no potential difference over an electric conductor. This means that in an electric wire (normally copper) there is no voltage drop, so we can interconnect electronic components with copper wires to supply them with supply voltages and signals.

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ELECTRONIC MEASUREMENTS

Electric Currents and Potentials

Potential differences are expressed with the unit of Volts. As a result, a potential difference is also referred to as the voltage. Charged particles, like electrons in a copper wire, will move under the influence of an electric potential difference. Electric current is defined as the change of charge as a function of time, so ൌ

ୢ୯ ୢ୲

(13)

with I the current in Amperes. The product of voltage and current has the meaning of electric power:  ൌ  ή (14)

which has the unit of Watts.

In electronic circuits there are several ways of drawing sources of voltage and current. In Figure 5 we can see: • • • • •

[a] The European symbol of an ideal voltage source [b] The American symbol of an ideal voltage source [c] The symbol of a voltage source that is an electrochemical battery [d] The European symbol of an ideal current source [e] The American symbol of an ideal current source.

There is a separate symbol for a battery to emphasize sources that are constant, like a power supply. With that, it clearly distinguishes from the basic voltage source that insinuates it is a signal input of a circuit: a voltage source that may be variable or dependent. Nevertheless, both provide a constant and ideal voltage.

Figure 5: Representations of current- and voltage sources

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Electric Currents and Potentials

3.2 AC AND DC SIGNALS Constant voltage and current sources as shown in Figure 1 are said to give Direct Current (DC): all charge is forced in a single direction. There are many time varying source signals possible, but one has special interest: a sinusoidal signal. Such signals, either current or potential, are referred to as Alternating Current (AC). The circuit symbol for a sinusoidal source is given in Figure 6. The sinusoidal signal itself can be described by the following equation: ୗ୧୬ୣ ሺ–ሻ ൌ ୓୤୤ୱୣ୲ ൅  •‹ሺɘ– ൅ ɔሻ (15)

which can be drawn as shown in Figure 7. The only four parameters describing a sinewave function are the offset UOffset, the amplitude A, the radian frequency Z, and the phase M .

Figure 6: Circuit symbol for an AC source

Figure 7: Elements in a sinusoidal signal

The relation between the characteristics in Figure 7 and equation (15) is explained by the four terms amplitude, phase, period and offset.

3.2.1 PERIOD

The period and phase describe the time behavior of the signal. The period can be expressed in the radian frequency Z of the equation as ଵ

‡”‹‘† ൌ ௙ ൌ

ଶ஠ ఠ

(16)

which also introduces f for the frequency in the unit of Hz=s−1. 31

ELECTRONIC MEASUREMENTS

Electric Currents and Potentials

3.2.2 PHASE

The signal repeats every 2/π period of the sine equation. An offset in radian frequency, results in a shift in time of the signal: Šƒ•‡ ൌ െɔ.(17)

This expression of phase is not completely correct in relation to Figure 7, where phase is expressed as a time-shift. The expression of phase as defined in equation (15) is more common: a shift in radians. We can say that a sine wave shifted by a phase of 2π becomes equal to the original wave. However, on an oscilloscope we will see a shift in time.

3.2.3 OFFSET

While the period and phase describe the time behavior of the signal, the offset and amplitude define the vertical variation in Volts (or Amperes for a current source). The offset, also bias, of the signal is ˆˆ•‡– ൌ ୓୤୤ୱୣ୲ (18)

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Electric Currents and Potentials

which is the average of the signal. Where the sine-wave has the “zero-crossings”. It is also correct to say that the offset is the DC-component while the amplitude is the AC-component.

3.2.4 AMPLITUDE

The amplitude is represented in the equation by ’Ž‹–—†‡ ൌ  (19)

which is how much the signal varies above and under the offset. In practical measurements, we measure the top-top value: the difference between the highest peak and the lowest peak. By understanding the sine function, we can see that is equal to two times the amplitude:

௣௣ ൌ ʹ.(20)

In power AC electronics, there is normally a large amplitude and zero offset. In practical electronic circuits for signal processing, we are dealing with relatively large offsets (“DC operational points”) and small signals (“AC modulations”). We will later see that we can calculate the circuit separately for the AC and DC point. This is because of the superposition principle. ሺ–ሻ ൌ ୓୤୤ୱୣ୲ ൅ —ሺ–ሻ. (21)

With this way of expressing the equation for the signal, we can clearly distinguish between DC large signal behavior (capital font) and AC small signal behavior (small font). So UOffset is the quasi-static DC operational point and u is the small signal which only has an amplitude, phase and frequency.

3.2.5 THE RMS VALUE

There is another way of expressing the signal strength besides the amplitude and the peakpeak value: the root-mean-square or RMS value. For a periodic signal U(t) the RMS value URMS is defined as ଵ

୘

ୖ୑ୗ ൌ ට୘ ‫׬‬଴  ଶ ሺ–ሻ†– (22)

with T the period. This equation describes nothing more than to take the square of all instantaneous signal levels, average them, and take the root to come back to the dimension of Volts. It is logical not to take the average, because that would reduce to the offset voltage. The RMS value has more a meaning of average power of the signal because power is also the square of potentials: P=U˜I=U2/R. 33

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Electric Currents and Potentials

When calculating the RMS voltage of a sine wave without an offset we find: ୘

ୖ୑ୗ

ͳ ଶ ൌ ඩ න൫ୗ୧୬ୣ ሺ–ሻ൯ †– 

ୖ୑ୗ

ͳ ൌ ඩ නሺ‫‹• ܣ‬ሺɘ–ሻሻଶ †– 

ୖ୑ୗ

ͳ ൌ ඩ න ଶ •‹ଶ ሺɘ–ሻ†– 

ୖ୑ୗ

ͳ ͳ ͳ ൌ ඩ න ଶ ൬ െ …‘•ሺʹɘ–ሻ൰ †–  ʹ ʹ

ୖ୑ୗ

ͳ ͳ ͳ ͳ ൌ ඩ න ଶ †– െ න ଶ …‘•ሺʹɘ–ሻ †–  ʹ  ʹ

ୖ୑ୗ

் ͳ ͳ ଶ ் ͳ ଶ ͳ ඨ ሺ ሻ ൌ ൤  –൨ െ ൤ •‹ ʹɘ– ൨  ʹ  Ͷɘ ଴ ଴

଴

୘

଴

୘

଴

୘

଴

୘

଴

୘

଴

ͳ ୖ୑ୗ ൌ ඨ ଶ െ Ͳ ʹ ଵ

ୖ୑ୗ ൌ ଶ ξʹ ήA

(23)

So: the RMS value of a sine-shaped signal which has no offset is ½√2≈0.707 times the amplitude A. The 230V value of our power supply system is an RMS value. This means that the amplitude of the power supply is actually 325V.

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ELECTRONIC MEASUREMENTS

Electric Currents and Potentials

3.3 VOLTAGE METERS AND CURRENT METERS Note that voltages (potential differences) are measured over a component and currents are measured through a wire or component. This is a direct consequence of the definition: voltage is a potential difference and current is the amount of charge conducted through a wire. How to measure is indicated in Figure 8. where a simple circuit of a voltage source connected to a light bulb is taken.

Figure 8: Measuring voltages [a] and currents [b] in a circuit

The current meter should not affect the circuit, and should therefore have a very low resistance (a concept which is introduced in a section below). A voltmeter should also not affect the circuit and must therefore have a very high resistance.

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ELECTRONIC MEASUREMENTS

Electric Currents and Potentials

3.4 RESISTANCE 3.4.1 OHM’S LAW

In Figure 9[a] a lamp is drawn which is connected to a constant voltage U: the voltage of the lamp is constant with UL=U. As a result, there is a constant current IL flowing through the lamp. For many electronic components, there appears to be a constant relationship between the voltage and current as shown in Figure 10[a]. Such components are said to be resistive or Ohmic. It is Ohm’s law that defines the relationship ୙

 ൌ ୍ (24)

more commonly written as

 ൌ ήR (25)

with R the resistance of the component. A unified resistive element is drawn in Figure 9[b]. The unit of resistance is Ohm [Ω].

Figure 9: A potential difference across a component induces a current

Figure 10: Resistance is defined from the I-U relation

36

ELECTRONIC MEASUREMENTS

Electric Currents and Potentials

3.4.2 NONLINEAR RESISTORS: THE DIFFERENTIAL RESISTANCE

In practical elements, the linear relation of equation (25) and Figure 10[a] is not observed over the full range. The [b] figure of Figure 10 shows the I−U curve of a non-linear component: current is not proportional to the potential, but to the square or power of the potential. In this case, we cannot speak of a single resistor value. However, what we can do is to define the resistance at a single point or over a small regime. In this case, we define a differential resistance r which is ”ൌ

ஔ୙ ஔ୍

௨

ൌ ௜ (26)

also indicated in Figure 10[b]. A small font is used for the same reasons as mentioned with equation (21): this is about the small variations on top of a large signal setting-point. It may be confusing that for a nonlinear line, the large signal operational point resistance R is not equal to the small signal differential resistance r. The large signal operational point resistance R represents the ratio between the voltage U needed to reach that operational point and the current I that is drawn in that point. The small signal differential resistance r represents the resistance experienced to put a small voltage on top of the operational point. We can compare this to walking up a mountain. If we are at 1000m above sea level on a mountain, but the sea is at 100km distance, we have to look down at an angle of 1% to see the sea. This is the operational point. But to make one step of 1m to walk up the mountain, it may take an increase of 10% depending on the steepness of the mountain at that operational point. This gives a much higher differential resistance. When solving electronic problems and designing circuits, we deal with the AC and DC component exactly in that way: small signal and large signal term of equation (21) may experience different resistances.

3.4.3 RESISTORS IN SERIES AND IN PARALLEL

Resistors can be placed in series or in parallel as shown in Figure 11. The question is how we can express the total equivalent resistance in the resistor values R1, R2 and R3. First, the series circuit of Figure [a] can be evaluated by ୗୣ୰୧ୣୱ ൌ ୖభ ൅ ୖమ ൅ ୖయ (27)

because voltages are potential differences between the nodes of the resistors. Next, we can replace the voltages by the Ohm’s laws of the individual resistors: ୗୣ୰୧ୣୱ ൌ ୖభ ଵ ൅ ୖమ  ଶ ൅ ୖయ  ଷ (28)

37

ELECTRONIC MEASUREMENTS

Electric Currents and Potentials

which is equal to ୗୣ୰୧ୣୱ ൌ ሺଵ ൅  ଶ ൅  ଷ ሻ ୗୣ୰୧ୣୱ (29)

because the currents through the resistors must be equal. As a result, we find

 ୗୣ୰୧ୣୱ ൌ ଵ ൅  ଶ ൅  ଷ ൌ σ௜  ୧ (30)

for the series equivalent resistance of the resistors.

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ELECTRONIC MEASUREMENTS

Electric Currents and Potentials

Figure 11: Resistors placed in series [a] and in parallel [b]

Similar, we can find for the parallel circuit of Figure [b] that ୔ୟ୰ୟ୪୪ୣ୪ ൌ ୖభ ൅ ୖమ ൅ ୖయ (31)

because currents must add-up since these are just flows of electrons which cannot be lost. Next, we can replace the currents by the Ohm’s laws of the individual resistors: ୔ୟ୰ୟ୪୪ୣ୪ ൌ

which is equal to

୙౎భ ୖభ

୙

୙౎య

ଵ

ଵ

൅ୖ ൅ మ

ଵ

ୖయ

(32)

୔ୟ୰ୟ୪୪ୣ୪ ൌ ቀୖ ൅ ୖ ൅ ୖ ቁ ୔ୟ୰ୟ୪୪ୣ୪ (33) భ

మ

య

because the voltages over the resistors must be equal. As a result, we find ଵ

ୖౌ౗౨౗ౢౢ౛ౢ

ൌ

ଵ

ୖభ

൅

ଵ

ୖమ

൅

ଵ

ୖయ

ൌ σ௜

ଵ

ୖ౟

(34)

for the parallel equivalent resistance of the resistors.

3.4.4 EXAMPLE OF RESISTORS IN SERIES AND PARALLEL

As an example, we will calculate the resistance the source of Figure 12 will “see”. When that resistance is knows, we can calculate the current that is drawn from the source.

39

ELECTRONIC MEASUREMENTS

Electric Currents and Potentials

Figure 12: An example of a source with resistors in series and in parallel

First, combine resistor R2 and R3: ͳ ͳ ͳ ൌ ൅  ଶଷ  ଶ  ଷ ୖ ୖ

ଷ଴ஐήଷ଴଴ஐ

 ଶଷ ൌ ୖ మାୖయ ൌ ଷ଴ஐାଷ଴଴ஐ ൎ ʹ͹Ǥ͵ȳ (35) మ

య

where the way of expressing the equivalent parallel resistance as the quotient of the product and the sum of the individual resonances is quite common. Next, calculate the series resistance of R1 and R23: ଵଶଷ ൌ ଵ ൅ ଵଷ ൎ ʹʹ͹Ǥ͵ȳ (36)

which will give a source current of 10V/227Ω=44.0mA.

3.4.5 THE VOLTAGE DIVIDER

The configuration of Figure 9 is commonly seen and is referred to as a Voltage divider.

Figure 13: The voltage divider

To calculate the output voltage Uout, we calculate the current through R2 first: ୖమ ൌ ୖ

୙౟౤

భ ାୖమ

(37)

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ELECTRONIC MEASUREMENTS

Electric Currents and Potentials

This current gives the output voltage Uout ୙ ୖ

୭୳୲ ൌ ୖమ  ଶ ൌ ୖ ౟౤ାୖమ (38) భ

మ

so, the overall equation of the voltage divider is: ୭୳୲ ൌ ୧୬ ୖ

ୖమ

భ ାୖమ

.(39)

This is an important relation because many configurations consist of this. The equation (39) is only true if no (or a very small) current is drawn from Uout, because we assumed IR1=IR2.

3.5 SUMMARY This chapter introduced the basic concepts of current and potential that are needed to understand signals and electronic networks. With respect to sensors and measurement technology these concepts are needed to quantitatively describe the sensor signals and how we process them. It is important to realize that an electric current flows through a material and has to me measured therefore by placing a current meter in series to the current. Electric potential is always between two points in space, and has to be measured by placing a voltmeter in parallel. The potential and current of charge are the two manifestations of electricity. The relation between them is described by Ohm’s law in resistive materials. The concept of resistors helps us to determine what happens with currents and potentials when components are placed in series or in parallel. A special set-up is where two resistors divide a potential into two lower potential differences: identifying this configuration helps us quickly assess circuits of multiple resistive elements. In signal analysis, direct currents (DC) are in most cases determining the setting point, referred to as large signal behavior. The smaller alternating currents (AC) determine the information content and are referred to as small signal behavior. These two elements which are available in a circuit at the same time, can be evaluated separately mathematically because of a principle called superposition.

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ELECTRONIC MEASUREMENTS

Network theory

4 NETWORK THEORY This chapter has the overall goal to give the required theory of electronic networks, needed to understand the electronic signal path of sensors. To be more specific, you will learn to: • Give the Kirchhoff voltage- and current-laws to calculate the potentials and currents in a circuit, • Apply the principle of superposition in electronic networks, and how it helps us to calculate potentials and currents, • Use Norton and Thévenin source transformations for understanding effective input and output resistances of circuits, • Apply the Wheatstone bride for sensor read-out, • Evaluate a capacitor as a time dependent element. After this chapter, the reader can • Solve an electronic network with resistors and sources using the Kirchhoff laws and the superposition principle: meaning he or she can calculate all potentials and currents,

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ELECTRONIC MEASUREMENTS

Network theory

• Calculate the source transformation of an active circuit using the Norton and Thévenin equivalent. An electronic network is the constellation of electronic components. It describes how passive components (like resistors, capacitors and inductors) and active components (like sources and semiconductor devices) are connected. The network theory is the mathematical toolbox to analyze such networks: meaning to derive all currents and potentials in the network.

4.1 KIRCHHOFF’S LAWS In the previous section, we have used some rules regarding voltages and currents for resistors in series and in parallel. For example, we used that two resistors in series will have the same current through them because no electrons are lost. Two resistors in parallel will have the same voltage across them. These rules are generalized in the Kirchhoff’s laws. Before introducing them, some terms and definitions have to be introduced.

4.1.1 NODES AND LOOPS

When we consider a circuit like the one of Figure 14 as a network of electronic components, the following terms can be introduced: • A component is an electronic element which has a specific current-voltage relationship between the pins • A node is the connection point of components • A branch is a non-split line between two nodes • A loop is any path we can walk from any node back to the same node.

Figure 14: Definition of nodes, branches, loops and components in an electronic network

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ELECTRONIC MEASUREMENTS

Network theory

4.1.2 KIRCHHOFF CURRENT AND VOLTAGE LAW

Now the Kirchhoff laws say: • Kirchhoffs current law: The sum of all currents flowing into a node equals zero (6Inode=0) o no charge is lost in a node • Kirchhoffs voltage law: The sum of voltages along a loop equals zero (6Uloop=0) o no energy is lost when travelling along a closed loop. We can use either law for solving the network completely: meaning, to determine all nodevoltages and branch currents. A network can be solved by three steps: • Write down all component equations: the equations determining the U - I characteristics of the components. This is Ohms law for a resistor, Ux=Constant for a voltage source, etc. • Write down the network equations: the equations that follow from either one of the Kirchhoff laws • Solve the set of linear equations. In case of Kirchhoff’s voltage law, the values to be solved are the loop currents, in case of the current law, these are the node potentials. This procedure is explained with the example of Figure 15. The question is to solve all network node-voltages and branch currents.

Figure 15: An example network to solve with Kirchhoff

4.1.3 USING KIRCHHOFF’S CURRENT LAW

With Kirchhoff’s current law we will solve the independent node-voltages first, and then we are able to solve any branch-current. The structural procedure is:

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• Give all nodes a name. Here are three remarks to make: • One node must be grounded to define it to 0V. Because voltages are defined as potential differences, we must create one absolute reference. It does not matter which one is grounded, so the most logical one can be taken. For example the one that is the lowest in the drawing or one negative pole of a source • Every fixed voltage source defines one dependent node-voltage because if one side is defined, the other side is defined as well by the voltage of the source • A current source does not add a new node because the voltage across it is zero • Give all branch currents a name and direction. The direction can be chosen arbitrary: we will find a minus sign when the real current appears to be in the opposite direction • Write down all component equations • Write down the Kirchhoff current laws for all independent nodes • Substitute the component equations in the Kirchhoff current laws and solve the set of equations. When there are N independent nodes, there will be N equations that can be solved.

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In Figure 16 this leaves two independent nodes (a and b) and five branch currents I1 to I5. The component equations are: ୖభ ൌ ୗ୭୳୰ୡୣ െ ୟ ൌ ଵ ଵ

ୖమ ൌ ୟ െ Ͳ ൌ ଶ  ଶ ୖయ ൌ ୟ െ ୠ ൌ ଷ  ଷ

ୖర ൌ ୗ୭୳୰ୡୣ െ ୠ ൌ ସ  ସ

ୖఱ ൌ ୠ െ Ͳ ൌ ହ  ହ (40)

and the Kirchhoff current equations are: Node a: ଵ െ ଶ െ ଷ ൌ Ͳ

Node b: ସ െ ହ ൅ ଷ ൌ Ͳ(41)

Figure 16: The example network annotated for Kirchhoffs current law

Substitution gives: Node a:

୙౏౥౫౨ౙ౛ ି୙౗

Node b:

୙౏౥౫౨ౙ౛ ି୙ౘ

ୖభ

െ

െ

ୖర

which can be solved by: Node a:

୙౏౥౫౨ౙ౛ ୖభ

୙

୙౗ ି଴୚ ୖమ

୙ౘ ି଴୚ ୖఱ

୙

െ

୙౗ ି୙ౘ

െ ୙

ୖయ

୙౗ ି୙ౘ ୖయ

୙

ൌͲ

ൌ Ͳ (42)

െ ୖ౗ െ ୖ౗ െ ୖ౗ ൅ ୖౘ ൌ Ͳ భ

మ

య

య

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Node b:

୙౏౥౫౨ౙ౛ ୖర

Network theory

୙

୙

୙

୙

െ ୖౘ െ ୖౘ െ ୖ౗ ൅ ୖౘ ൌ Ͳ(43) ర

ఱ

య

య

These are two linear equations with two variables Ua and Ub. By substituting all values from Figure 15 we find Ua=6.1277V and Ub=7.6596V. Once these are known, we can calculate any branch current with the component equations (40). For example, the current I3 through resistor R3 can be found using ୖయ ൌ ୟ െ ୠ ൌ ଷ  ଷ

ଷ ൌ

୙౗ ି୙ౘ ୖయ

ൌ

଺Ǥଵଶ଻଺୚ି଻Ǥ଺ହଽ଺୚ ଺୩ஐ

ൎ െͲǤʹͷͷ͵݉‫( ܣ‬44)

which has a minus sign because apparently, the current is in the opposite direction as assumed in Figure 16.

4.1.4 USING KIRCHHOFF’S VOLTAGE LAW

Now we solve the same circuit with Kirchhoff’s voltage law. With Kirchhoff’s voltage law we will solve the loop currents first, and then we are able to calculate the node-voltages. The structural procedure is: • Give all loops a name and direction. Here are two remarks to make: • Only the loops containing a single maze are needed • The loop direction can be chosen arbitrary: we will find a minus sign when the real current appears to be in the other direction • Write down all component equations • Write down the Kirchhoff voltage laws for all chosen loops • Substitute the component equations in the Kirchhoff voltage laws and solve the set of equations. In Figure 17 there are three open mazes that are defined as the independent current-loops D, E and J. We can fill-in the component equations directly in the three Kirchhoff voltage laws: Substitution gives: Loop D: ୗ୭୳୰ୡୣ ൅ ൫ ஒ െ ஑ ൯ଵ ൅ ൫ ஓ െ ஑ ൯ ଶ ൌ Ͳ

Loop E: ൫ ஑ െ ஒ ൯ଵ ൅ ൫െ ஒ ൯ ସ ൅ ൫ ஓ െ ஒ ൯ ଷ ൌ Ͳ

Loop J: ൫ ஑ െ ஓ ൯ ଶ ൅ ൫ ஒ െ ஓ ൯ ଷ ൅ ൫െ ஓ ൯ହ ൌ Ͳ (45)

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What is important here, is the sign of the currents and sources: • When stepping along a voltage source from − to + there is a positive voltage step (the potential increases) • When stepping over a passive element like a resistor, there is a voltage drop if we step in the direction of the current. For example, when stepping along loop D and passing by resistor R1, the current ID gives a voltage drop: −IDR1. Along the same loop and same resistor, current IE gives a voltage increase of +IE R1. As a result, the first term in loop D of equation (45) is equal to +(IE −ID)R1[Volt].

Figure 17: The example network annotated for Kirchhoffs voltage law

Manipulation of equation (45) gives: Loop D: ୗ୭୳୰ୡୣ െ ሺଵ ൅  ଶ ሻ ஑ ൅ ଵ ஒ ൅  ଶ ஓ ൌ Ͳ

Loop E: ଵ ஑ െ ሺଵ ൅  ଷ ൅  ସ ሻ ஒ ൅  ଷ ஓ ൌ Ͳ

Loop J:  ଶ ஑ ൅  ଷ ஒ െ ሺ ଶ ൅  ଷ ൅  ହ ሻ ஓ ൌ Ͳ (46)

which can be solved for ID, IE and I ஓ . By substituting all values from Figure 15 we find ID =8.043mA, IE =2.170mA and I ஓ =1.915mA. Once these are known, we can calculate any branch current (by adding the loop-currents involved) and any node voltage with the component equations (40). For example, the current I3 through resistor R3 can be found using I3=I ஓ −I ≈−2.553mA, which is equal to the result of Kirchhoff’s current law. To calculate a node E voltage, for example for node a of Figure 16 with respect to ground, we can calculate the voltage over R2: ୟ ൌ ୖమ ൌ ൫ ஑ െ ஓ ൯ ଶ ൎ ͸Ǥͳʹ͹͹ (47)

which is also equal to the result of Kirchhoffs current law. 48

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4.2 SUPERPOSITION The superposition theorem claims that: The voltage over, or the current through, a branch in a network is equal to the sum of all contributions of the sources (current and voltage) in the network. In other words, we can calculate the voltage on a certain node by setting all sources to zero one by one and adding the contributions. When doing this, we have to realize that: • setting a voltage source to zero means that it is replaced by a short-circuit (U=0) • setting a current source to zero means that is removed to make an open-loop (I=0). This is best explained by an example. Consider Figure 18: a network with one current source and one voltage source. If we are interested in the voltage over R2, we can used Kirchhoff (there is one independent node for Kirchhoff’s current law or two independent current loops for Kirchhoff’s voltage law). The alternative is to use the superposition principle.

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Figure 18: A network with one current source and one voltage source: we are interested in the voltage over R2

First, set all sources to zero, except for U1. This is shown in Figure 19[a]: the current source is set to zero (left out). The partial voltage UcR2 across R2 is ୖᇱ మ ൌ ଵ ୖ

ୖమ

భ ାୖమ

(48)

because this is a voltage divider.

Figure 19: Same network, but now with I1=0 [a] and U1=0 [b]

Next, set all sources to zero, except for I1. This is shown in Figure 19[b]: the voltage source is set to zero (short-circuit). The partial voltage Ucc R2 across R2 is ୖ ୖ

ୖᇱᇱమ ൌ ଵ ୖ భାୖమ (49) భ మ

where we use that the voltage over R2 equals the voltage over R2 parallel to R1.

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In total we find ୖమ ൌ ୖᇱᇱమ ൅ ୖᇱᇱమ ୖమ ൌ ଵ

ଶ ଵ  ଶ ൅ ଵ ଵ ൅  ଶ ଵ ൅  ଶ

ୖమ ൌ ሺଵ ൅ ଵ ଵ ሻ

ଶ ଵ ൅  ଶ

ୖమ ൌ ሺͳͲ ൅ ͸ ή ͵ȳሻ

଻ஐ

ଷஐା଻ஐ

ൎ ͳͻǤ͸ܸ (50)

4.3 NORTON AND THÉVENIN SOURCE TRANSFORMATIONS The theorems of Norton and Thévenin are helpful to model a complex source as two ideal components. The theorems are: • Thévenin: Any linear network of current sources, voltage sources and resistors, measured between two pins, is equivalent to a single ideal voltage source with a resistor in series. • Norton: Any linear network of current sources, voltage sources and resistors, measured between two pins, is equivalent to a single ideal current source with a resistor in parallel. The equivalent circuits are given in Figure 20. In these theorems, “equivalent” means that they behave the same with respect to voltage and current. For example, if we load the circuit of Figure 20[a] with a 1kΩ resistor, we will observe a certain voltage over the resistor and a certain current through the resistor. This voltage and current is the same of the complex network of which these are the equivalent.

Figure 20: The Thévenin equivalent circuit [a] and the Norton equivalent circuit [b]

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This is once more explained with Figure 21: the Figures [a], [b] and [c] will behave the same for any load resistance RL.

Figure 21: Thevenin equivalent circuit

The question is, how to determine the equivalent resistance and the Thévenin equivalent source (or Norton source)? There are in fact two methods. The first one is illustrated by Figure 22. First we define the short-circuit current and the open-circuit voltage. The ShortCircuit Current ISC can be determined by ୭୳୲ ሺ ୐ ൌ Ͳȳሻ ൌ Ͳ

୙

୭୳୲ ሺ ୐ ൌ Ͳȳሻ ൌ ୗେ ൌ ౐ ൌ ୒ (51) ୖ ౐

.

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and the Open-Circuit Voltage UOC is ୭୳୲ ሺ ୐ ൌ λȳሻ ൌ ୓େ ൌ ୘ ൌ ୒  ୒ ୭୳୲ ሺ ୐ ൌ λȳሻ ൌ Ͳ

(52)

Figure 22: Determining the Norton and Thévenin equivalent components from the open-circuit voltage and the short-circuit current

By combining the equations (51) and (52) for ISC and UOC we can conclude that  ୘ ൌ  ୒(53)

so the Norton and Thévenin equivalent resistors are the same. If both the Norton and Thévenin theorems are true, they must be interchangeable. So that is the case because RT=RN and UT=RTIN. In practice, the method above cannot be done: short-circuiting a circuit may damage the circuit, or may at least bring the circuit in a non-linear regime. What is normally done, is that the circuit is measured first open loop (with a high-resistive voltmeter) and next the output voltage is measured with a known load-resistance RL. In that case, the equivalent components can still be calculated. The second method is not applicable at all in practice, but can only be used to calculate the equivalent components from a known circuit: • The open-circuit voltage is equal to UT, just as with the previous method • The equivalent resistance RT or RN is equal to the resistance of the circuit when all sources are switched to zero. The Thévenin equivalent circuit as shown in Figure 20[a] has a consequence which may not be intuitive. In case the circuit is unloaded (U=∞Ω), there is no voltage drop over the resistor, and Uout=UT. This can be understood by Ohms law: when there is no current through a resistor, there is no voltage drop. 53

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4.4 THE WHEATSTONE BRIDGE Many sensors are resistors where the resistance is dependent on a certain environmental value like temperature or light. For example, a thermistor is a temperature dependent resistor. With such a sensor, the voltage divider of Figure 13 can be used to convert the resistance into a voltage. This is needed because most meters are essentially voltmeters and not resistance meters. In Figure 23 a voltage divider is prepared to convert a change in resistance into a voltage. This is done by using a constant supply voltage Us. The output voltage Uout can now be calculated with the equation for a voltage divider: ୭୳୲ ሺሻ ൌ ୱ ୖ

ୖమ

భ ሺ୘ሻାୖమ

. (54)

Figure 23: A voltage divider with a resistive sensor

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The response of equation (54) is plotted in Figure 24[a]. There are a few shortcomings of this response: • The output is not linear. However, over a short range it may be sufficient. • There is an offset-voltage on top of which there is only a small T-dependent variation. The second shortcoming may be a big issue. In some typical applications (like Pt100 temperature sensors) we see less than 1% variation. For other applications (like an LDR to detect only light and dark) there may be enough signal. Also electronic amplifiers (“differential amplifiers” and “instrumentation amplifiers”) may not perform well when the offset signal is more than a hundred times the signal of interest. In addition, the small variation of the output signal cannot be amplified because that would amplify the huge offset as well.

Figure 24: The output of a voltage divider with a resistive sensor [a] and of a Wheatstone bridge with a resistive sensor [b]

A solution is to use a Wheatstone Bridge as shown in Figure 25. This is in fact a voltage divider with the sensor, compared to a second voltage divider without a sensor. The resistor values R2, R3 and R4 are normally chosen equal, and equal to the nominal value of R1(T).

Figure 25: The Wheatstone bridge with a resistive sensor

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Now the output signal Uout(T) is shifted to a level around 0V: the offset is removed. Now we can amplify the signal without any problems. The equation describing the output voltage is ୭୳୲ ሺሻ ൌ ୱ ൬ ୭୳୲ ሺሻ ൌ െ

ଶ ͳ െ ൰ ଵ ሺሻ ൅  ଶ ʹ

୙౩ ୖభ ሺ୘ሻିୖమ ଶ ୖభ ሺ୘ሻାୖమ

.

(55)

where R3 and R4 are not visible because they are equal to each other. The sensitivity around the nominal value of the resistive sensor is డ୙౥౫౪ డୖభ

୙

ൌ െ ସୖ౩ .(56) భ

The response is shown in Figure 24[b]. We can see that the curve shape has not changed: it is still non-linear. In some applications we can replace R4 with a temperature sensor as well: in that case, the response becomes an s-shape which is quite linear around R1(T)=0.

4.5 DIMENSIONING A VOLT METER AND CURRENT METER Basic electric sensing elements are either a volt-meter or current-meter. For example, the Analog to Digital Converter (ADC) input of a microcontroller is normally a voltmeter. The more classical moving coil meter is in essence a current meter.

4.5.1 THE NOT-IDEAL VOLTAGE METER AND CURRENT METER

The ideal voltage meter and current meter of Figure 8 can be expanded to practical meters in a similar way as the Norton and Thévenin equivalents are adding resistive elements. In Figure 26 we can see that practical meters have an internal resistance Ri: • In case of a voltmeter (Figure 26[a]) this resistor should be modelled in parallel to the meter, • In case of an current meter (Figure 26[b]) this resistor should be modelled in series to the meter.

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Figure 26: The not-ideal voltage meter [a] and current meter [b]

To affect the measured circuit minimally, the Ri must be low for a current meter and high for a voltmeter.

4.5.2 MAKING CUSTOM VOLTMETERS AND AMPEREMETERS WITH AN AMPEREMETER

One of the basic building blocks is the classical moving coil meter of Figure 27. Internally, such a meter is an inductive coil in a permanent magnet. Depending on the applied electric current, the coil experiences a force in the permanent magnet. This force is counterbalanced by a mechanical spring. This system makes the position of the needle proportional to the applied current: so in essence, this meter is a current meter. Based on the used copper wire of the coil, such a meter has a constant internal resistance Ri.

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Figure 27: The classical moving coil meter

To use this meter as either a voltmeter or current meter over a custom range, we have to add a series resistor or parallel resistor respectively. In Figure 28, the current meter will become a custom-range volt meter when the appropriate series resistor Rseries is chosen.

Figure 28: Dimensioning the amperemeter as a custom voltmeter

For example, if we want to configure the current meter for a range of 0V..10V, and the meter is a 100PA meter (full range is 100PA) with an internal resistance of Ri=20Ω, we can find:  ୱୣ୰୧ୣୱ ൌ  ୱୣ୰୧ୣୱ ൌ  ୱୣ୰୧ୣୱ ൌ  ୱୣ୰୧ୣୱ ൌ

ୖ౩౛౨౟౛౩ ୖ౩౛౨౟౛౩

୘୭୲ୟ୪ െ ୖ౟ ୘୭୲ୟ୪

୘୭୲ୟ୪ െ ୖ౟  ୧ ୘୭୲ୟ୪ ୘୭୲ୟ୪ െ ୧ ୘୭୲ୟ୪ ଵ଴୚

 ୱୣ୰୧ୣୱ ൌ ଶ଴ஜ୅ െ ʹͲȳ ൎ ͷͲͲȳ(57) 58

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which is based on all full scale values for U and I. So, the final configuration has a relatively high internal resistance as should be the case for voltmeters. In Figure 29, the current meter will become a custom-range practical amperemeter when the appropriate parallel resistor is chosen. Such a parallel resistor is called a shunt resistor, so we prefer to use the symbol Rshunt for this parallel resistor.

Figure 29: Dimensioning the amperemeter as a custom amperemeter

For example, if we want to configure the current meter for a range of 0V..100mA, and the meter is a 100PA meter (full range is 100PA) with an internal resistance of Ri=20Ω, we can find:  ୱ୦୳୬୲ ൌ  ୱ୦୳୬୲ ൌ  ୱ୦୳୬୲ ൌ

ୖ౩౞౫౤౪ ୖ౩౞౫౤౪

୘୭୲ୟ୪ ୘୭୲ୟ୪ െ ୖ౟ ୖ౟  ୧ ୘୭୲ୟ୪ െ ୖ౟ ଶ଴ஜ୅ήଶ଴ஐ

 ୱ୦୳୬୲ ൌ ଵ଴଴୫୅ିଶ଴ஜ୅ ൎ Ͷȳ (58)

which has a relatively low internal resistance as should be the case for current meters.

4.6 TIME DEPENDENT CIRCUITS So far, all circuits only had resistors and sources and where therefore constant or quasistatic. When including time-dependent components like capacitors and inductors, the mathematics to evaluate them have to be expanded with differential equations. This requires new mathematical tools to solve differential equations and to transform to the frequency domain. An extensive tutorial of time-dependent components and circuits is not part of this book. The reason is that it is more complex because the mathematics are more difficult. The mathematical consequence is the need of differential equations and transform methods

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like Fourier and Laplace. The intention of this section is to give some basic insights in order to understand the complexity of this matter and to give some terminology related to time- and frequency methods.

4.6.1 THE CAPACITOR

A metal plate carrying a potential U can induce a charge Q on a second plate:  ൌ  ή (59)

where the proportional factor C is called the capacitance. Capacitance has the unit of Farad [F] with 1F=1Coulomb/Volt. From an electronic circuit perspective, it is more interesting to take the derivative with time of the charge: ൌ

ப୕ ப୲

ப୙

ൌ  ப୲ (60)

which says that the current I through a capacitor is proportional to the change of voltage U. As a result, a capacitor is a component that does not pass DC voltages (dU/dt=0). However, an AC voltage can be passed because the first derivative of a sine wave is a cosine wave. So we can see a capacitor as a component that passes fast changes in voltage, but blocks DC components. We call this high-pass behaviour. Take for example the simple RC Network of Figure 30.

Figure 30: An RC network as a low pass filter

To capture such a system in a mathematical equation, we first write down the component equations as we have seen with the Kirchhoff laws. In this case, that is Ohm’s law for the resistor and the voltage-current relation for the capacitor: େ ൌ 

ப୙ి ப୲

(61)

ୖ ൌ େ .(62) 60

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Next, we need the network equations. These follow from Kirchhoff’s current and voltage laws: େ ൌ ୖ

୭୳୲ ൌ େ ൌ ୧୬ െ ୖ (63)

which becomes

୭୳୲ ൅ 

ப୙౥౫౪ ப୲

ൌ ୧୬ (64)

After substitution of the component equations. Such an equation is called a differential equation and can be solved analytically for a step-input at t = 0 sec as ౪

୭୳୲ ሺ–ሻ ൌ ୧୬ ቀͳ െ ‡ି౎ి ቁ.(65)

The circuit of the RC response, and the transient response to a step-input as found in equation (65) are illustrated in Figure 31.

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Figure 31: Time domain evaluation of a circuit

4.6.2 COMPLEX IMPEDANCES

Laplace transforms and Fourier transforms give easier mathematics than differential equations. As an example we look at the same RC network as used in Figure 30. After introducing the complex number i all sinusoidal signals become simple linear functions. In electronics, we prefer to use the symbol j for the complex number i to avoid confusion with the smallsignal current i. After Fourier transform, the complex component equations become େ ൌ

ͳ Œɘ େ

ୖ ൌ ୖ . (66)

We could combine the component equations with the network equations (from Kirchhoff), but because all components have become simple impedances, we can also work with simple voltage dividers. Using a voltage divider, we can see that ୭୳୲ ሺŒɘሻ ൌ

୞ి ሺ୨னሻ

୞ి ሺ୨னሻା୞౎ ሺ୨னሻ

୧୬ ሺŒɘሻ(67)

which can be substituted easily with the component equations in impedance form ଵ

୭୳୲ ሺŒɘሻ ൌ ଵା୨னୖେ ୧୬ ሺŒɘሻ. (68)

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The result is an expression of the frequency transfer function of the RC circuit as a filter. It has a shape as shown in Figure 32. The frequency transfer function shows for each frequency (expressed in radians Z) the modulus of the relation between input and output, meaning how much the input signal is amplified. This explanation is far from complete, because we also have to take the phase difference into account.

Figure 32: Frequency transfer function of the RC filter

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4.7 SUMMARY When electronic components form a circuit to make relations between potentials and currents, we call this electronic networks. Notion of networks is needed to calculate potentials, currents and signals in general when sensors are connected in circuits. We have seen some techniques to assess the networks: • Kirchhoff’s voltage and current laws define network equations. These equations can be used together with the component equations to solve a network: meaning to calculate each node voltage and each branch current • The principle of superposition can be used to solve networks mathematically when multiple sources are present. This can also be done with Kirchhoff laws only, but sometimes superposition makes the work more easy or gives more insights. Superposition is also needed to address DC and AC behavior (sources) independently • All networks of passive components only, so comprising voltage and current sources and passive resistors, can be modelled as a single source and a single resistor. This is defined in the Thévenin and Norton source transformations. The modelled source must be seen as an ideal source (either voltage or current) and a single resistor that have a behavior similar to the complex circuit when loaded with a resistive load. For resistive sensors, the Wheatstone bridge can be a solution to eliminate offsets before a small signal is amplified. While the Norton and Thévenin source transformations are concepts to understand source resistances and the impact on resistive loads, this chapter also explained how to optimize the resistors in moving coil meters to create current and voltage meters in the desired range. The interaction between source resistances and meter internal resistances is important in designing measurement applications. Finally, a first preview on time-dependent circuits was given by introducing the capacitor. We have seen that the mathematical tools needed to assess such systems are different. As a consequence, the terminology and characteristics differ from DC circuits. In the electronic world, we distinguish analysis in the amplitude, time and frequency domain. Each domain has its own vocabulary, toolset, mathematical models and application areas.

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Domain

Network theory

Characteristics

Equipment

Mathematics

Amplitude

DC gain, hysteresis, RMS values, offset, Ubias

Multimeter

Kirchhoff

Time

rise time, peaks, events, period, delay

Counter, interval analyzer, oscilloscope

differential equations

Frequency

phase, gain, transfer functions, bandwidth

Oscilloscope, analyzer

Fourier, Laplace, FFT

spectrum

Table 8: Every measurement domain has its own tools, mathematics and vocabulary

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Basic Sensor Theory

5 BASIC SENSOR THEORY In this chapter we will go more into detail about the sensor. You will learn: • A sensor can be split up in the function of conversion and a subsequent step for electronic registration, • There are several classes of quantities a sensor can measure, • The concepts of sensor transduction and the associated characteristics. After this chapter, the reader is able to: • Classify the nature of the measured quantity as a vector or scaler, state or rate variable, extensive or intensive variable, dependent or independent variable, a property or variable • Pinpoint the physical phenomenon making a sensor selective and the electronic device comprising the sensor for transduction, which can be resistive, capacitive, inductive, piezoelectric or semiconductor device • Determine the offset, sensitivity and non-linearity of a sensor • Determine whether a sensor is a modulating or self-generating sensor • Understand the concepts transfer-curve, cross-sensitivity, drift, reference, hysteresis and motion artefacts. 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.

5.1 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 of the basic books about sensors are (Cobbold 1974), (Middelhoek and Audet 1989), (Fraden 2010), and (Regtien 2012). The Internet application notes can be found at the IC manufacturers, for example (Analog Devices 1999) for signal conditioning and (Analog Devices 2008) for sensors.

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5.2 A SENSOR IN THE MEASUREMENT CHAIN To come back to the terminology of section 1.5, a sensor is one of the elements in a measurement chain of Figure 2. 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-sensitivity artefacts. • The sensor-head is the transducer that converts information from a physical domain to the electrical domain. • Transduction cannot 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.

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5.3 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 33. • 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 selectivity (Middelhoek and Audet 1989). A remark about terminology 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 sensor-head.

Figure 33: 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 (Olthuis, Böhm, et al. 2000). 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.

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5.4 QUANTITIES The information a sensor probes is in fact a (physical) quantity. There are many different classifications of quantities (Regtien 2012). 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 dependent 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.

Class x Vector State variable Variable

Class y

Quantity has direction Quantity is static

Scalar Rate variable

Quantity is associated with an energetic phenomenon

Property

Dependent variable

Quantity is response or effect

Independent variable

Extensive

Size and mass included

Intensive

Quantity has no direction Quantity is dynamic Quantity is not associated with an energetic phenomenon Quantity is cause

Size and mass not included

Table 9: Classifications of quantities

These definitions of classes of quantities make an interesting link to a generalized method of modelling across the domains. In a technique called 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.

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Extensive variable ∂/∂t Extensive variable Intensive variable

State variable Rate variable = flow effort

Table 10: Relation of quantities to lumped element terminology

5.5 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 transducer (Middelhoek and Audet 1989). 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 transducers, examples are the pH sensing transistor (referred to as ISFET) and the thermistor temperature sensor.

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5.6 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 34, 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.

Figure 34: 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 34) 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.

5.7 REFERENCE The reference defines the quantitative meaning of the offset. A reference can be a reference electrode in an electrical probe, the definition by calibration of one g in an accelerometer, or the comparison with melting 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 quantity (Olthuis, Langereis and

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Bergveld, The merits of differential measuring in time and space 2001) (Olthuis, Böhm, et al. 2000). 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 bridge (Cobbold 1974). The advantage of bridge setups 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.

5.8 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-weight 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.

5.9 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 34. For example, in Figure 35 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.

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Figure 35: 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.

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If the non-linearity over a specified input range is acceptable, 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 36 it is indicated how we can define the level of non-linearity.

Figure 36: A sensor transfer curve with non-linearity

Non-linearity is defined as ‘Ž‹‡ƒ”‹–› ൌ

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୊୳୪୪ୗୡୟ୪ୣ

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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 37.

Figure 37: Hysteresis in a magnetic material

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Basic Sensor Theory

5.10 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.

5.11 TYPES OF SENSORS The separation of sensors into a selector part and a transducer part, as explained with Figure 33 in the previous subsection, makes us aware that several properties 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

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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 applications (Catrysse, et al. 2004). In other prototypes, the electrodes are placed in the fixed world, for example a car seat or a bed (Ishijima 1993). 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 (Richardson 1967). Only recently, the technology was integrated into textiles (Ouwerkerk, Pasveer and Langereis 2007) (Linz, Gourmelon and Langereis 2007) (Gourmelon and Langereis 2006) (Langereis, de Voogd-Claessen, et al. 2007).

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5.12 THE SMART SENSOR In the 1990’s, the word Smart Sensor was introduced to address the next generation of sensors. The vision at that time was that a sensor is not only a transducer, but comes together with a reference, calibration and signal processing system. In a smart sensor these are all facilitated in single hardware. Consider Figure 38 which generalizes the elements that are normally in and surrounding a smart sensor. In this case, the sensor is the boxed area which comprises a physical housing, mounting mechanism, bus interface and processing tools besides the sensor head itself.

Figure 38: A sensor with all signal processing, signal conditioning and self-calibration inside, is referred to as smart sensor

The blocks are: • Sensor head: the transducer itself which converts an environmental quantity into an changing property of an electronic device, • Read-out electronics: an analog electronic circuit to convert the changing electronic property into a signal like a quantity dependent voltage or frequency, • Pre-processing: first removal of noise, offsets, etc. in order to optimize the signal for analog to digital conversion, • AD-Conversion: a module to make the measured signal digital with a bit-width and bandwidth optimized for the specific purpose, • Post-processing: digital signal processing to retrieve the targeted quantity from the digitized signal, • Low-end network: a physical bus to put the measured signal on a wire to be transported to a main processor,

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• Meta-data: Since the smart sensor works with digital data, we can store calibration data, an identification number or other sensor specific data within the sensor, • Auto calibration: a subcircuit to perform (part of the) calibration procedure from within the sensor. Some smart sensors have only part of the generalized system. The blocks drawn after the sensor (the conversion between a high-end bus to a low-end bus) is normally seen in industrial automation applications where we see Field buses.

5.13 SUMMARY We discussed the sensor as the transducer of the measurement system that converts a quantity into an electronic or digital signal or value. For designing or selecting the best sensor, it is important to understand the transduction principle, how the sensor is calibrated and what the reference is. Categorizing sensors is a means to understand how they work and what the limits are.

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Sensor-Actuator Systems

6 SENSOR-ACTUATOR SYSTEMS While the previous chapter discussed the sensor as a transducer, this chapter discusses the concepts of a sensor in a system. The topics addressed are: • How sensors in combination with algorithms and signal processing convert physical data into meaningful knowledge, • The added value of sensors in combination with one or more actuators, • The added value of multiple sensor data, • Trends in the world of sensor constellations. With the knowledge of this chapter, the reader is able to: • Understand qualitatively the added value of sensor constellations, specifically sensoractuator systems, stimulus-response systems, multivariate analyses, redundancy, polygraphy, differential measurements, surface response methodology, • Understand how the stages in a measurement chain convert raw physical quantities into meaningful high-level data, • Position more complex sensor systems on the global trends for microsystems and learning systems. When we combine several sensors, with or without actuators, there is an advantage. More sensors can help to make the system more robust, or more sensors can be used to get more information about the subject of interest. This chapter describes Sensor-Actuator Systems and focuses on the topology to make robust implementations. The tricks and examples are mainly from the perspective of ambulant health applications and systems, but can be used in other fields as well.

6.1 FROM MEASURED QUANTITIES TO KNOWLEDGE In Figure 39 this is indicated by three steps. The quantities we can measure are normally the physical or (electro)chemical quantities. With signal conditioning these quantities are cleaned-up to represent quantities with a meaning. For example, temperature, strain, acceleration, optical transmission, pressure and skin potentials are quantities that are still in the physical world.

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Figure 39: How raw quantities can be converted into knowledge

Based on these physical quantities, we can determine physiological quantities that have a clinical meaning. For example, the skin potentials have become ECG or EMG and represent heart or muscle activity. The pulsation in the optical transmission of a bodily part can be interpreted as blood pulses (PhotoPlethysmoGraphy, PPG), and with some wavelength variation into oxygen saturation in the blood. This is still a deterministic step and needs signal processing like envelope detection or calibration towards a meaningful physiological property. The third step is normally no longer deterministic, but needs some sort of property estimation. An example is to correlate burned calories to accelerometry. To do this, we need knowledge about the use case because we cannot calculate calories from motion. If the estimation is not accurate enough for the application of interest, it may help to include a second or third property in the estimation like body temperature, heat flux etc. However, this is the most fascinating stage of signal analysis. Who could expect you can estimate subjective properties like stress and pain? The technological shift that is needed is conform the Data-Information-Knowledge-Wisdom (DIKW) triangle (Rowley 2007) expressing how raw data can be combined into a higher value resource of information. We need multiple sensor sources (polygraphy) to enable estimation of behavior, which could never be measured with a single sensor in ambulant situations. This triangle is shown in Figure 40.

Figure 40: How raw quantities can be converted into knowledge and wisdom represented in a triangle

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First, textiles can monitor a person and send the information to the outside world. This is a scenario applied to working people at risk, to monitor their condition and provide reports to a central control room. It is also an important data direction for personalized healthcare systems where a patient at risk or under test is being monitored. Our healthcare system has reached a level were the type of conditions we want to observe are occurring infrequently (epileptic seizures, early uterus contractions during pregnancy, heart failures, etc.), or are associated with behavioral and psychological aspects (stress, compulsive disorders, sleep, etc.). Therefore, in this data stream direction, the person being monitored should be unaware of the presence of sensors and electronics, in order to conduct his life or occupation as normally as possible: we don’t want to affect the life of the person using or wearing the system. In the second information flow direction, the environment is monitored and the information or response is given to a person by means of intelligent textiles. This direction of data flow opens opportunities for augmented senses: we can make people experience information content that is normally not conceivable with our human senses. In other words: data from a source that cannot be sensed because it is out of our reach (sensory-wise or location-wise) is converted into an information modality that can be sensed. Unlike the previous category currently there is not an application example available. However, it is used a lot by designers to explore the option of bringing emotions and intimacy from one person to another over larger distances. Many of us are already close to it when carrying our mobile phone in our trouser pocket. By this we can feel when the phone rings or when a message is received. The third information stream that is observed is where an individual person is monitored, and his own bodily data is offered in another modality to the same human body. Examples of such a short loop data stream is found in (bio)feedback systems. For example the watches receiving data from a textile chest belt that measures heart rate. This enables professional and leisure sportsmen to improve their training program. Heart rate is not easily sensed by our human senses. With such a feedback concept we can improve and perform better. Note that not all product scenarios will fit into the categories represented by the three data flow directions, simply because in some cases, the electronic system does not carry meaningful information - it just actuates some energy. Examples are electronic clothes or stockings for heating the human body and textiles with built in light for curing light-affected diseases or for wound healing.

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6.3 SENSOR/ACTUATOR NETWORK CONCEPTS When combining sensors and actuators in systems, the most common principle to be used is feedback. In a feedback loop the result of an actuator is measured, and the sensor reading is used to evaluate the effect of the actuator. Based on the outcome, the actuator operation is adjusted. The most common example is the thermostat in our houses controlling the temperature. Based on the desired temperature setting and the measured temperature, the heating system is switched on or off. This results into a room temperature with accuracy close to the accuracy of the temperature sensor, which would not be an option with a feedforward system without a sensor. In fact every system that should be safe and stable consists of feedback loops. The physiology in our body, like our oxygen control, our temperature, our blood pH, everything is controlled by feedback. It is therefore a good design choice to implement feedback loops in the sensor-actuator systems for protection and safety. The measurement of core body temperature by means of heat flux is an example of a feedback method (Fox and Solman 1971). The feedback loop may include the user, in which case we speak of biofeedback. An example of biofeedback is found in systems for psychological stress estimation and control for relaxation (Feijs, van Boxtel and Langereis 2010). With sensor arrays, we can do differential measurements. The advantage of a differential measurement is that common influences are invisible: the output is only determined by differences between sensors. When conceptualizing a differential measurement as a set-up where two sensors are measuring at the same time but at a different location, it is also possible to measure at the same location at two different events. In a Stimulus-Response measurement (Olthuis, Böhm, et al. 2000) (Olthuis, Langereis and Bergveld, The merits of differential measuring in time and space 2001), a sensor reading is compared to a sensor reading of the same sensor, but after an actuator has changed one single condition in the vicinity of the sensor. In that case, it can be assumed that disturbing factors changing slower than the interval between the two measurements are cancelled out. Sometimes, we can learn new parameters by monitoring the dynamic time response on the applied disturbance. The most straightforward advantage of multiple sensors with respect to safety critical systems is the redundancy. By having more statistics about a subject, the extra information results into information about the validity or can be used to detect whether one of the sensors has become unstable. However, there is a deeper advantage of multiple sensors that may lead to the estimation of new parameters. This is studied in the field of multivariate analysis and explained in the next subsection.

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6.4 MULTIVARIATE ANALYSIS While the aspects of the previous section are mainly on the sensor-head level, we also have a signal processing or mathematical layer where we can apply some multi-modal sensor concepts. Dittmar demonstrated the power of multivariate analysis with a shooting experiment using polygraphy where many physiological body parameters are measured (Dittmar 1995). Based on the combination of responses of physiological parameters, it was possible to distinguish between a successful and an unsuccessful shot. So, a parameter like ‘hit’ or ‘miss’, which cannot be measured with any single sensor on the human body, can be estimated using multivariate analysis. To make decisions based on many parameters, the mathematical toolbox is obtained from a technique called Surface Response Methodology (Myers and Montgomery 2002). With this technique, a set of strategically chosen measurements is used to explore the multi parameter space. The optimum location is subsequently chosen form the estimated model of the parameter space. This technique was successfully applied to find new parameters in complex multi-sensor systems (Olthuis, Böhm, et al. 2000).

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Fuzzy logic was developed by Lofti Zadeh in 1965, and has evolved to an alternative to the binary logic of the classical propositional logic. Fuzzy logic uses soft decisions, which means that together with the parameter of interest the validity of the parameter is reported (Zadeh 1996). In situations where we have a mix of decisions with different levels of importance, Fuzzy logic can be interesting.

6.5 NETWORK TOPOLOGIES Besides the interaction model between sensors and actuators, the topological mapping of the network should be considered. The snowboard jacket called ‘The Hub’ by O’Neill in the winter of 2004/2005 was mainly a design exercise to deal with network issues in electronic textiles. The jacket had partially integrated buttons and wires for an MP3 player, but made use of a Bluetooth connection for communicating with a mobile phone. The Nike ACG CommJacket of 2004 had a similar approach. The Levi Strauss RedWire DLX Jeans which is iPOD compatible is still on the market just like the Scottevest Revolution jacket (Scottevest n.d.). However, although these multi-device apparels are all conceptually strong for carrying entertainment products, it will be a big challenge to find the right network topology for textile safety and protective systems.

6.6 TRENDS The next steps in the developments of sensor-actuator systems for ambulant health applications should come from two sides: the technology push (materials and methods) and the concept development for applications. Electronic technology is in principle low-power and small enough to fit everywhere on the body and in our natural environment. The parties involved have only recently found each other: • Health methods are moving from the professional environment (hospital) to our living environments • Textile designers and electronic equipment manufacturers have always created competing technology: clothes should be washed and are personal, electronics was not personal until recently.

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6.7 TRENDS ON MATERIAL AND DEVICE LEVEL At the end of the 1970s and the beginning of the 1980s, the field of micro electromechanical systems (MEMS) originated from silicon technology. It was seen as the next logical step from integrated electronics towards surface-micromachining and bulk-micromachining integrated with control electronics. Already in an early phase, people managed to make micropumps in silicon for moving fluids. This initiated the trend towards microplumbing on wafer-scale by the creation of fluidic channels. Since then the term microsystem technology (MST) has become more common than solely MEMS. Terminology of microsystems can be explained based on a time line of Figure 42. It all started with the invention of the transistor in 1947 by Shockley, Bardeen, and Brattain. It took over ten years to make the first Integrated Circuit (IC) in 1958 by Jack Kilby of Texas Instruments. These inventions resulted into the product line of microelectronics. In the 1970s, people started to make mechanical structures in silicon using the technology of microelectronics. Such devices were called Micro Electro Mechanical Systems (MEMS). Soon after, the first silicon micropumps were reported and we could observe microplumbing in silicon. This has resulted into the field of Micro System Technology (MST). In 1990 we saw the next expansion to fully integrated systems including mixers, chemical sensors and onwafer separation columns. This was called Lab-on-a-Chip or µTAS, a concept introduced by Andreas Manz of Imperial College, London (Manz, Graber and Widmer 1990). Nowadays, we don’t bother so much on which physical or chemical domain is in the system, the problem is how to package it. This approach is called System in Package (SiP). A definition of a SiP would include that it is a consumer ready which means that the combination of microelectronics with mechanical or chemical interfaces is offered to the customer. SiP’s are multi-domain and cheap due to batch processing (Langereis, Microsystem technology based sensors for Ambient Intelligence 2006).

Figure 42: Development of microsystem technology over the past decades

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The result is that devices like microphones, accelerometers, biochemical reaction chambers, fluid pumps, fluid mixers, etc. have become extremely small, cheap and fast. This has its impact on our society where sensor and analysis systems are everywhere. In principle, all of our body information is available in the gases, fluids and electromagnetic waves on the surface of our body. MEMS technology can analyze this body information and use it to assist in making decisions to lower the risk of becoming ill or injured. With nanomaterials the level of integration can be even a step further and more sophisticated. So the trend of integration and miniaturization does not stop at the micron level. The integration level will be higher and higher until technology is completely integrated in our body and daily environment.

6.8 TRENDS AT SYSTEM TOPOLOGY LEVEL The shift towards wireless technologies makes it possible to connect easily to other objects. This cloud concept will transfer information from human to human, but also from any human to any physical object. The ‘Internet of Things’ is coming (Kranenburg 2008), and the impact of having all Internet data available at any low-profile node will change our life. What once started with tags in anything will expand to a world where the position and state of any object is known. It will not only be known for a scanner close to the object, but also to any other object wherever in the world. Data is not anymore stored at localized computers but data will be decentralized and distributed. Just like we could estimate new immeasurable parameters from polygraphic multi-sensor networks, the global internet of objects and data will give opportunities we cannot yet even imagine. Especially for protection and safety this must make a big difference. Until now, epidemiological data on diseases could only be evaluated on demand, after people got ill and after the disease was identified. With a cloud of objects and information about many people, we can not only identify diseases in an earlier phase, we can also immediately correlate the patterns to all other events in our society. The current trend towards monitoring mental conditions will expand in future to concepts where mental conditions of multiple people can be mapped. This will be used for risk analysis because stress correlates immediately to accidents, and so, accident prediction becomes feasible.

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6.9 SUMMARY In Chapter 5 we have seen that sensors have intrinsic difficulties with their reference and calibration because the sensed value comes from a different physical domain than the electrical target domain. In this chapter we have seen that the uncertainties of the sensor can be addressed by retrieving knowledge from other sources: for example other sensors or well defined steps from a known actuator. These sensor-actuator principles are clever solutions that can only be designed specifically for a certain context. As a result, there is no generic design method and we can only understand the concepts and discover whether one of them gives a feasible solution in the given application.

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7 SIGNAL CONDITIONING AND SENSOR READ-OUT The learning goals of this chapter relate to the signal conditioning element in the measurement chain. To be more specific: the electronic modules that bring a sensor in the right bias, and convert the sensor variation into a voltage for further processing. The topics to be addressed are: • • • •

Understanding how the type of sensor relates to the type of read-out that is needed, Several sensor read-out (biasing) circuits for contact, resistive, capacitive sensors, The conversion and filtering of a signal with simple operational amplifier circuits, Interfacing for wiring.

After this chapter, the reader is able to: • Select the elementary interface circuit for read-out of a sensor-head, • Find basic circuits for primary filtering and signal conversion, • Select an appropriate set-up for long wires using active shielding, balanced drivers, twisted pairs and/or 2-wire/4-wire configurations. A sensor can be described as a selector element and a modulated electronic device. This means that, from the perspective of the recording electronics, the sensor looks either as an electronic component, or a current/voltage source. For example, many sensors are modulated resistors: the Light dependent Resistor (LDR), the Pt100 temperature sensor, a magneto-resistor (MR sensor) and the strain gauge. When we want to connect such a sensor to electronics, we have to convert the value of the resistor to a voltage. Only then we can sample with an Analog to Digital Converter (ADC) or send the value as a signal over a wire. The conversion of the sensor value to a voltage is the first step of signal conditioning, also referred to as “biasing”. After biasing, we normally need extra steps to prepare the signal for sampling or transmission. This can be amplification or filtering. These steps are called analog signal processing. This chapter gives some examples of biasing and signal conditioning as needed with the most common sensors.

7.1 SENSOR INTERFACES An interesting view on sensor interfaces is given in the book of Fraden (Fraden 2010). In Figure 43, sensors are classified based on their location.

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Figure 43: Depending on the location and function, a sensor has a specific interface (Fraden 2010)

• A non-contact sensor, for example a magnetic sensor or a capacitive probe, has an inherent problem with the reference voltage since there is no DC path • A contact sensor that can be mounted to the subject under test, is more invasive but has less issues with defining a reference. A problem can be the physical coupling network of the sensor by which the measured quantity is affected • A contact sensor that is relative can in some cases eliminate a problem, because we are only interested in variations of a parameter, and not the absolute value • An active sensor is a sensor that needs an excitation signal. So this is part of a stimulus response measurement. It can be a measurement system where a frequency sweep is applied to find cut-off frequencies • An internal sensor can be a temperature sensor that monitors the temperature of the interface chip.

7.2 THE PUSH BUTTON A push-button, and many mechanical contacts, can only define a short-cut. Such push buttons are used as an input for a microcontroller. The mechanical action consists of an open/close contact. Something is needed to turn the open/close states into two well defined voltage levels. Therefore, a resistor is needed to pull the input pin to a second well defined level besides the closed state. The configuration using a pull-up or pull-down resistor is indicated in Figure 44. The pull-up configuration is preferred because in that case one side of the switch is connected to ground which is better when placing multiple controls in a front panel.

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Figure 44: Read-out of a push button using a pull-up resistor (left), or a pull-down resistor (middle). Such mechanical switches may have the phenomenon of contact bounce (right)

Because the use of a pull-up resistor is such a common configuration, most microcontrollers have internal pull-ups that can be initialized in software. Switches are mechanical components where two metal contacts are used to close an electrical connection. Because of the metal to metal contact, there may be some noise for a few milliseconds when closing the switch. This is called contact bounce and is indicated in the right-hand picture in Figure 44. When using the switch or another mechanical contact as an input (for example for a counter or a time critical time interval analyzer), this contact bounce may result into mis-counts.

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7.2.1 ELIMINATION OF CONTACT BOUNCE

Contact bounce can be removed in hardware or software/firmware: • With firmware we can confirm the closure of a switch some milliseconds after a first closure is detected. The firmware can track a state variable and count a noisy transition as a single state change from off to on • It can be removed by adding an RC-circuit that removed the high frequent bouncing When implementing the debouncing in hardware, the circuit should be a low-pass filter, for example a simple RC network. The low pass filter cut-off can be calculated as fcut−off=(2πRC)−1. Frequencies above will pass. To decide on the frequency, it is easier to think in terms of the time constant which must be in the order of 30ms: ɒ ൌ  ή  ൎ ͵Ͳ•(70)

If the button is connected to the input of a microcontroller, we may find in the datasheet of that controller that a logical “low” is defined as Vlo0.6Vcc. When pushing the button, the action is detected as a low to high transition when U(t) equals 0.6Vcc. With the exponential response of an RC circuit, we can now calculate W ͳെ

୲౞౟ౝ౞ ି த ‡

ൌ ͲǤ͸

୲౞౟ౝ౞

ɒ ൌ െ ୪୬ሺ଴Ǥସሻ ൎ ͳǤͲͻ– ୦୧୥୦ (71)

For W=30ms and taking a capacitor of 1PF, this means a resistor of 30kΩ.

Some circuits to implement the RC contact bounce are given in Figure 45.

Figure 45: Implementing a low pass RC filter for contact bounce in a pull-up (left) and pulldown (middle) configuration

The contact bounce on a time scale with the RC response is indicated in the right-hand picture. 92

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7.3 RESISTIVE SENSORS A strip or bar of resistive material can be used as a resistive sensor. The configuration of Figure 46 has a resistance R [Ω] of ୪

 ൌ ɏ ୅ (72)

with ࣁ the resistivity of the material [Ω/m], l the length of the bar and A the cross-sectional surface area.

Figure 46: A resistor as a resistive geometry

Any change in ࣁ, l or A will result in a variation of the resistance. For example, light sensors or temperature sensors make use of a variation in the resistivity ࣁ, while a strain gauge makes use of variations in the length l and the surface A. Typical resistive sensors we know are Light Dependent Resistors (LDR), strain gauges, the Pt100 resistive temperature sensor and Magnetoresistive sensors. Also the semiconductor temperature sensors with a positive or negative temperature coefficient (PTC’s and NTC’s respectively) are resistive. To convert information that is captured in the resistor value into a voltage, the basic method is to place it in a voltage divider.

7.3.1 A VOLTAGE DIVIDER FOR SIMPLE RESISTIVE SENSOR READOUT

A voltage divider is a basic electronic structure as shown in the left-hand picture in Figure  4. The idea is that the output voltage relates to the input voltage in the same ratio as R1 relates to R1+R2. So: ୭୳୲ ൌ ୧୬ ୖ

ୖభ

భ ାୖమ

.(73)

When resistor R2 (the upper one) is a resistive sensor as shown in the middle part of Figure 47, then the output voltage will be a function of the resistance value. The same will happen when the lower resistance (R1) is the sensor.

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Figure 47: Resistive sensor readout in a voltage divider

The question is what the sensitivity of the output voltage is for changes in the resistive sensor value. We take the situation where the sensor is in the top part, so R2 is the sensor like the middle situation of Figure 47. For simplicity, we write R1=R0 and R2=Rsens=R0+ΔR indicating that R1 is chosen equal to the nominal value of the sensor. The sensor has a certain nominal value R0 at zero input (or a resting state) and an extra value ΔR due to an input variable being sensed. ΔR can be both positive and negative and is zero at rest. For completeness, this alternative convention is drawn in Figure 48.

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Figure 48: Resistive sensor readout in a voltage divider with different definition of the resistor values

This way of writing in terms of ΔR results into an alternative form of the voltage divider equation: ୗ ൌ ୚ ଶୖ

ୖబ

.(74)

బ ା୼ୖ

By series expansion of (74) we can find the sensitivity in the linearized region (ΔR‫ا‬R0): ୼୙౏ ୼ୖ

ൎെ

୙౒

ସୖబ

.(75)

The sensitivity is equal for the situation where the resistor is in the lower part of the voltage divider, only the sign will change. The advantage of a voltage divider for sensor read-out is that it is simple and passive. However, there is a disadvantage. The nominal output voltage is half the power supply because the sensor value is roughly equal to R0. On the other hand, the variation ΔR of the sensor due to the sensed quantity, is much smaller for most sensors. For example for a strain gauge ΔR is a hundred times smaller than R0. This means that we have a huge offset and only some mV’s of signal. With such a large offset, we cannot amplify the signal after the voltage divider, because the output voltage would clip to the power supply. This issue can be solved by using a resistive bridge setup.

7.3.2 THE WHEATSTONE BRIDGE

When the output of the voltage divider is not measured with respect to ground, but with respect to a second voltage divider, we have created a resistive bridge. Such a bridge is known as a Wheatstone bridge and a single sensor version is represented in Figure 49. The single sensor version is referred to as quarter bridge.

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Figure 49: Single-sensor Wheatstone bridge

One of the advantages is that we can sense the unbalance in the bridge by a differential amplifier which has to deal with a signal around 0V (for ΔR=0Ω, the output is 0V). The only condition is that the measuring amplifier should have a sufficiently high Common Mode Rejection Ratio, because the offset can be in the range of Volts, while the signal is millivolts. For ΔR‫ا‬R0 we find that the response around zero is ୼ୖ

୕୳ୟ୰୲ୣ୰୆୰୧ୢ୥ୣ ൎ െ

ସୖబ

୼୙్౫౗౨౪౛౨ా౨౟ౚౝ౛

୙

୚ (76)

resulting in a sensitivity equal to the sensitivity of the voltage divider with a single sensor: ୼ୖ

ൌ െ ସୖ౒ .(77) బ

7.3.3 HALF BRIDGE AND FULL BRIDGE

In many cases, we have the opportunity to implement a second sensor, where the second sensor has an opposite response. So if sensor one has the response R=R0+ΔR, the second one has the response R=R0−ΔR. In this way, we can create a differential measurement (see chapter 6 on Sensor-Actuator Systems). In this differential measurement, the intended response on the quantity of interest is ΔR and has opposite sign for the two sensors. Some interfering effects will be in the R0 offset value, and have the same sign. This means that if we measure the difference between the two sensor readings, the common effects cancel out, while the differential intended effect results into a strong response. In a Wheatstone bridge this can be easily implemented with two or even four sensors (see Figure 50).

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Figure 50: Half bridge (left) and full bridge (right) set-up

Sensors that are especially suitable for half-bridges and full-bridges are strain gauges because we can freely position the resistive structures in mechanical structures. An example is given in where four strain gauges are positioned such that two have a positive sign, while the other two have a negative sign.

Figure 51: Four strain gauges placed on a mechanical structure in such a way that they form a full resistive bridge

The sensitivities become: ୼୙ౄ౗ౢ౜ా౨౟ౚౝ౛

and

୼ୖ

୼୙ూ౫ౢౢా౨౟ౚౝ౛ ୼ୖ

୙

ൌ െ ଶୖ౒ (78) ൌെ

୙౒ ୖబ

బ

. (79)

7.4 CAPACITIVE SENSORS The parallel plate capacitor structure of Figure 52 has a capacitance of ୅

 ൌ Ԗ଴ Ԗ୰ ୢ .(80)

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when ignoring fringing fields on the edges. Here, A is the surface area of a plate, d the gap distance, ࣅ 0 the permittivity of vacuum and ࣅ r the specific permittivity of the material in the gap (ࣅ r=1 for air).

Figure 52: Parallel plate capacitor

In principle, any variation in A, d or ࣅ r can be detected as a change in capacitance. • A variation of the overlapping plate area A can be used to make a translation sensor • Variation of the air gap distance d is used in a microphone and pressure sensor • Variation of the relative permittivity ࣅ r can be the basis of a sensor that can detect a dielectric material that is placed in the field lines of the parallel plate configuration. A humidity sensor is based on measuring variation in ࣅ r.

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So based on translation, pressure or material variation the capacitor value will be modulated. There are several read-out methods.

7.4.1 CONSTANT VOLTAGE READ-OUT

The relation between electrical capacitance C [F], charge Q [C] and voltage U [V] is defined as  ൌ  ή .(81)

The variation in capacitance is equal to ப୕ ப୲

൅

பେ

ൌ ൌ ୠ୧ୟୱ

பେ

ൌ

ப୙ ப୲

ப୲

(82)

where it can be noted that GQ/Gt is equal to the electric current I. When biasing the capacitor with a fixed voltage Ubias, for example by connecting through a high resistor, the output signal becomes ப୕ ப୲

ப୲

.(83)

which means that the measured current is defined by the variation in capacitance. This is more or less the method with which condenser microphones are biased. Electret microphones are also capacitive microphones, but there the biasing voltage is generated by a permanent charge element: an electret.

7.4.2 USING AN OSCILLATOR

For many other capacitive sensing principles (like used in accelerometers and some pressure sensors), circuits based on oscillators and timers are used. They have a frequency modulated output. The easiest way for readout is to place them in a resonating circuit where the capacitor is one of the frequency determining elements. Two examples are in Figure 53. The resulting frequency is equal to: ଵ

ˆ୰ୣୱ ൌ ଶ஠ୖେ. (84)

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The 1/C relation is not always a problem. Equation (80) indicated that the capacitance has a 1/d relation with the gap size d. This means that if the modulated value is d (like in pressure sensors), the frequency becomes proportional to d.

Figure 53: Oscillator methods to read out a capacitive sensor

7.4.3 OTHER METHODS

There are many other methods that convert a capacitor value to a change in a periodic signal. The result can be amplitude modulation, frequency modulation, or phase shift. In these cases, the response time is limited by the period of the carrier signal.

Figure 54: Image from the NXP-Freescale white paper on capacitive read-out

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7.5 PHOTO DIODE A simple circuit for determining the current generated by a photo diode can be found in Figure 55. A photo diode has a reverse current that is modulated by the amount of light. The circuit places the diode in a reverse voltage and converts the current to a voltage.

Figure 55: Circuit for determining the reverse current of a photo diode

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7.6 SIGNAL CONVERSION The last step in conditioning the sensor signal is converting the signal to a form that is suitable for the AD convertor. This means: • amplification of the voltage to the window of the AD converter • cutting down to the relevant bandwidth to prevent aliasing • additional filtering to remove noise and removing DC if needed.

7.6.1 AMPLIFIER

To amplify a signal, we need amplifiers with high quality, because they should not influence the linearity, noise, and offset of the sensor signal. An Operational Amplifier (OpAmp) can be configured as a differential amplifier (but take care of the Common Mode Rejection Ratio), but there are also dedicated instrumentation amplifiers like the INA114.

7.6.2 INPUT BUFFER

Based on a single OpAmp like the PA741 we can make a unity gain amplifier. Such an amplifier is a good impedance converter: it can track a high impedance sensor signal and convert it into a low impedance voltage suitable for transmission over linger wires. The function is: ୭୳୲ ൌ ୧୬ .(85)

Figure 56: Unity gain amplifier with a single OpAmp

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7.6.3 COMPARATOR AND SCHMITT TRIGGER

To make a digital signal out of an analog input (sensor) in a case where we are only interested in the crossing of a certain threshold, we can use a comparator circuit. ͷ ୭୳୲ ൌ ൜ Ͳ

ˆ‘” ˆ‘”

୧୬ ൐ ୰ୣ୤ (86) ୧୬ ൏ ୰ୣ୤

Figure 57: Comparator circuit

For a signal with some noise, the plain comparator circuit may result into some transition noise in the output signal which is similar to contact bounce. In Figure 58 we can see the comparator output for a clean signal. What happens with a noisy signal can be seen in the right-hand figure.

Figure 58: Output of a comparator when the input is clean (left-hand) and output of a comparator when the input is noisy (right-hand)

This response is quite logical because on a mV scale the signal does cross the decision level multiple times before it becomes “high” in the end. Just like contact bounce, a solution is to use a low-pass filter or software way of transition detection. However, a more appropriate solution for this problem is to use a higher decision level for a low-to-high transition and a lower decision level for a high-to-low transition. The comparator that does this is called a Schmitt Trigger. A Schmitt trigger is a comparator circuit that incorporates positive feedback.

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7.6.4 FILTERS

The simplest low pass filter is based on the RC network of Figure 59. The response is a first order low-pass behavior, so ൜

୭୳୲ ൌ ୧୬ ୭୳୲ ‫ ا‬୧୬

ˆୡ୳୲ି୭୤୤ ൌ

ଵ

ˆ‘” ˆ ‫ˆ ا‬ୡ୳୲ି୭୤୤ ˆ‘” ˆ ‫ˆ ب‬ୡ୳୲ି୭୤୤

ଶ஠ୖେ

(87)

Which is represented in Figure 60.

Figure 59: First order RC low-pass filter

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Figure 60: Response of a first order low-pass filter

The first order low-pass filter can be integrated with an amplifier as the inverting low-pass first order active filter of Figure 60. The response is ‰ƒ‹ ൌ െ

ଶ ଵ

ଵ

ˆୡ୳୲ି୭୤୤ ൌ ଶ஠ୖ

మ େభ

(88)

while the DC content is removed (additional high pass filter) by the R1Cin product, which must be higher than R2C1.

Figure 61: Active inverting first order low-pass filter

For more OpAmp circuits, see several internet sites from chip manufacturers.

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Similar to the low-pass filter, we can implement an RC high pass filter as shown in Figure 62. The response is ൜

୭୳୲ ൌ ୧୬ ୭୳୲ ‫ ا‬୧୬

ˆୡ୳୲ି୭୤୤ ൌ

ଵ

ˆ‘” ˆ ‫ˆ ا‬ୡ୳୲ି୭୤୤ ˆ‘” ˆ ‫ˆ ب‬ୡ୳୲ି୭୤୤

ଶ஠ୖେ

(89)

and is shown in Figure 63.

Figure 62: First order RC high-pass filter

Figure 63: Response of a first order high-pass filter

7.7 BANDWIDTH The bandwidth of the total sensor system determines the fastest signal that can be measured. The bandwidth is determined by the slowest stage in the chain and depends on • • • •

Coupling network Transducer properties Amplifier quality Wiring (sometimes long)

Based on the properties of the sensor and the read-out signal, the output has a DC content or not.

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7.8 LONG WIRES Sensors are normally placed at remote locations. As a result, the long wires can pick up noise due to emf and signals become affected due to cable resistances and capacitances. High-ohmic sensors (especially capacitive) are the most susceptible to pick up noise. To minimize noise we can • • • •

Transform to digital (CAN, SPI) as close to the sensor as possible Do an impedance conversion Use a 3- or 4-wire technique Use twisted pair, bi-phase cables

Figure 64: Sensors are normally placed at remote locations: in a car there are many lines from the central controllers to sensors to noisy locations

7.8.1 IMPEDANCE CONVERSION AND BOOTSTRAPPING

The output resistance of a unity gain amplifier is low, while the input resistance is high. So this is an excellent method to connect a sensor with a high impedance to a low impedance input or to connect to long wires. The circuit was already given in Figure 56. Bootstrapping is the technique to do active shielding: cable capacitance becomes ineffective (Figure 65). The cable capacitance is still there, but because the shield has the same potential as the signal carrying core, it has no influence. The capacitive coupling is removed by a strong amplifier delivering the current.

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Figure 65: A unity gain amplifier can be used to do active shielding. By this method called Bootstrapping, the effect of the cable capacitance removed

7.8.2 3 OR 4-WIRE CONNECTION

Measuring a resistor value can be done by applying a current, while measuring voltage. The ratio gives the resistor value. However, if the measured voltage is not the voltage over the resistor (or resistive sensor element), there will be an error in the perceived resistance. A possible cause of such an error is the voltage drop over the connecting wires.

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In Figure 66 this situation is indicated. The current source I drives a current through the resistor. Although the real current through the resistor is equal to the imposed current, the sensed voltage is probably not equal to the real voltage over the sensor. The reason is that the drive current results into a voltage drop over the wires. If the wires have a lead (wire) resistance RL we find: ୱୣ୬ୱୣୢ ൌ ୖ ൅ ʹ୐ ൌ  ൅ ʹ ୐ (90)

and so the estimated sensor resistance is  ୣୱ୲୧୫ୟ୲ୣୢ ൌ

୍ୖାଶ୍ୖై ୍

ൌ  ൅ ʹ ୐. (91)

Figure 66: A method to determine the resistance of a resistive sensor. An electric current is applied and the voltage is measured

The result that the measured resistance is equal to the resistance of interest plus the cable resistance is obvious, but highly undesired because it reduces the sensitivity of the overall system. The solution, however, is relatively simple: we should avoid that the imposed current affects the voltage drop over the cables. This can be done with the four wire solution of Figure 67. The current is driven through the outer wires, but the voltage sensing is done with the inner wires. When using a high impedance voltage measurement device (which they normally are), there will be only a few PA of current in the voltage sensing wires, and therefore hardly any voltage drop. This means that the sensed voltage is almost equal to the voltage over the sensor. There is a voltage drop over the current carrying wires, but that is irrelevant for the measurement.

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Figure 67: An improved method to determine the resistance of a resistive sensor. The wires for driving a current are separated from the wires for sensing the voltage over the sensor

The method of using 4-wires, is fundamentally equal to bootstrapping. With bootstrapping, the cable capacitance is physically still there, but the impact on the measurement is eliminated because there is no voltage drop over the capacitor. With the 4-wire measurement, the cable resistances are physically still in place, but they do not affect the measurement because there is no current through them.

7.8.3 TWISTED PAIR/BALANCED OUT

When we realize what is in fact the noise picked up by a long wire, then it must be electro motoric forces (emf ) due to induction from fluctuating magnetic fields. These are common for all wires. A solution to remove common noise that is generated in the cables is to put the signal on the cables as a differential signal as shown in Figure 68. A sensor where the output is driven as a bipolar signal is said to have a balanced out.

Figure 68: A balanced out driving method of wires, reduces common noise in the cables

Another method is to create alternating loops that will pick up emf with alternating sign. This can be done by twisting the wire. With a constant wire thickness, this will result into equally distributed current loops. The method of using twisted pairs of wires is illustrated in Figure 69.

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Figure 69: A twisted pair of cables eliminates voltages due to emf

The two methods can be combined as shown in Figure 70. Differences in the ground potential will result into the need of input amplifiers with a good Common Mode Rejection Ratio.

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Figure 70: A single wiring system using both balanced out and twisted pairs of wires

7.9 SUMMARY After a sensor was characterized as a modulated electronic device in chapter 5, the next step is to create a useful signal form that device. This chapter gives example circuits for several types of sensor devices: • The device is a galvanic switch: Examples are on/off switches. In this case, a simple pull-up resistor to a power supply makes a signal that is either 0 Volt or equal to the power supply depending on the button. The concern is contact bounce, especially when the electronic function is to count events, • The device is a modulated resistor: examples are NTC’s, PTC’s, Pt100, LDR and strain gauges. Depending on the relative variation and the need for a linear response, either a single voltage divider or a Wheatstone bridge is the preferred circuit, • The device is a modulated capacitor: capacitive microphones, pressure and humidity sensors are the most typical examples. Read-out is done by using oscillation or RC-charging, • The device is a diode: for example a photo-diode. Good circuits are for example current to cottage converters. If the sensor is not a modulated electronic device but a self-generating sensor, no conversion is needed. After the modulated electronic device is converted into a voltage, an analog signalconversion may be needed to adjust bandwidth, to amplify, to shift a level or to remove a bias. Several basic OpAmp circuits for this purpose are given in this chapter, but the real solution depends on the specific context and problem. To send a sensor signal over connection wires to a processing unit, some precautions may be needed to prevent noise or interference. Differential wiring can help to eliminate common noise, especially in combination with twisted wires. Bootstrapping is an active shielding method.

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8 ADC AND DAC The learning goals of this chapter are • Explain the process of digital to analog conversion, and from analog to digital, • Introducing several systems of analog to digital conversion and show what the relation between the number of bits, range, resolution and conversion time is, • Introduce the Nyquist criterium for sample rate, and the calculation of resolution from the number of bits. After studying this chapter, the reader is able to • Choose quantitatively the number of bits and sample frequency in an analog to digital converter which is appropriate for the system, • Distinguish between conversion systems for analog to digital converters with respect to resolution, complexity, conversion time and number of bits, specifically for converters of the integration type, • Choose the number of bits and range for a digital to analog converter, • Understand how the single bit technology of pulse-width modulation works. Analog to digital converters (ADC) and digital to analog converters (DAC) convert a signal which is continuous in time and amplitude to a signal which is discrete in time and amplitude and vice versa. This is a lossy process which has to be fast and efficient. Therefore, some insights are needed in how the conversion is done. After conversion however, digital signals are more robust. They can be stored and mathematically processed.

8.1 FROM ANALOG TO DIGITAL AND BACK In modern sensor systems there is in almost all cases a node in the system where we go from analog signals to digital signals. In the end, the human readout is analog again, and assumed to represent the original analog value of the sensor. It is therefore a logical approach to think of all digital electronics as an equivalent analog circuit. The reasoning is sketched in Figure 71. The analog signal that comes in is x(t). This signal is transformed by an Analog to Digital Converter (ADC) to a digital signal X(z) which is discrete in time and amplitude. Discrete in time means the signal is sampled at fixed timeslots. Discrete in amplitude means that only a finite number of voltage levels can be represented in the digital domain (quantization). So, information is lost in de ADC. The digital filter will do some manipulation towards a digital signal Y(z). The Digital to Analog Converter (DAC) converts it back to an analog signal y(t). 113

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Figure 71: The Shannon-Whitaker theorem describes when a digital path via sampling and DA-conversion is equal to an analog path

Although information is lost in the ADC because of quantization in time and amplitude, we may think of the signal y(t) as an analog filtered representation of x(t). The question rises under what criteria this reasoning is valid. The Shannon-Whitaker theorem describes this for the aspect of discretization in time. It says that the output signal is still a good representation when the sampling frequency is equal to at least twice the highest frequency content in the signal x(t). The minimum allowable sampling frequency is also known as the Nyquist rate.

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8.2 ANALOG TO DIGITAL CONVERSION In Figure 72 a symbolic representation is given of an analog to digital converter as a building block. What goes in is an analog signal. The ADC converts it to a binary representation that come out as a serial stream, or as some parallel bits (as drawn). A serial stream can be in an SPI or I2C protocol. The ADC is an active component and so it needs a power supply Udd.

Figure 72: An analog to digital converter with parallel out

To convert and analog signal, two steps are needed: • Sampling at given moments in time or intervals • Conversion to discrete amplitude levels With these two steps, the signal becomes discrete in time and amplitude.

8.2.1 DISCRETIZATION IN TIME: SAMPLING

The first step is represented in Figure 73. When the hold-switch is activated, the actual voltage Vin is copied and frozen on the capacitor. The switch is opened immediately afterwards, resulting in a switch action which is call latching. The capacitor voltage is equal to Uin and will remain equal until the latch is activated again. At any moment after activation of the switch, the voltage is available for read-out as Uhold. There is a moment of sampling after which there is a hold phase.

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Figure 73: A sample and hold circuit

There are two conventions that refer to the most common implementation in ADC circuits and the associated timer configurations in microcontrollers that have ADC’s. • On demand sampling - where a command activates the latch whenever we need a sample • Continuous sampling - where a certain clock activates the latch Continuous sampling is normally done at equidistant intervals. This means that between two samples there is always a period Δt=T. We call T the sampling interval based on which we can speak of the sample rate fs ଵ

ˆୱ ൌ ୘ (92) 8.2.2 DISCRETIZATION IN AMPLITUDE: QUANTIZATION

After sampling (and holding the value), there is the phase of conversion in which the captured voltage level is converted to a certain digital representation. This is in almost all cases at equidistant levels. For example, we can subdivide the amplitude range of interest, let’s say 0V to 5V into 256 voltage levels. This would mean that every step represents roughly 19.5mV. The number 256 is not a random number: it means we can represent all numbers as an 8-bit value. In general, the number of levels n that can be represented by N bits is equal to n=2N.

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These numbers will represent voltage levels. When mapping the input range onto a voltage range from Uref−to Uref+, each interval ΔU will be ȟ ൌ

୙౨౛౜శ ି୙౨౛౜ష ଶొ

(93)

which becomes ΔU≈19.5mV for Uref−=0V, Uref+=5V and N=8 bits as seen in the example before. In Figure 74 there is an example for N=3 bits resulting into 8 levels. In that case ΔU becomes 1/8=0.125 of the full range. The first level, captured by the binary value 000, can be the result of any input value between 0Uref and 0.125Uref. The binary value 001 (we speak of the least significant bit or simply lsb) represents input voltages between 0.125Uref and 0.250Uref. We can also say that the lsb is 0.125Uref. There is also a most significant bit or msb that represents whether an input voltage is in the range above or below 0.5Uref. The msb can also be seen as a sign-bit when we have a symmetrical input window.

Figure 74: How an analog signal (horizontal axis) is mapped onto a digital representation (vertical axis)

There is always a residual error in the quantization process. We map the analog value onto discrete levels. The error is called quantization noise and the maximum instantaneous value is between −0.5 ΔU and +0.5 ΔU. So the amplitude of the quantization noise is equal to the resolution ΔU and as such defined by the least significant bit. In Figure 74 the horizontal axes is expressed relative to Uref−, and normalized to Uref+−Uref− (the full scale voltage range EFSR which is also called span). Any input voltage outside the range Uref− to Uref+ will result into either the lowest or highest binary representation respectively. We say that the signal is outside the ADC range and it has clipped.

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The optimization of the signal range to the available sample window of an ADC is a process we do manually in a digital SLR camera. In Figure 75 we see the histograms as available in Photoshop or on the camera display in more professional cameras. What we do in fact, is to map the light intensity distribution of the current situation to the range that the CCD sensor can handle.

Figure 75: Determining the appropriate input range using the histograms in a digital SLR camera

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8.2.3 CONVERSION TECHNIQUES

To pick the right bit sequence for every analog level sounds easy: we could use N comparators for an N bit ADC and define N decision levels with resistor arrays. The msb is generated by a comparator with a decision level exactly in the middle of the input range. This technique would be a flash converter or parallel converter. However, this is not the right procedure. Although such an ADC would be fast, the complexity goes up with N and we will have serious amplitude noise from imbalances in the resistor arrays. Therefore, many techniques have been developed that have a better scalability. A better solution is the integrating ADC of Figure 76. The principle is based on the integration of a reference voltage Uref while measuring the time Tint it takes to reach a level equal to Uin. This is called a single slope integration. The integrator voltage at time t after the integration started is ୧୬୲ ሺ–ሻ ൌ

୙౨౛౜ த

ή – (94)

with W the characteristic integration time defined by R and C. So, when Uint=Uin we have found Tint which is then the actual t. So the relation between the measured integration time Tint and the voltage Vin that must have been at the input is ୧୬ ൌ ୰ୣ୤ ή

୘౟౤౪ த

.(95)

Figure 76: Single-slope integrating ADC

All ADC’s will have a certain conversion time. For the single-slope integration ADC we can understand that the conversion time we have to take into account is equal to the longest integration time which is W. The basic advantage of the integrating ADC is that time can be measured relatively easy and with low timing noise (jitter) in standard microcontrollers. Besides the single-slope ADC there are improved techniques like the dual-slope ADC, which is slower but has a higher precision. Other well-known ADC techniques worth investigating are: • Sigma-delta conversion • Successive approximation

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To summarize, what is important in ADC selection and configuration is • The number of bits that defined the resolution • The sample rate that must satisfy the Nyquist rate fsample≥fhighest • The signal range that must optimally use the input window for the best resolution • Sample and hold behavior to make constant voltage during conversion • Stable reference voltages • Conversion time taken into consideration with respect to sample rate • Stable conversion timing (low jitter)

8.3 DIGITAL TO ANALOG CONVERSION Digital to analog conversion is needed to close a measurement loop in a sensor/actuator system. In such a system, the DAC can function as a trigger or function generator to disturb the environment while measuring the response on this disturbance. The system of Figure 77 is needed then. Now we have a closed-loop system which is different from an open loop, or feed-forward system.

Figure 77: The need of a DAC in a measurement setup with a stimulus/response mechanism

The block scheme of a digital to analog converter (DAC) is given in Figure 78. These elements are needed in for example outputting digital audio (CD or MP3), motor control or signal generators.

Figure 78: A digital to analog converter with parallel in

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To do this, we need an element that outputs an analog voltage based on a binary input. The voltage range must be defined, so this element needs two reference inputs: Uref− and Uref+. We also have to make an agreement on how to map the binary code onto the output range. In Figure 79, the eight levels that are available because of the proposed three bit system are mapped onto a voltage range from 0V up to 10V by the simple conversion ୭୳୲ ൌ ୰ୣ୤ି ൅  ή

୚౨౛౜శ ି୚౨౛౜ష ଼

. (96)

Figure 79: Three bits digital to analog conversion has eight levels

The digital input can be supplied by a parallel input (8, 10, 12, 14 bits at once), or by a serial input using SPI or I2C. In the serial case, the serial clock must be significant higher than the requested DAC conversion rate. There are however completely different mechanisms to do digital to analog conversion. One example is pulsewidth modulation (PWM) as visualized in Figure 80. The advantage of this method is that it only needs a single bit on a pin that can handle 0’s and 1’s only. Consider a microcontroller that has an output pin on which we put a square wave. If the square wave is symmetrical, meaning it is high for the same time it is low, we speak of a duty-cycle of 50%. The average voltage on that pin is 50% of the level we would have when outputting a high level only. When connecting an LED to that pin, we would see half of the intensity. So, by tuning the duty cycle, we can dim the LED. We call this a pulse-width modulated output. In combination with a low-pass filter, we can create voltage levels at many analog levels. The number of levels is defined by the precision in timing.

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Figure 80: Pulsewidth modulation

In normal cases, PWM is implemented with a constant period, but with controllable dutycycle. If the timing is controlled with an 8-bits counter, the PWM will have 256 levels because the duty cycle can be controlled in 256 steps. With optimum low-pass filtering, the output voltage for an N-bit PWM DAC is ୭୳୲ ൌ ୰ୣ୤ି ൅  ή

୙౨౛౜శ ି୙౨౛౜ష ଶಿ

. (97)

because the duty cycle is controlled as n/2N.

8.4 SUMMARY A digital signal is discrete in time and amplitude. Analog electronic signals are converted to digital for easier storage, prevent loss after conversion and for mathematical signal processing. The conversion from analog to digital is a process of choosing the optimum number of levels (bits) and the range, because these two determine the resolution. The conversion is also a matter of choosing the optimum sample rate. It is the Nyquist rate that determines the theoretical lowest frequency of sampling.

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9 BUS INTERFACES This last chapter summarizes the most important structures of how sensors are connected to a central processing unit. The goal is to explain • Digital and analog buses and their advantages, • Trends in such bus systems. After this chapter, the reader is able to • Reproduce some names and characteristics of the most common bus interfaces, • Distinguish between internet based, digital and analog bus formats and explain their benefits. Sensors have to be at specific locations to pick up the required physical quantity correctly. As a result, a sensor system may result into many, long wires to the sensors or sensor modules. While many sensors are still connected by analog wires (either with or without a pre-amplifier), the overall trend is to move to digital, standardized bus systems.

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9.1 THE ADVANTAGES OF BUS INTERFACES AND NETWORKS The reasons to use a bus or bus system are: • Reduce the number of wires and simplify the wiring plan by using star-based or chain networks • Reduce the influence of electronic noise by going digital • Implementing channel coding • Adding metadata for error management • Adding metadata for real-time systems • Adding metadata for safety and security purposes • Standardization of physical and electrical interfaces for cost reduction and easy serviceability There are three examples where digital buses were introduced after electronic systems evolved for decades. These are discussed in the following paragraphs.

9.1.1 I2C, SPI AND ELECTRONIC APPLIANCES

In the 1970’s, TV sets had become complex and in the meantime, digital controllers were introduced. The wiring architectures had become very sophisticated and there became clear different wires for control- and setting data opposite to data channels for (digital) video information. Therefore, Philips introduced in 1979 the Inter IC Bus, short I2C bus. This is a serial, synchronous bus which was originally intended for rates up to 100kb/s over a maximum range of 1m. This has changed the architecture of TV sets, and also other equipment, dramatically. Sub-modules within a TV architecture got their own status, linked only by the I2C bus, and remote units, like the control panel, were easier to wire. Although the I2C bus is not intended for sensors, it is used for sensor chips. An alternative, which is similar in the sense that it comprises a clock-line and a serial data line, is the Serial Peripheral Interface (SPI) as introduced by Motorola in the late 1980’s. Also this bus protocol is implemented in many sensor chips.

9.1.2 CAN BUS IN CAR INDUSTRY

Similar to what happened in the architecture of appliances due to I2C, the automotive industry experienced a shift towards digital buses in the late 1980’s. The Controller Area Network bus (CAN bus) is a vehicle bus standard developed at Robert Bosch GmbH. It has changed the car’s architecture of the control system into central microcontrollers, communicating to remote units over a serial bus.

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Later on, more dedicated buses for safety critical subsystems were developed like FlexRay and the AUTOSAR (AUTomotive Open System ARchitecture) has formalized the software architecture. The CAN bus is also used as industrial fieldbus in general automation environments, primarily due to the low cost of some CAN controllers and processors.

9.1.3 INDUSTRIAL ETHERNET

In industrial machines for manufacturing, logistics (conveyer belts) and packaging, we see the integration of business and technical processes. Manufacturers want to control and monitor their production in a central place. As a result, we see the introduction of IT technologies and methods on the manufacturing floor. To be more specific, the technical consequences are: • Standard Operating Systems on PC automation systems (like Windows, Windows Embedded, Linux and Linux Embedded) • Use of Ethernet • Standard internet protocols like http (remote control, etc.) • Technologies like OPC, XML or TCP/IP Current sensor modules for industry, are available with one of the Ethernet based industry buses.

9.2 BUSES OPTIMIZED FOR SENSOR NETWORKS The most common protocols to implement communication from sensors to the main controller in electronic microsystems are I2C and SPI. A good overview is given in the paper by Zhou (Zhou and Mason 2002). Also the webpage of Byte Paradigm (Byte Paradigm Introduction to I2C and SPI protocols n.d.)gives a good comparison of I2C and SPI.

9.3 INDUSTRIAL BUSES In fact, an industrial machine is normally a distributed computer. This means there is a central controller (either PLC or Industrial PC as a master) which is connected to distant slave couplers by means of a field bus. From the coupler slices may address sensors and actuators by means of low-end field buses in a slave-slave or master-slave relation.

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The current topics of interest are: • • • • • • • •

Shift towards Ethernet based protocols (PROFINET, EtherCAT, EtherNet/IP) Integration of ICT services on the same bus Ring redundancy Power over Ethernet Real-time features (because they are part of a distributed computer) Integrated cyber security systems (detect authenticity) Integrated safety systems (guaranteed and predictable stop conditions) Extension to lower-end interconnect like point-to-point IO-Link

Recommended reading about EtherNet based buses is: • Industrial EtherNet Facts, System comparison - the 5 major technologies (EtherNet Powerlink Standardization Group 2016) • EtherCAT Communication (EtherCAT Technology Group (ETG) 2018) and • Industrial Ethernet Technologies (EtherCAT Technology Group (ETG) 2014)

9.4 SUMMARY This chapter gave a brief overview of bus systems: the protocols used for (digital) interconnect. The reason of using protocols is to make data transfer more safe and reliable, or to add metadata. A consequence of a protocol is the need for a description, because the sender and receiver must unambiguously understand each other. This results into standards, which are normally dependent on the field of origin, for example automotive, telecom or industrial automation.

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References

10 REFERENCES Analog Devices. “Analog Devices: Education.” Linear Circuit Design Handbook, Chapter 3: Sensors. 2008. https://www.analog.com/en/education/education-library/linear-circuit-designhandbook.html (accessed August 19, 2019). —. Practical Design Techniques for Sensor Signal Conditioning. 1999. https://www.analog. com/en/education/education-library/practical-design-techniques-sensor-signal-conditioning. html (accessed August 19, 2019). Bartneck, C., and M. Rauterberg. “HCI Reality - An ‘Unreal Tournament’?” International Journal of Human Computer Studies, no. 65(8) (2007): 737-743. Byte Paradigm - Introduction to I2C and SPI protocols. n.d. http://www.byteparadigm.com/ applications/introduction-to-i2c-and-spi-protocols/. Catrysse, M., R. Puers, C. Hertleer, L. Van Langenhove, H. van Egmond, and D. Matthys. “Towards the integration of textile sensors in a wireless monitoring suit.” Sensors and Actuators A, no. 114 (2004): 302–311.

.

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Cobbold, R.S.C. “Transducers for biomedical measurements: principles and applications.” New York: John Wiley & sons, 1974. Dittmar, A. “A Multi-sensor system for the non invasive measurement of the activity of the autonomic nervous system.” Sensors and Actuators B: Chemical 27, no. 1-3 (1995): 461-464. EtherCAT Technology Group (ETG). “EtherCAT Communication, Communication Principles.” EtherCAT Technology Group. December 10, 2018. https://www.ethercat.org/en/downloads/ downloads_4A8B20A0EDC348888CC85417677A359F.htm (accessed August 19, 2019). —. “Industrial Ethernet Technologies: Overview and Comparison.” EtherCAT Technology Group. April 4, 2014. https://www.ethercat.org/download/documents/Industrial_Ethernet_ Technologies.pdf (accessed August 19, 2019). EtherNet Powerlink Standardization Group. “Industrial EtherNet Facts, System Comparison  the 5 Major Technologies.” Ethernet-Powerlink. Maart 2016. http://www.ethernet-powerlink. org/en/downloads/industrial-ethernet-facts/ (accessed August 19, 2019). Feijs, L., G. van Boxtel, and G.R. Langereis. “Designing for the Movements of Heart Rate and Breath.” 6th International Workshop on Design & Semantics of Form & Movement: Design Semantics in Context, DeSForM 2010. Lucerne, Switzerland, 2010. Field, Andy. Discovering statistics using IBM SPSS statistics. Sage, 2013. Fox, R.H., and A.J. Solman. “A new technique for monitoring the deep body temperature in man from the intact skin surface.” J. Physiol. 212 (1971): 8-10. Fraden, J. In Handbook of Modern Sensors - Physics, Designs, and Applications. Springer, 2010. Gourmelon, L., and G.R. Langereis. “Contactless sensors for surface electromyography.” IEEE-EMBC ’06, 28th Annual International Conference IEEE Engineering in Medicine and Biology Society (EMBS). New York: IEEE, 2006. 2514-2517. Ishijima, Masa. “Monitoring of electrocardiograms in bed without utilizing body surface electrodes.” IEEE Transactions on Biomedical Engineering 40, no. 4 (1993): 593-594. Kranenburg, R. “The Internet of Things: A critique of ambient technology and the all-seeing network of RFID.” In Network Notebooks 02. Amsterdam: Institute of Network Cultures, 2008.

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L.A. Zadeh, et al., Fuzzy Sets, Fuzzy Logic, Fuzzy Systems, World Scientific Press, ISBN 9810224214, 1996. 1996. Langereis, G.R. “Microsystem technology based sensors for Ambient Intelligence.” In Microsystem Technology for Ambient Intelligence, by S. Mukherjee, R.M. Aarts, R. Roovers, F. Widdershoven, & M. Ouwerkerk, 151-177. Springer, 2006. Langereis, G.R., L. de Voogd-Claessen, A. Sipilä, C. Rotsch, A. Spaepen, and T. Linz. “ConText Contactless sensors for body monitoring incorporated in textiles.” Portable ‘07, IEEE Conference on Portable Information Devices. IEEE, 2007. 1-5. Linz, T., L. Gourmelon, and G.R. Langereis. “Contactless EMG sensors embroidered onto textile.” 4th International Workshop on Wearable and Implantable Body Sensor Networks BSN2007. Berlin, Heidelberg, Germany: Springer, 2007. 29-34. Manz, A., N. Graber, and H.M. Widmer. “Miniaturized total chemical analysis systems: a novel concept for chemical sensing.” Sensors and actuators B: Chemical (Elsevier) 1, no. 1-6 (1990): 244-248. Martin, Paul, and Patrick Bateson. Measuring behaviour: an introductory guide. Cambridge University Press, 1993. Middelhoek, S., and S. Audet. “Silicon sensors, microelectronics and signal processing.” London: Academic Press Limited, 1989. Myers, R.H., and D.C. Montgomery. “Response Surface Methodology (2nd ed.).” New York: Wiley, 2002. Olthuis, W., G.R. Langereis, and P. Bergveld. “The merits of differential measuring in time and space.” Biocybernetics and Biomedical Engineering 21, no. 3 (2001): 5-26. Olthuis, W., S. Böhm, G.R. Langereis, and P. Bergveld. “Selection in system and sensor.” In Chemical and biological sensors for environmental monitoring, by A. Mulchandani, & O. Sadik, 60-85. Washington D.C.: Oxford University Press, 2000. Ouwerkerk, M., F. Pasveer, and G.R. Langereis. Unobtrusive sensing of psychophysiological parameters: some examples of non-invasive sensing technologies. Vol. 8, in Probing Experience, by J.H.D.M. Westerink, M. Ouwerkerk, T.J.M. Overbeek, W.F. Pasveer, & B. de Ruyter. Springer Science & Business Media., 2007. POLAR. n.d. https://www.polar.com/en.

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Regtien, Paul P.L. “Sensors for Mechatronics.” Elsevier, 2012. Richardson, P.C. “The insulated electrode: A pasteless electrocardiographic technique.” 20th Conf. on Engineering in Medicine and Biology. 1967. 15-17. Rowley, Jennifer. “The wisdom hierarchy: representations of the DIKW hierarchy.” Journal of Information and Communication Science 33, no. 2 (2007): 163–180. Scottevest. Scottevest. n.d. http://www.scottevest.com (accessed August 19, 2019). The scale of the universe. n.d. http://scaleofuniverse.com/. YouTube. Powers of Ten (1977). n.d. http://youtu.be/0fKBhvDjuy0. Zadeh, L.A. “Fuzzy Sets, Fuzzy Logic, Fuzzy Systems.” World Scientific Press, 1996. Zhou, J., and A. Mason. “Communication buses and protocols for sensor networks.” Sensors, no. 2(7) (2002): 244-257.

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APPENDIX A CIRCUITS, GRAPHS, TABLES, PICTURES AND CODE

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APPENDIX A CIRCUITS, GRAPHS, TABLES, PICTURES AND CODE The learning goals of this appendix are • Introduce appropriate drawing of graphs, tables and documenting code, • Understand that visuals and documentation have a communicative goal which has to be optimized for the convenience and trust of the reader. After this chapter, the reader is able to • • • •

Annotate an electronic circuit, Draw a graph according to the common standards, Draw a table according to the common standards, Understand why structuring and annotating computer code is essential.

The meaning of a table or graph is the representation of the measurement (and/or simulation) in a compact way. There are certain rules to represent graphs and tables in such a way that the reader can trace what has been done and verify the conclusions. However, before representing the measurement data, the reader should be able to understand what has been done. What is essential to do that, is an appropriate drawing of the measurement set-up. In case of electronic circuits, this is the schematic, if applicable in combination with code.

A.1 The electronic circuit and set-up The schematic must follow the following rules: • The set-up description includes type numbers of used equipment, • The time and date is important, and so is a version number. Also other conditions (temperature) that may be relevant are described, • Ground points and power origins are visible in the circuit, • The whole circuit is drawn, • Indicate inputs and outputs, • Wires and connectors have descriptive names.

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In Figure 81 we can clearly see how a National Instruments myDAQ interface module is used to measure the potential difference over a resistor while monitoring the current (by means of a shunt resistor).

Figure 81: A good measurement set-up circuit drawing

A.2 Writing good code Also the code (if applicable) needs to communicate what it does, and how it is constructed. It does not matter whether it is text based code or graphical code like National Instruments LabVIEW. Rules are: • Use comments. In text-based programming languages this can be /* like this */ • Use functional variable names • Use subroutines

A.3 A good table There are certain conventions for good tables. See the left hand of Figure 82 for an example.

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Figure 82: A good graph and table

The common rules are: • A table has a header explaining the meaning of the data, • The table has a number for referencing: this number must be mentioned in the text, • In the table header, there are symbols and [units] of the corresponding quantities,

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• The comma in the numbers in the table are at a logical position. Reduce the number of leading zeros, which can be done by including a multiplier 103 in the header of the table, • The significance (digits) of the numbers are represented well, • The structure of the table is logical (cause and result), • Errors are indicated in the table. Evaluate somewhere the nature and size of the error.

A.4 A good graph The right hand of Figure 82 shows a graph. The basic good habits are: • A graph has a meaningful and numbered footer. The number is used to refer to the graph from the text, • Along the axes, there are symbols and [units] of the corresponding quantities, • On the horizontal axis there is the independent variable and on the vertical axis the dependent variable, corresponding intuitively to the cause and effect of the experiment, • The axes have a scale with appropriate numbers (similar to the table conventions), • The axes are chosen properly to indicate a linear relationship. This may be lin-lin, lin-log or log-log. This is needed to support the underlying model and to see outliers. The model can be plotted as well, for example as a simple straight line if applicable, • Errors are indicated if possible with error bars in x and y direction. The ultimate evidence is when the straight line of the model falls within the error bars.

A.5 Graph, table or just text? Always represent data in a form that makes a clear statement: • If you want to express just up to three numbers or insights, just write text or a statement, • When representing between 4 to 20 numbers, a table is probably the right form of presenting, • For more numbers, use graphs in combination with statistical measures.

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Appendix B: Common Mode Rejection Ratio (CMRR)

APPENDIX B: COMMON MODE REJECTION RATIO (CMRR) The goal of this appendix is to • Explain the concept of common-mode rejection ratio in a differential amplifier, • Substantiate the calculation of the common-mode amplification and differential-mode amplification of a differential amplifier based on a single OpAmp. After this chapter, the reader is able to • Calculate the common-mode rejection ratio from a circuit and from a measurement, • Understand and calculate the quantitative impact of a mismatch in resistors on the common-mode and differential-mode amplification of the single OpAmp based differential amplifier. The Common Mode Rejection Ratio is the ratio between the output of an amplifier due to a common signal with respect to a differential signal. Consider a certain amplifier with two inputs UA and UB. We define • the gain of the amplifier as A - meaning that differential voltages are amplified A times ୭ ൌ  ή

ሺ୙ఽା୙ా ሻ ଶ

. (98)

• an amplification of common signals of B - meaning that the average of UA and UB is amplified B times ୭ ൌ  ή ሺ୅ െ ୆ ሻ, (99)

The Common Mode Rejection Ratio is defined as ୅

 ൌ ȁ୆ȁ.(100)

How does this work with the simple OpAmp amplifier of Figure 83?

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Appendix B: Common Mode Rejection Ratio (CMRR)

Figure 83: Simple OpAmp based amplifier

We can start with superposition of the output UO due to voltage UB and UA. To simplify the problem, we start with the output UO due to voltage UA and U+: ୭ ൌ  ା ή ୭ ൌ  ା ή

ସ ൅ ଷ ସ െ ୆ ή ଷ ଷ

ή൅ ή െ ୆ ή  

୭ ൌ  ା ή ሺ ൅ ͳሻ െ ୆ ή 

(101)

where U+ can be calculated as ା ൌ ା ൌ

ଶ ή ଵ ൅  ଶ ୅

ή ή ൅ή ୅ ୫

 ା ൌ ଵା୫ ή ୅ .

(102)

Which yields

୭ ൌ

 ή  ή ሺ  ൅ ͳ ሻ െ ୆ ή  ͳ൅ ୅

୭ ൌ  ή ሺ୅ െ ୆ ሻ.

(103)

This means that in the ideal situation, the gain A is equal to m, and there is no effect of a common signal. However, we made an assumption that the factor m in R4 is equal to the factor m in R2. What would happen if we distinguish between m4 and m2?

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Appendix B: Common Mode Rejection Ratio (CMRR)

Start again with ସ ൅ ଷ ସ െ ୆ ή ଷ ଷ

୭ ൌ  ା ή

ସ ή  ൅  ସ ή  െ ୆ ή  

୭ ൌ  ା ή

୭ ൌ  ା ή ሺସ ൅ ͳሻ െ ୆ ή ସ

(104)

where U+ can be calculated as ା ൌ ା ൌ

ଶ ή ଵ ൅  ଶ ୅

ଶ ή  ή  ൅ ଶ ή  ୅ ୫

 ା ൌ ଵା୫మ ή ୅ .

Now we find

୫

మ

(105)

୭ ൌ ଵା୫మ ή ୅ ή ሺସ ൅ ͳሻ െ ୆ ή ସ (106) మ

which can no longer be simplified. However, we can bring it to the form of ୆

୭ ൌ  ή ሺ୅ െ ୆ ሻ ൅ ሺ୅ ൅ ୆ ሻ. (107) ଶ

The factors A and B (the differential gain and common gain respectively) are ൌ

ൌ

ଶ ൅ ʹଶ ସ ൅ ସ ʹ ή ሺ ͳ ൅ ଶ ሻ ୫మ ି୫ర ଵା୫మ

.

(108)

Here is A the desired differential gain and B the residue due to m2≠m4. This means ୫

 ା ൌ ଵା୫మ ή ୅ . (109) మ

where we can see that if m2 becomes equal to m4, the CMRR becomes infinite.

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APPENDIX C: A SCHMITT TRIGGER FOR SENSOR LEVEL DETECTION

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APPENDIX C: A SCHMITT TRIGGER FOR SENSOR LEVEL DETECTION The learning goal of this final appendix is to • Explain the need of a comparator with an state dependent decision level, • Introduce the circuit of a Schmitt trigger which has that function. After reading this chapter the reader is able to • Understand how hysteresis the decision level can smooth the output signal of a comparator circuit, • Calculate the resistors in an OpAmp based Schmitt trigger to define the two decision levels. We have seen hysteresis in sensors: we can use hysteresis in the comparator to eliminate noise. A Schmitt trigger is a comparator circuit that incorporates positive feedback. As a result, the comparator decision level for a low-to-high transition is higher than the comparator decision level for a high-to-low transition.

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For a signal with some noise, a plain comparator circuit may result into some transition noise in the output signal which is similar to contact bounce. This effect was observed in chapter 7 with Figure 58. This response is quite logical because on a mV scale the signal does cross the decision level multiple times before it becomes “high” in the end. Just like contact bounce, a solution is to use a low-pass filter or software way of transition detection. However, a more appropriate solution for this problem is to use a higher decision level for a low-to-high transition and a lower decision level for a high-to-low transition. The comparator that does this is called a Schmitt Trigger and has the symbol of Figure 84.

Figure 84: Schmitt trigger symbol

The circuit is the comparator of Figure 85 with a positive feedback loop. As a result, the comparator level on Uin+ is different for a high Uout than for a low Uout. Based on these two positions, we can calculate the relation between the resistors and the two decision levels.

Figure 85: Comparator with positive feedback

In Figure 88 the effective position of the resistors is drawn for two situations: where the output is high and where the output is low.

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Figure 86: Effective resistor connection in comparator with positive feedback for two states

The final response and effect is drawn in Figure 87 and Figure 88.

Figure 87: Input-output relation for Schmitt trigger

Figure 88: Illustration of how a double decision level system (Schmitt trigger) results into a noise free response

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Index

INDEX Coulomb’s law 28 Coupling network 15, 67 Criterion validity 22 Cross-sectional research 9 Cross-sensitivity 22, 72 Cross-sensitivity artefacts 67 Current 31, 52, 125 Current meter 56 Current meters 35 Current source 30

A Absolute error 19, 26 Accuracy 20 ADC. See Analog to digital converter Addition of quantities 27 Adjustment 16 Alternating Current (AC) 31 Amount of Substance 35, 101 Ampere 30 Amplifier gain 135 Amplitude 31, 33 Analog to digital conversion 115 Analog to Digital Conversion (ADC) 17, 115 Angle 37 Average 20 Averaging 20

D Dependent variable 10, 134 Derived scale 12 Differential equation 62 Differential measurement 20, 83 Differential resistance 37 Differential set-up 72 Digital to analog conversion 120 Direct Current (DC) 31 Direct transducer 70 Distinct categories 82 Division of quantities 27 Drift 72 Dual-slope ADC 119

B Bandwidth 106 Battery 30 Biasing 15, 67 Biasing circuit 17 Binary scale 85, 115, 117, 121 Biofeedback 83 Bootstrapping 107 Branch 43 Bureau international des poids et mesures (BIPM) 10

E Electrical engineering units 11 Electric current 28 Electric potential 28. See Potential Electronic network 43 Elementary charge 28 Error propagation 26 Errors 18 Ethernet (industrial) 126 Experimental research 9, 22 Extensive variables 69

C Calibration 16 CAN 107 Capacitance 60 Capacitive sensor 75 Capacitor 60 Cardinal scale 14 cd (Candela) 120 charge 28 Common Mode Rejection Ratio (CMRR) 96, 102, 135 Common signal 135 Comparator 103 Compensation 20 Component 43 Concurrent validity 22 Contact bounce 91, 92 Content validity 22 Continuous levels of measurement 116 Controller Area Network bus (CAN) 124 Conversion 116 Conversion time 119 Correlational research 9 Coulomb 28

F Farad 60 Feedback 83 Feedback system 20 Feed-forward 20 F (Farad) 60 Field bus 78, 125 Filtering 15 Fourier transform 63 Frequency 17 Full scale voltage range (EFSR) 117

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G Gauss curve 24 Gauss law 29

H Help units 11 High-pass 61 Hysteresis 74, 138 Hz (Hertz) 75

I I2C 15, 115 i (Complex number) 63 Ideal current source. See Current source Ideal voltage source. See Voltage source Independent variable 10, 134 Inductive sensor 75 Instrumentation amplifiers 102 Integrated Circuit (IC) 86 Integrating ADC 119 Inter IC Bus. See I2C International System of Units 10 Interval scale 83, 91, 116, 117

Index

Measuring 9 Micro Electro Mechanical Systems (MEMS) 86 Micro System Technology (MST) 86 m (Meter) 11 Modulating sensors 70 Most significant bit 117 Motion artefacts 75 Moving coil meter 56, 57 msb. See Most significant bit Multiplication of quantities 27 Multivariate analysis 20

N Network. See Electronic Network Node 43 Noise 8, 19, 21 Nominal scale 55, 56, 94, 95 Non-linearity 74 Nonlinearity errors 20 Nonlinear transfer curve 73 Normal distribution 24 Norton theorem 51 NTC 93 Nyquist rate 114

J

O

j. See i (Complex number)

Observed quantities 10 Offset 31, 32 Ohmic behavior 36 Ohm’s law 36 Ohm [Ω] 11, 36 Operational Amplifier (OpAmp) 102 Operational point 33 Ordinal scale 14 Oscillator 99 Outcome variable 10

K Kirchhoff Current Law (KCL) 44 Kirchhoff ’s laws 43 Kirchhoff Voltage Law (KVL) 44

L Lab on a chip 86 Laplace transform 63 Large signal behavior 33 Latch 115 LDR light sensor 75 Least significant bit 117 Length 11 Levels of measurements 13 Longitudinal research 9 Loop 43 lsb. See Least significant bit

M Magnetic sensor 75 Magnetoresistive sensor 93 Mass 69 Mean 20 Measurement chain 14, 67 Measurement errors 19 Measurement scales 13

P Parallel plate capacitor 97 Parallel (resistors) 38 period 31 Phase 31, 32 Photo diode 101 Piezoelectric sensor 75 Polygraphy 80 Potential 29 Potential Difference 12, 29 Powers of 10 12 Precision 20 Predictive validity 22 Predictor variable 10 Pre-processing 67 Product design cycle 8 Pt100 temperature sensor 75 PTC 93

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Index

Qualitative verification 9 Quantitative verification 9 Quantities and units 10 Quantity 10 Quantization 116 Quantization noise 117

Square root of quantities 27 Standard deviation 20, 21 State variables 69 Stimulus-response measurement 20 Strain gauge 75 Subtraction of quantities 27 Superposition principle 33, 49 Superposition theorem. See Superposition principle Systematic error 19, 21 System in Package (SiP) 86

R

T

Radian frequency 31 Random error 19, 21, 25 Range 16 RC Circuit 64 RC Network 61 Redundancy 83 Relative error 20, 26 Reliability 22 Repeatability 21 Reproducibility 21 Resistance 36 Resistive behavior. See Ohmic behavior Resistive sensor 75, 93 Resistivity 93 Resolution 21 RMS value 33 Rounding errors 20

Temperature 11 Thévenin theorem 51 Time 60, 62 Time dependent circuits 60 Tolerance 24 Tolerances 21 Transducer 15, 17, 68 Transducing part 68 Transduction 67 Transient response 62 Transistor 86 True value 21

Pulsewidth modulation (PWM) 121 Push button 90

Q

S Sampling 115 Scalar 69 Schmitt Trigger 103, 138 Scientific cycle 8 Selector part 68 Self-generating sensors 70 Semiconductor sensor 75 Sensor 17, 67 Sensor arrays 83 Sensor-head 67, 68 Series (resistors) 38 Shannon-Whitaker theorem 114 Shunt resistor 59 SI derived units 11 Signal conditioning 15, 67 Signal leveling 15 Significant digits 25 Single slope integration 119 Sinusoidal signal 31 SI Units. See International System of Units Small signal behavior 33 Solid Angle 37 SPI 15, 107, 115 SPSS 128

U Unit 10 Unit check 13 Unity gain amplifier 102 USB 15

V Validity 22 Variables 10 Voltage 30 Voltage divider 40, 93 Voltage source 30 Volt meter 35, 56 V (Volt) 56, 58

W Watt 30 Wheatstone bridge 54, 72, 95

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